a theoretical approach for polymeric dispersant action

8/3/2019 A Theoretical Approach for Polymeric Dispersant Action

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JOURNAL OF COLLOID AND INTERFACE SCIENCE 22, 269-284 (1966)

A The oretical Ap proach for Polym eric Dispersant ActionI. C a lcu la t ion o f Ent rop ic R epu ls ion Ex er ted b y R andom Polymer C ha ins

Termina l ly A dsorbed on P lane Sur faces and Spher ica l Par t ic lesE . J . C L A Y F I E L D a N D E . C . L U M B

"Shel l" Research Ltd. , Thornton Research Centre, P.O. Box 1, Chester, EnglandReceived September 30, 1965

At3STRACTA quantit ative treatmen t of an entropic repulsion mechanism by which polymericdispersants may protect colloidal suspensions from adhesion to plane surfaces, orfrom flocculation, has been developed. The dispersants are terminally adsorbedmacromolecules, treated as random chains constructed on a cubic lattice accordingto a four-choice model with a bond angle of 90; a Monte Carlo method is used to ob-tain the distribution of thicknesses.The results show that increasing polymer size can lead to irreversible adhesion orflocculation of particles of a given size. Alte rnat ively there is a minimum to the par-ticle size stabilized by a given macroraolecule. The minimum particle radius sta -bilized decreases only slowly with increasing molecular weight of polymer; the maxi-

mum radius increases almost linear ly with molecular weight.The largest particles prevented from flocculating by a given polymer molecule aretwice the radius of the largest ones prevented from adhering to a plane surface. Otherdifferences between flocculation and adhesion behavior are small. The greater thefreedom of movement of the polymer chain in all respects, provided it remains ad-sorbed, the more effective it is as a dispersant.I N T R O D U C T I O N

T h e e x i s t e n c e o f a s t e r i c h i n d r a n c e r e p u l -s i o n , i n w h i c h m e c h a n i c a l i n t e r f e r e n c e o f t h ea d s o r b e d l a y e r s o n d i f f e r e n t s u r f a c e s p r e -v e n t s t h e c l o s e a p p r o a c h o f t h e s u r f a c e s , h a sb e e n r e c o g n i z e d f o r o v e r t e n y e a r s (1), a n d am e c h a n i s m f o r i t s a c t i o n w a s e x a m i n e dt h e o r e t i c a l l y b y M a c k o r a n d v a n d e r W a a l s( 2 ) . T h e y c o n s i d e r e d r i g i d r o d s , a n c h o r e d a to n e e n d t o t h e s u r f a c e b y a f r e e l y h i n g e dj o i n t , a n d c a l c u l a t e d t h e r e d u c t i o n i n e n -t r o p y r e s u l t i n g f r o m r e s t r i c t i o n o f t h e i rm o v e m e n t b y a s i m i l a r o p p o s e d s u r f a c e .O n l y p l a n e , p a r a l l e l s u r f a c e s w e r e c o n s i d e r e d .T h e i r c a l c u l a t i o n a l s o t o o k i n t o a c c o u n t d e -s o r p t i o n c a u s e d b y c l o s e a p p r o a c h o f t h es u r f a c e s a n d t h e c o n s e q u e n t l o c a l l y h i g h c o n -c e n t r a t i o n o f a d s o r b a t e . T h e y s h o w e d t h a tpotent ial energy maxima of 15 kT to 25 kTwere possible in systems of carbon black

c o a t e d w i t h p a r a f f i n c h a i n s o f a b o u t 1 6c a r b o n a t o m s , i n a c c o r d a n c e w i t h t h e s t a -b i l i z a t i o n o b s e r v e d e x p e r i m e n t a l l y . T h es a m e a u t h o r s a p p l i e d t h e m e t h o d t o a c h a i no f f o u r l i n k s , b u t i t s u s e f o r c h a i n s m a r k e d l yl o n g e r t h a n t h i s s e e m s i m p r a c t i c a b l e .

I n t h e w o r k d e s c r i b e d i n t h i s p a p e r a m o r et r a c t a b l e m e t h o d h a s b e e n u s e d t o c a l c u l a t et h e r e d u c t i o n i n e n t r o p y o n c o m p r e s s i o n o f at e r m i n a l l y a d s o r b e d p o l y m e r m o l e c u l e , a n dt h e r e s u l t s h a v e b e e n a p p l i e d t o e s t i m a t e t h ei n t e r a c t i o n s b e t w e e n a s p h e r e a n d a p l a t e ,a n d b e t w e e n t w o s p h e r e s . T h e s e s y s t e m sf o r m m o d e l s o f t h e p r e v e n t i o n o f a d h e s i o n o fp a r t i c l e s t o a p l a t e a n d o f t h e p r e v e n t i o n o ff l o c c u l a t i o n o f p a r t i c l e s , r e s p e c t i v e l y . T h ec a l c u l a t i o n a p p l i e s t o a n i r r e v e r s i b l y a d -s o r b e d m o l e c u l e w i t h a n a d s o r b a b l e g r o u pa t o n e e n d o n l y , c o n t a i n e d i n a s o l v e n t h a v -i n g n o n e t i n t e r a c t i o n w i t h t h e u n a d s o r b e d

2 6 9


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2 7 0 C L A Y F I E L D A N D L U M B

p a r t o f t h e m o l e c u le ; a h y d r o c a r b o n s o l u t io no f a b l o c k e o p o l y m e r c o m p r i s in g a h y d r o -c a r b o n c h a i n a n d a p o l a r c h a i n i s a n a p p r o -p r i a t e b a s i s f o r t h e m o d e l .T h e r e t a r d a t i o n e f f e ct is n o t t a k e n i n t oa c c o u n t in t h e c a l c u la t io n o f v a n d e r W a a l s -L o n d o n a t tr a c t i o n . F u r t h e r w o r k , as y e ~u n p u b l i s h e d , h a s s h o w n t h a t t h i s e f f e c t a l -t h o u g h b y n o m e a n s n e g l i g i b l e d o e s n o tc h a n g e t h e g e n e r a l c o n c l u s i o n s e x p r e s s e dh e r e .

METHODENTROPY CHANGE ON COMPRESSION OF A

SINGLE RANDOM CHAINT h e e n e r g y r e q u i r e d t o c o m p r e s s a si ng l er a n d o m c h a i n i s f i rs t c a l c u l a te d . A n i d ea l , r a n -d o m c h a i n , o n e e n d o f w h i c h a d h e r e s t o ap l a n e s u r f a c e , i s c o n s i d e re d , a n d a d e t e r m i n a -

