Dilute heteroaggregation: A description of critical gelation using a cluster—cluster aggregation model

Dilute Heteroaggregation: A Description of Critical Gelation Using a Cluster-Cluster Aggregation Model E. M. S E V I C K 1 AND R. C. B A L L Cavendish Laboratory, University of Cambridge, MadingleyRoad, Cambridge CB3 0HE, UnitedKingdom Received October 11, 1990; accepted December 19, 1990 We investigate the critical behavior of binary heteroaggregation for dilute systems using a generalized cluster-cluster aggregation model which spans the conventional models of DLA (Diffusion Limited Aggregation) and RLA (Reaction Limited Aggregation). A new algorithm, based upon the time reversal technique, is used to investigate the reaction surface, namely its size and the extent to which it is kinetieally screened, i.e., not equally accessible under diffusion, over a range of compositions, including compositions very close to (and at) critical gelation where conventional algorithms are limited by "kinetic slowing down." We show that the reaction surface and its associated properties exhibit distinctive behavior in the gelling region where stoichiometry and kinetics permit formation of a network structure, as well as in the nongelling region where stoichiometry limits clusters to forming inert oligomers. © 1991Academic Press, Inc.

I. INTRODUCTION This paper addresses the aggregation of fine particles, focusing u p o n binary systems where the particles are o f different composition. The systems studied m i m i c particles e m b e d d e d in a c o n t i n u o u s liquid phase, such as opacifying pigment particles dispersed to obtain efficient scattering properties in surface coatings, and in dry mixtures o f solids, such as pyrotechnic composites where stoichiometries must be preserved at small length scales to ensure stable flame fronts. Such particle arrangements dep e n d in a complicated way u p o n the processing conditions, e.g., the flow field, electrostatic or h y d r o d y n a m i c interactions, and reversible or irreversible clustering. We focus u p o n the particular role irreversible clustering plays in structure formation. D o large scale aggregative structures develop at all, and h o w sensitive is kinetic aggregation to composition? We consider the simplest o f selective binary systems: volume-excluding particles, desigJ To whom correspondence should be addressed at present address: Department of Chemical Engineering, University of California, Berkeley, CA 94720.

nated as species A and B, having no interparticle or h y d r o d y n a m i c interaction and irreversibly aggregating via A - B b o n d i n g only. While we have in m i n d colloidal applications, such a system can nevertheless be viewed in chemical terms as a polycondensation reaction o f the form • ..A+B...

--~ . . . A - B . . . ,

where the functionality or n u m b e r o f possible bonds available to a single " A " m o n o m e r is a and to a " B " monomer,/3. F o r a,/3 > I, the cluster can propagate consecutively to form larger structures. A n i m p o r t a n t parameter is the relative concentration o f functionalities, f = N J ( N ~ + N~), where N , and N~ are the n u m b e r s o f a and/3 functionalities in the system. A critical composition f~ can be defined: abovefc a gelled structure where all m o n o m e r s are contained in a single cluster m a y be formed after some time, whereas belowf~ only disperse oligomers will ever be formed due to a lack o f stoichiometry. The structure o f aggregates follows well defined scaling with cluster mass or radius behavior: one such measure is, for example, the

