Diffusion in Heterogeneous Media - American Chemical Society

Ciba-Geigy Corporation, Ardsley, NY 10502. The solution to three ... tists for over one hundred years. .... Copyright 1984 American Pharmaceutical Ass...
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Chapter 3

Diffusion in Heterogeneous Media 1

1

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

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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.

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3.

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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

x

t

n

r

o

t ) =

u

g

L

n

t

n

e

l

a

s

d

t

T

j

b

2n-l

laminate.

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

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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

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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

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

BERNER ET AL.

Diffusion in Heterogeneous Media

1

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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

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

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CONTROLLED-RELEASE TECHNOLOGY

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*)

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

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Diffusion in Heterogeneous Media

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

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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

(

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

8

)

CONTROLLED-RELEASE TECHNOLOGY

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

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

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Diffusion in Heterogeneous Media

BERNER ET AL.

b

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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.)

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

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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

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

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BERNER ET AL.

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

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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.

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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

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