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. 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