Polymers in Disperse Systems: An Overview - ACS Symposium Series

1 Polymers in Disperse Systems: An Overview B. VINCENT

Downloaded by 203.64.11.45 on May 5, 2015 | http://pubs.acs.org Publication Date: February 10, 1984 | doi: 10.1021/bk-1984-0240.ch001

School of Chemistry, University of Bristol, Cantock's Close, Bristol BS8 1TS, England

In this paper some of the current thinking in three closely-related areas is highlighted: polymer adsorption; the effect of polymer on the pairwise interaction between particles; and the effect of polymers on dispersion stability.

I t would be an impossible task to summarize i n one short review the many f a c e t s of t h i s s u b j e c t . This has been more than adequately attempted i n s e v e r a l other recent reviews of the f i e l d s of polymer adsorption (1-4) and d i s p e r s i o n s t a b i l i t y i n the presence of polymers (1_, 5-7). My o b j e c t i v e , t h e r e f o r e , i s p r i m a r i l y to s e t the scene f o r the papers that f o l l o w : to h i g h l i g h t current t h e o r e t i c a l and experimental work, and to i n d i c a t e where f u t u r e research e f f o r t s might conceivably be d i r e c t e d . I t i s convenient to d i v i d e t h i s t o p i c i n t o three areas, which f o l l o w on from each other i n a l o g i c a l sequence : i) polymers a t a s i n g l e i n t e r f a c e : a d s o r p t i o n and d e p l e t i o n ii) i n t e r a c t i o n s between two p a r t i c l e s i n the presence of polymer: establishment of the p a i r p o t e n t i a l . i i i ) d i s p e r s i o n s t a b i l i t y i n the presence of polymer: thermodynamic and k i n e t i c c o n s i d e r a t i o n s . Polymers a t a S i n g l e I n t e r f a c e Our understanding of polymer adsorption has followed i n the wake of developments i n the theory of adsorption of small molecules and that of polymer s o l u t i o n s . I t i s u s e f u l , at the outset to introduce some of the ideas that have been developed i n recent years, p a r t i c u l a r l y with regard to the l a t t e r t o p i c . The c h a r a c t e r i s t i c f e a t u r e of a macromolecule i n s o l u t i o n i s i t s high degree of conformational freedom. The simplest p o s s i b l e model f o r an i s o l a t e d macromolecule i s the random walk (or 0097-6156/84/ 0240-0003S06.00/0 © 1984 American Chemical Society

In Polymer Adsorption and Dispersion Stability; Goddard, E., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

4

POLYMER ADSORPTION AND DISPERSION STABILITY

d i f f u s i o n ) model, i n which a polymer c h a i n i s regarded as a s e r i e s o f r l i n e a r l i n k s , e a c h o f l e n g t h 1, w i t h no r e s t r i c t i o n o n t h e angles between s u c c e s s i v e l i n k s . This leads t o the well-known e x p r e s s i o n f o r t h e r o o t - m e a n - s q u a r e (r.m.s.) end-end d i s t a n c e , *,

