Diffusion limited aggregation without branching: A brief overview

220

Nuclear Physics B (Proc. Suppl.) 5A (1988) 220 224 North-Holland, Amsterdam

DIFFUSION LIMITED AGGREGATIONWITHOUT BRANCHING: A BRIEF OVERVIEW Giuseppe ROSSI Materials Department, College of Engineering, University of C a l i f o r n i a , Santa Barbara, CA 93106. Diffusion l i m i t e d aggregation without branching is a model of i r r e v e r s i b l e growth where the accreting p a r t i c l e s move d i f f u s i v e l y and the rules determining the growth of the c l u s t e r force a very simple structure on the r e s u l t i n g aggregates. The s i m p l i c i t y of the model makes i t possible to give a more complete t h e o r e t i c a l treatment than in other d i f f u s i v e growth processes. Here a b r i e f overview is given of the a v a i l a b l e simulation results and of the t h e o r e t i c a l work: these results are compared with those obtained f o r ordinary DLA. I . INTRODUCTION

branching, this slows down the radial growth

I r r e v e r s i b l e growth models have been an area 1 of intense study during the l a s t ten years .

of the c lus t e r and accounts f o r i t s highly

Among these models d i f f u s i o n l i m i t e d

t i p s p l i t t i n g is forbidden, so that only the

aggregation2 (DLA) plays a special r o l e : this

f i r s t mechanism is at work. The rules

is because DLA or DLA-type patterns are found

c o n t r o l l i n g growth in t h i s model are as

in a wide v a r i e t y of seemingly unrelated

follows: one starts with a l i n e of absorbers

branched nature. In the model described here

physical s i t u a t i o n s I. In DLA p a r t i c l e s are

(sticky sites) located at the bottom of a

launched one at a time and walk randomly u n t i l

s t r i p - l i k e portion of a square l a t t i c e .

they reach any s i t e adjacent to the growing

Particles are launched one at a time from a

aggregate, at which point they accrete.

s i t e chosen at random above the location of

Extenslve computer simulations 3-6, have

the absorbers: they perform a random walk

uncovered the scaling structure of clusters

u n t i l they reach an absorber s i t e . When this

grown in this way. However from a t h e o r e t i c a l

occurs the s i t e reached by the p a r t i c l e

point of view many problems remain unresolved.

becomes occupied and the absorber is moved to

In p a r t i c u l a r , there seems to be no r e l i a b l e

the s i t e immediately above i t .

way to p r e d i c t the value of the exponent

grows needles or columns of p a r t i c l e s . Since

c o n t r o l l i n g the growth of an ordinary DLA

the sides of these needles are not sticky (in

cluster.

the computer simulations whose results are

Here I shall consider a model of c l u s t e r

In t h i s way one

reported below, the sides of the needles were

growth 7-9 where as in DLA, the accreting

taken to be r e f l e c t i n g ) incoming p a r t i c l e s can

p a r t i c l e s move d i f f u s i v e l y ; however, the rules

only attach themselves to the top of the

c o n t r o l l i n g growth in t h i s model force the

needles and the number of absorbers remains

r e s u l t i n g aggregates to have a much simpler

constant throughout the growth.

structure than that found in ordinary DLA. Two mechanisms appear to contribute to the

Figure 1 shows what a c l u s t e r grown in thls way looks l i k e at various stages of i t s

growth of an ordinary DLA c l u s t e r : one is

growth. I n i t i a l l y

competition among d i f f e r e n t t i p s to trap

growing; however, as more and more p a r t i c l e s

incoming walkers and advance in the r a d i a l

are added, most absorbers become screened, so

d i r e c t i o n ; the other is t i p s p l i t t i n g or

that only a few of the needles keep advancing;

0920-5632/88/$03 50 © Elsevier Science Pubhshers B V (North-Holland Physics Pubhstung Dxvis~on)

several needles begin

G Rosst / Dtffuslon hrmted aggregatton wtthout branchmg

221

e v e n t u a l l y , on a s t r i p of f i n i t e width, a saturation regime w i l l be reached where one needle outgrows a l l the others. In other words, at each stage of the growth, there is a c h a r a c t e r i s t i c distance, such that portions of the c l u s t e r separated by a distance l a r g e r

