RANDOM RAIN SIMULATIONS OF DENDRITIC GROWTH

FRACTALS IN PHYSICS L. Pietronero, E. Tosatti (editors) © Elsevier Science Publishers B.V.,

279 1986

RANDOM RAIN SIMULATIONS OF DENDRITIC GROWTH

B . C A P R I L E , A . C . L E V I and L . L I G G I E R I Universita del

d i Genova, D i p a r t i m e n t o

CNR, V i a Dodecaneso

33,

di F i s i c a and Gruppo N a z i o n a l e d i S t r u t t u r a

16146 Genova,

dell a Materia

Italy

Two-dimensional growth s i m u l a t i o n s are d e s c r i b e d f o r a "random r a i n " m o d e l , where the c a n d i d a t e s f o r s t i c k i n g approach the g r o w i n g c l u s t e r a l o n g random s t r a i g h t l i n e s . Both i s o t r o p i c growth from a c e n t r a l seed and growth on a base l i n e on to which the "atoms" f a l l o b l i q u e l y from a p a r a l l e l l i n e are s t u d i e d . The r e s u l t i n g c l u s t e r s appear to be h i g h l y r a m i f i e d , a l t h o u g h l e s s so than f o r DLA, and t h e i r H a u s d o r f f - B e s i c o v i t c h dimension i s c o n s i d e r e d . I n t e g r o - d i f f e r e n t i a l equations for the l o c a l d e n s i t y a s a f u n c t i o n o f p o s i t i o n and time are a l s o d e r i v e d . F u r t h e r the s i m u l a t i o n i s m o d i f i e d by i n c l u d i n g d i f f e r e n t p h y s i c a l e f f e c t s , namely: 1) e v a p o r a t i o n ; 2) s u r f a c e t e n s i o n ( i n the form o f d i f f e r e n t i a l s t i c k i n g p r o b a b i l i t i e s ) ; 3) heat d i f f u s i o n i n the s o l i d ; 4) s u r f a c e d i f f u s i o n a l o n g the b o r d e r . I n t h i s way a p a r t i a l l y r e a l i s t i c p i c t u r e o f t w o - d i m e n s i o n a l c r y s t a l growth i s a p p r o a c h e d .

1.

INTRODUCTION

sticking,

Although d e n d r i t i c

crystal

growth has been

f o r a l o n g time a model f o r the g e n e r a t i o n ramified

t h i s phenomenon are f a r theory

of

o b j e c t s , microscopic simulations of from abundant and the

has remained to a l a r g e e x t e n t 1

scopic .

Diffusion-limited

Brownian p a t h s c h a r a c t e r i s t i c the r e s u l t i n g model the

a g g r e g a t i o n (DLA)

i s not a l t o g e t h e r

s t a r t i n g from M a r j o r i e 4 and i n v o l v i n g the work o f S u t h e r l a n d ;

B e s i d e s , DLA s i m u l a t i o n s are f a i r l y

mon e t a l .

expensive

computationally

(because o f the c o s t o f random

walk).

s i n c e i n our l a b o r a t o r y

atom-

i s s t u d i e d , we were more

i n growth from vapour than i n growth

5

and two

of the a u t h o r s

for

recently,

. In

the

candidates

for

first

a

then the "atoms"

a g g r e g a t i o n a r e made t o

starting

s i m u l a t i o n s where the " a t o m s " , c a n d i d a t e s

6

both i n two d i m e n s i o n s . I n the f o r m e r , seed i s p l a c e d a t the c e n t r e ;

DLA. perform

has 3 Void

p r e s e n t work two g e o m e t r i e s were c o n s i d e r e d ,

inwards from the c i r c u m f e r e n c e

these r e a s o n s , we chose to

It

has been c o n s i d e r e d i n s i m u l a t i o n s by B e n s i -

from s o l u t i o n , which would be b e s t modeled by

For a l l

new.

a long h i s t o r y ,

it

interested

(RR) mo-

2. THE RANDOM RAIN MODEL

cept i n s p e c i a l c a s e s ) than DLA c l u s t e r s .

surface scattering

"random r a i n "

similarity

w i t h c r y s t a l g r o w t h ; but d e n d r i t i c c r y s t a l s are much more compact and l e s s r a m i f i e d ( e x ­

Finally,

o f DLA. We c a l l

del.

The RR model to bear a mathematical

the

macro­

2 was shown

" r a i n " on t o the g r o w i n g c l u s t e r a l o n g

random s t r a i g h t l i n e s r a t h e r than a l o n g

start

of a large c i r c l e ,

from random p o i n t s and moving a l o n g

random c o r d s . made to f a l l

I n the l a t t e r , on t o a l i n e ,

the

"atoms" are

i n random d i r e c t i o n s ,

Β. Caprile et al.

280

from a p a r a l l e l

meet e i t h e r the base l i n e o r the growing ter.

