摘要:

FARADAY DISCUSSIONS OF THE CHEMICAL SOCIETY NO. 76 1983 Concentrated Colloidal Dispersions THE FARADAY DIVISION THE ROYAL SOCIETY OF CHEMISTRY LONDONFARADAY DISCUSSIONS OF THE CHEMICAL SOCIETY NO. 76 1983 Concentrated Colloidal Dispersions THE FARADAY DIVISION THE ROYAL SOCIETY OF CHEMISTRY LONDONA GENERAL DISCUSSION ON Concentrated Colloidal Dispersions 14th, 15th and 16th September, 1983 A GENERAL DISCUSSION on Concentrated Colloidal Dispersions was held at Lough- borough University of Technology on 14th, 15th and 16th September, 1983. The President of the Faraday Division, Professor P. Gray, FRS, was in the chair: about 200 fellows of the Faraday Division and visitors from overseas attended the meeting. Among the overseas visitors were: Prof. B. J. Ackerson, U.S.A.Mr. H. Aijala, Finland Mr. J. L. Arauz-Lara, Mexico Mr. C. W. J. Beenakker, The Netherlands Prof. B. H . Bijsterbosch, The Netherlands Dr. K . Bridger, U.S.A. Dr. W. Brown, Sweden Dr. L. Canova, Italy Dr. D. Y . C. Chan, Australia Dr. J. Chappuis, France Dr. M. Corti, Italy Dr. M. D. Croucher, Canada Dr. N. C. Degaeger, Belgium Prof. V. Degiorgio, Italy Dr. C. G. De Kruif, The Netherlands Dr. A. Diez, West Germany Prof. H-F. Eicke, Switzerland Dr. J. Eisenlauer, West Germany Dr. D. Fairhurst, U S A . Prof. B. U. Felderhof, West Germany Dr. R. Finsy, Belgium Dr. M. Forster, Switzerland Dr. A. P. Gast, U.S.A. Dr. G. E. Gerhardt, U.S.A. Dr. L. Gillberg, Sweden Mr. P. J . Goetz, U.S.A. Mr. J. W. S. Goossens, West Germany Mr. J. Green, U.S.A. Mr. P. Guering, Sweden Dr.J. Hayter, France Dr. M. M. Henriet, Belgium Dr. C. G. Hermansky, U.S.A. Dr. R. E. Highsmith, U.S.A. Prof. H. Hoffmann, West Germany Mr. J . W. Jansen, The Netherlanh Mr. H . Kamphuis, The Netherlands Dr. K. P. Kehrer, U.S.A. Prof. R. Klein, West Germany Dr. R. Kubik, Switzerland Dr. D. Langevin, France Dr. B. Larsson, Sweden Dr. B. 0. Larsson, Sweden Dr. H. M. Laun, West Germany Dr. J. Lebell, Finland Prof. H. N . W. Lekkerkerker, Belgium Dr. J. J. Leucker, Switzerland Dr. G. Ley, West Germany Dr. J . W. Th. Lichtenbelt, The Netherlands Prof. B. Lindman, Sweden Dr. H. Ljusberg-Wahren, Sweden Prof. M. Medina-Noyola, Mexico Dr. M. Miquel, France Dr. J. Moonen, The Netherlands Dr. E. Moreels, Belgium Mr. P-G. Nilsson, Sweden Dr. L. Odberg, Sweden Dr. A.Parentich, Australia Dr. G. Parsiegla, The Netherlands Dr. W. Prange, West Germany Dr. H. M. Princen, U S A . Mr. B. Pugh, Sweden Dr. P. N. Pusey, U.S.A. Prof. W. B. Russel, U.S.A. Dr. P. C. Scholten, The Netherlands Dr. J. Sjoblom, Sweden Dr. I. Snook, Australia Prof. H. N. Stein, The Netherlands Dr. J . Tabony, France Dr. W. van Megen, Australia Dr. J. Van Nieuwkoop, The Netherlands Prof. V. L. Vilker, U.S.A. Prof. A. Vrij, The Netherlands Mr. H . Walderhaug, Sweden Mr. T. Warnheim, Sweden Dr. R. G. Willis, U.S.A. Dr. T. Wolff, West Germany Dr. D. T . Wu, U.S.A. Dr. H. Zecha, East Germany Dr. M. R. Zulauf, FranceOrganising Comm it tee Professor R. H. Ottewill (Chairman) Dr. E. Dickinson Mrs. Y. A. Fish Dr. J. W. Goodwin Dr. M. J. Jaycock Dr. Th. F. Tadros Dr.P. N. Pusey Dr. A. L. Smith Dr. D. A. Young ISBN: 0-85186-648-4 ISSN: 030 1-7249 Printed in Great Britain by Fletcher & Son Ltd., NorwichCONTENTS Page 7 19 37 53 65 77 93 123 137 151 165 179 189 203 219 Concentrated Colloidal Dispersions Viewed as One-component Macrofluids by J . B. Hayter Light Scattering of Colloidal Dispersions in Non-polar Solvents at Finite Concentrations. Silica Spheres as Model Particles for Hard-sphere Interactions by A. Vrij, J. W. Jansen, J. K. G. Dhont, C. Pathamamanoharan, M. M. Kops-Werkhoven and H. M. Fijnaut Properties of Concentrated Polystyrene Latex Dispersions by D. J . Cebula, J. W. Goodwin, G. C. Jeffrey, R. H. Ottewill, A. Paren- tich and R. A. Richardson Neutron-scattering Studies of Concentrated Oxide Sols by J.D. F. Ramsay, R. G. Avery and L. Benest Electrical Double-layer Interactions in Concentrated Colloidal Systems by B. Beresford-Smith and D. Y. C. Chan Light Scattering and Neutron Scattering from Concentrated Dispersions of Small Unilamellar Vesicles by A. G. Muddle, J. S. Higgins, P. G. Cummins, E. J. Staples and I. G. Lyle GENERAL DISCUSSION Hydrodynamic Interactions and Diffusion in Concentrated Particle Suspensions by P. N . Pusey and R. J. A. Tough Mass-diffusion and Self-diffusion Properties in Systems of Strongly Charged Spherical Particles by R. Klein and W. Hess Diffusion in Concentrated Monodisperse Colloidal Solutions. The Hard Sphere- Thermodynamic or Hydrodynamic? by W. van Megen and I. Snook Simulation of Particle Motion and Stability in Concentrated Dispersions by J.Bacon, E. Dickinson and R. Parker Diffusion in Hard-sphere Suspensions by B. U . Felderhof and R. B. Jones Phase Separations Induced in Aqueous Colloidal Suspensions by Dissolved Polymer by A. P. Gast, C. K. Hall and W. B. Russel Rheological Investigations of the Effect of Addition of Free Polymer to Concentrated Sterically Stabilised Polystyrene Latex Dispersions by D. Heath and Th. F. Tadros Cross-correlation Intensity Fluctuation Spectroscopy Applied to Colloidal Suspensions by B. J . Ackerson and N. A. ClarkGENERAL DISCUSSION 229 26 1 277 29 1 305 317 331 353 363 375 381 383 Temperature Dependence of the Sheer Viscosity of Sterically Stabilised Polymer Colloids by M. D. Croucher and T. H. Milkie Sedimentation and Viscous Flow of a Weakly Flocculated Concentrated Dispersion. A Comparative Study by R. Buscall and I. J. McGowan Percolation and Critical Points in Microemulsions by A-M. Cazabat, D. Chatenay, D. Langevin and J. Meunier Concentrated Dispersions of Aqueous Polyelectrolyte-like Microphases in Non-polar Hydrocarbons by H . F . Eicke and R. Kubik Fourier Transform Carbon-13 Relaxation and Self-diffusion Studies of Microemulsions by B. Lindman, T. Ahlnas, 0. Soderman, H. Walderhaug, K. Rapacki and P. Stilbs GENERAL DISCUSSION Electric Birefringence Studies on Concentrated Aqueous Polyoxyethylene Surfac tan t Sy s terns by P. G. Neeson, B. R. Jennings and G. J. T. Tiddy Viscoelas t ic Detergent Soh t ions by H . Rehage and H. Hoffmann GENERAL DISCUSSION LIST OF POSTERS INDEX OF NAMES

ISSN:0301-7249    

DOI:10.1039/DC9837600001

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1983, 76, 7-17 Concentrated Colloidal Dispersions Viewed as One-component Macrofluids BY JOHN B. HAYTER Institut Laue-Langevin, 156X Centre de Tri, 38042 Grenoble Cedex, France Received 9th May, 1983 The problem of obtaining a quantitative statistical-mechanical description of the structure of concentrated colloidal dispersions is greatly simplified if the only significant correlations are those between the colloidal particles themselves. The medium in which the particles are dispersed may then be treated as a uniform background, which has the physical characteristics necessary to ensure electroneutrality and appropriate dielectric properties, but is otherwise featureless. The problem thus reduces to calculating correlations within a single-component fluid of finite particles having a specified pairwise interaction. This paper will discuss the use of current liquid-state theories to calculate the static structure of such one-component macro- fluids, and will consider some of the experimental evidence which confirms the validity of the approach.In particular, examples will be taken from light- and neutron-scattering experiments on concentrated dispersions of neutral, charged or magnetic particles of approximately spher- ical shape, and the extension to other systems will be briefly considered. The most general statistical-mechanical problem associated with describing the structure of a concentrated colloidal dispersion is to predict the properties of the dispersion, given the structure and laws of interaction of all the particles of which it is composed.Thus formulated, the direct theoretical problem is at present intrac- table. Computer simulation can (at least in principle) offer numerical answers, but the technique is not generally of use to the experimentalist interested in analysing many systems under different experimental conditions. We may, however, pose a more tractable problem by concentrating on the component of most interest, namely the colloidal particles themselves, and renouncing for the present any questions about what happens in the dispersion on a much finer level of structural detail than the colloidal particle size. The purpose of this paper is to discuss the significant recent progress which has been made in solving this simpler problem.We shall see that questions about fine detail within the colloidal particle may be resolved concur- rently with understanding the correlations between the particles, but questions such as how the solvent is locally structured near a particle will be disallowed a priori, although this type of information can sometimes be inferred a posteriori in certain sys tems. To set this approach on a quantitative basis, consider a scattering experiment designed to measure the structure of a concentrated colloidal dispersion. Radiation (light, X-ray or neutron) of momentum ki impinges on a sample and is scattered through angle 28 with final momentum kf. The experiment performed is to measure the scattered intensity I ( Q ) as a function of the momentum transfer Q = ki - kf; for elastic scattering IQI = (4n/IE)sin 8.Each atom in the suspension has an associated scattering amplitude b, where b depends on the nuclear type and electronic moment of the atom for neutrons, and on electron density for electromagnetic radiation. The8 DISPERSIONS AS ONE-COMPONENT MACROFLUIDS scattered intensity is thus calculated by summing all the scattered waves, taking into account amplitudes due to scattering type and phase shifts due to scattering position, and averaging the result over all possible configurations weighted by their proba- bility of occurrence: I ( Q ) = ( l z b j ex~(iQ R j ) 1 2 ) j i k = (1 1 b j b k exp[iQ ( R j - &)I) (1) where Rj is the position of thejth atom, the sum is over all atoms in the suspension and ( ) indicates taking the ensemble average. It is the calculation of this latter average which poses the central (and generally unsolved) statistical-mechanical problem.We now take the colloidal particles to be the only macroscopic objects present in the suspension; i.e, the solvent and any other dissolved species are taken to have no structure as such, and no pairwise correlations either with each other or with the colloidal particles. Further, we assume that any contribution they make towards the structural correlations between the colloidal particles may be absorbed into an ap- propriate pairwise interaction potential between those particles, for example by screening the potential. The problem is then reduced to that of calculating the en- semble average for a one-component macrofluid (OCM) interacting through a speci- fied pair potential.While no exact theory exists for solving even this simplified problem, exact solutions to several useful approximate theories have become available in recent years. We shall now consider their application to several types of concentrated colloidal dispersion for which reasonable pair potentials may be postu- lated, considering first the case of isotropic dispersions. CORRELATION FUNCTIONS IN ISOTROPIC DISPERSIONS Eqn (1) represents all possible spatial correlations in the dispersion. For the OCM we may immediately exclude all atoms other than those in the colloidal par- ticles from the summation, and we may further split the sum into a sum with j and k both in the same particle, plus a sum overj and k belonging to different particles. On the further assumption that the particles are independently oriented, the result is (2) where ItQ) = S(Q)(W)>& + < F 2 ( Q ) > ~ - (F(Q>>t! F(Q) = C b j exp(iQ r j ) j is the single-particle form-factor (the sum running over all distances V j within a particle relative to the particle centre) and ( )Q indicates an angular average over all orientations of the particle with respect to the direction of Q.Here we have assumed the particles are monodisperse for simplicity; specific cases of polydispersity will be considered later. The interparticle structure factor S(Q) specifies the correlations between the cen- tres of different particles. While F(Q) and the related angular averages can be cal- culated easily (although perhaps numerically) for any given particle model, the cal- culation of S(Q) requires the evaluation of an ensemble average over all particle configurations: S(Q) = 1 + p( C exp (iQ R m n ) ) (3) n # mJ.B. HAYTER 9 where p is the number density of particles in the suspension and R,, is the distance between the centres of particles m and n. It is important to note at this point that, in reducing the problem to one for which solutions may be found, we have already made two strongly restrictive assumptions: particles are monodisperse and they are independently oriented. While polydispersity can be introduced in certain cases, there is no way at present to handle the general problem of correlated orientations of non-spherical particles in a way which is useful to the experimentalist; specific sys- tems in which there is total correlation of orientation will be discussed in a later section of this paper.Excellent texts are available on the statistical mechanics of f l ~ i d s , ~ ? ~ and we shall only outline necessary results to situate the notation. In real space the radial distri- bution function g(r) is a measure of the probability of finding another particle at distance r, given a reference particle centred at the origin; it is clear that g(r) must be zero for r less than the distance of closest approach of two particles. The total correlation function, h(r) = g(r) - 1, measures the total correlation between two particles separated by a distance Y. There are two contributions to the latter. First, the direct influence of one particle on another, described by a direct correlation function, c(r), and second, an indirect influence of one particle on another, trans- mitted via a third particle at distance ~ 1 3 from the first and ~ 3 2 from the second.The latter influence must be integrated over all possible positions of the third particle, leading to (4) which is the well known Omstein-Zernike (OZ) e q u a t i ~ n , ~ r12 being the distance between particles labelled 1 and 2 in the isotropic dispersion. Now h(r) is the weighting function needed to calculate the ensemble average for S(Q), so that eqn (3) takes the form h ( r 1 2 ) = dr12) + Pjc((y13) h ( r 3 2 ) dY3 where &Q) is the Fourier transfrom of h(r). Further, Fourier-transforming eqn (4) using the convolution theorem yields an equivalent form s(Q) = 1/[1 - ~ 4 Q ) l (6) so that if we can solve eqn (4) for either h(r) or c(r) we may calculate S(Q) from the Fourier transform k(Q) or Z(Q) of the solution.[For this reason it is current prac- tice3S4 to consider eqn (4) as the definition of c(r) in terms of the more physical quantity h(r).] To solve the integral eqn (4) we also require some ‘boundary con- ditions’ on the solution. These are known as closure relations, and it is the closure relations which introduce the interparticle potential specific to any given system. SPHERICALLY SYMMETRIC POTENTIALS A variety of closure relations has been proposed to supplement the OZ equation for potentials with spherical ~ y m m e t r y . ~ . ~ We shall restrict our attention throughout the rest of this paper to the so-called mean spherical approximation (MSA),6 which has proved particularly fruitful.The MSA closure relations take the form where d is the distance of closest approach between particles and V(r) is the inter-10 DISPERSIONS AS ONE-COMPONENT MACROFLUIDS action potential between a pair of particles. Eqn (7a) is exact for particles having a hard core. The MSA approximation [eqn (7b)l only gives good results for structural calculations at moderate to high number densities, when a significant proportion of space is filled by the exact condition (7a); a modification which extends its validity to all concentrations will be discussed later. HARD SPHERES In the special case when V(r) = 0 for r > d, eqn (4) and (7) reduce to the well known Percus-Yevick approximation 7 1 * for hard spheres of diameter d.This has proved a useful method of calculating S(Q) for many neutral systems such as micro- emulsion^.^ The important feature is not that the particles need to be infinitely hard, but that they cannot pass through one another; the ‘hard-sphere’ description then provides a reasonable estimate of the correlations due to purely excluded-volume effects. In the case of hard spheres, polydispersity may specifically be taken into account and the ensemble average (1) calculated numerically for various distributions. This has been studied in detail by Vrij et al., lo* l1 by Pusey et a1.12 and by Salacuse and Stell.13 In the case of monodisperse spheres eqn (2) reduces to I(Q) = P(Q) s(Q> (8) where P(Q) = (F(Q))a = ( F 2 ( Q ) ) ~ .Van Beurten and Vrij l1 assumed that the re- sult of evaluating eqn (I) for a polydisperse system could be written in the form of eqn (8) provided S(Q) was replaced by an effective value Ser (Q),obtained by divid- ing l(Q) by a size-averaged P(Q). For many purposes, however, retaining the form of eqn (2) may be preferable since it shows that polydispersity ‘switches on’ a diffuse-scattering term2 added to the basic form, eqn (8). CHARGED HARD SPHERES The MSA for charged hard spheres interacting through an unscreened Coulomb potential was solved by Palmer and Weeks.14 It can be shown generally that such a system has zero osmotic compressibility,15 so that this problem has no direct re- levance to charged colloidal dispersions (some potential screening always being pre- sent), but it raises the thermodynamic limit question of why a fluid interacting through a long-ranged repulsive potential should be stable, and not explode.Lebowitz and Lieb l 6 showed that such a fluid could be decomposed into a ‘Swiss cheese’ of non-overlapping regions which were each electrically neutral and thus had no Coulombic interaction with each other by virtue of Newton’s theorem; the inter- ested reader is referred to the review by Baus and Hansen s for a discussion of this important point and of one-component plasmas (OCP) in general. The formal problem of solving the MSA for particles interacting through a screened Coulomb potential was reduced from an integral equation to a set of simul- taneous non-linear algebraic equations by Waisman.Using a different approach, Hayter and Penfold l8 were able to calculate a fully analytic solution in a convenient form which is very rapid to compute, and they used this as the basis for a complete analysis of scattering data from concentrated solutions of charged micelles under different screening conditions. This not only opened the way to later systematic studies of such systems,2,20-22 but also allowed confirmation of the validity of the OCM approach by showing that the results correctly predict certain dynamic behaviour in the same systems.lgJ . B. HAYTER 1 1 Eqn (5) or (6) shows that S(Q) -+ 1 as the density tends to zero, so that eqn (8) may be used to measure the scattering function P(Q) for colloidal particles in dilute solution, and then S(Q) may be derived from measurements on concentrated sys- tems, as long as the particles are spherical.The solution structure may then be obtained in the form of g(r) by Fourier transformation of S(Q), using the inverted form of eqn (5). The particular problem posed by micellar solutions is that not only may the micelle size vary with surfactant concentration, but the micelles cease to exist below the critical micellar concentration (c.m.c.), so that measurements are often not possible on dilute solutions. The approach taken originally by Hayter and Penfold l9 was to calculate S(Q) from the MSA solution for the screened Coulomb potential l 8 and P(Q) from a model of the micelle. An advantage of this procedure is that only two parameters are required, namely the charge on the micelle and the aggregation number; these parameters are obtained by fitting the theory to the data.I I I I 0 1 2 3 5 Q/nm - I Fig. 1. Scattered neutron intensity plotted as a function of momentum transfer for (a) 0.6 and (b) 1.2 mol dm-3 sodium octanoate solutions at 301 K. 0 , Experiment;20 (-) fit to eqn (8); dashed lines show the relative contributions from P(Q) and S(Q). The typical quality of agreement obtained between experiment and a theoretical OCM calculation is shown in fig. 1 . The values of charge and aggregation number required to fit the theory agree well with those obtained from other methods when comparison is possible 2, 2 o (e.g. near the c.m.c.), but the technique is unique in its ability to analyse very high concentrations quantitatively. A feature of the data is that the observed peak does not correspond to the first12 DISPERSIONS AS ONE-COMPONENT MACROFLUIDS peak in S(Q). This is often the case, and may be understood by considering the standard statistical-mechanical result 3, S(0) = pk,TKT (9) where KT is the isothermal compressibility of the OCM.The primary effect of intro- ducing repulsive interactions is to reduce this compressibility and thus to suppress intensity as Q --+ 0. If the Coulomb potential in a charged system is screened by increasing the ionic strength of the solution, the primary effect seen is an increase in the scattering at small Q as the OCM becomes more compressible.19 DILUTE INTERACTING CHARGED HARD SPHERES A case of particular interest in colloidal dispersions is when the particles are numerically dilute but nevertheless strongly interacting. A well known example is provided by the data of Brown et al.23 on charged polystyrene spheres in dilute suspension at low ionic strength.As stated earlier, the MSA is of no direct use in this case,18 but the problem may be solved by treating it via an equivalent high-density calculation in which the length scales and the potential have been renormali~ed.~~ This rescaled MSA (RMSA) calculation preserves the analytic form of the Hayter-Penfold solution. The physical argument is based on the idea that once the potential has risen to several kBT, its detailed form no longer matters in calculating the dispersion structure (since high-energy configurations are rarely sampled); it leads to excellent results, as may be seen in fig.2. 2.0 1 . o Q 0.0 g 2.0 1 .o 0.0 I I 0.0 0.5 1.0 1.5 Qd Fig. 2. Scattered light intensity plotted as a function of momentum transfer for dispersions of charged polystyrene spheres at volume fractions of (a) 4.8 x lop4 and (b) 1.7 x 0, Experimental data of Brown et al.; 23 (-) fit to eqn (8) using RMSA to calculate S(Q)."J. B. HAYTER 13 An interesting aspect of colloidal dispersions which are concentrated in the sense of strong interaction, although numerically dilute, is that the particle shape is no longer important in calculating correlations between particles, so that the RMSA theory should allow a variety of both chemical and biochemical charged colloids to be studied in dispersions of low ionic strength.ATTRACTIVE INTERACTIONS The formal theory of Hayter and Penfold'* places no restriction on the sign of the potential between particles in the OCM, and it may be used in a phenomenolog- ical way to study systems in which there is a short-ranged attractive interaction. An interesting feature of such systems is the build-up of very long-ranged correlations in the dispersion through particle clustering, so that a short-ranged potential can gen- erate correlations which eventually become infinite and the dispersion changes phase. As the phase transition is approached, the divergence of the compressibility is 4 .O 3 .O n E c .- $ 2.0 W n 1.0 0.0 1 . 6 n v L 8 0 0.0 0.6 1.2 1.8 2 .L Q/nm- ~ 0 1 2 3 1, 5 6 rid Fig.3. (a) Scattered neutron intensity plotted as a function of momentum transfer for a 7% solution of C,E, at the following temperatures: (i) 298.1, (ii) 31 1.1, (iii) 317.1, (iv) 323.1 and (v) 327.6 K. , Experiment; (-) fit to eqn (8) using an attractive potential to calculate S(Q). (b) Calculated radial distribution functions g(r) corresponding to (a); note that the distance is scaled to the particle diameter.14 DISPERSIONS AS ONE-COMPONENT MACROFLUIDS accompanied by a corresponding divergence of the forward scattering through eqn (9). A class of systems which exhibits this phenomenon is provided by the micellar solutions of non-ionic detergents with polyoxyethylene head groups. Fig. 3(a) shows data for n-octylpentaoxyethylene glycol monoether (C, E 5 ) at fixed concentration as the temperature is increased towards the critical demixing temperature (T, = 329.7 K).These data are consistent with the presence of an increasingly attractive interaction between micelles of fixed diameter d = 4.7 nm as the temperature increases, and the solid lines in fig. 3(a) are calculated on this basis.25 The theoretical S(Q) has been transformed to the radial distribution function g(r) between micelle centres in fig. 3(b), which exhibits a particularly interesting feature: the second-neighbour peak develops at a separation of exactly two diameters. This indicates the formation of contact clustering, and is very different from the g(r) for an equivalent hard-sphere system, where the second-neighbour shell would be at larger r and less well defined. As the phase transition is approached [curve (v)], strong correlations are observed over many diameters, although the potential used in this example had a range of only d/16.ANISOTROPIC DISPERSIONS The OCM approach has been remarkably successful in describing isotropic dis- persions interacting through spherically symmetric potentials, and it is natural to try to extend it to dispersions with more general interactions. Here the state of the art is much less well advanced, and very little can be said at all at a quantitative level about isotropic dispersions of anisotropic particles, such as ellipses or rods. The problem arises mainly through the difficulty of calculating the probability of a given pairwise orientation at a specified separation, and for many such systems one cannot even write down the potential of interaction in an analytic form.One way around this difficulty is to impose an alignment on the system, for example by shearing the suspension or applying an electric or magnetic field, so that all particles take up the same orientation. FERROFLUIDS One anisotropic dispersion for which the OCM approach yields an analytic theory is the case of a ferrofluid in a saturating magnetic field.26327 The theoretical model particle is taken to have a spherical core of magnetic material (which may have a magnetically inactive surface layer), surrounded by a stabilising surfactant layer, and interacting through a spherically symmetric hard core plus ‘sticky’ poten- tial, together with an anisotropic magnetic dipolar interaction. The feature which allows analytic solution of such a complex system is that the geometry of the particle is spherically symmetric, even though the total interaction potential is not.Fig. 4 shows typical theoretical predictions for S(Q) and g(r) for a concentrated cobalt ferrofluid. Two features are noteworthy. First, the applied magnetic field H imposes an axis on the system, and Q and Y must now be treated as vector quantities. Second, the peaks in g(r) are at integral multiples of the diameter d for Y along the field, whereas there is little structure in the perpendicular direction. This indicates the formation of chains of particles in contact along the field direction. Magnetic colloidal dispersions are particularly well suited to study by the polarisation analysis technique of neutron scattering,2 and typical results from theJ.B. HAYTER 15 3.0 n 2 1.5 0.0 6 L 2 n v bo 0 - 2 -4 0 6 12 18 2L Qd 0 1 2 3 L 5 6 rld Fig. 4. (a) Theoretical structure factors for Q perpendicular (i) and parallel (ii) to a saturating applied magnetic field H for a concentrated suspension of surfactant-stabilised cobalt par- ticles. The total volume fraction is 30% and the total particle diameter is twice that of the magnetic core. (b) Calculated pair correlation functions corresponding to (a). [(i) r I H, (ii) r // H.] first such study29 are shown in fig. 5. Neutron polarisation analysis provides un- ambiguous, model-free separation of nuclear and magnetic scattering, and further allows direct measurement of the degree of magnetic alignment in the dispersion.The fluctuation scattering shown in fig. 5 indicates that the ferrofluid is not quite fully aligned in the applied field of the experiment, but even so the general agree- ment between theory and experiment is highly satisfactory. DISCUSSION AND OUTLOOK The OCM approach to describing the structure of monodisperse concentrated colloidal dispersions of spherical particles is now confirmed experimentally and ap- pears to be a generally useful technique. This very success raises some interesting questions, however. One such question is why it provides such a good description of many real systems which in detail are unlikely to be either perfectly spherical or16 DISPERSIONS AS ONE-COMPONENT MACROFLUIDS 1600 * I 800 --.h 0, * 0 I A t \ 0 10 20 Qd Fig. 5. Results of a neutron-polarisation analysis experiment29 (e) on a cobalt ferrofluid of 28% total volume fraction compared with theory26-27 (-)at a field H = 0.5 T. (a) Co- herent scattering, Q I H, (b) Q // H. (c) Magnetic fluctuation scattering, which would be zero if magnetic saturation were complete. monodisperse. For the specific case of charged dispersions this aspect has recently been examined by Hayter and Penfold.2 They concluded that provided the particles were relatively globular and not too polydisperse, the strong repulsive potential kept particles from sampling geometrical anisotropy, thus meeting the theoretical require- ment that there be no correlation of orientation with position. The initial impact of a departure from sphericity or monodispersity was found to be the addition of a diffuse background to the scattering from strongly interacting charged dispersions.Note that for non-spherical particles, the single-particle scattering function modu- lated by S(Q) in eqn (2) is no longer the scattering function for the isolated particle in dilute solution [when S(Q) = 13, unlike the case for spheres, so that measurements on dilute systems are only indirectly useful. Another interesting question is to ask to what extent the DLVO potential is confirmed by the success of the OCM theory based on screened Coulomb inter- actions. Cheq30 for example, has found that the theory of Hayter and Pen- fold, which is based on the small-screening form of the DLVO potential, still works very well in the presence of strong screening.The experience gained from studies on liquid alkali metals, which to some extent may also be described by the OCM ap- proach, is that the indirect problem of inferring detailed pair potentials from struc- tural measurements alone is A probable reason for this is that the mea- sured density (and therefore the mean interparticle spacing) is required as an input to the theory, so that a great deal of information about the true pair potential is al- ready included in a fundamental fashion in the calculation before a specific form for the potential is introduced. All of the examples discussed in this paper have been for particles with hard cores. It is worth noting that the theory may often be carried over directly to the case of permeable spheres, which have a finite probability of passing through each other.Salucuse and Stell have proposed this for neutral systems, but all of the isotropic models quoted above may be rescaled quite simply to allow for permeability if desired. This may well provide a useful basis for studying dispersions in which colliding droplets can coalesce briefly and then separate; microemulsions could exhibit this behaviour. There are some interesting systems for which the OCM approach is clearly in-J. B. HAYTER 17 sufficient, e.g. dispersions of polyelectrolytes. Here, the detailed question of the very local counterion distribution, which is ignored in the OCM approximation, becomes important. For the present, little progress has been made in this area, although some results can be explained using a ‘correlation hole’ For many interacting systems the OCM provides a useful approach, however, and it is certainly worth pursuing the theoretical effort to extend the OCM description to anisotropic col- loidal particles in concentrated dispersion, particularly following the recent success with anisotropic ferrofluids.I have enjoyed numerous discussions with colleagues on many of the ideas pre- sented here, and particular thanks are due to Prof. J-P. Hansen and R. H. Ottewill and Drs J. Penfold, P. N. Pusey and R. Pynn. H. L. Frisch and J. L. Lebowitz, The Equilibrium Theory of Classical Fluids (W. A. Benjamin, New York, 1964). J. B. Hayter and J. Penfold, Colloid Polym. Sci., in press. J-P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic Press, London, 1976).R. 0. Watts and I. J. McGee, Liquid State Chemical Physics (Wiley, New York, 1976). L. S. Ornstein and F. Zernike, Proc. K. Ned. Akad. Wet., 1914, 17, 793. This paper is reprinted in ref. (1). J. L. Lebowitz and J. K. Percus, Phys. Rev., 1966, 144, 251. N. W. Ashcroft and J. Lekner, Phys. Rev., 1966, 145, 83. D. J. Cebula, R. H. Ottewill, J. Ralston and P. N. Pusey, J. Chem. Soc., Faraday Trans. 1, 1981,77, 2585. P. van Beurten and A. Vrij, J. Chem. Phys., 1981, 74, 2744. ’ J. K. Percus and G. J. Yevick, Phys. Rev., 1958, 110, 1. l o A. Vrij, J. Chem. Phys., 1979, 71, 3267. l 2 P. N. Pusey, H. M. Fijnaut and A. Vrij, J. Chem. Phys., 1982, 77, 4270. l 3 J. J. Salacuse and G. Stell, J. Chem. Phys., 1982, 77, 3714. l 4 R. G. Palmer and J. D. Weeks, J. Chem. Phys., 1973, 58, 4171. l 5 M. Baus and J-P, Hansen, Phys. Rep., 1980, 59, 1. l 6 J. L. Lebowitz and E. H. Lieb, Phys. Rev. Lett., 1969, 22, 631. l 7 E. Waisman, Mol. Phys., 1973, 25, 45. l 8 J. B. Hayter and J. Penfold, Mol. Phys., 1981, 42, 109. l 9 J. B. Hayter and J. Penfold, J. Chem. SOC., Faraday Trans. 1 , 1981, 77, 1851. z o J. B. Hayter and T. Zemb, Chem. Phys. Lett., 1982, 93, 91. 2 1 D. Bendedouch, S-H. Chen and W. C. Koehler, J. Phys. Chem.,.1983, 87, 153. 2 2 R. Triolo, J. B. Hayter, L. J. Magid and J. S. Johnson Jr, J. Chem. Phys., in press. 2 3 J. C. Brown, P. N. Pusey, J. W. Goodwin and R. H. Ottewill, J. Phys. A, 1975, 8, 664. 24 J-P. Hansen and J. B. Hayter, Mol. Phys., 1982, 46, 651. 2 5 J. B. Hayter and M. Zulauf, Colloid Polym. Sci., 1982, 260, 1023. 26 J. B. Hayter and R. Pynn, Phys. Rev. Lett., 1982, 49, 1103. 27 R. Pynn and J. B. Hayter, J. Magnetism and Magnetic Materials, 1983, 31-34, 955. 2 8 J. B. Hayter, Polarised Neutrons, in Neutron Dffraction, ed. H. Dachs (Springer, Berlin, 1978). 29 R. Pynn, J. B. Hayter and S. W. Charles, to be published. 30 S-H. Chen, personal communication. 31 J. B. Hayter, R. Pynn and J-B. Suck, J. Phys. F, 1983, 13, L l . 3 2 J. B. Hayter, G. Jannink, F. Brochard-Wyart and P. G. de Gennes, J. Phys. (Paris), 1980,41, L451.

