Effects of random-walk size on the structure of diffusion ...ps24/PDFs/Effects of Random-Walk Size on the... · A diffusion-limited aggregation (DLA) model was in-troduced by Witten

PHYSICAL REVIEW A VOLUME 36, NUMBER 9 NOVEMBER I, 1987

Effects of random-walk size on the structure of diffusion-limited aggregates

Yi-Bin Huang and P. SomasundaranSchool of Engineering and Applied Science, Columbia University, New York, New York 10027

(Received 24 June 1987)

The step size of random walks is shown to exhibit an important role in controlling the structureof clusters produced ,by the diffusion-limited aggregation (DLA) processes. In this study, thestructural compactness is found to increase significantly as the random-walk size is increased. Thestep-size effects are precisely quantified by the radius of gyration evaluated during the clustergrowth process. It is also observed that by using a very long random walk the DLA model ap-proaches the Void-Sutherland ballistic aggregation model.

INTRODUcri.ON COMPUTER SIMULAllON: RESULTS AND DATAANALYSIS

A diffusion-limited aggregation (DLA) model was in-troduced by Witten and Sander! to simulate cluster for-mation processes based on the concept of a sequence ofcomputer-generated random walks on a lattice emulatingdiffusion of a particle. Implicitly, the step size of eachrandom walk was fixed as one lattice unit. A nonlatticeversion of DLA simulation was carried out by Meakin2,3where the step size was limited to one particle diameter.Diffusion in real cluster formation processes may, how-ever, exceed thi~ limitation with the step size dependingon such physical parameters of the system as tempera-ture, pressure, concentration, and particle size. It be-comes, therefore, very important to determine the step-size effects on the cluster formation processes.

On the other hand, the Void-Sutherland ballistic ag-gregation model (VS) uses random linear trajectories toadd particles to a growing cluster.4,s Since the only re-gion of interest in cluster formation is the neighborhoodof the growing cluster, a linear trajectory may be viewedas diffusion with an infinitely large step size. The VSmodel may therefore be approached by the DLA modelwith a very large step size. In this paper, we focus onthe step-size effects on the structure of clusters generatedby the DLA model and also attempt to establish a linkbetween the DLA and VS models.

The two-length-scale concept pro~sed by Bensimonet al.6 is also considered. One length scale, required forall aggregation models, is the "sticking length" a thatdefines the sufficient and necessary condition of an ag-gregation event. A diffusing particle is considered tostick to an aggregate when the center of the diffusingparticle is at a distance a from that of any member parti-cle of the aggregate. Usually, the particle diameter istaken as the sticking length on the basis of hard-sphereconcept. The other length scale, specific for diffusion-related processes, is the mean free path 10 of thediffusing particle which is called the step size of randomwalks in this paper. Since the constraint of I 0 ~ a maynot be valid for all physical systems, we examine thecase of 10 >a in addition to the extreme case of 10 »ainvestigated by.Bensimon el al.6

In this work nonlattice DLA computer simulationswere carried out with various step sizes ranging from 1to 64 particle diameters. Figures l(a)-l(d) show thestructure of 5000-particle clusters generated by the DLAmodel with step size of 1, 4, 16, and 64 particle diame-ters, respectively. It is clear from these figures that clus-ter structure becomes more compact as the step size isincreased.

Since the cluster in Fig. 1 (a) has a very open structure,one may imagine that a motion with a small step size(say, one particle diameter) can easily move a particleinto the central region of the cluster; however, the direc-tion of each movement is purely random so that it ishighly unlikely for a particle to keep its incident direc-tion over several steps, instead it will travel around in acertain domain and generate a substantially wide in-cident beam. Hence, a particle following short randomwalks will generally touch the outer branches of thegrowing cluster instead of penetrating into the clustercore. With a larger step size, the particle may move alonger distance in a single direction, i.e., with aminimum beam width of one particle diameter. Thisparticle has thus a much larger opportunity to avoidcontacts with the outer branches and consequently toyield a more compact structure.

For the purpose of comparison, VS simulation wasalso perfprmed to generate a cluster (shown in Fig. 2)which can be seen to be more compact that those inFigs. I (a)-I (c) and is comparable to that in Fig. 1 (d). Inthe VS model, particles move with linear trajectories.According to the above argument, the VS model (withan infinite step size) always generates denser clustersthan the DLA model (with a finite step size). Closecompactness seen in Fig. 1 (d) and Fig. 2 suggests that astep size of 64 particle diameters is "effectively infinite"for up to 5000 particle clusters.

In order to quantify the effects of the step size on clus-ter structures, the radius of gyration (Rg) during thecluster growth processes is calculated. Figure 3 showsthe dependence of Rg on the cluster size which is

36 4518 @1987 The American Physical Society

36 BRIEF REPORTS 4519

represented as the number of particles, N. To reduce theextent of data scattering so that trends, if any, can beclearly recognized, many runs of computer simulationwith different random number seeds were performed andonly the averaged data are shown. For N up to llXX>,data points are the averaged results for 100 runs, whilefor N above IIXX> in which data scattering isinsignificant, only ten runs were carried out to save com-puter processing time.

At a given N, Rg shows a decrease with increasingstep size indicative of the increase in structural compact-ness. Overlapping of the 64 step-size curve with the VSmodel further suggests that the step size of 64 particlediameters is indeed "effectively infinite" for clusters witha size up to 10000.

