Diffusion limited aggregation with directed and anisotropic diffusion

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Diffusion limited aggregation with directed and anisotropic diffusion

R. Jullien, M. Kolb, R. Botet

To cite this version:

R. Jullien, M. Kolb, R. Botet. Diffusion limited aggregation with directed and anisotropic diffusion.

Journal de Physique, 1984, 45 (3), pp.395-399. �10.1051/jphys:01984004503039500�. �jpa-00209768�

Diffusion limited aggregation with directed and anisotropic ^diffusion

R. Jullien, ^{M. Kolb and R. Botet}

Laboratoire de Physique ^des Solides, Université Paris-Sud, ^Centre d’Orsay, ⁹¹⁴⁰⁵ Orsay, ^France

(Reçu ^{le 20} septembre 1983, accepté le 14 novembre 1983)

Résumé.

On étudie des agrégats ^{à structure} fractale obtenus à partir d’un processus de croissance par diffusion limitée. On présente des simulations numériques bidimensionnelles faites dans le cas où un agrégat pousse ^{sur une}

ligne de base. Pour une diffusion isotrope, ^{ce modèle se} comporte ^comme le modèle de croissance sphérique ^de

Witten et Sander. On trouve qu’une diffusion directionnelle rend l’agrégat compact ^alors qu’une diffusion ani- sotrope semble conduire à un exposant fractal variant continûment.

Abstract.

Fractal aggregates obtained from diffusion limited growth processes ^are studied. Numerical simula- tions are reported ^{for a} two-dimensional cluster growing ^{on a basal line. For} isotropic diffusion this model behaves

as the spherically growing model of Witten and Sander. It is found that directed diffusion renders the cluster compact ^while anisotropic ^diffusion apparently yields ^a continuously varying ^{fractal dimension.}

Classification Physics ^Abstracts

68.70

05.40

82.70R

1. Introduction.

The formation of clusters by aggregation ^of particles

is a quite general phenomenon occurring ⁱⁿ many scientific areas such as physics (dendritic growth), chemistry (flocculation ^of colloids, formation of gels, polymerization), ^medicine (growth ^of tumors) ^etc...

In most cases the resulting clusters have interesting

« fractal » structures [1]. The irreversible character of the phenomenon yields peculiar scaling properties

which are probably different from those of equilibrium

systems currently encountered in statistical physics.

Many theoretical models have been developed ^to reproduce the main features of the experiments. ^{In the}

Eden model [2] ^{the extra} particle ^is added with equal probability ^on any neighbouring site of the cluster’s surface. The resulting cluster is always compact, ^i.e.

its fractal dimension is equal ^to the usual Euclidian dimension. This model has been invoked to describe the growth ^{of tumors. In the} diffusion limited _aggrega- tion (DLA) model of Witten and Sander [3] single particles ^{stick one} by ^{one on an immobile} growing

cluster after diffusing ^{in a} purely random walk fashion.

The resulting cluster has a characteristic fractal dimension smaller than the Euclidian dimension.

Recently, ^{it has been} proposed ^{that it can describe}

two fluid displacements in porous media [4]. ^{In the}

« clustering of cluster » model of Kolb et al. [5] ^and

Meakin [6], ^{clusters of} particles ^{as well as} single particles ^{are allowed to diffuse} together ^and any kind

of clusters stick on contact. The resulting ^{clusters are}

much more stringy ^{than in DLA} and their fractal dimension is smaller. This model seems better adapted

to describe flocculation and gelation. ^{In the last two}

models the diffusion _process is assumed to be always purely isotropic. ^Such hypothesis ^is clearly unphysical

and in most cases, the diffusive motion of a given particle, ^or cluster, ^is certainly influenced by ^the

presence of the other clusters. This effect is complex

in the clustering of cluster model. On the contrary ⁱⁿ the DLA model the study ^{can be done in} a,simple way since there is only ^{one diffusive} particle ⁱⁿ presence of

one dendrite. The aim of this _paper is to study ^{the effect}

of directed and anisotropic diffusion for the single particle in the DLA model. By directed diffusion we mean that the particle ^{has more} probability ^to go in a

given direction, ^for example ^the direction of the cluster. By anisotropic ^{diffusion we mean that the} probability ^of moving along ^{one axis is} greater ^than

along ^the perpendicular axis; however the probabi-

lities to move in one direction and the opposite ^are rigorously equal ^on each axis. This study ^{has been} performed ^{on a new} version of the DLA model which considers a different geometry. Here, the cluster _grows

on a surface instead from a single ^seed particle. ^After introducing ^{the model} (part II) ^we show that it gives

the same results as the Witten and Sander model in two dimensions in the case of isotropic ^diffusion (part III). ^{Then we} study ^{the cases} of directed and

anisotropic ^diffusion (part IV).

