VOLUME 75, NUMBER 20 PH YS ICAL REVIEW LETTERS 13 NOVEMBER 1995

Thorn and Seesselberg Reply: In a recent Letter [1],we used a stochastic simulation method to investigate thekinetics of an experimental system of colloidal aggrega-tion of polystyrene particles under both diffusion-limited(DLA) and reaction-limited (RLA) conditions [2,3]. Forthe DLA case, we have simulated the kinetics with theSmoluchowski kernel

( ) (1/1.75 + .1/1.75) (

—t/1. 75 + —1/1.75)

The simulated data are in qualitative agreement with theexperimental results (Figs. 1 and 2 in Ref. [1]). Weconjectured that gravitational settling which has beenobserved experimentally [3] may cause the remainingquantitative discrepancy. In the RLA case, we suggestedthe kernel

which fits the experimental data perfectly both in thescaled size distribution and in the temporal evolution ofthe size distribution (Figs. I, 3, and 4 in Ref. [1]).

In their Comment [4], Hasmy, Jullien, and Botet claim(a) that quantitative comparisons of experimental resultswith simulated data need more precise models involvingthe relevant geometry of the clusters. Furthermore, theyassert that (b) their standard cluster-cluster aggregation(CCA) model reproduces all the temporal evolution forDLA and RLA without the assumption of gravitationalsettling.

We reply as follows. (a) The standard CCA model ofHasmy, Jullien, and Botet takes the cluster geometry intoaccount by means of a lattice model. At first sight, thisseems to be a better approximation than our stochasticmodel, which takes the geometry into account in a mean-field way through the kernel K(i, j). However, the resultsof Ref. [5] suggest that the CCA model is equivalent toa mean-field model with a specific kernel K(i, j). Thestochastic method has the advantage of great efficiency.The simulation time and the storage requirement is severalorders of magnitude less than that of the CCA model,and the implementation of the algorithm is very simple[6]. Thus much better statistics can be achieved and thesimulations can reach much longer times than those of theCCA model. Also, if necessary, other processes such asfragmentation can be incorporated in the simulations [7].Therefore the stochastic method is of great practical use.The problem is to find the kernel which corresponds tothe cluster geometry. An appropriate kernel yields resultsof the stochastic simulations which agree with those ofthe CCA model. This is the case for RLA, which can beseen by comparing Figs. 1 and 3 of Ref. [1] with Figs. 1

and 2 of Ref. [4]. In the DLA case we did not find

the correct kernel which is clearly indicated in Ref. [1].However, one cannot conclude from this that the mean-field approach fails to describe quantitatively such kinetics[4]. This is clearly wrong, at least in the RLA limit.

(b) The CCA model presented in Ref. [4] can indeedreproduce the DLA kinetics. This indicates that gravi-tational settling does not play an important role in theexperimental data. This is the new insight of Ref. [4].It suggests that the Smoluchowski kernel Ko(i, j) doesmodel the kinetics of the experimental system [2,3] onlyqualitatively. This point deserves further investigations.On the other hand, for the RLA case, the agreement ofthe CCA simulations with the experimental data seems tobe good, but there are two shortcomings of the method.First, the calculation of the charge density is somewhatarbitrary and second the calculations cannot exceed thetime T = 10 as can be seen from Fig. 2 in Ref. [4]. Thelatter is a consequence of the immense computational ef-fort of this method.

However, the stochastic method and the CCA modelboth have their advantages. It would be an interestingproject to combine them. For example, one can determinethe kernel K(i, j) of an aggregation process by the CCAmodel and then perform longtime simulations with astatistically sufficient amount of data by the stochasticmethod.

Matthias Thorn and Markus SeesselbergAlbert-Ludwigs-UniversitatFakultat fiir Physik, Hermann-Herder strass 3D-79104 Freiberg i. Br., Germany

Received 14 July 1995PACS numbers: 82.70.Dd, 05.40.+j, 82.20.Wt

[1] M. Thorn and M. Seesselberg, Phys. Rev. Lett. 72, 3622(1994).

[2] M. L. Broide and R. J. Cohen, Phys. Rev. Lett. 64, 2026(1990).

[3] M. L. Broide and R. J. Cohen, J. Colloid Interface Sci.153, 493 (1992).

[4] A. Hasmy, R. Jullien, and R. Botet, preceding Comment,Phys. Rev. Lett. 75, 3777 (1995).

[5] R. M. Ziff, E.D. McGrady, and P. Meakin, J. Chem. Phys.82, 5269 (1985).

[6] M. Thorn, H. P. Breuer, F. Petruccione, and J. Honerkamp,Macromol. Theory Simul. 3, 585 (1994).

[7] M. Thorn, M. Broide, and M. Seesselberg, Phys. Rev. E51, 4089 (1995).

3778 0031-9007/95/75(20)/3778(1)$06. 00 1995 The American Physical Society