Colloidal Dispersions and Micellar Behavior - ACS Publications

10 Kinetic Theory for the Slow Flocculation at the Secondary Minimum PRANAB BAGCHI

Downloaded by UNIV OF PITTSBURGH on May 3, 2015 | http://pubs.acs.org Publication Date: June 1, 1975 | doi: 10.1021/bk-1975-0009.ch010

Research Laboratories, Eastman Kodak Co., Rochester, N.Y. 14650

Introduction The stability of lyophobic suspensions in the p r e s e n c e o f ionizing m o l e c u l e s is a d e q u a t e l y i n t e r p r e t e d in terms o f t h e b a l a n c e o f van d e r Waals attraction and t h e electrical d o u b l e - l a y e r r e p u l s i o n by t h e Deryagin-Landau-Verwey-Overbeek (DLVO) t h e o r y ( 1 ) . The total interaction potential as a f u n c t i o n o f t h e dis tances o f s e p a r a t i o n o f t h e s u r f a c e s o f two particles is o b t a i n e d by a d d i n g t h e electrical d o u b l e - l a y e r re pulsion and t h e van d e r Waals (vw) attraction f o r vari ous d i s t a n c e s o f s e p a r a t i o n . U s u a l l y it is o b s e r v e d t h a t such total potential c u r v e s g i v e rise t o two po tential minima. The e x t r e m e l y deep minimum a t t h e p o i n t o f c o n t a c t between t h e two s u r f a c e s has been de f i n e d as t h e p r i m a r y minimum and t h e s h a l l o w minimum a t l a r g e r d i s t a n c e s o f s e p a r a t i o n as t h e secondary minimum ( 2 ) . However, t h e situation is somewhat different in the c a s e o f t h e stabilization of l y o p h o b i c s u s p e n s i o n s by polymers o r n o n i o n i c s u r f a c t a n t s . I t has been shown by v a r i o u s a u t h o r s (3-20) t h a t t h e n o n i o n i c i n t e r a c t i o n p o t e n t i a l s between two p a r t i c l e s w i t h adsorbed non i o n i c polymers o r s u r f a c t a n t s owing t o t h e o s m o t i c (3-20) and t h e volume r e s t r i c t i o n (3-20) e f f e c t s i n good s o l v e n t s g i v e r i s e t o l a r g e r e p u l s i o n p o t e n t i a l s . On t h e a d d i t i o n o f t h e vw a t t r a c t i o n , t h e t o t a l po t e n t i a l s i n such systems show o n l y a secondary minimum a t l a r g e d i s t a n c e s o f s e p a r a t i o n between t h e p a r t i c l e s but no deep p r i m a r y minimum as i n t h e c a s e o f i o n i c s t a b i l i z a t i o n (except i n t h e c a s e o f a poor s o l v e n t ) . The s t a b i l i t y l i m i t s o f such systems have been q u a n t i t a t i v e l y i n t e r p r e t e d by B a g c h i i n p r e v i o u s p u b l i c a t i o n s (19,21_) i n terms o f t h e depths o f such secondary m i n i ma. Doroszkowski and Lambourne (22) , by measuring t h e

145

In Colloidal Dispersions and Micellar Behavior; Mittal, K.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.


146

C O L L O I D A L DISPERSIONS

AND

MICELLAE

BEHAVIOR

r e p u l s i o n energy between polymer c o a t e d p a r t i c l e s u s i n g a f i l m b a l a n c e , have i n d e e d shown t h a t i n the t o t a l i n t e r a c t i o n c u r v e s o f such systems, t h e r e i s no deep p r i m a r y minimum. S i n c e t h e p r i m a r y minimum i s i n f i n i t e l y deep, the a g g r e g a t i o n o f s u s p e n s i o n s a t the p r i m a r y minimum i s irreversible. Such i r r e v e r s i b l e a g g r e g a t i o n has been termed as c o a g u l a t i o n by LaMer (23). However, t h e depth o f the secondary minimum can be o f any magnitude so b o t h k i n e t i c a l l y r e v e r s i b l e o r i r r e v e r s i b l e aggre g a t i o n may t a k e p l a c e a t the secondary minimum. Such r e v e r s i b l e a g g r e g a t i o n a t the secondary minimum has been termed by LaMer (23) as f l o c c u l a t i o n . Since a g g r e g a t i o n r a t e i s a measure o f c o l l o i d s t a b i l i t y , v a r i o u s i n v e s t i g a t o r s have d e v i s e d methods o f d e t e r m i n i n g such a g g r e g a t i o n ( c o a g u l a t i o n o r f l o c c u l a t i o n ) r a t e c o n s t a n t s (24-26). The r a t e c o n s t a n t , k~, f o r f a s t b i m o l e c u l a r aggre g a t i o n o f monodisperse s p h e r i c a l p a r t i c l e s o f r a d i u s R i n t h e absence o f any r e p u l s i v e o r a t t r a c t i v e po t e n t i a l s i s g i v e n by t h e Smoluchowski (2^7,2£) t h e o r y as k

Q

«

8TT

(2D)R

,

(1)

where D i s t h e d i f f u s i o n c o n s t a n t f o r monodisperse s p h e r i c a l p a r t i c l e s o f r a d i u s R. According to E i n s t e i n , D = kT/βττη R f o r s p h e r i c a l p a r t i c l e s , where k i s t h e Boltzmann c o n s t a n t , Τ i s t h e a b s o l u t e tempera t u r e , and η i s t h e v i s c o s i t y o f the medium. Thus k may be r e w r i t t e n as k = **kT (9) 0 3η * Q

κ

K Z )

I t i s important to n o t i c e t h a t k i s independent o f the p a r t i c l e r a d i u s as l o n g as t h e l a t t e r i s l e s s than 0.1/xm such t h a t the c o n d i t i o n f o r p e r i k i n e t i c aggrega t i o n i s p r e v a l e n t (2,27,28). The v a l i d i t y o f Equa t i o n (2) has been e s t a b l i s h e d by v a r i o u s a u t h o r s (2931). One o f the most i m p o r t a n t assumptions o f the Smoluchowski t h e o r y i s t h a t a l l e n c o u n t e r s between p a r t i c l e s l e a d t o i r r e v e r s i b l e c o n t a c t s . Such an assump tion, unlike i n diffusion-controlled reaction kinetics where a l l c o l l i s i o n s do n o t l e a d t o r e a c t i o n , i s com p l e t e l y j u s t i f i e d i n the case o f unprotected c o l l o i d s because o f t h e p r e s e n c e o f t h e v e r y deep p r i m a r y m i n i mum a t t h e p o i n t o f c o n t a c t . T h i s i s why E q u a t i o n (2) d e s c r i b e s t h e r a t e c o n s t a n t o f f a s t a g g r e g a t i o n so well. I n t h e case o f s u s p e n s i o n s p r o t e c t e d by i o n i c ad s o r p t i o n , however, t h e s i t u a t i o n i s q u i t e d i f f e r e n t . In such a c a s e , c o a g u l a t i o n i n t h e p r i m a r y minimum Q


10.

