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

DIFFUSION LIMITED AGGREGATION WITHOUT BRANCHING: A BRIEF OVERVIEW

Giuseppe ROSSI

Materials Department, College of Engineering, University of Ca l i fo rn ia , Santa Barbara, CA 93106.

Dif fusion l imi ted aggregation without branching is a model of i r revers ib le growth where the accreting par t ic les move d i f f us i ve l y and the rules determining the growth of the c luster force a very simple structure on the resul t ing aggregates. The s imp l i c i t y of the model makes i t possible to give a more complete theoret ica l treatment than in other d i f fus ive growth processes. Here a b r i e f overview is given of the ava i lab le simulation results and of the theoret ical work: these results are compared with those obtained for ordinary DLA.

I . INTRODUCTION

I r revers ib le growth models have been an area 1 of intense study during the last ten years .

Among these models d i f fus ion l imi ted

aggregation 2 (DLA) plays a special ro le: th is

is because DLA or DLA-type patterns are found

in a wide var ie ty of seemingly unrelated

physical s i tuat ions I . In DLA par t ic les are

launched one at a time and walk randomly un t i l

they reach any s i te adjacent to the growing

aggregate, at which point they accrete.

Extenslve computer simulations 3-6, have

uncovered the scaling structure of clusters

grown in this way. However from a theoret ica l

point of view many problems remain unresolved.

In par t i cu la r , there seems to be no re l i ab le

way to predict the value of the exponent

con t ro l l i ng the growth of an ordinary DLA

cluster.

Here I shal l consider a model of c luster

growth 7-9 where as in DLA, the accreting

par t ic les move d i f f us i ve l y ; however, the rules

cont ro l l ing growth in th is model force the

resul t ing aggregates to have a much simpler

structure than that found in ordinary DLA.

Two mechanisms appear to contr ibute to the

growth of an ordinary DLA cluster: one is

competition among d i f f e ren t t ips to trap

incoming walkers and advance in the radia l

d i rec t ion ; the other is t i p s p l i t t i n g or

branching, this slows down the radial growth

of the c luster and accounts for i ts highly

branched nature. In the model described here

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

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

cont ro l l ing growth in th is model are as

fol lows: one starts with a l ine of absorbers

(st icky si tes) located at the bottom of a

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

Part ic les are launched one at a time from a

s i te chosen at random above the locat ion of

the absorbers: they perform a random walk

unt i l they reach an absorber s i te . When this

occurs the s i te reached by the pa r t i c le

becomes occupied and the absorber is moved to

the s i te immediately above i t . In th is way one

grows needles or columns of par t ic les . Since

the sides of these needles are not st icky ( in

the computer simulations whose results are

reported below, the sides of the needles were

taken to be re f lec t ing) incoming par t ic les can

only attach themselves to the top of the

needles and the number of absorbers remains

constant throughout the growth.

Figure 1 shows what a c luster grown in thls

way looks l i ke at various stages of i ts

growth. I n i t i a l l y several needles begin

growing; however, as more and more par t ic les

are added, most absorbers become screened, so

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)

G Rosst / Dtffuslon hrmted aggregatton wtthout branchmg 221

eventual ly , on a s t r ip of f i n i t e width, a

saturat ion 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

character is t ic distance, such that portions of

the c luster separated by a distance larger

than this do not inf luence (screen) each

other 's growth: i . e . , over these length scales

the growth process is local .

The model was introduced in an attempt to

improve our theoret ica l understanding of scale

invar lant d i f fus ive growth. Nonetheless,

physical s i tuat ions where un id i rect ional forces

are responsible for aggregation and d i f fus ion

is the rate l im i t i ng step should lead to

needles of the type described above.

Experimental studies of the accretion of

In teract ing magnetic holes in fe r ro f lu ids in

such a regime are current ly being considered I0

2. COMPUTER SIMULATIONS IN TWO DIMENSIONS

Fai r ly extensive computer simulations using

the rules descrlbed above have been performed.

