ISSN:0301-5696    

DOI:10.1039/FS98318FX001

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

Faraday Symp. Chem. SOC.,1983 18 7-27 Viscoelastic Properties of Entangled Flexible Polymers BY WILLIAM W. GRAESSLEY Exxon Research and Engineering Co. Corporate Research -Science Laboratories Annandale New Jersey 08801 U.S.A. Received 18th August 1983 Viscoelasticity in polymeric liquids is a natural consequence of macromolecular structure. The distribution of configurations is readily displaced from equilibrium and the configurations relax much more slowly than in monomeric liquids. As a result recoil effects and other types of elastic-like behaviour are easily observed and the stress depends in general on the history of deformation rather than simply the current or rate of strain. These properties are related systematically to molecular size and structure and to the concentration.The experimental laws are now rather well established for concentrated solutions and melts of nearly monodisperse flexible polymers. The chains in this case interpenetrate one another’s domains extensively and their dynamics in that tangled environment appear to depend in a universal way on mutual ‘uncrossability ’.The proposal that linear chains in this circumstance move to new conformations primarily by diffusion along their own lengths has recently been developed into a full theory of entangled-chain dynamics. Tests of this theory and proposals for its modification to include other motions are discussed. Viscoelasticity and chain dynamics are closely related properties in polymer liquids. All molecular theories of viscoelasticity begin from that viewpoint and it is well supported by experimental evidence.l At the very least it serves as a useful qualitative guide for understanding the complex mechanical behaviour -the mixture of rubber-like and liquid-like properties -that characterizes polymer liquids.As a simple example consider the effect of a sudden strain E imposed on the liquid at rest displacing the distribution of configurations from equilibrium. An increase in free energy is produced and this generates a restoring stress t~.To prevent recoil forces must be applied to hold the liquid in its new shape. That excess free energy is dissipated and the associated stress and capacity to recoil fade away as the Brownian motion of the chains act to return the configurations to equilibrium.For a simple shear strain y E= [Eii]= 0 1 0 r OI Lo o iJ. The shear stress 021= a(t,y)can be written as Y) = yG& Y) (2) where G,( t,y) the shear-stress relaxation function reflects the structure interactions and motions of the chains. Differences in the normal components of stress are also produced oll-02,= Nl(t,y) and o,,-033= N2(t,y) and these also return to zero at equilibrium. Such effects are of course not unique to polymer liquids. Small-molecule liquids are also viscoelastic,2 but only on scales of time which are extremely small (t g lo-* s). Departures from Newtonian behaviour (stress dependent on rate of strain alone) are difficult to detect except under rather special conditions (such as near the glass-transition 7 VISCOELASTIC PROPERTIES temperature G).What sets polymer liquids apart are their long relaxation times even well away from Tp and the ease of achieving large displacements from con- figurational equilibrium.Both features are natural consequences of macromolecular structure and they lead to important and easily observable viscoelastic phenomena which depend systematically on that structure. The interplay of configuration distortion resulting from deformation and Brownian motion acting to restore equilibrium illustrated above for the special case of a step shear strain also governs the mechanical response for more elaborate deformation historie~.~~~ Thus if the liquid is sheared at a constant rate j the two effects are eventually brought into balance and that balance determines the steady-state stress a(j).The shear stress a(?) and normal stress differences Nl(j) and N2(j)are increasing functions of shear rate and 49) = ?z7(j) (3) where ~(j) is the steady-state viscosity of the liquid.Stress growth functions o(t,j) Nl(t,j) and N,(t,y) describe the transition from rest to the steady state and there are corresponding functions for relaxation from steady state as well as for example the total shear-strain recoil from steady state YR = O(i))JS(?) (4) where Js(j)is the steady-state recoverable shear compliance function for the liq~id.~ Various functions are also used to describe response in extensional All these properties are assumed merely to be different manifestations of the same underlying competition between configuration distortion and Brownian motion.A great simplication occurs in the limit of small departures from configurational equilibrium. The response is linear and it depends only on the structure interactions and dynamics of the molecules at equilibrium. The stress in this regime of linear viscoelasticity can then be written as the sum of contributions from past deformations in terms of a single time-dependent property of the 1iquid.l There is some latitude of choice for this property but a convenient one for our purposes is the shear-stress relaxation modulus G(t) = Gs(t 0). Thus following any history of sufficiently slow or small shear deformations y(s) the normal stress differences are negligible in comparison to the shear stress a(t) and [Im a(t) = G(t -s) j(s) ds.Expressions for limiting steady-state flow properties such as qo= q(O) the zero- shear-rate viscosity and Jt = Js(0),the recoverable shear compliance can be obtained direct1y 1 t2(g) tG(t)dt = -dt. (7) J =-r7 s" 0 2t7 0 Both properties are sensitive to the behaviour of G(t)at long times and their product defines a weighted mean relaxation time for the liquid:3 W. W. GRAESSLEY The information in G(t)can be obtained in terms of frequency ofrom small-deformation oscillatory measurements. These provide in-phase and out-of-phase components of the dynamic shear modulus G*(w) = G’(o) + iG”(co) where G*(o) = G(t) exp (-icot) dr (9) io lom and G’(o) and G”(o) are the storage and loss moduli of the liquid.Given the connection between mechanical response and chain dynamics what role can we reasonably expect molecular theory to play? The theory of dynamics is incomplete even for ordinary liquids so it appears that we must begin with a compromise of some sort. What saves the situation is the very size of the molecules being considered. Many properties of interest depend on relaxation at long times -the slow dynamics of the liquid -which mainly reflects the large-scale motions of the chains. For these motions the local dynamics the central theoretical problem in monomeric liquids become averaged over long chain distances and can be represented by for example an average frictional coefficient per unit length. Its value must be determined empirically for each system and temperature but otherwise the difficulties of monomeric liquid dynamics need not be considered further.The first objective of any molecular theory of viscoelasticity is to express G(t)in terms of molecular motions and interactions of the chains at equilibrium. Relationships between molecular structure and properties such as qo and JZ would follow auto- matically from eqn (6) and (7). It should also provide expressions for non-mechanical dynamic properties such as D the self-diffusion coefficient of the chains. Non-linear viscoelastic properties such as G,(t,y) q(j),Js(j) etc. are necessarily more difficult because they also include the effects of finite displacements from the equilibrium configuration. Of these the step strain functions such as G,(t,y) would seem to be easiest to deal with because only the transition between two states is involved.Properties which describe the response to a continuous deformation history depend on contributions from an infinity of past shapes. The ultimate objective is to develop a constitutive equation or rheological equation of state for the liquid encompassing the response to any deformation history in a manner analogous to eqn (5)for the shear stress in linear viscoelasticity. These preliminary remarks about viscoelasticity apply broadly to polymer liquids of all types and levels of concentration. The remainder of this paper will focus on flexible non-associating polymers and will in particular examine the experimental observations for melts and concentrated solutions in the light of some recent molecular theories.Viscoelastic behaviour for nearly monodisperse linear chains will be empha- sized although some effects of polydispersity will be considered near the end. Other papers in the Symposium will deal with the effects of chain stiffness and branching. In dilute and semi-dilute solutions the excluded-volume and hydrodynamic inter- actions between chain units are important.l? Both interactions are increasingly 8l screened by other chains as the concentration increases but where the screening becomes effectively complete is not known. Recent estimates suggest that excluded volume has little effect on chain dimensions above a polymer volume fraction (b of ca. 0.15 in typical cases.l0* l1 Neutron-scattering studies are still inconclusive.12 According to a recent theory the static and dynamic screening lengths have the same concentration dependence but the dynamic screening length is larger,13 suggesting that a higher concentration is required to cancel the hydrodynamic interaction.For purposes of discussion it will be assumed that the polymer concentration is high enough to give complete screening. Thus the chains are assumed to have random conformations with mean-square VISCOELASTIC PROPERTIES end-to-end distance (R2)proportional to molecular weight A4 and equal to their the ta-solvent (unperturbed) dimensions where Lois the contour length of the chains and 1is the Kuhn step length I = C lo where lois the bond length and C is the characteristic ratio of the species.14 Likewise the basic dynamic model is the Rouse chain.I5 The polymer molecules are assumed to behave like strings of Stokes friction (monomeric friction coefficient co) freely flexible down to small distances and with all parts responding independently (no hydrodynamic interaction) to the random local impulses of Brownian motion.For a concentration of v long chains per unit volume each with n-mers and many Kuhn steps in length moving freely in a purely viscous medium and for times which are long enough that the effects of local stiffness are negligible the diffusion coefficient and stress relaxation modulus are given by kT DR=-40 where the longest relaxation time is Thus from eqn (6) and (7) 21 (J,") = -5 vkT' These equations give a surprisingly good description of behaviour in concentrated systems where the chains are not too long.Beyond a certain length however corresponding to a characteristic molecular weight M which varies with both polymer species and concentration the stress relaxation modulus G(t)separates into two main dispersions1 (fig. 1). Relaxation at short times the transition region is independent of chain length and appears to reflect only local rearrangements of chain conformation. At the lower end of this region approaching the plateau the relaxation resembles that of a network of Rouse chains. Relaxation at long times the terminal region reflects the rearrangement of large-scale conformation. Its location and shape depends strongly on chain architecture :chain length chain-length distribution and long-chain branching.The separation in time of these two dispersions increases rapidly with chain length (At cc kP4),but the modulus in the intermediate plateau G; is insensitive to architecture and depends only on polymer species and concentration. The variation of qo and J," with chain length changes from Rouse-like behaviour ~occM and Jt cc M to 'loxkP4and JZ cc Mofor long chains (fig. 2). These characteristics are called entanglement effects because they appear to derive essentially from topological restrictions on the chain motions.3$ It is certainly reasonable to expect such effects at high concentrations. The chain contours are extensively intermingled so each chain is surrounded all along its length by a mesh of neighbouring chain contours.Like the Rouse chain it can flex and relax freely W. W. GRAESSLEY 1oI0 -i E --2 lo* --0 3 W lo6 -U 5 5lo4 -W u lo2 -/ L ,IIIIIIII ,I loo lo2 104 lo6 to8 1o'O 4 Fig. 1. Relaxation modulus (A) and dynamic modulus master curves (B) for flexible linear polymers with nearly monodisperse distributions of chain lengths. 0 F bn log M log M Fig. 2. Viscosity (A) and recoverable compliance (B) plotted as functions of molecular weight for nearly monodisperse linear polymers. The dashed lines indicate Rouse-model behaviour. but only up to some chain distance which depends on the size of the mesh. Rearrangement on larger scales is retarded because the chain cannot cross through its neighbours.To relax completely the chains must in some sense diffuse around one another. At intermediate times or frequencies the liquid acts like a network with modulus G:. The theoretical problem suggested by this picture to calculate the dynamic properties of a dense collection of mutually uncrossable Rouse chains seems VISCOELASTIC PROPERTIES intractable at the present time. A manageable theory has been achieved however by focusing on one possible motion for the chains which does not violate the uncrossability constraints and which requires no interchain cooperation. In the regime of slow dynamics each chain is assumed to diffuse independently along the random path its contour makes through the mesh.The idea of dominance by this simple snake-like motion called reptation was introduced by de Gennes for the relaxation of unattached chains in a network16 and developed by Doi and Edwards into a full theory of dynamics for entangled liquids of linear The Rouse chain remains the basic model but now subject to spatial constraints in the form of a tube to represent the mesh. Direct numerical comparisons with data are possible because the equations contain only two parameters which must be established experimentally. One is the monomeric friction coefficient co which sets the basic time scale. The other is the primitive path step length a (see below) a measure of the mesh size (the ‘topology’ of the liquid) that varies only with the polymer species and concentration.The theory does not deal with species and concentration dependence of either c, or a. Comparisons with theory will be taken up following a summary of the current experimental situation. EXPERIMENTAL SUMMARY -NEARLY MONODISPERSE LINEAR CHAINS In undiluted polymers the plateau modulus is essentially a property of the species independent of chain length and insensitive to temperature.l Its variation with polymer volume fraction 4 is essentially universal :21* 22 G; cc bd,2.1 c d c 2.3. (16) The exponent may vary slightly with the polymer species but it is insensitive to the choice of solvent (fig. 3). Few data exist for 8-solvents however and values below 4 z 0.2 are uncertain. Even the dependence on polymer species seems to be part of some rough but fairly general law relating G; simply to the total concentration of chain contour in the liquid.Thus data from many species concentrations etc. reduce approximately to a universal power law G 13 = K(vL 12)d kT where 2.0 < d < 2.3 and K z for d = 2.22 The plateau modulus can be expressed as an entanglement molecular weight Me giving a length scale for the topological interaction. With the equation for the modulus of Gaussian networks1 we have where p is the density of the undiluted polymer. A liquid is highly entangled and the terminal dispersion is well resolved when M/Me is large. The terminal dispersion appears to approach a universal form for sufficiently long chains :21 G(t) = G&U(t/.ro) M B Me (19) whereby the particulars of individual liquids can be absorbed completely in GS and the mean relaxation time 7,.The loss modulus G”(o)in the terminal dispersion for various values of M/Meis shown in fig. 4. The universal form is apparently established W. W. GRAESSLEY 10' t7 lo6 E c -0 \ n a v 02 u 1o5 lo4 I I I I I 0.OL 0.1 0.2 0.L 1.o 4 Fig. 3. Plateau modulus as a function of polymer volume fraction for polybutadiene. The various symbols indicate different diluents including short-chain polybutadienes with GoN obtained from Gk as described in ref. (21). rather quickly in the leading edge (frequencies below w at the loss peak Gk) but much larger entanglement densities (M/M = 100) are required to resolve the trailing edge.With eqn (19) it follows from eqn (6)-(8) that the product J,"G; a measure of the terminal dispersion breadth,3 must also be universal. Experimentally JZ az WqVd for M > Mi = 6 Me and JZG; z 2.4 (20) for different species concentrations and chain lengths. Even allowing for some downward revision to a value for truly monodisperse chains (correcting J for the effects of residual polydispersity in the experimental polymers) the value lies significantly above J,OGg = 1 corresponding to a single exponential form for U(t). The analysis of viscosity is complicated by uncertainties about the monomeric friction coefficient^.^^ Ferry has established values of co and their temperature dependence for many undiluted species by applying the equations for networks of Rouse chains to data on response in the transition region.' He obtains similar values from measurements of diffusion coefficient Dofor small foreign molecules in melts and networks of the polymer species using c0= kT/D,.Using these with eqn (17) he obtains values for q0below M which although less than observed by a factor of about three are clearly of the right magnitude. Thus to a reasonable approximation the magnitude of viscosity for long chains (q0cc kP4)can be expressed as VISCOELASTIC PROPERTIES where M denotes the intersection of short-chain and long-chain behaviour. Typically M z 2 Me although there appears to be some variation in the ratio among species. That dependence on chain length has been established in many species beyond M z 10 M and a few cases as high as M z 100 Information on diffusion coefficient for long chains at high polymer concentrations is limited but the available results suggest D cc M-2.24-26 Surprisingly none of these studies has established a lower limit below which the chains switch to Rouse behaviour D cc M-l.From the available data that limit must lie well below Me. On the other hand separate studies on very short-chain diffusants suggest D cc M-l,l as expected. Data on non-linear response in nearly monodisperse liquids are unfortunately quite limited. Such experiments particularly those where the time dependence of stress is involved are inherently more diffi~ult.~~~ 28 Beyond the instrumental problems flow instabilities always intervene to restrict the range of variables.Only the shear-rate dependence of viscosity has been thoroughly explored with well characterized polymer^,^ although there is now a growing body of information on step strain response. Time and deformation dependences have been measured for a variety of shear and extensional histories; however with a few exceptions the work has been done on commercial polymers of uncertain struct~re.~ The properties of the viscosity function for M >> Mecan be stated quite simply. The viscosity starts off at qoat low shear rates then begins to decrease monotonically near some characteristic shear rate j,. The form of the function appears to be essentially universal (fig. 5) W)= q V(j/jO) (22) and it goes over smoothly to a power law at high shear rates V(j) 7i-p(7i + 7io) (23) where the exponent p is in the range 0.804.85.The form of V(j/jo)is sensitive to polydispersity but not to chain architecture the same function fits data from nearly monodisperse samples of both linear chains and stars. 29 The characteristic shear rate depends only on linear viscoelastic quantities jo z (qOJ,O)-l=z; l. Thus the beginning of departures from qo is governed solely by the mean relaxation time of the terminal dispersion jz z I. We are on less firm ground with Gs(l,q),but thanks to the pioneering studies of Osaki Kurata and coworkers3o* 31 certain features are now becoming well established. They have shown that under certain circumstances the strain and time dependences are factorable (fig.6). Thus they find GS(f = G(t)h(y) ’zk (24) where h(0) = I and zkand is characteristic time for the liquid which is typically much smaller than the mean relaxation time z,. Its value varies systematically with concentration and chain length moving closer to zo at smaller values of M/Me.On the other hand the strain dependence h(y) is quite insensitive to those variables. Strangely this latter feature and indeed factorability itself seem to break down at high entanglement densities (M = 100 Me).32, 33 Aside from these well documented properties of q(j) and G,(t,y) there are other characteristics of non-linear response in flexible-chain liquids which seem quite general and thus must be considered in judging molecular theories.One very important feature is the stress-optical law. The flowing liquid is birefringent and the optical tensor remains directly proportional to the stress tensor even in the non-linear regime.34 The W. W. GRAESSLEY 1.2 d 1 .o 0.8 d eE d u =' 0.6 d 52 u cld 0.4 0.2 /-\ I I 'k. \ 0 .o -I 1 o-~ lo-* lo-' 1oo 10' lo2 1 o4 o/o Fig. 4. Reduced dynamic loss modulus in the terminal region as functions of reduced frequency for melts of linear-chain polymers of several species. The numbers indicate values of M/M as explained in ref. (21). The dashed line is G(o)/Gk plotted against CL)/CL) for a single exponential terminal region G(t)= GoN exp (-t/z,,). The solid line is an estimation of the universal form for reduced loss modulus in the long-chain limit as explained in ref.(21). 1.o 0 F \ F 0.1 0.01 0.001 0.01 0.1 1.0 10.0 100.0 Yli.0 Fig. 5. Reduced viscosity as a function of reduced shear rate for polystyrene solutions with various molecular weights and concentrations and for various temperatures. See ref. (3) chap. 8. VISCOELASTIC PROPERTIES Fig. 6. Shear-stress relaxation function G,( t y) at several strains for a nearly monodisperse polystyrene solution. The molecular weight (M = 670000) and concentration (d x 0.375) correspond to M/M x 13. The shear strain y ranges from < 0.70 to 4.0 increasing with the clockwise rotation of the pips. The curves superpose by vertical shifting alone for t > zkx 100 s.See ref. (52). constant of proportionality the stress-optical coefficient depends on the optical anisotropy of the chain units suggesting that the stress is primarily a function of the anisotropy of chain orientations induced by the flow. Both birefringence and direct measurements of indicate that the steady-state extensional viscosity qT(i) of linear-chain melts remains relatively constant (near its limiting value 3q0)for some distance beyond k0. and melt This is in direct contrast to dilute-solution behavio~r~~ behaviour for branched for which the tensile stress rises precipitously with ti. Also the normal stress functions in shear flows seem clearly related by some general law Nl(j)is always positive but N2(j)is negative and3* 0.1 < -N2(j)/Nl(p)< 0.3 (25) for many systems.The ratio seems to be constant even when the individual values are changing with time.39 Finally relationships among properties for many deformation histories seem broadly consistent with certain constitutive forms of the single-integral strain- dependent type. These include special cases of the Lodge model for Gaussian networks with arbitrary strand creation and destruction rates40 and the BKZ equation modelling the relaxation of an arbitrary elastic network.41 The factorized form of BKZ 42 [reducing to eqn (24) for single-step strains] has been used dG(t -t’) H[E(t t’)]dt’ = j-mdtr 17 W. W. GRAESSLEY A-Fig. 7. Sketch illustrating the tube and primitive-path ideas in the Doi-Edwards theory.(A) The chain surrounded along its length by other chains (B) the surrounding chain constraints represented by a tube and (C) representation of the tube trajectory -the primitive path of the chain -by a sequence of N independently directed path steps of step length a. where E(t t’) is the deformation between times t’ and t. Systematic departures from even this form are sometimes noted at high strain rates43 and are perhaps related to the t parameter [eqn (24)]. Departures in non-linear response are also observed when the strain is not simply an ever-increasing function of time -recoil experiments or double-step strains of opposite sense. The latter effects have been accommodated by allowing for some rupture of network strands (the ‘irreversibility ’ hyp~thesis).~~ Although both the Lodge and BKZ forms are motivated by molecular arguments they contain functions as opposed simply to parameters which must be established experimentally.MOLECULAR THEORY -THE TUBE MODEL Doi and Edwards develop their theory of dynamics for entangled linear chain liquids from the following idea~.l~-~O (1) At high concentration the chains provide a random mesh-like structure in the liquid. They have random conformations at equilibrium and mean dimensions which are much larger than the mesh size. (2) Each chain moves freely except being forbidden to cross the mesh lines that surround it. Its large-scale conformation is defined by its trajectory through the mesh. The lifetime of the mesh constraints is long enough to restrict the range of lateral excursions of the chain.Each chain thus acts as if it were confined in a tube (fig. 7) and over some range of short times it mainly moves around an average trajectory the primitive path of the chain a random curve through the mesh characterized by some persistence length of the order of the mesh size.45 (3) Over longer times each chain moves as a whole along its primitive path creating new path of random orientation as one end or the other advances into the mesh while abandoning old path at the other end. If the mesh constraints are effectively permanent the large-scale conformation after some time interval t -t is made up of those parts of the path at t which are still occupied at t, plus those parts which have been created during the interval.Each element of path is created by advancement of a chain end and later abandoned by the first subsequent return by either chain end. (4) Solution of that ‘first-return’ problem for chains diffusing along tubes gives an VISCOELASTIC PROPERTIES expression for the average fraction of initial path length which is still occupied after a time t = t2-tt, 81 F(t) =-Z -exp (-tp2/td) (27) 712 oddp2 P in which td is the disengagement time L2 td =-n2D* L is the path length and D* is the diffusion coefficient along the path. The path length is proportional to chain length Lo.Fluctuations in L are omitted the surplus Lo-L is assumed to be constant and distributed uniformly along the path; the ratio L/Lo depends only on the mesh size.Rearrangement of large-scale conformation then takes place by pure reptation. Motions along the path are assumed to be governed by Rouse chain dynamics so D* =D [eqn (1 l)] and therefore td cc M3. (5) The macroscopic diffusion coefficient of random coils moving by reptation alone follows simply and directly from a consideration of the centre of gravity displacements caused by randomly directed chain ends emerging from the ends of the current paths so D cc M-2.Similarly the time course of stress relaxation in the terminal region from chains moving by reptation alone is given by G(t) = G",(t). (30) This assumes only that the chain motions are unaffected by an infinitesimal strain and that the stress tracks the fraction of chain segments which still retains the orientation given by the Eqn (30) in fact should apply to large random chains of any architecture moving in an effectively permanent mesh.The fraction of still occupied path F(t),would of course be different for abandonment mechanisms other than pure reptation. (6) From eqn (6),(7) (27) and (30) -2 J,"G; =6/5 (32) so ylo cc M3,J,"G; is a universal constant and G(t) has a universal form in the terminal region. (7) To provide a more detailed description of the mechanical response the primitive path is represented by N independently directed steps of length a. The chain segment in each path step is assumed to be long enough to behave as a random coil (a % I) and to contribute independently to the stress.Thus at equilibrium L =Na (33) and (R2) = Na2 (34) since both path and chain are assumed to be random walks with ends that coincide. The displacements from a strain E carrying the paths into distorted shapes are assumed to be affine down to distances of the order of a. W. W. GRAESSLEY (8) If N is large enough the relaxation following a sudden strain goes in three stages well separated in time (a) Equilibration of chain segments within each step in the manner of strands in a permanent network and corresponding to the viscoelastic transition region (time independent of M). (6) Equilibration of stretches along each path returning L/L to equilibrium but leaving the distortions of path shape unchanged. Since Rouse chain dynamics apply to motions along the path the time to complete that process the equilibration time z, should be of order zR K M2[eqn (1 3)].(c) Equilibration of large-scale conformation by reptation into random paths corresponding to the terminal region with a characteristic time zdK M3. (9) A general expression for stress c in the terminal region is obtained as the sum of contributions from oriented but unstretched chain segments. The reptation rate is assumed to be independent of deformation a(E,t) = 3vNkTQ(E)F(t) t > z (35) in which Q is a universal tensor with components and ( ) means the average taken over all directions of the unit vector u. The tube length immediately after deformation is L( IE*ul),which is larger than the equilibrium length L. The factor (IE-u~)-~ in eqn (36) accounts for the loss of path steps that occurs as the chain retracts inside the tube when L/Lo returns to equilibrium (t zze).Its influence is negligible in the limit of small strains. From eqn (35) the plateau modulus is G&= 2 vNkT. (37) (10) The model is extended to other deformation histories for cases when the flow is slow enough that equilibration along the path can be taken to be essentially instantaneous the path length is always at its equilibrium value L. A simplifying as- sumption the independent-alignment approximation (IAA) is also used. Its effect is to make the distribution of path-step lifetimes independent of deformation history. New path steps are created at a constant rate (the reptation rate is independent of deformation) and the stress is simply the sum of contributions from path steps whose orientations have been changed affinely since their creation.The result is a factorized form of the BKZ constitutive equation aij(t) = 3vNkT dF(t -t’)Q& t’)dt’ dt’ where Q$=( [E(t t’)*U]i [E(t, t’)’U]j IE(t t’) uy >* (39) The IAA gives a slightly different response for single-step strains [eqn (35) with Q* in place of Q],leading to G&= gNkTinstead of eqn (37). More importantly it converts the behaviour from a relaxing-network response with partial rupture [associated with the (IE*ul)factor in eqn (36)] to a purely relaxing network response and leads for example to quite different predictions for double-step strains of opposite sense. The instant homogenization of path-step lifetime implied by the IAA is inconsistent with other properties of the model it does not correctly reflect the interior ordering of path steps according to age.The practical importance of this for non-reversing strain VISCOELASTIC PROPERTIES histories is unknown. Its effect is certainly small in the predicted strain dependence of G& y). Curtiss and Bird47* 48 have analysed the reptating-chain problem by a different theoretical method. They arrive at an expression for the stress which is the sum of two terms. One is a purely orientational contribution which is identical to eqn (38) but with the factor of 3 replaced by unity. The other is essentially a frictional contribution reflecting a tension in the chains that comes from their motion relative to the surroundings.The latter is scaled by a new parameter the chain tension coefficient E. One might perhaps interpret that term as a contribution from the incompleteness of equilibration at high strain rates omitted by Doi and Edwards but that is still unclear. Computer simulations of chain dynamics in a rigid lattice of uncrossable lines leave little doubt that reptation is the dominant motion for that case at least.4g The situation is less clear in more realistic simulations with dense collections of moving and uncrossable chains.50 The latter studies are necessarily limited to relatively short chains perhaps too short to display a fully developed reptation behaviour. The trend of global properties such as D with increasing chain length51 suggests however an approach to the reptation law (Doc M-2).Little in the way of mechanical properties (qo,G&,J,")has been reported as yet from the simulation studies.COMPARISONS WITH EXPERIMENT The theoretical predictions of Doi and Edwards are consistent with many of the observations discussed earlier for entangled liquids of near-monidisperse linear chains. The product G&J,Ois essentially universal although the experimental value [eqn (20)] is larger than predicted [eqn (32)] the experimental terminal spectrum is broader perhaps caused in part by residual polydispersity . Conformance to the stress-optical law is a natural consequence of the predicted dependence of stress on chain orientation alone.The strain dependence of G,(t y) is factorable beyond a certain time zk and zk appears to be of the order of zR,52suggesting an origin in the Doi-Edwards equilibration process. Moreover except for the puzzling behaviour beyond M/M w 100 h(y)has a form very similar to that calculated from the theory (fig. 8). Values of D calculated from the theory with friction coefficient 5 estimated from qo for short chains [eqn (1411 and path step length a obtained from G; [eqn (33) (34) and (37)] agree with experimental values in both chain-length dependence and magnitude.53 Thus D = 0.26 M-2cm-2 s-l for undiluted polyethylene at 176 0C,24 and the prediction is 0.42 M-2;D z 0.6/M2for undiluted polystyrene at 225 0C,26 and the prediction is 0.2/M2.The special BKZ form [eqn (38) and (39)] gives a reasonable representation of many non-linear properties and its prediction for example of N2(p)/Nl(p) = -2/7 at low shear rates is remarkably close to observations [eqn (25)].The qualitative departures from BKZ for recoil and opposed double step strains are given a natural explanation by the network-rupture character of eqn (36). Perhaps the greatest disappointment especially in view of the apparently good results for D,is the lack of agreement in the viscosity [eqn (21)]. From the theory (taking M = 2MJ Vo = I~(VO)R(E)~ M3 (40) so the predicted magnitude is too large for some considerable distance beyond M, and the chain-length dependence is too weak. The form of q(j)calculated from eqn (38) and (39) also does not agree with experiment.Departure from qo occurs near jox (qoJ:)-l as observed but the predicted decrease at high shear rates is too rapid. W. W. GRAESSLEY 21 ; I00 10' Y Fig. 8. The strain dependence of G,(t,y) for a concentrated polystyrene solution. The strain dependenceh(y)for the data in fig. 6 at long times is compared with the Doi-Edwards prediction [eqn (36)' solid line] and the IAA form [eqn (38) and (39)' dashed line]. The steady-state shear stress a(?) in fact is predicted to pass through a maximum while experimentally a(j)grows monotonically [eqn (23)) Incidentally it is possible to obtain a rather good fit to experimental data on ~(j) by an appropriate choice of the chain tension parameter in the Curtiss-Bird formulation (E x 0.375).However that value gives quite unrealistic behaviour at high frequencies in linear visc~elasticity.~~ Those results and the requirement that the stress-optical law is obeyed even at high strain rates34 suggest that E must be very small (E < The appearance of departures from G,(t y) factorability at high entanglement densities (M 2 100 Me)is disturbing since the theory should become most valid in that limit. It is also disconcerting to find no evidence for departure from D cc MV2 for short chains (except perhaps the observation by Ferry of D cc M-l for very short chains'). The Rouse and reptation expressions for D intersect at N = 1/3 (MI = 4/16 Me). Surely one would expect to observe at least the beginnings of departure near Me yet D cc M-2 seems to persist well below that Data from related experiments suggest that motions other than pure reptation are important.54 The finite lifetime of the tube 56 may be responsible for the observations (1) that the relaxation rate of unattached chains are slower in a network than in a liquid of the same chains5' and (2) that the effects of chain-length distribution on properties such as qoand J are much weaker than predicted.53 Constraint release (the 'tube renewal' process) results from the reptation of the neighbouring chains that supply the constraints and should be absent in networks.Each release allows the primitive path to reorient locally resulting in a Rouse-like relaxation of conformation which competes with the reptation of the chain.55 To a first approximation the lifetime of a constraint should correspond with the mean lifetime of a primitive path step for reptating chains5* VISCOELASTIC PROPERTIES averaged over all chains in the liquid.Thus each chain in a mixture would reptate at its own rate (zdof M3),but would also relax by a superimposed motion of its path the local ‘jump’ frequency being in the simplest instance the same for all chains and dictated by t,. The chains no longer relax independently and the predicted effects of polydispersity are weakened (see Appendix). The effect of including constraint release in the theory reduces qoand increases JZ for monodisperse thus moving both in the direction of better numerical agreement with observations. The chain-length dependences and the expression for Gg are unchanged; the predicted effect on D is negligible so the rough numerical agreement with such data noted earlier is unaffected.Thermal fluctuations in the primitive path length L may also be important in some properties of linear chain liquids. Although treated as a constant for simplicity by Doi and Edwards L should in reality be a fluctuating quantity and these fluctuations assume a major role in theories about branched-chain 59 The same tube picture with disengagement from each part of the tube occurring at the first visit by a free end is retained in those theories. The branch point is assumed to prevent reptation so the lifetime of each part is governed solely by the fluctuations in L i.e. by the higher harmonics of chain motion in the tube.Thus abandonment of a part of the tube located at path distance X from the branch must wait for the first fluctuation of L(t)to a value less than X. This leads in a very natural way to an exponential dependence of viscosity on arm length behaviour observed experimentally in the case of star 6o However a well defined exponential dependence and clearly enhanced viscosity (relative to linear chains) only sets in when the branch lengths are well beyond M,. Thus it may be that fluctuations can compete rather effectively with reptation even at moderate entanglement densities suggesting that fluctuations may also be important even for linear chains in that range. Doi has recently considered the combination of reptation and fluctuations for linear chains,61 finding that now q has about the right magnitude at moderate entanglement densities and goes approximately as M3.4 over the typical experimental range (M < M < 30MJ.The M3 dependence is recovered asymptotically. The higher harmonics should have no direct effect on the global diffusion rate so again the rough agreement of D with pure reptation is undisturbed. Aside from omission of constraint release and fluctuation effects and from the uncertainties described earlier about the independent-alignment approximation the departures from the constitutive relation [eqn (38) and (39)] may come in part from the assumption that equilibration along the tube is instantaneous. Thus the theory gives and although the equilibration time is certainly much smaller than the longest relaxation time for highly entangled liquids that does not mean its effects are negligible in tests of the theory that involve high strain rates.Values of M/M z 20 or less are typical of many studies of ~(j), and the shear rates defining the power-law region [eqn (23)] range up to two or three orders of magnitude beyond j zzgl. Accordingly 7jze values of unity and beyond are commonly encountered so on that basis alone it is not too surprising to find a q(j) form which differs from the theory and that the theoretical stress is too small. A similar explanation may account for departures in other flow properties. Thus for example eqn (38) and (39) predict that starting from rest the first normal stress difference N should rise monotonically with time to its steady-state value for any W.W. GRAESSLEY constant shear rate. Experimentally N,(t j) passes through a maximum before reaching steady state when j 9 Yo. The onset of that behaviour however seems to correspond roughly with a shear rate of the order of l/z for the suggesting again that incompleteness of equilibration may be important. FURTHER COMMENTS ON CONCENTRATION DEPENDENCE Theories of viscoelasticity for concentrated solutions and melts make no direct statements about concentration dependence per se. Response in the terminal region shifts with polymer concentration in both the time scale q0J and the modulus scale GO,[eqn (1 9)]. As noted earlier the variation of G& and JZ are essentially universal going as powers of concentration which are opposite in sign but virtually identical in magnitude for different polymer and solvent species.The exponent for G; is clearly larger than the value of 2 which one expects based on a simple counting of intermolecular contacts in random systems3 or a geometrical expansion of the mesh with dilution.49 Computer simulations suggest that the step length a goes as 4-1/2,49 which corresponds to G&a #2 in the Doi-Edwards theory [eqn (34) and (37)]. The observed behaviour in fact is remarkably near the prediction of de Gennes et al. for semi-dilute solutions of good G$ cc d9l4,but the conditions assumed in that calculation should only apply at relatively small 4. In any case the data suggest a sc 4-0.s[from d z 2.2 in eqn (17)] and this dependence is unexplained.The time-scale variation with concentration depends on the viscosity which in turn depends on the monomeric friction coefficients co.Although proportional to solvent viscosity in dilute solutions and perhaps even for some distance beyond coil overlap To(#) depends finally on local composition and temperature at higher concentrations in a manner which is specific to each polymer-solvent combination. Presumably the viscosity also carries some additional dependence on 4 which is universal [qocc #g where g = 1 +(2.4 x 1.2) E 3.9 according to eqn (2 1) assuming M cc Me cc #lPd and d = 2.21 but it is extremely difficult to separate [,(4) from qo and thus confirm that inference with the required precision.Those uncertainties about lo(#) also complicate the search for universal behaviour in the semi-dilute region. CONCLUSIONS So where does the molecular theory of viscoelasticity stand for entangled flexible polymers? It seems to me that the idea that reptation is the dominant motion for linear chains at high concentrations is holding up rather well. The Doi-Edwards theory embodies this idea in a model which adds to the Rouse model a single parameter to represent the solution topology the primitive path-step length a. The theory accounts qualitatively for many observations and none seems to contradict its conclusions in any fundamental way. However it also seems clear that competing motions such as constraint release and fluctuations along the path must be included to achieve good quantitative agreement with experiment.The chain-length dependence of viscosity differences in relaxation rates for unattached chains in networks and in liquids and the effects of polydispersity and long-chain branching seem to require such motions. The real challenge now is to develop a model which contains the three contributions yielding not only good agreement in the various dynamic experiments but also reducing to an adequate theory of equilibrium elasticity for networks where the global mobility of chains is quenched by crosslinking. VISCOELASTIC PROPERTIES Fig. 9. A chain and the lattice of surrounding chains representing the tube idealization used to model the influence of constraint release. I o3 I I I I I o2 d n0 c W -.-0 $= 10’ 10 I 1-1 0.o 0.2 0.4 0.6 0.8 1.0 @B Fig.10. Viscosity plotted as a function of volume fraction of the long chains in a mixture of short and long chains of polybutadiene (Me= 1700). The short-chain component (A) has molecular weight M = 39000; the long-chain component (B) has molecular weight M = 181000. The points represent experimental values which turn out to obey qo= K(Hw)3.4 very closely. The lines are calculated from the for various values of the constraint release parameter z (z = 0 corresponds to the prediction of Doi’s theory for reptation and path-length fluctuations alone6I) (a)z = 0,(b)z = 1 ; (c)z = 3; (d)z = 6. W. W. GRAESSLEY 20 I 1 I I 24 20 16 < h I g 0 s" 12 8 4 0 0.0 0.2 0.4 0.6 0.8 1 .o @B Fig.11. Recoverable compliance plotted as a function of volume fraction of the long chains in a mixture of short and long chains of polybutadiene. See caption to fig. 10 for explanations. APPENDIX REPTATION AND CONSTRAINT RELEASE FOR BINARY MIXTURES OF CHAINS A method has been proposed recently for including the effect of constraint release in stress relaxation for monodisperse reptating chains.54 The constraints are pictured in a highly simplified way -as a lattice of lines with z effective constraints for each step of the primitive path (see fig. 9). The Orwoll-Stockmayer bond-flip models3 (bond length equal to the path step length a)is used to represent the motion of the path a flip occurring each time the first of the zconstraints is removed by reptation of the chain providing it.The average constraint lifetime is given by eqn (41) and the mean waiting time the time for release of the first of z constraints is z = jam [F(t)Jz dt (A 1) where F(r) is the fraction of surviving steps for chains moving in a rigid lattice eqn (27) for the case of linear chains moving by pure reptation. The expression for the stress relaxation modulus is G(t) = c",F(r) R(N,t/t,) (A 2) where 1N WN,t/7,> = -Z exp [ -t/27,] (A 3) Nk-1 VISCOELASTIC PROPERTIES and &(N) =4 (sin-2(E1J2 from Orwoll and Stockmayer. That calculation can be extended to mixtures of chains of two lengths NA and N path steps and and 4 volume fractions re~pectively.~~ The mean waiting time for this situation is where FA(?)and FB(t)are the corresponding fractions of surviving steps for each component.The stress relaxation modulus is then which reduces to the Doi-Edwards binary-mixture form for z = 0 (permanent constraints18) Values of %/(%)A and q/(J&can now be calculated as functions of composition for various values of z using the reptation and fluctuations expression for FA(?)and F,(t) given by Doi.61 The value z = 3 provides a fairly good fit for both qo and JZ (fig. 10 and 1 l) and that same value seems to work rather well for a variety of NA/NB = MA/M combination^.^^ It also yields J,"eN= 2.15 for long-chain monodisperse systems which agrees rather well with the experimental observation [eqn (20)].J. D. Ferry Viscoelastic Properties of Polymers (Wiley New York 3rd edn 1980). G. Harrison The Dynamic Properties of Supercooled Liquids (Academic Press London 1976). W. W. Graessley Adv. Polym. Sci. 1974 16 1. R. B. Bird 0.Hassager R. C. Armstrong and C. F. Curtiss Dynamics of Polymeric Liquids (Wiley New York 1967) vol. 2. G. C. Berry B. L. Hager and C-P. Wong Macromolecules 1977 10 361. R. B. Bird R. C. Armstrong and 0.Hassager Dynamics of Polymeric Liquids (Wiley New York 1977) vol. 1. ' A. S. Lodge Body Tensor Fielis in Continuum Mechanics (Academic Press New York 1974). P. J. Flory Principles of Polymer Chemistry (Cornell University Press Ithaca 1953). P. G. de Gennes Scaling Concepts in Polymer Physics (Cornell University Press Ithaca 1979).lo W. W. Graessley Polymer (London) 1980 21 258. l1 G. C. Berry H. Nakayasu and T. G. Fox J. Polyrn. Sci. Phys. Ed. 1979 17 1825; 1982,20,911. l2 J. S. Higgins in Treatise on Materials Science and Technology (Academic Press New York 1979) vol. 15 pp. 381-422; R. W. Richard A. Maconnachie and G.Allen Polymer 1978 19 266. l3 M. Muthukumar and S. F. Edwards Polymer 1982 23 345. l4 P. J. Flory Statistical Mechanics of Chain Molecules (Interscience New York 1969). l5 P. E. Rouse J. Chem. Phys. 1953 21 1272. l6 P. G. de Gennes J. Chem. Phys. 1971 55 572. l' M. Doi and S. F. Edwards J. Chem. SOC. Faraday Trans. 2 1978 74 1789. lH M. Doi and S. F. Edwards J. Chem. Soc. Faraday Trans. 2 1978 74 1802. l9 M. Doi and S. F. Edwards J.Chem. SOC. Faraday Trans. 2 1978 74 1818. *O M. Doi and S. F. Edwards J. Chem. Soc. Faraday Trans. 2 1979 75 38. 21 V. R. Raju E. V. Menezes G. Marin W. W. Graessley and L. J. Fetters Macromolecules 1981 14 1668. 22 W. W. Graessley and S. F. Edwards Polymer 1981 22 1329. 23 G. C. Berry and T. G. Fox Adtl. Polym. Sci. 1968 5 261. 24 J. Klein Nature (London) 1978 271 243; also Philos. Mag. Sect. A 1981 43 771. 25 L. Leger H. Hervet and F. Rondelez Macromolecules 1981 14 1732. 26 G. Fleischer Polym. Bull. 1983 9 152. 27 J. Meissner J. Appl. Polym. Sci. 1972 16 2877. 2H K. Walters Rheometry (Chapman and Hall London 1975). 29 T. Masuda W. W. Graessley J. Roovers and N. Hadjichristidis Macromolecules 1976 9 127. 3o Y. Einaga K. Osaki and M. Kurata Polym.J. 1971 2 550. 31 K. Osaki K. Nishizawa and M. Kurata Macromolecules 1982 15 1068. 32 K. Osaki and M. Kurata Macromolecules 1980 13. 671. W. W. GRAESSLEY 33 C. M. Vrentas and W. W. Graessley J. Rheology 1982 26 359. 34 H. Janeschitz-Kriegl Polymer Melt Rheology and Flow Birefringence (Springer-Verlag Berlin 1983). 35 H. Miinstedt J. Rheology 1980 24 847. 36 A. B. Metzner and A. P. Metzner Rheol. Acta 1970 9 1974; K. Gardner E. R. Pike M. J. Miles A. Keller and K. Tamaka Polymer 1982 35 1435. 37 M. H. Wagner T. Raible and J. Meissner Rheol. Acta 1979 18,427; H. M. Laun and H. Munstedt Rheol. Acta 1978 17 415. 3H M. Keentok and R. I. Tanner J. Rheology 1982 26 301. H. W. Gao S. Ramachandran and E. B. Christiansen J. Rheology 1981 25 213.4o A. S. Lodge Rheol. Actu 1968 7 379. L. J. Zapas and J. C. Phillips J. Res. Nut1 Bur. Stand. Sect. A 1971 75 33; B. Bernstein E. A. Kearsley and L. J. Zapas Trans. SOC. Rheol. 1963,7 391 ;A. Kaye CoA Note 134 (College of Aeronautics Cranfield 1962). M. H. Wagner Rheol. Acta 1979 18 33. 43 E. M. Menezes and W. W. Graessley J. Polym. Sci. Polym. Phys. Ed. 1982 20 1817. M. H. Wagner and S. E. Stephenson Rheol. Acta 1979 18 463. 45 S. F. Edwards Proc. Phys. SOC. 1967 92 9. 46 M. Doi Chem. Phys. Lett. 1974 26 269. .I7 C. F. Curtiss and R. B. Bird J. Chem. Phys. 1981 74 2016. JH C. F. Curtiss and R. B. Bird J. Chem. Phys. 1981 74 2026. J9 K. E. Evans and S. F. Edwards J. Chem. SOC. Faraday Trans. 2 1981 77 1891; 1913; 1929. so A. Baumgartner and K.Binder J. Chem. Phys. 1981 75 2994. j1 A. Baumgartner 'Dynamics of Macromolecules' Workshop Santa Barbara December 1982 J. Polym. Sci. Part C in press. j2 K. Osaki S. Kimura and M. Kurata J. Polym. Sci.,Polym. Phys. Ed. 1981 19 517. See also ref. (31). 63 W. W. Graessley J. Polym. Sci. Polym. Phys. Ed. 1980 18 27. 54 W. W. Graessley Adv. Polym. Sci. 1982 47 67. j5 S. F. Edwards and J. Grant J. Phys. A 1973 6 1169. 56 J. Klein Macromolecules 1978 11 852. j7 H.-C. Kan J. D. Ferry and L. J. Fetters Macromolecules 1980 13 1571. j* P. G. de Gennes J. Phys. (Paris) 1975 36 1 199. jg M. Doi and N. Y.Kuzuu J. Polym. Sci. Polym. Lett. Ed. 1980 18 775. 60 W. W. Graessley Ace. Chem. Res. 1977 10 332. M. Doi J. Polym. Sci. Polym. Phys. Ed. 1983 21 667.62 M. Daoud J. P. Cotton B. Farnoux G. Jannink G. Sarma H. Benoit C. Duplessix C. Picot and P. G. de Gennes Macromolecules 1975 8 804; P. G. de Gennes Macromolecules 1976,9 587. 63 R. A. Orwoll and W. Stockmayer Adu. Chem. Phys. 1969 15 305. M. J. Struglinski Doctoral Thesis (Northwestern University 1984).