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fractal dimension of a cluster, d f , defined by M ~ R de where M and R are the mass and radius of the cluster. In this paper other scaling relations are investigated, namely the reaction surface, describing the number of possible reactive configurations available for bonding two clusters together, and the kinetic screening, describing the competition among the many reactive configurations available to a cluster pair. Using an analogy to equilibrium and other renormalizable critical phenomena, we describe the scaling and crossover behavior of these quantities near the critical regime. As we show here, within the gelling regime, heteroaggregation is essentially the same as unselecrive homoaggregation (single species aggregation) with scaling law behavior identical to that of homoaggregation. However, near the critical composition, f~, new critical behavior with different scaling laws for both structure and kinetics becomes apparent. As the composition approachesf~ the aggregation follows the scaling laws of the homoaggregation regime up until some typical cluster radius R ] f - f c I-"; beyond R the aggregation crosses over toward gelling behavior or comes to a halt depending upon the sign of f - fc. Analogies suggest that the range of radius over which such critical behavior extends should depend only weakly upon the sign of f - f ~ . A number of theoretical and computer simulation based approaches have been used to investigate the critical regionf~ and the kinetics ofgelation, but little attention has been given to such scaling behavior near the critical regime. Mean field approaches, beginning with thef~ determination of Flory ( 1 ), and including the kinetic description of the Smoluchowski equation (2, 3 ), consider the particle aggregation to proceed uniformly, i.e., all available functionalities are assumed to have equal reactivity throughout the aggregation process. Such approaches ignore cluster structure and are valid if the clusters are compact or alternatively, mutually transparent to one another. However, aggregative clusters are fractal (see Fig. 1 ), and consequently possess a range of reactivities, being hindered and Journal of Colloid and Interface Science, Vol. 144, No. 2, July 1991

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hence unreactive in the interior and accessible and reactive near the "fingering" fronts. A more appropriate description of cluster structure has been accounted for in simulations based upon percolation theory: two such examples include the bond percolation model (4), where monomers are fixed in space for all time and bonds form among neighboring monomer pairs, and the kinetic gelation model (5, 6), where bond formation is incited by mobile radicals that impart a kinetic time scale to the problem. However, such percolation models describe dense heteroaggregation, where distances between clusters are small and mobility of the monomers and aggregated monomers or clusters is assumed to be unimportant. For dilute systems, the kinetics of reaction and diffusion of species play an important role in determining the cluster structure and have been studied in the context of homoaggregation using computer simulation. While homoaggregation does not, of course, provide a critical regime, we may nevertheless glean simulation techniques to investigate the proposed scaling behavior of heteroaggregation. In cluster-cluster homoaggregation both DLA (Diffusion Limited Aggregation) and RLA (Reaction Limited Aggregation) have been extensively explored and contrasted by computer simulation (for a review, see Meakin (7)). In DLA, clusters join irreversibly at first contact and, consequently, the distribution of probabilities of realizing these contact points in a diffusion process is nonuniform. Easily accessible reactive sites on fingering fronts will kinetically screen the more hindered reactive sites embedded in deep 0ords; see Fig. 1. In contrast, the reactive sites on clusters generated in RLA will be equally probable or reactive; i.e., the sites are kinetically unscreened. The cluster-cluster DLA model was first applied to heteroaggregation by Meakin and Djordjevic (8) whose basic model is also adopted in this paper: clusters (initially a dilute collection of monomers) are rigid but mobile, diffusing on a simple cubic lattice and forming bonds between all contacting A-B pairs so as

DILUTE HETEROAGGREGATION

563

F1G. 1. Heteroaggregateclustergenerated from simulation using square lattice and f = 0.50.

to produce fewer but larger clusters. More recently, Meakin and Miyazima (9) constructed a cluster-cluster RLA model for heteroaggregation. Both models present distinct gelling and nongelling regimes defined similarly to the more conventional percolative criteria of gelation. In the gelling regime, f~ < f < 1 - £ , aggregation proceeds indefinitely until a single cluster or network structure results. By contrast, within the nongelling regime, f < f ~ and f > 1 - f c , the aggregation will cease leaving inert clusters of only limited size, the available bonding sites being saturated or rendered sterically inaccessible. Conventional DLA and RLA simulations of the aggregation model become highly inefficient as f approaches f~ since aggregation events become exceedingly rare. The increasing time required for clusters to sample these rare events results in a "kinetic slowing down" of the algorithm prohibiting any structural or kinetic investigation of the critical region. In this work, we investigate the scaling behavior of the reaction surface and kinetic screening of dilute heteroaggregation using a novel simulation technique applied to the cluster-cluster aggregation model. The new simulation approach is based upon a time reversal technique which allows us to sample aggregation events when they become exceedingly rare thereby allowing efficient simulation near (and at) the critical composition. The restriction to dilute systems implies that ag-

