GENERAL DISCUSSION Prof. B. J. Ackerson (R.S. R.E., Malvern) addressed Dr Medina-Noyola: You have presented a theory for the diffusion of particles which treats one particle explicitly and the other particles as an effective fluid. In this theory a memory kerneI G ( p , p ’ ) appears which is treated in an approximate way using microscopic expressions, mode coupling theory etc. Is it possible to use fluctuation dissipation arguments to find or approximate G(p, p’)? Can one assume that the effective fluid will behave like a normal hydrodynamic fluid and produce a ‘Stokes drag’ on the tagged particle interacting with the fluid with a potential cp in the steady state? Presumably this drag would be in terms of an effective viscosity which would have to be eliminated self-consistently. Dr M.Medina-Noyola (Cinvestav, Mexico City) replied: Let me refer to the memory kernel G(r, r’; t ) in eqn (3.5) of my paper. This function contains essentially the same information as the dynamic correlation function (8n(r, t ) s n ( r ’ , 0)). I do not know of any fluctuation-dissipation arguments which, by themselves, lead to the determination of either of these quantities. However, as I have shown in my paper (section 3), this type of argument can be employed to define the general structure of G(r, r’; t ) , namely that in eqn (3.6). Although this only seems to substitute our ignorance concerning G( r, r ’ ; t ) by our lack of knowledge of the generalized friction tensor, G( r, r’; t ) , which describes the dissipative relaxation of the local macrofluid current, the fluctuation- dissipation theorem assures us, however, that this quantity may be expressed as the correlation function of a random force [see the equation following eqn (3.4)].This fact can be used in the construction of approximate expressions for G(r, r’; t ) . Very much in the spirit of your suggestion, Hess and Klein’ proceeded along these lines in the implementation of the mode-mode coupling approximation for the collective properties of a colloidal suspension. Such a method allows the self-consistent determination of G(r, r’; t ) , which is indeed related to an effective viscosity of the macrofluid. I would say, however, that in the time- and length-scales which we are interested in, this macro- (or effective) fluid cannot be assumed to behave as a normal hydrodynamic fluid.We may, on the other hand, calculate the analogue of the Stokes drag on a tagged particle with the stationary-state method that you describe. I will refer to it as the relaxation-effect (R.E.) method, as opposed to the fluctuation-theory (F.T.) approach that I employed in my paper. The equivalence between both methods is expected, at least in principle, on the basis of the (‘first’) fluctuation-dissipation theorem. Let me briefly illustrate this in the following way. Consider eqn (4.2) when the tracer is subjected to a constant force Fext: MT du(t)/dt=-{$u(t)+ [V+(p)]n’(p, t ) d3p+fT(t)+Fext. (A. 1 ) I I An average of this equation over all possible realizations of f T ( t ) leads to MT d(u(f))/dt = -S”ru(t))+ “v+(P)l(n‘(P, t ) ) d3P+Fext (A.2) which, under stationary-state conditions, reads F e x t = ~ $ ( ~ ) s h - [V+(p”{n‘(p))’’d’p (A.3) i.e.the external force needed to pull the particle at a constant velocity (u)” must balance 8788 General Discussion the friction produced by the solvent and the friction originating from the collisions of the tracer with the other diffusing particles. ( n ’ ( r ) ) s c contributes to the integral in eqn (A.3) only through its departure, An(p), from its radial equilibrium value, n e q ( p ) = ng(p). An (p) vanishes with Fxt, and is the solution of a (linearized) diffusion equation, which we shall write as t , = [Vneq(p)] - ( u ( t ) > + 1 d’p’D*V neq(p)Va&p, p’) An(p’, t ) . (A.4) a t This is essentially the average of eqn (4.4) over all the realizations of the random diffusive fluxes f’(p, t ) , and with G’(p, p’; t ) approximated by its modified Fick’s diffusion expression, eqn (4.16).The stationary solution reads Anss(p) = - %(p, p’)[V’neq(p’)] * (u)’” d ” p ’ (A.5) I where 3 is the Green’s function of eqn (A.4) under stationary conditions, i.e. it is the solution of d3p’ D*V n e q ( p ) V a i d ( ( p , p’)%(p’, p”) =S(p-p”). (A.6) I Substituting eqn (AS) in eqn (A.3), we get Fex‘= {LOTI+ d3p d”p’[V+(p)]%(p, p’)[V’neq(p’)]} - (u)”” (A.7) I I A . E { 607.1 - Glnt( z = 0)) - (2-))”. From this equation it is clear that the total friction coefficient on the tracer is the sum of the solvent friction, I[;, plus the contribution due to the direct interactions, which we denoted asAGin‘(z = 0).The reason for this notation is clear if we compare the expression for W”(z = 0) in eqn (A.7), which is the result of the R.E. method, with the corresponding result of the F.T. approach, i.e. the z = 0 limit of the Laplace transform of Gint ( t ) defined in eqn (4.10) and (4.1 l ) , together with eqn (4.16). From this comparison it is clear that both approaches should yield in principle the same results. Let me mention that if the external force F‘“‘ in eqn (A.l) is considered to be time-dependent, the R.E. method may still be applied along the lines just indicated, leading to the determination of a time-dependent friction function. Such a quantity is also given by eqn (4.10), (4.11) and (4.16). Furthermore, the use of Fick’s diffusion law, eqn (4.16), was meant to illustrate the relationship between the two methods.Other approximations for this diffusion equation, which might include memory effects, could also be considered. To the extent that the two approaches to calculate the friction coefficient are based on the same diffusion equation, identical results are to be expected. Let me finally remark that in establishing the equivalence between a result of the theory of spontaneous fluctuations, and a result derived with a linear response approach, the proper external field in the latter must be identified. In our example, since we are describing tracer diffusion, such an external field corresponds to a force acting only on the tracer, and not on the other diffusing particles. A more general discussion of the equivalence between the two theoretical approaches mentioned above has been made by Hess and Klein’ in the context of the properties of colloidal systems.1 W. Hess and R. Klein, Ado. Phys., 1983, 32, 173. Prof. I. Qppenheim ( M U , Cambridge, MA) then asked: Does it make a difference whether the tagged particle is heavy or light as compared to the other particles? Where does it enter that the other particles are big or small?General Discussion 89 Dr M. Medina-Noyola (Cinuestau, Mexico City) replied: The results for the friction function in eqn (4.9)-(4.13) are based, essentially, on eqn (4.2)-(4.4). The latter equations only define the structure of the coupled equations of motion of the tracer and of the local concentration of the surrounding particles, and are hence rather general.If the tagged particle is not much heavier than the solvent particles, one might have to replace the friction term -l:u(t), in eqn (4.2) by a temporally non-local friction. I would expect, however, that the Markovian limit in that equation should continue to be a very reasonable approximation for times comparable with the relaxation time of the local structure of the other particles around the tracer. On the other hand, considerations concerning the size of the other particles should enter when approximating the propagator ~ ’ ( p , p’; t ) or the relaxation kernel G’(p, p’; t ) appearing in the equations just referred. The applications mentioned in my paper, however, are based for simplicity on a Fick’s diffusion level of description for the dynamics of such particles, both, when they are as big (self-diffusion), or much smaller (electrolyte friction), than the tracer.Prof. H. N. W. Lekkerkerker (Utrecht, The Netherlands) said: I would like to raise the following point with Dr Medina-Noyola. In your paper you mention that the mode-mode coupling approximation and the modified Fick diffusion law [ eqn (4.16)] coincide in the low concentration limit. For hard spheres their result is given by D , = Do( 1 -$q) rather than the exact value D, = Do( 1 - 277). You say that you can show that if the inhomogeneous Fick’s diffusion approximation in eqn (4.16) is employed the correct result should follow. In view of the fact that the two approximations coincide in the low concentration limit, does this mean that you can also obtain the exact result from the mode-mode coupling approximation? If not what is wrong with the mode-mode coupling approxima- tion? Indeed can one trust the mode-mode coupling approximation at all if even for the simplest case it gives wrong results? Dr M.Medina-Noyola (Cinuestau, Mexico City) replied: Let me stress, first of all, that the most general results in my paper are eqn (4.9)-(4.13). These results still require a specific definition of G’(p, p’; t ) , i.e. of the diffusion equation in the reference frame of the tracer, and it is here that additional approximations must enter. We must say, however, that in the limit of infinite dilution of diffusing particles, and if the tracer is one such particle, the exact expression for G’(p, p’; t ) may be written, based on the two-particle Smoluchowski equation, as eqn (4.16) with neq(r) = n exp [-gb(r)/kT] o-iJ(r, r‘) = S( r - r ’ ) / neq( r ) .(L.1) (L.2) and Under these conditions, the properties of the hard-sphere (H.S.) system are known exactly after the work of Hanna et al.’ and of Ackerson and Fleishman.