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01984004503039500

396

2. A new version of the DLA model.

In the original ^{DLA model as studied} by ^{Witten and}

Sander [3] and Meakin [7] the dendritic cluster _grows

spherically ^{around an} initial seed particle. Later, Meakin [8] (see ^{also Racz and Vicsek} [9]) studied the

growth of dendrites by ^{DLA on} fibres and surfaces but with the limitation that the thickness of the deposit ^is

much smaller than the width of the substrate, ^{in order}

to avoid significant influence of the boundary ^condi-

tions. On the contrary, ^{in our} model, the dendrites grow from a basal plane, ^as tregs ^{on the} ground, ^but

with a height ^much greater than the width of the plane,

to recover the same feature as in the Witten and Sander’s _case, as explained below. Our numerical simulations are restricted to two dimensions. We consider a regular square lattice, the lattice constant

being considered as the unit of length. ^{The basal} plane ^{on which the} dendrites grow is here ^a line of

ordinate y

^0. Practically, ^this line is finite and contains L lattice constants. Then the added particle

moves on a semi-infinite ribbon of width L with

periodic boundary conditions in the x-direction (i.e.

x

L is equivalent ^{to x}

0). ^{At each} step ^{of the} iterative growth process ^a single particle is released

on a site chosen at random _among the L different sites of a row at ordinate y

yo far above the cluster.

(Practically ^{it is} enough ^{to choose} yo

ym + 1 where _yM is the maximum ordinate of the cluster.)

Then the particle undergoes ^a random walk on the lattice until it reaches an unoccupied ^site adjacent ^to

a site of the cluster and becomes part of the cluster

(see Fig. 1). (By definition we consider as the cluster

or the dendrite the base line and all the particles sticking ^to it.) As in the Witten and Sander model the particle is lost if it _goes too far _away from the cluster (practically y > 5 yM is enough). ^{After the} particle ^{sticks to} the cluster a new particle is released

Fig. ^1.

Sketch of the growth process.

and the process is repeated. ^{In order to} include the different possibilities of directed and anisotropic

diffusion we label differently ^the probability ^of jump- ing ⁱⁿ the different directions. _p is the probability ^to jump ^{to a} neighbouring ^site just ^{below the} original ^one.

q is the probability ^to jump ^{to the} right ^{or to the left.}

Then the probability ^to jump ^{to a} neighbouring ^site

above is 1

p - 2 q. The case of isotropic ^diffusion corresponds ^to p = q

1/4. ^We call directed diffu- sion the case when p > 1 - p - 2 q, ^i.e. p + q > 1/2

when the particle jumps preferentially in-the direction of the cluster. We call anisotropic diffusion the case

when p = 1 - p - 2 q ^i.e. p + q = 1/2 ^but p is

generally different than 1/4. ^{In that case one axis is} preferred ^{but the} probabilities ^to jump ^{to one direction}

or the opposite ^{on each axis are the same. To determine}

the speed ^of growth of the cluster quantitatively,

one can follow its maximum height yM or, alternatively,

its effective height ym defined by :

where the summation covers all the N occupied ^sites.

In the usual geometry (cluster growing ^{from a seed} particle), ^{one has}

where N ws is the number of particles ⁱⁿ the Witten and Sander’s cluster of radius RWS ^and of fractal dimension D. Now if we consider the volume R r R + ym and a 0 a + Aa where yM R ^{and Aa} 1, the number of particles ^N in this volume is :

Fig. ^2.

Definition of the jumping probabilities (a) ^and

sketch of the two cases of directed (b) ^and anisotropic (c)

diffusion.

In terms of the width L - R ea of the basal plane,

one gets :

Thus, ^{when a} steady ^{state is} reached, ^{N becomes} proportional ^to yM, and defining A(L)

N/ym, ^one

has for given Aa ¹ (1/Aoc ^{is a measure of the curva-}

ture of the basal plane) :

Thus in the spirit ^{of finite size} scaling ^{methods one}

can evaluate A(L) for different successive widths L and extract D from a log-log plot ^of A(L) ^{versus L}

for large ^L.