Slow Flocculation

BAGCHi

147

involves the crossing of the s t a b i l i z i n g p o t e n t i a l b a r r i e r , and E q u a t i o n (2) i s i n a p p l i c a b l e . Fuchs (2^32) s o l v e d t h e d i f f u s i o n e q u a t i o n i n a p o t e n t i a l f i e l d and showed t h a t t h e c o a g u l a t i o n r a t e c o n s t a n t kp i n such cases may be w r i t t e n as ι k k dH ' (3) 0 I exp - kT (2R+H ) Jo Λ

2R /" — P V V

Q

2

Q

where AG (H ) i s t h e summation o f t h e vw a t t r a c t i o n and the d o u b i e - x a y e r r e p u l s i o n p o t e n t i a l as a f u n c t i o n o f the c l o s e s t d i s t a n c e s o f s e p a r a t i o n o f t h e s u r f a c e s (H ) o f t h e two s p h e r i c a l p a r t i c l e s . I n t h e case o f u n p r o t e c t e d p a r t i c l e s (absence o f any s u r f a c e c h a r g e ) , the i n t e r a c t i o n p o t e n t i a l i s n o t z e r o , as assumed by the Smoluchowski t h e o r y , owing t o t h e p r e s e n c e o f vw attraction. So t h e e x p e r i m e n t a l l y o b s e r v e d t r u e f a s t f l o c c u l a t i o n r a t e constant, s h o u l d be c o r r e c t l y w r i t t e n i n t h e f o l l o w i n g manner as p o i n t e d o u t by McGown and P a r f i t t ( 3 3 ) :


Q

Q

k

o

-

H

d H

r-._pk< o>i

2 R

J

- Ρ | Τ Γ -

0

o

( 4 )

'

(2R+H )

2

Q

where A G ( H ) i s t h e vw a t t r a c t i o n p o t e n t i a l as a function of H . Since AG i s negative, the value o f A

Q

Q

A

dH Γ

2R

I exp 0

;

Δ

6

Α

(

Η

kT

Q

Ο >

(2R+H )

2

Q

i s u s u a l l y between 1 and 1/2 as p o i n t e d o u t by McGown and P a r f i t t (33J, which a c c o u n t s f o r s m a l l d i f f e r e n c e s between e x p e r i m e n t a l f a s t c o a g u l a t i o n r a t e s and t h e Smoluchowski c o a g u l a t i o n r a t e . I t i s important t o note t h a t Equations (2), ( 3 ) , and (4) c o r r e s p o n d t o a g g r e g a t i o n a t t h e p r i m a r y m i n i mum, i . e . , t h e p o i n t o f c o n t a c t i s a t H = 0. However, i n t h e c a s e o f secondary minimum f l o c c u l a t i o n , t h e equilibrium distance of separation a t contact i s a t a v a l u e o f H where t h e secondary minimum o c c u r s . Thus the f l o c c u l a t i o n r a t e c o n s t a n t a t t h e secondary minimum a c c o r d i n g t o t h e Smoluchowski,theory w i t h F u c h s cor r e c t i o n s h o u l d be g i v e n by k as Q

Q

1


148

C O L L O I D A L DISPERSIONS A N D M I C E L L A R

k

BEHAVIOR

(5) S

2R

where 2L i s t h e v a l u e o f H a t which the secondary minimum o c c u r s . Hence, the e x p e r i m e n t a l l y o b s e r v e d s t a b i l i t y r a t i o , W ' , f o r secondary minimum f l o c c u l a t i o n , d e f i n e d as t n e r a t i o o f the f l o c c u l a t i o n r a t e c o n s t a n t o f u n p r o t e c t e d p a r t i c l e s t o t h a t i n the p r e s e n c e o f p r o t e c t i o n , s h o u l d be g i v e n by Q


p

ΔΟ (Η ) Α

2R

1

exp

J2L

dH

0

kT

0

(2R+H )

2

0

(6) A G

f ° / exp

2R J

0

dH

(

" A V" kT

Q

(2R+H )

2

Q

I t has a l r e a d y been p o i n t e d o u t t h a t the two i n t e g r a l s i n E q u a t i o n (6) a r e s m a l l because A G i s n e g a t i v e . A l s o the c o n t r i b u t i o n s of these i n t e g r a l s f o r small v a l u e s o f H where A G i s s u b s t a n t i a l l y n e g a t i v e , a r e e x t r e m e l y s m a l l compared t o the v a l u e o f t h e s e i n t e g r a l s (34). In o t h e r words, t h e c o n t r i b u t i o n s t o t h e s e i n t e g r a l s m a i n l y come from A G . a t l a r g e d i s t a n c e s o f s e p a r a t i o n , where A G . i s s t i l l n e g a t i v e b u t s m a l l i n magnitude. Thus the i n t e g r a t i o n s o f the above f u n c t i o n s from 0 t o oo o r 2L t o oo s h o u l d make v e r y i n s i g n i f icant difference. ( T h i s i s easy t o r a t i o n a l i z e c o n c e p t i o n a l l y i n terms o f the f a c t t h a t when two p a r t i c l e s a r e s e p a r a t e d by v e r y s m a l l d i s t a n c e s , t h e p r o b a b i l i t y o f c o l l i s i o n by d i f f u s i o n i n a g i v e n l e n g t h of time i s so h i g h t h a t the p r e s e n c e o f a l a r g e a t t r a c t i v e p o t e n t i a l r e a l l y does n o t enhance t h e p r o c e s s t o any e x t e n t . Whereas, f o r p a r t i c l e s s e p a r a t e d by l a r g e d i s t a n c e s t h e p r o b a b i l i t y o f c o l l i s i o n by d i f f u s i o n i n the same l e n g t h o f time i s much s m a l l e r t h a n i n the p r e v i o u s c a s e and hence any a t t r a c t i v e p o t e n t i a l , even i f s m a l l , enhances t h e p r o c e s s o f b r i n g i n g the two particles together.) Thus, a c c o r d i n g to the foregoing arguments, a

Q

A



10.