* In particular, D,/ Do = 1 - 277. As explained in my answer to Prof. Ackerson, the use of the general results in eqn (4.9)-(4.13) yields equivalent results, regarding the calculation of A l ( t ) , to the use of the R.E. method. In fact, under stationary conditions, eqn (A.4) (of my response to Prof. Ackerson), with eqn (L.l) and (L.2), is identical to the equation that had to be solved in the derivation of the exact result for D , at infinite dilution via the R.E.method by yourself and Dhont3 (for hard spheres) and by Van den Broeck4 (for H.S. plus square-well potential). This explains my statement that the use of the inhomogeneous Fick’s diffusion approximation, i.e. eqn (4.16), with eqn (L.1) and (L.2), should yield the exact infinite-dilution limit. Unfortunately, except in this limit, even the writing of G’(p, p’; t ) must involve approximations. Eqn (4.16) still constitutes an interesting approximation, but eqn (L. 1)90 Genera 1 Discussion and (L.2) must be replaced by respectively, where g( r ) is the radial distribution function of the diffusing particles around the tracer, and c(l)(r, r‘) is the direct correlation function of such particles in the field of the tracer, i.e. effectively a three-body correlation function.Thus, even within Fick’s diffusion approximation, some simplifications would be desirable. The ‘homogeneous fluid approximation’ is one such simplification, which replaces eqn (L.3) and (L.4) by neq( r ) = n ( Lh5) and (L.6) - 1 oo0 = S ( r - r ’ ) / n - c ( l r - r’l) where c ( r ) is the bulk direct correlation function of the macrofluid. This leads,’ for example, to the following expression for D,: - 1 d3r[g( r ) - 1j2] . (L.7) It is not a simple matter to estimate a priori, and in a general fashion, the degree of accuracy of this type of approximation. As stated above, the result in eqn (L.7) originates from our inability to calculate exactly the properties of systems away from the infinite- dilution limit. Hence, applying it to infinitely dilute systems has only the purpose of establishing a comparison with exact results, and in this sense we have pointed out that eqn (L.7) yields DJD” = 1 - (4/3)77 for the hard-sphere potential to linear order in volume fraction.On the other hand, with situations explicitly different from the infinite dilution limit in mind, one could think of introducing the homogeneous fluid approximation at the outset, without specifically defining G’(p, 9’; t ) otherwise. This led to eqn (4.14) of my paper. Fick’s diffusion law provides one possible closure for this equation. The mode- mode coupling approximation (MMCA) provides still another. The version of the MMCA considered in my paper has already built-in the homogeneous-fluid approxima- tion, and hence, similar comments as above should be made concerning its infinite- dilution limit.If, however, one is particularly interested in incorporating the exact low-concentration results, it is then clear that the homogeneous-fluid approximation has to be relaxed, in the MMC or in other approaches. We should mention that a related problem concerning the applicability of the MMCA at low densities has been faced in the field of simple liquids, where approaches have been developed‘ that essentially interpolate between the exact low-density results and MMC-type results at high densities. Whereas it is surely advisable to implement analogous developments for colloidal systems, it is also interesting to have an idea of the accuracy of the current MMC theory when applied to those conditions for which it was suggested in the first place, i.e.for systems in which the interaction of the tracer with the macrofluid’s collective fluctuations, rather than with individual particles during binary collisions, is the dominant relaxation mechanism. Here another interesting limit should also be considered, which is rigorously satisfied by the mode-mode coupling approximation, namely, the weak coupling limit.’ One could mention, for example, that the results of the mode-mode coupling theory, as applied to coulombic systems,’ are consistent in the Debye-Huckel limit with Onsager’s limiting law. On the other hand, away from the weak coupling limit, i.e. for highly coupled colloidal systems, we do not have simple exact results toGeneral Discussion 91 1 0.8 0.6 13 \ h * v 0 .L 0.2 0 1 1 1 2 3 .4 5 tlms Fig. 1. Time-dependent self-diffusion coefficient, D( t ) = (ArT( t))2)/(6t), in units of the free- diffusion coefficient, Do = 9.5 x lo-” m2 sC1 as a function of time, for a suspension of particles interacting through a hard-sphere plus a highly repulsive Yukawa tail. The dots are the computer- simulation data of ref. (8) and the solid line indicates the MMCA results. guide us, other than computer simulations. Hence, in fig. 1 I present a comparison of a MMC-mean-field approximation, defined by’ eqn (4.14), with x ’ ( k t ) = x A k t ) X ( k , t ) = exp ( - k 2 D D , t ) exp [ - k ’ D ” t / S ( k ) ] (L.8) as applied to the hard-sphere plus Yukawa system defined in section 5, with Brownian- dynamics simulation results’ for a screening parameter z = 0.149, volume fraction 7 = 4.4 x and coupling constant K = 801.That this is indeed a highly coupled system is indicated by the fact that for the same values of z and 7, it freezes at K =924. It seems to me that this comparison indicates that the MMCA is a quantitatively useful and reliable approximation, at least under those conditions for which it was conceived. 1 S . Hanna, W. Hess and R. Klein, Physica A, 1982, 111, 181. 2 B. J. Ackerson and L. Fleishman, J. Chem. Phys., 1982, 76, 2675. 3 H. N. W. Lekkerkerker and J. K. G. Dhont, J. Chem. Phys., 1984, 80, 5790. 4 C. Van den Broeck, J. Chem. Phys., 1985, 82, 4248. 5 A. Vizcarra-Rendon, H. Ruiz-Estrada, M. Medina-Noyola and R. Klein, J. Chem. Phys., 1987, 86, 2976. 6 G. F. Mazenko and S .Yip, in Statistical Mechanics. Part B : Time-dependent Processes, ed. B . Berne 7 G. Nagele, M. Medina-Noyola, R. Klein and J. L. Arauz-Lara, to be published. 8 K. J. Gaylor, I. K. Snook, W. J. van Megen and R. 0. Watts, J. Chem. SOC., Faraday Trans. 2, 1980,76, 9 W. Hess and R. Klein, Adv. Phys., 1983, 32, 173. (Plenum Press, New York, 1977)- chap. 4, pp. 181-231. 1067. Dr B. Cichocki ( R . W. T. H. Aachen, Federal Republic of Germany; Warsaw University, Poland) said: I have a comment related to Dr Medina-Noyola’s paper and Prof. Lekkerkerker’s remark. First I would like to stress that one should be very careful when applying kinetic theory methods originally derived for fluids to suspensions. An example is the Mori-Zwanzig projection operator technique.Insufficiently careful application of the technique to the generalized Smoluchowski equation has led to the well known controversy between the configuration space description of suspensions and the descrip- tion on the Fokker-Planck level.’ This point has recently been clarified.292 General Discussion Fig. 2. Dynamical events of two hard spheres: ( a ) in gas, ( b ) in suspension. Another example is the mode-mode coupling theory. To explain the discrepancy mentioned by Prof. Lekkerkerker let us consider two hard spheres in a dilute gas and in a dilute suspension. In the gas, when we neglect the presence of other particles, two hard spheres can collide only once [fig. 2 ( a ) ] . In a suspension they can collide many, even infinitely many, times [fig.2( b ) ] . This fact indicates clearly the difference between both systems. In the mode-mode coupling theory of suspensions, as presented by Hess and Klein' and also by Dr Medina-Noyola, the final expressions have the structure corresponding to the so-called one-ring events, i. e. with two collision operators separated by two one-particle propagators. This means that in the low-density regime the two- particle dynamics is approximated by that part in which two particles can collide only twice. This approximation is the cause of the discrepancy between the exact result for the self-diff usion coefficient for dilute hard-sphere suspensions and the result derived within this theory. One can hardly imagine that this theory leads to better results for higher densities. A proper way to construct an approximation of the mode-mode coupling type is first to perform the so-called binary collision expansion3 to obtain an adequate description of two-particle dynamics and next to apply the mode-mode coupling ~ c h e m e .~ Then in the low-density regime one reproduces the exact results for the self-diffusion coefficient of hard spheres. 1 W. Hess and R. Klein, Adu. Phys., 1983 32, 173. 2 B. Cichocki and W. Hess, J. Chem. Phys., 1986, 85, 1705; Physica A, 1987, 141, 475. 3 R. Zwanzig, Phys. Rev., 1963, 129, 486. 4 B. Cichocki, to be published. Dr M. Medina-Noyola commented in conclusion: I wish to point out that in deriving the main results of my paper I made no use of methods pertaining specifically to kinetic theory. I think, however, that a description of the dynamics of colloidal systems within the framework of kinetic theory will be most useful.Prof. B. U. Felderhof ( R W H , Aachen, Federal Republic of Germany) turned to Prof. Mazur: In your treatment of hydrodynamic interactions you advocate the use of Cartesian tensors. I would like to point out that this may be advantageous to low order in the method of reflections, but that in higher orders the use of spherical tensors may be preferable. In fact this approach has led to the higher-order terms in the work of Schmitz and myself on hydrodynamic interactions between two spherical particles with general scattering properties.' For the case of hard spheres with stick boundary conditions Jeffrey and Onishi* have derived hundreds of terms in the two-sphere mobility series in this way.The point is that in this manner one employs the results on spherical harmonics derived in quantum mechanics which make maximal use of the group theoretical symmetry properties.General Discussion 93 1 R. Schmitz and B. U. Felderhof, Physicu, 1982, 116A, 163. 