ISSN:0301-5696    

DOI:10.1039/FS9831800007

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

Faraday Symp. Chem. SOC.,1983 18 29-35 Concentration Dependence of the Viscoelastic Properties of Polymer Solutions BY KARLF. FREED* The James Franck Institute and the Department of Chemistry Chicago Illinois 60637 U.S.A. AND ANGELO PERICO Centro di Studi Chimico-Fisici di Macromolecole Sintetiche e Naturali Consiglio Nazionale delle Ricerche Istituto di Chimica Industriale Universita di Genova 16132 Genova Italia Received 3rd August 1983 A heuristic summary is provided of the multiple-scattering representation of the concentration dependence of the viscoelastic properties of polymer solutions in the form of both a concentration cluster expansion and an effective medium theory. The essential role of and complications introduced by the dynamics of the bead-bead correlation function are described and are studied by a full treatment of the leading concentration dependence of the friction coefficients the polymer relaxation rates and the polymer normal-mode autocorrelation function.Specific behaviour is illustrated by exact solutions for two and three bead models while numerical solutions are given for chains of up to twenty beads. The classic theories of Kirkwood and Risemanl and of Rouse and Zimm2 describe the viscoelastic properties of polymer molecules only in the limit of infinite dilution low frequencies and vanishing shear rate. This rather small range of validity of the theories is nevertheless very important3 since it enables the characterization of the molecular properties of polymers at infinite dilution.Once determined these properties provide input parameters in principle for describing the concentration-dependent viscoelastic properties of polymer solutions. By analogy with other polymer properties it is to be expected that a theory of the concentration dependence of the viscoelastic properties of polymer solutions is most easily developed by beginning at the extremes of low and high concentrations. For low concentrations the theory need only consider the dynamics of one two three etc. typical fairly isolated polymers and a molecular-type approach is feasible albeit mathematically complicated4 because of the long-range nature of hydrodynamic interactions. At higher concentrations well within the semidilute region the polymers penetrate each other such that one chain is near many others so an average or mean-field approach is applicable.This leads to an effective medium theory5 in which a single chain undergoes dynamics in the medium provided by the solution of all the remaining chains. This effective medium-type theory then also provides a useful interpolation between the low- and high-concentration domains. At higher concentrations and high molecular weights the chains are known to become macroscopically entangled leading to dramatic changes in their viscoelastic properties. Edwards’ has shown however that these entanglement constraints are exactly preserved by the hydrodynamic eq~ations~-~ provided these equations are 29 VISCOELASTIC PROPERTIES OF POLYMER SOLUTIONS solved exactly.Since this is generally not possible it has been most convenient to describe the entangled regime through the use of models so we confine our attention to the non-entangled regime where individual polymers may suffer temporary entanglements without macroscopic manifestations such as the 3.4 power-law behaviour of the viscosity on the molecular weight M. The effective medium theory provides a general qualitative de~cription~-~ of the concentration dependence of the steady-state viscoelastic properties of polymer solutions explaining the transition from the individual polymer-like behaviour at low concentration c where the relative specific viscosity is a Zimm-like function of CM; to the higher-concentration domain where it is a Rouse-like function of cM.The theory has also been applied to the leading concentration dependence of the polymer relaxation times z with predictions of the form8 z = zZ[l +c[q]aa-"+O(c2)] where zz is the infinite-dilutior relaxation time for mode a,[q]is the intrinsic viscosity a is a specified constant8 and K depends8 on solvent quality through the effective exponent v in the M dependence of the radius of gyration (S2)cc M2v.Experiments by Lodge and Schragg show that eqn (1) provides an excellent description of the concentration shift of the relaxation spectrum throughout much of the dilute region. Deviations from the predicted a-K dependence however are observed9 at higher frequencies and remain to be explained quantitatively. The theory is sufficiently good that Lodge and Schrag suggest that it may be used to provide the extrapolation to infinite dilution obviating some laborious experiments.The treatment of the effective medium theory and of the leading concentration dependence of the relaxation spectrum has been pursued through the introduction of severe mathematical approximations to simplify the terribly complicated expressions of the 5+ The most severe approximation centres on the dynamical bead-bead correlation function gijm = ([Mt)-Rj(0)l2) (2) where R,(t) denotes the position of the ith segment at time t. Thisg,(t) can be obtained for a single chain at infinite dilution from the Rouse-Zimm theory,1° but only in the form of a slowly convergent series expansion in terms of the polymer normal modes.It is completely unknown at higher concentrations yet it is an essential ingredient required by the full hydrodynamic theory. The earlier theory introduces the 'static' approximation of replacing gij(t)by its t = 0 equilibrium average ([R,-Rjl2)which enormously simplifies the complicated mathematics to render it fairly tractable. This static approximation is however now understood to be quantitatively poor despite its being qualitatively correct. For instance the static approximation to the intrinsic viscosity is in error by a factor of two.8v11 Hence we have undertaken the difficult problem of retaining the full gii(t) to produce a quantitative theory to search for new effects in the concentration-dependent dynamics and to understand the higher- frequency corrections to eqn (1).In the next section we briefly provide a physical outline of the basic theoretical concepts while the following section describes some of our recent results for the leading concentration dependence of the viscoelastic properties when the full gij(t)is retained. THEORETICAL BACKGROUND We consider the polymers in a continuum fluid governed by the linearized Navier-Stokes equation. For low-frequency motions and a pure solvent the fluid may K. F. FREED AND A. PERICO be described in the steady-state limit so its dynamics with the polymers present is given . ~0 V2v(r t)+VP(r,t) = x6[r-R,-( t)]b,i( t) (3) ai where qa is the solvent viscosity v(r,t) is the fluid velocity at the point r at time t, P is the pressure and aai(t)is the force exerted by the ith segment of polymer a at time t.The fluid is taken to be incompressible so that V v = 0. By Newton’s law aai(t) is also the hydrodynamic force acting on ai,so a force balance for the polymer gives the inertialess-limit equation x~Aij R,j(t) = -~,i(t) (3) i with the no-slip hydrodynamic boundary condition r? v[R,,(t) t] = at R,,(t) (5) where for simplicity the Rouse chain model is employed with K the force constant and Aij the familiar Rouse matri~.~ The theory has been applied with non-uniformly scaled force constants to describe excluded-volume effectsa or with a stochastic linearization justification12 thereof but recently an exact formal incorporation of excluded-volume effects has been deve10ped.l~ For simplicity we restrict attention here to the Rouse model with hydrodynamic interactions.The formal mathematical solution of eqn (3)-(5) is quite in~olved,~ but the final form can readily be motivated heuristically as follows Consider the case of a single polymer in solution and let the imposed velocity field be vo(r).Note that vo(r)must contain the random velocity fluctuations always present in the fluid. The hydrodynamic boundary conditions eqn (9,couple these fluctuations to the polymer motion and drive the Brownian motion of the chain.4 The full velocity field of the fluid must then be a superposition of the unperturbed velocity vo(r)along with a contribution due to the perturbation of the fluid flow by the full polymer chain. This is written symbolically as v(rl t) = vo(r)-JJ dr’ dr”J dt’ G(r -r’) T(r’?r”;t t’ I (Ri(t))) vo(r”) with G(R)the familiar Oseen tensor3 and T the complicated operator depending on the full chain dynamics describing how the incident velocity field v,(r) is converted into the full force -T v that produces the perturbed velocity field -G T v in symbolic notation.Given the existence of T representing the response of a dynamical chain to an arbitrary incident flow field the general concentrated case readily follows by considering the tortuous meanderings of a typical fluid element through the polymer solution. This fluid element may be unperturbed providing a contribution of vo to v. It may also have its velocity field perturbed by a chain a giving the contribution -C G T v where T is the operator appropriate to chain a.The fluid element may have its flow sequentially perturbed by pairs of chains a and /3 thereby producing the perturbed velocity field C Cg+,G T G Tg v,. Inclusion of all possible processes produces the multiple-scattering expansion derived by more lengthy procedures14 to give v(r,t) = vo(r)-E G T V,+Z 2 G T G Tp V a a B#a -Z C Z G*T,*G.Tp*G*T,*v,+ ... (7) a BZay zs where the lengthy derivation provides the explicit form for the operator T,. We should VISCOELASTIC PROPERTIES OF POLYMER SOLUTIONS note that an analogous multiple-scattering expansion exists for the description of the concentration dependence of the hydrodynamics of suspensions and applications have been given to suspensions of spheres.15 The instantaneous velocity field is a horrendously complicated object but fortunately we only desire its average over all conformations of the polymer4 at some initial time and (v(r,t)) = u(r,t) is much simpler to handle.Transport properties like polymer viscosity can be obtained by noting that the average of the friction force density on the right-hand side of eqn (3) can be written in the linear regime as proportional to u with the proportionality factor providing the polymer contribution to the solution vis~osity.~ More generally this enables us to extract general correlation function expressions for the average polymer contributions to the stress tensor16 even in the non-linear domain.The multiple-scattering expansion eqn (7) provides an approach to developing a concentration expansion of viscoelastic properties which is most useful at lower concentrations where the individual chains are fairly isolated with respect to each other such that hydrodynamic disturbances propagate through the pure fluid between encounters with the chains. At higher concentrations the chains overlap and this simple picture becomes inadequate. Consider however the effective hydrodynamic interaction between a pair of separated chains a and p. Fluid elements flowing between these chains encounter a number of other chains so the hydrodynamic propagation from a to /?is best described through some as yet unknown screened concentration- dependent Oseen tensor e describing the average effects of these intervening chains.Since e describes the average perturbation of the fluid flow by the polymers there is an operator fadescribing the perturbation of the fluid flow due to chain a relative to this average. Following the physical arguments given above we are then led to the same multiple-scattering expansion with-e and f everywhere replacing G and T. The condition that f describe the average scattering implies that (Ta) =O (8) which produces rather complicated self-consistent non-linear integral equa-tion~~-~~ 139 l4for the screened concentration-dependent Oseen tensor e. Here we restrict our attention to the simplest leading concentration dependence emerging from terms in eqn (7) containing only a pair of polymer chains.This enables us to study the higher-frequency corrections to eqn (1) resulting from including the full form of eqn (2) and to develop the necessary methods for attacking the effective medium equations. In the next section we summarize some of our recent resultsll for this leading concentration dependence. LEADING CONCENTRATION DEPENDENCE OF VISCOELASTIC PROPERTIES The essential complication enters into the theory through the emergence of gij(t) of eqn (2) in the preaveraging type approximation to (Ta). gii(t)is known for a single Rouse-Zimm chain at infinite dilution within the preaveraging approximation. It is expressed in terms of the transformation Qia between segment positions R,(t) and normal coordinates c,(t) the eigenvalues of A of HA with H the Zimm-matrix and the eigenvalues pa of A aslo n-1 gu(t) = Z2 I i-jl -2Z2 Z pi1Q, Qia [exp( -a& t)-11 (9) a-1 with a = C-l tc,C-I the bead friction coefficient and I the Kuhn step length.The full K. F. FREED AND A. PERICO 33 gc,(t)function is not presently known even in a preaveraging approximation at higher concentrations. The summation over ain eqn (9) is oscillatory and slowly convergent and the lowest few a must be treated by explicit summation because of the pi1 factor the sum cannot accurately be replaced by an integration. (Sums over bead indices such as i and j etc. can often be converted to integrations producing the continuous-chain limit used in earlier work~.~q The discrete-bead notation is utilized14 because of its greater familiarity despite the fact that long-wavelength viscoelastic measurements cannot distinguish between the discrete-bead or continuous-chain models.) The leading multiple-scattering contribution to the order cportion of both the self and cooperative friction coefficient can be shown to be identical to each 0ther.l' Higher corrections exist for the cooperative friction coefficient but they are smaller by 8factor of the inverse cube of the interpolymer separation so they are ignored.Using the definitions of the friction coefficients f withf the c + 0 limit ofJ the exact representation of kfin terms of normal coordinates is given by" where N is Avogadro's number M is the molecular weight n is the number of effective segments and the superscript T designates a matrix transpose.Eqn (11) has been evaluated exactly" for n < 20 and an extrapolation to n-l+ 0 yields kf = 0.62 in reasonable agreement with the theta-point data summarized by Mulderije.ls Note that the evaluation of eqn (1 1) with exact eigenvalues p and 1 involves a five-fold summation plus one for each g and an integration which must be evaluated numerically. This expression is simplified enormously if we invoke the static approximation gfj = gij(t = 0) = 12 I i-jl previously used39 47 8,l2to analyse the concentration dependence of polymer viscoelastic properties because now the integral may be performed analytically and summations are not required for the gs.However the resultant value kf= 1.27 is rather poor displaying errors typical of this static approximation. The concentration dependence of the relaxation rates of the polymer modes can be extracted from the multiple-scattering representation for the Langevin equation describing the dynamics of chain a in the presence of all the remaining polymers.17 This series may be represented in the compact physically transparent ford7 -a Ra,(t) +J dt' I [Ra,(t),~aj(t/)] KAj Rg,(t') = vJa)[Ra,(t) t] (13) at jm where e(.) is the screened Oseen tensor for the whole polymer solution (written as a series in G and Tp for all /?# a17)in the absence of chain a and v&@ is the fluid velocity field in the absence of a. Replacing e@) by its average over all chains gives (G@)) as the quantity whose eigenvalues are the concentration-dependent relaxatioh rates.In terms of the polymer normal modes this equation takes the form" to order c. Note the presence of the memory kernel d,(t-t') which arises because 2 FAR VISCOELASTIC PROPERTIES OF POLYMER SOLUTIONS hydrodynamic disturbances are produced by polymers that are undergoing dynamical motion with a whole relaxation spectrum. Hence eqn (14) implies that generally the concentration-dependent relaxation rates Z;' = 01 -01,[q] cd,(cu) + O(C2) (15) must be frequency-dependent quantities. Consequently the normal-mode autocorre- lation function Z,(O = (ta(t) ' t,(O)>/( ItU(0) 12> (16) need not have a simple exponential decay. The quantities 6,(cu) are more involved to evaluate than is eqn (1 l) and it is useful to consider simple models to display the basic physics.The two-bead model n = 2 may be solved analyticallyll for d,(t- t') to show that d,(t-t') has all harmonics (but not the fundamental) of the relaxation rate oAl. Hence to order c Z,(t) for n = 2 has a simple exponential decay. In fact it may be shown" that Z,(t) has a single exponential decay for any n. However Z2(t)displays a multiple-exponential decay and this is demonstrated by our exact numerical solutionll for n = 3. This complicated decay arises from the non-linear coupling between the modes of a chain which are induced through indirect hydrodynamic interactions mediated by other polymers in the solution that have a similar relaxation spectrum to the chain under consideration.0.3 \ 0.2 n s I \ Ln.5 f.0 0.1 15 20 Fig. 1. The dimensionless steady-state limit correction to the ath-mode relaxation rate 6,(0) plotted against a/n with n the number of beads on the chain. The dashed lines give the extrapolation of 6,(0)to n +a. The expression for S,(w) simplifies in the zero-frequency limit o-+0 where d,(O) has been evaluated again for n d 20 using exact eigenvalues for partial and non-draining conditions.ll The non-draining limit of S,(O) is presented in fig. 1 for n = 5 8 10 15 and 20 as a function of a/n. Note that for small a/n S,(O) varies as a-g in accord with the theta-point behaviour of eqn (1) as obtained with the static-limit approximation eqn (12).However at larger a/n the dimensionless relaxation rate shift d,(O) tends to a constant value. This overall behaviour is in general accord with recent flow birefringence measurements by Lodge and S~hrag.~ The dashed line in fig. 1 provides K. F. FREED AND A. PERICO the extrapolation of S,(O) to n -+a, and it is clear that larger n values are necessary for a quantitative extrapolation for comparison with the experimental data. Since (G(@) describes the effective concentration-dependent hydrodynamic inter- actions within a single polymer it provides information concerning 'hydrodynamic screening' on length scales comparable to the polymer's size. Long-wavelength screening Darcy's-law behaviour,lg exists only when there are impediments14 to the flow but the above screening is a net reduction in intrachain hydrodynamic interactions due to the presence of other interspersed polymer chains.The forms eqn (10) and (15) describe the leading effects of this screening. DISCUSSION The quantitative treatment of the full frequency dependence of the concentration dependence of the viscoelastic properties of polymer solutions is complicated by the need to include the concentration-dependent dynamical bead-bead correlation function eqn (2). This correlation function is only available for c -+ 0 yet even there it is very difficult to handle in numerical computations. The simple static approximation yields poor numerical results while other approximations tried to date are even worse. This c -+ 0 limit correlation function however suffices for an analysis of the leading concentration dependence of the friction coefficients and relaxation times.The calculated friction coefficient is in good agreement with experiment while the relaxation times display a diminution of shift of the higher-frequency modes in general accord with experiments. This leading concentration dependence is useful to provide necessary input data for the treatment of higher concentration solutions with the effective medium theory. This research is supported in part by N.S.F. grants DMR78-26630 (polymers programme) and INT82-18620 and by a CNR-NSF Italy-U.S. joint study. J. G. Kirkwood and J. Riseman J. Chem. Phys. 1948 16 565. B. H. Zimm J. Chem. Phys. 1956,24 269. For a review see H.Yamakawa Modern Theory of Polymer Solutions (Harper & Row New York 1971). K. F. Freed in Progress in Liquid Physics ed. by C. A. Croxton (Wiley London 1978) p. 343. K. F. Freed and S. F. Edwards J. Chem. Phys. 1974 61 3626. K. F. Freed and H. Meitu J. Chem. Phys. 1978 68 4604. 'I S. F. Edwards Proc. R. SOC.London Ser. A 1982 385 267. M. Muthukumar and K. F. Freed Macromolecules 1978 11 843. T. P. Lodge and J. R. Schrag Macromolecules in press. lo A. Pecora P. Piaggio and C. Cunberti J. Chem. Phys. 1975 62 491 1 ; B. J. Berne and R. Perico Dynamic Light Scattering (Wiley New York 1976). l1 A. Perico and K. F. Freed to be published. l2 M. Muthukumar and S. F. Edwards Polymer 1982 23 345. l3 K. F. Freed Macromolecules in press. I4 K. F. Freed and A. Perico Macromolecules 1981 14 1290. K. F. Freed and M. Muthukumar J. Chem. Phys. 1982,76,6186;M. Muthukumar and K. F. Freed J. Chem. Phys. 1982 76 6195; 1983,78,497 511. M. Jhon and K. F. Freed unpublished work. A. Perico and K. F. Freed J. Chem. Phys. 1983,78,2059. l8 J. J. H. Mulderije Macromolecules 1980 13 1207. I9 K. F. Freed and M. Muthukumar J. Chem. Phys. 1978 68 2088. 2-2