0, where ζ i s t h e d i s t a n c e normal to the s u r f a c e . v i i ) the net segment-surface i n t e r a c t i o n parameter, X . A rigorous d e f i n i t i o n o f X has b e e n g i v e n by F l e e r and L y k l e m a ( 4 ) . The o b j e c t i v e o f any t h e o r y i s t o p r e d i c t how v a r i o u s p a r a meters which c h a r a c t e r i s e the adsorbed polymer v a r y w i t h these system v a r i a b l e s . T h e r e a r e a number o f p a r a m e t e r s w h i c h c h a r a c t e r i s e t h e a d s o r b e d p o l y m e r , and w h i c h s h o u l d , i d e a l l y , a l l be m e a s u r e d f o r any g i v e n s y s t e m : i) t h e a d s o r b e d amount. T h i s i s b e s t e x p r e s s e d as a d i m e n s i o n l e s s q u a n t i t y ^ Θ, t h e r a t i o o f t h e t o t a l number o f a d s o r b e d segments/ maximum number o f segments i n a m o n o l a y e r a t t h e s u r f a c e ( i . e . with every surface " s i t e " f i l l e d ) . The e x a c t c o r r e l a t i o n b e t w e e n an a d s o r p t i o n s i t e and t h e s u r f a c e s t r u c t u r e o f t h e a d s o r b e n t i s not always o b v i o u s . I f a l a t t i c e model i s used, t h e n c o n s i d e r a b l e d i f f i c u l t i e s a l s o a r i s e i n c o r r e l a t i n g the s i z e of the l a t t i c e element w i t h the s t r u c t u r e of the polymer c h a i n , the s o l v e n t m o l e c u l e s and t h e s u r f a c e s t r u c t u r e o f t h e a d s o r b e n t . A t h i g h p o l y m e r c o n c e n t r a t i o n s , φ > φ , one a l s o has t o d i s t i n g u i s h b e t w e e n segments w h i c h a r e p r e s e n t i n t h e a d s o r b e d l a y e r r e g i o n and b e l o n g i n g t o c h a i n s w h i c h a r e a c t u a l l y a d s o r b e d ( i . e . h a v e a t l e a s t one segment i n c o n t a c t w i t h t h e s u r f a c e ) , and t h o s e segments w h i c h a r e p r e s e n t i n t h i s r e g i o n b u t b e l o n g t o f r e e , ρ

g

s

ρ

ρ


8


overlapping chains which have no segments i n contact with the s u r f a c e . Scheutjens and F l e e r (15,16) i n t h e i r l a t t i c e model of polymer adsorption have computed the r e l a t i v e c o n t r i b u t i o n s to ®total ^ these two types of segments. In f i g u r e 3, 0 ^ is shown as a f u n c t i o n of log r , together with the two c o n t r i b u t i o n s (depletion) and 0 (excess) which are defined below, and inter preted i n the i n s e t r o r a

t o t a


e x

Σ (φ. i

φ ) Ρ

(4)

Σ(φ

φ?·*')

(5)

where φ i s the bulk polymer (or segment) volume f r a c t i o n (Scheutjens and F l e e r (15,16) use φ rather than φ as the symbol f o r bulk polymer c o n c e n t r a t i o n ) , φ^ i s the t o t a l volume f r a c t i o n of segments i n layer i , and φ * i s the volume f r a c t i o n of segments i n l a y e r i a r i s i n g from non-adsorbed chains. From f i g u r e 3, where φ = 10"^, i t can be seen that θ i exceeds θ only f o r r £ 10 , i . e . where φ > φ*. For lower values of r , θ ^ In r . In f i g u r e 4, l o g θ i s p l o t t e d as a f u n c t i o n of l o g φ f o r two values of χ (χ = 0, athermal; χ = 0.5, theta solvent) and f i x e d r and x . The l o g - l o g s c a l e i s used to h i g h l i g h t the main f e a t u r e s . The isotherm may be d i v i d e d i n t o three regions a) the Henry s law r e g i o n , where l o g θ increases s t e e p l y and l i n e a r l y with log φ . Here adsorption occurs as i n d i v i d u a l i s o l a t e d chains. b) beyond some c r i t i c a l value of θ, chain overlap occurs on the surface and θ increases very much more slowly with φρ. This c r i t i c a l value, Θ* may be conveniently defined by the i n t e r s e c t i o n of the two s t r a i g h t l i n e s as i n f i g u r e 4. C l e a r l y Θ* f o r the surface has strong analogies with φ* f o r bulk s o l u t i o n . Note that Θ* occurs at i n a c c e s s i b l y low φ v a l u e s . The experimental range of φ values (^ 10"^ to MO ) corresponds to the pseudo-plateau r e g i o n of the isotherm. c) beyond the l i m i t of the d i l u t e bulk s o l u t i o n region (φ > φ ) c o n t r i b u t i o n s from a r i s e and θ increases r a p i d l y up to the value f o r φ = 1. Tp Experimentally, the adsorbed amount i s u s u a l l y expressed as Γ i . e . mass polymer/area of s u r f a c e . This i s u s u a l l y obtained from a mass balance technique, af ter a n a l y s i n g the e q u i l i b r i u m s o l u t i o n . Γ θ , but an exact c o r r e l a t i o n i s d i f f i c u l t to e s t a b l i s h . F i n a l l y , i n t h i s s e c t i o n we consider the case where χ = 0, i . e . where no adsorption takes p l a c e . There i s then a d e p l e t i o n of segments i n the region near the i n t e r f a c e ( f i g u r e 5). I t i s u s e f u l to define an e f f e c t i v e d e p l e t i o n zone of thickness, 6^, d e f i n e d such that the two shaded areas are equal, and given by, ρ