Of _course, it is also possible ^{to evaluate D from the} particle-particle correlation function [3, 7] given by :

where the density ^function p(r), ^{defined on} each lattice

site, ^is equal ^{to one} if the site is occupied, ^{zero if not.}

In practice, ^we have calculated C(r) along ^{the x and} y-directions. ^{It is} known that in a fractal structure

C(r) ^{must behave as}

This is obviously ^valid only ^{for r L.}

3. Results in the case of isotropic ^diffusion.

In order to determine D precisely ^we have estimated the coefficient A(L) by waiting ^until ym reaches 20 L and by averaging ^over twenty samples. Figure ³

shows the top ^{of a} typical cluster obtained by ^this

method for L

48. The quantitative ^{results as a} plot

of log ^{A versus} log ^{L for L}

3, 6, 12, 24, ^{48 are} given

in figure ^{4. The error} bars have been estimated from the calculated standard deviation of the results. The dashed and full curves correspond respectively ^to ym and _{ym. From} the asymptotic slope ^{of both curves}

one can estimate :

This result is _very similar to Witten and Sander [3]

and Meakin [7]. ^{One can} conclude, ^as expected, ^that

the geometry (a ^{seed or a basal} plane) ^{does not} change

the fractal dimension of the structure.

In figure ^{5 one} gives the results for the correlation functions Cx ^and Cy ^{for L}

^{12, 24, 48. Cx and} Cy are quantitatively different, ^since considering ^{a finite} length ^{L in the} x-direction artificially introduces an

anisotropy. However, when L -> oo the difference

disappears gradually. Moreover, log Cx goes through

a maximum at r

L/2 ^as expected ^{from the} periodic boundary conditions. The scaling regime ^{is better}

defined for Cy ^{where log} Cy ^{shows a quite linear}

behaviour as a function of log ^r up ^to r > ^{L where it} reaches a constant value. In order to estimate the

Fig. ^3.

^The top ^{ten rows of a} typical cluster of total

height yM

20 L where L

48 is the width of the strip,

obtained in the case of isotropic ^diffusion p = q

0.25.

Fig. ^4.

^{Plot of} log A (L) ^versus log ^{L in the} isotropic ^case.

The continuous and dashed lines correspond ^{to the case}

where A(L) ^is calculated from the effective height ^{and the}

maximum height, respectively. ^For comparison ^the slope

0.66 is indicated.

exponent ^a

^{d - D one must} extrapolate ^the slopes

to the infinite L limit. This is done in figure ^{6 where}

we report ^the slopes estimated from log Cx ^and log Cy

as a function of 1/L. ^The slopes extrapolate quite ^well

to the common value :

which gives :

a value consistent with, ^{but less} precise, than the value

extracted from figure ^4.

398

Fig. ^5.

The correlation function in the isotropic ^{case :} log C(r) ^is plotted ^versus log ^{r for L}

^{12, 24, 48. The}

continuous and dashed lines correspond ^to the y ^{and x} directions, respectively.

4. Results ^{in the case} of directed and anisotropic ^dif-

fusion.

We have learned from the preceding sections that the finite size results from log ^{A versus} log ^L give ^{a better}

estimate of D than the correlation function. Therefore

we will only report the results for log ^{A versus} log ^L

in the case of directed and anisotropic ^diffusion.

Let us first discuss the case of directed diffusion.

We have studied the case where the probability ^to

diffuse to the bottom is larger than the others. A typical example ^obtained for q

0.25, p

0.3 (L

48) ^is

shown on figure ^{7. When} comparing ^this figure ^with figure 3, ^{it is} really ^{difficult to} appreciate ^the differences.

However, ^when looking ^{at the} quantitative results,

the conclusion is clear. In figure ^{8, we} report ^the log ^A

versus log ^L plot in the cases q

0.25 and p

0.3, 0.4.

One can see that, ^{after a finite L} crossover, the asymp- totic slopes ^become equal ^{to one for all} p values different from 0.25. For _p

0.3, ^{the crossover bet-}

ween the isotropic regime ^{and the directed} regime

occurs around L - 12 while for _p

0.4, the directed

regime ^is already reached for L - 6. From these results

one concludes that directed diffusion always yields ^a

compact structure, i.e. ^{a structure} in which D

d.

This is confirmed by calculations done for other _p values and not reported ^{here : even} if the directional character of the diffusion is small, ^the resulting ^cluster

is always compact ^{if we look at} sufficiently large ^scale.

This is _very similar ^to the conclusion of a recent work

by ^Meakin [10] who studied the effect of particle

drift in the original ^{DLA model. Even if the} geometry

Fig. ^6.