,.I,.L,., I.,. .L_.I J ,.Ld._J I IJ JL _L,.I._-I,.,

than this do not influence (screen) each other's growth: i . e . , over these length scales the growth process is l o c a l . The model was introduced in an attempt to improve our t h e o r e t i c a l understanding of scale

,,L.. L,.., I.,

i n v a r l a n t d i f f u s i v e growth. Nonetheless, physical s i t u a t i o n s where u n i d i r e c t i o n a l forces

1.,_., , .,

J , ,j .IL _.[,.I._J..,

are responsible f o r aggregation and d i f f u s i o n ,,L,.L,_~ I. IL ..... 1,,., j , , . . i k

is the rate l i m i t i n g step should lead to

1.1._

.I..J

needles of the type described above. Experimental studies of the accretion of

• .L,.L,

•

I.

=L

....

J__~

,

, . . . .

J.,

-I.

,

I n t e r a c t i n g magnetic holes in f e r r o f l u i d s in such a regime are c u r r e n tl y being considered I0 2. COMPUTERSIMULATIONS IN TWO DIMENSIONS F a i r l y extensive computer simulations using the rules descrlbed above have been performed.

FIGURE 1 Evolution of an aggregate grown on a s t r i p of width 128. The aggregate is shown (from bottom to top) a f t e r 200, 400, 600 and 800 p a r t i c l e s have accreted. Each p a r t i c l e is represented by a small rectangle.

The density p(y,N/L) and the d i s t r i b u t i o n of absorbers D(y,N/L) have been measured, as

D(y,N/L) = y-B

(2b)

functlons of the height y and of the r a t i o N/L between the number of p a r t i c l e s N and the

hold in a larger and larger range of y. The

width L of the s t r i p : one is interested in the

value of the exponents found from the

form of these functions in the l i m i t of large

slmulations are approximately ( f o r a d e t a i l e d

s t r l p widths. Since a l l the absorbers are

discussion and e r r o r bars see r e f . 9)

located at the top of the needles ( i . e . ,

at the

sites where the density changes from 1 to zero)

= .82

(3)

B = 1.82

one has D(y,N/L) : p ( y - l , N / L ) - p(y,N/L)

(I)

Note

t h a t , in the scaling regime, eq. ( I )

m p l l e s ~= ~+I. Typical data f o r the average namely, the d e r i v a t i v e of the density with

density as a function of the height y are shown

respect to y gives the d i s t r i b u t i o n of

in f i g u r e 2 f o r various values of the r a t i o

absorbers. I t is found th a t , as the r a t i o N/L

N/L. These data were obtained from a set of I00

increases, scaling laws of the form

clusters grown on a s t r i p of width L=128. The shoulder appearing at large y in the data f o r

p(y,N/L) = y-m

(2a)

N/L=IO is due to the fact that saturation effects are becoming important.

G Ross1 /Dtffuston hrmted aggregatton without branching

222

3. MEAN FIELD TREATMENTAND COMPUTER SIMULATIONS IN DIMENSIONS HIGHER THAN TWO ! •

For ordinary DLA i t is possible to w r i t e continuum equations 2 describing the average

T

o

growth of the c l u s t e r . I t has been argued that a continuum approximation which retains only

!

:l

" !~,,, • " ° •Q,14 m • ° °,°,°,,

>,,

"%, ".~

the l i n e a r terms in an expansion in powers of the density is e f f e c t i v e l y a mean f i e l d

(',i

treatment. Numerical solutions of continuum

o

I

equations of thls type can be f o u n d l l : they predict f o r the f r a c t a l dimension the value D = d - l . For ordinary DLA there is no evidence f o r the existence of an upper c r i t i c a l

I

o

•- 1 0 0

........

