Controlled Release Pesticides

2 Structural and Chemical Factors Controlling the Permeability of Organic Molecules through a Polymer Matrix

Downloaded by PURDUE UNIVERSITY on August 5, 2013 | http://pubs.acs.org Publication Date: June 1, 1977 | doi: 10.1021/bk-1977-0053.ch002

C. E. ROGERS Department of Macromolecular Science, Case Western Reserve University, Cleveland, Ohio 44106 The use o f polymeric m a t e r i a l s in c o n t r o l l e d r e l e a s e application systems almost always i n v o l v e s c o n s i d e r a t i o n o f the solubility and diffusivity of the active agent in the polymer m a t r i x . The c o n f i g u r a t i o n of the r e l e a s e system, be it r e s e r v o i r - t y p e o r a simple d i s p e r s a l o f d i s s o l v e d agent (we exclude any specific d i s c u s s i o n of polymer degradation r e l e a s e systems), is of little s i g n i f i c a n c e in this c o n t e x t . In certain systems there may be a rate-controlling step r e l a t e d t o interfacial processes which a f f e c t the attainment o f e q u i l i b r i u m across a reservoir-membrane phase boundary apart from normal boundary l a y e r phenomena. Except f o r such uncommon situations, the r e l e a s e r a t e characteristics of the system will reflect the diffusion r a t e s , s o l u t i o n levels, and r e s u l t a n t g r a d i e n t s o f d i s s o l v e d and diffusing agent w i t h i n the controlling polymer membrane. The inherent nature of t r a n s p o r t processes in polymeric media can be e x p l a i n e d and p r e d i c t e d in terms o f e s t a b l i s h e d models, t h e o r i e s , and phenomenological o b s e r v a t i o n s . There is an e x t e n s i v e literature which d e a l s w i t h the solution, diffusion, and permeation o f low molecular weight gases, vapors, liquids, and i o n s in polymer f i l m s (1-6). U n f o r t u n a t e l y , there a r e relatively few s t u d i e s i n v o l v i n g the t r a n s p o r t characteristics of l a r g e r o r g a n i c molecules in polymers. T h i s , in l a r g e p a r t , is due t o the experimental difficulties i n v o l v e d in such i n v e s t i g a t i o n s . P e r t i n e n t data relating t o the solubility and m i g r a t i o n of l a r g e r molecules in polymers can be gleaned from t h e literature d e a l i n g w i t h c o n t r o l l e d r e l e a s e systems (7-16), encapsulation (17-23), and plasticizer technology. The nature o f s o l u t i o n , diffusion, and permeation behavior in all membrane systems is common i n s o f a r as those processes and other p r o p e r t i e s of the m a t e r i a l s a r e governed by the p h y s i o chemical composition and s t r u c t u r e of the components under g i v e n environmental c o n d i t i o n s . The phenomena o f c o n t r o l l e d r e l e a s e are s i m i l a r ( o f t e n identical) t o those i n v o l v e d i n plasticizer technology, c o a t i n g s technology, environmental r e s i s t a n c e of polymers, and r e l a t e d areas of polymer technology. The e x t e n s i v e 17

In Controlled Release Pesticides; Scher, H.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.


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i n v e s t i g a t i o n s i n those areas can serve as both t h e o r e t i c a l and p r a c t i c a l guides i n the development of c o n t r o l l e d r e l e a s e systems. An important c o n s i d e r a t i o n l e a d i n g to an e l u c i d a t i o n of s o l u t i o n and d i f f u s i o n behavior i n many c o n t r o l l e d r e l e a s e a p p l i c a t i o n s i s that the polymeric m a t r i x o f t e n i s m o d i f i e d upon exposure to the a p p l i c a t i o n environment. Examples of common m o d i f i c a t i o n s are changes i n polymer s t r u c t u r e and morphology due to changes i n temperature or pH and changes i n polymer d e n s i t y and homogeniety of s t r u c t u r a l packing due to s w e l l i n g caused by an imbided e n v i ronmental s p e c i e s such as water. Other major c o m p l i c a t i o n s a f f e c t i n g c o n t r o l l e d r e l e a s e systems, such as boundary l a y e r phenomena and/or membrane f o u l i n g , are w e l l recognized. Consequently, i t i s a c h a l l e n g e to e i t h e r a l l e v i a t e the seriousness of these e f f e c t s o r , perhaps even b e t t e r , to t u r n them to some advantage. In any case, an understanding of the nature of s o l u t i o n and d i f f u s i o n processes and t h e i r dependence on penetrant, polymer, and environmental f a c t o r s i s a key element i n the design of v i a b l e polymer membrane c o n t r o l l e d r e l e a s e systems. With these concepts i n mind, we w i l l d i s c u s s aspects of the g e n e r a l dependence of d i f f u s i o n and s o l u t i o n processes on c e r t a i n s t r u c t u r a l f a c t o r s . We wish to emphasize a few u n d e r l y i n g p r i n c i p l e s and phenomena common to many types of a p p l i c a t i o n and to suggest avenues to approach the s o l u t i o n of remaining problem areas. In p a r t i c u l a r , we want to c a l l a t t e n t i o n to aspects of the g e n e r a l c o n c e n t r a t i o n dependence of s o l u t i o n / d i f f u s i o n p r o cesses which seem to be both neglected (or misunderstood) and promising f o r the development of e f f e c t i v e c o n t r o l l e d r e l e a s e systems. F a c t o r s A f f e c t i n g Membrane P e r m e a b i l i t y The mechanism of t r a n s p o r t of a penetrant agent w i t h i n a polymeric m a t e r i a l can be c l a s s i f i e d i n terms of the presence or absence of a) a gross porous (or h i g h l y swollen) s t r u c t u r e and b) f i x e d i o n i c charges. In the f i r s t case, the t r a n s p o r t behavior i s c l o s e l y governed by the r e l a t i v e molecular dimensions of the penetrant as compared to the pore diameter and c o n n e c t i v i t y . T h i s mechanism i s dominant i n d i a l y s i s and u l t r a f i l t r a t i o n membrane systems. The e l e c t r o s t a t i c i n t e r a c t i o n s between the penetrant molecule and the polymer-fixed i o n i c groups, as modif i e d by the a c t i o n of other sorbed s p e c i e s , i s a dominant f a c t o r i n i o n exchange and some b i o l o g i c a l membrane systems. In the absence of the above o v e r r i d i n g f a c t o r s , the most common mechanism of t r a n s p o r t i s the s o l u t i o n - d i f f u s i o n model. In t h i s case, the o v e r a l l t r a n s p o r t process depends on a m u l t i tude of f a c t o r s r e l a t i n g to such aspects as the s i z e , shape, composition, and c o n c e n t r a t i o n of the penetrant and the polymer composition, s t r u c t u r e , and morphology. Although the number of f a c t o r s , and i n t e r a c t i o n s between f a c t o r s , may l e a d to complex r e l a t i o n s h i p s to d e s c r i b e the s o l u t i o n - d i f f u s i o n behavior, i t i s