2 D. J. Jeffrey and Y. Onishi, J. Fluid Mech., 1984, 139, 261. Prof. P. Mazur (Leiden, The Netherlands) replied: I am not in the least advocating the use of ‘Cartesian tensors’, but I myself use irreducible tensors because I find them convenient. Somebody else may of course prefer to use spherical harmonics. The difference between both formulations may be compared to the difference between abstract vector notation and a notation using components relative to a given basis: it is largely a matter of convenience, and sometimes also of taste, which formulation one prefers.The formulation we chose helped us to calculate explicitly three-, four- etc. sphere hydrodynamic interactions which as we have shown, and as was confirmed experi- mentally, play a dominant role in more concentrated suspensions, owing to the essential non-additivity of these interactions. In that light it is of secondary importance whether, and in which way, one calculates another hundred terms for the two-sphere problem. Now it is certainly so, in principle, that one might also calculate explicitly many-body hydrodynamic interactions with the help of spherical harmonics, or spherical tensors, but it had not been done, and, as far as I am aware, still has not been done. Of course we should not forget that Kynch some thirty years ago had exhibited expressions for three- and four-body interaction terms, which remained unused because they were not well understood.He did not use spherical harmonics: but that does not mean anything. Prof. Felderhof then continued: In your formulation of the sedimentation problem you employ a large spherical container. One is interested here in a transport property, namely the friction coefficient giving the difference of the local average particle and average fluid velocity in terms of the applied gravitational force. Since the transport property is a local one and is independent of the shape of the sample I would prefer a formulation for general sample shape. Prof. Mazur replied: I beg to differ concerning what I was interested in when I studied sedimentation in a spherical container. Had T been interested solely in the value of the mean mobility, or friction coefficient, as a local property, and up to linear order in the volume fraction, I would have been perfectly happy with, in particular, Batchelor’s theory for this quantity.However, I was also interested in a possible dependence of the sedimentation velocity in a homogeneous suspension on container shape. Burgers noted such a possibility but states that it ‘does not appear to be readily acceptable’.’ For inhomogeneous suspensions it is well known that huge buoyancy-driven convection can occur in sedimentation, in particular near inclined boundary walls. To give an answer, however, to what I would like to call for brevity’s sake, Burger’s problem, it seemed necessary to us within our ‘microscopic’ formulation of the problem, to take into account (long-range) hydrodynamic interactions with container walls.We were able to perform this calculation for a spherical container, as well as for sedimentation towards a plane wall. For the spherical container we then found in homogeneous suspensions what we call essential convection under the influence of gravity. It is true that this essential convection is small compared to, and rapidly masked in a spherical container by, buoyancy-driven convective flows; and that the sedimentation velocity with respect to the non-vanishing volume velocity has Batchelor’s value, as one would hope. Again, however, as far as I am aware, even though one has sometimes wondered about essential convection, nobody has as a result of theoretical analysis explicitly mentioned its existence, or calculated its magnitude for a particular shape, whatever the formulation of the problem.1 J. M. Burgers, Proc. Kon. Ned. Acad. Wet., 1941, 44, 1177.94 General Discussion Prof. Felderhof commented: The situation is analogous to the theory of the dielectric constant in electromagnetism. Some years ago I formulated the sedimentation problem by analogy with electromagnetism, emphasizing the role played by the local field.' 1 B. U. Felderhof, Physica A, 1976, 82, 596, 611. Prof. Mazur replied: An analogy with the theory of the dielectric constant certainly exists for properties of suspensions, in particular for the effective viscosity. I know that you have very skillfully exploited this analogy when considering, for instance, collective diffusion.However, I have the impression that, owing to hydrodynamic interactions with container walls (and stick boundary conditions on such walls) one has in the case of sedimentation, a different situation from the one encountered in the conventional problem of shape dependence, or shape independence, of quantities in dielectrics. Dr M. La1 (Unilever Research, Port Sunlight Laboratory) asked Prof. Mazur: Can the method of induced forces discussed in the paper be extended to the evaluation of the hydrodynamic interaction between non-spherical particles as a function of interpar- ticle separation and orientation? Prof.Mazur replied: Formally the method can be applied also to ellipsoidal particles. I have not succeeded, however, in evaluating the resulting integrals for more than one particle. Prof. M. Fixman (Colorado State University) turned to Prof. Mazur: I have a comment and a question. The comment concerns the slow convergence of power-series expansions in the treatment of hydrodynamic interaction. The truncation of the expansion sometimes has the consequence, which is particularly unpleasant in Brownian simulations, that the friction matrix has negative eigenvalues for some part of the configurational space. This can be avoided, and an incidental improvement of convergence obtained, if the problem is given a variational formulation.' The question concerns the expansion of the diffusion constant in powers of the volume fraction. The convergence of the analogous dielectric expansion is improved if the Lorentz-Lorenz function rather than the dielectric constant itself is expanded, and this choice may be physically motivated.Is there a corresponding alternative for the diffusion constant? 1 See e.g. J. Rotne and S . Prager, J. Chem. Phys., 1969, 50, 4831 and M. Fixman, J. Chem. Phys., 1982, 76, 6124. Prof. Mazur replied: The analogue of the Lorentz-Lorenz (or Clausius-Mossotti) formula for the hydrodynamic case is Saito'e expression for the effective viscosity of a suspension, which is also a mean-field-type result. Thus a Saito function can be defined corresponding to the Clausius-Mossotti function. If one compares the virial expansions for the dielectric constant I of a dispersion of metallic spheres, and for the effective viscosity T of suspension of hard spheres, to the virial expansions of ECM, the Clausius-Mossotti value of E and of ~ s , the Saito value of 7, respectively, one finds to second order in the volume fraction 4 ~=10(1+3++4.51+~+* * - )General Discussion 95 In both cases the mean field results give the linear order correctly.In second order there are deviations arising from the fact that the mean-field theory does not handle adequately the two-sphere correlation problem. Now it is not clear to me at all what the 'mean-field' or Clausius-Mossotti-like formula for (self-)diffusion should be. One might be tempted to assume a Stokes- Einstein-like formula with a viscosity given by the Saito formula.One then has DMF= Do( 1 - 4)( 1 +$+), with Do the one-particle Stokes-Einstein diffusion coefficient. The virial-expansion of this quantity leads to DMF=Do(1-2.54+3.75$*+* * * ) while in reality one has (with a 5% error in the coefficients). D = Do( 1 - 1.73 4 + 0,8 8 4 * + - * . ) Thus already the linear term is ca. 50% out, and the second-order term is out by a factor of 4. However, perhaps, if one defines with DMF the function which corresponds to the Clausius- Mossotti function, this would nevertheless provide the alternative to which you are referring. Dr R. B. Jones (Queen Mary College, London) said: I have a comment followed by a question for Prof. Mazur and also for Prof. Bossis and Prof. Brady, who are present in the audience. In the successful application of Prof.Mazur's many-body hydrodynamic interactions to tracer diffusion in highly concentrated suspensions,' the lubrication theory results for almost touching spheres are not included because the mobility expansions in the quantities a / & , where a is a particle radius and R, is an interparticle distance, are cut off at low order. On the other hand, in a recent simulation study of the viscosity of a sheared concentrated suspension due to Prof. Bossis and Prof. Brady,' the lubrica- tion theory results for two-body hydrodynamic interactions at close separation were included through the use of accurate friction coefficients but the many-body hydrody- namic interactions were not included. As it seems desirable to treat all of the transport coefficients in the same framework I wish to ask Prof.Mazur as well as Prof. Bossis and Brady if one should include both the lubrication theory two-body results and the many-body interactions in order to get a consistent picture of the transport properties of a concentrated suspension? 1 C. W. J. Beenakker and P. Mazur, Physica A, 1983, 120, 388. 2 J. F. Brady and G. BOSSIS, J. Fluid Mech., 1985, 155, 105. Prof. Mazur replied to Dr Jones: I am not completely sure I understand your question in as far as it refers to the work of Beenakker and myself on self-diffusion. It is only in the calculation of the first few virial coefficients that the mobility expansions are cut-off at low order in R,' and in that case the prime objective was to establish the relative importance of three-body interactions.For more concentrated suspensions we completely resum, within the framework of a fluctuation expansion, many-body hydro- dynamic interactions. The lowest order in this fluctuation expansion thus contains the two-body problem to all orders in the inverse interparticle distance, but not only the two-body problem, also the three-, four- etc. sphere problem. Of course we pay a price for this: particle correlations are not correctly taken into account, and for that fact one corrects in the next order of the fluctuation expansion. In this correction one then limits oneself to contributions of low order in R,' of renormalized interaction terms. So I would say that we have precisely done to a degree (within the limitations encountered in such a problem) what Dr Jones suggests should be done.Perhaps this contributes to the fact that our treatment is relatively successful, as he is kind enough to note.96 General Discussion Prof. J. F. Brady (Caltech, Pasadena) added: Concerning the need to include both many-body interactions and lubrication forces in treating the hydrodynamic interactions in suspensions, the