Chapter 4
Concentration Dependence of the Polymer Diffusion Coefficient 1
Wender Wan and Scott L . Whittenburg Department of Chemistry, University of New Orleans, New Orleans, LA 70148
A theoretical expression for the concentration dependence of the polymer diffusion coefficient is derived. The final result is shown to describe experimental results for polystyrene at theta conditions within experimental errors without adjustable parameters. The basic theoretical expression is applied to theta solvents and good solvents and to polymer gels and polyelectrolytes.
One measure of t h e dynamics of m o t i o n o f p o l y m e r s i s the r a t e o f d i f f u s i o n of the polymer. Due t o the complex n a t u r e of the polymer m o l e c u l e the dynamic phase diagram i s a l s o complex. The r a t e of d i f f u s i o n depends o n the c o n c e n t r a t i o n o f the polymer, the tempera t u r e , the p o l y m e r - s o l v e n t i n t e r a c t i o n and t h e c h a r a c t e r i s t i c l e n g t h a s s o c i a t e d w i t h the experiment. To u n d e r s t a n d t h e concept of t h e c h a r a c t e r i s t i c l e n g t h of the measurement c o n s i d e r a t y p i c a l q u a s i e l a s t i c l i g h t s c a t t e r i n g (QELS) experiment. The i n t e n s i t y o f t h e l i g h t s c a t t e r e d by t h e polymer i s a n a l y z e d as a f u n c t i o n of a n g l e between the i n c i d e n t l a s e r beam and the s c a t t e r e d d i r e c t i o n w h i c h i s d e t e r m i n e d by two p i n h o l e s . U s i n g t h e s c a t t e r i n g a n g l e , Θ, t h e s c a t t e r e d w a v e v e c t o r , q, i s c a l c u l a t e d v i a ςΚ4τΓη/ λ)βΐηθ/2, where η i s the i n d e x o f r e f r a c t i o n o f the s c a t t e r i n g medium and λ i t h e w a v e l e n g t h o f the l a s e r l i n e . Thus, the e x p e r i m e n t e r chooses a p a r t i c u l a r v a l u e of q by a s u i t a b l e c h o i c e of s c a t t e r i n g a n g l e . The c h a r a c t e r i s t i c l e n g t h o f the experiment i s q " , w h i c h can be seen by dimensional a n a l y s i s . For l a r g e q the c h a r a c t e r i s t i c l e n g t h i s v e r y s m a l l , c o r r e s p o n d i n g t o the m o t i o n of i n d i v i d u a l segments. This regime i s termed the i n t e r n a l m o t i o n regime (1). For s m a l l e r v a l u e s of q the c o r r e l a t e d motions o f more and more segments are sampled u n t i l the experiment can be i n t e r p r e t e d as p r o b i n g t h e m o t i o n o f e n t i r e polymer m o l e c u l e s . T h i s t r a n s i t i o n o c c u r s a t ςξ l , where ξ i s the c o r r e l a t i o n l e n g t h . The n a t u r e o f t h e system i n the m a c r o - . s c o p i c regime i s determined by the r a t i o , q / q , where q ( T D ) ' 8
s
+
+ B
r
g
1Current address: Department of Chemistry, Brown University, Providence, RI 02914 0097-6156/87/0350-0046S06.00/0 © 1987 American Chemical Society
?
