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Universality of Colloid Aggregation in the Reaction Limit: The Computer Simulations Agustin E. Gonzalez

Instituto de Fisica, Universidad Nacional Autonoma de Mexico, Apdo. Postal 20-364, Del. Alvaro Obregon, 01000 Mexico, Distrito Federal, Mexico

(Received 26 March 1993)

Understanding aggregation phenomena is fundamental in colloid science. Recent experimental work has shown that colloid aggregation is universal, independent of the chemical details of the particular col­loid system. Although it had been possible to obtain this universality in the diffusion limit via computer simulations, the reaction limit remained elusive to simulators. Here is described how the results of different algorithms have led us to consider the correct one, for which the universality is obtained. The important mechanisms of reaction-limited colloid aggregation are therefore fully identified.

PACS numbers: 64.60.Qb, 02.70.-c, 05.40.+J, 81.10.Dn

The kinetic aggregation of small particles to form large aggregates is a widespread phenomenon in nature [1]. Among the examples, we can cite soot and smoke forma­tion, aerosol growth, droplet formation in clouds, poly­merization processes, and colloid aggregation. Recently, the study of aggregation phenomena has attracted a great deal of interest, stimulated by the fact that the aggre­gates exhibit dilation symmetry and scaling behavior [2-6] and therefore can be characterized as fractals [7]. It was also found that the dynamics of the aggregation exhibited scaling behavior [8-11]. Later on, researchers identified two limiting regimes of the irreversible aggre­gation process. Rapid, diffusion-limited colloid aggrega­tion (DLCA) occurs when the aggregation is limited by the time taken for the clusters to encounter each other by diffusion. Slow, reaction-limited colloid aggregation (RLCA) occurs when there is a substantial potential bar­rier between the particles, the aggregation rate being lim­ited by the time taken for two clusters to overcome this barrier by a thermal fluctuation. Although for DLCA there is a clear correspondence [12] between experimen­tal results and computer simulations, for RLCA there have been serious discrepancies between the two sets of results. Basically, the RLCA experiments yield the fol­lowing three essential results: (a) The fractal dimension [7] of the clusters is typically around 2.1. (b) The weight-average cluster size S(t) = [Ess

2Ns(t)]/[^ss *Ns(t)]—where Ns(t) is the number of clusters of size s at time t—grows exponentially with time according to most workers [13-16]; however, some of them have found [17,18] that this exponential growth crosses over to an algebraic growth at the later stages of the aggregation. (c) Ns(t) scales as follows: Ns(t)^N0S(t) ~2f(s/S(t)), where TVo is the initial number of colloidal particles and / is a universal function which behaves a s / C x ) — x ~TgGc), g(x) being a cutoff function exponentially decaying for x > l . The majority of researchers [10,13-15,18-20] agree on values for r close to 1.5, although some others [21-23] have obtained values near 2.

The only theory known to the author [24] for RLCA is based on the Smolochowski equation [25] and results also in r =1 .5 and an exponential growth of the mean cluster size. The computer simulations in three dimensions

[26-28], on the contrary, have yielded a number of ex­ponents T different from 1.5. Moreover, in many of the cases their values were varying with time, which prevent­ed scaling. Also, it was not possible to fit the curves for the weight-average cluster size to an exponential. The fractal dimension was perhaps the only quantity with a reasonably good agreement between experiments and simulations. More recently, the author started a series of investigations on computer simulations of RLCA, with an emphasis on algorithms that reflect the experimental situ­ation as closely as possible. Although the first two models used [29-31] failed to give results in accordance with the experiment, their outcomes guided us to consider the correct model (reported here), whose results are fully consistent with the observed universality.

In the research effort started there were four factors that we wanted to test in the algorithms: (1) a decrease in the sticking probability, (2) a decrease in the concen­tration, (3) a shortening in the step length of the diffusional motion of clusters and particles, and (4) the introduction of small angle rotational diffusion of the clusters. The first two factors were chosen with the thinking that perhaps previous simulations had not reached the limiting asymptotic behavior of RLCA, due to a not very low sticking probability or concentration. In the model used for testing these two factors [29,30], one considers a three-dimensional cubic lattice with periodic boundary conditions, where at some intermediate time a collection of clusters made of nearest-neighbor lattice cells diffuse randomly. One of the clusters is picked at random and moved by one lattice unit in a random direc­tion, only if a random number X uniformly distributed in the range 0 < A r < l satisfied the condition x<D(s)/ Z)max, where D(s)-*~s ~l/D is the diffusion coefficient for the selected cluster of size s and Z)max is the maximum diffusion coefficient for any cluster in the system. Here D =2.1 is the accepted value for the fractal dimension of RLCA clusters. After each cluster has been selected the time is incremented by l/NcDmax, where Nc is the num­ber of clusters in the system at that time, whether or not the cluster is actually moved. If the cluster attempts to invade the lattice cells occupied by another cluster (signi­fying an encounter), the move is not permitted and the