gregations occur pairwise and not through multiple cluster contacts, and that there is no cluster-cluster screening. The reaction surface and kinetic screening are measured in the simulation at specified time intervals and an analysis of their scaling behavior and crossover in the critical regime is constructed. The organization of the paper is as follows: in Section 2 we outline the new time reversal approach to simulating slow aggregation. Sections 3 and 4 discuss the quantities whose scaling behavior is investigated: the reaction surface of heteroaggregation and the kinetic screening, defined in terms of an escape probability. Section 5 discusses the relevance of these two quantities to the true clustering kinetics and a simple consequence for the kinetic screening. In Section 6 we discuss the simulation evidence for crossover in the scaling behavior within the critical regime. 2. SIMULATION The challenge in simulating dilute aggregation processes is that the aggregation events themselves are rare, and one must seek, in an unbiased way, to preferentially sample them. This problem was first solved by Brown and Ball (10) for the case of unselective but reaction limited homoaggregation, where all possible bonding events could occur with very low but equal sticking probability. Under such conditions, it is appropriate to simply sample Journal of Colloid and InterJkzce Science, Vol. 144, No, 2, July 1991

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reactive configurations completely at random, and to accept them with probability represented by a rescaling of time. Such random sampling of reactive configurations can also be exploited to circumvent the kinetic slowing down of aggregation due to selective heteroaggregation near the critical composition. Instead of following the diffusive trajectories of clusters searching for rare A-B contacts, we simply choose random reactive configurations, i.e., pairwise configurations having at least one A-B pair between the clusters, and irreversibly bond or aggregate the clusters if the reactive configuration is sterically acceptable. Two correction factors are required to properly sample the probability of a given reactive configuration. First the true rate of sampling reactive configurations between two clusters in a diffusion limited process is proportional to the cluster mobilities: thus, our random selection of reactive configurations must be weighted with probability proportional to relative mobility of the clusters. The relative mobility is, in principle, a function of the particular pairwise configuration sampled; but in neglect of hydrodynamic interactions, we need only consider the sum of the diffusion coefficients of the two clusters involved. A randomly chosen reactive configuration is weighted by (#1 + tz2)where/~j is the diffusivity of cluster j. We have investigated both the convenient approximation of ~tj = constant and also the more physical case of #j ~ 1/Rj, where Rj was taken as the circumscribing cluster radius: these gave indistinguishable results for the quantities studied. The second and more challenging correction factor is the kinetic screening. This factor takes into account that some configurations are less likely to be sampled by diffusion than others because of competing configurations that are encountered in the diffusive approach of the clusters. The kinetic screening is simply incorporated by examining the time reversed diffusion trajectories: starting from the reactive configuration of interest we follow the two clusters as they diffuse apart backward in time. Journal of Colloid and Interface Science, Vol. 144, No. 2, July 1991

The probability that the clusters successfully "escape" to large separation (taken to be 10(Rl + R2)) without reaction is directly proportional to the probability that the configuration would be sampled as they diffuse together, forward in time. Similar time reversal techniques have been used to simulate other growth processes described by a field that satisfies Laplace's equation: examples include viscous fingering simulation of Maloy et al. ( 11 ) and the simulation of dielectric breakdown proposed by Matsushita et al. (12). The pairwise aggregations are continued until only one cluster remains (suggestive of a gelled state), or until the reactive configurations are entirely depleted (a nongeUed state). We implement the algorithm using N = 1000 to 10,000 monomers, where the monomers and clusters are constrained to translate on cubic lattices, rotations being disallowed. The logic is general and could be applied to monomers of arbitrary shape and dimension having continuum motion. The algorithm recovers conventional simulations used in homoaggregation and moreover introduces features necessary to simulation of more complicated processes. In the limits of fully screened and unscreened aggregation events, the algorithm recovers the homoaggregative models of DLA and RLA, respectively, bridging a wide class of intermediate models. The time reversal technique offers an approach to a range of aggregatio n problems where the nonlocality of sampling configurations by diffusion is coupled critically to subtleties of local detail. Other examples include systems with hydrodynamic interactions, those having low but not negligible sticking or reaction probability, and systems of particles with complex shape of near contact forces. 3. REACTION SURFACE Brown and Ball (10) defined the contact surface for homoaggregation, denoted by Sc, as the set of pairwise configurations where clusters are in contact but not overlapping. It can be visualized geometrically in terms of