answer is that it depends on the quantity to be studied, i.e. some properties are more sensitive to the near-field and others to the far-field physics, and this sensitivity changes as the volume fraction of particles increases, but in general both must be included. As a simple example, consider a linear chain of closely spaced spheres. For a sufficiently large number of spheres the chain should behave as a slender body, for which the behaviour is well known.If the same force is applied to each particle in the chain, then the far-field many-body interactions will generally suffice to give the qualitatively correct instantaneous mobilities. This is because the solution for a translating slender body has, apart from the very ends, a constant force density along its length. If, however, the forces are not the same on all spheres, so that there would be a tendency for relative particle motion, then lubrication forces are essential. If you push only one sphere at the end of the chain, clearly all particles will move with the same translational velocity because of the lubrication forces and the connectivity of the chain; far-field interactions alone will not capture this effect.In order to obtain the proper behaviour, both many-body interactions and lubrication forces are needed. We have recently devised a computationally efficient scheme for calculating both far-field many-body interactions and near-field lubrication forces.' The results of this method for finite numbers of interacting particles compare very favourably (< 1 % error) with all available exact (usually numerical) calculations, and the method also reproduces slender-body theory. The procedure is readily incorporated into our Stokesian dynamics method for simulating infinite suspensions,' and a general development of Stokesian dynamics along with several applications will appear in the near f ~ t u r e . ~ 1 i. Diirlofsky, J.F. Brady and Or. Bossis, J. Fluid Mech., 1987, 180, 21 2 J. F. Brady and G. Bossis, J. Fluid Mech., 1985, 155, 105. 3 J. F. Brady and G. Bossis, Annu. Rev. Fluid Mech., 1988, 20, 111. Dr E. Dickinson ( University ofLeeds) turned to Dr van Megen: My question relates to the curves corresponding to effective volume fractions 0.44, 0.49 and 0.54 in fig. 2 of your paper. Increasing the volume fraction from 0.44 to 0.49 in the disordered phase leads to a substantial reduction in mean-square displacement over the whole range of quoted delay times. On the other hand, further increasing the volume fraction by the same amount, and at the same time changing the state of the system from liquid-like to crystalline, leads to very little change in the mean-square displacement over the same range of delay times.Since the curvature of the plots is attributed to the transition from short-time to long-time diffusive motion, I am a little surprised that curves for volume fractions 0.49 and 0.54 are so close, in view of the fact that the short-time diffusion coefficient is very dependent on volume fraction at these particle densities, and that the long-time diffusion coefficient should be effectively zero for a dense crystalline colloidal system. Does Dr van Megen have any comment on these observations? Dr W. van Megen (RMIT, Melbourne) replied: At a given volume fraction one would expect the local mobility, or diffusion coefficient D i , of a particle in the crystalline phase to be larger than that in the disordered phase. This feature, which is supported by calculations of D; for charge-stabilized dispersions [see fig.1 of ref. (l)], expiains, at least qualitatively, the similarity of the mean-squared displacements (m.s.d.) at short times in the coexisting disordered (& = 0.49) and crystalline (b, = 0.54) phases. A better indication of the difference between the m.s.d. in the coexisting disordered and crystalline phases at longer times may be seen in fig. 6 of ref. (2), where further details of this experiment are discussed. Whilst the true long-time limiting behaviour has not been reached, owing to the large wavevectors at which tracer system T2 wasGeneral Discussion 97 studied, the m.s.d. in the crystalline phase does appear to be approaching a constant value. I might add that this value is roughly consistent with the Lindemann melting criterion. 1 I.Snook and W. van Megen, J. Colloid Interface Scr., 1984, 100, 194. 2 W. van Megen, S. M. Underwood and 1. Snook, J. Chem. Phys., 1986, 85, 4065. Prof. A. Vrij (Utrecht, The Netherlands) then asked: The first five measured points of the short-time and long-time self-diffusion coefficients shown in the fig. 3 and 4, respectively, are rather different. Theory tells us, however, that the slope of Ds/Do against the volume fraction is nearly the same: -1.83 and -2.1 1 for short-and long-time self-diffusion, respectively. The points in fig. 3 follow the slope (-1.83) rather closely, but this can obviously not be said about the points in fig. 4. Have the authors an explanation for the different behaviour shown in fig.4? Prof. R. H. Ottewill ( University of Bristol) and Dr P. N. Pusey (R.S.R.E., Malvern) replied: We agree that, for monodisperse hard spheres, the predicted concentration dependences, to first order in 4, of DS and DL are very similar. That this is apparently not found to be the case for the data of fig. 4 is probably due to the significant polydispersity of the tracer particle PMMA/l (see table 1 of our paper). In particular, even in the limit 4+0, the short- and long-time slopes of In FM(Q, t ) [eqn (13)] will, because of polydispersity, be different. Thus in fig. 4 the data for DL/Do should not have been extrapolated to unity but to a lower value, implying thereby a weaker concentration dependence and a better agreement with theory. Dr A. K.Livesey ( D A M P , Cambridge) considered both the Bristol and the Utrecht papers: I note that you (both) report a certain degree of polydispersity in your samples and yet make no attempt to fit your data using an algorithm which is capable of fitting broad distributions or otherwise accounting for this polydispersity. Would you like to comment further on your fitting procedures? Dr A. van Veluwen (Utrecht, The Netherlands) replied: The basic-reason for not analysing our measured correlation functions as if they were the Laplace transform of a spectrum of decay rates, is that such a description may be inadequate. Whereas for a dilute polydisperse sample it can be shown that the electric field autocorrelation function is a multiexponential decaying function, such a demonstration has not been given, as far as we know, for a concentrated interacting polydisperse system.The general polydisperse case is too complex for a detailed calculation, and the theoretical argument that the electric-field autocorrelation function of a concentrated sample will be composed of two decoupled modes, associated with collective and exchange diffusion, still needed experimental verification. This is essentially what we have done, and one does not need a very sophisticated data-fitting algorithm to conclude that the correlation function consists of a fast and a slow decaying part with well separated decay times. However, the time dependence of these modes is not necessarily (multi)exponential, and therefore analysing the correlation function that way may be misleading.Therefore apart from all the difficulties associated with Laplace inversion, the so-found relaxation time distribution may lack physical significance. Prof. B. U. Felderhof ( R . W. 7'. H., Aachen, Federal Republic of Germany) said: in fig. 2 of your paper you plot the normalized short-time self-diffusion coefficients of charged and uncharged silica spheres as a function of volume fraction and find that the experi- mental curves coincide. I do not understand the remark at the end of paragraph 3 of your paper, where you write that this indicates that the short-time self-diffusion is determined mainly by hydrodynamic interactions and apparently is affected little by90 General Discussion direct particle interactions. Is it not true that in the case of charged spheres the Debye clouds keep the particles separated and that therefore to a good approximation the hydrodynamic interactions may be neglected? Dr W.van Megen (R.M.I.T., Melbourne, Australia) added: I am also surprised by the independence of DEho* on the presence of electrostatic interactions between the particles (fig. 2 of your paper). Intuitively one expects the additional electrostatic repulsion between the singly coated silica particles to increase the mean interparticle spacing, ie., the first peak in the radial distribution function moves to a larger spacing, resulting in an increased local mobility or Dghort. This picture is supported by calculations of Diho* for charge stabilized dispersions, based albeit on a empirical screened pair mobility tensor.' 1 I.Snook and W. van Megen, J Colloid Inlet-fuce Sci., 1984, 100, 194. Dr van Veluwen replied: At very low volume fractions this appears to be the case. However, at the high volume fractions we used, the mean interparticle separation is inevitably small and so hydrodynamic interactions are important. If direct particle interactions had a pronounced influence on the short-time self-diff usion coefficient, we would have found different values for this quantity for the two systems considered. This not being the case indicates that the short-time self-diffusion coefficient is not very sensitive to these direct particle interactions, even though such interactions give rise to different particle functions. We may conclude from this that the detailed form of the particle distribution functions has no significant effect on the short-time self-diff usion coefficient.This conclusion can also be drawn from the work of Beenakker and Mazur.' 