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V O L U M E 71, N U M B E R 14 P H Y S I C A L R E V I E W L E T T E R S 4 O C T O B E R 1993

moving cluster either sticks (and is merged) to the other with a small probability PQ or remains side by side with the other with probability 1 — fV It was noticed [29] that in a real, continuum (off-lattice) aggregation system an encounter occurs only between two colloidal particles at the same time. In the model on a lattice, with particles made of perfect cubes of exactly the same size, collisions at several pairs of colloidal particles can occur exactly at the same time. As this model is not meant to be identical to the physical system but just to represent it, only one test for sticking is done notwithstanding that two or more cells of the moving cluster try to overlap the cells of the second cluster.

The essential results of those simulations [29,30], be­sides the value 2.1 for the fractal dimension at low PQS, are the following: (a) The weight-average cluster size starts increasing exponentially with time, but only for a very small fraction of the whole aggregation time, cross­ing over to an algebraic growth at later times, (b) After some transient time, the log-log plot of Ns(t) vs s starts to develop a straight line. However, it was noticed that the exponent r (initially 1.5 for low / V s ) diminishes with time, therefore preventing scaling. In summary, in this first model we are starting with a true RLCA regime, which crosses over to something different as the clusters grow bigger.

In Ref. [31] a continuum model of aggregation of spherical particles of diameter d was used, in order to take into account diverse step lengths /, varying from dl 1.4 all the way down to d/20.0. Although for DLCA this factor should not matter because there is bonding at first contact, in RLCA there are many encounters be­tween two colloidal particles—belonging to two clus­ters— before they go away, coming from the very short steps of the Brownian movement [32], which may change the results. As in these models one tests for overlapping whenever a cluster moves, huge amounts of computing time would be spent in this testing when using very short steps, and improvements in the algorithms were necessary [31]. The clusters were divided into two categories: (1) those which have one or more nearby neighboring clusters and (2) those which have not. These last clusters were allowed to move with steps of a diameter or longer in size, even if the fundamental basic step is very short, by invok­ing a central limit theorem [31]. Nonetheless, even with the improvements the computation time was getting very large for the very short step length simulations. Unfor­tunately, the same essential behavior as in the first model was seen, namely, a diminishing exponent r and a very short time interval for the initial exponential growth. However, it was noticed that the overall effect of decreas­ing the step length was similar to the effect resulted from an increase in the bonding probability; that is, the aggre­gation time decreased substantially. This indeed indicat­ed that, if / is small, two nearby colloidal particles be­longing to different clusters perform many collisions be­

fore they go away. A moment of meditation shows that, if we want a con­

stant exponent r and not a decreasing one, we need a mechanism that would speed up the aggregation of medi­um size and big clusters (or that would slow down the ag­gregation of small clusters). However, given that the above models start with a true RLCA regime, the as­sumption will be made that the aggregation kinetics of small clusters was treated correctly. The same mecha­nism would also make the mean cluster size grow faster, perhaps exponentially fast for a longer range of time. When two big clusters get close, they do so not only at one pair of approaching particles but at several pairs. However, if only short step translational diffusion is al­lowed, geometric restrictions may impede the clusters from touching at some of those pairs of particles. In Fig. 1 there is a situation in which the two clusters can touch through the particles at zone a, which at the same time hinders any collision between the particles at zone b. In general, one of the pairs of approaching particles should be privileged for collisions, hindering at the same time any encounters at the other pairs. The only missing im­portant factor, according to the author, was the rotational diffusion of the clusters, whose inclusion would make all those pairs fully accessible for encounters. Moreover, some more thought reveals that if the two clusters get close through two pairs of particles, the combined effect of short step translational and small angle rotational diffusion would make the rapid hitting between the two clusters about twice as fast than when they get close through one single pair, which may be the desired mecha­nism. However, the development of such an (efficient) al­gorithm would be prohibitively expensive, both in com­puting time as well as in its complexity. Nonetheless, as we now know that the probability for bonding between the two clusters is proportional to the number of pairs of approaching particles, we can elude the difficulty by con­sidering a "poor man's RLCA algorithm," almost identi­cal to the first model on a lattice except that now the test for sticking with probability PQ, after the moving cluster overlaps another, is made not only once but for each pair of overlapping particles between the two clusters. It is needless to say that in this new model, each overlap be-

b

a ^-^

FIG. 1. A two-dimensional representation of two clusters ap­proaching through two sets of particles; one set at zone a and the other set at zone b.

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tween occupied unit cells represents a certain number of collisions between the corresponding particles.