DILUTE

HETEROAGGREGATION

fixing the location of the first cluster, and finding all the locations of the center (or other reference point) of the second such that the two clusters are in contact, Fig. 2. Because it is a completely enclosing surface, the "area" of Sc lies between bounds of order R 2 and R 3, respectively, where R is a representative cluster radius. We may alternatively express the scaling of Sc in terms of cluster mass, M, as

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Sc ~ M xc, where )tc is the contact surface exponent, bounded by (d - 1)/df<~ XC ~< d/dr with d the Euclidean dimension and df the fractal dimension of the cluster. O f the set of points composing the contact surface S o there exist a subset of points denoted SR corresponding to configurations

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FIG. 3. T h e reaction surface, SR, versus twin cluster mass, M , m e a s u r e d from s i m u l a t i o n ( p o i n t s ) a n d plotted l o g a r i t h m i c a l l y for h o r n • a g g r e g a t i o n (filled circle); f

= 0.50 (open circle); f = 0.70 (open triangle); and nongelling composition f = 0.80 (×'s). Included is critical behavior(line)predictedfromthe scalingof contactsurface density with mass.

leading to aggregation; we refer to SR as the reaction surface. The reaction surface can also be described using scaling arguments similar to that of Sc: SR ~ M xR.

FIG. 2. Contact surface generated by a cluster and its twin image. The virtual cluster, represented as the darkened cluster, is fixed in space and its twin image, outlined in dotted boundary, is free to translate without rotation so as to realize all possible contactable pairwise configurations. The contact surface corresponds to the clark line circumscribing the virtual cluster and is generated by marking the location of the center of mass (or some other reference point) of the image cluster in its contactable configuration. The contact surface density, Pc, is found by the size of the contact surface divided by the number of contactable sites on the virtual cluster. The contact surface density describes the amount of the contact surface attributable to a single contactable site.

Since SR e So, then XR has an upper bound: XR ~ XC. A geometric lower bound on XR does not exist since SR is not constrained to be a closed, bounding surface. However, one might anticipate that there exists a minimal gelling bound on )'R- The reaction surface exponent for marginal systems, i.e., those gelling systems on the brink of becoming nongelling, must at least be XR > 0, ensuring that arbitrarily large clusters retain reactive configurations and participate in the formation of infinite gel structure. Figure 3 displays SR measured from computer simulation versus cluster mass, M, for heteroaggregation with f = 0.50, 0.70, and 0.80, and homoaggregation. To resolve the problem of finding pairs of clusters of consistently comparable masses, the measurements are made for the reaction of a cluster with a twin copy of itself. Each point represents SR averaged over all clusters of mass M at a parJournal of Colloid and Interface Science, Vol. 144, No. 2, July 199l