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t h i s complexity which can a l l o w us to d e s i g n and develop unique and u s e f u l membrane systems. Regardless of the s t a t e of aggregation of the m i x t u r e , the d i f f u s i o n f l o w or f l u x , J , of a substance i n a mixture w i t h other substances can be d e f i n e d as the amount passing d u r i n g u n i t time through a s u r f a c e of u n i t area normal t o the d i r e c t i o n of f l o w . I n many cases of i n t e r e s t f o r r e l e a s e a p p l i c a t i o n s , i t i s neces sary t o r e a l i z e t h a t the t o t a l f l u x may be a combination of a pure d i f f u s i v e f l u x and other f l u x e s due to mass f l o w , f l o w through d e f e c t s , or f l o w due t o e x t e r n a l l y imposed s t r e s s or other d r i v i n g f o r c e . S u i t a b l e c o r r e c t i o n s to account f o r the frame of r e f e r e n c e f o r such systems have been d i s c u s s e d (2,3,5). The concepts and procedures of i r r e v e r s i b l e thermodynamics are g e n e r a l l y a p p l i c a b l e . These c o r r e c t i o n s seldom have been made i n p r a c t i c e so t h a t the i n t e r p r e t a t i o n of most p u b l i s h e d data i s s u b j e c t to r e a p p r a i s a l i n terms of i n t e r f e r e n c e s which are made on the b a s i s of concepts and t h e o r i e s f o r d i f f u s i v e mechanisms of transport. The f l u x i n the s t e a d y - s t a t e (or pseudo-steady-state over r e l a t i v e l y short time p e r i o d s ) f o r c o r r e c t e d pure d i f f u s i v e f l u x can be expressed as: J - -DK(dc/dx) = -D*(dlna/dlnc)(dc/dc)(dc/dx)

(1)

where DK = J?, the p e r m e a b i l i t y constant. The d i f f u s i o n c o e f f i c i e n t D* i s a measure of the average m o b i l i t y of penetrant mole c u l e s w i t h i n the d i f f u s i o n medium. The d i s t r i b u t i o n f a c t o r (solubility coefficient) Κ = (dc/dc)

(2)

i s a measure of the p a r t i t i o n i n g of penetrant between a polymer s o l u t i o n phase, "c, and an ambient penetrant phase, c. The l a t t e r phase may be the phase e x t e r n a l to the membrane, the enclosed penetrant r e s e r v o i r , or d i s p e r s e d phase-separated penetrant " m i c r o - r e s e r v o i r s " w i t h i n the polymeric m a t r i x . The Nernst-type d i s t r i b u t i o n f u n c t i o n i s a thermodynamic parameter c h a r a c t e r i z i n g the ρenetrant-polymer system which can be a f u n c t i o n of pressure or c o n c e n t r a t i o n as w e l l as temperature. The apparent F i c k ' s Law d i f f u s i o n c o e f f i c i e n t , D, i s the product of the non-negative m o b i l i t y f a c t o r and a thermodynamic f a c t o r r e l a t e d to the i d e a l i t y of the penetrant-polymer m i x t u r e : D = D*(dlna/dlnc)

(3)

where a i s the a c t i v i t y and "c i s the c o n c e n t r a t i o n of penetrant i n s o l u t i o n i n the polymer phase. When the mixture i s a thermodynamically i d e a l s o l u t i o n , dlna/dlric i s u n i t y . However, most mixtures e x h i b i t d e v i a t i o n s from i d e a l behavior of a magnitude


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p r o p o r t i o n a l t o the c o n c e n t r a t i o n . The thermodynamic term a l s o can be expressed as


d l n a / d l n c - 1 + (dlny/dlnc)

