TRANSIENT GRATING EXPERIMENTS IN PERCOLATION FRACTALS

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TRANSIENT GRATING EXPERIMENTS IN PERCOLATION FRACTALS

P. Evesque, J. Duran, A. Bourdon

To cite this version:

P. Evesque, J. Duran, A. Bourdon. TRANSIENT GRATING EXPERIMENTS IN PER- COLATION FRACTALS. Journal de Physique Colloques, 1985, 46 (C7), pp.C7-45-C7-49.

�10.1051/jphyscol:1985709�. �jpa-00224957�

Colloque C7, supplément au n°10, Tome 46, octobre 1985

TRANSIENT GRATING EXPERIMENTS IN PERCOLATION FRACTALS

P. E v e s q u e , J . Duran and A. Bourdon

Laboratoire d'Optique de la Matiere Condensee, UA 800 du C.N.R.S., Tour IS, 4 -place Jussieu, 75230 Paris Cedex OS, France

R é s u m é

Nous p r é d i s o n s l e s r é s u l t a t s d e s e x p é r i e n c e s de r é s e a u t r a n s i t o i r e sur un s y s t è m e d é s o r d o n n é en u t i l i s a n t le concept de l ' e s p a c e f r a c t a l et la t h é o r i e de la p e r c o l a t i o n . Nous m o n t r o n s , en p a r t i c u l i e r que le r é s u l t a t de c e s e x p é r i e n c e s dépend du r a p p o r t e n t r e l ' i n t e r f r a n g e du r é s e a u t r a n s i - t o i r e et de la longueur de c o r r é l a t i o n du d é s o r d r e . Ainsi, une e x p é r i e n c e effectuée avec un i n t e r f r a n g e plus petit que la longueur de c o r r é l a t i o n r e f l é t e r a l e s p r o p r i é t é s de l ' e s p a c e f r a c t a l de la diffusion à t r a v e r s une dépendance a n o r m a l e du t e m p s de déclin en fonction de l ' i n t e r f r a n g e du r é s e a u .

A b s t r a c t

Using both the concept of the f r a c t a l space and the p e r c o l a t i o n t h e o r y in t h e c r i t i c a l region we p r e d i c t t h e behaviour of the t r a n s i e n t grating e x p e r i m e n t s in the two e x t r e m e c a s e s when the i n t e r f r i n g e i s much l a r g e r or s m a l l e r than the c o r r e l a t i o n length of the d i s o r d e r . In p a r t i c u l a r we show that when p e r f o r m e d in the s m a l l i n t e r f r i n g e l i m i t , the r e s u l t of these e x p e r i m e n t s will d i r e c t l y reflect the p r o p e r t i e s of the f r a c t a l space of the diffusion through an a n o m a l o u s i n t e r f r i n g e dependence of the decay t i m e of the t r a n s i e n t grating.

I - INTRODUCTION

It h a s been d e m o n s t r a t e d r e c e n t l y , both t h e o r e t i c a l l y and e x p e r i m e n t a l l y , that s e v e r a l p r o b l e m s a r i s i n g in d i s o r d e r e d m a t e r i a l s could by fruitfully tackled

through a convenient mapping of the d i s o r d e r e d m a t e r i a l on a p e r c o l a t i o n model /Z, 5, 6 / . The main advantage of such a p r o c e d u r e c o n s i s t s in allowing to d e s c r i b e m o s t of the dynamic p r o p e r t i e s using the s p e c t r a l exponent d defined by Alexander and Orbach / l / . As pointed out by t h e s e a u t h o r s , this exponent c o n t r o l s as well the d e n s i t y of s t a t e s of the infinite c l u s t e r at the p e r c o l a t i o n t h r e s h o l d as the k i n e t i c s of a r a n d o m walk p r o c e s s on this c l u s t e r . Consequently a s far a s the c h a r a c t e r i s - tic length scale of the e x p e r i m e n t will be s m a l l e r than the c o r r e l a t i o n length of the d i s o r d e r , any d y n a m i c p r o p e r t y of a r a n d o m s y s t e m will be governed by the s p e c - t r a l exponent d . F o r l a r g e r e x p e r i m e n t a l length- s c a l e s , on the other hand, de Gennes / 4 / and Gefen et al. / 8 / p r e d i c t e d a c l a s s i c a l behaviour.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985709