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ticular time interval in the aggregation. There is no averaging of the data over time and a series of time intervals where there exist from N/2 clusters to a single cluster is represented. The scaling behavior of SR does not vary with time of aggregation, except for the f = 0.80 case where the reaction surface diminishes with time, possibly indicating the nongelling regime. As f a p p r o a c h e s ~ within the gelling regime (compare data for f = 0.50 to f = 0.70), a spread in SR results due to fluctuations from the average cluster configuration. This is simply a finite size effect; but as this is a rather unconventional simulation, let us briefly describe the fluctuation. Consider an ensemble of clusters generated after some time interval and a member cluster having all of its contactable sites filled with majority species. If this configuration were typical, then the cluster would remain inert with SR = 0 for the remainder of the aggregation. However, for the case of f = 0.70, such a configuration is not typical and at some short time later, this cluster will disappear from the ensemble, having been aggregated into a more representative cluster having minority sites. As a result, a smaller than normal value of SR appears in the data but disappears at later times. Such fluctuations occur more often and over a larger range of cluster sizes as fapproachesf~. We expect these fluctuations to diminish with increased number of monomers, N. Figure 3 shows that the behavior of SR scaling with M for the optimal gelling composition is identical to that ofhomoaggregation. A leastsquares fit analysis indicates Xc = 1.34 for homoaggregation and that XR = Xc for the heteroaggregating systems at progressively larger cluster masses as fincreases from 0.50 to 0.75 (not shown). The equivalent scaling behavior for homoaggregation and for heteroaggregation of large cluster mass can be explained in the following way: reasonably, the reaction surface of very large clusters is uniformly distributed over the contact surface. Since the contact surface grows according to Xc, then the reaction surface for large cluster Journal of Colloid and Interface Science, Vol. 144, No. 2, July 1991

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pairs must scale similarly, i.e., XR = Xc. As the stoichiometry becomes unbalanced, there are fewer contactable sites of the minority species and, consequently, the contact surface or cluster mass must be larger in order to realize a uniform distribution of the reaction surface on the contact surface. The result is that the onset of homoaggregative scaling behavior is postponed to larger cluster masses. The reaction surface at f = 0.80 provides a different scaling behavior which we take as indicative of the marginally gelling case. Simulations at f = 0.80 can fail to produce a single, final cluster, but the same is also true of f = 0.75 (due, we presume, to fluctuations of finite system). The two cases differ most clearly at late times or large cluster masses: not only does f = 0.80 fail to show any upturn in SR versus M or homoaggregative scaling behavior for its large clusters, but it also then exhibits small clusters of persistently, anomalously low SR. Neglecting the low cluster mass reaction surfaces obtained at late aggregation times, a least-squares fit shows SR ~ M 0.5. We assume this to be critical behavior, indicative of a marginally gelling system. Such marginal behavior would persist for large cluster masses and aggregation times until stoichiometric limitations halted the aggregation process. In the critical regime, on the margin between gelling and nongelling, it is at first sight surprising that SR should increase with cluster mass. However, recall that the reaction surface is a set of mutual configurations between two clusters and that just one available minority component site on one cluster may offer many different possibilities to react with majority component sites on other clusters. Pursuing this suspicion we have measured in homoaggregation, the contact surface density, pc, i.e., the amount of contact surface per contactable site on one of the two clusters (see Fig. 2). A least-fit analysis indicates 0c ~ M °5, scaling behavior that is within statistical uncertainty of that for the reaction surface in critical heteroaggregation. The interpretation of the critical reaction surface in terms of (on the order of) one re-

DILUTE HETEROAGGREGATION

actable site per cluster is consistent with the need for one bond per cluster aggregated into an eventual gel. The comparison with homoaggregation supports this but relies on two assumptions: first that the few reactable sites are equally likely to be located on any contactable site and second that the large scale geometry does not differ significantly between the gelling and marginally gelling regimes (an assumption prompted by the relatively constant fractal dimension of clusters within the gelling regime, see Meakin and Djordjevic (8)). 4. K I N E T I C S C R E E N I N G A N D ESCAPE PROBABILITY

The probability of escape, p~, from a reactable configuration to infinite separation (without intervening reaction) is proportional to the probability of two clusters reacting in that way when initially separated. Thus the escape probability is a precise measure of screening, the screened regime in which there is significant competition between reactive configurations corresponding to pe < 1 and the unscreened regime with little competition corresponding to Pe ~ 1. Figure 4 displays computer simulation measurements of the escape probability, pe, versus cluster mass, M, for heteroaggregation