ISSN:0301-5696    

DOI:10.1039/FS9831800029

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

Faraday Symp. Chem. SOC.,1983 18 37-47 Dynamics of Entangled Flexible Polymers Monte Carlo Simulations and their Interpretation BY ARTUR BAUMGARTNER AND KURT BINDER* KURTKREMER Institut fur Festkorperforschung der Kernforschungsanlage Julich D-5170 Julich Postfach 1913 West Germany Received 1st August 1983 Monte Carlo simulations have been reviewed for several models of dense polymer systems (i) a system of very short (N = 16 links) freely joined chains with Lennard-Jones interactions in the continuum (ii) self-avoiding walks on a diamond lattice with up to N = 200 links and (iii) pearl-necklace chains with up to N = 98 hard spheres. The cases of both a mobile and a frozen-in environment have been considered. The dynamics of displacements and the structure factors have been studied and interpreted in terms of the Rouse model and the reptation model paying particular attention to the crossover between these models.Only for the frozen- environment case is reptation fully verified while for models (i) and (ii) simple Rouse behaviour is found and for model (iii) relaxation with a diffusion constant D cc N-2-o*o.2and disengage- ment time zdcc 1v3.4*0.4is found but the monomer displacements are in disagreement with reptation laws. A brief comparison with pertinent experiments has been made and the crossover between the Rouse model and the Zimm model is considered briefly. 1. INTRODUCTION CURRENT MODELS FOR CHAIN DYNAMICS Static properties of long flexible polymers in dilute and semi-dilute solutions are understood by renormalization group methods,l and in concentrated solutions or melts they follow trivial random-walk statistics (radius R x IN; for chains of N links of length 1).2 Dynamic properties in contrast are less well understood and current theoretical discussions rest on very simplified model~.~-l~ In the Rouse the forces of the environment on a chain are represented by a heat bath inducing local conformational changes.The characteristic time zq over which fluctuations characterized by a wavevector q decay is then3? 7* where W is the rate at which links change their orientation and v x 0.59 for chains which are swollen due to excluded-volume interactions.' Eqn (1 a) describes the ~ relaxation for times t shorter than the chain-relaxation time T~ oc/ W-1N2v+1 while /~ for t > T~ diffusive behaviour eqn (1 b) dominates the motion for qR > 1.20 Similarly the monomer displacements (r%(t))behave as3 37 DYNAMICS OF ENTANGLED POLYMERS In the Zimm the hydrodynamic backflow forces are also included.This is important for dilute solutions where W cc qol,qo being the solvent viscosity and Instead of eqn (2a)for the monomer displacements we now have (rt(t)) x P(Wt); for t < z,/~. These displacements show up in the 'incoherent scattering factor' &,(q t) cc (1/N)Z (exp {h [(ri(t)-ri(O)ll<% ~XP [-+q2(ra(t))l* For both the Rouse and the Zimm model the coherent structure factor Sco,(q 0 cc N-2c (exp {iq [ri(t)-q(0)lD (54 ij takes the simple form &oh(q t) cc exp [-const (qZ)2(Wt).] Wt 9 1 (5b) where the exponent n = 1 in the diffusive regime while for qR > 1 and t < z,,~ one has n = (Rouse model) or n = (Zimm model).The 'reptation model' of de Gennes97 l23 l5 and EdwardslO? l3considers dense systems where the chains are entangled. It is thought that the effect of these entanglements is to create for each chain a tube of diameter dT,such that the chains are restricted to snake-like motions along the tube axis. On the local scale inside the tube i.e. for (ra(t)) 5 d& the chain performs Rouse-like motions [eqn (2)].For larger times one has insteadg idT( Wt)' W-'(dT/l)* < t <ZRouse OC W-lP (64 (rt(t)) % /dT(Wt/N)i t~~~~~ < t < zd cc (Z/~T)~N~ (6 b) [Dt t > zd D cc Wd+NP2. (64 While this behaviour also shows up in Si,,(q t) [eqn (4)] &h(q t) is more complicated as there is no single time zq dominating the scattering under wavevector q.15For R-l < q < G1it is predicted that Scoh (q,t)decays non-exponentially towards a constant value15 Scoh (4,t)/scoh (q' O) -q2 d+/36 < zd (7a) while later the relaxation should be indep endent of q Scoh (4,t)/scoh (4,O) exp (-t/zd ) > zd.(7b) All these models oversimplify the actual dynamics which are those of a dense fluid in the local environment. It is not completely clear under which conditions these models are valid. In addition one needs a better microscopic understanding of parameters such as W and dT.In fact while reptation seems to be a nice framework for bulk viscoelastic properties,13 the precise microscopic meaning of the entanglements is not so clear.We would also like to study the crossover between the various models which occurs when N or the concentration c of a polymer solution is varied.11y12q21 Even the derivation of some of the results is rather qualitative and doubts can be raised about their validity; e.g. it has been suggested20 that instead of eqn (7b) one should have s,,h(q t)/s,,h (q,O) cc exp (-Dq2 t) for t 9 zd as in the case qR < 1. Since Monte Carlo simulations have been useful for understanding static chain properties and the asymptotic laws valid for N +co could be verified for short chains,22 one expects simulations to be useful for chain dynamics also. Such A. BAUMGARTNER K.KREMER AND K. BINDER 39 simulations have indeed been performed ~ecently~~-~~ and have a bearing on the above models as well as on neutron-scattering experiment^.^^^ 28 In the following we have summarized these studies and pointed out some questions which still need answers. 2. SIMULATIONS OF SHORT CHAINS IN THE CONTINUUM A simple model consists of rigid links of length 1 freely jointed together at arbitrary angles. Interactions can be introduced for example by postulating a Lennard-Jones potential U(rij)= 4&[(a/rij)l2-(c~/r~~)~] between any pairs of beads at points ri and ri [rij= ri-ri3. While E determines the temperature scale a= 0.41 was chosen23 since then the static properties follow asymptotic laws down to very small values of N.29 Dynamics are introduced by randomly choosing a bead of chain i and moving it through a randomly chosen angle 4 on a circle while keeping all other bead positions fixed.This trial move is accepted only if the transition probability W(ri-r;)exceeds a random number q-with 0 < q-< 1 otherwise it is rejected and another move is tried. A transition probability which both satisfies detailed balance with the equilibrium probability distribution Po cc exp (-Z/kT) where X = ZU(rij)is the Hamiltonian of the system and simulates entanglement restrictions is [SZ = Wj # i ri) -Z(rj # i 4)l exp(SX/kT) if dX' < 0 no intersection if &%>O (0 if the move would require link intersection. (8 c> In our simulation of melts eqn (8 c) reduces the rate of accepted moves to ca.one-fifth of what it would be with eqn (8a)or (8b)alone. In the related work of Bishop et ~1.~~ longer chains [but only eqn (8a)and (8b)lare used; otherwise their results are similar to those of ref. (23). As a first step this model was studied in the dilute limit of isolated single chains with N = 16 links. It was found that eqn (l) (2) (4) and (5) account for the data. Only for very large q (41 1)does the gaussian approximation for ri(t)-r,(O)involved in eqn (4) become invalid as expected. As a second step 10 such chains were put in a box of size L = 41 and periodic boundary conditions were applied to simulate a macroscopic system. Again the results are in agreement with the Rouse model cf. fig. 1. Displacements (rf(t)) and SinC(q, t) show distinct the Rouse behaviour [eqn (2a)l.No intermediate reptation regime [eqn (6a)l is seen. Also Sc,,(q t)is in quantitative accord with Rouse behaviour. At longer times one sees crossover to diffusion of the chains as a whole. (Note the difference between Scohin the laboratory system and in the centre-of-gravity system in fig. 1; the crossover occurs at t z zIlRindependent of q in accord with theory.20) As a third step after equilibrating this system the configurations of all but one of the chains were frozen-in and only one chain allowed to move. Thus we simulated the situation of one chain moving in the presence of randomly fixed obstacle^.^ Now the predicted behaviour (rg(t)) cc ti [eqn (6a)lwas readily seen over at least three decades of time and in the same time interval the centre of gravity rcgfollowed the law9(r&(t)) cc ti.These results prove that in spite of the shortness of the chains there must be many entanglements along each chain. Furthermore the geometry of these entanglements at the time when we stop the movements of all other chains does not change substantially. Hence the fact that in the frozen-in case we clearly see reptation and in the mobile case we do not show that dT should not be linked exclusively to 40 DYNAMICS OF ENTANGLED POLYMERS 30 2 10 1.3 1 0.7 - n k W 01 a I& .5 0, Y c ‘i 1 lo-’ 0.7 1 0.L -2 v c \ 216’ n 4 G W c0 6 5 0 0 1 1o3 lo4 * 1o5 lo6 Fig.1. Log-log plot of S,,,(q t) (upper part) and &,h(q f)/&& 0) (lower part) against time (in units of attempted moves per bead) for various q. Solid curves represent the Rouse model prediction where the rate W[eqn (l)] was adjusted (W x 0.025 in our time units). Open circles represent the structure factor in a coordinate system in which the centre of gravity of the considered chain would be at rest. c = 2.5 kT/&= 3. From ref. (23). geometrical constraints of chain configurations but should also take into account the local chain mobility. Of course one may quote many reasons why melt simulations fail to see reptation (i) chain lengths N too short (ii) chain density too low (iii) temperature too high or (iv) reptation model inadequate.In order to check for (ii) and (iii) simulations were also performed at a four times higher density and at kT/c =0.4 respectively. In both cases the chains were found to be frozen into a sort of glass-like state with displacements (rf(t)) < 12 over the timescales of interest. 3. SIMULATIONS OF LONG LATTICE CHAINS Self-avoiding walks (SAW) on the diamond lattice with up to N =200 steps were simulated2s at a concentration of c x 0.344. The analysis of static properties revealed that on a small scale (within blobs of size NBx 20) the chains are still ‘swollen’ while on a larger scale the chains behave like ideal random walks. However simple Rouse A. BAUMGARTNER K. KREMER AND K. BINDER t h' 1oo l"l(l r IIIIIIIII loo lo2 104 lo6 t Fig.2. Log-log plot of gl(t) = (1/20) Z& (rf(t)) against time the latter measured in attempted moves per bond for chains with N = 200 at the tetrahedral lattice and a concentration c = 0.344.g2(r)is the same mean-square displacement of inner monomers of the chain measured in the centre-of-gravity systems of each chain while g3(r)= (r&(t)).From ref. (25). 1.0 .op, I 0.o 10.0 Fig. 3. Coherent structure factor s,,h(q t)/&,h(q 0) for a system of chains with 200 links moving on a tetrahedral lattice at a concentration c = 0.344. Data are plotted in semilog form against the variable (1/6)q21 2/ Wr to verify the asymptotic decay proportional to exp (-const q2I22/Wr)predicted by the Rouse model. The parameter W is set equal to unity.k = 0,0.1; 0.3; 0 for poly(dimethy1 V 0.2; + 0.25; 0 0.4; x ,0.5; A,0.6. Experimental data (0) siloxane) melts2' and the simulation data of fig. 1 (+)are also included fitting W = 4.3 for both. From ref. (25). behaviour is again observed even for the longest chains (fig. 2). It is seen that eqn (2a) holds until the mean-square displacement becomes of the order of the square of the gyration radius (RL) of the chain; then diffusion of the chain as a whole takes over. Also S,,,(q t) (fig. 3) confirms the Rouse-like behaviour. The simulation shows that in the regime where one probes the internal motions of the chains the normalized structure factor depends on qand t asq22/tonly. Deviations from this behaviour which lead to quicker decay are seen at long times and are due to the diffusive motion of the chains.By contrast according to the reptation model one would expect a DYNAMICS OF ENTANGLED POLYMERS 0 .A n + 4 .y v t. 0.5 t 0.01lIIrIll111IrlIIIlIlIrlll1 0.o 0.5 1.O 1.5 2.0 2.5 (&k212fh)-I Fig. 4.Coherent structure factor &,h(q t)/&,h(q 0) plotted against (&q212dWt)-l for the case of a single mobile chain in a frozen-in environment. Intercepts are used to estimate the tube diameter dT from eqn (74. From ref. (25). k S(k,t =a) dT N 0.3 0.50f0.05 275 45f 10 A 0.4 0.33f0.05 250 41 f5 + 0.5 0.21k0.02 225 38f5 0 0.6 0.13f0.02 205 34f5 l'"'~'~'~~'"'1 0 50 100 150 Fig. 5.Planar projections of 'snapshots' of chain configurations on the tetrahedral lattice confined to straight tubes of various diameters dT:(a) 32 (b) 24 and (c) 16 (N = 400,1= d3).From ref. (31). A. BAUMGARTNER K. KREMER AND K. BINDER 43 crossover to slower decay and the data should not scale as there is an additional dependence on the parameter 4dTe2' Such behaviour occurs if the environment of the moving chain is frozen-in (fig. 4). The resulting tube diameter dT =10 lattice spacings is of the same order as the length over which the SAW interaction is screened out in rough agreement with the expectation12 that there should be only one characteristic length in the system. Obviously this is not true in melts the screening length is of the order of a few A (monomer distances). Estimates for dTextracted from viscoelastic propertiesl3? 30 are dT =30-80 A.If dT were so large for real systems the experimental data27 included in fig. 3 would satisfy qdT 2 2 and might then be affected only weakly by the tube constraints. Thus it is necessary to study in more detail the crossover from Rouse to reptation behaviour. As a first step simulations of chains in straight tubes of various diameters have begun (fig. 5).31 7 4 3.0 I111111 I I11l11ll I I,11111 1 IIIIIII 1o3 104 105 106 107 Fig. 6. Log-log plot of gl(t) against time for the single mobile chain (N =200) in a frozen-in environment built up by similar chains at c =0.344. g2(t)is the mean-square displacement measured in the centre-of-gravity system. From ref. (25). While there are still uncertainties about coherent scattering from one reptating chain,l592of 239 25 the predictionss [eqn (6a)and (6b)l for the displacements are verified when one considers a chain in a frozen-in environment (fig.6):Rouse behaviour occurs for (ri(t)) 5 d+,while (rt(t)) Kti holds until (rt(t)) reaches (&),where one again finds (rp(t)) K&until at zd diffusive behaviour sets in [eqn (6c)l. Estimates for dT extracted from the various regimes are consistent with each One may again suspect of course that the failure to see reptation in the case where all chains are mobile might be due to too short chains and/or too small c~ncentration.~~ This is not the case for the experimental data in fig. 3 the molecular weight used (M =60000) is safely in the regime where the viscosity behaves as q KMk4 (this regime starts at M =1500027) and hence the chains are long enough to be in the strongly entangled regime.DYNAMICS OF ENTANGLED POLYMERS lo2 -I 1 I ,I,![ I I I I I I,lI I I I I I III 10 7 c w-b o-1 1 I I I1 1111 I I I I Illll I 1 I I1 I11 1o3 1o4 t 1o5 106 Fig. 7. Log-log plot of g(t) against time for a system of 24 chains of 'pearl-necklace' chains with N = 72 hard spheres each. Intersections of the straight lines serve to define various crossover times 7, 7; and 7d. From ref. (26). 10 N 100 I I I I IIIll1 / II 10 n c W bo" 4I'. .. . . .... .. .. .. .. . ..,,. *..* I I 1 10 102 1o3 1o4 1 t Fig. 8. Log-log plot of the mean-square displacement in the centre-of-gravity system plotted against time for several N (a) 72 (b) 32 and (c) 18.The chain-length dependence of the asymptotic value g!") is plotted in the upper left-hand corner. From ref. (26). 4. SIMULATIONS OF LONG CHAINS IN THE CONTINUUM The most extensive simulations to date26 concern ensembles of Np chains each consisting of N hard spheres of diameter h/Z = 0.9 freely joined together put in a box of size L chosen such that the concentration c= NpN/(L/Z)3= 0.7 for N up to 98 (and Npup to 28). In this case the diffusion constant for N 2 40 is indeed consistent with the reptation law [eqn (6c)l D cf N-2 as also found e~perimentally.~~ However the mean-square displacements do not follow the reptation predictions (fig. 7 and 8) A. BAUMGARTNER K.KREMER AND K. BINDER 0.3 ' I I I I I I I* 0.02 0.04 0.06 0.08 0.10 0.12 0.14 41A-I Fig. 9. Exponent n [defined in eqn (5b)]plotted against wavevector q for poly(dimethy1 siloxane) dissolved in deuterated benezene at concentrations c = 0.18 (triangles) and c = 0.45 (circles). From ref. (28). both g(t) and the squared displacement in the centre-of-gravity system g,(t) increase proportionally to F until t x zRouse where g,(t) saturates at gr(co)cc N. Note the contrast between g,(t) in fig. 8 and g2(t)in fig. 6 g,(t) does not show dependence on tf,in contrast with the case of the frozen-in environment. Since the saturation value satisfiesg,( co)cc N the time z where crossover to this value occurs must be identified as the relaxation time rRousecc W"2.However while in fig. 2 diffusion sets in at this time [and therefore D cc 1/N since DrRousex g,(aO) a N] in fig. 7 the displacements increase more slowly after this time and diffusion sets in at a much later time rd which varies with chain length as2' zd cc IV3.4*0*4. If this displacement behaviour in polymer melts persists for N +00 it implies that reptation theorye9l5 cannot apply. At present it cannot yet be excluded that the behaviour seen in fig. 7 and 8 is particular to the regime for crossover from Rouse to reptative behaviour. If so fig. 4 and 6-8 imply that this crossover behaviour in melts is different from the crossover in the case of the frozen-in environment. Standard theoryetl5 does not distinguish between these two cases. However various additional relaxation mechanisms might be important even if a tube model applies to melts the tube may locally expand or or the tube itself may show local Rouse-like diffusive motion together with the chain it contains; the topological constraints would be left invariant1°J3 under both these types of motion.It is not obvious to us that these mechanisms can be completely absorbed in an increase in the effective tube diameter dT. 5. CROSSOVER FROM ZIMM TO ROUSE BEHAVIOUR IN CONCENTRATED SOLUTIONS The hydrodynamic interactions important for dilute solutions4 are not easily incorporated into Monte Carlo simulations. It is believedll? 12,21 that hydrodynamic interactions are screened in semidilute and concentrated solutions and the Rouse model should then be applicable over a wide range of molecular weights.Then the simulations discussed in sections 2-4 might be compared to such systems. Experiments on poly(dimethy1 siloxane) dissolved in deuterated benzene at various concentrations have been carried out with the neutron spin-echo method.28 Analysing the scattering with eqn (5b)shows a crossover from (unscreened) Zimm relaxation (n = f)seen at large q to Rouse behaviour (n = &)at smaller q for low concentrations (fig. 9) as expected. But surprisingly at higher concentrations a second crossover is 46 DYNAMICS OF ENTANGLED POLYMERS detected from Rouse behaviour back to Zimm behaviour upon decreasing q. This phenomenon has been interpreted by incomplete screening of the hydrodynamic interaction.28 The average hydrodynamic forces are then modelled asz8 where ylo is the pure solvent viscosity ~(c)some effective solution viscosity and th(c) the hydrodynamic screening length.While the first term on the right-hand side of eqn (10) is standard,12 the last term is new. It leads to behaviour described by the Zimm model eqn (3a),in a regime R-l <q <[qo/q(c)]t;l(c) but with a reduced rate [W cc q-l(c)] screened Zimm relaxation.28 It is unclear how this hydrodynamic response over intermediate-length scales interferes with entanglement restrictions. 6. CONCLUSIONS Both neutron-scattering experiments and simulations reveal that the simple Rouse model has a wide range of applicability in understanding the time dependences of displacements and scattering intensity even in cases where entanglements affect the centre-of-gravity motion and/or viscous response of the polymer solution or melt.The crossover from Rouse behaviour to entangled behaviour in melts seems to be different from cases with a frozen-in environment; only in the latter case do simulations verify the reptation concepts. The need for more detailed theories for this crossover regime emerges since this regime also seems to be relevant for experiments. P. G. de Gennes Scaling Concepts in Polymer Physics (Cornell University Press Ithaca New York 1979). P. J. Flory Principles of Polymer Chemistry (Cornell University Press Ithaca New York 1967). P. E. Rouse J. Chem. Phys. 1953 21 1272. B. H. Zimm J. Chem. Phys. 1956 24 269.F. Bueche The Physical Properties of Polymers (Interscience New York 1962) and references therein. W. W. Graessley J. Chem. Phys. 1965 43 2696; 1967 47 1942. ’P. G. de Gennes Physics 1967 3 37. E. Dubois-Violette and P. G. de Gennes Physics 1967 3 181. P. G. de Gennes J. Chem. Phys. 1971 55 572. lo S. F. Edwards and J. M. V. Grant J. Phys. A 1973 6 1169 1 186. l1 K. F. Freed and S. F. Edwards J. Chem. Phys. 1974 61 3626. l2 P. G. de Gennes Macromolecules 1976 9 587 594. l3 M. Doi and S. F. Edwards J. Chem. Soc. Faraday Trans. 2 1978 74 1789; 1802; 1818; 1979 75 38. l4 J. D. Ferry Viscoelustic Properties of Polymers (Wiley New York 1980). l5 P. G. de Gennes J. Chem. Phys. 1980 72 4756; J. Phys. (Paris) 1981 42 735. l6 C. F. Curtiss and R. B. Bird J.Chem. Phys. 1981 74 2016 2026. l7 L. Leger and P. G. de Gennes Annu. Ret.. Phys. Chem. 1982 33 49. W. W. Graessley Adu. Polym. Sci. 1982 47 68. l9 M. Doi J. Polym. Sci. Polym. Phys. Ed. 1983 21 667. 2o K. Binder J. Chem. Phys. 1983 in press. 21 M. Muthukumar and S. F. Edwards Polymer 1982 23,345. 22 For a review see A. Baumgartner in Monte Carlo Methods in Statistical Physics II,ed. K. Binder (Springer Berlin 1983 in press). 23 A. Baumgartner and K. Binder J. Chem. Phys. 1981 75 2994. 24 M. Bishop D. Ceperley H. L. Frisch and M. H. Kalos J. Chem. Phys. 1982 76,1557. 25 K. Kremer Macromolecules October 1983 in press. 26 A. Baumgartner Proceedings of the Workshop on Dynamics of Macromolecules Santa Barbara 1982 to be published. 27 D. Richter A.Baumgartner K. Binder B. Ewen and J. B. Hayter Phys. Rev. Lett. 1981 47 109; 1982 48 1695. ** B. Ewen B. Stuhn. K. Binder D. Richter and J. B. Hayter to be published. 29 A. Baumgartner J. Chem. Phys.. 1980 72. 871; 1980 73 2489. A. BAUMGARTNER K. KREMER AND K. BINDER 47 30 W. W. Graessley J. Polym. Sci. Polym. Phys. Ed. 1980 18 27. 31 K. Kremer and K. Binder unpublished results. 32 Evidence for reptation in the simulations of lattice chains at higher concentrations is claimed by J. Deutch Phys. Rev. Lett. 1982,49 926. These simulations however are inconclusive as the SAW condition is strictly obeyed for links of different chains while overlaps of a chain onto itself are allowed and thus an artificial mobility of each chain along itself is created such a model leads to reptation trivially but seems rather unphysical. For a more detailed discussion see K. Kremer to be published. 33 J. Klein Nature (London) 1978 271 143; Philos. Mag.,1981 A43 771.

ISSN:0301-5696    

DOI:10.1039/FS9831800037

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

Faraday Symp. Chem. Soc. 1983 18,49-56 Viscoelasticity of Concentrated Solutions of Stiff Polymers BY MASAODOI Department of Physics Faculty of Science Tokyo Metropolitan University Setagaya Tokyo Japan Received 0th August 1903 A solution of rodlike polymers forms a liquid-crystalline phase above a certain concentration. In this paper we derive the constitutive equation for such a system under a weak velocity gradient from the molecular-kinetic theory. The equation agrees with Leslie’s phenomenological equation in the liquid-crystalline phase. The Leslie coefficients are expressed by molecular parameters such as the concentration molecular weight and the order parameter. 1. INTRODUCTION The viscoelastic properties of solutions of rodlike polymers depend strongly on concentration.At very low concentration the solution is nearly a Newtonian fluid with low viscosity. As the concentration increases the viscosity increases sharply and marked viscoelasticity appears. With a further increase in concentration the solution forms a liquid-crystalline phase1 and shows characteristic rheological properties which are quite distinct from those of isotropic solutions. In this paper we discuss how such behaviour may be understood on the molecular level. In particular we discuss the characteristic distinction between the isotropic phase and the liquid-crystalline phase. From the viewpoint of rheology the basic distinction between the isotropic phase and the liquid-crystalline phase is behaviour under low velocity gradients.For isotropic fluids the stress tensor tr for a steady flow is always given by Newton’s law if the velocity-gradient tensor K = (Vu)? is sufficiently small c = ~(K+K+). (1.1) (In this paper the isotropic part of the stress tensor is omitted for simplicity.) Since eqn (1.1) is the first term in a power-series expansion of c with respect to K it is generally correct as far as the stress tensor is an analytic function of K. From eqn (1.1) one can conclude for example that the elongational viscosity is generally three times larger than the shear viscosity at low elongational rate. On the other hand such a relation does not hold in the liquid-crystalline phase. In this case the relation between cr and K becomes singular at K = 0. This is seen in the phenomenological constitutive equation proposed by Leslie2 for the nematic liquid crystals.In nematics the system has uniaxial symmetry around a certain direction specified by a unit vector n called the director. If n is fixed by some external field the relation between cand K is analytic. However if there is no external field the direction of n itself is determined by the velocity gradient tensor K. As a result the relation between e and K becomes non-analytic. The problem discussed here is how such a ‘singular’ constitutive equation is derived from molecular theory. We shall show that Leslie’s phenomenological equation is naturally derived from the molecular kinetic equation and express the phenomeno- logical parameters appearing in the equation by molecular parameters.49 VISCOELASTIC PROPERTIES OF STIFF POLYMERS 2. BASIC EQUATION The polymer we consider here is a rodlike polymer of length L and diameter b(L + b). First we briefly review the static theory for the formation of the liquid- crystalline phase. Let flu) be the orientational distribution function which is the probability that an arbitrarily chosen polymer is in the direction of the unit vector u. According to on sage^,^ the free energy per unit volume of solution is expressed as a function offlu) as d[f]= ckB T(sduflnf+-2‘idu du’p(u u’)flu)flu’)) (2.1) where c is the number of polymers per unit volume k T is Boltzmann’s constant multiplied by the temperature and p(u,u’) is the excluded volume for a pair of polymers in the directions u and u’ D(u u’) is calculated from the interaction energy u(r,u;r’,~’) between the two polymers in the configuration (r,u)and (r’,~’)(r and r’ denoting the positions of the centres of mass) as Jr D(u,u’) = dr{1 -exp [-u(r,u; 0 u’)/kgT]}.For a rigid rod P(u,u’) = 2bL21ux u’(.~ In general B(u,u’) takes minimum when u is parallel (or antiparallel) to u’. The equilibrium distribution function is determined from the condition Sd/Sf = 0 which leads to fO(u) = exp -hCF(U; [f0l)lkB r) (2.3) where C is a constant and I/SCF(u [fJ) = ck T du’p(u,u’)flu’) (2.4) i is termed the mean field potential. Eqn (2.3)is a non-linear integral equation for fo and its solution generally requires the numerical calculation. However if B(u,u’) is approximated as4 cD(u,u’) = c-; U(U*U’)2 (2.5) where U is a parameter proportional to c eqn (2.3)is easily solved as fo(u)= C exp [ -US(~*U)~] (2.4) where n is an arbitrary unit vector and S is the solution of the equation du [(~=n)~-#] exp[-$US(n*u)2] S= (2.7) Cdu exp [-4 US(u n)2] If U is small eqn (2.7)has only one solution S = 0.This corresponds to the fact that the system is isotropic at low concentrations. On the other hand if U is sufficiently large eqn (2.7)has another solution for positive values of S. This solution represents the liquid-crystalline phase in which polymers are oriented towards n. To discuss the viscoelasticity we must know the time-evolution equation for the distribution functionflu t).In dilute solution such an equation is given by Kirkwood et a1.5as (2.8) at M.DO1 51 a where B=UX-(2.9) au is the operator of rotation 5 is the potential of the external field and D,,is the rotational diffusion constant which is related to the rotational friction constant [ro through Einstein’s relation Dro = k T/Cro* (2.10) The friction constant crois calculated by hydrodynamics as5? (2.11) where qSis the viscosity of the solvent.In concentrated solutions eqn (2.8) is modified reflecting various types of interaction between the polymers. Two interactions are particularly important (i) The excluded-volume interaction. As is shown in the static theory the excluded- volume interaction acts by aligning the polymers in the same direction. This effect is taken into account by adding the mean-field potential &cF(u,~u; t)]) to & in eqn (2.8).7-1 O (ii) The entanglement interaction.The constraint that polymers cannot pass through each other becomes a serious restriction for rotational Brownian motion of polymers in a concentrated solution. As a result the rotational motion is significantly slowed down. According to ref. (1 1) and (12) the effective rotational diffusion constant Dr is estimated as Dr = pD,0(cL3)-2 du’JTu; t)fTu’; t)lu x u’I (2.12) where 8’ is a numerical coefficient. Thus the time evolution equation in concentrated solution is obtained as (2.13) Two comments may be added. (a) In eqn (2.13) the hydrodynamic interaction is neglected. For flexible polymers this interaction is known to be important since the hydrodynamic interaction becomes screened as the concentration l4 For rodlike polymers however this effect is not very important because the effect of the hydrodynamic interaction is already weak in dilute solution.In fact the hydrodynamic interaction gives only a small correction to D,,in eqn (2. l6 (b)Although eqn (2.12) predicts correct molecular weight and concentration dependence,I7 its validity has been questioned18-20 since the experimentally obtained value of p is much larger than the naive expectation which is to say that p is of the order of unity.12 A recent careful estimation of j?’was made by Hayakawa,21 who showed that can be of the order of lo3 if the distribution of disengagement times is correctly taken into account.Therefore the discrepancy in the magnitude of Dr is not very large although the agreement between the theory and experiment is not yet complete. If the distribution function is known the stress tensor may be calculated by the procedure discussed in ref. (22). The stress tensor of polymer solutions is generally written as d = aS+aE+aV (2.14) where dS = qs(K+K+) (2.15) VISCOELASTIC PROPERTIES OF STIFF POLYMERS is the stress of the pure solvent and oEand trV represent the contributions of the polymers which are obtained as follows. We consider a hypothetical deformation BE which instantaneously displaces the material point at Y to ~+BE*Y.This deformation changes the free energy from Lcam to LcaLf].The stress oEis related to the change in Lca by this deformation as t7E BE = d[ f '1-dlfl.(2.16) On the other hand oVis given by the hydrodynamic energy dissipation caused by the deformation while the system is deformed. For the rodlike polymers the deformation BEchanges the distributionfas [see eqn (2.1311 f +f' =f-4E-[ux(BE-u)fJ. (2.17) From eqn (2.1) (2.16) a nd (2.17) we get tYE = 3Ck~T(UU-gI) C(U[UX 3(J'&F K)]) (2.18) On the other hand uVand nSare shown to be much smaller than cE.Hence the stress can be evaluated by eqn (2.18). 3. PERTURBATION EXPANSION Eqn (2.13) and (2.18) apply for both the isotropic phase and the liquid-crystalline phase. However a crucial difference appears in the mathematical handling of the equations for the two phases.For simplicity we consider the steady state in a weak velocity gradient. The equation to be solved is &-l+Glfl=O (3.1) (3.2) GW= -9*UX(K'Uf). (3.2) (For simplicity we assume =0.) The operator $is a non-linear operator while G is a linear operator. To solve eqn (3.1) we assume the solution in the following form f =fo+f1+ ... where fois the solution at equilibrium and fl is the first-order perturbation in K. Substituting eqn (3.4) into eqn (3.1) and comparing the first-order term in K we get ~flI+G[fOl= 0 (3.5) where the linear operator H is the first-order perturbation in $ i.e. +f1l-mo mo1 = m11+~(f12)* (3.6) In the isotropic phase fois equal to 1 /4n,and the linear equation (3.5) is easily solved. On the other hand in the liquid-crystalline phase this procedure encounters a difficulty because the equilibrium state foin the liquid-crystalline phase is not unique as the director n is arbitrary.Thus in the perturbation expansion for the liquid-crystalline phase we have to determine both the unperturbed state as well as the perturbed state. This is done as follows Let yAi) and qW be the right-hand-side and the left-hand-side eigenfunctions for the operator H Aw(i) @@i) = -~(z)~(i) = -~(i)@i). (3.7) M.Do1 53 These functions can be assumed to be orthonormal Sij = (@i)l~(j))E du @i)(u)V/~)(U). s We look for the solution of eqn (3.5) in the form fl= C ai yAi). (3.9) i Substituting eqn (3.9) into eqn (3.5) and using eqn (3.8) we get L@)ai= (qVi)lG[fo]). (3.10) This equation has solution if > 0 for all i.However the operator H has a zero eigenvalue. This is a direct consequence of the fact the equilibrium state in the liquid- crystalline phase is continuously degenerate. In fact iffo andfi denote the equilibrium states which have directors nand n’ = n+an,respectively then Sfo =fi -focorresponds to the eigenfunction of zero eigenvalue since fiSfO]= PLfJ-P[fO]= 0. (3.11) Therefore to obtain the steady-state solution we must have (Plmol) = 0 (3.12) for the eigenfunction qVi) corresponding to the zero eigenvalue. This equation determines the unperturbed state fo. 4. THE CONSTITUTIVE EQUATION FOR WEAK VELOCITY GRADIENT We now show that Leslies’s equation is actually derived from the above formulation. Although it is possible to work out eqn (2.13) and (3.12) we take a simplified ap- proachlO here for the purpose of demonstration.Instead of eqn (2.13) we construct an approximate equation for the following quantity Q f(UU-$1) (4.1) which is called the orientational-order parameter tensor. To obtain an equation for Q we multiply both sides of eqn (2.13) by uu-1/3 and integrate over u. Using the Hermitian property of i9 i.e. J du A(u)gB(u) = -J du[9B(u)]A(u) we can write the result as a -Q = -60 Q+ 60 U(Q*(UU) -(UUUU) Q) at By the use of the following decoupling approximation (uuuu) :Q = (uu) (UU) :0 (4.4) (UUUU) K = (UU) (UU) K eqn (4.3) is rewritten aslo a -Q= F+G (4.5) at VISCOELASTIC PROPERTIES OF STIFF POLYMERS where F = -6Dr[(1 -y) Q-U [Q.Q-g(Q Q) I]+ U(Q:Q) Q3 (4.6) G = ;(K + K?) + KOQ + Q xf -#(K :Q) I-2(~:Q) Q. (4.7) If G = 0 the equilibrium solution of eqn (4.5) is written as Q(O) = S(nn-$ I) ('4.8) where S is a solution of the equation -$Us+;Us2 =o1 (4.9) i.e. for U< 3 (4.10) This equation explicitly shows that the liquid-crystalline phase appears for large values of u. To obtain the perturbed solution we write Q = Q(O)+Q(l) and expand the equation F+G = 0 as 0 = F,p(Q) + G,p(Q) = Hap,pv Q$ + G,p(Q(O')+ 8(1c2) (4.1 1) where a,/?(= x,y z)denote the component of the tensors and summation is implied oyer the repeated Greek indices. The matrix Has,pv,which corresponds to the operator H is calculated from eqn (4.6) as Hap,pv= [(1 -y)+$US-$US2 1dapdpv -US(~gpnanV+~,,npnp) -2US2(nang-&p) (npnv-Qdpv).(4.12) The eigenvector @),which corresponds to the eigenfunction qW (u)of zero eigenvalue now satisfies Hpv,ap4g = 0. (4.13) The solution of eqn (4.13) is q5$) = n rnf)+ ngm&Q (4.14) where mci)is an arbitrary vector perpendicular to n.The condition (3.12) is now written as d$ G,g(Q'O') = 0 (4.15) From eqn (4.7) (4.8) (4.14) and (4.19 it follows that [(I -~)~,n~+(1+2S)1~,~ng]rn~) (4.16) = 0. Since rn@) is perpendicular to n this condition is rewritten as [(l-S)~t'n+(1+2S)rc*n]xn=0. (4.17) Eqn (4.17) determines the direction of the alignment in the unperturbed state. This result was already obtained in ref. (10) by a different method. Given the unperturbed state it is easy to calculate the stress tensor.For the potential (2.5) eqn (2.18) is written as t~ = ck T[3Q+(Q*(uu)-(uuuu):Q) U] (4.18) M.DO1 55 By use of the decoupling approximation eqn (4.4) and the steady-state condition F+G = 0 eqn (4.18) is rewritten as So far we assumed that there is no external field. In order to compare the present theory with Leslie's theory we have to introduce an external magnetic field H. The potential & for the magnetic field is given by I/e = -ia,(u*H)2 (4.20) where a = al,-al is the difference in the susceptibility of the polymer for the two cases that u is parallel to H and the case that u is perpendicular to H. Repeating the same calculation we obtain the following equations in place of eqn (4.5) i3 -Q= F+G+M (4.21) at where F and G are given by eqn (4.6) and (4.7)' and M is defined by M=-Da ['(HH-:PI) -$H2Q+HH*Q+Q-HH-'(HH Q)l -2(HH Q)Q].kBT (4.22) The stress tensor is calculated from eqn (2.18) and (4.20) as ck T[F(Q)+M(Q)] +;Zcaa(HH* g = -~ Q-Q.HH) 2Dr CkBT --G(Q) +ica,(HH*Q-Q*HH) 2Dr c =-ck T &(K+K~) +S(~mnn+nn.~t)-22S2(~:nn)nn] +-a S(H*n)(Hn-nH). 2Dr 2 (4.23) Eqn (4.15) is now replaced by d$[ Gap(0"))+Map(Q(O))] = 0 (4.24) which leads to (4.25) Introducing the symmetric and antisymmetric tensors A = ~(K+KT); Q = ~(Ic-K~) (4.26) and the 'molecular field' h = ca S(H*n)H (4.27) we can rewrite eqn (4.23) and (4.25) in the form of the Leslie equation o = a,(nn:A)nn+a,nN+a,Nn+a,A+a,nn.A+a,A*nn (4.28) and nx(h-y,N-y2A*n)=0 (4.29) dn where N = -fl*n.(4.30) dt VISCOELASTIC PROPERTIES OF STIFF POLYMERS The coefficients are obtained as a = -2S2q:a2= -S (I+-;fs)if:a3=-s (I-2:Sb a = f(l -S)q as = 2Sq a = 0 6s2 -q:y2 = -2sq (4.31) Y’-2+s where q = ck T/2& Leslie’s relation y = a3 -a2 y2 = a -a and Parodi’s relation22y2 = a3+a2are satisfied. Eqn (4.31) coincides with the result already obtained by Marrucci21 based on eqn (4.21). However the reason for this coincidence is not clear since Marrucci’s calculation is for the special case that the director is determined by the magnetic field. Note that the constitutive equations (4.28) and (4.29) also hold for the isotropic solution if S = 0 eqn (4.29)is automatically satisfied and eqn (4.28)reduces to b=-2ck T (K +K+).(4.32) 3Dr Eqn (4.9) and (4.31)predict the chracteristic behaviour of rodlike polymers i.e. the concentration dependence of the shear viscosity and the relative magnitude of the Leslie coefficients.lO* However due to mathematical approximations such as eqn 24 (2.5)and (4.4) the result it perhaps not very accurate. More accurate calculations are now being and will be published soon. This work was supported by a Grant-in-Aid for Scientific Research by the Japanese Government. P. G. de Gennes The Physics of Liquid Crystals (Clarendon Press Oxford 1975). * F. M. Leslie Arch. Ration. Mech. Anal. 1968 28 265. L. Onsager Ann. N. Y. Acad. Sci. 1949 51 627. 4 W. Maier and A. Z. Saupe Z. Naturforsch.Teil A 1959 882. J. G. Kirkwood and P. L. Auer J. Chem Phys. 1959 19 281; J. G. Riseman and J. G. Kirkwood J. Chem. Phys. 1949 17,442. G. B. Jefferey Proc. R. SOC. London Ser. A 1922 29 161. P. L. Nordio G. Rigatti and U. Segre J. Chem Phys. 1971 56 21 17. S. Hess 2.Naturforsch. Teil A 1976 31 1034. J. H. Freed J. Chem Phys. 1977 66 4183. lo M. Doi J. Polym. Sci. 1981 19 229. l1M. Doi J. Phys. (Paris) 1975 36 607. l2 M. Doi and S. F. Edwards J. Chem. SOC. Faraday Trans. 2 1978 74 568; 918. l3 S. F. Edwards and K. F. Freed J. Chem. Phys. 1974,61 1189. l4 K. F. Freed and S. F. Edwards J. Chem. Phys. 1974 61 3626; see also M. Muthukumar J. Phys. A 14 2129. l5 M. Doi Rheology of Concentrated Macromolecular Solutions in Theoryof Dispersed Multiphase Flow ed.R. E. Meyer (Academic Press London 1983). l6 M. Muthukumar preprint. l7 J. F. Maguire J. P. McTague and F. Rondelez Phys. Rev. Lett. 1980 45 1891 ; 1981 47 148. l8K. M. Zero and R. Pecora Macromolecules 1982 15 82. lB Y. Mori N. Ookubo R. Hayakawa and Y. Wada J. Polym. Sci. Polym. Phys. Ed. 1982,20,2111. 2o S. Jain and C. Cohen Macromolecules 1981 14 759. 21 R. Hayakawa personal communication. 22 M. Doi J. Chem. Phys. 1983 in press. 23 0.Parodi J. Phys. (Paris) 1968 31 581. 24 G. Marrucci Mol. Cryst. Liq. Cryst. 1982 72 153. 25 N.Kuzuu and M. Doi J. Phys. Soc. Jpn 1983 in press.