4. WAN AND WHITTENBURG
Polymer Diffusion Coefficient
47
where T i s t h e r e l a x a t i o n time f o r k n o t s i n the g e l entanglement network and D i s the g e l d i f f u s i o n c o e f f i c i e n t . B e l o w t h i s c u r v e the polymer system has a l i q u i d - l i k e (L) response t o p e r t u r b a t i o n s , w h i l e above t h i s c u r v e the response i s g e l - l i k e (G). These d i f f e r e n t dynamic regimes are shown i n the dynamic phase diagram shown below. The form o f t h e c o n c e n t r a t i o n dependence of the polymer d i f f u s i o n depends on the c o n c e n t r a t i o n of t h e polymer r e l a t i v e t o t h e c r o s s o v e r c o n c e n t r a t i o n , c , between the d i l u t e and s e m i d u l u t e regimes. A t c o n c e n t r a t i o n s below c and below q£ =1 the d i f f u s i o n i s of a s i n g l e polymer molecule. A t c o n c e n t r a t i o n s above c t h e polymer c h a i n s e n t a n g l e and the d i f f u s i n g s p e c i e s a g a i n becomes t h e segments of the p o l y m e r m o l e c u l e . These c o n c e n t r a t i o n regimes a r e shown i n F i g u r e 1. A s the polymer c o n c e n t r a t i o n i s i n c r e a s e d t h e f r i c t i o n c o e f f i c i e n t i s i n c r e a s e d , thus l e a d i n g t o a s l o w i n g down o f the m o t i o n of i n d i v i d u a l m o l e c u l e s . A measure of t h e r a t e of d i f f u s i o n i s the d i f f u s i o n c o e f f i c i e n t . The decrease i n t h e ^ r a t e o f d i f f u s i o n as the t h e c o n c e n t r a t i o n i s i n c r e a s e d below c i s marked by a decrease i n the d i f f u s i o n c o e f f i c i e n t , D. A m i c r o s c o p i c , h y d r o dynamic theory e x p l a i n s t h i s b e h a v i o r q u i t e a d e q u a t e l y (1). A s t h e p o l y m e r c o n c e n t r a t i o n approaches c * the d i f f u s i o n c o e f f i c i e n t v e r s u s c o n c e n t r a t i o n c u r v e f l a t t e n s out and, a t c o n c e n t r a t i o n s above c * b u t s t i l l i n the β em i d i l u t e regime, the d i f f u s i o n c o e f f i c i e n t becomes a l i n e a r l y i n c r e a s i n g f u n c t i o n of the polymer concentrât i o n ( 2^5). T h i s b e h a v i o u r i s shown i n F i g u r e 2 f o r p o l y s t y r e n e of v a r i o u s m o l e c u l a r weight s as a f u n c t i o n of c o n c e n t r a t i o n i n c y c l o h e x a n e a t t h e t h e t a temperature, 345°C, and i n c y c l o p e n t a n e a t i t s t h e t a p o i n t , 20.4°C T y p i c a l l y , s c a l i n g approaches are employed t o e x p l a i n the behav i o r i n the s e m i d i l u t e regime. By examining s t a t i c c o r r e l a t i o n s near t h e temperature, Daoud and J a n n i c k ( 6 ) have e x p r e s s e d t h e de ne i t y - d e n s i t y c o r r e l a t i o n f u n c t i o n i n terms of a c o r r e l a t i o n l e n g t h t h a t i s i n v e r s e l y p r o p o r t i o n a l t o c o n c e n t r a t i o n . Since t h e d i f f u s i o n c o e f f i c i e n t i s i n v e r s e l y p r o p o r t i o n a l t o the c o r r e l a t i o n length i t i s d i r e c t l y proportional t o the concentration. Combining the above d e s c r i p t i o n s l e a d s t o a p i c t u r e t h a t d e s c r i b e s the e x p e r i m e n t a l l y o b s e r v e d c o n c e n t r a t i o n dependence of t h e p o l y m e r d i f f u s i o n c o e f f i c i e n t . A t l o w c o n c e n t r a t i o n s the decrease of the t r a n e l a t i o n a l d i f f u s i o n c o e f f i c i e n t i s due t o hydrodynamic i n t e r a c t i o n s t h a t i n c r e a s e the f r i c t i o n c o e