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with f = 0.50, 0.70, and 0.80, and homoaggregation. The escape probability is constructed from twin cluster pairs and averaged over many escape attempts starting from different points of the reaction surface. Like the SR data, a set of points is constructed for each ensemble and there is no averaging of data over ensembles of time. Again, the escape probability displays little if any time dependence, except for f = 0.80. As in the reaction surface scaling, the escape probability at optimal gelling composition scales identically to that of homoaggregation, showing screened behavior as Pe ~ M -~ • As with the reaction surface, this scaling is approached at progressively larger masses as the stoichiometry becomes less favorable. It is difficult to discriminate from the simulation data whether Pe decreases as a weak power of the cluster mass. Nevertheless, it appears that the onset of screening occurs at larger cluster masses as the composition tends to f~. 5. R E A C T I O N K I N E T I C S A N D C O N N E C T I O N S B E T W E E N RLA A N D D L A

The aggregation kinetics can be interpreted in terms of the reaction surface and escape probability Pe. It is convenient to think in terms of the lattice model so that cluster diffusivities can be realized as local site-to-site j u m p rates and we will not consider either rotational motion or hydrodynamic coupling effects. These simplifications correspond to our simulations. The probability per unit time for two clusters of diffusivities #~ and tz2, respectively, to aggregate in any particular pairwise configuration G will simply be given by

r[G] ~ (#l + #2)Pe[G]/V,

o

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FIG. 4. The escape probability, Pe, versus twin cluster mass, M , measured from simulation (points) and plotted logarithmically for homoaggregation (filled circle); f = 0.50 (open circle); f = 0.70 (open triangle); and nongelling composition f = 0.80 ( × ' s ) .

where Vis the total number of sites or volume of the system in which the clusters diffuse. The factor of pe[G] takes screening into account and the other factors are simply the unbiased rate of encounter with any particular configuration. The total rate of reaction between two particular clusters is then given by K~ V, where the reaction rate coefficient is Journal of Colloid and Inter~bee Science, Vol. 144, No. 2, July 1991

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SEVICK AND BALL

K ~ (g, + g2)
[t1

and (p~) is the escape probability uniformly averaged over their reaction surface. The above expression for the reaction rate directly parallels expressions based on RLA where (p~ > would be replaced by a barrier activation probability. Nevertheless we are considering a diffusion limited model and can also view the reaction rate in terms related to DLA. Consider following the relative diffusion of two randomly chosen clusters with time until they approach with their centers closer than some fixed distance L apart, where it will be convenient to choose L to be some multiple of their maximum separation as the homoaggregation of spheres. The probability per unit time of such a separation is

Ps "~ SR/ L. Taken together these imply that if there is to be a simple crossover from unscreened to screened behavior then it must occur when SR ~L.

A crossover in the screening at SR ~ L is also consistent with simple fractal analysis. Consider the random walk trajectory (dimensionality 2) of one cluster relative to the other as they separate in an escape attempt. This is expected to have a nonempty intersection with a reaction surface of dimension dR if dR + 2 > 3 (the right hand side being the dimension of space), giving a dimension dR = 1 for the marginal case. 6. C R O S S O V E R

BEHAVIOR

We have indicated two arguments that put the threshold for screening at SR ~ L (or SR Now we introduce a cluster sticking probabilM ~/af) but it is not at all clear that we can ity Ps: the probability that the two clusters arexpect this to coincide with our postulated riving at separation L will subsequently agmarginally gelling behavior, determined from gregate before separating again to infinity. It the contact surface density o f homoaggregafollows that we can write the reaction rate coeftion as SR ~ M °5. The value of 1/dr evaluated ficient as from heteroaggregate clusters of low to interK ~ (#1 + #2)Lps [2] mediate mass within the gelling region is indistinguishable from 0.5. Although this indiwhich parallels the expression for simple DLA cates that the transitions scale similarly, we which would be given by the ratio of diffumust nevertheless consider the possibility of sional capture diameter to L. The cluster two separate time-dependent transitions in sticking probability Ps reflects screening in that aggregations which are only just gelling. Ps ~ I achieves the fully screened reaction We can directly test the possible equivalence rate ofhomoaggregation, whereasps < I would of crossovers by comparing the horizontal imply much less that the fully screened reacshifts in cluster mass required to construct tion rate and hence an unscreened regime. universal scaling in large cluster mass behavior Note that these regimes are in the reverse roles of SR/~/M and of pc. A lack of convergence to those for (p~). to a universal Pe curve might indicate that the Comparing the expressions [1 ] and [ 2] we threshold of screening and the departure from see that Ps and (pC) are related through marginal gelation do not necessarily coincide. Figure 5 displays Srd V M for the heteroaggre, SR ~ Lps. gate systems shifted to achieve a universal In the screened regime we have p~ ~ 1 and curve and the Pe data translated using the hence equivalent set of horizontal shifts. The shifts required to produce a universal pC curve appear (po> L/S to be slightly larger than those of SR, suggesting whereas in the unscreened regime we have that there exists a window of compositions (pe) ~ 1 and hence where marginally gelling clusters undergo un-