Λ

ρ

η , Ε

ρ

t o t a

5

e x

ρ

ρ

s

1

ρ

ρ

ρ

ρ

œ

β χ

8



VINCENT

Polymers in Disperse Systems

1

10

3

10

6

a n c

F i g u r e 3. θ t o t * i t s components, 6 and θ^, as a f u n c t i o n o f c h a i n l e n g t h , f o r φρ = 10"^ ( h e x a g o n a l l a t t i c e X = 1, χ = 0 . 5 ) . The i n s e t g i v e s a q u a n t i t a t i v e p i c t u r e o f t h e segment c o n c e n t r a t i o n p r o f i l e i n t h e a d s o r b e d l a y e r ( R e p r o d u c e d w i t h p e r m i s s i o n f r o m R e f . 16. C o p y r i g h t 1 9 8 2 , Academic P r e s s (London).) e x

s




10 In Polymer Adsorption and Dispersion Stability; Goddard, E., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

1.

VINCENT

Polymers in Disperse Systems e

d

=

Φ

ρ

6

11 (6)

d

ii) t h e bound f r a c t i o n o f s e g m e n t s . One may d e f i n e t h e f r a c t i o n o f segments i n t r a i n s , p, b y t h e r e l a t i o n s h i p , Ρ = Β /Β

(7)