C7-46 JOURNAL DE PHYSIQUE

F r o m the e x p e r i m e n t a l standpoint, the s o - c a l l e d t r a n s i e n t g r a t i n g technique h a s been proved repeatedly to provide a convenient and d i r e c t insight i n the diffu- sion p r o p e r t i e s of the investigated m a t e r i a l s 1 7 , 3 / . B a s i c a l l y i t c o n s i s t s in i n t e r - secting two c o h e r e n t and simultaneous l a s e r p u l s e s a t an angle 8 in a s a m p l e . In t u r n the r e s u l t i n g sinusoYdal i n t e r f e r e n c e p a t t e r n d i f f r a c t s a t h i r d delayed probe b e a m which provides an e s t i m a t e of the r e s i d u a l c o n t r a s t of the t r a n s i e n t grating a t any d e s i r e d moment a f t e r the exciting pulse. A s will be reminded s h o r t l y a f t e r - w a r d s , it i s quite intuitively understood t h a t any diffusion p r o c e s s will tend to f i l l in the z e r o s of the initially printed grating. Also the t i m e dependence of the intensity of the diffracted b e a m will l a r g e l y depend on the i n t e r f r i n g e A of the grating which t u r n s out to play the r o l e of the c h a r a c t e r i s t i c length of the m e a s u - r e m e n t .

T h i s w o r k i s aimed a t predicting the extinction c h a r a c t e r i s t i c of the f i r s t o r d e r of the diffracted probe b e a m in a t r a n s i e n t grating e x p e r i m e n t on a d i s o r d e - r e d s y s t e m which c a n be mapped on a percolation model.

I1 - TRANSIENT GRATING EXPERIMENTS ON A PERCOLATION FRACTAL

MODEL

A complete a n a l y t i c a l solution to the t r a n s i e n t g r a t i n g e x p e r i m e n t i n an homo- geneous diffusing medium h a s been put f o r t h by Salcedo et al. /10/. However,*

keeping in mind that our purpose i s t o r e l a t e d i s o r d e r and t r a n s i e n t g r a t i n g e x p e - r i m e n t s , we propose an a l t e r n a t i v e d e m o n s t r a t i o n to the b a s i c r e s u l t of t h i s study.

We c o n s i d e r the c h a r a c t e r i s t i c t i m e of the extinction of the f i r s t o r d e r diffracted probe b e a m which c o r r e s p o n d s to the t i m e needed by a r a n d o m walking excitation t o t r a v e l on a length s c a l e of A . _If 5 s t a n d s f o r the length of an e l e m e n t a r y jump, the number of n e c e s s a r y s t e p s will be proportional to A / g and t h i s quantity i s r e l a t e d to the t i m e T needed by the r a n d o m w a l k e r to execute this t r i p v i a a c l a s - s i c a l equation given by Alexander and Orbach / l /

Where To is the e l e m e n t a r y jump t i m e . So that we find :

Let u s t u r n now to the c o n s i d e r a t i o n of a t r a n s i e n t grating e x p e r i m e n t in a medium which exhibits bond o r (and) s i t e d i s o r d e r . In such a s y s t e m the jumps will b e e i t h e r possible o r not, depending on the pairs of sites c o n s i d e r e d s o t h a t i t i s expected t h a t the p r o b l e m will not be controled by the single A p a r a m e t e r but a l s o by a c o r r e l a t i o n length of d i s o r d e r 5 . T h i s p a r a m e t e r 5 may e i t h e r d e s c r i b e the c h a r a c t e r i s t i c length of the d i s o r d e r in the infinite c l u s t e r o r the length of the l a r g e s t likely finite c l u s t e r s . Considering the n a t u r e of the p r o b l e m , we a r e now led t o introduce two l i m i t c a s e s following an a p p r o a c h s i m i l a r t o t h e one d i m e n - sional c a s e / 3 / . A s we show in the following the solution of the p r o b l e m may be d e r i v e d e a s i l y in what we c a l l the s m a l l and the l a r g e i n t e r f r i n g e l i m i t s which a r e n a t u r a l l y defined a s

s m a l l i n t e r f r i n g e l i m i t

A << 5 l a r g e i n t e r f r i n g e l i m i t : A >>

In the following we s h a l l examin s u c c e s s i v e l y t h e s e two l i m i t c a s e s .

In t h i s p a r a g r a p h we will d e a l e s s e n t i a l l y with the diffusion of the excitation on the infinite c l u s t e r which i m p l i e s that the s y s t e m i s above the percolation t h r e s - hold. Be c c the c r i t i c a l concentration of the percolation p r o c e s s and c the con- c e n t r a t i o n which c o r r e s p o n d s to the r e a l s y s t e m when mapping i t on percolation model. We define A c a s t h e r e l a t i v e deviation of the c r i t i c a l concentration a c c o r d i n g t o :

C - C c

A c = -

C

c

The percolation t h e o r y p r e d i c t s that the probability P that a given s i t e p e r t a i n t o the infinite c l u s t e r i s

E ^{= O} f o r A c < O

Z N A c P f o r o r

dc 1

Where /3 i s the c l a s s i c a l c r i t i c a l exponent of the percolation t h e o r y a s defined by Stauffer / l l / .