rL ~ (u~ +

t.t2)L/V.

Journal of Colloid and InterfaceScience, Vol. 144, No. 2, July 1991

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569

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FI~. 5. (a) SR/![-Mversus twin cluster mass, M, plotted logarithmically and shifted so as to produce best estimate of universal contact scaling for homoaggregation (filled circles); f = 0.50 (open circles); f = 0.60 (open squares); f = 0.70 (open triangles); and f = 0.75 (stars). Shifts correspond to 2x(ln M) = 1.71, 2.25, 2.30, and 4.20; for f = 0.50, 0.60, 070, and 0.75. (b) Escape probability, Pc, versus twin cluster mass, M, plotted logarithmically and shifted using the same symbols and ~ (In M) shifts as in (a). On first glance it appears that larger 2~(In M) shifts are required to obtain universal scaling of the escape probability. This would indicate that the marginal gelation transition precedes the crossover from unscreened to screened aggregation. However, the required shifts for universal scaling of SR and Pe are sufficiently similar to render this conclusion doubtful.

screened aggregations. However, these shift factors are sufficiently similar to render this conclusion doubtful. 7. DISCUSSION W e find heteroaggregation t h r o u g h o u t the gelling, critical, and nongelling compositions, can be described by two scaling regimes described by SR ~ M xR where ~'R ~ 1 / 2, f o u n d from the scaling behavior o f the contact surface density, and XR = Xc = 1.34. The first, the critical or marginally gelling regime with XR ~ 1/2, is indicative o f clusters having on the order o f one minority reactive site and is significantly higher than our initial expectation o f the lower b o u n d on the reaction surface exp o n e n t for gelation o f XR > 0. This initial estimate o f a lower b o u n d corresponds to clusters where the contact surface density is mass independent, as for example in nonrotating hard spheres where there exist only one pairwise contact for any given point o f one sphere. Larger values o f the critical exponent correspond to a larger n u m b e r o f available pairwise

reactive configurations per contactable site or, equivalently, m o r e ramified cluster structure. Moreover, larger critical exponents might imply that the w i n d o w o f gelling compositions is wider. This critical regime is indicative of all clusters at f = f~, irrespective o f mass, and it also extends to a scaling description o f clusters o f low mass within the gelling regime. The second regime is homoaggregative scaling regime, ),R = )tC = 1.34, where heteroaggregates have large enough cluster masses that their reaction surface is uniformly distributed over the contact surface and, hence, the surfaces scale similarly. F o r gelling compositions, f < f~, there is a crossover from XR ~ 1 / 2 critical scaling behavior to 7tc = 1.34 homoaggregative behavior at large cluster masses. The onset o f the homoaggregative scaling behavior is postp o n e d to larger cluster masses as f a p p r o a c h e s f~. F o r f > f c , the scaling behavior follows XR 1 / 2 critical form, never attaining the h o m o aggregative scaling behavior, until termination where clusters o f all masses shift systematically to lower reactivities in a m a n n e r that is dependent u p o n reaction time. dotwnal of Colloid and Interface Science,

