摘要:

Introductory Lecture Inertia coarsening and ¢©uid motion in binary mixtures M. E. Cates,a V. M. Kendon,a P. Bladona and J-C. Desplatb a Department of Physics and Astronomy University of Edinburgh King©«s Buildings May¡©eld Road Edinburgh UK EH9 3JZ b Edinburgh Parallel Computing Centre University of Edinburgh King©«s Buildings May¡©eld Road Edinburgh UK EH9 3JZ Received 19th April 1999 Symmetric binary ¡ªuids quenched into a regime of immiscibility undergo phase separation by spinodal decomposition. In the late stages the ¡ªuids are separated by sharply de¡©ned but curved interfaces the resulting Laplace pressure drives ¡ªuid ¡ªow. Scaling ideas (of Siggia and of Furukawa) predict that ultimately this ¡ªow should become turbulent as inertial eUects dominate over viscous ones.The physics here is complex mesoscale simulation methods (such as lattice Boltzmann and dissipative particle dynamics) can play an essential role in its elucidation as we describe. Likewise it is a matter of experience that immiscible ¡ªuids will mix on some lengthscale at least if stirred vigorously enough. A scaling theory (of Doi and Ohta) predicts the dependence of a steady state domain size on shear rate but assumes low Reynolds number (inertia is neglected). Our preliminary simulation results (three-dimensional so far only on small systems) show little sign of the kind of steady state envisaged by Doi and Ohta; they raise instead the possibility of an oriented domain texture which can continue to coarsen until either inertial eUects or (in our simulations) ¡©nite size eUects come into play.I. Introduction When an incompressible binary ¡ªuid mixture is quenched far below its spinodal temperature it will phase separate into domains of the two ¡ªuids. The simplest case (in theory at least) is when the two ¡ªuids are thermodynamically and kinetically symmetrical ¡ªuids A and B have identical properties in all respects except that they are mutually phobic. For a deep quench (to a temperature well below the critical point) the thermodynamic equilibrium state then has the two ¡ªuids completely demixed. One remaining control parameter is the volume fraction /; only for a 50 50 mix (/\0.5 which with thermodynamic symmetry is a quench through the critical point) is the system totally symmetrical.Other signi¡©cant parameters are the surface tension p viscosity g and the ¡ªuid mass density o. There is also a mobility parameter M which controls the collective diUusion of concentration ¡ªuctuations this can remain important at late times but only if / is small enough (less than about 0.15¡í0.20 for 3-D) that the ¡ªuid domains depercolate. The minority droplets can then continue to coarsen by a ripening mechanism (controlled by M) or by diUusion and coalescence of the droplets themselves ; both give a mean droplet size that scales with time as t1@3. If instead the domains remain connected then the late-stage growth is driven by capillary forces (Laplace pressure) arising from curvature of the (sharply de¡©ned) interface between the two Faraday Discuss.1999 112 1¡í11 1 —uids; these drive —uid —ows from regions of tight curvature (necks narrow liquid bridges) into those of low interfacial curvature (large domains). Note that for coarsening of a bicontinuous structure to proceed this way one also requires discrete ììpinch-oÜœœ events to continually occur each of these allows the topology to change discontinuously in time. For a theory-oriented review of the late stages of spinodal domain growth see Bray;1 there are many relevant experimental studies as well.2 In what follows we address two issues concerning domain growth in three dimensional binary —uid systems. (The two dimensional case has some special features of its own; see ref. 3.) The –rst issue concerns the role of inertia in the late stages ; this is a matter of continuing interest and controversy,4 that our recent simulation work5 illuminates.The second concerns the eÜect of an applied shear —ow on the coarsening process. Here we have only preliminary simulation results but these are enough to suggest that there is more to this problem than a recent scaling theory6 suggests ; again our discussion focuses on inertial eÜects. Our simulation work uses two diÜerent mesoscale simulation methods. One is the lattice Boltzmann (LB) method in which a velocity distribution function f (ø) and composition variable / are de–ned at discrete lattice points. The rules for updating these quantities represent a discretization of the (zero temperature) Navier»Stokes equation for —uid —ow coupled to a Cahn»Hilliard type thermodynamic free energy functional F[/] which provides the driving force for phase separation.The second method is DPD (dissipative particle dynamics) which is a long-time step noisy molecular dynamics algorithm (oÜ lattice) involving soft repulsive interaction potentials between two types of particle. (The repulsive interactions are chosen to favour demixing.) This is a –xedtemperature algorithm in which stability is ensured despite the long time-step by introducing local damping and noise terms in accord with a suitable —uctuation dissipation theorem. Crucially both damping and noise act pairwise on particle velocities and hence conserve momentum locally recovering the isothermal Navier»Stokes equation at large length scales. For further information on LB for binary —uids see Swift et al.7 and on DPD see Groot and Warren;8 details of our algorithms and parameter settings are recorded elsewhere.5,9,10 Note that in both methods we have run tests to check that under the particular simulation conditions we adopt —uids are behaving incompressibly.(1) II. Binary demixing at high Reynolds number For simplicity we restrict attention to fully symmetric mixtures (/\0.5) for which the —uid domains comprise in three dimensions a fully bicontinuous structure. The late-time evolution of this structure remains incompletely understood despite theoretical,1,4,11,12 experimental2 and simulation10,13,14,15 work over recent years. A. Scaling expectations As emphasized by Siggia,11 the coarsening of a bicontinuous spinodal texture involves capillary forces (governed by the interfacial tension p) viscous dissipation (governed by the —uid viscosity g) and —uid inertia (governed by the mass density o).Out of these three physical parameters (p g and o) only one length L 0\g2/op and one time T0\g3/op2 can be constructed which allow us to describe the time evolution of the coarsening system in unique dimensionless length and time measures. For convenience we de–ne the lengthscale L (T ) of the domain structure at time T via the structure factor S(k) as L \(1/2n) / S(k)dk/ / kS(k)dk. Provided no other physics except that described by the three macroscopic parameters p g and o is involved in late stage growth this leads us to the dynamical scaling hypothesis :11,12 l\l(t) l4L /L t4(T [T where we use reduced time and length variables int)/T0 .Since dynamical and 0 scaling should hold only after interfaces have become sharp and transport by molecular diÜusion T ignorable we have allowed for a nonuniversal oÜset thereafter the scaling function l(t) should int; in principle approach a universal form the same for all (fully symmetric deep-quenched incompressible) binary —uid mixtures. It was argued further by Furukawa12 that for small enough t —uid inertia is negligible compared to viscosity whereas for large enough t the reverse is true. Dimensional analysis then Faraday Discuss. 1999 112 1»11 2 requires the following asymptotes (2) l]bt ; t@t* (3) l]ct2@3; tAt* where if dynamical scaling holds amplitudes b c and the crossover time t* (de–ned for example by the intersection of asymptotes on a log»log plot) are universal.Several earlier numerical studies claim to see one or other of these scaling regimes but of these few provide accurate values of g o p as required if the various datasets are to be compared on a scaling plot.15 Those that do so and which appear to con–rm eqn. (2) include the work of Laradji et al.,13 for which 2OlO20 that of Bastea and Lebowitz14 (1OlO2) and that of Jury et al.10 (20OlO2000). Jury et al. performed a careful comparison of their own (DPD) and these othersœs datasets ; while linear scaling was reported they found no consistency in the value of b. Instead a systematic trend appeared in which b drifts downward between datasets as one moves to larger t and l (roughly bDt~0.2).Despite this within each DPD dataset the linear t dependence is clearly better than a –t to lDt0.8. Jury et al. proposed that a non-scaling behaviour of this kind could perhaps be explained if some nonuniversal physics (that is not contained in p o g) were to intervene they suggested a candidate involving the physics of topological reconnection a process that even at late times could involve a molecular (or discretization) scale small compared to L . This possibility remains open although our own more recent LB data5 suggest the following alternative explanation (a) the linear law is indeed obeyed for lO20 (with b^0.07) but the b coefficients of Laradji et al. and of Bastea and Lebowitz are both overestimated due to residual diÜusion eÜects ; (b) the data of Jury et al.lie within a broad crossover region between eqn. (2) and eqn. (3) where the local slope on a log»log plot is around 0.8 ; (c) the preference for linear –ts in these DPD datasets is partly caused by –nite size corrections within each dataset. To better eliminate the latter in our LB coarsening data we insist that L OK/4 with K the linear system size (Jury et al. allowed L OK/2). A comparison between representative LB and DPD datasets in a regime where both are available is shown in Fig. 1. Though not identical it is hard to be sure that the remaining discrepancies do not arise from –nite size corrections. B. Inertial eÜects The relative importance of inertial to viscous terms in —uid mechanical problems is traditionally measured by a Reynolds number.In the spinodal context this is usually de–ned as Re\Res4 (oL /g)dL /dT \ll5. In the linear regime it is of order b2t. Note that since b is rather small the maximum Re achieved by Jury et al. (for t^2000) is only about 20 even with t of order 104»105. L 0\0.15 lattice units DPD datasets (solid) have L 0\0.29 0.19 0.13 Fig. 1 Comparison of individual l(t) datasets on a log»log plot generated by the DPD10 and the LB5 algorithms. LB dataset (crosses) has 0.077 DPD units (left to right). 3 Faraday Discuss. 1999 112 1»11 Fig. 2 L vs. T (in lattice units) for runs with L 0\5.9 (circles) and 0.0003 (diamonds). Dotted lines delineate the range of data points used for –tting (17\L \64) and the –ts to t1 and t2@3 respectively have been projected back to show the intercepts Tint .Indeed this is roughly where we now believe t* to lie ; a Reynolds number of 20 is large enough for inertial eÜects to be non-negligible but by —uid mechanics standards still modest. Note also that the scaling ideas clearly predict that Re should increase without bound (as t1@3) within the inertial regime eqn. (3). However in a recent paper4 Grant and Elder have argued that the Reynolds number cannot in fact continue to grow inde–nitely. If so eqn. (3) is not truly the large t asymptote which must instead have lDta with aO1 Essentially Grant and Elder argue that at 2. large enough Re turbulent remixing of the interface will limit the coarsening rate,4 so that Re stays bounded to a level which they estimate as ReD10»100.Our recent LB work5 represents the –rst large-scale simulations of 3-D spinodal decomposition to unambiguously to attain a regime in which inertial forces dominate over viscous ones allowing a test of this idea. We –nd direct evidence for Furukawaœs lDt2@3 scaling eqn. (3). Although a further crossover to a regime of saturating Re cannot be ruled out we –nd no evidence for this up to Re^350. Two of our LB datasets are shown in Fig. 2. The –rst has high L 0 (high viscosity) and corresponds to l values around unity (t^10) for the –tted part of the data. It is well –t by Fig. 3 Scaling plot in reduced variables (l t) for eight LB datasets. Second and third from left overlap and cannot be distinguished on the plot. Faraday Discuss.1999 112 1»11 4 L Fig. 4 Velocity maps for 0\5.9 3.0]10~4 (LB) and for freely decaying isotropic turbulence (pseudospectral dns). Each map projects a 32]32]2 thin section taken from a 1283 grid. The interface appears wide where it cuts the section at a glancing angle ; each site shows two arrows which are the projected velocity vectors in the two layers of the section. eqn. (2). The second has low L and corresponds to l around 105 (or t^107) ; it is well –t by eqn. 0 (3). The Reynolds number is about 0.1 near the end of the –rst run and about 350 near the end of the second. It goes without saying that no single simulation run could possibly cover this range of length or time scales since that would require a lattice of around (106)3 sites. Our lattices are 2563 but by changing the LB parameter values we can access both regimes and everything in between.5 Similar exploration is possible (to a lesser extent) within DPD.10 Such mesoscale methods have strong advantages over (say) MD13 since after proper calibration and subject to speci–ed range limitations they allow one to ìì dial in œœ ones own choice of thermodynamic (p) and kinetic (g) parameters.Thus one can build up the l(t) curve section by section ; if it is universal this is enough. (For a fuller explanation of the procedure see Kendon et al.5) This is done for our LB data in Fig. 3; note that the rightmost two datasets (both in the inertial regime) are almost contiguous as one would hope on a universal scaling plot. (Each has a slope close to 2/3.) However computational restraints prevent us so far from covering the entire l(t) curve with data; it remains possible that some mismatches between runs of the type reported by Jury et al.10 might still be observed at intermediate l t.Note the broad crossover between viscous (three leftmost runs) and inertial (two 5 Faraday Discuss. 1999 112 1»11 rightmost runs) regimes this crossover covers four decades in t (or three in l). But it is less spectacular when expressed in terms of Re; our data span 0.1OReO350 and the crossover region is roughly Re\1[100. It is important now to ask whether at the largest Re values we can reach there is signi–cant turbulence in the —uid —ow. One quantitative signature of turbulence is the skewness S of the longitudinal velocity derivatives ; this is close to zero in laminar —ow but approaches S\[0.5 in fully developed turbulence.16 A space-averaged value close to S\[0.5 would imply turbulence everywhere and presumably signi–cant remixing at the interface.4 We do detect increasingly negative S as Re is increased but reach only S^[0.35 for Re^350.17 This suggests that at our highest Re values partially but not fully developed turbulence is present.This view is consistent with Fig. 4 which shows velocity maps for low and high Re runs and (for comparison) single —uid freely decaying isotropic turbulence. In a low Re —ow one expects the only relevant lengthscale in the system to be the domain size L ; at high Re there should be a cascade of structure in the velocity –eld below the domain size.Some internal structure is plausibly though not conclusively visible in the plot for our lowest L 0 value. But since we do not yet see fully developed turbulence it remains an open issue whether such eÜects would lead beyond the observed t2@3 inertial scaling regime to a –nal turbulent remixing regime (of saturating Re) as proposed by Grant and Elder.4 If it does the limiting value of Re must signi–cantly exceed their estimate of 10»100. (4) III. Demixing under —ow Consider now the situation where a symmetric binary —uid undergoes a deep quench in conditions where a steady shear —ow is applied externally. In simulations there are two ways to achieve this. One is by Lees»Edwards (moving periodic) boundary conditions which have been implemented in our DPD codes;18 the other is to have sliding solid boundaries and the LB results reported below use this method.For binary —uids under shear the two approaches give qualitatively similar results. For related work in two and three dimensions see ref. 19 and 20. The questions are as follows does the presence of —ow arrest the coarsening process giving a steady state with a –nite domain size and if so what is the character of this steady state ? Also does this steady state depend on starting conditions»for example if the binary —uids are allowed to demix –rst and the —ow is then switched on is the same –nal state achieved? Perhaps surprisingly (despite the obvious importance of mixing and emulsi–cation in many technological areas) well-founded theoretical answers to these questions are almost nonexistent.A. The DoiñOhta theory An important theoretical inroad into this problem was made by Doi and Ohta.6 (Its rheological consequences are reviewed in the book by Larson,21 where comparisons to experiment can be found). The Doi»Ohta work is entirely limited however to a regime of vanishingly small Reynolds number. Note that for this problem the relevant Re is the larger of Res\(oL /g)dL /dT (the value extracted from the coarsening process which must vanish in any true steady state) and Ref\oc5 L 2/g which is the value found by considering the eÜect of the external shear rate c 5 on a droplet of scale L . Both de–nitions assume perhaps implausibly that there is only one relevant length L which requires for example that domains are not strongly anisotropic ; we return to this point below.Doi and Ohta assumed that for nonzero c 5 a steady state would be reached at negligible Re (so that inertial eÜects can be ignored). The applied —ow stretches and deforms the interfaces (tending also to orient these along the —ow direction) ; this is opposed by interfacial tension. The com- Ca\T petition is measured by the capillary number 0 c 5 L /L 0\c 5 L g/p; at small Ca a spherical droplet of size L is weakly perturbed but at large Ca it should be elongated to the point of rupture. By estimating the magnitude of each eÜect in the absence of the other and balancing these Doi and Ohta made the intriguing prediction that in steady state the —uid should have a c 5 -independent viscosity of the form g f (/).That is the behaviour should be independent of p and o but dependent on / (values of / close to 0.5 were assumed.) This form follows purely by dimensional analysis if one insists that the only relevant variables are c 5 p g and / (the absence of o from this list is because inertia is neglected.) Doi and Ohta went on to develop speci–c Faraday Discuss. 1999 112 1»11 6 approximation schemes that allowed them for example to study transient (shear startup) behavior. In the steady state the departure of f (/) from unity must come from the presence of interfacial stresses which if the structure is characterized by a single domain length L scale as p/L . Comparing this with the excess shear stress gc5 ( f[1) we obtain a length estimate in steady state (5) L ^ p gc5 which corresponds to the system arranging itself in steady state so that the capillary number is of order unity.This argument is of course somewhat oversimpli–ed. If more than one length scale is present (for example describing the structure along diÜerent directions relative to the —ow) then one has to determine which one dominates the interfacial contribution to the viscous stress. The largest Laplace stresses arise in regions of strongest curvature ; however if the strongest curvature is axial about the —ow directions (for example thin cylinders lying along the —ow) it cannot contribute to the shear stress at all.6 Doi and Ohta did not much explore these geometrical aspects of their work. Note that implicit in their whole description is an assumption that a steady state can be achieved without inertial terms coming into play.Our preliminary results on sheared spinodal systems given below cast some doubt on this assumption. B. Results We have simulated sheared binary —uids using both DPD and LB algorithms. The results presented here are for relatively small systems (50 000 DPD particles and 643 or 963 lattices) and thus are preliminary but we think they point towards some interesting physics. In future work we hope to address larger systems; among other things this will allow reliable statistics for the interfacial stresses and hence for the rheological properties of the sheared systems to be obtained. Our –rst result is that shear does inhibit coarsening. If one de–nes a single lengthscale via the total surface area A in the system (so that A\K3/L ) then indeed this lengthscale increases more slowly when shear is applied than without it (Fig.5). On the other hand this one-lengthscale analysis fails to re—ect the extreme anisotropy that can develop. Indeed by examining the eigenvalues j1 j2 j3 of the interfacial curvature matrix,20 ; La / � Lb / � (6) Dab\grid ; / � 2 grid lengths for the three diÜerent directions can be estimated (7) L 3 1\ j 1 L 2\ j 1 L 3\ j 1 1 2 0\0.14 in Fig. 6. In these expressions / � is the local compositional order parameter de–ned on the lattice. The time evolutions of these three lengths are plotted for a system with c 5 T These data show strong anisotropy developing with elongation of domains along the —ow direction.Under these conditions –nite size eÜects become greatly emphasized:22 domains elongated along the —ow soon reach the size of the simulation box and connect up to their periodic images. A sheared LB con–guration during the initial phase before this happens (such that L 1,2,3 are all small compared to the simulation size K) is shown in Fig. 7(a). However this con–guration is not close to a steady state. The structure we observe on approach to the steady state is in all cases we have so far studied clearly –nite-size in—uenced. For / diÜerent from 0.5 the appearance is like that of a ìì string phaseœœ23 in colloids eÜectively in–nite tubes of the minority —uid oriented along the streamlines.A typical example of this found using DPD is shown in Fig. 7(b). Although no longer bicontinuous in the transverse direction this structure can continue to coarsen very slowly by diÜusive 7 Faraday Discuss. 1999 112 1»11 c 5 T0\0.07 0.14 0.21 compared to zero shear. Using the system size to estimate the maximum Fig. 5 Overall coarsening of the domains as estimated from the amount of interface in an LB system for shear rates of capillary number for the system due to the shear gives Ca\0.4 0.8 1.5. transport (ripening) it seems likely that for this system size the –nal state will be a single cylinder of —uid. Fig. 7(c) shows an LB simulation for /\0.5 viewed along the streamlines ; this has coarsened to the point where –nite size eÜects are obvious even transverse to the —ow direction.The structure is strongly oriented with continuous domains of the two —uing right through the sample from front to back; however it remains connected transverse to the —ow with thin necklike bridges clearly visible. In fact for /\0.5 such transverse connectivity is inevitable unless the symmetry between the two —uids is spontaneously broken. Thus the structure remains capable of coarsening further by a non-diÜusive hydrodynamic mechanism. For this system size the –nal state could be total separation just as it would be without shear —ow such a state might be stable so long as the –nal interface between the two —uids is parallel to the streamlines. C. Discussion In our simulations on these (fairly small) systems we see no evidence of a steady state structure emerging that is independent of system size.This is intriguing. If the Doi»Ohta scaling is correct Fig. 6 Time evolution of three length scales given by the eigenvalues of the curvature matrix for the LB system with c 5 T0\0.14. Faraday Discuss. 1999 112 1»11 8 Fig. 7 (a) LB simulation with /\0.5 prior to steady state being achieved. Shown is a central 323 section from a 643 system with the shear applied through the top plane moving left and the bottom plane moving right. Light blue denotes the —uid/—uid interface. The horizontal plane shows a slice through the order parameter pro–le with solid colours representing the two —uids this shows the interfacial sharpness. (b) DPD simulation with /\0.2 and c 5 T0\0.32.Tubes are oriented along the streamlines. (c) LB simulation with /\0.5 and c 5 T0\0.14. This view is along the streamlines ; the top and bottom edges are the moving boundaries. The block colours (as in (a)) show the identity of the two —uids now on a vertical plane towards the back of the simulation cell. Passing in front of this one observes narrow —uid necks connecting blocks of similar —uid. we should obtain a lengthscale given by eqn. (5) ; so long as this is small compared to K (and the steady state structure is not too anisotropic) the simulation should achieve a proper steady state representative of that of an in–nite sample. Put diÜerently this will arise if the maximum capillary number attainable in a system of size K Ca\c 5 Kg/p is much larger than one.This is true for our largest DPD runs with CaD7 yet we see no sign of any steady state not dominated by –nite size eÜects ; larger runs would be useful to clarify this issue. An important possibility which these results suggest is as follows. By neglecting inertia and yet assuming a steady state Doi and Ohta implicitly assume that this steady state can be characterized by a notional capillary number Ca\c 5 L g/p (based on a single measure of domain size L ) of order one. This might be incorrect without inertia there may simply be no obstacle to complete separation of the two —uids in simple shear. Indeed one can instead postulate a fully phase separated state with two blocks of —uid separated by an interface in the plane of shear (or in fact any other orientation with the surface normals everywhere perpendicular to the —uid velocity).9 Faraday Discuss. 1999 112 1»11 For such a —ow the notional capillary number is in–nite (since L is in–nite) but the eÜective value is zero the interface is oriented parallel to the streamlines and there is no stretching of it by the —ow. This structure is likely to be destabilized by inertial contributions since it involves steady shear of a laminar interface between two —uids (although this may require a threshold of shear rate such as c 5 T0^1 to be exceeded). Thus it remains possible that a steady state with a nonin –nite domain scale can arise only when inertial terms are signi–cant. If so the Reynolds number is just as important as the capillary number and the implicit criterion Ca^1 of the Doi»Ohta theory is not adequate.To test this point one needs to run simulations under shear at high Re. We intend to pursue this using our parallel LB code,9 in the near future. Meanwhile it is straightforward to check that at the system sizes and shear rates used here a starting con–guration with oriented slabs of —uid (with an interface parallel to the shear planes) remains essentially unperturbed by the —ow. However if the initial interface normal is not perpendicular to the streamlines the slabs are immediately broken up and the late stage coarsening (with a slowly-evolving tubular structure along the streamlines) resembles that which we found above for the case of an initially homogeneous phase quenched to form demixing domains in the presence of steady shear.IV. Conclusions We have described two problems in the physics of immiscible binary —uids where inertial eÜects are important. One is the late stages of demixing (in the absence of applied —ow) where the internal dynamics of the system drives it to high Reynolds numbers. Whether the –nal Re is self-limiting as recently suggested by Grant and Elder4 remains to be seen ; however we have observed5 clear evidence of an inertial-dominated regime in which the Furukawa (lDt2@3) scaling is found. The second problem is in the steady state behaviour of binary —uids in simple shear. Our preliminary simulation results though far from conclusive inspired us to speculate that the very existence of such a steady state is itself dependent on inertial eÜects.If so the scaling analysis that underlies Doi»Ohta theory,6 is in doubt. Future simulations on much larger systems should allow us to settle this point. We thank Simon Jury Patrick Warren and Julia Yeomans for valuable discussions and Alistair Young for the turbulence dns simulation code. Work funded in part under the EPSRC E7 Grand Challenge. References 1 A. J. Bray Adv. Phys. 1994 43 357. 2 See e.g. K. Kubota N. Kuwahara H. Eda and M. Sakazume Phys. Rev. A 1992 45 R3377; S. H. Chen D. Lombardo F. Mallamace N. Micali S. Trusso and C. Vasi Prog. Colloid Polym. Sci. 1993 93 331; T. Hashimoto H. Jinnai H. Hasegawa and C. C. Han Physica A 1994 204 261. 3 A. J. Wagner and J. M. Yeomans Phys. Rev. L ett.1998 80 1429. 4 M. Grant and K. R. Elder Phys. Rev. L ett. 1996 82 14. 5 V. M. Kendon J.-C. Desplat P. Bladon and M. E. Cates Phys. Rev. L ett. in press cond-mat 9902346. 6 M. Doi and T. Ohta J. Chem. Phys. 1991 95 1242. 7 M. R. Swift E. Orlandini W. R. Osborn and J. Yeomans Phys. Rev. E 1999 54 5041. 8 R. D. Groot and P. B. Warren J. Chem. Phys. 1997 107 4423. 9 P. Bladon and J.-C. Desplat in preparation. 10 S. I. Jury P. Bladon S. Krishna and M. E. Cates Phys. Rev. E. 1999 59 R2535. 11 E. D. Siggia Phys. Rev. A 1979 20 595. 12 H. Furukawa Adv. Phys. 1985 34 703. 13 M. Laradji S. Toxvaerd and O. G. Mouritsen Phys. Rev. L ett. 1996 77 2253. 14 S. Bastea and J. L. Lebowitz Phys. Rev. L ett. 1997 78 3499. 15 The linear law has been reported by a number of groups for which reliable parameter values are unavailable T.Koga and K. Kawasaki Phys. Rev. A 1991 44 R817; S. Puri and B. Dué nweg Phys. Rev. A 1992 45 R6977; F. J. Alexander S. Chen and D. W. Grunau Phys. Rev. B 1993 48 634; linear –ts were not oÜered by W. Ma A. Maritan J. R. Banavar and J. Koplik Phys. Rev. A 1992 45 R5347; A. Shinozaki and Y. Oono Phys. Rev. L ett. 1991 66 173 although we believe these data to be at low Re; C. Appert Faraday Discuss. 1999 112 1»11 10 and S. Zaleski Phys. Rev. L ett. 1990 64 1 claimed to see the inertial region but we doubt this ; see ref. 5. 16 A. S. Monin and A. M. Yaglom Statistical Fluid Mechanics ed. J. Lumley MIT Press Cambridge MA 1975 vol. 2. 17 V. M. Kendon J.-C. Desplat P. Bladon and M. E. Cates in preparation. 18 S. I. Jury P. Bladon M. E. Cates S. Krishna M. Hagen N. Ruddock and P. B. Warren Phys. Chem. Chem. Phys. 1999 1 2051. 19 Simulations in 3-D T. Ohta H. Nozaki and M. Doi J. Chem. Phys. 1990 93 2664; J. F. Olson and D. H. Rothman J. Stat. Phys. 1995 81 199; simulations in 2-D P. Padilla and S. Toxvaerd J. Chem. Phys. 1997 106 2342; F. Corberi G. Gonella and A. Lamura Phys. Rev. L ett. 1998 81 3852. 20 A. J. Wagner and J. M. Yeomans Phys. Rev. E 1999 59 4366. 21 R. G. Larson T he Structure and Rheology of Complex Fluids Oxford University Press New York 1999. 22 This is the main reason for the large wiggles ; see also Y. Navot and M. Schwartz Phys. Rev. L ett. 1997 79 4786 for a discussion of periodic breakup and reconnection of droplets. 23 B. J. Ackerson and N. A. Clarke Phys. Rev. A. 1984 30 906. Paper 9/03105G 11 Faraday Discuss. 1999 112 1»11

ISSN:1359-6640    

DOI:10.1039/a903105g

出版商:RSC    

年代:1999

数据来源: RSC

摘要:

A scattering study of nucleation phenomena in polymer crystallisation Anthony J. Ryan,*ab J. Patrick A. Fairclough,a Nicholas J. Terrill,b Peter D. Olmstedc and Wilson C. K. Poond a Department of Chemistry T he University of Sheffield Sheffield UK S3 7HF b CCL RC Daresbury L aboratory W arrington UK WA4 4AD c Department of Physics T he University of L eeds L eeds UK L S 9JT d Department of Physics and Astronomy T he University of Edinburgh Edinburgh UK EH9 3JZ Received 7th January 1999 The mechanism of primary nucleation in polymer crystallisation has been investigated experimentally and theoretically. Two types of experiments have been performed on polypropylene polyethylene and poly(ethylene terpthalate). Crystallisations with long induction times studied by small and wide angle X-ray scattering (SAXS and WAXS) reveal the onset of large scale ordering prior to crystal growth.Rapid crystallisations studied by melt extrusion indicate the development of well resolved oriented SAXS patterns associated with large scale order before the development of crystalline peaks in the WAXS region. The results suggest pre-nucleation density —uctuations play an integral role in polymer crystallisation. A theoretical model has been developed which qualitatively describes the experimental results. (1) G(T p)\(U]pV )[T S Introduction Processing of semicrystalline thermoplastics relies on the shaping of molten material in either moulds or dies and the stabilisation of the shape produced by crystallisation.1 During crystallisation a microstructure develops which can control the mechanical and aesthetic properties of the polymer.To produce useful materials it is essential to understand and predict this process. Polymers in solutions and melts can be regarded as random objects whose size and shape is governed by inter- and intra-molecular interactions but is dominated by entropy. In the crystal this is no longer true and the behaviour of the chain is now in—uenced by the proximity of the neighbouring chains and the van de Waals forces which act between them. The Gibbs free energy G is a balance between entropy S and the enthalpy H\U]pV where U is the internal energy p the pressure and V the volume of the system thus In the melt entropy dominates and the polymer has a Gaussian (random) structure.Crystallisation is a process involving the regular arrangement of chains and is consequently associated with a large negative entropy change. For the free energy change upon crystallisation to be 13 Faraday Discuss. 1999 112 13»29 favourable there must also be a large negative enthalpy change generally associated with an increase in density and a reduction in internal energy The creation of a stable 3-D structure from a disordered state (i.e. a polymer melt) is generally considered a two step process. The –rst step is called nucleation and involves the creation of a stable nucleus from the entangled polymer melt. Most technical processes involve secondary or heterogeneous nucleation either from specially added surfaces nucleating agents or adventitious surfaces such as dust particles.In primary (homogeneous) nucleation creation of a stable nucleus is brought about by the ordering of chains in a parallel array stimulated by intermolecular forces. As a melt is cooled there is a tendency for the molecules to move toward their lowest energy conformation and this will favour the formation of co-operatively ordered chains and thus nuclei. However two factors impede the ordering required for nucleation ; cooling which reduces diÜusion coefficients and chain entanglements. In fact the thermal motion needed for diÜusion may be enough to cause the incipient nuclei to melt. The second step is growth of the crystalline region by the addition of other chain segments to the nucleus. This growth is impeded by low diÜusion coefficients at low temperatures and thermal redispersion of the chains at the crystal/melt interface at high temperatures.Thus the crystallisation process is limited to a range of temperatures between the glass transition temperature T and the melting point Tm . The alignment of polymer chains at speci–c g distances from one another to form crystalline nuclei will be favoured when intermolecular forces are strong. The greater the interaction between chains and the easier they can pack the greater the energy change will be. Thus symmetrical chains and strongly interacting chains are more likely to form stable crystals. Despite the maturity and market penetration of the polymer industry one aspect of polymer processing primary nucleation is little understood.Whereas the growth of polymer crystals is well established in literature and there are usable theories to predict the kinetics of crystallisation understanding of the initiation or nucleation step remains somewhat of a mystery. The available theories are complicated and somewhat unphysical.2 There is good reason for this experimental access to the nucleation step is very difficult whereas studies of crystal growth are simple enough to be used in undergraduate laboratory classes.3 Upon cooling a crystallisable polymer melt a hierarchy of ordered structures emerges. Elegant experimental and theoretical work (on melt and solution crystallised materials especially single crystals) in the 1960s and 70s allowed useful models to be developed.4,5 First there are crystalline ìlamellaeœ comprising regularly packed polymer chains each of which is ordered into a speci–c helical conformation.These lamellae interleave with amorphous layers to form ìsheavesœ which in turn organise to form superstructures e.g. spherulites. These structures may be probed by various techniques wide-angle X-ray scattering (WAXS) is sensitive to atomic order within lamellae (giving rise to Bragg peaks) while small-angle X-ray scattering (SAXS) probes lamellae and their stacking. Electron microscopy is useful for visualising the crystal structure (by selected area electron diÜraction) the lamellae morphology (by transmission microscopy of surface replicas or direct scanning electron microscopy) and spherulites (by scanning electron microscopy) and this has been reviewed in some detail.6 The growth of crystals by secondary nucleation have been explained by various versions of ììregimeœœ theories.2 In the classical picture of polymer melt crystallisation we expect and observe Bragg peaks in WAXS after an induction period qi .SAXS accompanies the WAXS corresponding to interleaved crystal lamellae and amorphous regions.7 No SAXS is expected during qi . However recent experiments8h12 have reported SAXS peaks during the induction period and before the emergence of Bragg peaks. Initially the SAXS peak intensity grows exponentially and it may be reasonably –tted to Cahn»Hilliard (CH) theory13 for spinodal decomposition; that is the spontaneous growth of —uctuations indicative of thermodynamic instability.The peak moves to smaller angles in time stopping when Bragg peaks emerge. By –tting to CH theory an extrapolated spinodal temperature (at which the melt –rst becomes unstable towards local density —uctuations) can be obtained.12 Spinodal kinetics have been reported in diÜerent polymer melts poly(ethylene terphthalate) (PET),8h10 poly(ether ketone ketone) (PEKK),11 polyethylene (PE) and isotactic polypropylene (iPP).12 An argument has been made against the spinodal crystallisation mode by Janeschitz- Kriegl,14 who invoked a variety of liquid»gas analogies and consideration of surface tensions in spite of considerable experimental evidence for spinodal modes. To explain recent results a spino- Faraday Discuss. 1999 112 13»29 14 dally assisted crystallisation (SAC) model has been developed,15 which deals with the contributions to the free-energy in terms of coupled order parameters of density and chain conformation and predicts a liquid»liquid phase diagram buried in the liquid»crystal coexistance region.Recent rheological studies16 show that gelation occurs in crystallising melts at very low degrees of crystallinity typically \3% shortly after qi . This means that there is a percolating network formed that is observed in the dynamic mechanical response of the system. This is not consistent with impingement of spherulites which only occurs at much higher (typically [40%) degrees of crystallinity. Furthermore there are also anomalies in the measurements of spherulite growth rates,17 whilst heat capacity volume change and X-ray experiments observe an induction time a plot of the variation in spherulite radius with time is seen to pass through the origin indicating that spherulites started to grow without an induction period being observed.Both pieces of evidence indicate that large-scale structure is formed prior to the appearance of Bragg peaks or evolution of appreciable enthalpy. In this paper we present new experimental evidence for precrystallisation density —uctuations in a range of polymer melts. In order to separate nucleation from growth two types of experiments have been performed on polypropylene polyethylene and poly(ethylene terepthalate). Rapid crystallisations studied by melt extrusion indicate the development of well resolved oriented SAXS patterns associated with large scale order before the development of crystalline peaks in the WAXS region.Crystallisations with long induction times studied by small and wide angle X-ray scattering (SAXS and WAXS) reveal the onset of large scale ordering prior to crystal growth. The experiments are compared to the SAC model and discussed in light of other experimental evidence for large scale structure prior to formation of crystals with Bragg peaks. Experimental Materials The polypropylene used was a commercial grade S-30-S (DSM) which was free from any additives. The number-average molar mass (GPC) and the polydispersity were 52 kg mol~1 and 2 respectively and the melting point (DSC) was 165^2 °C. The as-received iPP pellets had a degree of crystallinity of 0.6 as determined from measurements of the heat of fusion.The HDPE was a commercial extrusion grade (Shell) which was free from any additives. The melting point (DSC) was 135^2 °C.18 The PET was a gift from ICI and was an experimental grade which was free from any additives. The melting point (DSC) was 265^2 °C.18 Fast cystallisations by SAXS/WAXS/extrusion Simultaneous SAXS/WAXS/extrusion measurements were made on beamline 16.1 of the SRS at the CCLRC Daresbury Laboratory Warrington UK. The details of the storage ring radiation and camera geometry and data collection electronics have been given in detail elsewhere.18 An extruder above the X-ray position was used to provide a steady stream of crystallising polymer past the X-ray beam.Tape extrusion is a steady-state process which shows post-die plug —ow therefore the distance down the spinline where the observation was made correlates with the time since the material left the extruder die. The material in the X-ray beam is continuously replaced by material with the same shear and temperature history. A tape of polymer melt was extruded from a die (of dimensions 0.5]3 mm at Tm]40 °C) and collected via a wind up mechanism below the X-ray beam. The extruder used was an AXON BX18 which operated in starve-feed mode to minimise the time the polymer spent in the melt. The experimental set-up is shown schematically in Fig. 1. The distance from the die head to the beam could be varied between 0.3 and 1.8 m. The camera is equipped with a multiwire area detector (SAXS) located between 3 and 8 m from the sample position with —ight path between the WAXS and SAXS detector being under vacuum.Two types of area detector were employed for the WAXS. In order to observe the development of structure during an extrusion experiment a second multiwire area detector was used which was oÜset from the centre line of the beam and located approximately 20 cm from the extruded tape. Time resolved SAXS/WAXS measurements were made with the multiwire area detector intersecting either the meridian or the equator. The spatial resolution of the electronic area detectors is 400 lm and they can handle up to D500 000 counts s~1. To probe a wide range of reciprocal 15 Faraday Discuss. 1999 112 13»29 Fig. 1 Schematic diagram of the SAXS/WAXS/extrusion experimental set-up.space an image plate with a hole was used to record —at plate WAXS patterns contemporaneously with SAXS. The exposure time was set at 30 s using a mechanical shutter. A4 size Fuji and Kodak image plates were read with a Molecular Dynamics image plate reader with a spatial resolution of 176 lm. The scattering pattern from an oriented specimen of wet collagen (rat-tail tendon) was used to calibrate the SAXS detector and HDPE aluminium and an NBS silicon standard were used to calibrate the WAXS detectors.19 Parallel-plate ionisation detectors placed before and after the sample recorded the incident and transmitted intensities. The experimental data were corrected for background scattering for sample thickness and transmission and for the positional alinearity of the detectors.Slow crystallisations by SAXS/WAXS/DSC Simultaneous SAXS/WAXS/DSC measurements were made on beamline 8.2 of the SRS at the CCLRC Daresbury Laboratory Warrington UK. The details of the storage ring radiation and camera geometry and data collection electronics have been given in detail elsewhere.20 The camera is equipped with a multiwire quadrant detector (SAXS) located 3.5 m from the sample position and a curved knife-edge detector (WAXS) that covers 120° of arc at a radius of 0.2 m. A vacuum chamber is placed between the sample and detectors in order to reduce air scattering and absorption. The WAXS detector has a spatial resolution of 100 lm and can monitor up to D50 000 counts s~1; only 60° of arc are active in these experiments with the rest of the detector being shielded with lead.A beamstop is mounted just before the SAXS exit window to prevent the Faraday Discuss. 1999 112 13»29 16 direct beam from hitting the SAXS detector which measures intensity in the radial direction (over an opening angle of 70° and an active length of 0.2 m) and is suitable only for isomorphous scatterers. The spatial resolution of the SAXS detector is 200 lm and it can handle up to D500 000 counts s~1. Disk specimens of polymer (thickness D1 mm diameter D6 mm) were cut from pre-moulded sheet. A disk specimen for SAXS/WAXS/DSC was encapsulated in a TA Instruments DSC –tted with mica windows (thickness D25 lm diameter D7 mm) and the pan was inserted into a Linkam DSC apparatus of the single-pan design that has been described in detail elsewhere.20 The cell comprises a silver furnace around a heat-—ux plate containing 3]0.5 mm slot for X-ray access and the sample is held in contact with the plate by a spring of low thermal mass.The temperature was calibrated using the melting points of high purity indium and tin. In the present study a multi-stage temperature programme was used as follows for iPP and HDPE (i) heat to 200 °C at 50 °C min~1; (ii) hold at 200 °C for 1 min; (iii) cool at 60 °C min~1 to the crystallisation temperature and hold for 10»120 min depending on the temperature. The PET samples were heated to 290 °C. The crystallisation time at each temperature was chosen so that at least half primary crystallisation kinetics were observed.Data accquisition strategies were chosen so that for the longer crystallisations the samples were only exposed to the X-ray beam for 10% of the time after crystallisation had started to limit beam damage. Data were reduced to intensity versus scattering vector using the CCP13 programme xotoko.21 The peak intensities and areas were calculated using the CCP 13 programme –t.22 For the SAXS data a Gaussian peak (whose position was a variable) was –tted on top of a Porod background. For the WAXS Gaussian peaks were –tted on top of a Gaussian background. In iPP the positions of the 4 Gaussian peaks were set according to the positions of the iPP re—ections at hkl values of 110 040 130 (111 131 041). Two Gaussian peaks were –tted at the positions of the 110 and 200 re—ections for HDPE and for 100 and 110 re—ections for PET.Results The crystallisation curve shown in Fig. 2 was obtained in a SAXS/WAXS/DSC experiment from iPP (ref. 23) and shows the classic features of primary crystallisation. The detailed molecular structure of the polymer the speci–c nature of the nucleation process and the degree of undercooling determine the magnitude of the lamellar thickness and the degree of crystallinity within the lamellar stacks. The crystallisation kinetics are analysed using the Avrami model,24 expressed Fig. 2 The degree of crystallinity X versus crystallisation time for PP at 110 °C. The solid circles are SAXS and the open squares are WAXS measurements. The solid line is a –t of the Avrami equation to the average degree of crystallinity.The inset shows a conventional Avrami plot of [ln(1[X)] versus ln t the line has a slope of 3. 17 Faraday Discuss. 1999 112 13»29 in terms of the equation (2) 1[Xs\ektn where k is a rate constant and n is an integer which is sometimes interpreted in terms of the growth dimension. Fig. 2 shows a plot of X versus t from SAXS and WAXS in which the experi- s mental data have been –tted with the Avrami expression over the whole of the primary crystallisation process. The –t to the data gives a value for the exponent n of 3.0^0.1 and the inset is the conventional form of the Avrami plot. The value of n obtained is consistent with random nucleation of spherulites and is in good agreement with the crystallisation kinetics data obtained from dilatometric and calorimetric studies on iPP.25 It is interesting to note that there are signi–- cant deviations of the model from the data at the extremes of crystallisation and that the evolution of the structure probed by SAXS and WAXS is also diÜerent at the beginning and end of the primary crystallisation process.The long time diÜerences have been reviewed in some detail26 whereas the initial diÜerences have been largely ignored. It is difficult to separate nucleation from growth in crystallisation experiments due to the low concentration of nuclei which gives rise to poor counting statistics in a scattering experiment. One potential method is to borrow from elementary chemical kinetics27 and use a —ow apparatus. An extruder operating at steady state provides such a set-up (see Fig.1). Polymer above the melting point is extruded from a die the tape or –bre cools in the air (in our case in a column of chilled nitrogen) prior to being wound up as a solid. Extrusion of tape or –bre is a steady-state process where the crystallisation time increases down the spin-line. This allowed long data collection times (minutes) for very early stages of crystallisation (milliseconds). Prior to the development of crystals well resolved oriented small-angle patterns could be observed with length scales (50»200 Aé ) and intensities that grew down the spin-line the corresponding WAXS showed no Bragg peaks due to crystals. Fig. 3 and 4 shows the scattering patterns collected during extrusion of iPP and HDPE.At short times these early patterns have two Fig. 3 Data taken during extrusion of iPP with a low wind-up speed. 2-D wire chamber to collect the SAXS data and an image plate to collect the WAXS data. (a) Close to the die head tear shaped scattering features are observed in 2-D SAXS with only an amorphous halo in 2-D WAXS. (b) At the furthest position from the die head when crystallisation starts isotropic rings from crystals can be seen in SAXS and in the WAXS. Faraday Discuss. 1999 112 13»29 18 Fig. 4 Data taken during extrusion of HDPE with a low wind-up speed. 2-D wire chamber to collect the SAXS data and an image plate to collect the WAXS data. (a) Close to the die head tear shaped scattering features are observed in 2-D SAXS with only an amorphous halo in 2-D WAXS.(b) At the furthest position from the die head when crystallisation starts isotropic rings from crystals can be seen in SAXS and there are oriented arcs in the WAXS (due to preferential chain orientation in the spherulites). SAXS peaks at –nite q and no WAXS peaks. We interpret this as a signature of density —uctuations. The orientation observed in the scattering is caused by coupling of density —uctuations with the slight elongational —ow-–eld (the take-up speed was approximately twice the extrusion speed). Once crystallisation had been observed in the wide-angle region the shape of the small-angle pattern changed from that characteristic of sinusoidal density —uctuations to that typical of lamellar crystals. Since the elongational —ow was weak the crystallisation process dominated and only weakly anisotropic crystals were produced.The SAXS peaks shown at early times in Fig. 3 and 4 were approximately 100 times weaker than the diÜraction ring observed in SAXS once spherulitic crystallisation had occurred. There are a number of other possible interpretations of the data (1) The SAXS peak could be due to the formation of oriented nuclei the precursor to the ììshishœœ in ììshish-kebabœœ crystals and row nucleation,28 but this should also lead to orientation in the WAXS which is not observed furthermore the elongation is less than a factor of 2 which is very low for formation of such structures.1,4 (2) Nucleation could have formed poorly ordered crystals which do not diÜract. This would account for the lack of Bragg peaks in the WAXS (peak broadening due to small crystallites) but does not account for the peak in SAXS the scattering from a low concentration of randomly oriented objects would give a peak at q\0 from the shape factor of the scatterers.29 Guinier» Preston zones which form in systems with conserved order parameters,30 have nuclei surrounded by a depletion layer.These would give a peak in SAXS at low concentration due to the shape factor. However in this system with a non-conserved order parameter the nuclei are not surrounded by a depletion layer as regions of high density grow from a background of low density with an overall increase in density as shown in Fig. 5 and the electron density pro–le does not have a peak at –nite q.[In crystallisation of real systems there is a non-conserved order parameter (the crystals) which is coupled to a conserved order parameter (the density).] 19 Faraday Discuss. 1999 112 13»29 Fig. 5 Schematic diagram of the electron density pro–le of a Guinier»Preston zone and a nucleus for a polymer crystal. (3) The most obvious alternative explanation of the observation is that there is an outer skin of crystalline material due to the temperature gradient across the tape which is giving rise to the SAXS. In this situation the crystals formed would be well ordered and one would expect them to diÜract at wide angles which is obviously not the case. Under the current experimental conditions it is possible to observe diÜraction from 1% by volume of a crystalline ole–n dispersed in oil.31 In order to check the temperature of the melt in the scattering volume a number of techniques were tested an optical pyrometer indicated that the data taken close to the X-ray position [in Fig.3(a)] had a temperature in excess of 120 °C. Whilst it is undoubtedly true that at long times crystallisation proceeds from the exterior of the tape is unlikely that it has occurred in the data presented here. (4) A caveat to the interpretation of the data concerns the intensity (solid angle) on the Ewald sphere. Assuming peaks of equal strength the measured intensities scale as 1/d2 typically 100 and 5 Aé for SAXS and WAXS respectively. The SAXS detector is 5 times further away from the sample so for the same detector efficiencies the intensities scale as the reciprocal of distance squared.Overall the measured WAXS intensity will be approximately 0.06 (\52]52/1002) of the SAXS intensity for peaks of equal strength. For semicrystalline polymers not withstanding the argument above when compared to the background intensity WAXS peaks are generally very strong and SAXS peaks are often quite weak. Previous SAXS/WAXS studies on polymer extrusion have concentrated on the growth and orientation of crystals.32,33 Interestingly both studies Cakmak et al.32 on the extrusion of PVDF tape (using synchrotron radiation source and electronic detectors) and Katayama33 on the extrusion of PET –bres (using a sealed tube source and –lm as a detector) showed SAXS before WAXS down the spin line but made no comment on its signi–cance.Slow crystallisations with long induction times have been studied by simultaneous SAXS and WAXS. These experiments on quiescent samples show a clear development of a SAXS peak due to electron density —uctuations prior to the presence of crystals identi–ed by WAXS. The peak area versus time data in Fig. 6 for iPP at 137 °C show unequivocally that the SAXS peak grows before the WAXS peak. This whole of this data set corresponds to the –rst 100 s in Fig. 2 and the crystallinity at 1200 s in Fig. 6 is B0.1. In the –rst 200 s of the experiment there is no scattering above the background intensity. Between 200 and 400 s there is a measurable SAXS intensity above the background with no WAXS after 400 s Bragg peaks are observed and after 800 s the growth in SAXS and WAXS map onto each other.The inset to Fig. 6 shows the logarithm of the peak intensity versus time for the period where there is SAXS without WAXS and it gives a good straight line –t. Similar behaviour has been reported previously for semi-rigid polymers crystallised by devitrifying a glass8h11 and the kinetics of crystallisation after devitri–cation were analysed in terms of the Cahn»Hilliard13 theory for spinodal decomposition. It has been shown that the general form of the variation in scattered intensity I(q t) following a quench is given by the following equation (3) I(q t)\I(q 0)exp[2R(q)t] Faraday Discuss. 1999 112 13»29 20 Fig. 6 Integrated intensity data for a crystallisation at 128 °C. The open symbols are WAXS and closed symbols SAXS intensities respectively.The inset is a plot of the logarithm of the SAXS peak intensity versus time. The variation in I(q) at a given time interval is determined by the scattering law P(q) in the homogeneous state. R(q) is termed the growth rate constant and is given by eqn. (4) (4) Lo2 ]2iq2D R(q)\[Mq2CL2G where M is the mobility term G is the Gibbs free-energy and i is a gradient free-energy term. Modi–cations to eqn. (4) have been discussed previously by Cook34 which take into account random thermal —uctuations (inclusion of a Brownian motion term). In employing eqn. (4) to analyse the data the extremums are not strictly correct. The q dependence of the Onsager coefficient L 0 relating the diÜusive —ux of polymer molecules to the local chemical potential has been neglected.This may be valid for the early stages of phase separation and a shallow quench. It should be noted that L generally does have a q dependence. This dependence has been calculated by Pincus35 for a polymer blend (L 0Pq~2) but not for a homopolymer. Neglecting the Onsager 0 coefficient R(q)/q2 can be taken as a measure of the dynamic driving force for the growth of the concentration —uctuation with wave vector q/2p. There is a region of q in which R(q) and thus R(q)/q2 are positive and the concentration —uctuations do not decay but grow and give rise to phase separation. These growing concentration —uctuations have upper and lower critical boundaries to their wavenumbers. Outside these limits the concentration —uctuations decay and do not contribute to the phase separation dynamics.As originally published the Cahn»Hilliard theory13 of spinodal decomposition is a macroscopic description and has no direct relation to events at molecular level. The thermodynamic driving force for the growth of the concentration —uctuation with wave vector q/2p R(q)/q2 becomes a maximum at q\qm\JGA/i. Thus the wavelength q/2p of the dominant Fourier component of the growing —uctuations in the early stages of phase separation is determined by the Fourier component that exhibits maximum dynamic driving force. q is time m independent in the early stages of phase separation and is controlled by thermodynamics.36 R(q) is further controlled by the transport properties. Deff\[ 2R q2 (q) (5) q]0 The eÜective diÜusion coefficient Deff can be determined from an extrapolation to q\0 of the straightline portion of R(q)/q2 during phase separation using eqn.(5) and the linearity holds for qm\q\)2qm . 21 Faraday Discuss. 1999 112 13»29 Values of the ampli–cation factor R(q) for the early stage of crystallisation where we observe SAXS but no WAXS were determined by plotting ln I versus t for discrete wave vectors and –nding the slope.36 Fig. 7(a) shows a typical plot of R(q)/q2 versus q2. The solid line is a –t to the D data to allow estimation of from the q\0 intercept. The inset is a plot of R(q) versus q which eff shows the errors in R. Similar R(q)/q2 versus q2 plots were constructed for each polymer at a range of temperatures and those for iPP are shown in Fig.7(b) the data have been truncated at low q as the down turn has been previously shown to be an experimental artefact.36 As the crystallisation temperature is increased it should be noted that the data get less noisy (as the kinetics slow down and counting statistics improve) and the linear part of the graph is reduced this is because qm moves to lower values. Fig. 8»10 show plots of Deff versus 1/T from which the spinodal temperature is determined by D the eff\0 intercept. For example in polypropylene at 410 K we could estimate both the dominant length scale L B175 Aé and the eÜective co-operative diÜusion coefficient Deff\[4.5 Aé 2 s~1. By conducting these experiments at a series of temperatures the stability limit could be found at 415^5 K (ref.37) by extrapolation of D to zero. The stability limits obtained in this eff way are compared to the melting point of an in–nite crystal T m 0 in Table 1. The stability limit is the temperature below which the polymer spontaneously separates into two phases. One of which is rich in polymer segments of the appropriate chain conformation to crystallise (trans»gauche arrangement of the carbon backbone in isotactic polypropylene all trans in polyethylene38) and the other is depleted in polymer segments with the appropriate chain conformation to crystallise and is concentrated in sequences near entanglements and other defects which cannot crystallise. The stability limit is 7 K below the measured melting point for poly- Cahn»Hilhard plots to estimate the eÜective diÜusion coefficient,D Fig.7 (a) Cahn»Hilliard plot to estimate the eÜect diÜusion coefficient D from the q\0 intercept of eff R(q)/q2 versus q2. The solid line is a linear –t to the data. The inset shows R(q) versus q with error bars. (b) eff for iPP at 120 125 130 135 and 140 °C. Table 1 Stability limits and thermodynamic melting points39 for the polymers studied T Ts/T m 0 m 0 Ts 0.98 0.90 0.87 HDPE iPP PET 417 459 573 408 415 499 Faraday Discuss. 1999 112 13»29 22 Fig. 8 Plot of Deff versus 1/T (for iPP) to allow calculation of the spinodal temperature from extrapolation to Deff\0. propylene with a long spacing of 175 Aé and 44 K below the thermodynamic melting point of isotactic polypropylene.39 The measured stability limits are given in Table 1.Once WAXS from crystals (atomic order on the 1 Aé scale) was observed the kinetics reverted to those of nucleation and growth that is Avrami kinetics with an exponent nB3 (see Fig. 2). Similar behavior has also been observed in devitri–ed glasses of PET by Kaji and co-workers,8h10 and PEEK by Ezquerra and co-workers11 however as the measurements are made close to the glass transition and the dynamics are dominated by the viscosity estimation of the stability limit is not possible as Deff increases with temperature. For each of the polymers studied the quiescent time-resolved SAXS/WAXS and extrusion suggest that a process that strongly resembles spinodal decomposition of chain segments with diÜerent average conformations is the nucleation step in polymer crystallisation.That polymer crystallisation occurs with phase separation is in no doubt since at the end of the process regions of well ordered crystalline polymer coexist with regions of disordered polymer in a layered morphology (lamellae) with a spherulitic super-structure. Sequences that can be oriented with the right conformation and incorporated into the crystal separate from sequences near entanglements and Deff versus 1/T (for HDPE) to allow calculation of the spinodal temperature from extrapo- Fig. 9 Plot of lation to Deff\0. 23 Faraday Discuss. 1999 112 13»29 Deff versus 1/T (for PET) to allow calculation of the spinodal temperature from extrapolation Fig. 10 Plot of to Deff\0. other defects which can not crystallize and can only be part of the amorphous regions.The transformation from the disordered phase to the better ordered partially crystalline phase proceeds continuously passing through a sequence of slightly more ordered states rather than building up a crystalline state instantaneously this is consistent with the evolution of SAXS but at some stage secondary nucleation must form crystals directly from the melt. A mechanism of continuous transformation could be consistent with a fast homogeneous nucleation process. However it is difficult to make a clear distinction between spinodal decomposition and nucleation and growth with nucleation barriers smaller than kB T . Polymer crystallisation like any other phase separation is kinetically controlled.The structure formed is the one with the highest growth rate. Once a crystallite is formed its lateral growth rate is much higher than that of the —uctuations and so dominates. In this case the growth mechanism of semi-crystalline polymer lamellae in the form of spherulites takes over because the lateral growth rate of crystals (typically lm s~1) is 104 faster than the growth rate of the —uctuations (typically Aé 2 s~1). Thus the combination of the steadystate extrusion and the high intensity synchrotron X-ray source allows nucleation phenomena to be observed. Theoretical model To understand these observations we have developed a ìminimalistœ phenomenological model which we believe accurately captures the physics involved. The essential observation is the existence of spinodal-like behavior in a supercooled melt.By analogy with similar observations from metallurgy40,41 and recent experiments in colloid»polymer mixtures,42 as well as supercooled water,43 we propose that a metastable liquid»liquid (LL) phase coexistence curve (or ìbinodalœ) lies buried inside the equilibrium liquid»crystal coexistence region as shown in Fig. 11. Quenching sufficiently below the equilibrium melting point Tm we may cross the spinodal associated with the buried LL binodal at temperature Ts\Tm . A possible mechanism leading to such a buried LL binodal is as follows.15 In order to crystallise polymer chains must adopt the correct conformation. For example chains in crystalline PE have the all trans conformation and the chains in crystalline iPP are alternating trans»gauche but in the melt the conformation is randomly trans or gauche.Generally the preferred conformation is a helix. Furthermore the radius of gyration of a (very long) chain changes little during crystallisation suggesting44 that neighbouring segments adopt the correct conformation and crystallise in situ. While it is commonly assumed that conformational and crystalline ordering occur simultaneously we suggest that these processes can occur sequentially. Moreover chains with diÜerent conformations have diÜerent densities and therefore also diÜerent Faraday Discuss. 1999 112 13»29 24 Fig. 11 Proposed generic phase diagram for a polymer melt with a buried liquid»liquid phase predicted by the SAC model.15 T and T are the melting and spinodal temperatures encountered along the constant m density quench path (dotted line).Inset shows the measured induction time as a function of temperature for s iPP. energy barriers for reorientation between rotational isomeric states (RIS).45 Such conformationdensity coupling can induce a LL phase transition. A phenomenological free energy which incorporates these eÜects is a function of the mean mass density o 6 the coefficients of the Fourier expansion of the crystal density in the reciprocal lattice vectors (essentially the intensities of Bragg peaks) and the relative occupancies of various RIS. Consider a free energy with a single Fourier mode o (corresponding to BCC symmetry46) and a pair of RIS with g the population of the * ground (trans) state (6) f\f0(o6 )]f * (o6 o* )]fg(g o6 o* ) f0 is the free energy of a melt with random chain conformations.47 The (bare) * (o6 o* ).fg describes how the distribution of chain The –rst term Landau free energy of crystallisation is given by f conformations varies smoothly from a random coil (g\0) to a helix (g\1) as the temperature is lowered.48 In isolation a polymer thermally populates its RIS with a Boltzmann distribution ; to incorporate the coupling between density and conformation we take the energy gap to have the phenomenological form (7) E(o6 o* )\E0]vo6 ]jo* 2 As more bond sequences occupy the ground state monomers can rearrange to pack tighter and reduce the excluded volume interaction (hence the perturbation vo6 ).A positive v encourages phase separation to take advantage of this density-conformation coupling. Similarly adjacent ground state sequences enhance crystallisation (hence the term jo* 2 ). The j term is quadratic in o* by symmetry.46 To calculate the phase diagram we must minimize the free energy over the mean polymer conformation and then compare the free energy of amorphous and crystal branches of the free energy –nally using the common tangent contruction to –nd the equilibrium state. Physically this calculation says that at low enough temperatures the system gives up conformational entropy to relieve packing frustration and separates into a dense more conformationally homogeneous liquid and a less dense and more conformationally disordered liquid.In practice this happens only at appreciable rates by spinodal decomposition giving rise to two coexisting liquids with a coarsening interconnected domain texture as shown in Fig. 12. The dense liquid is closer in density and conformation to the crystal phase than the original melt with a lower energy barrier D to crystallisation. We expect D to decrease with increasing quench depth below Ts . The induction time q is then a sum of the time to coarsen into an intermediate spinodal texture and i an exponential activation time determined by q D. The strong temperature dependence of should o o6 )OkB To . This has o\Ts T i change over to a much weaker dependence at some D(T where been found in iPP (inset Fig. 11).12 25 Faraday Discuss. 1999 112 13»29 Fig.12 Schematic diagram illustrating the continuous transformation from a Gaussian chain through a microphase separated liquid to a semi-crystalline polymer. Coexisting liquid phases with diÜerent conformations showing a single chain; thin line\disordered conformation thick line\correct (helical) conformation for crystallisation. Each chain is a ìconformational copolymerœ. Our arguments so far have been based on conformation-density coupling. An analogous argument may be made in terms of a liquid»crystalline coupling by which density-orientation eÜects become more important as the polymer stiÜens upon cooling into the preferred helical conformation. This approach was adopted by Imai and co-workers,10 and probed by light scattering. Indeed the two mechanisms have in the main the same physical content.Until recently spinodal scattering was mainly observed in polymer melts crystallising under shear.7,32,49 This may be understood in a natural way within the present framework. Shear (and extensional) —ow couples principally to the orientation of polymer segments hence straightening chains and enhancing g thereby biasing the tendency towards LL separation. A simple way to E[v incorporate this is to renormalise the activation energy E as 0 p where p is the stress. It is highly suggestive that for appropriate values of stress under strong —ow (the plateau modulus G0) and volume (v above) the LL binodal of Fig. 11 is shifted upward signi–cantly (by dTsD 0 0.01E0/kB). Flow will shift the liquid»solid coexistence curve much less because the regions with crystalline order will resist deformation.This simple theory suggests several interesting experiments. First conformational —uctuations just above T could be detected and studied e.g. by s Raman spectroscopy,7 perhaps simultaneously with depolarised light scattering (to monitor orientational —uctuations). Second upon approaching a spinodal line various properties (e.g. correlation length) should exhibit power-law divergences. Third the LL spinodal line can be modi–ed by pressure. In particular it may be possible to access the LL critical point T recent simulations c T suggest a massive enhancement of the nucleation rate in the vicinity of c .50 The eÜect of strain on the crystallisation of PET close to the T has been studied using elegant time resolved scattering g experiments by Blundell et al.51 Crystallisation (determined by WAXS) occurred after the extension of 4 1 and followed –rst order kinetics (i.e.Avrami n\1 which is equivalent to spinodal). Close to the T the rate of the transformation was temperature insensitive as a reduction in tem- g perature caused an increase in the dynamic driving force but a reduction in mobility. These authors interpret this data as the LL energy barrier being reduced by chain orientation in extensional —ow. More generally the coupling of density to (molecular) structural order parameters is an emerging generic theme in the study of supercooled liquids (water»amorphous ice ;43 polymer melts near the glass transition52). Balsara and co-workers53 have described interesting behaviour in hexagonal rod forming block copolymers subjected to a deep quench.The microstructure formation in the liquid and crystal directions is not correlated the growth of crystalline order occurs before the development of a coherent structure along the liquid direction and they argue that this may be a signal of spinodal decomposition in liquid crystals. The ratio of Ts/Tm observed is 0.977 which is in the range described here for polymer crystallisation. The analogy for the case presented here would be chains pack locally in straight sections prior to becoming oriented along their length into lamellae. Faraday Discuss. 1999 112 13»29 26 Fig. 13 Data from Ratajski and Janeschitz-Kriegl17 showing that a plot of spherulite radius versus time passes through the origin for a wide range of crystallisation temperatures in iPP.o o . This is in general agreement with our theoretical Discussion It is obviously very difficult to make a clear distinction between spinodal decomposition and nucleation and growth with nucleation barriers smaller than kB T . Crystallisations with long induction times studied by small and wide angle X-ray scattering (SAXS and WAXS) reveal the onset of large scale ordering prior to crystal growth. Rapid crystallisations studied by melt extrusion indicate the development of well resolved oriented SAXS patterns associated with large scale order before the development of crystalline peaks in the WAXS region. The results suggest prenucleation density —uctuations play an integral role in polymer crystallisation.An interesting observation is made by Ratajski et al.17 Extrapolated spherulite size curves for temperatures below T show no induction time that is the line through the radius versus time curve (Fig. 13) passes through the origin despite the fact that X-ray heat and density measurements show a clear induction time (see Fig. 2). Recent rheological studies16 show that gelation occurs in crystallising melts at very low degrees of crystallinity typically \3% shortly alter qi . This means that there is a percolating network formed that is observed in the dynamic mechanical response of the system. Both pieces of evidence indicating large-scale structure is formed prior to the appearance of Bragg peaks or evolution of appreciable enthalpy.It is tempting to consider setting up an instantaneous structure that in the conformation and density that comprises isotropic sine waves with random phase and direction but –xed wavelength. This sets the subsequent lamellar thickness or long spacing. The crystallisation process which forms the perfected crystals which show Bragg peaks is a perfection and formation of grains the growth of which we see as spherulites. In a technical context Galilietner and co-workers have shown54 that nucleating agents are most eÜective in the ììmetastable supercooling zoneœœ. Nucleating agents cease to work in iPP at around 140 °C which is close to the measured T and that homogeneous nucleation takes over at temperatures below 110 °C which is close tos T model and experimental results the crystallisation rate is vanishing above T and homogeneous s nucleation is very fast below To .Acknowledgements The experimental work originated from an EPSRC ROPA to Professor R. J. Young and A.J.R. at UMIST. W.C.K. and P.D.O. started the theoretical work whilst W.C.K.P. was on sabbatical in Leeds. The many heated discussions with and constructive criticism from Tom Mcleish Frank Bates Herve Marand Andrew Keller Paul Phillips Gerhard Eder Julia Higgins and John Blackwell are most appreciated. 27 Faraday Discuss. 1999 112 13»29 References 1 A. N. Wilkinson and A. J. Ryan Polymer Processing and Structure Development Kluwer Amsterdam 1998. 2 (a) J. I. Lauritzen and J. D. HoÜman J. Res. Natl.Bur. Stand. Sect. A 1960 64 73; (b) J. D. HoÜman J. I. Lauritzen E. Passaglia G. S. Ross L. J. Frohlen and J. J. Weeks Kolloid. Z. Poly. 1969 231 564; (c) J. D. HoÜman G. T. Davis and J. I. Lauritzen T reatise on Solid State Chemistry Plenum New York 1976; (d) J. Petermann R. M. Gohil J. M. Schultz R. W. Hendricks and J. S. Lin J. Polym. Sci. Polym. Phys. Edn. 1982 20 523; (e) D. M. Sadler Polym. Commun. 1986 27 140; ( f ) D. M. Sadler Polymer 1987 28 1440. 3 R. J. Young and P. A. Lovell Introduction to Polymers Chapman and Hall London 2nd edn. 1991. 4 See for example A. Keller and H. W. H. Kolnaar in Materials Science and T echnology ed. R. W. Cahn P. Haasen and E. J. Kramer Wiley-VCH Weinheim 1997 vol. 18 ed. H. E. H. Meijer; P. Barham in Materials Science and T echnology ed.R. W. Cahn P. Haasen and E. J. Kramer Wiley-VCH Weinheim 1993 vol. 12 ed. E. L. Thomas. 5 G. Eder and H. Janeschitz-Kriegel in Materials Science and T echnology ed. R. W. Cahn P. Haasen and E. J. Kramer Wiley-VCH Weinheim 1997 vol.18 ed. H. E. H. Meijer. 6 D. C. Basset Principles of Polymer Morphology Cambridge University Press Cambridge 1981. 7 G. Strobl T he Physics of Polymers Springer-Verlag Berlin 1996. 8 M. Imai K. Kaji T. Kanaya and Y. Sakai Phys. Rev. B 1995 52 12696. 9 M. Imai K. Mori T. Mizukami K. Kaji and T. Kanaya Polymer 1992 33 4451; ibid 1992 33 4457. 10 M. Imai K. Kaji and T. Kanaya Macromolecules 1994 27 7103. 11 T. A. Ezquerra E. Loç pez-Cabarcos B. S. Hsiao and F. J. Balta` -Calleja Phys. Rev. E 1996 54 989. 12 N.J. Terrill J. P. A. Fairclough E. Towns-Andrews B. U. Komanschek R. J. Young and A. J. Ryan Polymer 1998 39 2381. 13 J. W. Cahn and J. E. Hilliard J. Chem. Phys. 1958 28 25; J. D. Gunton M. San Miguel and P. S. Sahni in Phase T ransitions and Critical Phenomena ed. C. Domb and M. S. Green Academic New York 1983 vol. 8. 14 H. Janeschitz-Kriegel Colloid Polym. Sci. 1997 81 1121. 15 P. D. Olmsted W. C. K. Poon T. C. B. McLeish N. J. Terrill and A. J. Ryan Phys. Rev. L ett. 1998 81 373. 16 N. V. Pogodina and H. H. Winter Macromolecules 1998 31 7103. 17 E. Ratajski and H. Janeschitz-Kriegel Colloid Polym. Sci. 1996 274 938. 18 I. W. Hamley J. P. Fairclough N. J. Terrill A. J. Ryan P. Lipic F. S. Bates and E. Towns-Andrews Macromolecules 1996 29 8835.19 W. Bras and A. J. Ryan Adv. Colloid Interface Sci. 1998 75 1. 20 W. Bras G. E. Derbyshire J. Cooke B. E. Komanschek A. Devine S. M. Clark and A. J. Ryan J. Appl. Cryst. 1994 28 26. 21 http://www.dl.ac.uk/SRS/NCD/manual.otoko.html 22 I. W. Hamley R. C. Denny M. Matsen B. Liao C. Booth and A. J. Ryan Macromolecules 1997 38 509; http ://wserv1.dl.ac.uk :800/SRS/CCP13/program/–t.html 23 A. J. Ryan J. L. Stanford W. Bras and T. M. W. Nye Polymer 1997 38 759. 24 M. Avrami J. Chem. Phys. 1939 7 1103; ibid 1940 8 212; ibid 1941 9 177; L. Marker F. M. Hay G. P. Tilley J. Polym. Sci. 1959 38 107. 25 Polypropylene ed. J. Karger-Kocsis Chapman & Hall London 1995 vol. 1. 26 R. K. Verma and B. S. Hsiao T rends In Polymer Science 1996 4 312; B. S. Hsiao B. B.Sauer R. K. Verma H. G. Zachmann S. Seifert B. Chu and P. Harney Macromolecules 1995 28 6931; M. Bark and H. G. Zachman Acta Polymerica 1993 44 259. 27 P. W. Atkins Physical Chemistry OUP Oxford 6th edn. 1997. 28 A. Keller and F. M. Willmouth J. Macromol. Sci. 1972 B6 493. 29 O. Glatter and O. Kratky Small Angle X-ray Scattering Academic Press NY 1982. 30 A. Guinier and G. Fournet Small Angle Scattering of X-rays Chapman & Hall London 1955. 31 B. Hsaio personal communication. 32 M. Cakmak A. Teitge H. G. Zachmann and J. L. White J. Polym. Sci. Polym. Phys. Edn. 1993 31 371. 33 K. Katayama Kolloid Z. Z. Polym. 1968 226 125. 34 H. E. Cook Acta Metall. 1970 18 297. 35 P. Pincus J. Chem. Phys. 1981 75 1996. 36 F. S. Bates and P. Wiltzius J. Chem.Phys. 1989 91 3258. 37 K. Binder in Materials Science and T echnology ed. R. W. Cahn P. Haasen and E. J. Kramer Wiley-VCH Weinheim 1993 vol. 5 ed. P. Haasen. 38 H. Tadokoro Structure of Crystalline Polymers John Wiley & Son 1979. 39 J. E. Mark Physical properties of Polymers American Institute of Physics NY 1996. 40 J. W. Cahn T rans. Metall. Soc. AIME 1968 242 166. 41 See J. W. Martin R. D. Doherty and B. Cantor Stability of microstructures in metallic systems Cambridge University Press Cambridge 1997. Faraday Discuss. 1999 112 13»29 28 42 W. C. K. Poon A. D. Pine and P. N. Pusey Faraday Discuss. 1995 101 65; M. R. L. Evans W. C. K. Poon and M. E. Cates Europhys. L ett. 1997 38 595. 43 S. Harrington R. Zhang P. H. Poole F. Sciortino and H. E. Stanley Phys. Rev. L ett. 1997 78 2409. 44 M. Dettenmaier E. W. Fischer and M. Stamm Colloid Polymer Sci. 1980 258 343. 45 L. R. Pratt C. S. Hsu and D. Chandler J. Chem. Phys. 1978 68 4202. 46 S. Alexander and J. McTague Phys. Rev. L ett. 1978 41 702. 47 J. Brandrup and E. H. Immergut Polymer Handbook Wiley NY 3rd edn. 1989. 48 P. J. Flory Statistical Mechanics of Chain Molecules Oxford University Press Oxford 1989. 49 Flow-induced Crystallization in Polymers Systems ed. R. L. Miller Gordon and Breach NY 1979. 50 P. R. ten Wolde and D. Frenkel Science 1997 277 1975. 51 D. J. Blundell D. H. McKerron W. Fuller A. Mahendrasingham C. Martin R. J. Oldman R. J. Rule and C. Riekel Polymer 1996 37 3303. 52 T. Kanaya A. Patkowski E. W. Fischer J. Seils H. Glaser and K. Kaji Acta Polym. 1994 45 137. 53 N. P. Balsara B. A. Garetz M. C. Newstein B. J. Bauer and T. J. Prosa Macromolecules 1998 31 7668. 54 M. Gahleitner J. Wolfschwenger C. Bachner K. Bernreitner W. Neissl J. Appl. Polym. Sci. 1996 61 649. Paper 9/00246D 29 Faraday Discuss. 1999 112 13»29

ISSN:1359-6640    

DOI:10.1039/a900246d

出版商:RSC    

年代:1999

数据来源: RSC

摘要:

Late stage coarsening in concentrated ice systems Ann-Marie Williamson,* Alex Lips Allan Clark and Denver Hall Unilever Research Colworth Colworth House Sharnbrook Bedford UK MK44 1L Q Received 26th January 1999 We have developed a dedicated automated cryomicroscope for the study of ice coarsening in binary aqueous systems. This together with new image analysis procedures can provide wide ranging characterisation capabilities for the study of isothermal coarsening of ice crystal ensembles. Single particle tracking has elucidated hitherto unattainable mechanistic detail of coarsening kinetics. Ice crystal faceting has been shown to be an important factor. Ice crystals with an average size in the mesoscopic range have a signi–cant tendency to evolve their initial kinetic growth forms towards the equilibrium WulÜ shape.By Fourier harmonic analysis the time evolution of the state of roughness of the prism plane of ice can be monitored both for growing and dissolving crystals. Results are presented for fructose/water at three temperatures [20 [19 and [17 °C. On the basis of recent high pressure studies on ice roughening the highest temperature may be near the onset of a (high order) thermodynamic roughening transition of the prism plane; pronounced faceting of both the prism and the basal plane of ice is expected at the lower temperatures. We describe coarsening of non-interacting faceted crystals at [19 °C and that of initially percolated networks of aggregated ice crystals at [20 °C. The onset of faceting of ice prism planes at [17 °C was monitored using Fourier harmonics to characterise the ìsharpnessœ of hexagonal contours of ice crystals viewed normal to their basal plane.A tentative analysis of the results suggests an estimate of the dimensionless step free energy of the prism face c of 7]10~3. The coarsening kinetics observed for dilute ice crystal ensembles did not conform with classical continuum theories such as the LSW treatment. This was indicated by the measured size distribution the scaling dependence of the mean radius and crystal number density with time and by single particle tracking showing that an asymptotic steady state was not reached. The lack of a sharply de–ned critical radius demarcating growing particles from those which dissolve does not then permit an assignment of excess chemical potentials to individual ice crystals on the basis solely of their observed curvature.While faceting and shape changes are important the overall kinetics are broadly consistent with diÜusion control. More concentrated ice systems at [20 °C form networks of aggregated faceted crystals under the action of van der Waals attractive forces. These initially percolated structures are linear chains of crystals with occasional branches. On ripening the chains thicken and progressively break. The tendency for linear chains may re—ect a preference for basal face ice contacts. This is being investigated further. 1 Introduction The coarsening of ice crystal ensembles from aqueous solutions can involve a rich phenomenology of complicated kinetic growth forms and interactions of ice crystals.It is generally held though 31 Faraday Discuss. 1999 112 31»49 not proven that dilute ice crystal systems show a broad diÜusion-limited coarsening regime consistent with classical LSW (Lifshitz Slyozov and Wagner) continuum kinetics. DiÜusion is also believed to be rate limiting at moderate to high ice phase volumes. Mean –eld treatments of overlapping diÜusion –elds can then provide a reasonable representation for the concentration dependence of the ripening kinetics.1h4 At very high ice phase volume,5 the growth of mean size with time no longer conforms with LSW physics and appears consistent with theoretical treatments for grain boundary coarsening.6 An additional factor is that the isothermal coarsening of ice in general proceeds by an initial aggregation dominated growth phase which plays an important role in determining kinetic crystal growth forms.The van der Waals forces between ice crystals across aqueous solutions can be strongly attractive especially in the sugar/water systems considered in this study. We describe the initial percolation of a moderately concentrated ice crystal network and the manner in which this coarsens. The faceting of crystal planes of ice during coarsening is a further complication. Kinetic rough- T of ca. [2 °C7,8 at ening induced by high initial driving forces for growth may give way to progressive faceting as curvature excess chemical potentials diminish and supersaturation falls. At present there is only a limited amount of data on roughening of ice planes.The thermodynamic roughening temperature TR for the basal plane is above the melting point of ice and cannot therefore be readily determined. The prism plane which is less prone to faceting is reported to have a R Fig. 1 Schematic of experimental apparatus. Faraday Discuss. 1999 112 31»49 32 the solid/vapour interface and between [13 and [17 °C9 at the solid/liquid interface. At present there is virtually no information on the thermodynamic roughening of ice in aqueous solutions and there is only very limited characterization of kinetic roughening. The studies presented here concern fructose/water in the temperature range from [20 to [17 °C. The lowest temperature ensures strong faceting for both the basal and prism planes of ice.At the higher temperature we observe the transition from kinetically roughened to faceted prism planes as the crystal ensemble ripens. Much research to date has been limited by the practical difficulty of studying kinetically evolving growth forms in ensembles of large numbers of ice crystals. For this reason we have invested in a new experimental capability. We employ automated cryomicroscopy with dedicated image analysis. The latter includes single particle tracking and a Fourier harmonics approach to characterize the ìsharpnessœ of hexagonal boundaries of ice crystals thereby providing information on their state of roughness. A combination of these approaches has enabled us to undertake a detailed study of ice coarsening kinetics which we discuss in relation to theories for coarsening and roughening.2 Experimental 2.1 Apparatus Most of the experiments were conducted using the automated cryomicroscope apparatus depicted schematically in Fig. 1. This consists of a Zeiss Axioplan 2 motorised microscope –tted with a specially commissioned Linkam cryo-stage. Two cameras are mounted on the microscope to allow the acquisition of monochrome or colour images the Adimec monochrome camera employed in this work oÜers a comparatively high resolution of 1024]1024 square pixels. The Linkam cryo-stage has been purpose designed to control the temperature across the sample to better then ^0.01 K over a de–ned temperature range and consists of a stage and a number of control modules.The airtight stage contains a motorised sample carrier which is used to vary the position of the sample relative to the passage of light through the microscope via the MDS 600 module. The sample is placed in a crucible approximately 7 mm in diameter and sealed from the environment by use of a cover glass. The thickness of the sample is controlled by employing a specially machined stainless-steel spacer. The crucible can be moved so that the sample either rests on a copper post designed to act as a thermal bridge to ambient temperature or rests on a silver block which is used to control its temperature. Temperature control of the silver block is achieved via the TMS 93 module which employs a proportional integral and derivative (PID) control mechanism with a platinum resistance thermometer (PRT) heaters and a heat exchanger through which liquid nitrogen is passed embedded within the silver block.A silver cover (with a hole to allow the passage of the light beam) is placed on top of the silver block covering the sample crucible to minimise any temperature gradient normal to the silver block. Liquid nitrogen is pumped through the heat exchanger by the LNP (liquid nitrogen pump) unit a special heater module having been designed to prevent condensation on the exterior of liquid nitrogen carrying pipes during the experiment. Since the experiment requires the maintenance of a sub-zero temperature for a number of days a mechanism for continually re–lling the 4 L Dewar supplying liquid nitrogen to the stage was developed.The latter comprises an Auto–ll module an Ohaus B10P05 balance and a pump. The 4 L Dewar rests on the balance and the measured mass is monitored by the Auto–ll module. When the mass of the Dewar falls below a speci–ed lowest value a control signal is sent from the Auto–ll module to activate a pump located on top of a 25 L supply Dewar. The pump remains activated until the required speci–ed upper mass value of the Dewar is attained. The Zeiss Axioplan 2 microscope is motorised and has been further developed in conjunction with Linkam and Zeiss to ensure that it can be automatically switched between bright-–eld polarised light phase-contrast and diÜerential interference contrast (DIC) modes of illumination. Most of the functions of the Linkam control system and the Zeiss microscope can be controlled by the PC via Image Analysis software (KS300).Macros have been written in the interpreter code supplied with KS300 to control all the functions necessary to conduct the experiment i.e. the Linkam cryo-stage the Zeiss microscope time controls image acquisition from either the Adimec 33 Faraday Discuss. 1999 112 31»49 34 Fig. 2 (a) Image 10743.tif and (b) crystals selected for tracking. Faraday Discuss. 1999 112 31»49 or JVC cameras and the writing of a data –le. The latter contains all relevant experimental information for each image i.e. its name and the time and temperature at which the image was taken magnifying optics employed and the (x,y) position of the sample when the image was acquired. This ability to ììtagœœ the image with relevant experimental information has ensured a signi–cant advance in the level of information that can be gained from each experiment.An uninterruptible power supply (UPS) is utilised to protect against temporary loss of power. Once the experimental data has been collected it is transferred to a second PC where it is analysed using KS400 Image Analysis software. The latter has been enhanced to enable Fourier analysis (using the radius-vector function) to be performed on selected crystals. The studies performed at [17 °C employed an earlier cryomicroscope which was equipped with a monochrome CCD video camera and a time lapse video recorder. The cryomicroscope consisted of a Leitz Ortholux II microscope which was operated in bright-–eld illumination and a Linkam THMS 600 cold stage.A frame-grabber was employed to obtain 720]512 digital images from the resulting video which were subsequently analysed using KS400 Image Analysis software. For this apparatus the temperature across the sample was controlled to better than ^0.1 K. 2.2 Experimental protocol The experimental protocol employed ensures controlled devitri–cation of the aqueous fructose solution followed by rapid nucleation of a large number of small ice crystals (at around [40 °C) which coarsen in the approach to the selected isothermal holding temperature. (a) Experiments performed at ‘19 and ‘20 ƒC. A 0.6 ll sample of 55% aqueous fructose solution was placed in the sample crucible (the fructose used was AR grade supplied by Fisons and the water was distilled and deionised).The sample thickness was –xed at 30 lm by use of a stainless-steel spacer and the sample sealed from the atmosphere by placing a 6.5 mm diameter No. 1 cover glass on top of this spacer. The crucible was retained in position by the sample carrier and moved so that it lay on top of the copper post. Dry nitrogen was repeatedly —ushed through the stage (to displace air) before sealing it so as to minimise condensation within the stage during the experiment. The silver block was set to cool at a rate of 60 K min~1 until the target temperature of [100 °C was attained. The sample was then rapidly translated onto the silver block where it was left to reach thermal equilibrium for 2 min. The temperature of the silver block was then raised at a rate of 5 K min~1 to the desired isothermal holding temperature which was maintained for a number of days.Time zero for the experiment is de–ned as the time when the silver block –rst reaches the isothermal holding temperature. A raster pattern was pre-set for the motion of the sample carrier during the experiment so that a number of selected spatial locations within the sample were regularly revisited. Since the aim of the experiments was to follow the evolution of a small number of crystals whilst simultaneously monitoring the behaviour of the ensemble a number of images were acquired at a range of magni–cations for each raster location during the experiment. (b) Experiment performed at ‘17 ƒC. Fructose/water with an initial concentration of 55% Fig.3 Variation in average mean radius of the ensemble as a function of time. 35 Faraday Discuss. 1999 112 31»49 fructose by mass provides a reasonably low ice phase volume (D0.04) at [17 °C. At this temperature the equilibrium shape of ice crystals is expected to show signi–cant faceting of the prism plane as well as the basal plane. A small volume (2.7 ll) of the sample solution was placed between two 16 mm coverslips and the assembly dropped onto the microscope stage which had been precooled to [150 °C. After allowing 10 min for the sample to reach thermal equilibrium the temperature of the stage was raised at a rate of 5 K min~1 to the target temperature of [17 °C where it was monitored isothermally for [50 h. 2.3 Image analysis Fourier analysis.In the early stages of coarsening especially at [17 °C the crystals display a high degree of circularity in their 2-D pro–le. Over time hexagonal symmetry evolves. Consequently the morphology of crystals oriented with their basal planes normal to the observation direction was characterised by Fourier analysis using the radius-vector method.10 The images were –rst segmented to obtain a binary image. Crystals were rejected if they were of an inappropriate shape (indicating the crystals were not aligned in the required orientation) were undergoing accretion or had a poorly de–ned interface. For the remaining crystals their centroid was determined and their boundary digitised. The boundary data was then transformed into polar coordinates (R h) with respect to the centroid of the crystal as origin.These co-ordinates were then –tted to the following equation using a digital integration method = R(h)\A0] ; An cos(nh[/n) n n/1 where A is the amplitude of the 0th harmonic A is the amplitude of the nth harmonic h is the 0 polar angle and / is the phase angle of the nth harmonic. n A number of checks and resolution tests were performed (i) a harmonic analysis was performed on model 2-D projected crystal shapes (hexagon circle rectangle) so as to obtain an appreciation of the boundary values of the harmonic amplitudes expected for extreme cases of either fully roughened or fully faceted ice crystals (assuming the crystal planes expressed are always normal to the viewing direction) ; (ii) since image analysis is performed in pixel space the results of the harmonic analysis will only be independent of object size above a certain minimum pixel threshold.In particular the ability to accurately pinpoint the centroid of an object will diminish signi–- cantly as the size of the object decreases to ultimately 1 pixel. In order therefore to understand the errors associated with harmonic analysis of data obtained in pixel space resolution checks were performed on images comprising a number of objects of a selected geometry (hexagon circle square) of widely varying size. Full results of these checks are outside the scope of this paper suffice it to say that all data obtained by Fourier analysis were screened and only data free of resolution artefacts accepted.In particular rejection of data for small crystals proved necessary (for a hexagon the amplitude of the normalised 6th harmonic is only essentially constant for objects [300 pixels for a circle the amplitude of all the normalised harmonics increases markedly as the object size decreases). Thereafter inaccurate centroid location and other resolution errors matter. Experiment performed at ‘19 ƒC. (a) Image analysis of ensemble data. To follow the evolution of the crystal ensemble as a whole images obtained at lower microscope magni–cation were analysed. In order to increase the statistical signi–cance of the data –ve images were analysed at each time point during the experiment. Each image corresponded to a speci–c (x,y) location in the sample and was taken using the ]32 objective of the microscope.The equivalent circular radius of each crystal in each image was determined and the average mean radius of the ensemble calculated by taking the average of all crystal radii for the –ve images. At t\4215 s 2888 crystals were averaged the number of crystals diminishing to 792 at t\32 735 s. For each time point a histogram of crystal size distribution was obtained. (b) Image analysis procedure for individual crystal tracking studies. A subset of crystals at one speci–c (x,y) location was selected for individual tracking and a reference image [image 10743.tif»see Fig. 2(a) obtained using the ]50 objective of the microscope at a spatial location of (65 [60)] selected. Other images corresponding to the same microscope magni–cation and Faraday Discuss.1999 112 31»49 36 Fig. 4 Variation in ln(number of crystals in ensemble) with ln(t). spatial location within the sample were identi–ed. All of the images were processed and segmented to obtain a binary image. A number of individual crystals in the reference binary image were then selected for study and their centroids determined [see Fig. 2(b)]. All the remaining images were sorted in chronological order. For each image in turn the crystals overlaying the determined centroids were analysed in the same sequence as was used for the reference image. In each case the parameters determined were the measured crystal area equivalent circular radius formfactor,§ aspect ratioî and perimeter. The result of this operation was the creation of a database for each crystal containing the measured data at each time point.Database manipulation was then performed so as to obtain a single database for each image (i.e. each time point) containing the data for the individual crystals in the same sequence as in the reference image. A dynamic data exchange (DDE) subroutine then translated the area data contained within these image databases into an Excel –le importing the data as a (number of crystals]1)](number of time points) array the –rst column of the array containing the time data the second column containing area data for crystal 1 the third column containing area data for crystal 2 etc. An analogous DDE subroutine was used to obtain an Excel –le containing the equivalent circular radius data for each crystal at each time point.To represent the variation in crystal shape with time we employed the shape descriptors of formfactor and aspect ratio as well as the harmonic analysis procedure previously described. Harmonic analysis was performed on each of the tracked crystals at each time step. Experiment performed at ‘17 ƒC. Fourier analysis was performed on 10 images obtained at elapsed times ranging from 30 min to 51 h. In this case the shape criterion used to reject crystals was formfactor O0.6. 3 Results and discussion 3.1 Ripening kinetics of non-interacting faceted crystals at ‘19 ƒC For the ensemble the time dependence of the average mean radius (r) shown in Fig. 3 implies a power law exponent l (rDtl) of 0.13^0.01 which is much smaller than the values of 1/3 expected for diÜusion-limited growth and 1/2 expected for (linear) interface-kinetics limited growth.The log»log plot of crystal number with time (see Fig. 4) is curved with an average slope of [0.64^0.03 signi–cantly smaller in magnitude than the value of [1 expected for diÜusionlimited ripening. Fig. 5 shows the crystal size distributions obtained for four of the eight time points considered. The crystal radii have been normalised to the average radius of the ensemble and the frequency to the total number of crystals in the ensemble. It appears that the distribution is approaching an asymptotic distribution at the longer times considered. Theoretical curves obtained from the work of Marqusee and Ross11 are overlaid onto the experimental data.The curves represent ripening which is either diÜusion limited (dl) [eqn. (4.10) of Marqusee and Ross] or controlled by interface kinetics (ik) [eqn. (4.6) of Marqusee and Ross]. Neither of these theoreti- § Measure of circularity of a sharp. Formfactor\(4p area)/perimeter2. î Aspect ratio\minimum object diameter/maximum object diameter. 37 Faraday Discuss. 1999 112 31»49 Fig. 5 Experimental and model crystal size distributions. c) cal curves seems to –t the distribution particularly well. The diÜusion-limited kinetic model –ts the left-hand side of the distribution quite well (i.e. the dissolving crystals) but gives a poor –t to the right-hand side (i.e. the growing crystals). The (linear) interface-kinetics limited model provides for a much broader distribution than is found experimentally but exhibits a more gradual reduction in frequency with increasing crystal radius for the right-hand side of the distribution which is more consistent with experiment.It is worth noting that the value of the normalised radius at which the experimental frequency falls to zero on the right-hand side of the distribution is intermediate between the values that pertain to diÜusion-limited and interface-kinetics limited growth. The growth of some of the individual crystal areas with time is shown in Fig. 6. This data particularly at the shortest times is not consistent with the LSW12,13 notion of a critical radius (r sharply demarcating growing and dissolving crystals. Crystals with their basal plane normal to the angle of observation appear either circular or hexagonal depending upon whether or not their prism plane is rough or faceted.The majority of the crystals designated as ììroughœœ dissolved only crystal 39 grew from an initial radius within the bound of radii one might assign as being critical at the –rst time point considered. Many crystals that appear hexagonal are seen to increase their area steadily with time. However the behaviour of crystals 42 47 62 and 59 is somewhat surprising. These crystals which are apparently larger than the critical radius at the –rst time point considered actually dissolve relatively rapidly. If they are tracked back in time then they appear to have once possessed greater hexagonal symmetry. It is possible that these larger crystals have a tendency to dissolve because they are much more plate-like than columnar possessing signi–cant curvature in their out of view dimension.Individual hexagonal crystals which grow are found to have an average exponent l (rDtl) of 0.16^0.05 which is not dissimilar to the value obtained for the ensemble. Dissolving crystals either hexagonal or rough do not show comparable power-law scaling. Polynomial –tting of crystal growth data enabled crystal growth velocity to be calculated as a function of crystal size at t\21 000 s (see Fig. 7) for hexagonal growing crystals initially hexagonal dissolving crystals and initially rough dissolving crystals. Crystal growth velocity as a function of size and supersaturation is potentially of value in calculating theoretical asymptotic crystal size distributions.Unfortunately rather more accurate data than that shown in Fig. 7 is required for this purpose. Faraday Discuss. 1999 112 31»49 38 Fig. 6 Variation in crystal area with time upon ripening. Crystals with their prism plane normal to the viewing direction provide additional information as this projection allows one to infer the rate of growth along both the prism and basal planes. The only crystal with its prism plane normal to the viewing direction that grows signi–cantly during the experiment is crystal 49. Other crystals such as crystal 41 initially appear to grow but then dissolve as the experiment proceeds to longer times. Aspect ratio is plotted as a function of crystal area for crystals 49 41 and 58 in Fig.8 the latter crystal dissolving as the experiment proceeds. Crystal 49 is observed to have a longer prism plane than basal plane with the asymmetry between the planes increasing as the crystal grows i.e. the crystal appears more columnar upon ripening. Crystal 41 is also observed to have a longer prism plane than basal plane with asymmetry initially increasing with crystal growth before ultimately decreasing as the crystal begins to dissolve. Crystal 58 is observed to have a longer basal than prism plane but the asymmetry between the two types of plane decreases as the crystal dissolves. The extent of the basal and prism planes of crystal 49 was determined as a function of time. The extent of the prism plane Fig. 7 Crystal growth velocity as a function of crystal size at t\21 000 s.39 Faraday Discuss. 1999 112 31»49 Fig. 8 Variation in aspect ratio with crystal area for crystals with their prism plane normal to the viewing direction. is found to increase faster than that of the basal plane i.e. ice is depositing faster onto the basal planes of this crystal than onto its prism planes. Power-law plots in Fig. 9 for the time dependence of the plane extents yield exponents of 0.39^0.02 and 0.12^0.01 for the prism and basal plane respectively. The latter is consistent with the above data for hexagonally symmetric crystals (whose increase in area represents deposition of ice onto the prism planes i.e. an increase in the extent of the basal plane). The value of 0.39 obtained is signi–cantly higher than values obtained for either individual hexagonally symmetric crystals or for the ensemble and is relatively close to the value of 1/3 which would be expected for diÜusion-limited growth.It appears that ripening of the basal and prism planes is not occurring synchronously. Harmonic analysis was performed on all crystals with their basal planes normal to the direction of observation. The crystals were then classi–ed according to whether they continually grew during the experiment dissolved or initially grew but subsequently dissolved. The results of this analysis at the –rst and last time points considered are shown in Fig. 10. Only at long times ([12 983 s) is there a sharp separation in the state of roughening between growing and dissolving crystals. Fig.9 Variation in ln(length of plane) with ln(time) for the prism and basal planes of crystal 49. Faraday Discuss. 1999 112 31»49 40 Fig. 10 Normalised sixth harmonic as a function of crystal radius for crystals with their basal plane normal to the direction of observation at t\8598 and t\22 556 s. It was decided to investigate the variation in ln(r with ln(t) for the subset of crystals individ- c) ually tracked. For the purposes of this analysis r was taken as the radius of the smallest growing c crystal at each time point considered. The gradient of the linear regression through the data obtained is 0.23^0.04 (see Fig. 11). This value is higher than the average value of 0.16^0.05 obtained for hexagonal crystals and is consistent with the hypothesis that the early data is taken at a time before true late-stage coarsening is established.For the same subset of crystals plotting ln(number of crystals) as a function of ln(t) yields a linear –t at longer time with a gradient of [0.85^0.08 (see Fig. 11). This value which is consistent with one obtained for the ensemble at longer time is much closer to the value of [1 expected for diÜusion-limited ripening. ln(r and (b) ln(number of crystals) with ln(t). c) Fig. 11 Variation in (a) 41 Faraday Discuss. 1999 112 31»49 In summary the above kinetic studies provoke a challenge for the models and theories we currently employ. Thus we –nd a lack of agreement with the predictions of the classical LSW theory of Ostwald ripening and related asymptotic continuum treatments. To illustrate our measured long time ice crystal size distribution in Fig.5 conforms neither with continuum theory prediction for diÜusion limited ripening nor with that for (linear) interface kinetics controlled growth. The measured exponent l\0.135 for the time dependence of the radius (rDtl) is also substantially less than the values 0.333 and 0.5 expected respectively for diÜusion limited and (linear) interface kinetics controlled growth. It is probably signi–cant that we do not obtain an obvious power law for the dependence of crystal number concentration with time (see Fig. 4) ; this lack of scaling could indicate that we are not observing asymptotic continuum behaviour. On the basis that the data (NDtd) in Fig. 4 would not be inconsistent at long times with an asymptotic power index d\[1 (diÜusion limited growth) we could consider that the coarsening is largely under diÜusion control but not yet fully evolved to the asymptotic scaling distribution.This is further supported by the scaling plots of Fig. 11 which track individual particles and demonstrate a progressive increase in l towards the diÜusion exponent of 0.333 and a progressive decrease in d towards the diÜusion exponent of [1. Clearly this shows the power of individual particle tracking for elucidating growth mechanisms. Other explanations however can be countenanced. For example the above classical continuum models imply a constraint of symmetry in the kinetics of dissolution and growth (both taken to the same power in local driving force). Relaxing this constraint and assuming for example that crystal dissolution is under diÜusion control whilst growth is under (linear) reaction control we can get a reasonable representation of the distribution in Fig.5. However this is a preliminary suggestion requiring signi–cant validation. Suffice it to say that model development of this type is under current investigation. It involves resolution of some formidable mathematical problems and asymptotic scaling is not expected. The –nding of Fig. 6 has major signi–cance indicating that the growth potential of an ice crystal in an ensemble is not uniquely speci–ed by size. There appears no sharply de–ned critical radius (as predicted by continuum theories) below which crystals dissolve and above which they grow.Of course the crystals are not spherical and present a variety of kinetic growth forms implying that their chemical potential is not isotropic (we will discuss this later). The stochastic viewpoint of Ostwald ripening developed by Bhakta and Ruckenstein14 could also be relevant here. Unlike the classical continuum theories this allows for microscopic —uctuations implying less sharp demarcation between growth and dissolution with the expectation therefore of some ììsmearingœœ of the critical radius. The measurement of the normalised sixth harmonic A6/A0 in Fig. 10 provides characterisation of the state of roughening of prism planes and shows the general behaviour of less faceting in crystals which dissolve than those which grow. However our time resolved classi–cation clearly shows that faceting per se is not the primary factor that determines whether a crystal grows or dissolves at least not in the initial stages of coarsening.It is also apparent from Fig. 10 that the overlap in size range between growing and dissolving crystals decreases with time which could suggest that the concept of a ììcritical radiusœœ is more meaningful at longer times. The stochastic treatment of Bhakta and Ruckenstein14 may also provide the basis of an explanation for the time dependence of crystal number density and average radius discussed earlier. This would require us to suppose that the overall kinetics are predominantly under diÜusion control but that there is a signi–cant component of reaction control the importance of which (relative to diÜusion) decays with time.Such a scenario is expected in any case the growth of small crystals under initial kinetic control progressively evolves to diÜusion control as larger crystals require relatively more of the available driving force *k for mass transport. This certainly applies to growth laws *kf with power index f\2 within the range of the continuous growth and screw dislocation mechanisms. The simulations of Bhakta and Ruckenstein for such a situation would indicate a dependence rDtl with l increasing from low values to an asymptotic value of 0.333. Two additional observations support the idea of ììnearœœ diÜusion-controlled kinetics. First plotting r3 vs. t yields an asymptotic slope at long times of 4]10~3 lm3 s~1 which if interpreted on the basis of the classical LSW theory gives an estimate for the mutual diÜusion coefficient D of fructose/water of 2.8]10~11 m2 s~1 (from Faraday Discuss.1999 112 31»49 42 slope\ 8Dpl2cequ(O) 9RT where p is the ice/solution interfacial energy v is the molar volume of ice R is the gas constant T is the temperature and cequ(O)is the molar concentration of water in solution at equilibrium2). Similarly forcing an asymptotic slope of [1 to the data in Fig. 4 for the change in crystal number with time yields an estimate for D of 9.4]10~11 m2 s~1 from N\ 3cin kBT 4pDplmol B c where in\[c(t\0)[cequ(t\O)]/cequ(t\O) c(t) is the concentration of water in solution at time t k is the Boltzmann constant and vmol is the molecular volume of ice.11 Both estimates are of the same order of magnitude as that calculated by extrapolating experimental data2 i.e.D\1.8]10~11 m2 s~1. Let us discuss then our observations on faceting of ice crystals and their shape development. Why when we suspect diÜusion control do we observe such distinct faceting of ice crystals ? Actually this is not altogether surprising. First curvature driving forces are weak *k/kBT (\2pvmol/rkBT ) is typically \10~3. If the kinetics are reaction controlled virtually all the available driving force is consumed by the interface attachment step. On the other hand diÜusion control implies that only a small fraction of the driving force is expended in surface integration mass transport then takes up most of the driving force. DiÜusion control thus implies relatively weak local dissipation of driving forces and therefore favourable growth conditions for faceting.It is a matter of detail how individual crystals in an ensemble balance the dissipation of curvature driving force between surface attachment and diÜusion although it is broadly true that the relative importance of surface attachment decreases with increasing crystal size. This together with the decrease in supersaturation as average crystal size increases explains why large crystals show a greater tendency to facet. Most treatments of Ostwald ripening consider the idealised case of a time-invariant spherical kinetic growth form for which it is possible to specify an isotropic chemical potential for each crystal and to represent in a relatively straightforward manner the spatial and time dependence of the chemical potential of crystallising solute in the nutrient phase of the ensemble.In general however we encounter non-spherical and time-dependent kinetic growth forms. This is illustrated for ice in Figs. 8 and 9 in the change from —at disc-like hexagons to columnar hexagons with a larger relative fraction of prism surface. Obviously the crystal chemical potentials are nonisotropic and the extent of the anisotropy is time dependent. Gibbs has provided a general treatment for the stress tensor normal to a crystal phase in a de–ned geometry. For hexagonal discs of ice the result is y *kPRISM\ 2pBASAL] 2pPRISM lmol J3 x and J3 x *kBASAL\ 4pPRISM lmol where p is the surface energy of the plane v is the molecular volume of water y the length of the mol prism plane and (x)3) the shortest extent of the basal plane with x the length of one side of a hexagon.We de–ne the ratio of plane lengths A as [x)3/y] and require at equilibrium by the A WulÜ theorem that equ\(pPRISM/pBASAL). Typically we study a-axis growth (normal to the prism plane) and observe the crystal in a direction normal to the basal plane. We therefore use the chemical potential of the prism plane as our reference and it is straightforward to show that 43 Faraday Discuss. 1999 112 31»49 44 Fig. 12 Three of the analysed images obtained at [17 °C. Faraday Discuss. 1999 112 31»49 HG BH *kPRISM kBT Aequ \G4lmol pPRISM 0.5A1] A J3 xkBT Note that the –rst bracketed term would de–ne the isotropic chemical potential of the crystal if it had attained its WulÜ shape.In general this is not the case and there is an anisotropy de–ned by the ratio of A for the kinetic growth form to that of the WulÜ form viz gC*k kCPRISM B *k T PRISM [ *k DWulff k BASAL BT h\0.5AA A equ [1B D kBT The data in Fig. 8 suggest a WulÜ aspect ratio at least as low as 0.70 (crystal 49) which is equivalent to a ratio of plane lengths Aequ of 0.85. Taking this as the value in the above equation suggests that crystals with initial (kinetic) aspect ratios of 0.85 (as in Fig. 8) i.e. A\1.40 have a signi–cant lower chemical potential (by 32%) of their basal planes than the prism plane. This appears to provide an explanation for the unexpectedly dominant c-axis growth of crystal 49.It is clearly an exciting prospect for us to be in a position to begin to characterise the WulÜ shape of ice crystals in sugar/water systems. 3.2 Progressive faceting of ice prism face at ‘17 ƒC Three of the 10 images used for the analysis are shown in Fig. 12. Many of the crystals in the images appear to display an ììinner ringœœ. This is interpreted as the maximum extent of the faceted basal plane of the crystal whilst the outer boundary of the crystal is interpreted as the maximum extent of the prism planes. We may consider whether the observed increase (see Fig. 13) in the normalised sixth harmonic A6/A0 with crystal size (and implicit time of coarsening of the ice crystal ensemble) re—ects an inverse kinetic roughening transition of the prism plane at [17 °C.Here we need to qualify that the data shown is not strictly time resolved though it is clear that we can monitor the progressive faceting of prism planes during coarsening. Data ììnoiseœœ is not primarily a consequence of image analysis resolution uncertainties discussed earlier but has two origins temperature ììnoiseœœ (considered later) and the lack of time resolution previously identi–ed. For non time-resolved data ììnoiseœœ re—ects the distribution A6/A0 with crystal area. Fig. 13 Variation in 45 Faraday Discuss. 1999 112 31»49 width of local excess chemical potentials in the ensemble and is more pronounced at smaller sizes since the driving force for growth in late stage coarsening is the relaxation of curvature excess potentials ; both the average potentials and their spreads around the mean decrease during ripening.The values of A6/A0 for the largest observed crystals (at long times) are signi–cantly smaller than that for a perfect hexagon. Three factors are relevant here. First there is the possibility that thermal —uctuations may have prevented complete faceting ; second the crystals may not all have presented perfect alignment of their basal plane normal to the observation direction ; and third the crystals may in any case be only partially faceted at equilibrium. We discuss each of these possibilities in turn. It is difficult to estimate temperature ììnoiseœœ precisely. We could hold external temperature across the cold stage to better than ^0.1 K and expect to achieve temperature constancy in the sample to within 0 (10~2 K).The chemical potential excess *kth/kBT [\(L f/kBT )*T /T ] associated with a temperature —uctuation *T of 0.01 K is 8]10~5 (L f/kBT \2.3 at T \273.2 K with L f the latent heat of fusion of ice). If we consider the crystal radius corresponding to the ììkneeœœ in Fig. 13 to be D9 lm this corresponds to an excess chemical potential *kcurv/kBT [\2pvmol/rkBT ] due to curvature of 5]10~5 (it is assumed that the interfacial energy between the prism plane ice and the sugar matrix is 30 mN m~1). It would seem then that above this radius temperature ììnoiseœœ dominates the driving force we wish to monitor. This clearly indicates the need for improved temperature control.Analysis noise introduced by the possibility of imperfect alignment of the hexagons normal to the observation direction is also difficult to quantify. Consideration of the absolute magnitude of A is of interest here. We –nd that pA shows the expected linear dependence the zeroth harmonic 0 2 0 on crystalline area of the basal plane however the slope of 0.9566 determined is signi–cantly lower than the predicted value of 0.9981 for perfect hexagons. If this diÜerence is not due to the nature of the crystallites themselves and rather arises from our procedure then we can consider two possible sources of analysis error (i) random image analysis noise in the perimeter of the analysed image and (ii) imprecise location of the centroid.The second possibility would obviously connect with variation in crystal alignment. 6/A0 should increase with increasing size but will fall short of the value for a perfect hexagon. rc\(3/F). Since the ìì critical radiusœœ de–nes the supersaturation (p) of water in the matrix curv/kT \F]1.5]10~4) our size class within that distribution would experience a net (F[1)*kcurv/kT The third factor to be considered is that the equilibrium growth form of ice at [17 °C does not correspond with full faceting of the prism plane. This could arise if the edges joining the prism faces are non-sharp so that for larger crystals the basal plane takes the form of a hexagon with rounded vertices. If the curvature of these edges does not depend greatly on crystal size then A The notion that the edges where prism faces meet are rounded receives some support from recent high pressure studies of ice/water which indicate that the prism faces of ice undergo an equilibrium roughening transition close to [17 °C.9 This observation also raises the intriguing A possibility that the decrease in 6/A0 with decreasing size could also be due to the onset of kinetic roughening as the crystal size is reduced.According to Fig. 13 the size at which this occurs is D3 (*k lm. The curvature chemical potential curv/kBT ) commensurate with this length scale is 1.5]10~4 (signi–cantly greater than our estimate of thermal noise). While recognising the lack of time resolution of the data let us suppose that the extrapolated crystal radius represents a size class within the ensemble distribution at the transition to faceting.We expect this size class to be greater than the ìì critical radiusœœ r of the ensemble say by a factor of F. If the ensemble distribu- c tion conforms reasonably with classical continuum predictions we expect an upper bound for F of ca. 2. F determines the critical radius of the ensemble in relation to our extrapolated size of 3 lm i.e. (p\F*k driving force for growth of which is 1.5]10~4 if we set F as the upper bound of 2. We further surmise that this available driving force for net growth is consumed predominantly by the attachment process at the interface of the prism plane i.e. we assume that the coarsening of the ice crystal ensemble is at this stage under reaction control having not yet evolved substantially towards diÜusion control.Cahn et al.,15 and more recently Elwenspoek and van der Eerden,16 have suggested criteria for the onset of kinetic roughening linking the chemical potentials driving growth with the step free energy of the interface. It is usually supposed that the transition from faceted growth (lateral spreading in transitional regime) to kinetically roughened Faraday Discuss. 1999 112 31»49 46 growth (continuous growth regime) occurs at the value of of the 2-D critical nucleus is of the order plane) are expected to show linear scaling with *k/kBT at which the work of formation kBT . Beyond this value growth rates (normal to the *k/kBT . Applying the simple relationship17 *k/kBT \pc2 with *k/kBT \1.5]10~4 (as above) would yield an estimate for the dimensionless step free energy c of 7]10~3 for the prism face at [17 °C.In view of our choice of F\2 we would expect this estimate to be an approximate upper bound for c whose value is made further less uncertain by the fact that we have applied a mean spherical argument for the representation of crystal chemical potentials and that we apply a kinetic roughening criterion which may be too simplistic for curved surfaces. Despite these reservations we consider in the following whether the above estimate for c is reasonable. Despite extensive studies on growth kinetics of ice from supercooled melts the literature is *k/kBT \5]10~3 and a step free energy c of 4]10~2. The thersomewhat confusing and inconsistent. For growth normal to the basal plane (c-axis growth) a reasonable understanding is in place largely as a result of studies based on a thermal wave technique.18 This can measure directly at a growing interface the temperature (or chemical potential) discontinuities inherent in the interface attachment process. From such studies the local supercooling required for the onset of continuous growth normal to the basal plane was found to be ca. 0.6 K which corresponds to modynamic roughening temperature of the basal plane is believed to be above the melting point. It is general experience that consistent with its high step free energy this plane can strongly resist roughened growth. In general ice surface integration is so rapid that heat or mass transfer can quickly become the rate limiting step in the overall growth thereby consuming the large majority of the available driving force.In such circumstances it is difficult to measure or estimate local chemical potentials at growing interfaces. Growth normal to the prism plane (a-axis growth) is known to be at least thirty times faster than that on the basal plane (see e.g. Cahn et al.15 for an overview) so the heat transfer problem is greatly exacerbated. To our knowledge it has not been possible to apply the thermal wave method to the study of a-axis growth which is simply proceeding too fast. The dominance of heat or mass transfer in comparison to interface attachment increases with crystal size. For this reason it has also not proved possible to study a-axis growth by the classical method of observing single large crystals under extremely low and precisely controlled supercooling.Instead much of the work in this area has had to involve growth studies in narrow geometries thin capillaries etc. to minimise heat transfer problems. Data interpretation of kinetic studies in these capillaries however is confusing since both heat and mass transfer processes show a dependence on driving force (*k2) similar to that of the screw dislocation mechanism for interface attachment. When it was assumed as e.g. in Cahnœs analysis that the a-axis growth observed in thin capillaries could be interpreted by the screw dislocation mechanism a surprisingly large estimate was obtained for the minimum supercooling necessary ([2 K) for continuous a-axis growth.This would suggest *k/kBT [2]10~2 and a step free energy c of 8]10~2 of 0 (10) times larger than our inference for the prism plane at [17 °C. It seems to us that the early thin capillary studies were not really free from complications of heat transfer and that their interpretation on the basis of screw dislocation growth could have been misleading. Several more recent observations support this prejudice. First it is known that ice grown from melts supercooled by less than 0.2 K shows the growth shape of a circular disc with faceted basal planes but rounded and therefore implicitly molecularly roughened prism planes.19 Second dendritic growth of ice is consistent with ììnear roughœœ growth at the interface with minor local consumption of driving force.20 Third studies on ice grown from the vapour phase indicate a roughening temperature for the prism plane at ca.[2 °C.7,8 Fourth ice crystals in a sugar matrix generally appear as rounded discs at temperatures greater than [10 °C. The –nal evidence and the most convincing to date is the recent high pressure work previously referenced. This suggests that the thermodynamic roughening temperature of the prism plane in relation to its melt is in the vicinity of [17 °C. If this is indeed the case then our estimate of c\7]10~3 at [17 °C indicating a relatively low step free energy would be quite reasonable. However it has very recently come to our attention that the presence of air in supercooled melts can aÜect the state of roughness of the prism plane.21 It appears that in the absence of dissolved gases there is greater stability for faceted prism planes and that the presence of dissolved air may 47 Faraday Discuss.1999 112 31»49 Fig. 14 Network formation and breakdown upon coarsening. induce roughening presumably by an adsorption mechanism. If this very recent –nding is more widely substantiated then the consideration of roughness of the prism plane requires a broader consideration than just temperature i.e. T may have to be speci–ed in relation to partial pressure R of gases etc. Clearly this is a priority issue for future work. 3.3 Coarsening of percolated ice crystal networks at ‘20 ƒC Fig. 14 shows the typical behaviour of ice crystal ensembles at higher ice phase volumes. There is Faraday Discuss.1999 112 31»49 48 weak network formation. While there is some branching the crystals display a tendency to form linear chain-like aggregates a feature which persists well beyond the stage of coarsening at which the ensemble ceases to percolate. At the selected temperature we expect individual crystals to be strongly faceted with a hexagonal disc-like shape. It is possible that the network could have been initially formed from such crystals and that their linear aggregation tendency could re—ect a preference for basal-to-basal plane adhesion contacts. In such a case the basal planes would be less involved in the initial coarsening of the network the aggregates retain a linear character but the chains ììthickenœœ by growth onto the more exposed prism planes.In general faceted crystals will eÜect stronger aggregation selectivity. They could thus provide ììtemplatesœœ for coarsening to large trapped kinetic growth forms which can not evolve their WulÜ shape. Further work on this is in progress. 4 References 1 R. L. Sutton I. D. Evans and J. F. Crilly J. Food Sci. 1994 59 1227. 2 R. L. Sutton A. Lips G. Piccirillo and A. Sztehlo J. Food Sci. 1996 61 741. 3 A. J. Ardell Acta Metall. 1972 20 61. 4 P. W. Voorhees and M. E. Glicksman Acta Metall. 1984 32 2001. 5 M. N. Martino and N. E. Zaritzky Cryobiology 1989 26 138. 6 J. E. Burke and D. Turnbull in Progress in Metal Physics ed. B. Chalmers Pergamon London New York 1952 vol. 3 p. 220. 7 M. Elbaum Phys. Rev. L ett. 1991 67 2982. 8 Y. Furukawa and S. Kohata J. Cryst. Growth 1993 129 571. 9 M. Maruyama T. Nishida and T. Sawada J. Phys. Chem. B 1997 101 6151. 10 A. G. Flook in Particle Size Analysis ed. N. G. Stanley-Wood and T. Allen Wiley Heyden Ltd. 1982 p. 255. 11 J. A. Marqusee and J. Ross J. Chem. Phys. 1983 79 373. 12 I. M. Lifshitz and V. V. Slyozov J. Phys. Chem. Solids 1961 19 35. 13 C. Wagner Z. Elektrochem. 1961 65 581. 14 A. Bhakta and E. Ruckenstein J. Chem. Phys. 1995 103 7120. 15 J. W. Cahn W. B. Hillig and G. W. Sears Acta Metall. 1964 12 1421. 16 M. Elwenspoek and J. P. van der Eerden J. Phys. Chem. 1987 A20 669. 17 X-Y. Liu P. Bennema and J. P. van der Eerden Nature (L ondon) 1992 356 778. 18 D. W. James in Crystal Growth»Proceedings of an International Conference on Crystal Growth Boston 20»24 June 1966 ed. H. S. Peiser Pergamon Press New York p. 767. 19 P. V. Hobbs in Ice Physics Clarendon Press Oxford 1974. 20 K. K. Koo R. Ananth and W. N. Gill Phys. Rev. A 1991 44 3782. 21 T. Schichiri J. Cryst. Growth 1998 187 133. Paper 9/00710E 49 Faraday Discuss. 1999 112 31»49

ISSN:1359-6640    

DOI:10.1039/a900710e

出版商:RSC    

年代:1999

数据来源: RSC

摘要:

Cascade nucleation in the phase separation of amphiphilic mixtures Jué rgen Vollmerab and Doris Vollmerc a Max-Planck-Institut fué r Polymerforschung Ackermannweg 10 55128 Mainz Germany b Fachbereich Physik Univ.-GH Essen 45117 Essen Germany c Inst. for Phys. Chemistry Univ. of Mainz W elder-W eg 11 55099 Mainz Germany Received 8th December 1998 Recently it was observed that oscillations arise in the turbidity and apparent speci–c heat of mixtures of water oil and surfactant when the phase separation of a phase of water droplets into a phase of smaller droplets coexisting with a water-rich phase is induced by constant heating. Here we develop a model for this oscillatory phase separation and compare it to data from detailed experimental studies. A scaling relation describing the composition and heating rate dependence of the period of the oscillations is established.Moreover it is pointed out that similar behaviour should also be observable in other phase-separating systems since the model is based on arguments of a general nature. I Introduction The formation and growth of nuclei leading to macroscopic phase separation when a single-phase system is driven into a coexistence region is a classical problem in statistical physics (see refs. 1»5 for overviews). This behaviour diÜers from other thermodynamic relaxation processes by the fact that small perturbations of an initially uniform state (an equilibrium single-phase structure which is metastable in the coexistence region) do not necessarily decay but might grow and lead to the formation of a macroscopically diÜerent state (the phase separated system).Non-linear kinetic equations are needed to describe such behavior.6,4 Even though this type of equation often leads to solutions which monotonously decay to the new equilibrium state there might also be situations where the system shows a very complicated (may be even chaotic) response to a change of external parameters like composition or temperature.6,7 Such behaviour is well-known from e.g. oscillating chemical reactions but we were not aware of studies addressing complex temporal behaviour of phase segregation in spite of the extensive experimental theoretical and numerical work on the subject. Therefore it was a major surprise to us when we –rst found evidence for such behaviour.8,9 The phase separation in amphiphilic mixtures leads to sustained oscillations in the turbidity and apparent speci–c heat when it is induced by a temperature ramp.In ref. 10 the energy barriers preventing nucleation were identi–ed and based on numerical work it was argued that the oscillations arise from an interplay between (i) a complex energy landscape where barriers can only be crossed by collective processes (ii) efficient relaxation mechanisms into local minima of the energy landscape and (iii) the driving due to constant heating. We suspected that this type of dynamics which we call cascade nucleation might show up whenever the phase separation is suÜering complex constraints e.g. due to conserved order parameters. In other words it should also be observable in a variety of other systems.Here we support this expectation by introducing a minimal model. On the one hand it may serve as a prototype to illustrate the mechanism 51 Faraday Discuss. 1999 112 51»62 leading to cascade nucleation and on the other it reproduces the salient features of the experimental observations. In Section II we give details of the investigated mixtures and revisit the experimental observations. Subsequently (Section III) a phenomenological model for cascade nucleation is introduced. The predictions of the model about the phase separation of the amphiphilic mixtures is worked out in Section IV and compared with the observations. Finally in Section V the main results are summarized. d V 4(/w]/s/2) V \ (1) (2) II Experimental observations revisited Amphiphilic mixtures contain water oil and surfactants.Water and oil are considered immiscible in the present work. A sample containing only water and oil will then phase separate into macroscopic water and oil domains and the surface tension will tend to minimize the area of the interface between the water and oil. Surfactants are molecules with a water-soluble polar head group and an oil-soluble apolar tail. These molecules preferentially enrich on the interface between water and oil thereby reducing the interfacial tension up to several orders of magnitude. 11h13 As a consequence mixtures of water oil and surfactants commonly organize in microphase separated structures with a –xed area of internal interface between water and oil.In the present paper we take H2O»octane»C12E5 (i.e. H[CH2]12[OCH2CH2]5OH) as an experimentally well studied11h21 model system. The thermodynamic state of these mixtures is characterized by temperature and the volume fractions of surfactant /s water /w and oil /o respectively where /s]/w]/o\1. To a very good approximation (except for 2»3%) the surfactant is located on the internal water/oil interface where it forms an incompressible monolayer. Its area can be expressed as the volume of surfactant divided by the eÜective thickness ls\1.3 nm of the monolayer. Consequently a sample with /s\0.13 forms an internal interface of about 100 m2 cm~3 of sample volume. A Phase behaviour The mixtures show a signi–cant temperature dependence due to a temperature dependence of the preferred curvature of the interface,13,14 which is induced by the temperature dependence of the interactions between the respective sides of the non-ionic surfactant molecules with water and oil.In Fig. 1 we show a section through the phase diagram for a –xed ratio /o//s . The solid line and the crosses denote respectively a theoretical prediction (cf. below) and the optically determined temperatures for the phase boundary Te (/s) between a single-phase structure of more or less monodisperse water droplets embedded in an oil matrix (T OT 1') and a phase of smaller sized e; water droplets coexisting with a water-rich phase (T [Te; 2'). For the two-phase region the phase predominantly consisting of small monodisperse droplets will in the following be denoted microemulsion phase.For the present system the parameter dependence of the radius of the droplets in both the single-phase region and the microemulsion phase has experimentally been studied in detail.13,14 The resulting dependences of the droplet size on composition and temperature within the 1' and 2'-regions respectively are sketched by the shaded circles in Fig. 1. Within the single-phase region the radius R of droplets is entirely determined by the conser- 1' vation of the partial volumes of water and oil i.e. by the volume / (4p/3)N1'R1' 3 and the surface area A4/s V /ls\4pN1'R1 2 ' enclosed by N1' droplets in a sample of volume V . This leads to R1'\ 3l / s /d s implying that typical droplet radii are of the order of 5»20 nm.Consequently about 1017 droplets are contained in a sample of volume 1 cm3. Within the two-phase region 2' the droplets take their optimal radius,20 which has experimentally been determined13 to be Ropt(T )\ a(T [T1 ) 1 Faraday Discuss. 1999 112 51»62 52 Fig. 1 Section through the phase prism for a mixture of water octane and C12E5 for a –xed ratio of oil to surfactant /o//s\5.71 and varying volume fraction of water and temperature. The solid line shows a prediction and the crosses represent optically determined values for the phase boundary. Close to the phase boundary the microstructure conforms to droplets. The shaded circles sketch the temperature and composition dependence of the droplet size in the single-phase (1') and two-phase (2') region respectively.(Optical data are taken from ref. 17.) where a\0.012 K~1 nm~1 and T1 \305.6 K. For each temperature there exists a unique value for the optimum radius which is independent of composition. Taken together the data suggest that a single phase structure is formed when there is too little water available to build droplets of the preferred size. Consequently a prediction for the phase boundary follows from the implicit equation Ropt(Te)\R1' that the radius of droplets changes continuously when crossing the phase boundary (see ref. 18 for more details). B Phase separation The phase diagram Fig. 1 constitutes an example of an upper miscibility gap. In equilibrium for T [Te all droplets have an average size R\Ropt(T ) and there is a single droplet which contains the surplus water.(In the thermodynamic limit its size diverges.) When quasistatically increasing temperature the small droplets continually shrink [see eqn. (2)] and the large one grows. Because of the conservation of the total volume of water and of the area of the internal surface the number of droplets is therefore continuously changing. Increasing the temperature enforces continuous topological changes in the structure of the interface between the water and oil. Since topological changes are slow processes one can already expect from this qualitative consideration an unusual kinetics of this phase separation when it is induced by constant heating. Indeed when applying a temperature ramp to drive the mixture into the two-phase region it periodically becomes turbid then transparent.8,9 The clouding and clearing can be quanti–ed by measuring the variation of the intensity of transmitted light I(T ).Fig. 2 (crosses left axis) shows a logarithmic plot of the extinction of light Irel ~1(T )PI~1(T ) as a function of temperature T for the heating rate vs\33 K h~1. Within the single-phase region the extinction of light is small and constant since the diameter of droplets is well below the wavelength of light. Slightly above the transition temperature T the turbidity increases strongly and subsequently the value of the e extinction varies periodically with temperature T or time t. (Note that the respective periods *T and *t are connected via the constant heating rate vs4*T /*t.) Up to nine oscillations can be resolved.In combination with videomicroscopy9 it can be inferred from these measurements that short time intervals where signi–cant numbers of large particles are formed alternate with periods in time where nearly no large droplets are formed. Interestingly similar oscillations also appear in the apparent speci–c heat of the mixtures as determined by microcalorimetric measurements (see refs. 9 and 22 for details on the experimental technique). When the experimental conditions are not changed oscillations in the turbidity 53 Faraday Discuss. 1999 112 51»62 Irel ~1 (crosses left axis) and variation of the Crel(T ) (solid line right axis) while passing the phase boundary (Te\319.3 K) with vs\33 K h~1. Both measurements are performed on identical mixtures (/w\0.13 v Fig.2 Temperature-dependent extinction of transmitted light apparent speci–c heat constant heating rate /o\0.13). For T [Te the signals show concordant oscillations under increasing temperature. (Data are taken from ref. 9.) coincide with the corresponding data determined by calorimetry (Fig. 2 solid line right axis). Moreover the oscillations in the speci–c heat persist even when there is no signal any longer that can be discerned optically. Since the period of oscillations is independent of the geometry of the container (optical and speci–c heat measurements were done in diÜerent geometries and for diÜerent sample volumes) one can exclude hydrodynamic instabilities due to the gravitational –eld as the origin of the oscillations.Instead we believe that they are the hallmark of nonlinearities in the kinetics of the phase separation which show up as a response to the slow change of temperature driving the system deeper into the two-phase region. As the last point concerning the experimental observations we address the parameter dependence of the oscillations. To this end the respective compositions of the metastable microemulsion ó phase at the maxima of the curve for the speci–c heat (i.e. immediately before another burst of nucleation arises) are plotted in Fig. 3 for the respective heating rates s\26.5 K h~1 (crosses) and v vs\3.8 K h~1 (boxes). The black line indicates the prediction for the phase boundary as o\0.57 /w\0.33 Fig. 3 Path of the composition of the metastable microemulsion phase as a function of the heating rate (see main text).The diÜerent symbols correspond to measurements on identical samples (/ /s\0.1) but using diÜerent heating rates vs\26.5 K h~1 for the crosses and vs\3.8 K h~1 for the boxes. Faraday Discuss. 1999 112 51»62 54 given in Fig. 1 but now plotted taking a logarithmic scale for / in order to be able to enhance w the distance between the data points and the phase boundary at higher temperatures. All data lie in the metastable regime to the top of the boundary and those corresponding to the higher heating rates lie further above i.e. the system is driven deeper into metastability. The oscillations typically decay after about twenty periods irrespective of the heating rate.On the other hand the amplitude as well as the period of the oscillations strongly depend on the heating rate and they slowly decrease in the course of phase separation. In ref. 9 a square root dependence of the period * v T on the heating rate was reported s (3) *T DJvs One of the objectives of the present paper is to develop a model for the origin of this behaviour which also allows the identi–cation of the composition dependent prefactors in this scaling relation. C Phenomenological picture of phase separation The experiments suggest the following picture for the phase separation,10 which is indicated by the grey lines connecting the data points in Fig. 3. When heating a metastable system closely above the phase boundary the composition does not change resulting in a vertical line segment.Upon progressing deeper into the coexistence region suddenly (close to the data point) large droplets appear which can be viewed as nuclei of the water-rich phase to be formed. Once assembled they rapidly grow in volume picking up the surplus water of the small droplets. The temperature is hardly changing during this fast relaxation resulting in a close to horizontal line segment. During the relaxation the small droplets shrink towards their equilibrium size Ropt and increase in number. Eventually the large droplets become visible by optical means causing the mixture to become turbid. At the same time they cluster and merge and if they become sufficiently large they eventually drop out the mixture due to gravitational forces.By this time the mixture relaxed to a point close to the phase boundary. Insigni–cant numbers of large droplets are then provided by nucleation so that the number of nuclei remains small and keeps decreasing. As a consequence the typical distance between (clusters of) large droplets increases and transport of water from the small to the large droplets becomes inefficient. Again the composition of the mixture is hardly changing and one moves upward in the phase diagram due to the applied heating. A new wave of nucleation arises when the system is driven sufficiently deep into the coexistence region (subsequent data point). Summing up the oscillations arise from an alternation of (a) periods of overheating until nucleation starts and (b) subsequent fast relaxation towards a close-to-equilibrium state which is again followed by overheating (a).III General model for cascade nucleation From a more abstract point of view cascade nucleation arises due to an interplay of processes connected to well-separated time scales.§ (a) Close to equilibrium the exchange of mass between the coexisting phases [i.e. the transport of water from the small droplets to the large ones] is exceedingly slow on the time scale of heating [since the distance between large droplets is large compared to the droplet»droplet distance]. The supersaturation grows. (b) This eventually enforces nucleation as a measure to allow a fast relaxation to a close to equilibrium structure with small supersaturation and a large number of nuclei.Further relaxation involves a coarsening [i.e. clustering and merging of large droplets] until again a structure similar to the one described in (a) is obtained. For appropriately chosen heating rates the structural changes due to nucleation and coarsening can be considered as fast processes. Consequently we expect cascade nucleation to occur at intermediate heating rates due to a switching between the comparatively slow process of mass exchange between coexisting phases and a faster process involving nucleation and coarsening which is only accessible for sufficiently § To provide the reader with a concrete example the respective processes for the amphiphilic mixtures are indicated in square brackets. 55 Faraday Discuss. 1999 112 51»62 large supersaturation of the sample.At very small heating rates the slow process (a) is still sufficient to keep the system close to equilibrium while at very high heating rates the system is quenched into the spinodal region leading to additional modes of relaxation which change for instance the dependence of the frequency of oscillations on the heating rate. From this perspective the scenario for cascade nucleation does not rely on particular properties of the mixtures described in the present work. We therefore strongly believe that it should be observable for a wider class of systems. In order to emphasize this expectation we develop now a generic model for cascade nucleation which does not make use of peculiarities of a speci–c system. To this end the time evolution is described by two parameters characterizing the instantaneous state of the system (i) the relaxation speed towards equilibrium C whose switching from very small values during period (a) to a much larger value during (b) captures the oscillating dynamics.In general it depends on the spatial and size distribution of growing nuclei [for the amphiphilic mixtures it will be taken to be the ratio of the number of large and small droplets]. (ii) the supersaturation R characterizing the total mass (or volume) that has to be exchanged between the coexisting phases in order to reach thermodynamic equilibrium. [For the amphiphilic mixtures this parameter is related to the ratio of the average value of the radius of droplets SRT and the optimal radius Ropt they should take according to thermodynamics.] The parameters are chosen such that they take close to constant values when the system is continuously following a change of temperature driving it deeper into the coexistence region while they oscillate in the case of cascade nucleation.In general the speed of segregation C grows when nuclei are provided which start to grow after nucleation while it decreases due to coalescence and segregation of the nuclei. On the other hand the supersaturation grows due to changes of an external control parameter (an increase in temperature) and its most prominent mechanism for decrease is the growth of nuclei. The functional form of these dependences on the parameters (C R) depends on the peculiarities of the system under consideration.On the other hand the mechanism leading to cascade nucleation should not depend on these peculiarities. To emphasize this observation we discuss the diÜerential equation for the time evolution for (C R) in the general setting (4a) C0 \f (C R) (4b) &0 \h[g(C R) It involves the most general expression for the change in C while the derivative of R has been split into a part h characterizing the external driving of the system due to a continuous change of a control parameter [heating] and a contribution g(C R) describing the in—uence of relaxation on the supersaturation [the relaxation of SRT towards Ropt]. For the purpose of the general model h is assumed to be constant.î We mentioned already that for weak driving [slow heating] the system should quasistatically adjust its state to the change of external parameters.In that case the time derivatives on the left of eqn. (4) vanish i.e. the equation allows for a –xed point (C1 (h) R1 (h)). (For convenience the functional dependence of R C1 and 1 on h will typically be suppressed in the following.) By de–nition (C1 R1 ) is the solution of the implicit equations f (C1 R1 )\0 (5a) g(C1 R1 )\h (5b) In the remainder of this section we discuss the stability of this solution in order to show that a mixture prepared in an initial state close to (C1 R1 ) will eventually approach this stationary behaviour and work out under which conditions this approach can be oscillatory. To this end we investigate the time evolution of the small deviations (6a) c\C[C1 î When applying the model one might have to keep track of a temperature dependence of the coupling of the heating to the state of the system e.g.due to changes of the speci–c heat. This however only gives rise to a re–nement of the general picture as explicitly discussed in the next section for the case of the amphiphilic mixture. Faraday Discuss. 1999 112 51»62 56 (6b) p\R[R1 Since c and p are small it is admissible23 to expand f (C R) and g(C R) in eqn. (4) to linear order in a Taylor series. Employing the condition in eqn. (5) for stationarity one –nds that the deviations evolve like (7) p5 Ac5 B B pB [LC g(C1 R1 ) \AC0 \A LC f (C1 R1 ) LR f (C1 R1 )BAc R0 [LR g(C1 R1 ) This is a linear diÜerential equation. Its solutions exponentially decay with the real part of the eigenvalues of the matrix it involves.23 Consequently when the eigenvalues are real there is a monotonous decay.If they are complex on the other hand the imaginary part amounts to the frequency of oscillations superimposed on the exponential decay. In the present case the eigenvalues are (8) j\i]iJw02[i2 with i\12[[LC f (C1 R1 )]LR g(C1 R1 )] u02\LR f (C1 R1 ) LC g(C1 R1 )[LC f (C1 R1 ) L& g(C1 R1 ) (9a) (9b) LC f (C1 R1 )\0 since C to From the physical picture of nucleation and relaxation° one expects that a large extend amounts to the number of growing nuclei and they aggregate faster if their mean distance becomes smaller. On the other hand the other derivatives are all positive. After all when the supersaturation is increased more nuclei are formed [i.e.LR f (C1 R1 )[0] and both an increase of supersaturation or in the relaxation rate lead to a faster decay of the supersaturation i.e. LR g(C1 R1 )[0 and LC g(C1 R1 )[0 respectively [note the negative sign in front of the term involving g(C R) in eqn. (4b)]. 0 2 [0 and i\0. The latter relation implies that the stationu For these signs of the derivatives u ary solution (C1 (h) R1 (h)) is approached for large times. This approach can be monotonous or oscillatory depending on the sign of the argument in the square root in eqn. (8). The oscillatory case which corresponds to 0 2 [i2 provides a generic mechanism for cascade nucleation. (10) C4 Nld Nsd IV Application to microemulsions In order to check under what conditions cascade nucleation occurs the functions f (C R) and g(C R) as well as their –rst derivatives evaluated at (C1 R1 ) have to be worked out for the particular system under consideration.Amazingly not very detailed information about the thermodynamic properties of the mixtures is needed to accomplish this task as we demonstrate now for the particular case of the phase separation of the droplet phase in amphiphilic mixtures. A Identifying dimensionless parameters The experiments revisited in Section II indicate an alternation between a rapid decreasing of the average size of small droplets and keeping the new size for some time. We argued in ref. 10 that the typical shape of the droplets remains spherical and the deviation from equilibrium of the heated mixtures is solely re—ected in the size distribution of the radii of droplets.Concerning the speed C of the relaxation of the phase separation numerical studies (see ref. 10) suggest that the most prominent dependence comes from the probability of small droplets to collide with a big one which takes away (part of) their surplus water. Up to compositiondependent prefactors this probability amounts to the ratio of the number of large droplets Nld over the number of small ones Nsd yielding ° In the following section these arguments will be illustrated for the amphiphilic mixtures by giving the explicit form of eqn. (7). 57 Faraday Discuss. 1999 112 51»62 In this picture C typically takes small positive values out of equilibrium and tends to zero when equilibrium is approached.The supersaturation R on the other hand should be related to the excess volume *V which has to be transferred from the small droplets with average radius SRT to the water-rich phase in N order to reach the thermodynamic equilibrium con–guration where Ropt are droplets of size opt formed. When the surface area of the small droplets is assumed to be –xed,“ i.e. A\ 4pNsdSRT2\4p Nopt Ropt 2 one –nds that *V amounts to (11) *V \ 4p 3 [NsdSRT3[NoptRopt 3 ]\ A 3 [SRT[Ropt] Consequently we take as dimensionless characteristics of the supersaturation the ratio of *V and the overall volume of the small droplets in equilibrium *V (12) \ SRT[Ropt4q[1 Ropt R\ 4p 3 NoptRopt 3 w(t)//wopt(T ) of the water /w(T ) over the value /opt w (T ) expected from thermodynamics.Consequentw( T ) and the corre- It vanishes in thermodynamic equilibrium and takes positive values for an overheated system. Note that in the spirit of the step-like behaviour indicated in Fig. 3 there is a simple graphical interpretation of R. According to eqn. (12) q corresponds to the ratio / content of the droplets ly RBln q corresponds to the distance in horizontal direction between / sponding value at the phase boundary when / is plotted on a logarithmic scale. In particular w the maximal value of R characterizes the width of the steps sketched in Fig. 3. B Time evolution In order to determine the time evolution of (C R) we observe [see eqn. (2)] that for constant heating (13) Ropt ~1(T )\a(T [T1 )\a(Te[T1 ]vs t) where the oÜset of the time scale has been chosen such that the phase boundary T is crossed at e t\0.Inserting this into eqn. (12) and taking the time derivative one obtains (14) R0 \SRTavs] L L t D CSRT Ropt Ropt/const The former term can be expressed as (15) SRTavs\q Te[T1 ]vs t 4(1]R)h vs where (16) vs h4 Te[T1 ]vs t s t?Te[T1 . plays the role of an eÜective heating rate. In the following discussion this parameter will be taken as constant since it has a negligible time dependence right after crossing the phase boundary v where s t>Te[T1 . Note however that the eÜective heating rate is suppressed like t~1 for large times where v The latter term of eqn. (14) characterizes the relaxation of supersaturation in the mixture at a –xed temperature.In the most naive approximation it is an exponential decay of R with a rate which in leading order is proportional to the density of large droplets i.e. to C. Combining these “ This holds to a very good approximation since excess droplets are much smaller in number than Nsd and larger in radius than SRT while the overall volume of small and large droplets is –xed. Faraday Discuss. 1999 112 51»62 58 expressions one obtains (17a) R0 B(1]R)h[GRCR where G is a constant rate depending on the details of the dynamics. We explicitly checked this dependence by more detailed calculations revealing a slight composition dependence of GR .p In order to determine C0 we observe that the contribution from the change of the number of N N and N typically change at the same time and (17b) N C0 \ N0 ld[C N0 sdB N0 ld\GK(R)[GC C2 N N sd sd sd is the (Kramers) nucleation rate of large droplets,5,24 which is a function of supersatu- C which will further be worked out in the following subsection.(18) p5 [GRR1 \A[2GC C1 GK @ (R1 ) BAc h[GRC1 where (19a) R1 ]2GC C1 D Ac5 B i\[ 1 u R small droplets may be neglected because sd sd ld the variation of the former is only of the order of at most 10% while typical changes ofN during ld a period involve several orders of magnitude. Consequently we obtain Here GK(R) ration only. The second term which involves the rate GC describes the decay of the number of large droplets by binary collisions leading to aggregation and merging.A more detailed derivation can be given for the latter term by relating the decay rate of the number of large droplets to the ratio of the Stokes»Einstein diÜusion coefficient D of large droplets and the square of their mean ld distance d. Using the conservation of surface and volume of all droplets to express d and the radius of large droplets appearing in D in terms of C R T and the composition of the sample ld one indeed recovers in leading order the term [GC C2. Moreover this derivation also reveals a composition dependence of G For eqn. (17) the linearized time evolution [see eqn. (7)] is obtained as C Comparison with experiment pB GK @ (R1 ) denotes the derivative LRGK of the Kramers-rate evaluated at R1 .Inserting the components of the matrix into eqn. (9) and using the condition for stationarity [i.e. the right-hand side of the eqns. (17) vanishes when evaluated at (C1 R1 )] one obtains 02\hG 2 C C1 GR 2 1 ](1]R1 ) G GK @ (R1 ) K(R1 )H u0 2 [0 such that cascade nucle- (19b) Faraday Discuss. 1999 112 51»62 Ch i2\u as explained below eqn. (9). Note that for the amphiphilic mixtures one always has i\0 and ation arises when 0 2 0 2 [i2 required for oscillations to occur [see eqn. (8)]. First we note that the slight dependence of the period on the number of preceding oscillations which had been reported in ref. 9 (see Figs. 7 and 9 of that paper) –ts very well to the expectation that the oscillation frequency depends parametrically on the composition which changes in the course of phase separation.Moreover in many experimental curves one has the impression of a cross-over from only slightly damped oscillations which after some time are strongly suppressed (see Fig. 2 for an example). This behaviour –ts the t~1 suppression of the eÜective heating rate for (T times larger than e[T1 )/vs [see eqn. (16)] which will rapidly lead to a violation of the condition u In order to check quantitatively the predictions on the composition dependence of the period we performed measurements on nine diÜerent sample compositions and various heating rates. Immediately after crossing the phase boundary a detailed comparison of the predictions with experimental data is then feasible because the composition is accurately known.According to p Note that eqn. (17a) is of a slightly more general nature than eqn. (4b) due to the additional term &h. This however only leads to a straightforward modi–cation of the matrix in eqn. (7) as shown below. 59 R\5.8 nm; ]/d\0.094 d\0.096 R\5.8 nm; R\8.2 nm; K / Fig. 4 (a) Composition dependence of the square of the period of the –rst oscillation *T 2 on v symbols represent experimental data obtained by microcalorimetric measurements and the straight lines s . The d\0.19 R\15 nm; = /d\0.38 R\15 nm; ] linear –ts to the data. (Sample compositions Ö / /d\ 0.096 R\5.8 nm; @ /d\0.19 R\5.8 nm.) (b) Data collapse to demonstrate the scaling dependence eqn. (20). The points show the results of microcalorimetric measurements on 9 diÜerent compositions and typically 4 diÜerent heating rates per sample.The straight line is a –t through the data points with slope 0.5. (Sample R\16 nm; d\0.19 compositions @ / Ö /d\0.19 | / R\6.8 nm; + /d\0.19 R\15 nm; > /d\0.38 R\6.0 nm; d\0.37 = / d\0.38 R\11 nm; ] /d\0.38 R\15 nm.) C C1 varies with the Dld/d2 as remarked under eqn. (17b). Given that the typical distance between large eqn. (19b) the leading order contribution to the frequency is indeedJh as observed in the experiments [see eqn. (3)]. Now the composition dependence of the prefactor has to be determined. To this end we note that the supersaturation and the Kramers rate depend only on the average size of droplets and the energy barrier which can both be expressed exactly in the reduced units introduced also in the present discussion (see ref.10). Hence the composition dependence of the terms in braces in eqn. (19b) may be neglected. On the other hand the factor G composition as droplets d is related to the radius of small droplets by Cd3\(4p/3)SRT3//d(t) and that the diÜusion coefficient does not signi–cantly change during the –rst oscillations one obtains *T 2(vs)\Cu vs Dvs(Te[T1 ]vs t) D d2 D2 (20) 0 \ vsSRT C4pD2@3 q6 D R1' ld 3C1 2@3 vs /d /2@3 d aDld In order to arrive at this prediction for the scaling of *T it was also used that for the –rst oscillation the radius of droplets SRT in the stationary state can be very well approximated by the droplet radius R in the single-phase region.It is our expectation that the dimensionless 1' scaled variables and the diÜusion coefficient of large droplets will show insigni–cant dependences on the composition and heating rates. Faraday Discuss. 1999 112 51»62 60 The experimental data agree remarkably well with this prediction. For a wide range of compositions the linear dependence of v *T 2 on has been veri–ed obtaining vastly diÜerent slopes of the straight lines [see Fig. 4(a) for typical examples]. Moreover when plotting *T against R1' vs//d2@3 s on a log»log-scale [Fig. 4(b)] the data for the respective compositions and scan speeds indeed collapse to a single line with slope 0.5 as predicted by the theory. V Conclusion We have worked out a model for cascade nucleation in the phase separation of a droplet-phase microemulsion driven across a phase boundary by constant heating.Beyond this phase boundary are relatively quiet periods where the composition of the sample hardly changes which are interrupted periodically by bursts where the droplets rapidly shrink and increase in number. To a good approximation the overall surface area of the droplets is preserved in this process ; water that is not needed to form the droplets is expelled into a coexisting water-rich phase which might lead to an oscillatory variation of the turbidity of the mixtures. The model presented here identi–es the competition between fast relaxation and slow driving due to heating as the origin of this behaviour. Moreover it provides a quantitative prediction for the parameter dependence of the oscillations as demonstrated by a data collapse of more than forty independent measurements covering a wide range of experimentally accessible compositions and heating rates.In this article we have restricted ourselves to a quantitative comparison of the theory with the experimental data obtained for the period of oscillation immediately after crossing the phase boundary since in that case the composition of the mixtures is well known. In order to discuss the weak composition dependence of the oscillation frequency due to changes in the composition of the samples in the course of phase separation an adiabatic approximation can be applied where the change of composition is monitored and the parameters entering the diÜerential equations describing the time evolution of the mixtures are self-consistently adjusted to these changes.The additional calculations needed to arrive at such an analysis are one of the next problems to be addressed. Self-consistent numerical integration of the evolution equations will have to be performed in order to make comparisons with the superstructures in the experimentally observed oscillations so as to verify that the present model also describes these aspects of the phase separation of the mixtures. Besides this more detailed analysis of the H2O»octane»C12E5 system it will be challenging to test whether data for other amphiphilic mixtures lie on the same scaling relations. This would be an experimental indication that cascade nucleation is a universal dynamical behaviour in the phase separation of amphiphilic systems.We strongly believe that the –ndings presented here are indeed universal since the proposed model is based on fairly general arguments. Only very few systemspeci –c thermodynamic dependencies are needed which all arise in dimensionless combinations. Therefore similar behaviour might even be observable in a variety of other systems besides amphiphilic mixtures. To sum up we have shed more light on the origin of the intriguing observation of cascade nucleation in the phase separation of amphiphilic mixtures and given an even stronger impetus to the question of also identifying similar behaviour for other phase-separation processes. Acknowledgements It is a pleasure to thank B.Dué nweg P. Gaspard and G. Nicolis for illuminating discussions and J. Krug and H. Pleiner for criticism on an earlier version of the manuscript. J.V. is grateful to G. Nicolis for the hospitality shown to him during a stay at the Center for Nonlinear Phenomena and Complex Systems of the Universite Libre de Bruxelles where part of this work has been carried out. D.V. acknowledges –nancial support of the Deutsche Forschungsgemeinschaft. References 1 J. S. Langer Rev. Mod. Phys. 1980 52 1. 2 J. D. Gunton M. San Miguel and P. S. Sahni in Phase T ransitions and Critical Phenomena ed. C. Domb and J. L. Lebowitz Academic Press New York vol. 8 1983. 61 Faraday Discuss. 1999 112 51»62 3 R. H. Doremus Rates of Phase T ransformations Academic Press New York 1985.4 H. Metiu K. Kitahara and J. Ross in Fluctuation Phenomena ed. W. Montroll and J. L. Lebowitz North Holland Amsterdam 1987 p. 259. 5 L. D. Landau and E. M. Lifshitz L ehrbuch der theoretischen Physik Band X Physikalische Kinetik Akademie Verlag Berlin 1983. 6 P. GlansdorÜ and I. Progogine T hermodynamic T heory of Structure Stability and Fluctuations Wiley London 1971; H. Haken Synergetics. An Introduction Springer Verlag Berlin 1983. 7 G. Nicolis Introduction to Nonlinear Science Cambridge University Press Cambridge 1995. 8 D. Vollmer J. Vollmer and R. Strey Europhys. L ett. 1997 39 245. 9 D. Vollmer R. Strey and J. Vollmer J. Chem. Phys. 1997 107 3619. 10 J. Vollmer D. Vollmer and R. Strey J. Chem. Phys. 1997 107 3627. 11 M. Kahlweit and R. Strey Angew. Chem. Int. Ed. Eng. 1985 24 654. 12 M. Kahlweit R. Strey and G. Busse J. Phys. Chem. 1990 94 3881. 13 R. Strey Colloid Polym. Sci. 1994 272 1005. 14 U. Olsson and H. Wennerstroé m Adv. Colloid Interface Sci. 1994 49 113. 15 S. Clark P. D. I. Fletcher and X. Ye L angmuir 1990 6 1301. 16 M. Kotlarchyk S.-H. Chen J. S. Huang and M. W. Kim Phys. Rev. A 1984 29 2054. 17 D. Vollmer and R. Strey Europhys. L ett. 1995 32 693. 18 D. Vollmer J. Vollmer and R. Strey Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. T op. 1996 54 3028; ibid. 1997 56 E7329. 19 D. Vollmer J. Vollmer R. Strey H.-G. Schmidt and G. Wolf J. T herm. Anal. 1998 51 9. 20 S. A. Safran Phys. Rev. A 1991 43 2903. 21 G. Gompper and M. Schick in Phase T ransitions and Critical Phenomena ed. C. Domb and J. L. Lebowitz Academic Press London vol. 16 1994. 22 D. Vollmer and P. Ganz J. Chem. Phys. 1995 103 4697. 23 C. M. Bender and S. A. Orszag Advanced Mathematical Methods for Scientists and Engineers McGraw- Hill Interational Editors Auckland 1987. 24 H. Risken T he Fokker-Planck Equation Springer Verlag Berlin 1989. Paper 8/09567A Faraday Discuss. 1999 112 51»62 62

ISSN:1359-6640    

DOI:10.1039/a809567a

出版商:RSC    

年代:1999

数据来源: RSC

摘要:

General Discussion Prof. Stamm opened the discussion of the Introductory Lecture Prof. Cates you mentioned brie—y the possibility of two length scales which might develop during phase separation. I would like to draw your attention to Fig. 3b in our manuscript (Gutmann et al.) where two length scales are also observed in our phase separated thin –lm structures. Those structures develop during spin coating of a polymer blend from solution. I therefore wonder if there is not some common and more general feature in those two phenomena. Prof. Cates responded There may indeed be common features. But in two dimensions there are special reasons for any assumption of a single length scale to break down see ref. 1. Since your –lms are eÜectively two dimensional their work may be more relevant than ours on this issue.1 A. J. Wagner and J. M. Yeomans Phys. Rev. L ett. 1998 80 1429. Dr Yeomans added Tanaka1 has described the phenomenon of double phase separation in two dimensions a pattern of circular domains within domains. It occurs in —uids because a fast hydrodynamic time-scale drives the initial domains to become circular while because of a slow diÜusion time the order parameter is still far from its equilibrium value. This leads to secondary spinodal decomposition with new domains appearing within those originally present. A similar morphology is seen in Fig. 3b of Gutmann et al. (at this meeting). 1 H. Tanaka Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. T op. 1995 51 1313. Prof. Stamm responded Concerning the reason for the occurrence of both small and large structures we observe during phase separation in spin coating from solution we only have a very qualitative explanation.We believe they are due to a quite complicated kinetic path of the system in the ternary phase diagram and may be also non-equilibrium structures depending for instance on the speed of solvent evaporation. So the PBr S/PBr S/toluene system is –rst miscible in solu- x y tion and might phase separate during evaporation of toluene forming the large structures. At this state two phases might be forming which both contain all three components in diÜerent concentrations. During further evaporation of toluene the diÜusion of the polymer components may become quite slow and the formation of large domains could be kinetically hindered.Thus the second polymer component will only phase separate very locally into smaller structures which are frozen in when the glass transition is reached with further evaporation of toluene. That the diÜusion length might play a role in the formation of those structures might be also concluded from the fact that the smaller domains usually are not found close to the boundary to another large domain. Simulations of this process would be of course very helpful for a better understanding. 3 Dr Yeomans asked If noise is added to a two-dimensional lattice Boltzmann simulation in the inertial regime the exponent 2 crosses over to 12 .1 Presumably this is because the driving force due to surface tension curvature is destroyed by the noise.Do you expect to see something in 3-D? Does this provide a possible mechanism for satisfying the Grant»Elder2 criterion ? 1 G. Gonella E. Orlandini and J. M. Yeomans Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. T op. 1999 59 R4741. 2 M. Grant and K. R. Elder Phys. Rev. L ett. 1999 82 14. Prof. Cates answered This is an interesting suggestion ; I think the issue remains open. Dimensionality may be important however. Indeed a crude estimate of the excess free energy in a domain of size L due to interfacial curvature is pL d~1 whereas the intrinsic (rms) scale of free 63 Faraday Discuss. 1999 112 63»80 energy —uctuations in such a domain is of order kBT L d@2. Based on this estimate thermal noise can swamp the driving force in two dimensions (just) but not not in three.Dr J. Vollmer said I have a question for Prof. Cates. It concerns the smaller scale droplets appearing in the –gure for the asymmetric quench. You argued that the smaller droplets are left over after the pinching oÜ of channels of the initially bicontinuous structure produced by spinodal decomposition. On the other hand for the break-up of liquid –laments in binary mixtures a hierarchy of satellite and sub-satellite droplets have been observed elsewhere (cf. e.g. Fig. 46 in ref. 1). For mixtures where both components have the same viscosity the size of the satellites rapidly decreases however when moving up the levels of the hierarchy. Eggers pointed out in his review1 that the hierarchy might be cut oÜ at the –rst or second level when the necks break early.Is it conceivable that a similar mechanism holds in the present situation ? In that case the relevant quantity to look out for is not so much a new length scale but the ratio of the typical size of necks at break-up and of the newly formed droplets. This ratio might be (at least to some extent) universal again involving as additional dimensionless quantities the typical geometry of the –laments at break-up and typical velocity gradients driving the break-up. 1 J. Eggers Rev. Mod. Phys. 1997 69 865. Prof. Cates replied Eggersœ paper is an important one. For a single —uid in vacuo he showed that the physics of pinchoÜ is universal so long as the length scale L 0 de–ned from the viscosity and the surface tension is much larger than a molecular length scale.This is true for some —uids (glycerol) but not others (water). The possible eÜects of this pinchoÜ physics on spinodal decomposition in 50/50 mixtures is discussed in Jury et al.1 But we have not thought about its relevance to oÜ-symmetry quenches where the satellites you refer to are seen. 1 S. I. Jury P. Bladon S. Krishna and M. E. Cates Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. T op. 1999 59 R2535. Prof. McLeish said With regard to your remark that the absence of the t1@2-scaling of l predicted by Grant and Elder1 may be due to ììoutpacingœœ of turbulent re-mixing by coarsening is this not what one would expect from scaling ? One can turn the lDt1@2 result around to describe the growth-time of a turbulent eddy of scale l so that teddyDl2.Now this is longer than the coarsening time of the system to size scale l in the late stage which is tgrowthDl3@2. So the eddies (even at l) do not have time to form let alone the larger ones which might be predominantly responsible (via cascading) for turbulence at length l. 1 M. Grant and K. R. Elder Phys. Rev. L ett. 1999 82 14. Prof. Cates said Yes thank you for this helpful remark. My own thoughts were along similar lines but less precisely formulated. Dr Frith asked It is known that the capillary number for drop break-up in shear is diÜerent from that for coalescence in shear. Would a good test of your simulations be to take a single drop in shear and break it and then reverse the shear –eld to re-coalesce the two drops? Given that there is a diÜerence in break-up and coalescence capillary numbers have you any thoughts on how this should in—uence the evolution of structure in your simulations ? Prof.Cates replied Yes it would be a good test we should try that. The diÜerence in these two critical capillary numbers should certainly show up in the detailed structure of a stirred emulsion. But it does not impact on the basic scaling argument of Doi and Ohta which can be summarized by stating that the dimensionless group de–ning the capillary number is of order unity in the steady state. Prof. Stamm opened the discussion of Prof. Ryanœs paper It is very interesting to see in your work a lot of the topics which have been discussed in Faraday Discussion no.68 in 1979. So I should mention that the crystallisation model you present in Fig. 12 has many similarities to the Faraday Discuss. 1999 112 63»80 64 ììsolidi–cationœœ model which we presented at that time.1 Based on small angle neutron scattering data from melt crystallized polyethylene we derived a model where Gaussian chains crystallise from the melt by straightening of the initially coiled segments without signi–cant rearrangement of the whole chain (solidi–cation process). This model was strongly under discussion since other more regular chain folding models could also explain the data. I believe that the chain conformation of melt crystallized polymers is still not very much better understood now although it seems to be clear that it depends very much on sample history.As a second comment I would like to draw your attention to some previous work on stress induced crystallization studies by synchrotron radiation.2 Also a spinodal crystallization model has been proposed based on kinetic SAXS experiments where a ììspinodal defect decompositionœœ could explain the scattering data very well. In particular a diÜerence in the time evolution of the small and wide angle scattering was recognized especially at small crystallization times which can be interpreted by an increasing density diÜerence between crystalline and amorphous regions starting with density —uctuations at a characteristic wavelength which still do not give rise to a wide angle scattering peak. Those observations are consistent with your –ndings only that our samples have been at least slightly oriented by stretching.1 M. Stamm E. W. Fischer M. Dettenmaier and P. Convert Faraday Discuss. R. Soc. Chem. 1979 68 263. 2 E. W. Fischer M. Stamm Q. Fan and R. Zietz ACS Polymer Preprints 1989 30 (2) 291; M. Stamm E. W. Fischer Q. Fan and R. Zietz Europhys. Conf. Abstr. 1988 12D 136. Prof. Ryan responded We thank Prof. Stamm for his supportive comments and for bringing our attention to his previous work. We had the ììsolidi–cationœœ model very much in mind during our discussion (see ref. 44) and note with interest the emphasis on our diÜerences in interpretation of the data. Prof. Allegra asked I appreciate very much the paper by Ryan et al. It clearly demonstrates the existence of a pre-crystalline structure in the polymer melt.Also the spinodally assisted crystallisation appears to be a very interesting model. I believe the picture of crystallisation in the paper under discussion could rather naturally be complemented by the ììbundleœœ model proposed by myself1 and with Meille2. Bundles are intramolecular associations of the dissolved or molten polymer determining a meta-stable precrystallisation structure (Fig. 1). Formation of bundles is favoured by attractive van der Waals interactions between parallel stems a few chain atoms long provided the loops connecting the stems are sufficiently short (a few tens of chain atoms). Due to their small size the relaxation time required by the bundlesœ conformational equilibrium is very short even in the melt.We think the pre-crystallisation state observed by Ryan and co-authors may be due to association among bundles each of which only has 4»5 parallel stems according to our statistical calculations on polyethylene (see Fig. 2). This ordering process may well take a long time as random —uctuations inside and between the bundles are needed to create sufficiently large aggregations. We suppose that the ìì tear shapedœœ SAXS maxima reported in the early stages of the extrusion of Fig. 1 A bundle with 4 stems in a hexagonal array. 65 Faraday Discuss. 1999 112 63»80 Fig. 2 Chain con–guration at T \T 0(T 0\limiting melting or dissolution temperature). Bundles and bundle aggregates are shown. The initial fold length resulting from either bundle association (primary nucleation) or deposition on a growing surface (surface nucleation) is proportional to the average value of a.isotactic polypropylene –lms could be interpreted in terms of associations among bundles oriented by the extrusion process. Incidentally we believe that this association process as the spinodally assisted nucleation may well imply very low energy barriers 1 G. Allegra J. Chem. Phys. 1977 66 5453; Ferroelectrics 1980 39 195. 2 G. Allegra and S. V. Meille Phys. Chem. Chem. Phys. to be submitted. Prof. Ryan and Dr Olmsted replied We welcome the comments of Prof. Allegra and see that there are some qualitative similarities between our models speci–cally the existence of a precrystallisation metastable state that is composed of more dense regions.ììBundleœœ formation indeed seems to be a complementary point of view. One caution is that a description in terms of the conformational statistics (trans gauche etc.) makes no reference to loops and applies whether or not one worries about inter- or intra-chain interactions. It would seem the looping/bundling picture might be slightly more appropriate for crystallization out of solution. Naturally some looping does occur in the melt but this might seem to be subsidiary to the fundamental coupling which we posit could exist between conformation and density. We are not convinced however that the low q structures are discrete agglomerates of bundles as suggested by Allegra but a continuous three dimensional structure of dense precrystalline structure. Dr Hanna commented In his presentation Prof.Ryan indicated that his simultaneous small and wide-angle scattering (SAXS and WAXS) measurements showed a small-angle peak some time before there was any indication of any wide-angle peaks. This was observed in both the quiescent crystallisation studies and the studies using a spin-line and could be viewed as the key piece of evidence on which the theory of crystal nucleation by spinodal decomposition is based. However I would like to express a note of caution. In the data that Prof. Ryan showed,1 the level of noise on the WAXS traces was much greater than that on the SAXS traces suggesting that the wide-angle peaks are of lower intensity than the small-angle peaks. This appears to be a general feature of such simultaneous measurements and is observed in other published data.2,3 I wondered therefore whether it would be possible for Prof.Ryanœs quiescent crystallisation data to be explained using a more traditional model of a growing stack of crystalline lamellae i.e. whether such a model would predict the wide-angle peaks to be sufficiently weak that they might not be observed given the sensitivity of the experimental apparatus. For this reason I have performed X-ray calculations based on a very simple model of a stack of isotactic polypropylene (iPP) crystals of constant thickness in the chain axis direction but with a varying lateral width such as might be expected during crystal growth. In the model the positions of the crystalline lamellae were modelled as an in–nite one-dimensional paracrystalline stack with Gaussian distance distribution function an average separation of 200 ” and a standard deviation of 20 ”.A temperature of 145 °C was assumed. Small and wide-angle traces were calculated by integrating over all of the unit cells in each crystal and then integrating over all possible orientations. The intensities were then corrected to reproduce Prof. Ryanœs experimental set-up. Faraday Discuss. 1999 112 63»80 66 The results of the calculations can be seen in Figs. 3 and 4. In Fig. 3 it is clear that the SAXS peaks are between 4 and 5 times as intense as the WAXS peaks. The peaks become visible in both sets of simulated data at a crystal width of 120 ”. However when statistical noise is added to the traces to approximately the level seen in the published experimental data (Fig.4) the wide-angle crystalline peaks are easily lost in the noise from the amorphous background. When noise is present the small-angle peak –rst becomes visible at a crystal width of 140»160 ” whereas the wide-angle peak cannot be seen until the crystal width reaches about 200»220 ”. Thus it would appear that this simple model based on traditional ideas of crystal growth is able to explain the delay in observing the wide-angle crystalline peaks compared to the smallangle peaks provided that the experimental data are noisy. We might argue that new theories based on spinodal decomposition are not required in order to explain the experimental data presented or alternatively that the current experiments are not capable of discriminating between traditional crystal growth and spinodal decomposition.I would be interested in Prof. Ryanœs comments. 1 N. J. Terrill P. A. Fairclough E. Towns-Andrews B. U. Komanschek R. J. Young and A. J. Ryan. Polymer 1998 39 2381. 2 H. G. Zachmann and C. Wutz in Crystallisation of Polymers ed. M. Dosie` re NATO-ASI-C series on Mathematical and Physical Sciences Kluwer Dordrecht Netherlands 1993 vol. 405 pp. 559»564. 3 B. S. Hsiao Z. Wang F. Yeh Y. Gao and K. C. Sheth Polymer 1999 40 3515. Prof. Ryan responded We thank Dr Hanna for taking our paper both the experimental and theoretical aspects so seriously. His model does indeed predict that for a paracrystalline system of lamellar stacks experimental data such as that presented may be found because the total SAXS intensities are four times stronger than the integrated crystalline WAXS peaks.We have considered in our paper many arguments to account for the behaviour observed; such as poorly Fig. 3 Simulated SAXS and WAXS curves for an in–nite stack of iPP as a function of lateral crystal width. Intensities are expressed in absolute units i.e. as a multiple of the scattered intensity from a single electron. Fig. 4 The same curves as in Fig. 3 but with the intensities scaled and Poisson noise applied. 67 Faraday Discuss. 1999 112 63»80 ordered crystals that do not diÜract ; inhomogeneous temperature pro–les ; the solid angle intercepted by the detectors and the detector efficiency and count rate. Admittedly in the quiescent experiments the SAXS detector sees B200 000 events per second and the WAXS detector only 10 000 events per second moreover the whole of the SAXS detector is integrated for the data in Fig.6. (of our paper) whereas peaks covering only 8% of the detector are integrated for the WAXS. This is most likely the noise eÜect you discuss. This diÜerence in solid angle and detector efficiency is not the case in the extrusion experiments though and as the two detectors are identical the WAXS detector will have 25 times the intensity as it is at 1 of the distance of the SAXS 5 detector. Dr Hanna continued In the X-ray calculations I described the data were corrected to account for the diÜerent geometries of the small and wide-angle detectors used by Prof. Ryan. In particular I applied diÜerent solid-angle corrections consistent with the use of a knife-edge detector for the wide-angle traces and a quadrant detector for the small-angle traces.The calculations were intended to model the data from the quiescent crystallisation studies only. I have not considered the data from the spin-line experiments but wonder whether the use of a —ow –eld might imply diÜerent physics in that case. Prof. Ryan Dr Olmsted and Prof. McLeish responded Thank you for providing the information concerning the modelling of the quiescent one-dimensional experiments. It is a pity you did not model the extrusion data in the same thorough manner. The maximum draw ratio for the iPP extrusion experiments presented in the paper is 1.4 that is 40% extension.Such small deformations in the melt are unlikely to change the physics enormously in our opinion for a system that is 200 K above its glass transition and able to locally relax. The reason 40% elongation is used is experimental. It created sufficient tension in the spin-line that the tape did not —op in and out of the X-ray beam. The –nal crystalline structure is relatively unoriented. An initial point to Dr Hanna concerns the model used to derive the scattering curves presented. It assumes in–nite paracrystalline stacks of constant thickness perfect iPP crystals that only change in their lateral dimension such a stack would have a square wave density pro–le. The whole point of this paper is to question whether such –rst order transitions with formation of perfect crystal nuclei that subsequently grow are appropriate.If Dr Hanna believes that a paracrystalline model is a good one for polymer crystallisation we would be pleased to know how the perfect crystal nuclei are initially formed in such a stack. A second point for Dr Hanna to consider is the kinetics of the growth process. Obviously in the paracrystalline model used to describe our experimental data kinetics are explicitly ignored and there is an arbitrary mapping on time of the width of the crystallites whose arbitrary shape and electron density pro–le does not change from a square wave throughout the growth process. The important feature of the experimental data that a model should capture is the wavevector dependence of the growth rate R(q). Dr Hannaœs results are clearly qualitatively similar to our experimental data.A more stringent test is to examine the detailed predictions for R(q). According to Cahn»Hilliard theory for spinodal decomposition with a conserved order parameter the growth rate should obey R(q)\Aq2(B[q2). To check this with Hannaœs calculations one must perform the following steps (1) Assume a suitable form for the time dependence of the width W . A linear dependence W (t) is consistent with the model of growing constant thickness lamellae. (2) Ascertain that the intensity I(q W (t)) is in fact exponential in time at early times. (3) Extract the growth rate R(q) for a range of q spanning the peak in the SAXS data. (4) Examine R(q) to determine whether or not it is consistent with the clear signature found experimentally of spinodal decomposition.This analysis is crucial for it is this speci–c form of the kinetics which leads us to suggest that spinodal or ììphase separationœœ eÜects play a role in polymer crystallization. A preliminary analysis indicates that R(q) extracted from Hannaœs calculations is not consistent with Cahn» Hilliard but we have been unable as yet to conclusively show this. We will report on the results of this procedure in the future. Faraday Discuss. 1999 112 63»80 68 If this kinetic analysis fails then Dr Hannaœs model although suggestive would be inconsistent with the observed kinetics and so by his own criterion would not be a candidate explanation of the observed phenomenon. Prof. Stamm said At initial stages of crystallization I believe you do not only have the noise but also a very intrinsic problem which you cannot avoid.This is due to the fact that a small number of parallel stem sequences only provide a very weak and broad signal which even with good statistics can hardly be resolved. We have calculated this contribution for diÜerent crystallisation models1 showing that at least –ve parallel stems are necessary to provide a reasonable signal. A small number of stems might be however already sufficient to give a density —uctuation which can be seen in SAXS. One should also keep in mind that the amorphous scattering is quite high in this regime and the separation of a peak and the amorphous background is not easy. 1 M. Stamm J. Polym. Sci. Polym.Phys. Ed. 1982 20 135. Prof. McLeish added Dr Hanna made two points the –rst was experimental to do with signalto-noise ratio and is well taken; the second has not yet been addressed»the deduction that one ought not to develop more ììcomplexœœ theories for polymer crystallization. I thought that that should not go unchallenged in a meeting such as this. Such expressed restrictions on what scientists ììoughtœœ to think about have a bad record in the history of science ! Indeed one could think from the theoretical point of view and ask whether one would expect any qualitatively new features in the way polymers crystallize. From this direction one does see that an additional order parameter (cf. in the paper) describing the Ising-like intra-chain order does naturally arise.It is this new order parameter that drives the ììhidden spinodalœœ. It is then entirely reasonable to look for such a thing which to the minds of many people goes a long way to explaining severe conceptual difficulties in the standard picture of polymer crystal nucleation that demand simultaneous local changes of density intra-chain order and crystallinity. Prof. Binder said Somewhat similar anomalous small angle scattering also occurs in undercooled polymer melts that do not crystallize but vitrify instead. These anomalous —uctuations near glass transitions sometimes referred to as ììFischer clusters œœ could be related to the small angle scattering you have observed. I would appreciate your opinion about this. Prof. Ryan responded This point has been raised previously in reference to small angle scattering and multi-dimensional NMR studies of vitri–ed amorphous polymers.In the case of PMMA I am convinced that the phenomenon is related to the results presented here. Moreover in PMMA there are subtle tacticity eÜects to drive the clustering or phase separation process. Dr Barham commented I would like to draw a note of caution on the concept that some form of liquid/liquid phase separation (llps) precedes crystallization at least in quiescent polymer melts. If true the resulting phase separation would aÜect either the nucleation or the growth of the spherulites or both. There is a wealth of data on nucleation and growth of spherulites at low supercoolings especially for polyole–ns. In particular nucleation of spherulites always occurs at the same sites (at a given crystallization temperature).Nucleation occurring simultaneously at all the separate sites with no further nucleation occurring before all the material has crystallized. The number of these ì active œ nucleation sites increases rapidly as the supercooling is increased. It seems obvious to me that if there is llps nucleation will only occur in the regions richer in the all-trans con–gurations. The probability that any given heterogeneous nucleus will lie within such a region will of course be less than unity so the spherulites should not always nucleate at the same places. Similarly the probability of nucleation occurring at any given heterogenity will increase as time progresses and the region of alltrans con–gurations around it grows in size (and perhaps in order) so one should expect from time to time to see new nuclei appearing during the isothermal growth of spherulites.The isothermal growth rates of spherulites are known to be constant. In a system undergoing llps the ììmore crystallizable œœ zones will increase in size (and possibly in order) as time passes. I would expect this to be re—ected in an increase in the overall growth rate»which is not observed. 69 Faraday Discuss. 1999 112 63»80 Prof. Oxtoby said I would suggest that the two-step nucleation of a phase transition found in this paper may be a relatively common process. If two order parameters both contribute the free energy surface may induce a curved path from one equilibrium state to the other in which one order parameter changes before the other does.Besides the example given in your paper we have given several other examples:1 density and crystal structure in protein crystallization bcc ordering along the path of fcc crystallization from the melt and density and composition in the cavitation of binary —uids. 1 D. W. Oxtoby Acc. Chem. Res. 1998 31 91. Prof. Ryan and Dr Olmsted replied We had considered using a two-dimensional energy map to illustrate the minima in the free-energy coming from the contributions of the two order parameters and were not fully aware of all your contributions in this area. For example the idea behind spinodal assisted nucleation is that the system runs ììdownhillœœ spinodally decomposing in the density variable until the barrier to crystallization in the relevant Fourier components becomes low enough for nucleation to proceed rapidly.Since our manuscript was submitted there has been a report1 of a transient metastable phase occurring on crystallization of hexadecane. The experiments were performed using synchrotron radiation on a homologous series of n-alkanes and a cross-over from stability to ììlong-livedœœ metastability to transient metastability was observed. In studies of crystallisation in iPP 2 gelation is observed to occur below 3% crystallisation in situations where spherulites are formed. Light scattering studies show large scale structures prior to gelation. Both these phenomena indicate that precrystallisation ordering occurs. 1 E.B. Sirota and A. B. Herhold Science 1999 283 529. 2 N. V. Pogodina S. K. Siddiquee J. W. van Egmond and H. H. Winter Macromolecules 1999 32 1167. Dr Barham addressed Prof. Ryan and Prof. McLeish In his response Prof. Ryan raised the issue of the occurrence of a ìì gel œœ point at low (B3%) crystallinities in a system crystallized at quite high supercooling suggesting that this could be indicative of some form of ììnetwork structureœœ within the melt re—ecting a spinodal decomposition. However with long polymer molecules caught up in diÜerent crystals and forming ììbridgesœœ between them a crystallinity of 3% is more than sufficient to allow a molecular network connected by small crystals acting as junction zones to be built up. Prof. McLeish argued quite rightly that we should never discourage theorists from dreaming up new models for anything»even polymer crystallization ! Certainly I would hope no-one would interpret any of my (or Dr Hannaœs) earlier remarks in that way.However I do believe that any such new models are likely to be useful only if that at least they are not inconsistent with already well established experimental data. As I noted in an earlier comment it seems to me that a model which proposes global liquid/liquid phase separation as a precursor to crystallization appears at –rst sight to be inconsistent with observations of spherulitic nucleation and growth at low supercoolings in polyole–n melts. Prof. Ryan Prof. McLeish and Dr Olmsted responded In the discussion Dr Barham raised the point of spherulite growth from the same nucleation sites in a repeated crystallisation and used this as evidence for the invalidity of our model.One needs to determine the size of the nucleants. If they are of the order of the size of a typical spinodal domain (say 50 ”) or larger then the odds are that they will span a domain the dense phase will hit it and nucleate. This is a good point however which should be checked. We are at pains to emphasize that this is a kinetic model and whichever structure grows fastest will win. The liquid/liquid (L/L) structure has slow lateral growth kinetics compared to secondary nucleation of crystallisation and once conventional crystallisation processes start we expect that they will wipe out (kinetically dominate) the L/L structure.Surface wetting is a phenomenon commonly observed in L/L phase separation and combined with the argument concerning nucleant size above in the context of our model could rationalise Dr Barhamœs observation of repeated secondary nucleation of spherulites from heterogeneities. Faraday Discuss. 1999 112 63»80 70 In raising the issue of mechanical gelation at very low degrees of crystallisation long before spherulites impinge and at crystal separations that are much bigger than the polymer radius of gyration Pogodina et al.1 show that isotropic light scattering passes through a maximum at the gel-point indicating volume –lling has occurred and that the magnitude of the gel-point is related to the stereochemistry of the polymers. As we have repeatedly argued our model of L/L phase separation prior to secondary crystallisation is not inconsistent with spherulitic nucleation and growth if one considers the surface segregation phenomenon associated with phase separation.In fact it is ignoring the vast experimental evidence for precrystallisation structure formation that is inconsistent with observations. 1 N. V. Pogodina S. K. Siddiquee J. W. van Egmond and H. H. Winter Macromolecules 1999 32 1167. Prof. McLeish added I should like to respond to the last comment of Dr Barham by pointing out in more detail the link between the quiescent and —ow-induced data. The objection he raises on the basis of the re-appearance of crystals could only be resolved by quite tightly determined quantitative versions of the theory that would need to be derived from density-functional formulations connecting local structure to coarse-grained variables.However the data on —owing melts are at least qualitatively suggestive. Dr Barham concedes that there is a real eÜect in this case»but what then might its continuation to zero-—ow rate be? In fact estimates of the shift in the spinodal under —ow made by coupling to the entanglement network with G0 the plateau modules do (in an unforced way) give shifts of just the right order to explain the data. This is at least suggestive of our model approach. Dr Frith added Regarding Prof. Ryanœs earlier comment on the observation by Winter1 of gelation in a melt containing 3% crystallinity that the crystallites could not form a network. It is not necessary that they form a network they simply need to cross-link the molten polymer by bridging of crystallites by polymer chains.1 N. V. Pogodina and H. H. Winter Macromolecules 1998 31 8164. Dr Blundell commented I would like to present experimental evidence of a transient precrystallisation state in oriented poly(ethylene terephthalate) (PET) that appears to be consistent with the proposals made in the paper by Prof. Ryan and colleagues. In contrast with their quoted examples of oriented polypropylene and polyethylene where the chain orientation is relatively low the chains in the PET example are highly oriented. The PET data I will present are part of an ongoing time resolved synchrotron study with Prof. Fuller and Dr Mahendrasingham at Keele University and with my colleagues at Wilton.Unoriented amorphous PET –lm has been drawn at 90 °C at a draw rate of 13 s~1 to a draw ratio of 3.7 1 during which the WAXS pattern is recorded every 40 ms. During the 0.2 s of the deformation stage the pattern changes from an amorphous halo to a highly oriented noncrystalline pattern. After the completion of the deformation stage oriented crystalline re—ections appear over a period of 0.5 s in what appears to be a –rst order transformation process. However I would like to highlight an additional sharp re—ection which appears on the meridian during the last stages of deformation and which then decays while the crystalline re—ections are growing. This meridional re—ection at ” B10 spacing corresponds to a monomer repeat length and indicates a rod-like chain con–guration extending over several monomer units.The limited lateral width of this re—ection indicates signi–cant correlation in the register of neighbouring chains. The re—ection is not associated with the triclinic crystal cell of PET but has been often cited in the literature as a characteristic of a PET mesophase. Fig. 5 shows the development of the crystalline phase based on the intensity of an equatorial re—ection together with the abundance of the mesophase based on the intensity of the meridional re—ection. The data are consistent with the creation of a mesophase during deformation followed by the transformation from a mesophase to a crystalline phase. In a general way this observation recalls the pioneering studies of Imai et al.on unoriented PET in which an intermediate ordered state was shown to form during an induction period as a precursor to crystal nucleation. However a degree of caution is needed in generalising a link with the unoriented case. We have so far only detected the meridional re—ection at the highest draw ratios and rates and at draw temperatures up to 90 °C where high network extension occurs as a result of restrictions in chain relaxation. 71 Faraday Discuss. 1999 112 63»80 Fig. 5 The development of crystallinity and the abundance of the mesophase for PET. Dr Ettelaie asked Prof. Ryan Is there a theoretical reason as to why the liquid/liquid phase coexistence curve is always embedded inside the liquid/crystal coexistence one? If not are there examples of polymeric melts in which a true equilibrium liquid»liquid (less dense/more dense) coexistence is observed? Such systems would provide very strong support for the theory presented in your paper.Dr Olmsted and Prof. Ryan replied Unfortunately we lack a proper density functional theory to answer such a question. There seems to be no reason in principle why such a spinodal should remain buried although as with many phenomena it may be that the numbers for typical systems are such that it is always buried. For example atomic systems almost always have a liquid»gas phase and one requires special tuning of potentials (as with a model colloidal system) to remove the liquid»gas coexistence phase. But we do note that the application of stress (or strain) could in principle be sufficient to pull the L/L spinodal into the stable region.This may be related to the extrusion experiments. Mr Kirov commented While I believe that under certain conditions i.e. the quiescent crystallisation case the spinodal decomposition may be relevant to the process of nucleation I would like to make a note of caution regarding the interpretation of data from stress-induced crystallisation experiments. In your paper the work of Blundell et al.1 has been adopted to ìextrapolateœ the relevance of the minimalist model suggested by you and your co-authors to the processes occurring in polymers under high strain above the glass transition temperature. I certainly believe that stress-induced crystallisation is a very complex process and therefore experimental data from such processes might be ìobscuredœ by totally kinetic eÜects.I hope you would agree that for an experiment which lasts 1.2 s (ref. 1) with a calculated half time of crystallisation B0.1 s i.e. almost three orders of magnitude faster than the maximum rate observed in unoriented PET this is not impossible. Furthermore I believe that the mechanism of stressinduced crystallisation is diÜerent from the quiescent case. It is interesting to note that Blundell et al.1 do not interpret their results the way you do. For the value of the Avrami exponent n\1 they have suggested two possible explanations (1) Growth of crystalline entities in one dimension from pre-existing nuclei conforming to a model initially proposed by Flory for –brous crystal growth in the direction of strain.(2) The explanation they –nd more probable is that strain induced crystallisation is associated with the sporadic appearance of uncorrelated crystal entities. Faraday Discuss. 1999 112 63»80 72 Regarding the time lag for the development of crystallinity during stretching at high rates in Blundellœs work1 I would like to quote a few lines from this publication ìAn alternative possibility is that irrespective of the temperature of draw the chains in the oriented network are stretched beyond their natural degree of extension and are therefore slipping through entanglements. In order for the crystallisation of these fully extended chains to take place it may be sufficient to rely on chain movements associated with local segment reorientation where there is relatively weak dependence on temperature.. . . Therefore signi–cant main-chain motions associated with the glass transition with their strong dependence on temperature may not be necessary to provide the mobility to enable sequences of monomer to move into crystallographic register. This scenario could also help to explain why crystallisation does not start until after the extension process. The ability of local segments to provide the mobility for chains to reorganise into crystallographic register is likely to be restricted while extension and chain slippage are still in progress. However when extension has been completed the onset of chain relaxation would restore the freedom of local organisation and allow crystal nucleation to occur.œ In my opinion this explanation for the delay in crystallisation under such conditions is very plausible but it does not agree very well with your model as it suggests that very localised mobility of molecules is sufficient to cause crystallisation but does not support large scale rearrangement and separation of portions of molecules.Here I would like to bring to your attention a review paper by Schultz2 which gives an interesting description of the eÜect of ìsoluteœ and ìheat —owœ in the development of polymer microstructure. I believe it is a good example of how applied stress can modify the crystallisation process of polymers and in this sense is relevant to our discussion. Schultz2 points out that in –bre spinning and for very thin –lms drawn from the melt crystallisation becomes diÜusionless that is it becomes impossible for the non-crystallisable molecules to diÜuse away fast enough to keep up with the growth front.The requirements for such a diÜusionless transformation are (Schultz2) (1) The chain repeat units must be already near their ultimate crystalline sites so that no longrange transport is necessary i.e. the system must be highly oriented. (2) The eÜective undercooling must be very great at least locally. (3) A molecular mechanism of such a diÜusionless transformation must be available. Interestingly for the latter requirement the suggestion is that for crystallisation in highly strained systems the chains need only ì jostle œ slightly to come into a near crystalline register.This suggestion is very similar to the one we –nd in Blundellœs work. May be it is also necessary to say that in previous work on crystallisation orientation-induced nucleation is considered separately. According to Long et al.3 there are three physical mechanisms for polymer nucleation (i) spontaneous homogeneous nucleation which occurs in a supercooled melt (ii) orientation-induced nucleation caused by alignment of macromolecules and spontaneous crystallisation and (iii) heterogeneous nucleation on the surface of a foreign phase. Wunderlich4 amongst others has also given the thermodynamic reasons for enhanced nucleation as a result of strain. According to him deformation of a macromolecule to a more extended chain conformation can lead to enhanced nucleation and crystallisation.A more extended chain would have a lower entropy but change only little in enthalpy so that the melting temperature expressed as Tm\*H/*S increases. At a given temperature a deformed macromolecule may thus have a larger supercooling and greater driving force towards nucleation and crystallisation. A more elaborate expression for the eÜective undercooling in strained materials can also be found in the review by Schultz.2 With all that in mind I would suggest that it is not entirely appropriate for the minimalist model to be extended to crystallisation induced by stress at least not for strain rates as the ones encountered in Blundellœs work. There is obviously a need for more evidence supporting the validity of the model for high strain regimes.1 D. J. Blundell D. H. MacKerron W. Fuller A. Mahendrasingam C. Martin R. J. Oldman R. J. Rule and C. Riekel Polymer 1996 37 3303. 2 J. M. Schultz Polymer 1991 32 3268. 73 Faraday Discuss. 1999 112 63»80 3 Y. Long R. A. Shanks and H. Starchushki Prog. Polym. Sci. 1995 20 651. 4 B. Wunderlich Macromolecular Physics Academic Press New York 1976 vol. 2 p. 66. Prof. Ryan and Prof. McLeish responded We thank Mr Kirov for his extensive comment. We do not agree however that the link between elongation and subsequent crystallisation cannot be explained by our model. Prof. McLeish argued (above) that the coincidence of the plateau modulus and a consequent shift of the spinodal of order 1 toward the binodal is sufficient to explain the reduction of the nucleationœs barrier to crystallisation of large deformation (ref.15 of Prof. Ryanœs paper). Furthermore when extensional or shear —ow is seen in q-space any growing Bragg peaks will be convected away from their equilibrium positions and crystallisation will be suppressed until the cessation of —ow as observed by Blundell et al. Finally —ow has been observed to aÜect the spinodal boundary in polymer mixtures and this can also suppress demixing. 1 1 N. Clarke and T. C. B. McLeish Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. T op. 1998 57 3731. Dr Ding said Fig. 11 (of your paper) assumes —uid/crystal coexistence in equilibrium at a temperature below Tm . This implies however that the degree of crystallinity cannot be 100% in principle.In contrast the present knowledge of polymer crystallization is that the crystallizable polymer is either a —uid or a crystal in equilibrium state unless something special happens. The actual degree of crystallinity is not 100% which is kinetics controlled rather than an in-principle result. I do think that the experimental design is clever and the theoretical interpretation of the primary nucleation kinetics is stimulating. But would you please give us further interpretation of the phase diagram and thus thermodynamics? Dr Olmsted and Prof. Ryan replied The phase diagram is certainly idealized ; the degree of crystallinity of the ììequilibriumœœ phase diagram is only \100% if the quench leaves the system inside the two-phase region.We know that kinetics will take over and folds and dangling ends will prohibit annealing into perfectly crystalline phases and the –nal state will be determined largely by kinetics competing with an initial structure ììfrozen in œœ during the quench. Prof. A. H. Clark said It is perhaps worth commenting that much of the present discussion of crystallisation from melts is equally relevant to the formation of gels on cooling dilute biopolymer solutions e.g. polysaccharide gels from agarose amylose etc. Similar scattering approaches at low and wide angle are adopted and mechanisms have also been proposed in terms of early spinodal demixing followed by a limited form of crystallisation. Structure formation in such systems at a mesoscopic level thus also involves a complex interplay between chain ordering solution phase separation and crystallisation.Have Ryan et al. any thoughts on how the solvent might in—uence the crystallisation of their synthetic polymers and can they say whether this would lead to analogous gel formation? œ Prof. Ryan and Dr Olmsted responded We purposely kept away from the solvent issue in order not to deal with these eÜects and to make the theory simpler. We have not yet had the opportunity to expand our model to deal with solvents or test it by experiments on polymer solutions. Should such an opportunity arise we would be happy to do so. Prof. Tiddy asked What are the in—uence of extrusion rate and temperature on the data obtained in the extrusion experiment? Is there an in—uence of polymer polydispersity ? Prof.Ryan replied We have just been awarded an EPSRC grant where we intend to study the eÜects of processing conditions and molecular weight distribution on a range of polymers. These results will be reported at a future meeting. Dr Meille commented The paper by Ryan et al. implies the development of order at the mesoscopic scale as a preliminary step towards crystallisation in line with work by Imai et al.1 on the crystallisation from the glassy state and with some other recent papers. I think that such partial ordering although in many instances difficult to evidence is highly probable and various experi- Faraday Discuss. 1999 112 63»80 74 mental facts point to its existence. A somewhat similar intermediate role has been suggested by Prof.Keller and collaborators2 in the case of polyethylene for the hexagonal phase; it can be envisaged also for the thermotropic mesophases of a number of diÜerent polymers. A general consideration is that during crystallisation the chain con–guration is likely to deviate from its unperturbed random coil state especially at a local level since a substantial supercooling is needed for polymer crystal nucleation and growth to occur. Some aggregation at the local and possibly on a broader scale will develop in such collapsed polymer chains if only for space occupation reasons. In the bundle theory of polymer crystallisation Allegra has given a description of the chain in terms of a metastable equilibrium which can help us understand the nature of such precrystalline semi-ordered states.According to this theory the chain con–guration depends only on the supercooling with respect to the equilibrium dissolution temperature (or the equilibrium melting temperature) i.e. the conditions under which the chain is an unperturbed coil. Again the Bristol group has beautifully demonstrated3 that the essential experimental factor aÜecting the initial fold length is supercooling indeed neither the mode of crystallisation (from solution from the melt from the glass) nor the speci–c features of nucleation aÜect the value of this key parameter related to the lamellar thickness. We can add that as long as it is not mesomorphic even the speci–c structure of the crystallising polymorph does not appear to aÜect the initial fold length.Thus it seems reasonable to search for a correlation with the thermodynamic properties of the crystallising chain rather than with the nucleation mechanism. Modelling studies along with experimental investigations of appropriate systems are likely to be essential in the clari–cation of these fundamental issues in polymer crystallisation. 1 M. Imai K. Kaji T. Kanaya and H. Sakai Phys. Rev. B Condens. Matter 1995 52 12696. 2 A. Keller and S. Z. D. Cheng Polymer 1998 39 4461. 3 P. J. Barham R. A. Chivers A. Keller J. Martinez-Salazar and S. J. Organ J. Mater. Sci. 1985 20 1625. Dr Frith opened the discussion of Dr Williamsonœs paper The diÜusion limited and reaction limited theories for ripening kinetics are valid for the dilute limit.Could you comment on the eÜect of the –nite volume fractions of ice present in your experiments? Dr Williamson responded The experiment was designed to operate with a low volume fraction of ice so as to allow meaningful comparison with theories developed for the dilute limit. The volume fraction of ice used (estimated from a knowledge of the fructose»water phase diagram) was approximately 8% for the experiment conducted at [19 °C. We have considered theoretical studies that speci–cally address the eÜect of –nite volume fraction of the dispersed phase upon ripening. In the theory of Marqusee and Ross1 the exponent for the power law plot of the recrystallization parameter (crystal dimension or number density) with time is predicted to be invariant to changes in volume fraction ; only the prefactor in this expression is predicted to show a volume fraction dependence.The asymptotic crystal size distribution however depends quite sensitively on crystal volume fraction. Following the approach developed by Marqusee and Ross we have determined the crystal size distribution for the case of an 8% volume fraction mixture (see Fig. 6 here showing the experimental and model crystal size distributions). We –nd that whilst the distribution is signi–cantly aÜected by the volume fraction correction and in particular shows an increased value for the upper cut-oÜ approaching the value found experimentally the overall –t to experimental data is not improved 1 J. A. Marqusee and J. Ross J. Chem. Phys. 1984 80 536. Prof.Binder commented This is a comment on the observation that the coarsening kinetics does not conform to the Lifshitz»Slyozov treatment. I also want to emphasize that the latter is only accurate for vanishingly small volume fractions of the new phase. Only then does the original theory apply and the many extensions to –nite non-zero volume fraction often use unjusti–ed aproximations some of which lead to questionable or even misleading predictions and should not be used by the experimentalists. Of course there are also valuable theories of coarsening at –nite volume fraction but they are very complicated and show that there are slow transients before one reaches the asymptotic regime (see ref. 1 for a review and reference to earlier work by Enomoto et 75 Faraday Discuss.1999 112 63»80 Fig. 6 Experimental and model crystal size distributions. al.2) In general one needs to observe growth over many decades in time to ensure whether the asymptotic regime was reached. In your paper Fig. 2(a) suggests that the high volume fraction of the crystals is indeed a problem. From a picture like this it is not even completely clear whether the new phase has been formed by nucleation. Compare to Fig. 10(a) of our paper this is a calculation where one also sees ììdropletsœœ but they have been formed by spinodal decomposition (the state is inside the spinodal curve and no thermal —uctuations indispensable for nucleation were included). 1 A. J. Bray Adv. Phys. 1994 43 357. 2 For example K. Kawasaki Y. Enomoto and M. Tokuyama Physica A (Amsterdam) 1986 135 326; M.Tokuyama and Y. Enomoto Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. T op. 1993 47 1156. Dr Williamson answered Continued equipment development has now ensured that we are in a position to run extended experiments to validate or otherwise the existence of an asymptotic distribution and consider in detail the eÜect of time transients. Whilst the importance of these are acknowledged other possibly more important concerns arise when relating our observations to current theoretical predictions. The majority of the latter assume that ripening can be considered to be essentially under either diÜusion control or reaction control. In actuality we believe that interface control may well dominate for small crystals but that diÜusion will become increasingly signi–cant and eventually dominate as ripening proceeds.Moreover the Lifshitz»Slyozov treatment and many recent modi–cations assume ripening of spherical particles. Our crystals are obviously non-spherical and express faces with diÜering properties which presents a further complication. The ice crystals studied in this work are formed by devitri–cation at low temperature. This can involve elements of dendritic and spherulitic growth which ultimately evolve a distribution comprising a large number of small crystals which undergo further coarsening. By the time the distribution has developed to crystals such as those shown in Fig. 2(a) however we are con–dent that a process of late-stage coarsening has been approached.Faraday Discuss. 1999 112 63»80 76 Prof. Widom opened the discussion of Dr Vollmerœs paper Have you considered possible affinities between the temporal instabilities you have described and the spatial periodicities seen in Liesegang ring formation? Your theoretical explanation is reminiscent of that which is often given for the Liesegang ring phenomenon. Dr D. Vollmer and Dr J. Vollmer responded Our approach of identifying pivotal variables for the time evolution and subsequently performing a stability analysis of –xed points lies at the heart of the analysis of many pattern forming systems such as for instance the Liesegang rings you mention. Currently we see the closest analogy to chemical oscillations in well stirred closed reactors which show an oscillatory approach towards (chemical) equilibrium.In contrast when such a reaction runs in a gel spatial concentration gradients build up and the coupling between diÜusion and the oscillatory dynamics can lead to spatio-temporal patterns (for instance spirals). We believe that Liesegang patterns are more closely related to this type of phenomenon since they also originate from the coupling of a chemical reaction to a diÜusive —ux through a gel. The distinguishing feature of the Liesegang rings is that the reaction product building up the rings is stable and immobile in the gel. One of our current interests is to induce the phase separation in the microemulsion by a diÜusive —ow through a well characterized temperature gradient. We hope to –nd a coupling of the oscillations to the diÜusion leading also for this system to complex spatio-temporal behaviour.This would be a direct experimental con–rmation of the close relation between these at –rst sight quite diÜerent phenomena. Prof. Gelbart asked ììEmulsi–cation failureœœ»the appearance of an excess water phase» depends on several assumptions which appear to be implicit in your work. First essentially all of the surfactant must lie at the interface between water droplets and oil ; second the optimum droplet size»the spontaneous curvature of the surfactant monolayer»must be sufficiently strongly preferred i.e. the size distribution should be highly monodisperse. In this context do you have independent experimental data on the average size and polydispersity of the water droplets in your microemulsions and what are the typical values of the size changes associated with your temperature changes (ììquenchesœœ) ? Dr D.Vollmer and Dr J. Vollmer replied For the system we used in the present study the critical micellar concentration (c.m.c. the amount of surfactant residing at the interface) and the temperature dependent size distribution of droplets have been studied independently by Strey and by Olsson and their respective coworkers (cf. refs. 13 and 14 of our paper). The c.m.c. is about 1% at room temperature1 and increases up to 3% at 50 °C i.e. the surfactant does indeed remain predominantly at the interface for the mixtures we investigated. On the other hand the c.m.c. seems to play no essential role for the modelling of the (equilibrium) phase behavior.In particular when one de–nes / as the fraction of the surfactant which resides at the inter- s face the given expressions remain valid for weak surfactants that have a much higher c.m.c. Also the oscillations have been observed for weak as well as for strong surfactants. Strey (ref. 13 of our paper) studied the average size and polydispersity of water droplets in the two-phase region by careful SANS measurements. These measurements con–rm that for C12E5 dissolved in octane and water (i) the droplets are spherical on average with a variance of about 15% in the radius (ii) the temperature dependence of the average radius of droplets is well described by the law given in eqn. (2) of our paper. Hence the droplet radii vary from about 15 nm when –rst crossing the phase boundary down to less than 5 nm at the end of the measurements i.e.the droplet volume typically changes by more than a factor of 25 in the course of the measurement. In a single oscillation on the other hand typically only a few percent of the water residing in the droplets is expelled. 1 See for instance R. Strey O. Glatter K.-V. Schubert and E. W. Kaler J. Chem. Phys. 1996 105 1175. Prof. Vincent asked I can understand how the nuclei of the big water droplets form by coalescence of w/o microemulsion droplets and that the big water droplets grow by diÜusion of water from the microemulsion droplets (such that they reduce to their optimum size at that 77 Faraday Discuss. 1999 112 63»80 temperature). However you also say that the total number of smaller microemulsion droplets increases (to preserve the total surface area as controlled by the amount of surfactant in the system).What is the mechanism by which new extra small droplets form? Dr D. Vollmer and Dr J. Vollmer answered First we would like to point out that no coagulation is needed to form the large droplets»redistribution of water and surfactant among the existing droplets is sufficient to that end. Moreover when assuming that the small droplets remain spherical while the number of droplets is not increasing the large droplets have to attain an elongated (or very —oppy and strongly —uctuating) shape irrespective of the mechanism of phase separation as can be checked from the conservation of the overall volume and surface area of the droplets.In this case however the large droplets can lower their free energy by expelling small droplets with a radius close to the optimum radius until they attain a close to spherical shape. To our understanding this is the dominant mechanism for the increase of the number of droplets you were asking for. We have con–rmed this picture experimentally by contrast enhanced video-microscopy (in collaboration with Rehageœs group at the University of Essen). When a large water droplet with a radius of about 10~6 m is immersed into a microemulsion phase of a phase-separated system its size increases when the microemulsion droplets try to expel water due to an increase of temperature and it decreases again when the temperature is lowered.In the whole cycle the shape of the large droplet remains close to spherical. Prof. Coveney commented I wanted to take up a few issues raised by some of the remarks I read in the introduction to your interesting paper. You make the observation that nucleation and growth processes require a description in terms of non-linear kinetic equations. This is certainly correct and indeed the description of these phenomena is a new area for non-linear dynamics. I would like to emphasize however that the theoretical analysis of nucleation and growth processes by these methods is both new and itself very challenging ; and secondly that other authors have recognised that it is possible to investigate more familiar manifestations of nonlinearity in related systems.1 In fact some simple macroscopic models of clock reactions which we have used to model nucleation and growth processes such as the scheme2 P »»’ kp A A »»’ k1 B A]2B »»’ k2 3B B]C »»’ ki D (in which P is a precursor A is an amorphous material B a crystalline form of the same substance C a nucleation/growth inhibitor and D a poisoned species) are directly related to simple nonlinear chemical kinetic schemes in this case the autocatalator of Gray and Scott,3 which supports limit cycle oscillations.In general however macroscopic non-linear kinetic schemes and their associated diÜerential equations are not adequate to handle the complexities and subtleties of nucleation and growth phenomena. We have advocated a more fundamental and systematic approach to the description of these processes starting out from a –ne grained microscopic model the intention is to derive progressively more mesoscopic and ultimately if necessary macroscopic descriptions.The reason for doing this is that the experimentalist will not often have access to all the –ne detail implied by the microscopic model or real system. r To illustrate this approach consider nucleation and growth via the so-called Becker»Doering mechanism,4 that is stepwise addition of a single monomer C to a growing cluster C of aggre- 1 Faraday Discuss. 1999 112 63»80 78 gation number r ar Cr]C1 EF Cr`1 br`1 (a mechanism which has some relevance to spinodal decomposition). A kinetic model of such a system would contain two times an in–nite number of generally unknown (forward and backward) rate coefficients ! To make useful progress in deriving viable theoretical models of nucleation and growth based on these equations it is necessary to coarse-grain the system; we do this by clustering clusters into clusters and so on in a hierarchical manner.In so doing the dimensionality of the dynamical system is greatly reduced with fewer parameters to be determined or needed for –tting to experimental data. Indeed we have recently shown that there is actually a dynamical renormalisation group underlying the Becker»Doering cluster equations ; this has the consequence that universality classes may be identi–ed for certain types of nucleation and growth processes and brings with it the considerable bene–t that in some cases of interest (late time asymptotic behaviour in the macroscopic limit) the behaviour is largely independent of the detailed nature of the multitudinous rate coefficients present in the original microscopic model.5 Our approach has been successfully applied to a growing range of nucleation and growth phenomena including (i) micelle formation and self-reproduction ;6 (ii) vesicle formation and selfreproduction; 7 (iii) cement setting ;8 and (iv) the origin of the RNA world.9 1 P.V. Coveney in Self-Production of Supramolecular Structures eds. G. Fleischaker S. Colonna and P. L. Luisi NATO ASI Series Kluwer Dordrecht 1994 pp. 157»176; T. Buhse R. Nagarajan D. Lavabre and J. C. Micheau J. Phys. Chem. 1997 101 3910. 2 J.Billingham and P. V. Coveney J. Chem. Soc. Faraday T rans. 1993 89 3021. 3 P. Gray and S. K. Scott Chem. Eng. Sci. 1983 38 29. 4 R. Becker and W. Doering Ann. Phys. (L eipzig) 1935 24 719. 5 P. V. Coveney and J. A. D. Wattis Renormalization group theory of the Becker»Doering cluster equations preprint (1999). 6 P. V. Coveney and J. A. D. Wattis Proc. R. Soc. L ondon Ser. A. 1996 459 2079. 7 P. V. Coveney and J. A. D. Wattis J. Chem. Soc. Faraday T rans. 1998 102 233. 8 J. A. D. Wattis and P. V. Coveney J. Chem. Phys. 1997 106 9122. 9 J. A. D. Wattis and P. V. Coveney J. Phys. Chem. B 1999 103 4231. Prof. Cates addressed Dr Vollmer The model in your paper is linear. Similar oscillations could arise in a completely non-linear mechanism where on cooling one crosses the spinodal rapidly creates equilibrium and then crosses it again on cooling further.How can these be distinguished ? Dr D. Vollmer and Dr J. Vollmer replied The model we propose for the time evolution of the overheating q and the relaxation speed C is non-linear. We demonstrated however that the origin of the oscillations and their parameter dependence can be understood from the linearized equations describing small deviations from the –xed point of the non-linear problem. For increasing values of the control parameter vsR1’/d~2@3 the amplitude and period of the oscillations grow such that eventually the linear approximation breaks down. The situation you describe can be viewed as such a limit. Denote the (composition dependent) diÜerence in temperature between the binodal and the spinodal as *Tsp. If the time required for the relaxation to equilibrium after crossing the spinodal is then short as compared to the time needed to increase the temperature by *Tsp the period of oscillations scales as *T D*Tsp sR1’/d~2@3 one expects a v Hence upon increasing v s. irrespective of the heating (or cooling) rate crossover from the scaling behavior *TDv1@2 s discussed in our article towards *T Dvs0 79 Faraday Discuss. 1999 112 63»80 corresponding to repeated crossing of the spinodal. From the energy barriers for phase separation in the investigated microemulsion (cf. ref. 9 of our paper) one infers that the repeated crossing of the spinodal may arise when *T D*T spB Te[ 2 T1 . d. This parameter range is experimentally accessible for small volume fractions of large droplets. Amazingly within the error margins the data points for periods smaller than *Tsp lie on the master curve indicating that for this particular system the non-linearities only signi–cantly aÜect the period of oscillations when quenching the system into the spinodal regime. The parameter value for the cross-over is solely determined by the droplet radius since *T sp(R) neither depends on v nor on / In the strong driving regime additional parameters are needed for s the description of the dynamics. Faraday Discuss. 1999 112 63»80 80

ISSN:1359-6640    

DOI:10.1039/a903991k

出版商:RSC    

年代:1999

数据来源: RSC

摘要:

Model of the hydrophobic interaction A. B. Kolomeisky§ and B. Widom Department of Chemistry Baker L aboratory Cornell University Ithaca NY 14853 USA Received 30th November 1998 The potential of mean force between interstitial solute molecules in Ben-Naimœs one-dimensional many-state lattice model (related to the one-dimensional q-state Potts model) is calculated. Since the model is exactly soluble all results are explicit and analytic. It is found that the magnitude of the eÜective attractive force between solutes and the range of that attraction vary inversely with each other the strength of the attraction as expected increases with increasing magnitude of the (entropically unfavorable) free energy of ììhydrogen-bondœœ formation but at the same time the range decreases.Conversely when the unfavourable entropy and favourable energy of ììhydrogen-bondœœ formation are nearly in balance the attraction between hydrophobes while then weak is of very long range. It is remarked that solubility in a one-dimensional solution model when the direct intermolecular interactions are of short range can only be de–ned osmotically. The solubility of the hydrophobe as so de–ned is calculated with the present model. It is found to decrease with increasing temperature as expected for a hydrophobic solute. 1 Introduction The hydrophobic eÜect (thinking of water as the solvent) is a manifestation of an unfavorable free energy of solvation. The volume of solvent that is unfavorably aÜected by the forced accommodation of two hydrophobic solute molecules is less when the latter are close together than when they are widely separated thus leading to an eÜective solvent-mediated attraction between them.This is superimposed on and often dominates the direct interaction between the solute molecules. In water the unfavorable solvation free energy arises from an unfavorable entropy change associated with the reorganization of the solvent structure (the formation or strengthening of hydrogen bonds). This unfavourable entropy change outweighs the accompanying energy change which by itself would favor that structural rearrangement. Ben-Naim1 has described a one-dimensional many-state lattice model (related to the ììq-state Potts modelœœ2 in one-dimension) which we now adapt to illustrate and illuminate the hydrophobic eÜect.The model is suited to this purpose because it incorporates the basic mechanism described above even though its picture of ììwaterœœ is highly unrealistic. There are very many interesting and important studies of the hydrophobic eÜect that are based on much more realistic pictures of water.3h16 The advantages of the present model are that it is analytically soluble and that its extreme simplicity allows some aspects of the hydrophobic eÜect that might otherwise be obscured by detail to be exhibited clearly. § Present address Institute for Physical Science and Technology University of Maryland College Park MD 20742 USA. 81 Faraday Discuss. 1999 112 81»89 In the following section we de–ne the model and outline the calculation of the solvent-mediated contribution to the potential of mean force between solute molecules.In Section 3 we calculate via the modelœs transfer matrix the elements called for in Section 2 and thus obtain explicit formulas for the mean force. Section 4 provides numerical illustrations of the results. In Section 5 we de–ne and calculate the soluteœs solubility in the model solvent. (1) (2) 2 Model and the potential of mean force The model one-dimensional solvent is shown in Fig. 1. Each molecule occupies one lattice site and may be in any of q states (orientations). The horizontal orientation is identi–ed as state number 1. A hydrophobic solute molecule (shown as the shaded circle in the –gure) can only be accommodated at an interstitial site between two solvent molecules that are both in state 1.Thus the presence of such a solute forces its two neighboring solvent molecules to be both in state 1. A pair of neighboring solvent molecules with both members of the pair in state 1 corresponds to a ììhydrogen-bond.œœ Only neighboring solvent molecules interact with each other and they interact also with any interstitial solute that may reside between them. (i j\1 . . . q) be the energy of interaction between a pair of neighboring solvent Let wij molecules when one is in state i and the other in state j and take wij\Gw u, when otherwise i\j\1 with u[w The positive diÜerence u[w may be identi–ed as the (magnitude of ) the favorable energy of ììhydrogen-bondœœ formation while the positive quantity k ln(q[1) where k is Boltzmannœs constant is the (magnitude of) the unfavorable entropy.They are the energy and entropy associated with restricting a solvent molecule to be in state 1 given that a speci–ed one of its neighbors is also in state 1. Let v be the energy of interaction of an interstitial solute molecule with its solvent neighbors. The model is then de–ned by three parameters u[w v and q. It will transpire that the solventmediated part of the potential of mean force between solutes depends only on the –rst and third of these (thus only on the properties of the pure solvent as already remarked by Pratt and Chandler5) while the solubility of the solute depends on all three. In this model the picture of ììwaterœœ is highly oversimpli–ed and in this simplest version of the model there is no provision for distinguishing solutes of diÜerent sizes.Much of the detail incorporated in more realistic models3h16 is therefore absent although the essential features of the hydrophobic interaction are still present. The calculation of the potential of mean force will be based on the potential-distribution theorem.17 Let /(r) with r measured in units of the lattice spacing be the direct interaction potential between two solute molecules a distance r apart and let g(r) be the solute»solute pairdistribution function. Also let P be the probability that the solvent molecules at a given pair of 11 consecutive sites be both in state 1 irrespective of the states of the molecules at the other sites and let P(r) likewise be the probability that the molecules at two such pairs of consecutive sites m m]1 and n n]1 with n[m\rP1 be all in state 1.Then in the dilute-solution limit where Fig. 1 The model one-dimensional solvent with each molecule centered at a lattice site and in one of q states (orientations). The horizontal orientation is identi–ed as state 1. A solute molecule (the shaded circle in the –gure) can only be accommodated at interstitial sites between two solvent molecules that are both in state 1. Faraday Discuss. 1999 112 81»89 82 the solute ìì test particle œœ and ìì test-particle œœ pair17 encounter only solvent but not other solutes (3) (rP1) (P g(r)e’(r)@kT\ P(r)e~2v@kT P(r) 11 e~v@kT)2 \ P11 2 at the temperature T . We note the cancellation of the parameter v in this expression.The total potential of mean force between pairs of solute molecules is [kT ln g(r) of which the solvent-mediated part W (r) is [kT ln g(r)[/(r) ; thus W (r)\[kT ln[g(r)e’(r)@kT] (4) \[kT ln[P(r)/P11 2 ] from eqn. (3). That this is independent of the parameter v and so depends only on the properties of P the pure solvent is as anticipated. Explicit expressions for and P(r) which are the two ingre- 11 dients in the formula for W (r) are obtained in the following. (5) (6) (7) (8) (9) (rP1) 3 Transfer matrix and calculation of W(r) P The quantities and P(r) required in eqn. (4) for the evaluation of W (r) may be obtained from 11 the eigenvalues and eigenvectors of the transfer matrix V of the model solvent.Let b\e~w@kT a\e~u@kT with u and w the interaction energies in eqn. (1). Then V is the q]q matrix in which the 1,1 element is b and all the other elements are a V\ a a a ……… a qCq a< < < a a a …… … … b a a … a a a ……… … a a a <b It is convenient only because it allows a slightly simpler notation in what follows to take the linear chain of Fig. 1 to form a closed circle of N sites so that sites 1 and N are neighbors. This has no eÜect on properties calculated in the thermodynamic limit N]O [the limit taken at –xed r in the case of P(r)]. Then by standard transfer-matrix methods18 one –nds that the quantity P11 de–ned in Section 2 as the probability that the molecules at a pair of consecutive sites be both in state 1 is P11\ Z 1 V11(VN~1)11 where Z is the partition function of the model solvent Z\trace VN (VN~1) V11 and and where in eqn.(7) the notation means the 1,1 element of the matrices V and 11 VN~1 respectively. Likewise P(r) the probability that the molecules at two such pairs of sites m m]1 and n n]1 with n[m\r be all in state 1 is P(r)\ 1 Z (V11)2(Vr~1)11(VN~r~1)11 t(1) Note the distinction between (Vij)p and (Vp)ij when p[1. The relation in eqn. (9) holds as indicated even down to r\1 where the sites m]1 and n are the same site. V as de–ned in eqn. (6) has two positive eigenvalues,1 j(1)[j(2)[0 the remaining q[2 eigenvalues being all 0. Let ti (l) be the ith component (i\1 . . . q) of the eigenvector that belongs to the eigenvalue j(l) (l\1 .. . q). Then from the structure of V in eqn. (6) it follows since j(1) and j(2) are non-zero that all the t(1) except t(2) except t(2); have a common value as do all the i 1 i 1 Faraday Discuss. 1999 112 81»89 83 i.e. (10) t(2)\t(2)\… … …\t(2) q t(1)\t(1)\… … …\t(1); q 2 3 2 3 Then the eigenvalues j(1) and j(2) of V and their associated eigenvectors satisfy1 (11) b a t t 1 ( 1 ( l l )] )] ( ( q q [ [ 1) 1) a a t t 2 ( 2 ( l l )\ )\ j j ( ( l l ) ) t t 1 ( 2 ( l l ) )H(l\1 2) From eqn. (11) j(1) and j(2) are respectively the larger and smaller of the two roots j (both positive) of1 (12) K Kb[j (q[1)a \0 a (q[1)a[j Take the eigenvectors to be normalized by &i/1 q ti (l)2\1 so that from eqn.(10) (13) 1 t1 (l)2](q[1)t2 (l)2\1 (l\1 2) t(1) t(1) and t(2) t(2) of the normalized eigenvectors follow from 1 2 2 Then the distinct components eqns. (11)»(13). The elements of any power Vp of V may be expressed in terms of the eigenvalues j(l) and normalized eigenvectors w(l) (l\1 . . . q) by (14) q (Vp)ij\ ; j(l)pti (l)tj (l) l/1 But we noted that j(1)[j(2)[0 and all other j(l)\0. It then follows from this and from eqns. (6)»(9) upon taking the thermodynamic limit N]O [at –xed r in the case of P(r)] that (15) P11\ j b (1) t1(1)2 and (16) P(rP1)\P11 2 C1]At t 1(2) (1)B2Aj j (2) (1)Br~1D 1 The formula in eqn. (15) for P will be required in Section 5 in the calculation of the solubility of 11 the solute in the model solvent.Meanwhile from eqns. (4) and (16) we obtain for the solventmediated part W (r) of the potential of mean force between two solute molecules separated by r (P1) lattice spacings in the dilute-solution limit (17) W (r)\[kT lnC1]At1(2) j(1)Br~1D t(1) 1 B2Aj(2) One sees from eqn. (17) that W (r) vanishes proportionally to exp([r/m) at large r with an exponential decay length m which we may identify as the range of the solvent-mediated force between solutes given by 1 (18) m\ ln j(1) j(2) This is the same formula as that for the correlation length in the Ising model when j(1) and j(2) are the largest and next largest eigenvalues of its transfer matrix.18 Approach to the critical point in the Ising model is characterized by a closing of the gap between j(1) and j(2) and a consequent divergence of m.In the present model it will transpire there is an analogous phenomenon when the favorable energy and unfavorable entropy of ììhydrogen-bondœœ formation are nearly in balance j(1) and j(2) are nearly equal and the potential of mean force is then long ranged (although weak). Faraday Discuss. 1999 112 81»89 84 1 (t1(2)/t(1))2 and j(2)/j(1). De–ne The explicit calculation of W (r) from eqn. (17) requires the ratios new quantities c x S and Q in terms of a b and q[1 by (19) c\ b a \e(u~w)@kT[1 (20) x\ q[1 c (21) S\S1[ 4x 1[ 1 (1]x)2 A cB (22) Q\Ssign (x[1) 1] (x[1)2c 4x where sign (x[1) means ]1 if x[1 and [1 if x\1.That c[1 is a consequence of the earlier assumption in (2). Then S and Q are manifestly real with 0\S\1 and [1\Q\1. Then from eqns. (11)»(13) we –nd for the required ratios (23) At(2) t(1) 1 B2 \ 1 1 ] [ Q Q j j (1) (2) \ 1 1 [ ] S S 1 Then from eqns. (17) and (23) with the de–nitions in eqns. (19)»(22) we obtain W (r) as an explicit function of r and of the modelœs two parameters u[w and q at any temperature T (24) 1]SBr~1D W (r)\[kT lnC1] 1]Q A1[S 1[Q It was remarked in Section 2 that u[w may be identi–ed as the (magnitude of) the favorable energy of ììhydrogen-bondœœ formation and k ln(q[1) the (magnitude of) the unfavorable entropy. Then the corresponding free energy of ììhydrogen-bondœœ formation *F (unfavorable when it is positive) is [(u[w)]kT ln(q[1) ; or from the de–nition of x in eqn.(20) *F\kT ln x (25) We may then anticipate that for this to be a model of the hydrophobic interaction we shall need x to be greater than 1. In the numerical illustration in Section 4 which follows it will be seen that realistic values of x are in the range 2»20. At the same time it will be seen in Section 4 that the favorable energy and unfavorable entropy are not far from being in balance; i.e. that the positive *F is typically much smaller than either of the positive quantities u[w or kT ln(q[1) separately ; and that as a consequence c as de–ned in eqn. (19) is typically thousands of times as great as x. Thus as a practical matter from the de–nitions of S and Q in eqns.(21) and (22) the quantities (1]Q)/(1[Q) and (1[S)/(1]S) required in eqn. (24) may be taken to be (26) 1 1 ] [ Q Q Bg( 0 x[ x 1)2 c x x [ \ 1 1 (27) 1 1 [ ] S S Bgx 1 x, x x [ \ 1 1 It was remarked earlier that 0\S\1 (whatever x and c) so we have 0\(1[S)/(1]S)\1. Therefore from eqns. (24) and (26) the solvent-mediated attraction between solutes is practically nil when x\1. This con–rms that x[1 [or *F[0 from eqn. (25)] is necessary for an attractive hydrophobic interaction as anticipated. 85 Faraday Discuss. 1999 112 81»89 When x does exceed 1 so that there is a non-negligible solvent-mediated hydrophobic attraction we see from eqns. (18) (23) (25) and (27) that its range is (28) mB 1 ln x \ * kT F The closer the otherwise positive *F is to 0 the weaker is the attraction but then also the longer ranged is it according to eqn.(28). Thus the strength and range of the attraction are inversely related. This will be very clear in the plots displayed in Section 4. Should this prove to be a general feature of the hydrophobic eÜect not limited to the present model it could have important implications in the interpretation of the hydrophobic eÜect in real systems. When the energy and entropy of hydrogen-bond formation are nearly in balance *F is near 0 and x is near 1. That from eqns. (23) and (27) makes the eigenvalues j(1) and j(2) nearly degenerate. This is the analogy with approach to a critical point that was mentioned earlier.Strictly however as noted above (1[S)/(1]S)\1 so unlike at a true critical point we cannot here have an exact degeneracy. 4 Numerical illustration For a realistic representation of hydrogen-bond energy one should think of the model parameter u[w as being around 25 kJ mol~1. For purposes of illustrating the workings of the present model we shall then take (u[w)/k\3000 K. There is no generally agreed on value of the free energy of hydrogen-bond formation except that at 300 K it is much less than 25 kJ mol~1; i.e. the favorable energy and unfavorable entropy are largely mutually compensating at that temperature. For purposes of illustration we shall take *F to be about 4 kJ mol~1 at 300 K which by eqn. (25) is equivalent to taking x to be around 5 at that temperature.Then from eqns. (19) and (20) with (u[w)/k\3000 K we conclude that q[1 is around 110 000 and that x varies from 2 to 20 as the temperature varies from 275 to 348 K. Over this temperature range then the quantity c de–ned in eqn. (19) is enormously greater than x as anticipated in Section 3. That q should be many thousands has as a precedent what Andersen and Wheeler –nd in an analogous lattice model of closed-loop coexistence curves.19 They have the parameters uB5000 and u*B(nu)1@2 sin h with h the tetrahedral angle arccos([1/3) and they identify the number of 1 orientations of each ììH 2uu*. In the notation of the present model that would 2Oœœ molecule as imply qB300 000. Both in that model and in the present one it is probably because the picture of ììwaterœœ is unrealistic that so large a value of q is required.20 Fig.2 shows W (r)/k as a function of r as calculated from eqn. (24) with eqns. (19)»(22) at the three temperatures T \275 K 300 K and 348 K with the parameter values (u[w)/k\3000 K and q[1\110 000. Using the approximations in eqns. (26) and (27) instead of the exact eqns. Fig. 2 W (r)/k (in K) with (u[w)/k\3000 K and q[1\110 000. The curves marked x\2 5 and 20 correspond to the temperatures T \275 K 300 K and 348 K respectively. Faraday Discuss. 1999 112 81»89 86 Fig. 3 [(1/k) dW (r)/dr (in K) with the same values of the parameters and at the same temperatures as in Fig. 2. (21) and (22) makes no discernible diÜerence. The analytic function of r in eqn.(24) was used to plot smooth curves for ease of visualization although for this lattice model only integer values of r (as integer multiples of the lattice spacing) are de–ned. The plots are made only for rP1; we choose not to de–ne W (r) at r\0 for any such de–nition would be arbitrary and unphysical. The solvent-mediated force (in units of the Boltzmann constant and of the reciprocal of the lattice constant) [(1/k) dW (r)/dr is plotted in Fig. 3 with the same values of the parameters and for the same three temperatures as in Fig. 2. One sees in both Figs. 2 and 3 that the strength of the solvent-mediated hydrophobic attraction increases with increasing temperature while the range of the attraction decreases. The strength increases because the free energy *F which is [(u[w)]kT ln(q[1) becomes increasingly positive hence increasingly unfavorable to ììhydrogen-bondœœ formation as T increases ; while the range decreases with increasing T because then by eqn.(28) (29) mB[[(u[w)/kT ]ln(q[1)]~1 The inverse relation between strength and range anticipated in Section 3 is apparent in the –gures. 5 Solubility De–ning or measuring solubility requires the coexistence of two phases one a reference phase and the other the solution in which the solubility of the solute is to be determined. The two phases are in equilibrium ; the solute is present in the two at equal chemical potential (activity). The reference phase may be pure liquid or solid solute ; it may be pure gaseous solute in which case one is de–ning the solubility of a gas; or it may be another liquid or solid solution in which case one is de–ning the solubility by the partition coefficient of the solute between the two phases.The case in which the reference phase is pure gaseous solute is a special case of the latter where the ììsolventœœ in the reference phase is vacuum. Because normal phase coexistence is not possible in one-dimensional systems with short-range forces the solubility of the hydrophobic solute in the model solvent can here only be de–ned osmotically. We imagine a semi-permeable membrane permeable only to the solute. (Note that this is opposite to the usual picture of osmosis where the membrane is permeable only to solvent not to solute.) In the present one-dimensional model the ììmembraneœœ is a point separating the saturated solution of interest on one side from the reference phase on the other.For de–niteness and for purposes of illustration we shall take the reference phase to be pure gaseous solute dilute enough to be an ideal gas. We are thus determining the solubility of a gaseous hydrophobe in the model solvent. Let ogas and osoln be the number densities of the solute in the reference phase and in the solution respectively at osmotic equilibrium. We than take as 87 Faraday Discuss. 1999 112 81»89 Fig. 4 Solubility R]105 as a function of temperature (in K) with (u[w)/k\3000 K q[1\110 000 and v\0. our de–nition of the solubility R the dimensionless ratio (30) R\osoln/ogas With the reference gas phase taken to be ideal and with the solubility in the solvent assumed to be low enough so that the saturated solution is also very dilute this R is the Ostwald absorption coefficient.It is also proportional to the Henryœs-law coefficient (or its reciprocal depending on how one writes Henryœs law). With the activity z of a species de–ned so as to become asymptotically equal to the number in the de–nition of R in eqn. density in the limit of an in–nitely dilute gas the denominator ogas (30) is also the common value of the activity z of the solute in the two phases. Then R is the ratio of the number density to the activity in the model solution. By the potential-distribution theorem,17 that ratio in the present model is precisely the P exp([v/kT ) that occurs squared in 11 the denominator of the –rst equality for g(r) exp[/(r)/kT ] in eqn.(3) ; thus from eqn. (15) (31) R\ b j(1) t1(1)2 e~v@kT Unlike the potential of mean force the solubility R depends on the modelœs third parameter v which is the energy of interaction of an accommodated solute molecule with its two neighboring solvent molecules. It is equivalently the energy of transfer of a solute molecule from the reference gas phase to an already available interstitial site in the model solvent. The energy of transfer without the stipulation ììalready available œœ when q[1 is very large so that very few sites are indeed available is v[(u[w). bt(1)2/j(1) of the exponential in eqn. (31) from eqns. (11)»(13) with 1 We may obtain the coefficient l\1 with the result that (1/c)e~v@kT (32) R\ [(x[1)(1[1/c)]12(x[1]2/c)[x]1](x[1)/Q] with c x and Q as given by eqns.(19) (20) and (22). With x[1 and c thousands of times as great just as before we have QB1 from eqn. (22) and (33) e~v@kT RB (x[1)2c The pre-factor [(x[1)2c]~1 of the exponential in eqn. (33) is the approximation to P11 the probability that the solvent molecules at a pair of consecutive sites be both in state 1. With u[w[0 this probability must decrease with increasing temperature and that this is so for Faraday Discuss. 1999 112 81»89 88 [(x[1)2c]~1 may be veri–ed from the de–nitions of c and x in eqns. (19) and (20). The additional energy parameter v may be positive or negative. In Fig. 4 there is plotted the solubility R as a function of temperature from eqn. (33) for v\0 and for the same values of the other parameters q[1 and u[w as in Figs.2 and 3. The temperature range in Fig. 4 is 275 to 348 K; these are the lowest (x\2) and highest (x\20) of the temperatures to which the three curves in Figs. 2 and 3 correspond. As expected R falls with increasing T . This is one of the signatures of the hydrophobic eÜect in real systems. The solubility is seen to be very low with these values of the parameters of the order of 10~6 or 10~5 over the temperature range of the –gure. For v\0 the solubility is greater than that in Fig. 4 by the factor exp([v/kT ) but it then decreases more rapidly with increasing T . When v[0 the solubility is less by that exponential factor but it then decreases less rapidly with increasing T and may even go through a minimum and then increase.This happens when v[u[w; i.e. when the energy of transfer of a solute molecule from the reference gas phase into the solvent (with q[1 very large) is positive. Such a minimum in the solubility of a hydrophobic solute in water as a function of temperature is often observed in experiment and in simulation ;6 but in the present model the solubility R at such a minimum could only be at most of order 1/(q[1)2 which is unrealistically low. Thus it may be that the mechanism by which the minimum occurs here is diÜerent from that by which it occurs in real systems. This question remains open. Acknowledgement This work has been supported by the U.S. National Science Foundation and the Cornell Center for Materials Research.Gr‘nbech-Jensen and S. Doniach J. Chem. Phys. 1996 104 8639. References 1 A. Ben-Naim Statistical T hermodynamics for Chemists and Biochemists Plenum New York 1992 pp. 220»223. 2 F. Y. Wu J. Appl. Phys. 1984 55 2421. 3 F. H. Stillinger J. Solution Chem. 1973 2 141. 4 J. C. Owicki and H. A. Scheraga J. Am. Chem. Soc. 1977 99 7413. 5 L. R. Pratt and D. Chandler J. Chem. Phys. 1977 67 3683. 6 B. Guillot and Y. Guissani J. Chem. Phys. 1993 99 8075. 7 J. Forsman and B. Joé nsson J. Chem. Phys. 1994 101 5116. 8 M. E. Paulaitis S. Garde and H. S. Ashbaugh Curr. Opin. Colloid Interface Sci. 1996 1 376. 9 M. Pellegrini N. 10 G. Hummer S. Garde A. E. Garcïç a A. Pohorille and L. R. Pratt Proc. Natl. Acad. Sci. USA 1996 93 8951. 11 S. Lué demann H. Schreiber R. Abseher and O. Steinhauser J. Chem. Phys. 1996 104 286. 12 F. M. Floris M. Selmi A. Tani and J. Tomasi J. Chem. Phys. 1997 107 6353. 13 N. A. M. Besseling and J. Lyklema J. Phys. Chem. B 1997 101 7604. 14 R. D. Mountain and D. Thirumalai Proc. Natl. Acad. Sci. USA 1998 95 8436. 15 K. A. T. Silverstein A. D. J. Haymet and K. A. Dill J. Am. Chem. Soc. 1998 120 3166. 16 K. Lum D. Chandler and J. D. Weeks J. Phys Chem. B 1999 103 4570. 17 B. Widom J. Chem. Phys. 1963 39 2808. 18 C. Domb Adv. Phys. 1960 9 149. 19 G. R. Andersen and J. C. Wheeler J. Chem. Phys. 1978 69 3403. 20 J. C. Wheeler personal communication. Paper 8/09308C 89 Faraday Discuss. 1999 112 81»89

