Diffusion in Heterogeneous Media - American Chemical Society

Chapter 3

Diffusion in Heterogeneous Media 1

1

Bret Berner, J. C. Keister , and Eugene R. Cooper

Downloaded by NANYANG TECHNOLOGICAL UNIV on September 26, 2017 | http://pubs.acs.org Publication Date: September 4, 1987 | doi: 10.1021/bk-1987-0348.ch003

Ciba-Geigy Corporation, Ardsley, NY 10502

The solution to three heterogeneous media permeation problems are discussed: (a) nonsteady-state diffusion through oil-water multilaminates, (b) desorption from an oil-water-multilaminate, and (c) steady-state permeation through a membrane with a thin discontinuous impermeable surface coating., In the nonsteady-state, oil-water multilaminates demonstrate a remarkable capacity to separate permeants exponentially based on partition coefficient and diffusion constant. In contrast, in the desorption problem, multilamination has l i t t l e effect. Permeation through films with discontinuous surface coatings depends on the ratio of the coating strip width to the membrane thickness as well as the area fraction of holes. The w i d e s p r e a d a p p l i c a t i o n o f problems i n v o l v i n g d i f f u s i o n i n h e t e r ogeneous media t o b i o l o g i c a l t r a n s p o r t and t o m a t e r i a l s s c i e n c e , i n p a r t i c u l a r , t o polymer s c i e n c e , has a t t r a c t e d t h e a t t e n t i o n o f s c i e n t i s t s f o r over one hundred y e a r s . An e x t e n s i v e body o f l i t e r a t u r e e x i s t s and a good summary o f t h i s f i e l d has been g i v e n by B a r r e r (_1). A t y p i c a l approach has been t o d e v e l o p a p p r o x i m a t i o n methods o r s o l u t i o n s f o r c l a s s e s o f heterogeneous media d i f f u s i o n problems. These methods a p p l i e d t o complex p r a c t i c a l problems a r e o f t e n slow o r l a b o r i o u s and t h e a p p r o x i m a t i o n methods o f t e n l a c k s u f f i c i e n t accuracy. By c o n c e n t r a t i n g on t h e method o f s o l u t i o n and t h e comp l e x i t y o f t h e problem, t h e s u r p r i s i n g p h y s i c a l p r o p e r t i e s and s i m p l i c i t y o f the s o l u t i o n s a r e o f t e n overlooked. In t h i s paper, s o l u t i o n s t o t h r e e i m p o r t a n t heterogeneous d i f f u s i o n problems a r e p r e s e n t e d , and t h e i r i m p l i c a t i o n s f o r t r a n s p o r t i n b i o l o g i c a l systems a r e d i s c u s s e d . While t h e d e t a i l e d methods o f s o l u t i o n s and s u b t l e t i e s a r e p r e s e n t e d i n o t h e r p a p e r s ( 2 - 6 ) , t h e a s y m p t o t i c s o l u t i o n s a r e e a s i l y d e s c r i b e d , and they d e f i n e t h e Imp o r t a n t p h y s i c s o f d i f f u s i o n f o r most o f t h e ranges o f i n t e r e s t . I n p a r t i c u l a r , a) n o n s t e a d y - s t a t e d i f f u s i o n through o i l - w a t e r m u l t i l a m i n a t e s ( 2 , 3 ) ; b) d e s o r p t i o n from o i l - w a t e r m u l t i l a m i n a t e s ( 4 ) ; and 'Current address: Alcon Laboratories, 6201 South Freeway, Fort Worth, T X 76134

0097-6156/87/0348-0034$06.00/0 © 1987 American Chemical Society

Lee and Good; Controlled-Release Technology ACS Symposium Series; American Chemical Society: Washington, DC, 1987.


3.

35

Diffusion in Heterogeneous Media

BERNER ET A L .