χ

where θ j i s t h e f r a c t i o n a l c o v e r a g e i n t h e f i r s t l a y e r . I n l a t t i c e t h e o r i e s , θ j = φ j . A l l t h e o r i e s p r e d i c t (J_) t h a t ρ i s a d e c r e a s i n g f u n c t i o n o f φ a n d h e n c e Θ, a t l e a s t beyond θ*. χ i s a n i m p o r t a n t parameter: t h e r e i s a c r i t i c a l v a l u e o f x , X g , b e l o w w h i c h ρ = 0, i . e . no segments a r e a d s o r b e d . F o r X > χ § , ρ i s a n i n c r e a s i n g f u n c t i o n o f χ . ρ g e n e r a l l y i n c r e a s e s a s χ d e c r e a s e s ( i . e . as t h e s o l v e n t " q u a l i t y " i s i n c r e a s e d ) . F o r X > ^ X§, f i x e d χ and θ, ρ g e n e r a l l y decreases as r i n c r e a s e s . ρ may be o b t a i n e d (1-4) u s i n g s p e c t r o s c o p i c (h.m.r., e . s . r . o r i n f r a r e d ) , c a l o r i m e t r i c o r electrochemical techniques, e i t h e r d i r e c t l y o r through t h e e s t i m a t i o n o f θ j (equation 7 ) . i i i ) t h e e x t e n s i o n o f segments n o r m a l t o t h e s u r f a c e . Ideally, one w o u l d l i k e t o b e a b l e t o p r e d i c t t h e o r e t i c a l l y a n d / o r e s t a b l i s h e x p e r i m e n t a l l y φ(ζ) ( o r φ£ f o r a l a t t i c e m o d e l ) , i . e . t h e f o r m o f t h e segment d e n s i t y p r o f i l e n o r m a l t o t h e s u r f a c e . C l e a r l y , t h i s o n l y h a s r e a l m e a n i n g f o r θ > 9*7~where a u n i f o r m d i s t r i b u t i o n o f segments p a r a l l e l t o t h e s u r f a c e may r e a s o n a b l y b e assumed. A l t h o u g h a number o f r e c e n t t h e o r i e s a r e a b l e t o p r e d i c t φ(ζ), t h e S c h e u t j e n s - F l e e r t h e o r y (15,16) i s a l s o a b l e t o p r e d i c t t h e s e p a r a t e c o n t r i b u t i o n s t o φ£ f r o m t a i l s a n d l o o p s . E x p e r i m e n t a l l y , t h e f o r m o f φ(ζ) h a s b e e n r e c e n t l y e s t a b l i s h e d f o r a d s o r b e d homopolymers a n d t e r m i n a l l y a n c h o r e d t a i l s b y t h e B r i s t o l group (17,20). K n o w i n g φ(ζ) one may t h e n c a l c u l a t e t h e r.m.s. t h i c k n e s s o f t h e a d s o r b e d l a y e r . P r e v i o u s measurements o f t h e " t h i c k n e s s " (1-4) h a v e u s u a l l y i n v o l v e d e l l i p s o m e t r y ( f l a t s u r f a c e s ) o r some h y d r o d y n a m i c t e c h n i q u e ( p a r t i c l e s ) . In neither c a s e c a n t h e c a l c u l a t e d t h i c k n e s s b e u n a m b i g u o u s l y r e l a t e d t o φ(z), a l t h o u g h r e c e n t t h e o r e t i c a l w o r k b y Cohen S t u a r t e t a l . ( 2 1 ) , t o be d i s c u s s e d a t t h i s m e e t i n g , h a s made a n a t t e m p t t o r e l a t e t h e hydrod y n a m i c t h i c k n e s s , δ^, t o φ(ζ). The t h e o r e t i c a l a n d (model) e x p e r i m e n t a l w o r k r e f e r r e d t o above h a s l a r g e l y b e e n c o n c e r n e d w i t h l i n e a r homopolymers a d s o r b e d o n r e g u l a r s u r f a c e s . However, t h e r e i s a v a s t l i t e r a t u r e o f e x p e r i m e n t a l s t u d i e s o n more c o m p l e x s y s t e m s . Unfortunately, i n many c a s e s t h e s y s t e m s a r e e i t h e r i l l - d e f i n e d a n d / o r o n l y a d s o r p t i o n isotherms have been e s t a b l i s h e d ; f o r drawing g e n e r a l c o n c l u s i o n s o r comparison w i t h theory such s t u d i e s a r e o f l i t t l e use. On t h e t h e o r e t i c a l s i d e , c l e a r l y t h e work needs t o b e e x t e n d e d t o w a r d s t h e s e more c o m p l e x s y s t e m s . I n p a r t i c u l a r , developments a r e r e q u i r e d i n t h e f o l l o w i n g areas ( s t a r t s have a l r e a d y b e e n made i n some c a s e s ) : 3

s


s

8

s


12 a) b) c) d) e) f) g)

POLYMER ADSORPTION AND DISPERSION STABILITY copolymers and branched polymers. p o l y e l e c t r o l y t e s ( l e a d i n g h o p e f u l l y and e v e n t u a l l y to bipolymers) polymer mixtures ( f r a c t i o n a t i o n ) adsorption onto n o n - f l a t surfaces ( e s p e c i a l l y spheres) adsorption onto heterogeneous surfaces adsorption at the l i q u i d / l i q u i d and liquid/vapour i n t e r f a c e s co-adsorption between p l a t e s or p a r t i c l e s , and adsorption i n c a v i t i e s (pores).