ISSN:0301-7249    

DOI:10.1039/DC9837600007

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

Faraday Discuss. Chern. Sue., 1983, 76, 19-35 Light Scattering of Colloidal Dispersions in Non-polar Solvents at Finite Concentrations Silica Spheres as Model Particles for Hard-sphere Interactions BY A. VRIJ, J. W. JANSEN, J. K. G. DHONT, C. PATHMAMANOHARAN, M. M. KOPS-WERKHOVEN AND H. M. FIJNAUT Van’t Hoff Laboratory, University of Utrecht, Transitorium 3, Padualaan 8, 3584 CH Utrecht, The Netherlands Received 12th May, 1983 Static and dynamic light-scattering studies are reported for spherical model particles in non-polar solvents. The model particles have a core of silica and a dense surface layer of octadecylalcohol chains which makes them ‘oil soluble’. The refractive-index difference be- tween particles and solvent is very small and so dispersions can be studied by means of light scattering up to high concentrations.Multiple-scattering effects are considered briefly. In cyclohexane the particles show repulsive forces which can be described by a hard-sphere interaction. The small refractive-index differences can also be used to detect differences in optical density in the silica core. The periphery of the core is found to be more dense than the centre. Furthermore, the small natural spread in refractive index of the particles can be used to differentiate between collective-diffusion and self-diffusion processes. Differences in refrac- tive indexes can also be obtained by variations in the particle synthesis. In this way it is possible to study self-diffusion by following the motion of tracer particles. 1. INTRODUCTION Since the last Faraday Discussion’ on Colloid Stability there has been a large increase in interest and activity in the study of concentrated colloidal dispersions in relation to interparticle forces.Such forces strongly influence the static and dynamic structural properties of the system and are found when the concentration is so high that the particles continuously experience each others’ force fields. A great stimulus for studying these systems has been the introduction of new experimental techniques like quasi-elastic light scattering and small-angle neutron scattering. In scattering studies multiple-scattering effects must be absent or corrected for in order to interpret the results in terms of particle interactions. This is a severe handi- cap of the light-scattering technique, which is therefore restricted in its use to very low particle concentrations (e.g.latex in dilute electrolyte) or to very small particle sizes (e.g. microemulsions or micelles). In our laboratory we have therefore explored the possibility of circumventing this problem by using model systems with particles having a refractive index very near to that of the solvent so that the scattering intensities become relatively low. In this paper we will report on the work with silica particles and discuss the merits of these (model) co!loids in light-scattering studies. The interactions appeared to be of the hard-sphere type.20 LIGHT SCATTERING OF COLLOIDAL DISPERSIONS 2. ORGANOPHILIC SILICA PARTICLES 2.1. SYNTHESIS OF PARTICLES Silica (SiO,) was chosen as the material for the colloidal particles because of its convenient refractive index [n, = 1.45, close to that of many (organic) solvents] and because a synthesis is available (the hydrolysation and polymerization of tetraethoxy- silane) which leads to spherical particles which are nearly monodisperse in size.2 Furthermore, a method is known by which silica particles can be made organophilic, i.e. soluble in organic solvents, by terminal bonding of long-chain alcohols to the particle surface.According to I l e ~ - , ~ who has studied this lyophilization thoroughly, a surface silanol group reacts chemically with the OH group of an alcohol by split- ting off a water molecule. This results in particles which are easily dispersible in non- polar solvents. The density of the surface layer is high and compatible with the picture in which the spherical particle core is covered with nearly close-packed, parallel alcohol chains sticking out into the solvent.In our laboratory Van Helden et al.4 combined the two synthetic procedures to obtain a novel variety of lyophilized silica particles. In the lyophilization the ethyl alcohol solvent medium, in which the SiO, cores are synthesized, gradually becomes a solvent medium of (molten) stearyl alcohol. We surmise that in this way a gradual change in stabilization mechanism will take place from charge-stabilized particles in the ethyl alcohol into sterically stabilized particles in the stearyl alcohol melt, with- out crossing an intermediate solvent medium in which the particles are unstable and become (permanently) aggregated.We surmise this because some attempts to obtain unaggregated particles by starting from aqueous SiO, dispersions dialysed against ethyl alcohol were unsuccessful. The synthesized particles have an overall mass density of Pd x 1.6-1.8 g cmP3. The density of the core was found to be Pd x 1.9-2.2 g ~ m - ~ , which is low com- pared with the value for amorphous silica, which is 2.2 g cm-3. This implies that the core may still contain material other than SiO,, e.g. water, alcohol or Si-0-C2H5, trapped during the synthesis. This will not lead to difficulties as long as these molecules remain permanently inside the core. In situations where the refractive- index difference between particle and solvent, An, becomes very small one may expect that leaching could cause trouble in the light-scattering experiments.The particles are reasonably monodisperse with a relative standard deviation of 30% for a radius of 10 nm to 5% for a radius of 80 nm. Recently we found that still larger particles with, say, a radius of 150 nm can be synthesized which, on standing in cyclohexane, give beautiful, transparent colloidal crystals. The same has been found with PMMA particles6 in toluene. Our further studies in this paper were all performed on silica particles lyophilized with stearyl chains. 2.2 OPTICAL PROPERTIES AND LIGHT SCATTERING OF PARTICLES Silica particles in dilute solution are characterized by measuring the scattered light intensity as a function of the refractive-index difference between particle and solvent brought about by variation of temperature or solvent composition (in the case of a solvent mixture). This gives information on the spatial distribution of scattering power inside the particles.The scattering intensity (in Rayleigh units) can be written as R ( K ) = P O W ) (2.1)A. VRIJ et al. 21 where p is the number of particles per unit volume (number density) and a(K) is the scattering cross-section of a particle. In the Rayleigh-Gans-Debye limit (An/no << l), a(K) is equal to (2.2) x* = 27t=nt3<4 (2.3) K = (4m/Ao)sinO/2 (2.4) a(K) = X*(n, - no)2 V2P(K)n where np, n and no are the refractive indices of particle, solution and solvent, V is the particle volume and P(K) is the particle scattering function, describing interference inside the particle.For small K P(K) z 1 - J-R,2K2 = exp(- +R,2K2). (2-5) For an optically homogeneous particle R,, the radius of gyration, is dependent only on the size and shape of the particle, and this situation prevails when np - no is not too small. In special cases, however, when np - no becomes very small, i.e. comparable to the small local variations in refractive index due to small local vari- ations in silica density inside the particle, the value of R, will not only depend on size and shape but also on the spatial distribution of refractive index inside the particle. This situation is found in the so-called contrast variation technique7 in X-ray and neutron scattering8 and was reconsidered by Van Helden et al.9 for the case of light scattering.The intensity can now be written, to order K2, as and The first terms in eqn (2.6) and (2.7) are comparable to the previous situation, i.e. Rgo is the radius of gyration of the homogeneous particle (in the limit where fi, - no is large) and Q = (Tip - no)V, with fi, the average refractive index of the particle and I/ the particle volume which is not accessible to the solvent medium. The second and third terms in eqn (2.6) and (2.7) are new. Their presence can be explained as fol- lows. The average scattering amplitude inside the particle can be written as s Y where Aqh and Aqy represent volume elements with a positive and negative, respec- tively, deviation of scattering amplitude from the average, with the condition When the position of the optical centre (2) of Vis rz and those of the sets Aqa and Aqy are rB and uc, respectively, then it can be shown that 5 and Ty are the ‘optical moments’ of the sets p and y with respect to 2 and that p is the ‘optical dipole moment’ of the particle, given by P = (CA4a)(rE3 - Vc).B For elements Aqa, Aqr spherically symmetric distributed around 2, p = 0 and22 LIGHT SCATTERING OF COLLOIDAL DISPERSIONS the situation simplifies considerably. One may write for a spherical particle of radius a and refraction index distribution n,(r) 4nr2[np(r) - ii,]dr. As in eqn (2.8), the last term is zero because it implies the definition of ii,. Furthermore, one has R2g z= Ri0 + 7 E (2.10) n p - no with Rgo = a(3/5)*. The quantity E is equivalent to (G + T,) in eqn (2.7), and is simply defined by E = I:4zr4[np(r) - ii,]dr -na3.1:: (2.1 1) SOME RESULTS Light-scattering intensities were measured O for stearyl-silica particles with a hydrodynamic radius of 71 & 2 nm, in dilute solutions (c = 5.6 g dmV3) at temper- atures varying between 10 and 35 "C. In this interval the intensity varied by a factor of 20,iip-- no by a factor 4.5 and R2g by a factor of 1.5. The matching temperature (where np - no sz 0) was found to be ca. 1 "C (A = 436 nm). From this a value fi, = 1.4465 (1, = 436 nm, 25 "C) was obtained. From plots of R2g against (2, - n0)-l, Rgo was obtained and from this a particle radius of 68 & 3 nm was calculated, in good agreement with the hydrodynamic value. The value of E (= 26 nm2), obtained from the slope in fig. 1, is positive, which implies that the refractive index at the periphery of the particle is larger than at the centre.This value of E is too large to be accounted for by the stearyl layer and must be due to an inhomogeneous silica core. An indication of the refractive-index variation can be obtained by taking a model function for n(r), e.g. a step function. From E and iip one may then find np = 1.40 for r < 43 nm and np = 1.46 for 43 < r/nm < 71. This difference in refractive index is substantial, which implies that the inner partial core must contain material other than Si02 with a lower refractive index. 1 1 t 0 Fig. 1. Square of the 1 1 1 I 50 100 150 200 (rip - no)-' (optical) radius of gyration, R,, as a function of the reciprocal of the contrast. 0, ,lo = 436 nm; a, Lo = 546 nm.POLYDISPERSITY EFFECTS 23 For a polydisperse system one has9 where j runs over different particles, e.g.particles with a variation in radius aj and refraction index np,{r), with p = Z pi. This leads to complex (mixed) averages such as Q , (Q2Ri0) etc., as follows from eqn (2.6), which cannot be disentangled without additional information about the particles. For instance, depends on variations - in np and V , quantities which may be coupled. In any case, will be positive when Q = 0. This implies that the contrast variation method must break down near Q = 0. It is difficult to investigate this region experimentally because of the low signal in dilute solutions, whereas at higher concentrations concentration effects and other complexities arise.-~ - 3. MULTIPLE-SCATTERING EFFECT As mentioned above, the experimentally determined scattered-light intensities must be free from multiple scattering effects in order to obtain the desired single- particle properties. Although in our silica systems the scattering power and therefore the multiple-scattering effect is small because of the selection of a suitable solvent refractive index, this is not always sufficient or feasible, in particular for larger particles. In that case there is no alternative than to correct for such an effect. This has been investigated theoretically and experimentally by Dhont,' * who has pro- posed, in the realm of the Rayleigh-Gans-Debye approximation, a theory that cor- rects for double scattering. The theory, adequate for turbidities (7) up to 0.5 cm- l , is rather complex so we refer the interested reader to the original article. Here, we will report some of the results.In fig. 2 the light-scattering intensity of a silica dispersion with particles of 2.5 1 0 0.5 1 1.5 k?'/lOl5 m-2 Fig. 2. Guinier plots for silica in cyclohexane. (- - 0 - -) Experimental plot at 39 "C; (- - -) double-scattering corrected curve; (----) double-scattering and attenuation corrected curve; ( 0 0 ) 15 "C line extrapolated to 39 "C.24 LIGHT SCATTERING OF COLLOIDAL DISPERSIONS 53 & 1 nm radius in cyclohexane (0.072 g cm-3) is shown at a temperature of 39 "C (--0--). Note that the plot is curved and that the scattering intensity level is fairly high: the Rayleigh ratio RW ( K = 0) = 0.067 cm- and the turbidity z = 0.67 cm- l .Both are indicators for double (or multiple) scattering. When the calculated double scattering is subtracted, a straight line is found (- - - -). The intensity obtained after correction for double scattering and attenu- ation [factor exp(-zL), L = diameter of cell] is also given (-). We can check the calculation for our system in the following way. Measuring at a temperature be- tween 9 and 15 "C, where the scattering intensity is much lower and where the plot is a straight line, no multiple-scattering effects are present. Accounting for the increase in ti, - no between 15 and 39 "C, the single scattering at 39 "C can be calculated from experiments at 15 "C. The result is shown in the same figure (- 0 ) . It is indeed in good agreement with the corrected plot (-) at 39 "C.4. LIGHT SCATTERING IN NON-DILUTE DISPERSIONS 4.1. LIGHT-SCATTERING EQUATIONS In non-dilute systems interparticle interference effects of the scattered light must be taken into account through multiplication by a so-called structure factor, S(K), (4.1) The factors in front of S(K) give the scattering of single particles as formulated previously in eqn (2.1) and (2.2) but now written in terms of the mass concentration of the particles, c, and the particle molar mass, M , with R(K) = LXcM P(K) S(K). Z = 2n2n2(dn/dc)2(i:N)-1 (4.2) where dn/dc is the refractive-index increment, which for small An = fi, - no is equal to An/cpd, and N is Avogadro's number. The structure factor is given by S ( K ) = 1 + pK(K) with K(K) = 4n [om r'h(.)(l>dr sin K r where h(r) = g(r) - 1 and g(r) is the radial distribution function which expresses the chance of finding the centre of a particle at a distance Y from the centre of a given particle.In particular as K + 0, one has aP an S(K -+ 0 ) = k T - where ll is the osmotic pressure. For low concentrations (4.5) and S(K -+ 0) = (1 + 2B2p + 3B,p2 + . . . ) - l (4.7) 4.2. HARD-SPHERE INTERACTIONS A useful model for steeply repulsive interactions among particles is the hard- sphere (pair) potential U(r), i.e.A. VRIJ et al. 25 for r < d for r 2 d U(r) = where d is the so-called hard-sphere diameter, giving the range of this potential. Even for this simple potential no exact theory is available to calculate g(r) and thus S(K). However, good approximate equations are known, as give_n by the so-called Percus-Yevick (PY) theory [see e.g.ref. (12) and (13)]. Then h ( K ) of eqn (4.3) follows from 1 S(K) = 1 + pL(K) = 1-PW9 where c ( K ) = 471 /:r2C(r)(.>d. sin K r (4.9) (4.10) and Cpy(r) = a + P(r/d) + ~ ( r / d ) ~ for 0 < r < d (4.11) C P Y ( d = 0 The coefficients a, P and y are simple functions of cp, the volume fraction of hard spheres 1 6 for r > d. (4.12) = -71d~p a = - ( I + 2 ~ p ) ~ ( 1 - ~ p ) - ~ ; p = 69(1 + i ~ ) ~ ( l - ( P ) - ~ ; Y = -+~p(l + 2 ~ ) ~ ( 1 - ~ p ) - " . The resulting expression for S(K) is still complex and will not be given here. For small K , however, one may apply a series expansion with and S(K) = S(K = O)[l + X(9)K2d2 + . . .] 4 11 1 5 20 5 -cp - -cp2 + -(P3 (4.13) (4.14) (4.15) The PY theory gives exact results for small concentrations, up to and including the third virial coefficient, (4.16) and is still good for, say, cp x 0.4.The PY expression for n for was semi-empirically improved to the so-called Carnahan-Starling equation, (4.17) For particles in a solvent medium, the hard-sphere (pair) potential is a model potential for the potential of mean force between the particles (i.e. the isothermal26 LIGHT SCATTERING OF COLLOIDAL DISPERSIONS work required to bring the particles, in equilibrium with the solvent environment, from r = GO to r = r). Experiments must decide to what extent such HS model is appropriate. The factor relating cp and the weight concentration c, i.e. q = qplc, is then a parameter which must be fitted.From q and M, the hard-sphere diameter can be calculated and compared, e.g. with the size of the particle. See also section 4.6. 4.3. NON-SPHERICITY We know that real colloidal particles always show deviations from the spherical shape and a distribution in size. We wish to know the expected influence of these effects . Expressions are known for the (osmotic) pressure of hard ‘convex’ particles as obtained by the so-called scaled particle theory (SPT).14 We have found that the derivative of I-I can be written as follows [l + (3k - l)qI2 (1 - 9)” (4.18) where cp = Vp and k = RS/3 V, with the radius averaged over all orientations, S the surface area and V the volume of the particles. The approximate eqn (4.18) applies to ellipsoids, tetrahedra, cubes and cylinders.l4 For spheres, k = 1 and eqn (4.18) becomes identical with the Percus-Yevick result, eqn (4.14). For moderate devi- ations from the spherical shape, e.g. a prolate ellipsoid with an axis ratio of 1.5, k equals 1.06, which will be hardly noticeable in practice, we presume, and which would manifest itself in a larger hard-sphere diameter. We concluded that 17 is not very sensitive for deviations from the spherical shape. 4.4. POLYDISPERSITY IN THE HARD-SPHERE DIAMETER This effect can be treated 5 9 l6 well in the Percus-Yevick approximation, because a closed (but complicated) solution is known for mixtures of hard spheres, not only for ll and for S(K = 0), but also for ‘S(K) at all K ’ . Some calculations l 7 of the Rayleigh ratio for a log-normal HS-diameter distribution are shown in fig.3. All 0.1 5 R 0‘10 0.05 0 0 0.1 0.2 0.3 0.4 0.5 cp Fig. 3. Intensity of scattered light of a polydisperse system of hard spheres for a log-normal distribution of hard-sphere diameters with relative standard deviations CT = 0, 0.1 and 0.3 (-). Also shown are plots for a monodisperse system but with adjusted prefactors (1.144 for CT = 0.1 and 3.20 for CT = 0.3, see text) to match the polydisperse curve at small cp.A. VRIJ et al. 27 curves have a maximum arising from the fact that the initially rising scattering, due to increasing particle numbers, is overcome by the decreasing compressibility of the system, leading to the peculiar situation that a concentrated dispersion ‘looks’ diluted. The lowest curve (a = 0) is for a monodisperse system.For a > 0 the scattering intensity becomes larger because the contributions of the particles with above-average size is relatively larger. When one scales on the initial slope (see dashed line), this shows that at high cp polydispersity gives an additional increase in scattering intensity. In practice, the HS scaling factor q that relates cp to c is usually not known a priori, so a best-fitting q must be selected. To obtain a first estimate of q we plot ln[c/Re=,] against c. For pure HS interactions such a plot appears to be nearly linear over a large concentration range, as is shown in a plot of In S - l ( K = 0) against cp in fig. 4 (a = 0). It is also apparent that a linear plot is found for 0 = 0.3 but with a 4 3 s II g - 2 ‘ 2 1 0 0.2 0.4 0.6 cp Fig.4. Logarithmic plot of the reciprocal structure factor S - ’ ( K = 0) as a function of cp. Points o are from the Percus-Yevick theory for cr = 0 and cr = 0.3. The dashed curve (- - - -) is an ‘effective’ hard-sphere interaction (see text). smaller slope. This implies that a good fit will also be obtained when the mono- disperse equation is used but with a smaller value of q. For the case CT = 0.3, for example, one calculates that cp should be reduced by a factor of 0.79, which implies that the calculated HS diameter will become J- 0 79 = 0.92 times smaller. Our conclusion is that it will be difficult to detect CT smaller than, say, 30% from the concentration dependence of S(K = 0) alone. On the other hand, one may28 LIGHT SCATTERING OF COLLOIDAL DISPERSIONS 300 expect that a much more sensitive way to detect polydispersity effects is to study S(K) at larger K where the oscillations of S(K)I6 are damped out.0 r 4.5. SOME EXPERIMENTAL RESULTS GUINIER SLOPE AS A FUNCTION OF CONCENTRATION For small wavenumber K, the scattered intensity is linearly dependent on K 2 , and the slope of such a linear (Guinier) plot gives the radius of gyration of the particle [see eqn (2.5)]. Van Helden and Vrij l 3 found that this slope depends on the particle concentration. Another example is shown in fig. 5, where an ‘apparent’ Rl is plotted against c. The origin of this concentration dependence stems from inter- particle interferences. With the equations for S(K) [eqn (4.13) and 4.16)] one finds upon Taylor expansion of K and cp (4.19) where cp = qc. From the plot in fig.5 one finds R, = 25.6 nm, from which a radius of aRg = 25.6J5/3 = 33.1 nm is found and, with q = 0.7 cm3g-l, a hard-sphere radius 4 4 2 = 24.7 nm, which is a very reasonable value, compared with a value of 26.5 for the hydrodynamic radius. 700 500 0 0.05 0.1 Fig. 5. Apparent radius of gyration squared as a function of silica concentration. S(K = 0) AS A FUNCTION OF CONCENTRATION Results with silica particles, up to high concentrations, were obtained first by van Helden and Vrij.I3 An example is shown in fig. 6 for cp = qc up to ca. 0.3. The experimental results can be fitted quite well by a hard-sphere potential with q = 0.73 cm3g-l. A more recent result l 8 with c = 0-15% and a value of q of 0.78 cm3g-l is shown in fig.7. Going to still higher concentrations, cp 2 0.4, van Helden and Vrij found anomalous behaviour at small scattering angles where scattering intensitiesA. VRIJ et al. 29 15 10 5 I I I 1 0 01 0.2 0.3 0.4 cp Fig. 6. Comparison of experiment and hard-sphere theory. (kT)- X I / a p = S - ( K = 0) as a function of volume fraction with q = 0.73 cm3g-l. 0, A = 546 nm; e, A = 436 nm. Hydro- dynamic radius x 22 nm. 3 1 I I 0 0.0 5 0.10 0.15 C/g cm - Fig. 7. Reciprocal structure factor as K -+ 0 as a function of weight concentration, obtained from light-scattering intensity measurements. The line for hard-sphere theory with q = 0.78 cm3g-l. Hydrodynamic radius = 23 nm.30 LIGHT SCATTERING OF COLLOIDAL DISPERSIONS - 1 . 5 1 h -3.0 0 0.5 1 .o 1.5 K2/101' cm- Fig.8. Scattering intensity of silica particles as a function of wavevector at high to very high concentrations. were larger than expected (see fig. 8). These deviations were attributed to a small number of large clusters of particles because at 8 = 150" no such deviations were found. In the explanation of van Helden and Vrij the large clusters of particles were thought to be reversible associates, because upon dilution the increase in scattering intensity at small angles disappears again. At the moment we surmise that the clus- ters may be permanent. This view is based on calculations for mixtures of hard spheres of two sizes [see, e.g., ref. (1311 that differ by a factor of ten. It appears that at high concentrations the scattering from the large number of small particles is suppressed with respect to the scattering from the small number of large particles.4.6 EFFECTIVE HARD-SPHERE INTERACTIONS The hard-sphere pair potential as used in the above discussion is a model for a steep potential of mean force. In our colloidal systems this potential orginates in the solvated layers of alkane chains which, when coming into close proximity, feel a steep repulsion over a short distance. This implies that when the particles are dis- solved in a liquid which is a 'poor' solvent towards the alkane chains, these chains will prefer contact with each other over contacts with solvent molecules, thus leading to an attractive potential of mean force. Indeed, it is found that our particles show this tendency in poor solvents. In benzene, a phase separation takes place when the temperature drops below 40 "C.A precise model for the potential of mean force is not available at present because it requires detailed knowledge of pair distribution functions in mixtures. The sim- plest, but not so realistic, model would be: a mixture of large hard spheres (2) in a 'solvent' of small hard spheres (1). This model system can be solved in the Percus-Yevick approximation. The pressure difference (= osmotic pressure) of the mixture in osmotic equilibrium with the small hard spheres (1) is calculated. In particular it can be shown that for d,/d, -c the result (within 1%) can be represented by the equationA. VRIJ et al. with 1 - 40; 1 + 2q; m = 31 (4.20) (4.2 1) where q; is the volume fraction of the small hard spheres in pure ‘solvent’ in osmotic equilibrium with the two-spheres mixture.The volume fraction of small hard spheres in the mixture is then q1 z qT( 1 - q2) for d,/d2 << 1. Note the simple form of this equation [which is, accidently, also equal to eqn (4.17)], from which it is immediately clear that the presence of a ‘solvent’ of small hard spheres has a mitigating effect on dITldp2. As an example, the case with q; = 0.4 (thus m = 1/3) is shown in fig. 4 (- - -). This can be compared with the case of pure hard-sphere particles [without ‘solvent’ (l)], the case = 0 in fig. 4. We may describe the situation approximately by saying that the diameter of the large spheres is effectively smaller (ca. 7.5%) and, furthermore, that the increase of S - with 40 is steeper than for a pure hard sphere.5. (HYDR0)DYNAMIC INTERACTIONS Our silica dispersions are also well suited for investigations of hydrodynamic interactions among particles of the hard-sphere type by means of light scattering. Because the refractive index of the particles, and therefore their scattering power, can be varied, it is even possible to obtain both the collective-diffusion coefficient and the self-diffusion coefficient in some cases. For instance, the mean refractive index of the particles can be varied by changing the temperature. This is shown schematically in fig. 9, where the refractive indexes of particles and solvent are plotted as functions of temperature. The line for the refractive index of the particles is accompanied by two broken lines indicating that not all particles have exactly the same refractive index but that there is a small spread around the mean value (see also section 2.1).This makes it possible to measure collective and self-diffusion, using the phenomenon of incoher- ent scattering. \ T Fig. 9. Schematic plots of refractive indexes of solvent and silica particles as functions of temperature.32 LIGHT SCATTERING OF COLLOIDAL DISPERSIONS 5.1. COHERENT AND INCOHERENT SCATTERING With dynamic light scattering it is possible to measure the so-called dynamic structure factor S "( K,z) where Q is the particle scattering amplitude as defined in eqn (2.8), rj(t) is the position of particlej at time t, (. . .) is the thermal average and N is the total number of particles. The overbar indicates an average over the distribution of scattering species: When all particles have the same size, but are spread over Qi, it is possible to average separately within the brackets (.. .) over species. This is sometimes called optical polydispersity. Clearly QiQj = Q2 when i = j and QiQj = Q2 when i # j so that Q iQ j = e' + (Q' - S*)Sij where dijis the Kronecker delta (Sij = 1 if i = j and 0 when i # j) and - - - - Performing this species average in eqn (5.1) leads to a sum of two expressions, one of which describes the motion of single particles. When the discussion is further restricted to small K , the following expression is ~ b t a i n e d : ' ~ ? ~ ~ Q 2 Q2 Q2 S"(K,z) = = s ' ( K ) exp( - D,K2z) + Q2 1' ' exp( - DsK2z). (5.2) The first term describes collective diffusion where D, is the collective-diffusion coeffi- cient and S'(K) is the static structure factor as given by eqn (4.3).The second term describes self-diffusion with a diffusion coefficient Ds. For z -+ 0, eqn (5.2) leads to a measured static structure factor Q' S"(K,O) = S'(K) + Q' - Q2 (5.3) In collective diffusion all particles move as a collection from spatial regions with high osmotic values to spatial regions with low osmotic values. When, in such a process, the refractive index does not change (i.e. when = 0) then the process cannot be detected with light scattering, as follows from eqn (5.2) and (5.3). On the other hand, in self-diffusion, exchange of particles (at constant osmotic value) takes place and this can only be detected when some of the particles have a 'contrasting' refractive index, i.e.- s2 > 0. From eqn (5.3) it appears that when e;! = e2, then Sm(K) = S'(K) and one has what is sometimes called 'coherent scattering'. On the other hand, when = 0, Y ( K ) = 1 and one has 'incoherent scattering'. This nomenclature stems from neutron scattering, where similar phenomena are known. This has been recognized by Weissman.lg The assumption that all particles have the same size is of course an idealization. This is discussed further elsewhere.'OA. VRIJ et al. 33 5.2. SOME EXPERIMENTAL RESULTS A spread in Qi may be obtained in two ways. First, by bringing the system near the matchpoint as shown in fig. 9, where use is made of the natural spread in the refractive index. Secondly, by synthesizing two particle preparations with a slightly different refractive index, which are then mixed in a so-called tracer experiment.Both routes were followed. TEMPERATURE VARIATION Particles with a hydrodynamic radius of 71 f 2 nm (a = 9%) and refractive index np = 1.434 (1 = 546 nm, 25 "C) were used in cyclohexane (no = 1.425, A = 546 nm, 25 "C). The difference, np - no, decreases with decreasing temper- ature, see fig. 9. The field auto-correlation function g(')(K,z) was determined by photon correlation experiments. This function is proportional to S"(K,z). 4. . 4. 't T 0 0.1 0.2 0.3 C/g ~ r n - ~ Fig. 10. Mean diffusion coefficient of silica particles as a function of concentration and at different temperatures: *, 34.4; A, 25 and 0 , 12 "C. The matching temperatures is ca.1 "C. For high particle concentrations, c > 0.3 g ~ r n - ~ , these functions could be fitted with a two-exponential function [see eqn (5.2)], which was no longer feasible for c < 0.3 g cm-3. In the latter case the data were fitted to a single exponential, with an average D = D,. Here we will only consider the lower concentration range. Results are shown in fig. 10, where D, as a function of c is shown at three temperatures. For low concentrations one has D,/Do = 1 + O!DC = 1 -k k,q (5.4) with q = qc and aD = qkD. One notes in fig. 10 that Do decreases with decreasing temperature. This can be fully accounted for by the increased viscosity of the sol- vent. Furthermore, the slope, and thus kD, changes from slightly positive to negative.This is further shown in fig. 11, where kD is plotted as a function of contrast and for34 LIGHT SCATTERING OF COLLOIDAL DISPERSIONS which several particle preparations were used. The dashed line is a least-squares fit. The limiting values of kD appear to be: k D = -2.7 f 0.3 for low An and k D = 1.3 & 0.2 for An = 0.83. TRACER EXPERIMENTS Two different silica cores were synthesized, one in ethyl and one in propyl alcohol. The particles were then lyophilized with stearyl alcohol. The hydrodynamic radii were a: = 33 k 1 nm and ui = 38 k 1 nm with n, = 1.455 and 321, = 1.430. Solutions were prepared in cycloheptane in which, at 25 “C, the scattering of particles a was very weak (60 times smaller than that of particles b). So particles b could be used as tracer particles in a dispersion of increasing concentration of ‘invisible’ particles a.The value of kD thus determined was equal to -2.7 & 0.3, with q = 0.67 cm3 g - l , determined from sedimentation. This is the point shown in fig. 11 on the kD axis (thus at An = 0). From theory it is known that kD for hard spheres is equal to + 1.45 for collective diffusion.23 -4, 0.010 0.020 0.030 An Fig. 11. Values for the constant kD from eqn (5.4) as a function of optical contrast An = ii, - no from different silica dispersions: 0, from temperature-variation experiments; 0, from tracer experiments. For self-diffusion a theoretical value for kD is known23 for short times, where it is - 1.83. For long times, however, no theoretical value has been published, as far as we know.B a t ~ h e l o r ~ ~ has given the value -2.68, which is indeed in accordance with our experimental value of -2.7. The long-time limit is more difficult to analyse than the short-time limit because in the former case the ‘relaxation of g(r)’ must be taken into account. This situation is analogous to the well known ‘relaxation effect’ in electrolyte theory, as studied by Onsager in the thirties. HIGHER CONCENTRATIONS For silica concentrations ~ 0 . 3 g cm-3 the two separate contributions to Sm(K,~), as given in eqn (5.2), may be disentangled under favourable circum- stances.21 We refer the interested reader to the original paper.A. VRIJ et al. 35 In the near future we hope to study the high-concentration case in more detail for mixtures of particles of different sizes. Theory predicts20 that for this case a separ- ation of S”(K,z) in two modes is also possible, at least when the concentrations are large enough. We thank Dr. C . G. de Kruif for his critical reading of the manuscript and Renske Kuipers for typing the manuscript. This work is part of the research programme of the Foundation for Fundamental Research of Matter (F.O.M.) with financial support from the Netherlands Organization for Pure Research (Z. W.O.). Faraday Discuss. Chem. Soc., 1978, 65. W. Stober, A. Fink and E. Bohn, J. Colloid Interfuce Sci., 1968, 26, 62. R. K. Iler, The Chemistry of Silica (Wiley, New York, 1979). A. K. van Helden, J. W. Jansen and A. Vrij, J . Colloid Interface Sci., 1981, 81, 354. J. W. Jansen, unpublished results. E. A. Nieuwenhuis and A. Vrij, J . Colloid Interfuce Sci., 1979, 72, 321. ’ H. B. Stuhrman, J. Appl. Crystallogr., 1974, 7, 173. * C. Taupin, I. P. Cotton and R. Ober, J . Appl. Crystallogr., 1978, 11, 613. A. K. van Helden and A. Vrij, J . Colloid Interface Sci., 1980, 76, 418. l o C. Pathmamanoharan and M. M. Kops-Werkhoven, Chem. Phys. Lett., 1982, 93, 396. l 1 J. K. G. Dhont, Physica (Utrecht), in press. l 2 N. W. Ashcroft and J. Lekner, Phys. Rev., 1966, 45, 33. l 3 A. K. van Helden and A. Vrij, J. Colloid Interface Sci., 1980, 78, 312. l4 R. M. Gibbons, Mol. Phys., 1969, 17, 81. l 5 A. Vrij, J . Chem. Phys., 1979, 71, 3267. l 6 P. van Beurten and A. Vrij, J . Chem. Phys., 1981, 74, 2744. l 7 A. Vrij, J . Colloid Interface Sci., 1982, 90, 110. l 8 M. M. Kops-Werkhoven and H. M. Fijnaut, J. Chem. Phys., 1981, 74, 1618. l9 M. B. Weissman, J . Chem. Phys., 1980, 72, 231. 2o P. N. Pusey, H. M. Fijnaut and A. Vrij, J. Chem. Phys., 1982, 77, 4270. 21 M. M. Kops-Werkhoven and H. M. Fijnaut, J. Chem. Phys., 1982, 77, 2242. 22 M. M. Kops-Werkhoven, C . Pathmamanoharan, A. Vrij and H. M. Fijnaut, J . Chem. Phys., 1982, 2 3 G. K. Batchelor, J. Fluid Mech., 1972, 52, 245, 1976, 74, 1. 24 G. K. Batchelor, personal communication. 77, 5913.

