Faraday Discuss. Chem. SOC., 1987, 83, 167-177 Brownian-dynamics Simulation of the Formation of Colloidal Aggregate and Sediment Structure Geoffrey C. Ansell and Eric Dickinson" Procter Department of Food Science, University of Leeds, Leeds LS2 9JT Brownian-dynamics simulation is used to investigate the effect of particle volume fraction and interparticle interactions on the structure of colloidal materials formed by irreversible aggregation. Short-range structure of aggre- gates and sediments is represented by the pair distribution function g ( r ) , and longer-ranged structure by the effective fractal dimension. For unstable DLVO-type particles with a secondary minimum, the simulated short-range aggregate structure resembles that in a stable concentrated dispersion; for particles without a secondary minimum, g ( r ) is relatively unstructured either in sediments or gels.Hydrodynamic interactions play a crucial role in the formation of dense sediments at high field strengths, but are much less important when Brownian motion is predominant. Studies of small-floe dissociation give insight into the likely consequences of neglecting multi- body hydrodynamic interactions in simulations of this type. Brownian motion is the primary process whereby particles come together to form colloidal aggregates and gels. Brownian motion plays a crucial role in the dissociation of small colloidal flocs, in the desorption of particles from surfaces, in the unfolding and denaturation of proteins, in the kinetics of biological reactions and in many other diffusive phenomena.In sedimentation or creaming, Brownian motion has a secondary role compared with the action of gravity; but it is nevertheless important in determining the final sediment or cream structure. The term colloidal particle is essentially synony- mous with the term Brownian particle. To a large extent, then, colloid science is the science of Brownian motion. To simulate a system containing solute particles and a very much larger number of solvent molecules (plus ions etc.), where one is interested only in the dynamics of the solute particles, it is convenient to use a stochastic-dynamics approach rather than the more conventional molecular-dynamics approach used in connection with simple liquids. Where inertial terms are negligible, the motion is entirely diffusive, and the technique is called Brownian dynamics.In a Brownian-dynamics simulation, each particle diffuses in a force field caused by the presence of neighbouring particles, with spatial correlations in the motion determined by position-dependent hydrodynamic interactions between the particles. In this paper we report on how the technique of Brownian-dynamics simulation can be applied to the problem of aggregate and sediment structure formation. We consider in particular how the colloidal structure is affected by such factors as particle size, dispersed-phase volume fraction, external field strength, hydrodynamic interactions and the nature of the interparticle colloidal forces as expressed through the classical DLVO theory. Emphasis will be placed on an analysis of short-range structural features which are not adequately described by simple lattice-based simulation models.And the possible importance of multi-body hydrodynamic interactions will be assessed by considering their effect on the kinetics of small floc dissociation. In a low-density sediment or gel formed from irreversibly aggregated spherical colloidal particles, three spatial scales of structure may be identified: ( a ) short-range order associated with packing and excluded-volume effects, ( b ) medium-range disorder 167168 Brownian -dynam ics Sirnula tion associated with the fractal characteristics of diff usion-limited aggregation and (c) long- range uniformity as expected for a material that is homogeneous on the macroscopic scale.These three spatial scales can be described' by three regions of the normalized particle-particle distribution function G( r ) for pairs separated by centre-to-centre dist- ance r : In eqn (1) a is the particle radius, g ( r ) is the short-range liquid-like pair distribution function which extends out to r = 5, D is the characteristic fractal dimension and 6 is the characteristic network length of the