t i o n is m a d e o f t h e p r o p o r t i o n o f t h e t o t a ln u m b e r o f c o n f i g u r a t i o n s o p e n t o i t t h a t i sl o s t w h e n a p a r a l l e l, i m p e n e t r a b l e p l a n e s u r -f a c e is p l a c e d a t a d i s t a n c e f r o m t h e f i rs tp l a n e . T h i s a s s u m e s , i n e f f e c t, t h a t m a c r o -m o l e c u l e s o n o p p o s i n g s u r f a c e s d o n o t i n t e r -p e n e t r a t e . T h e w o r k o f B lu e s t o n e a n d V o l d( 3 ) o n t h e g e n e r a t i o n o f t w o c h a i n s i np r o x i m i t y i n d i c a t e s t h a t t h i s is s u b s t a n -t i a l l y t r u e .T h e i n f l u e n c e o f e x c l u d e d v o l u m e i s c o n -s i d e r a b l e w h e n t h e c h a i n i s g r e a t l y c o m -p r e s s e d . P r e s e n t a n a l y t i c a l m e t h o d s f o rr a n d o m c h a i n c a l c u l a t i o n s o m i t c o n s i d e r a -t i o n o f t h is e f f e c t a n d a r e t h e r e f o r e in a p p r o -p r i a t e f o r t h e p r e s e n t p u r p o s e ; n o r w a s t h ei n t r o d u c t i o n o f a c o r r e c t i o n f a c t o r t o a l l o wf o r e x c l u d e d v o l u m e e f f e c t c o n s i d e r e d s a f e .A m o r e d i r e c t a p p r o a c h w a s t h e r e f o r e m a d ei n w h i c h a M o n t e C a r l o t e c h n i q u e w a s u s e dt o b u i ld u p a n u m b e r o f r a n d o m c h a i ns o n ac u b i c l a t t i c e , t o m e a s u r e t h e h e i g h t s ( o rt h i c k n e s se s ) o f t h e c h a i n s a n d t o o b t a i n t h ed i s t r i b u t i o n o f h e i g h t s s t a t is t i c a l ly . T h i sw o r k w a s c a r r i e d o u t w i t h a d i g i t a l c o m -p u t e r a n d i s t h e s u b j e c t o f a s e p a r a t e p a p e r( 4 ) . T h e c h a i n s w e r e b u i l t o n a c u b i c l a t t i c ea c c o r d i n g t o a f o u r - c h o ic e , 9 0 b o n d a n g l em o d e l, a n d n o t m o r e t h a n o n e m o n o m e r u n i tw a s a l l o w e d t o o c c u p y a l a t t i c e s i t e , i . e . , t h ee x c l u d e d v o l u m e e f f e c t w a s t a k e n i n t o a c -c o u n t . T h e M o n t e C a r l o t e c h n i q u e u s e d ,w h e n a p p l i e d t o fr e e ( n o t a d so r b e d ) r a n d o m

c h a i n s , g i v e s c o n f i g u r a t i o n a l d i m e n s i o n s i na g r e e m e n t w i t h t h e a n a l y t i c a l e x p r e s s i o n( 5) f o r e n d - t o - e n d d i s t a n c e s o f l o n g c h a i n s ,w h i c h i s w e l l c o n f i r m e d b y e x p e r i m e n t a lm e a s u r e m e n t s ." T h e r e s u lt s w e r e e x p r e ss e d i n t h e f o r m o ft h e p r o p o r t i o n , W ( l l ) , o f t h e t o t a l n u m b e r o fc o n f i g u r a t i o n s f o r w h i c h t h e d i s t a n c e o f t h ef u r t h e s t e x t r e m i t y o f t h e c h a i n f r o m t h ea d s o r b i n g p l a n e i s l e ss t h a n ll. B y c o n v e n -t i o n a l t h e r m o d y n a m i c s , t h e d e c r e a s e ine n t r o p y , A S , r e s u l t i n g f r o m t h e r e d u c e df r e e d o m o f t h e c h a i n is

1A S = k In - -W ( l l ) 'w h e r e k is B o l t z m a n n ' s c o n s t a n t ; t h e f r e ee n e r g y c h a n g e p e r m o l e c u l e , A F R , w h i c h i nt h e a b s e n c e o f a n y o t h e r e n e r g y c h a n g e s ish e r e e q u a l t o t h e c h a n g e i n p o t e n t i a l e n e r g y( P E ) , i s g i v e n b y

~FR- I n W ( l l ) .k T

F i g u r e i s h o w s [ - l n W ( /1 )] p l o t t e d a g a i n s tL , t h e r a t i o o f l~ t o l ~ , t h e r o o t - m e a n - s q u a r e( r .m . s .) v a l u e o f l, f o r v a r i o u s n u m b e r s o fl i n ks i n t h e c h a i n , a n d f o r t h e a n a l y t i c a l e x -p r e s s i o n ( 5 ) m e n t i o n e d a b o v e .ENTROPIC REPULSION FOR RANDO~I POLY-MER CHAINS ADSORBED ON 32 SPHERICAL

PARTICLE AND A PLANE SURFACER a n d o m c h a in s a d s o r b e d o n a s u r fa c e

w i l l h a v e t h e i r e x t r e m i t i e s a t a v a r i e t y o fd i s t a n c e s , l , f r o m t h e s u r f a c e ; t h e r . m . s .v a l u e o f l d e n o t e d b y l r , i s t a k e n a s a m e a s -u r e o f t h e t h i c k n e s s o f t h e u n d i s t u r b e da d s o r b e d f i l m .O n a v e r a g e , a fr e e r a n d o m c h a i n h a s t h es a m e d i a m e t e r i n a n y d i r e c t io n , a n d t h i s isa p p r o x i m a t e l y t r u e f o r a n u n c o n f i n e d a d -s o r b e d c h a i n a s w e l l . T h e r . m . s , h e i g h t s a n dw i d t h s o f t h e c h a i n s o b t a i n e d f r o m t h eM o n t e C a r l o c a l c u l a t i o n s a r e s h o w n i nT a b l e I . T h e s u r f a c e c o v e r a g e , 0 , c a n b et a k e n a r b i t r a r i l y a s u n i t y w h e n t h e c h a i n so c c u p y a n a v e r a g e a r e a o f lr~ e a c h . I f a d s o r p -t i o n is s tr o n g , c l o s e r p a c k i n g t h a n t h i s c a no c c u r , w i t h 0 > 1 , a n d w i t h c o n s e q u e n ts t r o n g l a t e r a l c o m p r e s s i o n o f t h e c h a i n s .W i t h t h i s c o n v e n t i o n , i t i s a s s u m e d t h a t ,


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A T H E O I~ Y O F P O L Y M E I ~I C D I S P E R S A N T A C T I O N . I

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0 0 .2 0 .4 0 . 6 0 .8 1 .0 1 ,2 1 .4 1 .6 1 .8 2 , 0lF IG . 1 . F r e e e n e r g y o f c o m p r e s s i o n f o r v a r i o u s r a n d o m c h a i n s

w h e n c h a i n s o f d i f f e r e n t l e n g t h s a r e c o rn - m o l e c u l e , a n d t h e f r e e e n e r g y o f c o rn r e s s l o np a r e d , a c o n s t a n t v a l u e o f 0 i m p l i e s c o n - p e r u n i t a r e a i s S x F ~ O / l~ 2 .s t a n t s t r e n g t h o f a d s o r p t i o n . I f t h e s u r f a c e o f t h e s p h e r e o f r a d i u s a i s aT h e f re e e n e r g y o f c o m p r e s s i o n i s A F ~ p e r d i s t a n c e H f r o m t h e p l a n e , t h e l e n g t h l~ o f


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2 7 2 C L A Y F I E L D A N D L U M BT A B L E I

ROoT-MEAN-SQuARE DIMENSIONS OF MODELPOLYMER CHAINS IN UNITS OF ONE BOND

LENGTHNo. of links in chain Height (thickness) Width

10 2.38 2.4730 5,67 5.33100 12,60 12.32

t w o e q u a l m o l e c u l e s j u s t i n c o n t a c t , o n e e a c ho n t h e t w o n e a r e s t p o i n t s of th e s p h e r e a n dp l a n e , i s H / 2 ( F ig . 2 ) . I f t h e m o l e c u l e o n t h es p h e r e m a k e s a n a n g l e 4) w i t h t h e a x i s o f t h es p h e r e p e r p e n d i c u l a r t o t h e p l a n e , t h e l e n g t hI1 o f t h i s m o l e c u l e a n d t h e o n e o n t h e p l a n ew h i c h i t t o u c h e s i s

ll - - - - - a( 1 - - cos 4)) - t- H1 + cos 4)T h e a r e a o f t h e a n n u l u s , o f r a d i u s x a n dw i d t h dx, a t t h e j u n c t i o n o f a l l s u c h p a i r s o fm o l e e u l e s s u b t e n d i n g a n a n g l e 4 ) , i s

2~rx dx ,w h e r e

x = ( a + l l ) s in 4)( 2 a + H )

- sin 4)1 + cos 4)f r o m w h i c h t h e a r e a b e c o m e s

S i n c e2~r (2a n - H ) 2(1 + co s 4)F

sin 4) d4).