1 C. W. J. Beenakker and P. Mazur, Physica A, 1984, 126, 349. Dr W. van Megen ( R . M.I. T., Melbourne, Australia) said: In addition to your finding that the singly coated 440 nm radius silica spheres show an electrophoretic mobility, have you any other evidence for the presence of screened electrostatic forces between the particles? The crystallization transition, for instance, can be easily measured' for an optically matched dispersion of near-micrometre sized particles such as yours, provided that the polydispersity is not too large. Further, the location of this transition is very sensitive to the range of the repulsive force between the particles.Have you observed this transition in either of your 440 nm silica systems? 1 P. N. Pusey and W. van Megen, Nature (London), 1986, 320, 340. Dr van Veluwen replied: Another indication of the presence of long-range forces between the single-coated particles is the observation that the sediment of these particles is significantly less dense than the sediment of the double-coated particles. The formation of crystallites is not only influenced by the direct particle interactions and polydispersity, but also by the sedimentation velocity. Silica particles have a higher density than the latex particles you refer to. This implies that in the solvent we used the large silica particles will have settled under gravity before crystallization occurs.After some time (up to 1 year?) crystallization takes place in the sediment, both for the single- as well for the double-coated particles. Of course this influence of sedimentation on the formation of crystallites is less pronounced for smaller particles. Indeed single-coated silica particles of 80 nm radius already show a crystallization transition at volume fractions of 4 = 0.25. This, together with the concentration dependence of the static structure factor, is clear evidence for the existence of long-range repulsive forces.General Discussion 99 Dr J. G. Rarity (R.S.R.E., Malvern) remarked: The data shown in your fig. 3 show a behaviour similar to the pure optical polydispersity case. Is it possible that your data could be better explained by a model that takes both optical and size polydispersities into account? What effect does the compressibility of the stabilizing polymer coating have on your calculation of volume fraction? Dr van Veluwen agreed: It is indeed likely that our experimental data can be better explained by a model that takes both optical and size polydispersity into account.The compressibility of the stabilizing polymer coating is so small that it has no significant effect on the calculation of the volume fraction. Prof. P. Mazur (Leiden, The Netherlands) said: Some years ago one did not have sufficient low-density data at one’s disposal for the short-time self-diff usion constant in a hard-sphere suspension to determine from these data the coefficient of the quadratic term in an expansion of this quantity in powers of the volume fraction.Now the situation is different: there are many low-density points available down to volume fractions of ca. 0.025. My question is: has anyone tried to determine this coefficient, in addition to comparing theoretical curves to experimental ones? We found theoretically for this coefficient, taking into account three-body hydrodynamic interactions, the value 0.88, as compared to -0.93 calculated on the basis of two-body interactions alone. Dr van Veluwen replied: From our measurements it follows that the coefficient of the quadratic term in the expansion of the short-time self-diffusion coefficient in powers of the volume fraction must be positive. Furthermore, if we try to determine this coefficient using the expression ,ihon/ D~ = 1 - 1.83+ + K ~ $ ~ we find K2 = 0.7k0.3, in agreement with your calculations. Prof.M. Fixman (Colorado State University) asked: The initial response of a flexible surfactant is probably to provide a slip boundary condition. Is it feasible to go to very fast time resolution and see such an effect? Prof. R. H. Ottewill replied: This suggestion is an interesting one, and this may well be possible. It would, however, require rather different measurements with a very fast time resolution at low concentrations. It certainly would be interesting to try this. Dr van Veluwen added: One would expect the phenomenon you mention to show up at the timescale at which the velocity of a colloidal particle relaxes, i.e.m/6vqa- s. We do not think it is feasible to probe the motion at such short timescales with dynamic light scattering. Dr A. van Veluwen and Prof. H. N. W. Lekkerkerker (Leiden, The Netherlands), said: van Megen et al. have argued that the non-Gaussian terms in the self-dynamic structure factor are unimportant in the limit of small values of the scattering vector Q. We would like to present experimental evidence that for values of Q which are not very small, but are still below the location of the first maximum in the static structure factor, the non-Gaussian terms do become appreciable. The system studied consisted of an optically matched host dispersion of sterically stabilised silica particles in cyclo-octane, to which a small amount of sterically stabilized poly(methy1 methacrylate) (PMMA) tracer particles had been added. Both types of particles have a diameter of 160 nm.100 General Discussion J f l It2 1 t3 t4 1 t5 0 0.5 1 1.5 2 2.5 3 delay time, t/ms Fig.3. Self-dynamic structure factor F, plotted against delay time t for 4 = 0.145 at three different scattering angles: (-) 8 = 40", Q = 0.98 x lo-* nm-l, (- - -) 8 = SO", Q = 1.85 x lo-' nm-', (- - -) 8 = 150", Q = 2.78 x nm-'. The arrows on the abscissa indicate the times at which -In F,/(Qa)' has been plotted as a function of Q2 (fig. 4). 0.LO r I I I 1 I I I 01 2 4 6 Q2/ 1 0-4 nm-* Fig. 4. Plot of -In F,( Q, t ) / ( Q c z ) ~ as a function of the square of the scattering wavevector Q' for the delay times t , to t , as indicated in fig. 3.Straight lines are first-order cumulant fits, according to eqn ( 1 ) . ( a ) tl=0.5ms, ( b ) r,=l.Oms, (c) t3=1.5ms, ( a ) r4=2.0ms, ( e ) tS=2.5ms. In fig. 3 we present -In F,(Q, t)/QLa' as a function of delay time t for various scattering angles. Here a is the radius of the particle. The volume fraction of the sample was # = 0.145, and the relative concentration of tracer to host particles was 1 : 50. If the non-Gaussian terms are unimportant, all these functions should coincide. This being not the case indicates the presence of non-Gaussian terms. Indeed we can write downGeneral Discussion 51 4 3 a 2 2 1 0 delay time, t/ms Fig. 5. Non-Gaussian behaviour of F, expressed as a2 plotted against t. a cumulant expansion of the self-dynamic structure factor: 0' 3 60 - ln K(Q, t ) - (ArLt)') Q2( 3(Ar(t)4) - 5(Ar(t)2)2 101 Here Ar( t ) = r( t ) - r ( 0 ) is the displacement of an individual particle.In order to extract the first non-Gaussian term we plot -In F,( Q, t ) / Q2a2 as a function of Q2 for various values of t (fig. 4). From the intercept at Q=O we can extract the mean-square displacement (Ar( 1 ) ' ) and from the slope [3(Ar( t)") - 5(Ar( t)2)2]. Following Rahman' we express the departure of F,(Q, t ) from a Gaussian form in terms of the function ct2 defined as In fig. 5 we plot a2 as a function of t. Note that the values of a2 experimentally obtained here are much higher than those obtained by Rahman in a molecular-dynamics simulation for liquid argon. 1 A. Rahman, Phys. Rev. A, 1964, 136, 405 Dr W. van Megen (R.M.I.T., Melbourne, Australia) commented on Prof.Lek- kerkerker's data: The non-Gaussian terms in the self-dynamic structure factor (or the self-space-time correlation function) have been calculated by computer simulation for a variety of systems at densities just below crystallization (freezing). The work of Rahman,' on a liquid of atoms interacting with the short-ranged Lennard-Jones pair potential, contrasts with the study of Gaylor et a1.2 for a dilute dispersion of particles interacting with a long-ranged weakly screened Coulomb pair potential. However, both systems show qualitatively similar non-Gaussian effects: the first non-Gaussian term a2, [as defined in ref. ( l ) ] has a maximum of ca. 0.1. Similar observations have also been made on liquid argon by Skold et al.' More recent Brownian-dynamics simulations on102 General Discussion a system of particles with a pair potential of intermediate range at a volume fraction of 0.25 (for which freezing occurs at 0.26) yield qualitatively similar results for a? (see fig.6). The addition of tensorial hydrodynamic interactions, represented by an empirical pair mobility as discussed in ref. (4), has little effect on the non-Gaussian terms. The magnitude of your value of cr2 and its apparent reluctance to converge to zero at short times are in stark contrast to the results discussed above. I find this even more surprising in view of the volume fraction of your (near) hard-sphere dispersion of only 0.15, which is well below the hard-sphere crystallization transition.1 A. Rahman, Phys. Rev. A, 1964, 136, 405. 2 K. Gaylor, 1. Snook and W. van Megen. J. Chem. Phys., 1981, 5, 1682. 3 K. Skold, J. M. Rowe, G. Ostrowski and P. D. Randolph, Phys. Rev. A, 1972, 6, li07. 4 I. Snook, W. van Megen and R. J. A. Tough, J. Chem. Phys., 1983, 78, 5825. Dr R. B. Jones (Queen Mary College, London) also commented: I wish to add to Prof. Oppenheim’s informally made remarks about the theoretical expectation of long- time tails and hence non-Gaussian behaviour in the intermediate scattering function. These non-Gaussian effects are contained in principle in the memory function and hence can be obtained from a careful calculation of F(k, t ) at high concentrations. Prof. Felderhof and I presented details of such a calculation based on two-body dynamics (without hydrodynamic interactions) at the 1983 Faraday Discussion on Concentrated Suspensions.It should be possible to obtain analytic expressions for the order k4 terms from our formalism to see how they compare with your experimental data. For low- density hard-sphere suspensions the memory function has been studied in detail by Prof. Ackerson’ and m y ~ e l f , ~ . ~ and there are clear non-Gaussian effects seen at intermedi- ate times. 1 B. J. Ackerson and L. Fleishman, J. Chem. Phys., 1982, 76, 2675. 2 R. B. Jones and G. S. Burfield, Physica A, 1982, 111, 562. 