connected gel or sediment. Experimental light-scattering measurements on coagulating polystyrene latices2 and a recent X-ray study of aggregated gold colloids' have both shown that short-range structure as measured by g ( r ) is sensitive to whether the aggregation is rapid or slow. The presence of an intermediate fractal scaling regime is indicative of a material produced from aggregate precursors; a recently reported example4 is the so-called silica aerogel ( D ==: 2.0, c / a = 10).In gels or sediments formed at high or intermediate particle concentration, the fractal regime could well be missing altogether ( 5 = 6). Brownian Dynamics with Hydrodynamic Interactions The algorithm used to simulate the diffusive motion of interacting colloidal particles was devised by Ermak and McCammon.' The translational displacements of N particles during a single time-step A t are given by where x, is the coordinate vector of particle i, FJ denotes the non-hydrodynamic force associated with particlej, D, is the diffusion tensor, k is Boltzmann's constant, T is the absolute temperature and the symbol O denotes quantities at the beginning of the time-step.The stochastic term R, has first and second moments: It is implicit that A t is much larger than the particle momentum relaxation time (m,D,,/ kT for a particle of mass m,), but small enough that FJ and D, remain sensibly constant over the interval. Extended forms of the basic algorithm can be written to allow for rotational Brownian motion6 and the effect of an external flow field.' Computer simulations of systems containing hundreds of particles are feasible only if the interactions between individual particles are of a relatively simple form. In practice, this means neglecting interactions altogether when their magnitude is small, making use of good approximations when their magnitude is large, and moving away from pairwise additivity only as a last resort.With charge-stabilized dispersions, the pairwise additivity assumption of the colloidal interactions is valid so long as the electrolyte concentration is not too low (i.e. K a >> 1, where K - ' is the Debye length). Pairwise additivity of hydrodynamic interactions is not such a good approximation, however, and its adoption in concentrated systems can lead to physically absurd predictions (e.g. negative diffusion coefficients') and computational difficulties in Brownian-dynamics simulations due to non-positive-definite diffusion matrices.' Hydrodynamic interactions are included in the simulation through the mobility tensor p, = D,/kT. For a pair of spheres of radius a, the mobility tensor can be written in the form' O (R,(At))=O, (R,(At)R,(At)) =2D;At.(3)G. C. Ansell and E. Dickinson 169 where r is the absolute scalar separation, $12 is the unit vector between particles 1 and 2, A,( r ) and B,( r ) are distance-dependent mobility coefficients, and 7 is the viscosity of the medium. Explicit expressions for the coefficients A, and B, (i, j = 1,2) in powers of ( a / r ) are available" to order r-I2. To order r-3, with stick boundary conditions, eqn (4) reduces to the Rotne-Prager tensor:12 r;'= (6.rrr7a)-"6,1+ ( 3 / 4 ) ( a / N - 6,)(1+ t 1 2 t 1 2 ) - ( 1/21 ( a / r13( 1 - 8, (3 31 2312 - 1 I. ( 5 ) Eqn ( 5 ) is a convenient tensor from the computational viewpoint, since it is algebraically simple and also physically well behaved. It always gives a positive-definite 3 N x 3N diffusion matrix for r > 2 a in N-particle systems, and the divergence term in eqn (2) can be omitted because we have V .r f' = 0. To go beyond Rotne-Prager hydrodynamics, one is compelled to confront the problem of multi-body interactions. Mazur and van Saar100s~~ have presented a general scheme for obtaining the mobility tensor for an arbitrary number of spheres to any desired order in ( a / r ) . Dominant contributions to translation from clusters of N spheres are shown13 to be of order r - ( 3 N - 5 ) , which means that three-body interactions first appear at order rb4, and four-body interactions at order r-'. The results including three- and four-body interactions were, in fact, anticipated by KynchI4 more than 25 years ago, but, because of their apparent complexity, Kynch's results have remained unused, although not ~nnoticed.'