c) s t e a d y - s t a t e p e r m e a t i o n through a membrane w i t h a d i s c o n t i n u o u s impermeable s u r f a c e c o a t i n g a r e examined ( 5 , 6 ) . N o n s t e a d y - s t a t e d i f f u s i o n through o i l - w a t e r m u l t i l a m i n a t e s has been used e x t e n s i v e l y as a model f o r the o p t i m a l b i o l o g i c a l response o f a s e r i e s o f congeners w i t h r e s p e c t t o p a r t i t i o n c o e f f i c i e n t (3,7, 8). A c t u a l s o l u t i o n o f t h i s model r e v e a l s the d e f i c i e n c i e s o f m u l t i l a m i n a t e s as a model f o r b i o l o g i c a l t r a n s p o r t , but i t does show the e x t r a o r d i n a r y s e p a r a t i o n f a c t o r s of t h e s e m u l t i l a m i n a t e s i n the n o n s t e a d y - s t a t e regime. The second d i f f u s i o n problem, d e s o r p t i o n from o i l - w a t e r m u l t i l a m i n a t e s , i s c o n s i d e r e d as a model f o r (a) c o n t r o l l e d r e l e a s e from liposomes and l i p i d m u l t i l a y e r s and (b) f o r t r a n s p o r t through b i o l o g i c a l l a m i n a t e s such as s t r a t u m corneum. In c o n t r a s t to n o n s t e a d y s t a t e t r a n s p o r t a c r o s s m u l t i l a m i n a t e s , d e s o r p t i o n from laminates depends o n l y on the outermost l a y e r s . F i n a l l y , we study the e f f e c t o f t h i n d i s c o n t i n u o u s c o a t i n g s on t r a n s p o r t a c r o s s membranes. Permeation through d i s c o n t i n u o u s imperme a b l e s u r f a c e c o a t i n g s i s p a r t i c u l a r l y important f o r (a) the use o f s u r f a c e c o a t i n g s i n p a c k a g i n g f i l m s , (b) p r e d i c t i n g the e f f e c t s o f o c c l u s i v e s k i n c o n d i t i o n i n g a g e n t s , (c) p r o t e c t i v e b a r r i e r f i l m s f o r s k i n , and (d) c o n t r o l l e d r e l e a s e d e v i c e s ( 9 ) . Perhaps e q u a l l y im p o r t a n t , t h i s d i f f u s i o n problem i s the s i m p l e s t case o f a s i n g u l a r i t y a t a r e - e n t r a n t c o r n e r (10,11). In r e c e n t y e a r s , p e r c o l a t i o n t h e o r y and e f f e c t i v e medium t h e o r y (12-14), have b e i n g s u c c e s s f u l l y applied to the polymer and c o n t r o l l e d - r e l e a s e areas. Accurate a p p l i c a t i o n s o f t h e s e t h e o r i e s t o complex heterogeneous systems c o n t a i n i n g s i n g u l a r i t i e s a t c o r n e r s w i l l r e q u i r e a d r o i t treatment of t h e s e s i n g u l a r i t i e s based on u n d e r s t a n d i n g of the s i m p l e r c a s e s (5, J_0). N o n s t e a d y - S t a t e Permeation Through O i l - W a t e r

Multilaminates

The remarkable c a p a b i l i t y of o i l - w a t e r m u l t i l a m i n a t e s t o s e p a r a t e permeants i n the nonsteady s t a t e can be b e s t demonstrated by s t u d y i n g the a s y m p t o t i c s o l u t i o n s o f the s i m u l t a n e o u s d i f f u s i o n e q u a t i o n s ( 2 , 3 ) . An a l t e r n a t i n g s e r i e s o f η o i l and n-1 water l a m i n a t e s ( F i g u r e 1) s e p a r a t e a w e l l - s t i r r e d , i n f i n i t e aqueous s o u r c e compartment of s o l u t e c o n c e n t r a t i o n C and an aqueous r e c e p t o r compartment of z e r o s o l u t e c o n c e n t r a t i o n . W i t h i n the i t h membrane phase, the s o l ute c o n c e n t r a t i o n , obeys F i c k ' s second law, 3C — 3t

2

3 D

C

'

— 3x

e q u a l s D o r D depending on the c o m p o s i t i o n o f t h a t phase. The permeant has an o ^ l - w a t e r p a r t i t i o n c o e f f i c i e n t , P, and the t h i c k ness o f each o i l and water l a m i n a t e i s 1 and 1 , r e s p e c t i v e l y . We s o l v e f o r C ^ ( t ) , the t o t a l amount t r a n s p o r t e d through the l a s t l a m i nate as a f u n c t i o n o f time t . That i s , q

V

where J ^ n - l

i s

t

h

e

f

l

u

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t

n

r

o

t ) =

u

g

L

n

t

n

e

l

a

s

d

t

T

j

b

2n-l

laminate.