We now t u r n to t h e a n a l y s i s of the concentration dependence of the d e c a y t i m e of the diffracted probe b e a m . According to a p r o c e d u r e r e p o r t e d by de Gennes /4/

we note that the infinite c l u s t e r exhibits a local f r a c t a l s t r u c t u r e . The f r a c t a l and

..l

s p e c t r a l d i m e n s i o n s of t h i s peculiar s p a c e a r e r e s p e c t i v e l y D and d /9/ a t l e a s t when the s y s t e m i s c o n s i d e r e d on a s m a l l length s c a l e R ( R < g ) . It h a s been

shown t h a t , under t h e s e c i r c u m s t a n c e s , the mean number of d i s t i n c t visited s i t e s between the origin of the t i m e and t i m e t , i s given by /4/ :

w h e r e a and t stand r e s p e c t i v e l y f o r the length and the t i m e c o r r e s p o n d i n g to an e l e m e n t a r y jump of the excita$on. As h a s been shown by Alexander and Orbach / l / , the s p e c t r a l dimension d l i e s v e r y n e a r 4/3 whatever the dimension of the s p a c e of the percolation i s .

As we mentioned previously, the diffusion p r o c e s s m u s t t u r n out to a c l a s s i c a l one a s soon a s the length s c a l e b e c o m e s l a r g e r than the c o r r e l a t i o n length of the d i s o r d e r 5 . If t h i s o c c u r s , we expect that the s y s t e m will be c o r r e c t l y d e s c r i b e d by E q . (1). If now we want t o achieve a continuity 'relationship between Eq (1) and E q . (4), that i s t o s a y between the m a c r o s c o p i c and the m i c r o s c o p i c r e g i m e s , we a r e compelled t o e x p r e s s the kinetic of the diffusion under the m a c r o s c o p i c r e g i m e through

Where R i s much l a r g e r than 5 and To i s a p a r a m e t e r which m u s t be adjusted in o r d e r to get the continuity between the m i c r o s c o p i c and the m a c r o s c o p i c r e g i m e s . T h i s can be done e a s i l y by considering E q . (4) a t the point when R t u r n s out to be exactly equal to the c o r r e l a t i o n length of d i s o r d e r 5

then t i s equal t o To and we get f r o m Eq. (4) :

&

If we r e a l i z e that the c h a r a c t e r i s t i c t i m e T of the extinction of the f i r s t o r d e r diffracted probe b e a m c o r r e s p o n d s to a diffusion length equal to an i n t e r f r i n g e A ,

we c a l c u l a t e T by r e p l a c i n g A! by A in Eq. (5a) and plugging the To value

C7-48 JOURNAL DE PHYSIQUE

e x t r a c t e d f r o m (5b), s o that we get :

Stauffer /11/ h a s r e p o r t e d that 5 c a n be e x p r e s s e d a s function of the c o n c e n t r a - tion A c via the c r i t i c a l r e l a t i o n s h i p :

Which finally l e a d s to the value of the t r a n s i e n t grating d e c a y t i m e i n the l a r g e

i n t e r f r i n g e l i m i t

- 2 J ( ~ / d - 1 )

T - (*l2 ^{( A c ) with A c > o (8}

Moreover a s the k i n e t i c s of t h e r a n d o m w a l k p r o c e s s i s quite c l a s s i c a l in t h i s m a c r o s c o p i c r e g i m e (A? >> .)f ) the decay law of the f i r s t o r d e r diffracted b e a m should be s i m p l y given by

B) S m a l l interfringe limit

As mentioned e a r l i e r , t h i s c a s e c o r r e s p o n d s to the existence of l a r g e finite c l u s t e r s the length of which being l a r g e r than the i n t e r f r i n g e J'L ( ~ i ~ . 1 ) . As a m a t t e r of f a c t , the complete calculation of the t i m e dependence of the t r a n s i e n t grating s i g n a l t u r n s out t o be quite i n t r i c a t e in t h i s c r i t i c a l r e g i m e , s o that we w i l l u s e v a r i o u s convenient simplifications in the p r e s e n t work. In p a r t i c u l a r , we will c o n s i d e r that only t h o s e c l u s t e r s exhibiting a s p a t i a l expansion l a r g e r t h a t the i n t e r f r i n g e d i s t a n c e w i l l contribute to the t r a n s i e n t grating decay. Another i m p o r - tant point c o n c e r n s the fact t h a t the l a r g e c l u s t e r s will, generally, c o v e r s m o r e than a single i n t e r f r i n g e s o that we c a n c o n s i d e r h e r e that the e q u i l i b r i u m which will be r e a c h e d on t h e s e l a r g e c l u s t e r s will be independent of t h e i r s i z e and s i m i l a r t o t h i s one r e a c h e d on the infinite c l u s t e r .