~ To order r-4, the mobility tensor has the formI3 Higher-order terms become increasingly more complicated, and therefore expensive to include explicitly in simulations. One way round the problem is to use an effective screened-pair mobility tensor:9716 it gives reasonable predictions of diffusive behaviour, but the approach has been criticized" as lacking in rigour.On the more pragmatic level, recent studies of fractal aggregate formation1s would seem to suggest that aggregate structure is insensitive to the exact form of particle and aggregate diffusion coefficients. So, in simulations of colloid coagulation kinetics, one may in practice be able to avoid the complication of multi-body hydrodynamics. And, if one is interested only in aggregate structure, it may not be unreasonable to neglect inter-aggregate hydrodynamic interactions altogether.Effect of Hydrodynamic Approximation on Small-floc Dynamics To assess quantitatively the effect of three- and four-body hydrodynamic interactions on the average dynamical behaviour of interacting colloidal particles, we describe the process of Brownian dissociation, whose simulation we have considered previ~usly'~ only at the level of two-body hydrodynamics. We consider groups of three or four DLVO-type particles flocculated in the secondary minimum. The van der Waals attraction UA(r) is calculated from UA( r ) = -(AH/ 12) { [ 4a2/ ( r2 - 4a2)] + ( 2 a / r ) 2 + 2 In [ 1 - ( 2 a / r)']} (7) where A, is the Hamaker constant. The screened Coulombic repulsion, UR(r), is calculated from ~ , ( r ) = 2 .r r ~ , ~ ~ a & In [I +exp ( - K S ) ] ; ( ~ a >> 1) (8) where E~ is the relative dielectric constant of the medium, E~ is the permittivity of free space, t+b0 is the surface potential, s = r - 2a is the surface-to-surface separation, and K170 Brownian-dynamics Simulation Table 1. Mean dissociation time f of small flocs as simulated for various levels of hydrodynamic approximation [up to terms of order ( a / r ) " ] equilateral linear n trimer trimer tetramer 0.37 0.49 0.54 0.32 0.42 0.39 0.42 0.5 1 0.58 0.35 0.50 0.58 0.39 0.53 0.47 estimated errorn k0.03 k0.04 *0.07 ' Taken as *2 standard deviations. is the inverse Debye length, defined for a 1 : 1 electrolyte by K~ = 2e*CN/~,.r,kT (9 1 where e is the electronic charge, N is Avogadro's number and C is the ionic strength.Potential parameters in eqn (8) and (9) are set as follows: a = 0.5 pm, q90 = 30.5 mV, T = 300 K, E , = 80 and C = 1.5 x lop4 mol dmP3. Dissociation events are studied, as a function of the level of hydrodynamic approxi- mation, for trimers starting from both linear and equilateral configurations, and for tetramers starting from a tetrahedral configuration. In trimer simulations the parameter AH is set at 0.83 x J, producing a pair potential with a high primary maximum (> 60 kT) and a shallow secondary minimum (0.8kT at rmin = 2 . 3 5 ~ ) . Initial pair separ- ations are rI2 = r23 = rmin and rI3 = 2rm,, for linear trimers, and r,? = r23 = r I 3 = rmin for equilateral trimers, and a trimer is deemed to have dissociated when each of the pair separations exceeds 1.5 p m ( 3 a ) .A total of 300 separate dissociation events are recorded for both linear and equilateral trimers. In tetramer simulations the parameter AH is set at 1.66 x J, corresponding to a potential with secondary minimum of 2.4kT at rmin = 2 . 3 ~ . Initial pair separations are rI2 = ~ 2 3 = r34 = rI3 = r,4 = r24 = rmin, and a tetramer is deemed to have dissociated when one particle gets to be at least 1.5 p m from the rest. Averages are taken over 100 separate tetramer dissociation events. The simulation time-step is set at 20 ps, and the medium viscosity is taken as 0.865 mPa s (water at 300 K). Table 1 lists values of the mean dissociation time t obtained by truncating the series expansion at various values of (air)".