ISSN:0301-7249    

DOI:10.1039/DC9837600019

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1983, 76, 37-52 Properties of Concentrated Polystyrene Latex Dispersions BY DERYCK J. CEBULA,~ JAMES W. GOODWIN, G. CHARLES JEFFREY, RONALD H. OTTEWILL, ANTHONY PARENTICH f AND RACHEL A. RICHARDSON School of Chemistry, University of Bristol, Cantock’s Close, Bristol BS8 1TS Received 24th May, 1983 In order to understand the properties of concentrated dispersions it is necessary to obtain the spatial and temporal locations of the particles in the system as a function of both the volume fraction and the electrolyte concentration and to correlate this with theoretical models. A useful experimental technique for examination of the static structure factor is provided by small-angle neutron scattering, using cold neutrons with a wavelength of ca. 10 A, since the scattering vectors then available are appropriate for systems containing small particles over a wide range of volume fractions.Polystyrene latices consisting of spherical particles of radius 157 A, with a narrow distribution of particle sizes, provide an excellent system for such studies. Structures are reported for these latices covering a range of volume fractions from 0.01 to 0.14 and electrolyte concentrations from very low values, ion- exchanged systems, up to 5 x mol dmP3 sodium chloride. The structure factors ob- tained are compared with various theoretical models for electrostatically interacting systems. An important factor in determining the properties of stable aqueous colloidal dispersions is the range over which the electrostatic field of one particle can be felt by another.For example, at very low electrolyte concentrations the range of particle-particle electrostatic interactions can be of the order of one micrometre, whereas in high salt concentrations it is only of the order of ten gngstroms or so. Although the form of the pair potential for the electrostatic interaction between colloidal particles has been examined in some detail,’ - 4 there are uncertainties as to how this can be used to predict the bulk properties of colloidal dispersions in which there is considerable interaction between the particle^.^ Consequently, detailed struc- tural information is required on this type of system to enable progress to be made in this field and this requires careful selection of both the system and the investigating technique.As the system we have chosen polystyrene latices since these can be produced as dispersions of spherical particles in which the spread of sizes is very narrow. Furthermore, the particles are rigid spheres which maintain a constant size and where the size can be well defined by a number of different techniques. In addition the basic surface charge density of the surface groups can be obtained by conduc- tometric titrations and the zeta-potential of the particles can be obtained in very dilute dispersions by electrophoresis. In earlier work7-12 it has been shown that the light-scattering technique could be used to obtain both structural and dynamic information on latices. However, its t Present address: Rutherford Appleton Laboratory, S.E.R.C., Chilton, Didcot, Oxfordshire OX1 1 $ Present address: The Western Australian Institute of Technology, South Bentley, 6 102 Australia.OQX.38 CONCENTRATED LATEX DISPERSIONS use is limited to dilute dispersions because multiple scattering becomes pronounced at concentrations in excess of ca. 0.01%. Moreover, the range of scattering vectors which can be obtained with light scattering restricts the range of reciprocal space which the technique can probe. A much wider range is available using cold neutrons since the wavelengths available (3-16 A) are much shorter than those available using visible light. In addition, the weak nature of the interaction of neutrons with matter (i.e. scattering by nuclei rather than electrons) allows deep penetration with little attenuation of the radiation.Consequently much more concentrated dispersions can be examined with a neutron beam than with light. In this communication we report a small-angle neutron-scattering examination of polystyrene latices containing particles of radius 157 A, over a wide range of volume fractions and electrolyte concentrations. EXPERIMENTAL MATERIALS The distilled water used throughout the investigation was doubly distilled from an all- Pyrex apparatus. The sodium chloride used was B.D.H. AnalaR material. The anion- and cation-exchange resins were supplied by Bio-rad Ltd in a specially pre- cleaned form suitable for analytical use. The resins were further cleaned by the method des- cribed by van den Hul and Vanderhoffs in order to remove any soluble polyelectrolyte species.Particular care was taken to ensure that the final wash water from the resins had low ultraviolet absorption at a wavelength of 280 nm. The resins were stored under double- distilled water which was changed at frequent intervals. The polystyrene latex used for the experiments was SLRR1, whose preparation has been previously described. The particle size of this latex was determined by transmission electron microscopy and also by time-average light scattering. The former technique gave a number- average particle radius of 156 %, with a standard deviation on the average value of 28 & and light scattering gave a weight-average radius of 161 5 A. The agreement between the number-average and weight-average results indicated that the latex was monodisperse and free of aggregated material.12 SURFACE CHARGE DENSITY OF THE POLYSTYRENE LATEX PARTICLES Following dialysis and ion-exchange treatment, samples of the latex were titrated with standard sodium hydroxide solutions.The titration was followed conductometrically. l 2 The surface charge density of the particle surface was found to be 4.2 pC cm-2. SMALL-ANGLE NEUTRON SCATTERING, SANS The SANS measurements were all carried out at the Institut Laue-Langevin (ILL), The intensity of scattering was measured over a range of scattering vectors (momentum Grenoble, France, using the small-angle neutron-scattering apparatus D 1 1. transfers) Q defined by Q = (47r/A) sin (8/2) M (2z/A)(rO/D) where 8 is the total scattering angle, A is the wavelength of the incident neutrons, ro IS the radial distance on the detector measured from the undeviated beam in the plane of the two- dimensional detector and D is the distance between the detector and the sample. An incident wavelength of 10 A was used in most of the measurements and all samples were examined at sampIe-to-detector distances of 2.66, 5.66 and 10.66 m. The smallest setting was utilised in order to obtain a good estimate of the incoherent background for the concen- trated samples by making measurements at high Q values.The procedure used will be de- scribed in detail elsewhere. l4D. J. CEBULA et al. 39 PREPARATION FOR SCATTERING MEASUREMENTS All the neutron-scattering measurements were carried out with the samples in optical- quality quartz cells having a path length of 1 mm.For some of the latex samples a small amount of mixed-bed ion-exchange resin was placed in the cell and then latex of known volume fraction was added. Other samples were prepared in sodium chloride solution. PTFE stoppers were inserted into the cells and then the top of the cell was covered with Parafilm to prevent evaporation. The samples containing ion-exchange resin were allowed to stand for several weeks before examination in order to ensure that exchange was as complete as pos- sible; it was found to be a slow process. The volume fraction of the latex was determined by taking a known weight of latex in a glass container and then drying to constant weight in an oven at 70 “C. THEORY SMALL-ANGLE NEUTRON SCATTERING The coherent scattered intensity at a scattering vector Q, i.e.I(Q), from a beam normally incident to the target is directly related to the target’s differential cross- section with respect to angle, and in the static approximation is given by where ( ) denotes the time-average value and Nt is the number of particles in the scattering volume; multiple scattering has been assumed to be absent. In eqn (2) rp and r4 are the position coordinates of the pth and qth scattering elements and ap and a, are the scattering amplitudes of the particles. For a monodisperse system, with non-interacting spherical particles, as shown by Guinier and Fournet,15 eqn (2) takes the form (3) where A is a constant containing instrumental factors, Np is the number of particles in the dispersion per cm3, Vp is the volume of each particle of radius R, pp is the coherent neutron-scattering length density of the particle, pm is the coherent neutron-scattering length density of the dispersion medium and P(Q) is the particle form factor, as given by I(Q> = A(pp - ~m)’ VP’ NP p(Q) P(Q) = [3(sin QR - QR cos QR)/(QR)3]2.(4) Vp = 4 nR3/3 ( 5 ) q~ = Np Vp. ( 6 ) For a system of monodisperse spheres Vp is given by and the volume fraction of the latex by In practice the latex is not completely monodisperse and this has to be taken into account in addition to the small-wavelength spread in the incident beam and the finite size of the detector elements. This procedure has been described previously. - * In concentrated dispersions, when particle-particle interactions are present, the scattering of the beam by the particles is no longer independent and the spatial correlations between the particles have to be allowed for by the introduction of a structure factor.The intensity of scattering at scattering vector Q is then given by40 CONCENTRATED LATEX DISPERSIONS where S(Q) is given by S ( Q ) = 1 + ~ 71NP j: [ g ( r ) - 11 sin (Qr) r dr Q where g(r) is a distribution function giving the radial distribution of the particles relative to a reference particle. In terms of volume fractions eqn ( 3 ) and (7) can be written in the form and where rp, is chosen as the volume fraction for the particles such that S(Q) = 1 at all the Q values investigated. For this system if the instrumental constants are deter- mined and I(Q) is expressed as an absolute intensity then P(Q) can be obtained directly.Alternatively, since I(Q) is a sensitive function of Q a fitting procedure can be used to obtain P(Q). In all cases qn, Vp and (pp - pm) are well known. Once P(Q) is known then in terms of absolute intensities S(Q) is given by eqn (7a). Alternatively, and conveniently for a system such as polystyrene latex, S(Q) can be written as It is important that the experiments in this case should be carried out at exactly the same conditions of sample container, beam intensity, beam aperture and wavelength and that the incoherent background corrections to the concentrated sample should be carefully determined.14 The results reported in this paper were obtained using this procedure. RESULTS SCATTERING RESULTS For a dilute dispersion of non-interacting monodisperse spherical particles the scattered intensity at a particular Q value is given by eqn (3) or (3a).Since the volume fraction of the latex was accurately known it was possible to obtain the radius of the latex using a fitting procedure. In order to do this it was assumed that the particle size distribution was a zeroth-order log-normal distribution. In ad- dition, the procedure of Chauvin l 7 was followed in order to allow for the small spread of wavelengths in the incident beam and the finite size of the detector ele- ments. Fig. l(a) shows the experimental points obtained with SLRRl at a latex volume fraction of 0.01 1 in a 1.3 x mol dm-3 sodium chloride solution. A good fit to the data was obtained using a particle radius of 157 f 14 A, a standard deviation on the mean of 9.0%.The values for the coherent neutron-scattering length densities of polystyrene and water were taken as pp = 1.41 x 1O1O cm2 and pm = -0.56 x 1Olo cm2, respectively.20 The value of R obtained by SANS, namely 157 A, was in good agreement with the values of 156 and 161 A obtained by electron microscopy and light scattering. In concentrated systems the effect of S(Q) becomes important and this has a substantial effect on the form of the I(Q) against Q data. This is illustrated in fig. l(b), where results are given for the latex at a volume fraction of 0.14 in molD. J. CEBULA et al. 200 41 0 - oo*oooo 0 0 I I -0bOoc 00 00 OO 4001 x " t 800 6ool 400 00 0 0 0 0 0 0 0 0 0.005 0.01 0.015 0.020 Q1A-l Fig.1. (a) Plot of I(Q) against Q for latex at cp = 0.011 in 1.3 x chloride solution. (b) Plot of I@) against Q for latex at 9 = 0.14 in mol dmP3 sodium mol dm-3 sodium chloride solution. dm-3 sodium chloride solution. A pronounced peak in the intensity is now apparent at Q = 0.0137 A - l . The two curves given in fig. l(a) and (6) represent examples of the basic data obtained for non-interacting systems, as represented by eqn (3a), and for interacting systems, as represented by eqn (7a). From results of this type using eqn (9) it was possible to obtain curves of S(Q) against Q. S(Q) AGAINST Q The results obtained under various conditions of volume fraction and electrolyte concentration are shown in fig. 2 and 3. Fig. 2(a) shows the results obtained with ion-exchanged samples at volume fractions of 0.04, 0.08 and 0.13.Fig. 2(b) gives the results obtained at an electrolyte concentration of mol dm-3 for volume frac- tions of 0.01, 0.04 and 0.13. Similar sets of results obtained at sodium chloride concentrations of mol dm-3 and 5 x In fig. 4 a comparison is given of curves of S(Q) against Q obtained at the same volume fraction, 0.04, at different salt concentrations. The decrease of the main peak in S(Q) is clearly discerned as well as an increase in the magnitude of S(Q) at low Q as the salt concentration is increased. rnol dm-3 are given in fig. 3. DISCUSSION The experimental curves of S(Q) against Q obtained indicate clearly that a structure is built up in dispersions of polystyrene latices which is dependent on both42 CONCENTRATED LATEX DISPERSIONS 2.0 1 .o G o G, 2.0 1 .o 0 d I I 0.01 Q1A-l 0.0 2 Fig.2. (a) Plot of S(Q) against Q for an ion-exchanged latex: @, cp = 0.04; A, cp = 0.08; 0, cp = 0.13. (b) Plot of S(Q) against Q for latex in lov4 mol dm-3 sodium chloride: @, cp = 0.01; A, cp = 0.04; 0, cp = 0.13. volume fraction and electrolyte concentration. This is indicated by the increase in magnitude of the peak in S(Q) and the decrease in S(Q) at low values of Q as the volume fraction is increased. The first peak also moves to higher Q values as the volume fraction is increased, the variation of peak position with Q being consistent with the formation of a regular array with the particles in lattice sites in the system. The effect of electrolyte is visible in the sequence of curves given in fig.2 and 3 and in the results presented in fig. 4 which show the effect of different electrolyte con- centrations at a constant volume fraction of 0.04. The softening of the peak with increase in electrolyte concentration and its movement to higher Q values (fig. 4) indicate that the repulsion between the particles becomes of shorter range and that a loosening of the structure occurs. The sharpening of the peaks with increasing volume fraction, which is commen- surate with an increase in the force of interaction between the particles, also suggests an increase in ordering with volume fraction, and consequently it is of interest to determine at which volume fraction of latex the array conforms to a regularD. J. CEBULA et al. 43 2.0 1 .o 6i K O 1.5c 0 l .o ~ * 5 0 0.01 QIA- 0.02 Fig. 3. (a) Plot of S(Q) against Q for latex in A , cp = 0.04; 0, mol dm-3 sodium chloride: e, cp = 0.01; mol dm-3 = 0.13. (b) Plot of S(Q) against Q for latex in 5 x sodium chloride: 0, cp = 0.01; 0 , cp = 0.04; A, cp = 0.08. 'crystalline' lattice. In many of the optical studies of iridescent latices2l, 22 it is es- sentially only the position of this first peak which can be observed, and hence the information which can be obtained from such studies is, of necessity, limited. How- ever, by the use of small-angle neutron scattering information can be obtained with a different range of particle sizes over a much wider range of scattering vectors and the data so obtained are not seriously affected by multiple scattering.The method of approach based on peak position alone is to consider that the primary peak in S(Q) constitutes the first Bragg peak.8i21,22 As shown previously8p22 for a face-centred cubic array the value of 8 at the peak position Omax is given by sin(O,,,/2) = J3 A(3cp/16~)'/~/2R (10) for the first-order reflection which corresponds to a reflection from a (1 11) plane. Hence Qmax is given by (1 1) Qmax = 47~ ~in(omax/2)/A-44 CONCENTRATED LATEX DISPERSIONS I I I 1 0 0 .OO 5 0.010 0.01 5 0.020 Q/A - ‘ Fig. 4. Plot of S(Q) against Q for latex at cp = 0.04: 0, IER; A, mol dm-3 NaC1; U, 5 x mol dm-3 NaCl. The reflection for an f.c.c. lattice in fact is close to that for the (1 10) reflection of a b.c.c. lattice and we will not discuss the distinctions here. From eqn (1 1) it is possible to calculate Qmax as a function of 9 and to compare these values with those obtained experimentally.This is shown in fig. 5(a). It can be seen from this plot that for the ion-exchanged samples all the measured values of Qmax lie on the calculated curve. As the salt concentration is increased, so there are deviations from the calculated curve at the lower volume fractions and correspon- dence with the calculated curve at the higher volume fractions. The point of intersec- tion can be taken as a possible transition point from an ‘ordered-liquid lattice’ to a 0 0.0 5 0.10 0 0.0 5 0.10 0.1 5 cp Fig. 5. (a) Plot of em,, against cp: 0, IER; A, NaCI; 0, 5 x mol dm-3 NaCI; 0 , loT3 mol dmF3 mol dm-3 NaC1. (b) Plot of log (salt concentration) against cp showing transition line.D.J. CEBULA et al. 45 ‘disordered liquid-like’ array. The transition volume fraction is plotted against electrolyte concentration in fig. 5(b). The words used to describe this effect have to be carefully chosen since the transition is a continuous one with the particles gradu- ally moving together until all the particles take up lattice sites. All the particles appear to be separated by the continuous phase and no nucleation effects have been observed as would be expected in the formation of an ordered solid phase. THE DISTRIBUTION FUNCTION g(r) The distribution function g(r) given in eqn (8) is related to the number con- g(r) = N(r)/Np (12) where N(r) is the number concentration of the particles at a distance Y from a reference particle.The quantity g(r) can be obtained directly by Fourier transform- ation of eqn (8), when we obtain centration of the system, Np, by This equation can be used in the form of a summation to obtain g(r) against r from the experimental curves of S(Q) against Q provided that certain precautions are taken. First, to extrapolate the experimental curve to small Q, and hence to Q = 0, which can be achieved if it is recalled that as Q --* 0 1 2 , 2 3 S(Q) = S(0) + b e 2 . . . . (14) Secondly, S(Q) at very small increments of Q must be taken from a smoothed curve. Thirdly, truncation of the summation is preferable at a point where S(Q) = 1. Curves of g(r) against r are shown in fig. 6 for volume fractions of 0.01, 0.04 and 0.13 in mol dmF3 sodium chloride solutions. In fig.6(a), the form of the g(r) against r curve appears to resemble closely that expected for a vapour-like system; the exclusion region is clearly visible and the first peak is rather small. With an increase in volume fraction to 0.04 the initial slope has increased, indicating a streng- thening of the repulsive force between the particles, and clear first and second peaks are visible. At q~ = 0.13 the initial gradient is very steep, suggesting a much harder interaction. The curve provides evidence for a particle arrangement in which there is considerable short-range order, as evidenced by the substantial peaks in g(r) in the range 400-1500 A, and long-range disorder, as suggested by the fact that g(r) is clearly tending to unity as r tends to values > 2000 A. The form of this curve suggests ‘liquid-like’ order of the particles in the system. THE POTENTIAL OF MEAN FORCE From eqn (12) it also follows that g(r) = W Y N P = exP[w)/kTl @(r) = W ) + W ) (15) (16) where CD is the potential of mean force between the particles such that where V(r) is the pair potential for interaction between the particles and $(r) is a perturbation potential which allows for the many-body interactions present in the s ys tem.46 CONCENTRATED LATEX DISPERSIONS 1 .o 0.5 0 1 .5 1 .o n Z 0 . 5 0 1.5 1 .o 0.5 0 I I I 500 1000 1500 2 000 rlA Fig. 6. Plot of g(r) against r for latex in mol dm-3 sodium chloride: (a) cp = 0.01, (b) cp = 0.04, (c) cp = 0.13. Fig. 7 shows the potential of mean force for a volume fraction of 0.13 at a sodium chloride concentration of lop4 mol dmP3.It is clear from this curve that the central particle is interacting with more than one layer of particles and that many particle interactions are involved. On the other hand, the results for a volume frac- tion of 0.01 appear to resemble very closely a simple pair potential. The steeply rising portion implies that very strong repulsion can occur at @/kT values of the order of 1 kT, and hence we conclude that it is the ‘tail’ of the potential-energy curve which strongly influences particle-particle interactions in colloidal dispersions.1 .O 0.5 0 -0.5 h s o 8 0.5 0 -0.5 -1 .o D. J. CEBULA et al. \ (4 I 1 I ( b ) 47 0 500 1000 1500 rlA Fig. 7. Plot of @/kT against r for latex in mol dm-3 sodium chloride: (a) cp = 0.01, (b) q = 0.13.COMPARISON WITH THEORETICAL MODELS In attempting to consider the theoretical aspects of particle-particle interaction one of the simplest models to use is the hard-sphere model, based on the idea of a hard-sphere potential. This has been utilised to explain the physics of simple liquids and also in recent work to explain the properties of microemulsions and colloidal d i s p e r ~ i o n s . ~ * ~ ~ The model, in colloidal terms, suggests that the particle has an effective radius & which is larger than its actual radius R and that when two particles are separated by a distance 2Rff the potential energy of repulsion becomes infinite. The particles then have an effective volume fraction given by q e K = 4zR& Np/3. (17) A straightforward solution in order to obtain S(Q), based on the original theory of Percus and Y e v i ~ k , ~ ~ has been given by Ashcroft and Lekner 26 in the form48 with CONCENTRATED LATEX DISPERSIONS where a, p and y are given by a = (1 + 2 (Ped2/(1 - p = -6 q e m (1 + 0.5 cpeff)2/(l - ( ~ e f f ) ~ Y = 0.5 (Peff (1 + 2 (Peff)2/(l - ( ~ e d ' .Fig. 8 shows the fits obtained to the experimental curves of S(Q) against Q by allowing the values of Reff to float until the best fit is obtained. It can be seen that at cp = 0.01 a reasonable fit is obtained using Reff = 435 A and at cp = 0.1 1 using Reff = 251 A. Although the fit is reasonable in terms of amplitude and position of the first peak, at low Q values the hard-sphere model gives values lower than those determined experimentally and at high Q values the modulation appears to be slight- ly out-of-phase with respect to the position of the second peak.The hard-sphere model, although qualitatively of interest, is formally incorrect for 2.0 1.5 1 .o 0.5 0 Ql.8- Fig. 8. Plot of S(Q) against Q showing fits using a hard-sphere model: 0, experimental points; (-) theoretical fit. (a) with = 435 A, (b) with Rerf = 251 A.D. J. CEBULA et al. 49 charged particles in that it ignores the essential features of the particle-particle interaction, i.e. the electrical double layer. However, as pointed out in an earlier paper,12 a link can be made via the theory of Barker and Henderson27-29 in order to take some account of the softness of the potential. In this approach the effective interaction radius of the particles can be written in the form R Rem = R + J: ( 1 - exp[ - V(r)/kT]) d(r/R) where V(r) is the pair potential for the interaction between two particles with their centres separated by a distance r.In the conditions we have used the repulsive interaction is dominated by electrostatic repulsion and hence as a first approach for the pair potential the form given by Verwey and Overbeek2 can be used, namely V(r) = 4 E ~ E , R2 $: exp(2~R) exp( - w ) / r (21) where Er the relative permittivity of the medium, E~ is the permittivity of free space, $s is the surface potential of the particles and K is the Debye-Huckel reciprocal double-layer thickness of the bulk electrolyte, which for a 1 : 1 electrolyte can be written in the form (22) where e is the fundamental charge on the electron, N is Avogadro's number and c is the concentration of the electrolyte in mol dm-3.From eqn (18) and (19) using ReK values of 435 and 251 A, i.e. the values ob- tained in fig. 7, and an electrolyte concentration of lop4 mol dmV3 the value of rl/s is found to change from 20 mV at cp = 0.01 to 1 1 mV at cp = 0.1 1. These appear to be low potentials compared with the [-potential of 58 & 10 mV obtained on a very dilute dispersion using laser electrophoresis. Low potentials would account for the fit obtained to the hard-sphere model 30* but it might be unwise from this evidence alone to conclude that the potential is changing with cp. This approach appears to be more appropriate for low volume fractions.12 An alternative method to obtain a theoretical estimate of the form of g(r) against r is to utilise eqn (21) in a Brownian-dynamics computation, an approach which has been used by several a ~ t h o r s .~ ~ - ~ ~ In the approach utilised by Ermak,36* 3 7 if hydro- dynamic effects are neglected, the charged particle equation of motion can be put in the form = 2e2 N c 103/&,~, kT where Ri ( t ) denotes a random displacement of the particle determined by a Gaus- sian probability function that has an average value of zero and a mean-square value given by (Ri ( A t ) 2 ) = 6 D kT (24) The term Fi ( t ) is the force component32 given by in which V(rij) is taken as the pair potential. values of r the diffusion coefficient D was calculated from In the computational procedure adopted to obtain values of g(r) at various50 CONCENTRATED LATEX DISPERSIONS where q is the viscosity of water and D(u) is the diffusion coefficient for the approach of one particle to another to a distance uR along the line of the centres.The quantity P(u) has been calculated by Honig et al.38 who found p -+ 1 as u + 00. In the computations values of Ri were generated using a random-number procedure 39 using a box containing 108 particles and taking a time step of lo-’ s. Two initial configurations, a face-centred cubic lattice and a random distribution, were used. g(r) against Y was calculated as a histogram of separations and plotted at various 1 .o 0.5 0 1.5 1.0 - 0.5 0 1.5 1 .o 0.5 0 0 3 P I I I ( c) 500 1000 1500 2000 rlA Fig. 9. Plot of g(r) against Y showing computations using Brownian dynamics: 0, comput- ations; (-) experimental data.(a) cp = 0.01, $s = 30 mV; (b) cp = 0.04, @s = 40 mV; (c) cp = 0.13, $s = 50 mV. mol dm-3 NaC1.D. J. CEBULA et al. 51 times until a constant result was obtained. The use of two starting configurations ensured that an equilibrium situation was reached. The data were then collected over several hundred steps to reduce the statistical noise. The results obtained using this procedure are shown in fig. 9 wherein they are compared with the experimentally determined values of g(r) against r obtained in mol dm-3 sodium chloride solution (fig. 7). The latter electrolyte concen- tration was used to calculate K and eqn (21) was used for the pair potential, @s, until the best fit was obtained. As can be seen from fig.9, this was 30 mV at q = 0.01,40 mV at cp = 0.04 and 50 mV at cp = 0.13. Again the value of 30 mV obtained at the lowest volume fraction is lower than the -58 mV obtained for the zeta-potential of the particles by laser electrophoresis using very dilute dispersions at the same electrolyte con~entration.~~ The apparent increase of potential suggested by this approach is interesting since, if correct, it would suggest the possibility of interaction occurring at constant charge. What is clear is that at a mol dmP3 sodium chloride concentration the effective interaction surface charge density of the par- ticles, corresponding to t,hs = 30 mV, is 0.21 pC cm-2, a value very substantially below the value of 4.2 pC cm-2 found for the actual surface charge density of the latex particles.It can therefore be concluded that it is the diffuse part of the electrical double layer which controls the interaction between the particles in concentrated dispersions. However, further wprk is required to determine whether the zeta- potential measured under dilute conditions is the appropriate potential to use for modelling the behaviour of concentrated dispersions. In this context it should be pointed out that there are a number of problems attached to the use of eqn (21) for strongly interacting systems. These include the assumption of constant surface potential, the assumption of constant electrolyte concentration with increasing volume fraction and the assumption that there is always present a reservoir of bulk electrolyte as a reference state for the calculation of K .It is clear that in concentrated systems these conditions are not fulfilled, par- ticularly in the absence of added electrolyte, and that the concentration of the counter-ions and its dependence on latex volume fraction should be taken into ac- count, as suggested by Beresford-Smith and Chan.41 Our preliminary calculations 42 indicate that their approach provides a clearer interpretation of the behaviour of ion-exchanged systems, and this work will be reported in due course. We thank the S.E.R.C. for support of this work and the Institut Laue-Langevin for neutron-beam facilities. One of us (R. H. 0.) thanks Dr. Derek Chan for a number of stimulating and useful discussions. €3. V. Derjaguin and L. Landau, Act0 Physicochim.U.R.S.S., 1941, 14, 633. E. J. W. Verwey and J. Th. G. Overbeek, Theory of Stability of Lvophobic Colloids (Elsevier, Amsterdam, 1948). R. H. Ottewill, Prog. Colloid Polym. Sci., 1980, 67, 71. J. N. Israelachvili, Furaday Discuss. Chem. SOC., 1978, 65, 20. H. J. van den Hul and J. W. Vanderhoff, Br. Polyrn. J., 1970, 2. 121. R. H. Ottewill and J. N. Shaw, Kolloid Z . Z . Yolym., 1967, 218, 34. J. C. Brown, P. N. Pusey, .I. W. Goodwin and R. H. Ottewill, J . Phys. A, 1975, 8, 664. * J. C. Brown, J. W. Goodwin, R. H. Ottewill and P. N. Pusey, Colloid Znterfuce Sci., 1976, IV, 59. A. Vrij, E. A. Nieuwenhuis, H, M. Fijnaut and W. G. M. Agterof, P-uruduy Discuss. Chem. Soc., 1978, 65, 101. A. K. van Helden and A. Vrij, J . Colloid Znterfuce Sci., 1980, 78, 312.l o A . Vrij, J . Chem. Phys., 1979, 71, 3267. l 2 R. H. Ottewill and R. A. Richardson, Colloid Polym. Sci., 1982, 260, 708.52 CONCENTRATED LATEX DISPERSIONS l 3 Neutron Beam Facilities at the H.F. R. Available for Users (Institut Laue-Langevin, Grenoble, l4 D. J. Cebula, R. H. Ottewill and R. A. Richardson, in press. l 5 A. Guinier and G. Fournet, Small Angle Scattering of X-Rays (John Wiley, New York, 1955). l 6 K. Ibel, J. Appl. Crystallogr., 1976, 9, 296. l 8 N. M. Harris, D . Phil. Thesis (Oxford University, 1980). l9 W. F. Espenscheid, M. Kerker and E. MatijeviC, J. Phys. Chem., 1964, 68, 3093. 2 o R. H. Ottewill, Colloidal Dispersions, ed. J. W. Goodwin (Royal Society of Chemistry, London, 2 1 P. A. Hiltner and 1. M. Krieger, J. Phys. Chem., 1969, 73, 2386. 2 2 J. W. Goodwin, R. H. Ottewill and A. Parentich, J. Phys. Chem., 1980, 84, 1580. 2 3 J. H. Nixon and M. Silbert, J. Phys. C, 1982, 15, L165. 24 D. J. Cebula, R. H. Ottewill, J. Ralston and P. N. Pusey, J. Chem. Soc., Faraday Trans. 1, 1980,77, 2 5 J. K. Percus and G. J. Yevick, Phys. Rev., 1958, 110, 1. 26 N. W. Ashcroft and J. Lekner, Phys. Rev., 1966, 45, 33. 27 J. A. Barker and D. Henderson, J. Chem. Phys., 1967, 47, 4714. 2 8 J. A. Barker and D. Henderson, Annu. Rev. Phys. Chem., 1972, 23, 439. 29 W. van Megen and I. Snook, Chem. Phys. Lett., 1975, 35, 399. 30 J. B. Hayter and J. Penfold, Mol. Phys., 1981, 42, 109. 3 1 K. Alexander, D. J. Cebula, J. W. Goodwin, R. H. Ottewill and A. Parentich, Colloids and 3 2 K. Gaylor, I. Snook and W. van Megen, J. Chem. Phys., 1981, 75, 1682. 3 3 C. T. Havens, Ph.D. Thesis (Case Western Reserve University, 1978). 34 K. Gaylor, I Snook, W. van Megen and R. 0. Watts, Chem. Phys., 1979, 42, 233. 3 5 J. Bacon, E. Dickinson, R. Parker, M. La1 and N. Anastasiou, J. Chem. SOC., Faraday Trans. 2, 36D. L. Ermak, J. Chem. Phys., 1975, 62, 4189. 37 D. L. Ermak, J. Chem. Phys., 1975, 62,4197. 38 E. P. Honig, G. J. Roebersen and P. H. Wiersema, J. Colloid Interface Sci., 1971, 36, 97. 39 NAG Library routine 505 DDF. 40 R. H. Ottewill, R. A. Richardson and P. N. Pusey, unpublished work. 4 1 B. Beresford-Smith and D. Y. C. Chan, Chem. Phys. Lett., 1982, 92, 474. 4 2 G. C. Jeffrey, R. H. Ottewill and R. A. Richardson, to be published. 1977). ’ C. Chauvin, ThPse de Doctorat (Universite de Grenoble, 1979). 1982). 2585. Surfaces, 1983, 7, in press. 1983, 79, 91.

ISSN:0301-7249    

DOI:10.1039/DC9837600037

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1983, 76, 53-63 Neutron-scattering Studies of Concentrated Oxide Sols BY JOHN D. F. RAMSAY, RONALD G. AVERY AND LANCE BENEST Chemistry Division, B.429, AERE, Harwell, Oxfordshire OX1 1 ORA Received 28th April, 1983 Small-angle neutron scattering (SANS) and light scattering have been used to study the interactions between particles (diameter < 20 nm) in silica and ceria sols covering a wide range of concentration. Structure factors, S(Q), have been determined which have been com- pared with those calculated for a model hard-sphere (HS) potential. At low sol concentrations ( 5 lo-, g ern-,) interactions are dominated by double-layer repulsion which is most pro- nounced at low electrolyte concentrations (ca. lop4 mol dm-,). At higher sol concentrations (2 10- g cm-3) the effects of electrolyte concentration become less important and the form of S(Q) is similar to that of a HS system, in which the effective interaction diameter is greater than the diameter of the sol particles. Preliminary SANS studies on mixtures of titania and iron oxide sols using the contrast variation technique are also described from which inform- ation on the compatibility and structure of each component has been obtained.A knowledge of the structure and nature of the interactions in concentrated oxide dispersions is important in many technological applications but also has a wider significance for a better understanding of colloid stability and the properties of oxide/water interfaces. * - Small-angle neutron scattering (SANS) has been used recently4 to study the structural changes which occur when silica and ceria sols, containing discrete and approximately spherical particles, are progressively concen- trated and finally converted into solid gels.A feature illustrated was the importance of interparticle repulsion forces in promoting short-range order and the formation of a regular porous gel structure. In this paper more detailed SANS investigations of the interactions in these sols are described, together with complementary light-scattering measurements made on less concentrated dispersions. The versatility of SANS is also demonstrated by pre- liminary studies made with mixtures of titania and a-FeOOH sols, where the dif- ference between the scattering length densities of the oxides is exploited. Possible applications of this technique in studies of structure and interactions in mixed sol systems are discussed.EXPERIMENTAL MATERIALS The concentrated silica sol was a commercial sample (LUDOX HS40) and the cerium oxide sol was prepared by peptising a hydrous oxide precipitate with nitric acid, as previously described. These stock sols were dialysed repeatedly against dilute electrolytes (NaNO, and KNO,, respectively) of different ionic strengths, having a fixed pH (ca. 8 for SiO,; ca. 3.6 for CeO,). Silica sols used for SANS were prepared in deuterium oxide (> 99% D20) to elimin- ate the background incoherent scattering from water; ceria sols were prepared in H 2 0 to achieve greater contrast. Individual samples of known concentration were prepared either by dilution with equilibrium dialysates or by concentrating further using an ultrafiltration cell.Particle-size distributions, determined by transmission electron microscopy as already de-54 INTERACTIONS IN OXIDE SOLS ~ c r i b e d , ~ showed discrete, almost spherical, particles. Mean diameters, d,, were 16.5 nm (stan- dard deviation 0.21) for silica and ca. 6 nm for ceria (majority in range 4.5-7.5 nm). Titania and iron hydroxide sols were prepared by hydrolysing solutions of TiC14 and Fe(N03)3, respectively. A cooled solution of TiC14 (2.0 mol dmP3) was neutralised rapidly with NH40H to pH z 3. The dispersion formed was dialysed repeatedly against water (pH ca. 3) to give a stock sol containing small (ca. 4 nm) particles of poorly crystalline anatase.A solution of Fe(N03)3 (2.0 mol dmd3) was dialysed to pH 3.1, concentrated by ultrafiltration and aged (7 months) to give a stock sol containing thin (ca. 5-10 nm) needle-shaped particles (length ca. 8 nm) of goethite. Samples for SANS were prepared from these stock sols by dialysing (five times) against H 2 0 + D,O mixtures (pH ca. 3.2). LIGHT SCATTERING Measurements (SOFICA, model 4200) were made with vertically polarised light with a wavelength of 546 nm as previously described.6 Scattered intensities on an absolute basis (viz. Rayleigh ratios) were obtained using a benzene standard for calibration. SMALL-ANGLE NEUTRON SCATTERING Measurements were made at wavelengths of 6 and 10 A on samples of sols contained in silica cells (path length 1 and 2 mm), using a multidetector instrument’ installed in the PLUTO reactor at AERE, Harwell.At 10 A the incident flux was considerably lower ( x 0.1) than at 6 A, although measurements could be extended to smaller Q (0.08 and 0.15 A- for 10 and 6 A, respectively). Data were analysed using standard programmes to normalise counter efficiencies and to correct for sample self-absorption and incoherent background. Absolute scattered intensities, expressed as macroscopic cross-sections, [dC/dlI],,h, were obtained using a water standard. DATA TREATMENT AND ANALYSIS NEUTRON SCATTERING For the case of coherent small-angle scattering from a concentrated colloidal dispersion of identical particles the scattered intensity is given by ItQ) = F(pp - pJ2Vp2npP(Q)S(Q) where Q is the scattering vector, defined as IQI = 4nsinO/A for a scattering angle 28 and wavelength A, F is an experimental factor, pp and ps are, respectively, the mean scattering length densities of the particles and solvent (which here is water), V, is the volume of each particle, np is the particle number density and P(Q) is the particle form factor, which for spheres of radius R is given by 3(sinQR - QRcosQR) = ( Q3R3 (3) The structure factor, S(Q), which describes the effects of interparticle interactions, is related to the particle pair-distribution function, g(r), by the Fourier transform to obtain details of the spatial distribution of the particles as a function of the mean inter- particle separation, r.In very dilute dispersions (viz. np -, 0) containing widely separated non-interacting particles, S ( Q ) = 1 .Hence S(Q) can be obtained experimentally by eliminating P(Q) from eqn (1) using the normalisation relationshipJ. D. F. RAMSAY, R. G . AVERY AND L. BENEST 55 LIGHT SCATTERING For incident unpolarised light, the scattered intensity, expressed in an equivalent form to eqn (l), is given by Re = K*McP(Q)S(Q)n (6) where Re is the Rayleigh ratio, M is the molecular weight of the particles with a mass concentration c and K* is an optical constant given by (7) K* = 2~~fio(dfi/d~)’&-~ N - ‘. Here fi0 is the refractive index of the solvent, dfi/dc is the refractive index increment and lo and N are the wavelength of the incident light and Avogadro’s number, respectively. The equivalent normalisation to eqn (S), from which S(Q) is obtained, is thus S(Q) = [&/cll[Re/clc-o. (8) HARD-SPHERE STRUCTURE FACTORS A very satisfactory interaction model for describing the properties of simple liquids,* which has more recently been applied to colloidal di~persions,~-’~ is the hard-sphere (HS) potential. In this model the particle is regarded as a non-attracting rigid sphere of diameter Q? with potential energy @(r) = a for r < Q and @(r) = 0 for r > Q.Here S(Q)Hs, solved using the Percus-Yevick approximation 1 3 , l4 for appropriate values of HS number density, nHS, and volume fraction, <pHs [where <DHs = (n/6)03nHs], has been compared with that obtained experimentally. RESULTS AND DISCUSSION INTERACTIONS AND STRUCTURE IN SILICA AND CERIA SOLS LIGHT SCATTERING Measurements of scattered intensity, Re, at eleven fixed angles (28 from 30 to 1500) were made on silica and ceria sols covering a wide range of concentration, c, from ca.5 x to ca. 3 x lov2 g ~ m - ~ . Over this range of c the attenuation of the transmitted beam was very low (c 573, even for the silica sol of highest con- centration, and complicating effects due to multiple scattering were not considered significant. Plots of RB/C against Q were horizontal, an indication that P(Q) x 1 in the range of Q/A-l covered here (ca. 8 x loM4 to 3 x Values of (RB/c)~-,o and other scattering data for the two sols are given in table 1. Table 1. Light-scattering results for sols sol (dn”/dc)/cm3 8-l (R&)c-to/cm2 g-’ K*/mol cm2 g-I M W silica 0.064 0.212 2.68 x 7.96 x lo6 ceria 0.157 0.822 1.61 x lo-’ 5.10 x lo6 Values of S(Q)Q-+~, obtained from eqn (8), are plotted for each value of c for both silica and ceria sols of two different ionic strengths in fig.1. The importance of interactions between the electrical double layers of the sol particles is apparent from the steep decline in S(Q)o which occurs at low ionic strength, where the effective56 INTERACTIONS IN OXIDE SOLS 0 2 4 0.01 0.05 iPHS 0 - 1 0.2 0.3 c/10p2g cm-3 Fig. 1. Dependence of S(Q)Q-.O and corresponding q H S on concentration, c, for sols of silica (0 and.) and ceria (0 and m) having electrolyte concentrations of (0 and 0) and 5 x mol dm-3 (@ and H). thicknesses of the diffuse double layers are correspondingly greater. This effect is more marked for the ceria sol, which might be expected, on account of its consider- ably smaller particle size.As c is increased, and the average particle separation is accordingly decreased, interactions in the sols of high ionic strength become increas- ingly important and S(Q) shows less sensitivity to differences in ionic strength and tends to approach a level which is similar for both sols. The particle interaction behaviour can be related to that for a HS potential by comparing the calculated l4 value of q H S , which corresponds with the experimentally determined S(Q)o for a given c and ionic strength, q, as is shown in fig. 1 . The effective hard-sphere diameter, a, is then given by where R and q p p ( = c S - l ) are the radius and volume fraction of the sol particles, respectively. Plots of a against ( p p in fig.2 reflect the influence of c, on the range of double-layer repulsion between the sol particles. This can be further demonstrated by the accord between (a - 2R)/2 and the Debye-Hiickel screening distances, K - ~J. D. F. RAMSAY, R. G . AVERY AND L. BENEST 57 (viz. 30.4 and 4.3 mm), for the two different electrolyte concentrations of 5 x and mol dm-3, respectively, where15 60 LO --- L3 20 0 \ \ \ \ \ 2 Fig. 2. Dependence of effective HS diameter, g, of particles on volume fraction, (pp, for silica [(i) and (ii)] and ceria [(iii) and (iv)] sols having different electrolyte concentrations, ci, of ca. low4 [(i) and (iii)] and 5 x [(ii) and (iv)] mol dm-3 SMALL-ANGLE NEUTRON SCATTERING The dependence of scattered neutron intensity, I @ ) , on momentum transfer, Q, for more concentrated silica sols (ci = 5 x mol drne3) is illustrated in fig.3. The development of the maxima in I(Q) is caused by interparticle interference effects and indicates that the particles are not arranged at random but have some short- range ordering. Thus the movement of the maxima to higher values of Q with increasing concentration, c, reflects a reduction in the equilibrium separation dis- tance between the particles as already discussed.4* ti The spatial distribution can be described by the radial distribution function, g(r), which defines the probability that the centres of a pair of particles will be separated by a distance Y. Values of Y at the first maximum in g(r), viz. r,,,, derived from eqn (4) are given in table 2.It can furthermore be shown that as c is increased the interparticle separation decreases inversely as c1/3 and approaches that of the particle diameter, 2R. From further SANS measurements it was shown that a reduction of ionic strength to mol dm-3 had no apparent effect on the development of structure in sols of similar concentration, a feature which can be demonstrated more clearly from a comparison of S(Q).58 INTERACTIONS IN OXIDE SOLS 100 ,- I L. I ,- 3 2 5 0 3 0 0.0 5 Q/A-l 0.1 Fig. 3. Small-angle neutron-scattering curves for silica sols of different concentrations, c, in 5 x lop3 mol dm-3 sodium nitrate solution: 0,0.014; 0,0.14; e, 0.27 and 0,0.55 g ~ m - ~ . Table 2. Hard-sphere simulation results for silica sols silica conc., ionic strength, clg cm-3 [Na+]/mol dm-3 n,/1016 rrnax/nm VHS a/nm 0.137 5 x 10-3 1.74 38 0.27 30.9 0.266 5 x 10-3 3.37 29 0.28 25.1 0.550 5 x 10-3 7.46 21 0.32 20.2 0.08 1 5 1 x 10-4 1.02 46 0.28 37.4 0.175 5 1 x 10-4 2.2 1 36 0.29 29.3 0.384 5 1 x 10-4 4.85 26 0.32 23.3 Two procedures were necessary to calculate S(Q).At high Q (Q/A- 2 0.02) eqn (5) was used to normalise intensity data against that for the sol of lowest con- centration (0.014 g cmW3) measured at 6 A. In this range of Q, interactions in the dilute sol had an insignificant effect on I@), viz. S(Q)dil x 1 [n.b. S(Q) + 1 as Q -+ a]. Such effects were, however, apparent for Q/A 5 0.02, and in the range 0.01 c Q/A- < 0.025 corrected intensity data were normalised against the theoretical P(Q) [cf. eqn (3)], giving a satisfactory range of overlap as shown (by the solid and open symbols) in fig.4, [The latter procedure was unsatisfactory for Q/A- 5 0.03 because both polydispersity and polychromaticity of the neutron beam resulted in a marked departure from the theoretical P(Q), calculated for a monodispersed system, as has been discus~ed.'~]J. D. F. RAMSAY, R. G. AWRY AND L. BENEST r--r-- 59 2 4 0 2 L 6 Q/IO-2 A-l Fig. 4. Structure factors, S(Q), for silica sols of different concentrations, c, and ionic strengths, ci: (i) 0.08, (ii) 0.17 and (iii) 0.38 g cm-3 (c; z mol dm-3); (iv) 0.14, (v) 0.27 and (vi) 0.55 g cmb3 (c, = 5 x mol dm-3). Full lines correspond to S(Q) for HS systems with qHS of 0.28, 0.29, 0.32, 0.27, 0.28 and 0.32 for (i)-(vi), respectively. The form of the plots of S(Q) against Q can be ascribed to a system of spherical particles which are maintained in a partially ordered structure by repulsion forces.Two features are also apparent: first, the ionic strength has no apparent effect on S(Q), and hence the interaction potential; secondly, increases in sol concentration, although resulting in shifts in S(Q)max, do not cause a marked increase in repulsive potential, as might at first be expected. These features are demonstrated more clearly by comparing the experimental S(Q) data with those calculated for a HS potential. A unique solution for the S(Q) of a HS system is obtained by fixing any two of the three interdependent parameters: 0, qHS and nHs; viz. qHS and nHS in the present case, where nHS is identical to np (=6qP/ndv3), which is already known. Results obtained (fig.4) by varying qHS to give the closest match with the experimental data in the region of S(Q)max show that qHS increases only slightly over the range of concentrations examined. This implies that for the more dilute sols, 0 is relatively large and corresponds to an effective diameter which is considerably greater than the particle diameter, d,, of 16.5 nm. As the sol concentration, c, is increased, o de- creases progressively (table 2) until in the gel it can be shown that the two are of comparable size.60 INTERACTIONS IN OXIDE SOLS 10 Although the HS model does not give a completely satisfactory simulation with the experimental S(Q), the similarity is nevertheless remarkable, because a slowly decaying potential would be expected to result from the interaction between the electrostatic double layers surrounding the particles.Indeed there is evidence that at the lower sol concentrations a 'softer' potential would be more suitable for simulat- ing the more gradual increase in S(Q) which occurs initially. This possibility has been examined more recently by applying the screened Coulomb potential model of Hayter and Penfold. * Although a detailed description is not yet appropriate,lg initial results show improved fits to experimental data compared with that obtained with the HS model. Satisfactory fits are, however, obtained with an effective surface charge on the particles which is considerably lower (viz. from ca. 0.3 to 0.7 ,uC cm-2) than the typical total surface charge density (> 2 ,uC cmP2) of silica sols, as determined under similar conditions by conductimetric titrations.20 This would sug- gest that a considerable uptake of counterions occurs, either at the oxide interface or within the Stern layer, which reduces the effective potential of the diffuse double layer markedly, so that the interaction becomes more similar to that of an HS system.The dependence of I(Q) on Q for ceria sols covering a range of concentration with an ionic strength of 5 x mol dmP3 is illustrated in fig. 5. Again marked - I 1 e 00 0 0 8 . 0 - 0 0 0 o""., . 0 0.0 5 0.1 QIA- 0.15 Fig. 5. Small-angle neutron-scattering curves for ceria sols of different concentrations, c, in 5 x mol dm-3 potassium nitrate solution: 0, 0.032; 0, 0.080; a, 0.16 and 0 , 0 .3 2 g C M - ~ . interference is observed, for reasons already described. The maxima in l ( Q ) occur at considerably larger Q values compared with those for the silica sols of similar vol- ume fraction, on account of the smaller size of the ceria particles. This feature is illustrated more clearly by the dependence of rg(r)max on c, which has been des- cribed previously for this ~ y s t e m . ~ Further measurements on sols having an ionic strength of mol dmP3 showed that the behaviour of I(Q) at corresponding ceria concentrations was very similar, an indication that the repulsive interaction was not appreciably enhanced, as already noted with the silica sols. The interaction properties of these concentrated sols are of particular interest because the particlesJ.D. F. RAMSAY, R. G. AVERY AND L. BENEST 61 are much smaller, although more polydispersed, than those of model latex disper- sions studied recently. Consequently the interparticle separation is considerably less than the screening length, K - even at relatively high electrolyte concentrations. SMALL-ANGLE NEUTRON SCATTERING OF MIXED OXIDE SOLS In principle an understanding of the interactions and homogeneity of the differ- ent components in mixed oxide sols may be obtained from SANS by exploiting the contrast variation technique, which has been employed extensively in the study of biological structures.21 In the present study of mixtures of iron hydroxide and titania sols, the oxides have a large difference in scattering length density, &xi& (see table 3), and thus under contrast match conditions for one component the scattering from the other will be considerable.This is important experimentally with very small colloidal particles, as studied here, because I(Q) is relatively weak [cf. eqn (l)]. Table 3. Molecular scattering lengths, Cibi for different oxides and corresponding scattering-length densities, p, for mass densities, 6 oxide Zihi/10-'2 cm S/g cmP2 p/lO'O cmP2 H2O -0.168 1 .oo - 0.56 D20 1.914 1.10 6.36 SiOz 1.575 2.20 3.47 Ce02 1.642 7.13 4.10 Ti02 0.825 3.84 2.39 FeOOH 1.737 4.28 5.04 FeOOD 2.778 4.28 7.96 To determine the contrast match condition for each oxide sol, measurements of I(Q) against Q were made with samples having different H 2 0 : D 2 0 solvent ratios, as shown in fig.6. For titania [fig. 6(a)] a match was obtained at ca. 50 volume % D 2 0 (data had poor statistics and are not shown) corresponding to a scattering length density, pTiO,, of 2.9 x 1 O l o cm-2. This is significantly larger than the calculated value of 2.39 x 1 O l o cmP2 for bulk anatase (cf. table 3) and suggests that the sol particles contain protons, possibly as hydroxyl groups, which are readily exchange- able. The contrast match condition for the iron hydroxide sol was never reached, even at 100% D 2 0 [fig. 6(b)]; however, from an extrapolation of plots of I(Q)* against psolvent a value of ca. 8.4 x 1 O l o cm cm-3 was obtained. This is close to that calculated for a-FeOOD and is an indication that exchange of structural protons occurs readily.The form of the scattering curve of the titania sol, which can be ascribed to a dispersion containing very small polydispersed spherical particles (diameter 6-8 nm), remains unchanged when the titania and FeOOH sols are mixed [cf. fig, 6(a), (i) and (ii)]. This shows that the dispersions are mutually stable, probably because both oxides are positively charged at this pH. An interparticle separation of ca. 30 nm can be estimated from the position of the interface maximum observed at ca. 1.9 x 10- A - 1 for the titania sol. The slight shifts in this maximum with the mixed sols (ca. 1.6 x l o p 2 and 2.2 kl, respective1y)~are dependent on the concentration of titania and suggest that the particles are grouped together around the larger a-FeOOH particles.1 0 INTERACTIONS IN OXIDE SOLS I I 1 '.0.1 0 0.1 0.2 QIA -- Fig. 6. SANS results for ( a ) titania and (6) iron hydroxide sols and their mixtures in water with different H20/D20 volume fractions. (a) Titania (ca. 0.028 g cmP3), volume % D20: U, 0; A, 20; V, 40; +, 70; e, 100; (i) and (ii) mixed sols (100% D20) with concentrations (Ti02 + FeOOH) of (0.014 + 0.015) and (0.1 1 + 0.12) g crnp3, respectively. (b) Iron hy- droxide (ca. 0.030 g cm-3), vol % D20: U, 0; 0 , 50; x , 80; a, 100; (iii) mixture (50% D20) concentrations as in (i). The scattering from the a-FeOOH sols [fig. 6 (b)] is consistent with that expected for thin rod-like particles, with diameter D, for which it can be shown22 that [I(Q)Q]cc exp( - Q2D2/16). Fits of this relationship gave D values in the range 4-6 nm, in reasonable accord with electron microscopy.The absence of any interference maximum suggests that these rods are either widely spaced or orientated at random, both in the single-component sol and also amongst the smaller anatase particles in the mixed sols. We thank Dr. J. Penfold for discussions and advice on the application of his mean spherical approximation model, Mr. B. Booth for experimental assistance and Mr. J. L. Woodhead for providing the sample of ceria sol. A. Breeuwsma and J. Lyklema, Faraday Discuss. Chem. Soc., 1971, 52, 324. G. R. Wiese, R. 0. James and T. W. Healy, Faraday Discuss. Chem. Soc., 1971, 52, 302. R. J. Hunter, Zeta Potential in Colloid Science (Academic Press, New York, 1981), p. 278 et seq. J. D. F. Ramsay and B. 0. Booth, J. Chem. SOC., Faraday Trans. I , 1983, 79, 173. J. L. Woodhead, Sci. Ceram., 1977, 9, 29. J. D. F. Ramsay, S. R. Daish and C. J. Wright, Faraday Discuss. Chenz. SOC., 1978, 65, 65.J. D. F. RAMSAY, R. G. AVERY AND L. BENEST 63 D. I. Page, Atomic Energy Research Establishment Report (AERE-R 9878, 1980). J. A. Barker and D. Henderson, Rev. Mod. Phys., 1976, 48, 587. A. K. van Helden and A. Vrij, J. Colloid Interface Sci., 1980, 78, 3 12. D. J. Cebula, R. H. Ottewill and J. Ralston, J . Chem. Soc., Faraday Trans. 1, 1981, 77, 2585. 1982), p. 197. l o A. Vrij, J. Colloid Interface Sci., 1982, 90, 110. l 2 R. H. Ottewill, in Colloidal Dispersions, ed. J. W . Goodwin (Royal Society of Chemistry London, l 3 J. K. Percus and G. J. Yevick, Phys. Rev., 1958, 110, I. l 4 N. W. Ashcroft and J. Lekner, Phys. Rev., 1966, 45, 33. E. J. W. Verwey and J. Th. G. Overbeek, Theory of Stability of Lyophobic Colloids (Elsevier, Amsterdam, 1948). l 6 J. D. F. Ramsay, Faraday Discuss. Chem. Soc., 1978, 65, 139. l 7 R. H. Ottewill, in Colloidal Dispersions, ed. J. W. Goodwin (Royal Society of Chemistry, London, 1982), p. 143. J. B. Hayter and J. Penfold, Mol. Phys., 1981, 42, 109. l 9 J. F. Penfold and J. D. F. Ramsay, unpublished work. 2 o G. H. Bolt, J. Phys. Chem., 1957, 61, 1166. 2 1 B. Jacrot, Rep. Prog. Phys., 1976, 39, 91 1. 2 2 A. Guinier and G. Fournet, Small-Angle Scattering of X-Rays (John Wiley, New York, 1955).