A r e p r e s e n t a t i v e example o f a

grown i n the former geometry 1 . The RR procedure tures

H . - B . d i m e n s i o n o f RR c l u s t e r s i s t r i v i a l ,

l i n e and t o s t i c k when they clus­

produces r a m i f i e d

Figure

creasing cluster

than near the

t o s t i c k on a branch i s

i s presumably not

fi­

size.

Growth on a l i n e

3.2.

struc­

( a l t h o u g h l e s s so than i n D L A ) , because

the p r o b a b i l i t y

1.86

n a l , and s h o u l d i n c r e a s e s l o w l y to 2 w i t h i n ­

cluster

i s shown i n

D=2. Thus the v a l u e

i.e.

Here D was measured by c o u n t i n g "atoms" w i t h

higher

centre.

i n s t r i p s o f l e n g t h 1 and i n c r e a s i n g width

z,

a c c o r d i n g t o the formula N ( z ) - ^ l z ° \

re­

The

s u l t i s a g a i n D = 1 . 8 6 ± . 0 2 , but a g a i n we expect D to approach 2 f o r l a r g e r

Ξ _ι

clusters.

0


CJ OvJ en

—ι

0

1

2

3

4

5

0

1

2

3

4

5

LN C D I STANCE FROM THE SEED 1 FIGURE 2 E v a l u a t i o n o f the H . - B . d i m e n s i o n a c c o r d i n g to the a l g o r i t h m D ( r ) = l n [ A - | N ( r ) / K ] / I n r where A^ i s the u n i t c e l l area and N ( r ) the number o f o c c u p i e d s i t e s w i t h i n a c i r c l e o f r a d i u s r, f o r the c l u s t e r s shown i n (1) F i g . 1 ; (2) F i g . 3 ; (3) F i g . 4 and (4) F i g . 5 .

FIGURE 1 C l u s t e r o b t a i n e d by s i m p l e RR attachment square

on a

lattice.

3 . HAUSDORFF-BESICOVITCH DIMENSION 3 . 1 . Growth from a seed 4.

The H a u s d o r f f - B e s i c o v i t c h dimension D o f a c l u s t e r was measured by c o u n t i n g "atoms" w i t h i n c i r c l e s o f i n c r e a s i n g r a d i u s , as shown i n The r e s u l t

i s D=1.86±.03.

However,

both

Fig.2.

extensi­

ve s i m u l a t i o n s by M e a k i n 7 and a mathematical gument by B a l l

and W i t t e n ^

indicate

that

the

a_r

INTEGRO-DIFFERENTIAL EQUATIONS The mean r a d i a l

density

^(r,t)

growing from a seed i n the RR model w i t h time a c c o r d i n g to the equation

of a c l u s t e r increases

integro-differential

:

—Τ = Α φ ^ t τ

d < y ( R 2 + r 2 - 2Rr cos y

)"* χ

Random rain simulations of dendritic growth

x exp { Jln[l - v p ( r \ t ) ] dl } where ( R , - ^ )

(1)

a r e p o l a r c o o r d i n a t e s o f the s o u r c e

p o i n t o f the c i r c u m f e r e n c e

o f the l a r g e

and d 1 i s the d i f f e r e n t i a l h o l d s p r o v i d e d the c l u s t e r

circle

path l e n g t h .

Eq.(1)

i s very l a r g e i n com

281

b) heat d i f f u s i o n

i n the s o l i d ;

c) macroscopic c r y s t a l d) s u r f a c e

symmetry

diffusion.

Attempts have been made i n more r e c e n t s i m u ­ l a t i o n s t o account f o r these

effects.

p a r i s o n t o the l a t t i c e s p a c i n g . The e x p o n e n t i a l d e s c r i b e s t h e o p a c i t y o f the c l u s t e r . Eq,

( 1 ) b e l o n g s t o an i n t e r e s t i n g

class of

e v o l u t i o n e q u a t i o n s whose s i m p l e s t i n s t a n c e i s : = [ f ( x ) + g ( x ) i f ( x , t ) ] e x p { - Γvp(x\t)dx'} (2) which can be e a s i l y s o l v e d a n a l y t i c a l l y

i n the

c a s e s : a ) g = 0 ; b) g = 1 , f = 0 . When t - > © o ,γ> tends to a functionvp

whose i n t e g r a l

diverges a t long

d i s t a n c e s . The o p a c i t y o f the o u t e r c l u s t e r becomes i n f i n i t e ,

p a r t s o f the

s o t h a t a t each

point

χ the d e n s i t y y s t o p s i n c r e a s i n g a f t e r a c e r ­ t a i n time t ( t depends on x ) .

5.