(Terms of order F2 and F 5 are identically zero even for liquid particles of finite viscosity.") The results show that the Rotne-Prager tensor ( n =3) leads to faster dissociation than the leading three-body term ( n = 4). Within the statistical uncertainties, however, the other approximations ( n = 1,6 and 7) give essentially the same times. The data in table 1 follow a pattern of behaviour that is consistent with a more detailed analysis of trimer trajectories*' in which the trend of inferred relative mobilities ( r P 3 > r-' > r-'> r-' > F4) is rarely broken. Overall, perhaps the most significant general trend is for the two-body approximations to give rise to more rapid dissociation than the multi-body approximations, although we would not wish to propose this as being a universal rule.Trajectory studies of sets of systems subject to exactly the same Brownian fluctuations (same sequences of random numbers used to compute { R, >) have demonstrated" that relative particle configurations are very insensitive to the value of n ; only the spatial scale of relative diffusion changes (at constant time) [or, alternatively, the time-scale changes (e.g. f) over constant spatial scale].G. C. Ansell and E. Dickinson 171 Fig. 1. Pair potentials of mean force A and B. The energy U (reduced by k T ) is plotted against centre-to-centre separation Y (reduced by a ) . These results on small-floc dynamics can be used to justify the neglect of multi-body hydrodynamic interactions in simulations of aggregation of DLVO-type particles, especially where one is primarily interested in the structure and size distribution of aggregates, as opposed to the coagulation rate constant.Pairs of particles spend most of their time at separations for which either hydrodynamic interactions can be ignored altogether ( n = 0) or the Rotne-Prager tensor [eqn ( 5 ) ] is adequate. Owing to the strong van der Waals attraction U,( Y), particles spend little time at close separations ( Y S 2 . 1 ~ ) before coagulating irreversibly, and so the nature of plJ at close separations is of little structural consequence. Moreover, at these close separations, the higher-order approxi- mations can give non-positive-definite diffusion matrices, thereby leading to computa- tional problems.Probably the best way to allow for multi-body interactions in a concentrated dispersion is to use the Rotne-Prager approximation, but to replace the viscosity 7 of the medium in eqn (5) by the effective viscosity of the dispersion.** In effect, this just means a simple scaling of the coagulation time-scale by the appropriate viscosity ratio. Effect of Colloidal Interaction and Field Strength on Sediment Structure A procedure of single-particle deposition has been used to simulate sediment formation in a very dilute system of coagulable DLVO spheres. The two pair potentials considered here, A and B, are plotted in fig. 1. Both are weakly attractive for moderately large separations ( r 2 3 a ) . The parameter values in eqn (7) and (8) are set as follows: a = 0.25 pm, Go = 20 mV, T = 300 K, E~ = 80 and AH = 1.94 x J.Curve A corre- sponds to C = 3 x lop4 mol dm-3 in eqn (9); it has a well defined secondary minimum at r = 2.3a, and a small primary maximum at r==2.la. Curve B corresponds to C = 6 x lop4 mol dmp3, and is attractive at all separations. With a sedimenting force contribut- ing to the systematic motion in eqn (2), Brownian particles were accumulated one at a time (so-called particle-cluster aggregation) until a sediment of 2500 particles had been deposited. A Rotne-Prager form was assumed for the hydrodynamic interaction between the moving particle and its nearest neighbour in the existing deposit. Statistical averages were derived from sets of three or five runs at each set of simulation conditions.Full details of the methodology have been reported e l ~ e w h e r e . ~ ~ Type A particles gave sediments with volume fractions of 0.12, 0.20 and 0.33 at field strengths of 500g 15OOg and 5000g, respectively, where g is the normal acceleration due172 Brown ia n-dynam ics Simulation I I L 2 4 6 8 01 r l a 0 ’ I I t 2 4 6 8 r / a Fig. 2. Plots of normalized pair distribution function g ( r ) as a function of reduced separation r / a . ( a ) Potential A: (-) sediment formed at 1500g; (- - -) 512-particle cluster formed at 4 = 0.2. ( b ) Potential B: (-) sediment formed at 500g; (- - -) 512-particle cluster formed at 4 = 0.1. to gravity. Fig. 2(a) shows g ( r ) for the sediment formed with potential A at 1500g. The high probability of pairs at r=2a corresponds to nearest neighbours in contact; the nearby peak at r Z 2 .3 ~ arises from non-bonded pairs in the secondary minimum; correlations disappear