ISSN:1359-6640    

DOI:10.1039/a809308c

出版商:RSC    

年代:1999

数据来源: RSC

摘要:

A simple o¢­ -lattice model for microemulsions V. Talanquera and David W. Oxtobyb a Facultad de Quic mica UNAM. Mec xico 04510 D. F. Mec xico b T he James Franck Institute T he University of Chicago 5640 S. Ellis Ave. Chicago IL 60637 USA Received 23rd November 1998 We have developed a simple oU-lattice density functional theory and applied it to ternary water¡íoil¡íamphiphile mixtures. Our approach is based on mean ¡©eld free energy functionals calculated from hard-sphere perturbation theory to which is added a contribution arising from molecular association between the water-like and the amphiphilic species. This latter term is treated using the Wertheim theory for associating liquid mixtures and it gives rise to a re-entrant binary phase diagram in a very natural way.The resulting ternary phase diagrams resemble experimental data both qualitatively and quantitatively. We also calculate the structure and free energy of interfaces between phases in this system and show that the presence of amphiphile dramatically lowers the surface tension between the water-rich and oil-rich phases. This simple model contains many of the features of real microemulsion systems and can be extended to study lamellar and other complex phases. I Introduction Ternary mixtures of water oil and surfactants (amphiphiles) have attracted considerable interest in recent years. This interest has several sources. First amphiphilic systems have potentially signi ¡©cant materials properties because they display extraordinarily varied phase diagrams in which lamellar (layered) phases hexagonal arrays of cylinders and complex bicontinuous phases compete for stability.The extent of spontaneous organization and the extremely low surface tensions found in these systems have suggested a number of applications.1 A second reason for interest is the fact that these systems serve as models for biological membranes; insight into simple amphiphilic mixtures may help to explain the structural basis of living cells. Finally such mixtures provide a fascinating theoretical challenge understanding how simple intermolecular interactions can yield such a range of behavior. Most theories of amphiphilic systems employ lattice models. Once a lattice geometry is chosen and the interactions between the three species are speci¡©ed the resulting statistical mechanical problem is well de¡©ned.One of the advantages of lattice theories is that they often can be mapped onto other problems for which the qualitative or even quantitative solutions are already known. For example one of the most widely studied such models due to Widom,2 employs a cubic lattice and prevents the ii oil ©«©« and iiwater©«©« components from occupying neighboring sites ; any interface must then be mediated by an amphiphile. This model can be mapped directly onto a spin-1/2 Ising model whose phase behavior is well understood. Other more elaborate lattice models are described in a recent review by Gompper and Schick.3 91 Faraday Discuss. 1999 112 91¡í101 Real amphiphilic systems exist in continuous space however so it would be desirable to develop microscopic theories that are free of lattice assumptions.One direction is to move directly to computer simulation via Monte Carlo methods or molecular dynamics as in the recent work of Smit et al.,4 in which amphiphiles are treated like short copolymers with hydrophilic and hydrophobic chains connected at the middle. Such simulations provide detailed information about particular state points but it is difficult to explore large ranges of parameter space or qualitative trends because of the length of each calculation A second type of continuous theory is a Landau» Ginzburg (phase –eld) model in which the free energy is taken to be a functional of one or more order parameters. Ref. 3 summarizes the results of a number of such calculations which average over molecular-level details to provide an overall view of the phase equilibrium.The disadvantage of these approaches is that the coefficients in the expansion of the free energy are arbitrary and their relationship to real interaction potentials is not evident. These observations have led us to the conclusion that statistical mechanical methods based on density functionals may provide a useful alternative to other theories of phase behavior in multicomponent mixtures. These approaches resemble Ginzburg»Landau theories in that they involve free energy functionals of order parameters but they diÜer in that the free energies are directly derived from (approximate) microscopic models of intermolecular interactions.They thus open up the possibility of a truly microscopic continuum theory of ternary systems and microemulsions in which questions such as the eÜect of amphiphile chain length can be explored qualitatively and quantitatively. To date the only existing density functional approaches to microemulsions are due to Telo da Gama and coworkers.5,6 These authors employed a model in which all three components of a ternary mixture are approximated as hard spheres that interact through isotropic attractive potentials. In addition each amphiphile sphere has at its center a vector pointing from the water-like to the oil-like end and interacting in an anisotropic fashion with surrounding water and oil molecules. Thus oil molecules are preferentially attracted to one ììendœœ of the spherical amphiphile and repelled from the other end; the opposite is the case for the water molecules.This model has been used to calculate surface properties phase diagrams and elastic constants of amphiphilic membranes. A similar model was used earlier by the two of us in a study of gas»liquid nucleation in ternary water»alkane»alcohol mixtures.7 In the present paper we employ a diÜerent microscopic model as the basis for a density functional theory. We retain the hard-sphere repulsions and long-range attractions of earlier approaches but we replace the dipolar interaction with a diÜerent and potentially more realistic model. We treat the water»amphiphile interaction using a theory of associating liquids with very strong directional and saturable short-range attractions.Among the striking consequences of such an approach is the natural emergence of re-entrant two-component phase equilibria for water»amphiphile mixtures in which a closed coexistence loop appears without any ad hoc assumptions about the temperature dependence of eÜective interactions. The phase diagrams and interfacial properties of our simple model have many of the features of real-world experiments. The outline of the paper is as follows. In Section II we describe our model and the theory behind it and in Section III we show the resulting phase diagrams interfacial pro–les and surface free energies. Section IV presents some brief conclusions. Our goal in this paper is not to explore the full range of parameter values possible for our model but simply to choose a reasonable set of parameters and explore the sensitivity of the results to those values.In particular the present paper looks only at amphiphilic ternary systems with equal oil and water concentrations. In addition we do not explore the possible existence of lamellar phases or other states that can appear in strongly interacting systems at high amphiphile concentrations. Other consequences of our simple theoretical model and additional comparisons with experiment are postponed to a future paper. II Model Let us consider a ternary mixture of spherical hard-core molecules A B and C representing a polar solvent (water-like) a non-polar solvent (oil-like) and an amphiphile. Molecules of type A and C are assumed to have an association site that allows the formation of dimers AC mimicking the presence of highly directional attractive forces between these particles such as hydrogen Faraday Discuss.1999 112 91»101 92 bonds. In a –rst-order thermodynamic perturbation theory the Helmholtz free energy density of this associating mixture can be written as a sum of separate contributions (1) Fu\fh]fbond]fatt where f is the Helmholtz free energy density of the hard-sphere reference —uid fbond is the mean-–eld contribution to the free energy due is the change h in the free energy due to association and fatt to the dispersion forces between diÜerent species in the mixture. In this simple approach we will consider all particles in the mixture to have the same diameter p; the hard-sphere free energy is then given (2) fh\kT ; oi[ln(oi)[1](4g[3g2)/(1[g)2] i i oi .where the Carnahan»Starling expression is used,8 and k represents the Boltzmann constant T is the absolute temperature o is the total density of particles of type i (i\A B C) and the packing i fraction g\(p/6)p3 ; The contribution to the free energy due to bonding between particles A and C is calculated using Wertheimœs theory for associating —uids.9h11 For molecules with only one attractive bonding site this theory leads to the expression (3) fbond\kT ; oj[ln Xj[Xj/2]1/2] ( j\A C) j where X is the fraction of molecules of type j that are not bonded. These quantities are given by j the mass-action equations (4) X XA\1/(1]oCDACXC) C\1/(1]oADACXA) and D is related to the equilibrium constant between bonded and unbonded species.In particu- AC lar we assume D to have the approximate form9 AC (5) DAC\4pgHS(p)KAC FAC gHS(p) is the contact value of the pair distribution function of the hard-core reference —uid where given by gHS\(1[g/2)/(1[g)3 (6) in the Carnahan»Starling approximation. The quantity K is a measure of the volume available FAC\exp(ew/kT )[1 is the Mayer f-function for the for bonding on molecules A and C and AC square-well interaction potential of depth [ew between two bonding sites. Long-range attractive interactions between particles in the mixture are treated in a mean-–eld fashion with a free energy density contribution of the form (7) fatt\[1/2 ; aij oi oj[dab oAXA oB i where (8) ij\[Pdr /ij(r) (i j\A B C) a with an isotropic interaction potential rOp /ij(r)\0 (9) r[p /ij(r)\[4eij(p/r)6 AXA).Faraday Discuss. 1999 112 91»101 In studying the phase behavior of our ternary mixtures we introduce an additional interaction term that takes into account in a mean-–eld approximation the ììscreeningœœ of the water»oil interaction due to the presence of the amphiphiles. The interaction parameter dab\aAB f [aAB in the second term of eqn. (7) introduces diÜerent free energy contributions for interactions between (a particles of type B and both bonded AB) ( and free aAB f ) particles of type A (the average concentration of the latter is given by o 93 (10) ki\(Lfu/Loi) (i\A B C) (11) P\; ki oi[fu i (12) P1(o ki1(oA oB oC T )\ki2(oA oB oC T) (i\A B C) A oB oC T )\P2(oA oB oC T ) AA P*\p3P/eAA eij *\eij/eAA ew * \ew/eAA and K*\KAC/p3.In particular T BC * in our model. Fig. 1(c) depicts the phase III Results The thermodynamic properties of our ternary mixtures are determined by the relative values of the energy parameters eij and the bonding parameter ew . The corresponding phase diagrams can be obtained by following the properties of the coexisting phases at diÜerent temperatures and compositions. At –xed temperature T –xed chemical potentials and –xed pressure the properties of a pair of coexisting phases 1 and 2 are uniquely determined by the conditions These equations can be solved numerically and the task is simpli–ed by working in reduced units T *\kT /e K*\1.0]10~5 in all our calculations.A Phase diagrams for binary mixtures The phase behavior of our model microemulsion is best understood by –rst studying the phase diagrams of the corresponding binary mixtures when one of the components is absent. They de–ne the topology of the three faces of the phase prism where the phase diagrams of ternary mixtures are usually mapped.12,13 Here we present as an example the properties of the three limiting binary mixtures that correspond to a particular model microemulsion. The binary mixtures A»B (water»oil) in our model are chosen to be highly immiscible exhibiting a liquid»liquid phase separation at low temperatures and high pressures ending at an T ucp * upper critical point (UCP) at for given P*. In real systems this critical point occurs at temperatures much higher than the liquid»liquid upper critical points for the water»amphiphile and oil»amphiphile mixtures.12,13 For symmetrical A»B mixtures (eAA * \eBB * \1.0) liquid»liquid e phase separation occurs in systems with AB * \1.Fig. 1(a) illustrates the phase behavior as a function of the mole fraction of component A (xi\oi/;j oj) for the particular case eAB * \0.45 whose UCP occurs at ucp * \1.9035 for a reduced pressure P*\1.0. The properties of the binary mixture A»C (water»amphiphile) depend on the values of eCC * eAC * The phase diagram at P*\1.0 for a binary mixture A»C with e T e e and the bonding parameter w * \0 and small values of e w* . For AC * the system shows the typical behavior of a segregating mixture with two diÜerent coexisting liquid phases up to a critical point.This behavior changes when the value of ew * is increased favoring the formation of the associated species AC. For temperatures close to the UCP the concentration of bonded species is in general small and the properties of the system are similar to those of a non-associating mixture. At lower temperatures however bonding between particles A and C induces mixing of these two components and the miscibility gap becomes narrower. If e is large enough a lower critical point w *(LCP) develops and the system exhibits re-entrant miscibility. At lower temperatures excludedvolume eÜects lead to phase segregation again and two diÜerent liquid phases coexist at temperatures below a low temperature UCP.CC * \2.0 eAC * \1.35 and ew * \ 8.25 is shown in Fig. 1(b) ; the value of the latter parameter has been chosen so the LCP is located at T lcp * \1.00. The relatively high value of the attractive term between species of type C shifts the location of the LCP to low mole fractions of this component mimicking the observed behavior in real water»amphiphile mixtures.14 The low and high temperature upper critical points are located at ucp * \0.587 and T ucp * \1.523 respectively. The last type of binary mixture to be considered is the B»C type (oil»amphiphile) which we assume to be segregating in nature. In general the UCP of an oil»amphiphile mixture tends to be located at lower temperatures than the LCP of the corresponding water»amphiphile mixture; the location of this critical point depends on the value of e Faraday Discuss.1999 112 91»101 94 e AC * \1.35 ew * \8.25. (c) B»C eBB * \1.0 eCC * \2.0 eBC * \1.45. AA * \eBB * \1.0 eAB * \0.45. (b) A»C eAA * \1.0 eCC * \ Fig. 1 Phase diagrams for binary mixtures. (a) A»B 2.0 e T ucp * \0.883 in this case. behavior of a B»C mixture with eBC * \1.45 and energy parameters eBB * and eCC * chosen to match those of the binary mixtures in Fig. 1(a) and 1(b) ; the UCP occurs at Figs. 1(a)»1(c) illustrate the type of limiting phase behavior that we impose on the microemulsion systems described in the next section. The interplay of the lower miscibility gaps for the A»B system and the B»C system with the upper loop of the A»C system determines the phase behavior of our ternary mixtures.B Phase diagrams for ternary mixtures Experimental results for the phase behavior of microemulsions are usually expressed in terms of the overall concentration of surfactant and the oil/water mass fraction or volume fraction in the mixture.15,16 We characterize the properties of our model systems using equivalent thermodynax mic quantities such as the mole fraction of component C C\oC/(oA]oB]oC) and the B/A mole ratio /\oB/(oA]oB). In particular we follow the properties of coexisting phases at constant / as a function of temperature. As in equivalent experimental studies which tend to be restricted to the temperature region between the freezing and boiling points of water we restrict our analysis to the temperature range where three-phase coexistence is displayed (intervals ranging from 0.1 to 0.3 in reduced temperature T * units).The energy parameters for our ternary mixtures were chosen to reproduce qualitatively the typical phase behavior of real microemulsions. Fig. 2 shows a schematic phase prism that summarizes the calculated variation of phases as a function of temperature. Three-phase coexistence 95 Faraday Discuss. 1999 112 91»101 Fig. 2 Schematic phase prism of a ternary mixture of components A B and C. The three-phase body extends from a lower critical endpoint at T l* to an upper critical endpoint at T u* . between an A-rich (water-like w) a B-rich (oil-like o) and a middle phase (m) always occurs between the low and high temperature critical endpoints at T l* and T u* respectively.When T * approaches T l* from above the middle and A-rich phases become critical and the three phase triangle leads to a lower critical tie line. Close to T u* however it is the B-rich phase that merges with the middle phase and the three phase region degenerates into an upper critical tie line. The separation into three phases is the consequence of the inversion of the distribution of component C between the A-rich and the B-rich phases within the narrow temperature range *T *\T u* [T l*. C Fig. 3 illustrates the kind of phase diagram obtained by taking a T *[x vertical section of the phase prism at a constant B/A mole ratio /\0.5 which is equivalent to one at equal concentrations of oil and water. This phase diagram shows the –sh-shaped phase boundaries commonly observed in nonionic amphiphilic systems;15,16 the contour of the –sh separates regions where only one phase or two or three coexisting phases can be found.Homogeneous mixtures close to the tail of the three-phase body represent microemulsion phases. The properties of the three-phase body can be characterized by the width of its temperature interval *T * the mean temperature of Fig. 3 Section at constant B/A ratio (/\0.5) of the phase prism in a ternary mixture with eAA * \eBB * \1.0 eCC * \2.0 eAC * \1.35 eAB *f \0.45 eBC * \1.45 dab/aAA\[0.35 ew * \8.25. The values of eAA * eBB * and eCC * are kept the same in all our calculations. Faraday Discuss. 1999 112 91»101 96 T1 *\(T u*]T l*)/2 the location of the –sh-tail point x (where the three-phase body C t *xC\xCt [xC h de–ned as the length of the –sh (xC h ) ( to ìì tail œœ xC t ).These properties depend strongly on the particular values the interval touches the one phase region) and its length pattern from ììheadœœ of the model parameters for a ternary mixture. Let us –rst consider the eÜect of the screening parameter dac/aAA; Fig. 4 depicts our results for several mixtures. The more negative the values of dab the stronger the screening of the A»B interactions due to the presence of associated species AC. Hence the efficiency of component C (amphiphile) in homogenizing A and B increases and xC t shifts to lower values. The width of the –sh pattern *T * increases in this process due to the stabilization of the middle phase while its length *x decreases.As the –sh pattern gets wider T l* decreases more rapidly than T u* increases C shifting the three-phase body to lower temperatures (T1 * decreases slightly). dab and the bonding parameter e are a measure of what we w * In our simple model the values of can call the ììamphiphilicityœœ of component C. These parameters determine the ability of surfactant particles to associate with water-like species and to screen the interaction between water and oil. Given a particular value of the screening parameter dab the phase behavior is then also aÜected by either favoring or disfavoring bonding between species A and C. Increasing the value (eAB *f ) ; the mixing parameter KAB * \eAA * ]eBB * AB *f can be thought of as a measure of the hydrophobicity of component B.As can be seen in T " and u* T l* fall with decreasing AB * of e barely changes the location of the –sh-tail xC t and the length of the –sh-pattern *x as shown w * C in Fig. 5. The mean temperature T1 * increases as the presence of the bonded species AC is favored in the system. The lower critical point however shifts to higher temperatures faster than the upper critical point and the width *T * diminishes. Fig. 6 illustrates the eÜect on the phase behavior of the interaction between particles of component B and nonbonded particles of component A [2e this –gure both Hence the mean temperature mixtures where the length of the oilœs carbon chain is taken as a measure of its hydrophobicity. 12,13 The length of the –sh-pattern also decreases with decreasing KAB * because smaller ; there is almost no eÜect however on the temperature at which the –sh-tail point occurs.but the former drops faster than the latter. T1 * and the width *T * decrease resembling the behavior of real xC t becomes Changing the hydrophobicity of the amphiphile also has a strong in—uence on the phase behav- AC * and eBC * in our model; the corresponding results are summarized in Figs. 7(a) and 7(b). Increas- (T drop; *T * and T1 * then decrease [Fig. 7(a)]. Smaller l*) but makes e ior of real microemulsions. This phenomenon can be studied by analyzing the eÜect of changing e ing the interaction between B and C species (oil»amphiphile) has almost no eÜect on the location of the lower critical point T u* attractive interactions between components A and C on the other hand shift the three-phase body to lower temperatures with little eÜect on the size of the –sh-pattern [Fig.7(b)]. The comeAC * bined eÜect of a larger leads us to expect as observed in real systems,12,13 BC * and a smaller sections at /\0.5 of the three-phase body of a ternary mixture with eAC * \1.35 eAB *f \0.45 Fig. e 4 T »x BC * \1.45 ew * \8.25 and two diÜerent values of the screening parameter dab/aAA . C 97 Faraday Discuss. 1999 112 91»101 section at /\0.5 of the three-phase body of a ternary mixture with eAC * \1.35 eAB *f \0.45 e Fig. 5 T »x BC * \1.45 dab/aAA\[0.35 and three diÜerent values of the bonding parameter ew* . C section at /\0.5 of the three-phase body of a ternary mixture with eAC * \1.35 eBC * \1.45 Fig.d 6 T »x ab/aAA\[0.35 ew * \8.25 and three diÜerent values of the mixing parameter KAB * \eAA * ]eBB * [2eAB *f . C Fig. 7 T »x [0.35 ew * \8.25 and diÜerent values of (a) eBC * and (b) e C sections at /\0.5 of the three-phase body of a ternary mixture with eAB *f \0.45 dab/aAA\ AC * . Faraday Discuss. 1999 112 91»101 98 Fig. 8 Density pro–les of the A»B interface at diÜerent temperatures of the ternary mixture with eAC * \1.35 eAB *f \0.45 eBC * \1.45 dab/aAA\[0.35 ew * \8.25. (a) T *\0.9375 (b) T *\1.0 (c) T *\1.04. that the width *T * will shrink and that the mean temperature T1 * will decrease as the hydrophobicity of the amphiphile increases. It is important to point out that most of the model interaction parameters have an asymmetric in—uence on the location of the lower and upper critical endpoints.Consequently the temperature at which the –sh-tail point occurs on the T *[x cuts of the phase prism for /\0.5 varies from C system to system and can be either higher or lower than the mean temperature T1 *. C Interfacial behavior The properties of the diÜerent interfaces between coexisting phases in our ternary mixtures can be explored within the framework of density functional theory. In a simple local approximation the Helmholtz free energy of the system can be written as a functional of the local densities oi(r) :17 PP (13) ]o F[oi(r)]\Pdr( fh[oi(r)]]fbond[oi(r)])]1/2 ; dr dr@ /ij(r[r@) i j\A B C) j(r)oi(r@)[PPdr dr@ dab(r[r@) ij oA(r)XA[oi(r)]oB(r) ( where fh[oi(r)] and fbond[oi(r)] are given by eqns.(2) and (3) and are considered to be functions of the local densities oi(r). Assuming that the law of mass-action is valid at the local level eqns. (4) Faraday Discuss. 1999 112 91»101 99 cwm com and cwo in the ternary mixture Fig. 9 Temperature dependence of the three interfacial tensions described in Fig. 8. Xj[oi(r)] ( j\A C) at every and (5) may be used to evaluate the fraction of unbonded molecules point throughout the system. Consider a planar interface between coexisting phases. The corresponding density pro–les for the various species are given by the minimization of the grand potential (14) X[o P i(r)]\F[oi(r)][; ki dr oi(r) i at constant chemical potentials k under appropriate boundary conditions ; the associated Euler» i Lagrange equations can be solved iteratively by standard numerical procedures.Once the equilibrium pro–les are found the interfacial tension c may be obtained by taking the diÜerence (15) u c\(X[oi(r)][Xu)/S where X is the grand potential of the uniform bulk phases and S represents the interfacial area. Figs. 8(a)»8(c) show typical density pro–les of the A»B (water»oil) interface at a temperature T l* (8a) close to T1 * (8b) and close to but below the upper critical endpoint at above but close to T u* (8c). As can be seen the density of component C is always high at this interface with a greater proportion of bonded species close to the A-rich phase. The A»B interface in our model tends to be very wide with an interfacial thickness that ranges from thirty to –fty molecular diameters p at intermediate temperatures.We calculated the interfacial tensions at the diÜerent interfaces between coexisting phases from temperatures lower than T l* to temperatures higher than T u* for several mixtures. The three interfacial tension curves between the A-rich phase the B-rich phase and the middle phase were determined along the width *T * of the three-phase body. Fig. 9 illustrates our results for the mixtures studied in this paper. As one can expect the interfacial tension between the A-rich phase and the middle phase (cwm) rises from zero at T l* and increases with the temperature. On the other hand the interfacial (c tension between the B-rich phase and the middle phase decreases with T * and vanishes at om) T u* .We –nd that the sum of these two surface free energies is always greater than the interfacial tension for the A»B interface (cwo\cwm]com along the three-phase body) which indicates that in the three-phase coexistence regime the middle phase partially wets this interface between and up to points very close to the critical endpoints. The interfacial tension between the A-rich (water-like) and the B-rich (oil-like) phases exhibits a minimum whose location seems to be linked to the in—ection point of the projection of the middle phase trajectory onto the T *»/ plane. Hence it does not necessarily occur at the mean temperature T1 * for all our model mixtures (see Fig. 9). The interfacial tension at the minimum is –ve Faraday Discuss.1999 112 91»101 100 to ten times smaller than that corresponding to the bare A»B interface at the same temperature. In this sense component C in our model behaves as a weak amphiphile.15 IV Discussion Our simple oÜ-lattice density functional model for ternary amphiphilic systems and microemulsions has many of the features of real mixtures including a qualitatively reasonable phase diagram and a signi–cant lowering of interfacial tensions between oil-rich and water-rich phases in the presence of small concentrations of surfactant molecules. In future work we will explore the range of parameter space of our model more extensively and compare its predictions directly to trends seen in experiment. To make such comparisons useful it will be necessary to connect the parameters of the present theory more explicitly with the quantities explored in experiments.For example Strey1 has systematically varied the length of the nonpolar (hydrocarbon) and polar (ether) chains connected to make each amphiphile molecule and has investigated the eÜect this has on the phase diagram and on surface tensions. Our approach may help to explain both the success and the limitations of the corresponding states descriptions that he –nds for such mixtures.16 A diÜerent avenue of exploration would involve a density functional model in which amphiphiles are modeled as diblock copolymers containing —exible chains of connected spheres. Here bond constraints ensure the proper connectivity and eÜects of chain length can be introduced directly.Such an approach has been initiated by Cherepanova and Stekolnikov for monolayer surfactant –lms at the liquid»vapor interface.18 A similar —exible chain density functional theory was used by one of us in a recent theory of nucleation of alkane liquids from the vapor.19 Some of the most interesting aspects of amphiphilic mixtures involve self-organization in lamellae or more complex morphologies. Additional order parameters that identify periodic oscillations in the density are needed to characterize such phases and will also be the subject of future work. Acknowledgements This work was supported by the National Science Foundation through grant CHE 98-00074. Support to VT from the Facultad de Quimica and DGAPA at UNAM is also gratefully acknowledged.DWO thanks Reinhard Strey for helpful discussions. References 1 For a general review of experiments on microemulsions and their applications see R. Strey Colloid Polym. Sci. 1994 272 1005; and R. Strey Curr. Opin. Colloid Interface Sci. 1996 1 402. 2 B. Widom J. Chem. Phys. 1986 84 6943. 3 G. Gompper and M. Schick in Phase T ransitions and Critical Phenomena ed. C. Domb and J. L. Lebowitz Academic New York 1994 Vol. 16. 4 B. Smit K. Esselink P. A. J. Hilbers N. M. van Os L. A. M. Rupert and I. Szleifer L angmuir 1993 9 9. 5 M. M. Telo da Gama and K. E. Gubbins Mol. Phys. 1986 59 227. 6 C. Guerra A. M. Somoza and M. M. Telo da Gama J. Chem. Phys. 1998 109 1152. 7 V. Talanquer and D. W. Oxtoby J. Chem. Phys. 1997 106 3673. 8 N. F. Carnahan and K. E. Starling J. Chem. Phys. 1969 51 635. 9 G. Jackson W. G. Chapman and K. E. Gubbins Mol. Phys. 1988 65 1. 10 J. K. Johnson and K. E. Gubbins Mol. Phys. 1992 77 1033. 11 J. M. Walsh and K. E. Gubbins Mol. Phys. 1993 80 65. 12 M. Kahlweit E. Lessner and R. Strey J. Phys. Chem. 1983 87 5032. 13 M. Kahlweit R. Strey and P. Firman J. Phys. Chem. 1986 90 671. 14 J. C. Lang and R. D. Morgan J. Chem. Phys. 1980 73 5849. 15 M. Kahlweit R. Strey and G. Busse Phys. Rev. E 1993 47 4197. 16 T. Sottmann and R. Strey J. Phys. Condens. Matter 1996 8 A39. 17 R. Evans Adv. Phys. 1979 28 143. 18 T. A. Cherepanova and A. V. Stekolnikov Mol. Phys. 1994 82 125. 19 C. Seok and D. W. Oxtoby J. Chem. Phys. 1998 109 7982. Paper 8/09149H 101 Faraday Discuss. 1999 112 91»101

ISSN:1359-6640    

DOI:10.1039/a809149h

出版商:RSC    

年代:1999

数据来源: RSC

摘要:

Surface-directed spinodal decomposition versus wetting phenomena Computer simulations Kurt Binder Sanjay Puri§ and Harry L. Frischî Institut fué r Physik Johannes Gutenberg-Universitaé t Mainz Staudinger W eg 7 D-55099 Mainz Germany Received 10th November 1998 If a binary mixture con–ned by hard walls that prefer one component is quenched from the one-phase region to the two-phase region phase separation near the wall is characterized by concentration waves with wavevectors oriented perpendicular to the walls. The dynamics of this surface-directed spinodal decomposition is investigated by numerical simulations. We consider cases with short-range and long-range forces due to the walls with special focus upon the interplay of this structure formation with the growth of wetting layers.Attention is also paid to the change of the equilibrium phase diagram of the mixture as a result of con–nement in the thin –lm geometry and the consequences that one expects for the dynamical behavior of phase separation. While most of the numerical results concern systems at critical composition some results on oÜ-critical quenches are also presented. (1) 1 Introduction In the processing of materials it is rather common that one can bring a binary (AB) mixture from a homogeneous initial state in the one-phase region to a state inside the miscibility gap by a sudden change of control parameters (e.g. temperature T pressure p etc.).Thermodynamic equilibrium inside the miscibility gap corresponds to the coexistence of macroscopic regions of the two phases.If the state reached immediately after the quench lies within the spinodal curve Fig. 1 the homogeneous initial state is unstable against long wavelength concentration —uctuations that exceed a critical wavelength jc{ . For these initial stages of spinodal decomposition1h3 one –nds that the amplitudes of the fastest growing modes (with wavelength jm\J2 jc) grow exponentially with time t after the quench MPexp(jm t)N. However in the later stages of phase separation one observes a coarsening of the structure jm(t)Pta and the scattering intensity due to these concentration inhomogeneities grows according to a power law,2,3 S(q t)\[jm(t)]dS3 Mqjm(t)N t]O where S(q t) denotes the scattering intensity at wavevector q d is the dimensionality of the system and S3 (x) is a scaling function.The growth exponent is a\1/3 for solid mixtures where the § Present and permanent address School of Physical Sciences Jawaharlal Nehru University New Delhi 110 067 India. î Present and permanent address Department of Chemistry State University of New York at Albany 1400 Washington Avenue Albany New York 12222 USA. 103 Faraday Discuss. 1999 112 103»117 Fig. 1 Schematic description of a quenching experiment that leads to spinodal decomposition of a binary mixture The system is initially at temperature T and is in thermal equilibrium for time t\0 at the chosen 0 average concentration c6 of species A. At time t\0 the system is suddenly cooled to a temperature T below c the coexistence curve (consisting of two branches coex (1) ccoex (2) that merge at a critical point (Tc ccrit)).If the state (c6 then the linearized theory1h9 predicts that long wavelength T ) lies inside the spinodal curve c point sp(T ) concentration —uctuations (exceeding a critical wavelength jc) are unstable and grow spontaneously in time (with the maximum initial growth rate occuring for jm\J2 jc). This is schematically indicated in the –gure where the growth of a single concentration wave in the x-direction is shown. Lifshitz»Slyozov4 evaporation»condensation mechanism applies ; a\1/d for oÜ-critical —uids where the droplet diÜusion-coagulation mechanism5 (ììBrownian coalescenceœœ) dominates; while in —uid mixtures with concentrations near ccrit (Fig. 1) hydrodynamic mechanisms yield6 a\1 in d\3 and7 a\2/3 in d\2.In the present paper we investigate the question how this mechanism of phase separation is aÜected when the —uid mixture is con–ned between hard walls with a preferential attraction for one component. In a semi-in–nite system (or a very thick –lm where the two walls are essentially non-interacting) this preferential attraction may lead to the phenomenon of wetting8h10 (Fig. 2). Then domain growth in the bulk will compete with the growth of a wetting layer at the surface.11h13 However it must be noted that in the thin –lm geometry typically studied in experiments, 14 the –nite –lm thickness D limits the growth of wetting layers drastically and wetting transitions are rounded.15,16 Also the coexistence curve of the binary mixture in the thin –lm geometry gets changed (Fig.3) through a subtle combination of –nite size and surface eÜects.15h18 Owing to the presence of all these eÜects the interpretation of the many experiments (for a review see ref. 14) that have appeared since surface-directed spinodal decomposition was discovered19 is difficult and also from the numerous theoretical studies (reviewed in refs. 20 and 21) no full consensus on the growth laws in the presence of surfaces has emerged. In the present work we investigate this problem through the numerical solution of a discretized nonlinear diÜerential equation for phase separation near surfaces. We consider the cases of both critical and oÜ-critical quenches. Particular attention is focused upon a proper description of the boundary conditions at the surface dealing with both short-range and long-range surface forces.We also clarify the conditions for the surface to be wet or non-wet in equilibrium. However we do not include hydrodynamic interactions in our present treatment and the eÜect of statistical —uctuations is also ignored. (The latter neglect is appropriate for deep quenches but is not good for quenches near the critical point of the mixture.) In view of the simpli–cations made care should be Faraday Discuss. 1999 112 103»117 104 b s~1\s b (2) (ììincomplete wettingœœ) while from / /s Fig. 2 (a) Schematic phase diagram of a semi-in–nite mixture exhibiting a –rst-order wetting transition on the A-rich side of the coexistence curve for an interaction parameter s\sw .In this case there exists a prewetting transition line in the one-phase region on the A-rich side beginning at the –rst-order wetting transition at the coexistence curve and ending in a prewetting critical point. Three paths (1) (2) (3) are indicated by dashed lines in the one-phase region where one approaches the coexistence curve at constant s by reducing the bulk volume fraction / of the mixture. (b) Variation of the surface excess / plotted vs. / for the three paths (1) b ]/coex (2) w~1 and / (2) (3). For s the surface excess reaches a –nite limit /s s~1[s for of w~1 the surface excess diverges at phase coexistence (ììcomplete wettingœœ). For /\/pre a –nite jump s,pre (1) to /s,pre (2) occurs (ìì–rst-order prewetting transition œœ). This jump /s,pre (2) »/s,pre (1) smoothly vanishes as the prewetting critical point is reached.(c) Variation of /(2) with s~1 at phase coexistence [i.e. / is kept at while a critical divergence of /(2) is encountered for the second-order wetting transition. s /coex (2) (s)]. At the –rst-order wetting transition /(2) s jumps from a –nite value /s,w (2) discontinously to in–nity b s exercised while comparing results from the present model or related ones22h34 with experimental data.14,35 This paper is organized in the following manner. In Section 2 we brie—y describe our model for the dynamics of binary mixtures near surfaces. Section 3 describes the numerical results obtained from our model for surface-directed spinodal decomposition resulting from critical quenches. Section 4 describes the preliminary results for the case with oÜ-critical quenches.Section 5 contains a discussion and outlook to experiments. 2 Theoretical background and modelling First let us consider in greater detail the static properties of a semi-in–nite binary mixture which has the surface phase diagram shown in Fig. 2(a). For simplicity we focus on the case with Faraday Discuss. 1999 112 103»117 105 Fig. 3 (a) Qualitative phase diagram of a symmetrical binary (A B) mixture both in a thin –lm of the thickness D and in a semi-in–nite geometry (D]O). The variables are s~1 (the inverse of the eÜective interaction parameter proportional to the temperature) and / 6 (the average relative concentration of A in the system). Assuming a symmetric mixture in the bulk (D]O) the critical concentration /crit\1/2 then for s~1\s Assuming a preferential attraction of one species (B) by the walls this symmetry is broken and both the crit ~1 the concentrations /coex (1) /coex (2) of the two coexisting phases are related as /coex (2) \1[/coex (1) .critical concentration / 6 crit(D) and the values / 6 coex (2) (D) at the A-rich branch of the coexistence curve are shifted towards smaller concentration as compared to the bulk. With short-range forces at the wall and unchanged pairwise interactions between atoms of the mixture near the wall one expects a second-order wetting tran- T sition at a temperature w\Tcb in the semi-in–nite system but this transition is rounded in the thin –lm geometry. (b) Schematic description of a state of the thin –lm for a concentration / 6 inside the coexistence curve / 6 coex (1) (D)\/ 6 \/ 6 coex (2) (D) [e.g.the point marked by a cross in the phase diagram part (a)]. Then the –lm is inhomogeneous in the x,y-directions parallel to the walls the A-rich part of the –lm in equilibrium being separated from the B-rich part by a single A»B interface running perpendicular to the –lm. The relative amounts of A-rich (x) and B-rich (1[x) phases are simply given by the lever rule / 6 \xcoex (2) (D)](1 [x)/ 6 coex (1) (D) with 0\x\1. Here we assume that the –lm thickness D is much larger than the interfacial thickness w so that the A-rich phase has enrichment layers of the B-rich phase ììcoatingœœ the walls of the thin –lm. (c) Concentration pro–le /(Z) in the z-direction across the –lm in the A-rich phase.Here the shaded area denotes the surface excess /s . (d) Same as (c) but for the B-rich phase. coex (2) [/crit [Fig. 2(a)]. The coordishort-range surface forces at present. Our description will be in terms of an order parameter –eld /(R Z) which is measured in units of the order parameter / nates R and Z are rescaled coordinates parallel and perpendicular to the surface R\(x/2mb y/2mb) Z\z/2mb cf. Fig. 3; where m is the correlation length in the bulk. In the mean –eld b approximation for a lattice of coordination number q we have mb\[2q(1[T /Tc)]~1@2 if lengths Faraday Discuss. 1999 112 103»117 106 are measured in units of the lattice spacing. The corresponding free energy functional for the system has the following form *F = (2) dZ2 1 [12 (+/)2[/2]12 /4]B[ hl c /l[ 1 g c /l2H 2 0 kB Tc (3) 2+2/(R Z q)[V (Z)N Z[0 (4) \PdRGAP /l(R)4/(R Z\0) is the local order parameter at the surface and the ììbareœœ surface free Here energy density has been assumed to be a quadratic function of / only with expansion coefficients l/c and 2g/c respectively,10 i.e.in eqn. (1) we have assumed a short-range interaction between the l h 1 wall and the molecules in the mixture. For the bulk free energy density a simple Ginzburg» Landau form has been assumed. With the chosen normalization of distances and the order parameter all parameters of this bulk free energy density are absorbed in this normalization. For the alternative case of long-range forces due to the wall on the preferred component of the mixture a potential term /(R Z)V (Z) is introduced in the integrand on the right-hand side of eqn.V (Z)\[(h (1). In this paper we consider power law potentials of the form 1Z~n for Z[1 and V (Z)4V (1) for Z\1. The results presented subsequently are for the cases n\1 2 and 3. The surface phase diagram that results from the minimization of eqn. (2) exhibits second-order h wetting transitions for lc/c\^g/c if g/c\[2 while –rst-order wetting transitions occur10 for g/c[[2. We now turn to the extension of eqn. (1) to dynamical phenomena. Using a lattice model of a binary mixture in the molecular –eld approximation a kinetic equation for /(R Z q) as well as associated boundary conditions at the surface have been derived36 (q is a scaled time27).The kinetic equation in the bulk is essentially the Cahn»Hilliard equation1 of spinodal decomposition L/(R Z q) \[+2M/(R Z q)[[/(R Z q)]3]1 Lq apart from the last term representing the long-range surface –eld. The –rst boundary condition represents the eÜect of the surface perturbation L/(R 0 q) \h Lq l]g/(R 0 q)]c L/(R L , Z Z q) Z/0 (5) K The second boundary condition is a no-—ux condition enforcing conservation of the order parameter at the surface 0\ L LZ M/(R Z q)[[/(R Z q)]3]12 +2/(R Z q)[V (Z)N oZ/0 Note that eqns. (3)»(5) are deterministic no —uctuating forces are included and the static equilibrium built into eqns. (2)»(5) is entirely of a mean-–eld character.With these assumptions we have to restrict attention to the later stages of phase separation and critical phenomena are excluded from consideration. Despite all the simpli–cations made eqns. (3)»(5) cannot be solved analytically and a numerical solution is required. We have simulated Euler-discretized versions of eqn. (3)»(5) on two- Nx\400 Nx]Nz with dimensional lattices of size Nz\300. The surface was located at and Z\0 and is modelled by the boundary conditions in eqns. (4) and (5). Flat boundary conditions were applied at the other end in the z-direction and periodic boundary conditions were applied in the x-direction. In this fashion we attempt to model semi-in–nite sytems with one free surface at Z\0 while bulk behavior is approached for large Z.The discretization mesh sizes typically used were *q\0.03 and *x\1.0. The initial conditions for each run consisted of the –eld / being uniformly distributed with small-amplitude —uctuations about a background value (which is zero for a ìì critical quenchœœ i.e. a symmetric binary mixture at critical concentration but non-zero otherwise). This initial condition mimicks the homogeneous state of the system before the quench. All statistical quantities presented here were obtained as averages over a large number (typically 200) of independent runs. Faraday Discuss. 1999 112 103»117 107 Fig. 4 Evolution pictures from a two-dimensional Euler-discretized version of our model in eqns. (3)»(5) for spinodal decomposition in the presence of surfaces that exert a long-ranged attractive force on one component (A marked in black) of the mixture.Regions rich in the other component (B chosen to correspond to a h negative sign of the order parameter) are unmarked. Shown is the case 1\8 g\[4 c\4 and n\2. The times q are shown above each frame. 3 Long-range vs. short-range surface forces The case of critical quenches l In our simulations the –xed parameters were g\[4 and c\4 (c is related to the bulk correlation length h 27). We vary (the surface –eld strength) to obtain thermodynamic states which are wet or nonwet in equilibrium. For example in the short-range case hl\4 corresponds to a /\/ Fig. 5 Laterally averaged order parameter pro–les /av(Z q) vs. Z for the evolution depicted in Fig. 4. They were obtained by averaging /(X Z q) in the x-direction for a run and then averaging over 200 independent runs.From this picture one can verify that the system evolves towards complete wetting of the surface note the two-step decay of the order parameter pro–le starting from a surface value /(Z\0)\/ considerably b\1 over some region of Z at late larger than the bulk value /b(/b41 in our units) and stays then at 1 times. Finally a decay from /\/b\]1 to /\[/b\[1 occurs corresponding to the interfacial pro–le between coexisting A-rich and B-rich phases at the interface between the wetting layer and the depletion zone. Deep into the bulk we expect that /av(Z q)\0 due to equal probability to –nd A-rich and B-rich domains in a critical quench. Faraday Discuss. 1999 112 103»117 108 Fig.6 Laterally averaged order parameter pro–les /av(Z q) plotted vs. Z for several diÜerent scaled times q after the quench showing only the region close to the wall. The surface potential is short ranged and parameter values used are c\4 g\[4 h1\4 (a) or h1\8 (b) respectively. In the numerical implementation of the continuum model eqns. (3)»(5) a rather –ne mesh is used *q\0.001 *X\0.4 unlike Fig. 5. This was done with the intention of clarifying the nature of the order parameter pro–le at and near the surface. From Puri and Binder.27 nonwet surface while hl\8 corresponds to a wet surface. A detailed phase diagram is presented in ref. 10 for the short-range case. As a typical example Fig. 4 shows the temporal evolution of a disordered initial condition for the case of a long-range surface potential V (Z)\[h1/Zn with h1\8 and n\2 (see ref.33 for the temporal evolution in (i) the more common case with n\3 corresponding to a nonretarded van der Waals interaction between the surface and the preferred component and (ii) the case with a short-ranged surface potential (n\O)). It is seen that the surface rapidly develops an A-rich layer that is followed by a B-rich layer and the thickness of these layers grows with time. Because of the ongoing accretion of the component A on the surface layer the region in the vicinity of the surface is depleted in A and there the domain morphology is droplet-like rather than bicontinuous as in the bulk. This is particularly evident at late times e.g. q\9000. Thus in the late stages the A-rich layer at the surface is followed by a B-rich zone and then a zone of A-rich droplets in a B-rich background before one obtains the characteristic twophase structure of the bulk.These observations are qualitatively similar to the cases n\3 and 109 Faraday Discuss. 1999 112 103»117 q) vs. q and (b) V (Z)\ Fig. 7 Log»log plot of (a) L A(Z q) vs. q for a long-range surface potential L M(Z [h1/Z3 and the choice of parameters h1\8 g\[4 and c\4. Three choices of Z are shown in each case as indicated in the –gure. The solid line indicates the corresponding behavior of a bulk system. From Puri et al.33 n\1 studied earlier,33 as expected. However when one examines the laterally averaged order parameter pro–le /av(Z q) Fig. 5 characteristic diÜerences emerge while for n\3 one has at short times (qO30) still well-developed damped concentration oscillations in the pro–le the hallmark of surface directed spinodal decomposition although the oscillations here are much weaker and rather irregular.The relatively rapid growth of the wetting layer seen in Fig. 5 only occurs for long-range forces. With short-range forces however the growth is very slow (Fig. 6) and the thickness of the wetting layer only increases as a logarithm of time. This behavior is similar to the growth of wetting layers at the coexistence curve.11,12 Even for a nonwet situation [Fig. 6(a)] one observes a slow growth of the thickness of the A-rich layer at the wall but this growth ultimately saturates unlike the case of a wet surface [Fig.6(b)] where the growth continues forever. A two-step decay of /av(Z q) is seen in the wet case only [Fig. 6(b)]. From Fig. 6 one can see that in the region where changes from /B]1 to /B[1 the pro–le is reasonably approximated by the familiar tanhpro –le G /av(Z q) /av(Z q)B[tanh[Z[R1(q)] where the zero R1(q) slowly increases with time. In order to obtain a more quantitative description for the growth of characteristic length scales we de–ne Z-dependent correlation functions parallel and perpendicular to the surface as A(X Z q)\S/(X1 Z q)/(X1]X Z q)T[S/(X1 Z q)TS/(X1]X Z q)T (6) Faraday Discuss. 1999 112 103»117 110 R1(q) of laterally averaged order parameter pro–les /av (Z q) for surface- 1/Zn and n\1 2 3. The parameter values Fig. 8 Log»log plot of the –rst zero directed spinodal decomposition choosing a potential V (Z)\[h h are 1\12 g\4 c\4.The solid lines are the best linear –ts to the respective data sets and yield best-–t growth exponents x\0.31 (n\1) x\0.21 (n\2) and x\0.16 (n\3). The error bars on these exponent are ^0.01. (7) GM(Z1 Z q)\S/(X Z q)/(X Z]Z1 q)T[S/(X Z q)TS/(X Z]Z1 q)T Characteristic length scales are then de–ned as the distance over which these correlations have decayed to one-half of their initial values (8) GA(L A(Z q) Z q)\GA(0 Z q)/2 (9) GM(L M(Z q) Z q)\GM(0 Z q)/2 Fig. 7 presents as an example log»log plots of the length scales L A(Z q) and L M(Z q) vs. q for the van der Waals surface potential V (z)\[h1/Z3 with h1\8. It is seen that diÜerent regimes occur for ln(q)O7 both L A(Z q) and L M(Z q) increase proportional to a power of time qa with an exponent consistent with 1/3.Then a cross-over occurs (the precise location of this cross-over depends on Z the smaller Z the earlier the cross-over sets in) where L A(Z q) grows faster (aB1/2) while further growth of L M(Z q) slows down drastically. Presumably this behavior does not re—ect the true asymptotic behavior but rather the changes in shape of the A-rich droplets when the B-rich depletion zone reaches the considered –xed value of Z. More work is clearly necessary to understand this behavior precisely. It is important to note that the length scale L M(Z q) shown in Fig. 7(b) represents a scale of the coarsening structures and must be distinguished from the scale R1(q) introduced above the thickness of the growing wetting layer (Fig.8). The log»log plot of R1(q) vs. q in Fig. 8 indicates that the R behavior is compatible with 1(q) P qx with x\0.16 (n\3) x\0.21 (n\2) and x\0.3 (n\1) respectively. These exponents diÜer from the predictions made by Lipowsky and Huse12 for the growth of wetting layers in equilibrium (x\1/4 1/6 and 1/8 for n\1 2 and 3 respectively). Of course one could argue that the numerical results are aÜected by some transient cross-overs so that the exponent x is an eÜective exponent depending on our choice of param- 1\2 h 1\8 h eters such as h However even for 1. and we have found almost exactly the same exponent values33 as those in Fig. 8 (h1\12). Thus we believe that these exponents have a general validity.At a qualitative level we expect that the growth under non-equilibrium conditions is faster than that in equilibrium since the conservation of concentration slows down the growth of the wetting layer considerably in equilibrium (compared to a non-conserved situation11). However in the present situation the conservation law does not appreciably hinder the growth of the wetting layer since it interacts with the spinodally decomposing bulk where the local concentration relaxes much faster (L A(q)PL M(q)Pq1@3). 111 Faraday Discuss. 1999 112 103»117 /av(Z q)B1 0\0.3. 4 Short-ranged surface forces The case of oÜ-critical quenches We have also conducted simulations for the oÜ-critical situation where the background value of the order parameter –eld is some non-zero value say / Figs.9»11 present some typical results 0 . for (i) the morphology that develops (visualized in snapshot pictures) ; and (ii) the corresponding laterally averaged pro–les. These results are obtained for a short-range surface potential only Mi.e. V (Z) in eqns. (3) and (5) is set to zero and hi\8 in eqn. (4)N. Figs. 9 and 10 refer to cases where the phase that is enriched at the surface is the majority phase in the bulk whereas Fig. 11 shows a case where the phase that is enriched at the surface is the bulk minority phase. One can observe a remarkable asymmetry with respect to the growth of the surface wetting layer. In Figs. 9 and 10 the system develops an enriched layer which equilibrates very fast and stays rather thin 1\8 g\[4 c\4.The dimensionless times q are shown above each frame. (b) Laterally averaged Fig. 9 (a) Evolution picture as in Fig. 4 but for a oÜ-critical quench where A (black) constitutes the majority component with an average order parameter /0\0.3. The potential was short ranged and parameter values were h concentration pro–le /av(Z q) obtained as an average over 200 independent runs with parameters as chosen in case (a). The horizontal straight line denotes the background value of the order parameter viz. / Faraday Discuss. 1999 112 103»117 112 Fig. 10 Analogous to Fig. 9 but for the case with /0\0.5. av(Z q)B/0 the average /av(Z q) with Z characteristic of surface-directed spinodal 0\0.5 (Fig. 10) since it corresponds to a /0sp\1/J3B0.577 in a Ginzburg»Landau model.1h3 According to the linearized m 0 0sp jmP(/0sp[/0)~1@2 and av(Z q)[/0 in Figs.9(a) and 10(a) we see that they are systematiis reached already at ZB1 and then /av(Z q) decreases further and at late times reaches /av(Z q)B[1 i.e. a well-de–ned depletion layer (rich in the minority phase) follows. This is followed by another surface enrichment layer Mwith / q)B]1N whose thickness clearly grows in time as av(Z does the thickness of the depletion layer. After this structure we have / value of the order parameter chosen for the oÜ-critical quench. Only for ììearlyœœ times (q\30) do we still see the regular oscillations of decomposition [Fig. 9(b)]. It is particularly relevant to consider the case with / value of the average order parameter which is rather close to the ììspinodal curveœœ (Fig.1) which occurs at theory of spinodal decomposition,1 the characteristic wavelength j of the concentration variations that initially grow fastest should diverge when / approaches / this behavior also carries over to the surface-directed wave.32 Indeed when we compare the location of the –rst few zeros of / cally shifted further inward for /0\0.5 i.e. the wavelength is larger in the case with a concentration closer to the spinodal in the phase diagram. We must add the caveat however that the linearized theory is only valid as long as o/av(Z q)[/0o> o/0sp[/0 o and this condition is not satis–ed for any of the times shown here ; while in the bulk this condition is satis–ed for very 113 Faraday Discuss.1999 112 103»117 Fig. 11 Analogous to Fig. 9 but for the case /0\0.3. early times (qO1) it breaks down at even shorter times at the surface since the local order parameter at the surface rapidly equilibrates to satisfy the boundary condition eqn. (4) which is of h relaxational rather than diÜusive character with a relaxation time of the order of 1~1 (\0.125 in the case shown here). Thus in our case the linearized theories of surface-directed spinodal decomposition32,34 have almost no time-window of applicability. It is also interesting to discuss the corresponding sequence of snapshots [Fig. 10(a)]. Note that we display cells in white if the order parameter in a cell is negative and in black if it is positive since /0\0.5 here the bulk for q\30 is still uniformly black as the amplitudes have not reached negative values anywhere in the bulk.In the depletion region adjacent to the surface enriched layer one sees a one-dimensional nearly regular arrangement of ììdropletsœœ. However one should not interpret the term ììdropletœœ here literally in the sense of nucleation as the frames in Fig. 10(a) correspond to the non-linear analog of a spinodal wave with a wavevector oriented parallel to the surface. In fact eqns. (3)»(5) would not even admit the description of nucleation phenomena due to the lack of a random force term on the right-hand side of eqn. (2) statistical —uctuations which could drive a growing droplet over a nucleation free energy barrier in phase space are not included. Therefore in a more general context the isolated ììdropletsœœ that are seen in Fig.10(a) for q\90 in the bulk should not be mistaken for nucleation events rather the randomness inherent in Faraday Discuss. 1999 112 103»117 114 Fig. 12 Time-dependent of the ììzeroœœ R1(q) of laterally averaged order parameter pro–les /av(Z q) for the case of oÜ-critical quenches. The surface potential was short ranged and parameter values were h1\8 g\[4 and c\4. The ììzeroœœ is de–ned as the –rst zero of (/av(Z q)[/0) vs. Z. We present numerical results for the cases /0\[0.3 [0.5 [0.8 and 0.0 (for comparison) and these are denoted by the indicated symbols. We do not present results for /0[0 where the growth is extremely slow. The solid lines superposed on the 0\[0.3 1(q)\a]bqx. The data sets for / [0.5 [0.8 correspond to the best non-linear –ts of the form R corresponding best –t exponents are indicated in the –gure and have an error bar of ^0.01.The data for /0\0.0 does not admit a reasonable non-linear –t of the above form. the time evolution due to the random initial condition /(X Z 0) leads to the phenomenon that negative values of /(X Z q) for q\90 are only reached at a few and randomly located spots. In addition the fact that no such spots are visible for q\30 yet and only a few for q\90 should not be misinterpreted as a time lag of nucleation rather what we see is that a –nite time q is needed ’0 until the local concentration can relax from initial values very close to / to a negative value. 0 These simple considerations show that one must be careful in drawing any connection between the morphology of growth phenomena [such as visualized in the snapshots of Figs.9(a) 10(a) 11(a)] and the underlying kinetic mechanism by which the growth starts (spinodal decomposition vs. nucleation). In fact observations such as described in Fig. 10(a) in the experimental literature are often taken as evidence of random droplets formed by nucleation after the corresponding time lag. In the present example we know that such an interpretation must be incorrect while in experiment (where one cannot turn on or oÜ statistical thermal —uctuations at will unlike our numerical calculations) one does not really know. Also the irregular size distribution of the droplets seen in Fig. 10(a) at times q\900 and q\9000 and their (nearly) irregular arrangement is therefore no proof of nucleation and growth; rather it is the result of non-linear spinodal decomposition.Of course the above comments are of general relevance for bulk-phase separation and not speci–c to the surface-directed problem studied here. Similar remarks also apply to the growth of ììdropletsœœ of positive order parameter in the case where / is negative [Fig. 11(a)]. However there is one important distinction and that refers to 0 the dynamics of the wetting layer which is seen to grow steadily with time and the growth is much faster than for the case /0\0. h1,g are chosen such that wetting of the phase with /\1 occurs at Note that our parameters the surface of a system whose bulk order parameter is /\1 (this corresponds to a mirror image of the schematic phase diagrams shown in Figs.2 and 3 of course). Thus it is rather at /\/crit natural that a wetting layer can grow only for / negative and not for / positive however the 0 /0[0 we have a thick depletion layer close to the surface and an unexpected feature is that for 0 enriched layer further inward Figs. 9(b) and 10(b) a feature absent in Fig. 11(b). 1(q) of the wetting pro–le for oÜ-critical quenches is 1(q) is initially largest for /0\[0.5 which is a remnant of the R1(q) vs. q is approximated well by /0\[0.3 the value of R1(q) at early times is The temporal evolution of the –rst ììzeroœœ R shown in Fig. 12. We see that R spinodal singularity discussed earlier. The numerical data for the power law form R1(q)Pqx with xB0.24.For 115 Faraday Discuss. 1999 112 103»117 smaller but growth of the wetting layer is faster. The data set for /0\[0.3 is approximated well by a power law form with exponent xB0.30. It is tempting to associate this exponent with Lifshitz»Slyozov growth in the bulk4 which then would also control the surface. 0\[0.8) in which the homogeneous bulk is Fig. 12 also includes a strongly oÜ-critical case (/ metastable. In this case the wetting layer at the surface exhibits a rapid diÜusive growth viz. R1(q)Pqx with xB0.49. We can understand this diÜusive behavior in the context of a linearized theory.36,37 However we should stress that the deterministic case is of limited physical relevance as we suppress bulk nucleation which is experimentally relevant in the metastable region of the phase diagram.For purpose of comparison Fig. 12 also includes numerical data for the case of a critical quench (/0\0). As we have commented earlier the growth of the wetting layer is compatible with a logarithmic growth law in this case R1(q)P ln q and the data set does not admit a power-law –t except over restricted time scales. 5 Discussion Our numerical results illustrate that the interplay between spinodal decomposition and wetting phenomena near the surface of a phase separating binary mixture is rather intricate. Typically there is a non-trivial dependence on the range of the forces due to the wall (which we have studied only for /0\0 corresponding to a critical quench in the bulk) and on the composition /0 of the mixture.The surface breaks the rotational symmetry of the system so that directions perpendicular and parallel to the surface are not equivalent. The morphology is characterized by various divergent length scales viz. L A(Z q) and L M(Z q) describing the pair correlation function at a distance Z from the surface and the thickness of the growing surface enrichment or wetting layer R1(q). Figs. 7 8 and 12 show however that an interpretation of these length scales in terms of simple power laws is generally inadequate (and if this seems to work as in Fig. 8 the exponents are not yet understood!). Also one has to be very careful in interpreting the resulting morphologies correctly in particular droplet morphologies can arise (Fig. 10) which have nothing to do with nucleation.Thus despite many years of work preceding the present study and contrary to some claims that one can –nd in the literature,24 one is still far from a full understanding of the problem. The situation is even more cumbersome if we turn to a discussion of experiments which are almost always conducted in a thin –lm geometry with two inequivalent surfaces. The –nite thickness of the –lm is expected to lead to a signi–cant distortion of the phase diagram even for two completely equivalent surfaces (Fig. 3). So the case of inequivalent surfaces is far more complicated even at the level of a static description. Furthermore the nature and strength of forces due to the walls are not fully known in addition to long-range van der Waals forces there are shortrange contributions re—ecting the ììpackingœœ of molecules near walls and surface corrugation or long-range roughness also plays a ro� le.One should also keep in mind that real binary mixtures do not have the nice fully symmetric bulk phase diagram depicted in Figs. 1»3 but have asymmetries both with respect to their static and dynamic properties not described by eqn. (3). Finally most experiments are done with —uid rather than solid binary mixtures and hence hydrodynamic forces are present which are known to have signi–cant eÜects on spinodal decomposition in the bulk.2,3,6,7 Such eÜects obviously play a ro� le both near surfaces of a semi-in–nite geometry and in thin –lms and there are many experiments demonstrating this.14 In view of all these caveats we must be cautious regarding the applicability of our results to experimental situations.14,19,35 Nevertheless there are gratifying similarities between the experimental results those reported here and those in related publications.20 It is our hope that this ongoing work will stimulate further experimental and numerical investigations of this fascinating problem.Acknowledgements This work was supported in part by the Deutsche Forschungsgemeinschaft (DFG) Sonderforschungsbereich 262/D2. S.P. is grateful to the Condensed Matter Theory Group at Mainz for the kind hospitality. H.L.F. is partially supported by NSF Grant DMR 9628224. Faraday Discuss. 1999 112 103»117 116 References 1 J. W. Cahn Acta Metall. 1961 9 795. 2 J. D. Gunton M. San Miguel and P.S. Sahni in Phase T ransitions and Critical Phenomena ed. C. Domb and J. L. Lebowitz Academic Press London 1983 Vol. 8 p. 267. 3 K. Binder in Materials Science and T echnology V ol. 5 Phase T ransformations in Materials ed. P. Haasen VCH Weinheim 1991 p. 405. 4 I. M. Lifshitz and V. V. Slezov J. Phys. Chem. Solids 1961 19 35. 5 K. Binder and D. StauÜer Phys. Rev. L ett. 1974 33 1006. 6 E. Siggia Phys. Rev. A Gen. Phys. 1979 20 595. 7 H. Furukawa Phys. Rev. A Gen. Phys. 1985 31 1103. 8 D. E. Sullivan and M. M. Telo da Gama in Fluid Interfacial Phenomena ed. C. A. Croxton Wiley New York 1986 p. 45. 9 S. Dietrich in Phase T ransitions and Critical Phenomena ed. C. Domb and J. L. Lebowitz Academic Press London 1988 Vol. 12 p. 1. 10 I.Schmidt and K. Binder Z. Phys. B Condens. Matter 1987 67 369; S. Puri and K. Binder Z. Phys. B Condens. Matter 1992 86 263. 11 R. Lipowsky J. Phys. A Math. Gen. 1985 18 L585. 12 R. Lipowsky and D. A. Huse Phys. Rev. L ett. 1982 52 353. 13 K. K. Mon K. Binder and D. P. Landau Phys. Rev. B Condens. Matter 1987 35 3683; K. Binder in Kinetics of Ordering and Growth at Surfaces ed. M. G. Lagally Plenum Press New York 1990 p. 31. 14 G. Krausch Mater. Sci. Eng. R. 1995 14 1. 15 H. Nakanishi and M. E. Fisher J. Chem. Phys. 1983 78 3279. 16 K. Binder Adv. Polymer Sci. 1999 138 1. 17 T. Flebbe B. Dué nweg and K. Binder J. Phys. II 1996 6 667. 18 Y. Rouault J. Baschnagel and K. Binder J. Stat. Phys. 1995 80 1009. 19 R. A. L. Jones L. J. Norton E. J. Kramer F.S. Bates and P. Wiltzius Phys. Rev. L ett. 1991 66 1326. 20 S. Puri and H. L. Frisch J. Phys. Condens. Matter 1997 9 2109. 21 K. Binder J. Non-Equilib. T hermodyn. 1998 23 1. 22 S. Puri and K. Binder Phys. Rev. A Gen Phys. 1992 46 R4487. 23 G. Brown and A. Chakrabarti Phys. Rev. A Gen Phys. 1992 46 4829. 24 J. F. Marko Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. T op. 1993 48 2861. 25 C. Sagui A. M. Somoza C. Roland and R. C. Desai J. Phys. A Math. Gen. 1993 26 11163. 26 W. J. Ma P. Keblinski A. Maritan J. Koplik and J. R. Banavar Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. T op. 1993 48 R2362. 27 S. Puri and K. Binder Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. T op. 1994 49 5359. 28 S. Puri and K. Binder J. Stat. Phys. 1994 77 145. 29 P. Keblinski W. J. Ma A. Maritan J. Koplik and J. R. Banavar Phys. Rev. L ett. 1994 72 3 30 G. Krausch E. J. Kramer F. S. Bates J. F. Marko G. Brown and A. Chakrabarti Macromolecules 1994 27 6768. 31 G. Brown A. Chakrabarti and J. F. Marko Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. T op. 1994 50 1674. 32 H. L. Frisch P. Nielaba and K. Binder Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. T op. 1995 52 2848. 33 S. Puri K. Binder and H. L. Frisch Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. T op. 1997 56 6991. 34 H. P. Fischer P. Maass and W. Dieterich Phys. Rev. L ett. 1997 79 893. 35 L. Sung A. Karim J. F. Douglas and C. C. Han Phys. Rev. L ett. 1996 76 4368. 36 K. Binder and H. L. Frisch Z. Phys. B Condens. Matter 1991 84 403. 37 H. L. Frisch S. Puri and P. Nielaba J. Chem. Phys. in press. Paper 8/08787C 117 Faraday Discuss. 1999 112 103»117

ISSN:1359-6640    

DOI:10.1039/a808787c

出版商:RSC    

年代:1999

数据来源: RSC

摘要:

In—uence of wetting properties on hydrodynamic boundary conditions at a —uid/solid interface Jean-Louis Barrata and Lydeç ric Bocquetb a Deç partement de Physique des Mateç riaux Universiteç Claude Bernard and CNRS 69622 V illeurbanne Cedex France b L aboratoire de Physique ENS-L yon and CNRS 69364 L yon Cedex 07 France Received 14th December 1998 It is well known that at a macroscopic level the boundary condition for a viscous —uid at a solid wall is one of ììno-slipœœ. The liquid velocity –eld vanishes at a –xed solid boundary. In this paper we consider the special case of a liquid that partially wets the solid i.e. a drop of liquid in equilibrium with its vapor on the solid substrate has a –nite contact angle. Using extensive non-equilibrium molecular dynamics (NEMD) simulations we show that when the contact angle is large enough the boundary condition can drastically diÜer (at a microscopic level) from a ììno-slipœœ condition.Slipping lengths exceeding 30 molecular diameters are obtained for a contact angle of 140° characteristic of mercury on glass. On the basis of a Kubo expression for d we derive an expression for the slipping length in terms of equilibrium quantities of the system. The predicted behaviour is in very good agreement with the numerical results for the slipping length obtained in the NEMD simulations. The existence of large slipping length may have important implications for the transport properties in nanoporous media under such ììnonwettingœœ conditions. I Introduction Properties of con–ned liquids have been the object of constant interest during the last two decades thanks to the considerable development of surface force apparatus (SFA) techniques.While static properties are now rather well understood (see e.g. ref. 1) the dynamics of con–ned systems have been investigated more recently.2h4 These studies have motivated much numerical and theoretical work5h10 and some progress has been made in giving a simple coherent description of the collective dynamics of con–ned liquids. Both from experimental and theoretical studies has emerged a rather simple description of the dynamics of not too thin liquid –lms i.e. –lms thicker than typically 10»20 atomic sizes. The hydrodynamics of the –lm can be described by the macroscopic hydrodynamic equations with bulk transport coefficients supplemented by a ììnoslip œœ boundary condition applied in the vicinity (i.e.within one molecular layer) of the solid wall. Hence in spite of the fact that the wall induces a structuring of the —uid into layers that can extend 5»6 molecular diameters from the wall the hydrodynamic properties of the interface are quite simple. It turns out however that all experimental and numerical studies of con–ned —uids have been carried out with —uid/substrate combinations that correspond to a total wetting situation. By this we mean that cLS]cLV\cSV where c is a surface energy and the indexes L V and S refer to the liquid its vapor and the substrate respectively.11 In this paper we investigate the structure and 119 Faraday Discuss.1999 112 119»127 cos the hydrodynamic properties of a —uid –lm that is forced to penetrate a narrow pore in a situation of partial wetting i.e. when cLS]cLV[cSV . This corresponds to the case where a drop of the liquid resting on the same substrate at equilibrium with its vapor has a –nite contact angle h\(c c which can be deduced from Youngœs law SV[cLS).12 LV (1) vij(r)\4eCAp rB12 [cijAp rB6D S cFS/oF cFF . FS Table 1 Dependence of the surface tensions (in units of e) and contact angle h (in degrees) of the tuning parameter cFS h cos(h) cSV[cLS cFS 137 133 121 111 99 [0.74 [0.68 [0.52 [0.36 [0.17 0.5 0.6 0.7 0.8 0.9 [0.71 [0.65 [0.50 [0.35 [0.16 The liquid/vapour surface tension was determined independently to be cLV\0.94e.II Model and results We –rst describe our model for the —uid and the substrate and some details of the simulation procedure. All interactions are of the Lennard-Jones type with identical interaction energies (e) and molecular diameters p. The index i j run over the species (—uid F or solid S). The surface energies will be adjusted by tuning the coefficients cij . In all the simulations that are presented in this paper the solid substrate is described by atoms –xed on a FCC lattice with a reduced density oS p3\0.9. As the atoms are –xed the coefficient c is in fact cFF\1.2 meaning that the irrelevant. The interactions between —uid atoms are characterized by SS —uid under study is more cohesive than the usual Lennard-Jones —uid.The —uid»substrate interaction coefficient c will be varied between 0.5 and 1. All the simulations will be carried out at a constant reduced temperature kB T /e\1. FS Finally we mention that the con–guration under study will be that of an —uid slab con–ned between two parallel solid walls. Typically a con–guration contains 10 000 atoms with a distance between solid walls h\18p and lateral cell dimensions L x\L y\20p. Periodic boundary conditions are applied in the x and y directions i.e. parallel to the walls. For each wall three layers of FCC solid (in the 100 orientation) will be modelled using point atoms a continuous attraction between the —uid and the wall in the direction perpendicular to the walls being added in order to model the in—uence of the deeper solid layers.B T /e\1) by coupling the —uid All simulations were carried out at constant temperature (k atoms to a Hooverœs thermostat.13 In —ow experiments only the velocity component in the direction orthogonal to the —ow was thermostatted. Before we discuss in detail the structure of a –lm we can roughly estimate the in—uence of the interaction parameters on the wetting properties of the —uid. Following the standard Laplace estimate of surface energies,12 we have cij\[oi oj /r=0 rvij(r) dr. Using Youngœs law we obtain for the contact angle cos h\[1]2o From this expression a variation of c of between 0.5 and 0.9 would be expected to induce a from variation of h between 100° and 50°. A more accurate determination of the surface tensions was carried out using the method of Nijmeijer et al.14 The surface tensions are de–ned in terms of an integral over components of the pressure tensors which can be computed in a simulation.We refer to ref. 14 for more details. The results are listed in Table 1. By tuning the solid»—uid interaction cFS\1.0 c strength cFS\0.5 we found the contact angle deduced from Youngœs law to FS Faraday Discuss. 1999 112 119»127 120 Fig. 1 A typical con–guration of a liquid droplet (1000 atoms) on a solid substrate in a ììnonwettingœœ case (cFS\0.5) in equilibrium with its vapor. This con–guration was prepared by starting from a homogeneous —uid slab of thickness 18p con–ned between two walls. The droplet is formed by simultaneously increasing the upper wall by 20p in the z direction and removing the —uid atoms that lie near the box boundaries.The bounding box has a size of 20p. varies from hB90° to hB140°. In Fig. 1 a typical con–guration of a liquid droplet (in coexistence with its vapor) on the solid substrate is shown for cFS\0.5 corresponding to hB140°. In the following we shall describe such large contact angles (i.e. larger than 90°) as corresponding to a ììnonwettingœœ situation. In order to force the —uid into a narrow liquid pore under such partial wetting conditions an P0\2(cLS[cSV)/h. For the —uid with cFS\0.5 we –nd for e\0.05 eV then external pressure has to be applied. A simple thermodynamic argument shows that for a parallel slit of width h the minimal pressure is h\18p P0\0.079e/p3 P0\0.018e/p3 when cFS\0.9.while If we use p\5 P0BMPa for h\9 nm in the ììnonwettingœœ case h\140°. Fig. 2 shows the density pro–les of the nonwetting —uid inside the pore for pressures corresponding to 2.8P and 16.4P 0 number of particles. It is seen in this –gure that the highest pressure structure strongly resembles what would be obtained for the usual case of a wetting —uid with a strong layering at the wall. ” 0 . The pressure is changed at constant pore width by changing the (c Fig. 2 Density pro–les of the ììnonwettingœœ —uid FS\0.5) con–ned between two solid walls separated by 20p. The positions of the –rst layer of solid atoms have been indicated by vertical dashed lines. Full line P/P0\16.4 ; dashed line P/P0\2.8. The latter curve has been shifted upwards for clarity.Faraday Discuss. 1999 112 119»127 121 Fig. 3 Velocity pro–le (in reduced units) of the ììnonwettingœœ —uid (cFS\0.5) in a Couette geometry. The reduced pressure is P/P0B7.3. The solid line is the simulation result the dashed line is a linear –t of the numerical results and the dashed»dot line is the velocity pro–le predicted by the no-slip boundary condition. The velocity of the upper wall is U\0.5. The structure at the lower pressure is markedly diÜerent with both a layering parallel to the wall and a density depletion near the wall. We now turn to the study of the dynamical properties of the con–ned —uid layer. Two types of numerical experiments corresponding to Couette and Poiseuille planar —ows were carried out. In the Couette —ow experiments the upper wall is moved with a velocity U (typically U\0.5 in reduced Lennard-Jones units).In the Poiseuille —ow experiment an external force in the x direction is applied to the —uid particles. In Figs. 3 and 4 we compare the resulting velocity pro–les to those that would be expected for a ììno-slipœœ boundary condition applied at one molecular layer from the solid wall. Obviously the velocity pro–les for the nonwetting —uid imply a large amount of slip at the solid boundary. As usual this slippage eÜect can be quanti–ed by introducing a Fig. 4 Same as in Fig. 3 but in a Poiseuille geometry. The external force applied on each —uid particle is fext\0.02e/p. Faraday Discuss. 1999 112 119»127 122 K cFS\0.5 cFS\0.6 cFS . From top to bottom the data correspond to c Fig.5 Variation of the slipping length d (in units of p) as a function of the reduced pressure P/P0 for several values of the interaction parameters FS\0.7 cFS\0.9. Solid lines are the theoretical prediction eqn. (17) (see text for details). ìì partial slip œœ boundary condition for the tangential velocity v at the solid liquid boundary (2) L L v z t \[ d 1 vt K t FS\0.9 corresponding to a contact angle h\100° the usual behaviour (i.e. small d) z/zw z/zw This boundary condition depends on two parameters the wall location z and the slipping length w d. By studying simultaneously Couette and Poiseuille —ow for the same —uid –lm both parameters can be determined if they are used as –t parameters for the velocity pro–les obtained in the simulation.The results of such an adjustment are shown in Figs. 3 and 4. It turns out that as was the case in earlier studies,8 the hydrodynamic position z of the walls is located inside the —uid w typically one atomic distance from the outer layer of solid atoms. Much more interesting is the variation of the slipping length d which in earlier work was always found to be very small. In Fig. 5 the variation of d as a function of the pressure is shown for several values of the interaction parameters. The pressures are normalized by the capillary pressure P de–ned above as the 0 minimal pressure that must be applied to the —uid in order to enter the pore. For an interaction parameter c is obtained. For an interaction parameter cFS\0.5 which corresponds to a contact angle h\150°,15 slipping lengths larger than 15 molecular diameters can be obtained at the lowest pressures ; even at relatively high pressures (10P0) the slipping length remains appreciably larger than the molecular size p.III Theory FS . In order to understand the relation between the hydrodynamic boundary condition and the wetting properties of the —uid on the substrate one needs to estimate the dependence of the slipping length d on the microscopic parameters of the system such as ììroughnessœœ temperature c From the ìì kinetic œœ point of view this is obviously a hard task to complete since the slipping length accounts for the ìì parallel œœ transfer of momentum between the —uid and the substrate. Explicit calculations can only be done in some model systems.16 Now in the general case of a dense —uid no explicit formula for d in terms of microscopic quantities is available in the literature.In the following we derive an approximate expression for the slipping length which will allow us to discuss qualitatively the relationship between wetting and boundary conditions. 123 Faraday Discuss. 1999 112 119»127 Our starting point is a Green»Kubo expression for the slipping length :8 (3) P`= SFx(t)Fx(0)T dt j\ g d \ AkB T 1 0 where g is the shear viscosity of the —uid A\L x L y is the lateral solid surface and F is the x x-component of the total force due to the wall acting on the —uid at equilibrium (x is a component parallel to the wall). The quantity j\g/d can be interpreted as the friction coefficient of the —uid/wall interface relating the force along x due to the wall to the —uid velocity slip (Vslip) at the wall (4) SFxT\[AjVslip By introducing the density»density correlation function eqn.(3) can be rewritten (5) 1 P`= Pdr2 Fx(r1)Fx(r2)So(r1 ; t)o(r2; 0)T g d \ dt Pdr 1 AkB T 0 The force –eld Fx(r1) derives from the —uid-substrate potential energy V (xyz). The latter has been computed by Steele for a periodic substrate interacting with the —uid through Lennard-Jones interactions.17 In the case of a (100) face the main contribution can be written in terms of the shortest reciprocal lattice vectors according to (6) V (xyz)\V0(z)]V1(z)Mcos(qA x)]cos(qA y)N qA a \2p/as and s is the lattice spacing in the FCC solid.The altitude z is the distance to the V0(z) and V 1(z) are given by Steele in ref. 17. In 1(z) are multiplied by the tuning factor cFS . where –rst layer of the atoms of the solid. The functions our case however the attractive parts of V0(z) and V The force along x is the derivative with respect to x of the interaction potential V (xyz) (7) Fx(x y z)\qA V1(z)sin(qA x) When inserted into eqn. (5) one obtains 1 `= dx 1 dy1 dz1 2 dy2 dz2 qA 2 V1(z2) g d \ dt Pdx P P (8) AkB T 0 ]sin(qA x1)sin(qA x2)So(r1; t)o(r2; 0)T We now introduce the Fourier transform of the density in the plane parallel to the substrate (9) ok(z)(t)\Pdx o(x z ; t)eik ’ x with x\(x y) and the vector k is parallel to the substrate.This allows us to rewrite the previous eqn. (8) as dz1 dz2 V1(z1)V1(z2) g d \ qA 2 2AkB T (10) ]P`=dt R(SoqA (z1)(t)o~qA (z2)(0)T) A A (11) So P P 0 with q in the x direction and R stands for the real part. Note that in deriving eqn. (10) the homogeneity of the system in the direction parallel to the substrate was taken into account. Owing to the presence of the con–ning solid we now assume that the main contribution of C(qA z1 z2; t) to the time-integral comes from the dynamics in the plane (x y) parallel to the substrate. Moreover these dynamics are probed at the –rst reciprocal lattice vector qA . Since q is close to the position of the –rst peak in the structure factor it is reasonable to assume a diÜusive relaxation of SoqA (z1)(t)o~qA (z2)(0)T,18,19 yielding qA (z1)(t)o~qA (z2)(0)T\exp([qA 2 DqA t)SoqA (z1)o~qA (z2)T Faraday Discuss.1999 112 119»127 124 where So DqA qA (z1)o~qA (z2)T is the static correlation function. is a collective diÜusion coefficient and The time integration can now be performed to obtain 1 (12) dz dz1 2 V1(z1)V1(z2)SoqA (z1)o~qA (z2)T g d B 2DqA AkB T 1 solid) de–ned as (13) T eiqA(xi~xj) d(zi[z1)U ; i j (14) dz1 o(z1)V1(z1)2 (15) 1(z)2 P latter can be written (16) P P qA (z1)o~qA (z2)TBASo(z1)T d(z1[z2) S(qA o z1). so that the slipping length is expressed in terms of static properties of the inhomogeneous system only. A further simpli–cation can be done by assuming that due to the strati–cation near the substrate the main contribution to the integrals in eqn.(12) arises from the z1Bz2 terms so that So S(q is the z-dependent structure factor in the plane z\z (parallel to the The quantity A o z1) S(qA o z1)\ ASo(z 1 1)T 1. ASo(z The factor 1)T (A being the lateral surface) normalizes the average by the number of —uid particles in the layer at the altitude z If no locking of the —uid occurs near the substrate one may S(qA o z1) by its value at the –rst layer S(q approximate A o z1)BS1(qA). Eqn. (12) thus reduces to g d B S1(qA) 2DqA kB T the behaviour around the –rst layer located at Pdz o(z)V p d D qA . P In order to have a practical estimate to compare with this formula can be further approximated.The integral term in eqn. (14) may be approximated by assuming that it is dominated by zcDp so that 1(z)2Doc p `=dz V with o the density at the –rst layer denoted in the latter as the ììcontactœœ density. Moreover one c expects the ììlong rangeœœ attractive part of V1(z) to contribute mainly to the integral. Since the cFS V 1att(z) with V att 1 (z) independent of cFS one gets DqA * S1(qA)cFS 2 oc p3 where DqA * \DqA /D0 and D0\kBT /3pgp is the Stokes»Einstein estimate for the bulk selfdiÜusion coefficient. All the quantities involved in eqn. (16) can be computed in equilibrium molecular dynamics simulations. The density at contact o can be measured from density pro–les such as in Fig. 2. On c D S the other hand 1(qA) can be computed from the correlations of density —uctuations in and qA S the –rst layer.In practice we introduce the function 1(qA t)\N1~1SoqA(t)oqA(0)T where N1 is the average number of particles in the –rst layer and oqA (t)\;k/1 N1 exp iqA xk(t) is restricted to S1(qA t) at time t\0 yields S atoms in the –rst liquid layer. The value of 1(qA) while DqA is S obtained in terms of the inverse relaxation time of 1(qA,t) according to eqn. (11). Let us note at S this point that the assumption of an exponential decay of 1(qA,t) is indeed veri–ed in the simulations allowing us to clearly de–ne D In Fig. 6 the ratio d/d* with d*\pDqA * /S1(qA)cFS 2 is plotted as a function of the inverse density 1/o at contact c p3. In these variables the theoretical estimate eqn.(16) predicts a linear dependence of d/d* as a function of 1/oc p3. As shown in Fig. 6 a linear behaviour d/d*\a(1/oc p3 [1/oshift p3) is indeed observed in agreement with the prediction. A least-square –t of the data in this plot gives a\3.04 and 1/oshift p3\0.47. The presence of a shift in the density 1/oshift p3 can be interpreted to account for the higher-order correction in the density at contact which has been neglected in deriving eqn. (16) [in particular in the rough approximation assumed in eqn. (15)]. In the interesting limit where the contact density o is small and d is large this shift does not contrib- c ute any more. 125 Faraday Discuss. 1999 112 119»127 A Fig. 6 Normalized slipping length d/d* with d*\pD* / qA S1(qA)cFS 2 as a function of the inverse contact density 1/oc p3.In this plot a linear dependence is expected according to the theoretical prediction eqn. (16). The dashed line is a least-square –t of the numerical data with slope a\3.04 and shift in inverse density 1/oshift p3\0.47. In Fig. 5 the full theoretical result for d (17) 1[ oc oshift d p \a S1(qA)cFS 2 oc p3 D*qA o B is plotted as function of P/P against the measured (out-of-equilibrium) results for d. The good 0 agreement obtained in these variables for all diÜerent pressures and interaction strength cFS emphasize the robustness of the previous theoretical estimate. Obviously this expression breaks down for very large contact density c[oshift\2.1p3 where d is expected to vanish anyway. This result calls for several comments.First eqn. (17) shows that for given —uid»substrate interaction cFS d decreases with the density and structuring of the —uid in the –rst layer. The slipping length is thus expected to be quite small in a dense —uid at high pressures as usually observed and measured experimentally.8,20 More speci–cally eqn. (17) predicts a strong dependence of d on the value of the structure factor in the –rst layer taken at the shortest reciprocal lattice vector qA . This result is in qualitative agreement with previous simulation results.6 Now if the —uid»substrate interaction cFS is decreased at a given contact density of the —uid o (by simul- c taneously increasing the pressure) the previous result predicts a strong increase of the slipping length.This explains why substantial slip may be obtained even if a strong structuring does exist in the —uid a fact which is a priori counter-intuitive. FS . Now as emphasized for example by the Laplace estimate of the contact angle h\[1]2os cFS/oF cFF the contact angle may be interpreted as a ììmeasureœœ of the —uid» Finally let us come back to the problem of the in—uence of the wetting properties on the slipping length d. As noted above eqn. (17) predicts that d is a decreasing function of the interaction strength c cos substrate interaction strength cFS . In particular one expects the —uid to approach a nonwetting situation (cos h][1) when cFS decreases to zero. The previous equation eqn. (17) thus predicts a strong increase of the slipping length d when cos h][1.In other words in the idealized situation of a nonwetting —uid h\p a perfect slip may be expected for the boundary condition of the —uid near the surface. The correct trend is observed in our simulation results. This result is in agreement with several experimental observations,21,22 reporting very large slipping lengths for nonwetting —uids. IV Conclusions Obviously the existence of such a large slippage eÜect should manifest itself in the dynamical properties of a liquid con–ned in a nanoporous medium. If one considers a single cylindrical capillary a straightforward calculation in the Poiseuille geometry shows that the existence of slip Faraday Discuss. 1999 112 119»127 126 eÜective permeability K on the boundaries increases the —ow rate in the tube as compared to the ììusualœœ no slip case by a factor 1]8d/h (with h the pore diameter and d the slip length).Thus in a porous medium the eff which relates according to Darcyœs law the —ow rate to the pressure (18) hB drop,23 is expected to increase by the same Keff factor \K0A 1]8 d K is the ììstandardœœ permeability obtained within the no slip assumption (i.e. when d is 0 effBK0 . However in a non- Keff is expected to be much larger than K (say more than one where zero). In a wetting situation d is obtained to be very small and K wetting situation (hB140°) the slipping length d may largely exceed the nanometric pore sizes h so that the eÜective permeability 0 order of magnitude in view of the prefactors). It can also be expected that the microscopic dynamics of the molecules could be rather diÜerent in a ììnonwettingœœ medium compared to what it is in the bulk or in a medium with strong solid/liquid affinity.In fact recent studies point towards the importance of the surface treatment for the reorientation dynamics of small molecules in nanopores.24 Correlating the wetting properties with such microscopic studies seems to be a promising area for future research. This work was supported by the Pole Scienti–que de Modeç lisation Numeç rique at ENS-Lyon the CDCSP at the University of Lyon the DGA and the French Ministry of Education under contract 98/1776. We would like to thank E. Charlaix and P.-F. Gobin for introducing us to this subject and Dr. S. J. Plimpton for making publicly available a parallel MD code,25 a modi–ed version of which was used in the present simulations.References 21 and 22 were pointed out to us by Dr. Remmelt Pit. ñreports/ References 1 J. N. Israelachvili Intermolecular and Surface Forces Academic Press London 1985. 2 Dynamics in small con–ning systems ed. J. M. Drake J. Klafter and R. Kopelman Materials Research Society Pittsburgh PA 1996. 3 H. W. Hu G. A. Carson and S. Granick Phys. Rev. L ett. 1991 66 2758. 4 D. Y. C. Chan and R. G. Horn J. Chem. Phys. 1985 83 5311. 5 J. Koplik J. R. Benavar and J. F. Willemsen Phys. Rev. L ett. 1988 60 1282. 6 P. A. Thompson and M. O. Robbins Phys. Rev. A. Gen. Phys. 1990 41 6830. 7 I. Bitsanis S. A. Somers H. T. Davis and M. Tirrell J. Chem. Phys. 1990 93 3427.8 L. Bocquet and J.-L. Barrat Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. T op. 1994 49 3079; ibid J. Phys. Condens. Matter 1996 8 9297. 9 C. J. Mundy S. Balasubramanian K. Bagchi J. I. Siepmann and M. L. Klein Faraday Discuss. 1996 104 17. 10 K. Koplik and J. R. Banavar Phys. Rev. L ett. 1998 23 5125. 11 We mention that in some numerical simulations purely repulsive interactions between the —uid and the substrate were considered. In that case however the pressure of the —uid is very high and the properties of the con–ned —uid are very similar to those obtained for a wetting —uid (with attractive interactions to the substrate) at a lower pressure. 12 J. S. Rowlinson and B. Widom Molecular T heory of Capillarity Oxford University Press Oxford 1989. 13 M. Allen and D. Tildesley Computer simulation of liquids Oxford University Press Oxford 1987. 14 M. J. P. Nijmeijer C. Bruin A. F. Bakker and J. M. J. van Leeuwen Phys. Rev. A Gen. Phys 1990 42 6052. 15 The contact angle of mercury on glass is typically 140°. 16 L. Bocquet C. R. Acad. Sci. Ser. II 1993 316 7. 17 W. A. Steele Surf. Sci. 1973 36 317. 18 P. G. de Gennes Physica 1959 25 825. 19 J. P. Boon and S. Yip Molecular Hydrodynamics Dover Publications New York 1980. 20 J. M. Georges S. Millot J.-L. Loubet and A. Tonck J. Chem. Phys. 1993 98 7345. 21 N. V. Churaev V. D. Sobolev and A. N. Somov J. Colloid Interface Sci. 1984 147 574. 22 T. D. Blake Colloids Surf. 1990 47 135 and references cited therein. 23 J. Bear Dynamics of —uids in porous media Elsevier New York 1972. 24 M. Arndt R. Stannarius W. Gorbatschow and F. Kremer Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. T op. 1996 54 5377. 25 S. J. Plimpton J. Comput. Phys. 1995 117 1; code available at http ://www.cs.sandia.gov/tech sjplimp. Paper 8/09733J 127 Faraday Discuss. 1999 112 119»127