f f i c i e n t and thereby s l o w down the m o t i o n of the polymer c h a i n . A t h i g h c o n c e n t r a t i o n s t h e system becomes an e n t a n g l e d network. The c o o p e r a t i v e d i f f u s i o n o f the c h a i n s becomes a c o o p e r a t i v e p r o c e s s , and the d i f f u s i o n of t h e c h a i n s i n c r e a s e s w i t h i n c r e a s i n g polymer c o n c e n t r a t i o n . This d e s c r i p t i o n r e q u i r e s two d i f f e r e n t e x p r e s s i o n s i n the two c o n c e n t r a t i o n regimes. A m i c r o s c o p i c , hydrodynamic theory s h o u l d be c a p a b l e of e x p l a i n i n g the o b s e r v e d b e h a v i o r a t a l l c o n c e n t r a t i o n s . r
Theory The d i f f u s i o n c o e f f i c i e n t i s r e l a t e d t h r o u g h a Green-Kubo r e l a t i o n D - (1/3)
to a microscopic v a r i a b l e
+
dt
( 1 )
where f o r d i f f u s i o n the m i c r o s c o p i c v a r i a b l e i s the f l u x o f t h e
American Chemical Society library 1155 16th St., N.W. Washington, O.C. 20039
48
REVERSIBLE POLYMERIC GELS AND RELATED SYSTEMS
number of p a r t i c l e s through an imaginary p l a n e of u n i t a r e a i n t h e fluid. Thus, f o r d i f f u s i o n A ( t ) » (dx/dN)N(t), where N(t) i s t h e f l u x of t h e number of p a r t i c l e s through t h e p l a n e a t time t. Equat i o n 1 i s a g e n e r a l e x p r e s s i o n f o r the d i f f u s i o n c o e f f i c i e n t . A d i f f u s i o n c o e f f i c i e n t may d e s c r i b e e i t h e r c o o p e r a t i v e o r non-coope r a t i v e p r o c e s s e s . F o r n o n - c o o p e r a t i v e p r o c e s s e s the c h e m i c a l s p e c i e s t h a t we a r e f o l l o w i n g w i t h our experiment moves i n t h e a v e r a g e f o r c e f i e l d of t h e i n s t a n t a n e o u s s t r u c t u r e of t h e f l u i d a t i t s instantaneous position. The average f o r c e i s u s u a l l y e x p r e s s e d i n terms of some average p r o p e r t y o f t h e f l u i d , t y p i c a l l y t h e v i s -
0
1
2
3
4
5
F i g u r e 1. The d i f f e r e n t d y n a m i c r e g i m e s f o r a p o l m e r system. The n o t a t i o n f o r t h e r e g i m e s i s g i v e n i n t h e text. (Reproduced from Ref. 20. C o p y r i g h t 1985 A m e r i c a n Chemical S o c i e t y . )
4.
WAN AND WHITTENBURG
Polymer Diffusion Coefficient
Figure 2. Concentration dependence of the d i f f u s i o n c o e f f i c i e n t for polystyrene i n two solvents at the theta point for various molecular weights. The l i n e i s the theoretical curve. (Reproduced from Ref. 12. Copyright 1983 American Chemical Society.)
49
REVERSIBLE POLYMERIC GELS AND
50
RELATED SYSTEMS
c o s i t y of the f l u i d . For n o n - c o o p e r a t i v e d i f f u s i o n we may v i s u a l i z e an experiment i n w h i c h we t a g one of the p a r t i c l e s i n the f l u i d and o b s e r v e i t s d i f f u s i o n t h o r u g h the f l u i d . T h i s i s r e f e r r e d t o a s e l f - d i f f u s i o n and i s approximated i n s c a t t e r i n g e x p e r i m e n t s i n w h i c h a dye m o l e c u l e i s embedded i n t o a polymer m a t r i x . Alterna t i v e l y , i f we do not l a b e l any of the m o l e c u l e s i n the f l u i d , t h e n we can o n l y measure the r a t e a t w h i c h m o l e c u l e s move past each o t h e r . T h i s i s r e f e r r e d t o as m u t u a l d i f f u s i o n . In experiments i n w h i c h n o n - l a b e l l e d p a r t i c l e s are employed the measured q u a n t i t i t y i s m u f Depending on the e x p e r i m e n t a l c o n d i t i o n s i t i s p o s s i b l e t h a t the m o t