^The slopes extracted from the log-log plot ^{of the}

correlation functions as a function of 1/L. ^The large ^error

bars for the correlations in the x-directions are due to the

difficulty ⁱⁿ defining ^the scaling regime ^{in that case.}

Fig. ^7.

^The top ^{ten rows of a} typical cluster of total height

yM

20 L, ^{where L}

⁴⁸ is the width of the strip, ^obtained

in the case of directed diffusion with _p

0.3, q

^0.25.

Fig. ^8.

^{Plot of} log A(L) ^versus log ^{L in the case} of directed diffusion with q

^{0.25 and two values} of p, p = 0.3 and p

0.4. The case p = q

0.25 is shown on the same figure

and the slopes 0.66 and 1 are indicated.

is completely different, there is also a crossover from

a fractal structure on short length ^{scales to a uniform}

structure (D

d ) ^on longer length scales. One can

conclude _very generally, using ^the renormalization group language, ^{that the DLA} is unstable with respect

to any directional perturbation.

The case of anisotropic ^diffusion appears ^to be much

more tricky. ^{We have not} reported ^the figure ^{of a} typical cluster in that case because it always ^{looks like} figure ^{3. In} figure ^{9 we} plot log ^{A versus} log ^{L in the}

case p + q

^{0.5 and} p

0.1, 0.2, 0.3, ^{0.4. Instead} of a clear crossover as in figure 8, ^we observe here a

slow continuous variation. The results for D, ^with

error bars, estimated from the asymptotic slopes ^{are I} reported ⁱⁿ figure ^{10. D varies} smoothly ^with p. There ⁱ

are three special ^{cases : for} p - 0.5 the horizontal

motion is practically suppressed and the cluster becomes compact. For p - 0, ^{on the} contrary, ^the diffusive particle ^moves primarily horizontally ^and prefers ^{to stick to the} top ^{of a cluster so that we must}

recover D

1. Finally ^for p

0.25, ^the previously

studied isotropic ^case is recovered. Figure ¹⁰ strongly suggests ^a continuously varying exponent (like ^{a line}

of fixed points ⁱⁿ the renormalization _group language)

but we cannot really ^{exclude a constant} exponent with finite size crossover effects to the different limiting

cases p

0 and _p

0.5. (This could be due to syste- matic errors due to size effects different from the statistical errors reported ^{in the} figure.)

5. Conclusion.

This work presents ^{a new} version of the DLA model in which the cluster _grows on a surface. We have shown that in two dimensions this model behaves as ’ the original ^DLA model. Then we have studied the ⁱ influence of modifications in the diffusion _process.

We have shown that directed diffusion has a dramatic effect on the fractal properties yielding ^a compact cluster, ^while anisotropic ^diffusion apparently changes ⁱ

the fractal dimension continuously though ^{we cannot}

exclude a jump ^{of the} exponent ^{D. It would be} really

Fig. ^9.

^{Plot of} log A(L) ^versus log ^{L in the case of ani-} sotropic diffusion with _p + q

^0.5 and for the following

values of p : p = 0.1, 0.2, 0.3, ^0.4.

Fig. ^10.

^{Plot of} the estimated extrapolated fractal expo- nant D as a function of p ^{in the case of} anisotropic ^diffusion.

interesting ^to extend such kind of study ^{to the case of}

the clustering of clusters models [5, 6] ^and try, also,

to include more realistic interactions between clusters.

This will be the subject ^of future work.

References

[1] MANDELBROT, ^B. M., ^The fractal geometry of ^nature (Freeman) ^1982.

[2] EDEN, M., Proc. Fourth Berkeley Symp. Math. Stat.

and Prob., ^J. Neyman (ed.) ^vol. 4, p. 233 (U.C.

Press) ^1961.

[3] ^WITTEN Jr., ^{T. A. and} SANDER, ^L. M., Phys. ^{Rev. Lett.}

47 (1981) ^1400.

[4] PATERSON, L., preprint.

[5] KOLB, M., BOTET, ^{R. and} JULLIEN, R., Phys. ^Rev.

Lett. 51 (1983) ^1123.

[6] MEAKIN, P., Phys. Rev. Lett. 51 (1983) ^1119.

[7] MEAKIN, P., Phys. ^{Rev. A 27} (1983) ^604.

[8] MEAKIN, P., Phys. ^{Rev. A 27} (1983) ^2616.

[9] RACZ, ^{Z. and} VICSEK, T., preprint.

[10] MEAKIN, P., preprint.