( 2 )

CONTROLLED-RELEASE TECHNOLOGY

36

The s o l u t i o n t o t h i s s e r i e s o f d i f f u s i o n e q u a t i o n s demonstrates ( F i g u r e 2) t h e e x t r a o r d i n a r y c a p a b i l i t y o f these o i l - w a t e r m u l t i l a m i n a t e s t o s e p a r a t e permeants based on p a r t i t i o n c o e f f i c i e n t . L e t P be t h e p a r t i t i o n coefficient f o r maximum t r a n s p o r t . For ^^JlAX* * l t r a n s p o r t C ( t ) depends e x p o n e n t i a l l y on, t h e number o f o i l l a y e r s ; f o r j j £ £ » Si depends e x p o n e n t i a l l y on n-1, t h e number o f water l a m i n a t e s . To u n d e r s t a n d t h e o r i g i n o f t h i s e x p o n e n t i a l s e p a r a t i o n , l e t us study t h e c o n c e n t r a t i o n p r o f i l e s a t t i m e s s h o r t e r t h a n t h e l a g t i m e . For s m a l l p a r t i t i o n c o e f f i c i e n t s , ( Ρ )> ^ ^- S tinie f ° sin g l e o i l l a m i n a t e i s s h o r t compared t o t n e time t o change t h e c o n c e n t r a t i o n o f t h e s u r r o u n d i n g water p h a s e s . C o n s e q u e n t l y , one e x p e c t s (a) t h e c o n c e n t r a t i o n p r o f i l e s a c r o s s each o i l b a r r i e r t o resemble s t e a d y s t a t e , ( i . e . , t h e c o n c e n t r a t i o n s h o u l d be a l i n e a r f u n c t i o n o f d i s t a n c e , ) and (b) t h e c o n c e n t r a t i o n i n each water phase s h o u l d almost be c o n s t a n t . In F i g u r e 3, a t y p i c a l c o n c e n t r a t i o n p r o f i l e f o r an n=2 o i l - w a t e r m u l t i l a m i n a t e i s shown t o demonstrate these two features. The assumption o f a s t e a d y - s t a t e p r o f i l e i n t h e o i l l a m i n a t e s and s m a l l c o n c e n t r a t i o n d r o p s i n the water l a y e r s may be used t o d e r i v e a s y m p t o t i c s o l u t i o n s f o r t h e p e r m e a t i o n problem. I t may be shown t h a t (2) f o r i (where t i s t h e time l a g f o r the m u l t i l a m i n a t e ) , M A Y

t

1 8

t

o

t

a

R

p > > P

< < Ρ

t

i e

a

ra


Μ Δ Υ

ρ

ρ

a

n

d

ϋ < ϋ

t

h

e

In an a n a l o g o u s f a s h i o n , f o r τ assumptions about o i l and water b a r r i e r s may be i n t e r c h a n g e d arm i t may be shown that, n-1 Μ

Α

χ

under the^above c o n d i t i o n s , t h e s e p a r a t i o n depends e x p o n e n t i a l l y on D and Ρ t o t h e power o f t h e number o f water b a r r i e r s . An e s t i m a t e o f Ρ » Ρ f o r o p t i m a l t r a n s p o r t , c a n be found from the i n t e r s e c t i o n o f t h e a s y m p t o t i c s o l u t i o n s ( F i g u r e 4 ) . In t h e l i m i t o f l a r g e n, Μ Δ Υ

P

(

MAX »

5

)

f o r a homologous s e r i e s o f compounds, t h a t s t r u c t u r e w h i c h s a t i s f i e s e q u a t i o n 5 w i l l be o p t i m a l l y t r a n s p o r t e d . The a s y m p t o t i c s o l u t i o n s agree q u i t e w e l l w i t h t h e n u m e r i c a l s o l u t i o n o v e r t h e range o f i n t e r e s t ( F i g u r e 4) and thus the v a l u e o f P from e q u a t i o n 5 i s q u i t e -, . , Ϊ MAX reliable. W A V


BERNER ET AL.


1


Source

2

Oil

2 n-1

Water Oil

Ρ Do a

3

D

Ρ D

w

b

a

Oil

Ρ D

0

b

a

0

b

Figure 1. Model for permeation through oil-water multilaminate of 2n-l membranes. (Reproduced with permission from Ref. 3. Copyright 1984 American Pharmaceutical Association.)