i o n of one c h e m i c a l s p e c i e s i s d r i v e n by the m o t i o n of o t h e r s p e c i e s . T h i s w o u l d be the case of the d i f f u s i o n of polymer s e g ments i n an e n t a n g l e d network. The d i f f u s i o n of a s i n g l e c h a i n must push o t h e r segments out of the way. The i s r e f e r r e d t o as c o o p e r a t i v e d i f f u s i o n . E q u a t i o n 1 d e s c r i b e e s e l f , mutual and c o o p e r a t i v e d i f f u s i o n because of the g e n e r a l i t y of the dynamic v a r i a b l e , N(t) i s a phase-space v a r i a b l e ; t h a t i s , the f l u x w i l l depend on b o t h the p o s i t i o n s and v e l o c i t i e s of the polymer segments. T h e r e f o r e , we can expand i t i n the v a r i a b l e s (x^, p^) and, u s i n g E q u a t i o n 1, o b t a i n D
D»
/q d t < v ( t ) v ( 0 ) > i
i
(dx /dN) (dN/dp )
fiat
2
2
i
i
i
i
() 2
where we have used the r e p e a t e d i n d e x c o n v e n t i o n . For i s o t r o p i c sy s tern s we c o u l d w r i t e each of the terms on the r i g h t hand s i d e of E q u a t i o n 2 as o n e - t h i r d of the component of the v e l o c i t y o r f o r c e i n each d i r e c t i o n . N o t i n g t h a t the v e l o c i t i e s of the i n d i v i d u a l s e g ments add up t o the o v e r a l l polymer v e l o c i t y , we see t h a t the f i r s t term i n E q u a t i o n 2 g i v e s s i m p l y the r e l a t i o n between the polymer d i f f u s i o n c o e f f i c i e n t and the v e l o c i t y - v e l o c i t y c o r r e l a t i o n function. A m i c r o s c o p i c approach y i e l d s the E i n s t e i n r e l a t i o n O ) D k T / f ( l + k c ) , where k i s Boltzmann's c o n s t a n t , Τ i s the a b s o l u t e temperature, ÎQ, i s the f r i c t i o n c o e f f i c i e n t at i n f i n i t e d i l u t i o n , k , i s a c o n s t a n t , and c i s the polymer c o n c e n t r a t i o n . Because the f o r c e s on the i n d i v i d u a l segments are not a d d i t i v e , the second term i n E q u a t i o n 2 must be l e f t i n terms of segment v a r i a b l e s . It is c l e a r t h a t the f i r s t term on the r i g h t hand s i d e of E q u a t i o n 2 d e s c r i b e s the s e l f d i f f u s i o n c o e f f i c i e n t . We are f o l l o w i n g the v e l o c i t y of an i n d i v i d u a l p a r t i c l e as a f u n c t i o n of time and the c o n t r i b u t i o n of i t s n e i g h b o r s i n the f r i c i t i o n a l d r a g e x p r e s s e d t h r o u g h the v i s c o s i t y of the f l u i d . The second term on the r i g h t hand s i d e of E q u a t i o n 2 d e c r i b e s the c o n t r i b u t i o n of b o t h m u t u a l d i f f u s i o n and c o o p e r a t i v e d i f f u s i o n . F o l l o w i n g F e r r e 11(8), the second term i n E q u a t i o n 2 can be e x p r e s s e d as a Green-Kubo i n t e g r a l o v e r a f l u x - f l u x c o r r e l a t i o n function. The t r a n s p o r t i s due t o a v e l o c i t y p e r t u r b a t i o n caused by two d r i v i n g f o r c e s , the B r o w n i a n f o r c e and f r i c t i o n a l f o r c e . The t r a n s p o r t c o e f f i c i e n t due t o the segment-segment i n t e r a c t i o n can be c a l c u l a t e d f r o m the Kubo f o r m u l a i 9 ) b
0
8
b
g
λ « (l/k T)///Q d r d r d ( t - t ) b
1
2
2
1
J(r ,t )J(r , t ) 2
2
x
(3)
x
where J ( r , t ) « S ( r , t ) V ( r , t ) and the t r a n s p o r t c o e f f i c i e n t Λ i s r e l a t e d t o the d i f f u s i o n c o e f f i c i e n t t h r o u g h λ «3(V/k T) D. S(r,t) i s the d e n s i t y f l u c t u a t i o n , and ¥(r,t) i s the l o c a l v e l o c i t y of the 2
B
s
4.