3 R B Jones, J. Phys. A, 1984, 17, 2305. Dr P. N. Pusey (R.S.R.E., Malvern) then added: I have some comments on Prof. Lekkerkerker’s very interesting data showing large non-Gaussian effects in the displace- ment of tracer particles in concentrated suspension.(1) As you pointed out, computer simulations of simple liquids show much smaller non-Gaussian effects than you find in particle suspensions. An important difference between the dynamics of atoms in a liquid and particles in suspension is the presence of hydrodynamic interactions in the latter case. As you know, we have recently shown that hydrodynamic interactions can cause non-Gaussian contributions in suspensions which are quite different in nature from those caused by direct interactions.’ While these theoretical calculations apply only at short times it seems possible that hydrody- namic interactions, with their complicated many-body nature, could cause large effects at longer times. These would not necessarily show up in the computer simulations of van Megen and Snook, since these authors used an effective pair form for the hydrody- namic interaction tensor.(2) Presumably you would still expect the non-Gaussian contributions to become small at short times. However, your data do not show this trend. (3) In view of the delicate manipulations required to derive these effects from the data it is clearly important to discount possible experimental artefacts. I do not see how your results could be caused by polydispersity of the particles. However, I wonder if you have considered multiple scattering and/or possible ‘breakthrough’ of coherent scattering which could, I think, cause spurious effects in the direction you observe. 1 R. J. A. Tough, P. N.Pusey, H. N. W. Lekkerkerker and C. van den Broeck, Mol. Phys., 1986, 59, 595.General Discussiori 103 0.4 a 0.2 0 0 X X X X X X x . ' X X ' 2 4 t/ms f Fig. 6. First (lower points) and second (upper points) non-Gaussian terms of the self-dynamic structure factor for a system of particles at d, =0.25. The pair potential in units of kT is 100 exp [ -20( r - 1)]/ r, with r expressed in units of the particle diameter. Freezing for this system occurs at d, = 0.26. The crosses exclude and the dots include the effects of tensorial hydrodynamic interactions. volume fraction Fig. 7. Di/ Do plotted against volume fraction for short-time self-diffusion of poly(methy1 methacrylate) particles in dispersions of refractive-index-matched poly( vinyl acetate). @, Experi- mental results; theoretical calculations from Beenakker and Mazur: (.- - a ) three-body, (- - -) multi-body and (-) two-body. Prof. Lekkerkerker and Dr van Veluwen replied: ( 1 ) We fully agree with you on this point. (2) We simply do not reach such short times in our experiments. An interesting question is, on what timescale would one expect the maximum of the non-Gaussian effect to occur. (3) We paid much attention to these problems. The refractive index and tracer concentration were chosen such that both coherent scattering and multiple scattering could be completely neglected. Besides, it can be argued that a possible 'breakthrough' of coherent scattering at large k values and/or multiple104 General Discussion scattering would lead to effects opposite to those observed, that is to say a negative value of a2 instead of a positive one.Prof. R. H. Ottewill (University of Bristol) said: In the work of Neal Williams and myself' we have shown that poly( vinyl acetate) particles can be conveniently matched by refractive index by using a mixture of cis- and trans- decalin. This, as reported in the paper, has enabled self-diffusion measurements to be made over a wide range of volume fractions using poly( methyl methacrylate) particles at tracer concentrations. The results so obtained can be compared with calculations of the expected results from the theories of Beenakker and Mazur2 as shown in fig. 7. As can be seen, the best agreement between experiment and theory for the short-time self-diffusion is obtained using their multibody approach.1 R. H. Ottewill and N. St. J. Williams, Nature (London), 1987, 325, 232. 2 C. W. J. Beenakker and P. Mazur, Physica A, 1984, 126, 349. Dr I. Markovie and Prof. R. H. Ottewill (University ofBristoZ) said: We should like to point out that in addition to photon correlation spectroscopy it is possible to determine the short-time self-diffusion coefficient of small colloidal particles by the use of the neutron spin-echo technique.' In recent work, using calcium carbonate particles stabilised by an alkylaryl sulphonic acid2 with an average particle diameter of 10nm we have investigated the small angle neutron scattering over a wide range of volume fractions and scattering vectors ( Q), thus obtaining information about the structure factor, S( Q), as a function of Q.It was then possible to determine the diffusion coefficient of the particles at high Q values, where S( Q) + 1, by neutron spin-echo. The sample times using the latter technique are of the order of nanoseconds and hence the diffusion coefficient obtained was the short-time self-diff usion ~oefficient.~ The results are plotted in fig. 8. Also plotted in this figure are the results obtained from tracer measurements of poly( methyl methacrylate) particles in dispersions of poly( vinyl acetate) matched by refractive index4 and results obtained on poly(methy1 methacrylate) particles in a dispersion medium of dodecane and carbon disulphide near the refractive index match point.5 The results presented cover a range of particle diameters from 10 nm to 1 ,urn, two orders of magnitude, by two different techniques.The agreement between the various sets of experimental data is remarkably good. Moreover, the results are in quite reasonable agreement with the theory of Beenakker and Mazur6 for the dependence of the self-diff usion coefficient on volume fraction with allowance for many-body interac- tions. 1 J. B. Hayter, in Lecrure Nores in Physics, ed. F. Mezei (Springer Verlag, Berlin, 1980). 2 I. MarkoviC and R. H. Ottewill, Colloid Polym. Sci., 1986, 264, 65; 454. 3 I. MarkoviC and R. H. Ottewill, Colloids SurJ, 1987, 24, 69. 4 R. H. Ottewill and N. St. J. Williams, Nature (London), 1987, 325, 232. 5 P. N. Pusey and W. van Megen, J. Phys. (Paris), 1983, 44, 258. 6 C. W. J. Beenakker and P. Mazur, Physica A, 1984, 126, 349.Dr I. Markovic and Prof. R. H. Ottewill (University ofBristoZ) (communicated): In many of the papers presented at this meeting measurements of a physical quantity, e.g. the self-diff usion coefficient, are plotted against volume fraction. In these experiments the particles used are often, using a general term, sterically stabilised, i.e. the particles have a hard core surrounded by a shell of polymeric or surface-active material, and in order to obtain effective colloid stability this layer is solvated; this means that there is penetration of the solvent molecules between the chains of the stabilising moeities. In recent work we have examined the determination of volume fraction by various experi- mental techniques for this type of system and have been forced to conclude that theGeneral Discussion 105 Fig.166 0.1 0.2 0.3 0.4 0.5 volume fraction 8. Di/ Do against volume fraction for various particles. A, poly(methylmethacry1ate) diameter nm;4 0, poly(methy1methacrylate diameter 1.18 ~ m ; ~ 0, calcium carbonate diameter 10 nm;3 (-) calculated from the multi-body equation6 precise determination of this is very difficult, if not impossible experimentally, without making a number of assumptions. Some of the methods which can be used include the following: ( 1 ) The weight/weight or weight/volume fraction can for many cases be determined precisely. However, for conversion to volume fraction a density is needed. This can also be determined accurately, hut, sn iisirig ihis to obtain the voliiiiie fraction, the quantity obtained is the ‘dry’ volume fraction and not the solvated volume fraction occupied by the particles in the dispersion. If there is non-ideal mixing of dispersion medium and the stabilising chains it becomes even more complicated, (2) Experimental techniques such as photon correlation spectroscopy and viscosity on very dilute dispersions are often used to obtain a ‘hydrodynamic radius’.Unfortu- nately this is not necessarily the same as the radius of the particle required to determine the actual volume of the solvated particle, since solvent outside the stabilising chains may be included. (3) In a number of cases the chemical formula of the stabilising molecule is known and hence measurements on molecular models can give an estimate of the likely extension of the molecule in the solvated state.This could be reasonable if the chains are monodisperse or fully extended. (4) In the case of concentrated monodisperse systems the theoretical freezing volume fraction (0.494) has been used to estimate the actual volume fraction. This, however, depends upon how accurate the theory of hard spheres is, and even if this is accurate it only fixes the volume fraction at one concentration. Thus the thickness obtained at this high concentration may not be valid for dilute systems. In a recent examination of poly( methyl methacrylate) particles by small-angle neutron scattering we have obtained some evidence for the compression of the stabilising layer106 General Discussion of poly(l2-hydroxystearic acid) as the concentration was changed from low to high values, suggesting that the volume of each particle was dependent on volume fraction.' It should be stressed that for large particles, e.g.1 pm, with a stabilising layer of 10 nm, the experimental error will be small even if the core volume fraction is used but for small particles; e.g. with a radius of 10 nm and the same thickness of stabilising layer, the accurate determination of volume fraction is non-trivial. 