AFR- - - ln W(ll)k T

t h e f r e e e n e r g y o f c o m p r e s s io n o v e r t h i s a r e aisAFR 0 2~r(2 a -t- H ) 2 sin 4) co s 4) d4)kTl~2 (1 + co s 4))2

0 2 v (2 a + H ) 2 s in 4) cos 4) d4)U (1 + cos 4))~

I - - I n W ( / 1 )] .T h e c os,4 ) t e r m i n t h e n u m e r a t o r r e s o l v e s t h er e p u l s i v e f o r c e s a l o n g t h e a x i s p e r p e n d i c u l a rt o t h e p l a n e . F o r t h e w h o l e s p h e r e a n d p l a n e

PLate

FIG, 2 . N ome nclature for der ivat ion of sphere/plane entrop ic repulsion.

t h e f r ee e n e r g y i s( 2 a H ) ~k T /~0 ( 1 + cos 4))2

. s in 4 ) co s 4 ) [ - ln W (L ) d4)].T h i s e x p r e s s i o n m a y b e i n t e g r a t e d h u m e r i -c a l l y t o o b t a i n F ~ / k T 8 a s a f u n c t i o n o fH/21~ f o r v a r i o u s v a l u e s o f a / l ~ a n d f o rv a r i o u s n u m b e r s o f l i n k s i n t h e c h a i n , t h ei n t e g r a t i o n b e i n g c o n t i n u e d u n t i l dFR/d4)b e -c o m e s n e g l i g i b l e .T h e v a n d e r W a a l s - L o n d o n a t t ra c t i o n ,V ~ , p , c a l c u l a t e d in t h e A p p e n d i x f o r a p l a t ew h i c h m a y b e c o n s i d e r e d i n fi n it e ly t h i c kc o m p a r e d t o t h e s iz e o f t h e s p h e r i c a l p a r t i c l e ,is g i v e n b y

A F 2 a ( H + a )V A ~

i n w h i c h A is t h e H a m a k e r a t t r a c t i o n c o n -s t a n t ( 6 ) i n e r g s .T h e r e s u l t a n t i n t e r a c t i o n e n e r g y , F z / k T ,is o b t a i n e d a s t h e s u m o f th e r e p u l s i o n


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A THEORY OF POLYMERIC DISPERSANT ACTION. I 273

FIG. 3. Nomenclature for derivation of sphere/sphere entropic repulsion.FR/I~T (positive) and t h e attraction V ~ / k T(negative).ENTROPIC REPULSION FOR I~ANDOM POLY-

MER CHAINS ADSORBED ON TWOSPHERICAL PARTICLESThe derivation of sphere/sphere entropicrepulsion may be carried out in a way essen-tially similar to that used for the sphere/plate system, the same results being em-ployed for the calculated entropy change oncompression of a single random chain andthe same conventions for surface coverageand adsorbed film thickness. The derivation

is as follows: Consider two spherical particlesof radius a and distance H apart (Fig. 3);the radius of the circle subtending a semi-angle at the centers of the spheres from aplane midway between the spheres is x =(a + H / 2 ) tan , and the area of the atom-]us formed by this circle, with its thicknesssubtending an angle de, is2~r a + ~ tan Cse e 2 d .

The length l~ of the two adsorbed moleculeswhich touch at this annulus is given bya + H

2l l - - a .COS ~

The repulsive (or compressive) free energyover this annulus is thenAFRO ( H )kTl-~J .2~r a -k ~ tan ~bsec 2 ~b COS ~b de(~rO a + tan~sec [ - - In W ( l l ) ] d

The cos term resolves the repulsiveforces along the axis of the spheres.

Integrat ing this expression from = 0 to = ~- /2 gives the total energy of repulsionFR between the spheres; thus

k~ - 2 tOf : i : ( a H ) 2"4 \ ~ + ~ t a I l C s e c + l n W ( L ) de .

This expression may be integrated numeri-cally to obtain F~/kTO, the integration beingcontinued until d F , / d is negligible.The London-van der Waals attract ionenergy for the two spheres, V~:, is given bythe Hamaker expression

A [ 2 2 s~ - 4 \VA~ = -~ [ s-T~_4+p+inT/ ,iin which s = ( H + 2a) / a . The resultantinteraction energy F s / k T is obtained as thesum of the repulsion F R / k T and the attrac-tion V A J k T .

' I '801" / _ _ _ f O / Ir : 5

/ r a / l~ = I \ \2 o [ \ e : l \ + \ \t .]~+~=b,~ . " ~ +[ i / + "o - -

_ _ _ t0 I 2H / 2 [ rFIG. 4. Curves for potential energy of inter-action between a sphere and plane coated withpolymer with a 30-1ink chain. A/k T = 25.


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274 CLAYFIELD AND LUMBIO 0

8 0E

6 0 +

4 0

2 2 Ig

N

y0. ~ ~ +

Param eters against ' ~ " ' ~curves show 0 .j~ - I 0a / ] r

FIG. 5 . Po ten t ia l energy max ima and min imaa s a f u n c t i o n o f a /1 4. T h i r t y - l i n k e h Mn. A/kT =12.5.

RESULTS AND DISCUSSIONSPHERICAL PARTICLE/PLANE SURFACE

INTERACTIONSize of Macrom olecule and Su rface Cover-age. Typical results for the potential energy

(PE) of interaction are shown in Fig. 4 as afunction of the dimensionless separation dis-tahoe H / 2 L for a single value of the Hamakerconstant A and a single value of the numberof links in the polymer chain. At low valuesof a/l~, i.e., for thick films or small particles,the maxima are small and the minimanegligible. Reduct ion of 0 lowers the maxi-mum, with little change in the minimum.With higher values of a/ l r , the maxima andminima both increase arithmetically.This behavior has an interesting practicalconsequence. Increasing the length of thepolymer molecule with the object of increas-ing the fihn thickness may actually increasethe likelihood of irreversible adhesion. Con-versely, there is a lower limit to the size ofparticle stabilized by a given polymer. Thisis because large molectfles, which pack to a

small number of molecules per unit area, pro-vide only a small number of interactingmolecules on close approach of a particleand a plane. On the other hand, small mole-cules, although preventing irreversible ad-hesion, may allow the reversible adhesionassociated with the PE minimum to occur.Thus there is an optimum size of moleculefor preventing particles of a given diameterfrom adhering.To allow ready interpolation, the informa-tion in Fig. 4 has been plotted in Fig. 5 asPE maxima and minima against a/ l r .This form of plot may be used to obtaincurves of constant PE maximum and mini-mum as a function of 0 and a/l~ ; Fig. 6, forexample, shows the 15 kT maximum and-1.5/cT minimum when A / k T is 12.5 for apolymer chain of 30 links The value of themaximum increases above and to the rightof the 15 kT line; conditions for protectionfrom irreversible adhesion are found on thisside of the line Similarly, the conditions forprotection from reversible adhesion arefound above and to the left~ f the - 1 5 k Tline.The diagram is divided by these lines intofour regions. In the lower left region,bounded by A B D , particles are protectedfrom reversible but not from irreversible ad-hesion; relatively low coverage and smallparticles (for a given film thickness) thuslead to irreversible adhesion. The lower rightregion with low coverage and larger particlesis subject to both forms of adhesion, andparticles will ultimately adhere irreversibly.With large particles at high coverage, in theupper right region, reversible adhesion isfound. Finally, within the region A B C , thesystem is protected fr om adhesion, withinthe limits implied by the values of themaximum, minimum, attraction constant,and polymer size.Attraction Constant and Po tential EnergyLimits . The effect of increasing the attrac-tion constant A is naturally to decrease thesize of the protected region. Figure 7 showsthe 15 ~T maxima and -1.5 /cT minimawhen A / k T = 12.5, 25, and 50. These val-ues of A / k T are chosen to cover the prob-able range of practical interest. The increasefrom 12.5 to 25 doubles the minimum cover-age for stabilization, and nearly halves themaximum particle size that can be stabilized


7/16

A T H E O R Y O F PO L Y/V IE I~ IC D I S P E R S A N T A C T I O N . I 2752 .0

I n c r e a s i n g p a r t i c l es i z e w i t h g i v e n I r R e v e r s i b l e a d h e s i o n .