WAN AND WHITTENBURG
Polymer Diffusion Coefficient
51
segment i n the p o l y m e r s o l u t i o n . For s i m p l i c i t y ( 8 ) , ve d e f i n e e q u a l time d e n s i t y c o r r e l a t i o n f u n c t i o n G(r) G(r
1 2
(4)
) « 1
the
2
and the t i m e - i n t e g r a t e d v e l o c i t y c o r r e l a t i o n t e n s o r A — ( r ) A
Substituting λ-
ij
(
r
12
)
β
dt i
G ( r ) and A - ( r )
(l/k T)
άτ
///
b
j
(5)
2
i n t o E q u a t i o n 3,
dr dt
χ
1
we
have (6)
S(r ,t)V(r ,t)S(r ,0)V(r ,0)
2
2
2
1
1
which s i m p l i f i e s to λ « (l/^D/Z/d^ λ-
( l / k T ) // G ( r b
1 2
)A(r
dr dt
G(r )V(r ,t)V(r ,0)
2
J L 2
)dr
1 2
dr
1
2
1
- (V/k T) / G(r)A(r)
2
b
dr
where A ( r ) i s the t r a c e of A — and V i s the v o l u m e of the system. The d i f f u s i o n c o e f f i c i e n t can be o b t a i n e d by E q u a t i o n 1, t h a t i s , the f l u x d i v i d e d by the thermodynamic d r i v i n g f o r c e v i a D - (1/3) k T/ b
V / G(r)A(r) dr
- (1/3)
(7)
Such a d e c o m p o s i t i o n of the d i f f u s i o n c o e f f i c i e n t has p r e v i o u s l y been n o t e d by P a t t l e e t a 1.(10) Mow we must e v a l u a t e . The timei n t e g r a t e d v e l o c i t y c o r r e l a t i o n f u n c t i o n A ^ j i s due t o the h y d r o dynamic i n t e r a c t i o n and can be d e s c r i b e d by the Oseen tensor. The Oseen t e n s o r i s r e l a t e d t o the v e l o c i t y p e r t u r b a t i o n caused by the hydrodynamic f o r c e , F. By c h e c k i n g u n i t s , we see t h a t A i s the Oseen t e n s o r t i m e s the energy term, k T, o r b
Ay(r) where 6 y
3
« (k T/87m)(6ij/r + r r j / r ) b
i s the K r o n e c k e r d e l t a . A ( r )
(8)
£
«
k
b
Obviously,
T / ( 2
ï
ï
n
the t r a c e of A ^ j ( r ) i s
r )
(9)
For now, we r e s t r i c t our d i s c u s s i o n t o the θ p o i n t , where the p o l y mers i n s o l u t i o n have a random c o i l d i s t r i b u t i o n * T h e r e f o r e , i t i s r e a s o n a b l e t o say t h a t i n the s e m i d i l u t e regime at the θ p o i n t the d i s t r i b u t i o n of the segments i n the s o l u t i o n i s s p h e r i c a l . With t h i s concept, the v e l o c i t y of the d i f f u s i n g segments a t the θ c o n d i t i o n can be e x p r e s s e d as 3
V ( t ) ( 4 7T /3) « /
dr V ( r , t ) (10) ij Then the d i f f u s i o n c o e f f i c i e n t due t o the hydrodynamic i n t e r a c t i o n between the segments i n the p o l y m e r s o l u t i o n , t h a t i s , the mutual d i f f u s i o n c o e f f i c i e n t , i s g i v e n by i
i
D
r
b
ο
ij
- 1
where we have used E q u a t i o n 9 and E q u a t i o n 10 and < Κ ^ > i s the i n v e r s e average d i s t a n c e between any two segments i n the s o l u t i o n * A l t h o u g h the n o t a t i o n i s perhaps m i s l e a d i n g , we have chose s η t o w r i t e the e x p r e s s i o n i n t h i s form t o c o r r e s p o n d t o the n o t a t i o n of I m a i ( l l ) . E q u a t i o n 11 i s the c o n t r i b u t i o n t o the d i f f u s i o n c o e f f i c i e n t due to m u t u a l o r c o o p e r a t i v e d i f f u s i o n . T h i s can be seen by r e a l i z i n g t h a t i t a r i s e s f r o m the f o r c e - f o r c e c o r r e l a t i o n f u n c t i o n or, from E q u a t i o n 11, the v e l o c i t y - v e l o c i t y c o r r e l a t i o n f u n c t i o n of d i f f e r e n t p a r t i c l e s . Thus, the dynamic v a r i a b l e i s the v e l o c i t y of two p a r t i c l e s r e l a t i v e to each other. T h i s d e s c r i b e s mutual d i f f u s i o n . Combining the e x p r e s s i o n s f o r the s e l f and m u t u a l d i f f u s i o n c o e f f i c i e n t s y i e l d s the f o l l o w i n g e x p r e s s i o n f o r the polymer diffusion coefficient D = k T / f ( l + k c) b
G
s
kbT/o^oXRy"^
+
( 1 2 )
T y p i c a l l y the e x p e r i m e n t a l d a t a on the d i f f u s i o n c o e f f i c i e n t are n o r m a l i z e d r e l a t i v e t o the d i f f u s i o n c o e f f i c i e n t at i n f i n i t e d i l u t i o n , DQ = k T / f , so t h a t E q u a t i o n 12 w i l l be most u s e f u l as b
Q
D/D
- (1
Q
+
k.c)"
1
+ V R i j " ^ s
where the hydrodynamic r a d i u s R g e n e r a l r e s u l t of t h i s chapter.