(4)

where γ i s the a c t i v i t y c o e f f i c i e n t . I f , f o r any reason, the thermodynamic term becomes negative i n s i g n , d i f f u s i o n occurs a g a i n s t t h e c o n c e n t r a t i o n g r a d i e n t . The presence o f u n s t a b l e phase r e g i o n s w i t h i n the d i f f u s i o n medium, due, f o r example, t o sudden l o c a l changes i n a p p l i e d s t r e s s o r temperature, w i l l l e a d to t h i s " u p h i l l " d i f f u s i o n behavior which i s phenomenologically s i m i l a r t o " a c t i v a t e d " t r a n s p o r t i n b i o l o g i c a l systems. The c o n c e n t r a t i o n dependence o f D i s seen t o a r i s e from two sources: the c o n c e n t r a t i o n dependence of the m o b i l i t y , u s u a l l y the dominant f a c t o r , and a c o n c e n t r a t i o n dependence a t t r i b u t e d t o the n o n - i d e a l nature of the system, per se. These f a c t o r s can be i n t e r p r e t e d and assessed i n terms of t h e o r i e s o r models concerned w i t h polymer s o l u t i o n behavior, the mode of s o r p t i o n , molecular f r i c t i o n f a c t o r s , c h a i n segmental m o b i l i t y , f r e e volume concepts, etc. C e r t a i n of these aspects w i l l be d e s c r i b e d i n a l a t e r section. I t i s w e l l t o note t h a t the d e f i n i t i o n s of Κ and D do n o t impose any r e s t r i c t i o n s as t o t h e f u n c t i o n a l dependence of t h e parameters on experimental c o n d i t i o n s . I n t e r p r e t a t i o n o f t h e s i g n i f i c a n c e o f the parameters must be made i n l i g h t o f r e f i n e d and r e a l i s t i c t h e o r i e s o r models which account f o r the nature of the system t o i n c l u d e the dominant t r a n s p o r t mechanism, frame of reference c o r r e c t i o n s , boundary l a y e r e f f e c t s , and other r e l e v a n t concepts. For engineering design purposes, the parameters can serve as phenomenological c o e f f i c i e n t s which d e s c r i b e the system under the g i v e n c o n d i t i o n s without need f o r c o r r e c t i o n o r i n t e r pretation. A g e n e r a l mechanism t o d e s c r i b e the m i g r a t i o n o f a penetrant molecule through a medium v i s u a l i z e s the process as a sequence of u n i t d i f f u s i o n steps or jumps under the i n f l u e n c e of a chemical p o t e n t i a l (concentration) g r a d i e n t by a cooperative a c t i o n o f the surrounding complex of molecules d u r i n g which the penetrant molecule passes over a p o t e n t i a l b a r r i e r s e p a r a t i n g one s i t e from the next. The magnitude o f the d i f f u s i o n c o e f f i c i e n t i s equated as a product o f a constant times the p r o b a b i l i t y o f a s u c c e s s f u l jump. That p r o b a b i l i t y can be r e l a t e d t o the ease o f h o l e forma t i o n which depends on t h e r e l a t i v e m o b i l i t i e s of penetrant mole c u l e s and polymeric c h a i n segments as they are a f f e c t e d by changes i n s i z e , shape, c o n c e n t r a t i o n , and i n t e r a c t i o n between components. Another major f a c t o r a f f e c t i n g t r a n s p o r t i s the number, s i z e , and d i s t r i b u t i o n of d e f e c t s t r u c t u r e s , such as v o i d s , c a p i l l a r i e s , and domain boundaries, w i t h i n the polymer m a t r i x . The inherent morphological nature of the polymer, coupled w i t h the p a r t i c u l a r sample f a b r i c a t i o n and p r o c e s s i n g c o n d i t i o n s , determine the de t a i l e d defect structure.


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I n r e l a t i v e l y homogeneous polymers the dependence of permea b i l i t y on the a n t i c i p a t e d c o n t r o l l i n g f a c t o r s can be s t a t e d q u i t e g e n e r a l l y . For example, the temperature dependence of d i f f u s i o n and s o l u b i l i t y over reasonable temperature ranges can be represented by the Arhennius-type equations D • D exp(-E /RT)

(5)

S - S exp(-AH /RT)

(6)

Q

Q

D

s

where Eq i s the apparent a c t i v a t i o n energy f o r d i f f u s i o n and AH i s the heat of s o l u t i o n . Consequently, any f a c t o r which a c t s to reduce the ease of h o l e f o r m a t i o n f o r d i f f u s i o n can be expected to decrease the o v e r a l l r a t e of permeation. These e f f e c t s a r e q u i t e n o t i c e a b l e i n s t u d i e s of d i f f u s i o n of a s e r i e s of penet r a n t s of i n c r e a s i n g molecular s i z e and shape s i n c e the o v e r a l l t r a n s p o r t process i s extremely s e n s i t i v e to the magnitude and s i z e d i s t r i b u t i o n of " h o l e s " a v a i l a b l e per u n i t time and volume f o r d i f f u s i v e jumps as determined by the i n h e r e n t o r m o d i f i e d polymer c h a i n segmental m o b i l i t i e s . The l o c a l segmental m o b i l i t y or c h a i n s t i f f n e s s may be a f f e c t e d by c h a i n i n t e r a c t i o n s a r i s i n g from hydrogen bonding, p o l a r group i n t e r a c t i o n s , or simple van der Waal's a t t r a c t i o n s . As the number of these groupings per u n i t c h a i n segment l e n g t h i n c r e a s e s , the degree of i n t e r a c t i o n i n c r e a s e s , the segmental m o b i l i t y decreases, and t h e r e f o r e the permeation r a t e a l s o decreases. These e f f e c t s a r e e s p e c i a l l y pronounced f o r the case of symmetrical s u b s t i t u t i o n of p o l a r groups s i n c e the packing of adjacent c h a i n segments i s somewhat f a c i l i t a t e d l e a d i n g to more efficient interactions. Other m o d i f i c a t i o n s ( 1 - 2 ) which serve to decrease c h a i n segmental m o b i l i t y , and t h e r e f o r e decrease permeation, are s u f f i c i e n t l y h i g h degrees of c r o s s l i n k i n g , the presence of s o l i d a d d i t i v e s ( f i l l e r s ) onto which the polymer i s s t r o n g l y adsorbed, and the occurrence of c r y s t a l l i n e domains w i t h i n the polymer i t s e l f . An i n c r e a s e i n d e n s i t y and c r y s t a l l i n e content r e s u l t s m a i n l y i n a corresponding decrease i n s o l u b i l i t y s i n c e c r y s t a l l i n e r e g i o n s are not g e n e r a l l y a c c e s s i b l e f o r s o r p t i o n . However, the conc u r r e n t p e r m e a b i l i t y decrease i s s u b s t a n t i a l l y g r e a t e r , i n d i c a t i n g that the d i f f u s i o n c o e f f i c i e n t a l s o i s decreased, p r e s u mably because of the r e s t r a i n i n g e f f e c t s of c r y s t a l l i n e r e g i o n s on l o c a l c h a i n segmental motion i n a d j o i n i n g n o n - c r y s t a l l i n e r e g i o n s and a more t o r t u o u s path through the m i x t u r e of amorphous and c r y s t a l l i n e domains. The l o c a l c h a i n segmental m o b i l i t y of a polymer i s enhanced by the presence of an added p l a s t i c i z e r , r e s u l t i n g i n a l o w e r i n g of the g l a s s t r a n s i t i o n temperature of the polymer. The permeat i o n of s o l v e n t s i n a polymer i s s i m i l a r i n t h a t the sorbed and d i f f u s i n g s o l v e n t a c t s to " p l a s t i c i z e " the polymeric system.