1 I. MarkoviC, R. H. Ottewill, S. M. Underwood and Th. F. Tadros, Langmuir, 1986, 2, 325. Prof. B. J. Ackerson (R.S.R.E. Malvern) addressed Prof. Felderhof: At a previous Faraday Discussion on Concentrated Dispersions Hess and Klein argued that the Fokker-Planck equation was the proper starting point for the calculation of light scattering functions in concentrated colloidal suspensions.' They argued that the general- ized Smoluchowski equation may lead to erroneous results.However, Cichocki and Hess2 have reinterpreted the memory function obtained for the generalized Smoluchowski operator. As a result the Fokker-Planck and generalized Smoluchowski equation approaches are believed to be equivalent descriptions on timescales of interest to us. However, in your present work on colloidal crystals you find a large effect due to counterion motion. The counterion distortion depends on the colloidal particle velocity and presumably the correlated velocities between different colloidal particles. Does this dependence on particle velocities mean that we must necessarily consider a Fokker-Planck approach in analysing the particle correlation functions or is the general- ized Smoluchowski equation still valid in general? 1 R.Klein and W. Hess, Faraday Discuss. Chem. Soc., 1983, 76, 137. 2 B. Cichocki and W. Hess, J. Chem. Phys., 1986, 85, 1705. Prof. B. U. Felderhof ( R . W. T. H., Aachen, Federal Republic of Germany) replied: As you know I have always been an advocate of the use of the Smoluchowski equation. I am not yet sure what effect the dipolar forces will have in the liquid state. I expect that it will still be possible to use the Smoluchowski equation on the timescale we see in light-scattering experiments, say ca. lOP4s. This is much longer than the momentum relaxation time of ca. lo-* s. Hence it should be possible to work in configuration space and not be necessary to go back to the more complicated Fokker-Planck equation, which describes the time evolution in phase space.What effect, if any, the dipolar forces have on the form of the Smoluchowski equation we hope to find out in the near future. Prof. M. Fixman (Colorado State University) said: You suppose that in the absence of ion retardation effects the friction constant associated with the relaxation of a transverse mode vanishes at q = 0; this makes the relaxation rate finite because the force constant also vanishes at q = 0, and the relaxation rate is, in rough terms, the ratio of force constant to friction constant. However, Hurd et al. (your references) make the point that their experiments were done in thin samples, and that a boundary condition on fluid velocity then changes the friction constant to a non-vanishing value at q = 0.They rationalize the vanishing relaxation rate in this way rather than by retardation effects, and suggest that the friction constant would really vanish for a large system. Although they did not consider the retardation effects, and they are surely significant for some problems, there is a physical reason to doubt their ability to make the friction constant finite for transverse modes at q = 0. If the transverse mode really consists of a coherent motion of solvent and colloidal particles, which is the reason that the friction constant vanishes if retardation is neglected, then convective terms that you discard would cause the ion atmosphere to move in phase with the solvent and colloidal particles.In other words, the electrophoretic effect would reduce the retardation force to zero.General Discussion 107 Prof. Felderhof responded: We do not think that the samples used by Hurd et af. were sufficiently thin for boundary conditions on the fluid velocity to have an appreciable effect. We believe that they have seen an effect at small wavenumbers characteristic of bulk samples. We claim that in bulk samples retardation affects the dispersion curves in the way we have discussed. Even for long wavelengths the actual flow pattern in a unit cell is much more complicated than you suggest. See for example Hasimoto's calculation' for viscous flow past a cubic array of spheres. The situation is analogous to that in a lattice of electric dipoles, the local crystalline field being quite different from the average Maxwell field.In our problem a complete calculation taking account of the lattice structure would have to include the effect of convection on the small ions, as well as the electrical force density acting on the fluid. The equations to be used are known [see our ref. ( 5 ) ] but the explicit calculation will be difficult. As we mention in the paper, we expect that the electrophoretic effects reduce the retardation force, but do not make a qualitative difference. Although the electric dipole moments may be small, their interactions are of long range and seem to have a drastic effect on the crystal dynamics. 1 H. Hasimoto, J. Nuid Mech., 1959, 5 , 317. Dr M. Medina-Noyola (Cinuestav, Mexico City) then addressed Dr van de Ven: In your experiments you measure the contribution AS"' to the friction on a single polyion originating from the spontaneous fluctuations of its electrical double layer around spherical symmetry.The theory of Ohshima et af.' that you employ in your analysis calculates the friction of an isolated polyion as measured in a sedimentation experiment. 1 hat in principle the results of such a theory are indeed applicable to your light-scattering measurements is expected from fluctuation-dissipation arguments such as those in my response to Prof. Ackerson's question on my paper. Nevertheless, a theory describing the electrolyte friction effects as the consequence of spontaneous fluctuations may also be elaborated as a particular application of the general approach presented in my paper.In a simple ~ e r s i o n ~ - ~ which neglects the coupling of the diffusive fluxes of the electrolyte ions with the hydrodynamic flow of the solvent, such a theory contains Schurr's results' for ALe' in the Debye-Huckel limit. Owing to the neglect of hydrodynamic interactions, our results are expected to be quantitatively less accurate than those of Ohshima et af., but the main qualitative features that you observe in your experiments seem already to be contained in such a simpler theory. Our methods, however, allow us3 to calculate A C ( t ) , and not only its time integral A[e'. In the Debye-Huckel approximation, the long-time asymptotic expression for A l ( t ) reads AL( t ) = ( Q * / / & F ) exp ( - ~ * D ' t ) l ( D0t)3'2 where Q is the charge of the polyion, F the dielectric constant of the solvent, Do the free-diffusion coefficient of the electrolyte ions and K is the inverse Debye screening length.If we take K - ~ = 20 nm and Do = 1.5 x m2 s-.', which correspond to conditions considered in your experiments, one estimates a relaxation time T~~ 2 0.27 ps. As seen in fig. 4 of your paper, in your experiments you seem to reach the microsecond regime. I wonder, then, to what extent the time dependence of A C ( t ) could be observed in measurements of the type that you have performed. I H. Ohshima, T. W. Healy and L. R. White, J. Chem. Soc., Faraday Trans. 2, 1983, 79, 1613. 2 M. Medina-Noyola and A. Vizcarra-Rendon, Phys. Rev. A, 1985, 32, 3596.3 H. Ruiz-Estrada, A. Vizcarra-Rendon, M. Medina-Noyola and R. Klein, Phys. Rev. A, 1986, 34, 3446. 4 A. Vizcarra-Rendon, H. Ruiz-Estrada, M. Medina-Noyola and R. Klein, J. Chem. Phys., 1987, 86, 2976. 5 J. M. Schurr, Chem. Phys., 1980,45, 119.108 General Discussion Dr T. G. M. van de Ven (McGill University, Montreal, Canada) replied: It is as yet not clear why we observe a short relaxation time of ca. 1 ps. It could be for the reason you mentioned. Another likely explanation is that it is due to rotary Brownian motion of a small fraction of doublets which are known to be present. Yet another possibility is that the effect is an experimental artifact. Dr P. N. Pusey (R.S.R.E., Maluern) said: Is it possible that the initial rapid decay of the correlation function [inset of your fig.4(a)J is caused by ‘after-pulsing’ in the photomultiplier tube? This is a well known artifact in PCS’ which arises from the ionization of a residual gas atom by a pulse of electrons, the drift of this positive ion back to the cathode resulting in a second, spurious pulse of electrons being detected a short time (ca. 1 ps) after a real pulse. Problems with after-pulsing can be largely circumvented by cross-correlating the outputs of two photomultipliers, both of which observe the same scattered light field. This is a procedure worth implementing if correlation delay times s 1 ps are important. I do not think it is necessary to invoke solvation of the latex to explain the difference in particle radii obtained by PCS and by electron microscopy.It seems to be caused largely by polydispersity. Consider the case of small particles where the intensity scattered by a particle goes as the sixth power of its radius a. Then it is easily shown that apcs, the radius derived by using in the Stokes-Einstein expression the average diffusion coefficient obtained from the first cumulant, is given by * -- aPCS = a ‘ / / ’ . ( 1 ) Similarly the second cumulant is A useful, not widely appreciated, property of a narrow relatively symmetrical distribution P ( a ) is that its moments are given by a universal function of its standard deviation ( T : ~ *2+. . . (3) regardless of the detailed form of the distribution; here - a 2 a a2 = -2- 1. Substitution of eqn (3) into eqn ( 1 ) and (2) gives aPCS =r a( 1 + 5a2) (4) and Q = (T2.( 5 ) For the polystyrene latex electron microscopy gave a = 32 nm and (T = 0.18. Thus eqn (4) predicts aPCS = 37 nm in reasonable agreement, when all uncertainties are accounted for, with the measured value of 40 nm. Eqn ( 5 ) predicts Q = 0.03, in excellent agreement with the value calculated by the more laborious procedure of assuming a Gaussian distribution of particle size. 1 C. J. Oliver, in Photon Correlation and Light Beating Spectroscopy, ed. H. Z. Cummins and E. R. Pike 2 J. C. Brown, P. N. Pusey, J. W. Goodwin and R. H. Ottewill, J. Phys. A, 1975, 8, 664. 