ISSN:0301-7249    

DOI:10.1039/DC9837600053

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

Faraday Discuss. Chem. Sac., 1983, 76, 65-75 Electrical Double-layer Interactions in Concentrated Colloidal Systems BY BRYAN BERESFORD-SMITH AND DEREK Y . C. CHAN Department of Applied Mathematics, Research School of Physical Sciences, Australian National University, Canberra, Australia Received 23rd May, 1983 In this paper we compare three methods for deducing the effective pair potential that describes the double-layer interaction between colloidal particles in a concentrated dispersion. The structure factors and small-time behaviour of the intermediate scattering function predic- ted by these models are compared with results obtained from laser and neutron-scattering experiments. 1. INTRODUCTION The study of the structure of colloidal suspensions has a long history which dates back to Langmuir,’ who put forward the idea that the electrostatic or electrical double-layer interactions between colloidal particles are responsible for giving rise to long-range structures or order in such systems.Kirkwood and Mazur2 made the first attempt to treat colloidal particles as an effective one-component ‘“fluid’ in which the colloidal particles interact via pair potentials (of mean force) which de- pend on the properties and ionic composition of the solvent. Using the Debye-Huckel expression for the double-layer interaction, they produced colloid-colloid pair correlation functions which resemble those found between atoms in a simple monatomic fluid. Their results demonstrated that the repulsive double- layer interaction between the colloidal particles can be responsible for the observed structure in colloidal systems.Recent experimental advances in light- and neutron-scattering techniques, to- gether with the application of modern concepts of liquid-state physics, have seen a revival of interest in the subject. In modelling the properties of a dispersion as an effective one-component fluid of colloidal particles, computer-simulation techniques 3, or the integral equation methods - of liquid-state physics have been used. However, both of these approaches require, as input, the form of the inter- action potential between the colloidal particles. The remaining task, therefore, is to construct the colloid-colloid interaction potential in terms of the physico-chemical properties of the particles and the composition of the dispersion medium such as pH, electrolyte concentration etc.In the conventional treatment of the interaction between colloid particles, the colloidal system is assumed to be in osmotic equilibrium with a large electrolyte reservoir of known fixed composition. The screening of the electrostatic interaction between the particles is controlled by the ionic strength of this reservoir. However, in order to maximize the electrostatic repulsion, colloidal systems are often treated with ion-exchange resins to remove all excess electrolytes. The concentration of added66 ELECTRICAL DOUBLE-LAYER INTERACTIONS electrolytes is then controlled by the addition of known amounts of salt. As a conse- quence, the properties of the electrolyte reservoir which is supposed to be in equili- brium with the system are left undetermined, although in principle they can be deter- mined by equilibrium dialysis.The theoretical analysis outlined in this paper is developed with a view to circumvent this problem. The basic idea is to regard the colloidal system as a collection of ions and colloidal particles of known valence and number concentration. The solvent is taken to be a dielectric continuum. The next step is to exploit the obvious asymmetry in charge and size between the ions and colloidal particles and formally regard the colloidal particles as a ‘solute’ in a ‘solv- ent’ of small ions. Using the McMillan-Mayer picture, we can obtain the free energy of a configuration of the system in which the colloidal particles have been held fixed but the small ions have been averaged over all allowed positions.Obviously some approximations must be made to effect this average. The resultant free energy of this configuration of colloidal particles is the N-body colloid-colloid potential of mean force. In general this is a many-body potential. However, within our approximation scheme this many-body potential of mean force can be written as a sum of pair interactions between the colloidal particles. This effective pair potential turns out to be dependent on the concentration of colloidal particles and it is the appropriate potential to employ if one wishes to regard the colloidal system as an effective one- component system. The usefulness of the effective pair potential will be tested against measurements on polystyrene dispersions based on dynamic light scattering and neutron scattering.The idea of treating a colloidal system as a highly asymmetric electrolyte com- prised of ions and colloidal particles has been used to model the dynamical pro- perties of dense colloidal sy~terns.~*~O The analyses were made in the linear Debye-Huckel limit, which is only capable of yielding ‘limiting-law’ behaviour. However, because of the high charge on the colloidal particles, the weak interaction assumption implicit in the Debye-Hiickel treatment cannot be applied to the colloid-colloid interaction. The equilibrium properties of highly asymmetric electro- lytes have been investigated by integral equation 1i and Monte Carlo methods for a charge asymmetry of up to 20 to 1.2. THE EFFECTIVE PAIR POTENTIAL By modelling a colloidal dispersion as a highly asymmetric electrolyte, we have obtained an analytical expression for the effective pair potential which describes the double-layer interaction between the colloidal particles. l4 As mentioned in the Introduction, this result is applicable to situations in which the composition of the colloidal system is known but the properties of the electrolyte reservoir that is in osmotic equilibrium with the dispersion remain undetermined. (As a consequence, coventional double-layer theory cannot be applied.) In the original derivation l4 the effective pair potential was obtained from an asymptotic analysis of the Ornstein-Zernike equations for the colloid-colloid, colloid-ion and ion-ion pair correlation functions.By replacing the ion-ion and ion-colloid direct correlation functions by their limiting forms which are valid for large separations, one readily obtains simple analytic expression for the effective colloid-colloid pair potential. The same result can be obtained from a derivation based on the McMillan-Mayer formalism for studying solutions. Here the colloidal particles are regarded as ‘solutes’ in a ‘solvent’ of small ions.’ The colloid-colloid potential of mean force was obtained by averaging over all configurations of the ions. This averaging processB. BERESFORD-SMITH AND D. Y. C. CHAN 67 is achieved by invoking the same approximations for the ion-ion and ion-colloid interaction as in the Debye-Hiickel theory of electrolytes.Our expression for the effective pair potential between two colloid particles, each having a charge (zoe), at a distance r apart is14 where E is the dielectric constant of the solvent and K is defined by h e 2 rc2 = - C niz: EkT i The summation in eqn (2.2) is to be taken over all species of counterions necessary to balance the colloidal charge as well as over all species of added salt (mean con- centration ni, valence zi). In the absence of added salt, where the system is made up of colloidal particles and counterions (a limit which has no counterpart in conven- tional double-layer theory), eqn (2.2) becomes where z1 is the valence of the counterions and nl, the concentration of counterions, is related to the concentration of colloidal particles, no, by the electroneutrality condition nOzO + n,zl = 0.(2.4) Thus we can see from eqn (2.4) or its generalization in the presence of added salt that the screening parameter K in Ueff will be a function of the number density of colloidal particles. In the limit of low concentration of colloidal particles or swamp- ing added salt, K will only be determined by the added salt concentration. To obtain an appreciation of the relative magnitudes of contributions to IC from the counterions and from the added salt, let us consider a colloidal dispersion of 8% volume fraction of spherical particles of 160 A radius, each carrying 840 elementary charges (this corresponds to a surface charge density of 4.2 pC cmP2) with univalent counterions (see later). Let there be loe3 mol dm-3 of 1 : 1 added salt.From eqn (2.2) the screening parameter for Ueffcan be written as K 2 = dountenon + K?dded salt. (2.5) For the above data we find Kzoun terion/K?dded salt = 3 * 3. In a previous paper l 4 the colloid-colloid structure factor and pair distribution function were obtained by solving the multicomponent Ornstein-Zernike equation for the asymmetric electrolyte in the hypernetted-chain approximation. By treating the colloidal particles as an effective one-component system an effective pair poten- tial was obtained from the colloid-colloid pair distribution function by inverting the one-component hypernetted-chain equation. The effective pair potential so obtained was in very good agreement with the result given by eqn (2.1) and (2.2). The colloid-colloid pair correlation functions calculated according to eqn (2.1) and the hypernetted-chain approximation are also in good agreement with the corresponding quantities obtained by a Monte Carlo simulation of the asymmetric 1 : zo electro- lyte.68 ELECTRICAL DOUBLE-LAYER INTERACTIONS In deriving eqn (2.1) we have assumed that the particle radius is small compared with the mean interparticle spacing and with IC- l .Moreover, the linearity assump- tion in treating the ion-colloid interactions imposes the condition (zoz1e2/ekTa) < 1 where a is the distance of closest approach between a colloidal particle and a coun- terion. For systems where the above assumptions do not hold we propose the follow- ing generalization to the effective pair potential given by eqn (2.1). s We model the colloidal dispersion as a multicomponent electrolyte in which the ions and colloidal particles all interact via r - coulomb potentials. The Ornstein- Zernike equations for the pair distribution functions gij(r) = 1 + hij(r) have the form (ij,k = 0,1,2 .. .) From eqn (2.7) we can write down a formal one-component equation for the colloid-colloid correlation function (component 0) lo0(k) = ~ i : ( k ) + no ~:\(k) h",,(k) (2.8) where the Fourier transforms are defined by (f= h or c) y ( k ) = $ j: dr r sin ( k r ) f ( r ) . The effective colloid-colloid direct correlation function ZfL(k) is related to the direct correlation in eqn (2.7) by C"'o"0 = zoo + c";f [I - E*]- 20 (2.10) (2.1 1) ( E * ) i j = ( n p j ) f Eij(k), i,j = 1,2,. . . (2.12) (2.13) where the column matrix to has elements (C"0)i = nf Eio(k), i = 1,2, .. . and the matrix The effective pair potential is then defined by Veff(r) = -kT lim cedf,(r). r-+ OD To obtain coo(r) via eqn (2.10) we replace the ion-ion direct correlation functions cij(r) by their asymptotic forms (2.14) which is reasonable for univalent ions. We obtain the ionsolloid direct correlation functions cio(r) by solving the Ornstein-Zernike equation hio(r) = cio(r) + 1 nj J ci& - sl)hjo(s)ds + no S cio(lr - s()hoo(s)ds (2.15) j = 1 using eqn (2.14) together with hio(r) = - 1, r < a (2.16)B. BERESFORD-SMITH AND D . Y. C. CHAN 69 The assumptions embodied in eqn (2.14)-(2.17) are identical to those needed to derive the Poisson-Boltzmann equation for the potential distribution near charged interfaces.The use of eqn (2.18) assumes that as far as determining c . ( r ) is * con- cerned, the remaining colloidal particles are treated as a uniform jellium. This is the simplest way to handle the last term in eqn (2.15) while maintaining electroneut- rality. For later reference we shall call this the jellium approximation. With the substitution eqn (2.15H2.18) are equivalent to solving the differential equation with boundary conditions Y - 0 , r + O (2.21) (2.22) where cp = 4na3no/3. The effective pair potential determined from eqn (2.13) still has the same functional form as eqn (2.1) but with the constant (zoe)2 replaced by a constant which is a function of a, zo and K , and hence the volume fraction of colloidal particles.3. THE CELL MODEL An alternative approach to evaluating the double-layer interaction in concentrated dispersions is to deduce the properties of the ‘fictitious’ electrolyte reservoir which would be in osmotic equilibrium with the colloidal system in terms of the compo- sition of the system. Conventional double-layer theory can then be applied to deter- mine the pair interaction. The properties of the electrolyte reservoir can be deter- mined using a cell model for the dispersion. l6 In this model each colloidal particle is assumed to be in the centre of a spherical cell of radius rs (47tr?Yt0/3 = 1) with one particle per cell. The cell also contains the average number of counterions and added electrolyte ions so that it is electrically neutral. The potential distribution within the cell is assumed to be given by the Poisson-Bol tzmann theory : where the ionic composition of the electrolyte reservoir is given by {nk}.The con- centrations {$} can be determined from the conditions that (i) the ionic compo- sition within the cell is known, (ii) the cell is overall neutral and (iii) there is no surface charge density at the cell boundary. In general, {nk) has to be determined by numerical iteration.16 We note, however, that the cell model approach cannot be applied to ‘salt-free’ dispersions comprised only of colloidal particles and counterions. If one further assumes that the exponential factor in eqn (3.1) may be linearized, the quantities {np) may be determined analytically. In this linear approximation the70 ELECTRICAL DOUBLE-LAYER INTERACTIONS result for the effective screening parameter for the ‘fictitious’ electrolyte reservoir is where rc2 is given by eqn (2.2).Thus we see that our expression for the effective pair potential Ueff given by eqn (2.1) is equivalent to the interaction between two point charges (zoe) in an electrolyte which has a screening parameter given by K . From the above observation one might expect the non-linear version of the cell model to give better agreement with experimental measurement. However, as we shall see in the next section, this does not appear to be the case. 4. COMPARISON WITH SCATTERING MEASUREMENTS STRUCTURE FACTOR The structure of a dispersion can be probed by scattering measurements. The relevant experimental quantity is the colloid-colloid structure factor where goo(r) is the colloid-colloid pair correlation function.The scattering vector, k , is related to the scattering angle, 8 by the usual formula ( 4 4 where II is the wavelength of the radiation in the dispersion medium. For a potential of the form given by eqn (2.1) the structure factor, S(k), obtained by solving the one-component Ornstein-Zernike equation in the hypernetted-chain approximation is, in general, found to be in good agreement with that obtained by Monte Carlo simulations l 7 for almost all values of k. However, when the height of the first peak in S(k) is large (ca. 2) the hypernetted-chain approximation tends to underestimate this peak height by ca. 15%. Nevertheless, there is still good agree- ment with regard to the location of the peak position.These observations are very similar to those found for one-component plasmas. While light-scattering studies have been carried out on numerous systems, few of these are sufficiently well characterized to permit detailed comparison with theory. One notable exception is the light-scattering data of Brown et al. on dispersions of polystyrene spheres. These particles carry 500 elementary charges and a mean radius of 230 A. The system was treated with ion-exchange resin to remove all excess electro- lyte, leaving only univalent counterions. In fig. 1 a comparison between the experi- mental and theoretical structure factors is shown. The theoretical results are ob- tained by solving the one component Ornstein-Zernike equation in the hypernetted- chain approximation using the effective pair potential in eqn (2.1).The system was taken to be salt free but the particles only carry 235 instead of 500 elementary charges. However, since the presence of small amounts of added 1 : 1 electrolyte (ca. mol dm-3) can significantly affect the theoretical results there is a fair amount of leeway in obtaining a fit between theory and experiment. Indeed the quality of fit is comparable to those obtained using different theoretical 6, 2oB. BERESFORD-SMITH AND D. Y. C. CHAN 71 2.0 1.5 1.0 0.5 s 1 .o 0.5 0 I I I ' 1 2 .o 4 .O 6.0 8.0 10.0 krs Fig. 1. Structure factor for a dispersion of polystyrene spheres, radius 230 I$, at particle num- ber concentrations: (a) 8.46 x 1 O I 2 ~ r n - ~ and (b) 1.67 x 10l2 ~ m - ~ .( 0 . 0 ) Experimental points ; l 9 (-) calculated according to eqn (2.1) and (2.2) for a particle charge of 235 (see text). It has been observed experimentally that the location (kmJ of the first peak in S(k) is related to the particle number density by the simple empirical result l 9 km,, rs = 4.6 (4.3) where rs = ( 3 / 4 ~ n ~ ) ' / ~ . In our calculations this product is k,,, rs = 4.4 (4-4) which is almost identical to the value obtained by an analysis based on Bragg diffraction assuming the colloidal particles are in a body-centred cubic lattice.2 Another comparison between theory and experiment is afforded by recent neutron-scattering experiments on dispersions of polystyrene spheres. Here the particles are of radius a = 160 A and carry 840 elementary charges.The dispersion was treated with ion-exchange resin to remove excess electrolytes, but subsequently sodium chloride was added to the dispersion to make up added salt concentrations of mol dm-3. An effective one-component pair potential of the form given by eqn (2.1) was used to calculate the structure factor. The screening parameter K was calculated according to eqn (2.2). However, because of the high charge and Ka (ca. 3) values of these systems, the pre-exponential factor in eqn (2.2) was replaced by (zoe)2 + A 2 . (4.5)72 ELECTRICAL DOUBLE-LAYER INTERACTIONS Fig. 2. Structure factor for a dispersion of polystyrene spheres, radius 160& with mol dmP3 added sodium chloride and 840 elementary charges per particle. Points, ex- perimental data [ref.(22)] (-) calculated according to the jellium model (see text). Volume fraction: (a) 2, (b) 8 and (c) 13%. 3 .o 2 . 5 2 .o h 2 1 . 5 1 .o 0.5 0 2 .o 4 .O 6.0 8.0 10.0 k r s Fig. 3. Structure factor calculated according to three different forms of U"': (a) as obtained from conventional double-layer theory assuming IC is determined only by added salt (- --); (b) as above but with K determined by cell model (--); (c) as obtained from the jellium model (- - --). The particle volume fraction is 8%, all other data as for fig. 2.B. BERESFORD-SMITH AND D. Y. C. CHAN 73 The coefficient A in eqn (4.5) is obtained by using the jellium approximation de- scribed in section 2. In fig. 2 a comparison is given for the structure factor obtained by using the jellium approximation and the hypernetted-chain approximation and that obtained by small-angle neutron-scattering studies.22 In fig.3 are shown the structure factors obtained from the present theory, the cell model and from a model which assumes that the screening parameter is determined solely by the added salt (i.e. neglect the contribution of the counterions to the screening). These results il- lustrate the need to account for the contribution of the counterions to the screening length in the effective colloid-colloid pair potential. In view of the comments in sections 2 and 3 the disagreement between the experiment and the cell model is unexpected. From the discussion so far, it is clear that it is fairly easy to obtain good fits to the structure factor especially if a theory has an adjustable parameter.Indeed, even a hard-sphere model with a suitably chosen hard-sphere radius can provide a reasonable fit to S(k). However, a more stringent test can be provided by examining temporal correlations. THE INTERMEDIATE SCATTERING FUNCTION From dynamic light-scattering experiments it is possible to extract the small-time behaviour of the intermediate scattering function G(k,t). When hydrodynamic in- teractions between colloidal particles are negligible, an acceptable approximation at low volume fractions, we have the result23 where (4.8) Here we see that the quantity Q(k) is given in terms of the colloid-colloid pair correlation function goo(r) and the effective pair potential, V f f ( r ) . In fig. 4 we see B(k) = (no/kT) J goo(r)( 1 - cask r)(k V)2 Uefl(r)dr.h .y, 0, 6.0 4.0 2 .o 0 2.0 4.0 6 .O 8.0 10.0 krs Fig. 4. Comparison of experimental values [ref. (19)] (-) and theoretical predictions(-) of Q(k) for the system described in fig. l(a).74 ELECTRICAL DOUBLE-LAYER INTERACTIONS that the same input data that were able to give a reasonable fit to S(k) in fig. 1 predict quite different forms for the function Q(k). In particular, the theoretical Q(k) contains a fair amount of structure with a dominant peak in the same position as the main peak in S(k). This observation is in agreement with that found for the one- dimensional harmonic lattice for which an exact analytic solution for S(k) and Q(k) have been obtained.23 A possible source for the disagreement between theory and experiment may perhaps be found in the magnitude of Q(k).The experimental values of Q(k) were obtained by a truncated cumulant analysis in which the intermediate scattering func- tion, G(k,t), is assumed to have the form ln[G(k,t)/S(k)] = 1 - T(k)t + +Q(k)[r(k)tI2. (4.9) In the original analysislg it was noted that this assumption was expected to be accurate for small Q(k) (< 0.5), but for larger Q(k) the truncation in eqn (4.9) will introduce systematic errors. 5. DISCUSSION In this paper we have outlined and compared approximate methods for deter- mining the effective pair potential which characterises the electrical double-layer interaction between particles in concentrated colloidal systems. The input data re- quired are the particle number concentration, the particle charge and the amount of added salt in the dispersion.These are quantities that experimentalists should deter- mine in the process of characterising their system in order to eliminate unknown or fitting parameters in a comparison with theory. From earlier work, as well as from the results given here, it is clear that a knowledge of the structure factor alone does not provide very stringent constraints on the theoretical model. However, when combined with temporal measurements there appear to be disagreements which require clarification. We thank Prof. R. H. Ottewill for allowing us access to his data on neutron- scattering measurements. I. Langmuir, J.Chem. Phys., 1938, 6, 873. J. G. Kirkwood and J. Mazur, in John Gamble Kirkwood Collected Works: Macromolecules, ed.P. L. Auer (Gordon and Breach, New York, 1967). W. van Megen and I. Snook, J. Chem. Phys., 1977, 66, 813. E. Dickinson, Faraday Discuss. Chem. Soc., 1978, 65, 127. R. P. Kelvey and P. Richmond, J. Chem. SOC., Faraday Trans. 2, 1976, 72, 773. J. P. Hansen and J. B. Hayter, Mol. Phys., 1982, 46, 651. D. W. Shaeffer, J. Chem. Phys., 1977, 66, 3980. E. J. W. Verwey and J. Th. G. Overbeek, Theory of the Stability of Lyophobic Colloids (Elsevier, Amsterdam, 1948). M. J. Stephen, J. Chem. Phys., 1971, 55, 3878. l o S. Harris, J. Phys. A , 1975, 9, 1895. l 1 D. Elkoubi, P. Turq and J. P. Hansen, Chem. Phys. Lett., 1977, 52, 493. l 2 F. J. Rogers, J. Chem. Phys., 1980, 73, 6272. l 3 P. Linse and B. Jonsson, J . Chem. Phys., 1983, submitted for publication. l4 B. Beresford-Smith and D. Y. C. Chan, Chem. Phys. Lett., 1982, 92, 474. l 5 B. Beresford-Smith and D. Y. C. Chan, J. Chem. Phys., 1983, submitted for publication. l 7 K. Gaylor, 1. Snook and W. van Megen, J. Chem. Phys., 1981, 75, 1682. J. A. Beunen and L. R. White, Colloid and Surfaces, 198 1, 3, 37 1 .B. BERESFORD-SMITH AND D. Y. C. CHAN 75 K-C. Ng, J. Chem. Phys. 1974, 41, 2680. l 9 J. C. Brown, P. N. Pusey, J. W. Goodwin and R. H. Ottewill, J. Phys. A., 1975, 8, 664. 2 o B. J. Ackerson, in Photon Correlation Spectroscopy and Velocimetry, ed. H. Z . Cummins and E. R. 21 J. C. Brown, J. W. Goodwin, R. H. Ottewill and P. N. Pusey, in Culluid and Interface Science, 22 R. H. Ottewill, in Colluidal Dispersiom, ed. J. W. Goodwin (Special Publication, Royal Society of 2 3 B. J. Ackerson, J. Chem. Phys., 1976, 64, 242. Pike (Plenum, New York, 1977). ed. M. Kerker (Academic Press, New York, 1976), vol. 4, p. 54. Chemistry, London, 1982), no. 43, p. 197.