EVAPORATION S t i c k i n g without

i n f i n i t e chemical tween f l u i d

evaporation corresponds to

potential

and s o l i d .

and atoms e v a p o r a t e .

differenceΔ^ι be­

I n real

lifeA^

This situation

is finite

i s simulated

FIGURE 3 C l u s t e r o b t a i n e d on a hexagonal l a t t i c e . The s t i c k i n g p r o b a b i l i t y o f an "atom 11 i s f i x e d a s 0 . 0 2 , 0 . 3 o r 1 i f i t s attachment g i v e s t h e f u l ­ f i l m e n t o f a segment, a t r i a n g l e o r a h e x a g o n . a) The e f f e c t s

of surface tension are p a r t l y

c o n s i d e r e d , s i m p l y by f a v o u r i n g attachment

at

by a l l o w i n g random detachment o f atoms from t h e

p o i n t s h a v i n g many n e i g h b o u r s . T a k i n g an under­

periphery

l y i n g hexagonal l a t t i c e , a d i f f e r e n t i a l

stick­

ing p r o b a b i l i t y

higher

o f the c l u s t e r .

along s t r a i g h t l i n e s u n t i l

S u b s e q u e n t l y they move they e i t h e r meet a n ­

i s assumed by a s s i g n i n g

o t h e r p a r t o f the c l u s t e r and s t i c k a g a i n o r

probability

d i s a p p e a r f a r from t h e c l u s t e r .

completes a t r i a n g l e

f o r s t i c k i n g when t h e added "atom" and s t i l l

h i g h e r when

completes a h e x a g o n . By f a v o u r i n g l i n e a r 6.

HEAT PROPAGATION AND DIFFERENTIAL S T I C K I N G

d e r s , t h i s procedure s i m u l a t e s s u r f a c e

The p r e v i o u s s i m u l a t i o n s l a c k many p h y s i c a l

b) L a t e n t

properties tal

t h a t a r e o f importance

i n real

crys­

g r o w t h . Among t h e s e p r o b a b l y t h e most impo£

it

bor­

tension.

heat i s produced when an "atom" s t i c k s ,

and i s p r o p a g a t e d t h r o u g h t h e c l u s t e r ,

with

heat f l u x p r o p o r t i o n a l

differen­

t o temperature

tant are:

ce between n e i g h b o u r i n g c e l l s . E x c e s s heat i s

a) s u r f a c e t e n s i o n ;

radiated

away o u t o f t h e p l a n e .

Β. Caprile et al.

282

H . - B . d i m e n s i o n s are e v a l u a t e d apparent

i n F i g . 2:

the

D tends to i n c r e a s e a s more p h y s i c a l

e f f e c t s are

included.

FIGURE 4 Cluster obtained

introducing

sticking probabilities; s i d e the c l u s t e r ; cluster;

4)

3)

2)

1)

differential

heat t r a n s p o r t

heat r a d i a t i o n

out o f

" s u r f a c e " d i f f u s i o n o f the

in­ the

sticking

"atom".

achieved

in the p r e s e n t

physical

information

t i o n s by a l l o w i n g

the

REFERENCES

s i m u l a t i o n s ; however

the

1 . J . S . L a n g e r , Rev. M o d . P h y s . 52 (1980)

concerned s h o u l d be a g a i n

i s included

until

i n the

simula­

it

finds a

1.

Phys.Rev.Letters

P h y s . R e v . B27 (1983) 5686.

3 . M . T . V o l d , J . C o l l o i d S c i . 18 (1963)

684.

the

site

4.

D.N.Sutherland, 25 (1967) 373.

J . C o l l o i d S c i 22 (1966)

300;

e i t h e r because o f

o r because o f ease to

dis­

5.

D . B e n s i m o n , E.Domany and A . A h a r o n y , L e t t e r s 51 (1983)

Phys.Rev.

1394.

(b).

S i m u l a t i o n s d i f f e r by i n c l u d i n g one or more of the e f f e c t s

(a)

-

(d)

and/or

s t i c k i n g only (Figure

3),

including or add­

i n g h e a t p r o p a g a t i o n and s u r f a c e d i f f u s i o n and e v a p o r a t i o n

(Fig.

5).

The

6 . A . C . L e v i and

L.Liggieri,Surf.Sci.148(1984)

212.

evaporation.

The examples shown a r e c l u s t e r s grown differential

and L . M . S a n d e r ,

47 (1981) 1400;

"atom" to wander a l o n g

where s t i c k i n g i s f a v o u r e d

s i p a t e heat

2. T.A.Witten

sticking.

perimeter o f the c l u s t e r ,

n e i g h b o u r number ( a )

evaporation.

not

crystal

r e l a t e d to d i f f e r e n t i a l d) S u r f a c e d i f f u s i o n

but i n c l u d i n g

symmetry i s

c) M a c r o s c o p i c hexagonal

4)

FIGURE 5 Same a s F i g . 4 ,

(Fig.

corresponding

7.

P.Meakin, private

communication.

8.

R . C . B a l l and T . A . W i t t e n , (1984) 2966.

P h y s . R e v . A 29