)'~ -o ~ P r o t e c t e d f r o m ~-3 I l ~

1 .0 2 ~ ' a d h e s i o n 7 -" ~ - C T /

o~u~e ad h . - - - - - - = 4 , z - - "eSlon 4_~ . - - - - - -~ - [ r r e v e r s , b l e+ - - - ~ a d h e s i o nDo o ', ~ ~ 4 ; ; f

d / I r

F l a . 6 . S t a b i l i t y d i a g r a m f o r a d h e s i o n , w i t h 3 0 -1 in k p o l y m e r c h a i n , A / k T = 12.52 .0

1.5

\ , , > . . . . ! . . . . . . . 7 + + _. . . . + . . . . . . . 4 _ _ _

. . . . . .

if

i r t I i I I

O 0 I 5 4 5 6 7a / I F

F IG . 7 . V a r i a t i o n o f s t a b i l i t y d i a g r a m f o r a d h e s i o n w i t h a t t r a c t i o n c o n s t a n t . T h i r t y - l i n k p o l y m e rc h a i n , P E m a x i m u m i s 1 5 k T , m i n i m u m - 1 .5 k T .at a given coverage. With A/hT = 50, 0needs to be greater than unity to achieveany stabilization.The 15 ]cT maximmn and the -1.5 le tminimum in PE are chosen on the basis ofthe Smoluchowski theory, as applied byVerwey and Overbeek (7), and of the aver-age translational energy of particles, respec-tively. Since the experimentally appropriatevalues for such stability criteria may vary,the effect of changing each of these limits bya factor of two is given in Fig. 8. Doublingthe limit of the minimum increases themaximum size of the stabilized particle by

1.5

, I ii .O + + + - 1. S k T ; - 3 k Ttkr j

+ j fo 2 X ~ 8 ~;

a / 1 rF IG . 8 . V a r i a t i o n o f s t a b i l i t y d i a g r a m f o r a d -

h e s i o n w i t h h e i g h ts o f m a x i m a a n d m i n i m a .T h i r t y - l i n k p o l y m e r c h a i n. A / k , T = 12.5.


8/16

2 7 6 C L A Y F IE L D A N D L U M B2 . 0

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(~ 1 .0

0 . 5

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1 .0

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l 2 3 4 5 6 7a / I r~ + . . . ~ .,- , t ( c ) A / k T = 5 0

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I 2 3 5 6 7a / l r

F IG . 9. V a r i a t i o n o f s t a b i l i t y d i a g r a m f o r a d h e s i o n w i t h n u m b e r o f l i n k s in p o l y m e r c h a i n . P E m a x i -m u m i s 1 5 k T , m i n i m u m - 1 . 5 k T . ( F i g u r es a g a i n s t c u r v e s sh o w n u m b e r s o f l i n k s. )5 0 % a n d d o e s n o t a ff e ct t h e m i n i m u mp a r t i c l e s i z e . H a l v i n g t h e l i m i t o f t h e P Em a x i m m n h a l v e s t h e m i n i m u m s i z e o fp a r t i c l e t h a t c a n b e s t a b i l i z e d a t a g i v e nv a l u e o f 0, a n d s l i g h t l y r e d u c e s t h e m i n i m u mv a l u e o f 0 r e q u i r e d f o r s t a b i l iz a t i o n ; t h e

c h a n g e d o e s n o t a f f e c t t h e m a x i n m mp a r t i c l e s i z e .Flexibility of Polymer Chain. T h e e f f e c t ofc h a n g i n g t h e n u m b e r o f l in k s i n ti le p o l y m e rm o l e c u l e is s h o w n i n F i g . 9 . T h i s f i g u r e m a yb e i n t e r p r e t e d i n t w o w a y s : w i t h a c o n s t a n t


9/16

A~THEORY OF POLY ME RIC DISPER SAN T ACTION. I 277

I0 0

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FIo. 10. Maximum and minimum radii of particles stabilized a gainst adhesion, in un its of bond length,as a function of film thickness.l i n k l e n g t h , o r b o n d l e n g t h , ~ n i n c r e a s e i nt h e n u m b e r o f l i n k s i s e q u i v a l e n t t o a n i n -c r e a s e i n m o l e c u l a r w e i g h t o f a g i v e n t y p eo f p o l y m e r ; t h i s p a r t i c u l a r v a r i a b l e i s di s-c u s s e d la t e r . H o w e v e r , i f t h e f i l m t h i c k n e s slr i s k e p t c o n s t a n t , a n i n c r e a se i n t h e n u m b e ro f l i n k s i s t a n t a m o u n t t o a n i n c r e as e i n t h ef l e x ib i l i t y o f t h e m o le c u l e . I n F ig . 9 , w h ic hc o v e r s t h e r a n g e o f A / k T v a l u e s e x a m i n e d ,i n c r e a s in g t h e f l e x ib i l i t y i n t h i s w a y i n c r e a s e st h e d e g r e e o f p r o t e c t i o n a g a i n s t a d h e s i o n .T h e e f fe c t o n t h e m i n i m u m s iz e o f s t a b i li z e dp a r t ic l e , a n d t h e m i n i m u m 0 r e q u i re d , i sg r e a te r t h a n t h a t o n t h e m a x i m u m s t a b il iz e dpar t ic le s ize .T h i s i s a r e f l e c t i o n o f t h e w a y i n w h i c h ,w h e n t h e c o m p r e s s i o n i s o n l y s l i g h t , t h es t i f f n e s s o f t h e c h a in m a k e s l i t t l e d i f f e r e n c et o t h e c o m p r e s s i b i l it y a n d t h e r e f o re t o t h ed e p t h o f t h e m i n i m u m ; a t g r e a t e r c o m p r e s -s io n s , t h e r e a r e r e l a t i v e ly f e w e r c o n f ig u r a -t i o n s a v a i l a b l e t o t h e m o r e f l e x ib l e c h a in ,a n d t h e P E m a x i m u m i n c re a se s w i t h i n -c r e a s i n g n u m b e r o f l i n k s (c f . Fig . 1 ) . Fo re x a m p l e , d i s c o u n t i n g t h e e x c l u d e d v o l u m ee f f e c t f o r s im p l i c i t y , w i th a c h a in l y in g