BAGCHi

Slow Flocculation

149

Therefore, W = 1 f o r s e c o n d a r y minimum f l o c c u l a t i o n a c c o r d i n g t o t h e p r e d i c t i o n s o f t h e Smoluchowski t h e o r y w i t h Fuchs' c o r r e c t i o n . I n terms o f t h e u n c o r r e c t e d Smoluchowski E q u a t i o n (2) such a p r e d i c t i o n i s e a s i e r t o s e e . I n t h e c a s e o f secondary minimum f l o c c u l a t i o n , the e q u i l i b r i u m d i s t a n c e o f s e p a r a t i o n a t c o n t a c t i s a t H = 2L. T h e r e f o r e , one c a n say t h a t t h e e f f e c t i v e r a d i i o f t h e p a r t i c l e s a r e R+L r a t h e r t h a n R i n such a case. A c c o r d i n g t o E q u a t i o n ( 2 ) , k i s independent o f R, which i n d i c a t e s t h a t such secondary minimum f l o c c u l a t i o n r a t e s s h o u l d be independent o f t h e p o s i t i o n o f the secondary minimum. The above d e d u c t i o n u s i n g Smoluchowski t h e o r y i s , however, n o t c o n s i s t e n t w i t h e x p e r i m e n t a l o b s e r v a t i o n s of s e c o n d a r y minimum f l o c c u l a t i o n . F r e n s and Overbeek (35-37) have o b s e r v e d i n c o l l o i d a l s i l v e r s o l s t h a t v a r i o u s degrees o f r e v e r s i b l e i n s t a b i l i t y a r e produced at various e l e c t r o l y t e concentrations. They have con c l u d e d t h a t such v a r i e d d e g r e e s o f i n s t a b i l i t y can be a t t r i b u t e d t o f l o c c u l a t i o n a t t h e secondary minima o f v a r i o u s depths. In n o n i o n i c s t a b i l i z a t i o n , the s t a b i l i t y o f A g i d i s p e r s i o n s i n water and i n t h e p r e s e n c e of p o l y ( v i n y l a l c o h o l ) [ i n v e s t i g a t e d by F l e e r e t a l . ( 3 £ , 3 9 Π and p o l y ( s t y r e n e ) (PS) l a t e x d i s p e r s i o n s s t a b i l i z e d by n - d o d e c y l h e x a o x y e t h y l e n e monomer (C-ioEg) i n water i n v e s t i g a t e d by O t t e w i l l and Walker (15) was i n t e r p r e t e d by B a g c h i (19,21) i n terms o f t h e depths o f the secondary minima. The b a s i c argument o f H e s s e l i n k , V r i j , and Overbeek (IB) i n such an i n t e r p r e t a t i o n i s t h a t i f t h e d e p t h o f t h e s e c o n d a r y minimum i s 1 kT o r l e s s , s u s p e n s i o n s would be i n f i n i t e l y s t a b l e as an energy o f 1 kT i s a v a i l a b l e t o a p a r t i c l e i n d i f f u s i o n a l motion. I n systems where t h e d e p t h o f t h e s e c o n d a r y minimum i s 5 kT o r l a r g e r , s u s p e n s i o n s would be Q

Q

The average k i n e t i c energy o f a p a r t i c l e a c c o r d i n g t o s t r i c t s t a t i s t i c a l t h e o r y i s 3/2 kT r a t h e r than 1 kT, as f r e q u e n t l y used i n an approximate sense and as used i n t h i s paper. As t h e u s e o f 3/2 kT i n p l a c e o f 1 kT would make o n l y v e r y minor d i f f e r e n c e s i n t h e computed r e s u l t s p r e s e n t e d i n t h i s paper, 1 kT would be used as the average k i n e t i c energy o f a p a r t i c l e .


150


BEHAVIOR

t o t a l l y u n s t a b l e as 5 kT i s about t h e maximum energy t h a t may be i m p a r t e d t o a p a r t i c l e owing t o t h e r m a l motions o r by s h a k i n g . C o n s e q u e n t l y , i f t h e d e p t h o f t h e secondary minimum i s l a r g e r than 5 kT, f a s t aggre g a t i o n would t a k e p l a c e g i v i n g r i s e t o a s i t u a t i o n which i s k i n e t i c a l l y s i m i l a r t o p r i m a r y minimum f a s t c o a g u l a t i o n . ^However, i f t h e d e p t h o f t h e secondary minimum, |ΔΘ^"| , i s between 1 and 5 kT, slow f l o c c u l a t i o n would o c c u r as o b s e r v e d e a r l i e r (4,16,17). I t s h o u l d always be remembered t h a t t h e v a l u e o f |AG | = 1 kT between s t a b l e and r e d i s p e r s i b l e systems i s a thermodynamic q u a n t i t y whereas t h e v a l u e o f | àG^. | = 5 kT f o r t h e boundary between r e d i s p e r s i b l e and uns t a b l e systems, i s an approximate p h y s i c a l e s t i m a t e (18). The v a l i d i t y o f t h e importance o f t h e d e p t h o f t h e secondary minima i n t h e i n t e r p r e t a t i o n o f t h e s t a b i l i t y o f n o n i o n i c s t a b i l i z a t i o n i s a g a i n i n d i c a t e d by t h e r e c e n t work o f Long e t a l . (4£) i n which they have shown q u a l i t a t i v e l y t h a t t h e weak f l o c c u l a t i o n o f l a t e x p a r t i c l e s i s d e t e r m i n e d by t h e depths o f such secondary minima. The r a t h e r l e n g t h y i n t r o d u c t i o n o f t h i s paper was meant f o r p o i n t i n g o u t t h e f o l l o w i n g two f a c t s : 1. The importance o f t h e secondary minimum i n s u s p e n s i o n s t a b i l i t y , e s p e c i a l l y i n t h e c a s e o f nonionic stabilization. 2. The Smoluchowski t h e o r y o r Smoluchowski t h e o r y w i t h Fuchs' c o r r e c t i o n i s i n c a p a b l e o f p r e d i c t i n g t h e r a t e o f slow f l o c c u l a t i o n a t t h e secondary minimum. Thus, i n t h i s paper a t h e o r y i s d e v e l o p e d f o r t h e p r e d i c t i o n o f t h e slow f l o c c u l a t i o n r a t e a t t h e s e c o n d a r y minimum i n l i g h t o f t h e e x p e r i m e n t a l work o f O t t e w i l l e t a l . (15).