ISSN:0301-7249    

DOI:10.1039/DC9837600065

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1983, 76, 77-92 Light Scattering and Neutron Scattering from Concentrated Dispersions of Small Unilamellar Vesicles BY ANDREW G. MUDDLE AND JULIA S. HIGGINS Department of Chemical Engineering and Chemical Technology, Imperial College, London SW7 2AZ AND PHILLIP G. CUMMINS, EDWIN J. STAPLES AND IAN G. LYLE Unilever Research, Port Sunlight Laboratory, Quarry Road East, Bebington, Wirral, Merseyside L63 3JW Received 16th May, 1983 Data are reported on the single-particle form factor P(Q) and the interparticle structure factor S(Q) for dispersions of small unilamellar vesicles at concentrations up to 4% (w/v) solids. The interparticle potential is investigated by varying the net charge on the vesicles (via the lipids used in the preparation) and by addition of electrolyte.The integrity of the vesicles with varying concentration allows S(Q) to be obtained by simple division of high- and low- concentration data. While the overall behaviour of S(Q) and S(0) is in agreement with model predictions, there are unresolved discrepancies. 1. INTRODUCTION In recent years a great deal of research has been done on the properties of phospholipid vesicles (liposomes), the main objective being either to gain some in- sight into the structure and function of biological membranes or, ultimately, to use the vesicles themselves as carriers for pharmaceutically active materials in the body. l , Liposomes are, however, by no means the exclusive domain of the bioscientist. Vesicular dispersions may be regarded in many respects as conventional colloids, and as such they are amenable to study by techniques like static and dynamic light ~cattering,~.~ neutron scattering, X-ray scattering,6 particle electrophoresis and analytical ultracentrifugation.8 The well known DLVO theory of colloid stability has been applied with some success to explain the aggregation of charged vesicles as a function of the ionic composition of the suspending medium.l o , In particular, the role of Ca2+ ions in promoting aggregation and fusion of negatively charged vesicles has been studied extensively because of its proposed relationship to pro- cesses which may occur during biological membrane f u ~ i o n . ~ ~ ' ~ ~ ' ~ These experiments, however, have tended to rely on determination of initial aggregation rates and are therefore normally restricted to unstable systems at low concentrations (< 1% phase volume).Although small unilamellar vesicles are not thermodynamically stable, they are certainly metastable, being kinetically stabilised by strong repulsive forces in low-ionic-strength media, particularly when charged phospholipids are included. The repulsive forces which give rise to the formation of lamellar phases in phos- pholipid + water mixtures and stabilise liposomes in colloidal dispersion have been studied extensively by Rand, Parsegian and coworkers. In all cases, a strong, -78 SCATTERING FROM UNILAMELLAR VESICLES exponentially decaying repulsion appears to dominate the interaction at bilayer sep- arations < 30 A. This has been tentatively ascribed to a 'hydration force' originat- ing from water structuring around the polar phospholipid headgroups.- * Repul- sion at larger separations is due to overlap of electric double layers and the results are qualitatively consistent with DLVO theory, although quantitative fitting in some cases requires adjustment of parameters such as the surface charge densityl6 or the Debye screening length. The stability of phospholipid bilayers and of liposomes which form spontaneous- ly in excess water has been explained in terms of packing constraints on the molecules and thermodynamic considerations. Ultrasonic disruption of multi- lamellar liposomes results in the formation of small, spherical single-walled vesi- cles.20*21 The size and polydispersity of vesicles prepared in this way depends on a number of factors, such as the size and shape of the sonication vessel, intensity and duration of sonication, temperature and concentration of the dispersion, and on the molecular properties of the surfactants used.Prolonged sonication above the phase- transition temperature yields vesicles which have a minimum size (usually ca. 200 A for phospholipids of natural origin) and a narrow size distribution. We now report measurements on concentrated and kinetically stable vesicular dispersions. The phospholipids used in this study were 1,2-diacyl-sn-glycero-3- phosphocholine (phosphatidylcholine; PC), which is zwitterionic at pH 3-1 1, and 1,2-diacyl-sn-glycero-3-phosphoserine (phosphatidylserine; PS), which has ionisable carboxylate, phosphate and ammonium groups and carries a net negative charge at neutral pH.Both phospholipids were of natural origin, and therefore possessed a distribution of acyl chain lengths and levels of unsaturation, resulting in phase- transition temperatures below 0 "C. The following discussion therefore relates to vesicles in the fluid state whose stability was not affected by proximity to a phase transition. The results of these experiments are explained in terms of interparticle interactions in these systems which cause some structuring even at fairly large separ- ations. The techniques used to give information on the particle structure factor P(Q) and the interparticle interference term S(Q) are dynamic light scattering 2 2 * 2 3 (photon correlation spectroscopy) and small-angle neutron scattering.24 In a scattering experiment the distance scale observed is governed by the wave- vector change on scattering, Q. For elastic and near-elastic scattering, Q is defined by 4nn 0 Q=- sin - A 2 where A is the wavelength of the probing radiation, 0 is the angle of scatter and n is the refractive index of the scattering medium (n = 1 for neutron-scattering experi- ments). The vesicles investigated here have a diameter of ca. 200 A. The use of light (A z 5 x lo3 A) and neutrons (A ss 10 A) thus allows exploration of a wide range of correlations within the scattering system, including long-range interactions and overall particle dimensions. The relative stability of the vesicle particle allows the single-particle and interparticle scattering terms to be separately determined in a way not possible for micellar systems.2 26 Previous neutron-scattering studies on vesicles have, however, been concerned with the bilayer structure itself rather than with the interparticle potentials.MUDDLE, HIGGINS, CUMMINS, STAPLES AND LYLE 79 2.THEORY We have calculated the interactions between vesicles by use of an extended form of the DLVO treatment.9~25 The net interaction potential between a pair of particles is given by the sum of repulsive and attractive contributions v = VR + VS VA where VR and V, are, respectively, the electrostatic and ‘hydration’ repulsion and VA is the energy of van der Waals attraction. VR is calculated using a modified form of the equation suggested by McCartney and Levine 27 where and 6 4 ~ a ( k T ) ~ ~ y ~ H -i- a a VR = z2e2 ---(I H + 2a +=exp(--k-H) y = [ exp (;:;) - - l]/[exp(%) + 11 where ni is the number of ions of species i carrying charge ei, a is the particle radius, E is the permittivity, H is the surface-to-surface separation, z is the number of charges, $0 is the surface potential and e is the electronic charge.The hydration force V, is calculated using the empirical formula of Rand and coworkers,28 which is derived from osmotic-pressure measurements for lamellar phases, modified for spherical geometry: Vs = .nat2Po exp( - H / t ) . (4) The decay length t is given as 2.56 A and the value of Po as 7.05 x lo9 dyn The attractive potential VA comes from the expressions of Vincent29 for attrac- tion between particles surrounded by a ‘single sheath’ adsorbed layer.In this case the ‘particle’ is water. The sheath (i.e. the vesicle wall) is taken to be a mixture of hydrocarbon and water, the precise composition of which is determined by the area per molecule (74 A2) and the bilayer thickness (40 A). The Hamaker constants for hydrocarbon and water were taken as 14kT and 8.9kT, respe~tively.~~ For the range of interparticle separations considered in this study (2-1500 A) and a typical surface charge density of 2 pC cm-2 calculation shows that the interac- tions are dominated by the electrostatic repulsion VR. The effective dispersed phase volume of such a system should therefore be much larger than its nominal phase volume, and structuring is to be expected.30 Assuming that the dispersion is essenti- ally monodisperse in the absence of added electrolyte, the condition of electrical neutrality permits interpretation of the coherent scattering data in terms of a pseudo-two-component s y ~ t e m .~ We adopt the notion of an effective hard-sphere diameter as suggested in a recent p ~ b l i c a t i o n . ~ ~ In this case the authors use the Barker-Henderson pairwise hard-sphere approach, which yields Cm-2 2 8 where V(x) is of the form of VR in eqn (2). x, is an arbitrary cut-off distance, in units of the particle radius, a.80 SCATTERING FROM UNILAMELLAR VESICLES In scattering terms the osmotic compressibility may be related to the intensity of radiation scattered into the forward angle. For a centrosymmetric system this is the Q = 0 limit of the well known structure factor30 S(Q) sin Qr S(Q) = 1 + 471p [g(r) - 11- r2 dr s Qr where g(r) is the pair radial distribution function, p is the particle number density and r is the centre-to-centre separation.The limiting expression (Q = 0) is related to the compressibility through S(0) = ( w ( a P / w where 7~ is the osmotic pressure and p is the particle number density. For vesicles, both the zero-Q limit and finite Q may be probed using light and neutrons, respectively. Percus-Yevick hard-sphere expressions for S(Q) have been applied to a variety of systems, including latex particles, silica particles and water-in-oil microemul- ~ i o n s . ~ ~ ~ ’ Recently, an analytical expression for S(Q) derived using a closed form of the Ornstein-Zernicke equation in the mean-field approximation has been applied to micelles and non-ionic surfactant~.**~~’~~ This method combines a hard-core repulsive potential with a coulombic repulsive term.Using such models, in the low-Q limit S(Q) is suppressed as the phase volume increases, and the peak in S(Q) sharpens. The position of the peak is related to the interparticle distance, d, through 2 4 Q z d. In a scattering experiment both this interparticle structure factor and the single-particle form factor P(Q) contribute to the scattered intensity. For spherical particles the intensity scattered into a given angle is given by where C is related to the optical refractive indices or the scattering-length densities for light or respectively, p is the number density of particles of molecular mass M and A is an instrumental constant.The particle form factor P(Q) for mono- disperse vesicles in which the bilayer is taken as having an isotropic polarisability is given by45.46 >’ 3 [sin x - sin(yx) - x cosx + yx cos(yx)] where and aI and aJ are the inner and outer radii, respectively. In the absence of any interparticle structuring [S(Q) = 11 the intensity should scale with P(Q).25p47 If the particle retains its integrity with increasing number density, a direct ratio of scat- tered intensities from the interacting and non-interacting system should yield S(Q)*25148,49 3. EXPERIMENTAL 3.1. MATERIALS Egg phosphatidylcholine (PC) Grade I and bovine spinal cord phosphatidylserine (PS), monosodium salt, Grade I were obtained from Lipid Products, South Nutfield, and the Sigma Chemical Co., Poole, and were subsequently used without further purification. Water wasMUDDLE, HIGGINS, CUMMINS, STAPLES AND LYLE 81 distilled from alkaline KMnO,.D20 was obtained from B.D.H. AnalaR-grade sodium chloride was obtained from Hopkin and Williams. 3.2. PREPARATION OF VESICLE DISPERSIONS Vesicles were prepared from pure PC and from mixtures of PC with PS containing 5% and 10% by weight of the charged component, PS. Ca. 500 mg of the required lipid mixture was dissolved in ca. 10 cm3 of a 2: 1 mixture of chloroform and methanol. The solution was placed in a 250 cm3 round-bottomed flask and the solvent removed by rotary evaporation; residual traces of solvent were removed by overnight storage in a desiccator under vacuum. 10 cm3 of water (or D20) was then added and the flask swirled to wash the lipid film from the walls.The resulting turbid suspension of multilamellar liposomes was transferred to a small glass beaker which was covered over with Nescofilm and flushed with nitrogen for 15 min. The sample vessel was immersed in an ice bath and kept under nitrogen during sonication, which was carried out using a Dawe soniprobe tuned for maximum cavitation. Sonication was continued in 5 min bursts, to prevent overheating of the sample, until the suspension clarified. Normally 15-20 min total sonication time was adequate; slightly longer was required for the pure PC, which did not disperse so easily. The sample was centrifuged at 120 OOOg for 2.5 h to remove titanium fragments (released from the sonicator tip) and remaining large liposomes.The upper fraction from the centrifuge tube was used in the case of dispersions in water; for those in D20 the lower fraction was taken. When experiments were to be done in the presence of salt, an appropriate volume of concentrated NaCl solution was added to the dispersion. After preparation the vesicle disper- sions were stored at 4 "C and were used as soon as possible. This stock dispersion (nominally 4% concentration) was diluted in simple ratios as required. Quantitative determination of phospholipid concentrations (PC + PS) was done by colorimetric phosphate analysis following the method of Morrison or by direct weighing of the dried samples. 3.3. SMALL-ANGLE NEUTRON-SCATTERING EXPERIMENT The dispersions were contained in 2 and 1 mm path-length quartz spectrometer cells (Hellma, Mulheim, Baden, Federal Republic of Germany). The D11 camera at the Institut Laue-Langevin in Grenoble was used for all the experiments.A wavelength of 10 h; was used in all experiments. Two detector-sample distances were used (10.66 and 2.66 m) which gave observed momentum transfer Q in the range 3.0 x to 8.7 x A - l , respectively. The scattered intensity per solid angle is given by a coherent contribution as in eqn (7), with C = (pv - ps)2 where pv and ps are the scattering-length densities of vesicles and solvent, respectively, together with an isotropic incoherent contri- bution. The incoherent scattering from the solvent is removed by subtraction in these experi- ments; that from the vesicles is small and was neglected at the concentrations used.The vesicle bilayer is considered to be of homogeneous scattering density since the variation between the hydrocarbon and the phospholipid head group is small, ca. & 1 x cm A-3 with re- spect to the difference between the scattering-length densities for the hydrocarbon (0.284 x cm and the D20 background (6.404 x lo-', cm k3). This gives (pv - ps) x 6.12 x cm A-3. This approach has been considered valid by other workers. The instrumental constant A in eqn (7) was removed by normalisation using the isotropic incoherent scattering from water. to 2.2 x and 1.2 x 3.4. PHOTON CORRELATION SPECTROSCOPY EXPERIMENTS The apparatus consisted of an E.M.I. 9863/100 KB photomultiplier and K7023 Malvern photon correlator with 48 delay channels.The optical source was a 5 mW Spectra Physics helium-neon laser which had a wavelength of 6328 A. All the experiments were carried out at82 SCATTERING FROM UNILAMELLAR VESICLES 25 k 0.5 "C. Detailed descriptions of photon correlation spectroscopy (P.c.s.) have been given elsewhere.22 However, a simple description of the important terms is warranted. The physical quantity that is measured in a P.C.S. experiment is the intensity autocorre- lation function which in its normalised form is g'2'(4 = ( I ( o ) w > / ( ~ > 2 (9) where I(0) is the intensity measured at some arbitrary time, I ( z ) is at some time z later and the average is taken over all starting times. For light that follows Gaussian statistics, this ex- pression may be related to the normalised field autocorrelation function Ig(')(z)l using the Siegert relationship (10) From statistical arguments, it has been shown that this expression applied to non-interacting system [ S ( Q ) = 13 reduces to (11) g'2'(z) = lg(*)(z)12 + 1.g(')(z) = 1 + exp[-z/(2DQ2)]. The mutual diffusion coefficient in the limit of infinite dilution is given by the well known D = kT/6nqR (12) where 9 is the viscosity of the medium and R is the equivalent sphere hydrodynamic radius of the particle. Eqn (1 I) predicts a linear form for a plot of lnfg(2)(z) - 11 against z. However, interparticle interactions may lead to deviations from linearity especially at long times. Even for non- interacting systems polydispersity and multiple scattering may lead to non-linear behaviour.In all instances the autocorrelation functions were fitted using the cumulant approach of K0ppe1.~~ (Only in the absence of multiple scattering and interactions can the resulting variance be related to the polydispersity.) For highly interacting systems the channel width z was varied and an effective D was obtained as z -, 0. Absolute intensity measurements were performed simultaneously with the correlation experiment. The usual dead-time correction was made and the solvent scattering removed. Stokes-Einstein equation 4. RESULTS AND DISCUSSION 4.1. NEUTRON SCATTERING Sample details are given in table 1 for uncharged (pure PC) and charged (PC with 10 and 5% by weight PS) vesicles. Experiments were carried out at concent- rations within the range 0.05-4.0% and with the addition of NaCl up to 0.1 mol dm-3. Fig.1 shows data for uncharged vesicles for which we expect little structure in S(Q) especially at high Q. The residual structure at high Q in fig. l(a) is therefore to be associated with the single particle form factor, P(Q). We can discard an explan- Table 1. Determined concentrations (mg cm-3) and lipid ratios technique neutron scattering light scattering sample PC 95:5 90: 10 90: 10" 90: loh concentration of nominal 4% material 25.2 & 0.5 21.6 & 4 41.1 k 4 48 k 3 43 _+ 3 ratio PC: PS 100: 0 96:4 91 :9 90: 10 91 :9 'Diffusion measurements; babsolute intensity measurements.MUDDLE, HIGGINS, CUMMINS, STAPLES AND LYLE 83 0.06 O*O8[ O x 0.04 0 5 10 15 20 25 Q ~ O - 3 A- I 0 0 I I I I 1 I I I I 1 0 5 10 15 20 25 0 5 10 15 20 25 o b Q/lO- A - p/10-3A-1 Fig. 1.Neutron-scattering data for uncharged PC vesicles. (a> Nominal 1% high Q. (6) Nominal 1% low Q. The solid line is the calculated P(Q) for a fit using the following para- meters: mean radius 95 & thickness 40 8, and h.w.h.m. 18 8,. (c) Comparison of low-Q data for nominal concentrations as shown ( x , 0.05%; 0, 1.0%). ( d ) As (b) with bimodal size distribution shown as solid line. ation in terms of interlamellar interference from multilamellar superstructures both because of the invariance of the structure with added salt and because the high-Q Kratky plots [In I(Q)Q2 against Q2] show a positive slope expected for unilamellar particles as opposed to the negative slope observed for polydisperse multilamellar particles.The high-Q data in fig. l(a) are therefore fitted by the single-particle form factor in eqn (8). Using a mean radius of 95 A and a bilayer thickness of 40 I$, a Gaussian distribution of radii 5 2 (half width at half maximum, h.w.h.m., 18 A) has been con- voluted with eqn (8) to give the solid line in the figure. At low Q the data deviate from this theoretical curve, as shown in fig. l(b). The high intensity at very low Q indicates the presence of very large particles, and closer examination of the data in fig. l(b) and ( c ) shows several oscillations which could arise from the Bessel function in the form factor of such large particles. Fig. l(c) also shows that a change in concentration by a factor of 20 produces negligible change in the scattering curve, confirming the absence of interparticle interactions and the integrity of both types of particles.If the oscillations in the scattering curves are associated with Bessel func- tions then this indicates that any large particles must be highly monodisperse. Indeed, fig. l(d) shows that the data could be approximately fitted by a bimodal84 SCATTERING FROM UNILAMELLAR VESICLES to2 - n .; Y Y 8 - 1 -E1: ._ 3 1 0 1 - W ._ Y ho @ 1 oo double Gaussian size distribution, comprising small particles as fitted for high-Q data and large particles with radius 900 A, membrane thickness 40 A and h.w.h.m. on the radius of 130 A. With this assumption of a vesicle form factor, in order to fit the data, the large particles must have a relative number concentration of 1 : 1200.Such trace contami- nants have been observed previouslys4 and, while their origin may remain obscure, their integrity allows us to handle the scattering in the same way as for the small particles. Fig. 2(a) and (b) show high-Q data for PC : PS 90 : 10 and 95 : 5 charged vesicle solutions, respectively. In this Q range S(Q) is expected to have reached its limiting 1 0 ‘lo - - I I I 1 - n U .“ 10’ $ 8 3 Y loo W A U .d lo-’ 0 0.025 0.05 0.075 0.1 QIA - ’ Fig. 2. Neutron-scattering data at high Q for charged vesicles. (a) 90: 10 PC:PS nominal concentration 4%. (b) 95:5 PC:PS nominal concentration 1%; 0, in the presence of 0.1 mol dm-3 salt; x , no added salt. The solid lines are the calculated P(Q) for a fit using the following parameters: (a) mean radius 77 A, thickness 40 A and h.w.h.m.13 A; (b) mean radius 71 A, thickness 40 A and h.w.h.m. 15 A. value of unity, and indeed fig. 2(b) shows little effect on the scattering curves when electrostatic interactions are screened by addition of 0.1 mol dm-3 salt. The single- particle form factor in eqn (8) again fits the data reasonably well, as shown by the solid lines in fig. 2(a) and (b). On the other hand, the values of the radii obtained (71MUDDLE, HIGGINS, CUMMINS, STAPLES AND LYLE 85 A for 90: 10 and 77 A for 95: 5) are unreasonably small compared with the other neutron results, which all give values > 90 A, and it seems possible that S(Q) is still contributing to the scattering curves. A negative slope in S(Q) would shift the Bessel function peaks to the right and cause eqn (8) to fit for an apparently diminished radius of the vesicles. Slight differences in the data for the 90: 10 sample could be observed after addition of salt (data not shown) and this would be consistent with a residual contribution of S(Q), incompletely screened by the salt.The effect cannot be seen for the 95: 5 data in fig. 2(b), however. The effect of concentration on S(Q) is clearly seen in the low-Q data for 90: 10 charged vesicles shown in fig. 3(a). On addition of 0.1 mol dm-3 salt the structure (except at low Q) disappears and data for a 1% solution become closely similar to that for 0.05% as shown in fig. 3(b). The oscillations at low Q in fig. 3(a) and (b) again indicate the presence of large particles.A relatively crude fit to the data in fig. 3(b) (better at high Q) using the method described for fig. l(d) agaiwindicated a bimodal distribution of particles with diameters of 180 and 2000 A. Fig. 4(a) and (b) show the effect on the data of progressively screening out the x x ( a ) x x x xx 40 0 X X X Io0I xx X 100 ':::i 0 0 1 . 4 - 1.2- 1 - .$ 0.8- 2 0.6- 2 0.4 - 0.2 F: W v1 - XX X x#*oOOoOO PXX&WO *O O O6 * +++++++++ + ++r* ++ 0000 ** +++++++ **** ****H**w**H* xxx xxxx x x m 0 9, +++ +++ ** ** + ++ I I I 1 1 5 10 15 20 25 ~110- 3 A-1 0 0 5 10 15 20 25 ~ / i 0- 3 A - 1 Fig. 3. Neutron-scattering data at low Q of 90: 10 PC : PS vesicles. (a) Intensity plotted against Q at various nominal concentrations: *, 0.5; +, 1; 0, 2; x , 4%. (b) Comparison of (0) nominal 0.05% and ( x ) nominal 1% in the presence of 0.1 mol dm-3 salt.86 22s n 9 's 150 75 SCATTERING FROM UNILAMELLAR VESICLES - - - 300 (a) O* 6 - n 9 3 4 .5 2 3 - ** ** **** 2 X X x x x xxxxxx 0 - 0 ,"** OW** 0 5 10 15 20 25 Qilo- 3 A - 1 gr x *** 7 . 5 1 * * (b) 0 1 I I I I 1 0 5 10 15 20 25 ~ / i o - 3 A- 1 Fig. 4. Neutron-scattering data at low Q for 90: 10 PC: PS vesicles in the presence of salt. (a) Nominal 4%. (b) Nominal 1%. x , 0.001; 0, 0.01; *, 0.1 mol dmP3 NaCl. interactions by addition of salt to 1% and 4% solutions of 90: 10 charged vesicles. Although the effect of salt on the data for the 1% solution [fig. 4(a)] is minimal above Q = 1.2 x A-1, this is not the case for the 4% solution [fig. 4(b)] in- dicating again (as suggested above in explaining the small particle sizes obtained from fig.2) that at high concentration S(Q) does not reach its limiting value of unity until rather high Q values are reached. The suppression in intensity at low Q at low ionic strength may be understood as an apparent reduction in the compressibility of the system arising from the increased effective diameter of the particles. A real increase in particle size is ruled out, in view of the invariance of the scattering at high Q shown in fig. 3(b). Relying on the evidence presented that both large and small particles retain their integrity with changing concentration, the structure factor S(Q) was obtained fol- lowing eqn (7) by dividing the high-concentration data by the whole curve for the lowest concentration (0.05% unstructured system).This assumes that the large par- ticles behave coherently with respect to the structuring and is formally justified if there is no correlation between size or orientation and position. Fig. 5(a) shows the results of the procedure for the 90: 10 system. The data follow the general predic- tions of the models described in section 2 normalising correctly to unity. The relia-MUDDLE, HIGGINS, CUMMINS, STAPLES AND LYLE 87 1.5 - t - Q s 0 . 5 - X 0 ** (4 *** ** ++++x ++ *+, 0 5 10 15 20 25 Q/ 10- 3 A- 1 Fig. 5. (a) Structure factor S(Q) plotted against Q for 90: 10 PC:PS vesicles for various nominal concentrations: x , 0.5; 0, 1; +, 2; *, 4%. (b) Interparticle spacing ( d ) obtained from peak position in (a) plotted against nominal concentration; gradient = - 0.321.bility of the low-Q data (and hence of our division involving the large particles) is confirmed by the light-scattering values of S(0) which are included in the figure (see section 4.2). However, the detailed shape of the curves cannot be fitted using analyti- cal expressions described in section 2.25 Riley and Oster 5 5 have shown that for a close-packed array e m a x 4 l a x - - 1.22. 271 However, Pusey4* has obtained an experimental value of 1.18 which we use here since our arrays may not be close packed but are certainly similar in organisation to those discussed by Pusey. Table 2 lists the interparticle spacings calculated from fig. 5(a), together with the particle number density pN calculated assuming cubic pack- ing. Fig. 5(b) shows that d,, scales closely with concentration’/3 as is necessary for any three-dimensional array of particles.A value for the number density can also be obtained from the exact concent- ration given in table 2. For a vesicle of radius 90 A, molecular volume 1480 A3,88 SCATTERING FROM UNILAMELLAR VESICLES Table 2. Interparticle spacing and number density for 90: 10 vesicles nominal actual number density number density W % ) c(wt%) spacing, d/A particle cm - particle cm- concentration, concentration, interparticle (4 PN/10l6 (4 pa/lo'6 0.5 0.51 74 1 0.24 0.25 1 1.03 59 1 0.48 0.49 2 2.06 469 0.97 0.98 4 4.11 378 1.85 1.96 ~ ~~ ~~ surface area per molecule ca. 74 A and lipid density 1.019 g cm-3,s6 values of 1345 and 359 molecules on the outer and inner layers of the bilayer, respectively,8,56 are calculated.The number densities pk calculated on this basis are shown in table 2 and are in good agreement with those calculated from the interparticle spacing. 4.2. LIGHT SCATTERING k1 and are hence at the Q -+ 0 limit of the neutron-scattering results. P.c.s. experiments were carried Light-scattering experiments fall in the Q range < 2 x 0 0 100 I I I I 1 0 1 2 3 4 300 c 250b 50 0 0 0 0 1 2 3 4 c(wt%) Fig. 6. Diffusion data for charged vesicles. Effective Stokes diameter plotted against nominal concentration. (a) 90 : 10 PC : PS vesicles; (b) 95 : 5 PC : PS vesicles.MUDDLE, HIGGINS, CUMMINS, STAPLES AND LYLE 89 out on freshly prepared D20 dispersions of PC : PS 90 : 10 and 95 : 5 charged vesicles. The sample details in table 1 show reasonable consistency with the neutron samples.The effective Stokes diameters (2R) obtained via eqn (10)-(12) are shown as a function of concentration in fig. 6(a) and (b) for 90: 10 and 95: 5 samples. (The viscosity of the D,O was determined separately using a polystyrene latex of well characterised size.) In both cases, there is a rapid increase at low concentrations to the infinite-dilution result. Addition of salt to the 4% samples increased the effective diameter back to the infinite-dilution result. No similar effect of salt or concen- tration was observed for uncharged PC vesicles (not shown). The data are sum- marised in table 3, where it is notable that the particle dimensions are larger than the Table 3. Effective diameters (A) and second cumulant 49 (in brackets) sample nominal 0.05% nominal 4% 4% + electrolyte PC 270 (0.1) 95:5 240 (0.2) 128 (0.7) 250 (0.1) 95: 10 245 (0.3) 175 (0.8) 238 (0.1) neutron results.The logarithmic plots of correlation functions [eqn (1 l)] were dis- tinctly non-linear at high concentration and this is reflected in the values of the variances included in table 3. No phase volume correction has been made to these values in table 3, but since cp < 0.07 this is not a large source of error and cannot explain the effect of concentration on the diameters. The intensity of scattered light followed the same reduction with increasing con- centration as the low-Q neutron data [fig. 4(a)] and again addition of salt to the concentrated dispersions restored the intensity.However, the addition of small amounts of electrolyte (resulting concentration < 5 x mol dm-3) produced little change in the compressibility of a 2% dispersion of 95 : 5 vesicles. This suggests an equivalent ionic strength for the dispersion of ca. 5 x mol dm-3, an ad- mittedly crude estimate which is consistent with a 50% dissociation of the PS. The addition of electrolyte to the charged vesicle solution should rigorously be treated in terms of a multicomponent s ~ l u t i o n . ~ 1, 58 The low-concentration intensity light- scattering results in fig. 7(a), however, indicate that the infinite-dilution limit is invariant with electrolyte concentration. Thus the intrinsic scattering cross-section of the particle remains unaltered despite the multicomponent nature of the dispersion.Fig. 7(b) shows intensity light-scattering data for 90: 10 vesicles over a wide concentration range, and at various concentrations of electrolyte. At these very low Q values where P(Q) tends to unity c / l is proportional to S(0)- [eqn (7)] and hence may be related to the phase volume of the sample. The data in fig. 7(b) have there- fore been normalised to unity at infinite dilution using the intercept in fig. 7(a). We have then applied a semi-empirical extension of the Percus-Yevick equation intro- duced by Carnahan and Starling57*58 to relate S(O)-l to the phase volume (14) 1 + 4q + 4cp2 + q4 - 4cp3 (1 - 0)" S(0) - 1 = where cp is the effective hard-sphere phase volume, related to the hard-sphere radius 0 and the number density p by q = 4/37tpa3.Curves (i) and (ii) in fig. 7(b) have been fitted to the data using an effective phase volume as a simple multiple of the nominal phase volume calculated using a particle model and the packing constraints for the90 SCATTERING FROM UNILAMELLAR VESICLES 15 12 o t I I I I I 0 0.05 0.1 0.15 0.2 0.25 8 7 6 5 2 4 3 2 1 9 I 1 I l l I I I l l 1 o - ~ lo-’ phase volume, 4p Fig. 7. Absolute intensity light-scattering data for 90: 10 PC: PS vesicles. (a) Inverse intensity as a function of low nominal concentration: *, no added electrolyte; 0, 0.01 mol dm-3 NaC1; +, 0.1 mol dm-3 NaCl; A = 632.8 nm. (b) Normalised inverse intensity against phase volume: +, no added electrolyte; *, 0.01 mol dm-3 NaC1; 0, 0.1 mol dm-3 NaC1. The solid lines are calculated curves as described in the text.lipid molecules outlined in section 4.1. The data at high salt concentration are fitted with a one-to-one ratio of the effective to nominal phase volumes, but in the absence of salt the ratio is 26: 1. It is, moreover, clear that eqn (14) cannot fit these data with a simple multiple of the phase volume. The dissociated Na + counterion’s contributions in the solution will, however, increase the effective ionic strength with increasing concentration. This in turn affects VR [eqn (2)] through the contribution of K [eqn (3)] and hence changes the effective hard-sphere radius in eqn (5). Curve (iii) in fig. 7(b) was obtained from eqn (14) using an effective phase volume calculated from eqn (5) for each nominal phase volume assuming arbitrarily that the PS species is 50% dissociated and that a surface potential of 30 mV persists.Adjusting the degree of dissociation and the surface potential produced fits of similar quality to curve (iii). A common factor in all the calculations was a divergence from the data at high concentration indicating that the sample is more compressible at these concen-MUDDLE, HIGGINS, CUMMINS, STAPLES AND LYLE 91 trations than the model suggests. Deformation of the particles themselves is partly ruled out by the invariance of the particle scattering factor in the high-Q (neutron) data as electrolyte is added. Consideration of only pairwise interactions appears justified as tc-' is typically < 0.25 times the intersurface spacing. Three possible reasons for the discrepancy are suggested: (a) reduction in surface potential (consis- tent with reducing the interparticle interaction); (b) the inadequacy of an equivalent hard-sphere approach for describing very soft potentials; (c) the sensitivity at high concentration of this approach to the assumed particle size. In the absence of hydrodynamic effects the apparent diffusion coefficient at finite concentration Deff (and hence the effective Stokes diameters in fig. 6 ) is directly re- lated to the infinite dilution Do value via S(0).23*25 Den = DOS(O)-l afid thus the effective diameter should directly follow S(0) as concentration increases.Com- parison of fig. 6 and 7 shows that this is clearly not the case. The differences in behaviour may be attributed to hydrodynamic effects in the diffusion data as- sociated with frictional drag.These effects have been calculated for hard spheres, and such corrections certainly improve the agreement between data in fig. 6 and 7. However, the corrections cannot explain the rapid change in apparent particle size at very low concentration, and until the exact form of the static behaviour has been clarified it is not fruitful to speculate on the details of the hydrodynamic interactions. 5. CONCLUSIONS (1) Both light- and neutron-scattering data confirm the integrity of the vesicles with changing concentration. (2) The extraction of S(Q) by simple division using the whole P(Q) curve including the large particles is well justified by the overall shape of the curves, which norinalise correctly to unity at high Q, by the variation in position of Qmax with concentration and by the agreement of the S(0) values with independent light-scattering results.The detailed shape of the curves remains to be explained. (3) Particle sizes for neutron and light scattering do not agree. This effect might partly be explained by failure of the assumption of a uniform scattering-length density through the bilayer, by difficulty in completely removing effects of S(Q) or the influence of polydispersity on measurements at different Q values. (4) Differences in the concentration dependence of S(0) and the effective diameters obtained from diffusion data indicate the importance of hydrodynamic effects in the latter. The very low concentration behaviour of Denremains to be explained. (5) At high con- centrations the samples are more compressible than predicted by hard-sphere models.D. A. Tyrell, J. D. Heath, C. M. Colley and B. E. Ryman, Biochim. Biophys. Acta, 1976, 457, 259. D. Papahadjopoulos, Ann. N.Y. Acad. Sci., 1978, 308. J. Goll, F. D. Carlson, Y. Barenholz, B. J. Litman and T. E. Thompson, Biophys. J., 1982, 38, 7. E. P. Day, J. T. Ho, R. K. Kunze and S. T. Sun, Biochim. Biophys. Acta, 1977, 470, 503. W. Knoll, J. Haas, H. B. Stuhrmann, H. H. Fuldner, H. Vogel and E. Sackmann, J. Appl. Cry- stallogr., 1981, 14, 191. V. Luzzati and F. Husson, J. Cell Biol., 1962, 50, 187. M. Eisenberg, T. Gresalfi, T. Riccio and S. McLaughlin, Biochemistry, 1979, 18, 5213. E. J. W. Verwey and J. T. G. Overbeek, Theory of the Stability of Lyophobic Colloids (Elsevier, New York, 1948).* C. Huang and J. T. Mason, Proc. Nut1 Acad. Sci. USA, 1978, 75, 308. l o S. Nir and J. Bentz, J. Colloid Interface Sci., 1978, 65, 399. l 1 S. Nir, J. Bentz and N. Diizgunes, J. Colloid Interface Sci., 1981, 84, 266.92 SCATTERING FROM UNILAMELLAR VESICLES l 2 M. Koter, B. de Kruijff and L. L. M. van Deenen, Biochim. Biophys. Acta, 1978, 514, 255. l 3 J. Lansmann and D. H. Haynes, Biochim. Biophys. Acta, 1975, 394, 335. l 4 J. Wilschut, N. Diizgiines and D. Papahadjopoulos, Biochemistry, 1981, 20, 3126. l 5 D. M. Le Neveu, R. P. Rand, V. A. Parsegian and D. Gingell, Biophys. J., 1977, 18, 209. I b A. C. Cowley, N. L. Fuller, R. P. Rand and V. A. Parsegian, Biochemistry, 1978, 17, 3163. l 7 R. P. Rand, Annu. Rev. Biophys. Bioeng., 1981, 10, 277. l 9 J.N. Israelachvili, D. J. Mitchell and B. W. Ninham, Biochim. Biophys. Acta, 1977, 470, 185. 2o E. G. Finer, A. G. Flook and H. Hauser, Biochim. Biophys. Acta, 1972, 260, 49. * C. Huang, Biochemistry, 1969, 8, 344. 2 2 B. J. Berne and R. Pecora, Dynamic Light Scattering (J. Wiley, New York, 1976). 2 3 P. N. Pusey and J. M. Vaughan, Dielectric and Related Molecular Processes (Specialist Periodical Report, The Chemical Society, London, 1975), vol. 2, chap. 2, pp. 48-105. 24 B. Jacrot, Rep. Prog. Phys., 1976, 39, 91 1. 2 5 J. B. Hayter and J. Penfold, J . Chem. Soc., Faraday Trans. I , 1981, 77, 1851. 26 D. Bendedouch, Sow-Hsin Chen, W. C. Koehler and J. S. Lin, J. Chem. Phys., 1982, 76, 5022. 2 7 L. N. McCartney and S. Levine, J . Colloid Interface Sci., 1969, 30, 345. 2 8 V.A. Parsegian, N. L. Fuller and R. P. Rand, Proc. Nafl Acad. Sci. USA, 1979, 76, 2750. 29 B. Vincent, J . Colloid Interface Sci., 1973, 42, 270. 30 P. A. Egelstaff, An Introduction to the Liquid State (Academic Press, London, 1967), chap. 6. 3 1 M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic Press, New 3 2 J. A. Beunen and L. R. White, Colloids and Surfaces, 1981, 3, 371. 3 3 R. H. Ottewill and R. A. Richardson, Colloid Polym. Sci., 1982, 260, 708. 34 A. Vrij, J . Chem. Phys., 1979, 71, 3267. 3 5 A. Vrij, J . Chem. Phys., 1978, 69, 1742. 3 6 H. M. Fijnaut, C. Pathmamanoharan, E. A. Nieuwenhuis and A. Vrij, Chem. Phys. Lett., 1978,59, 3 7 P. van Beurten and A. Vrij, J . Chem. Phys., 1981, 74, 2744. 38 M. M. Kops-Werkhouen and H. M. Fijnaut, J. Chem. Phys., 1982, 77, 2242. 3 9 M. M. Kops-Werkhouen, C . Pathmamanoharan, A. Vrij and H. M. Fijnaut, J . Chem. Phys., 1982, 40 D. J. Cebula, D. Y. Myers and R. H. Ottewill, Colloid Polym. Sci., 1982, 260, 96. 41 J. B. Hayter and J. Penfold, Mol. Phys., 1981, 42, 109. 4 2 J. B. Hayter and M. Zulauf, Colloid Polym. Sci., 1982, 260, 1023. 43 J. B. Hayter and T. Zemb, Chem. Phys. Lett., 1982, 93, 91. 44 J. S. Higgins and R. S. Stein, J . Appl. Crystallogr., 1978, 11, 346. 4 5 D. 0. Tinker, Chem. Phys. Lipids, 1972, 8, 230. 46 R. Sergio, S. Aragon and R. Pecora, J. Colloid Interface Sci., 1982, 89, 170. 47 J. C. Brown, P. N. Pusey, J. W. Goodwin and R. H. Ottewill, J . Phys. A , 1975,8, 664. 48 B. J. Ackerson, J . Chem. Phys., 1976, 64, 242. 49 J. C. Brown, P. N. Pusey and R. Dietz, J. Chem. Phys., 1975, 62, 1136. 5 0 W. R. Morrison, Anal. Biochem., 1964, 7 , 218. 5 1 K. Ibel, J. Appl. Crystallogr., 1976, 9, 296. 5 2 S. Komura, Y. Toyoshima and T. Takeda, Jpn J . Appl. Phys., 1982, 21, 1370. 53 D. E. Koppel, J . Chem. Phys., 1972, 57, 4814. 5 4 F. C . Chen, A. Chrzeszczyk and B. Chu, J. Chem. Phys., 1976, 64, 3403. 5 5 D. P. Riley and G . Oster, Faraday Discuss. Chem. Soc., 1951, 11, 107. 5 6 B. A. Cornell, J. Middlehurst and F. Separovic, Biochim. Biophys. Acfa, 1980, 598, 405. 5 7 N. F. Carnahan and K. E. Starling, J . Chem. Phys., 1969, 51, 635. 58 A. Vrij, E. A. Nieuwenhuis, H. M. Fijnaut and W. G. N. Agterof, Faraday Discuss. Chem. Soc., 5 9 B. J. Ackerson, J . Chem. Phys., 1976, 64, 242. M. E. Loosley-Millman, R. P. Rand and V. A. Parsegian, Biophys. J., 1982, 40, 221. York, 1969), chap. 9, p. 504. 351. 77, 5913. 1978, 65, 101.