,

,

101

:,

A,,

,

10 2

10 s

dimension where such behavior is reached. Continuum equations describing the growth of the c l u s t e r can also be w r i t t e n down f o r the model described here 12. In this case, the equations can be solved exactly: in p a r t i c u l a r one finds f o r the density p(y,t)

~ l _ e(t)

for y < I / e(t ) (4)

p(y,t)

= 0

FIGURE 2 Average density as a function of y obtained from lO0 clusters grown in two dimensions on a s t r i p of width L=128. The curves r e f e r (from l e f t to r i g h t ) to N/L=2,3,5,8 and lO.

for y > I / e ( t )

4. "REAL SPACE RECURSION" TREATMENT For the model described above i t is possible to give a t h e o r e t i c a l treatment which

Here continuum time t plays the same role as

is in many ways s i m i l a r to real space

the parameter N/L in eq.(2) and e ( t ) is a

renormalization group methods. Thls is in

function which decreases r a p i d l y as t

contrast with ordinary DLA, where so f a r i t has

increases. In other words, according to this

proven impossible to formulate a conslstent

mean f i e l d treatment, as t gets l a r g e r , scaling

treatment of this type.

laws of the form (2) with an exponent mmf = l

I t was stressed above t h a t , as the c lus t e r

(Independent of s p a ti a l dlmension) are obeyed

grows, absorbers become screened so that the

in a larger and larger range of y.

corresponding needles stop growing. Suppose

Simulations in three and four dimensions have been done9 (the basic rule is s t i l l

the

that the f r o n t of the c lus t e r has reached a certain height ho; one wants to estimate the

one described in the i n t r o d u c t i o n , but the

distance k that the f r o n t of the c lus t e r has

initial

to cover in i t s advance in order f o r a

seed is now a d-I dimensional surface

of absorbers): i t is found that (within

f r a c t i o n f of the absorbers to become

s t a t i s t i c a l error) both in three and four

screened. Since the density at the f i n a l helght

dimensions the exponent m f o r the density is I:

ho+k w i l l be reduced by a corresponding

i.e.,

f r a c t l o n with respect to the density at ho one

i t coincides with the mean f i e l d

pre d i c t i o n .

can obtain an estimate f o r the density e x p o n e n t ~:

G. Rosst /Dtffuston hmzted aggregation wtthout branching

223

a s u f f i c i e n t l y large number of configurations.

-lnf =

(5)

In ((ho+k)/h o)

The results found f o r the exponent m in this way agree well with those obtained from the simulatlon (see ref. 9); as expected, the

In the scaling regime the value of a obtained

agreement improves when c e l l s wlth a larger

in this way must be independent of ho.

number of absorbers are used.

In order to get results out of this scheme

Cells of this type containing only two

consider an averaged description of the

absorbers constitute one of the simplest (non

c l u s t e r where the L' absorbers l e f t at height

trivial)

ho are taken to be equally spaced and at the

problem the existence of a simple recurrence

same helght. Using the f a c t that the growth

r e l a t i o n between the posslble h i s t o r i e s

process is l o c a l , one divides the system in

leading to a certain configuration makes i t

c e l l s with each c e l l containing only a small

possible to perform the s t a t i s t i c a l sum

number P of absorbers. Instead of f o l l o w l n g

exactly. Of course to do t h i s , one must know

the evolution of the whole system one follows

the p r o b a b i l i t y P(M,K) of a p a r t i c l e landing

the evolution of one such c e l l as p a r t i c l e s

on the t a l l e s t needle: here M is the wldth of

are added. Since the c e l l problem involves only

the c e l l and K is the difference between the

a small number of degrees of freedom, i t is

ordinates of the two absorbers. For d l f f u s i n g

r e l a t i v e l y easy to estimate the s t a t i s t i c a l sum

p a r t i c l e s P(M,K) is found by solvlng the

d i f f u s i v e growth problems. For this

and f l n d the average advance of the t a l l e s t

Laplace equation f o r the c e l l and thls has in

needle when i t has outgrown the other needles

general to be done numerically.

in the c e l l ( s u f f l c i e n t l y precise c r i t e r i a f o r

While I t is clear that in ordinary DLA the

this can be given: see ref° 8). At the next

Laplace dynamics conspires with noise (due to

stage one considers the same problem f o r a c e l l

the f i n i t e slze of the accreting p a r t i c l e s ) to

which is P times as wide as that of the

y i e l d f r a c t a l behavior, there is l i t t l e

previous stage, but st111 contains only P

q u a n t i t a t i v e understanding of the mechanism

absorbers. In this way one can obtain

leading to t h i s r e s u l t , and much work has

estimates f o r k at d i f f e r e n t stages of the

focussedon how small modifications to the

growth; one can check that indeed there is

growth process (such as l a t t i c e 6 or external 4

scallng: i . e . ,

anisotropies) change t h i s behavior. I t is

that the r a t i o between values of

k r e l a t i v e to successlve stages is a constant,

therefore of some i n t e r e s t to study how the

and one can obtain an estimate f o r m which in

exponent~ obtained from a sequence of two

p r i n c i p l e can be s l s t e m a t i c a l l y improved (by

absorber c e l l problems changes wlth P(M,K). I

using larger values of P). Note th a t , at each

considered9 the class of P(M,K) given by

stage of this procedure, only the p a r t i c u l a r length scale over which growth is local is treated. The procedure o u t l i n e d above has been