W

Theory The p r i m a r y r e a s o n why Smoluchowski t h e o r y does n o t work i n t h e c a s e o f secondary minimum f l o c c u l a t i o n i s because o f i t s b a s i c assumption t h a t a l l c o l l i s i o n s l e a d t o permanent c o n t a c t . As t h e secondary minimum i s n o t i n f i n i t e l y deep, such an assumption f o r t h e f l o c c u l a t i o n a t t h e secondary minimum i s i n c o r r e c t . Also i n t h e c a s e o f Fuchs' t h e o r y , i t i s c o n s i d e r e d t h a t a g g r e g a t i o n n e c e s s a r i l y means c o n t a c t between t h e p a r t i c l e s u r f a c e s , whereas s e c o n d a r y minimum f l o c c u l a t i o n imp l i e s t r a p p i n g o f two p a r t i c l e s a t t h e secondary m i n i mum, which o c c u r s a t d i s t a n c e s o f s e p a r a t i o n l a r g e r than the p o i n t o f p a r t i c l e - p a r t i c l e contact.


10.

BAGCHi

Slow Flocculation

151

F i g u r e 1 shows a t y p i c a l c u r v e o f t o t a l p o t e n t i a l as a f u n c t i o n o f H ~ , the minimum d i s t a n c e o f s e p a r a t i o n o f the p a r t i c l e s u r f a c e s , w i t h a secondary minimum a t H = 2 L . In t h i s f i g u r e the depth o f the secondary minimum i s d e s i g n a t e d as | A G | / k T . The average energy o f d i f f u s i o n i s 1 kT w i t h a a i s t r i b u t i o n o f e n e r g i e s v a r y i n g from z e r o t o i n f i n i t y a c c o r d i n g t o the Boltzmann d i s t r i b u t i o n o f e n e r g i e s . Thus the e n e r g i e s o f a l l b i m o l e c u l a r c o l l i s i o n s a r e a l s o g i v e n by a Boltzmann d i s t r i b u t i o n o f e n e r g i e s as f o l l o w s ( £ 1 ) : Q


w

df =

exp

(-E/kT)

dE ,

(8)

where d f i s the f r a c t i o n o f c o l l i s i o n s w i t h energy E . Thus, i n a c o l l i s i o n i n v o l v i n g a secondary minimum o f d e p t h | A G I , i f the c o l l i s i o n energy i s l a r g e r than ( | Â G | / k T j - 1 , the p a r t i c l e s w i l l bounce back and no attachment w i l l take p l a c e . An energy o f 1 kT i s s u b t r a c t e d from ( A G - J / k T because an energy o f 1 kT i s a v a i l a b l e to a p a r t i c l e i n t r a n s l a t i o n a l or d i f f u s i o n a l motion. A l s o i f | A G . J = 1 kT and c o l l i s i o n s t a k e p l a c e w i t h an energy Ε l e s s than 1 k T , the p a r t i c l e s w i l l not s t i c k t o g e t h e r because r o t a t i o n a l o r t r a n s l a t i o n a l d i f f u s i o n energy a v a i l a b l e t o such p a r t i c l e s w i l l p u l l them a p a r t . I f the c o l l i s i o n e n e r g y , E , i s l e s s than ( | A G | / k T ) - l , permanent c o n t a c t w i l l t a k e p l a c e . Hence t h e f r a c t i o n s o f c o l l i s i o n s , A f , w i t h energy l e s s than ( | A G - J / k T ) - l l e a d i n g t o permanent c o n t a c t w i l l be g i v e n by the f o l l o w i n g i n t e g r a l , W

W

w

à

f

=

kl f

e

where Ë = ( | A G | / k T ) - l w

x

p

("~E/kT) dE ,

(9)

f o r | A G | £ l kT and W

Ε = 0 f o r I A G U l kT w

Hence, G

A f = 1 - exp

l* wl

(1 - i - j g ^ ) .

(10)

I t i s i m p o r t a n t t o note t h a t i n the c a s e o f d i f f u s i o n c o n t r o l l e d b i m o l e c u l a r r e a c t i o n r a t e s one c o n s i d e r s the overcoming o f a p o t e n t i a l b a r r i e r i n o r d e r t o make permanent c o n t a c t . C o n s e q u e n t l y , one has t o c o n s i d e r o n l y t h o s e c o l l i s i o n s w i t h energy l a r g e r than the v a l u e o f t h e p o t e n t i a l maximum f o r r e a c t i o n t o take p l a c e . In such a c a s e the i n t e g r a t i o n o f E q u a t i o n (9) i s c a r r i e d o u t from 0 t o t h e v a l u e o f the p o t e n t i a l b a r r i e r (or t h e a c t i v a t i o n e n e r g y ) . However, i n the c a s e


152


BEHAVIOR

o f secondary minimum f l o c c u l a t i o n , the c o l l i d i n g p a r t i c l e s a r e t r a p p e d i n a p o t e n t i a l t r o u g h , i n which c a s e i n t e £ r a t i o n o f E q u a t i o n (9) i s c a r r i e d o u t between 0 and E . The b i m o l e c u l a r f l o c c u l a t i o n r a t e i s g i v e n by t h e general equation, -

af -

k

s'

(

Δ

£

)

*

2

*


3 where ψ i s the number o f p a r t i c l e s p e r cm , t i s t i m e , and k i s t h e c o l l i s i o n f r e q u e n c y g i v e n i n t h e most r i g o r o u s form by t h e Smoluchowski e q u a t i o n w i t h F u c h s ' c o r r e c t i o n f o r the secondary minimum by E q u a t i o n ( 5 ) . S i n c e A G . , i s c o n s t a n t f o r a g i v e n s i t u a t i o n , one can w r i t e t h e f l o c c u l a t i o n r a t e c o n s t a n t , k , a t the secondary minimum as f o l l o w s (42): g

(12) The r a t e c o n s t a n t f o r the c o a g u l a t i o n o f the u n p r o t e c t e d p a r t i c l e s w i l l be g i v e n by E q u a t i o n (3) as i n such a c a s e a g g r e g a t i o n t a k e s p l a c e i n t h e p r i m a r y minima. So t h e e x p e r i m e n t a l s t a b i l i t y r a t i o , as d e f i n e d p r e v i o u s l y , i n t h e c a s e o f secondary minimum f l o c c u l a t i o n would be g i v e n by W as: p Y P