ISSN:0301-7249    

DOI:10.1039/DC9837600077

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

GENERAL DISCUSSION Dr. P. N. Pusey (RSRE, Malvern) (communicated): These comments concern the analysis of elastic scattering data (neutron, X-ray and light) obtained from disper- sions of strongly interacting polydisperse spherically symmetrical particles. Thus they are relevant to several papers e.g. those by Hayter, Cebula et al., Ramsay et al. and Muddle et al. The starting point is eqn (1) of Dr. Hayter’s paper; after summing over the atoms within the particles this can be written: Here F(Q) is a particle scattering amplitude, defined by Hayter’s eqn (2)’ Rm is the position of the centre of particle m and N is the number of particles. Within the limits of a classical description eqn (1) is an exact result. However, for polydisperse particles, rigorous reduction of eqn (1) to a simpler form is not, in general, possible.This is because the scattering amplitudes and particle positions are usually corre- lated; for example, small weakly scattering hard-spherical particles can pack closer than large strong scatterers. Of course, faced with high-quality scattering data some analysis must be attempted. An obvious course, implicit in Hayter’s treatment and discussed in more detail by Pusey et al.,l Kotlarchyk and Chen2 and Hayter and P e n f ~ l d , ~ is to neglect the correlation between scattering amplitudes and particle positions and to factorize the average in eqn (I), i.e. m n where the brackets (. . .)p indicate an average over the particle size distribution. With use of the obvious identities (FmFn)p = ( F 2 ) p , if m = n if m = n (FmFn)p = ( F ) ; , (3) this ‘decoupling approximation’ leads to where S(Q) is an ‘effective monodisperse’ structure factor defined by Hayter’s eqn ( 3 ) .By normalizing I(Q) to its low-concentration value, N ( F ( Q ) ) P , (as one would if treating a monodisperse system) we can define an apparent structure factor Here the first term can be called ‘coherent’ [and is present for a monodisperse system where (F2(Q))p = (F(Q))a and the second term ‘diffuse’ or ‘incoherent’. The main purpose of this comment is to point out that there are three cases (at least) where the validity of the decoupling approximation implicit in eqn (2) can be tested. The results are summarized in table 1. We consider only scattering at small Q for systems where S(Q) << 1 and the relative importance of the diffuse (second) term in eqn (5) is greatest.94 GENERAL DISCUSSION The first case concerns polydisperse homogeneous hard spheres, for which a theory which takes account of the correlation betwen scattering amplitudes and particle positions has been developed by Vrij and and coworker^.^-^ This theory is based on the Percus-Yevick approximation, which is generally accepted to be quite accurate for hard spheres.We consider the case of volume fraction cp = 0.2 and a Schulz particle-size distribution of standard deviation 0 = 0.2 [see section IV of ref. (l)]. From table 1 we see that the incoherent scattering (column 3) calculated using the decoupling approximation [the second term in eqn (5) or eqn (4.2) of ref. (l)] is about four times greater than that (column 4) calculated using the more realistic Percus-Yevick approximation [ A - from eqn (4.9), (7.9) and (7.13) of ref.(l)]. Furthermore, the total scattering calculated from the decoupling approximation [eqn (9, first entry, column 23 is about twice that calculated from the Percus-Yevick approximation [ref. (l), eqn (4.9) and second entry, column 21. Thus, as we con- cluded previously, the decoupling approximation ‘is entirely inadequate to describe size polydispersity in a mixture of hard spheres’ [ref. (l), section IVJ. The second case concerns a dilute suspension of charged polystyrene ~pheres,~ for which the particle-size distribution was measured by electron microscopy.6 It was possible to measure the total scattering5 (column 2) and to separate it into incoherent (column 4) and coherent contributions by dynamic (photon correlation spectroscopy) measurements.(An analysis of quasi-elastic scattering, ‘1’ - similar to that given above for elastic scattering, shows that coherent scattering reflects collective particle motions while incoherent scattering reflects single-particle motions. For strongly interacting systems, collective and single-particle motions studied in the low-Q limit have very different time dependences, thus allowing the separ- ation of the two components.) Here the incoherent scattering calculated from the decoupling approximation (column 3) is a factor of two greater than that mea- sured (column 4). Recently Hart1 and Versmoldl* have studied a mixture of two individually monodisperse dilute dispersions of different-sized charged polystyrene spheres (table 1, case 3).By comparing scattering data from the individual dispersions with those obtained from the mixture they derived total scattering (column 2) and incoherent scattering (column 4). In this case the incoherent scattering calculated using the decoupling approximation (column 3) is a factor of two or so less than that measured. Thus I make the following points: (i) The decoupling approximation is bad for hard spheres in the Q + 0 limit, although it is probably better away from this limit [cf. fig. 2 of ref. (2), based on the decoupling approximation, with fig. 5 of ref. (1 l), based on the better Percus-Yevick approximation]. (ii) For particles showing longer-ranged repulsive interactions (cases 2 and 3) the decoupling approximation appears to be better.A possible interpretation is that the Coulombic interaction between the smaller weakly charged particles is less well shielded than that between the larger particles, so that the correlation between scat- tering amplitudes and particle positions is red~ced.~ Nevertheless it must be realized that complications can also arise from ‘charge polydispersity’ even for identically sized particles. (iii) In summary, while the decoupling of eqn (1) is an obvious route to take and may work reasonably well in some cases, it must always be remembered that it is an approximation. Theoretical models based on this approximation must naturally be viewed with some caution. The fact that experimental data can be made to fit such aGENERAL DISCUSSION 95 model does not necessarily imply that the model and parameters derived from it provide a correct description of the system.Clearly computer simulation of polydis- perse systems would provide valuable further tests of the decoupling approximation. P. N. Pusey, H. M. Fijnaut and A. Vrij, J. Chem. Phys., 1982, 77, 4270. M. Kotlarchyk and S. H. Chen, J. Chem. Phys., 1983, 79, 2461. J. B. Hayter and J. Penfold, Colloid Poiym. Sci., in press. A. Vrij, J. Chem. Phys., 1978, 69, 1742. P. N. Pusey, J. Phys. A , 1978, 11, 119. J. C. Brown, P. N. Pusey, J. W. Goodwin and R. H. Ottewill, J. Phys. A, 1975,8, 664. P. N. Pusey, in Light Scattering in Liquids and Macromolecular Solutions, ed. V. Degiorgio, M. Corti and M.Giglio (Plenum Press, New York, 1980). D. J. Cebula, R. H. Ottewill, J. Ralston and P. N. Pusey, J. Chem. SOC., Faraday Trans. I , 1981,77, 2585. P. van Beurten and A. Vrij, J. Chem. Phys., 198 1, 74, 2744. 13 M. B. Weissman, J. Chem. Phys., 1980, 72, 231. l o W. Hart1 and H. Versmold, J. Chem. Phys., in press. l2 G. D. J. Phillies, Macromolecules, 1976, 9, 447. Table 1. Tests of the decoupling approximation (F(0));l incoherent 1 -~ Sapp(0) (total scattering) < F Z ( W P contribution (1) hard spheres O.43Ob 0.234' Schulz distribution d = 0.2, cp = 0.2 (2) charged spheres 5 - 7 ~ 0 . 1 6 (3) charged spheres" 0.41" 0.280 0.075' 0.19 < O.Ogd 0.11 0.24" Incoherent contribution calculated from the particle-size distribution and based on the decoupling approximation.* Calculated from the decoupling approximation, eqn (5). Calculated from Percus-Yevick approximation, ref. (1). Determined from photon correlation measurements. Measured from fig. 1 of ref. (10). Prof. A. Vrij and Dr. J. Moonen (University of Utrecht, The Netherlands) said: Our comment considers questions similar to those discussed by Dr. Pusey concern- ing the analysis of interacting polydisperse spherically symmetrical particles. For convenience we will use his nomenclature. We did some model calculations using the program of van Beurten and Vrijl to compare the structure factor as defined by them with the 'effective monodisperse' structure factor as defined by Hayter. These structure factors are designated as S(Q)app and S(Q), respectively, in the comment of Pusey.By rewriting Pusey's eqn ( 5 ) one obtains In the special case that (F(Q)) = (F(Q))2, all particles have the same scattering power and S(Q) equals S(&,,,. One could say that this structure factor, which we call S(Q)ref, is the pure reflection of the structure, caused by the (polydisperse) hard- sphere interactions, with which S(Q) and S(Q)app could be compared. We performed calculations on a hard-sphere size distribution as given in fig. 1 which mimics a silica system surrounded by a non-scattering hydrocarbon layer 15 A thick. This distribution was determined by means of SAXS and electron micro-96 GENERAL DISCUSSION h v d I .o 0.8 0.6 0.4 0.2 0 50 I00 150 200 2 50 4 Fig. 1. Particle-size distribution function as determined by (---) small-angle X-ray scattering (a = 15%) and (- ) electron microscopy ( D = 18%). a is the radius of the silica core.~copy. In the first calculations the scattering power of all particles was taken to be identical [i.e. (P(Q)) = (F(Q))2 = 1 for all Q]. In the second calculation the scat- tering material is taken to be homogeneously distributed over the particles. For all particles a non-scattering outer layer of 15 A was assumed. Both S(Q) and S(Q)app are calculated. 1 . 5 I 0 5 10 QdO Fig. 2. S(Q) plotted as a function of Qdo; do is most probable diameter (do = 375 A); (----) S(Q),ef; (- - -) S(Q), according to Hayter; (- . . - .) S((&,,, according to van Beurten and Vrij; the volume fraction, q, is 0.235. All three structure factors are shown in fig. 2. One observes that for small Q values S(Q)app is in much better accordance with S(Q)ref than S(Q).Other deviations arise around the minimum of the form factor. This is true both for S(Q) and S(Q)app but the deviations are stronger for S(Q). That this is to be expected can be seen from eqn (1). For a polydisperse system ( F ( Q ) ) , will become zero around Qdo = 8.5 (where do is the most probable hard-sphere diameter). On the other hand (F(Q)) will always be greater than zero. This implies that S(Q) may become very large or small. We are aware of the fact thatGENERAL DISCUSSION 97 [S(Q)app - 13 is also very small around the minimum of the form factor. Nonetheless it seems to differ significantly from zero. Our conclusion is that, although at the maximum S(Q) and S(Q)app are in good accordance with S(Q)ref for hard spheres, the structure factor as defined by van Beurten is in better accordance with S(Q)ref both for smaller and larger Q values.* P. van Beurten and A. Vrij. J. Chem. Phys., 1981,74, 2744. Prof. H. Hoffmann (University of Bayreuth, West Germany) said: Could Dr. Hayter comment on how large S(Q) can become at the scattering peak for a system consisting of spheres under repulsive conditions with the system in the liquid state? Dr. J. B. Hayter (ILL, Grenoble, France) said: This question has been studied in detail by Hansen and Schiff using computer simulation techniques applied to a variety of repulsive potentials. In the specific case of the Coulomb potential the liquid freezes when the amplitude of the first peak in S(Q) reaches 2.55; for hard spheres the corresponding value is 2.85.J-P. Hansen and D. Schiff, Mol. Phys., 1973, 25, 1281. Dr. R. B. Jones (Queen Mary College, London) said: Dr. Hayter has said that polydispersity can be handled well by the use of e.g. eqn (2) together with the RMSA procedure in the case of charged suspensions with strong long-range forces. As the range of the forces is reduced, at what range will the procedure cease to work adequately and hence need to be replaced by an explicit hard-sphere-like treatment? Dr. J. B. Hayter (ILL, Grenoble, France) said: The problem of polydispersity is far from being resolved, and I can only indicate some currently known limits. At one extreme we have highly charged spheres in dilute suspension at low ionic strength, where there is significant Coulombic interaction over average interparticle distances of the order of ten diameters.Eqn (2) of my paper describes the measured static scattering from such systems very well at size polydispersities at least up to 2Ox.l Even in this case, however, the use of the same approximation (i.e. no correlation between size and position) to calculate the dynamic scattering gives results at vari- ance with experiment.2 At the other limit of very short-ranged interaction, com- parison of eqn (2) with exact numerical results for polydisperse hard spheres shows that the approximation is poor in this case, especially at low Q. I conjecture that the cross-over will occur when the width of the size distribution becomes an appreciable fraction of the range of the potential, but we clearly need more extensive numerical results for polydisperse systems with potentials of various ranges before a more quantitative statement can be made.J. B. Hayter, in Physics of Amphiphiles: Micelles, Vesicles and Microemulsions, ed. V. Degiorgio and P. N. Pusey, in Light Scattering in Liquids and Macromolecular Solutions, ed. V. Degiorgio, M. Corti M. Corti (Italian Physical Society, Bologna, in press). and M. Giglio (Plenum Press, New York, 1980), pp. 1-29. Dr. G. J. T. Tiddy (Unilever Research, Port Sunlight) said: Dr. Hayter’s micellar model for the lower consolute behaviour of the C8E,/water system is very convinc- ing. In the 7% solution the micelles occupy a large volume fraction. Assuming a hydrocarbon radius of 1.0 nm, with a total radius of 2.35 nm and a hydrocarbon98 GENERAL DISCUSSION volume fraction of ca. 2.6%, the volume occupied by micelles is ca.34%. Is this relevant for the well defined peak at r/d = 2 in fig. 3b? Also, can one estimate the maximum degree of anisotropy possible for the micelles if non-spherical shapes occur? Dr. J. B. Hayter (ILL, Grenoble, France) said: The volume fraction may be calculated without recourse to any model of the micelle geometry, provided the degree of hydration is specified. If there are x molecules of solvent bound in the micelle for each surfactant monomer, the volume fraction cp occupied by micelles in the solution is where V, and V, are the solvent and monomer volumes, respectively, and q m is the volume fraction of dry surfactant in the micelles.(cpm is known from the solution composition and the c.m.c.) In our experiments cpm = 0.069 and Vs/Vm = 0.059. The usual solvation value of about two solvent molecules per oxygen would give x = 10, but we found the absolute intensity measurements we performed were better fitted by taking x = 6, yielding cp = 0.093. This value was used in the calculations of S(Q). Dr. Tiddy's value of cp = 0.34 would imply x = 66. This very high figure probably results from using an unsuitable geometrical model to calculate cp; as stated above, cp may be calculated without invoking any geometrical model. The well defined peak in fig. 3(b), on the other hand, relies on taking the micelles as spherical. The relevance of the peak position at r = 2d is that the spherical micelles are predicted to be in contact in the clusters through at least two neighbours.The question of sensitivity to anisotropy is related to the problem of polydisper- sity. For example, the scattering from a dispersion of monodisperse ellipsoids may always be modelled as if the particles were suitably polydisperse spheres.' We have been able to model the system as if the particles were spherical with a low degree of polydispersity. If this apparent result were in fact due to rather more monodisperse but anisotropic particles, numerical estimates indicate that the major-to-minor axial ratio would be ca. 1.5 : 1. J. B. Hayter, in Physics of Amphiphiles: Micelles, Vesicles and Microemulsions, ed. V. Degiorgio and M. Corti (Italian Physical Society, Bologna, in press).Dr. Th. F. Tadros (ICI, Jealott j . Hill) (communicated): In the calculation for attractive interactions in which Dr. Hayter used non-ionic surfactants as a model system, the assumption was made that the micelles have a fixed diameter as the temperature is increased. This is clearly not the case as demonstrated in the paper by Neeson et al. (this Discussion) which shows that the micelles increase in size with increasing temperature. Did Dr. Hayter consider taking the size increase into ac- count for comparing the experimental results with the theory based on the attractive in terac ti ons? Dr. J. B. Hayter (ILL, Grenoble, France) (communicated): The paper by Neeson et al. at this Discussion refers mostly to micellar solutions adjacent to cubic, hexa- gonal or lamellar phases.The system I describe is more dilute (7% C,EO,) and was chosen specificially to avoid working near to any mesophase boundaries, so that a direct comparison with the results of Neeson et al. is otiose. The 7% C,EO, scatter- ing results are quantitatively described by the temperature-dependent clustering of fixed-diameter spherical micelles. Note that such a system could exhibit birefringenceGENERAL DISCUSSION 99 even though the particles are isotropic. An applied electric field will induce a dipole parallel to the field in each particle. Although there will be no further interaction with the field, there will now be an anisotropic induced-dipole-induced-dipole inter- action between particles, and the clusters will tend to become chain-like along the field.Only a careful quantitative analysis would distinguish this behaviour from the alignment of anisotropic particles. We have still to resolve the question of size variation with concentration, and work is currently under way to study this point, as well as the influence of meso- phase boundary proximity in similar systems with longer-chain surfactants. Dr. M. Warner (Rutherford-Appleton Laboratory, Chilton, Oxfordshire) said: Dr. Hayter’s paper does raise some interesting questions. Thus one wonders what is the nature of the phase change alluded to in the discussion of attractive interactions? Is the resulting phase like a solid in any way, i.e. would it have a shear modulus like a colloidal crystal? How should one assess the validity of Dr.Hayter’s theory (based on particular molecular potentials) in describing such critical aspects as ‘the build-up of very long-ranged correlations in the dispersion’? Prof. V. Degiorgio (University of Pavia, Italy) said: Fig. 3(b) of Dr. Hayter’s paper shows that the radial distribution function g(r) for non-ionic micelles in aqueous solution presents a second-neighbour peak which becomes stronger as the phase-transition temperature is approached. This is an unexpected result for a crit- ical binary mixture because it is known that the onset of long-range order in the critical region is not accompanied by a modification of the short-range order. Is it possible that this result is an artifact, due to the approximations involved in the MSA, or does Dr. Hayter think it is a real effect which differentiates the critical behaviour of colloidal solutions from that of normal binary solutions? Dr.J. B. Hayter (ILL, Grenoble, France) said: I think it is a real effect. The basic point of my paper is that the correlations between particles in colloidal solutions are effectively those of a pure one-component fluid. It is thus not surprising that the critical behaviour differs from that of a binary mixture in which both components play similar roles in the physics of the isotropic phase. In the OCM calculation of fig. 3(b) it is precisely the onset of short-ranged order, due to increased attraction between the particles in the fluid, which leads to the formation of clusters and hence to correlations at long range.Prof. B. U. Felderhof (RWTH Aachen, West Germany) said: Dr. Hayter states in his paper that a system with an unscreened Coulomb potential has zero osmotic compressibility and moreover that such a system has no direct relevance to charged colloidal dispersions. These statements seem to me to be incorrect. A system with unscreened Coulomb interactions has a perfectly well defined equation of state and hence a non-vanishing compressibility. Of course in a one-component system one does need a neutralizing background to make the system stable. I also thought that a system of charged polystyrene spheres without added salt is reasonably well de- scribed by the above model. Dr. J. B. Hayter (ILL, Grenoble, France) said: Prof. Felderhof s comment on the equation of state of the one-component plasma (OCP) is quite correct.The usual relation, eqn (9), is not applicable to a fluid with an interaction potential as long-100 GENERAL DISCUSSION ranged as the (non-integrable) Coulomb potential. In the latter case the compressi- bility may remain finite (although not necessarily positive in the case of the OCP l ) even though S(Q -+ 0) = 0. The question of the relevance to charged colloidal dispersions of calculations based on a pure Coulomb (unscreened) potential has been addressed by Hansen and myself.2 The general rule is that an unscreened potential may be used whenever AD 2 2a (1) where AD is the Debye screening length in the dispersion and a is the ion sphere radius, defined for number density p by 4na3/3 = l/p. (2) Condition (1) applies to sufficiently dilute charged polystyrene spheres without added salt, the data of fig.2 providing a well known example. For many of the dense charged dispersions currently being studied, however, the counterions present from the ionized particles provide significant screening, which must be taken into account even in the absence of added salt. In the particular case of charged micelles, the concentration of ionized monomers is usually such that an unscreened Coulomb calculation would only be relevant in the case of an exceptionally low c.m.c. * P. Vieillefosse and J-P. Hansen, Phys. Rev. A, 1975, 12, 1106. J-P. Hansen and J. B. Hayter, Mof. Phys., 1982, 46, 651. Prof. B. U. Felderhof (RWTH Aachen, West Germany) said: The curve given in fig. 11 of Prof.Vrij’s paper seems to me to contain information on the polydispersity of the system. Has an attempt been made to extract this information? Prof. A. Vrij (University of Utrecht, The Netherlands) replied: Yes. We write for the scattering amplitude for a particle i, Qi = (ni - n0)K where ni is the (mean) refractive index and & the volume of particle i and no is the refractive index of the solvent, and assume that all particles have the one can show that Using the equations for e2 and [below eqn (5.1) and eqn (2.8)] - ( i i i - no)2 - (An)’ - - Q 2 -- - - Q2 (ni - fii)’+ (iii - no)’ 0; -I- (An)2 with same size. Then one obtains 0; = (ni - i i i ) 2 the variance of the refractive index of the particles. and it looks reasonable. A calculated plot is drawn in fig.3 for kD, = 1.45, kD, = -2.68 and on = 0.01, Dr. R. B. Jones (Queen Mary College, London) said: I have a question and a com- ment. I would like to ask Prof. Vrij for confirmation that in the low-concentrationGENERAL DISCUSSION 101 - 4 0 0.010 0.020 0.030 All Fig. 3. Calculated plot of kD as a function of An. measurements from which he obtains the self-diffusion coefficient he always sees a single exponential rather than two exponentials. My comment is that this would be evidence that the silica particles behave as good hard spheres in the sense that the hydrodynamic radius Rh is nearly equal to the hard-sphere potential radius R,. For the ideal case when R, = Rh there are three calculations of memory effects -3 in the self-diffusion at low concentration and low wavenumber, including hydrodynamic interactions. All three calculations now agree agree that memory contributions are quite small by comparison with the short-time result Do (1-1.839), suggesting that experimentally one should see single-exponential behaviour.In contrast, if the potential radius of the hard-sphere R, were consider- ably larger than the hydrodynamic radius Rh then memory effects become large. For example, using the formalism of ref. (2) for the case R, = 2Rh, I have calculated that the short-time self-diffusion coefficient becomes Do (14.929) while the long-time result is Do( 1-0.929-6.429) = Do( 1-7.349). (Here cp = n%nRz with n the number density.) In such a case I would expect clear deviations from a single exponential which Prof.Vrij apparently does not see. S. Hanna, W. Hess and R. Klein, Physica, 1982, 111A, 181. R. B. Jones and G. S. Burfield, Physicu, 1982, 111A, 562, 577. G. K. Batchelor, .I. Fluid Mech., in press. Prof. A. Vrij (University of Utrecht, The Netherlands) said: Please refer to fig. 10 of our paper. At the lowest temperature, where self-diffusion prevails, the second cumulant has a small value as found in very dilute solutions. This is consistent with a single exponential (at K = 0). For the highest temperature in fig. 10 the cumulant was appreciably larger, so a pure exponential is not then found. This is attributed to the presence of a second exponential due to collective diffusion, which becomes perceptible farther away from the matching point, where the contribution of ‘coher- ent’ scattering becomes larger.102 GENERAL DISCUSSION Prof.R. Klein (University of Konstanz, West Germany) said: In the Discussion Dr. Jones raised the question whether the autocorrelation function which describes self-diffusion is a pure exponential function of time in the limit as k --+ 0. The answer is no. Whereas the memory term in the case of collective diffusion vanishes faster than k 2 , this does not apply to self-diffusion. Denoting the autocorrelation function for self-diffusion by G(k, t ) one can show that in the hydrodynamic limit k -+ 0, t -+ co, k2t constant, G(k, t ) = exp(-D, k 2 t ) , where D, is the true self-diffusion coefficient. The short-time behaviour is, however, given by the following first cumu- lant: p l ( k ) = 0," k 2 , where D," is given by D," = k B T / ( [ i ~ ) o .Here the denominator is the configuration-space average of the (22)-component of the friction tensor of the tagged particle. Without hydrodynamic interaction one has simply 0," = Do, the free diffusion coefficient of a non-interacting Brownian particle. Summarizing, the initial decay is characterized by 0," and the decay at large times by D,. For the case of hard spheres with hydrodynamic interaction the mem- ory contributions are not large; according to Batchelor 0," = Do(l - 1.8340) and D, = Do(l - 2.11~) to lowest order in the concentration. For the highly charged spherical particles the difference can be much larger. For analysing experiments it is of course of interest to know at which correlation times the long-time behaviour is reached.This crossover time z, above which D, characterizes the decay of G(k, t ) , is the time where the mean-square displacement reaches its final behaviour ( ( r i ( t ) - ~ ~ ( 0 ) ) ~ ) = 6 D,t. We have estimated z for the highly charged polystyrene spheres and find z to be of the order of 10-3-10-2s. Dr. W. van Megen (Royal Melbourne Institute of Technology, Australia) said: Aside from the fact that some controversy ex'sts concerning the value of kD in eqn the paper of Pusey and Tough], is it really meaningful to compare experimental data at volume fractions up to q x 0.2 with theoretical results valid only in the con- centration regime linear in cp? (5.4) of Prof. Vrij's paper, for the long-time se \ f-diffusion constant [see eqn (2.44) in Prof.A. Vrij (University of Utrecht, The Netherlands) said: We used two methods for the experimental measurements of the diffusion coefficient for the long-time self- diffusion. The first one [see ref. (21) of our paper] is based on incoherent scattering in an optically polydisperse system. The measurements in the concentration range 4p 5 0.2 were linearly fitted. This leads to a value of kD = -2.7 & 0.5 (not +0.3 as originally given in the text). The concentration range is rather large but so also is the uncertainty in kD. The second method [see ref. (22) of our paper] is based on tracer measurements in the concentration range cp 5 0.09. This leads to kD = -2.7 0.3, so in both independent measurements we find the same values although the concentration range is different.Prof. V. Degiorgio (University of Pavia, Italy) said: I would like to know how good is the assumption that Prof. Vrij's particles behave as hard spheres. In parti- cular, I am referring to fig. 11, which shows kD as a function of T. In principle it should be possible to explain (at least partially) the behaviour of kD by assuming a small attraction plus a small repulsion, which compensate each other at the high temperature, but become unbalanced as T is decreased. Prof. A. Vrij (University of Utrecht, The Netherlands) replied: To check if the decrease in kD with decreasing temperature might be caused by attractive forces atGENERAL DISCUSSION 103 the lower temperatures, sedimentation velocity experiments were performed (as a function of particle concentration) at different temperatures.[See ref. (2 1) of our paper, fig. 9.1 For low concentrations the sedimentation velocity is For hard-sphere repulsive forces ksed = -6.5. When attractive forces are present ksed should increase (become less negative). We found, however, no influence of ksed on temperature. Another indication is that data of D, ( K = 0) when plotted against concentration in the lower concentration range c < 0.39 ~ m - ~ fit smoothly with data of D, at the higher concentration range. The D, in the higher concentration range, however, are determined by a quite different analysis. Prof. H. Hoffman (University of Bayreuth, West Germany) said: I assume that attractive interactions in Prof. Vrij’s model system can also become important when the concentration of the particles is increased, and it is conceivable that the system can go to a condensed or liquid-crystalline state.Have such transitions been ob- served, and what might be the nature of any such new phases? Prof. A. Vrij (University of Utrecht, The Netherlands) answered: In good solvents (like cyclohexane) the dispersion of colloidal particles can form colloidal crystals when the concentration is large enough. (See the last part of section 2.1 of our paper.) This is believed to be the so-called Kirkwood-Alder transition, which is also found in computer simulations of hard-sphere fluids. In ‘poorer’ solvents (e.g. benzene) attractive interactions can occur. Here we find a separation into two phases (a dilute and a concentrated phase) when the temper- ature is decreased below ca.40 “C. This type of separation can be compared with an ordinary liquid-gas transition of low-molecular-weight fluids. In the colloidal dis- persion the silica particles then play the r6le of ‘supramolecules’. Prof. B. J. Ackerson (Oklahoma State University, U.S.A.) said: Prof. Ottewill makes a point of comparing the position of the maximum height of the liquid static structure factor with the corresponding position for a face-centred cubic crystal; however, in fig. 9 of his paper the comparison of the experimental g(r) with that generated by Brownian dynamics is off systematically. Is it known what sort of packing is generated by the Brownian dynamics? Do the experimental and Brownian dynamics results represent different packing states? Prof.R. H. Ottewill (University of Bristol) replied: I do not know what packing the Brownian dynamics represents. The experimental packing seems to be more ‘loose’ and may represent a simple cubic structure. Inversion of the experimental data to produce g(r) may also introduce errors. Prof. B. J. Ackerson (Oklahoma State University, U.S.A.) said: While represent- ation of the liquid as an ‘ordered’ or ‘disordered’ state may seem peculiar to many of us, it is interesting to note that the cross-correlation function (see the paper by Ackerson and Clark) for a computer simulation of a two-dimensional hard-disc liquid is not significant until the liquid is ‘ordered’. Before the local packing is great enough, the liquid is very random and exhibits few three- and four-body effects in the cross-correlation function.As the packing increases the maximum value of S(Q)104 GENERAL DISCUSSION at Qmax shifts to the position seen in the solid phase and the cross-correlation then shows significant three- and four-body effects. Dr. E. Dickinson (University of Leeds) said: I address these remarks to Prof. Ot tewill . (1) I should like to attempt to clarify the interpretation of the potentials of mean force given in fig. 7 by Cebula et al. The potential of mean force @(r) = - kT In g(r) has the character of a free-energy function rather than just a potential energy. Oscill- ations in @(r) for a concentrated system are primarily of entropic origin, associated with the fact that, owing to packing considerations, there is a spatial periodicity in the weighted probability of finding two particles a distance r apart.Curves similar to that shown in fig. 7(b) are found with systems having no attractive forces at all (hard spheres). This means that the oscillations in @(r) have little to do with the form of the interparticle pair potential V(r) at separations r corresponding to second or third ‘layers’ of particles. I think that it is a little misleading to say that ‘it is the “tail” of the potential curve which strongly influences particle-particle interactions in col- loidal dispersions’. (2) In fig. 9(c) of the paper by Cebula et al. there is a comparison between g(r) from computer simulation and that obtained experimentally by neutron scattering. While the overall agreement is generally good, there is a small but clear discrepancy in the shape and position of the second peak in g(r) for the two cases.Could Prof. Ottewill please comment on three possible reasons for the difference. First, there is the effect of the distribution of particle sizes in the polystyrene latex sample. Our computational evidence 1r suggests that the effect of polydispers- ity is to broaden the peaks in g(r). The experimental curve in fig. 9(c) shows a deeper first trough and sharper second peak than the simulation g(r). Therefore it would seem that the discrepancy between experiment and simulation cannot be entirely attributed to polydispersity in the latex sample. Secondly, it is well known that there are inherent uncertainties in calculating g(r) from S(Q) owing to the fact that scattering intensities are measurable only over a finite range of Q.To determine whether the difference in the position of the second peak in g(r) is experimentally significant or not, can the authors give some estimate of the maximum numerical error of the Fourier transformation? Thirdly, the question arises as to whether there was any partial solid-like order- ing in the experimental colloidal dispersion. While the radial distribution function from the simulation is clearly liquid-like, the stronger oscillations in the experi- mental g(r) are perhaps indicative of some solid-like ordering. Was there any irides- cence from the latex samples containing mol dm-3 sodium chloride, and was there any gradual change in structure with time following addition of electrolyte to the ion-exchanged latex? ( 3 ) In connection with the Brownian-dynamics computations, the authors quote a scalar diffusion coefficient D(u) which depends on the reduced pair separation u.It is not clear exactly what form of diffusion tensor was used in the simulation. In particular, I should like to know (a) whether or not each particle has the same diffusion coefficient during a particular time step and (b) whether u refers to some average pair separation in the dispersion or the instantaneous closest separation between each pair in the system. Strictly speaking, it seems inconsistent to use a ‘moving-on’ routine applicable in the absence of hydrodynamic interactions [eqn (23) of the paper] in conjunction with a diffusion coefficient which includes hydrodynamic interactions through the para- meter P(u). However, as the authors are interested here only in time-averagedGENERAL DISCUSSION 105 equilibrium behaviour, the assumed form for the diffusion tensor is essentially immaterial.E. Dickinson, Faraday Discuss. Chem. Soc., 1978, 65, 127. E. Dickinson, R. Parker and M. Lal, Chem. Phys. Lett., 1981, 79, 578. Prof. R. H. Ottewill (Uniuersity of Bristol) said: (1) In our experience although the hard-sphere model can be used to fit many of the data on colloidal dispersions it is more appropriate, particularly in dilute disper- sions at low electrolyte concentrations, to use a soft-sphere model. This allows more correctly for the exponential decay of the electrostatic repulsion with distance. Indeed our recent work,' using light scattering with small latex particles of radius 16 nm, showed that the hard-sphere model only works if an effective hard-sphere radius is used which includes the particle and the electrical double layer out to a distance where the interaction energy is of the order of 1 kT. The latter is in the soft tail of the pair potential.(2) The polystyrene latices used for the work reported in the paper had a stan- dard deviation on the particle radius (1 6 nm) of 15% as determined by small-angle neutron scattering. However, in most of our work the range of the electrical double layer (ca. 2 / ~ ) was greater than the particle radius and hence the particles were well spaced apart. This must to a large extent offset the effect of polydispersity in S(Q), although it will be present in P(Q).The latter effect, however, is allowed for by using a dilute non-interacting sample as the reference state. The effect of polydispersity would, of course, be much more pronounced in systems where the interaction (hard- sphere?) radius was comparable to the physical radius of the particle. The direct experimental measurement in our work was the curve of S(Q) against Q and the experimental uncertainty in the data was of the order of f 5%. The curves of g(r) against r were obtained by Fourier transformation of the S(Q) against Q data. These are thus subject to some uncertainty but we did our best to minimise this. First, the curves of S(Q) against Q were smoothed and at the low Q end we checked that the data were of the form S(Q) = S(0) + C Q2, where C is a constant, a well known relation at low Q.The Fourier transformation was carried out using data from the smoothed curves by numerical integration. Tests were carried out using various intervals of Q, until it was found that these were small enough to give (g(r), r ) results which were independent of the interval in Q. At the high-Q end of the spectrum the Q value was taken to a point where S(Q) was equal to unity, either actual or extrapolated. Recent tests using a fast Fourier procedure have given essen- tially identical curves. The position of the second peak does, of course, rely on data at low Q and the obtaining of data in this region is limited by the design of the instrument and the time available on it.The neutron-scattering data presented were obtained using three different detector positions, 2.66, 5.66 and 10.66m. With a larger allocation of in- strument time results could have been obtained at 20.66 and 40.66 m and this would have increased the degree of precision. The next generation of instruments with a zoom detector in a 40 m vacuum tube may help in this respect. There was no iridescence in the samples studied and I would not have expected any. Iridescence only occurs when the distance between the particle centres is com- parable with the wavelength of visible light. We were using neutrons with a wave- length of 1 nm and hence the peaks in the structure factor occurred in the range of scattering vectors associated with a small-angle neutron instrument.I am not sure quite what partial solid-like ordering means unless this is what we have termed an106 GENERAL DISCUSSION ‘ordered liquid’, i.e. a system where the peak positions in the structure factor obey the Bragg equation for an f.c.c. lattice. It must also be remembered that the latices we have used in this work are approximately an order of magnitude smaller than those used for optical studies of iridescence and hence the structures will be less condensed, i.e. the measurements in our systems could be above the critical temper- ature. This is a topic which needs more detailed examination. R. H. Ottewill and R. A. Richardson, Colloid Polym. Sci., 1982, 260, 708. Prof. B. U. Felderhof (RWTH Aachen, West Germany) said: Prof. Ottewill states in his paper that in the Brownian-dynamics calculation of g(r) he used the diffusion coefficient of eqn (26), which includes hydrodynamic interactions.It seems to me that he could just as well have used a constant diffusion coefficient, since the hydro- dynamic interactions cannot possibly influence the static pair correlation function. Dr. G. C. Jeffrey and Prof. R. H. Ottewill (University of Bristol) replied: We agree that the equilibrium properties of dispersions, including g(r), are independent of the velocity-dependent (hydrodynamic) forces. However, the diffusion term in our Brownian-dynamics computations was modified, as in eqn (26), in order to enhance the sampling efficiency upon close approach of the particles. This procedure allowed the use of a larger time step, which is effectively reduced by the use of eqn (26), when particles come sufficiently close together so that the (pair force, distance) curve is steep, to assure that the constancy of the latter is maintained as required by the Brownian-dynamics algorithm of eqn (23).Prof. A. Vrij (University of Utrecht, The Netherlands) said: Under eqn (1 3) of his paper, Prof. Ottewill mentions that S(Q) could be extrapolated to Q = 0. One won- ders if this extrapolation was actually performed because of the relatively large gap in Q which must be overbridged. From the extrapolated value of S(Q) a value of the osmotic compressibility could be obtained which can be compared with theoretical (osmotic) equations of state. Dr. I. Snook (Royal Melbourne Institute of Technology, Australia) said: I have performed some Brownian-dynamics (BD) simulations using a simple screened Coulomb potential [Prof.Ottewill’s eqn (21)] for parameter values appropriate to his systems. Several interesting features arise from these calculations which he may wish to comment on. ( a ) One can get a reasonably close fit between experimental and theoretical S(Q) values, except at low Q. This later discrepancy is, I believe, the result of polydisper- sion. Does Prof. Ottewill agree? (6) The surface potential t,bs required to fit the data increases with (i) decreasing salt concentration, c, and (ii) increasing volume fraction, cp, the exception to (ii) being for c = (c) ll/s is a much more important parameter in determining S(Q) than is salt concentration, c. I believe this insensitivity of S(Q) [and g(r)] to c can also be infer- red from fig.4 in the paper. Is this so? (a) I find that the Andersen, Weeks and Chandler2*3 hard-sphere perturbation theory is accurate only for the highest salt concentration (c = 5 x mol dm-3). Thus for most of the systems examined by Prof. Ottewill and his colleagues the interaction potential is too soft to allow the use of hard-sphere modelling. mol dm-3. Could we have a comment on these results?GENERAL DISCUSSION 107 (e) For the ion-exchange-resin-treated system, S( Q) may be more reasonably fitted to that of a classical one-component p l a ~ m a . ~ ? ~ Also the value of $s deduced from this model is in reasonable agreement with that found by the BD method. K. Gaylor, I. Snook and W.van Megen, J. Chem. Phys., 1981,75, 1682. H. C . Andersen, J. D. Weeks and D. Chandler, Phys. Rev. A, 1971,4, 1597. ’ W. van Megen and I. Snook, J . Colloid interface Sci., to be published. 4S. G. Brush, H. L. Sahlin and E. Teller, J . Chem. Phys., 1966, 45, 2102. I. Snook and W. van Megen, J . Colloid Interface Sci., to be published. Prof. R. H. Ottewill (Uniuersity of Bristol) replied: We are extremely interested in Dr. Snook’s calculations and in reply to the points you raised would like to make the following comments: (a) The question of polydispersity in general is covered in my reply to Dr. Dickinson, i.e. that it should be a smaller effect for electrostatically stabilized sys- tems where the double-layer extension is large than for systems of hard spheres with the hard-sphere radius close to that of the actual particle radius.Certainly at low Q one would expect polydispersity to have the largest effect. We have tried to correct for the incoherent background and to eliminate multiple-scattering effects in our experimental procedures, but they may still make some contribution. We anticipate trying to make a more detailed exploration of this region. It would be interesting to make and use an even more monodisperse polystyrene latex, which is difficult, or to make a more deliberately polydisperse latex, which is not quite so difficult. (b) In our Brownian-dynamics analysis we also find that to obtain a fit with the experimental data we need to increase the surface potential as a function of both salt concentration and latex volume fraction.I do not at present have an explanation for the mol dm-3 salt concentration. (c) I agree with Dr. Snook’s conclusion that surface potential plays a more major role in determining S(Q) than does salt concentration. However, it must be remem- bered that at these latex concentrations the counter-ions from the latex make a significant contribution and the procedure suggested by Beresford-Smith and Chan’ should be used for the calculation of K. ( d ) We have tried some calculations using the procedure of Andersen, Weeks and Chandler and run into difficulties. Dr. Snook’s explanation sounds correct. (e) We would be very interested to see Dr. Snook’s computations using the one- component plasma model. B. Beresford-Smith and D. Y . C. Chan, Chem.Phys. Lett., 1982, 92, 474. Dr. W. van Megen (Royal Melbourne Institute of Technology, Australia) said: On page 59 of his paper Dr. Ramsay comments that the ionic strength has no apparent effect on S(q). I doubt whether the same can be said for the pair potential and other properties derived from it, such as the equation of state. Had it been possible to measure S(4) accurately at very low wavevectors, approaching the hydrodynamic limit, Dr. Ramsay might well have found considerable sensitivity to variations in the electrolyte concentration. He attempts to analyse his data using the hard-sphere model for charge- stabilized dispersions in which rcd <- 1 [K is defined by eqn (10) in the paper and d is the particle diameter], i.e. for which the pair potential is a very soft and long-ranged repulsion.Under these conditions the very concept of an effective hard-sphere diameter must be meaningless, as is indicated by the strong cp dependence of the108 GENERAL DISCUSSION hard-sphere diameters in fig. 2 of the paper. A better reference system and a more suitable way to analyse the data may be provided by the one-component plasma. Could Dr. Ramsay explain how he actually calculated the hard-sphere diameters shown in fig. 2 and table 2 and what we can expect to learn from these? Dr. J. D. F. Ramsay (Harwell, Oxfordshire) said: In the analysis of the neutron- scattering results hard-sphere structure factors S(Q)Hs were calculated, by the pro- cedure described by Ashcroft and Lekner,' for particular values of hard-sphere number concentrations, nHS, and volume fraction, qHS.Simulations with S(Q)exp for oxide dispersions of known number concentration were made by fitting qHS and thus obtaining the corresponding oHS. In a more recent analysis2 the mean spherical approximation (MSA)3*4 has been applied. With this model a softer screened coulomb potential is fitted to the experi- mental data from which an effective surface charge is obtained. An example of such a fit to the [Ice), Q] data for one of the concentrated silica sols (0.55 g cm-3) refer- 0 5 10 Q. 15 Fig. 4. Small-angle neutron-scattering data for silica sol (0.55 g ml- l ) in 5 x sodium nitrate solution. Full line is fitted MSA model corresponding to an effective surface charge of 0.48 pC cm-2. mol dm- red to in our paper is shown in fig.4, from which an effective surface charge of 0.48 pC cm-2 is obtained. This charge is low compared with that measured by con- ductimetric titration, and is a feature illustrated by the results for the other silica sol concentrations given in table 2. In light scattering the range of Q is much more restricted (Qd << l), and it is only possible to obtain S(0) as a function of particle volume fraction, qp. We can cal- culate the equivalent hard-sphere volume fraction, qHS, which would correspond to these particular values of S(0) and thence derive the equivalent oHS as described. Values of (cHS - 4 2 ) are surprisingly similar to the screening length, K - ' , and thusGENERAL DISCUSSION 109 Table 2. Results obtained from fitting the MSA model to neutron-scattering data obtained on silica sols sol volume ma+] K / A - diameter,” charge concentration fraction, /mol dm- 4 /pC cmP2 / g c111-~ t7 0.08 1 0.037 ca.loP5 I x 197 0.26 0.175 0.080 ca. 3.2 x 187 0.38 0.384 0.174 ca. 3.2 x 175 0.43 0.14 0.062 5 x lod3 2.3 x 197 0.75 0.266 0.121 5 x 10-3 2.3 x 10-2 185 0.59 0.550 0.25 5 x 10-3 2.3 x 10-2 167 0.49 a Fitted diameter it is suggested that oHS can be considered as an indication of the effective range of interparticle interaction. Under conditions for which Kd << 1 and d << r (where r is the mean interparticle separation), we would, however, expect the hard-sphere approximation to be unreal- istic as Dr. van Megen points out, and a softer screened coulomb potential to be a more appropriate model for describing S(Q).To examine this aspect more fully SANS measurements would be required on more dilute dispersions, comparable to those studied by light scattering, expecially in a range of low Q(Qd << 1) where the form of S(Q) is more sensitive to the interaction potential. N. W. Ashcroft and J. Lekner, Phys. Rev., 1966, 45, 33. J. Penfold and J. D. F. Ramsay, unpublished work. J. B. Hayter and J. Penfold, Mol. Phys., 1981, 42, 109. J. B. Hayter, Furaduy Discuss. Chem. SOC., 1983, 76, 7. Dr. B. Beresford-Smith and Dr. D. Y. C. Chan (Australian National University, Canberra) said: We would like to note the qualitative similarities between the mea- sured structure factors S(Q) for silica sols given in fig. 4 of Dr. Ramsay’s paper and the predictions of our theory.(1) The fact that the peak height of S(Q) is almost independent of colloid particle number concentration over a 5-fold variation in concentration strongly suggests that the effective interaction between the particles in Dr. Ramsay’s system is volume- fraction dependent. A volume-fraction independent pair potential will exhibit larger variations in S(Q) over a 5-fold change in number concentration. Under appropriate conditions our theory also predicts a similar constancy in the variation of the peak of S(Q) with number concentration. Unfortunately a more quantitative comparison is not possible because the required inputs for the theory do not appear to be noted during the course of Dr. Ramsay’s measurements. (2) The peak positions em,, are related to the number concentration by Qmaxrs NN 4.0 - 4.5 (as read from fig.4 of the paper). (See also section 4 of our Discussion paper.) (3) Dr. Ramsay observed in the paragraph following fig. 4 in his paper that S(Q) appears to be independent of ionic strength. If one takes a surface charge density of 2pC cmP2 (it is in fact noted in the paper that typical values are in excess of 2 pC cm--2) then the concentration of counterion needed to balance the colloidal charge would be in the range 8.6 x 10F4-4.7 x mol dm-3 as the SO2 concen-110 GENERAL DISCUSSION tration varies between 0.1 and 0.55 g ~ m - ~ . In our theory, we find that both of these counterions as well as the added salt contribute to the screening of the electrical double-layer interaction between colloidal particles (see section 2 of our Discussion paper).Therefore, when the salt concentration is varied between and 5 x mol dmP3, this variation is, to a significant extent, masked by the high concentration of counterions. In contrast, the DLVO potential only allows added salt to contribute to the screening and so it would predict larger variations in S(Q) with salt concentrations. Finally, the methods Dr. Ramsay has adopted to prepare dispersions of various concentrations (by dilution with the equilibrium dialysate or by ultrafiltration) have the consequence that the exact ionic composition of the final dispersions remains unspecified. This renders a detailed theoretical analysis of the results very difficult if not impossible. A possible alternative is to remove all excess salt by treatment with ion-exchange resin, as in the Discussion paper by Cebula et al., or dialyse against pure water and then make up the final system by adding a known amount of salt to the dispersion.Dr. J. D. F. Ramsay (Harwell, Oxfordshire) replied: The additional counterions required to balance the colloid charge, and their effect in modifying the electrostatic screening, is an important aspect which we have considered recently, and we do agree this would help explain the apparent insensitivity of S(Q) to electrolyte con- centration in dispersions of high volume fraction (cp 2 0.1). This would seem to be particularly important for dispersions containing very small particles (< 10 nm) at high volume fraction as studied here, since for these conditions the ionic compo- sition of the ‘interstitial electrolyte’ would be more difficult to specify.As Dr. Chan suggests, one possible way of overcoming this problem is to make additions of known amounts of salt to the dispersion; however, one tacitly assumes that no adsorption occurs either at the surface or within the Stern layer surrounding the particles. Dr. W. van Megen (Royal Melbourne Institute of Technology, Australia) said: Dr. Chan expresses surprise at the failure of the cell model. Whilst this model, in its simplest form, provides an adequate description of the ordered (crystal-like) phase, it is totally inadequate for the disordered (fluid-like) phase. To describe the latter phase by means of a cell model, multiple occupancy of the cell as well as intercell correlations need to be included as corrections to what amounts to a perturbation expansion about a perfect lattice.The results may show considerable sensitivity to the shape of the cell. Would Dr. Chan care to comment on the possible significance of these effects in the context of the approach taken by Beunen and White? Prof. W. B. Russel (Princeton University, U.S.A.) said: My comments address not Dr. Chan’s treatment of the statistical mechanics of these systems but the mis- impression about continuum theories created by the statements concerning ‘cell models’. One can indeed derive, with purely continuum arguments, a result for the effective Debye which reduces to eqn (2.5) of Dr. Chan’s paper at low volume frac- tions q of partic1es.l Indeed the derivation requires no assumption about the spatial distribution of particles.Within the fluid the potential must satisfyGENERAL DISCUSSION 111 with the ci unknown constants of integration. In a closed volume V the total con- centrations ni of each ionic species are known, so that Vfl with Vii = (1 - q)V. The Ci can be evaluated by linearizing, not about zero, but as $ = ($) + $' with Then Substitution into eqn (1) yields with (4) ( 5 ) For overall electroneutrality so that with zp and np the valence and number density of the particles, respectively, ni, the concentration of added electrolyte and z, the valence of the counterions. Hence xeff equals that in eqn (2.5) as q~ + 0 and accounts for the reduced volume of fluid at finite q. Clearly this derivation requires linearizing the Poisson-Boltzmann equation and sheds no light on the pairwise additivity of electrostatic forces.On the other hand it applies to arbitrary volume fractions and could prove a useful input to structural theories which assume pairwise additive potentials. W. B. Russel, in Theory of Dispersed Multiphase Flow, ed. R. E. Meyer (Academic Press, New York, 1983) pp. 1-34. Dr. B. Beresford-Smith and Dr. D. Y. C. Chan (Australian National University, Canberra) said: It is clear that the term 'cell model' can mean very different things to different people. In our Discussion paper we were only concerned with the cell model of Beunen and White. In this approach the rationale is to use the cell model (one particle per spherical cell) to determine the effective screening parameter, IC, of a concentrated dispersion.This value of K is then used in the classical DLVO potential to characterize particle interaction. The structure of the dispersion is obtained by treating the particles as a one-component fluid and using liquid-state theory. In our calculation we use the hypernetted-chain approximation to determine the structure factor.112 GENERAL DISCUSSION In reply to Dr. van Megen’s comment: The cell model in our Discussion paper is used as described above. The structure factor was not determined using a cell model, so that questions regarding multiple cell occupancy, intercell correlations or cell shape are not relevant. We can only speculate as to the reason why the cell model (as used in our paper) gives such different results (see fig.2). The choice of a cell shape and the assumption that particles outside the cell are uncorrelated with the particle within the cell is equivalent to a very special choice for the form of the particle-particle correlation function. This is the most likely source of discrepancies. In reply to Prof. Russel’s comment: We have also made the observation that an appropriate linearization of the Poisson-Boltzmann will also yield the effective screening parameter, K , given in eqn (2.5) of our Discussion paper [see the paragraph surrounding eqn (3.2)J. Prof. B. U. Felderhof (RWTH Aachen, West Germany) said: Surely the fact that Dr. Chan’s effective Hamiltonian is a sum of pair interactions is a result of the approximations he has made.In fact there will be many-body interactions and one must ask oneself at which concentrations these become important relative to the effective pair interactions. Dr. B. Beresford-Smith and Dr. D. Y. C. Chan (Australian National University, Canberra) (communicated): The effective Hamiltonian, obtained by averaging over the coordinates of the ions, consists of a sum of one- two-, three- . . . up to N-body terms. Our theory gives an approximate expression for the two-body term. It is not a trivial exercise to estimate the magnitude of three- and higher-body terms. However, we see, a posteriori, that the two-body term alone seems to give a reasonably good account of available experimental data. Because counter-ions are included in screen- ing the electrostatic interaction between the colloidal particles, an increase in the particle concentration will also increase the screening or decrease the range of the interaction.Consequently, unlike unscreened hydrodynamic interactions, it is not obvious that many-body effects will become important at high concentrations. Dr. E. Dickinson (University of Leeds) said: In simulating dispersions of electro- statically stabilized particles with Ka 2 10, we have taken DLVO pctentials to be pairwise additive. Could Dr. Chan tell me if his calculations give any information about the range of rca over which pairwise additivity is a good approximation for modelling a concentrated dispersion of specified volume fraction and particle surface-charge density. Dr. B. Beresford-Smith and Dr.D. Y. C. Chan (Australian National University, Canberra) said: Owing to the nature of Dr. Dickinson’s question, it is not possi- ble to provide a simple direct answer: a little preamble is necessary. The DLVO theory for the electrical double-layer interaction between colloidal particles is valid only in the limit of infinite dilution of colloidal particles, when one only needs to consider the interaction between two colloidal particles in a large (infinite) reservoir of electrolyte of known composition. Because the colloidal par- ticles are present only at ‘zero’ particle concentration as implicit in the DLVO theory (two particles in an infinite volume of electrolyte), it is only sensible to think of pairwise additive interactions, irrespective of the Ka value. (K in this instance is determined only by the ionic composition of the dispersion medium and is independ- ent of colloid particle volume fraction or particle surface charge.)GENERAL DISCUSSION 113 In a concentrated colloidal dispersion in which colloidal particles are present at a finite concentration, it is no longer possible to think in terms of the DLVO picture.The appropriate conceptual starting point is to consider the whole colloidal system, consisting of colloidal particles, intrinsic counterions (needed to balance the charge on the particles) and the other cations and anions (that make up the added electro- lyte) as an asymmetric electrolyte. That is, the colloid particles are thought of as large ions of very high valence. Now if one wishes to treat the colloid dispersion as an one-component system consisting of colloidal particles, what is the form of the effective colloid-colloid interaction? Starting from the picture of the colloid dispersion as a highly asymmetric electrolyte, it is possible to demonstrate rigorously using statistical mechanics that the effective colloid-colloid interaction will consist of a sum of one-body, two-body, three-body etc.interactions. While such a result is formally exact, it is not very useful in practice. The aim of our calculations is to produce an effective pair potential to character- ize the electrical double-layer interaction between colloidal particles in a concen- trated dispersion. Like the DLVO potential, this effective pair potential also has a screened Coulomb form but the Debye parameter K now depends on the intrinsic counterion concentration as well as the concentration of added salt in the system.As a consequence, the parameters in this effective pair potential will vary with the volume fraction of the colloidal particles. In contrast, the DLVO pair potential is independent of the volume fraction of colloidal particles. The utility of our effective pair potential has been demonstrated in our Discussion paper as well as in ref. (14). One interesting limit of our calculation is that if we let the colloidal particles have the same size and valence as the small ions (for instance, we can assume all charged species to be univalent) we can, using our effective pair potential between colloidal particles, recover the Debye-Hiickel limiting law for the thermodynamic behaviour for the total system (colloidal particles and ions).From the above we conclude that (1) the DLVO pair potential for electrical double-layer interactions is not appropriate for concentrated systems, (2) we have an approximate theory for the effective pair potential which will take the place of the DLVO potential and (3) our pair potential includes many-body effects (because it is volume-fraction dependent), and because we can recover the Debye-Huckel limiting law this potential appears to be valid even in the limit Ka + 0. (The definition of K in this instance is given in our Discussion paper.) Dr. P. N. Pusey (RSRE, Malvern) said: I have a few comments on the section of the paper by Beresford-Smith and Chan which contains fig. 4.There they consider some old measurements of ours1 and compare their theoretical predictions for the quantity Q(k), a measure of the departure from a single exponential of the corre- lation function G(k,t) of the scattered light, with values derived from the experi- mental data [and quoted in ref. (1) of this comment]. In fig. 4 substantial disagree- ment between theory and experiment is found and these comments address possible reasons for this disagreement. For values of kr, 2 4, i.e. at and beyond the peak in the theoretical prediction, I think the disagreement is almost certainly due largely, as Beresford-Smith and Chan suggest, to the use of a truncated cumulant analysis. It is perhaps worth elaborating this point. Quite generally one can expand ln[G(k,t)/S(k)] as an infinite power series in time: ln[G(k,t)/S(k)] = 1 - T(k)t + *Q(k)[r(k)tl2 + @(k)[T(k)tl3 + .. .114 GENERAL DISCUSSION Experimentally one is faced with the task of extracting the cumulants I'(k), Q(k), R(k) etc. from a set of data. Clearly one cannot fit an infinite polynomial. The approach most commonly used is to exploit the fact that as t + 0 the higher-order terms in eqn (1) become relatively less important. Thus, at a given value of k, one makes a measurement of G(k,t) over a range o f t , 0 < t < t,,,, say. Effective cumu- lants can then be obtained from a truncated fit, e.g. fitting lnG(k,t) to a quadratic or cubic function of t . The measurement is then repeated using a smaller value tmax and new values for the effective cumulants are obtained. On repeating this procedure with decreasing values of t,,, one should ultimately obtain unbiased estimates of the low cumulants, I'(k), Q(k) etc.The drawback of this approach is, of course, that the random error in these quantities grows as a decreasing range of t is sampled and the extrapolation to t = 0, implied in the above procedure, becomes ambiguous. Rather than attempt this time-consuming analysis, Brown et al.,l in what was intended as a preliminary study, truncated eqn (1) at the t2 term and chose a fixed value of t,,, given by T(k)tm,, a 0.5. As pointed out by Beresford-Smith and Chan (and stated in the original paper), we expected this analysis to be reasonably ac- curate for Q(k) 50.5, and it is gratifying that in fig.4 at kr, = 10 where Q a 0.6 there is reasonable agreement between experiment and theory. In subsequent years I have, on several occasions, attempted the full extrapolation procedure outlined above on strongly interacting systems. Then I found that, as was reduced, the effective second cumulant Q(k) could, in the region of the peak in S(k), become very large (>>1) (but with large random error), as predicted by Beresford-Smith and Chan. Below the peak in S(k), kr, -c 3, say, a second process, not understood until 1980,2 can contribute to the measured second cumulant. This is the incoherent scat- tering arising from polydispersity (see the paper by Hayter and the discussion com- ment by Pusey). In the k + 0 limit there is considerable evidence3 that G(k,t) can be written as G(k,t) = S(O)[( 1 - x) exp( -D,k2t) + x exp( -D,k2t)] (2) where x is a complicated function of the particle-size distribution, the particle inter- actions, etc.(x = 0 for a monodisperse system) and D, and D, are, respectively, the collective and long-time self-diffusion coefficients. At small k, where S(k) is small (< l), the relative effect of the second (incoherent) term in eqn (2) can become appreciable. Then, since, in a strongly interacting system, D, can be much smaller than D,, the incoherent term can contribute significantly to the non-exponentiality of G(k,t) and hence to Q(k). This almost certainly explains why the data in fig. 4 lie above the theory (which considered a monodisperse system) at small k . I have two further comments.First, theoretical expressions for the third cumu- lant [R(k) in eqn (l)] of an interacting system have recently been ~ b t a i n e d . ~ , ~ Thus an energetic theorist could evaluate these, either for intrinsic interest or to provide some estimate of the truncation errors introduced by the analysis of Brown et a1.l Secondly, it was shown by Ackersoq6 using a projection-operator analysis, that G(k,t) should be a single exponential in the k --+ 0 limit; the validity of this predic- tion seems to be widely accepted. However, it can be seen from Beresford-Smith's and Chan's fig. 4 [or by expanding the cosine in their eqn (4.8)] that iim Q(k) # 0 implying a non-exponential G(k, t ) as k + 0. It is possible that this apparent con- k + OGENERAL DISCUSSION 115 tradiction arises from the inadmissibility of taking the k -+ 0 limit after the t --+ 0 limit implied in eqn (4.6).One would like to understand this in detail. J. C. Brown, P. N. Pusey, J. W. Goodwin and R. H. Ottewill, J. Phys. A, 1975, 8, 664. M. B. Weissman, J. Chem. Phys., 1980, 72, 231. P. N. Pusey, H. M. Fijnaut and A. Vrij, J. Chem. Phys., 1982, 77, 4270. P. N. Pusey and R. J. A. Tough, J. Phys. A., 1982, 15, 1291; 16, 2889, corrigendum. B. J. Ackerson, J. Chem. Phys., 1978, 69, 684. ’ J. L. Arauz-Lara and M. Medina-Noyola, Physica, 1983, 122A, 547. Dr. B. Beresford-Smith and Dr. D. Y. C. Chan (Australian National University, Canberra) said: We should add that Ackerson [ref. (20) of our paper] was the first to attempt to calculate the cumulant Q(k). However, he chose to compare the function B(k) [see eqn (4.8) in our Discussion paper], which did not appear to emphasise the discrepancy between theory and experiment.The fact that Q(k + 0) # 0 does not seem to be in conflict with the observation that G(k,t) should be a single exponential in the limit k -+ 0. The first cumulant, T(k), vanishes like k2 as k -+ 0, thus the coefficient of the t 2 term in Dr. Pusey’s eqn (1) will vanish faster, like k4 as k -+ 0, even though Q(0) is ’ non-zero. Prof. R. Klein (University of Konstanz, West Germany) said: With regard to the calculation of the second cumulant of the intermediate scattering function G(k,t ), eqn (4.6), and its comparison with experiment, fig. 4, I would like to make the following comment. We have shown (in our paper at this Discussion) that the sec- ond cumulant can be related to the longitudinal high-frequency elastic modulus E(k) of the liquid of interacting Brownian particles.This quantity is defined as in the theory of simple liquids and is a measure for the response of the Brownian system to a high-frequency external force. At high frequencies the system behaves elastically, so that it seems obvious that the short-time behaviour of the concentration autocor- relation function G(k,t) is related to the elastic properties at high frequencies. We have calculated E(k) by using a screened Coulomb interaction for the one- component macrofluid and the Hansen-Hayter method for the radial distribution function g(r). For system parameters describing the dispersions investigated by Gruner and Lehmann we obtain for E(k) the results shown in fig.1 of our paper. Taking the limit k -+ 0, we obtain the long-wavelength limit of the longitudinal elastic modulus as a function of concentration cp in fig. 2 of our paper. It is found that E(k -+ 0) - p2. This relationship can be compared with experiment, since E(k) is related to the second cumulant. Gruner and Lehmann have determined the second cumulant by fitting their measurements of G(k,t) to the sum of two exponentials. Using the relation between the second cumulant and E(k), the experiments show that indeed E(k -+ 0) - (p2, F. Gruner and W. Lehman, J. Phys. A, 1982,15, 2847. Prof. A. Vrij (Uniuersity of Utrecht, The Netherlands) said: I would like to ask Dr. Chan whether it is possible to calculate the negative adsorption of electrolyte with his theory? This would make it possible to check the theory with experiments of Moller on Donnan equilibria with bovine serum albumine (BSA) in several electro- lyte concentrations. At the highest electrolyte concentration Moller found that the negative adsorp- tion was proportional to the BSA concentration, whereas at the lowest electrolyte concentration the negative adsorption as a function of BSA concentration reached a116 GENERAL DISCUSSION maximum value.In the latter case there is a large overlap of the electrical double layers . W. J. H. M. Moiler, Dissertation (University of Utrecht, 1959), p. 85. Dr. B. Beresford-Smith and Dr. D. Y. C. Chan (Australian National University, Canberra) replied: Yes, it is possible to calculate negative adsorption of electrolyte in a Donnan equilibrium experiment.Our theory will include overlap of electrical double layers due to particle interactions. Dr. M. Corti (Uniuersity of Milan, Italy) said: Does Dr. Cummins have evidence that the bimodal distribution is an intrinsic feature of the system (like a dynarnical equilibrium process, for instance) and that it is not due to the preparation technique? Dr. Th. F. Tadros (ICI, Jealott’s Hill) said: I address my remarks to Dr. Cummins. (a) Did you attempt to separate the large vesicles from the small ones in order to simplify your system? Hydrodynamic chromatography could be applied in this case and the problem of metastability could be overcome by carrying out the measure- ments immediately after separation.(b) Did you attempt to measure the electrophoretic mobility of the vesicles? This could be carried out using the Doppler technique. Such measurements, which could be used to calculate t+bo, could be useful in removing some of the uncertainties in calculation of VR. Dr. P. G. Cummins (Unileuer Research, Port Sunlight) replied: (a) After sonication the vesicle dispersions were centrifuged at 120 OOOg. for 24 h. This removes most of the large vesicles but inevitably a few will remain in suspen- sion. Our results indicate that this number is very small, only amounting to ca. 1 in lo6 of the total population. Nevertheless, even this small number appeared to have a significant effect on the scattering behaviour of the dispersion, and it would certainly be desirable to remove the ‘contaminating’ large vesicles in some way.Gel- permeation chromotography (e.g. using Sepharose 2B or 4B) has been used by some workers to ‘purify dispersions of small vesicles’, but this method is not without its problems: it has been shown that phospholipid is lost onto the bed during filtration and that the size distribution of the small vesicles may even be altered. Also, gel filtration causes dilution of the sample which would then have to be reconcentrated in some way, a procedure which itself would probably lead to fusion and growth of the vesicles (i.e. the final suspension is likely to be more polydisperse than the original). We have not tried to apply hydrodynamic chromatography to vesicle fractions; this technique may indeed be useful if it avoids the problems normally associated with gel-permeation chromatography and if the technique is capable of fractioning vesicle particles in the range 200-2000 A.Experiments using latices would indicate that this is quite feasible. We do try to make measurements as soon after sample preparation as possible in order to minimise problems of metastability, but it is unlikely that a highly concen- trated dispersion completely free of larger vesicles could be prepared, owing to the intrinsic thermodynamic instability of highly curved bilayers. ( b ) In our laboratory we are well versed in the technique of Doppler electro- phoresis and have made such measurements of other systems. However, for anGENERAL DISCUSSION 117 unsonicated sample of our system values obtained using conventional microelectro- phoresis are available from the literature. The author obtains zeta potentials and surface potentials of ca.20-25 mV for a 90: 10 sample in 0.145 mol dm-3 NaCl. They also indicate that electrokinetic data greatly underestimate the surface poten- tial at higher surface charge densities. L. Ginsberg, Ph.D Thesis (University of London, 1981) Dr. I. Snook (Royal Melbourne Institute of Technology, Australia) said: Whilst studying aqueous dispersions of dihexadecylphosphate (DHP) vesicles 1, (used in the solar photolysis of water) we came to similar conclusions as Dr. Cummins about the size distribution of vesicles. Photon correlation spectroscopy measurements, using both cumulant analysis methods (the latter giving the smoothest non-negative distribution that is consistent with the data to within the noise level) showed that the majority of vesicles are relatively small but significantly polydispersed and that there is a very small fraction of much larger particles present.Furthermore, we find that the size and stability of the vesicles, their adsorption and entrapment capabilities and the permeability of the vesicular membrane are strongly affected by pH and ionic strength. and Provencher's Y-M. Tricot, D. N. Furlong, W. H. F. Sasse, P. Daivis and I. Snook, Austr. J. Chem., 1983,36, 609. * Y-M. Tricot, D. N. Furlong, W. H. F. Saw, P. Daivis, I. Snook and W. van Megen, J. Colloid. Interface Sci., to be published. J. C. Brown, P. N. Pusey and R. Dietz, J.Chem. Phys. 1975, 62, 1136. S. W. Provencher, Makromol. Chem., 1979, 180, 201. Dr. P. N. Pusey (RSRE, Malvern) said: Is it possible that some of the excess scattering seen at low Q is due to diffuse scattering arising from the small particles in addition to the contribution arising from the large species? Dr. P. G. Cummins (Unileuer Research, Port Sunlight) replied: Following sub- mission of our paper we have attempted to analyse the data using the model of Hayter and Penfold.' Using this procedure we have observed that their model will mimic the experimental data at lower concentrations (fig. 5 ) but at the higher-phase volumes the peak in S(Q) and the compressibility limit S(0) may not be simulta- neously modelled (fig 6). The discrepancy between the model S(0) and the light- scattering S(O), whilst maintaining a good quality fit for the peak, increases with concentration.One explanation for such a deviation could arise from the incoherence/diffuse scattering associated with the polydispersity of the material. An immediate candidate for such a contribution is the trace amounts of large particles present in the sample (as observed by us and Dr. Snook) whose scattering intensity would have linear dependence on the concentration in direct opposition to the coherent contribution which yields the compressibility limit S(0). As a result, pro- vided there is no correlation between position and scattering amplitude, this incoher- ent contribution [i.e. the P(0) of the large particles] should increase linearly with concentration.Within this simplistic model, the difference between the model S(0) and the light-scattering S(0) should generate an incoherent/diffuse intensity linear with concentration. Fig. 7 shows such a plot for the data exhibited in the paper. For a further increase in the diffuse scattering one would expect the P(0) func- tion for the large particles to become identifiable as an upturn in S(Q) at low Q. This feature does not appear to be present, however, and the overall profile of S(Q), in118 0 . 2 5 , d GENERAL DISCUSSION 1.5 1 . 2 5 1 n 0.75 0.5 0 5 10 15 2 0 25 10-3 Q Fig. 5. Plot of S(Q) against Q for PC: PS = 90: 10, 1% vesicles. (-) Hayter-Penfold model: cp = 0.011, diameter = 187, 1/K = 152, V = 31 mV. 1 . 5 * 1.25 l t 9 0.75 0.5 c 0 5 10 15 20 25 10-3 Q Fig. 6. As fig. 5, but for 4% vesicles. (-) Hayter-Penfold model. relation to the theoretical model will, as suggested by Dr. Pusey, also reflect a contri- bution from the polydispersity in the size distribution of the smaller species. J. B. Hayter and J. Penfold, J. Chem. Soc., Faraday Trans. I , 1981, 77, 1851. Dr. W. van Megen (Royal Melbourne Institute of Technology, Australia) said: In the theoretical analyses of the data does Dr. Cummins ignore the attractive contri-GENERAL DISCUSSION 119 500 L O O n c c .- ' 300 e 8 200 W x +- .- c U c 0 1 2 3 L 5 concentration (wt. %) Fig. 7. Plot of incoherent scattering intensity as a function of concentration. PC: PS = 90: 10. bution to the total interparticle potential? If so, can this be justified, particularly for those systems with added salt? Eqn (9, as Dr. Cummins has stated it, is not strictly correct. For pair potentials with a (secondary) minimum X, is not arbitrary but represents the point where the pair potential passes through zero, whilst for purely repulsive pair potentials the upper limit on the integral [in eqn (5)] has to be determined variationally.' J. A. Barker and D. Henderson, Rev. Mod. Phys., 1976, 48, 587. Dr. P. G. Cummins (Unileuer Research, Port Sunlight) said: In the paper only the repulsive contribution to the interaction potential is considered. Justification for this is given, for a single sheath vesicle, by fig. 8 and 9. Fig. 8 represents the attractive 30 t . - separationlii -20 -3 0 - L O y -50 Fig. 8. Vesicle interactions: attractions only.120 GENERAL DISCUSSION E - 2 - 6 - 8 - 10 35 70 105 1LO 175 separationlA 1 Fig, 9. Vesicle interactions: attraction and electrostatic repulsion. contribution represented by the Hamaker relationship using the parameters indi- cated in the paper and fig. 9 gives the full attractive + electrostatic repulsive contri- bution (fig. 10) for the ionic strengths 0.1 and 0.001 mol dm-3. There is a misprint in the original form of eqn (2), it should of course read In (1 + [a/H + A)] exp It is clear from fig. 8 and 9 that the attractive contribution plays little role within the limits of the experiment. Fig. 10 shows a comparison of the highest concentration we studied in mol dmV3 NaCl with the theoretical model of Hayter-Penfold. The parameters used are indicated in the caption. The quite fair agreement rep- resents further justification for our choice of potential and indicates that problems (-W. 1 . 2 5 1 . 5 1 0.75 0 . 5 0.25 t - 0 5 10 15 20 25 10-3 Q Fig. 10. Plot of S(Q) against Q for PC:PS = 90: 10,4% vesicles in 0.01 mol dmP3 salt. (-) Hayter-Penfold model: cp = 0.53, diameter = 193, l/K = 24, V = 38 mV.GENERAL DISCUSSION 121 associated with X, [eqn (5)] that could occur from a secondary minimum do not arise. Since the nature of the attraction (fig. 8) yields only a primary minimum for our dispersions, the systems are not at thermodynamic equilibrium. However, our concern with the essential kinetic stability of the particles requires us to consider only that part of the potential accessible within the time scale of the experiment. Our calculation in eqn ( 5 ) was thus carried out only over the positive contribution of V(r). X, is essentially arbitary between the peak of the potential barrier and the position at which the attractive term makes the potential zero. With regard to the upper limit of the integral, strictly speaking Dr. Megen’s comment is correct. In our case this was done empirically by increasing the width of the integration up to, when required, the limit of half the average interparticle spacing.