The density p(y,N/L) and the d i s t r i bu t i on of

absorbers D(y,N/L) have been measured, as

functlons of the height y and of the ra t io N/L

between the number of par t ic les N and the

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

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

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

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

sites where the density changes from 1 to zero)

one has

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

,,L.. L,.., I . , 1.,_.,,,,., J , ,j .IL _.[,.I._ J..,

,,L,.L,_~ I. IL ..... 1,,., j , , . . i k 1.1._ . I . .J

• . L , . L , • I . = L . . . . J _ _ ~ , , . . . . J . , - I . ,

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

D(y,N/L) = y-B (2b)

hold in a larger and larger range of y. The

value of the exponents found from the

slmulations are approximately ( for a deta i led

discussion and error bars see ref . 9)

= .82 (3)

B = 1.82

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

namely, the der iva t ive of the density with

respect to y gives the d i s t r i bu t i on of

absorbers. I t is found that, as the ra t i o N/L

increases, scaling laws of the form

p(y,N/L) = y-m (2a)

Note that , in the scaling regime, eq. ( I )

mpl les ~= ~+I. Typical data for the average

density as a function of the height y are shown

in f igure 2 for various values of the ra t io

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

clusters grown on a s t r ip of width L=128. The

shoulder appearing at large y in the data for

N/L=IO is due to the fact that saturat ion

effects are becoming important.

222 G Ross1 /Dtffuston hrmted aggregatton without branching

3. MEAN FIELD TREATMENT AND COMPUTER

SIMULATIONS IN DIMENSIONS HIGHER THAN TWO

For ordinary DLA i t is possible to wr i te

continuum equations 2 describing the average

growth of the c luster. I t has been argued that

a continuum approximation which retains only

the l inear terms in an expansion in powers of

the density is e f f ec t i ve l y a mean f i e l d

treatment. Numerical solutions of continuum

equations of thls type can be foundl l : they

predict for the f racta l dimension the value

D = d- l . For ordinary DLA there is no evidence

for the existence of an upper c r i t i c a l

dimension where such behavior is reached.

Continuum equations describing the growth

of the c luster can also be wr i t ten down for the

model described here 12. In this case, the

equations can be solved exact ly: in par t i cu la r

one finds for the density

l _ e ( t ) p ( y , t ) ~

p ( y , t ) = 0

for y < I / e ( t )

for y > I / e ( t )

(4)

Here continuum time t plays the same role as

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

function which decreases rapid ly as t

increases. In other words, according to this

mean f i e l d treatment, as t gets larger, scaling

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

(Independent of spat ia l dlmension) are obeyed

in a larger and larger range of y.

Simulations in three and four dimensions

have been done 9 (the basic rule is s t i l l the

one described in the int roduct ion, but the

i n i t i a l seed is now a d-I dimensional surface

of absorbers): i t is found that (within

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

dimensions the exponent m for the density is I :

i . e . , i t coincides with the mean f i e l d

predict ion.

>,,

T o

(',i I

o

I o

• - 1 0 0

! • !

: l " !~,,,

• " ° Q,14 m • °, • ° °, °,,

"%, " . ~

. . . . . . . . , , A , , :, ,

101 1 0 2 1 0 s

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

4. "REAL SPACE RECURSION" TREATMENT

For the model described above i t is

possible to give a theoret ical treatment which

is in many ways s imi lar to real space

renormalization group methods. Thls is in

contrast with ordinary DLA, where so far i t has

proven impossible to formulate a conslstent

treatment of th is type.

I t was stressed above that , as the c luster

grows, absorbers become screened so that the

corresponding needles stop growing. Suppose

that the f ront of the c luster has reached a

certain height ho; one wants to estimate the

distance k that the f ront of the c luster has

to cover in i ts advance in order for a

f rac t ion f of the absorbers to become

screened. Since the density at the f ina l helght

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

f ract lon with respect to the density at h o one

can obtain an estimate for the density

exponen t ~:

G. Rosst /Dtffuston hmzted aggregation wtthout branching 223

- l n f = ( 5 )

In ((ho+k)/h o)

In the scaling regime the value of a obtained

in this way must be independent of h o.