P l e a s e n o t e t h a t i n E q u a t i o n (13) the e f f e c t o f t h e hydrodynamic d r a g on t h e c l o s e approach o f the two p a r t i c l e s , f o r m u l a t e d by Spielman (53) and Honig e t a l . (54) and r e c e n t l y e x p e r i m e n t a l l y s u b s t a n t i a t e d by H a t t o n e t a l . (55), has been n e g l e c t e d . The r e a s o n behind t h i s i s t h a t the c o r r e c t i o n f a c t o r s f o r R = 30 nm and R = 30 + δ nm ( w h e r e δ ~ 0 — 5 nm) a r e not t o o d i f f e r e n t from each o t h e r such t h a t i n the c a l c u l a t i o n o f W p , which i n v o l v e s the l o g o f the r a t i o s o f t h e two c o r r e c t i o n f a c t o r s , c o n t r i b u t e i n s i g n i f i c a n t l y to the value of l o g W _ , c a l c u l a t e d n e g l e c t i n g the hydrodynamic d r a g e f f e c t T 5 6 ) . E X

p x


10.

BAGCHi

Slow Flocculation

153

AG (H ) r oo 2R / exp kt ^2L L i

W. EXP

p A G

r

2R

e x

J

( H

0

o

l-exp(l

r

dH, (2R+H ) Q

dH, (2R+H )'

P[-kT-

Q


A

0

(13)

I AG. Wi kT

Since the approximation i n d i c a t e d i n Equation s t i l l v a l i d , one g e t s

W. EXP exp(l

-

(7)

is

(14)

J3ç w kT

E q u a t i o n (14) i n d i c a t e s t h a t the r a t e o f f l o c c u l a t i o n i s s o l e l y d e t e r m i n e d by the d e p t h o f the s e c o n d a r y minimum, IAG^I. I t i s o b s e r v e d by i n s p e c t i o n o f E q u a t i o n (14) t h a t as | AG-J g e t s l a r g e r W approaches 1; t h a t i s , f o r a deep secondary minimum f l o c c u l a t i o n r a t e approaches the Smoluchowski r a t e o r the f a s t f l o c c u l a t i o n s i t u a t i o n . χ

C a l c u l a t i o n s and

ρ

Results

F i g u r e 2 shows a p l o t o f W „ as a f u n c t i o n o f AG c a l c u l a t e d using Equation ( i f ; . I t i s observed t h a t f o r A G = -1 kT, W = oo. As A G d e c r e a s e s , EXP ^ s e s and becomes one a t a v a l u e o f AG., o f aêout -5.5 kT, i n comparison w i t h the t o t a l i n s t a b i l i t y l i m i t o f 5 kT as e s t i m a t e d by H e s s e l i n k , V r i j , and Overbeek (18). I t has been p o i n t e d o u t p r e v i o u s l y t h a t the s t a b i l i t y l i m i t s o f PS l a t e x d i s p e r s i o n s i n water and i n the p r e s e n c e o f C E as i n v e s t i g a t e d by O t t e w i l l and Walker (16) can be r e a s o n a b l y w e l l i n t e r p r e t e d i n terms o f the d e p t h s o f the s e c o n d a r y minima (21) . Total s t a b i l i t y i n the s c a l e o f time means a s t a b i l i t y r a t i o o f i n f i n i t y and t o t a l i n s t a b i l i t y i n the s c a l e o f time means a s t a b i l i t y r a t i o o f u n i t y . For c o n d i t i o n s i n t e r m e d i a t e between t h e s e two l i m i t s , slow f l o c c u l a t i o n was o b s e r v e d f o r t h e PS-LatexC,-Eg-water system by O t t e w i l l and Walker (15) . The s t a b i l i t y r a t i o s as x p

W

W

W

χ

ρ

W

e c r e a

1 2

g



154

C O L L O I D A L DISPERSIONS A N D M I C E L L A E

Figure 1. Secondary Minimum

Schematic diagram of a sec ondary minimum

£-5 -

-2

-3

-4

-5

-6

-7

-8

-9

-10

-II

Depth of the secondary minimum in kT (AG ) W

Figure 2.

Plot of W XP as a function of AG according to Equation (14) E

BEHAVIOR

W



10.

BAGCHi

Slow Flocculation

155

measured by them f o r t h i s system i n the p r e s e n c e o f e l e c t r o l y t e beyond the c r i t i c a l f l o c c u l a t i o n concen t r a t i o n was p r e v i o u s l y (21) termed as t h e l i m i t i n g s t a bility ratio. These e x p e r i m e n t a l s t a b i l i t y r a t i o s a r e d i r e c t l y comparable t o the t h e o r e t i c a l v a l u e s p r e d i c t e d by E q u a t i o n (14) s i n c e such e x p e r i m e n t a l v a l u e s c o r r e spond t o the s t a b i l i t y due t o the n o n i o n i c s t a b i l i z e r , which a r e known t o g i v e r i s e t o o n l y s e c o n d a r y minima. I t i s a d v i s e d t h a t the r e a d e r examine r e f e r e n c e (21_) i n o r d e r t o u n d e r s t a n d the a p p l i c a b i l i t y o f E q u a t i o n (14) t o t h e d a t a d e r i v e d i n r e f e r e n c e (21) from t h e paper o f O t t e w i l l and Walker (15) . T a b l e I l i s t s the v a l u e s o f the a d s o r p t i o n l a y e r t h i c k n e s s e s of C 6 Latex (radius =30 nm), the depths o f the secondary minima as c a l c u l a t e d by B a g c h i (21), t h e e x p e r i m e n t a l limiting s t a b i l i t y r a t i o s as d e t e r m i n e d by O t t e w i l l and Walker (15), and t h e t h e o r e t i c a l l y c a l c u l a t e d v a l u e s o f t h e s t a b i l i t y r a t i o s from t h e v a l u e s o f |AG-J using E q u a t i o n (14) a t v a r i o u s l o g molar e q u i l i b r i u m concen t r a t i o n s of C , c Latex-C-. 6 ~ system i n v e s t i g a t e d by O t t e w i l l and Walker TI5;. In t h i s c a l c u l a t i o n the Hamaker c o n s t a n t used f o r the PS L a t e x water system was 5 χ 1 0 ~ e r g as used by O t t e w i l l and Walker (15). F i g u r e 3 shows a d i r e c t comparison o f t h e experimental W t o the t h e o r e t i c a l l y c a l c u l a t e d W as a f u n c t i o n or the l o g molar e q u i l i b r i u m c o n c e n t r a tions of C E . The e r r o r l i m i t s i n T a b l e I and F i g u r e 3 c o r r e s p o n d t o an e r r o r o f +1 nm i n the d e t e r m i n a t i o n o f t h e a d s o r p t i o n l a y e r t h i c k n e s s as i n d i c a t e d by O t t e w i l l and Walker ( L 5 ) . E