ISSN:0301-5696    

DOI:10.1039/FS9831800049

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

Faraday Symp. Chem. SOC.,1983 18 57-81 An Experimental Approach to the Molecular Viscoelasticity of Bulk Polymers by Spectroscopic Techniques Neutron Scattering Infrared Dichroism and Fluorescence Polarization BY LUCIEN MONNERIE Laboratoire de Physic0 Chimie Structurale et Macromoleculaire Ecole Superieure de Physique et de Chimie Industrielles de Paris 10 rue Vauquelin 75231 Paris Cedex 05 France Received 29th July 1983 Small-angle neutron scattering (s.a.n.s.) infrared dichroism (i.r.d.) and fluorescence polariz- ation (f.p.) are techniques which provide information on the molecular orientation of stretched polymers; they can thus be used to study the viscoelastic behaviour of bulk polymers on a molecular scale. The basic principles the experimental requirements and the molecular parameters available from experiments are described for each technique.Thus s.a.n.s. leads to overall dimensions of deuterated chains in a mixture of deuterated and hydrogenated polymers. In contrast i.r.d. yields the chain segment orientation averaged over all the chains of the sample whereas orientation determined from f.p. only deals with labelled sequences (central or end sequences) of labelled chains mixed with normal chains. Experimental results obtained by i.r.d. and f.p. on polystyrene samples stretched at a constant strain rate above are discussed. The influence of the molecular weight of either labelled chains or polymer matrices leads us to consider first a topological coupling between the relaxation processes of the labelled chain and those of the surrounding chains and secondly an orientation effect arising from the anisotropy of the strained polymer medium.An i.r.d. study of the orientation of compatible blends of polystyrene and poly(2,6- dimethyl-l,6phenylene oxide) is reported. Each polymer component exhibits different orien- tation behaviour. A recent renewal of interest in the viscoelasticity of entangled bulk polymers has arisen from the theoretical approach of chain relaxation developed by de Gennes Edwards and Doi. These theories are based on a molecular description of chain motions which involves reptation of the chain along the 'tube' formed by neighbouring chains. Although such theories can be tested through their predictions of macroscopic viscoelastic properties of bulk materials such as viscosity and elastic and loss moduli it is very tempting to obtain direct information on chain dynamics on a molecular scale.For this purpose spectroscopic techniques such as small-angle neutron scattering (s.a.n.s.) infrared dichroism (i.r.d.) and fluorescence polarization (f.p.) are particularly suitable. S.a.n.s. yields information either on the molecular dimensions of the whole chain or on the structure of part of the chain whereas i.r.d. and f.p. applied to stretched samples lead to information about the chain segment orientation. In this paper each technique will be presented with a particular emphasis on its application to studies of the molecular viscoelasticity of bulk polymers. Two examples are discussed the first dealing with polystyrene the second concerning 57 MOLECULAR TECHNIQUES AND VISCOELASTICITY the compatible blends of polystyrene and poly(2,6-dimethyl- 1,4-phenylene oxide) and these are compared with predictions of molecular viscoelasticity theories of polymer me1 t s.SMALL-ANGLE NEUTRON SCATTERING Small-angle neutron scattering is an experimental technique which has been used during the last ten years to investigate polymer conformation over the entire concentration range from dilute solution to the melt. Indeed owing to the wavelength of the thermal neutrons used (2-20 A) and to the low-angle spectrometers which are available it is possible to investigate sizes in the range from 1 to 5 x lo3A corresponding to the characteristic dimensions of polymers.A number of excellent reviews are available which deal with neutron-scattering theory1** and its application to ELASTIC NEUTRON SCATTERING THEORY In a scattering experiment involving any type of radiation (light X-rays y-rays or neutrons) an incident beam with a well defined wavelength A and wavevector k (k is a vector in the direction of travel of magnitude Ik I = 271/1) reaches the sample under investigation. The radiation scattered in a direction 8 is characterized by a wavevector k. If no energy transfer occurs between the incident radiation and the system i.e. the system does not undergo any motion on the timescale of the experiment the scattering is purely elastic and A = A,. Then the momentum transfer hq defined as hiq = k-k is simply expressed by 471 .8 q = -sin-. A0 2 Thus an elastic-scattering experiment consists of simply measuring the scattered intensity at various values of the scattering angle 8. A quantity commonly used to describe radiation scattering is the differential of the scattering cross-section da(q)/dR which gives the ratio of the flux scattered in the q direction to incident flux per unit solid angle i2. The method applied to calculate da/dR is the same for the various types of radiation including neutrons. If one considers the interaction potential between the radiation and the sample under investigation V(r),where r is the position vector of the interacting species da/dR is written as the spatial Fourier transform of the pair correlation function of the interaction potential = (V(q) V(-4)) dR with V(q)= d r exp (iq r) V(r) the angular brackets indicating a thermodynamic average.However the particular characteristics of neutrons lead to important differences. A neutron has a mass m = 1 a.u. which is much larger than that of an electron (m= 1836 me).Furthermore it is uncharged and has a spin of 1/2. The main interaction of a neutron with the sample under investigation and the only one considered hereafter is that involving the nuclei. Such an interaction is very different from that involved in light or X-ray scattering which concerns electrons. The interaction characteristic distance between neutron and nucleus is exceedingly L. MONNERIE short range ( A) and therefore is very small compared with the wavelength of the neutron.Thus the scattered wave is isotropic whereas it is anisotropic for X-rays and is characterized by its amplitude of -b where b is termed the scattering length for dimensional reasons. b is complex; its real part can be positive or negative and its imaginary part is a measure of the neutron absorption which can occur. Owing to this nuclear interaction the scattering length is different for each isotope as well as each element unlike the X-ray scattering length which depends only on the atomic number of the element and not on the isotope. Instead of the scattering length b another commonly used quantity is the cross-section ratio CT of the scattered neutron flux to the incident flux which is given by CT = 4nb2.First let us consider an array of N nuclei corresponding to the same isotope and without any spin; the nuclei are denoted by subscripts i and j. The only interaction of a neutron with such nuclei occurs via purely nuclear forces and there is a scattering length b which is identical for all nuclei. The interaction potential is V(Ri) = bd(R-Ri). Thus the scattering cross-section is expressed by = b2 C (exp [iq(R -Itj)]) = b2IX exp (iq Ri)I2 i i i where Ri and Riare the position vectors of nuclei i and j. Thus da/dR is a coherent scattering equal to the square of the sum of the scattering amplitudes of each atom. Let us now look at what happens for an array of N nuclei corresponding to the same isotope but with a spin I.Owing to the spin of 1/2 of the neutron in addition to the previous interaction there is a magnetic interaction between the spins of the neutron and the nucleus which will be different whether the spins are parallel or antiparallel i.e. whether the total spin state is I+ $ or I-4. This leads to two different scattering lengths b+ and b- respectively and the mean scattering length (b) can be derived taking into account the degeneracy of total spin states I+ 1 (b) =---b++-I b-21+1 2r+1 and in the same way (b2) = I+ 1 -(b+)2+-(b-)2. I 2r+ 1 2I+ 1 The above expression of do/dR can be separated into two terms corresponding to i =j and i # j leading to = N(b2)+ 2 (bibjexp[iq(Ri-Rj)]). dR i #j As there is no correlation between the position of the nuclei and their spin one obtains MOLECULAR TECHNIQUES AND VISCOELASTICITY which can be rewritten as Thus two fundamentally different cross-sections are obtained where the coherent contribution is = (b)2Cexp [iq(Ri-Iti)]= b& I Cexp (iq Ri) l2 (g)coh i i i and the incoherent part is = ((b2)-(b)2)N =(b-(b))2N = b:ncohNn (&) incoh At this stage it is worthwhile to indicate several points (1) The coherent scattering depends on the mean value of the scattering interaction and has the same form as for other types of radiation.It is phase dependent and leads to information on the structure of the investigated system. For this reason it will be the only contribution considered hereafter.(2) The incoherent scattering arises from fluctuations in the scattering length away from the mean value (b). It is isotropic therefore no information can be obtained about the relative positions of the nuclei. It leads only to a background over which the coherent scattering is superposed. (3) The incoherent scattering is unique to neutrons. It arises from the fact that the neutron has a spin which interacts with the spin of the nucleus. Of course incoherent scattering does not exist for atoms without spin such as 12Cor lSO. (4) Table 1 gives the values of coherent incoherent and absorption cross-sections for various elements. Note that the values of bcoh are different for H and D in such a way that it is possible to obtain a contrast between hydrogenated and deuterated molecules.Furthermore the value of oincoh for H is almost an order of magnitude greater than that for any other nucleus and it is also considerably larger than all the other coherent cross-sections. Therefore the scattering from any sample containing hydrogen will be dominated by the latter's high incoherent scattering. We now consider the case of systems containing different types of atoms. As each type of atom has a different scattering length it is worth considering a local scattering-length density defined by b(R)= I:b,6(R-Ri) or b(q)= C biexp (iq Ri). i i Thus the coherent-scattering cross-section is simply expressed by When dealing with a binary mixture of particles a and p if one assumes that the local fluctuations of the number of atoms are negligible (i.e.that the system is incompressible) it can be shown that b(q)is given by where pa is the local density of particles of type a and Vaand Vpare the partial molar L. MONNERIE Table 1. Scattering lengths incoherent cross-sections and absorption cross-sections for some elements H D 1/2 1 -0.374 0.667 79.7 2 0.19 -0 'ZC 0 0.665 0 -0 14N 1 0.94 0.3 1.1 lS0 0 0.58 0 -0 28Si 0 0.42 0 0.06 c1 3/2 0.96 3.4 19.5 volumes of particles a and /I,respectively. Therefore b(q) is the Fourier transform of the density of particles a only weighted by an apparent scattering length K termed the contrast factor Thus one obtains (g) = ~2 (Pa(q)Pa(-q))* coh This result is very important for it shows that for a binary mixture one can extract information about the relative positions of atoms of one species.This is the basis of the deuterium-labelling technique which allows one to study deuterated species in a matrix of hydrogenated species. This approach has been extended to the case of a solution of labelled and normal polymers leading to interesting and unique contrast possibilities. POLYMER STRUCTURE STUDIES OF ISOTROPIC SAMPLES PRINCIPLES As pointed out previously only the q(or angular) dependence of coherent scattering contains information on the structure of the sample. Thus in a study of mixtures of deuterated and protonated polymers it is necessary to subtract the large incoherent scattering arising from the hydrogen atoms. In the previous section all the systems considered were arrays of atoms in which each atom was a scattering particle.In the case of polymers which consist of long chains of chemically bound monomer units the whole macromolecule is too large compared with the neutron wavelength to be considered as a unique scattering particle. However on the scale of a monomer unit or of a statistical segment provided that their dimensions are smaller than the q-l values investigated interference between the waves scattered by the various constituent atoms is negligible and the centres of mass of these units can be considered as scattering points. Then the coherent-scattering length associated with the centre of mass of a unit is the sum of the scattering lengths of the individual atoms.For example this leads to bcoh= -0.166 x 10-l2and 3.998 x 10-l2 cm for C2H and C2D, respectively and to bcoh= 2.328 x 10-l2 and 10.656 x 10-l2 cm for hydrogenated and deuterated styrene. The contrast factor K is obtained from the expression given previously and the coherent scattering can be expressed as a function of the deuterated species only (denoted by D): (g) = K~Z (exp[iq(~,~-~?)]). coh i,j MOLECULAR TECHNIQUES AND VISCOELASTICITY In this expression the sum involves all deuterated monomers both those in the same chain and those belonging to different chains. Low concentrations of labelled polymers If the concentration is low enough to neglect the interchain correlation the single-chain correlation function of the labelled polymer can be obtained.Then the treatment is identical to that developed for light scattering by polymer solutions. Thus (&) = K2sE coh (4) coh with sy coh (4) exp (k* R)P(R)dR where P(R)is the probability of having two scattering centres at a distance R. For an ideal Gaussian chain of N scattering centres P(R)is well known and leads to 2 sycoh(q) = N:fD = N:,--"exp(-q2(R~))-1+q2(R~)1 4 (R&) wheref, is the classical Debye function and (I?& ) represents the mean-square radius of gyration of the chain. For a concentration c (with the dimensions of weight per unit volume) of a labelled polymer of molecular weight M the intensity of scattering per unit volume Z(q) is where N is Avogadro's number. Depending on the q values investigated different types of information can be obtained on the structure of the labelled chain.The characteristic molecular parameters of a chain are the radius of gyration RG,the persistent length a and the statistical step length 1. The values of q compared with the reciprocal of these molecular parameters define four q regions. (A) q c RG~, called the Guinier range corresponds to a long-range correlation between monomers and gives information on the overall chain dimensions. In this range the exponential term infD can be developed leading to Therefore from measurements performed at various q values the molecular weight and the value of RG for the labelled polymer are obtainable. (B) RG~ < q < a-l. In this range termed intermediate correlations at short distances between monomers inside a chain are observed.However unlike the Guinier range no characteristic distance can be derived. Only the type of q dependence can give information about the chain structure on this scale. Thus for a random Gaussian chain Z(q) varies as q-2 whereas for more or less rigid filaments Z(g) varies as q-l. (C) a-l c q c I-l. In this range the chain behaves as a rod and thus Z(g) presents a q-l dependence. (D) I-' < q. In this high-q range the scattering is governed by the structure of the chain segments. L. MONNERIE High Concentrations of labelled polymers The above requirements of low concentrations of labelled polymers to obtain information on single-chain structure implies in practice that experiments should be performed at several concentrations and then extrapolated to c+O using the well known Zimm plot.Such a procedure needs the use of the neutron beam for a rather long time. Recently it has been shown theoreti~ally~~ that the single-chain scattering function can be obtained from measurements at concentrations as high as 50% (wt/vol). The calculations are based on the assumption of incompressibility and for the case of hy- drogenated and deuterated chains with identical molecular-weight distributions they lead to where b, is the volume fraction of labelled polymer and S,(q) the single-chain scattering function of the labelled chain. Thus information on the deuterated chain structure can be derived from only one scattering experiment at any concentration.If the normal and labelled polymers have different molecular-weight distributions5 both S,(q) and S,(q) are involved and one obtains where x is the Flory interaction coefficient between H and D polymers. Thus two or three measurements are required to determine SD(q) and S,(q) depending on whether x is neglected or not. EXPERIMENTAL RESULTS A description of the neutron-scattering apparatus is beyond the purpose of the present paper. Relevant information can be found in ref. (2) and (6). The main problem in coherent neutron scattering studies on hydrogenated and deuterated polymer mixtures arises from the subtraction of the hydrogen incoherent scattering. This implies that one should make similar measurements on a blank which contains the same relative proportions of D and H atoms now randomly distributed over all the molecules.In the case of low concentrations of deuterated polymers the results obtained in the Guinier range on several polymers have proved that in the bulk state the chains are Gaussian and their dimensions are identical to those under 8 conditions. In the intermediate q range satisfactory results have been obtained but it appears that in this range it is necessary to take into account the conformational structure of the polymer.' Furthermore for high q values (qR > 5) corresponding to correlation distances comparable to the size of a few monomers the cross-section of the chain should be considered as shown experimentally.8 Thus the test of a given q dependence is delicate.For high concentrations of deuterated polymers the theoretical previsions have been successfully checked5. in the Guinier and intermediate q ranges when dealing with normal and labelled polymers with close molecular-weight distributions. In the case of very different molecular weights the problem of extracting the information on single-chain structure remains. For high-molecular-weight polymers (M > 109 it is more and more difficult to reach the Guinier range with the low q values experimentally achievable. MOLECULAR TECHNIQUES AND VISCOELASTICITY APPLICATION OF S.A.N.S. TO MOLECULAR VISCOELASTICITY STUDIES As previously shown s.a.n.s. yields information on polymer structure in various ranges. It may therefore be used for studying deformed (e.g.stretched) samples in order to gain a better understanding of the molecular mechanisms involved in many processing techniques as well as to test molecular theories of either rubber elasticity or chain relaxation.The time required to perform an s.a.n.s. experiment is too long to allow direct measurements during stretching or relaxation when dealing with uncrosslinked polymers. In the case of crosslinked rubbers for which the equilibrium state can be achieved at each elongation ratio (A) the problem is different and studies can be carried out directly. For uncrosslinked polymers only materials with a glass-rubber transition temperature (r,>higher than room temperature can be studied. The method consists of stretching the sample at the chosen temperature until a given A value quenching it at room temperature and subsequently performing the s.a.n.s.measure- ments. The same procedure is used for relaxation studies after stretching at a given A value; here the samples are simply quenched after various relaxation times. st retching direction Fig. 1. Schematic view of the small-angle scattering of a strained sample. The q vector lies in the detector plane perpendicular to the incident neutron beam of wavevector k,. The characteristic directions 11 and Iare defined with respect to the sample stretching direction. Curves of equal intensity are represented as ellipses in the q plane. It is obvious that to get sufficiently large changes in polymer structure it is necessary to use A values larger than 2.Thus the s.a.n.s. measurements are performed on significantly deformed polymer chains. Using the whole plane of q shown in fig. 1 one can consider curves joining all the points (detector cells) which correspond to the same given scattered intensity. Of course an isotropic sample gives circles whereas a uniaxially strained sample leads to curves which should be ellipses if the polymer chains are affinely deformed relative to the overall sample dimensions. For uniaxial stretching one can study the scattered intensity along directions either parallel or perpendicular to stretching direction (qI1and ql respectively).T\us both the Guinier range and the intermediate q range can be studied for ql,and ql. In the same way as for isotropic samples measurements in the Guinier range lead to the mean-square radius of gyration of the chain but now there are two values (R&) and (I?&,) corresponding to qtland ql respectively.For the intermediate q range it has been ShownlO that the q dependence of S(q) provides information on the mechanism of deformation. Thus for an affine deformation L. MONNERIE the q dependence is changed compared with the isotropic case whereas for a mechanism in which chain deformation is achieved by end-to-end pulling the q dependence is identical to isotropic chains but with a lower value of S(q,,).Experiments performed on stretching polystyrenell show that the actual behaviour lies between these two extreme cases. The affinity is more pronounced for deformations carried out at temperatures closer to q.A more elaborate deformation model has been developed recently.12 Thus the s.a.n.s.technique offers two different ways of studying the viscoelastic behaviour of uncrosslinked polymer chains. First one can look at the evolution of the radius of gyration as a function of either the elongation ratio or the relaxation time at constant strain and compare the resulting curves with the predictions of molecular viscoelasticity models. The second way is to consider the intermediate-q range and to study the behaviour under various experimental conditions (temperature strain rate relaxation time etc.). The s.a.n.s. measurements provide a unique means of studying the overall chain behaviour. However note that owing to the q values that are available practically in low-q region and owing to the fact that (RGII) increases nearly as A these molecular characteristics can be determined as a function of R only for polymers with molecular weight ca.lo5Dal. For higher molecular weights only (RGI)can be derived. Another interesting feature of neutron scattering deals with the technique of labelling using deuterium. Indeed in amorphous polymers there is no segregation between hydrogenated and deuterated chains. Thus it is possible to obtain information on the structure of a labelled chain which can be directly related to normal polymer behaviour. Furthermore it is possible to use a labelled chain with molecular characteristics (uw, chain branching etc.) very different from those of the matrix.At present such studies are only valid for low concentrations of labelled chains. Finally polymers can be labelled either all along the chain or in particular sequences (internal or end sequences) yielding the possibility of examining the behaviour of particular chain sequences. Although these studies require further theoretical develop- ment to derive structural information from s.a.n.s. they constitute for the future the most powerful experimental test of molecular-viscoelasticity theories. In contrast to these advantages an important problem arises from the lack of flexibility in the experiments owing to the use of high-flux neutron reactors. From a more technical point of view difficulties appear in thermoplastic samples of avoiding any microvoids both in isotropically moulded samples and in stretched samples.These requirements are greater for samples with a high content of deuterated polymer awing to the higher contrast between air and deuterium than with hydrogen. These microvoids give additional scattering at low qvalues which can affect the determination of R,. ORIENTATION DISTRIBUTION FUNCTION Before considering the spectroscopic techniques which lead to a measurement of the orientation distribution of characteristic vectors we introduce some convenient quantities to describe the orientation of uniaxially symmetric systems. Each vector is characterized by the angle 6 that its direction makes with the symmetry axis. The orientation distribution is represented by a function f(6) which can be expanded in terms of Legendre polynomials in cos 8 as follows with b = (1/2 n)(21+ 1)/2 (P,(cos 8)) 3 FAR MOLECULAR TECHNIQUES AND VISCOELASTICITY where (~(COS 8) averaged over the distribution 8)) is the value of ~(COS (p2(C0s8)) = (112)(3 cos2 8-1) 8)) = (1 /8) (35 COS~8-30 COS~ (~~(cos a+3) (cosn8) = s"f(9)cosn 8sin 8dO.0 It has recently been shown13 that for an uniaxial distribution the determination of (4)and (p4) is sufficient in most cases. INFRARED DICHROISM PRINCIPLE OF ORIENTATION MEASUREMENTS Adsorption in the infrared region of the spectrum deals with vibrational motions of the various atoms of a molecule and corresponds to a change in the vibrational energy level of the system.For the harmonic-oscillator approximation this energy change can be correlated with the structure and force constants of the molecule by performing a classical analysis of a vibrating system. This leads to a set of normal vibration frequencies associated with the normal modes of oscillation of the molecule. In order to be active in the infrared a vibrational mode must imply a change in the dipole moment of the molecule. Although a normal mode involves to some extent all the atoms in a molecule a large number of these vibrations are localized to a high degree of approximation in small groups of atoms. Thus some absorption bands can be assigned to the stretching of specific bonds the bending of bond angles or the wagging twisting and rocking motions of a given group of atoms.Owing to the change in the dipole moment of the molecule during the infrared-active normal vibration each mode will have a transition moment M with a definite orientation in the molecule. The intensity of an infrared absorption band depends upon the angle the electric vector E in the incident radiation makes with the transition moment M. The in- dividual absorbance a is given by a = log, I,/I = IMI2(M E)2 where I and I are the incident and transmitted intensities. This equation forms the basis for the use of polarized infrared radiation as a tool to study the orientation of a molecule in an anisotropic material. Let us consider a single molecule in an oriented sample with a preferential orientation axis A. The transition moment M of the considered mode makes an angle a with an axis of the molecule the direction of which is at an angle 8 from the orientation axis A (fig.2). For incident radiation polarized along A the related absorbance all will be all = I MI2(cos2acos28+4 sin2a sin28). In the same way for incident radiation perpendicular to A we get al = IMI2[+cos2asin28++ sin2a(1+cos28)]. If we deal with a set of identical molecules characterized by an orientation distribution function f(6) relative to the axis A the parallel and perpendicular overall absorbances will correspond to All = J,rn ailf(8)sin 8d8 L. MONNERIE preferential orientation axis M lar Fig. 2. Positions of molecular axis and transition moment with respect to the reference axis A.A convenient quantity is the dichroic ratio R defined as R=AII/Al. The value of R can range from zero (no absorbance in the parallel direction) to infinity (no absorbance in the perpendicular direction). For an isotropic sample no dichroism occurs and R = 1. For a set of molecules perfectly aligned along A the resulting dichroism is R = 2cot2a. Using this value leads to the following expression for R jon[1+(R,-1) cos20]f(0)sin 0 d0 R= i," [1+;(R~-1) sin2 elj-(e)sin 0 de 1 +(R,-1) (COS~0) or R= 1 +;(R,-1)(1 -(COS20))' Thus the measurement of the dichroic ratio of an infrared absorption band for which R is known yields the second moment of the orientation distribution function (cos2 e). The orientation function (P2(c0s0)) is obtained by APPLICATION TO THE ORIENTATION OF POLYMERS Infrared dichroism has been applied for many years to orientation measurements involving polymers.These studies have been reviewed re~ent1y.l~ Dichroism measurements are easily performed either on dispersive instruments or more conveniently on a Fourier-transform apparatus. Measurements can be carried MOLECULAR TECHNIQUES AND VISCOELASTICITY out either on stretched samples or during stretching. The main practical problem in the case of Fourier-transform measurements arises from the requirement of band absorbance lower than ca. 0.7 absorbance units in order to permit use of the Beer-Lambert law. This means one must obtain sufficiently thin films. Depending on the extinction coefficient of the considered band the required thickness can range from 1 to 200 pm.From this point of view polymers with strong absorption bands (e.g. polycarbonate) are difficult to study. The main difficulty in using infrared dichroism to study orientation is the necessity to find infrared absorption bands of the polymer which are sufficiently well assigned to normal vibrations of specified atomic groups. Such an assignment can be achieved by making a normal-coordinate analysis and experimentally by looking at deuteration effects and dichroic behaviour. Furthermore it is necessary that these well defined vibrational bands do not overlap with other bands resulting from another normal mode a harmonic or a combination of other modes. 