Pairwise I n t e r a c t i o n Between P a r t i c l e s i n the Presence of Polymer The s t a r t i n g point f o r any theory of p a r t i c l e i n t e r a c t i o n s i s the D.L.V.O. theory ( 2 2 ) , which considers the p a i r p o t e n t i a l between two charged p a r t i c l e s across a continuous medium (solvent plus ions, where a p p r o p r i a t e ) . The t o t a l i n t e r a c t i o n f r e e energy i s s p l i t i n t o the c o n t r i b u t i o n from the van der Waals (electrodynamic) f o r c e s (G^), and the e l e c t r o s t a t i c f o r c e s a r i s i n g from the overlap of the e l e c t r i c a l double l a y e r s around the charged p a r t i c l e s (Gg). Both G^ and Gj? are a f f e c t e d by the presence of adsorbed polymer l a y e r s . Moreover, a d d i t i o n a l i n t e r a c t i o n s are introduced. The s i t u a t i o n i s complex even f o r n e u t r a l polymers, but may be con v e n i e n t l y r a t i o n a l i s e d i n terms of f i g u r e 6. Two regions of i n t e r a c t i o n may be d i s t i n g u i s h e d : ( i ) h > 26; ( i i ) h < 26. For h > 26 (b > 0 ) , the DLV0 theory operates: one simply has to i n v e s t i g a t e how the presence of the adsorbed polymer modifies G^ and Gg. To attempt to do t h i s e x a c t l y one would need to know the form of φ(ζ) and a l s o the charge d i s t r i b u t i o n i n the e l e c t r i c a l double l a y e r . However, v a r i o u s s i m p l i f y i n g approximations may be made. For G^ (b) one may e i t h e r assume that the average segment c o n c e n t r a t i o n i n the adsorbed l a y e r i s so small that G i s not perturbed by the adsorbed l a y e r ( v a l i d f o r h i g h M.W. polymers), or use the Void type of approach (23,24), which regards the p a r t i c l e plus i t s adsorbed polymer sheath e s s e n t i a l l y as a composite p a r t i c l e , assigned two Hamaker constants : one f o r the core and one f o r the sheath. For Gç (b), a reasonable (although not s t r i c t l y c o r r e c t ) procedure i s to replace the Stern p o t e n t i a l i n one of the standard equations f o r Gg by the zeta p o t e n t i a l of the polymer-coated particles; t h i s assumes that the plane of hydrodynamic shear corresponds to the p e r i p h e r y of the adsorbed l a y e r . For h < 26, the s i t u a t i o n i s much more complex. One not only needs to know φ(ζ) f o r each l a y e r , but how φ(ζ) changes as the two p a r t i c l e s approach, i . e . φ(z,h); t h i s may w e l l depend on the times c a l e of the approach, i . e . the e q u i l i b r i u m path may not be f o l l o w e d . Scheutjens and F l e e r (25) i n an extension of t h e i r model f o r polymer adsorption have analysed the s i t u a t i o n f o r two i n t e r a c t i n g uncharged p a r a l l e l , f l a t p l a t e s c a r r y i n g adsorbed, n e u t r a l homopolymer, i n t e r a c t i n g under e q u i l i b r i u m c o n d i t i o n s . Only a s e m i - q u a n t i t a t i v e p i c t u r e w i l l be presented here. A


1.

VINCENT

13

Polymers in Disperse Systems

One may d i v i d e t h e i n t e r a c t i o n s i n t o t h r e e c o n t r i b u t i o n s ( i . e . i n a d d i t i o n t o G^) a) t h e e l a s t i c term ( G ^ ) : t h i s a r i s e s from the l o s s i n con formational entropy o f the chains. Depending on t h e coverage (Θ) and t h e c o n f i g u r a t i o n o f t h e a d s o r b e d c h a i n s , t h i s may o r may n o t be a s i g n i f i c a n t c o n t r i b u t i o n f o r δ < h < 2δ, b u t c l e a r l y becomes v e r y i m p o r t a n t when h < δ, s i n c e t h e c h a i n s a r e r e s t r i c t e d by t h e o p p o s i n g s u r f a c e and a l s o p o l y m e r b r i d g e s f o r m w h i c h have a v e r y l o w c o n f o r m a t i o n e n t r o p y compared t o l o o p s o r t a i l s . Napper (26) h a s g i v e n t h e f o l l o w i n g e x p r e s s i o n f o r G , f o r two p a r a l l e l flat plates e