S


22


The net r e s u l t of the much higher sorbed c o n c e n t r a t i o n s of good s o l v e n t s i n a polymer (a h i g h Κ value) times the attendant p l a s t i c i z i n g a c t i o n which i n c r e a s e s the corresponding d i f f u s i o n c o e f f i c i e n t i s a marked i n c r e a s e i n the o v e r a l l permeation r a t e . T y p i c a l l y , f o r low c o n c e n t r a t i o n s (up to about 10 percent by weight) of a sorbed penetrant, where Henry's Law i s reasonably v a l i d , the d i f f u s i o n c o e f f i c i e n t v a r i e s w i t h c o n c e n t r a t i o n as:


D = D(c=o)exp(ac)

(7)

where D(c=o) i s the e x t r a p o l a t e d v a l u e of D a t zero c o n c e n t r a t i o n and α i s a c h a r a c t e r i s t i c parameter which can be r e l a t e d , f o r example, to the Flory-Huggins i n t e r a c t i o n parameter. For wider ranges of sorbed c o n c e n t r a t i o n s , b e t t e r r e p r e s e n t a t i o n s are ob t a i n e d i n terms of s o l v e n t a c t i v i t y , a: f

D = D(c=o)exp(a a)

(8)

T h i s i n c l u d e s the r e g i o n s of sorbed c o n c e n t r a t i o n where Henry's Law i s no longer obeyed, but r a t h e r the s o r p t i o n f o l l o w s the Flory-Huggins equation or the r e l a t e d e x p r e s s i o n ( 2 , 3 ) : Κ = K(c=o)exp(orc)

(9)

Combination of Equations (8) and (9) f o r the case when σ ap proaches zero leads to Equation ( 7 ) . D e t a i l e d t h e o r i e s have been proposed to r a t i o n a l i z e the observed c o n c e n t r a t i o n depen dence of d i f f u s i o n , mainly i n terms of f r e e volume concepts, and to account f o r the phenomena of penetrant c l u s t e r formation w i t h i n the polymeric m a t r i x . We w i l l consider some of these aspects i n a l a t e r s e c t i o n . I n most i n v e s t i g a t i o n s of d i f f u s i o n and s o l u t i o n i n p o l y mers, the t a c i t assumption has been made t h a t the a c c e s s i b l e r e g i o n s are s t r u c t u r a l l y homogeneous so that d i f f u s i o n can be considered to occur by a s i n g l e a c t i v a t e d mechanism w i t h i n the continuum. R e c e n t l y , however, evidence has been presented f o r the presence of a microporous s t r u c t u r e i n c e r t a i n amorphous polymers below or near t h e i r g l a s s temperature, and i n semic r y s t a l l i n e polymers above t h e i r g l a s s temperature. The d i s t r i b u t i o n of v o i d s i z e and shape, dependent on the manner of membrane p r e p a r a t i o n and f a b r i c a t i o n , may range from submicrovoids of the order of u n i t - c e l l dimensions to p o r o s i t i e s and cleavages of much g r e a t e r dimensions w i t h non-random c o n f i g u r a t i o n s . These v o i d s are to be d i s t i n g u i s h e d from the f r e e volume a s s o c i a t e d w i t h l i q u i d s or amorphous s o l i d s and t h e i r magnitude i s not a thermodynamic q u a n t i t y . I n g l a s s y amorphous polymers as the temperature i s lowered below the g l a s s tempera t u r e , the a c t u a l t o t a l volume occupied by a polymer becomes pro g r e s s i v e l y g r e a t e r than the e q u i l i b r i u m volume of an e q u i v a l e n t l i q u i d . Since segmental m o b i l i t y i s low, t h i s volume d i f f e r e n c e