A, 5 ( 8 ) A <<S (4

F i g u r e 1

( a ) Small i n t e r f r i n g e l i m i t ( 5 b A ) , (S*= cD). F i n i t e c l u s t e r s of s i z e l a r g e r than e x i s t .

(b) L a r g e i n t e r f r i n g e l i m i t (g 4 A ), (S*= 5 D ^{) .} Only finite c l u s t e r s

of size much s m a l l e r than A e x i s t in the sample

the diffusion can

then only o c c u r in the infinite c l u s t e r i f i t e x i s t s .

p r o b e b e a m w i l l be m o s t l y governed by a typical c l a s s of c l u s t e r s with s i z e A

T h e c h a r a c t e r i s t i c t i m e of t h e d e c a y T will be e s t i m a t e d by calculating the t i m e needed by a r a n d o m w a l k e r t o e x p l o r e one of t h e s e t y p i c a l c l u s t e r s .

As e a c h finite c l u s t e r i s m-apped on a f r a c t a l s p a c e with f r a c t a l and s p e c t r a l d i m e n s i o n s r e s p e c t i v e l y D and d . the kinetics of the diffusion p r o c e s s i s given by Eq. (4), s o that we get the c h a r a c t e r i s t i c t i m e T f r o m

d

which g i v e s :

111. C ONC LUSION

As shown i n t h e preceding p a r a g r a p h , the study of the diffusion i n a d i s o r d e -

r e d m e d i u m through a t r a n s i e n t g r a t i n g e x p e r i m e n t involves two different r e g i m e s . The f i r s t one, which c o r r e s p o n d s t o a m a c r o s c o p i c r e g i m e w i l l o c c u r when the i n t e r f r i n g e A i s l a r g e r than the c o r r e l a t i o n length of the d i s o r d e r 5 . In t h i s c a s e t h e c h a r a c t e r i s t i c d e c a y t i m e of t h e g r a t i n g w i l l s c a l e through t h e c l a s s i c a l law ( T f 1 2 ) .

Now, if the i n t e r f r i n g e i s r e d u c e d t i l l i t b e c o m e s s m a l l e r than t h e c o r r e l a t i o n length of the d i s o r d e r 6 , one should expect a c r o s s - o v e r between t h e p r e v i o u s l y mentioned m a c r o s c o p i c r e g i m e through a c r i t i c a l one. T h i s l a s t r e g i m e i s c h a r a c -

t e r i z e d by a d e c a y t i m e of the d i f f r a c t e d probe b e a m which w i l l s c a l e u n c l a s s i c a l l y with the i n t e r f r i n g e d i s t a n c e .

Achnowledgements

The a u t h o r s a r e indebted t o D r . Daoud and P r o f . P . G. d e Gennes f o r s t i m u - lating d i s c u s s i o n s .

R e f e r e n c e s

/ l / Alexander, S. and O r b a c h , S. J. de Physique L e t t r e s par is)^ (1982) L-625 / 2 / A r g y r a k i s , P.and Kopelman, R., J.Chem. P h y s . & (1977) 3301

/ 3 / Bourdon, A . , D u r a n , J . , P e l l k , F. and d e V i r y , D . , Phys. Rev. B 30 (1984) 7105

/4/ d e G e n n e s , P. G , C . R . Acad. S c . P a r i s 296 (1983) 881 /5/ E v e s q u e , P . , J , d e Physique ( p a r i s ) (1 983) 12 17

/6/ E v e s q u e , P . and D u r a n , J . , J. C h e m . P h y s . 80_ (1984) 3016

/7/ F a y e r , M. D . , Picosecond Phenomena 111, Springer s e r i e s i n C h e m i c a l P h y s i c s 23, ed. F. Shafer Springer V e r l a g B e r l i n 1982

/8/ G e f e n , Y . , Aharony, A . , A l e x a n d e r , S . , P h y s . Rev. L e t t . 0 (1983) 77 /9/ R a m m a l , R. and Toulouse, G, , J , de Physique L e t t r e s ( p a r i s ) 44 (1 983) L1 3 / 1 0 / ~ a l c e d o , J. R . . Siegman, A. E . , Dlott, D, D, and F a y e r , M. D. , P h y s . Rev.

L e t t . 41 (1978) 131

/l l / s t a u f f e r , D. , P h y s , Rev. 4 (1979) 1