6


f

G _ = 2ΜΓ In = 2kTr R (h) el Ω(°°) el

(8)

Λ

where Γ i s t h e number o f a d s o r b e d c h a i n s p e r u n i t a r e a , and 0 ( h ) and Ω(°°) a r e t h e number o f c o n f o r m a t i o n s a v a i l a b l e t o t h e a d s o r b e d c h a i n s a t h = h and h = °°, r e s p e c t i v e l y ; R ^ i s a geometric f u n c t i o n w h i c h depends o n t h e f o r m o f φ(ζ,η). G ^ i s a l w a y s a repulsive contribution. b) the m i x i n g term ( G £ ) : t h i s a r i s e s f r o m t h e n e t segment/ s o l v e n t i n t e r a c t i o n s and t h e l o c a l changes i n segment c o n c e n t r a t i o n i n t h e i n t e r a c t i o n r e g i o n between t h e p a r t i c l e s . I t comes i n t o p l a y f o r a l l v a l u e s o f h < 2δ. Napper (26) h a s a l s o d e r i v e d a n a p p r o x i m a t e e x p r e s s i o n f o r t h i s t e r m f o r two p a r a l l e l flat plates, E

e

m

x

2kT G . mix

=

ν

2

2 ν s

Γ

2 (I -

x

)

. ( ) mix

R

(9)

h

where Vp a n d v a r e t h e m o l a r v o l u m e s o f t h e p o l y m e r and s o l v e n t , respectively. R £ i s a g a i n a g e o m e t r i c f u n c t i o n w h i c h depends o n t h e f o r m o f φ(ζ,η). χ i s t h e F l o r y i n t e r a c t i o n p a r a m e t e r . Clearly, Gjnix c a n be p o s i t i v e o r n e g a t i v e ( i . e . r e p u l s i v e o r a t t r a c t i v e ) d e p e n d i n g o n t h e m a g n i t u d e o f χ. F o r a θ-solvent (χ = | ) , G £ = 0. c) t h e a d s o r p t i o n t e r m ( G j ) : t h i s r e s u l t s f r o m t h e change i n the n e t number o f s u r f a c e / s e g m e n t c o n t a c t s , p, a s h d e c r e a s e s . G j ( l i k e G ^) may o r may n o t be s i g n i f i c a n t f o r δ < h < 2δ, d e p e n d i n g o n t h e c o n f i g u r a t i o n a l changes t h a t o c c u r i n t h i s r e g i o n ( i . e . w h e t h e r changes i n ρ o c c u r ) a s h d e c r e a s e s , b u t c l e a r l y becomes v e r y i m p o r t a n t when h < δ s i n c e segments c a n now become adsorbed on the surface o f t h e opposing core p a r t i c l e s o r p l a t e s , i n a d d i t i o n t o a n y changes i n ρ a t t h e o r i g i n a l s u r f a c e . One may write the following expression for G ^ for p a r a l l e l f l a t plates s

M

X

m

x

ac

a(

e

G ^ « - 2kTrr .Ap(h)

(10)

X o

where Ap(h) i s t h e n e t change i n ρ a s a f u n c t i o n o f h . S i n c e , i n g e n e r a l , as h d e c r e a s e s Δρ w i l l b e p o s i t i v e , t h e n G j w i l l c o n s t i t u t e an a t t r a c t i v e c o n t r i b u t i o n . a c