Oh

Log Ρ

Figure 2. Log C (t) versus log Ρ for n=2 (A), 3 (B), and 4 (C). t=7142 s and D = / x 10" cm /s. (Reproduced with permission from Ref. 3. Copyright 1984 American Pharmaceutical Association.) 6

2




Oil

Water

Oil

Figure 3. Concentration profile across oil-water multilaminate for n=2, P=10" , and t=71390 s. (Reproduced with permission from Ref. 2. Copyright 1983 Elsevier.) 4

_12 '

1

1

1

1

1

- 4 - 3 - 2 - 1 0 Log Ρ

ι

1

'

2

'

3

•

4

Figure 4. Comparison of the asymptotic and numerical solutions for n=3 and t=7142 s. The intersection of the two asymptotes is P ^ . (Reproduced with permission from Ref. 2. Copyright 1983 Elsevier*)


3.

39


BERNER ET AL.

Although the e x p o n e n t i a l s e p a r a t i o n c a p a b i l i t y p e r s i s t s only f o r times s h o r t e r than t h e l a g t i m e , t , t h e l a g time f o r such l a m i n a t e s i s g r e a t l y extended by the p a r t i t i o n c o e f f i c i e n t (_1 ). That i s , i n the l a r g e η l i m i t f o r < < » T

P

P

M A X

2

l

Ο


r

MAX* PI

2

W where 1 i s the t o t a l t h i c k n e s s o f t h e m u l t i l a m i n a t e . C o n s e q u e n t l y , the l a g time f o r t h e s e l a m i n a t e d systems may be o f t h e o r d e r o f months t o y e a r s , w h i l e t h e l a g time f o r t h e same t h i c k n e s s o f mem brane m a t e r i a l s a r r a n g e d i n a form which i s n o t l a m i n a t e d may be on the o r d e r o f a day. The e x p o n e n t i a l s e p a r a t i o n o f t h e s e m u l t i l a m i n a t e s might be used t o e x c l u d e some permeant almost t o t a l l y f o r a r e l a t i v e l y l o n g time o r t o p u r i f y m a t e r i a l s . P e r m e a t i o n t h r o u g h a s e r i e s o f o i l - w a t e r m u l t i l a m i n a t e s has been used as a model f o r t h e o p t i m a l b i o l o g i c a l r e s p o n s e o f a s e r i e s of congeners w i t h r e s p e c t t o p a r t i t i o n c o e f f i c i e n t ( 3 , 7 , 8 ) . While the n o n s t e a d y - s t a t e d i f f u s i o n model o f m u l t i l a m i n a t e s p r e d i c t s expo n e n t i a l s e p a r a t i o n based on p a r t i t i o n c o e f f i c i e n t t o t h e power o f the number o f b a r r i e r l a m i n a t e s , i . e . , a power much g r e a t e r than one, t y p i c a l b i o l o g i c a l r e s p o n s e c u r v e s e x h i b i t exponents on t h e o r d e r o f one (3,15). The s i m p l e m u l t i l a m i n a t e t r a n s p o r t model i s a poor a p p r o x i m a t i o n f o r t h e treatment o f b i o l o g i c a l r e s p o n s e phenom ena, and the i n c l u s i o n o f shunt pathways t h r o u g h t h e m u l t i l a m i n a t e s might e x p l a i n d i s c r e p a n c i e s between t h e model and t h e d a t a (16,17). D e s o r p t i o n From An O i l - W a t e r M u l t i l a m i n a t e D e s o r p t i o n from an o i l - w a t e r m u l t i l a m i n a t e s h o u l d be an a c c u r a t e model f o r c o n t r o l l e d r e l e a s e from l i p o s o m e s and l i p i d m u l t i l a y e r s and may be h e l p f u l t o u n d e r s t a n d t r a n s p o r t t h r o u g h n a t u r a l l y o c c u r r i n g b i o l o g i c a l l a m i n a t e s such as s t r a t u m corneum. A s y m p t o t i c s o l u t i o n s based upon s i m p l e assumptions about t h e c o n c e n t r a t i o n p r o f i l e may a l s o be used t o u n d e r s t a n d t h e d e s o r p t i o n p r o p e r t i e s . The model f o r d e s o r p t i o n from an o i l - w a t e r m u l t i l a m i n a t e i s shown i n F i g u r e 5. Only the boundary and i n i t i a l c o n d i t i o n s change from t h e e a r l i e r d i f f u s i o n problem. Both s o u r c e and r e c e p t o r com partments a r e now m a i n t a i n e d under s i n k c o n d i t i o n s . At time z e r o , each o i l l a y e r c o n t a i n s i n i t i a l c o n c e n t r a t i o n PC o f s o l u t e and t h e c o n c e n t r a t i o n o f each aqueous l a y e r i s C . To determine t h e amount of m a t e r i a l p e r u n i t a r e a , Ν , t h a t has l e f t t h e l a m i n a t e a t time t , t h e f l u x a t x=o, J ^ , i s i n t e g r a t e d o v e r t i m e , t