o

n

t

e

p

s

1 2

E

f

o

r

t

n

e

p

2

s

E

w

a

r

2

1 4

p

1 2

p

6

F i g u r e 5 shows the p a r t i c l e s i z e dependence o f the s t a b i l i t y r a t i o o f PS l a t e x d i s p e r s i o n s as a f u n c t i o n of the e q u i l i b r i u m c o n c e n t r a t i o n of j 2 6 The v a l u e s o f W_ i n F i g u r e 5 were c a l c u l a t e d f o r p a r t i c l e r a d i i r a n g i n g between 10 and 480 nm w i t h a q u i t e l e g i t i m a t e assumption t h a t the a d s o r p t i o n l a y e r t h i c k n e s s e s o f CjoE/: as a f u n c t i o n o f i t s e q u i l i b r i u m c o n c e n t r a t i o n a r e i d e n t i c a l f o r PS l a t i c e s o f a l l sizes. c

E

e

xp

Discussions The agreement o b s e r v e d i n F i g u r e 3 between t h e e x p e r i m e n t a l l i m i t i n g s t a b i l i t y r a t i o and t h e o r e t i c a l l y c a l c u l a t e d s t a b i l i t y r a t i o using Equation (14) can be c o n s i d e r e d t o be r a t h e r good c o n s i d e r i n g the u n c e r t a i n t i e s i n v o l v e d i n the e s t i m a t i o n o f t h e Hamaker c o n s t a n t used i n r e f e r e n c e (21) t o c a l c u l a t e the v a l u e s o f IAG | . As p o i n t e d o u t p r e v i o u s l y , the t h e o r e t i c a l v a l u e o f W^^ i s s o l e l y dependent upon the



c

a

•5.7 •5.5 •5.2 •4.6

-5.3 -4.8 -4.4 -3.9 -3.5 -3.1

E

12 6

0( 0— 0.8( 0— 2.8(1.8— 8.0(7.0— 8.6(7.69.2(8.2—

erg.

U s i n g E q u a t i o n (14)

5 χ 10~

1 3

21.

W

-3.13(- « -0.66(-1.18— -0.13(-0.16— -0.1K-0.14— -0.10(-0.12-

g

*

0 0 1.4

Q

3

W

3

1.2

0 (0 0.06(0 — oo (0.77— 00 00 00

(1 - | AG / k T | ) = 0 f o r G / k T < l .

c

0.12) 0.80) 00 )

W

the c o r r e s p o n d i n g values of | A G |

E X p

Theoretical log W using

System

used f o r t h e PS L a t e x - w a t e r system

-2.42) -1.18) -0.43) -0.10) -0.09) -0.08)

l

Experimental EXP FlocDepths o f t h e Floccula- culasecondary minima, t e d by t e d by AG , in k r La(N0 ) HNO-

Hamaker c o n s t a n t

and t a k i n g

Calculated i n reference

15.

1.0) 1.8) 3.8) 9.0) 9.6) 10.2)

Twice the adsorp tion layer thicknessesj 2L i n nm

T a k e n from r e f e r e n c e

d

2

V a l u e s o f 2 L , A G . . , and S t a b i l i t y R a t i o s f o r t h e PS L a t e x - C , E - W a t e r a t V a r i o u s Log M o l a r C o n c e n t r a t i o n s o f C ,

Log molar equilibrium concentra tions of

T a b l e 1.


10.

BAGCHi

157

v a l u e o f | A G - J [ E q u a t i o n ( 1 4 ) ] . Hence, the a c c u r a c y w i t h which p r e d i c t i o n s can be made u s i n g E q u a t i o n (14) w i l l depend e n t i r e l y on t h e a c c u r a c y w i t h which t h e depths o f t h e secondary minima can be c a l c u l a t e d . S i n c e the o b j e c t i v e o f t h i s paper i s b a s i c a l l y t o d e v e l o p the t h e o r y and t o p r e d i c t t h e o b s e r v e d t r e n d , no e x t e n s i v e e f f o r t was made on an a c c u r a t e e s t i m a t i o n o f t h e Hamaker c o n s t a n t . Nor was the V o i d e f f e c t (43) , whose importance i n e s t i m a t i n g t h e l o n g - r a n g e i n t e r a c t i o n s has been shown r e c e n t l y (44-46), c o n s i d e r e d i n the c a l c u l a t i o n o f t h e vw a t t r a c t i o n f o r t h e PS L a t e x 12 6~ system. However, i t was shown p r e v i o u s l y (21) t h a t t h e i n c l u s i o n o f the V o i d e f f e c t t o c a l c u l a t e the vw a t t r a c t i o n i s w e l l w i t h i n t h e e r r o r o f e s t i m a t i o n o f the Hamaker c o n s t a n t f o r the PS L a t e x - w a t e r system. I t i s e x t r e m e l y i n t e r e s t i n g t o note t h a t t h e l i m i t s of s t a b i l i t y (W = 1 for unstable-redispersib l e and ^ = oo f o r r e a i s p e r s i b l e - s t a b l e ) a r e i n e x c e l l e n t agreement w i t h the t h e o r e t i c a l p r e d i c t i o n s as o b s e r v e d i n F i g u r e 3. T h i s i s t r u e f o r the f o l l o w i n g reasons. In t h e s t a b i l i t y l i m i t o f W = «> f o r s t a b l e d i s p e r s i o n s t h e e q u i l i b r i u m v a l u e o f ÈJ[=2L) i s large. I t i s w e l l known (21_) t h a t the vw a t t r a c t i o n s a t l a r g e d i s t a n c e s a r e much b e t t e r r e p r e s e n t e d by c o n v e n t i o n a l t h e o r i e s than a t s m a l l e r d i s t a n c e s of s e p a r a t i o n . Also a t t h e l i m i t o f i n s t a b i l i t y and r e d i s p e r s i b i l i t y (W = 1) where | A G . J i s about 5 kT, t h e r a t e o f dec r e a s e o f t h e vw a t t r a c t i o n i s so sharp compared t o t h a t f o r l a r g e r d i s t a n c e s o f s e p a r a t i o n (H ) t h a t the e r r o r i n t h e v a l u e o f 2L has v e r y l i t t l e e r f e c t on W ; as f o r | â G - J l a r g e r than 5 kT, W i s essent i a l l y e q u a l t o u n i t y ( F i g u r e 2 ) . Thus f o r a r e d i s p e r s i b l e system where | A G . J i s between 5 kT and 1 kT, a more q u a n t i t a t i v e agreement between E q u a t i o n (14) and e x p e r i m e n t a l r e s u l t s may be o b s e r v e d o n l y i f a more a c c u r a t e e s t i m a t i o n o f the vw a t t r a c t i o n can be made. The P a r s e g i a n and Ninham (48) t y p e o f c a l c u l a t i o n s h o u l d be p a r t i c u l a r l y u s e f u l f o r such e s t i m a t i o n s . I t i s i n t e r e s t i n g t o note t h a t i f a v a l u e o f 2.3 χ 1 0 ~ e r g i s used f o r t h e Hamaker c o n s t a n t o f t h e PS L a t e x water system (shown i n F i g u r e 4) a much b e t t e r f i t o f the theory with the experimental r e s u l t s i s observed. However, t h i s . d o e s n o t , i n any way, mean t h a t the v a l u e o f 2.3 χ 10 f o r t h e Hamaker c o n s t a n t i s any b e t t e r than t h e v a l u e o f 5 χ 1 0 ~ e r g as used by O t t e w i l l and Walker (15). In F i g u r e 5, i t i s i n t e r e s t i n g t o note the t h e o r e t i c a l p r e d i c t i o n s o f the p a r t i c l e s i z e dependence o f t h e s t a b i l i t y r a t i o o f t h e PS l a t e x d i s p e r s i o n s as a f u n c t i o n of the e q u i l i b r i u m c o n c e n t r a t i o n of C, E . It C