w h o l l y i n t h e s u r f a c e t h e r e a r e t w o c h o ic e sf o r e a c h l i n k , a n d f o r t h e f r e e c h a i n , f o u rc h o ic e s p e r l i n k. T h u s t h e p r o p o r t i o n o fc o n f i g u r a ti o n s ly i n g w h o l l y i n t h e s u r f a c e i s21/41 = 2 - l fo r a t en - l ink cha in , an d21 /41 = 2 -1 for a hu nd re d- l in k cha in .Molecular W eight of Polymer. T h e r e l a t i o nb e t w e e n t h e m o l e c u l a r w e i g h t ( n u m b e r o fl i n k s ) a n d t h e f i l m t h i c k n e s s i s k n o w n f r o mt h e M o n t e C a r l o r e s u l t s . T h e i n t e r a c t i o nc a l c u l a t i o n s a r e e x p r e s s e d i n t e r m s o f t h er a t i o o f t h e p a r t ic l e r a d i u s t o t h e f il m t h i c k -n e s s , a n d s o t h e l im i t i n g r a d i i o f s t a b i l i z e dp a r ti c le s m a y b e o b t a i n e d i n t e r m s o f al i n k ( o r b o n d ) l e n g t h .F i g u r e 1 0 s h o w s r e s u l ts i n t h i s f o r m ; t h er a d i u s a i s p l o t t e d a g a i n s t t h e f i l m t h i c k n e s s1 ,, b o t h b e i n g i n u n i t s o f b o n d l e n g t h . T h em a x i m u m p r o t e c t e d p a rt i c le siz e w i t h 0 = 1in c r e a s e s a lm o s t l i n e a r ly w i t h 1, ; t h e m in i -m u m p r o t e c t e d p a r t i c l e s i z e , a l s o w i t h0 = 1 , i s i n d e p e n d e n t o f lr w h e n A / k T =1 2 . 5 a n d a t h ig h e r v a lu e s o f A / k T d e c r e a s e sa s l , i n c r e a s e s . T o t h e l e f t o f t h e i n t e r s e c t i o no f e a c h p a i r o f li n e s is t h e r a n g e o f s m a l l f i lm


10/16

2 . 0

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c b i . o

2 7 8 C L A Y F I E L D A N D L U M B

+

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R e s t r i c t e d /

+

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N o t/restricted/+ / - ~ / + / N o t r e s t r i c te d

- + ' " + - ~ R e s t r ic t e di 2 5 4 5 6 7

a / I t ,Fro. 11. Stabil i ty diagram for adhesion for a polymer chain of 100 l inks, with restr icted ares. PEmaximum is 15 kT , m i n im u m - 1 . 5 k ,T ; A /k T = 12.5.

t h i c k n e s s e s w h i c h w i l l n o t s t a b i l i z e a n yp a r t i c l e .T h e m a x i m u m p r o t e c t e d p a r t i c l e s i z e d e -c r e a s e s w i t h i n c r e a s i n g v a l u e s o f t h e a t t r a c -t i o n c o n s t a n t ( F i g . 1 0a ) a n d w i t h a r i t h -m e t i c a l l y d e c r e a si n g P E m i n i m u m ( Fi g.10 b) b u t i s i n d e p e n d e n t o f P E m a x i m u m .T h e m i i d i ) n u m p a r t i c l e s i z e d e c r e a s e s w i t hi de e re a si li g P E m a x i m u m , b u t is i n d e p e n d e n to f P E m i i ~ m u m . T h e p a r t i c le s iz e f o r w h i c h0 i s a m i n i m u m ( d a s h e d l i n e s ) i s a l i t t l e l e sst h a n t h e m a x i m u m p a r t i c le siz e, a n dd e c r e a s e s ~ s l i g h t ly w i t h d e c r e a s i n g P Em a x i m u m .

Resul t s Expressed in Ac tua l Par t i cl e Ra di i .I n t h e p r e v i o u s s e c t i o n p a r t i c l e s i ze s a r e e x -p r e s s e d i n t e r m s o f t h e b o n d l e n g t h o f t h ' ~p o l y m e r m o d e l . T o e x p r e s s t h e s e i n, s a y ,a n g s t r o m u n i t s , i t is n e c e s s a r y t o c o n s i d e r t h er e l a t i o n b e t w e e n t h e m o d e l o f a p o l y m e rm o l e c u l e a n d a r e a l m o l e c u l e .T h e e f f e c t o n a c o m p l e t e l y r a n d o m c h a i no f r e s t r ic t i n g t h e b o n d a n g l e t o 9 0 is t o l e a v et h e e n d - t o - e n d d i s ta n c e u n c h a n g e d , a n d t h er e s t r ic t i o n o f r o t a t i o n a b o u t t h e b o n d t om u l t i p l e s o f 9 0 a l so r e s u l t s i n n o c h a n g e ( 8 ).T h e p r o p e r t i e s o f t h e c o m p l e t e l y r a n d o mc h a i n a n d t h e 9 0 b o n d a n g le , f o u r - c h o i c em o d e l a r e t h e r e f o r e i d e n t i c a l .I f t h e b o n d a n g l e o f a c o m p l e t e l y r a n d o mc h a i n i s r e s t r i c t e d t o t h e a n g l e o f t h e c a r b o n -c a r b o n b o n d , t h e m e a n - s q u a r e e n d - t o - e n d

d i s t a n c e i s d o u b l e d . I f t h e a n g l e ~ b e t w e e nt h e p l a n e s o f t w o p a i r s o f a d j a c e n t b o n d sw i t h o n e b o n d i n c o m m o n i s r e s t r i c t e d s ot h a t t h e a v e r a g e v a l u e o f c o s ~ i s c o s ~ , t h ee n d - t o - e n d d i s t a n c e i s i n c r e a s e d b y t h efa c to r (1 + eo-o-s-q~)/(1 - co s ) ; w it h nor e s t r i c t i o n , c o s q~ i s z e r o a n d t h e f a c t o r i s 1 .T h i s f a c t o r i s u s u a l l y d e t e r m i n e d e x p e r i -m e n t a l l y .

I n a r e a l , r e s t r i c t e d p o l y m e r c h a i n w i t h l i n k s o f l e n g t h f l a n d m e a n - s q u a r e e n d - t o -e n d d i s t a n c e ~ ,= 2t~2. 1 + _ _ c s . .

1 - cosA r a n d o m c h a i n o f t h e s a m e e n d - t o - e n d d is -t a n e e a n d t h e s a m e n u m b e r o f l in k s m a yt h e n b e d e f i n e d s o t h a t

w h e r e h i s t h e l e n g t h o f a s t a t i s t i c a l s e g m e n te q u a l t o ( 2 . 1 4 - c o s .~ .1 - - co sS i n c e t h e 9 0 m o d e l i s e q u i v a l e n t t o t h ec o m p l e t e l y r a n d o m c h a in , a r e a l p o l y m e rm o l e c u l e w i t h z l i n ks o f l e n g t h ~ i s e q u i v a -l e n t t o a m o d e l m o l e c u l e w i t h z l in k s o fl en g th ~1 F o r c a r b o n - c a r b o n b o n d s , ~ i s 1 . 54 A .F o r , s a y , p o l y i s o b u t y l e n e i n b e n z e n e , t h e


11/16

A T H E O R Y O F P O L Y M E R I C D I S P E R S A N T A C T I O N . I 2 79T A B L E I I

R O O T - M E A N - S Q u A R E D I M EN S IO N S O F U N R E -S T R I C T E D 1 0 0 L I N K C H A I N A N D O F 1 0 0 L I N KC H A I N R E S T R I C T E D T O A N A R E A O F 1 2 X 1 2L I N K L E N G T H S , IN U N I T S O F O N E B O N D L E N G T H

(th~e~gnhets W idth V o l u m eU n r e s t r i c t e d 1 2 .6 0 1 2 . 3 2 1 90 7R e s t r i c t e d 1 4 . 1 0 7 . 7 0 8 3 5

T A B L E I I I~ I A x I M U M A N D M I N I M U M S T A B I L I Z E D P A R T I C L E

S I Z E S ~ I N B O N D L E N G T H S ~ F O R R E S T R I C T E DAND UNRESTI%ICTED CI-IAIN

Minimum MaximumUn res t r ic ted 5 .4 81W idth res t r ic ted 6 .1 78

e x p e r i m e n t a l v a l u e ( 9) f o r ~1 / ~ i s 2 . 60 , s ot h a t f l l i s 4 .0 A.