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It is instructive to compare values of the reaction surface exponent with those of other models namely that of the homoaggregative RLA as found from simulation, XR = 1.06 (10). At first glance, one might compare this value with that of the critical exponent of XR 1/2 and conclude that since RLA proceeds via unscreened aggregation events (by definition), then marginally gelling systems are well within the unscreened regime. However, our measurements of the escape probability show that the crossovers in kinetic screening and homoaggregative scaling of the reaction surface are not easily discernible. Moreover, in the RLA model, aggregation events are unscreened by construction whereas in the heteroaggregation problem, aggregation remains diffusion controlled and events are treated as kinetically screened. The kinetic unscreening naturally evolves in the heteroaggregation because the number of available reactive configurations is diminishing as fapproachesf~ and the events become unscreened on a local scale. This local scale of uncompetitive reaction sites grows as f a p p r o a c h e s f~; however, it is not clear whether this loss of screening occurs uniformly over the reaction surface. Thus, a direct comparison of the unscreening events in homoaggregative RLA and a diffusion controlled heteroaggregation model cannot be easily made. Although crossover behavior is evident in the reaction surface, it may not be so strong as to produce crossover behavior in the kinetics. Critical kinetics corresponds to SR increasing with cluster mass, but is consistent with on the order of one distinctive reactable site per cluster. The consequence for kinetics for mobility independent clusters is that Me t ~/~/-xcl for Xc < 1 as compared with Mc e x p ( t / t c ) for Xc = 1, where Mc is a characteristic cluster mass. The measured value for Xc for the critical regime is at the threshold condition for screening in the reaction surface; i.e., as compared with the measured values of Xc ~ 0.50 and d f ~ 2.0. It is therefore not

Journalof Colloidand lnterjaceScience.Vol.144,No. 2, July 1991

surprising that we have difficulty seeing a clear transition in the screening. This also means that it would be difficult to see any corresponding crossover in the kinetics, since including mobility ts ~ L - ~would given Mc ~ t for DLA or unscreened aggregation and Mc t 1/(l-xc+l/aO for unscreened critical RLA. Our model here, where clusters are confined to translate and aggregate on a cubic lattice, provides indistinguishable Xc and dr; however, it may be possible to construct other models which distinguish these exponents. Alternatively, one might consider whether the values of Xc and 1 ~dr are constrained relative to one another through simple geometric arguments. ACKNOWLEDGMENTS E.M.S. and R.C.B. acknowledge funding provided by 1CI Corporate Colloids Group at Runcorn, UK. E.M.S. also thanks P. Warren, R. B. S. Oakeshott, and R. Wessel for discussions. REFERENCES 1. Flory, P. J., "The Principles of Polymer Chemistry," Cornell Univ. Press, Ithaca, NY, 1953. 2. yon Smoluchowski, M., Z. Phys. Chem. 92, 129 (1917). 3. For a review see R. M. Ziffin "Kinetics of Aggregation and Gelation" (F. Family and D. P. Landau, Eds.), North-Holland, Amsterdam, 1984. 4. Stauffer, D., Coniglio, A., and Adam, M., Adv. Polym. Sci. 44, 103 (1982). 5. Herrnann, H. J., Landau, D. P., and Stauffer, D., Phys. Rev. Letl. 49, 412 (1982). 6. Hermann, H. J., Staufer, D., and Landau, D. P., J. Phys. A 16, 1221 (1983). 7. Meakin, P., Phase Transitions 12, 336 (1988). 8. Meakin, P., and Djordjevic, Z. B., J. Phys. A 19, 2137 (1986). 9. Meakin, P., and Miyazima, S., J. Phys. Soc. Japan 57, 4439 (1988). 10. Brown, W. D., and Ball, R. C., J. Phys. A 18, L517 (1985.). l 1. Maloy, K. J., Boger, F., Feder, J., Jossang, T., and Meakin, P., Phys. Rev. A36, 318 (1987). 12. Matsushita, M., Honda, K., Toyoki, H., Hayakawa, Y., and Kondo, H., J. Phys. Soc. Japan 55, 2618 (1986); Hayakawa, Y., Kondo, H., and Matsushita, M., J. Phys. Soc. Japan 55, 2479 (1986).