3 P. N. Pusey, H. M. Fijnaut and A. Vrij, J. Chem. Phys., 1982, 77, 4270. (Plenum Press, New York, 1974).General Discussion 109 Dr T. G. M. van de Ven (McGill University, Montreal, Canada) replied: After-pulsing is one of the possible explanations for the rapid initial decay in the autocorrelation function.As discussed before, other explanations are possible. I agree with your comments on particle size. In the case of gold sols, the distribution in particle size is narrow, but is complicated by the presence of doublets. Here the difference between particle size from electron microscopy and PCS is small, but the second cumulant is large. In the case of latex particles it is of interest to remark that the radius from electron microscopy provides a better fit between our experiments and recent numerical calculations, kindly made for us by Dr R. W. O'Brien. Assuming latex particles of radius 40nm, the theory predicts no minimum in the diffusion coefficient in the electrolyte concentration range 10-4-10-3 mol dm-3 (for @ = -3), while for a radius of 32 nm a minimum is predicted for K a == 0.5.Furthermore, the numeri-cal results predict the observed ion size effects: for the three salts used the friction coefficient is predicted to be maximum for (CH3CH2)4NCl, followed by NaC6H5C02H and NaCl. The main difference between the theory and observation is, besides the minimum being at K a = 0.5 rather than K a = 1, that the observed effects are larger than the predictions. This implies that zeta potentials determined from diffusion coefficients are larger than- potentials determined from electrophoresis. Dr D. Weitz (Exxon, Annandale, NJ) said: As suggested by Dr Pusey, the effects of phototube after-pulsing may lead to the short time decay observed in these experi- ments (fig.4), and this should certainly be checked. However, I would like to suggest that rotational diffusion may also be quite important. This is due to the particular optical properties of gold colloids. Since gold is a nearly-free-electron metal, it possesses a pronounced electronic plasma resonance in the optical frequencies. This results in the characteristic wine-red colour of the unaggregated colloid. However, upon aggrega- tion, the resonances of the particles are shifted, which is reflected by the change in colour of the aggregates. In particular, a dimer will possess two distinct resonances: one when the incident electric field is aligned normal to the axis joining the two spheres, and a second when the field is aligned parallel to this axis.The former occurs at nearly the same frequency as that of an isolated sphere, while the latter occurs at a lower frequency. In fact, for gold, this lower-frequency resonance is quite close to the wavelength of the HeNe laser used in these experiments. As a result, at this wavelength the scattering from a dimer will be considerably more intense than from a monomer. Furthermore, the polari~ibi!iiy of a dimer will be quite anisotropic. Indeed, one calculation of this effect' (performed for silver but also applicable to gold), suggests an anisotropy of as much as two orders of magnitude for excitation along the axis of the two spheres compared to normal to it, when the exciting frequency exactly corresponds to the lower frequency resonance. While one can not expect quite so large an anisotropy here, it should nevertheless be significant.The combination of the large anisotropy and increased magnitude of the polarizability would lead to a substantial contribution from rotational diffusion in the dynamic light scattering. However, I would expect this effect to be caused by the dimers in the solution, rather than by the slight asphericity of the monomers, as is suggested in your paper. Finally, I note that this effect is due to the particular optical properties of gold colloids, and one can not predict the asymetry using the calculations for the much larger fractal clusters presented in fig. 4 of our paper. 1 P. K. Avarind, H. Metiu and A. Nitzan, SurfSci., 1981, 110, 189. Dr D.S . Home (Hannah Research Institute, Ayr) said: You rightly point out that previous studies of pH or salt effects on the diffusion coefficients of colloidal species neglected frictional effects associated with the electrostatic double layer. However, I think it would be equally wrong to dismiss the possibility that such changes, particularly110 General Discussion with proteins, were not the results of conformational or configurational changes in these molecules without independent evidence of the absence of the latter effects. I say this as one who in the past has ascribed increases in diffusion coefficient to configurational collapse of protein molecules on the surface of casein micelles following the introduction of ethanol (Horne and Davidson, Colloid Polym.Sci., 1986, 264, 727). On the model you propose, introduction of alcohol would vary the medium dielectric constant and through this the Debye length of the double layer. As well as varying the ethanol level, the studies of Horne and Davidson also considered the effect of added salt. If the increases they observed in the diffusion coefficient were due only to electrostatic effects on the double layer then the results of those experiments should all fall on the same line when plotted against K . I cannot make this happen. The increases in D, however, are influenced by the ionic strength of the medium, and this I think is a manifestation of secondary electrical effects on the interactions between the surface molecules or hairs rather than the primary effect on the diffusional motion you are suggesting here. Dr T.G. M. van de Ven (McGill University, Montreal, Canada) replied: I fully agree that conformational or configurational changes in proteins or polyelectrolytes can cause changes in their effective diffusion coefficients. All we are saying is that the effects of the surrounding double layers must be taken into account as well. Prof. B. J. Ackerson (R.S.R.E., Maluern) also addressed Dr van de Ven: I am pleased to see that you are able to go to sufficiently low colloidal particle concentrations to see only self-diffusion effects and still have enough intensity to perform the experiment. This is evidenced by seeing the measured diffusion coefficient approaching the Stokes value at high and low salt concentrations.The largest self-diffusion reduction is observed at Ka = 1. Evidently, at high salt concentration the colloidal particle charge is screened and not many ions participate in the drag reduction. At low salt concentration there are not many ions to participate in the drag reduction. Thus the maximum effect occurs at some intermediate salt concentration. However, dynamic light scattering measures a mutual diffusion constant. The effect of the small ions on mutual diffusion must be considered, in general, to be sure one knows what is being measured in a given experiment. The initial motion of the colloidal particle produces an asymmetry in the ion cloud surrounding the particle. The net charge separation between the particle and cloud retards the particle motion and lowers its self-diffusion in general. The net charge separation also produces a dipole which couples to other colloidal particles and modifies their motion. Unlike the self-diffusion effect, this dipole coupling continues to increase as the salt concentration decreases. The mutual diffusion coefficient is decreased by this dipole coupling, while it is increased by direct interactions, as the ionic strength is reduced. At finite colloidal particle concentrations a measurement of the first cumulant of the time decay of the intensity correlation function will have contributions from both self- and dipole-coupling effects, even after multiplying by the static structure factor to eliminate the direct interaction effect. Presumably at low enough concentrations self-diffusion is measured as shown in your results. However, at higher concentration the effect of dipole coupling may be to give a much more pronounced reduction in the measured effective diffusion constant which does not ever increase with decreasing salt concentration. The effect of dipole coupling may explain the differences seen between your self-diffusion measurements and the ‘self-diffusion’ measurements of Gorti et al.’ 1 S. Gorti, L. Plank and B. R. Ware, J. Chem. Phys., 1984, 81, 909. Dr T. G. M. van de Ven (McGill University, Montreal, Canada) replied: You point to an interesting singular limit, namely Ka + 0. For a fixed concentration of particles,General Discussion 111 no matter how small, as KU -+ 0 the double-layer thickness approaches infinity and double layers will start to interact. Hence it is not surprising that the dipole coupling continues to increase as the salt concentration decreases. However, we do not think that in our experiments these effects were present. At the lowest salt concentration, KU = 0.3, the double layer is about three times the particle radius, while at volume fractions of ca. the average distance between the particles is ca. 300 particle radii. At higher (but still low) volume fractions, dipole coupling can become important. Prof. V. Degiorgio (Pauia, Italy) commented: Since your experiment is performed at a fixed particle concentration and it is known that the diffusion coefficient of solutions of charged particles depends very strongly on concentration at very low ionic strength, it is possible, in principle, that the measured D contains at low salt concentration some effect of interparticle interactions. It would be interesting to evaluate whether such an effect is significant in your case. Dr J. G. Rarity (R.S.R.E., Malvern) also remarked: You interpret the short time data shown in fig. 4 of your paper as a rotational signal from the anisotropic polarisability of a small number of doublets in your system. A theoretical treatment of this type of contribution to the correlation function is given in the book by Berne and Pecora.' Your data imply that depoiarised scattering has a greater magnitude than the polarised scattering. For this to be true, with only 4% doublets, implies a strong resonant enhancement of scattering along the doublet axis and large phase differences between resonant and non-resonant scattered components. A measurement of the depolarised scattering ( I V H ) would quickly confirm or disprove your theory. 1 B. J. Berne and R. Pecora, in Dynamic Lighr Scattering (Wiley-Interscience, New York, 1976), chap. 7.