Modeling of diffusion and electric current flow through disordered thin layers (membranes)

~

SOlid State Communications, Vol. 77, No. i, pp. 1-3, 1991. Printed in Great Britain.

0038-i098/9153.00+.00 Pergamon Press plc

MODELING OF DIFFUSION AND ELECTRIC CURRENT FLOW THROUGH DISORDERED THIN LAYERS (MEMBRANES) Mendeleev I n s t i t u t e

A.Yu. T r e t y a k o v ( U n i v e r s i t y ) o f Chemistry and Technology, Moscow 125820, USSR

Institute

S.F.Burlatsky o f Chemical Physics, Moscow 117334, USSR

(Received 15 October 1990 by V . M . A g r a n o v i c h )

Numerical c a l c u l a t i o n s i n t h e framework o f a percolation model a r e performed t o study stationary diffusion and conduction in t h i n l a y e r s of disordered m a t e r i a l s . It is shown t h a t when t h e f r a c t i o n o f unbroken bonds on a square l a t t i c e in two dimensions i s l e s s than 0.25 the diffusion coefficient (conductivity) of s strip cut out of the l a t t i c e depends e x p o n e n t i a l l y on t h e s t r i p w i d t h .

One o f t h e m o s t common approaches to the modeling of diffusion in disordered media is approximation of percolation theory ~,2. It is assumed t h a t d i f f u s i n g p a r t i c l e s are l o c a t e d a t the s i t e s of a r e g u l a r l a t t i c e and t h e p r o b a b i l i t y of t r a n s i t i o n of a p a r t i c l e from one s i t e t o a n o t h e r a t a u n i t time i s expressed e x p l i c i t l y . As a r u l e , o n l y the transitions between the next-neighboring sites are t a k e n into account. If the probability of t r a n s i t i o n between t h e two n e i g h b o r i n g s i t e s is zero, the bond between t h e s e s i t e s i s assumed t o be broken. In most p u b l i c a t i o n s devoted to percolation t h e o r y t h e b e h a v i o r o f such systems i s i n v e s t i g a t e d i n a narrow c o n c e n t r a t i o n s range in t h e v i c i n i t y o f t h e p e r c o l a t i o n t h r e s h o l d , i . e . near such fraction of broken bonds a t which t h e probability for a particle to m o v e a l o n g unbroken bonds t o i n f i n i t y becomes z e r o . Here t h e approximation of scaling is applicable which yields a number o f analytical results~,2, 3. However f o r t h e t h e o r y o f d i f f u s i o n in membranes4 p a r t i c u l a r l y interesting i s t h e case w h e n t h e concentration of unbroken bonds is essentially smaller than their concentration at the p e r c o l a t i o n t h r e s h o l d , w h e n t h e mean size of c l u s t e r s of unbroken bonds is small and d i f f u s i o n t h r o u g h membrane i s p o s s i b l e o n l y due t o seldom o c c u r r i n g fluctuations, w h e n the c l u s t e r size becomes comparable w i t h t h e membrane t h i c k n e s s . I t would be n a t u r a l in this case t o e x p e c t n o n - t r i v i a l dependencies between membrane permeability and t h i c k n e s s 5 , d i f f e r e n t from s c a l i n g laws that describe mass transfer in random

lattices near the percolation threshold. In the present work we study stationary diffusion in two dimensions through a strip cut out of a square lattice. The p r o b a b i l i t y of transition (per unit time) between neighboring sites is ~ (unbroken bond) or zero (broken bond). The concentration of particles on b o u n d a r y s i t e s i s P on one side of the strip a n d z e r o on t h e o t h e r , One can present this problem as a problem of conductivity determination for a strip of random resistors ~ • Resistors with conductivity b correspond to unbroken bonds, those with zero conductivity to broken bonds. The boundary sites at one side of the strip have a constant potential U, those at the opposite side have zero potential. The e f f e c t i v e coefficient of diffusion through the membrane can be expressed via specific conductivity of the corresponding random resistor strip in the following way: D = Z ~ 12 / b, (I) where Z is specific conductivity and l is the period of the lattice. Further we will discuss the conductivity problem bearing in mind that it is equivalent to the diffusion problem. We w i l l measure the lattice conductivity in the units of b, t h a t is let b = I. To c a l c u l a t e the conductivity of strips of random resistors, we u s e d the transfer-matrix method 7 . We made calculations for the strips with the widths from 2 to 14 bonds while the fraction of unbroken bonds was between 0.15 and 0.5 (the percolation threshold for the bond problem on plane square lattice is 0.5). The c a l c u l a t i o n results are listed in Table I. For the unbroken

MODELING OF DIFFUSION AND ELECTRIC CURRENT FLOW Table 1. function

p 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

Specific conduct vity of a s t r i p of of its width n at different fractions (square lattice).

n = 2 2.62X10 4.86X10 7.70X10 1.14X10 1.59X10 2.07X10 2.59X10 3.15X10

n = 4 -2 -2 -2 -1 -1 -~ -~ -~

9 . 5 2 x 1 0 -4 3.69X10

-3

9.90x10 2.30X10 4.62X10 7.85x10 1.26x10 1.83x10

-3 -z -z -z -~ -~

Vol. 77, NO. i

random resistors as of unbroken bonds

n = 6

n = 8

n =

10

3 8 2 x 1 0 -5 2 8 6 x 1 0 -4 1 48x10-3 5 50X10-3 1 54x10-2 3 67x10-2 7 33x10-2 1 27x10-~