into the statistical noise for rb6a; there is no evidence for a fractal scaling regime. The sediment volume fraction for potential B at 500g is 0.09, almost exactly the same value as that obtained with pure Brownian trajectories (no colloidal or hydrodynamic forces), but significantly lower than that found with potential A (0.12). Fig. 2 ( b ) shows g ( r ) for the sediment formed with type B particles. The structure near r=2.3a is now lost owing to the absence of a secondary minimum. Sediments simulated with simple ballistic trajectories (no Brownian motion and no colloidal or hydrodynamic forces) were of volume fraction 0.14, in good agreement with the pioneering simulations of V ~ l d .~ ~ The simulations show that sediment structure is sensitive to the form of the coagulat- ing potential. Specifically, the presence of a secondary minimum leads to denser sediments. Also, the volume fraction increases with increasing field strength, as new particles tend increasingly to fill up holes in the existing structure, instead of sticking immediately to protrusions at the surface of the sediment. Hydrodynamic interactions have little effect on sediment structure at low field strengths, but this is not the case at high field strengths, where the Brownian component of the motion is swamped by the systematic component.’ At high field strengths the timescale for the moving particle to ‘slip past’ a fixed particle in the sediment becomes lower than the Brownian coagulation time, and so immediate ‘sticking’ does not occur, unless the line of approach is so close as to lie inside the critical capture t r a j e ~ t o r y .~ ~G. C. Ansell and E. Dickinson 173 Table 2. Half-life, t 1 / 2 , and total coagulation time, tc, as a function of volume fraction 4 and colloidal potential (A or B) potential A potential B 0.05 90* 10 1100 f 300 10*2 1200 f 400 0.10 65*7 520 * 60 1.2k0.3 200 * 30 0.15 43+6 120 * 30 0.36 f 0.04 49*9 0.20 40*4 110*30 0.22 * 0.04 6.4 * 0.5 0.25 28*3 90 * 20 0.13 * 0.03 4.2 * 0.4 0.30 17*3 82 f 20 0.10*0.02 1.8 f 0.2 Effect of Colloidal Interaction and Particle Concentration on Aggregate Structure In the final section of this paper, we report simulation results for the Brownian coagula- tion of a non-dilute system of 512 DLVO particles in a cubic box with periodic boundary conditions.Here we ignore inter-aggregate hydrodynamic interactions and aggregate rotation, and we focus attention on the nature of the coagulating potential and the overall particle volume fraction. (Previously, in a small two-dimensional s i r n ~ l a t i o n ~ ~ using a Brownian dynamics constraints algorithm,27 we found that, except for very small clusters, aggregate rotation is of minor importance,) The two types of DLVO particle investigated in the coagulation study are identical to those examined in the sedimentation study (see fig. 1). Based on the arguments discussed already, one would expect the secondary minimum in potential A to cause pairs to associate loosely at separations r == 2.3~1, before thermal motion either induces dissociation or causes irreversible aggregation into the primary minimum ( r + 2a) after jumping over the primary maximum.We shall assume that the scalar translational diffusion coefficient D, of an aggregate of m particles is given by D,= (kT/6rr7a)mY where the exponent y is taken to be -0.54 as estimated by Meakin et ~ 1 . ~ ’ from a cluster-cluster aggregation model incorporating Kirkwood-Riseman theory at the level of Rotne-Prager hydrodynamics. The use of eqn (10) enables us to move to larger systems and longer simulation times without losing the essential physics of the problem. It does, however, make an a priori assumption about the fractal nature of the aggregates to be formed, since D, scales as m-”” for aggregates of fractal dimension D.29 Nevertheless, we do not expect the structural conclusions drawn below to be significantly affected by the choice of y ; it has been shown elsewhere3’ that, so long as D, is indeed a decreasing function of m (as must be the case in reality), it matters little to the final long-range aggregate structure what values the diffusion coefficient takes.Coagulation has been simulated for potentials A and B at each of the volume fractions 4 = 0.05,0.10,0.15,0.20,0.25 