fl
( 1 3 )
ÎQ/OTUIQ.
E q u a t i o n 13 i s the
A p p l i c a t i o n to Theta S o l v e n t s The term has been e v a l u a t e d by Imai f o r hydrodynamic i n t e r a c t i o n s a t the θ p o i n t Q J ) . The t r a n s p o r t i s due t o a v e l o c i t y p e r t u r b a t i o n caused by some d r i v i n g f o r c e . D i f f u s i o n of the segments i s a r e s u l t of the hydrodynamic f o r c e t h a t can be d e s c r i b e d i n terms of the Oseen t e n s o r (11) w i t h the s o l v e n t modeled as a continuum w i t h v i s c o s i t y , η Q. Thus, Imai has shown t h a t the second term i n Equa t i o n 13 can be w r i t t e n as the energy, k^T, times the t r a c e o v e r the Oseen t e n s o r . A v e r a g i n g o v e r the d i s t r i b u t i o n of segment p o s i t i o n s yields 1
2
c * w o u l d d e c r e a s e i n good s o l v e n t s and i n c r e a s e i n poor s o l v e n t s , i n agreement w i t h p r e d i c t i o n s f r o m w
w
54
REVERSIBLE POLYMERIC GELS A N D RELATED SYSTEMS
s c a l i n g theory (16^17). This i s only s u p e r f i c i a l , since expansion parameters a r e f u n c t i o n s of t h e polymer c o n c e n t r a t i o n and, t h e r e f o r e , f o r ηοη-θ s o l v e n t s t h e d i f f u s i o n c o e f f i c i e n t c u r v e a t concen t r a t i o n s above t h e c r o s s o v e r c o n c e n t r a t i o n may no l o n g e r be l i n e a r (18). These p r e d i c t i o n s a r e i n agreement w i t h t h e r e s u l t s on p o l y s t y r e n e i n THF (19) ( F i g u r e 3 ) . A p p l i c a t i o n t o t h e Ordinarν-Extraordinary T r a n s i t i o n The above approach a l s o c a n e x p l a i n the o r d i n a r y - e x t r a o r d i n a r y t r a n s i t i o n o b s e r v e d i n p o l y e l e c t r o l y tes. The o r d i n a r y - e x t r a o r d i n a r y t r a n s i t i o n i n d i l u t e s o l u t i o n s o f mononucleosomal DNA fragments has been s t u d i e d by B l o o m f i e l d and t h e r e s u l t s r e p o r t e d i n a n o t h e r c h a p t e r i n t h i s monograph. E q u a t i o n 2 shows t h a t b o t h t h e v e l o c i t y v e l o c i t y c o r r e l a t i o n f u n c t i o n and t h e f o r c e - f o r c e c o r r e l a t i o n f u n c t i o n add t o t h e measured d i f f u s i o n c o e f f i c i e n t , D. I n t h e p o l y e l e c t r o l y t e s t h e f o r c e , F^, i s t h e c o u l o m b i c f o r c e due t o t h e charge on t h e polymer. A s t h e i o n i c s t r e n g t h o f t h e s o l u t i o n i s l o w e r e d the apparent t r a n s l a t i o n d i f f u s i o n c o e f f i c i e n t , D(c), i n c r e a s e s due to t h e l o w e r e d s h i e l d i n g of t h e r e p u l s i v e p o l y e l e c t r o l y t e i n t e r a c t i o n . A t a s u f f i c i e n t l y low s a l t c o n c e n t r a t i o n t h e v e l o c i t y v e l o c i t y c o r r e l a t i o n f u n c t i o n begins t o c o n t r i b u t e t o t h e measured d i f f u s i o n c o e f f i c i e n t . Thus, a t v e r y low s a l t c o n c e n t r a t i o n s t h e e x p e r i m e n t a l c o r r e l a t i o n f u n c t i