1 P(M,K) = - (I + (2K/M)a) 2

(6)

For a = 2 eq. (6) resembles c los e ly the

carried out at length f o r c e l l s containing two,

p r o b a b i l l t y P(M,K) that one would obtain

three and four absorbers. For c e l l s with more

solving the Laplace equation; f o r a small any

than two absorbers i t is d i f f i c u l t

to evaluate

K # 0 w i l l make P(M,K) nearly equal to I , while

the s t a t i s t i c a l sum exactly and one has to

f o r a large P(M,K) w i l l remain very close to

content himself with Monte Carlo sampling over

I / 2 u n t i l K = M/2. One flnds (note that the

G Rosst /Dtffuston hrntted aggregation wtthout hranchmg

224

problem is being solved e x a c t l y ) :

analyze the i n t e r p l a y between d l f f e r e n t factors contributing to growth in slmple model

= I

for a < 1

a + 1 -

situations. (7)

for a > 1

ACKNOWLEDGEMENTS

2a

This work was funded in part by the

(at a=l, m is 1 but there are logarithmic

Department of Energy under grant DE-FGO3-

corrections to the power behavior). In other

87ER45288.

words, f o r a < 1 noise is the dominating f a c t o r and the p a r t i c u l a r form of P(M,K) is unimportant; f o r a > I , one is in a regime where " d r i f t "

overcomes noise and m changes

continuously with a. Of course i t is an open question whether a picture of t h i s type w i l l apply to less contrived growth processes and, in p a r t i c u l a r , to ordinary DLA. Note t h a t , fo r a=2 eq.(7) gives m=.75, which is reasonably

REFERENCES I. For an overview see T.A. Witten and M.E. Cates, Science 232 (1986) 1607. 2. T.A. Witten and L.M. Sander, Phys. Rev. Lett. 47 (1981) 1400 and Phys. Rev. B27 (1983) 5686. 3. P. Meakin, Bull. Am. Phys. Soc. 30 (1985) 222.

close to the value reported in eq.(3).

4. R.C. B a l l , R.M. Brady, G. Rossi and B.R. Thompson, Phys. Rev. Lett. 55 (1985) 1406.

5. CONCLUSION

5. T.C. Halsey, P. Meakln and I. Procaccia, Phys. Rev. Lett. 56 (1986) 854.

In summary, the model described here e x h i b i t s non t r i v i a l

scaling behavior and t h i s

occurs s o l e l y as a product of screening r e s u l t i n g from competition among d i f f e r e n t absorbers. Simulations in three and four

6. P. Meakln, R.C. B a l l , P. Ramanlal and L.M. Sander, Phys. Rev. A35 (1987) 5233. 7. P. Meakin, Phys. Rev. A33 (1986) 1984. 8. G. Rossi, Phys. Rev. A34 (1986) 3543.

dimensions give f o r the exponent c o n t r o l l i n g growth a value consistent with that found from mean f i e l d arguments based on the continuum approximation. I t is possible to give a t h e o r e t i c a l treatment s i m i l a r to real space renormalization group methods used f o r

9. G. Rossi, Phys. Rev. A35 (1987) 2246. I0. R. Pynn, p r iv at e communicatlon. II.

R. B a l l , M. Nauemberg and T.A. Witten, Phys. Rev. A29 (1984) 2017.

12. M.E. Cates, Phys. Rev. A34 (1986) 5007.

e q u i l i b r i u m s t a t i s t i c a l systems. The treatment provides estimates fo r the exponent c o n t r o l l i n g growth which agree well with the simulation r e s u l t s ; the treatment can also be used to

13. An extension of the model described here, which allows f o r f i r s t order branching has been recently put forward by P. D e v i l l a r d and H.E. Stanley, Phys. Rev. A36 (1987) 5359.