ISSN:0301-7249    

DOI:10.1039/DC9837600093

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 76, 123-136 Hydrodynamic Interactions and Diffusion in Concentrated Particle Suspensions BY PETER N. PUSEY AND ROBERT J. A. TOUGH Royal Signals and Radar Establishment, Malvern, Worcestershire WR14 3PS Received 20th May, 1983 We extend our earlier Langevin treatment of the dynamics of interacting Brownian par- ticles to include hydrodynamic interactions, couplings of the particles’ motions transmitted through the solvent. We consider the short-time behaviours of the self and full dynamic structure factors and a particle’s mean-square displacement. Effects due to the finite propag- ation time of hydrodynamic interactions (‘long-time tails’) are also discussed briefly. Finally, experimental measurements of the short-time self-diffusion coefficient of a particle in a con- centrated suspension are reported.In agreement with recent theories, these experiments de- monstrate the importance of irreducible many-particle effects. 1. INTRODUCTION In this paper we discuss aspects of the Brownian motions of interacting colloidal particles suspended in a liquid, with special attention being paid to the hydro- dynamic interaction between the particles, transmitted by fluctuating velocity fields in the liquid. It is a truism that a full classical description of the dynamical properties of a particle suspension requires the solution of Newton’s equations of motion (or, equivalently, the Liouville equation) for the N Brownian particles, the much larger number of liquid molecules and any other components such as small ions which may be present.However, since the Brownian particles are usually much bigger and more massive than the liquid molecules, interest frequently centres on their motions on temporal and spatial scales much larger than those characterizing the microscopic motions of the liquid molecules. Starting with the work of Einstein, Langevin, Smoluchowski and others, this realization has motivated the development of coarse-grained descriptions of the suspension in which the detailed properties of the liquid molecules are considerably ‘averaged over’. A large literature now exists which establishes from first principles the validity of these In the study of concentrated (interacting) particle suspensions, probably the most widely used such description is the N-particle diffusion (or Smoluchowski or Kirkwood-Riseman) equation, which gives the evolution in time of the probability distribution of particle positions.Recently approaches based on the Langevin * and Fokker-Planck equations, which take more explicit account of particle velocities, have been exploited. Although in many cases the Langevin approach leads to the same conclusions as does the Smoluchowski equation, it is our view (or prejudice) that it displays the physical principles and assumptions involved more clearly. In the next section we show how our earlier work8 on the Langevin approach can be generalized to include a full treatment of hydrodynamic interactions. We calculate, to second order, Taylor expansions in time of such quantities as the mean- square displacement of a single particle and the dynamic structure factors measured124 HYDRODYNAMIC INTERACTIONS IN CONCENTRATED PARTICLE SUSPENSIONS in scattering experiments.We also discuss further the similarities and differences between the dynamics of particles in liquid suspension and of atoms in a simple classical liquid (section 2.2) and calculate explicitly the mean-square displacement of a particle in a dilute hard-sphere suspension (section 2.3). In section 3 we outline the theory underlying the possible detection by dynamic light scattering of ‘long-time tails’ in the velocity cross-correlation functions of par- ticles in concentrated suspension. One of the most important developments of recent years in the field of hydro- dynamic interactions has been the systematic theoretical treatment of many-particle effects.In section 4 we draw attention to this work and describe a recent experiment which demonstrates unambiguously the importance of many-particle hydrodynamic interactions in concentrated particle suspensions. Because of length restrictions, much of the treatment in this paper is, perforce, sketchy; more details either have been or will be given elsewhere. 2. LANGEVTN DYNAMICS WITH HYDRODYNAMIC INTERACTIONS 2.1. THEORY AND TAYLOR EXPANSIONS The Langevin equation for particles in concentrated liquid suspension has been derived by a number of authors. Our approach follows closely that of Ermak and McCammon.lo It will be described in detail elsewhere; here we only give an outline. For a suspension containing N identical spherical particles we find,- by the many-particle Langevin e q ~ a t i o n , ~ that one Cartesian component placement of particle j in time z can be written as Arjl(z) rjl(7) - rjl(0) = ArBjl(7) where rj(t) is the position vector of particle j at time t , roman subscripts integrating of the dis- (2.1) are particle labels which run from 1 to N and greek subscripts indicate Cartesian components and run from 1 to 3.The displacement Arjl(z) has divided naturally into three components. The first, Brownian, part, AYBjl(Z), is that caused by the rapidly fluctuating ‘solvent’ forces and can be written where uBj(t) is the Brownian component of the particle velocity; uB fluctuates rapidly on timescale TB zB = mlf (2.3) f = 6rcyR (2.4) where rn is the particle mass and f the ‘free particle’ friction coefficient y being the liquid viscosity and R the particle radius.For submicron particles zB 5 s. Since we are mainly interested in processes slower than this (see, how- ever, section 3) we limit consideration to timescales z >> z,; (2.5a)P. N. PUSEY AND R . J. A. TOUGH 125 in fact this inequality has already been assumed in the derivation of eqn (2.1). The third term in eqn (2.1) represents the ‘drift’ displacement caused by the direct inter- particle forces, e.g. Coulombic, hard-sphere etc. These forces are derived from a potential energy of interaction, U[(ri(t)}],which, in the general case, depends on the instantaneous spatial configuration (or positions) ( r i ( t ) ) of all the particles. The ‘drift’ velocities associated with these potential-driven displacements thus fluctuate on a timescale zI characteristic of significant changes in the particle positions;ll in the Brownian or Smoluchowski limit considered here >> z B .(2.5b) D k a , l p [ ( d t ) ) ] is a component of the configuration-dependent hydrodynamic mobility or diffusion tensor which gives the drift velocity of particle k induced by a force (fluctuating slowly compared to zB) on particle I (and vice versa). The quantity p, which should not be confused with the subscript p, is p = ( k T ) - ’ , where k is Boltzmann’s constant and T is the temperature. The origin and significance of the second term in eqn (2.1), which depends on the gradients of the diffusion tensors, will be discussed elsewhere.It represents the tendency of a particle to move in the direction of increasing local mobility. There is an important connection between the Brownian velocity components vB and the diffusion tensors which, on the long timescale of interest, can be written ( ~ ~ k a ( t ) ~ ~ i p ( t ’ ) ) v = 2h,ip[(rXt))]8(t - t’) (2.6) where the brackets ()v represent an average over the rapidly fluctuating Brownian velocities but not over the positions of the particles in the suspension. Integration of eqn (2.6) provides a Kubo relationship j:di. (UBka(t)UBip(t’))v = In dilute enough suspension, where the than the particle radius, the correlation particles can be neglected and typical interparticle spacing is much greater between the Brownian velocities of different where Do is the ‘free-particle’ diffusion constant Then eqn (2.1) takes the simpler form (2.10) Eqn (2.10), which was the starting point of our earlier work,* applies to the case where direct interparticle interactions (represented by U ) are significant but hydro- dynamic interactions can be neglected. In practice, this will only occur in very dilute suspensions with potentials (e.g.Coulombic) which are long-ranged compared with the particle radius. However, the hypothetical case of short-ranged (e.g. hard sphere) forces, but negligible hydrodynamic interactions, is of some theoretical interest (see section 2.3). Our aim now is to use eqn (2.1) to calculate power-series expansions in time z of various dynamic properties of the particle system. NaYvely one might attempt such an expansion by writing, for example,126 HYDRODYNAMIC INTERACTIONS IN CONCENTRATED PARTICLE SUSPENSIONS (2.1 1) where ~ ( t ) = drl(t)/dt is the ‘full’ velocity of particle I and we have adopted the notation However, it is immediately apparent that eqn (2.11) is only valid on a timescale which is short compared with all fluctuations in the particle velocities ( D l ( t ) ) , i.e.for t << zB. Since we are interested only in behaviour for times t >> zB, the appropriate expansion is in terms of the (total) particle displacements: U = U [ { r i ( O ) ) ] . (2.12) (2.13) and (2.14) D j 1 , k a D j l , k a [ ( r i ( O ) f ] - (2.15) + c 1 a 2 D j l , k a j l , k a [ { r i ( t ) } ] dD j l , k a arks arks 1 /3 a r k a d r l p Arlp ( t ) + . .. - where Iterative use of eqn (2.1), (2.13) and (2.14) then gives d2D j l , ka a 2 u a D j l , k a au f l D j l , k a - - - pp- +EX(-- k,l a$ arkc&l/I a r k a a r l p arlp d r k a x Jomdt b B l p ( t ) + . . . . (2.16) The use of expansion (2.13), rather than expansion (2.1 I), which allows subsequent averaging over the rapid particle motions (see below), is a crucial step in the analysis; this point was probably not emphasized strongly enough in our earlier paper.8 With use of eqn (2.16) we can now calculate the quantities of interest. As discus- sed previously [ref. @), section 31, the necessary averages are carried out in two stages. We exploit the fact that the Brownian components (oBi(t)} of the particle velocities fluctuate rapidly on a timescale zB over which the particle positions are essentially constant (i.e.over which they change by only a small fraction of a particle radius). Thus we first average over the rapid velocity fluctuations and use eqn (2.2), (2.6) and (2.7) to give functions of the random displacements {ArBi) in terms of the configuration-dependent diffusion tensors. Then the configurational ensemble average, denoted by ( ), is performed over all possible particle positions, frequent use being made of the ‘Yvon identity’ l 2 (2.17) where G is any regular function of particle positions. ponent of the mean-square displacement of particlej in time z is given by Proceeding in this manner we obtain the following results to order z2. One com-P. N. PUSEY AND R. J. A. TOUGH 127 The self dynamic structure factor (or intermediate scattering function) is given by FS(K, Z) = <e~p[iKArjl(~)]) + K4(Dj1,,?1)] + .. . (2.19) where K (modulus K ) is the scattering vector, taken to be in the ‘1’ direction. Finally, the full dynamic structure factor is where S(K) is the static structure factor S(K) = F(K,O) (2.2 1) and rij ri(0) - Vj(0). (2.22) The value of the results embodied in eqn (2.18), (2.19) and (2.20) is that they follow exactly from the Langevin equation in the z >> zB limit (or, equivalently, from the Smoluchowski equation). Thus they play the same role in the theory of particle suspensions as do the ‘sum rules’ in the theory of liquids.13 For example, the equiva- lent results for suspensions without hydrodynamic interactions are being used as guidelines for the construction of the memory functions describing the full time evolution of F(K, 2).14 As noted elsewhere, 8v9 there are several similarities between the dynamics of particles in suspension and atoms in a simple liquid which are considered further in the next section.Then, in section (2.3), we evaluate, as an example, eqn (2.18) for the mean-square displacement of hard spherical particles at low concentration. 2.2. COMPARISON WITH SIMPLE LIQUIDS The motion of an atom (of mass m) in a classical monatomic liquid is described by Newton’s law128 HYDRODYNAMIC INTERACTIONS IN CONCENTRATED PARTICLE SUSPENSIONS which gives, on integration, (2.23) The similarity between this result and eqn (2. lo), which describes particle dynamics in a suspension in which hydrodynamic interactions are neglected, is immediately apparent.In each case the first term can be identified as a 'random' displacement which does not depend either on the random displacements of other particles (or atoms) or on the instantaneous spatial configuration {ri(t)}. The second terms, the systematic displacements, are also similar in form. In a liquid the interatomic forces cause accelerations of the atoms; in a particle suspension, on timescale z >> zB, the forces cause 'drift' velocities of the particles. This accounts for the similar roles played by the atomic mass rn and the particle friction coefficient$ It is clear from the structures of eqn (2.10) and (2.23) that the Taylor expansions will contain terms of similar form but with different powers of z.For atoms, eqn (2.1 l), applicable in the true z + 0 limit, is the appropriate form of expansion. Use of eqn (2.1 I), (2.23) and the equipartition theorem then leads to whereas eqn (2.10), (2.13) and (2.9) give (2.24) (2.25) (particles, without hydrodynamic interactions). Eqn (2.24) and (2.25) show the similarities anticipated above. However, the inclusion of hydrodynamic interactions considerably complicates the situation [compare eqn (2.1) and (2.10) or eqn (2.18) and (2.25)]. Put simply, this is because not only do the interparticle forces depend on the instantaneous spatial configuration of the particles (as in a simple liquid) but, in addition (and in contrast to the case of simple liquids), each particle's motion in response to these forces also depends on the positions of all the particles through the configuration-dependent mobility tensors.Not surprisingly this leads to an unusually complicated dynamic many-body problem. An interesting point concerns the statistics of the particle displacement at short times. We showed previously that if hydrodynamic interactions are neglected there are no non-Gaussian contributions to the moments of the displacement of order less than z4. Expansion of the first part of eqn (2.19) gives K 2 K4 F ~ ( K , T ) = 1 - - (Arj:(z)) + -(Arj;(T)) + . . . 2 4! . (2.26) Comparison of eqn (2.18), (2.19) and (2.26) then gives (ArjLf(7)) = 12~~(Djl,j:> + t (r3) (2.27) so that with use of eqn (2.18) the non-Gaussian contribution to the fourth moment of the displacement becomes (Arj:(T)) - 3(Arj:(~))~ = 12~~[(Djl,j:> - (Dj1,j1)2] + C' (z3).(2.28)P . N . PUSEY AND R. J. A. TOUGH 129 Thus when hydrodynamic interactions are taken into account the particle displace- ment is not Gaussian-distributed even at short times. Since PDjl,jl[{ri}] can be identified as the short-time 'self' mobility of particle j it is perhaps not surprising that the lowest-order non-Gaussian contribution is simply proportional to a measure of the spread of mobilities that a particle experiences in all possible environ- ments in the suspension. Explicit evaluation of eqn (2.28) shows that the non- Gaussian contribution is small in relatively dilute suspensions; however, one might expect a larger effect in a concentrated suspension. It should be emphasized that, by cen tral-limi t - t heorem arguments, we still expect Gaussian statistics for the (long- time) displacement of a particle over distances large compared with the typical par- ticle separation. 2.3.MEAN-SQUARE DISPLACEMENT AT LOW CONCENTRATION As an example of the use of the results derived in section 2.1 we evaluate eqn (2.18) for the mean-square displacement of a particle to first order in volume fraction q (and to order T ~ ) . We write (ArjZl(7)) = 2 7 ~ 1 + r2(p2 + p3) + . . (2.29) where the definitions of the ,u are obvious by comparison with eqn (2.18). At low concentrations the probability of finding two particles close together is much greater than the probability of finding three or more in the same small volume. Thus the interparticle potential can be written in terms of a pair potential U(lrjk1) and we need only consider pair diffusivities.Following Batchelor we write the latter in the form (2.30) where rjk = Irjkl, 1 is the unit dyadic and numerical values of A and B are tabulated by Batchelor l 6 or given, approximately, by a series expansion in powers of In this low-concentration limit we then get, after performing the orientational part of the average, and For hard-spherical particles (radius R) these results can be evaluated. In the low- concentration limit the radial distribution function can be written = for ' 2R} hard spheres. = 0 for r < 2 R (2.33) For p1 we get the well known resultHYDRODYNAMIC INTERACTIONS IN CONCENTRATED PARTICLE SUSPENSIONS 130 where PI (Dj1,jI) = Do(1 - 2~9) the volume fraction cp is 4 3 cp = -nR3 P (2.34) (2.35) p being the number density of particles.For AA Batchelor l 6 obtained 1.83 from ‘exact’ numerical data for Ajj and Bjj whereas Felderhof obtained 2 A = 1.73 using series expansions up to order r - 7 . For hard spheres the first term in p2 [eqn (2.31)] is zero since Ajj - Ajk = 0 for touching spheres,16 corresponding to zero relative mo- bility along their line of centres. To evaluate the second term in eqn (2.31) we note that = 6(r - 2R), for hard spheres (2.36) and use the fact that Bjj - B j k = 0.40116 for touching spheres, to get Finally, use of Felderhof’s expansions (to order r-’) of the A and B gives (2.37) (2.38) The use of more exact hydrodynamics to evaluate p3 would probably change the numerical coefficient in eqn (2.38), but we still expect Ip3I << p2. Collecting contri- butions to eqn (2.29) we get, in terms of a dimensionless distance r/R, (1 - 1.83q) + T~ (:$2 - 0.61q + .. . . (2.39) ( ArjZT) ) R2 These results can be discussed in the light of other calculations. Ackerson and Fleishman I 9 and Hanna et a1.20 solved the Smoluchowski equation exactly for the (hypothetical) case of a dilute hard-sphere suspension with no hydrodynamic inter- actions. They found that the short-time effective self-diffusion coefficient, defined by [see eqn (2.18) and (2.29)] is simply D,”ff = Do. This reflects the fact that the probability of finding two hard spheres actually touching is vanishingly small so that in the absence of hydrody- namic interactions they diffuse freely over distances much smaller than R.The long- time self-diffusion coefficient, defined by (2.4 1) was found to be Ds = Do(l - 2q); (2.42) thus Ds is smaller than Do, which presumably reflects a hindering effect of particleP. N. PUSEY AND R. J. A. TOUGH 131 encounters on long-distance diffusion. In our treatment (above) the neglect of hydro- dynamic interactions corresponds to setting Ajj = Bjj = 1 and Ajk = Bjk = 0 SO that p1 = Do, p3 = 0 and (2.43) a result which can be obtained directly from eqn (2.25). For hard spheres we im- mediately get p2 = - 00. This result (which also implies a slowing down of the diffusion away from the z = 0 limit) is not inconsistent with the calculations of Ackerson and Fleishman and Hanna et al., who find fractional powers of z in the time regime intermediate between very short and very long times.Hanna et a1.20 also considered the effects of hydrodynamic interactions as did Jones and Burfield (who incorporated more detailed hydrodynamic interactions into Ackerson’s 22 projection operator analysis of the Smoluchowski equation). They found Den as given by eqn (2.34) and a long-time self-diffusion coefficient. Ds = Do(1 - A*(p + acp) (2.44) with c1 z -0.1, i.e. small and negative. Recently BatchelorZ3 has calculated Ds by a different method, effectively calculating the sedimentation velocity 24 of a few tagged particles in the presence of an excess of otherwise identical neutrally bouyant particles and then applying Einstein’s ‘dynamic equilibrium’ approach. Batchelor obtained a = +0.27, a result surprising (to us at least) at first sight since it implies a faster diffusion over long than over short distances.We point out, however, that our result, eqn (2.39), with a positive coefficient of z2, is consistent with Batchelor’s findings. We have discussed the case of hard spheres because it is conceptually simple and commonly considered.2 Unfortunately, many of the effects described above are small and will be hard to detect experimentally. Furthermore, it should be apparent that hard spheres constitute a pathological case; they cannot strictly be treated by the methods of section 2.1 which assume weak forces to provide the timescale separ- ation implied by eqn (2.5b). It is nevertheless interesting that a proper inclusion of hydrodynamics in our approach removes the infinity in the coefficient of z2 in the mean-square displacement. Finally, we note that, with soft repulsive interparticle potentials, the first term in p2 [eqn (2.31)] can dominate all other terms in eqn (2.31) and (2.32), making p2 + p 3 negative and implying a slowing down of diffusion away from the z -+ 0 limit, 3.LONG-TIME TAILS In our discussion so far we have taken the Brownian velocity components of the particles in the suspension to be effectively delta-correlated in time [see eqn (2.6)], as we have been concerned with observations made over times much greater than zB, characteristic of the exponential decay of the velocity correlations predicted by class- ical Langevin theory.’ However, it has been accepted for some time that the velocity autocorrelation function of an isolated Brownian particle should decay algebraically (as z - 3 / 2 ) at long times.Recent experiments 26 - have revealed this ‘long-time tail’ feature quite convincingly at times which are still short compared with that in which the particle diffuses a significant fraction of its radius R. Several have recognized that similar algebraic behaviour should be found in the velocity (cross) correlations of different particles in a concentrated suspension with decay times1 3 2 HYDRODYNAMIC INTERACTIONS IN CONCENTRATED PARTICLE SUSPENSIONS corresponding to the time of propagation of a hydrodynamic disturbance between the particles. These effects should, in principle, be observable in light-scattering experiments.When it is recognized that, as in the isolated particle case, these long- time tails should still have decayed completely in a time much shorter than zI, which characterizes a significant change in the spatial configuration of the particles, we see that our formalism provides a convenient route to a theoretical description of these experiments. For identical particles dynamic light scattering measures the dynamic structure factor F(K, z) defined by the first equality of eqn (2.20). Double differentiation of F(K, z), use of the stationarity condition and an integration gives, as an exact ex- pression for classical particles in equilibrium,8. 3 2 Since the effects of interest still decay rapidly compared with zI (see above) we can set (rJ(t)} = {rJ(0)} in eqn (3.1).Furthermore, provided the interparticle potential is not too 'hard', the velocities in eqn (3.1) can be replaced by their Brownian compo- nents ( ~ 8 ) ~ and the 'fast' average OV, discussed in section 2.1, can be performed. On this timescale, therefore, eqn (3.1) becomes We now define an effective time-dependent diffusion coefficient by where S(K) = F(K,O) is the static structure factor. Use of eqn (3.2), (3.3) and (2.7) then gives where is the usual effective diffusion coefficient obtained from the slope of F(K,z) on time- scale zB << z << zs.8733 In order to evaluate eqn (3.4) we need explicit expressions for the velocity correl- ations (vBil(0) vBjl(t))v.Recently, van Saarloos and Mazur 31 have developed a gen- eral scheme for calculating these (in frequency rather than time).For simplicity we keep here only the lowest-order term 8, (in an expansion in powers of Y; ') giving (3.6) (3 * 7) Do T 1 -312 lim (vBiI(0) vBjl(t))V = - L2 t t-'Q 2 f i where and p L is the density of the liquid; note that the time TL is of the same order of magnitude as zB [eqn (2.3)]. It is interesting that, at this (low) level of approximation, the i # j velocity cross-correlation, essentially the time-dependent Oseen tensor, has the same z - 3 / 2 long-time tail as the i = j auto~orrelation.~' Substitution of eqn (3.6) into (3.4) then gives TL = PLR2/VP. N. PUSEY AND R. J. A. TOUGH 133 In some respects the result embodied in eqn (3.8) is disappointing since it re- sembles closely the behaviour of a non-interacting (dilute) particle suspension [see eqn (2.17) of ref. (27)] for which DeK(K) -+ Do.We stress, however, that eqn (3.4) and (3.5) apply quite generally (for identical particles) so that substitution of the full complicated expressions for the velocity correlation functions will very likely still reveal some interesting effects. 4. MANY-PARTICLE HYDRODYNAMIC INTERACTIONS In the general case the diffusion tensors Dj,k are functions of the positions { r i ( t ) ) of all the particles in the suspension (see section 2.1); the new results given in this paper, eqn (2.18), (2.19), (2.20) and (3.4), retain this generality. Until recently, how- ever, detailed understanding of the structure of the tensors has been limited largely to the case of isolated pairs of particles [however, see ref.(34)]. As mentioned in section 1, this has changed in the last two years with the development by Mazur and coworkers - 3 8 of a systematic method for calculating the hydrodynamic inter- actions between an arbitrary number of spheres [see also ref. (39)]. This theory gives the Dj,k as a series in powers of reciprocals rjk', r6' etc. of the instantaneous interpar- ticle separations. With increasing powers of r - ', terms involving increasingly large clusters of particles contribute. For example, for j # k they find the usual two- particle terms of order rjkl and rS3, three-particle contributions of order r j 2 r i 2 and rjr rK4, a four-particle term of order r j rL3 r;: and so on. By coincidence we recently performed an experiment, described in detail elsewhere,40 which provides a partial test of this theory.We studied by dynamic light scattering a suspension of poly(methy1 methacrylate) colloidal spheres of diameter ca. 1.2 ym, suspended in an index-matched mixture of liquids, at volume fractions up to cp z 0.44. We measured the effective diffusion coefficient Deff(K) on a timescale zB << z << zI), defined by eqn (3.9, at scattering vectors K well above any oscillations in the structure factor S(K). In this high-Klimit, S(K) -+ 1 and i # j terms in eqn (3.5) average to zero. Thus we derive from our measurements the quantity Dzff can be identified as the short-time self-diffusion coefficient which describes the average diffusion of a particle over distances much smaller than its radius R.This differs from Do only because of hydrodynamic interactions (assumed to act instanta- neously on this timescale) of the particle with its neighbours and is probably the simplest interesting dynamic property of a concentrated suspension. The experiment- ally determined values of Desff are plotted as points in fig. 1. In ignorance of many-particle effects one could assume hydrodynamic inter- actions to be pairwise additive. Thus substitution of eqn (2.30) for the two-particle mobility tensor into eqn (4.1) gives Use of the low-concentration expression [eqn (2.33)] for the radial distribution func- tion g(r) of hard spheres leads immediately to the result eqn (2.34) quoted in section 2.3. At higher concentrations the effect of pair hydrodynamic interactions on Ifff1 34 HYDRODYNAMIC INTERACTIONS IN CONCENTRATED PARTICLE SUSPENSIONS i- 0.2 0 0 0.1 0.2 0.3 0.4 0.5 volume fraction, cp Fig.1. Short-time self-diffusion coefficient & (divided by the free-particle value Do) plotted as a function of suspension volume fraction cp. The data points are experimental results, curve A is the theoretical result for pairwise additive hydrodynamic interactions, curve B is the density expansion of Beenakker and Mazur [eqn (4.3)] and curve C is their full theory. can be calculated by numerical integration of eqn (4.2) with use of approximate (but accurate) expressions for the radial distribution function of hard spheres. This cal- culation has been performed by Pusey and van Megen40 using Felderhof’s series expansions for A(r) and B(r) and by Glendinning and Russel 41 using exact numer- ical data.The result of the latter calculation is shown as the lower solid line (A) in fig. 1. The inadequacy of assuming pairwise additivity of hydrodynamic interactions is immediately apparent, both when the experimental data and curve A are com- pared and when one realises that a hard sphere must retain some mobility within its nearest-neighbour shell up to close packing (cp = 0.64 - 0.74) whereas curve A predicts D& = 0 at ( ~ ~ 0 . 4 3 . ~ ~ Beenakker and Mazur37i38 have taken two approaches to the problem. First37 they evaluated D& as a virial expansion and obtained (4.3) Three-body effects contribute to the q2 term; if hydrodynamic interactions had been taken to be pairwise additive they would have obtained -0.93q2.The dashed line, curve B, in fig. 1 is eqn (4.3), which is limited in validity to relatively low cp by neglect of four-particle and higher-order terms. Se~ondly,~ they have, remarkably, been able to sum formally the full expression for D j , j which includes many-particle effects. A fluctuation expansion then provides predictions for D2f, expected to be quite accurate to high volume fraction. These predictions are plotted as curve C in fig. 1, the error bars being estimated from information given in section 7 of ref. (38). The differences between this theory (curve C ) for hard spheres and the experi- mental results for cp 2 0.3 are significant. Beenakker and Mazur suggest that the experimental particles may not be (in fact cannot be) strictly hard spherical. Indeed in the experiments 40 we noticed some ‘crystallization’ of the spheres into solid-like clusters at cp = 0.44, whereas computer simulations of hard spheres predict the onset D& = Do (1 - LAcp + 0 .8 8 ~ ~ + . . .).P. N. PUSEY AND R. J. A. TOUGH 135 of the transition to be at cp z 0.49.42 We suggested that, because of ioosely packed polymer coatings on the particles, the effective hard-sphere volume fraction might be greater than the calculated value. However, it seems equally possible that interaction of the polymer coatings could result in a small interparticle attraction which could cause early crystallization and reduce Nevertheless, we emphasize that both the theory and the experimental results demonstrate the importance of irreducible many-particle hydrodynamic interactions in concentrated particle suspensions.We thank Prof. G. K. Batchelor and Prof. P. Mazur for communicating their work prior to publication. The experiments described in section 4 were performed in collaboration with Dr. W. van Megen. Selected Papers on Noise and Stochastic Processes, ed. N. Wax (Dover, New York, 1954). P. Mazur and I. Oppenheim, Physica, 1970,50, 241. J. Albers, J. M. Deutch and I. Oppenheim, J. Chem. Phys., 1971, 54, 3541. J. M. Deutch and I. Oppenheim, J. Chem. Phys., 1971,54, 3547. T. J. Murphy and J. L. Aguirre, J. Chem. Phys., 1972, 57, 2098. G. Wilemski, J . Stat. Phys., 1976, 14, 153. W. Hess and R. Klein, Physica, 1978, 94A, 71. R. Klein and W. Hess, Lecture Notes in Physics (Springer-Verlag, Berlin, 1982), vol.172, p. 199. l o D. L. Ermak and J. A. McCammon, J. Chem. Phys., 1978,69, 1352. l 1 P. N. Pusey and R. J. A. Tough, in Dynamic Light Scattering and Velocimetry: Applications of l 2 P. G. de Gennes, Physica, 1959, 25, 825. l 3 J. P. Boon and S. Yip, Molecular Hydrodynamics (McGraw-Hill, New York, 1980). l4 J. L. Arauz-Lara and M. Medina-Noyola, Physica A, in press. l S B. R. A. Nijboer and A. Rahman, Physica, 1966, 32, 415. l6 G. K. Batchelor, J, Fluid Mech., 1976, 74, 1. B. U. Felderhof, Physica, 1977, 89A, 373. l 8 B. U. Felderhof, J. Phys. A, 1978, 11, 929. l9 B. J. Ackerson and L. Fleishman, J. Chem. Phys., 1982, 76, 2675. 2o S. Hanna, W. Hess and R. Klein, Physica, 1982, 111A, 181. 2 1 R. B. Jones and G. S. Burfield, Physica, 1982, 111A, 562 and 577. 2 2 B. J. Ackerson, J. Chem. Phys., 1978, 69, 684. 23 G. K. Batchelor, J. Fluid Mech., 1983, 131, 155. 24 G. K. Batchelor, J. Fluid Mech., 1982, 119, 379; G. K. Batchelor and C. S. Wen, J. Fluid Mech., 1982, 124, 495. ‘Hard-sphere’ suspensions have also been studied extensively by static and dynamic light scattering: see A. Vnj, J. W. Jansen, J. K. G. Dhont, C. Pathmamanoharan, M. M. Kops-Werkhoven and H. M. Fijnaut, Faraday Discuss. Chem. SOC., 1983, 76, 19. 2 6 J. P. Boon and A. Bouiller, Phys. Lett. A, 1976, 55, 391; A. Bouiller, J. P. Boon and P. Deguent, J. Phys. (Paris), 1978, 39, 159. 27 G. L. Paul and P. N. Pusey, J . Phys. A, 1981, 14, 3301. 2 8 K. Ohbayashi, T. Kohno and H. Utiyama, Phys. Rev. A, 1983, 27, 2632. 29 T. S. Chow and J. J. Hermans, Kolloidn. Zh., 1972, 250, 404. 30 A. R. Altenberger, J. Polym. Sci., Polym. Phys. Ed., 1979, 17, 1317. 31 W. van Saarloos and P. Mazur, Many-sphere Hydrodynamic Interactions 11: Mobilities at Finite 32 P. N. Pusey, J. Phys. A, 1975, 8, 1433. 33 Eqn (3.5) was first obtained for polymers, e.g. R. Zwanzig, J. Chem. Phys., 1974, 60, 2717; Z. 34 G. J. Kynch, J. Fluid Mech., 1959, 5, 193. 35 P. Mazur, Physica, 1982, llOA, 128. 3 6 P. Mazur and W. van Saarloos, Physica, 1982, 115A, 21. 37 C. W. J. Beenakker and P. Mazur, Phys. Lett. A, 1982, 91, 290. 3 8 C. W. J. Beenakker and P. Mazur, Self Diffusion of Spheres in Concentrated Suspensions * P. N. Pusey and R. J. A. Tough, J. Phys. A, 1982,15, 1291; 1983, 16, 2289, corrigendum. Photon Correlation Spectroscopy, ed. R. Pecora (Plenum, New York, in press). Frequencies (preprint). Akcasu and H. Gurol, J. Polym. Sci., Polym. Phys. Ed., 1976, 14, 1. (preprint).1 36 HYDRODYNAMIC INTERACTIONS IN CONCENTRATED PARTICLE SUSPENSIONS 3 9 M. Muthukumar and K. F. Freed, J. Chem. Phys., 1983, 78, 497 and 51 1 . 40 P. N. Pusey and W. van Megen, J. Phys. (Paris), 1983, 44, 285. 4 1 A. B. Glendinning and W. B. Russel, J. Colloid Interface Sci., 1982, 89, 124. 4 2 B. J. Alder, W. G. Hoover and D. A. Young, J. Chem. Phys., 1968, 49, 3688.