and 0.30. The time-step was varied between 2.5 and 20 ps depending on the conditions. Initially, particles are positioned at random, but subject to the condition r > 2.4a for each pair in the system.A simulation is run until all particles are incorporated into a single 5 12-particle cluster. Averages are taken over five separate runs at each volume fraction. Although we are not primarily interested in the coagulation kinetics here, some useful insight into mechanism can be derived by first considering the set of coagulation times listed in table 2. Quantities tabulated are the half-life t l / * (time to reduce number of particles by 50% ) and the total coagulation time t , (time to produce a final 5 12-particle aggregate). Owing to the neglect of inter-aggregate hydrodynamic interactions, and174 Brownian-dynamics Simulation Table 3. Number distribution n ( m ) of m-particle aggregates after one half-life for potentials A and B at 4 = 0.1 and 4 = 0.3 as compared with Smoluchowski theory; mmax denotes largest simulated aggregate; N ( m 3 8) denotes number of particles existing in aggregates with m 3 8 4 =0.1 4 = 0.3 theory m 1 2 3 4 5 6 7 2 8 mmax N ( m 2 8 ) 147 * 5 54*4 22k3 15*2 6*3 4*2 3*2 6*2 15 5 2 i 12 139*4 54*7 28*4 16*4 5*2 3*2 4*2 14 32* 14 8*2 140*5 47*6 23 *3 8*3 12*4 4*2 3*2 7*2 26 70* 18 147 * 5 49*5 25*4 13*2 8*2 4 5 2 4*2 3*2 15 54* 15 128 64 32 16 8 4 2 2 18 uncertainty about the most appropriate choice of starting configuration, the absolute values of t l l Z and t , are to be treated with caution, but their relative values contain useful information.We note that t l , Z is much more strongly dependent on the form of DLVO potential than is t , , and this is especially true at low volume fraction, where t , appears to become independent of the potential.The reason for this is fairly clearly understood:26 at short times the rate is controlled by the potential-energy barrier to coagulation (‘reaction-limited’), whereas at long times the rate-determining factor is the speed at which the (large) aggregates move around (‘diffusion-limited’). Despite the wide variation in values of t , / 2 , table 3 shows that the distribution of aggregate sizes is relatively insensitive to volume fraction and pair potential, although there does seem to be a definite tendency towards more particles in larger aggregates ( m 2 8) with potential A and at the higher volume fraction. This trend becomes stronger as the coagulation proceeds: after two half-lives at 4 = 0.3, the largest aggregate in the system contains more than twice as many particles with potential A as with potential B (52, 44, 52, 50 and 36 for the five separate runs with A, as compared with 12, 16, 20, 18 and 18 for B).In every case, simulated distributions were broader than calculated from Smoluchowski theory (see table 3), which is expected to hold only as 4 -+ 0. Deviations from theory are of a similar order. to those reported3’ for aggregating polystyrene latex particles. For each set of simulation conditions, effective fractal dimensions were calculated from plots of log m against log RG( m ) , where RG is the radius of gyration of an aggregate of m particles ( R G - rn””). Values of D are collected in fig. 3 as a function of volume fraction.The data show a slight tendency for type A particles to give more compact structures than type B particles, although the difference is hardly significant compared with the statistical uncertainty (k0.05). It is difficult to make a reliable extrapolation to 4 = 0, but the limiting value of D at infinite dilution does seem to lie below that given by the reaction-limited clustering of clusters Fig. 4 shows the effect of colloidal potential on g ( r ) for the final 512-particle aggregates generated at 4 = 0.15. The plot for potential A has a strong peak at r=2.3a corresponding to non-bonded pairs in the secondary minimum, and a broader peak at r =: 4a corresponding to the ‘second shell’ of liquid-like structure characteristic of stable concentrated dispersion^.'