In order to get results out of this scheme

consider an averaged descript ion of the

c luster where the L' absorbers l e f t at height

h o are taken to be equal ly spaced and at the

same helght. Using the fact that the growth

process is loca l , one divides the system in

cel ls with each cel l containing only a small

number P of absorbers. Instead of fo l lowlng

the evolut ion of the whole system one fol lows

the evolut ion of one such cel l as par t ic les

are added. Since the cel l problem involves only

a small number of degrees of freedom, i t is

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

and f lnd the average advance of the t a l l e s t

needle when i t has outgrown the other needles

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

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

stage one considers the same problem for a ce l l

which is P times as wide as that of the

previous stage, but st111 contains only P

absorbers. In this way one can obtain

estimates for k at d i f fe ren t stages of the

growth; one can check that indeed there is

scallng: i . e . , that the ra t io between values of

k re la t i ve to successlve stages is a constant,

and one can obtain an estimate for m which in

pr inc ip le can be s ls temat ica l ly improved (by

using larger values of P). Note that , at each

stage of this procedure, only the par t i cu la r

length scale over which growth is local is

treated.

The procedure out l ined above has been

carr ied out at length for ce l ls containing two,

three and four absorbers. For cel ls with more

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

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

content himself with Monte Carlo sampling over

a su f f i c i en t l y large number of conf igurat ions.

The results found for the exponent m in this

way agree well with those obtained from the

simulatlon (see ref . 9); as expected, the

agreement improves when ce l ls wlth a larger

number of absorbers are used.

Cells of th is type containing only two

absorbers const i tute one of the simplest (non

t r i v i a l ) d i f fus ive growth problems. For this

problem the existence of a simple recurrence

re la t ion between the posslble h is tor ies

leading to a certain conf igurat ion makes i t

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

exact ly. Of course to do th is , one must know

the p robab i l i t y P(M,K) of a par t i c le landing

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

the cel l and K is the di f ference between the

ordinates of the two absorbers. For d l f fus ing

par t ic les P(M,K) is found by solvlng the

Laplace equation for the ce l l and thls has in

general to be done numerically.

While I t is clear that in ordinary DLA the

Laplace dynamics conspires with noise (due to

the f i n i t e slze of the accreting par t ic les) to

y ie ld f racta l behavior, there is l i t t l e

quant i ta t ive understanding of the mechanism

leading to th is resu l t , and much work has

focussedon how small modif icat ions to the

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

anisotropies) change th is behavior. I t is

therefore of some in terest to study how the

exponent~ obtained from a sequence of two

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

considered 9 the class of P(M,K) given by

1 P(M,K) = - (I + (2K/M) a) (6)

2

For a = 2 eq. (6) resembles c losely the

p robab i l l t y P(M,K) that one would obtain

solving the Laplace equation; for a small any

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

for a large P(M,K) w i l l remain very close to

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

224 G Rosst /Dtffuston hrntted aggregation wtthout hranchmg

problem is being solved exact ly) :

= I for a < 1

a + 1 (7) - for a > 1

2a

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

corrections to the power behavior). In other

words, for a < 1 noise is the dominating factor

and the par t i cu la r form of P(M,K) is

unimportant; for 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 th is type w i l l

apply to less contrived growth processes and,

in par t i cu la r , to ordinary DLA. Note that , for

a=2 eq.(7) gives m=.75, which is reasonably

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

5. CONCLUSION

In summary, the model described here

exhib i ts non t r i v i a l scaling behavior and th is

occurs sole ly as a product of screening

resul t ing from competition among d i f fe ren t

absorbers. Simulations in three and four

dimensions give for the exponent cont ro l l ing

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

theoret ica l treatment s imi lar to real space

renormalization group methods used for

equi l ibr ium s t a t i s t i c a l systems. The treatment

provides estimates for the exponent con t ro l l i ng

growth which agree well with the simulation

resul ts; the treatment can also be used to

analyze the in terp lay between d l f fe ren t factors

contr ibut ing to growth in slmple model

s i tuat ions.

ACKNOWLEDGEMENTS

This work was funded in part by the

Department of Energy under grant DE-FGO3-

87ER45288.

REFERENCES

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3. P. Meakin, Bul l . Am. Phys. Soc. 30 (1985) 222.

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

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I0. R. Pynn, pr ivate communicatlon.

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

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