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Matrix

23

must r e s u l t i n the formation of d i f f e r e n t d e n s i t y regions on the m i c r o s c a l e . The l e s s densely packed r e g i o n s then correspond i n e f f e c t to v o i d s w i t h i n the surrounding more densely packed m a t r i x . This phenomenon may be even more pronounced f o r c e l l u l o s i c polymers which e x h i b i t v e r y low r a t e s of conformational rearrangement due to t h e i r inherent low segmental c h a i n m o b i l i t y . The c o n d i t i o n s of p r o c e s s i n g , such as c a s t i n g temperature, s o l u t i o n composition, and subsequent annealing treatments, w i l l have a marked e f f e c t on the m i c r o s t r u c t u r e of the membrane. In many cases, the r e s u l t a n t s t r u c t u r a l h e t e r o g e n e i t i e s can be cons i d e r e d as v o i d s w i t h i n the terms of the above d i s c u s s i o n . The v o i d content w i l l be c h a r a c t e r i z e d by a magnitude and s i z e d i s t r i b u t i o n which w i l l then change w i t h time as the membrane i s subjected to more extreme environmental c o n d i t i o n s during i t s use. The v o i d d i s t r i b u t i o n i s d i r e c t l y r e l a t e d to the polymer c h a i n conformation s t a t i s t i c s . The e f f e c t of a microporous s t r u c t u r e on the s o l u b i l i t y and t r a n s p o r t p r o p e r t i e s depends on the c o n t i n u i t y of path a f f o r d e d by the d i s t r i b u t i o n of microvoids and on the nature of the penet r a n t contained w i t h i n such v o i d s . The presence of i n t e r c o n nected micropores, s m a l l channels, c r a c k s , or other flaws i n polymer s t r u c t u r e permits convection of penetrant to occur through the medium i n a d d i t i o n to a c t i v a t e d d i f f u s i o n . Such c a p i l l a r y f l o w does not show v e r y pronounced d i f f e r e n c e s f o r v a r i o u s penetrants unless the d i f f u s i n g molecule i s of a dimens i o n comparable w i t h that of the c a p i l l a r y . For the case of a homogeneous d i s t r i b u t i o n of n o n - i n t e r c o n nected microvoids the o v e r a l l r a t e of t r a n s p o r t would be expected to i n c r e a s e somewhat owing to the smaller s t r u c t u r a l packing d e n s i t y a f f o r d e d by the presence of the lower d e n s i t y v o i d r e g i o n s . However, when the cohesive f o r c e s between penetrant molecules are g r e a t e r than the a t t r a c t i v e f o r c e s between penet r a n t and polymer, the incoming penetrant tends to c l u s t e r w i t h i n the polymer. With reference to the o v e r a l l d i f f u s i o n f l u x , a molecule w i t h i n a c l u s t e r g e n e r a l l y w i l l be l e s s mobile than an i s o l a t e d f r e e molecule owing to the a d d i t i o n a l energy r e q u i r e d to break f r e e from the c l u s t e r . A d e t a i l e d d i s c u s s i o n of the dependence of the d i f f u s i o n f l u x on c l u s t e r formation as i t v a r i e s w i t h penetrant c o n c e n t r a t i o n , v o i d content, time, and other f a c t o r s has been presented (2,3). A more fundamental and comprehensive approach to the general problem of d i f f u s i o n i n multicomponent systems i s a f f o r d e d by the theory of i r r e v e r s i b l e thermodynamics. The r a t e of f l o w of a substance i n such a system i s dependent not o n l y on i t s own g r a d i e n t of chemical p o t e n t i a l ( i . e . , c o n c e n t r a t i o n g r a d i e n t ) but a l s o on the g r a d i e n t s of chemical p o t e n t i a l of the other components as w e l l as e x t e r n a l f o r c e g r a d i e n t s ( s t r e s s , e l e c t r i c f i e l d s , temperature, e t c . ) . For systems not f a r removed from e q u i l i b r i u m , t h i s interdependence may be assumed to be l i n e a r .


24


The r e s u l t a n t c r o s s terms have been neglected o r considered n e g l i g i b l e i n almost a l l past i n v e s t i g a t i o n s of d i f f u s i o n . However, these terms c e r t a i n l y are s i g n i f i c a n t i n many cases, such as i n s o - c a l l e d " a c t i v e " b i o l o g i c a l t r a n s p o r t , and t h e r e f o r e should be i n c l u d e d to o b t a i n a more complete understanding of d i f f u s i o n phenomena.


C o n c e n t r a t i o n Dependence of D i f f u s i o n A major f a c t o r which may a f f e c t the performance of con t r o l l e d r e l e a s e systems i s the p r o g r e s s i v e enhancement of d i f f u s i o n r a t e s by the d i s r u p t i o n of the polymer m a t r i x due to s w e l l i n g f o r one cause or another. Related e f f e c t s , to e i t h e r i n c r e a s e or d i m i n i s h the r a t e of t r a n s p o r t , are due to changes i n the inherent m o b i l i t y of the penetrant molecule caused by v a r i a t i o n s i n the mode of s o r p t i o n . These f a c t o r s can be con s i d e r e d under the g e n e r a l concept of c o n c e n t r a t i o n dependence of d i f f u s i o n . As mentioned p r e v i o u s l y , the thermodynamic term ( d l n a / d l n c ) i n equations ( 1 ) , ( 3 ) , and (4) can be assessed d i r e c t l y from the s o l u t i o n theory equation which best r e p r e s e n t s the s o r p t i o n i s o therm f o r the p a r t i c u l a r system. I n most cases, t h i s term i s of r e l a t i v e l y minor s i g n i f i c a n c e as compared to the e f f e c t s on the m o b i l i t y parameter. The m o b i l i t y term, D*, i n equations (1) and (3) can be f u r t h e r d e f i n e d i n terms of two concepts: a) i m m o b i l i z a t i o n of penetrant molecules, and b) the segmental m o b i l i t y of polymer c h a i n s . The f i r s t concept i n v o l v e s c o n s i d e r a t i o n s of the e f f e c t s of d i f f e r i n g modes of s o r p t i o n , such as s p e c i f i c s i t e s o l v a t i o n , chemisorρtion, or c l u s t e r f o r m a t i o n , on the a c t u a l m o b i l i t y of the i n d i v i d u a l penetrant molecules. The other con cept c o n s i d e r s p l a s t i c i z a t i o n and other e f f e c t s of s w e l l i n g , u s u a l l y i n terms of f r e e volume treatments. An understanding of the nature of polymer s o l u t i o n s sug gests t h a t one should a n t i c i p a t e a spectrum of penetrant mole c u l a r m o b i l i t i e s , D t ( c i ) , r e l a t e d to a spectrum of modes of s o r p t i o n determined by the experimental time s c a l e and the i n t e r a c t i o n e n e r g e t i c s of v a r i o u s s o r p t i o n s i t e s i n the system. Penetrant molecules i n v o l v e d i n s p e c i f i c s i t e s o r p t i o n or i n p h y s i c a l c l u s t e r s ( p e n e t r a n t - r i c h domains below l i q u i d n u c l e a t i o n s i z e ) can be considered as l o c a l i z e d i f the r a t e of ex change of molecules between those modes of s o r p t i o n and f r e e l y d i f f u s i n g s p e c i e s i s slower than the r a t e of the u n i t d i f f u s i o n process. I n the s i m p l e s t case, we can consider the d u a l s o r p t i o n mode approximation which d e f i n e s the t o t a l sorbed c o n c e n t r a t i o n , "c, as composed of f r e e l y d i f f u s i n g s p e c i e s of c o n c e n t r a t i o n , "c-, and bound s p e c i e s of c o n c e n t r a t i o n c" . With the assumption that the bound s p e c i e s i s t o t a l l y immobilized (D^ 0 ) , equaβ