Slow Flocculation

E

w a t e r

E X p

Ε

χ

ρ

χ

ρ

E

Q

1 4

1 4

?


fi

158

COLLOIDAL

DISPERSIONS

A N DM I C E L L A R

BEHAVIOR

I**- Present Theory

Experimental values of limiting log W

EXP

x Flocculated by La(N0 ) Ο Flocculated by HN0 3

3


Ι ι Theoretically calculated ^ valves of log W^p

Figure 3. Comparison of the experimen tal limiting stability ratios for the PS Latex-C E -water system of Ottewill and Walker with the theoretical values of the stability ratio calculated using Equation (14) and the values of \AG \ from Table I as a function of the log molar equilibrium concentration of C E . The error bars correspond to an error of ±1 nm in the values of the adsorption layer thicknesses. Hamaker constant used for the calculation of \AG \ is 5 X 10r erg as used by Ottewill and Walker. Particle radius = 30 nm. 12

6

W

i2

Smoluchowski Theory and / Smoluchowski Theory with Fuchs Correction

6

u

W

-5

-4

-3

Log molar equilibrium concentration of C, Eg 2

Present theory

4

Experimental values of limiting log W

EXP

x Flocculated by La(N0 ) Ο Flocculated by HN0 3

3

ι Theoretically calculated valves of log W EXP

Smoluchowski Theory and / Smoluchowski Theory with Fuchs Correction -6

-5

-4

-3

Log molar equilibrium concentration of C E )2

Figure 4. Same plot as Figure 3 except the Hamaker constant used for these cal culations of \AG \ is 2.3 χ 10 erg 14

6

W



10.

BAGCHi

159

Slow Flocculation

Log molar equilibrium concentration of C | E 2

6

Figure 5. Size dependence of the stability ratio of dispersions of PS latex in water as a function of the equilibrium concentration of


160

COLLOIDAL DISPERSIONS AND

MICELLAR BEHAVIOR

i s observed t h a t f o r p a r t i c l e s of r a d i i l e s s than 1 2 0 nm one i s a b l e t o v a r y the s t a b i l i t y r a t i o between 0 and oo as a f u n c t i o n o f C, r a d s o r p t i o n such t h a t | A G | ranges between 5 . 5 ana I kT. However, f o r p a r t i c l e s o f r a d i i l a r g e r t h a n 1 2 0 nm, the depths o f the secondary minima a t s a t u r a t i o n a r e between 5 . 5 and 1 kT. C o n s e q u e n t l y i n such c a s e s , a f t e r s a t u r a t i o n , one has d i s p e r s i o n s w h i c h f l o c c u l a t e a t c o n s t a n t r a t e s s l o w e r t h a n the Smoluchowski r a t e and never r e a c h com plete s t a b i l i t y . F o r p a r t i c l e s l a r g e r t h a n 4 8 0 nm the depths o f t h e s e c o n d a r y minima, even a t s a t u r a t i o n , are l a r g e r than 5 . 5 kT. Hence, under a l l c o n d i t i o n s such systems would undergo f a s t f l o c c u l a t i o n . So f a r the d i s c u s s i o n has been on the i n t e r p r e t a t i o n o f the p e r i k i n e t i c s e c o n d a r y minimum f l o c c u l a t i o n u s i n g the d e p t h s o f such minima. Hiemenz and V o i d (49) have o b s e r v e d s i m u l t a n e o u s f l o c c u l a t i o n d e f l o c c u l a t i o n i n carbon black - p o l y ( s t y r e n e ) - t o l u e n e and c a r b o n b l a c k - p o l y ( s t y r e n e ) - c y c l o h e x a n e systems obeying the f o l l o w i n g k i n e t i c e q u a t i o n : E