^ (In aggregates formed at 4 = 0.3, the ‘third shell’ is clearly evident.) The plot for potential B, on the other hand, has little short-range structure, in qualitative agreement with theG. C.Ansell and E. Dickinson 2.4 D 2.0 175 - / 7’ I I I I 0 0.1 0.2 0.3 d Fig. 3. Effective fractal dimensions D for simulated aggregates at various volume fractions #: 0, potential A; e, potential B. W 2 4 6 8 Fig. 4. Influence of colloidal pair potential on short-range structure of aggregates generated at # = 0.15. The normalized pair distribution function g( r ) is plotted against reduced separation r / a for ( a ) potential A and ( b ) potential B.176 Brownian-dynamics Simulation Table 4. Number Np of final percolating clusters in the simulations for various values of the shell parameter u [ Np (maximum) = 51 4 potential Np (u = 2 .5 4 N p ( u = 3 a ) Np ( u = 3 . 5 ~ ) Np ( D = 4a) ~ ~~ 0.05 A B 0.10 A B 0.15 A B 0.20 A B 0.25 A B 0.30 A B 2 0 4 3 4 5 0 0 0 0 4 5 5 5 4 5 5 5 5 5 earlier sedimentation data. Indeed, it is instructive to compare the short-range sediment structures with the structures of the final aggregates simulated here at the same overall volume fraction. The latter g ( r ) plots are the dashed curves in fig. 2 ( a ) (potential A, 4 = 0.20) and fig. 2 ( b ) (potential B, 4 = 0.10). While agreement is reasonably good, there are clearly differences outside the statistical error. This suggests that a sediment and a gel, although of the same overall volume fraction, may have rather different micro-structures, and therefore different mechanical properties.In part, this represents the difference between particle-cluster aggregation (sediment) and cluster-cluster aggre- gation (gel). A system is said to have gelled when aggregates of the order of the system size have been produced. This will occur if the original particle volume fraction exceeds some critical percolation threshold &. Computer simulation is a blunt instrument for estimat- ing 4,: in small systems with periodic boundary conditions, there are large fluctuations in structure, which means that percolating configurations may be observed both above and below the true percolation threshold. As pointed out recently by Seaton and Glandt,34 it is possible for the infinite system of replicas to be spanned, but not the small system itself, and vice versa.Nevertheless, even in this preliminary study, it still seems worthwhile trying roughly to estimate 4p, if only to give a handle on the approximate size of the parameter 6 in eqn (1). Table 4 records the number of percolating 512-particle clusters produced in the various simulation runs. Percolation is registered as occurring when the cluster in the basic cell ‘connects’ to its nearest image clusters in each of the x, y and z directions, a ‘connection’ between two particles being deemed to exist if r < a, where a is a conductive shell parameter.” Numbers in table 4 indicate that the final clusters generated at C$ = 0.05 and 0.10 are below the gelation threshold; and, while N , is clearly sensitive to a, most of those generated at 4 2 0.15 are above the gelation threshold.The values of Np at 4 = 0.15 suggest that the form of the colloidal potential has an influence on the gel point, but it would be imprudent to make any generalizations based on these few preliminary pieces of information. We note, however, that a value of +,=0.15 in the simulation corresponds to a characteristic network length of == 25a. This means that the value of D calculated from the simulations at C$ = 0.20, 0.25 and 0.30 are based partly on data in the sol-gel crossover region, and should not therefore be interpreted as true fractal dimensions in the self-similar sense.G. C. Ansell and E. Dickinson 177 One factor which probably distinguishes the model calculations reported here from many real colloidal systems is the absence from the simulations of any allowance for consolidation or relaxation of sediment or gel structure following initial sticking together of the constituent particles.Such considerations will have to be included in the next generation of models if Brownian dynamics simulations are to fulfil their potential as a predictive tool in colloid and material science. E.D. acknowledges support from S.E.R.C. G.C.A. acknowledges receipt of a Studentship from the Ministry of Agriculture, Fisheries and Food. References 1 E. Dickinson, J. Colloid Interface Sci., 1987, 118, 286. 2 D. Giles and A. Lips, J. 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