N

out

-

2

\ ]

d

t

l

J

I

(


8

)


40

In an a n a l o g o u s manner t o the n o n s t e a d y - s t a t e problem, f o r P /

M

Ζ

, = upper bound

^lower

bound

π + S {1η2

7Γ - π/2

π + S {1η2

- InAy}

- 1ηΑ„} Η

(17)

(18) -3

These bounds and Ζ f o r - 0.1 and Ay = 10 are p l o t t e d i n F i g u r e 11. While the r e s u l t i s sandwiched even a t Ay = 0.1, the a p p r o x i m a t i o n s a r e good to w i t h i n a c o u p l e o f p e r c e n t by Ay = 0.001. As Ay d e c r e a s e s the h o l e chokes up because i t cannot accommodate the corner flow, i . e . , Ζ decreases. In f a c t , Ζ approaches z e r o as —ir/SlnAy. N e v e r t h e l e s s , t h i s approach t o z e r o i s l o g a r i t h m i c and even a t s m a l l a r e a f r a c t i o n s , the r o l e o f f i l m c o n t i n u i t y i s s t i l l important. A l t h o u g h Ζ approaches z e r o f a s t e r than a t l a r g e r A^, the c o n t r i b u t i o n o f Ζ to the reduced f l u x , H, i s p r o p o r t i o n a t e l y l a r g e r at s m a l l Ay. That i s , f o r s m a l l Ay (from e q u a t i o n 15), Η ~ Ζ + Ay


(19)


BERNER ET AL.

b


C =

1

Figure 9. Flow diagram of the diffusion problem. (Reproduced with permission from Ref. 5. Copyright 1986 Elsevier.)

.0I I I I

8 10

.8 1.0

SFigure 10. Ζ and C versus S for 0.3 < A < 0.99. (Reproduced with permission from Ref. 5. Copyright 1986 Elsevier.)

1.0-1

1

.1

.2

1

1

.4

1—I τ ι τ ι

.6 .8 1.0 Si-

1

2

1

1

4

1—ι—|'"^ t τ

6

8 10

Figure 11. Ζ versus S for small A . The actual values are represented by the solid lines. Other symbols represent the lower and upper bounds of Z. (Reproduced with permission from Ref. 5. Copyright 1986 Elsevier.)



46


C o n s e q u e n t l y , f i l m d i s c o n t i n u i t y e f f e c t s can be s i z e a b l e even a t s m a l l A^. A T i l m can appear c o m p l e t e l y c o a t e d even though the f l u x i s v i r t u a l l y u n a f f e c t e d i f S f o r the f i l m i s v e r y s m a l l . Perhaps the s i m p l e s t way t o s t u d y the r o l e o f f i l m c o n t i n u i t y i s t o determine as a f u n c t i o n o f Ay the v a l u e o f S f o r which c o r n e r f l o w changes from b e i n g n e g l i g i b l e to i m p o r t a n t . As a measure o f t h i s t r a n s i t i o n , we p l o t S (Z=0.5) .versus l o g A^ i n F i g u r e 12. Note o v e r f o u r o r d e r s o f magnitude (10 £ Ay £ 1), S (Z=0.5) s h i f t s o n l y from 0.4 to 1.5. That i s , f o r most p r a c t i c a l Ay, f i l m c o n t i n u i t y becomes an important f a c t o r whenever the c o n t i n u o u s r e g i o n s o f the f i l m a r e s m a l l compar ed to the f i l m t h i c k n e s s . To d e s i g n e f f e c t i v e c o a t i n g s which reduce the f l u x t o c l o s e t o A „ , the c o n t i n u o u s r e g i o n s o f the c o a t i n g s s h o u l d be much g r e a t e r than the f i l m t h i c k n e s s . I f the f i l m t h i c k ness i s much g r e a t e r t h a t the s t r i p w i d t h , the c o a t i n g w i l l g e n e r a l l y be t o t a l l y i n e f f e c t i v e . In the i n t e r m e d i a t e S r e g i o n , t h e r e can be s i g n i f i c a n t a l t e r a t i o n s i n the c o n c e n t r a t i o n p r o f i l e w i t h i n the membrane w i t h o n l y s m a l l a l t e r a t i o n s i n f l u x . R e s u l t s i n Ref. 6 have shown t h a t the p a r t i c u l a r d i f f u s i o n p r o b lem o f i n t e r e s t can be modeled as e l e c t r i c a l a n a l o g s , t h a t i s , by measuring v o l t a g e s and c u r r e n t s u s i n g a tank of water t o "mock up" the geometry o f the problem. In t h i s a n a l o g model, c u r r e n t s c o r r e spond to the f l u x e s w h i l e v o l t a g e s c o r r e s p o n d to c o n c e n t r a t i o n l e v els. The a p p a r a t u s c o n s i s t e d o f aluminum p l a t e s i n a tank o f w a t e r , as shown s c h e m a t i c a l l y i n F i g u r e 13. The c u r r e n t and v o l t a g e mea surements were n o r m a l i z e d to produce e x p e r i m e n t a l v a l u e s o f Ζ and C^, w h i l e p o s i t i o n s and l e n g t h s were measured to produce v a l u e s f o r Α^ and S. A comparison o f these e x p e r i m e n t a l r e s u l t s w i t h the p r e v i o u s l y d i s c u s s e d t h e o r e t i c a l r e s u l t s i s shown i n F i g u r e 14. As can be seen, the agreement i s good to w i t h i n 3-5% a c c u r a c y . This analog approach may be the s i m p l e s t method o f s o l v i n g more complex d i f f u s i o n problems i n h e t e r o g e n e o u s media. 5(2=0.5) 2

