Colloidal Dispersions and Micellar Behavior

13 Rheological Studies of Polymer Chain Interaction ROBERT J. HUNTER, PAUL C. NEVILLE, and BRUCE A. FIRTH

Downloaded by MONASH UNIV on April 14, 2016 | http://pubs.acs.org Publication Date: June 1, 1975 | doi: 10.1021/bk-1975-0009.ch013

Department of Physical Chemistry, University of Sydney, Sydney, N.S.W. 2006, Australia

Introduction In t h i s paper we examine some a s p e c t s o f the flow behaviour of latex particles coated w i t h a h y d r o philic polymer. The core particle is a poly(methyl m e t h a c r y l a t e ) sphere and the adsorbed trib l o c k copolymer i s p o l y - ( e t h y l e n e o x i d e - b - m e t h y l m e t h a c r y l a t e ) i n which the end h y d r o p h o b i c m o i e t i e s a c t as anchors and p e r m i t the e t h y l e n e oxide u n i t s to form a loop which s t r e t c h e s i n t o the s u r r o u n d i n g solution. These systems are t h e r m o d y n a m i c a l l y s t a b l e (1) below a c e r t a i n t e m p e r a t u r e , c a l l e d the critical flocculation temperature (c.f.t.), because the c l o s e approach o f the core particles i s p r e v e n t e d by a r a p i d rise in Gibbs Free Energy as the adsorbed polymer c h a i n s on the colliding particles b e g i n to i n t e r a c t (2-4) . The total energy o f interaction, V , c o n s i s t s o f the van der Waals energy o f a t t r a c t i o n between the core particles, V, and the polymer interaction energy V = ΔG = ΔH - TΔS where ΔH and ΔS are the e n t h a l p y and entropy changes which o c c u r when the adsorbed polymer c h a i n s i n t e r p e n e t r a t e . Both are p o s i t i v e i n t h i s system so t h a t at the c.f.t, ΔG has presumably been reduced to a v a l u e comparable with V . Tne r h e o l o g i c a l b e h a v i o u r o f a c o l l o i d a l s u s p e n s i o n i s d i r e c t l y a f f e c t e d by the i n t e r a c t i o n e n e r gy between the p a r t i c l e s . Systems which are u n s t a b l e or o n l y m a r g i n a l l y s t a b l e , e x h i b i t p s e u d o p l a s t i c or p l a s t i c b e h a v i o u r (5_,6J and the t o t a l energy d i s s i p a t i o n , E ^ , c o n s i s t s of a purely viscous p a r t , E ^ , and a p a r t which may be a t t r i b u t e d to the overcoming of i n t e r p a r t i c l e i n t e r a c t i o n s , E j . L i k e w i s e the shear s t r e s s at h i g h shear r a t e s can be d i v i d e d i n t o a v i s c o u s p a r t , τ ^ , and an a d d i t i o n a l t e r m , τ , c a l l e d the Bingham y i e l d v a l u e s i n c e : T

A

R

R

R

R

R

R

R

A

β

193

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

194

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

= Ε

v

+

E

i

BEHAVIOR

(1)

•

F i g u r e 1 shows the b e h a v i o u r which i s a n t i c i p a t e d . To c h a r a c t e r i s e t h i s curve we need to be a b l e to i n t e r p r e t (a) the y i e l d v a l u e , τ , (b) the s l o p e o f the l i n e a r p a r t o f the graph ( u s u a l l y c a l l e d the p l a s t i c v i s c o s i t y , η ) , (c) the shear r a t e at which the r e l a t i o n becomes^Iinear and (d) the d e t a i l e d shape o f the n o n - l i n e a r p a r t o f the τ-D c u r v e . In some cases the systems showed at low shear r a t e s , a more c o m p l i c a t e d , Ostwald type (7) flow b e h a v i o u r ( F i g u r e 2 ) , the o r i g i n o f which i s d i s c u s s ed below.