2.

Permeability

ROGERS

through

a Polymer

25

Matrix

t i o n (1) may be r e w r i t t e n as: J = -D (dlna /dlnc )(dc /dc)(dc/dc)(dc/dx) f

f

f

f

The e x p e r i m e n t a l l y determined

d i f f u s i o n c o e f f i c i e n t i s then:

D = D (dlna /dlnc )(dc /dc)


f

f

(10)

f

(11)

f

The context of the assumption i s such t h a t the f r e e l y d i f f u s i n g species can be considered t o form an i d e a l s o l u t i o n mixture so that equation (11) can be s t a t e d as simply: D = D (c /ïï) f

(12)

f

Thus any process which tends to l o c a l i z e penetrant molecules w i t h i n the d i f f u s i o n medium w i l l cause t o be l e s s than c" and the e x p e r i m e n t a l l y determined v a l u e of D w i l l be l e s s than the " t r u e " v a l u e Df. T h i s phenomenon has been e x t e n s i v e l y confirmed and the model treatments extended (4)· There have been s e v e r a l model r e l a t i o n s h i p s proposed t o account f o r the e f f e c t s of sorbed low molecular weight m a t e r i a l to p l a s t i c i z e the polymer and so i n c r e a s e the r a t e of d i f f u s i o n . The general method has been t o estimate the p r o b a b i l i t y of a s u c c e s s f u l d i f f u s i o n jump i n terms of e i t h e r the energy r e q u i r e d f o r a c r i t i c a l volume d i s t u r b a n c e o r f r e e volume e f f e c t s (3,5). Those based on f r e e volume concepts have gained the widest acceptance but the energy c o n s i d e r a t i o n s show g r e a t e r promise f o r extended treatments to e l u c i d a t e the nature of the d i f f u s i o n mechanism. In the f r e e volume model proposed by F u j i t a and Kishimoto (24-26), the p r o b a b i l i t y of h o l e formation i s r e l a t e d to the p r o b a b i l i t y of o b t a i n i n g a f r e e volume domain of a s i z e s u f f i c i e n t to a l l o w a d i f f u s i o n jump (as d e f i n e d f o r other processes by Cohen and T u r n b u l l and by D o o l i t t l e ) . The d i f f u s i o n c o e f f i c i e n t i s s t a t e d as D* = D(dlricYdlna) = Aoexp(-Bo/f)

(13)

where Arj i s a constant a t a g i v e n temperature, f i s the f r e e volume f r a c t i o n , and i s a parameter, c h a r a c t e r i s t i c of the system, which g i v e s a measure of the e f f i c i e n c y of use of a v a i l a b l e f r e e volume by the d i f f u s i o n process. According to the WLF equation, the temperature dependence of f r e e volume i n u n d i l u t e d polymer a t Τ > Tg i s f(o,T) = f ( o , T ) + a ( T - T ) g

where T

g

2

g

(14)

i s the g l a s s temperature of pure (dry) polymer and a


2

26

CONTROLLED

R E L E A S E PESTICIDES

i s the d i f f e r e n c e between the thermal expansion c o e f f i c i e n t s above and below Tg. Provided the c o n c e n t r a t i o n o f d i l u e n t i s not too h i g h , the c o n c e n t r a t i o n dependence a t a g i v e n sorbed volume f r a c t i o n , φ-^, o f penetrant a t a g i v e n temperature Τ may be g i v e n as f(φ ,Τ) - f(o,T) + 3φ = fο + βφχ


χ

χ

(15)

where β i s a parameter r e p r e s e n t i n g the c o n t r i b u t i o n of the d i l u e n t toward i n c r e a s i n g the f r e e volume. Β i s a f u n c t i o n o f Τ but, t o a f i r s t approximation, i s independent o f φ·^. The con c e n t r a t i o n dependence o f d i f f u s i o n a t a g i v e n temperature i s then ln[D*/D(o)] - B B ^ j / i f g + f l % ] 0