9 0 0 / , / 0 0 0 0 \ I I 0 I I I /'-0 $3,.' (b) Fig.3. Local chain axis (a) in polystyrene and (b)in poly(2,6-dirnethyl-l,Cphenyleneoxide). Another problem arises from the choice of a local chain axis which is convenient to describe the chain orientation. When dealing with vibration modes which are independent of the local conformation of the main chain (trans or gauche) the in- frared dichroism will lead to the average orientation of chain segments and the local chain axis must be chosen in such a way that the same value of the orientation function is obtained when various absorption bands of the same type corresponding to different a values are used to calculate Pz.Examples of such local chain axes are given in fig. 3.Some vibration modes and absorption bands are characteristic of specified chain conformations (ttt for instance) and thus infrared dichroism will lead to the L. MONNERIE orientation value of these particular chain conformations. This can be done in the same way for absorption bands characteristic of the crystalline structure. Finally note that infrared dichroism studies do not require any labelling; however consequently the derived orientation refers to an average over different chains. Some experiments using deuterium-labelled polystyrene chains blended with hydrogenated polystyrene have been performed recently by J. F. Tassin in our laboratory. Some specific bands do not overlap but allow a measurement of the orientation of each species. Unfortunately this method requires concentrations of deuterated chains in the blend > lo% which makes studying the effect of the molecular weight of the labelled chains or of the polymer matrix difficult.In the case of blends if specific bands which do not overlap can be found for each polymer infrared dichroism is a good tool to study the orientation of each component in the blend. An example of this type is given later. FLUORESCENCE POLARIZATION PRINCIPLES Fluorescent molecules have the property of re-emitting in the form of visible light part of the energy acquired by the absorption of luminous radiation. After illumination by a very short pulse at time to the fluorescent light emitted at time t,+u is proportional to exp (-u/z) where z is the mean lifetime of the excited state (usually called the fluorescence lifetime).The most frequent z values range from 1 to 100 ns. When absorbing light of a suitable wavelength a molecule behaves as an electric dipole oscillator with a fixed orientation with respect to the geometry of the molecule. Such an equivalent oscillator is termed an absorption transition moment M,. In the same way for the fluorescence emission we have an emission transition moment M. When such a molecule receives an incident beam polarized along the P direction (fig. 4) the absorption probability is proportional to cos2a,. In the same way the fluorescence intensity measured through an analyser A is proportional to cos2p. Thus for the P and A directions of polarizer and analyser the observed luminescence intensity is proportional to cos2a0cos2~.Owing to the lack of phase correlation between t observation Fig. 4. Polarized absorption and fluorescence emission P,polarizer; A analyser. MOLECULAR TECHNIQUES AND VISCOELASTICITY excitation and emission lights fluorescence emission can be described as resulting from three independent radiations respectively polarized along the X,Y and 2axes with intensities Ix Iy and I,. The Curie symmetry principle applied to excitation light polarized along 2,leads to Ix =Iy. The fluorescence polarization is characterized by the emission anisotropy r =(111-Id/(lIl +w where Illand Zl correspond to the fluorescence intensity obtained with an analyser direction parallel and perpendicular respectively to that of the polarizer.When dealing with an isotropic set of fluorescence molecules in such conditions that the relaxation times of the molecular motions are in the range of the fluorescence lifetime f.p. yields information on the mobility of the molecules; such an application has been reviewed re~ent1y.l~ ORIENTATION OF UNIAXIALLY SYMMETRIC SYSTEMS For our present purpose we are mainly interested in the use of f.p. to look at the orientation distribution of fluorescence molecules. The main results which can be derived are presented below; more details can be found in a recent review15 and in the original paper.l6 7 Fig. 5. Illustration of the angles which define the orientation of molecular axis Moat time to and M at time to+u with respect to the fixed frame OXYZ.In the following we will assume that the transition moments in both absorption and emission coincide with a molecular axis A4of the molecule whose direction is specified by the spherical polar angle R =(a,8) in the reference frame (fig. 5).Let us introduce the angular functions N(Ro,to) the orientation distribution of M at time to (Moin fig. 5),and P(R tIno,to),the conditional probability density of finding at position R at time t a vector M which was at position R at time to. After illuminating the sample by a linearly polarized short pulse of light at to,the intensity emitted at time (to+u) for the P and A directions of polarizer and analyser is given by i(P,A,t,+u) =~~~~(~o,to)~(~,t,+~l~o,to) x cos2(P, M,)cos2(A,M)exp (-u/z)dRo dR L.MONNERIE where K is an instrumental constant. In this expression to corresponds to the macroscopic evolution of the sample for example in a rheological experiment whereas u corresponds to a microscopic reorientational motion in the scale of the fluorescence lifetime z. In most cases the todependence of N and P can be ignored within the time z (ca. s) and the fluorescence intensity emitted under continuous excitation is given by i(P,A toz) = i(P,A,to+U) du. Jo* In the case of a uniaxial symmetric distribution of the molecular axes M the intensities corresponding to the P and A directions lying along the fixed-frame axes (2 corresponds to the symmetry axis) can be conveniently expressed through the following quantities Gg) = $(3cos28,-1) Gg) = i (3 COS~8-1) Gg)= ~((3C0S260- 1)(3cos28- 1)) G& = (sin 80 cos 8 sin 8cos 8cos (/3 -Po)) G$i) = s9a (sin2 80 sin2 8cos 2(p-/30)).Thus for example I(Z,2)= (K/9)(1 +2Gg) +2Gg) +4G#) i(2,X)= (K/9)(1 +2Gh;) -G# -2Gi:)). UNIAXIAL FROZEN SYSTEMS In such cases no molecular motion occurs during the fluorescence lifetime z and the quantities G# (= Gh;)) and GLF) can be rewritten with only two independent quantities cos2 8and COS~8. All the information on the fluorescence intensities may be displayed in a 3 x 3 tensor I ’ # (sin48) (sin48) 4 (sin28cos28) I = K b (sin4 6) (sin4 8) 4 (sin2 t?cos2 8) . (sin28cos28) i (sin28cos28) (COS~8) It is easy to see that the second and fourth moments of the orientation distribution (cos20) and (COS~S) respectively can be derived from fluorescence intensity measurements Ixx I, and Izx.UNIAXIAL MOBILE SYSTEMS In this case both orientation and mobility contribute to the fluorescence polariza- tion and the two effects have to be separated from the measured intensities. This is possible16if we assume that during the fluorescence lifetime (ca. s) the orientation distribution does not change. Nevertheless special optical equipment” is required and only (cos26) can be obtained from the measurement of five intensities i(P,A) corresponding to P and A directions which are not contained in the same plane. On the other hand the mean amplitude of the motion performed during the fluorescence lifetime is available from the data.MOLECULAR TECHNIQUES AND VISCOELASTICITY The same information on frozen and mobile systems can be obtained even if the absorption and emission transition moments are no longer parallel. APPLICATION OF FLUORESCENCE POLARIZATION TO ORIENTATION MEASUREMENTS IN POLYMERS Since f.p. is an optical technique only samples that are sufficiently transparent can be studied; i.e. amorphous polymers of any thickness or thin films (< 100pm) of semi-crystalline polymers. The use of f.p. to derive the orientation functions [Pz(cos 8) and p4 (cos e)]of a set of fluorescence molecules requires that there is no energy transfer between the fluorescent molecules implying a concentration of fluorescent species in the sample below 100 ppm. Thus the intrinsic fluorescence of a monomer unit cannot be used e.g.phenyl groups in polystyrene. Q Fig. 6. Labelled chain containing an anthracene fluorescence group (the arrow represents the transition moment). In order to correlate unambiguously the orientation of the transition moment of the fluorescence molecule with that of the polymer chain it is necessary to carry out covalent labelling. This can be achieved by performing an anionic polymerization and deactivating the living chains by 9,1O-bis(bromomethyl)anthracene.In this way the resultant polymer contains a centrally located fluorescence group in which the transition moment lies along the local chain axis (fig. 6).The labelled polymer is then incorporated into normal polymer at a concentration of 0.5-1 %.End-chain labelling can be obtained by using a monofunctional anthracene derivative. Such a method has been successfully applied in our laboratory to polystyrene and various polydienes. The main interest of f.p. is the great sensitivity of the fluorescence intensity measurements which allows the use of a very small amount (< 1%) of labelled polymer. In this way it is easy to look at the orientation of a labelled chain surrounded by normal chains and to vary the molecular weight of either the labelled chain in a given matrix or the matrix with a given labelled chain. In tmcase of polymers mobility studies performed by f.p. on isotropic bulk polymerslg have shown that a significant motion during the fluorescence lifetime only occurs at temperatures more than 5OoC above Tg.Thus polymers below this temperature range can be considered as frozen systems. F.p. measurements can be performed either on samples stretched above and then quenched or during stretching. For this last case special equipment has been developed in our laboratoryl7? l9 which allows measurements on both frozen and mobile systems. L. MONNERIE MOLECULAR-VISCOELASTICITY STUDIES OF BULK POLYSTYRENE . The techniques described above were first applied to the molecular behaviour of polystyrene chains during either stretching or relaxation at constant strain. As the s.a.n.s. results are presented and discussed in the paper by Boue we will focus on the experiments performed by i.r. dichroism and f.p.; most of these have been obtained recently in our laboratory.They deal with the influence of the experimental conditions of stretching (temperature strain rate and molecular weight) on the orientation of uniaxially stretched atactic polystyrenes (PS) the characteristics of which are reported in table 2. Table 2. Number-average molecular weight and polydispersity of poly-styrene samples polymer PS 1 149 800 1.70 PS 100 105 000 1.12 PS 160 160 000 1.16 PS 200 190 000 1.17 PS 400 420 000 1.24 PS 600 660 000 1.15 PS 900 855 000 1.19 PS 1300 1 300 000 1.28 INFRARED DICHROISM (I.R.D.) STUDIES2' Thin films suitable for i.r. spectroscopy have been obtained by solution casting. The stretching was performed on a hydraulic machine developed in our laboratory,19 operating at a constant strain rate (i) in the range 0.008-0.115 s-l up to 600% deformation of a sample of 6 cm length between its jaws.It is equipped with a special oven to obtain good temperature stability ( 0.02 "C)and a temperature homogeneity along the stretching axis of at least 0.03 OC. The PS films were stretched at a given II and then suddenly quenched at room temperature. 1.r.d. measurements were later performed using a Nicolet 7 199 Fourier-transform spectrometer. The resulting orientation function (P2(c0s 8)) refers to the local chain axis of PS shown in fig. 3. The 1028 and 906 cm-l absorption bands used are not sensitive to chain conformation and thus lead to a mean orientation of PS chains. Measurements in either the temperature range (1 10-128.5 "C) or the strain-rate range (0.008-0.115 s-l) yield a variation of (P2(cos8)) with draw ratio (1 = l/l,,,where 1 is the initial length of the sample and 1 is its length after drawing) which is nearly linear up to = 4 within the accuracy of our measurements.In order to illustrate the orientation behaviour the value of (Pz(cos 8)) at 1 = 4 is plotted as a function of T and log k-l in fig. 7 for PS 100 and PS 900 samples. For a stretching temperature close to the two polymers behave in the same way. An increase in Tor a decrease in i results in a greater orientation for PS 900 than for PS 100. These results suggest that a relaxation of orientation occurs during stretching. Therefore it is of interest to treat the orientation data in a more quantitative way in order to derive an orientation relaxation function 8(t)in a similar manner to that MOLECULAR TECHNIQUES AND VISCOELASTICITY Fig.7. Orientation function (Pz(cos 6)) at 1= 4 as a function of temperature and strain rate for monodisperse polystyrene samples. (-) PS 900 from i.r.d. ;(- -) PS 100 from i.r.d. ;(- -) PS 400-PAP 370 from f.p. -1.5-Fig. 8. Log 6(t) plotted against log t for PS 100. (1) 110 (2) 113 (3) 116.5 (4) 122 and (5) 128.5OC. commonly carried out in viscoelasticity studies for the relaxation modulus E(t). Thus the data have been analysed according to the method developed by Lodge2' for birefringence using the constitutive equation proposed by this author to describe the orientation during deformation.This leads for PS 100 to the variation of log O(t) as a function of log t shown in fig. 8 for the five temperatures studied. From these curves L. MONNERIE Fig. 9. Master curves of log O(t) plotted against log t for monodisperse samples between 110 and 128.5 "C. Reference temperature T,= 115 OC. (1) PS 900; (2) PS 100. Fig. 10. Comparison between log O(t) and log E(t) plotted as a function of log t for PS 1. Reference temperature T = 115 OC. (1) 1.r.d. orientation relaxation (2) mechanical relaxation. it is possible to obtain a master curve using the W.L.F. shift factor with the coefficients obtained by Plazeck22 for polystyrene. Master curves corresponding to PS 100 and PS 900 at a reference temperature of 115 "C are shown on fig.9. In order to compare our results with the relaxation modulus E(t) viscoelasticity measurements have been performed using dynamic shear experiments. In fig. 10 E(t) and O(t) curves are compared for sample PS 1. The two curves are very similar indicating that the relaxation of the orientation is closely related to the plateau region and the beginning of the terminal zone. The molecular-weight dependence of the orientation master curve (fig. 9) is clearly explained by the extent of the plateau region at long time with the increase of molecular weight as found from the mechanical viscoelasticity studies. MOLECULAR TECHNIQUEs AND VISCOELASTICITY FLUORESCENCE POLARIZATION STUDIES Unlike the i.r. dichroism experiments which yield only the chain segment orientation averaged over all the polymer chains in the sample the labelling required for f.p.measurements allows us to look at the orientation behaviour of only labelled chains. In f.p. studies on PS performed in our laboratory the anthracene fluorescent group was located in the middle of the polymer chain (fig. 6)and < 1 % (wt/wt) of labelled chains were mixed with normal PS chains. The molecular weights of the labelled chains used denoted PAP are reported in table 3. Stretching was carried out on the same machine as for i.r.d. studies but now f.p. measurements are performed during the stretching. The samples moulded under pressure and annealed were 8 cm long 2 cm wide and 0.2 cm thick. Table 3. Molecular weight of anthra-cene-labelled polystyrene chains labelled polymer M PAP 17 17 000 PAP 33 33 000 PAP 77 77 000 PAP 140 140 000 PAP 287 287 000 PAP 370 370 000 PAP 500 500 000 PAP 930 930 000 COMPARISON OF F.P.AND I.R.D. ORIENTATIONS The first point is to compare the orientation evolution with T and 6 given by f.p. and i.r.d. measurements. Results derived from f.p. for PAP 370 in a PS 400 matrix are presented in fig. 7 with the data for PS 100 and PS 900 obtained from i.r.d. At temperatures far from there is a satisfactory agreement the orientation of PS 400 lying between those of PS 100 and PS 900 as expected. On the other hand at 113 OC f.p. leads to a much lower degree of orientation than i.r.d. Such behaviour has been confirmed by the fact that the orientation observed by f.p.does not seem to depend on the glassy part of the stress but is related to the rubber-like comp~nent.~~ A possible origin of this is discussed later. INFLUENCE OF THE MOLECULAR WEIGHT OF LABELLED CHAINS Another interesting feature is the influence of the molecular weight of the labelled chain when that of the polymer matrix is kept constant. Thus experiments have been recently performed by J. F. Tassin and C. Ayrault at 128.5 OC with a PS 160 matrix. Results are presented in fig. 1 1 at a strain-rate value i = 0.115 s-l; similar behaviour is observed at other strain rates. First there is an increase in orientation with the molecular weight of the labelled chain proving that the orientation determined by f.p. is sensitive to overall chain relaxation.However a more surprising result is the rather high orientation obtained for PAP 17 chains. Indeed as the molecular weight of this labelled polymer is around the mean molecular weight between entanglements for PS (Me 1 16000) one would have expected a rather low orientation or even no orientation if only topological constraints L. MONNERIE / A Fig. 11. Orientation function (P2(cos 0)) plotted against draw ratio for various molecular weights of the PAP-labelled chains in a PS 160 matrix. Stretching temperature T= 128.5OC. Strain rate d = 0.115 s-l. (1) PAP 17 (2) PAP 33 (3) PAP 77 (4) PAP 140 (5) PAP 370 (6) PAP 500 (7) PAP 930. were considered. Such chains are too short to be oriented efficiently by the deformation of the physical network.On the other hand it seems that their orientation comes from the anisotropy of the surrounding medium. Note that similar orientation effects have been observed for pendant polyisoprene chains in a chemically crosslinked The pendant chains labelled inside the chain or at the end of the chain exhibit an orientation which increases with A although always remaining lower than the orientation of the labelled chain involved in the permanent network. Furthermore free fluorescent probes made up of 9,lO-dialkylanthracene with 16 CH groups are oriented at > 3.23Thus from these results it appears that in addition to orientation arising from topological effects (a physical entanglement network) and which could be described by molecular-viscoelasticity theories based on the tube concept there is another contribution arising from the interactions with the surrounding anisotropic medium.Although it could be argued that this effect is specific to f.p. and originates from the perturbation introduced by the fluorescent label the fact that the f.p. orientation behaviour observed at 128.5 *Cis similar to that found from i.r.d. measurements indicates that the contribution from the anisotropic medium is also involved to some extent in i.r.d. Indeed the statistical unit of a polymer chain corresponds to an anisotropic object which can also be oriented by interaction with the strained surroundings. It is now possible to account for the fact that f.p. orientation unlike that of i.r.d. MOLECULAR TECHNIQUES AND VISCOELASTICITY does not depend on the glassy part of the stress which means that f.p.orientation only appears over some values of A (A > 1.2-1.3). Indeed the label inside the chain should induce a longer anisotropic object than the statistical segment of the polymer chain and it has been shown e~perimentally~~ that the longer an anisotropic flu- orescent probe is the larger is the value of A required to orient it. INFLUENCE OF THE MOLECULAR WEIGHT OF THE POLYMER MATRIX The last study which has been performed concerns the influence of the molecular weight of the polymer matrix on the orientation of PAP 287.25The stretching was performed at 128.5 OC for various constant strain rates. PAP orientations obtained at A =4 are plotted in fig.12 as a function of the ratio of molecular weights of the polymer matrix and the labelled chain M,,/MpAp.From this figure three main features can be pointed out (1) there is a limiting value of (P2(c0s8)) which is independent of ,4 (P2,max 0.04); (2) the molecular weight of the matrix required to = reach this limit depends on d and increases as d decreases; (3) for matrices smaller than PS 900 an influence of d is observed P2 increases as d increases up to the limiting value P2,max. 0.051 Fig. 12. Orientation function (P2(cos 0)) measured at A. =4 at various strain rates plotted against the ratio Mn(matrix)/MPAP.t =0.115 s-l; A,d =0.059 s-l; **, 0 d =0.029 s-l. These results are very interesting for they raise a question concerning the validity of the theory of Doi and Edwards.26 Indeed from the latter’s slip-link model at a given d the same orientation of the labelled chain should be obtained independent of the molecular weight of the matrix as far as Mmatrix>MPAP,i.e.if the environ- ment is fixed relative to the motion of the labelled chain. Such behaviour is qualitatively observed at the higher t (0.1 15 s-l) but for lower values the orientation increases with increasing molecular weight of the matrix. Among the three relaxation processes considered by DO^,^' it appears that the disengagement of the chain from its original tube is not involved under the experimental conditions used since it has been shown19 that deformation of the sample is completely recoverable by annealing treatment above q.Thus it seems that the orientation is mostly affected by the shrinking of the chain into its deformed tube characterized by a relaxation time zBwhich should scale as M2,where M is the chain molecular weight.Indeed if the data of fig. 12 are plotted against the quantity all the points lie close to a single curve shown in fig. 13. At first sight such qualitative agreement would *be surprising since it has been shown above that the f.p. orientation contains two contributions one related to topological effects and tentatively described by the slip-link model the other reflecting the anisotropy of the surrounding medium. However it is clear that this medium L. MONNERIE I I 0 9 18 10-10 EM2 n Fig. 13. Orientation function (Pz(cos8)) measured at 2 = 4 plotted against the parameter -Data obtained from .i = 0.115 s-l; A,.i = 0.059 s-1; * i = 0.029 s-1; 0, i:M:(matrix).0 data obtained for M(matrix) = MpAp.anisotropy is governed by the same topological relaxation effects. Thus in the experiment concerned in which the shrinking of the chain is the dominant relaxation process a Mi(matrix) dependence should be expected. Although these results qualitatively support the M2 dependence of zB they nevertheless prove unambiguously that an improvement of the Doi-Edwards theory is required to take into account the effect of the polymer matrix molecular weight. This has been recently done28 by taking into account that in strained polymer melts unlike free chains in a deformed permanent network there is a coupling between the relaxation of a labelled chain and the relaxation of the matrix chains.Thus in the slip-link model a self-consistent treatment has to be made which leads to a new concept of tube relaxation i.e. partial disappearance of the topological constraints on the labelled chain resulting from the relaxation of the surrounding chains. Of course the tube relaxation effect depends on the relative molecular weights of the labelled and matrix chains; the theoretical predictions are in qualitative agreement with the experimental results. NEW INSIGHT INTO POLYMER MELT RELAXATION The studies reported above on the orientation of uniaxially stretched polymer melts show that there is a good agreement between the relaxation curve of the average chain orientation determined from i.r.d.and the modulus relaxation curve. However the f.p. results show that although the M2 dependence of the chain- shrinkage relaxation time predicted by Doi2’ seems to be confirmed qualitatively it is necessary to improve the slip-link model first by considering a topological coupling between the chains in the melt and secondly by taking into account the anisotropy of the strained medium on the orientation of chain segments and its consequence with regard to chain relaxation. At the present time only the topological coupling has been considered.28 ORIENTATION- AND CHAIN-RELAXATION BEHAVIOUR OF COMPATIBLE POLYMER BLENDS As mentioned above i.r. dichroism can be a powerful technique for examining the chain orientation of each partner in a polymer blend.Such studies have been carried out in our laboratory29 on mixtures of atactic polystyrene PS 1 and poly(2,6- MOLECULAR TECHNIQUES AND VISCOELASTICITY dimethyl-l,4-phenylene oxide) (PDMPO M = 23000 M = SOOOO) and they lead to compatible blends over the entire concentration range. The local chain axis used in PDMPO is shown in fig. 3 and the convenient vibration bands are 906 and 865 cm-l for PS and PDMPO respectively. Films from various blends with PDMPO concentrations in the range 0-35 % (wt/wt) have been obtained from solution and then stretched at constant strain rate at a temperature T = Tg = 11.5 OC where Tg refers to the glass-rubber transition temperature of the blend under investigation.-3 I I I The first unexpected result concerning orientation is that the two polymers do not behave in the same way when the composition of the blend is varied. The PDMPO orientation is high and does not depend on its concentration in the blend unlike the PS orientation which is affected by the amount of PDMPO present. Orientation data for each polymer have been treated according to the Lodge method to obtain orientation-relaxation functions O(t) in a similar way as indicated earlier for pure PS. Only one master curve is obtained for PDMPO orientation independent of the composition of the blend. In contrast each composition leads to a different master curve for PS orientation as shown on fig. 14. At short times the PS orientation increases wth PDMPO content up to a concentration of lo% beyond which it remains constant.For long times different behaviour is observed the PS orientation relaxation becoming increasingly slow as the DPMPO concentration is raised. It is reasonable to assign the orientation behaviour observed in PS/PDMPO blends to the high stiffness of the PDMPO chains. Because of this stiffness PDMPO chain relaxation occurs on a timescale longer than that which is involved during stretching in the experimental conditions used. Owing to entanglements one can consider that after deformation PDMPO chains participate in a highly oriented physical network. Interaction between PS and PDMPO chains hinders PS relaxation over the entire timescale studied. For short relaxation times the interaction factor certainly prevails in hindering local motions of the PS chains.For longer relaxation times besides the interaction term topological constraints arising from the highly oriented and slowly relaxing PDMPO network hinder large-scale motions of the PS chains (such an L. MONNERIE 81 interpretation is supported by the influence of the molecular weight of the PDMPO chains). In some ways the effects of PDMPO chains in the matrix are similar to those reported above for high-molecular-weight polystyrene matrices. G. Allen and J. S. Higgins Rep. Prog. Phys. 1973,36 1073. C. Picot in Static and Dynamic Properties of the Polymeric Solid State ed. R. A. Pethrick and R. W. Richards (N.A.S.I. Series D. Reidel Dordrecht 1982) pp.127-172. A. Maconnachie and R. W. Richards Polymer 1978 19 739. A. Z. Akcasu G. C. Summerfield S. N. Jahshan C. C. Han C. Y. Kim and H. Yu J. Polym. Sci. Polym. Phys Ed. 1980 18 863. F. Boue M. Nierlich and L. Leibler Polymer 1982 23 29. J. P. Cotton D. Decker H. Benoit B. Farnoux J. Higgins G. Jannink R. Ober C. Picot and J. des Cloizeaux Macromolecules 1974 7 863. ' D. J. Yoon and P. J. Flory Polymer 1975 16 645. M. Rawiso and C. Picot to be published. M. Beltzung J. Herz C. Picot J. Bastide and R. Duplessix ZUPAC Znf. Bull. 1981 275. lo H. Benoit R. Duplessix R. Ober M. Daoud J. P. Cotton B. Farnoux and G. Jannink Macro-molecules 1975 8 451. l1 F. BouC and G. Jannink J. Phys. (Paris) C2 1978 39 183. l2 F. Bod J. Bastide M. Nierlich and K.Osaki to be published. l3 D. I.Bower J. Polym. Sci. Polym. Phys. Ed. 1981 19 93. l4 B. Jasse and J. L. Koenig J. Macromol. Sci.,Part C,1979 17 61. l5 L. Monnerie Static and Dynamic Properties of the Polymeric Solid State ed. R. A. Pethrick and R. W. Richards (N.A.S.I.Series D. Reidel Dordrecht 1982) pp. 383414. l6 J. P. Jarry and L. Monnerie J. Polym. Sci. Polym. Phys. Ed. 1978 16 443. J. P. Jarry P. Sergot C. Pambrun and L. Monnerie J. Phys. E 1978 11 702. J. P. Jarry and L. Monnerie Macromolecules 1979 12 927. l9 R. Fajolle J. F. Tassin P. Sergot C. Pambrun and L. Monnerie Polymer 1983,24 379. 2o D. Lefebvre B. Jasse and L. Monnerie Polymer in press. 21 A. S. Lodge Trans. Faraday SOC. 1956 52 120. 22 D. J. Plazeck J. Phys. Chem. 1965 69 3480. 23 J.P. Queslel Thbe Docteur-Znginieur (Paris 1982). 24 J. P. Jarry ThPse Docteur-6s Sciences (Paris 1978). 25 J. F. Tassin and L. Monnerie J. Polym. Sci. Polym. Phys. Ed. in press. 26 M. Doi and S. F. Edwards J. Chem. SOC. Faraday Trans. 2 1978 74 1789; 1802; 1818. 27 M. Doi J. Polym. Sci. Polym. Phys. Ed. 1980 18 1005. 28 J. L. Viovy L. Monnerie and J. F. Tassin J. Polym. Sci. Polym. Phys. Ed. in press. 29 D. Lefebvre B. Jasse and L. Monnerie Polymer in press.