I f w e t a k e t h i s v a l u e f o r t h e b o n d l e n g t ha n d A / k T = 1 2.5 , t h e r a d i u s o f t h e s m a l l e s tp r o t e c t e d p a r t i c l e i n F i g . 1 1 o f 5 .5 b o n dl e n g t h s b e c o m e s 22 A . W i t h a n y l i k e l y v a r i a -t io n s in th e b o n d l e n g t h o r th e P E m a x i m u m ,i t se e m s i m p r o b a b l e t h a t t h e s m a l l e s t p r o -t e c t e d p a rt i c l e w o u l d b e m o r e t h a n a b o u t ah u n d r e d a n g s t r o m s i n d i a m e t e r. H o w e v e r ,i t w o u l d s e e m t h a t e n t r o p i c r e p u l s i o n i s i n -h e r e n t l y u n a b l e t o p r e v e n t t h e i r r e v e r s i b l ed e p o s i t i o n o f v e r y s m a l l c o l l o i d a l p a r t i c l e s .Closely Pa cke d Adsorption. T o s i m u l a te t h eb e h a v i o r o f a m o l e c u le a d s o r b e d a t a c o v e r -a g e c o n s i d e r a b l y g r e a t e r t h a n u n i t y , a M o n t eC a r l o c a l c u l a t i o n w a s c a r r i e d o u t f o r m o l e -c u le s o f 10 0 l i n k s r e s t r i c t e d t o a n a r e a o n t h es u r f a c e a li t t le l es s t h a n t h e s q u a r e o f t h er . m . s, w i d t h o f t h e u n r e s t r i c t e d a d s o r b e dm o l e c u l e . T h i s g a v e a v a l u e f o r 0 o f 2. 56 a n dt h e m o l e c u l a r d i m e n s i o n s s h o w n i n T a b l eI I . T h e I n W c u r v e i s s h o w n i n F i g. 1, a n d as t a b i l i t y d i a g r a m i n F i g . 1 1 .

S i n c e t h e l a t e r a l r e s t r i c t i o n o n t h e c h a i ni n c r e a s e s i t s h e i g h t , o r f i l m t h i c k n e s s , t h er e p u l si o n m i g h t b e g r e a t e r t h a n f o r t h e ( u n -r e a l i z a b l e ) s i m i l a r c h a i n a d s o r b e d a t t h es a m e c o v e r a g e b u t w i t h o u t l a t e r a l r e s tr i c -t i o n . T h e m e a n v o l u m e o c c u p i e d b y t h e r e -s t r i c te d c h a i n i s a b o u t h a l f t h a t o f t h e u n r e -s t r i ct e d o n e , h o w e v e r , so t h a t m a n y o f t h e

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FIG. 12. Var ia t ion of s tab i l i ty d iagram forf loe e u la t io n w i th h e ig h t s o f P E ma x ima a n d m in -ma. A/kT = 12.5.e x p e c t e d " t h i c k e r " c o n fi g u ra t io n s a r e v e r yr a r e l y f o u n d . T h e r e s u lt is t h a t t h e r e -s t r i c t e d c h a i n i s m o r e c o m p r e s s ib l e a tm o d e r a t e c o m p r e s si o n s t h a n t h e u n r e s t r ic t e do n e , b u t t h a t b o t h a r e e q u a l l y c o m p r e s s i b l ea t h i g h e r c o m p r e s s i o n s ( F i g . 1 ) . I n c o n s e -q u e n c e , t h e P E m a x i m a a r e cl o se ly s i m i l ar


12/16

280 CLAYFIELD AND LUMB

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+ " - - 7 + = . 2 5 4, " ~ - - " - - ~ - L _ _ + _ /. ~ . ~ + J t 3Ibkl ~ I ' A / I ~ T = 1 2 .5 ~ 7

6 ,+/ #/ // / / /1 _ + ~ ~ ~ + -

o & & If> 1 '2 1 4a/ . I r( b ) 5 0 l in k p o l y m e r c h a i n

+

' ~ + ~ ,, A / k T = 1 2 . 5 - - ~ /~ + _ #" / _ _/ . , . b i . . . i / - -

a l l r( c ) I 0 0 l i n k p o l y m e r c h a in

I I i

A / k T = 5 0+ . . . . .

- r / " c _ ~ . . . . . . . + -~ - . . . . . . . . .

a / ! r

FIG. 13. Variation of stabili ty diagram for floceuIation with attraction constant. PE maximum is15kT , minimttm - 1.5 kT .for the two chains, whereas the more re-stricted chain has the arithmetically greaterminimum.The stability diagram in Fig. 11 reflectsthis behavior. The lines for the 15 kT maxi-

mum, and hence the minimum particle sizeratios a/ l~ , are almost identical for the twocases. The line for the --1.5 kT minimum ismoved so that the maximum particle sizeratios are reduced by 20 % by the restriction.


13/16

A T H E O R Y O F P O L Y M E R I C D I S P E R S A N T A C T I O N . I 281T A B L E I V

COMPARISON OF FLOCCULATION OF PARTICLES AND ADIcIESION O17 PARTIC LES TO A PLAT E2S = two spher ica l pa r t ic les .S P = s p h e r ic a l p a r t i c l e a n d p la te :]

N o , o f l i n k s , A / k T

1 2 . 51 0 2 5

5 0] 1 2 . 5

3 0 i 2 5i 5 0

1 2 . 51 0 0 2 55O

PE limits ,kT un i t s

i 5 / - 1 . 57 .5 / -1 .53o1 5 / - 3

I 7 . 5 / - 3[

M i n i m u m 0 M a x i m u m a / l r ( 0 = 1 ) M i n i m u m a / l r ( e = 1)

R a t i o R a t i o R a t i o2 S S P S P / 2 S 2S S P 2 S / S P 2 S S P 2 S / S P

0 . 5 7 ' 0 . 6 2 1 . 0 9 1 3 . 7 6 . 4 2 . 1 4 3 . 1 2 . 3 1 . 3 50 . 9 2 1 . 0 5 1 . 1 4 8 . 9 ~ - - 6 . 2 5 ~ - -1 . 3 6 1 . 7 4 1 . 2 8 . . . . . .0 . 3 2 0 . 3 2 1 . 0 1 4 . 6 6 . 9 2 . 1 2 1 . 0 3 0 . 9 5 1 . 0 80 . 5 4 0 . 6 0 1 . 11 9 . 6 4 . 4 2 . 1 8 1 . 4 1 . 3 1 . 0 80 . 8 2 1 . 0 1 . 2 2 6 . 7 5 2 . 8 5 2 .37 2 . 3 2 . 8 5 0 . 8 10 . 1 9 0 . 2 0 1 . 0 5 1 3 . 9 5 6 . 4 5 2 . 1 6 0 . 5 2 0 . 4 2 1 . 2 40 . 2 9 0 . 3 4 1 . 1 7 9 . 1 5 4 . 1 2 2 . 2 2 0 . 6 0 0 . 4 7 ' ' 1 . 2 80 . 5 4 0 . 6 7 1 . 2 4 6 . 3 2 . 8 2 . 2 5 0 . 7 0 0 . 6 9 1. 01

0 . 3 2 0 .3 2 1 . 0 0 1 4 . 6 6 . 9 2 . 1 2 1 .0 3 0 . 9 5 1 .0 80 . 2 5 0 .2 6 5 1 . 0 6 1 4 . 6 6 . 9 2 . 1 2 0 . 3 5 0 . 5 0 . 7 00 . 2 8 5 0 . 3 0 5 1 . 0 7 2 1 . 7 1 0 . 2 2 . 1 2 1 . 0 3 0 . 9 5 1 . 0 80 . 2 4 0 . 2 5 1 . 0 4 2 1 . 7 1 0 . 2 2 . 1 2 0 . 3 5 0 . 5 0 . 7 0

No te : F o r t h e f i rs t n i n e l i n e s , t h e P E m a x i m u m i s 1 5 leT, t h e m i n i m u m , - 1 . 5 k T . F o r t h e l a s t f o u rl i n e s , A / k T = 1 2 . 5 .