ISSN:0301-7249    

DOI:10.1039/DC9837600123

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

Faraday Discuss. Chem. Soc., 1983, 76, 137-150 Mass-diffusion and Self-diffusion Properties in Systems of Strongly Charged Spherical Particles BY RUDOLF KLEIN AND WALTER HESS Fakultat fur Physik, Universitat Konstanz, D-7750 Konstanz, West Germany Received 6th May, 1983 Using the Fokker-Planck equation as the basic description, various collective and single- particle properties of highly charged spherical macroparticles in solution are derived. The dynamic structure factor S(k,t) for finite k is found to be non-exponential in t due to the viscoelastic properties of the liquid of interacting macroions. By using a mode-coupling pro- cedure the dynamical properties can be completely reduced to the static structure factor S(k), which is calculated according to the method developed by Hansen and Hayter.At short times the system exhibits elastic behaviour and the corresponding high-frequency elastic moduli have been calculated. The bulk modulus is found to increase with c 2 , where c is the con- centration. With regard to the single-particle properties, the velocity autocorrelation function is calculated and from it are obtained the mean-square displacement and the self-diffusion coefficient D, as a function of concentration and of the coupling strength between macroions. 1. INTRODUCTION Colloidal suspensions can be considered as binary liquids consisting of two very different components. The solvent is a simple liquid and the macroparticles are much larger and heavier than the solvent molecules. In concentrated solutions each macroparticle experiences not only frequent collisions with solvent molecules but also the presence of other macroparticles through direct deterministic interactions. In this paper, highly charged spherical particles are considered, where the main interactions are the hard core (excluded volume) and the fairly long-ranged screened Coulomb interactions.Such systems are realized by polystyrene spheres in aqueous solutions. - 4 It is known experimentally that these systems exhibit highly correlated behaviour for volume fractions of lop4 and less. Under such circumstances the neglect of hydrodynamic interactions seems to be a good approximation. Since the solution consists of two very different components the dynamical behaviour of the total system will take place on two separated time scales.Most experiments such as light scattering look only at the subsystem of macroparticles; therefore, it is sufficient to develop a theoretical description for the dynamics which is applicable on a time scale on which the fast dynamics of the solvent molecules plays no role. Such a description is provided by the Fokker-Planck equation, which is the equation of motion for the distribution function f(r, t ) = f(pl, . . . ,pN,rl, . . . ,rN,t) of the momenta pi and coordinates r j of the centres of mass of the spherical macroparticles. On the basis of the Fokker-Planck equation the solvent is described as a continuous medium which produces friction and mediates a hydrodynamic interaction bet ween different macro particles. Static light scattering shows a liquid-like structure factor S(k) with a pronounced first peak.Since the scattering arises practically only from the macroparticles one can speak of the subsystem of macroparticles as a ‘liquid of interacting Brownian138 STRONGLY CHARGED PARTICLES particles’. The main difference from an ordinary so-called ‘simple’ liquid is that each constituent of the Brownian liquid experiences friction due to the presence of the solvent. In this paper the dynamical properties of the Brownian liquid are developed in a similar way to the generalized hydrodynamics of simple liquids. For the mass- or collective-diffusion properties the dynamic structure factor S(k,t) is calculated by a two-step memory function procedure, the first memory function being a generalized diffusion function and the second the longitudinal dynamic viscosity of the liquid of Brownian particles. For the case of single-particle properties such as the velocity autocorrelation function and the self-diffusion coefficient it is the self-dynamic structure factor G(k,t) which is treated similarly.To calculate the second memory function using this procedure a mode-coupling method is used which reduces the dynamics essentially to static properties. The static property of most interest is the static structure factor S(k). Following the work of Hayter and Penfold and of Hansen and Hayter we have calculated S(k) for the polystyrene solutions investigated by Gruner and Lehmann.3 The solution is described as a one-component plasma of macroions of finite size. The counterions are treated as a uniform neutralizing background, which determines the screening of the ion-ion interactions, taken to be a repulsive screened Coulomb interaction.The attractive van der Waals interactions are assumed to be negligible since the Coulomb interactions are in most cases larger than kBT. The input for the rescaled mean-spherical approximation (MSA) method developed by Hansen and Hayter is the Verwey-Overbeek two-particle potential where E is the dielectric constant of the solvent, a is the radius of the macroions, $o is the surface potential and K is the Debye-Huckel screening parameter. The value of t,bo is determined by fitting the height of the main peak of S(k) as obtained from the MSA calculation to the experimental result at one concentration.Keeping this value fixed for all other samples which were investigated by Gruner and Lehmann, good agreement was obtained with the experimentally determined S(k), with the exception of small scattering angles. The theoretical results are always smaller for k -+ 0 than the extrapolations of the experimental curves to k -, 0. Since later results are quite sensitive to the values of S(k) at long wavelengths, we have corrected7 the long- wavelength part of S(k) by a phenomenological m e t h ~ d . ~ In this way static structure factors are available, if the system parameters in eqn (1.1) and the volume fraction q~ are specified. In the next section the theory for the collective properties is outlined. It is shown that S(k,t) deviates from a simple exponential function outside the hydrodynamic regime because of the frequency and wavevector dependence of the dynamic longi- tudinal viscosity of the liquid of interacting macroions.Satisfactory agreement with experimental results from dynamic light scattering is obtained. The viscoelastic behaviour of the system is further illustrated by calculating the short-time or high- frequency elastic moduli. In the third section, one-particle properties such as the velocity autocorrelation function of a tagged particle and the self-diffusion coeffi- cient as a function of concentration and of the interaction strength are obtained.R. KLEIN AND W. HESS 139 2. MASS- OR COLLECTIVE-DIFFUSION PROPERTIES As mentioned in the introduction, the transport equation of a Brownian liquid is the Fokker-Planck equation a -fTr,t) = fif(r,t) (2.1 ) at where O(r) is the Fokker-Planck operator where pi denotes the forces which all other macroparticles exert on macroparticle i through direct interactions and lo is the friction coefficient at infinite dilution.The first part of eqn (2.2) is identical to the Liouville operator of a simple liquid and the second part originates from integrating out the fast solvent variables in the Liouville equation. O , The basic phase-space variables are the concentration fluctuations N ~ ( k ) = 1 exp( - ik ri) - (1 exp( - ik ri))o (2.3) i = 1 1 where the subscript zero indicates an equilibrium average, and the current fluctuations Because of the non-hermitean character of 6 the time derivatives of phase-space functions are given by operating with the hermitean adjoint operator Q+.One obtains a+P(j) = -ik - j ( k ) (2.5) In the last line the pressure fluctuations kB T 1 S(k) N $(k) = - P(k) ; S(k) = - (P(k) i'( -k))* and the fluctuating force density f ( k ) = ik a(k) - coj"(k) have been introduced, where a@) is the viscous stress tensor of the 'liquid of inter- acting Brownian particles'. The latter is formally identical to the expression obtained in the theory of simple liquids.12 The main difference between our Brownian liquid, in which every particle experiences friction by the solvent, and a simple liquid is the presence of the second tezm in eqn (2.8). It originates from the second part (the 'Fokker-Planck part') of SZ in eqn (2.2). For a study of collective properties of a system a quantity of central importance is the dynamic structure factor S(k,t), which can experimentally be obtained from scattering experiments. It is defined by140 STRONGLY CHARGED PARTICLES I S(k,t) = T(P(k) exp(fit)t( -k))oO(t).(2.9) Results for S(k,t) for a system of particles described by a Fokker-Planck equation are now derived using a Mori-Zwanzig projection operator formalism. Taking the time derivative of eqn (2.9) and using eqn (2.5) d i at N - S(k,t) = - k - ( j ( k ) exp@t)?( -k))oO(t) + S(k) &I). (2.10) Defining the Laplace transform by fa, 1 f(k,z) = dt exp( - z t ) S(k,t) = A (?(k) [z - a]- ?( - k))o J O N eqn (2.10) becomes 1 zS(k,z) = S(k) - k * - ( J k ) [Z - 61-l ?(--k))o. (2.12) N Since according to eqn (2.5) the time derivative of the concentration fluctuations vanishes in the long-wavelength limit (k -+ 0), which exp_resses the conservation of the number of Brownian particles, a projectiqn operator P, will be introduced which projects an arbitrary phase-space function A(k) on the concentration fluctuations, which are the slow variables of our system 1..1 P,A(k) = ~ ( 4 k ) ?( - W o (c(k)- (2.13) N S(k) Using in eqn (2.12) the operator identity [z - 61-1 = [z - sz Q,y (1 + SiP,[z - a]-1) where & = 1 - p,, leads to (2.14) i N zf(k,z) = S(k) - k - (i(k) [Z - Si Qc]- ?( -k))o 1 - k--(Ak)[z - SiQc]-lSiPc[~ - 61-l P(-k))o. (2.15) N The se_cond term vanishes since Qc projects on the subspace orthogonal to ?(k) and since j ( k ) is orthogonal to t(k). Inserting eqn (2.13) in eqn (2.15), the last term factorizes, one term being $(k,z) according to eqn (2.1 1).Therefore S(k) S(k,z) = z + &k,z) k2 (2.16) wherej,,(k) is the component ofj’(k) parallel to k. The hydrodynamic limit of D(k,z) defines the collective- or mass-diffusion coefficient D, = D(0,O). (2.18)R. KLEIN AND W. HESS 141 In this limit the dynamic structure factor S(k,t) becomes a simple exponential func- tion of the time S(k,t) + S(0) exp( - D,kzt). (2.19) The function D(k,z) for non-zero k and z generalizes the diffusion coefficient to a wavevector- and frequency-dependent diffusion function. Since the light-scattering experiments on polystyrene spheres also cover the non-hydrodynamic regime, this generalized diffusion function is important. It can be further analysed by applying the Mori-Zwanzig technique to the Forrelation function in eqn (2.17) with a projec- tion operator Pj which projects on j ( k ) Using the same procedure as for s(k,z), one obtains where Q -= 1 -A Pi.Since cu_rrent _and c_oncentration fluctuation are orthogo_nal, QC Pj = Pj and Qc Qj = 1 - P, - Pj = Q. Using the definition eqn (2.20) for Pj in the second term in eqn (2.21) and an operator identity similar to eqn (2.14) in the third term yields (2.22) where r , , ( k , z ) is a generalized friction function and eqn (2.22) can be considered as the generalization of the Stokes-Einstein relation to finite k and z. In the hydro- dynamic limit and for non-interaEting particles [S(O) = 13, eqn (2.22) becomes D, = Do = kBT/co, where io = c(0,O). For interacting systems the mass- diffusion coefficient becomes, from eqn (2.18) and (2.22), and the dynamic structure factor becomes, from eqn (2.20) and (2.22), (2.24) (2.25) where cT(k) = [kBT/rn S(k)]* is the isothermal velocity of sound.142 STRONGLY CHARGED PARTICLES The dynamical friction function [ , ( k , z ) can be further simplified. Using eqn (2.3)-(2.8) in eqn (2.23) (2.26) The total friction function consists of two contributions, the first term is the single- particle friction due to the presence of the solvent and the second term is just the contribution from the longitudinal viscosity of the 'liquid of Brownian particles' (2.27) where k has been chosen in the z direction.It follows from eqn (2.25) and (2.26) that a measurement of the dynamical structure factor gives information about the viscos- ity function of the liquid which consists of the interacting macroions.For poly- styrene spheres of diameters of the order of several hundreds of Angstroms the relaxation time zB = m/C0 is of the order of lop8 s, whereas the shortest correlation times of a dynamical light-scattering experiment are ca. 10- 5-10-6 s. Therefore, eqn (2.25) can well be approximated by Experimental results for S(k, t ) are often characterized by following short-time expansion (2.28) cumulants, which is the S(k,t) = S(k) exp n! (2.29) The first two cumulants of eqn (2.28) are pcL1(k) = D"(k) k 2 ; D"(k) = Do/S(k) (2.30~) P 2 ( 4 = - P 1 W k2 VlI(k,O)/(~ lo). (2.30b) Therefore, the initial slope of loglS(k,t)/S(k)l against t determines the static structure factor, which is a well known r e s ~ l t .' ~ Including hydrodynamic interactions, D"(k) is changed by a k-dependent factor. If Do k2/pu,(k) agrees with the determination of S(k) from static light-scattering experiments, hydrodynamic interactions are unim- portant. This seems to be the case in the experiments. Since ,u2(k) and all higher cumulants are of higher order in k2 the dynamic structure factor S(k,t) for small scattering angles is a simple exponential of t. The initial value q,,(k,t = 0) of the longitudinal viscosity enters the second cumulant; it is essentially determined by the total interaction potential UN exp[ -ik(zi - zj)] )o - - kBT). (2.31) SW The short-time behaviour of correlation functions can also be characterized by a moment expansion or a continued fraction representation. This method can be used to show that the system of interacting macroions exhibits elastic behaviour at short times.Following Zwanzig and Mountain who treated the visco- elastic behaviour of simple liquids, we have introduced high-frequency elastic shear and Schofield,R. KLEIN AND W. HESS 143 and bulk moduli G"(k) and K"(k) for the Brownian fluid. They are given by a combination of the first three moments of the transverse and longitudinal current correlation functions. One obtains 1 - cos kz a2 U(r) k2 ___ ax2 d3r) (2.32) 4 1 - cos kz d2U - G"(k) + K"(k) = c 3kBT + c g(r) 3 ( 1 k2 dz2 - d3r). (2.33) The moduli are therefore given in terms of the radial distribution function g(r) and the two-particle potential U(r).Using eqn (1.1) and the results for the structure factor as discussed in the first section, one obtains from eqn (2.32) and (2.33) the curves displayed in fig. 1 and 2. The high-frequency moduli are qualitatively similar to those of simple liquids; l 7 they decrease with increasing wavevector and are in- creasing functions of concentration. kais Fig. 1. High-frequency elastic moduli as a function of ka,, = ak/q1'3for four volume con- centrations: q = 0.5, 1.5, 3.0 and 7.5 x Full lines: g"(k) = G"(k) 47ca3/(3kBT); dashed lines: e"(k) = E(k) 4na3/(3kBT). The longitudinal modulus E(k) = (4/3)G"(k) + K"(k) can be obtained from the second cumulant, since eqn (2.30b) can be rewritten as l 6 This procedure was used by Griiner and Lehmannls to determine E(k -+ 0) as a function of concentration.They obtained E(k -+ 0) - c2, which is in agreement with our result from eqn (2.33), shown in fig. 2. The shear modulus G"(k + 0) is found by us to be proportional to c4I3. Going back to the time dependence of S(k,t) it is convenient to define a mean relaxation time (2.34)144 STRONGLY CHARGED PARTICLES 101 1 OC n s 8u -0 5 lo-' h = a0 1 6 ' 1 o - ~ I 10-4 1 o - ~ (P Fig. 2. Long-wavelength limit of high-frequency elastic moduli em(0) and g"O(0) as a function of concentration. Note eoo(0) - (p2 and g"(0) - As a measure of the deviation of S(k,t) from a simple exponential function of t we introduce (2.35) where in the second line use was made of eqn (2.28). The experimental determination of A(k) therefore gives the time integral of the dynamic longitudinal viscosity func- tion.The latter has been calculated7# l 6 from eqn (2.27) by a mode-coupling ap- proximation. In the spirit of this approximation one replaces Q in eqn (2.27) by a projection operator which projects onto bilinear products of the slow variables, which are the concentration fluctuations. As a result, qll(k,t) can be completely expressed by the static structure factor and the unknown dynamic structure factor so that the mode coupling expression together with eqn (2.25) and (2.26) give a closed set of equations. From an approximate numerical solution of these equations, using the phenomenologically corrected static structure factor, A(k) has been calculated for four different concentrations. The results are shown in fig.3 together with theR. KLEIN AND W. HESS 145 'I 0 0.5 1 1.5 2 2.5 k / k m Fig. 3. A(k) plotted against k/k,, eqn (2.35). Curves are theoretical results for cp = 0.1, 1.5, 3.0 and 4.5 x Data points are the experimental results of Gruner and Lehmann3 for 40 between 1.5 x and 7.5 x experimental results of Gruner and Lehmann.3 Noting that no adjustable parameter enters except the surface charge, which was already fitted to the static structure factor at one concentration, the agreement is quite satisfactory. The experimental results do not show a definite concentration dependence of A(k) within the investigated range of volume fractions cp. We have calculated A(k) as a function of cp for those values of k for which A(k) reaches its maximum. This value 1 1 I 1 1 0 0.2 0.4 0.6 0.8 1 1029 Fig.4. A(k x 2kJ3) as a function of concentration.146 STRONGLY CHARGED PARTICLES is roughly at k = (2/3) k , = 5.5/(2ais), where k , denotes the position of the main maximum of S(k) and 4, = (3/4~c)'/~is the ionic sphere radius. The result in fig. 4 shows that A(5.5/2ais) increases from zero to a nearly constant value of ca. 0.7 in the concentration range which was investigated e~perimentally.~ The weak increase with cp above cp = 1.5 x explains the observation of Griiner and Lehmann that A(k) seems to be independent of concentration. However, fig. 4 shows that this does not hold at concentrations lower than those used in the experiments. 3. SINGLE-PARTICLE PROPERTIES Recently various attempts have been made to measure single-particle properties in colloidal systems.Therefore we'present here some results of calculations of such properties, which were obtained on the same basis as the collective results. The dynamics of the tagged particle (index I ) is described by the correlation function (3.1) G(k,t) = ( 2 , ( k ) exp(fit)2,( -k))" O ( t ) where 2,(k) = exp( -ik r , ) . From it one can calculate the mean-square dis- placement and the velocity autocorrelation function 1 a 2 1 v(l) = - ( o l ( t ) u1(0)) = - lim - - G(k,t). 3 k - 0 at2 k2 The self-diffusion coefficient is given by D, = lim W(t)/t = 2-03 (3.3) (3.4) Using procedures similar to those which resulted in eqn (2.16) and (2.22), one ob- tains for the Laplace transform of eqn (3.1) (3.5) 1 z + D,(k,z) k2 G(k,z) = Therefore, the velocity autocorrelation function is simply V(t) = Ds(O,t) and the self- diffusion coefficient D, = B,(O,O) = kBT/~,,(O,O).The projection operator formalism gives the longitudinal dynamic self-friction function as L(k4 = 5" + A L , , ( W AL,,(k,z) = p(filz exp( -ik rl)[z - Flz exp(ik rl))O (3.7) where pl denotes the direct interaction forces acting on the t_agged particle and Q1 is the projector onto the subspace perpendicular to E,(k) and jl(k). Since we are againR. KLEIN AND W. HESS 147 restricting ourselves to times larger than the relaxation time zB ;5: simplifies to s, eqn (3.6) D,(k,z) = Do - AB,(k,z) Therefore, the velocity autocorrelation function becomes V(t) = 2 Do d(t) O(t) - AD,(O,t) and the mean-square displacement becomes dt’ ( t - t’) AD,(O,t’).Thus for small times for large times. W(t) = (3.9) (3.10) (3.11) Since AB,(O,O) 3 0, the self-diffusion coefficient D, is always smaller than the free- diffusion constant Do. Eqn (3.10) and (3.11) show how the motion of the tagged particle is slowed down by interactions with other particles. It remains to calculate ATs,,(kLz). This was again done l9 in a mode-coupling approximation by approximating Q by bilinear products of the one-particle fluctu- ation i , ( k ) and of the concentration fluctuations of the other particles. The result of this calculation gives AC,,(k,t) in terms of S(k), G(k,t) and S(k,t) (3.12) where c,(k) = [S(k) - l]/[c S(k)] is the direct correlation function. The integral has been evaluated with the mean-field correlation function GMF(k,t) = exp( - Dok2t) and SMF(k,t) = S(k) exp[ - Dok2t/S(k)].With this result various single-particle pro- perties have been calculated. From eqn (3.8) and (3.9) the dynamic part AD,(O,t) of the velocity autocorrelation function is obtained. The result is shown in fig. 5. With increasing concentration its initial value increases and its decay becomes more rapid. Moreover, the velocity autocorrelation function is not a simple exponential in time. The mean-square dis- placement follows from eqn (3.10). Its initial slope according to eqn (3.11) is Do. With increasing time the neighbouring particles are felt and W(t) grows more slowly than linear. This deviation from the initial behaviour starts earlier for the more concentrated system. The quantity of most practical interest is the self-diffusion coefficient D,, which is calculated from V(t) using eqn (3.4).In fig. 6 D, is shown as a function of con- centration for the type of polystyrene spheres investigated by Griiner and Lehmann.3 One finds a rather sharp decrease at very low concentrations, but for the interval of concentrations investigated by these authors our calculation predicts an only slightly decreasing value of D,/Do between 0.7 and 0.6. Fig. 7 shows D, for a fixed concentration as a function of a dimensionless coupling parameter y7 which essentially measures the surface potential, y = (fl/2) E a $; exp(2alc). Numerical simulations of single-particle properties by Gaylor et aL20 show similar qualitative behaviour. A detailed comparison is, however, not possible, since148 STRONGLY CHARGED PARTICLES lo” 10‘ 6 6 d d T ’O 1 16’ 1 I 1 t Fig.5. Normalized dynamic part of the velocity autocorrelation function as a function of z = Dot/(2a,J2 for four volume concentrations: cp = 0.1, 0.5, 1.5 and 7.5 x 1 0 9 0.5 9 0 0.2 0.4 0.6 0.8 1 I O * q 10-3. Fig. 6. Normalized self-diffusion coefficient as a function of volume concentration for poly- styrene spheres of diameter 900 A, surface potential I,!I~ = 73 meV and screening length K - = 5000 A.R. KLEIN AND W. HESS 149 0 1000 2000 3000 Y Fig. 7. Normalized self-diffusion coefficient as a function of the dimensionless couplin para- meter y for polystyrene spheres of diameter 900 A, screening length c1 = 5000 1 at a volume concentration of cp = 5.6 x lop4. the simulations have been performed for different system parameters. Earlier theoretical work on self-diffusion properties of strongly correlated macroparticle systems is based on the Smoluchowski equation, which is the strong-friction limit of the Fokker-Planck equation.Using the Mori-Zwanzig technique and a mode-coup- ling approximation together with an experimentally measured structure factor, we have given results for W ( t ) and D,. For the particular systems whose S(k) was used, the value of D, turned out to be lower than the value obtained on the basis of the Fokker-Planck equation. 4. CONCLUSIONS The generalized hydrodynamic theory for a Brownian liquid, developed on the basis of the Fokker-Planck equation, leads to the following picture of the dynamics of systems of charged colloidal particles: the strongly correlated system has a short- range structure [reflected in S(k)], consisting of particle clusters of some lifetime Z.For times smaller than Z the system exhibits elastic behaviour, whereas for longer times it is liquid-like. This viscoelastic behaviour is reflected in the time dependence of the viscosity functions. By assuming that the interactions among the macroions are of the form of eqn (1. l), the collective diffusion properties such as S(k,t) and the high-frequency elastic moduli can be calculated and are found to agree with experi- mental results. This agreement suggests a similar treatment for the single-particle properties, the results of which are given in section 3. We thank Dr. J. Hayter for providing us with the computer program to calculate S(k)- J.C. Brown, P. N. Pusey, J. W. Goodwin and R. H. Ottewill, J . Phys. A , 1975,8, 664. P. S. Dalberg, A. Bere, K. A. Strand and T. Sikkeland, .I. Chem. Phys., 1978, 69, 5473.150 STRONGLY CHARGED PARTICLES F. Griiner and W. Lehmann, J. Phys. A , 1979, 12, L-303. P. N. Pusey and R. J. A. Tough, in Dynamic Light Scattering and Velocimetry: Applications of Photon Correlation Spectroscopy, ed. R. Pecora (Plenum Press, New York, 1982). J. B. Hayter and J. Penfold, Mol. Phys., 1981, 42, 109. J. P. Hansen and J. B. Hayter, Mol. Phys., 1982, 46, 651. R. Klein and W. Hess, in Ionic Liquids, Molten Salts and Polyelectrolytes, ed. K . H. Bennemann, F. Brouers and D. Quitmann (Springer-Verlag, Berlin, 1982), pp. 199-21 1. E. J. W. Verwey and J. T. G. Overbeek, Theory of the Stability of Lyophobic Colloids (Elsevier, Amsterdam, 1948). H. Minoo, C. Deutsch and J. P. Hansen, J. Phys. (Paris) Lett., 1977, 38, L-191. l o J. M. Deutch and I. Oppenheim, J. Chem. Phys., 1971, 54, 3547. T. J. Murphy and J. L. Aguirre, J. Chem. Phys., 1972, 57, 2098. l 2 J. P. Boon and S. Yip, Molecular Hydrodynamics (McGraw-Hill, New York, 1980). l 3 B. J. Ackerson, J. Chem. Phys., 1976, 64, 242. l4 R. Zwanzig and R. D. Mountain, J. Chem. Phys., 1965, 43, 4464. l 6 W. Hess and R. Klein, Ado. Phys., to be published. l 7 A. Z. Akcasu and E. Daniels, Phys. Rev. A, 1970, 2, 962. F. Gruner and W. Lehmann, J. Phys. A , 1982, 15, 2847. l 9 W. Hess and R. Klein, J. Phys. A , 1982, 15, L-669. 2o K. J. Gaylor, I. K. Snook, W. J. van Megen and R. 0. Watts, J. Phys. A , 1980,13,2513; J . Chem. 21 W. Hess and R. Klein, Physica, 1981, 105A, 552. P. Schofield, Proc. Phys. SOC. London, 1966, 88, 149. SOC., Faraday Trans. 2, 1980, 76, 1067.

ISSN:0301-7249    

DOI:10.1039/DC9837600137

出版商:RSC    

年代:1983

数据来源: RSC