2


w

- II

= kN

2

- 0N

,

(15)

where k i s the f l o c c u l a t i o n r a t e c o n s t a n t and β i s the d e f l o c c u l a t i o n rate constant. In l i g h t o f the p r e s e n t d i s c u s s i o n s , the q u e s t i o n i s whether one can equate k with k and i n t e r p r e t d e f l o c c u l a t i o n i n terms o f the unsuccessful c o l l i s i o n s . In t h e work o f Hiemenz and V o i d ( 1 0 , 2 3 ) the k i n e t i c u n i t s t h a t underwent f l o c c u l a t i o n and d e f l o c c u l a t i o n were l a r g e f l o e s o f r a d i u s above 2 0 0 t o 3 0 0 nm. F o r such l a r g e f l o e s i z e s , as has been p o i n t e d out p r e v i o u s l y , c o n d i t i o n s f o r o r t h o kinetic flocculation prevails. Thus the Smoluchowski c o l l i s i o n f r e q u e n c y as used i n the d e r i v a t i o n of E q u a t i o n (12) i s inapplicable. C o n s e q u e n t l y , f o r such l a r g e f l o e s t h e f l o c c u l a t i o n r a t e c o n s t a n t k may not be compared w i t h k . The f l o e s i n such systems a r e o f random s t r u c t u r e as t h e o r e t i c a l l y deduced by V o i d (50) and M e d a l l i a ( 5 1 , 5 2 ) and e l e c t r o n microscopically demonstrated f o r graphon s u s p e n s i o n s i n heptane by B a g c h i and V o i d ( 1 0 ) . The d e f l o c c u l a t i o n o f such l o o s e l y bound and randomly s t r u c t u r e d f l o e s i n t h i s system has been r i g h t l y i n t e r p r e t e d by Hiemenz and V o i d ( 2 4 , 4 9 ) t o be due t o s h e a r i n g i n t h e r m a l g r a d i ents. Thus such a d e f l o c c u l a t i o n p r o c e s s cannot be r a t i o n a l i z e d i n terms o f u n s u c c e s s f u l c o l l i s i o n s . How e v e r , the o b s e r v a t i o n t h a t such f l o e s d e f l o c c u l a t e i n t h e r m a l g r a d i e n t s perhaps i n d i c a t e s t h a t the b i n d i n g energy o f such f l o e s comes from a g g r e g a t i o n a t s e c o n d a r y minima o f depths l e s s t h a n 5 . 5 kT. g

g


10.

BAGCHi

Slow Flocculation

161


Acknowledgments The a u t h o r would l i k e t o g r a t e f u l l y acknowledge the t h o u g h t f u l comments o f P r o f e s s o r Egon M a t i j e v i c ' o f C l a r k s o n C o l l e g e o f T e c h n o l o g y , Potsdam, Ν. Υ . , on t h i s work. Thanks a r e a l s o g i v e n t o D r . G . J . Robersen o f the v a n ' t H o f f L a b o r a t o r i u m , U t r e c h t , f o r v e r y k i n d l y p r o v i d i n g the hydrodynamic d r a g c o r r e c t i o n s mentioned i n c o n n e c t i o n w i t h E q u a t i o n (13) and p o i n t i n g o u t an e r r o r i n E q u a t i o n (2). The a u t h o r would a l s o l i k e t o v e r y g r a t e f u l l y acknowledge the t h o u g h t f u l s u g g e s t i o n s o f P r o f e s s o r s R o b e r t D. and M a r j o r i e J . V o i d d u r i n g the p r e p a r a t i o n o f t h i s p a p e r . Summary I t i s w e l l known t h a t the i n s t a b i l i t y o f a s u s p e n s i o n can be due t o slow f l o c c u l a t i o n a t a secondary minimum i n the p a r t i c l e - p a r t i c l e p o t e n t i a l p r o f i l e , f o r b o t h i o n i c and n o n i o n i c s t a b i l i z a t i o n . In most c a s e s o f i o n i c s t a b i l i z a t i o n , the p o t e n t i a l b a r r i e r t o the p r i m a r y minimum i s n o t h i g h enough and c o n s e q u e n t l y one o b s e r v e s f a s t a g g r e g a t i o n a t t h e p r i m a r y minimum. However, i n the n o n i o n i c c a s e , where the d i s p e r s i o n medium i s a good s o l v e n t f o r the s t a b i l i z e r , c a l c u l a t i o n s i n d i c a t e t h a t u n s t a b l e systems always e x h i b i t a secondary minimum, which i s the p r i m a r y cause o f i n s t a bility. The measure o f such i n s t a b i l i t y i s t h e a g g r e gation rate. The Smoluchowski t h e o r y o f a g g r e g a t i o n i s n o t g e n e r a l l y v a l i d f o r secondary minimum f l o c c u l a t i o n , s i n c e i t makes the i m p l i c i t assumption t h a t e v e r y c o l l i s i o n l e a d s t o permanent c o n t a c t . Such an assumption i s c o r r e c t o n l y i n the c a s e o f p r i m a r y minimum c o a g u l a t i o n where the a t t r a c t i o n p o t e n t i a l , a t the p o i n t o f c o n t a c t , goes t o i n f i n i t y . Fuchs theory i s i n a p p l i c a b l e t o secondary minimum f l o c c u l a t i o n b e cause i t c o n s i d e r s o n l y t h o s e c o l l i s i o n s w i t h e n e r g i e s h i g h enough t o overcome the p o t e n t i a l b a r r i e r t o make c o n t a c t a t the p r i m a r y minimum as s u c c e s s f u l c o l l i sions. In t h e c a s e o f secondary minimum f l o c c u l a t i o n s , o n l y t h o s e c o l l i s i o n s w i t h e n e r g i e s l e s s t h a n the d e p t h o f the secondary minimum w i l l l e a d t o permanent c o n t a c t o r , i n o t h e r words, s u c c e s s f u l f l o c c u l a t i o n . On t h i s a s s u m p t i o n , a k i n e t i c t h e o r y f o r the secondary minimum f l o c c u l a t i o n i s d e v e l o p e d i n t h i s paper and a p p l i e d t o the p o l y ( s t y r e n e (PS) l a t e x - n - d o d e c y l h e x a o x y e t h y l e n e monomer ( Ο , - Ε ^ ) w a t e r system as i n v e s t i g a t e d by O t t w i l l and W a l k e r . Tne t h e o r e t i c a l l y p r e d i c t e d dependence o f the l i m i t i n g s t a b i l i t y r a t i o due t o the n o n i o n i c s u r f a c t a n t as a f u n c t i o n o f t h e e q u i l i b r i u m 1


162

COLLOIDAL DISPERSIONS AND MICELLAR BEHAVIOR

c o n c e n t r a t i o n o f C , E seems t o agree f a i r l y w e l l w i t h the e x p e r i m e n t a l o b s e r v a t i o n s o f O t t e w i l l and W a l k e r . 2

Literature 1.


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