ISSN:0301-5696    

DOI:10.1039/FS9831800057

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

Furuday Symp. Chem. SOC.,1983 18 83-102 Dynamics of Molten Polymers on the Sub-molecular Scale Application of Small-angle Neutron Scattering to Transient Relaxation BY F. BouEi,* M. NIERLICH AND K. OSAKI S.R.M. C.E.N. Saclay 91 191 Gif-sur-Yvette France Received 2nd September 1983 The application of small-angle neutron scattering (SANS) to transient viscoelastic relaxation is advantageous in that it can for example give the form factor of a single chain in a polystyrene melt undergoing transient relaxation namely restoration (stress relaxation) at constant shape after a sudden deformation. The deformation is typically a uniaxial extension of ratio = 3 and the form factor is measured instantaneously on samples quenched from deformation temperature at different times t after the extension.Complete data are given here for one partly reported experiment [F. Boue et al. J. Phys. (Paris) 1982 43 137; J. Phys. (Paris) Lett. 1982 43 L585/L591 and L593/L600] covering an increased range of scattering vector q and time t. These data permit comparison of calculated form factors for the Rouse model and the tube model (reptation) confirming the following. (a) At small times the data agree both with the Rouse model and with the tube model which assumes a three-dimensional Rouse motion over lengths smaller than the tube diameter. At larger times the data disagree with the Rouse model. (b)A previously reported discrepancy with the tube model concerning the process of contraction of the deformed chain in its ‘tube’ persists [F.Boue et al. J. Phys. (Paris) 1982 43 137 M. Doi and S. F. Edwards J. Chem. Soc. Faraday Trans. 2 1978 74 1789 1802 and 18181. (c) At larger times data are not in disagreement with the tube model (disengagement process) if one accepts that contraction does not occur but that the chain has already relaxed by this stage up to the scale of tube diameter. Supportive results from other experiments which we have carried out are briefly presented. 1. PRINCIPLE OF THE EXPERIMENT In this work one measures the elastic neutron scattering at small angles from polystyrene melt samples containing some perdeuterated chains (PSD) in a matrix of non-deuterated chains (PSH). Scattering is elastic (which however gives information on chain dynamics) because the samples examined correspond to successive states depending on time of a melt undergoing transient relaxation.In practice the sample is made from a strip of PSH +PSD mixture which has been carefully moulded and annealed. This strip is then clamped in a stretching machine and heated at a constant temperature T above the glass-transition temperature. Then it is extended during a relatively short time interval t and thereafter maintained at constant deformed length and temperature. These are the conditions required for transient states approaching equilibrium at infinite time. Since the samples are not cross-linked the equilibrium value of the tensile strength is zero. For experimental purposes the sample is quenched at time t measured from the end of the deformation period t,.Samples are prepared in this way for each value of t and the whole set is subsequently examined in the quenched condition by neutron scattering. From each sample at a given t one obtains the static form factor of one embedded chain:? t The small-angle neutron scattering (SANS) technique is explained by Prof. Monnerie in his paper at this Symposium. Also see ref. (1) for pioneer work and ref. (2) for the use of high PSD concentrations as employed here. 83 DYNAMICS OF MOLTEN POLYMERS where i and j index the N monomers of the chain ri(t)and ri(t) are their positions at the same time t and q is the scattering vector. This approach has the advantages that it introduces the wavevector q which permits exploration in space and the establishment of a (q t)relation isotopically labelled chains (unlike chemically labelled species) do not display a propensity to demix in the bulk the available q-range easily covers submolecular lengths from monomers (5 A; q FZ 2 x 10-l A-l) to chains of M = los (300 A; q x 7 x A-l) and finally the techniques can be applied over a large t-range.The upper time limit is related to defects in the sample but can be 1/10 of the terminal time for stress relaxation in contrast to the case of inelastic scattering from melts at rest where smaller ratios are obtained for the most rapidly relaxing melts (PDMS at 100 0C).3[Values for characteristic times such as the terminal time are given in ref. (4).] POTENTIAL AND LIMITATIONS It is now necessary to discuss critically the potential and limitations of the technique in terms of the practical ranges of the parameters (q and t) and of the uncertainties in temperature deformation and derived form factor.STATE OF DEFORMATION Extension and relaxation can either be performed successively in the same heating medium (air or oil) or separately should it be necessary to examine the stretched sample before conducting the relaxation in another device. Both the first5-’ and second procedures have been employed.8 If the relaxation is very long inhomogeneity will ultimately appear but as for the extension the essential conditions are the use of a well annealed isotropic strip which is initially constant in thickness and width effective clamps but most of all an oven temperature constant throughout the heating medium.Good homogeneity of the final strip is the best guarantee of processing and indeed is required for proper characterisation of the deformation. A geometrical test is to measure the initial thickness eo(r)at a point r and the distance between dots painted on the surface in the two directions parallel zo(r),and perpendicular yo(r),to the extension. The same quantities are then measured in the final state ef,zf,yf,at the same point in the material (whose position vector is now r’). We write The scatter of the three quantities &,A and A, which ideally should be equal gives the uncertainty AA. We obtained AA < 0.05 for all samples which had relaxed for a period t less than one hundredth of the terminal time.We believe that AA < 0.02 is achievable by careful work. For larger values of t only AA < 0.15 was obtained but it should be possible to reduce this spread by better annealing and an improved temperature distribution down to perhaps 0.05 for t < T,,,/5. It is also possible if e is both sufficiently constant and less than 0.3 mm to use an optical birefringence test using white light where one seeks uniformity of colour. DEFORMATION HISTORY If the final state appears homogeneous by the above and other tests there is good reason to believe that the deformation history was identical at all points. It is measured by the variation with time of the total length [(t)(A[/[ < 1%) the tensile strength (AF/F < 5% ) and the light transmission between crossed nichols [cos2 e(t)(n,-n,) (t)] F.BOUE M. NIERLICH AND K. OSAKI n (n,) being the refractive indices in directions z 01).Repeated extension under the same conditions showed these quantities to be reproducible to within 5% .7 TEMPERATURE One has to distinguish spatial and time variations of temperature for each sample and also reproducibility between each sample for perturbations due to all variations of the external temperature ;this can modify space distributions before each extension. The spatial variation arises mainly from vertical temperature gradients in the equipment but is also affected by impulsive draughts during and following the extension because the relative positions of some metallic components are changed. Tests with a sensor in place of the sample7 indicated that spatial and time variations were bounded within & 0.5 "C.However by careful adjustment of a@ air oven or by using a thermostatted room it proved possible to obtain variations < 0.1 "C.Such a value is easily attainable with an oil bath if one removes the vertical gradient; a practical limit would be 0.05 "C since an uncertainty of f0.05 "C is caused from heat produced or absorbed on stretching7 TIMES The first two relevant times concern the act of extension.This was conducted at a constant speed gradient s and thus lasted a time t = (1/s) (In A). As s was varied appropriately to the chosen temperature between 0.06 and 0.18 s-l t ranged for A= 3 between 7 and 20 s; the inverse of s spans 5-17 s.These numbers illustrate the rapidity of the deformation. The duration of the quenching must be added but was found7 to be equal to 15 s. These times are to be compared with the characteristic times of the polystyrene melt variation of which led to the time-temperature superposition4 given previ~usly.~ Quenching times could roughly be reduced to < 5 s. t and l/s are already the lower limits for the corresponding temperatures at which they were used (see table 1). Were faster stretching employed plastic deformation or breaking would occur. We have compared the relaxation of two strips one stretched at a given rate so the second at a rate 2s,. For relaxation of duration t > 2tso,no significant difference appears either in recorded curves of stress relaxation or in the form factors of samples quenched at t.It is thus possible for t > 2t, to disregard the exact elongation history and thus to compare the present results with models which assume a sudden deformation. Even for t "N 2t, it remains true that the elongational histories were very similar for different samples (scatter in stress was believed to be < 5%). In practice the minimum value of t must be set at the minimum of t + t, equal to 10+ 15 = 25 s at 117 "C. t cannot be reduced to < 5 s by reducing the temperature because the sample would break so one can only achieve a reduction in t, the quenching time. The maximum value of t is easy to obtain by increasing the temperature which can rise to 140 "C. However here the essential limiting factor is the deterioration of the strip; it is difficult for t to exceed one tenth of the terminal stress relaxation time of the matrix.SCATTERING VECTOR RANGE The range of 141 presently available is 7 x 10-3-2 x 10-1 A-l. The lower limit is larger than the lowest value of q handled by the most powerful small-angle machine the spectrometer D11 at the I.L.L. which is ca. 1 x A-l. However at this q-vector counting times become very long because of the large sample-detector distance. In addition scattering by microvoids becomes important for 141 < 5 x A-1. Since these voids are difficult to suppress the lower limit of the spectrometer is not a serious DYNAMICS OF MOLTEN POLYMERS problem. The upper limit is imposed by the problem of subtraction of the incoherent background2.(see also Prof. Monnerie's paper at this Symposium). Thus for a high concentration (C = 15%) of labelled chains the ratio of coherent to incoherent scattering will be for the most oriented sample (no. 71) 1280/840 in the perpendicular direction and 834/772 in the parallel direction for 141 = 2 x 10-1 A-l. If the thickness and the neutron transmission of the sample necessary to subtract the substantial incoherent background are known within lo% the uncertainty of the form factor AS,(q)/S,(q)will be > 30% whereasit is < 10% at 141 = 1 x 10-1 A-l. Thisessentially determines the maximum value of 141. TREATMENT OF NEUTRON MEASUREMENTS Assuming that the classical treatment we used2 is correct the essential uncertainties arise as follows.First is the subtraction of the incoherent background at large 141 the error decreases very rapidly as (41 decreases. For 1q1 = 3 x A-l it is < 2 x Second is the normalisation of the form factor. This quantity has to be compared for different samples of different volumes (thicknesses) and different neutron transmissions so it is necessary to divide the coherent scattering by these quantities. This leads to a normalisation uncertainty of ca. lo% although actual experiments appeared to be better than this. By using homogeneous samples of the same thickness this uncertainty could be reduced to less than a few percent. SCIENTIFIC BACKGROUND The present experiment operates at the microscopic scale (5-300 A) yet in the same field as various macroscopic techniques which have been compared with or even inspired by modern the~ries.~? lo Thus in the regime of large times near the terminal time for stress relaxation there have been measurements of the stress terminal time (qercc lW4), the self-diffusion coefficientll and fracture welding,l2? l3all of which were to be in good agreement with the de Gennes theory of reptation (disengagement).In the regime of smaller times macroscopic experiments have involved measurements of overstress in the regime of non-linear deformation and of its relaxation; these were found to be in agreement with the Doi-EdwardslO prediction. In an endeavour to link the macroscopic and microscopic approaches an attempt is made in this paper to discuss the results of one experiment in terms of the Rouse and reptation models9~ lo at the microscopic level in the two regimes of long and short times.Finally other experiments will be introduced into this comparison. 2. EXPERIMENTAL Some of the data presented below have already been reported namely variation of the transverse radius of gyration R,,(t) extracted at small q,5and observation of S,(q)in the parallel direction for large t.5Following those papers and comments above the experimental description will be brief. There then follows an improved comparison with models using all the data and calculated functions S,(4). Samples were made from a mixture of PSD and PSH of molecular weights given in table 1. They were moulded in vacuum (2 x Torr) and annealed; the stretching conditions are given in table 2.The speed gradient was held constant and the oven was heated by air.* Five different temperatures were used with the uncertainty explained above allowing coverage of a large range of the ratio t/T,,,(T) (for polystyrene Tgz 100 "C).The tensile force and the birefringence {in fact cos2[e(t) An( t)])were measured during the stretching as well as during the * We used a machine in Prof. Monnerie's laboratory [a description is given in ref. (6)]. F. Bod M. NIERLICH AND K. OSAKI Table 1. Preparation characteristics of the samples (isotropic) rate of duration of sample number stretching /s-l stretching /s duration of relaxation T/"C 71 0.189 7 30 s 113 10 and 7 0.06 20 10 s 117 19 0.06 20 20 s 117 18 0.06 20 1 min 117 6 0.06 20 4 min 117 9 0.06 20 20 min 117 24 0.115 10 10 s 122 23 0.1 15 10 1 min 122 25 0.115 10 20 min 122 42 0.189 7 30 s 128 44 0.189 7 1 min 128 43 0.189 7 8 min 128 46 0.189 7 30 min 128 49 0.07 18 1 min 134 48 0.189 7 1 min 134 50 0.189 7 16 min 134 relaxation [see ref.(5) and (7)]. Birefringence An,and stress appeared to vary in the same way during the relaxation for t > 22,; for I < 2t and during stretching that was the case only for the highest temperatures 128 and 134 "C. For lower Tthe stretching was indeed much faster at a given s and the stress curves were vertically shifted by a quantity A(s T).The relaxation of this part A took < 1 s following the end of stretching. The concentration C, of labelled species was set high (1 5 % ) allowing more precise neutron data and the material handled as described earlier in this The two spectrometers D11 and D17 (1,L.L.) gave data on three partly overlapping ranges.Data were corrected in order to superimpose in the overlapping domains.6 The total q range was then 7 x 10-3-2 x 10-l A-l. Comparison of t-values with the characteristic times of the sample employs the W.L.F. superp~sition,~ for which we obtained5 T,,,, a 5 x lo3 s and q,,z 5 x lo5s but with large uncertainty5 while the temperature-reduced values of I P7,lie between 20 and lo5 s. 3. RESULTS The data for S,(q) are presented in fig. 1 in the representation q2St(q)as a function of log, (qRgiso),Rg being the radius of gyration measured for the isotropic sample.The q2St(q)data in this figure are presented in two sections. Thus in the lower part of the figure are the curves for different values of t in the parallel direction. For each time t as q increases the curve rises so becoming closer to the isotropic curve. This means that the chain is more isotropic and thus more relaxed at small distances. An increase in t also moves the curves systematically closer to the isotropic limit. For large t an inflexion point appears. In the upper part of the figure are plotted the curves for the perpendicular direction. They all display a maximum which comes from the quantity q2St(q).For small q this quantity which can be regarded as the linear density of the chain increases just as the isotropic case but lies above the isotropic curve because the chain contour is more dense and more compressed than the isotropic at this scale.For large q the chain DYNAMICS OF MOLTEN POLYMERS ....... OO 0.5 1 1.5 2 lOg,o qRg is0 Fig. 1. Data for the single-chain form factor S,(q) in the representation q2St(q)plotted against log, qRgisofor the following samples e,71 (reduced time at 1 17 "C = 10 s) ; + 18 (reduced time at 117 "C= 60 s); A,49 (9000 s) and 0, 50 (130000 s). The dotted line represents affine deformation and the dash-dotted line represents the isotropic Gaussian conformation. as with the parallel direction becomes more isotropic. The form factor returns to the isotropic case the linear density decreases and a maximum is produced.The lines in fig. 1 denote the theoretical value for the isotropic chain [i.e.the Debye function for a Brownian chain of the same R, D(q2),multiplied by q2]and for a chain deformed in a totally affine way i.e. S(q) = D(qBq),where B is the Finger tensor A 0 B= (0 l/z/A 0 0 0 l/dA (the curves are given for the two directions). F. Bod M. NIERLICH AND K. OSAKI 89 One can see that in the parallel direction all the experimental data except at the smallest t,have departed from the affine curve. In the perpendicular direction for small t and at small q the data coincide with the affine curve before departing from it at a higher q. The small-q region corresponds for the affine curve to the upper boundary of its Guinier range (qRgI< l) where the behaviour of the form factor can be characterized entirely by the value of the radius of gyration.Thus the coincidence of the data and the affine curve in this range is equivalent to the fact that the transverse radius R,,(t) is the affine one at small t this was reported previously in ref. (5) for L = 3 as well as for L = 2 and 1.5. In contrast for the parallel direction one can see that the Guinier range of the affine curve is well below the experimental q-range; it is then impossible to extract the value of RgII.However we have already remarked that at the smallest t the data are close to the affine parallel curve we can deduce that they would coincide at smaller q supporting an affine value of R,,, (t)for this very small value of t.TIME-TEMPERATURE SUPERPOSITION One new result of this experiment is to show within experimental uncertainty that the time-temperature superposition works for S,(q) on a submolecular scale. Two samples prepared at two different temperatures with the same value of reduced time tTo= (aT0/aTl) t,(T,) lead to the same form factor. This has already t,(T,) = (aTO/aT2) been reported for R,(t); it has now been checked for the whole form factor for several values of reduced time as in fig. 2. 4. DISCUSSION COMPARISON WITH MODELS THE ROUSE MODEL In this classical mode114 the chain behaves as if were free in a viscous medium. This gives a set of modes zp = (p/W2TRouse corresponding to the motion of sequences of p sub-units. The rate of relaxation from a deformed shape will thus depend on the length scale at which it is observed the characteristic time at scale r will be z(r)cc r4 [as r K d(zp)for Brownian chains].Consequently at time t the form factor will vary with q in the following way. (i) At high values of q corresponding to small values of r the chain conformation appears to be relaxed i.e. the form factor will be close to the one for the isotropic Brownian chain. (ii) At small values of q the chain conformation will remain close to the initial deformed shape at t =0. As t increases the onset of relaxed behaviour (return to the isotropic form factor) will be obtained at a smaller value of q proportional to t-lI4. We have calculated Spouse(q), taking as the initial condition an affine deformation of the chain at all scales.This calculation is given elsewhere;15 the results are the dotted and solid lines of fig. 3 again in the q2S,(q)against log,,q representation for different t/TROuSe ratios. The relaxation of the form factor is as qualitatively described above in both directions the curves move from the affine case back to the curve for the isotropic material rising for the parallel direction and falling for the perpendicular direction which in that case produces a maximum in the curves. The abscissa of the onset of this departure depends on time. Let us first compare predictions and experiment for the smallest value of t e.g. t/ TRouse= 5 x lop5compared with data for t = 10 s (a)and t = 60 s (+).If we make a very qualitative comparison we can say that the shape of the curves is similar in both directions.If we increase the accuracy of the comparison we observe some DYNAMICS OF MOLTEN POLYMERS 0.23 h v) c. O+ .d E +a 1 0 + e v 0 h CIt b v O+ h” + n c7. 0 + 0 + 0 + 0.115 0 + 0 + 0 + I 1 1 0 0.5 1 1.5 2 log10 qR,, Fig. 2. Agreement with the time-temperature superposition showing a comparison of two samples stretched and relaxed at two different temperatures in order to have the same temperature-reduced time f1170C = 80 s + 18 T = 117 “C;0,24 T = 122 “C. systematic differences. The filled circles (0,t = 10 s) lie on the theoretical line in the parallel direction only at low q; they depart from it by staying closer to the affine line (i.e.more oriented than predicted).In the perpendicular direction they always lie above the theoretical line. Points marked by a cross (+ t = 60 s) lie above the theoretical line in the parallel direction; in the perpendicular they lie slightly below the theoretical line at low q and slightly above it at large q (i.e.again more oriented at high q than predicted). If we now take the theoretical line for t/ TRouse= 10-4 we see an agreement in the parallel direction with data for t = 60 s (+).In contrast the data for the same volume of t in the perpendicular direction lie well above the line except at low q. In summary the difference in shape between experimental and theoretical curves permits F.BOU& M.NIERLICH AND K. OSAKI ....*-.....I. ........ I 1 3 I 1 2.5 2 h *v, .-e e 2 1.5 n b v 4-n b 1 0.5 Fig. 3. Calculated form factor S,(q)for a Rouse model in the [q2St(q),log, qR,,,,] representation for t/TRouse= 5 x lop3 5 x lop2 lop1 2 x 10-l and 4x 10-l (full lines). Extrapolations following eqn (2) in the text are plotted for 5 x and as the dotted lines. The line made up of large dots represents affine deformation and the dash-dotted line represents the isotropic Gaussian conformation. The symbols (which are the same as those in fig. 1) show a comparison with the data. DYNAMICS OF MOLTEN POLYMERS 0.23 h U .d E e W h a vp. hi a+x 0 0 N cl. a+' +* 0 0.115 -8 + x 0 0 + x 0 0 0 I log, (4f;) Fig.4. (a) For legend see opposite. only a cruae owervation me experimental aara ror IU <t/s <ou overlap wirn theoretical lines for 5 x <t/TROuSe < For larger values of t/TROuSeno overlapping is possible. From this an experimental value of TRouse can be obtained as lo4 <Tg&se/s<5xlo5 at T =117 "C.This can be compared to (i) the value of the terminal time that we obtained from the literature4 and from our stress measurements:5 5 xlo5 <Ter/s <5 xlo6 at 117 "C and (ii) the value of the Rouse time of a free chain extrapolated from data for low M,* assuming TRousecc M2:5xlo3 <TRouse/s<5 xlo4. Tkx&, lies between the maximum for (i) and the minimum for (ii).For larger values of t/TRousewe have then to retain this estimate of T"R"opuse. The t/TRouseline must be compared to data for 50 <t/s <2500. However even data for t =9000 s appear much less close to the isotropic line than the Rouse line! The disagreement is still larger at larger q. F. BOUE M. NIERLICH AND K. OSAKI I I I (b ... 0 0 0 0 0 0 0.5 1 1.5 2 25 log, (4& Fig. 4. (a) Test of a qtl/*superposition for different values of t1170C using a horizontal shift of OC curves in the [q2St(q),log,,qRgi,,] representation for small times t117 = 10 (O), 20 (+) 60 (x) and 240 s (0).(b) Test of a qf1/4 superposition for different values of f1170C using a OC horizontal shift of curves in the [q2St(q),log, qRgi,,] representation for large times t117 = 9000 (@) and 130000 s (0).Thus the evolution of S+(q)with t appears qualitatively to be equally fast at small t but slows down at large t and the Rouse model appears much too fast. The comparison that we have just made has some disadvantages which arise from the difference in shape between the experimental and theoretical curves and from the need for a value for T,,,,,. This can be avoided as follows. THE qf114SUPERPOSITION LAW This alternative test of the Rouse model is based on the following theoretical prediction the fact (see above) that a return to the isotropic form factor begins at a value q* cc t-l14 leads more precisely to the asymptotic law15 at high q the behaviour Sis,(q)cc l/q2 corresponding to the plateau of q2Sis,(q)at high q (fig.1). Eqn (2) is a (q,t) superposition law. By plotting q2St(q)against log, qt114it should be possible to superimpose the curves; fig. 4(a) shows that this is indeed the case in the perpendicular and parallel directions for small ratios t/TRouse, but not for higher DYNAMICS OF MOLTEN POLYMERS 0 t (117°C) Fig. 5. Values of the horizontal shift log, q obtained by superimposition of plots as in fig. 4(a) and (b) plotted against log, t. Solid lines have the slope 1/4. ratios confirming earlier findings. (In the perpendicular direction the superposition can only occur after the maximum in order for it to be in the asymptotic regime.) However a second apparent superposition appears [fig. 4(b)] for the largest times reached it has already been reported for the parallel dire~tion,~.but is not found in the perpendicular direction. This superposition can also be achieved by measuring the horizontal shift necessary to overlap two plots of q2St(q)when plotted against log, t. Fig. 5 shows (i) a slope 1/4 at small t in agreement with the Rouse model (ii) a departure from that slope at higher t with a weaker slope corresponding to the slowing down of the real process and (iii) a second part at high t also of slope 1/4 although not on the same straight line. This is the one previously rep~rted,~? and it will be discussed below. To summarise the Rouse model starting from an affine deformation agrees with present data at small t but predicts too rapid a relaxation as t increases.However a different qt1/4superposition seems to appear at large times. THE TUBE MODEL This model considers the chain to be embedded in the matrix as confined in a tube made by the surrounding chains. Three motional processes are included (i) a wriggling three-dimensional motion inside the tube at scales smaller than its diameter D; (ii) a one-dimensional Rouse motion along the tube of maximum time Tes= T~,,,,cc N2 which is the time fluctuation of the length of the tube; (iii) a consecutive back-and-forth motion of the whole chain along the tube axis leading to a progressive disengagement of the chain from the original tube creating by the two ends a new tube within a time Tdiscc N3. These concepts were first developed by de Gennes9 and then extended by Doi and Edwards, especially for deformed systems.This work was the basis for the calculation of S,(q).16 The three processes are regarded as strictly consecutive because the chain is long enough. Recall that in the first process the form factor accords with the Rouse model up to a time t = T, which is the Rouse time of a chain of N monomers of size D (D cc Z/N, T cc NE). This process then stops. The second process which for a melt at rest leads to a fluctuation of tube length L leads for a deformed melt systematically to a contraction of L because the value after affine deformation Laffis larger than the equilibrium value Lo.During that contraction the form factor at small q related to the density of monomers in the chain will increase whatever the direction of q F.BO~, M. NIERLICH AND K. OSAKI because the contorted tube axis encompasses three dimensions; at the end of the second process the form factor increased for a given q by a factor Laff/L,,. The third process will start only once this situation has been attained and is desirable as disengagement from the deformed tube thereby creating two new isotropic ends of the tube the centre-deformed part decreasing in size with time. A detailed calculation by Dr Ball shows that the result is close to the form factor of a three-block copolymer isotropic-still deformed-i~otropic.~* l5 This is equivalent at high q to simply adding the contribution from the Np(t)monomers of the centre part and the M1 -p(t)] monomers of the end parts99 l5 l6 St(q) = ~(t) Saniso(q) + 1 -~(t)Siso(q)* (3) In eqn (3) the rate of relaxation is the same for different q values given the condition that q is high enough this is very different from the Rouse model and gives a much slower relaxation.At small angles eqn (3) is not valid; at the length scales of the radius of gyration the relaxation is again independent of q but with a different function. These questions are developed in ref. (1 5) with numerical evaluations. COMPARISON FOR THE FIRST PROCESS There is in this case no difference from the three-dimensional Rouse process which agrees with experimental data at small t. However as t increases this Rouse behaviour should stop and must be replaced with the form factor ofa chain relaxed on this length scale of a monomer sub-chain i.e.D.The corresponding time T, can be estimated if one knows N, it is classically the number of monomers between entanglements and is extracted from the plateau modulus value.3 For PS an acceptable value is N z 200 which gives Te= (Ne/N)2TRouse= (200/7000) x 5 x lo3 = 5 s at 117 "C. In fig.6are compared form factors for this timescale with an equivalent representation of the result of a Rouse relaxation. This avoids the difficulty of estimating T,explained above. The concept is of a Brownian chain simply obliged to pass affinely displaced points which are separated by N monomers along the chain:15 below the length scale of the distance between those points (ca.0)the chain is relaxed. Fig. 6 shows that best agreement is achieved for N/N 90 thus N = lo2 for samples 10 and 19.However for sample 18 (?17 = 70 s > q)relaxation does not seem to have stopped. One reason could be that another process has already started at this time. Let us now test if it is the second process of the Doi-Edwards model. SECOND PROCESS 'CONTRACTION ' This process should occur between Te and Tdis.According to our estimates4 the experimental t range covers all times when it could occur. One should observe an increase in the form factor in the parallel direction as well as in the perpendicular direction in the region q c 1/D. D can be estimated as the radius of gyration of a Brownian chain of N monomers (molecular weight Me):D = 0.275 2/M z40 A (from neutron-scattering data). For q < 1/40 = 2.5 x A-l the increase in &(q) should move the curve q2St(q)further from the isotropic one for that process.For the experimental data for the perpendicular direction only a continuous return down towards the isotropic curve and never up is observed as t increases. Thus in fig. 7 experimental data are contrasted with the calculated form factor after the end of contraction q c l/D (dotted line) the data lie below the graph. This result has been noted before using instead of S,(q) an extracted value of Rg(t).Note however that the variation with t is the opposite if St(q)increases R,(t) decreases. The same discrepancy with theory appears. It is apparent that the contraction could be masked I I 1 n Y ." E d W n P W t4- U b 3 2 1 .-r 1 I -1 -0.5 0 0.5 1 1.5 2 log, qR Fig.6. Comparison of data for the factor of a chain relaxed between entanglements S(q) = EC,Zjexp(-q2r:j/6) with i-j > N,,affine values for r$i i-j < N, isotropic value for r& (dashed line) and r&= (i-j)2/NZ,(R2-1) Nea2+(i-j) a2(full line) for N = N/30 N/60 N/90 N/120 (N = 7000). a,10 + 20 and x ,60 s [see ref. (7) and (16)]. F. BOUE M.NIERLICH AND K. OSAKI ">-' /' / / / I I I I 1 I Ii' /--1 0 1 2 log10 qRgiS0 Fig. 7.Comparison of data (same symbols as in fig. 1) with a form factor calculated by assuming the disengagement of a chain contracted in a tube of diameter zero. Chosen times are such that t/qer=0.1 0.2 0.4 and 0.8.4 FAR DYNAMICS OF MOLTEN POLYMERS by the beginning of disengagement this question will be discussed in section 5 with reference to newly available data. THIRD PROCESS DISENGAGEMENT The solid lines in fig. 7 show the results of the theoretical calculation16 assuming that contraction has occurred and is completely achieved and that the tube diameter Dis sufficiently small that qD 61. The figure allows a comparison with experimental data which shows that the effect of the disengagement on the form factor does not permit us to keep both assumptions and to account for the data. The disengagement induces an over-slow relaxation at low q in the perpendicular direction; at large q in both directions it always leads to a plateau as it averages [eqn (3)] between Saniso and Siso,which both give a plateau.Accordingly S,(q) was calculated taking account of our observations about the first and second processes we include D# 0 and we exclude the contraction. The initial state for disengagement then corresponds to the theoretical form factor plotted in fig. 6. The same calculation as was done for an initial contracted state in fig. 7 is then applied to the form factors of fig. 6. A corresponding comparison with the data is illustrated in fig. 8(a)and (b).Here it was necessary to choose the same value of N (here N,”,) for all the calculated curves however at least fig. 8(a)-(c) show that a common value of N/Ne = 120 does lead to a satisfactory agreement with t/Tter t/Tdis-DISENGAGEMENT AND FACTORISATION LAW We now provide a direct test of the disengagement removing the preceding modifications needed to calculate the form factor.As remarked above a law of this type expressed in eqn (3) is implied by the theoretical calculation of the disengagement. Agreement between the calculation and experimental data means that eqn (3) is obeyed within experimental uncertainty. Eqn (3) can be termed a factorization law since it can be written as St(q)-siso(q) = [saniso(q) -siso(q)l dt) (4) where the q-and t-dependences can be factorised. Thus for different t,the curves of log,,[S,(q) -Siso(q)] against qshould superimpose at high qby a q-independent vertical shift. The variation with tof this shift has been compared with two other quantities log,,[Ri,(t) -and log,,a(t) where CT is the stress.At large tthe three functional dependences coincide and are close to a linear variation as reported in ref. (6). However higher values of t/qer> 1 are necessary to check this variation. To avoid sample destruction at such values an experiment is planned in which long labelled chains are dissolved in a matrix of molecular weight three times larger than the already large molecular weight of the labelled chain. 5. COMPARISON WITH OTHER EXPERIMENTS The results discussed above show discrepancies with both the Rouse and the tube models. However a comparison with the Rouse model leads to the same conclusions as do macroscopic observations. At small times (or for small molecular weights) agreement is observed which vanishes at long times (see for example the variation with z of the retardation spectrum H(1n T)~ which follows the Rouse prediction (ccIn c at small z).On the other hand comparison with the tube model leads to discrepancies which are not observed at the macroscopic level. These discrepancies are discussed here. For this purpose supplementary SANS experiments were conducted on samples I Y " -1 -0.5 0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 1.5 log qR 1% qR N e Fig. 8. Comparison of data with a form factor calculated by assuming the disengagement of non-contracted chain with a non-zero tube diameter such as N = N/30 N/60 N/90 N/120 as in fig. 6. (a) Data from sample 50 compared with model for t/T = 0.4. (6) Data from sample 46 compared with model for t/ T = 0.1.DYNAMICS OF MOLTEN POLYMERS Table 2. Molecular-weight distribution of polymer samples PSD 784000 1.41 PSH 760000 1.51 PSD 95 500 1.17 PSH 117000 1.11 PSD 2 600 000 1.15 PSH 2 640 000 1.28 where the molecular weight of the chains is varied. Data for one mixture of very long PSD and PSH chains of about the same weight (M z 5 x los) and for one mixture of short chains (M z lo5) dissolved in a matrix either of the same molecular weight or one much higher (M > lo6) are categorised in table 2. HIGHER-MOLECULAR-WEIGHT CHAINS AND THE CONTRACTION PROCESS This experiment was planned to check the lack of contraction. The chosen values for t lie from 0 to above the estimated value of TRouse.Two extension ratios were used A= 3 and a higher one A = 4.6 which is judged to increase the contraction effect. The results are reported in detail in a forthcoming paper but the present conclusion is that while all the experimental orders of magnitude would in accord with theory lead to a contraction effect the latter is not observed. In particular the difference between TRouse and GiSis for this molecular weight large enough to prevent a complete masking of contraction by the commencement of disengagement. SMALL-MOLECULAR-WEIGHT CHAINS TUBE DIAMETER AND DISENGAGEMENT This experiment covers the usual q range and a time range of 10 s at 1 17 "Cto 20 min at 117 "C (qerx 5 x lo3 s at 117 "C for M = 95000). The essential results reported in ref.(6) are as follows. At small times agreement with the curve calculated from the Rouse model is found as for the mixture 780000 (PSD)+ 760000 (PSH) described earlier. At time t z Te,satisfactory agreement is found with the form factor of a chain relaxed at the length scale of the tube diameter with N = 2 x lo4,which is the same value of N as in the preceding experiment. From this given state one can calculate the effect of a disengagement process and it is found to agree with the data at the largest time (t117 = 20 min i.e. t/qerz 2/10). Let us recall that the divergences from Rouse behaviour would in any case be weak because the chains are short. EFFECT OF THE MATRIX For each value of t at different temperatures one sample has been made for the two matrices M x lo5 and M x 1.5 x log to enable a comparison of the two form factors to be made.For t/qerx (t/TRousez 5 x no difference appears; this is in agreement with the Doi and Edwards model. For the largest times (t/cerz 2/ 10 t/ TRousez l) one can see a difference of the order of the experimental uncertainty in the parallel direction where the form factor for the matrix of smaller molecular weight is more relaxed. However this smaller molecular weight could provoke sample deterioration as the ratio of t to the terminal time of the matrix is ca. 1/10 while it is lo3 times more for the other. A more suitable experiment has to be done for large t. F. BOUE M.NIERLICH AND K. OSAKI 6. GENERAL REMARKS AND CONCLUSIONS Our observation of the return of S,(q) to the isotropic form factor with time owing to a time-temperature superposition leads to systematic behaviour each stage of which is characterized by a different variation of the function S,(q) the rate at a given 4.When t is small a strong dependence upon q is observed such that only the small distances relax; the variation is close to that of the Rouse model [S,(q)x q4]. Then a second stage appears where the relaxation slows down very much while large distances now relax as well as short ones. This evolves to a third stage where it is possible to observe a q-independent rate [S,(q) = c,,]. This succession of stages almost exactly corresponds to the relaxation of stress with time (in linear regime) for a similar stepstrain experiment.The first stage corresponds to the transition zone at the end of this zone the slowing down of the stress relaxation ~. ~ starts for a time z~The time of~onset of the slowing down of S,(q) is of same order as T,~.Consecutively the Rouse model works for t,, for SANS as for stress. Evidently this is the same small-t region of the inelastic neutron-scattering experiment3 which also leads to the suggestion of Rouse motion. For t,, the Rouse model fails for S,(q) it also fails for stress relaxation for long chains. We are then led to test the tube theories which explain this slowing down. The disengagement process which is proposed leads to a very large terminal time. The q-independent rate (factorisation law) observed in the third stage of evolution of S,(q) (t/?& < 0.I) is characteristic of this disengagement.These conclusions in favour of the tube model must be modified by two remarks. First the test of a q-independent rate must be extended to larger t/qerratios. This may solve the paradox6 that S,(q) in the parallel direction seems to accord both to a superposition law and to a factorisation law.? The second remark concerns the second stage ofevolution of S,(q).For the non-linear regime the Doi and Edwards prediction agrees with the stress relaxation experiment the stress is higher at t 2Ttr than the linear value [i.e.0= (A2-l/A) G; G; being the plateau modulus4] and returns to the linear value for t x M2.In our SANS experiment the corresponding effect (contraction of the dimensions of the chain) is not observed.It then remains to be explained how both the stress behaviour and the lack of contraction can be consistent. We therefore propose to observe what motions occur when the tube model motions cannot be present i.e.when reptation is hindered this is the case for cross-linked chains. The same experiment would be performed. A strip of cross-linked polystyrene would be suddenly stretched and observed instantaneously by quenching using SANS during transient relaxation. This experiment has been started. Preliminary results are plotted in fig. 9. As the time t after the sudden deformation increases the form factor goes t The following explanation could be advanced starting from low q the onset of the behaviour exemplified by eqn (3) is encountered when it is possible simply to add the new and old tube contributions the calculation shows that this is possible when I/q is higher than the sizes of the different parts of the ‘triblock copolymer’.The relevant size is the smaller giving the smaller q; here it is the size of the new part rnew(t)zc J(:[I -p(f)]). From the expression of p(f)given in ref. (9) and (lo) we can develop it for f/Tdls < 1/10 81 ~(t) = X 7~XP(-p2t/GiJ z 1 -t (t/Tdls). PL This gves rnewa t1I3 an onset of eqn (3) at q a leading to an apparent superposition in the range of the onset and factorisation at higher q. DYNAMICS OF MOLTEN POLYMERS 1 oL I 1 1 I I 0 0.02 0.04 0.06 0.01 0.1 q1A-l Fig. 9. Data for preliminary experiment of transient relaxation upon cross-linked melt from a mixture of PSD M = 600000 in PSH MH = 600000.Approximate mass between cross-links M x 50000. Stretching temperature 117 "C.0, t = 10 s; x ,t = 100 s; + isotropic sample and 0,isotropic melt. back to the one at t infinite. This shows that it is possible to cover the time range where rearrangements occurs in a deformed crosslinked rubber before reaching the plateau behaviour. J. P. Cotton D. Decker M. Benoit B. Farnoux J. Higgins G. Jannink R. Ober C. Picot and J. des Cloizeaux Macromolecules 1974 7 863. F. Boue M. Nierlich and L. Leibler Polymer 1982 23 29. D. Richter A. Baumgartner K. Binder B. Ewen and J. B. Hayter Phys. Rev. Lett. 1981 47 109. J. D. Ferry Viscoelastic Properties of Polymers (Wiley New York 3rd edn 1982).F. Boue M. Nierlich G. Jannink and R.C. Ball J. Phys. (Paris) 1982 43 137. F. Boue M. Nierlich G. Jannink and R. C. Ball J.Phys. (Paris) Lett. 1982,43 L-585; L-591; L-593; L-600. ' F. Boue Thtse d'Etat (University of Orsay 1982). F. Boue M. Nierlich J. Hertz and K. Osaki to be published. This paper will report tests of the tube model especially the Doi and Edwards contraction for very long chains. @ P. G. de Gennes J. Chem. Phys. 1971,55 572. lo M. Doi and S. F. Edwards J. Chem. SOC., Faraday Trans. 2 1978 74 1789; 1802; 18 18. l1 J. Klein Macromolecules 1981 4 460. l2 R. P. Wool and K. M. O'Connor J. Appl. Phys. 1981,52 5953. l3 K. Judd H. H. Kaush and J. C. Williams J. Muter. Sci.,1981 16 204. l4 P. E. Rouse Jr J. Chem. Phys.1953,21 1273. See also ref. (3) or P. G. de Gennes Scaling Concepts in Polymer Physics (Cornell University Press Ithaca 1979 USA). l5 F. Boue K. Osaki and R. C. Ball accepted for publication in J. Polym. Sci.,Polym. Phys. Ed. It is shown in ref. (6) that in spite of the apparent intention of the author to use a reptation model the calculation is as given in S. Daoud J. Phys. (Paris) 1977 38 751. lo P. G. de Gennes and L. LCger Dynamics of Entangled Chains to be published. l7 K. Osaki and K. Kurata Macromolecules 1980 13 671. l8 L. Leger H. Hervet and F. Rondelez Macromolecules 1981 14 1732.