N o t s t a b l e w h e n 0 = 1 .W h e n t h e g r e a t e r f i l m t h i c k n e s s o f t h e r e -s t r i c t e d c h a i n i s t a k e n i n t o a c c o u n t , t h em a x i m u m a n d m i n i m m n p a r ti c le s iz es s ta -b i li z ed w h e n 0 = 1 a r e sh o w n i n T a b l e I I I .

T h u s l a t e r a l c o m p r e s s i o n o f t h e c h a i n r e-d u c e s t h e r a n g e o f p a r t i c l e s iz e s s t a b i li z e d ,c o m p a r e d w i t h a h y p o t h e t i c a l u n c o m -p r e s s e d c h a in a t t h e s a m e c o v e r a g e .S P H E R I C A l , P A R T I C L E / S P R E R I C A L P A R T I C L E

I N T E R A C T I O NT h e m e t h o d o f c r o s s - p l o t t i n g d e s c r i b e d

a b o v e a l l o w s t h e i n t e r a c t i o n s b e t w e e n t w os p h e r e s t o b e s u m m a r i z e d c o n v e n i e n t l y i n" s t a b i l i t y d i a g r a m s " a s s h o w n i n F i g . 1 2 ;c u r v e s o f c o n s t a n t P E m a x i m u m a n d m i n i -m u m d i v i d e t h e d i a g r a m i n t o r e g i o n s r e p r e -s e n t i n g p r o t e c t i o n o r l a c k o f p r o t e c t i o n f r o mf l o c c u l a t i o n , j u s t a s F i g . 6 d o e s f o r a d h e s i o n .Qualitative Similarity to Particle~PlaneAdhesion. F i g u r e 1 2 s h o w s s t a b i l i t y d i a g r a m sf o r c h a i n s o f 1 0 , 3 0 , a n d 1 0 0 l i n k s w h e n

A / k T = 1 2.5 , f o r t w o v a l u e s o f t h e P Em a x i m u m a n d m i n i m u m . F i g u r e 13 s ho w ss t a b i l i t y d i a g r a m s f o r v a r i o u s v a l u e o f A / k Tw h e n t h e P E m a x i m u m a n d m i n i m u m a r e1 5 k T a n d - 1 .5 k T , r e s p e c t i v e l y . T h e r e s u l t sa r e v e r y s i m i l a r i n f o r m t o t h o s e f o r a p a r -t i c le a n d p l a t e , a n d t h e d i s c u ss i o n o f t h e s ea b o v e i s r e l e v a n t h e r e t o o .Quan titative Com parison of Criteria forFlocculation and A dhesion. I n g e n e r a l , f l o c cu -l a t i o n o f t w o p a r t i c l e s i s m o r e e a s i l y p r e -v e n t e d t h a n a d h e s i o n o f a p a r t i c l e t o a su r -f a c e . I n d e t a i l t h e c a l c u l a t e d r e s u l t s s h o wt h e f o l l o w i n g c o m p a r i s o n s ( T a b l e I V ) .

(a) Size of macromoleeule. T h e t h i n n e s tf i h n ( a t 0 = 1 ) w h i c h p r e v e n t s f l o c c u l a t i o no f p a r t i c l e s o f a g i v e n s i z e i s 0 . 4 4 t i m e s a st h i c k a s t h e f il m p r e v e n t i n g a d h e si o n . T h er a t i o d e c r e a s e s s l i g h t l y w i t h i n c r e a s i n g a t -t r a c t i o n c o n s t a n t .T h e t h i c k e s t f i l m ( a t 0 = 1 ) w h i c h p r e -v e n t s f l o c c u l a t i o n o f p a r t i c l e s o f a g i v e n s i ze


14/16

2 8 2 CL A Y F I E L D A N D L U M Bi s 0. 8 t i m e s a s t h i c k a s t h e f i lm p r e v e n t i n ga d h e s i o n . I n o t h e r w o r d s , s t a b i l i z a t i o na g a in s t i r r e v e r s ib l e a d h e s io n i s a l i t t l e e a s i e rt h a n a g a i n s t i r r e v e r s i b l e f l o c c u l a t i o n . T h i st r e n d i s r e v e r s e d f o r c h a in s o f a f e w l i n k sw h e n A / k T is 50.(b )~ Pa rt icle s ize . T h e m a x i m u m p a r t i c l es iz e ( a t 0 = 1 ) p r e v e n t e d f r o m f l o c c u l a t i n gw i t h a g i v e n f il m t h i c k n e s s i s a b o u t 2 .2 t i m e st h a t p r e v e n t e d f r o m a d h e r i n g .T h e m i n i m u m p a r t i c l e si ze (a t 0 = 1) p r e -v e n t e d f r o m f l o c c u l a ti n g w i t h a g i v e n f i lmt h i c k n e s s i s 1 . 2 t i m e s t h a t p r e v e n t e d f r o ma d h e r i n g .

(c) Surface coverage. T h e m i n i m u m c o v e r -a g e f o r p r e v e n t i n g f l o c e u l a t i o n is a l i t t l e l e sst h a n t h a t f o r p re v e n t i n g a d h e s i on . T h e r a t ioi s n e a r u n i t y a t l o w v a l u e s o f A , b u t i n -c r e a s e s t o a b o u t 1 . 2 5 w h e n A / k T = 50 . T hec h a n g e i n r a t i o w i t h c h a n g e i n n u m b e r o fl i n k s i s v e r y s m a l l ]( d) C r it er ia f o r P E m a x i m u m a n d m i n i -m u m . A c h a n g e i n t h e l i m i t on t h e P E m i n i -m u m f o r s t a b i li z a t i o n do e s n o t a f f e c t t h er a t i o o f p a r t i c l e s i z e s p r e v e n t e d f r o m f l o c c u -l a t i n g a n d a d h e ri n g .

A c h a n g e i n t h e l im i t o n t h e P E m a x i m u mf o r s t a b i l i z a t i o n f r o m 1 5 k T t o 7 . 5 k Tc h a n g e s t h e r a t i o o f m i n i m u m p a r t i c l e s i z e sp r e v e n t e d f r o m f l o c c u la t i n g t o t h o s e p r e -v e n t e d f r o m a d h e r i n g f r o m 1 .0 8 t o 0 .7 ; a t 15k T , s m a l l e r p a r t i c l e s a r e p r e v e n t e d f r o m a d -h e r i n g t h a n f r o m f l o c c u la t i n g ; a t 7 .5 k T t h erever se i s t rue .T h e c h a n g e s i n m i n i m u m 0 f o r st a b il i za -t i o n w i t h c h a n g i n g P E l i m i t s a r e s m a ll .T o s u m u p t h e c o m p a r i s o n , t h e m a i n d i f -f e r e n c e i s t h a t , i n g iv e n c o n d i t i o n s , t h el a r g e s t p a r t i c l e s t h a t a r e p r o t e c t e d f r o mf l o e c u l a ti o n a r e t w i c e t h e d i a m e t e r o f t h o s et h a t a r e p r e v e n t e d f r o m a d h e r i n g t o f l a t s u r-f ac e s. I n p r a c t i c e t h i s m e a n s t h a t i f t h ed e g r ee of p r o t e c t i o n i s s u c h t h a t p a r t ic l e sa r e o n l y j u s t p r e v e n t e d f r o m a d h e r i n g t oo n e a n o th e r , t h e l a r g e r p a r t i c l e s w i l l a d h e r eto a n y a v a i l a b l e s o l i d s u r f a c e .Fi l m T h i ckn es s a n d Pa r ti c le Ra d i i i nUni t s o f Bond Leng th . F i g u r e 1 4 s h o w s t h ep a r t i c l e s iz e s t a b i l i z e d a g a in s t f l o c c u l a t i o n a sa f u n c t i o n o f fi lm t h i c k n e s s , b o t h i n u n i t s o fb o n d l e n g t h . T h e f i g u r e i s d e r i v e d i n t h es a m e w a y a s F ig . 1 0 , t o w h ic h i t i s s im i l a r