o n s h o u l d d i s p l a y two e x p o n e n t i a l decays. One i s due t o F^, i s p r e s e n t a t a l l i o n i c s t r e n g t h s , and i s t h e r e f o r e the o r d i n a r y c o n t r i b u t i o n . The second i s due t o v ^ , i s p r e s e n t o n l y a t low s a l t c o n c e n t r a t i o n s , and i s t h e e x t r a o r d i n a r y contribution. These assignments a r e i n agreement w i t h t h e s u g gestions of B l o o m f i e l d . A p p l i c a t i o n t o Polvmer G e l s A p o l y m e r g e l c o n s i s t s o f a c r o s s - l i n k e d p o l y m e r network t h a t g i v e s e l a s t i c i t y t o the g e l and a l i q u i d o c c u p y i n g the space between t h e network c h a i n s . The f r i c t i o n c o n t r i b u t i o n of t h e l i q u i d w i l l g i v e the same c o n t r i b u t i o n t o the d i f f u s i o n c o e f f i c i e n t as i n t h e l i q u i d regime d i s c u s s e d i n t h e " A p p l i c a t i o n t o T h e t a C o n d i t i o n s " s e c t i o n above. The e l a s t i c f o r c e between the k n o t s i n t h e g e l network i s independent of t h e p o l y m e r c o n c e n t r a t i o n and w i l l t h e r e f o r e c o n t r i bute a non-zero i n t e r c e p t t o t h e D(c) v e r s u s c o n c e n t r a t i o n curve ( F i g ure 4). Thus, E q u a t i o n 2 p r e d i c t s t h a t t h e D(c) v e r s u s c o n c e n t r a t i o n i n t h e g e l regime s h o u l d be l i n e a r w i t h a s l o p e e q u a l t o t h a t o f t h e l i q u i d regime, b u t w i t h a p o s i t i v e i n t e r c e p t . T h i s p r e d i c t i o n i s i n agreement w i t h the measurements on p o l y s t y r e n e i n t h e g e l and l i q u i d regimes. Summary We have shown t h a t t h e m i c r o s c o p i c e x p r e s s i o n f o r t h e polymer d i f f u s i o n c o e f f i c i e n t , E q u a t i o n 2, i s t h e s t a r t i n g p o i n t f o r a d i s c u s s i o n of d i f f u s i o n i n a wide range of polymer systems. F o r t h e example worked out, polymer d i f f u s i o n a t t h e t a c o n d i t i o n s , t h e r e s u l t i n g expresssion describes the experimental data without a d j u s t a b l e para meters. I t s h o u l d be p o s s i b l e t o d e r i v e e x p r e s s i o n s f o r d i f f u s i o n
WAN AND WHITTENBURG
4.
1
10
55
Polymer Diffusion Coefficient
100
1
200
5
10
Figure 3. Concentration dependence of the diffusion coeff i c i e n t of polystyrene in THF at 25°C. (a) c o l l e c t i v e modes, (b) cumulant values and classical gradient diffusion, (c) cooperative mode. (Reproduced from Ref. 19. Copyright 1985 American Chemical Society.)
Figure 4. Concentration dependence of the liquid, D , gel, D , diffusion coefficients. The various symbols refer to different molecular weights. (Reproduced from Ref. 20. Copyright 1985 American Chemical Society.) c
56
REVERSIBLE POLYMERIC GELS A N D RELATED SYSTEMS
i n p o l y m e r g e l s and p o l y e l e c t r o l y t e e s t a r t i n g from E q u a t i o n 2. are working i n t h i s d i r e c t i o n .
We
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