ISSN:0301-5696    

DOI:10.1039/FS9831800083

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

Faraday Symp. Chem. SOC.,1983 18 103-114 Viscoelastic Studies of Reptational Motion of Linear Polydime thylsiloxanes J. LAMB,G. HARRISON BY R. R. RAHALKAR,~ AND A. J. BARLOW Department of Electronics and Electrical Engineering The University Glasgow G12 SQQ Scotland AND W. HAWTHORNE AND J. A. SEMLYEN Department of Chemistry University of York Heslington York YO1 5DD AND A. M. NORTHAND R. A. PETHRICK* Department of Pure and Applied Chemistry University of Strathclyde Thomas Graham Building 295 Cathedral Street Glasgow G1 1XL Scotland Received 23rd August 1983 Viscoelastic measurements are reported on linear polydimethylsiloxanes with molecular weights greater than the critical value for entanglement (M,). Data covering a frequency range from to lo8 Hz are reported for five samples and the frequency dependence of the modulus and shear viscosity in the terminal region are compared with the predictions of the shifted Rouse and Doi-Edwards models.Provided that allowance is made for the molecular-weight distribution of the polymer sample being studied it is found that for broad-molecular-weight-distribution samples both approaches are adequate. However for high-molecular-weight narrow-fraction samples it is found that the Doi-Edwards theory provides a better fit than the shifted Rouse model. Comparison of the data at high frequency with experiment indicates that the use of a Rouse model to describe the intermediate region of the viscoelastic spectrum is inappropriate. Suggestions are put forward as to the origins of the difference between experiment and theory.Viscoelastic measurements of a large number of polymers have indicated that for high-molecular-weight materials the low-frequency contribution to the relaxation spectrum is independent of the polymer type and is simply a function of its molecular weight.l? Studies of the variation of viscosity with molecular weight have further indicated that the behaviour of polymer melts can be subdivided into at least two regions polymers with molecular weight below Mc and those above. For the lower-molecular-weight polymers the viscosity is observed to vary approximately as the first power of the molecular weight whereas above Mc a 3.5power law is obeyed. A review by Graessley in 19743highlighted the poor theoretical understanding of the viscoelastic behaviour of polymers and subsequently prompted consideration of this problem by de Gennes4 and later by Edwards and DO^.^ In 1976 de Gennes4 proposed that the viscoelastic spectrum could be divided into two parts; a lower-frequency part associated with the diffusion-reptation of a polymer molecule through a matrix of polymer molecules and a higher-frequency contribution which corresponds to the motion of the chains between entanglement points formed as a consequence of polymer-polymer contacts.A rigorous analysis of this type of motion was developed t Present address Agricultural Research Council Food Research Institute Colney Lane Norwich NR4 7U4. 103 VISCOELASTIC STUDIES OF POLYDIMETHYLSILOXANES by Doi and Edwards5 in 1979 and it describes the reptation-diffusion motion of a polymer with a molecular mass above M within an imaginary tube formed by the entanglement points.This motion models the so-called terminal relaxation3 region and constitutive equations describing viscoelastic response of such a system were deri~ed.~ These equations do not include contributions associated with relaxation of the chains between entanglement or the motion of local elements of the chain associated with the glass-liquid transition. These latter contributions to the overall motion have not been considered theoretically and semi-empirical relations are usually invoked to describe the higher-frequency contribution to the viscoelastic spectrum.3 In this paper an attempt is made to compare the results of isothermal viscoelastic experiments with the predictions of the reptation theory.Use of isothermal conditions rather than the superposition conditions usually used in viscoelastic studies1 avoids uncertainties with regards the effects of temperature on the form of the relaxation distribution. Data will be presented on a series of polymers with varying molecular weights and molecular-weight distributions. We believe this to be the first systematic attempt to test the ,validity of the reptation theory using viscoelastic measurements. EXPERIMENTAL MATERIALS A number of polydimethylsiloxane (PDMS) samples were used in this study. The first group of materials designated A had a broad molecular-mass distribution and were obtained from Midland Silicones Ltd.These samples were originally used by Dr G.Harrison for a study of the viscoelastic response of polydimethylsiloxanes at frequencies above lo4 HZ.~ The second group of samples designated B were obtained from Dr D. Meier of Midland Macromolecular Institute Michigan U.S.A. Fractions of broader-molecular-weight materials were obtained using gel-permeation chromatographic equipment at York University and are designated C. The apparatus has been described previ~usly,~ the columns used for the fractionation were obtained from Polymer Laboratories PLC. The column was eluted with toluene and the injection aliquot was typically 1-5 g ~m-~. The total elution volume was split into fractions and ca. 0.01-0.2 g of material were obtained in each fraction.Because of the severe practical limitations set by the small volumes which could be collected viscoelastic studies of the narrow fractions were limited to the low-frequency range. One of the samples designated D was prepared by ring-opening polymerisation of the cyclic trimer ;the conditions used are described CHARACTERISATIONOF THE POLYMERS The molecular masses of the polymers used in this study were determined using gel-permeation chromatography. The calibration of the columns used has been described previously.l0? l1This involved both osmotic pressure and light-scattering measurements of lower-molecular-mass samples and allowed an absolute calibration of the g.p.c. columns. A detailed discussion of the characterisation of linear and cyclic PDMS has been presented elsewhere.12 VISCOSITY MEASUREMENTS The viscosities of the samples were determined using a falling-ball method13 and are considered to be accurate to f5%.All measurements were made at 296.2 K. RESULTS The molecular masses molecular-mass distributions and viscosities are presented in table 1. The variation of viscosity with the number average molecular weight is shown in fig. 1 which also contains earlier results of the group at York UniversitylO. l2 R.R. RAHALKAR et al. Table 1. Molecular mass molecular-mass distributions and viscosities of linear pol ydimet hylsiloxanes code designate 10-3 M 10-3 M viscosity/Pa s s1 A 28.3 65.5 2.31 12.5 s2 A 44.9 84.4 1.88 34 s3 A 66.0 123.4 1.87 94 s4 B 84.7 150.2 1.88 387 s5 B 132.9 227.6 1.71 713 S6 D 93.5 108.2 1.15 1066 s7 C 92.6 124.0 1.34 1830 S8 C 84.6 114.7 1.36 47.6 s9 D 189.0 248.1 1.31 1430 3 2 1 h m CL .t v 0 go -1 -2 -3 1% M Fig. 1. Viscosity-molecular-mass relationship for linear polydimethylsiloxanes. Code as table 1. Undesignated points taken from ref. (1 1) and (1 2). VISCOELASTIC STUDIES OF POLYDIMETHYLSILOXANES -1 0 2 4 6 8 log CfIW Fig. 2. Variation of modulus with frequency for sample S3. 0,G’(cu);0, G”(o);(-) Doi and Edwards theory and Rouse theory combined; (--) Rouse theory two-block model; (* * -* a) Doi and Edwards theory and Rouse theory combined monodisperse polymer; (-.-.) Doi and Edwards theory and modified Rouse theory combined.for the viscosity-molecular-mass relationship for PDMS of narrow molecular mass distribution and masses below M,. EXPERIMENTAL DYNAMIC VISCOELASTIC RESPONSE A torsional rheometer was used to make measurements of the components G’(o) and G”(co)of the shear modulus in the frequency range 10-1-102 Hz. The results for five of the polymers are shown in fig. 2-6. A description of the instrument has been published previo~s1y.l~ It consists essentially of an oscillatory cone-and-plate visco- meter. The errors in the values of G’(o) and G”(o)obtained are estimated to be < +5%. For certain of the samples the components R and XL of the shear impedance were determined at frequencies of 38 and 77 kHz by means of torsional quartz crystals; the method and apparatus used15+ l6have been described previously.Errors in the values of R and X are estimated to be < f5%. The components of the shear modulus were calculated using the equations G’(co)= (Ri-XE)/p (1) G”(co)= 2RLXL/p (2) where p is the density of the liquid. Where appropriate results obtained from the previous studys were combined with the data obtained in this study. Inclined incidence R. R. RAHALKAR et al. 107 7 c 5 4 h a" -. %? -M 1 1 0 -1 Fig. 3. Variation of modulus with frequency for sample S2. (-) Doi-Edwards and Rouse theory combined; (--) Rouse theory two-block model. 1 I I I I -1 0 2 lr 4 0 log CflW Fig. 4. Variation of modulus with frequency for sample S4.Symbols as fig.3. VISCOELASTIC STUDIES OF POLYDIMETHYLSILOXANES 2 1 0 -1 Fig. 5. Variation of modulus with frequency for sample S7. Symbols as fig. 3. 't Fig. 6. Variation of modulus with frequency for sample S9. Symbols as fig. 3. R. R. RAHALKAR et al. I I I I I 0 2 4 6 8 log CfIHd Fig. 7. Variation of viscosity with frequency for sample S3. Symbols as fig. 2. measurements were performed on certain of the samples between 6 and 78 MHz together with R measurements using a normal incidence wave reflection technique at 450 MHz.” Within the limits of experimental error (ca. +6%) these results for the narrow- fraction materials are the same as those found previously,s confirming the observation that at this temperature (296.2 K) the viscoelastic behaviour of long-chain poly- dimethylsiloxanes is independent of chain length for frequencies > ca.lo7Hz. In making this comparison the method of reduced variables was used to allow for the small temperature difference between previous and present results i.e. values of R,/dp and XL/dpwere plotted as a function OfflV296.2 K/27303.2 K). To allow a later comparison with theoretical curves the variations of G’(o) and G”(w)have been calculated from the smoothed shear-impedance data. Previous resultss obtained for polymers having a broad molecular-mass distribution may not be applicable for the narrow-molecular-mass-distribution samples and are not included in the plots for the fractions.Only values of R are obtained from the measurements made at 450 MHz. In using eqn (1) to calculate G’(co)at this frequency it has been assumed on the basis of previous work that Xi < RL giving G’(o) xRL/p. Extrapolation of the results obtained at the lower frequencies indicates that XLat 450 MHz is probably ca. 0.4R.If this estimate is included in the calculation of G’(w),log G’(w) is reduced by only 0.08. Also for XL= (d2-1) R, G”(co)= G’(o); therefore at this frequency it is reasonable to assume that G”(co)xG’(cu). The variation of the dynamic viscosity with frequency ~’(cu) = G“(w)/2nf for the polymers studied is shown in fig. 7-1 1. As before the earlier measurementss for the broad-molecular-mass-distribution samples are included where appropriate.VISCOELASTIC STUDIES OF POLYDIMETHYLSILOXANES I I I I I 0 2 4 6 8 log WHz) Fig. 8. Variation of viscosity with frequency for sample S2. Symbols as fig.3. 2 1 h LQ a -. 0 F W 00 --1 -2 I I 1 1 I 0 2 4 6 8 log CflW Fig. 9. Variation of viscosity with frequency for sample S4. Symbols as fig. 3. R. R. RAHALKAR et al. 1 1 I I I I 0 2 4 6 8 log CfIW Fig. 10. Variation of viscosity with frequency for sample S7. Symbols as fig. 3. -1 \ t 1-1 0 1 2 1 log CfIW Fig. 11. Variation of viscosity with frequency for sample S9. Symbols as fig. 3. DISCUSSION MOLECULAR-MASS DEPENDENCE OF THE VISCOSITY The molecular-mass dependence of the viscosity (fig. 1) can be subdivided into three regions.In region (i) low-molecular-mass polymers M < lo3 exhibit an M2 dependence. This behaviour is typical of non-polymeric liquids.8 There is evidence from previous studies’ that a broading of the viscoelastic relaxation occurs with increasing molecular mass. However at least ten repeat units are necessary in the chain for a Gaussian distribution of the end-to-end distance of segments. In region (ii) VISCOELASTIC STUDIES OF POLYDIMETHYLSILOXANES 2 x lo3 <Zn<2.1 x lo4 the chain length is sufficient for the valid application of Gaussian statistics and the viscosity is observed to depend on the first power of the molecular mass. In region (iii) for polymers above 2 x lo4 the line is fitted to the data for polymers with values of Hw/Mn< 1.35.Broader-molecular-mass samples tend to lie to the left of this line. Taking this data selection we find that the viscosity varies as Mnz3.5. This dependence is in contrast with the Doi and Edwards theory which predicts an 3.0 beha~iour.~ The value of M from these data is ATn =21 000. This value is used in the subsequent discussion. DYNAMIC VISCOELASTIC BEHAVIOUR OF PDMS The total spectrum can be sub-divided into two regions the terminal region lying between and lo4 Hz and the intermediate region between lo4 and los Hz. The higher-frequency region associated with the glass-liquid transition is experimentally inaccessible for this polymer and lies above los Hz.’ This paper will concentrate on a discussion of the terminal and intermediate regions.A more complete presentation of the data and discussion of the theory used in modelling these experiments is to be published elsewhere.8 TERMINAL-REGION RELAXATION The Doi-Edwards theory5 predicts that the frequency dependence of the complex shear viscosity should be described by the following relationship 8 q*(iw) =-Go c -Td W (3) 5 PODD p4z21 i-iw G/p2 where Gtv is the average value of the equilibrium modulus wiis the mass fraction of molecules having molecular mass Mi, Gi is the diffusion time for the polymer within the tube and p is the mode number of the motion. This equation has been discussed fully by Doi and Edwardss and its application to viscoelasticity has been considered elsewhere.8 The above relationship does not allow for a molecular-mass distribution and as we can see from fig.2 the predictions of the theory based on the above relationship are unable to describe the observed experimental response. If we assume that where p =M/Mnthen eqn (3) can be transformed into m)is the mass fraction of molecules having molecular-mass ratios in the interval p-(p +Ap). Information on the distribution of p is directly obtained from g.p.c. data and evaluation of eqn (5) may be performed by summation of the mode contributions for each of the molecular-mass components present. In the computation we recognise that G cc M3in which case G in eqn (5) can be written in the form T,p3.Using the relationship that G*(iw) =G’(w)+iG”(w) =iwq*(iw) (6) we can obtain and =coq’(w).R. R. RAHALKAR et al. The above equations describe the terminal region. To the values predicted by eqn (7) and (8) we require to add a contribution due to the motion of the chains between entanglement and this is usually described by a Rouse mode with molecular mass M,. A detailed discussion of this calculation together with the equations for the shifted Rouse mode are presented elsewhere.8 It is important to realise that the simple Doi and Edwards model does not describe the complete viscoelastic response of a polymer and that terms associated with motions between entanglements and local motions of the polymer backbone have to be added to the terminal-region contribution in order for the whole spectrum to be described correctly. The contribution from these higher-frequency modes is very small in the frequency range 10-1-103 Hz; however they dominate the response observed at higher frequencies.The existence of these higher-frequency modes must not be forgotten when attempting to model the time-dependent behaviour of a polymer melt whatever the technique being used. It can be seen from fig. 2-4 that for these broad-molecular-mass-distribution samples the fit of the data is good using either the shifted Rouse or the Doi and Edwards model. Similarly the narrow-molecular-mass fraction S7 (fig. 5)can be fitted by either theory adequately. It is only with the very-high-molecular-weight sample S9 (fig. 6) that it is possible to differentiate between the two approaches. It is clear from fig.7-1 1 that the terminal-region relaxation is more accurately described by the Doi and Edwards theory. Unfortunately the experimental frequency range available without resort to time-temperature superposition is limited and it has not been possible to cover the complete relaxation region in this study. Efforts are currently being made to extend these measurements in an attempt to explore whether or not the theory can describe the form of the curve in the tail of the terminal region. In summary these experiments indicate that the Doi and Edwards theory modified as necessary to allow for the molecular-mass distribution is capable of describing the relaxation of polymers covering a molecular-mass range from M,% 21OOO (= M,) to M % 190000. INTERMEDIATE-RELAXATION REGION Relaxations above ca.lo4 Hz can be associated with the motion of the chain trapped between entanglements. Usually the frequency dependence of the viscoelastic relaxation in this region has been described by a Rouse type of equation with characteristic molecular mass equal to Me.It is clear from fig. 2 and 7 that this type of approach is unable to predict correctly the magnitude of the viscoelastic response in the frequency range 106-107 Hz. Attempts to obtain better agreement by using a lower molecular mass for the effective value of M used in the Rouse contribution lead to the prediction of too low an amplitude in the frequency range 104-105 Hz. In an attempt to explain the origins of this discrepancy we have developed a model in which the polymer chain is allowed to adopt an asymmetric distribution about the entanglement points rather than the symmetric one assumed by the simple Rouse approach (fig.12). The analysis of this situation will be presented elsewhere.* It is clear from fig. 2 and 7 that allowing an asymmetric distribution does increase the viscoelastic contribution in the megahertz frequency range with only a minor reduction in amplitude for the contribution in the kilohertz region. However the change in the distribution is greater than is required experimentally. The asymmetric distribution reduces the length of certain of the polymer tails which extend beyond an entanglement and increasing others. The data presented here would indicate that the distribution used overestimates the effects of these short chains and underestimates the number of the longer tails.A more complete discussion of this topic will be presented elsewhere.8 VISCOELASTIC STUDIES OF POLYDIMETHYLSILOXANES (b) Fig. 12. Schematic representation of polymer motions. The dotted lines indicate interacting polymer chains. (a) Symmetrical distribution (b)assymetrical distribution. CONCLUSIONS This present study of polydimethylsiloxanes indicates that the Doi and Edwards model can describe the viscoelastic behaviour of polymer with molecular weights above M,. It must however be stressed that accurate modelling of experimental data is only possible when the effects of molecular-mass distribution and the presence of additional higher-frequency contributions are included correctly.It is also clear that the factors controlling the higher-frequency relaxation processes are not completely understood and it is probable that a more precise modelling of these motions may help to resolve the long-standing problem of the discrepancy between theory and experiment regarding the power law for the viscosity. R. R. R. and W. H. thank the S.E.R.C. for support during the period of this study. Provision of financial support allowing the purchase of certain items of electronics and also the columns used in the sample fractionation is also gratefully acknowledged. The assistance of Dr C. Pearce of Dow Coming Barry is gratefully acknowledged both in supplying cyclic trimer and also with the ring-opening polymerisation.The authors also acknowledge the gift of samples of polydimethylsiloxanes from Dr Dale Meier of MMI. J. D. Ferry Viscoelastic Properties of Polymers (Wiley New York 1970). J. Lamb Molecular Basis of Transitions and Relaxations ed. D. J. Meier (Gordon and Breach London 1978) vof. 4 p. 25. W. W. Graessley Adu. Polym. Sci. 1974 16 1. P. G. de Gennes Macromolecules 1976 9 587 594. M. Doi and S. F. Edwards J. Chem. Soc. Faraday Trans. 2 1978,74 1789; 1802. (I A. J. Barlow G. Hamson and J. Lamb Proc. R. SOC.London Ser. A 1964 282 228. 'I K. Dodgson D. Sympson and J. A. Semlyen Polymer 1978 19 1285. R. R. Rahalkar J. Lamb G. Harrison A. J. Barlow W. Hawthorne,J. A. Semlyen A. M. North and R. A. Pethrick in preparation. US.Patents 1,217,335; 1,220,686; 231,000.lo K. Dodgson D. J. Bannister and J. A. Semlyen Polymer 1980 21 663. l1 K. Dodgson and J. A. Semlyen Polymer 1977 18 1265. l2 C. J. C. Edwards R. F. T. Stepto and J. A. Semlyen Polymer 1980 21 281. l3 British Standard 188. l4 G. Hamson Rheol. Acta 1974 13,28. l5 A. J. Barlow G. Harrison J. Richter H. Seguin and J. Lamb Lab. Pract. 1961 10 786. l8 G. Hamson and A. J. Barlow Methods Exp. Phys. 1981 19 137. J. Lamb and J. Richter J. Acoust. Soc. Am. 1967 41 1041.

ISSN:0301-5696    

DOI:10.1039/FS9831800103

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

Faraday Symp. Chem. Soc. 1983 18 115-129 Molecular Viscoelasticity of Xanthan Polysaccharide BY SIMON B. ROSS-MURPHY* Unilever Research Colworth House Sharnbrook Bedford MK44 1LQ AND VICTORJ. MORRIS ARC Food Research Institute Colney Lane Norwich NR4 7UA AND EDWIN R. MORRIS National Institute for Medical Research Mill Hill London NW7 1AA Received 14th July 1983 Previous studies of xanthan polysaccharide using molecular probes of local chain geometry such as optical rotation and n.m.r. have demonstrated a cooperative disorder-order transition in aqueous solution on cooling or on addition of salt. In the present work we have investigated chain geometry of the ordered species at a 'macromolecular level' using quasi-elastic light scattering transient electric birefringence and elongational flow birefringence and have used viscoelastic measurements to probe the 'supramolecular' organisation responsible for the 'weak-gel' properties of xanthan solutions.Solution viscoelasticity at fixed xanthan concentration was drastically modified by changing the counterion to the polyelectrolyte and by treatment with urea although in all cases the local ordered structure was unaffected. 'Macromolecular' studies under conditions which minimise intermolecular interactions (Na+ salt form in the presence of urea) and at substantially lower concentrations indicate a persistence length comparable to that of other highly persistent biopolymers such as double helical DNA and the triple helical polysaccharide schizophyllan. We conclude that in aqueous solution xanthan may be regarded as a highly extended worm-like chain interacting by non-covalent association to develop a weak-gel network which is readily reversible under shear.Xanthan is an anionic polysaccharide produced commercially by fermentation of the bacterium Xanthomonas campestris which in nature occurs as a plant pathogen. Aqueous solutions of the polymer show distinctive 'weak-gel ' properties' which form the basis of its technological utility. The primary structure2* is based on a linear chain of 1,4-linked/3-~-glucopyranosyl residues as in cellulose but with charged trisaccharide sidechains attached to alternate residues of the polymer backbone to give a pentasaccharide repeating sequence (fig. 1). Characterisation of the xanthan macromolecule is for a number of reasons much more complex than for most synthetic polymers.In particular being a polyelectrolyte its chain dimensions would be expected to change in response to changes in ionic strength. Under conditions of elevated temperature and/or comparatively low ionic strength this is indeed the case and the local chain flexibility (as monitored by for example the timescale of n.m.r. rela~ation)~ is very similar to that of other coil-like cellulose derivatives. On cooling or on addition of salt however the molecule undergoes a cooperative conformational transition to a rigid ordered struct~re.~-~ The disorder-order transition (fig. 2) may be monitored by a variety of physical techniques including optical r~tation,~-~ differential scanning circular dichr~ism,~? calorimetry,* solution viscosity6$ and loss of detectable high-resolution n.m.r.~ignal.~ 115 MOLECULAR VISCOELASTICITY OF XANTHAN POLYSACCHARIDE backbone y2-I OH I sidechain II 0 ) Fig. 1. Pentasaccharide repeating sequence of xanthan. The proportion of pyruvate and acetate substituents present may vary appreciably from sample to sample. 1 .o . 0.8-0.6 -f$ 0.4-0.2 -0.0 I -A The transition (which is fully reversible and shows no thermal hysteresis) obeys first-order kinetics** and seems to occur without an associated change in molecular weight.'? From this and other evidence it is proposed* that the fundamental structural unit is a single helix stabilised intramolecularly by ordered packing1* of sidechains along the polymer backbone and that the 'weak-gel' properties arise from higher levels of non-covalent interaction between helically stabilised species." Depending upon the nature of the growth medium the xanthan polyanion is associated with mixed proportions of the cations Na+ K+ and Ca2+.At the same time different growth conditions also alter the levels of pyruvate and acetate substituents S. B. ROSS-MURPHY V. J. MORRIS AND E. R. MORRIS 117 on the sidechains.l2? l3 Both factors are known to modify the viscoelastic behaviour in solution. Moreover the molecular-weight distribution is believed to be broad and fractionation of samples by sonication and/or size-exclusion chromat~graphy~~ l5 presents serious difficulties.Molecular characterisation of xanthan solutions is further frustrated by the presence of a microgel fraction the amount of which seems to be sensitive to for example the proportion of cations the ionic strength and the time after dissolution. In this paper we discuss the application of physicochemical techniques acting over longer distances than the ‘molecular probes’ such as optical rotation and n.m.r. which have been discussed previously. These we have arbitrarily divided into ‘macromolecular probes ’ acting over tens of nanometres in dilute solution including in particular quasi-elastic light scattering transient electric birefringence and extensional flow birefringence and ‘supramolecular probes’ such as oscillatory and steady-shear viscoelasticity for more concentrated solutions.In all cases reported here considerable care was taken to assess and apply a consistent regime of sample preparation in view of the complications outlined above. Results from ‘macromolecular’ and ‘ supra-molecular’ techniques are assessed in terms of the available information on local conformational rigidity from ‘molecular ’ probes. EXPERIMENTAL SAMPLE PREPARATION Commercial xanthan (Sigma) was freed of cellular debris by centrifugation of aqueous dispersions (ca. 76000 g; 1-3 h). Clarification was assessed by phase contrast and electron microscopy. The resulting clarified dispersions were ion-exchanged (Dowex SOX-W8) to single-salt forms (Na+ K+ Ca2+) and freeze dried. Solutions were prepared by adding the freeze-dried material to deionised water and stirring at room temperature until homogeneous (ca.12 h). Supramolecular aggregates (microgel particles) were removed by the techniques described in a subsequent section. Where heating was employed evaporation losses were corrected by addition of solvent after cooling. All samples contained 0.02 % sodium azide as preservative and were stored at 4 OC. RHEO-OPTICAL TECHNIQUES QUASI-ELASTICLIGHT SCATTERING (QELS) Measurements were made at 25 OC on a Malvern 4300 goniometer (Precision Devices Malvern) using homodyne detection. Single clipped autocorrelation was employed over the angular range 30-1 50°. The background was calculated from measurements of the clipped and unclipped counts and the number of samples.Results were expressed in terms of the normalised function C(t)= [g‘2’(t) -l]/Eg‘2’(0)-11 where g(2)(t)is the second-order correlation function.1s TRANSIENT ELECTRIC BIREFRINGENCE (TEB) The TEB apparatus was built to a standard design,17 which has been fully described elsewhere.1B Linear detection” was used to determine the sign of the birefringence whilst studies of the field dependence of the equilibrium birefringence and of the time decay were made using quadratic detection. ELONGATIONAL FLOW BIREFRINGENCE (EFB) Elongational flow fields were generated using two opposed jets operating under ~uction.~~~ 2o Strain rates (i) were calculated from the measured flow velocity and the geometry of the apparatus. The induced flow birefringence was monitored using quadratic detection.” The light transmitted by the analyser was detected by a photomultiplier and displayed on an oscilloscope.For all three rheo-optical techniques the incident wavelength was 633 nm. MOLECULAR VISCOELASTICITY OF XANTHAN POLYSACCHARIDE VISCOELASTIC MEASUREMENTS These were performed using a Rheometrics RMS-605M mechanical spectrometer (Rheo- metrics Inc. New Jersey U.S.A.).Both sensitive (ST-10) and mid-range (TC-2000) transducers were used and experiments were performed under both oscillatory and steady shear using cone- and-plate geometry. Sample temperature was maintained at 25 k0.5OC; in all experiments reported here a fixed polymer concentration of 0.5%w/w was used. RESULTS DETECTION AND ELIMINATION OF MICROGEL ’ BY ULTRAFILTRATION Filtration of aqueous dispersions of Na+ xanthan (which had previously been freed of cellular debris by centrifugation) through 0.22 pm filters proved virtually impossible indicating extensive association of the polymer into supramolecular aggregates (‘microgel ’).On centrifugation a gel-like deposit was obtained which contained a substantial fraction of the total polymer mass (e.g. centrifugation of a 0.1 % disper-sion at 100000g for 1 h resulted in deposition of ca. 15% of the polysaccharide). The supernatant solution could then be filtered although with some difficulty. Dispersions which had undergone prolonged heating (90 OC 3 h) no longer gave gel-like deposits on centrifugation and could be filtered although again with difficulty (filtration of a 0.1 % w/v sample through a 0.22pm membrane resulted in a polymer loss of ca.5%). After a similar heating regime but in the presence of urea (2-8 mol dm-3) there was no detectable loss of polymer on centrifugation or filtration. Solutions of xanthan prepared directly from fermentation broth with no precipitation or drying stage show5 the filtration characteristics that we obtained only after removal of microgel. It therefore appears that the formation of stable aggregates is promoted by chain-packing in the solid state. RHEO-OPTICAL MEASUREMENTS QUASI-ELASTIC LIGHT SCATTERING The results obtained from QELS were very sensitive to sample history (fig. 3). For example ‘dispersions ’ (solutions from which polymer could be extracted by filtration or centrifugation) gave rise to extremely slowly decaying correlation functions [even when plotted as In C(t)];heat treatment of aqueous ‘dispersions’ produced a faster decay whilst ‘dispersions’ which had been centrifuged and/or filtered after heat treatment or which had been heat treated in the presence of urea produced C(t) functions which decayed to zero within ca.30 ms. Heat treatment was insensitive to urea concentration in the range 2-8 mol dm-3 (fig. 4). TRANSIENT ELECTRIC BIREFRINGENCE TEB studies of heat-treated xanthan solutions (4 mol dm-3 urea) have been de- scribed elsewhere,18 and data on heat-treated aqueous b dispersions ’ will be reported in detail at a later date. In both cases the sign of the birefringence was positive and at low-electric-field amplitude the Kerr law was obeyed,17 but at sufficiently large- electric-field amplitude the birefringence saturated the saturated birefringence then varying linearly with polymer concentration (c) for 0.02 < c (% w/w) < 0.08.Semi-logarithmic plots of the field-free decay were curved indicating a broad distribution of relaxation times. At given c this spectrum was dependent upon the magnitude IEI of the applied-electric-field amplitude but at given 1 El decay curves were relatively insensitive to c. Similar relaxation data were obtained for samples heat treated in water or 4 mol dm-3 urea. Fig. 5 shows normalised curves for cases in which [El was sufficient to induce saturation of the birefringence.S. B. ROSS-MURPHY V. J. MORRIS AND E. R. MORRIS I 1 1 1 I 0.0 1 2 3 4 5 tl10-2 s Fig. 3. Time dependence of the normalised autocorrelation function C(t) for an aqueous dispersionof xanthan (0.1 % w/v) initially (a),after heat treatment (m)and after centrifugation and filtration (A).Scattering angle = 40'; temperature = 25 OC. 0 0.01 1 1 1 1 0 1 2 3 4 5 tl10-2 s Fig. 4. NormaIised autocorrelation function C(t)for xanthan (0.1% w/w) after heat treatment in the presence of the following concentrations of urea/mol dm-3 2 (a) and 4 (a),6 (0) 8 (0).Conditions as in fig. 3. ELONGATIONAL FLOW BIREFRINGENCE These measurements proved difficult for samples heated in urea so only data for filtered heat-treated aqueous samples are reported here.These showed a progressive increase in birefringence (fig. 6) (An,) with increasing i rather than the sudden onset of An at a critical strain rate normally observed for flexible polymer c0i1s.l~ Turbulence at high velocity gradients prevented the measurement of the saturation birefringence so the rotatory diffusion coefficient could not be determined from the initial slope of Anf against 6. MOLECULAR VISCOELASTICITY OF XANTHAN POLYSACCHARIDE 0. OL 1 1 1 I 1 0 1 2 3 1 5 t/10-3 s Fig. 5. Normalised birefringence decay curves for xanthan (0.066% w/v) after heat treatment in water (A) and for the following concentrations of xanthan (% w/v) after heat treatment in 4 mol dm-3 urea 0.04 (@) 0.06 (0) The amplitude of the applied electric and 0.08 (u).field was sufficient to induce saturation of the equlibrium-induced birefringence An,(O) at elapsed time t= 0. The solid lines show the best theoretical fits based on a log-normal distribution of particle size. I 1 & I 1 2 3 strain rate e/i03 s-l Fig. 6. Dependence of the induced elongational flow birefringence (An,) on the applied strain rate (i)for a 0.02% w/w solution of xanthan in water following heat treatment at 90 OC for 3 h. S. B. ROSS-MURPHY V. J. MORRIS AND E. R. MORRIS VISCOELASTIC STUDIES ‘NATIVE ’ XANTHAN Viscoelastic measurements for native xanthan heated with and without urea have been published previously,113 21 but for completeness they are summarised briefly below. 1 OL 10’ m a -.u 1oo m a \ u lo-’ 10-2 10” 1oo 10’ lo2 o/rad s-’ Fig. 7. Frequency dependence of dynamic storage and loss moduli G’ and G for xanthan (0.5%w/w; 0.02 mol dm-3 KCl) in the presence of the following concentrations of urea/mol dm-3 0.02 (0, 0) and 4 (m m). (Reproduced with permission from J. Polyrn. Sci. Polym. Lett. Ed.). In the absence of urea the mechanical spectrum of 0.5% w/w solutions is such that over the frequency range 10-1-102 rad s-l G’ is greater than G and neither shows much frequency dependence over this range (‘gel’ spectrum) (fig. 7); the linear strain region extends to y x 0.05-0.20 for both native and single-cation samples. Essentially identical spectra are given for ‘native’ samples heat treated with 0.02 mol dm-3 urea.For the heat-treated 4 mol dm-3 urea samples the spectrum is quite different. At low frequency G” > G’ and both moduli approach the dependence G” oc w and G’ cc w2 as w becomes lower. At higher o (> 30 rad s-l) G’ > G consistent with physical 122 MOLECULAR VISCOELASTICITY OF XANTHAN POLYSACCHARIDE entanglement and G” is similar to that observed in 0.02 mol dm-3 urea. Intermediate urea concentrations produce intermediate effects. Fig. 8 shows the shear-rate dependence of steady-shear viscosity measured for solutions which had been treated with progressively increasing urea concentrations. In 4 rnol dm-3 urea typical pseudoplastic polymer solution behaviour is observed with a low shear rate Newtonian plateau (qox 0.4 Pa s) for i < 1 s-l.At this urea concentration q* values calculated from the data of fig. 7 are in close agreement with q at equivalent co and i (Cox-Merz superposable),22whereas with no added urea q* > q over the same range. This lack of superposition is at least partly an effect of the strain deformation applied to the system (low in oscillatory shear ‘high’ in steady shear). loi \ \ \ \ \ \ 10‘ \ \ \ \ ‘\ \ CA ‘\ 2 \ c-loc \ \ \ \ \ \ lo-’ \ \ \ \ \ \. \ 10-2 10” loo 10’ lo2 103 -+-’ Fig. 8. Shear rate (1;) dependence of viscosity (q) for ‘native’ xanthan (0.5% w/w; 0.02 mol dm-3 KC1) in the presence of the following concentrations of urea/mol dm-3 0.02 (O),0.2 (a), 2 (A)and 4 (a).The dashed lines indicate the approximate upper and lower shear-stress limits of the instrument for the test configuration used.(Reproduced with permission from J. Polym. Sci.,Polym. Lett. Ed.). The progressive disruption of structure by urea treatment is seen in fig. 8; with decreasing urea concentration q at low i shows a progressive increase until by 0.02 mol dmh3there is little sign of a Newtonian plateau and the j dependence is much greater than that normally observed for polymer solutions (viz.d log q/d log i 7 -0.9 compared with ca. -0.7),23 including disordered polysaccharides. At high y all the curves appear to converge to a common value of q(j).All the results for q against j and G’ G” against w are completely reversible i.e.independent of whether i is S. B. ROSS-MURPHY V. J. MORRIS AND E. R. MORRIS I23 increased or decreased. Xanthan solutions might therefore be regarded as ‘shear-reversible gels ’. Under the same conditions of low urea the proportionality constant between G’(o) and Nl(j),with Nl the primary normal stress at equivalent CL) and j (i.e. the normal stress analogue24 of the Cox-Merz rule) is significantly greater than ca. 2-3 usually observed for flexible polymers (and for disordered polysaccharides) and approaches ca. 6. Further discussion of this effect will be published later. 10-2 ,;-I 100 10’ lo2 103 1 7h-l Fig. 9. Shear-rate dependence of viscosity for Na+ xanthan before urea treatment (A)and after treatment with 0.02 mol dm-3 urea (0)and 2 mol dm-3 urea (m).The solid line is the data for ‘native’ xanthan in 0.02 rnol dmP3 urea (fig. 8). ‘SINGLE-CATION ’ XANTHAN The data reported above show that increasing urea concentration gradually reduces the ‘gel-like’properties of xanthan solutions at low deformation rates. This tendency is substantially affected by the nature of the cation for ion-exchanged samples. Fig. 9 gives the ~(j) dependence for Na+ xanthan without urea treatment and after treatment with 0.02 and 2 mol dm-3 urea. The behaviour of Na+ xanthan is quite different from that of the mixed-cation ‘native’ form with some evidence that a Newtonian plateau would be reached at lower shear rates than were experimentally accessible. Even this behaviour is so modified by treatment with 0.02 mol dm-3 urea that there is little further change when 2 mol dm-3 urea is used.By contrast K+ xanthan (fig. 10) behaves in the absence of urea very similarly to the ‘native’ form but there is a considerable decrease in low-shear viscosity on treatment with 0.02 mol dm-3 urea which has no detectable effect on ‘native’ xanthan. At higher concentrations of urea (2 mol dm-3) where the ‘gel-like’ properties of ‘native’ xanthan are diminished little further change is observed for the K+-salt form and the ~(9) profiles for ‘native’ and K+ xanthan in 2 mol dm-3 urea are again quite similar. Finally fig. 11 shows that the Ca2+ form of xanthan has at the same concentration a much greater increase in low-shear viscosity (i.e.ca. 8 x as great at 3= 0.1 s-l) compared with the native form and that this is scarcely altered by urea treatment. I24 MOLECULAR VISCOELASTICITY OF XANTHAN POLYSACCHARIDE lo2-10' -v) ." 0 a \ c loo -lo-' -If-2 16-1 160 161 102 1d3 ?Is-' Fig. 10. As fig. 9 but for K+ form in KC1. I t. lo2 ' 10' P .-0 a \ c loo. \ lo-'/ I 10-,;-2 10-1 100 101 102 103 ?is-' Fig. 11. As fig. 9 but for Ca2+ form; urea effect is negligible. Qualitatively the slow but progressive disruption in structure for 'native ' xanthan may be rationalised in terms of the relative contributions of the pure Ca2+ Na+ and K+ forms. DISCUSSION 'MACROMOLECULAR ' STRUCTURE Evidence from the rheo-optical techniques employed in this paper re-emphasised the importance of a consistent regime of sample preparations when studying xanthan (and other biopolymers).Furthermore the importance of using a range of physico- chemical techniques (preferably probing over different 'length ' scales) is reasserted S. B. ROSS-MURPHY V. J. MORRIS AND E. R. MORRIS 125 particularly when attempting to interpret rheological data on the basis of ‘molecular’ models. For example QELS studies on dispersions of xanthan (sodium-salt form) in water or urea are characterised by both ‘fast ’(macromolecular) and ‘slow’(‘ supramolecular’ or ‘gel’) modes of 25 However at the same polymer concentration (c) the slow mode was extremely sensitive to the method of sample preparation and could be eliminated by heat treatment in the presence of urea or heat treatment followed by filtration.Without the filtration step intermediate results were obtained (cf. fig. 3). Loss of the slow mode in QELS occurred (at lower c) under the same preparative conditions as those which drastically modify the viscoelastic mechanical spectrum from that typical of a polymer gel to that of a viscoelastic polymer solution (see fig. 7). Nevertheless the optical rotation (and n.m.r.) data for both native xanthan and the Na+ form demonstrate that the ordered helical structure is retained after heat treatment for both ‘aqueous’ and urea-treated samples.l19 21 The evidence thus suggests that the ‘slow’ mode in the QELS spectrum and the predominantly elastic mechanical response (to o < 0.1 rad s-l) result from inter- molecular association of the polymer chains.The irreproducible and irreversible results obtained for example on xanthan aqueous ‘dispersions’ without heat treatment in both QELS and mechanical measurements strongly imply that such ‘dispersions’ involve supramolecular aggregates which only slowly ‘dissolve’. Measurement for example of MWz1 by integrated light scattering suggests that heat-treated aqueous samples still contain some very high Mwaggregates (microgel) which may be removed by prolonged centrifugation or filtration whilst measurements of Mw for samples heated in the presence of urea suggest that these are true solutions. The presence of microgel would account for the slight mass loss observed on filtration of aqueous solutions and fluctuations in the concentration of such aggregates in the sample volume would explain the tail in the correlation function and its removal on filtration.Thus heat treatment in the presence of urea or heat treatment followed by filtration may be used to prepare ‘molecular solutions’ of the Na+-salt form of xanthan. The choice of associated cation ionic strength and level of pyruvate and acetate substituents would of course also modify these effects. However the above evidence confirms that suitable pretreated solutions of xanthan in the sodium-salt form may be used to investigate the size shape and flexibility of the isolated macromolecule. For example preliminary information may be deduced from the course of the strain-rate dependence of birefringence under elongational flow.As mentioned earlier there was no evidence of the sudden increase in Anf above a critical strain rate which is typical of random coil polymers;lg rather the data suggest that xanthan behaves in the manner characteristic of a highly extended rod or persistent worm-like chain. TEB studies provide further information on the hydrodynamic properties of the xanthan macromolecule. The field-free decay curves (fig. 5) suggest that the relaxation process for heat-treated ‘aqueous’ and 4 mol dm-3 urea-treated solutions were quite similar and the magnitudes of the relaxation times furnish a guide to the local rigidity of the polymer.2s For flexible chains the electric dipole moments associated with individual monomeric units may orientate or disorientate virtually independentl~.~’? 28 For such coils electrical polarisation processes are insensitive to Mwand are virtually identical to those for solutions containing equivalent concentrations of the monomers.29-31 The relaxation behaviour ofmore persistent chains is morecomplicated.For example for cellulose derivatives (e.g. sodium carboxymethylcellulose and hydroxyethylcellulose) where the Kratky-Porod persistent length q is ca. 10-30 nm (depending upon conditions) the relaxation time z = 1-2 ps.32933 For rod polymers I26 MOLECULAR VISCOELASTICITY OF XANTHAN POLYSACCHARIDE z is proportional to L3,where L is the rod length (contour length); thus for worm-like chains z cc L3 for L xq (rod limit) but becomes independent of L for L % q (coil limit).34 In the present measurements the initial slopes in fig.5 yield z = 1600ps (4 mol dm-3 urea) and z = 580 ps (aqueous solution). Since the xanthan backbone (1,4-B-~-glucan) is the same as in the above cellulose derivatives we can estimate q for xanthan to be ca. 10 times as great as that for these derivatives i.e. ca. 100-300 nm. More precise estimates cannot be made from the present measurements but the results are comparable with other systems which are known to be extremely persistent chains. For example DNA (double helical) has q x 150 nm at low ionic strength decreasing to ca. 50 nm at high ionic strength,35- 36 and the triple helical polysaccharide from the bacterium Schizophyllum commune has q x 150-200 nm.379 38 Evidence from X-ray fibre diffraction suggests that xanthan exists as a five-fold helix with a pitch of ca.4.7 nm,l07397 40 which yields a mass per unit length (ML)of ca. 1000 dalton nm-l. Since optical rotation results suggest that this helical conformation is retained in solution we can calculated the contour length for a rod of known M,. The distribution of relaxation times observed in fig. 5 may be attributed either to a limited amount of flexibility or to a distribution of contour lengths. Of course in practice both effects would contribute simultaneously but by considering each in turn we can examine their relative contributions. For example if we assume that the relaxation-time spectrum is solely due to the distribution of rod lengths (molecular- weight distribution) and select a model for this distribution we can invert the relaxation data to obtain the length distribution.21 In this case we assume that the molecular-weight distribution follows a log-normal distribution (compared to the more general Schulz-Flory distribution); inversion of the relaxation data yields a length distribution (fig.12) with Mw = 1.06 x lo6. This is in very good agreement with the value from integrated-intensity light scattering uiz. (1.1 &-0.1) x lo6. Alternatively we can use the measured root-mean-square radius of gyration from integrated light scattering21 (250 & 10 nm) to calculate the persistence length assuming a monodisperse molecular-weight distribution.In more detail since (P)x250 nm and M z 1.1 x lo6,L (= (S2)i/d12) x870 nm; from the value of M and M we can also calculate the contour length as M/M = (1.1 x 106/103) nm = 1100 nm; the lower value from (S2)would imply some degree of flexibility. Using the Benoit-Doty expansion for (9)for the Kratky-Porod chain we can also estimate q from M/M, as suggested recently by Norisuye and F~jita.~l Using M = 1000 nm-l the optimum number of statistical segments per chain nk [= M/(2q M,)] was ca. 1.45 giving q x380 nm. This is probably too high when compared with DNA and schizophyllan but interestingly using a larger M, as was suggested by Paradossi and Brant,14 would produce a still larger value of q. In practice a number of possibilities exist for gaining more reliable estimates for q although most require the preparation of a number of samples of very narrow molecular-weight distribution.For example one could perform combined light- scattering/intrinsic-viscosity measurements as was done for the schizophyllan 38 or polysa~charide,~~~ using narrow-molecular-weight samples perform the Hagerman-Zimm analysis34 of TEB data as a function of (L/q)for the worm-like model. Perhaps the most reliable measurements are those for ‘infinitely dilute’ viscoelastic properties for example those by Hvidt and coworkers for the persistent rod-like protein myosin.42 The use of the worm-like chain model for xanthan has been discussed by H01zwarth.~~ S. B. ROSS-MURPHY V. J. MORRIS AND E. R. MORRIS /-\ effective rod length 1/10” m Fig.12. Calculated length distributions based on a log-normal distribution function fir) = (27r)-4 (al>-lexp { -0.5[In(l/rn)/~]~}>. The results shown provide the best fits to the experimental data in fig. 5 for xanthan after heat treatment in water (-) and in 4 mol dm-3 urea (--); the fitted parameters are Q = 0.6 rn = 0.58 pm and Q = 0.7 rn = 0.60 pm respectively. ‘SUPRAMOLECULAR’ STRUCTURE Results from viscoelastic measurements cast further light on the nature of the non-covalent intermolecular interactions which are believed to confer ‘weak-gel’ properties to xathan solutions. The earlier sections have dealt with probes of local chain flexibility and order (optical rotation and n.m.r.) or studies of macromolecular species over longer distances (TEB and QELS); the viscoelastic measurements are primarily concerned with supramolecular structure at higher concentration (0.5% w/w).In fact [q]for ‘native’ xanthan was found to be ca. 3 dm3 g-l so that the overlap parameter c [q]z 15. Nevertheless previous QELS studies of xanthan placed great emphasis on the presence of ‘slow ’ modes of relaxation.44* 45 As the concentration is increased these have been attributed to either hindered rotational effects or diffusional processes in a weak-gel network. The TEB data shown in fig. 5 do not show the marked concentration dependence predicted for hindered rotational effects.46 In addition the saturation birefringence for heat-treated solutions in water and 4 mol dm-3 urea varies linearly with polymer concentration.21 Both factors are consistent with independent non-interacting particles for the more dilute solutions.Hence the slow modes observed in QELS result from association of the polymers. The extent of this interaction is here shown to depend on the method of sample preparation the nature of the cations and the ionic strength of the medium. The present datal1g2l do not support previous ~uggestions~~-~~ that heat treatment in the presence of urea results in a disordered ‘random-coil’ xanthan at low temperatures. As far as the supramolecular (rheological) measurements are concerned previous results have shown how differing proportions of acetate and pyruvate groups on the sidechains (again introduced by subtle changes in the bacterial growth medium) can modify the shear-rate-viscosity pr0fi1e.l~ The present data show that quite drastic MOLECULAR VISCOELASTICITY OF XANTHAN POLYSACCHARIDE changes in the viscoelastic response may be induced by modifying the balance of associated cations.Unfortunately we cannot unequivocally characterise the nature of the intermolecular interactions; what is clear is that the effect is much greater than would be expected for non-specific entanglement coupling. The sensitivity to urea might suggest a contribution from hydrogen bonding or modification of dispersive forces due to the change in dielectric constant of the medium while ionic interactions are modified by counterion concentration and type and by pyruvate content. Again these effects are much more pronounced than would be expected purely from the polyelectrolyte character of the sidechains.Another possibility is that the structuring is associated with the separation of an ordered anisotropic phase (isotropic-nematic phase transition). According to the argument of Fl~ry,~O this would occur at a volume fraction CD given by where yn is the (number average) length/diameter ratio of a rigid rod in an assembly of such rods. Using the molecular data presented earlier yn x 400 and thus CP x 0.02. Of course both the curvature associated with large q and the effect of aggregation would tend to increase CDc (by reducing yn) but for xanthan the partial specific volume51 is ca. 0.62 so that the weight fraction of polymer (w*)at CDc is given by w* x 1.6 CD, i.e.w* x 0.032. Two report an apparent 0,transition at ca. 2.9%w/w and ca. 3.5% w/w respectively. In neither case are sufficient details of sample electrolyte or solution preparation given so that no rigorous comparison with results predicted above may be made but qualitatively the agreement with the above calculation is reasonable and the quantitative prediction is in keeping with results for synthetic rod-like Nevertheless w* is greater than the concentrations reported here uiz. 0.005 and weak-gel properties of xanthan have been reported1 down to 0.1% i.e. w = 0.001 so that the evidence seems to rule out liquid-crystalline structuring in this regime. We conclude therefore that overall properties of xanthan in dilute and semi-dilute solution are governed by non-covalent association of worm-like chains (which may be modified by solvent quality and ionic environment) to produce at concentrations down to ca.0.1% a weak-gel network but one which is readily reversible under shear. We thank C. Turner and D. Franklin (Physics Department Brunel University) K. I’Anson G. R. Chilvers and S. R. Ring (FRI) and R. K. Richardson S. A. Frangou and L. A. Linger (URL Colworth) for their help and M. J. Miles (FRI) for providing unpublished flow birefringence data (fig. 6). P. J. Whitcomb and C. W. Macosko J. Rheol. 1978 22,493. P. E. Jansson L. Kenne and B. Lindberg Carbohydr. Res. 1975 45 275. L. D. Melton L. Mindt D. A. Rees and G. R. Sanderson Carbohydr. Res. 1976,46 245. E. R. Morris D.A. Rees G. Young M. D. Walkinshaw and A. Darke J. Mol. Biol. 1977 110 1. D. A. Rees Biochem. J. 1972 126 257. G. Holzwarth Biochemistry 1976 15 4333. M. 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ISSN:0301-5696    