In the simulations described here, three values of the volume fraction p were used: 0.01, 0.003, and 0.001. For each of the concentrations, six values of the sticking prob­ability were employed: 0.1, 0.05, 0.01, 0.005, 0.001, and 0.0005. Finally, two simulations were done for each case, just to check for reproducibility. The number of initial particles N0 used were 40960, 24000, and 13144 for p equal to 0.01, 0.003, and 0.001, respectively. The simula­tions were stopped when a cluster reached a maximum size of 2000 (p=0 .01) , 1600 (p =0.003) , and 1500 (p =0.001) . Each simulation with P0 =0.0005 and p equal to 0.01, 0.003, and 0.001 took roughly 31, 63, and 103 h of a Cray Y-MP processor, respectively.

From a least squares fitting of the log-log data for the radius of gyration versus size, the values of the fractal di­mension for all the simulations were obtained. It was found that the asymptotic RLCA value of about Z) =2.1 was attained with a sticking probability of 0.01 or smaller for the three concentrations considered.

In Fig. 2 are shown the log-log plots of the weight-average cluster size versus time, for p =0.003 and the six values of the sticking probability. Sit) appears to grow exponentially with time at the beginning, while at the end it grows as a power law: Sit)~~tz. The value of the ex­ponent z obtained for sticking probabilities in the reaction limit (curves c, d, e, and / ) settles down at about 2.8. However, this exponent is concentration dependent, and its value for p =0.001 is around 2.6, while for p = 0 . 0 1 stays at around 3.0. To check that the initial behavior is really exponential, a semi-log plot of Sit) vs t was made for all concentrations. In Fig. 3 for p = 0 . 0 1 we see that, although for Po=OA the exponential growth occurs only

FIG. 2. A log-log plot of the weight-average cluster size vs time for a volume fraction of p =0.003. The different curves correspond to sticking probabilities of (a) 0.1, (b) 0.05, (c) 0.01, (d) 0.005, (e) 0.001, and (f) 0.0005. The limiting slopes of each of the curves are (a) 1.9, (b) 1.9, (c) 2.8, (d) 2.8, (e) 3.0, and (f) 2.9.

for a very small fraction of the aggregation time, for Po^O.OOOS this fraction of time increases considerably. We therefore have an exponential growth of Sit) in RLCA, crossing over to an algebraic growth at later times. The situation for other values of Po is intermedi­ate. In the same figure the semi-log plots of the number of clusters Ncit) vs time are shown. That the initial slope for Sit) is roughly twice the negative of the slope for Ncit)—and this was checked in all the simulations performed—indicates that Sit)— Snit)

2 at the begin­ning, where Snit) is the number-average cluster size.

After a transient time, the log-log plots of Ns vs s were starting to develop a straight line. Contrary to the previ­ous results, now the slope does not diminish with time but maintains at the value of around —1.5, in agreement with most experiments. This occurred at all concentra­tions studied for sticking probabilities within the reaction limit. To test scaling, a log-log plot of f=Sit)2Nsit)/N0

vs x=s/Sit) was made for different times. In Fig. 4 we see such a plot for six values of the aggregation time, cor­responding to one of the p = 0 . 0 1 and Po^O.OOOS simula­tions. The collapse of the data for different times into a single master curve indicated that the data for the num­ber of clusters indeed scaled. More surprising than this was the superposition of the experimental data by Broide and Cohen with our data. The discontinuous curve pass­ing through our data point comes from a similar plot made in Ref. [18]. The discontinuous straight line, with

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V O L U M E 71, N U M B E R 14 P H Y S I C A L R E V I E W L E T T E R S 4 O C T O B E R 1993

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slope —1.5, is just a guide to the eye. It is important to stress that this coincidence of experimental and simula-tional data occurs without the aid of any adjustable pa­rameters.

What is remarkable about the results presented here is the way nature picks the mechanisms of the different pro­cesses, such that in many occasions they would lead to universality and scaling. For example, to get the scaling in the DLCA process [33] we just need to have Stokes-Einstein translational diffusion of the particles and clus­ters plus rigid aggregation at first contact. Notwithstand­ing that the clusters in the real system are rotating, we do not need to consider it because the aggregation at first contact makes the introduction of this factor irrelevant (assuming isotropic clusters). In the same way, the short step length factor is again irrelevant. When we go to the opposite limit of RLCA, where the short step length fac­tor should make a difference, universality is not obtained unless we introduce an additional factor, which is the small angle rotational diffusion of the clusters. The surprising thing is that, although there are many mecha­nisms that will not lead to universality and scaling (e.g., the first two models proposed by the author), nature in many occasions picks the one that leads to this universal behavior. This sympathy of nature for the universal pro­cesses is perhaps linked more profoundly to the funda­mental laws of nature.

I have benefited from discussions with D. Weitz, F. Leyvraz, and L. Mochan. I am also grateful to E. Lopez-Pineda from Cray Research for his help with the computer programs. This work was supported in part by UNAM-DGAPA (Grant No. IN300191).

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