An examination of the DLA curves shows the ex-istence of three zones of different behavior. At smallN's, all DLA curves coincide with the VS curve (zone 1).

1-250 PARTICLE DIAMETERS ~'-'25() PARTfCLE DIAMETERg...~- "'"

..' 250 PARTICLE DIAMETERS .. j.. 250 PARTICLE DIAMETERS . -I

FIG. I. Random cluster of 5(xx) particles generated using the DLA model with the step size of (a) I particle diameter, (b) 4 par-ticle diameters, (c) 16 particle diameters, and (d) 64 particle diameters.

4520 BRIEF REPORTS 36

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The DLA curve deviates from the VS one at intermedi-ate Ns (zone 2). At large Ns, all DLA curves followstraight lines which are parallel to one another but notparallel to the VS line. In fact, the differences in theslopes of the two' curves reflect the variation in fractaldimension of flocs generated by these two models.7 Atthe extreme of step size of 64, only zone 1 appears inFig. 3; zones 2 and 3 are expected to appear at evenlarger Ns. At the other extreme of step size of 1, zones1 and 2 are so small that most data points fall in zone 3.For DLA with an intermediate step size, e.g., 8 particlediameters, the three zones become quite evident (Fig. 4).This indicates that in a cluster growth process based oila DLA model with an intermediate step size, the clustergrows following the VS mechanism in the beginning, andthen during a transition period the growth process shiftsto the DLA mechanism, and finally, the growth behaviorreflects the DLA mechanism totally.

There may be a critical transition point in the transi-tion zone such that above it the DLA mechanismpredominantly controls the growth process while belowit the VS mechanism dominates. This point was ob-tained . hereby by extrapolating the zone 3 portions ofDLA curve to intersect the VS curve. As shown in Fig.4, the transition point was found at N = 182 andRg = 7.9 for the case of step size = 8. The value of Nand Rg at the transition points obtained for all cases aresummarized in Table I. Consistently the transition pointlocates at larger Rg for the case of larger step size, andfor the step size greater than 4, transition points locateat Rg very close to the step size. This suggests that theVS linear trajectory model predominantly determines thegrowth process when the growing cluster's radius ofgyration is smaller than the step size, while the DLA

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cluster size N for the VS model and the DLA model with vari-

ous step sizes.

model takes control of the process when the cluster's Rgbecomes greater than the step size. Qualitatively, for asmall cluster, the particular step size is sufficiently largeto be viewed as "infinite" by the cluster; it will no longerbe sufficient to view it as infinite when the cluster growscontinuously. Quantitatively, this result points out thatthe radius of gyration is a good measure of cluster for-mation for such systems.

SIGNIFICANCE OF THIS WORK

The step size of random walks is shown to exhibit animportant role in controlling the nature of cluster struc-tures. The structural compactness can be significantly

10 100 1000

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FIG. 2. Random cluster of 5<XX> particles generated usingthe VS model.

FIG. 4. An example of the determination of a transitionpoint.

36 BRIEF REPORTS

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her of particles for identifying the transition point. Thusfor a three-dimensional floc one will need an even largersize to clarify its controlling mechanism. \

From the viewpoint of fractal analysis,8 there is al-ways a minimum length scale for a real-world object toclaim as a self-s~r fractal. Here we found that forcl~ters produced with DLA this minimum is character-ized well by the step size of random walks. A DLA flocmay be called a fractal with a self-similar structure onlywhen the measuring scale is well above the average stepsize of diffusion involved in its growth process.

increased by enlarging the random-walk step in theDLA model. It is also the step size which brings togeth-er the two distinct models-diffusion-limited aggrega-tion and VS ballistic aggregation. Our results impliesthat the ideal ballistic aggregation model can be physi-cally approximated by DLA with very long randomwalks. In other words, the VS model may be viewed asa special case of DLA with step size approachinginfinity. In fact, our detailed quantitative analysis of RgversU-""N confirms the crossover effect from 10 ~a to10 »a proposed by Bensimon et al.6

This work shows that DLA simulation with an inter-mediate step size generally produces clusters with atwo-level structure which contains a compact core in thecentral region as well as sparse, dendritic branches in theouter region [see Fig. l(c), for example]. This providesan alternative interpretation for two-level structure flocsfound experimentally which have generally been inter-preted by a combination of at least two different mecha-nisms.

It is also suggested that the experimental data forsmall clusters is not suitable for determining whether thegrowth process is controlled by DLA or VS mechanism.For example, if in a flocculation system the average stepsize of diffusion is 8 particle diameters, a cluster mustcontain at least 182 particles for it to be used to distin-guish between these two mechanisms. Keeping in mindthat the present simulation is for two-dimensional pro-cesses, we expect that the three-dimensional simulationwill give similar trends but will require even larger num-

ACKNOWLEDGMENT

The authors would like to acknowledge the support ofthe International Fine Particle Research Institute.

IT. A. Witten and L. M. Sander, Phys. Rev. Lett. 47, 1400(1981).2 .

P. Meakm, Phys. Rev. A 17, 604 (1983).3P.Meakin, Phys. Rev. A 17,1495 (1983).4M. J. VoId, J. Colloid Sci. 18,684 (1963).sO. N. Sutherland, J. Colloid Interface Sci. 22. 300 (1966).

60. Bensimon, E. Domany, and A. Aharony, Phys. Rev. Lett.51, 1394 (1983).

7p. Meakin, J. Colloid Interface Sci. 96, 415 (1983).8B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman,

San Francisco, 1982).