14


S i m i l a r expressions to equations (8) to (10) may be derived for s p h e r i c a l p a r t i c l e s . The form of t h e t o t a l i n t e r a c t i o n ( i . e . A e l mix ad> assuming a d d i t i v i t y ) i s c l e a r l y complex and depends very much on φ(z,h), which i n turn depends on θ (or Γ ) . For s i m p l i c i t y and by way of i l l u s t r a t i o n , we s h a l l j u s t consider two p o s s i b l e l i m i t i n g cases : ( i ) at low θ (θ < θ * ) ; ( i i ) at high θ (psuedo-plateau r e g i o n of the isotherm). These are i l l u s t r a t e d i n f i g u r e 7 f o r the case of a good s o l v e n t (χ < J ) . At low coverages ( f i g u r e 7a), there i s a s i g n i f i c a n t minimum (^min) » r e s u l t i n g from the dominance of G j and at h < ^ δ, and G (» G i + G £ ) f o r h « 6. This r e s u l t s i n s o - c a l l e d " b r i d g i n g f l o c c u l a t i o n " . At high coverages ( f i g u r e 7b), on the other hand, a d °° 0 and only a very shallow minimum e x i s t s at h > ^ 26 due to the r e s i d u a l c o n t r i b u t i o n from G^ at these d i s t a n c e s . C l e a r l y , provided G £ i s small enough, the system w i l l be s t a b l e to flocculation; t h i s s i t u a t i o n corresponds to s o - c a l l e d " s t e r i c stabilisation". The question of j u s t how deep G £ has to be f o r f l o c c u l a t i o n to be observed w i l l be discussed l a t e r i n the paper. The above a n a l y s i s a p p l i e s to the case of p o s i t i v e adsorption. What happens f o r example when χ = 0, and d e p l e t i o n zones e x i s t near the surface? This s i t u a t i o n i s d e a l t with i n d e t a i l i n a l a t e r paper (27) here. The f i r s t a n a l y s i s was by Asakura and Oosawa (28,29), who showed that the overlap of the d e p l e t i o n zones ( f i g u r e 5), r e s u l t s i n a net a t t r a c t i o n , and hence d e p l e t i o n f l o c c u l a t i o n . The o r i g i n of t h i s a t t r a c t i v e term may be viewed i n terms of d i s p l a c i n g solvent molecules from a r e g i o n of higher chemical p o t e n t i a l ( i n the d e p l e t i o n zone) to one of lower chemical p o t e n t i a l ( i n the bulk polymer s o l u t i o n ) . There i s a s i m i l a r i t y here with the mixing term ( G £ ) r e f e r r e d to above, where the reverse s i t u a t i o n holds ( i n a good s o l v e n t ) , i . e . solvent i s d i s placed f r o m a r e g i o n of lower c h e m i c a l p o t e n t i a l (the polymer sheaths) to high chemical p o t e n t i a l (the bulk medium). Note, a s we show i n the l a t e r paper here (27), that as φρ increases beyond φρ, the thickness of the d e p l e t i o n zone, 6^, decreases and hence tne a t r r a c t i o n term e v e n t u a l l y decreases again, and the system i s r e s t a b i l i s e d (depletion s t a b i l i s a t i o n ) . The types of i n t e r a c t i o n that can occur between p a r t i c l e s immersed i n polymer s o l u t i o n s has been d e a l t with here on a rather ad hoc b a s i s . Even f o r n e u t r a l systems, the s i t u a t i o n i s complex, although recent e q u i l i b r i u m analyses such as those given by Scheutjens and F l e e r (25) f o r adsorbed homopolymers are beginning to shed some more l i g h t , b u t time-dependent e f f e c t s need to be considered a l s o . As with polymer adsorption per se, one expects the t h e o r i e s to be extended to deal with g r a d u a l l y more complex systems. In t h i s r e s p e c t , the i n t r o d u c t i o n of charge e f f e c t s C h a r g e d s u r f a c e s , p o l y e l e c t r o l y t e s ) w i l l be a major step forward; at best our understanding at present here i s only q u a l i t a t i v e . The u l t i m a t e goal must be to d e r i v e general equations f o r the p a i r p o t e n t i a l , without the a r b i t r a r y (and p o s s i b l y i n c o r r e c t ) s e p a r a t i o n i n t o v a r i o u s terms. G

+

G

+

G

+

G

a

Polymers in Disperse Systems: An Overview - ACS Symposium Series

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