ISSN:1359-6640    

DOI:10.1039/a809733j

出版商:RSC    

年代:1999

数据来源: RSC

摘要:

Dynamics of a drop at a liquid/solid interface in simple shear �elds A mesoscopic simulation study Janette L. Jones Moti Lal J. Noel Ruddock and Neil A. Spenley Unilever Research Port Sunlight Bebington W irral UK L 63 3JW Received 16th February 1999 The dynamics of a surface-con–ned drop in a simple shear –eld has been studied pursuing the dissipative particle dynamics (DPD) simulation approach. The shear –eld induces contact angle hysteresis in the drop the degree of hysteresis increasing with the shear rate. At shear rates exceeding a critical value the drop acquires the tendency to lift oÜ the boundary leading to its removal. In the equilibrium contact angle range he[120° the drop preserves its integrity as it escapes from the boundary whereas at lower contact angles the drop assumes a distinctly elongated shape prior to its removal which develops ììnecksœœ at subsequent times.The drop breaks up as the necks are ruptured upon thinning with some fragments escaping into the bulk phase and some remaining at the surface. Under certain hydrodynamic conditions the moving drop sheds a trail of tiny droplets on the surface. The simulation results are in qualitative agreement with experimental studies on the corresponding systems published in the literature. 1 Introduction The motion of drops at liquid/solid interfaces actuated by external forces constitutes an important component of the mechanisms that drive key processes in detergency personal products application oil recovery and surface coating among many other technologies.The physics of this motion and its consequences for the shape and stability of the drop is not well understood due partly to the inadequacy of conventional theoretical and computational approaches for treating multiphase systems involving moving boundaries and partly to the lack of accurate experimental studies on relevant systems. The result is insufficient progress in the development of new fundamental insights into the role played by the gravitational centrifugal hydrodynamic and impact forces in processes like the removal of oily soil from fabrics in washing spreading of creams and lotions on skin and break-up of emulsion drops in the regions close to mixer blades. The past few years have witnessed the emergence of computer simulation approaches involving time and length scales that are neither atomistic nor phenomenological but lie in a regime residing between the molecular and colloidal domains the mesoscopic regime.1 With considerable success achieved in the study of the kinetics of domain growth in aggregation processes,2 spinodal decomposition, 3,4 —ow through complex microporous media5 and rheology of polymers6 and colloids,7 such approaches are showing signi–cant promise in the modelling of the dynamics of complex systems under hydrodynamic conditions well beyond the Stokes regime.It is thus opportune to consider the application of mesoscopic simulation in the study of the problem of our present concern. Mesoscopic simulation methods are mainly of two types lattice based and particle dynamics based.The former represent essentially the revival of the lattice gas (LG) models introduced in the 129 Faraday Discuss. 1999 112 129»142 early years of computer simulation.8 In such models the structural elements of the systems are assumed to be located at the nodes of a lattice the elements moving from node to node. A move involves two stages collision and propagation governed by a set of rules consistent with the requirement of mass and momentum conservation at each step. The LG models have been shown to obey the classical Navier»Stokes hydrodynamics equation.9 A more powerful simulation approach based on the idea of solving the Boltzmann transport equation on a lattice (the lattice» Boltzmann method) was proposed a decade ago by McNamara and Zanetti10 and also by Higuera and Himenez.11 Further advances have been made in the lattice»Boltzmann (LB) approach leading to important applications with remarkable success.12 In the particle dynamics based methods the models assumed are free from lattice structures the structural elements moving in free space under a variety of forces acting on them.The dynamics of the system is computed at a large number of successive time steps by solving numerically Newtonœs equation of motion for each structural element. In our study we have pursued the dissipative particle dynamics (DPD) method»a particle dynamics based approach»for modelling the dynamic behaviour of a drop in a simple shear –eld located at a solid/—uid boundary. Our objective in these simulations is to establish the mode of deformation motion and possible break-up of the drop at —ow rates corresponding to Reynolds numbers of the order of 1»10 the hydrodynamic regime conveniently accessible by the DPD method.In this paper we give a brief account of the simulation methodology describe the present model and present our results delineating the new eÜects revealed by the simulations. 2 A brief description of the dissipative particle dynamics (DPD) model The structural element in mesoscopic models is a momentum carrier a conceptual entity whose motion is assumed to represent the collective dynamic behaviour of a large number of molecules in the system. Thus a mesoscopic model is a highly coarse-grained description of matter with each structural element standing for an unde–ned but a very large number of molecules.Originally advanced by Koelman and Hoogerbrugge,13 DPD is based on the idea of thermostating a simulation ììboxœœ by invoking dissipative and —uctuation forces counter-balancing each other to ensure constant thermal energy in the system. In addition conservative i.e. static forces can be included. Thus the total force acting on a momentum carrier i at any instant is the iD) sum of three forces dissipative iR) ( and conservative FiC) 2.1 Dissipative forces The dissipative i.e. hydrodynamic force between a pair of particles is assumed to be directly related to their relative velocity in analogy with colloidal systems.14 Thus the dissipative force Fij D exerted on i by a momentum carrier j may be expressed as (F Fi\FiD]FiR]FiC D\[nij … (øi[øj) where n is the dissipation (or resistance) tensor a function of r ij ij the vector representing the separation between ø i and j.ø and are respectively the velocities of i and j. The negative sign in j i front of n signi–es that the dissipation force acts in the direction opposite to that of the relative ij velocity. To render the model computationally efficient the functional form for n assumed in ij DPD simulations is extremely simple where c is a scalar coefficient characteristic of the system. uij D is the weight function which speci–es the dependence of n on rij the distance between i and j. ij Substitution of eqn. (3) into eqn. (2) gives Fij ij\cuij D eij eij n 2.2 Fluctuation forces 130 The —uctuation forces are assumed to be Brownian-like i.e.they are random conforming to a Faraday Discuss. 1999 112 129»142 Fij —uctuation (F ij e D\[cuij D[eij … (øi[øj)]eij is a unit vector coincident with (1) (2) (3) ij . r (4) Gaussian distribution with zero mean and the second moment proportional to the kinetic energy kT . Fij R the —uctuation force on i due to j may therefore be written as (5) FijR\pfijuij R eij where f is a random number drawn from a Gaussian distribution whose –rst moment is equal to ij zero and variance equal to unity. u is the weight function corresponding to the —uctuation forces. ij The coefficient p de–nes the variance of the —uctuation force which must scale as 1/Jdt where dt is a small time interval during which the force would remain constant.2.3 Conservative forces Since the time average of the dissipative and —uctuation forces is zero th do not feature in the equilibrium behaviour of the system which is governed solely by the conservative forces. A simple but realistic functional form of Fij C universally assumed in mesoscopic models is (6) FijC\auij C … eij where uij D uij R and uij C . The relationship uij C is the weight function and a is a parameter. In respect of both dynamic and conservative forces a DPD model is completely characterised in terms of the parameters c p and a and the weight functions between a and c is furnished by the —uctuation-dissipation theorem:15 p2/2c\kT with (uij R)2\uij D Taking kT as the unit of energy eqn.(7) gives (7) (8) (9) p2\2c In the interest of preserving simplicity in the model u values are assumed to have the follow- ij ing functional form (10) \0 uijR\(uij D)1@2\uijC\uij\[1[rij/rc] for rij\rc for rij[rc rc serving as the unit of length is the ììcut-oÜœœ distance above which all the three forces vanish. Eqn. (10) represents a repulsive spring-like truncated conservative force which means that the momentum carriers are assumed to be soft spheres. A further assumption is that the forces are pairwise additive permitting the force acting on i in a system of N momentum carriers to be expressed as i\; (FijD]FijR]Fij C) j (11) \; [[cuij 2Meij … (øi[øj)N]pfij(uij)]auij]eij j The summation (12) j F ; runs over all momentum carriers except i.Eqn. (11) when combined with j Newtonœs second law of motion gives *øi\*t/mC; [[cuij 2Meij … (øi[øj)N]pfij(uij)]auij]eijD where *ø is the change in the velocity of the momentum carrier i during a small time interval *t i and m is the particle mass which is taken as the unit of mass. Eqn. (12) constitutes the equation of motion for a DPD system. The key task in the simulation is to solve this equation for all momentum carriers successively over a large number of time steps to compute the dynamic as well as the equilibrium properties such as correlation functions and thermodynamic quantities. 131 Faraday Discuss. 1999 112 129»142 3 The present systems The present systems are each comprised of three phases two immiscible —uids (1 2) and a solid (3).In principle eighteen parameters would be needed to characterise all the like and unlike interactions in such a system; they are c11 c22 c33 c12 c13 c23 p11 p22 p33 p12 p13 p23 a11 a22 a33 a12 a13 and a23 . This number is however reduced to twelve by virtue of the —uctuation-dissipation theorem [eqn. (9)]. The equation of state for the DPD model enables the value of a for the like interactions to be readily calculated from the compressibility of the —uid. For a pair of partially miscible liquids a12 can be calculated from the solubilities of liquid 1 in 2 and vice versa. 3.1 The liquid phases The two immiscible liquids in our systems are water (1) and oil (2). Ideally we would like to model incompressible liquids but in DPD —uids are necessarily compressible to some degree.The greater the density of particles and the like interaction parameter the more incompressible is the liquid. However because the interparticle forces are then greater the time-step must be smaller and the algorithm is less efficient. The like interaction must therefore be chosen as a compromise between these two requirements. The miscibility of the two —uids is controlled mainly by the unlike interaction parameter a12 a12B(a11]a22)/2 and almost entirely immiscible a12 exceeds the mean value appreciably. We defer the discussion on the determination of the c the two phases would be completely miscible if if and p parameters to the section on simulation details. 3.2 The solid phase In the model the solid phase is ììcreatedœœ by freezing the motion of the momentum carriers so they are locked in a certain con–guration which may be crystalline or random.The absence of dynamics in the solid phase means that in the simulation runs the equation of motion is solved only for the momentum carriers in the —uid phases taking account of their static and dynamic interactions with the solid. The parameters corresponding to the —uid/solid conservative intera a actions are a (water/solid) and (oil/solid) which together with determine to a large 13 23 12 extent the structure and thermodynamics of the three interfaces featuring in the system. 4 Simulation details Occupying the bottom section of the simulation ììboxœœ which was subjected to the usual periodic boundary conditions the particles constituting the solid phase were arranged either in a random con–guration (in our initial studies mainly) or in a face-centred cubic structure maintained in a stationary rigid state at all times.The oil phase was introduced as a drop of volume equal to 113 (r (equivalent spherical radius of 3 units) located at the solid boundary. Water occupied the units c3) remaining space in the box. All three phases were assumed to be of the same density equal to 6. Depending on the properties to be calculated three box sizes were used 40]20]20 40]10]10 and 10]10]10 containing respectively 96 000 24 000 and 6 000 particles. All simulations started from random con–gurations of the momentum carriers in the —uid phases. The conservative forces were chosen as follows where phases 1,2,3 are water oil and solid respectively.We took a11\a22\12.5 (this means that water and oil are assumed to be of identical isothermal compressibility and density). The water/oil interaction a12 was taken to be 80 (given a11 and a22 this –xes the water/oil interfacial tension at 13.5). The water/solid and oil/solid parameters a13 and a23 were varied between 20 and 40 to control the corresponding surface tensions and thus the contact angle. The equation of motion was integrated numerically for each particle in the box using the ììleapfrogœœ approach often pursued in MD simulations. The basic algorithm was developed by Groot and Warren16 who modi–ed the original version to ensure that the acceleration will feature in the position update of the momentum carriers at each step.Presented succinctly by the set of following equations the algorithm is accurate to second order in the time step permitting much longer steps to be assumed than in the original Verlet algorithm. Faraday Discuss. 1999 112 129»142 132 Fig. 1 Computed velocity pro–les of the —uid in contact with a solid boundary. The spikes represent the particle layers in the solid substrate. The dashed line is the velocity showing slip (the velocity does not vanish at the surface). The heavy line is the shear velocity with the modi–ed algorithm.(The units along both axes are arbitrary.) Fi(t)\f [ri(t) øi(t[12 dt)] øi(t)\øi(t[12 dt)]12 dtFi(t)]O(dt)2 ri(t)\ri(t[dt)]dtøi(t)]12(dt)2Fi(t)]O(dt)3 (13) øi(t]12 dt)\øi(t)]12 dtFi(t)]O(dt)2 where dt is a small time step during which the force on each momentum carrier would remain virtually constant.Starting from the initial con–guration of the momentum carriers the force on each carrier was calculated. The forces velocities and positions of the carriers at the next step were computed using the above scheme which involves the computation of the velocities at half steps (leap-frog). The computation was carried out for a large number of steps sufficient to cover the duration of the process of interest and to ensure convergence of the time averages of the properties of the system. Positions velocities and forces of each momentum carrier were recorded at convenient time intervals. From this basic information on the time evolution of the dynamics and con–guration of the system such properties as viscosity pressure and interfacial tensions were readily obtained.Groot has carried out an extensive investigation of the eÜect of the magnitude of dt on the temperature stability of the system.16 He concluded that the time step of one DPD unit used by Koelman and Hoogerbrugge is much too large to preserve the requisite temperature. The instability problem can only be overcome by reducing the time step to 0.05 DPD units or to even smaller values if practicable. In the same computations Groot established through trial and error that the optimum value of p meeting the requirements of fast temperature equilibration rapid convergence and stability is around 3.35 for dt in the range from 0.04 to 0.05.In the present simulations we have followed Grootœs recommendation in selecting the values for p. Thus p11\ p22\p12\p13\p23\3.35. Hence c11\c22\c12\c13\c23\5.6. The viscosity is determined by the like repulsion a the dissipation parameter c and the particle density ; for the values chosen it was equal to 1.74 (for both oil and water). 4.1 Introduction of a shear –eld A simple shear —ow can be conveniently imposed by subjecting the simulation box to the well known Lees»Edwards boundary condition,17 whereby the periodic images of the box in the zdirection are moved with velocity equal to c c 5 L /2 where 5 is the shear rate and L the box height. The direction of imposed velocity is positive for the upper image box and negative for the lower image box.The velocities thus imposed propagate into the simulation box producing a linear velocity pro–le in due course with the plane of zero velocity located in the middle of the box. This boundary condition can be readily adapted to our systems where the surface of the solid boundary 133 Faraday Discuss. 1999 112 129»142 de–nes the plane of zero velocity. The velocity imparted to the upper image box will therefore be c 5 (L [*L ) where *L is the thickness of the solid phase since the system exists in the equal to zP0 semi-in–nite space it includes no lower image boxes. 4.2 Stick boundary condition at the solid/—uid interface Since the solid phase is maintained in the stationary state the momentum carriers of the —uid in immediate contact with the solid boundary must realistically assume zero velocity ensuring no slip at the —uid/solid interface.However examination of the velocity pro–les obtained in a series of preliminary simulations did show the occurrence of slip ; a typical pro–le is given in Fig. 1. The origin of the slip resided in the repulsive nature of the conservative force depleting the region close to the boundary with the maximum particle density occurring at zBrc . Thus most —uid particles remained beyond the range of the —uid/solid dissipative as well as other forces. To eliminate slip the particles in the region z\r were allowed to move freely only in the normal direction. Their c velocities in the transverse directions were randomised periodically in such a way that the Maxwell»Boltzmann distribution was preserved with a mean velocity equal to that of the interface.With this modi–cation the velocity pro–le (Fig. 1) indicated no slip. This procedure also has the eÜect of changing the diÜusion coefficient of particles near the surface (the surface diÜusion coefficient will be somewhat smaller than that in the bulk). (14) C12\P[pzz[12(pxx]pyy)] dz (15) 5 Results and discussion We pursued two separate sets of simulations. In the –rst set the solid phase was formed by freezing a random con–guration of the particles in a slab of 2 unit thickness and in the second set the solid phase assumed the face-centred cubic structure with the boundary in the 111 plane the density of both variants of the solid phase being 6. Signi–cant geometrical irregularities in the boundary of the randomly structured solid phase meant that separate simulations had to be carried out over a large number of the frozen con–gurations to ensure satisfactory convergence in the values of the —uid/solid interface properties.This proved computationally too expensive to be pursued beyond the preliminary stage. Most of the subsequent simulations were therefore performed on systems where the solid phase assumed the crystalline structure. The parameter space investigated here corresponds to a11\a22\12.5 (water and oil are assumed to be of identical isothermal compressibility and density) with a particle density of 6. In this region of the parameter space the changes in the structure and properties of the —uid/—uid and —uid/solid interfaces are governed entirely by the unlike parameters a12 a13 and a23 .Properties of our immediate interest are those that essentially determine the con–guration of the oil drop at the boundary. These include the various interfacial tensions and the viscosity of the —uid phases. 5.1 Calculation of the interfacial tensions In simulations the —uid(1)/—uid(2) interfacial tension C12 is conveniently computed using the Irving»Kirkwood equation18 where pxx pyy and pzz are the three diagonal components of the pressure tensor. The interface is parallel to the x»y plane. In terms of the interparticle forces eqn. (14) is expressible as C12\1/AC; Fijz rijz[12(Fijx rijx]Fijy rijy)D i[j i,j (Fijx Fijy Fijz) ( and rijx rijy rijz) are respectively the x y and z components of the force and the separation vectors between a pair of particles i,j the summation carried over all particle pairs.A is the interface area. Faraday Discuss. 1999 112 129»142 134 The liquid/solid interface is somewhat more subtle. Assuming as in the present work that the interface is modelled by using particles to represent the liquid and these are acted on by an external potential which represents the eÜect of the solid the interfacial tension is given by the expression (16) i CLS\1/AC; Fijz rijz[12(Fijx rijx]Fijyrijy)D]1/A ; Fiz ex riz i,j The second sum is taken over all particles. Fiz ex is the force on i due to the external potential. This result was originally derived for a potential constant in the x and y directions19 but was later shown also to be valid for a potential which is periodic in x and y.20 For systems involving two interfaces we need to include a third summation in eqn.(16) to account for the second external potential i CLS\1/AC; Fijz rijz[12(Fijx rijx]Fijy rijy)D]1/A ; Fex1 iz riz1 i,j (17) ]1/A ; Fex2riz2 iz i where Fex1 and Fex2 are respectively the forces acting on i due to the external potentials corre- iz iz r riz2 sponding to boundary 1 and boundary 2. are the distances of i from the two bound- and iz1 aries. A is now the combined area of both surfaces. For a hard wall the position of the interface is de–ned unambiguously. For any other potential an arbitrary choice must be made. This corresponds to the usual situation with potential energies which are always de–ned relative to some –xed point.Since only diÜerences of free energies are physically meaningful there is no fundamental ambiguity. In the present model where the solid boundary is constituted of immobilised particles the forces on the —uid particles deriving from the external potentials are substituted by those a i arising from their C interaction F F ; 1/ ijy ] [1 r ijz LS\1/AC; with the ijz boundary 2(Fijx rijx particles. Fijy r Eqn. )D] (17) A thus ; modi–es iaz riaz to i,j (18) ]1/A ; ; Fibz ribz b i Here the variable a runs over all immobile particles in interface A and b over interface B. In the simulations on the —uid/—uid and the —uid/solid systems forming planar interfaces ensemble averages of the various summations appearing in eqns.(15) and (18) were computed to obtain the respective interfacial tensions. The results are presented in Fig. 2 where C and C are plotted 12 13 (\a23). a 13 a against a12 a13\10 the interfacial tensions are negligibly small rapidly and For 12 rising to plateau values as the values of these parameters approaches 100. As one would expect the —uid/solid interfacial tension is always greater than the —uid/—uid interfacial tension for the a12 a13. = C12; Ö C13 . (The units along both axes are Fig. 2 Plots of computed C12 C13 against arbitrary.) 135 Faraday Discuss. 1999 112 129»142 and same values of a13 at a12\a13\80 C13 exceeds C12 a12 the calculations on drops were done with by a factor of more than two.All a12\80 corresponding to an interfacial tension of 13.5. 5.2 Shear viscosity g Since the values of the parameters for like»like interactions are assumed to be the same for the two —uids their shear viscosities will be identical. The shear stress Pxz resulting from the applicai, j tion of a shear —ow P was r (v iz ) [ xz\ computed (1/V )T; taking ijxFijz into U] account (m/V )Tboth ; ix virial vix s and … (v kinetic [viz s ) contributions U B B i (19) kinetic virial varied linearly with the shear rate establishing that the model —uids are Newtonian. vs. shear rate plot was found to be 1.74 for the V is the volume of the system and m is the particle mass (equal to 1 unit in the model). v and v ix iz are the x and z components of the particle velocity ; vix s and viz s are the contributions of the shear –eld to the x and z components of the particle velocity.The angular brackets denote the ensemble averages. Pxz The value of g equal to the slope of the Pxz present —uids. 5.3 Drop pro–les and contact angles in the absence of —ow [(R2[z0 2)]2zz0[z2]. R z is the distance of the centre of the sphere from the surface. o is 90]sin~1(z0/R) degrees. Under equilibrium conditions a drop exists as a spherical cap on a planar solid surface in the absence of gravity. We can determine the contact angle by –tting a spherical segment to the simulated shape. The easiest way to achieve this is to calculate in a simulation run the average number of particles in a cross-section of the drop as a function of height z above the surface.For a sphere the number of particles per unit height should be equal to po is the radius of the sphere and 0 the particle density. The contact angle is simply equal to Fig. 3 shows simulation data and the –t. Note the oscillations caused by the layering of particles near the surface. It is necessary to make an arbitrary choice for the position of the interface ; we have taken this to be the –rst layer of particles in the substrate. Another obvious choice would be the position of the –rst density maximum in the liquid but in practice it seems to make little diÜerence. Table 1 presents the values of contact angles obtained from simulations and predicted by Youngœs equation from interfacial tensions for several diÜerent sets of parameter values.In general the agreement is quite good except for the very lowest contact angle (30°) for which the boundary layer at the substrate probably causes a large distortion. Fig. 3 Number of particles as a function of height for the drop on the surface (111 plane) of a face-centred cubic solid. The dotted line is the simulation data the solid line is the –t. The spikes represent the positions of the particles in the solid substrate. Parameter values a11\a22\12.5 ; a13\a23\40. (The units along both axes are arbitrary.) Faraday Discuss. 1999 112 129»142 136 Table 1 The drop contact angles measured from the simulation and calculated from Youngœs equation Contact angle (Young)/degrees Contact angle (simulation)/degrees a a23 b 13 a 44 (39c) 31 20 20 40 60 90 120 68 91 114 40 80 149 145 80 40 40 20 20 a Repulsion strength for water/solid interaction.b Repulsion strength for oil/solid interaction. c Computations performed for a drop of an equivalent radius of 9 units. 5.4 Flow-induced deformation and lateral motion of the drop A drop was placed on a surface and exposed to a shear —ow. In each case the drop was allowed to reach its equilibrium contact angle before shear was applied. We varied the contact angle and shear rate with the other quantities –xed as above. The capillary and Reynolds numbers are usually de–ned as Ca\gac 5 /C and Re\oc5 a2/g where a is the equivalent spherical radius of the drop.For the quantities used here Ca\0.39c 5 and Re\31c 5 . We found that at low shear rates the drop deforms and travels slowly across the surface. Fig. 4 shows the shape of a longitudinal cross-section of the drop (de–ned as a contour of constant density) indicating how the drop progressively deforms as the shear rate is increased. It may be noted that the advancing and receding dynamic contact angles are no longer equal i.e. the system displays shear-induced contact angle hysteresis in agreement with experimental observations. The distortion of the shape caused by the density oscillations at the surface is also visible. It would be interesting to measure the contact angles as a function of shear rate but the distortion makes this rather difficult. We have attempted to extrapolate the shape of the drop through the oscillating region to the surface but this proved to be rather inaccurate.The modi–cation of Youngœs equation to account for the presence of an external force on the drop seems to have been –rst made about seventy –ve years ago by Adam and Jessop.21 They recognised that the action of a force parallel to the boundary would render the advancing and receding contact angles h and hr unequal the imbalance in the surface forces occasioned by the a changes in the contact angles being exactly equal to the applied force at equilibrium. The imbalance is positive at the advancing front and negative at the receding end which means that the external force per unit length f ex must be subtracted from and added to Youngœs equations corresponding to the advancing and receding angles respectively C C13[C12 … cos ha[C23[f ex\0 13[C12 … cos hr[C23]f ex\0 (20) or (cos hr[cos ha)\2f ex/C12 (cos hr[cos ha) is proportional to the external force per unit Thus the degree of hysteresis length of the drop-boundary contact perimeter.It has been observed in experimental studies that below a certain value of the capillary number the deformed drop remains stationary.22 In such a Fig. 4 Cross-section of the drop at Ca equal to 0 (a) 0.04 (b) and 0.08 (c). The equilibrium contact angle is 90°. 137 Faraday Discuss. 1999 112 129»142 Fig. 5 Removal of a 150° contact angle drop from the surface. The frames are from top to bottom at 320 440 460 and 500 time units after ììswitching onœœ the —ow –eld.situation the whole of the external force acting on the drop must be balanced by the extra force on the contact line and therefore eqn. (20) is strictly applicable. In the simulations the deformed drop moved at all the shear rates applied resulting in a loss of a fraction of the external force due to viscous dissipation at the drop-boundary interface ; so f ex is correspondingly reduced. It follows that the shape of the drop and the extent of the contact angle hysteresis are functions of the drop velocity which in turn depends on the contact line motion controlled by a non-classical —uid mechanics traditionally interpreted in terms of a slip length. The friction coefficient between the drop and the surface in the present model is quite large so at Ca\0.1 (the regime in which the shape pro–les in Fig.4 were computed) the drop moves only slowly. The stress due to the shearing motion is therefore quite small and the shape should be approximately the same as if the drop were stationary. Based on their lattice»Boltzmann simulations Yeomans et al.23 have proposed that for particles moving on solid boundaries diÜusion dominates the contact line motion. In our model the diÜusion coefficient near the contact line is determined by the modi–ed boundary condition described above; interestingly this opens the possibility of changing the surface diÜusion coefficient independently of the other parameters in the system. 5.5 Removal of the drop oÜ the boundary Our simulations show that with increasing shear rates the drop continues to deform and it moves on the surface with increasing velocity.As the shear rate exceeds a certain value the critical value Faraday Discuss. 1999 112 129»142 138 Fig. 6 The –tted parameters q (open circles) and Sq 0 times as a function of shear rate. (The units along both axes are arbitrary.) rT (–lled circles) which de–ne the distribution of removal Fig. 7 Break-up and partial removal of a 90° contact angle drop (note that the detached sub-group re-enters from the left through the periodic boundary condition). The snapshots were taken at 155 200 210 and 230 time units (from top to bottom). 139 Faraday Discuss. 1999 112 129»142 Fig. 8 Trail of droplets left behind a 90° contact angle drop (note periodic boundary condition). The snapshots were taken at 150 230 380 and 540 time units (from top to bottom).the drop develops a tendency to detach itself from the surface leading to its removal. At shear rates only slightly above the critical value the deformed drop remains adhered to the surface for a considerable time before being removed. To investigate the removal process we carried out a large number of simulations at diÜerent shear rates. These calculations were performed in a rather small cell (10]10]10) since it was necessary to carry out many repeated trials to obtain reliable estimates of the mean time of removal. Fig. 5 presents a series of 4 snapshots showing the lift oÜ process for a drop with equilibrium contact angle equal to 150°. We found that the removal process possesses a strong stochastic feature whereby the removal times are randomly distributed with a mean value that depends on the shear rate.We have performed a statistical analysis of the data assuming that the mechanism involves two time regimes Fig. 9 Observed behaviour of the simulated drop at diÜerent contact angles and shear rates. Ö Drop remains on surface ; L drop removed from surface ; ) + drop partially removed and partially retained ; ì trail of dropsœ (as in Fig. 8). Faraday Discuss. 1999 112 129»142 140 one relating to the establishment of the —ow in the simulation cell and subsequently inside the drop (q0) with attendant deformation of its shape; and the other corresponding to the detachment process (q process. q is assumed to follow an exponential distribution.q and Sq r). It is the second process which underlies the stochastic nature of the overall removal r rT are plotted against shear 0 q varies only weakly with shear rate while SqrT seems to rate in Fig. 6. As might be expected 0 diverge as the critical shear rate is approached. In explaining the removal process one may invoke the concept of an energy barrier that the drop must overcome in order to escape from the boundary. In the absence of —ow the barrier is too large for thermal —uctuations to overcome. The —ow –eld lowers the barrier ; at the critical value the barrier height becomes comparable to the thermal energy so the thermal —uctuations are sufficient to drive the drop away from the surface. 5.6 Drop break-up In the simulation studies of the drops in the contact angle regime O120° it was observed that they have a strong propensity to deform leading to the development of ììnecksœœ in their shape.The necks become unstable upon thinning and are eventually ruptured. This behaviour is similar to that exhibited by drops suspended in strong shear —ows.24 The drop may thus break up into several fragments some remaining on the surface and some escaping into the bulk phase. Fig. 7 depicts the break-up process where a time sequence of the con–guration snapshots from the simulation of a drop at 90° contact angle is presented. Experimental studies on con–ned drops in shear –elds reveal a remarkably similar picture of the break-up process.25 The fraction of the drop removed depends on the contact angle ; for example at 30° contact angle only a tiny fraction of the drop is removed whereas at 120° most of the drop leaves the surface.At a contact angle of 90° and intermediate shear rates we see the drop being dragged along the surface leaving a trail of small droplets behind it Fig. 8. This behaviour is of common occurrence and has been studied systematically in several experimental investigations e.g. ref. 26. Fig. 9 summarises our simulation results on the state of surface-con–ned drops in a range of shear rates and contact angles. Each point corresponds to a simulation run. The lines are conjectured boundaries between the diÜerent regions. 6 Concluding remarks In the present simulation study we have successfully modi–ed the DPD algorithm to ensure the occurrence of the stick boundary condition at a solid/—uid interface a necessary step in the development of a credible mesoscopic model for the three-phase solid/drop/—uid system.In the absence of —ow the computed values of the equilibrium contact angles he are found to be in satisfactory agreement with those given by Youngœs equation for he[90°. The determination of he for values less than 90° proved problematic due to poor de–nition in the drop pro–le close to the solid boundary a result of oscillations in the —uid structure in the interfacial region. In this range of he the large departures observed in the DPD values from those calculated from Youngœs equation may be possibly ascribed to the lack of requisite accuracy in the former. The problem may be alleviated by increasing the drop size in the simulations so the magnitude of the pro–le distortion at the interface is reduced relative to the drop size leading to better estimates of the contact angles.Alternatively the quality of the drop pro–le close to the boundary may be improved by smoothing the density in the interfacial region by modelling the wall as a potential chosen to compensate for the oscillations.27 The disadvantage of this approach is that there is only one such potential and therefore only one possible value of the interfacial tension. The simulations unequivocally establish the occurrence of shear-induced contact angle hysteresis in agreement with various experimental results reported in the literature. However the problems originating from the distortion of the drop pro–le at the interface precluded accurate computation of the degree of hysteresis (cos hr[cos ha).As stated above simulation of large drops would hopefully permit the desired accuracy to be achieved enabling validity of the DPD model to be tested through comparison with the results from the Adam»Jessop equation. The mechanism of the removal of the drop from the boundary involves two time scales corresponding respectively to the attainment of steady shear —ow and the escape of the drop. The 141 Faraday Discuss. 1999 112 129»142 latter is stochastic in nature. The characteristic time for the stochastic process has been shown to diverge in the limit of the critical shear rate. The simulation results presented here although preliminary are quite encouraging providing sufficient motivation for extending our modelling studies to systems such as large drops at solid boundaries drops at rough boundaries and interfaces involving surface tension gradients.Acknowledgements The authors acknowledge with pleasure the help and practical advice given generously by Rob Groot and Patrick Warren in the course of the extension of the DPD code to treat solid boundaries and —ow –elds. Discussions with Julia Yeomans and Daan Frenkel are also acknowledged. We are grateful to Owen Griffiths for carrying out a number of the simulations for us. Paper 9/01273G References 1 Cellular Automata and Modelling of Complex Physical Systems ed. P. Manneville N. Boccara Y. G. Vichniac and R. Bidaux Springer Heidelberg 1990.2 F. W. J. Weig P. V. Coveney and B. M. Boghosian Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. T op. 1997 56 6877. 3 A. J. Wagner and J. M. 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G. Jacquin and P. M. Adler J. Colloid Interface Sci. 1988 126 314. 23 J. M. Yeomans unpublished work. 24 M. Lal J. N. Ruddock and N. A. Spenley unpublished results. 25 B. J. Carroll Colloids Surf. A 1996 114 161; S. Basu R. Kumar and K. S. Gandhi Chem. Eng. Sci. 1997 52 2849. 26 D. Quere and E. Archer Europhys. L ett. 1993 24 761. 27 R. D. Groot unpublished results. Faraday Discuss. 1999 112 129»142 142

ISSN:1359-6640    

DOI:10.1039/a901273g

出版商:RSC    

年代:1999

数据来源: RSC