(16)

where D(o) i s the v a l u e of D* a t φ^ = 0 . Under c o n d i t i o n s such t h a t ΒΦ]/£ « I t t h i s equation reduces t o 0

ln[D*/D(o)] - B B ^ j / f g

(17)

and lnD*(and InD) i s approximately l i n e a r w i t h φ^ (correspon d i n g t o equation ( 7 ) ) . These r e l a t i o n s h i p s can represent data over ranges of c o n c e n t r a t i o n and s w e l l i n g corresponding t o those found i n a p p l i c a t i o n s o f p r a c t i c a l membrane systems. They a l l o w u s e f u l p r e d i c t i o n s t o be made of a n t i c i p a t e d behavior based on a v a i l a b l e m a t e r i a l parameters. The f r e e volume approach t o d i f f u s i o n behavior a l s o i s of i n t e r e s t i n t h a t i t b r i n g s t o the f o r e f r o n t the correspondence between mass t r a n s f e r and momentum t r a n s f e r i n polymeric s y s tems (27-29). R h e o l o g i c a l and mechanical p r o p e r t i e s of p o l y mers, such as v i s c o s i t y , creep, s t r e s s - r e l a x a t i o n , and dynamicmechanical b e h a v i o r , a r e l i k e w i s e determined by the nature of the c h a i n segmental motion processes i n polymers, both i n the absence and presence of a d i l u e n t . Many s u c c e s s f u l model treatments t o e x p l a i n and p r e d i c t v a r i o u s r e l a x a t i o n processes a r e based on f r e e volume concepts. The correspondence between d i f f u s i o n processes and s t r e s s r e l a x a t i o n processes i n a s e r i e s o f random copolymers o f i s o prene and methylmethacrylate has been measured u s i n g the d e r i v e d r e l a t i o n s h i p , i n terms o f f r e e volume as above, i n the form: ln[D*/D(o)] = ( B / B ) l n ( a ) o D

where:

n

c

(18)

( a ) « π Φ2/η(Φΐ) η = v i s c o s i t y ( o r modulus) a t φ^ = ο η(φ^) = v i s c o s i t y a t φ^ = φ^ c

0

0

0



2.

ROGERS

Permeability

through

a Polymer

27

Matrix

The parameters Bip and Β η c h a r a c t e r i z e the e f f i c i e n c i e s by which the processes of penetrant d i f f u s i o n and s t r e s s r e l a x a t i o n u t i l i z e the a v a i l a b l e f r e e volume i n the system. I t i s found that the r a t i o ( Β β / Β η ) tends to u n i t y as the s i z e of the penetrant molecule i n c r e a s e s . T h i s i s i n accord w i t h the r e a l i z a t i o n t h a t f o r s u f f i c i e n t l y h i g h molecular weight pene t r a n t s , the d i f f u s i o n process corresponds w i t h t h a t f o r s e l f d i f f u s i o n of polymer segments and c h a i n s . Since many agents i n c o n t r o l l e d r e l e a s e a p p l i c a t i o n s are of moderately h i g h molecular weight, i t seems a p p r o p r i a t e to assess t h e i r t r a n s p o r t behavior i n analogy to the v i s c o u s f l o w behavior of the polymeric m a t r i x . Thus, an a p p r a i s a l of the log modulus versus time-temperature-concentrâtion master curve g i v e s a t l e a s t q u a l i t a t i v e i n d i c a t i o n of the behavior to be expected i n a system w i t h v a r i a t i o n s i n those parameters. As time, temperature, or d i l u e n t c o n c e n t r a t i o n i n c r e a s e , the r a t e of d i f f u s i o n ( i n v e r s e l y r e l a t e d to v i s c o s i t y ) should i n c r e a s e i n accord w i t h the decrease i n modulus. The decrease i n modulus w i t h added p l a s t i c i z e r i s w e l l known and w e l l represented by model treatments. A c o m p l i c a t i o n i n p r e d i c t i o n by the above c o r r e l a t i o n method i s due to the p o s s i b i l i t y of non-uniform s w e l l i n g of a polymer by a d i l u e n t . I n such a case, the unswollen domains d i s t r i b u t e d through an otherwise s w o l l e n m a t r i x would c o r r e spond i n e f f e c t to a d i s p e r s e d impermeable f i l l e r . I f the degree of s w e l l i n g i s e x t e n s i v e , the system would resemble a porous d i f f u s i o n medium. C o r r e c t i o n f o r these e f f e c t s can be made on the b a s i s of model treatments d e r i v e d f o r those cases as i n d i c a t e d e a r l i e r . The permeation of l a r g e molecules i n r e l a t i v e l y non-swollen media can be t r e a t e d by the r e a l i z a t i o n t h a t the d i f f e r e n c e s i n d i f f u s i o n c o e f f i c i e n t s w i l l be r e l a t i v e l y s m a l l between homologous members due to the r e l a t i v e l y s m a l l number of s u i t a b l e h o l e s or f r e e volume domains of a p p r o p r i a t e s i z e f o r d i f f u s i o n jumps. Therefore, the net f l u x w i l l be determined more by the s o l u b i l i t y l e v e l d i f f e r e n c e s between penetrant s p e c i e s than by m o b i l i t y , per se. The s o l u b i l i t y can be estimated by use of t y p i c a l polymer s o l u t i o n theory expressions such as the s i m p l i f i e d form of the Flory-Huggins equation * lna

x

- 1ηφ

+ φ

χ

The c h a r a c t e r i s t i c parameter the Hildebrand equation X l

+

Χ ι

φ|

(19)

can be estimated, i n t u r n , by

- v /RT(6 1

£

1

-

δ )

2

2


(20)

28


The s o l u b i l i t y parameters, δ, can be estimated by group c o n t r i b u t i o n methods, heat of v a p o r i z a t i o n data, o r other methods. T h i s general approach has been used and extended t o p r e d i c t the r e l e a s e r a t e c h a r a c t e r i s t i c s of s t e r o i d s from polymers w i t h c o n s i d e r a b l e success (30).