r

1 . Ε

Ι .2-

.θ -

-J 2 --LOG

1

J -

1

4 CRRF.R

1

I

6 F'RRCT I ON

1

1

8 OF

HOLES)

Figure 12. That value of S for which Z=0.5 versus -log A


I

10



BERNER ET AL.

Figure 14. A comparison of the experimental electrical values with the diffusion theory.

American Chemical Society Library

1155Controlled-Release 16th St., N.W.Technology Lee and Good; ACS Symposium Series; American Chemical Washington. D.C. Society: 20036Washington, DC, 1987.

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References 1. Barrer, R.M. In "Diffusion in Polymers"; Crank, J . ; Park, G.S.; Eds.: Academic Press: New York, 1968: pp. 165-217. 2. Berner, Β.; Cooper, E.R.; J. Memb. Sci. 1983, 14, 139-145.


3 Berner, Β.; Cooper, E.R.; J. Pharm. Sci. 1984, 73, 102-104. 4. Berner, Β.; Cooper, E.R.; J. Controlled Release. 1984, 1, 149-152. 5. Keister, J.C.; Berner, Β.; J. Controlled Release. 1986, 3, 155-166. 6. Keister, J.C. Manuscript in preparation. 7. Penniston, J.T.; Beckett, L.; Bentley, D.L.; Hansch, C. Pharmacol. 1969, 5, 333-341.

Mol.

8. Higuchi, T.; Davis, S.S.; J. Pharm. Sci. 1970, 59, 1376-1383. 9. Kuu, W.Y.; Yalkowsky, S.H.; J. Pharm. Sci. 1985, 74, 926-933. 10. Bell, G.Ε.,; Crank, J . ; J. Chem. Soc. Faraday Trans.II. 1974, /0, 1259-1273. 11. Whiteman, J.R.; Papamichael, N.; J. Appl. Math. Phys. 1972, 23, 655-664. 12. Winterfeld, P.H.; "Percolation and Conduction Phenomena in Dis ordered Composite Media"; University Microfilms International: Ann Arbor, 1982. 13. Mohanty, K.K.; Ottino, J.M.; Davis, H.T.; Chem. Eng. Sci. 1982, 37, 905-924. 14. Sax, J.E.; "Transport of Small Molecules in Polymer Blends: Transport-Morphology Relationships"; University Microfilms Inter national: Ann Arbor, 1985. 15. Kubinyi, H.; J. Med. Chem. 1977, 20, 625-629. 16. Hwang, S.; Owada, E.; Yotsuyanagi, T.; Suhardja, L.; Ho, N.F.H.; Flynn, G.L.; Higuchi, W.I.; J. Pharm. Sci. 1976, 65, 1574-1578. 17. Yalkowsy, S.H.; Flynn, G.L.; J. Pharm. Sci. 1973, 62, 210-217. 18. Cooper, E.R.; Berner, Β.; In "Methods in Skin Research"; Skerrow, D.; Skerrow, C.J.; Eds. John Wiley and Sons: New York, 1985, pp. 407-432. RECEIVED November 6, 1986


Diffusion in Heterogeneous Media - American Chemical Society

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