250 (a) PE maximum 15kTPE minimum -I.5kT r..-02 0 0 o

150 Z "

I00 t . ~ / /

5 o ~

o 0 5 I0I r

5/kT~50 25 12,~2Y. _ _ . - - - k _ _15 ---20 0 5 I0 15 20I

FIG. 14. M aximum and m inimu m radii of particles stabilized agains t f loeeulation, in units ofbond length, as a functio n of f ilm thickness.


15/16

A THEORY OF POLYMERIC DISPERSANT ACTION. I 283

dR

I II II I

I II I

" ~ I _ _ .

d

- - -t

FIG. 15. Derivation of London-van der Waals attr acti on between spherical particle and flatplate.e x c e p t t h a t t h e m a x i m u m p a r t ic l e s iz es ar er o u g h l y d o u b l e d , i n a c c o r d a n c e w i t h t h ef i n d in g s o f t h e p r e v io u s s e c t i o n .

REFERENCES1. VAN I)E~ WaAI~DEN, M., J. Col loid Sci . 5, 317(1950).2 . MACKOR, E. L., A N D V AN D E R W A A L S , J . ] : I .,

J. Col loid Sci . 6, 492 (1951); ib id . 7, 535(1952).3. BLUESTONE, S., AND VOLD, M . J . , J . P o l y m e rS c i . , Part A, 2,289 (1964).4. Part II.5. For example, TANFORD, C., "Physica l Chemis-try of Macromolecules," pp. 170-174. Wiley,New York, 1961.6. HAMAKER,J-I. C., P h y s i c a 4, 1058 (1937).7. VEnWEY, E. J. W., AND OVE~BEEK, J. Tm G.,"Theory of the Stability of LyophobicColloids," p. 164 . Elsevier, Amsterdam,1948.8. TANFORD, C., "Physica l Chemist ry of Macro-molecules," pp. 156-157. Wiley, New York,1961.9. TANFORD, C., "Physica l Chemist ry of Macro-molecules," p. 403. Wiley, New York, 1961.

1 0 . L O N D O N , F., Z. P h y s i k 63, 245 (1930).11. VE~wEY, E. J. W., AND OVER:BEEK, J-. TH. C-.,"Theory of the Stab ility of LyoprobicColloids," p. 101 . Elsevier, Amsterdam1948.APPENDIX

C A L C U L A T I O N O F LONDON-VAN D E R W ' A A L SATTRACTION ENERGY BETWEEN SPHERICAL

PARTICLE AND FLAT PLATET h e L o n d o n - v a n d e r W a a l s a t t ra c t i o ne n e r g y b e t w e e n t w o e q u a l a t o m s d i s t a n c e da p a r t i s V ~ = X/d 6, i n w h i c h X is t h e L o n d o n

c o n s t a n t ( 1 0 ) . T h e c o r r e s p o n d i n g a t t r a c t i o nfo rce i sf o~ = 6 / d 7,

a n d t h e r e f o r e , w h e n F ig . 1 5 i s r e f e r r e d t o ,a n d a n a t o m n e a r t o a n i n f i n i te l y l a rg e p l a t eo f th i c k n e s s t c o n s id e r e d, t h e a t t r a c t i v e f o rc eis f o u n d b y s u m m a t i o n o f t h e c o m p o n e n t s o ft h e t o t a l a t t r a c t i v e f o r c e s e x e r c i s e d b y t h ea t o m s o f t h e p l a t e s u b s t a n c e , i n a d i r e c t io np e r p e n d i c u l a r t o t h e s u r fa c e p l a n e ( 11 ):6X cos 0t h e f o r c e c o m p o n e n t f ~ - d~ ;

6hx( r 2 + x D 4"

I f t h e p l a t e i s c o n s i de r e d to b e b u i l t u p o fr i ng s o f ra d i u s r a n d i n f i n i t e s im a l l y s m a l lc ro ss sec t ion dr b y dx, w i t h q t h e n u m b e r o fa t o m s c o n t a i n e d i n 1 c u . c m . o f t h e p l a t em a t e r i a l , t h e t o t a l f o r c e f~ s b e t w e e n t h e a t o ma n d t h e p l a t e i s g i v e n b yf o "~ f R + t x . d xf ~ = 12~-qX r . d r ~ 0 .2 x2)4 ,

i .e. ,~rqX [ 1 1 ]f ap = ~ - ~4 (R-~- t ) i "

F o r t h e d e r i v a ti o n o f t h e a t t r a c t i o n b e-t w e e n t h e s p h e r i c a l p a r t i c l e a n d t h e f l a tp l a t e , w h e r e b o t h a r e c o m p o s e d o f t h e s a m em a t e r i a l , a d i s c - sh a p e d e l e m e n t o f t h ep a r t i c l e , o f t h i c k n e s s dR , r a d i u s y , d i s t a n c eR f r o m t h e p l a t e s u r f a c e i s c o n s i d e r e d . T h e


16/16

284 CLAYFIELD AND LUMBn u m b e r o f a t o m s c o n t a i n e d i n t h i s e l e m e n t= q~ry~.dR, a n d s i n c e e v e r y a t o m i n t h ee l em en t w i ll b e a t t r ac t e d t o t h e p l a t e w i t h afo r ce f a~ , t h e t o t a l f o r ce f ,~ b e t w e en t h i sd i sc e l ement and the p la te wi l l bef ,~, = q~ [a2 - { (H + a) - R }2] .dR ~ X

T h e t o t a l a t t ra c t i o n f o rc e f ~ b e t w e e n t h espher ica l par t i c l e and the p la te wi ll be

fH-]-2a

f sp - - - f ep_ q 21r2x r H + 3 ae 2a)

a

3(H -}- 2a -[- 0 2

H - a3 H 2a

3(H + t) 2

1 "3(H -F t )

T h e co r r e sp o n d i n g en e rg y o f a t t r ac t i o n ,Vas~ , is, if A = r2q2X,A V 2 a ( H + a )Va~ 6

_ ]n ( H ~ 2 a ) aH + 2 a + ta + l n ( H + 2 a - ~ - t ) 1H W t H + t "

W h e r e t h e p l a t e m a y b e c o n s i d e r e d t o b ein f in i t e ly th ick com pared to th e s i ze o f thespher ica l par t i c l e , t he above express ion re -d u ces to a f o rm s i mi l a r t o t h a t o b t a i n ed , i n ad i ff e r e n t m a n n e r , b y H a m a k e r (6 ) fo r t h ecase o f two spher i ca l par t i c l es in which thed iameter o f one i s t aken to be in f in i ty , i . e . ,V a.~ = _ A V 2 a ( H + a )

6 L H ( H - t - 2a)- - In ( H ~ / 2 a ) ]

a theoretical approach for polymeric dispersant action

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