Conclusion The d i f f u s i o n - s o l u t i o n model f o r understanding and p r e d i c t i n g c o n t r o l l e d r e l e a s e r a t e s can be extended t o i n c l u d e many other aspects and f a c t o r s . Of p a r t i c u l a r i n t e r e s t i s t h e strong dependence o f t r a n s p o r t behavior on polymer composition, s t r u c t u r e , and morphology. M o d i f i c a t i o n s of these system v a r i a b l e s can be achieved by c a r e f u l s e l e c t i o n , f a b r i c a t i o n , and p o s t - f a b r i c a t i o n chemical and/or p h y s i c a l treatments. The use of multicomponent polymer media o f f e r s advantages f o r en hanced and b e t t e r c o n t r o l l e d f l u x behavior. The development of s p e c i f i c polymeric f o r m u l a t i o n s f o r p a r t i c u l a r agents and a p p l i c a t i o n c o n d i t i o n s i s not only d e s i r a b l e but a l s o p o s s i b l e . The b a s i s f o r t h a t development r e s t s , i n p a r t , on o b t a i n i n g and ex tending our knowledge of the dependence o f t r a n s p o r t processes on the nature of the system components and t h e i r i n t e r a c t i o n w i t h the a p p l i c a t i o n environment.

Literature

Cited

1.

Crank, J. and Park, G. S., eds. "Diffusion in Polymers", Academic Press, New York, 1968. 2. Rogers, C. Ε., in "Physics and Chemistry of the Organic Solid State", D. Fox, M. Labes, and A. Weissberger, eds., Vol. II, Wiley-Interscience, New York, 1965, Chap. 6. 3. Rogers, C. E. and Machin, D., CRC Critical Reviews of Macromolecular Science, (1972), 1, 245. 4. Hopfenberg, H. B., ed., "Permeability of Plastic Films and Coatings to Gases, Vapors, and Liquids", Vol. 6 of Polymer Science and Technology, Plenum Press, New York, 1974. 5. Crank, J., "The Mathematics of Diffusion", 2nd Ed., Claren don Press, Oxford, 1975. 6. Flynn, G. L., Yalkowsky, S. Η., and Roseman, T. J., J . Pharmaceutical Sci., (1974), 63, 479. 7. Tanquary, A. C. and Lacey, R. Ε., Eds., "Controlled Release of Biologically Active Agents", Vol. 47 of Advances in Experimental Medicine and Biology, Plenum Press, New York, 1974. 8. Cardarelli, N. F., Ed., "Controlled Release Pesticide Sym posium", The University of Akron, 1974. 9. Harris, F. W., Ed., "Proceedings 1975 International Con trolled Release Pesticide Symposium", Wright State University, Dayton, Ohio, 1975. 10. Williams, Α., "Sustained Release Pharmaceuticals", Noyes Development Corp., Parkridge, N. J., 1969.



2.

ROGERS

Permeability

through

a Polymer

Matrix

29

11. Allan, G. G., Chopra, C. S., Friedhoff, J. F., Gara, R. I., Maggi, M. W., Neogi, Α. Ν., Roberts, S. C., and Wilkens, R. Μ., "Pesticides, Pollution, and Polymers", Chemtech., (1973), 3, 171. 12. Colbert, J. C., "Controlled Action Drug Forms", Noyes Data Corp., Park Ridge, N.J.,1974. 13. Cardarelli, N. F., "Concepts in Controlled Release-Emerging Pest Control Technology", Chemtech., (1975), 5, 482. 14. Baker, R. W. and Lonsdale, Η. Κ., "Controlled Delivery-An Emerging use for Membranes", Chemtech., (1975), 5, 668. 15. Cardarelli, N. F., "Controlled Release Pesticides Formula tions", CRC Press, Cleveland, Ohio, 1976. 16. Paul, D. R. and Harris, F. W., eds., "Controlled Release Polymeric Formulations", ACS Symposium Series 33, Am. Chem. Soc., Washington, D. C., 1976. 17. Chang, T. M. S., Artifical Cells, C. C. Thomas, Springfield, Ill., 1972. 18. Gutcho, Μ., "Capsule Technology and Microencapsulation", Noyes Data Corp., Park Ridge, N.J.,1972. 19. Vandegaer, J. E., Ed., "Microencapsulation-Processes and Applications", Plenum Press, New York, 1974. 20. Fanger, G. O., "What Good are Microcapsules", Chemtech., (1974), 4, 397. 21. Goodwin, J. T. and Somerville, G. R., Microencapsulation by Physical Methods", Chemtech., (1974), 4, 623. 22. Chang, T. M. S., "Artificial Cells", Chemtech., (1975), 5, 80. 23. Thies, C., "Physicochemical Aspects of Microencapsulation", Polymer-Plast. Technol. Eng., (1975), 5(1), 1. 24. Fujita, H., Fortschr. Hochpolym. Forsen., (1961), 3, 1. 25. Fujita, H. and Kishimoto, Α., J. Chem. Phys., (1961), 34, 393. 26. Fujita, H. and Kishimoto, Α., J. Polym.Sci.,(1958), 28, 547, 569. 27. Machin, D. and Rogers, C. E.,J.Polym.Sci.A2, (1972), 10, 887. 28. Machin, D. and Rogers, C. E.,J.Polym.Sci.,Phys., (1973) 11, 1535, 1555. 29. Rogers, C. Ε., Semancik, J. R., and Kapur, S., Polym. Sci. Tech., (1973), 1, 297. 30. Michaels, A. S., Wong, P. S. L., Prather, R. and Gale, R. M., AlChE J., (1975), 21, 1073.


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