2, 35x10 -4 1.37x10-3 5.90x10-3 2.00x10-2 4.90x10-2 9.75x10-z

2. 28x10 -3 1 . 0 5 x 1 0 °2 3.41xlO-Z 8. 00x10 -2

a p

n = 12

n =

14

2.47x10-z 6 . 7 9 x 1 0 -2

1.84x10 -z 5.92x10 -z

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

bonds fraction from 0.4 to 0.5 we made 5x104 iterations, for 0.35 105 iterations, from 0.25 to 0.3 2.5x105 iterations, from 0.15 to 0.2 5x105 iterations. According to our estimates the error never exceeded 8%. The data corresponding to the percolation threshold coincide with those obtained by Derrida and Vannimenus 7 and, as one should expect from scaling considerations 7,8 , are straight lines in double logarithmic coordinates corresponding to power dependence between conductivity and strip thickness. The tangent of the slope that according to scaling must be t /~ = 0.96 (t and are critical indexes) in our case has a slightly different value (0.8) which can be explained by inaccuracy of the scaling hypothesis in the case of narrow strips. (It was shown by Derrida and Vannimenus that strips with the widths from 10 to 40 bonds give much better approximation for the t / ~ value). For the strips with the fraction of unbroken bonds less than 0.25 the correlation radius (characteristic size of clusters of unbroken bonds) which is calculated 2 by R = a~ (where a is a value of order unity and % is the ratio of the difference between percolation threshold and the fraction of unbroken bonds to percolation threshold) has a value of order unity, In this case the region of self-similarity is absent and the scaling approximation is invalid, In fact, a c c o r d i n g to o u r c a l c u l a t i o n s , the dependencies between conductivity and strip width for the fractions of unbroken bonds 0.15 and 0.2 are not linear in double logarithmic coordinates but are linear in semi-logarithmic coordinates (Fig.l). Similar exponential dependence between conductivity and strip width can be obtained as a consequence of a simplistic assumption that the current f l o w s o n l y t h r o u g h " d i r e c t " paths, i.e. o n l y the b o n d s w h i c h a r e p a r a l l e l to the f l o w of the current and assembled in g r o u p s of n b o n d s (n is the s t r i p w i d t h ) are t a k e n into a c c o u n t . It can be e a s i l y

~

01

T--

,

p = 0,50

001

153

16-4 tu 10-5 0

i

i

I

L

h

~

J

2

4

6

8

I0

12

14

n

Fig.

1.

Specific

conductivity

of

a

r a n d o m r e s i s t o r s s t r i p ~ as a f u n c t i o n of its w i d t h n at d i f f e r e n t f r a c t i o n s of u n b r o k e n b o n d s p.

g I

0.I

0.01

2o 10-3

10-4

p = 015

~

10-,5

10-6

i

0

2

n

i

J

4

6

Fig. 2. Comparison of the true conductivity d e p e n d e n c e on the s t r i p width with approximation given by e q u a t i o n (21. A p p r o x i m a t i o n is p r e s e n t e d with dark points.

B

Vol. 77, NO. 1

MODELING OF DIFFUSION AND ELECTRIC CURRENT FLOW

shown t h a t such c o n d u c t i v i t y through the " d i r e c t " paths at the unbroken bonds f r a c t i o n p is p"

= exp( n

ln(p)

) .

(2)

Probably m o r e accurate analytical approximation might be obtained if besides " d i r e c t " paths one takes i n t o account paths w i t h branches.

Comparing this dependence with the c a l c u l a t e d one ( F i g . 2 ) one can see that in the l a t t e r c a s e the tangent of the slope is smaller in the absolute value, the discrepancy decreasing w i t h the decrease of the unbroken bonds f r a c t i o n .

Acknowledgments - Discussions A.Yu. Grosber9 are gratefully acknowledged.

1. D . S t a u f f e r , Phys. Rep. 45. 1 ( 1 9 7 9 ) . 2. J.W.Essam, Rep. P r o g . Phys. 43, 833 (1980). 3. I.M.Sokolov, Uspekhi f i z i c h e s k i k h nauk 150, 221 (1986). 4. A.A.Ovohinnikov, 8.F.Timashov, A.A.Bely, K i n e t i c s of d i f f u s i o n - c o n t r o l l e d processes (Moscow: Khimia), 1986.

5. S . F . B u r l a t s k y , G.S.Oshanin, S.F.Timashov, Khimicheskaya fizika, in print. 6. J . P . S t r a l e y , J . P h y s . C 13, 2991 (1980). 7. B . D e r r i d a , J . V a n n i m e n u s , J . P h y s . A 15, L557 ( 1 9 8 2 ) . 8. C . D . M i t e s c u , M . A l l a i n , E.Guyon, J.Clerc, J . P h y s . A 15, 2523 ( 1 9 8 2 ) .

with

Dr.