DOI:10.1039/FS9831800115

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

Faraday Symp. Chem. SOC.,1983 18 131-143 Dynamics of Xanthan Biopolymer in Semi-dilute Aqueous Solution Studied by Photon Correlation Spectroscopy Comparison with Solution Viscosities BY A. M. JAMIESON J. G.SOUTHWICK AND J. BLACKWELL Department of Macromolecular Science Case Western Reserve University Cleveland Ohio 44106 U.S.A. Received 7th September 1983 Recent theoretical treatments of molecular motion in entangled solutions of rod-like polymers lead to results which describe the rheological behaviour in terms of the concentration- dependent rotational diffusion coefficient D,. Thus a comparison between experimental values for D,and rheological data is possible and provides an exacting test of such theory. Here dynamic light scattering (DLS) studies of semi-dilute aqueous solutions of xanthan gum are presented and compared with independent rheological data.Despite extensive precautions unambiguous interpretation of these data is precluded by the known tendency of xanthan for self-association. Such problems appear to be common when dealing with rod-like polymers and indeed many features of the observed DLS and viscometric data for xanthan in water are strikingly similar to published results on other rod-like polymers. In particular strong correspondences are found when comparing numerical estimates of D,determined by DLS and shear-rate-dependent viscosities q(j) with a similar analysis of D,and q(j) for semi-dilute methanesulphonic acid solutions of poly(p-phenylene-2,6-benzobisthiazole),a rod-like polymer with a reportedly weak potential for self-association.For each system the D,and q(j) data are qualitatively consistent with recent modified versions of the Doi-Edwards dynamical theory for entangled solutions of rod-like polymers. However large ( x 100) quantitative discrepan- cies are found when comparing q(0)and D via this theory. A survey of rod-like polymer solvent systems for which D and q(j) are available indicates comparable discrepancies and none produces a completely self-consistent interpretation in terms of the above theory. INTRODUCTION Photon correlation spectr~scopy~-~ (PCS) generates useful information regarding the molecular dynamics of polymer molecules in solution in the form of the dynamic structure factor S(q,t) S(q7 0 = cc (exp [iqR,(t)l exp [iqRj(o)l> (1) ij where R,(t) denotes the time-dependent position of the ith chain segment.PCS provides numerical data in the form of characteristic relaxation times which describe the motion of R,(t)relative to Rj(0).Since these relaxation times are also determinants for the viscoelastic behaviour of the solutions PCS provides a means of testing molecular theories of viscoelasticity. In this paper we are concerned with the application of PCS to determine S(q,t) for entangled solutions of stiff or rod-like polymers and the comparison of these experimental data with appropriate theory. We review in detail results of a PCS study by Southwick et aZ.5-9of xanthan biopolymer in two solvent systems uiz. deionized water containing 0.003 mol dm-3 Na sodium 131 5-2 DYNAMICS OF XANTHAN BIOPOLYMER azide and water containing 4 mol dm-3 urea in the absence of salt after thermal cycling from 25 to 70 OC.These data are compared with similar experimental studies taken from the more recent 1iterature.lo*l1 A further comparisong of our PCS relaxation spectrum for xanthan in water with rheological data12 is presented Striking similarities are found with a recent independent study13 in which rheological experi- ments on solutions of rod-like polymers were subjected to an analysis in terms of molecular-dynamic relaxation times. Xanthan polysaccharide has a repeat pentasaccharide unit consisting of a back- bone of (1 -+ 4)-B-~-glucose linked to a trisaccharide side-chain at the 3 position on al- ternate glucose residues.The side-chain is p-D-mannopyranosyl-( l -+ 4)-a-~-glucuro-pyranosyl-( 1 -+2)-b-~-mannopyranoside 6-0-acetate. The terminal mannose exhibits a variable degree of pyruvate substitution. Each repeat unit therefore has an average of 1.3-1.5 charges. The native structure of xanthan appears to be a fivefold helix14 and the question of whether xanthan is a single- or multi-stranded helix remains ~nsett1ed.l~ A major difficulty in solution of studies of xanthan is that in common with many other rod-like polymers,13 it tends to form aggregate structures by a self-association mechanism.16 The stability of the native xanthan structure17~ and the degree of aggregationl69 l9are in each case enhanced by addition of salt. Since slow self-association of xanthan occurs after ultrafiltration of the sample preparation history must be carefully controlled.EXPERIMENTAL SOLUTION PREPARATION Xanthan polysaccharide ('Kelzan') was obtained from the Kelco Co. and purified by the method described by Ho1zworth.20 A stock solution containing 0.45% xanthan was prepared in deionized water prefiltered through 0.1 pm Millipore filters and dialysed for 4 days against deionized water. Light-scattering experiments were carried out on two solvent systems (a) deionized water containing 0.02% sodium azide (3 mmol dm-3) and (b) deionized water containing 4 mol dmP3 urea. The method of preparing solutions for light scattering was a critical parameter which strongly affected the quality of data obtained from PCS experiment^.^ Solutions of xanthan were obtained by dilution of the stock solution to the approximate desired concentration with pre-filtered (0.1 pm Millipore) deionized water containing 0.02 % sodium azide or 4 mol dm-3 urea.The solutions were adjusted to pH 7 if necessary by adding a small quantity of NaOH and filtered directly into the scattering cell. The filter size used strongly influenced the light-scattering results. Filtration through 0.45 or 0.8 pm filters resulted in solutions with unsatisfactory light-scattering characteristics because of the presence of large xanthan aggregates. Only filtration through 0.22 pm Millipore filters produced solutions with useful i.e. reproducible scattering proper tie^.^ It is important to note that the latter procedure resulted in significant concentration losses e.g.6% loss for a nominal 0.0115% solution and 35% loss for a nominal 0.115% solution. The concentrations of xanthan solutions were therefore determined after light-scattering analysis by evaporating the solutions to dryness and weighing the residual film. Solutions of xanthan in 4 mol dm-3 urea were heated to 90 "C for three hours then cooled to room temperature prior to light-scattering analysis. Filtration of these solutions through 0.22 ,um filters was much easier than for the first system and negligible concentration losses were detected. In 0.02 mol dm-3 sodium azide xanthan retains its native rod-like structure. In 4 mol dm-3 urea after heat treatment we discovered based on polarimetry and intrinsic viscosity measure- ments that the denatured form of xanthan was stabilized at room temperat~re.~.~.~~ Subsequent studies in other laboratoriesle* 21 have shown that the principal effect of thermal cycling in 4 mol dm-3 urea is to aid dissolution of xanthan aggregate structures and that 4 mol dm-3 urea does not in fact stabilize the denatured form.It appears that our isolation of denatured xanthan at room temperature was primarily because of the extreme salt depletion A. M. JAMIESON J. G. SOUTHWICK AND J. BLACKWELL [a1 / 20 40 1 I 1 I I I 20 40 60 80 temperature /"C Fig. 1. Temperature dependence of optical rotation of xanthan in deionized water 0,increasing temperature; 0,decreasing temperature; vertical arrow at T = 22 OC indicates decrease of rotation on addition of salt.of the 4 mol dmP3 urea solutions which did not contain azide and were derived from the deionized stock solution. Thus it is possible,33 as shown in fig. 1 to denature xanthan irreversibly in deionized water in the absence of added salt as shown by the irreversible loss of optical rotation after thermal cycling between 25 and 90 OC. The original optical rotatory characteristics of the xanthan deionized water solution indicative of a renaturation is recovered only after the addition of salt. The observations shown in fig. 1 essentially parallel those reported earlier for thermal cycling of xanthan in 4mol dm-3 urea. For heat-treated xanthan solutions in 4mol dm-3 urea we determined the molecular weight to be M = 2.16 x lo6 daltons.This value is in excellent agreement with independent determinations for samples of comparable history studied in both 4 mol dm-3 urea and in aqueous NaCl. It is finally noted that the PCS data reported below were in each case obtained from solutions subjected to filtration through 0.22 pm Millipore and centrifugation at 4500 g for 30 min immediately prior to light-scattering analysis. PHOTON CORRELATION SPECTROSCOPY OF ROD-LIKE POLYMERS As noted above PCS produces information regarding molecular dynamics of polymer solutions in the form of the dynamic structure factor S(q t) [eqn (l)]. The characteristics of S(q,t) are strongly dependent on the magnitude of the inverse scattering length 1qI-l relative to the radius of gyration of the macromolecules R,.For small gR one can neglect terms in eqn (1) reflecting motion of chain segments relative to others on the same chain and consider only the relative motion of pairs of molecules. The latter process is described by Fick's diffusion equation and one S(q,1) = exp ( -D,s") (2) where D is the rotationally averaged translational diffusion coefficient of the polymer. Thus for monodisperse solutes S(q t)is a single exponential. For polydisperse systems S(q,t)= G(D)exp (-Dq2t)dD (3) jOm DYNAMICS OF XANTHAN BIOPOLYMER where G(D)is the distribution of diffusion coefficients. It is possible to analyse such data by the method of 1 P2 2!r2lnS(q,t) = -i+t+-_( -rt)2+ ... (4) where the first moment is related to the z-average diffusion coefficient I? = D,,,q2 and the second moment gives the variance in r p2 = i=2-(f)2.(6) When qR 2 1.5 i.e. q-L & 6 PCS detects contributions to S(q,t) reflecting internal m~tions,l-~ uiz. rotational motion in the case of rigid rods. These are manifested for monodisperse rods as the appearance of a second exponential decay in addition to eqn (2)of the form Sr,t(q t) = exp -(0,q2+60,)t (7) where D is the rotational diffusion coefficient. The relative amplitude of S,,,(q,t) increases as q is increased beyond q*Lx6. It is also pos~ible~-~~~~ to determine D and D from PCS studies of depolarized light scattering if the rod polymers have sufficient anisotropy in their molecular polarizability. The characteristics of S(q,t) are strongly dependent on the concentration of macromolecules.In dilute solution where the centres of mass of the solute molecules are on average separated by distances which are large compared with R, the numerical values of Dt and D may vary with c~ncentration,~ reflecting the effect of direct and indirect (hydrodynamic) forces between molecules. Furthermore above a characteristic concentration c* where the domains of individual chains overlap it may be anticipated that the physical entanglement of rods will produce changes in S(q t)because of the coupling of rotational motions with anisotropic translational diffusion. At least three theoretical discussions have been given for the case of monodisperse rod-like molecules which imply that the effect of entanglements is to change S(q,t) from a single-exponential decay to a markedly non-exponential form.Lee et aZ.24deduce that S(q t) will be the sum of two exponential decays with time constants and relative amplitudes which depend strongly on the magnitude of D,q2 relative to D,. A generalized form of this theory has been given recently by Zero and Pecora.l0 For the weak coupling limit defined as y 5 10 where y = AD,g2/D,,with ADt = Dll-DI the diffusional anisotropy Zero and Pecora derivelo T2 = Dtq2-&Dry2 where D = $(Dll +2D1) is the average translational diffusion coefficient. Doi and 26 have presented an alternative derivation using a similar initial equation of motion which deduces for the strong coupling case ADtq2/D,% 1 that S(q t) contains a continuous distribution of relaxation times.The initial and long-time decay regions of S(q,t) can be approximated however as single exponentials with time constants rl= to,,, q2 = iDtoq2 (10) r2 = (q2DrDtoY (1 1) A. M. JAMIESON J. G. SOUTHWICK AND J. BLACKWELL where D, is the infinite-dilution value of D,. The discussion of Doi and Edwards includes a prediction2' of the concentration dependence of D where D, is the infinite-dilution value p* = 1/L3 and p** = l/dL2 are the overlap concentration and the liquid-crystal transition concentration respectively p is a constant of order unity and f(c/c**)= (1 -Bc/c**)-2 (13) with B a constant slightly smaller than unity. For c* 4 c 4 c** evidentlyf(c/c**) = 1. Since the dynamical analysis of Doi and Edwards26 also predicts that the rheological properties of the solution will scale with concentration and molecular weight in a manner that is dictated uniquely by the behaviour of D,,a numerical comparison between S(q t) and rheological experiment is in principle possible.More recently certain deficiencies in the original Doi-Edwards theory have been recognized and improved versions 29 The changes lead to significant numerical differences in the predicted rheological results e.g. the onset of entanglement interaction occurs at a much higher polymer concentration i.e. p M 100/L3 and hence in eqn (12) /?M O(lo4). However certain of the qualitative scaling predictions of the original Doi-Edwards theory are still valid for the revised theories.For example r, the zero-shear viscosity for c > c* is given by28 Also q(0)/qsis expected to fall on a master curve independent of concentration when plotted against the reduced shear rate j/D,. The revised dynamical theories do not appear to imply a modification of the predicted shape of S(q,t) but certainly suggest that the onset of non-exponential behaviour will be seen at a much higher concentration. The modified versions of the Doi-Edwards theory have been found to fit available rheological data on semi-dilute solutions of stiff-chain molecules semi-quantita- tively.28* 29 Comparisons of theory with experimental S(q,t) data have also been described for such systems. Bimodal exponential decays which are qualitatively .~~ consistent wth the theory of Lee et ~1have been observed in DNA PCS data qualitatively in agreement with Doi-Edwards theory have now been reported for several rod-like polymer^.^-^^^ 30 RESULTS We begin by re-examining our dynamic light-scattering studies of xanthan solutions.PCS data from xanthan in 0.003 mol dm-3 sodium azide showed a slow decrease in D,, and an accompanying increase in p2/r2when observed over a period of 100 h after filtration through 0.22pm Millipore.' This indicates slow self-association behaviour.' Table 1 summarizes the hydrodynamic parameters deduced from dilute freshly filtered solutions of xanthan in aqueous NaCl and 4moldrnF3 urea. The data in table 1 indicate that [q] and DY,* each decrease with increasing ionic strength.Furthermore the filterability of the solutions decreases with ionic strength and filtration losses increase.16 Thus it appears likely that the screening of intermolecular electrostatic repulsions at higher ionic strength results in enhanced predominantly DYNAMICS OF XANTHAN BIOPOLYMER Table 1. Summary of hydrodynamic data for xanthan in various solvent systems solvent H,O (0.003 mol dmF3 azide) > 7000 -2.42 1005 0.01 mol dm-3 NaCl 5150 0.633 0.3 rnol dm-3 NaCl 1.94 1226 1.0 mol dm-3 NaCl 4700 0.724 1.55 1436 4 mol dm-3 urea (heated) 2000 0.083 2.75 800 ~~ a Huggins constant. 0 0 =I 0.01 0.02 I I I 0.25 0.5 time/s Fig. 2. Effect of concentration on photon correlation function of light scattered by xanthan in 0.003 mol dm-3 sodium azide (a) c = 0.026% w/v (b) c = 0.069%w/v (c) c = 0.092%W/V, 0 = 40° R = 6328 A.side-by-side association. The larger Dt and smaller [q] observed for xanthan in 4 mol dm-3 urea presumably reflect partial flexibility in the denatured state. It should be noted however that the values of [q]and D:,z indicate a hydrodynamic volume considerably larger than calculated for a xanthan random coil.lS Evidently the denatured form in 4 mol dm-3 urea is highly extended presumably because of the A. M. JAMIESON J. G. SOUTHWICK AND J. BLACKWELL xxX xX X X X X X X X I 5 0 10 20 time/ms Fig. 3. Photon correlation function of light scattered by xanthan in 4 mol dm-3 urea at c = 0.15%W/V,e = 400 A = 6328 A.1 0.04 0.08 0.12 0.16 concentration (% w/v) Fig. 4. Concentration dependence of average PCS relaxation frequencies for light scattered from xanthan in 0.003 mol dm-3 sodium azide. For c < 0.046% w/v data plotted are first moments from cumulant fits; for c >/ 0.069% w/v points plotted are rl and Tzfrom bi-exponential fits. bulky side-chains and intramolecular electrostatic repulsions. Note that for freshly filtered dilute xanthan solutions in all solvents p2/T2z0.3-0.4. Thus even in dilute solution PCS observes non-exponential decays and the xanthan molecules are polydisperse. Based on a discussion of Kubota and Chu," we estimate M,/M z 1 +p2/T2z 1.35 k0.05. DYNAMICS OF XANTHAN BIOPOLYMER \ \ \ \ \ 1 \ \ \ \ \ .--I r-.--,--a 0 0.1 0.2 0 -3 concentration (% w/v) Fig.5. Concentration dependence of average PCS relaxation frequencies for light scattered from xanthan in 4 mol dm-3 urea. For c < 0.075%w/v data plotted are first moments r from cumulant fits; for c 2 0.15%w/v points plotted are rl and Tzfrom bi-exponential fits. As the xanthan concentration is increased in both 0.003 mol dm-3 sodium azide (fig. 2) and in 4mol dm-3 urea (fig. 3) a significant increase in the deviation from exponentiality occurs over a narrow concentration range. The latter coincides accurately with the concentration regime where independent measurements of the macroscopic viscosity6 l2and microviscosity9 of the solution show sudden increases associated with the onset of entanglement interactions.At concentrations above 0.069% w/v for 0.003 mol dm-3 sodium azideg and above 0.15% w/v for 4 mol dmP3 urea,5 the experimental S(q,t) can be approximately interpreted as the sum of two exponential decays as indicated in fig. 4 and 5. DISCUSSION As we have discussed elsewhere,s it is not possible to achieve a definitive molecular- dynamic interpretation of the available PCS data on concentrated solutions of xanthan. Similar difficulties have been noted in interpreting rheological data on solutions of rod p01ymers.l~ Because of the intrinsic polydispersity of xanthan and its tendency to form aggregates one possibility is that at higher concentrations association occurs to form slowly diffusing Although such a process would be expected to decrease solution viscosity if side-by-side association is present,13 it may be that at sufficiently high concentration the aggregates act as junction zones in a gel Also if a random association process occurs enhancement of the viscosity and gel formation may res~1t.l~ Even for the case of xanthan in 4 mol dm-3 urea where evidence suggestslg~ that aggregation does not occur at lower concen- 21 trations we cannot definitively exclude the possibility that the appearance of a slow decay process above c = 0.15% w/v is due to a weak residual tendency for self-association.Morris et a1.l9 have observed a slow mode in PCS data of concentrated xanthan solutions which disappeared upon ultracentrifugation and ultrafiltration and therefore was assigned to the presence of aggregates.The aggregate structure is A. M. JAMIESON J. G. SOUTHWICK AND J. BLACKWELL Table 2. Dynamics of xanthan in water 0.069 ~ 832 42.0 3.03 670 14 0.092 560 4.0 2.18 8.2 9 0.14 536 2.8 2.07 4.1 3.6 a Assumes DYlc = DF:lc (c*/c)~,where c* = 3 x w/v. interpreted19 as due to side-by-side association of 47 units. One potential difficulty of this approach for polydisperse solutes however is that such treatment is likely also to fractionate the polymer even if aggregates are not present and thus will also dramatically alter the hydrodynamic properties of the solutions which are very sensitive to rod length [see eqn (12)]. As noted above we have attempted to exclude aggregates by ultrafiltration immediately prior to light-scattering analysis.On the other hand since a small degree of side-by-side association produces aggregates of the same approximate length as a single molecule PCS and rheological data on polydisperse solutions of rod-like chains may be expected to retain the principal dynamical features discussed by Doi and Ed~ards.~~-~~ Thus the rheological studies of Whitcomb and Macosko31 show evidence for the existence of conventional shear-thinning viscometric behaviour in concentrated solutions of xanthan in water. Approximate estimates of q(0) from these data9 scale as c3.0 for concentrations 0.1 5 c(wt%) 5 0.20 consistent with Doi-Edwards the~ry.~~-~~ Similarly the PCS data for c > 0.1% observed in our experiments show features qualitatively consistent with such theory.Thus the initial slope becomes concentration independent while the long-time decay varies as c-l in accord with eqn (10) and (1 1). Table 2 shows a comparison of experimental data for xanthan in water with the modified Doi-Edwards theory uiz. eqn (10) and (1 l) based on the values L x d = 15000 A x 20 A p M lo3 and c* = @/L3N" 3 x This comparison is amplified in fig. 6 which compares experimental values of q(j)/q(O) against j/D with the Doi-Edwards theory and subsequent modifications. Bearing in mind the intrinsic polydispersity of xanthan and that the PCS and rheological data were performed on different xanthan samples it is interesting to note several features of the above results which are remarkably similar to observations made independently on other rod-like polymers (1) our estimates of D for c > 0.1% w/v are numerically very similar to those reported in other PCS experimentslO7 11,30 and indicate #? z 103-104; (2) utilization of our experimental relation for D, viz.D,(s-') = 6.6 x (conc.% w/v)-~ (15) results in reduced shear viscosity curves q(j)/q(O)against j/D (fig. 6) which are very consistent with theoretical expectation28v 29 and experimental analyses;l3? 29 (3) the 28v observed values of Dtl,are much larger than the predicted result Dtll = $Dto and there is no formal basis in theory for the observed maximum in rl(fig. 4); (4) the value of p deduced from D predicts values of q(0)based on eqn (13) which are much larger (ca.x 100) than the experimental q(0).Note that a similar maximum in the short-time decay constant of S(q,t)appears in the PCS study of concentrated solutions of rod-like poly(y-benzyl glutamate) (PBLG) by Kubota and Chu. l1 Also comparable internal numerical discrepancies in the scaling of q(0)against D and ~(j) to against (?/Or) DYNAMICS OF XANTHAN BIOPOLYMER 0.0 -*.O t Fig. 6. Reduced plots of shear viscosity from ref. (12) v(j)/q(O) as functions of j/Dr where D is estimated from eqn (15); solid lines are theoretical predictions; DE ref. (26); JC ref. (28); FD ref. (29); 0,0.1% w/v; x 0.2%w/v; 0, 0.35% w/v. those referred to under (4) above are encountered in much of the literature data on rod-like polymers.1o$ l1 139 28 At least two factors are likely to contribute to the existence of a maximum in rl.First we note that the PCS experiment probes the mutual diffusion coefficient and that this quantity can exhibit significant concentration dependence even in dilute solution because of thermodynamic and hydrodynamic forces. For a system of rods interacting through predominantly repulsive forces one might anticipate a significant increase in D,as one nears the concentration where the equivalent hydrodynamic spheres overlap. Since the effect of entanglement coupling is seen at such comparatively high concentrations these effects cannot be neglected. Conversely rotational self- diffusion is driven by Brownian collisions and is expected to stay essentially constant or slowly decrease as c increases towards c*.A maximum in rl is thus plausible at c* when rotation couples to the enhanced mutual diffusion process. Secondly the onset of entanglement interaction is gradual and there is clearly a range of concentrations which coincide with the observed maxima9?l1 in rl,where a comparatively weak coupling of translational and rotational motions occurs. In this regime as indicated by eqn (8) and (9) theory SuggestslO that rl >Dtq2and Tz<D,q2 are reasonable. As the concentration increases the coupling strength y increases and since in the y +co limit the Doi-Edwards theory [cf. eqn (10) and (I l)] implies rl <D,q2 and T2 6 Dtq2 it appears that a maximum in rl must occur even in the absence of thermodynamic and hydrodynamic contributions to D,.Note comparing fig.4 and 5,that xanthan in 4 mol dm-3 urea shows behaviour similar to that for xanthan in water albeit at a higher concentration. This is consistent with the idea that denaturation of xanthan at low ionic strength produces an extended stiff coil which will however have a smaller axial ratio because of the extended side-chains and a smaller hydrodynamic volume because of the enhanced chain flexibility. The precise origin of the numerical discrepancies in comparing experimental estimates of ~(0) against D via eqn (14) is more difficult to identify. Similar problems were encountered in a rheological study by Chu et aZ.13of several rod-like polymers A. M. JAMIESON J. G. SOUTHWICK AND J. BLACKWELL in which experimental estimates of D were calculated from ~(0)and the limiting recoverable compliance R(0) via the equationI3 z = q(O)R(O)= (6Dr)-'.Chu et aZ.13observed systematic numerical disparities for these polymers in comparing ~(0) against eqn (14) which correlate closely with the experimental variation in observed values for c** the concentration for onset of the ordered anisotropic phase for each polymer. Thus they suggest the deviations may be due to systematic variations in chain flexibility or interchain association which alter ~(0) and c** in the same way.31 For example ~(o),~(1;)and D values for the semi-flexible polymer poly(pphenyleneterephtha1amide) (PPTO) in methanesulphonic acid (MSA) fall in the concentration range c* 6 c 6 c** and are found to give a good self-consistent fit when tested against the Doi-Edwards theory.However the rheological data for the more rigid poly(p-phenylene-2,6-benzobisthiazole)(PBT) in MSA show strong discrepancies which are remarkably similar to those noted above in our observations on the xanthan + water system. Specifically the experimental D produce q(j/Dr) curves which accurately fit the modified 29 of Doi-Edwards and predict ~(0) values from eqn (14) which show the anticipated scaling behaviour against concentration but which are 100 times larger than experiment. The rheological data reported for PBT in MSA fall in a narrow concentration range near c** 0.8 c** 5 c 5 c**. For xanthan preparations similar to those studied in our work a value c** = 2.5 g dm-3 has been reported;32 thus it is evident that our analyses indeed fall in a concentration regime very similar to that referred to for PBT in MSA.Further as we have noted el~ewhere,~~ the viscosity of xanthan in water appears to increase faster than c3-0in this concentration range in agreement with the deduction that the functionf(c/c**) 6 1. Chu et aZ.13 interpret their data for PBT in MSA as suggesting a relatively low degree of (side-by-side) association and a high rigidity for PBT. However PCS studies of dilute solutions of PBT in MSA34 indicate that while most of the polymer is present in a low state of aggregation (1-2 chains per aggregate) a small percentage is in fact associated in very large loosely organized aggregates. Also slow time-dependent increases of viscosity have been observed for PBT in MSA.13 It seems worth noting that in the xanthan+water and PBT+MSA systems the aggregation processes occur even though in each case we are dealing with a highly charged polyion in a solvent of low ionic strength.It is also pertinent that smaller but significant (x 10) differences are found for semi-dilute solutions of helical poly(y-benzyl-L-glutamate) (PBLG) between values of determined28 from ~(0)via eqn (13) and those estimated from D data either uia reduced ~(j) curves28 or from PCS studies,lo the latter again being smaller. These discrepancies may derive from one or more of several factors. For example the polymers studied generally have rather broad molecular-weight distributions and different rheological quantities may represent different molecular-weight averages.Moreover for a given mass/volume concentration side-by-side association may have a relatively minor effect on the average rod length and hence D, but decrease ~(0) significantly by reducing the average number of particles. For the particular examples of the xanthan + water and PBT + MSA systems studied near c** where the discre- pancies are rather large the influence of strong electrostatic repulsions between molecules and/or the possible presence of small numbers of giant aggregates in such highly congested systems may significantly modify the viscometric properties through some as yet unexplained mechanism. Finally near c** as noted by DO^,^' the dynamical model from which eqn (12) and (1 3) are derived is an approximation and may be inadequate.For example the situation near and above c** may be analogous 142 DYNAMICS OF XANTHAN BIOPOLYMER to the onset of a glass-transition 33 As a result the width of the relaxation spectrum may be increased and individual rheological quantities may be influenced by different portions of this spectrum. In summary correlation of experimental data for D,and q(p) represents an exacting test of theories of molecular dynamics in entangled solutions of rod-like molecules. In view of the extreme sensitivity of such data to rod length and particle number density it is difficult however to realize in practice an appropriate experimental test of such theory because of the intrinsic polydispersity partial flexibility and the strong tendency for self-association.In our PCS study of xanthan in water a definitive interpretation of the long-time portion of S(q,t) is precluded because of the possible formation of small concentrations of large aggregate structures. However the short-time decay shows behaviour similar to that reported in comparable studies in other congested solutions of rod-like polymers. These observations are qualitatively consistent with the molecular-dynamic theory. Furthermore estimates of D derived from the long-time decay compare with ~(9) data for xanthan in water in a fashion consistent with theory. A survey of systems for which D,and q(j)data exist indicates reasonable qualitative agreement with theoretically predicted scaling laws for D and ~(0)via eqn (12) and (14).Also experimental estimates of D appear to produce reduced viscosity curves ~(f)/q(O)in reasonable harmony with theory. However systematic numerical dis- crepancies appear to exist when comparing q(0)against D,via eqn (14) which range from values of ca. 100 for highly congested systems (c x c**) such as xanthan in water or PBT in MSA to comparatively minor differences in the system PPTA+MSA. Paradoxically however PPTA appears to be a comparatively flexible molecule with a persistence length approximately four times smaller than the contour length for the polymers studied.13 We thank the National Science Foundation for support through grant CPE 8017821. We also thank Prof. G. C. Berry for discussions which aided our under-standing of the rheological properties of entangled solutions of rod-like polymers.I B. 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ISSN:0301-5696    

DOI:10.1039/FS9831800131

出版商:RSC    

年代:1983

数据来源: RSC