β

Theory (i) Ostwald Flow B e h a v i o u r . The g r a d u a l r e d u c t ion i n v i s c o s i t y w i t h i n c r e a s e i n shear r a t e i s u s u a l l y a t t r i b u t e d to a breakdown i n " s t r u c t u r e " i n the system and v a r i o u s attempts have been made to quantify this idea. We propose t h a t Ostwald b e h a v i o u r o c c u r s when the shear f i e l d has to d e s t r o y not o n l y the s h e a r - i n d u c e d d o u b l e t s , but a l s o some o f those c r e a t e d by Brownian m o t i o n . Doublets which s e p a r a t e by Brownian motion ( i r r e s p e c t i v e o f how they are formed) make no c o n t r i b u t i o n to the i n t e r a c t i o n energy. Thus Ε = n ( D ) E D

(2)

g

where n ( D ) i s the number o f d o u b l e t s d e s t r o y e d by shear per u n i t volume p e r second at shear r a t e D, and Ε i s the energy which the shear f i e l d must s u p p l y i n o r d e r to s e p a r a t e the p a r t i c l e s i n the d o u b l e t . The t o t a l energy d i s s i p a t i o n per u n i t volume per second i s then n

E

2

T

= D n (l s

+ 2 . 5 φ ) + n (D) D

E

$

(3)

where η „ i s the s o l v e n t v i s c o s i t y and φ i s the volume f r a c t i o n o f p a r t i c l e s . Here we have assumed t h a t Ε i s g i v e n by the s i m p l e E i n s t e i n r e l a t i o n . Smoluchowski (8) showed t h a t the t o t a l number o f d o u b l e t s produced by both Brownian motion and shear was (4)

+ D where a i s

the p a r t i c l e r a d i u s .

In o r d e r to

determine



13.

HUNTER ET AL.

Polymer Chain Interaction

195

Industrial and Engineering Chemistry

Figure 1. The basic shear diagram for a plastic ma terial showing the contributions to the total stress, as suggested by Michaels and Bolger (21)

Figure 2. Ostwaldflowcurves for thermo dynamically stable suspensions of the same radius (116 nm), showing the effect of temperature on the curved region. The tem peratures were (a) 35°C and (b) 43°C respectively and the theta temperature 47 °C for MgSO, (0.39M). t


196


BEHAVIOR

the f r a c t i o n o f these which are d e s t r o y e d by the shear f i e l d , we must know the average l i f e t i m e o f a doublet i n the shear f i e l d . If this is greater than the Brownian motion l i f e t i m e , we w i l l assume t h a t t h a t d o u b l e t i s not broken by the shear f i e l d . Smoluchowski showed t h a t the Brownian motion l i f e t i m e was t

B = 3n /4kTN

(5)


s

where Ν i s the number o f p a r t i c l e s per u n i t volume. The s h e a r - f l o w l i f e t i m e t , depends on the a n g l e , Φ , between the d o u b l e t a x i s p r o j e c t e d onto the shear p l a n e and the d i r e c t i o n o f s h e a r . Goldsmith and Mason (9) d e r i v e d the r e l a t i o n g

t

s

= (5/D)

tan"

1

(6)

(% tan Φ)

Thus t h e r e e x i s t s a c r i t i c a l angle Φ below which the shear l i f e t i m e i s s h o r t e r than t and t h i s w i l l be g i v e n (from E q u a t i o n s 5 and 6) by: 0

g

Φ

= tan"

0

(2 tan (3n

1

D/20 kT N))

g

(7)

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 angle l e s s than Φ was shown by G o l d s m i t h and Mason (10) to be g i v e n by sin Φ . Hence the f r a c t i o n , B, o f c o l l i s i o n s d e s t r o y e d by the shear f i e l d i s : 0

2

0

2

Β = sin (tan" The

v a l u e o f n^(D)

π a and

1

is

, „

V

1

(8)

J

g

w i t h E q u a t i o n s (1)

.

„

. _ § * ! . (, .

At h i g h shear r a t e s straight line:

this

τ = DTigCl

φ)

+

D/20 kT N)))

§

then

M n a

combining t h i s τ

(2 tan ( 3 η

2.5

"

and (3) )

gives

Β Β,

e q u a t i o n degenerates to a E

+

s

ΤΓ a since

(

k T / 4 η a ΏΉ) and B-*l . s


1

0

)

13.

HUNTER

Extrapolation of this straight the Bingham y i e l d v a l u e : τ

Β

197


ET AL.

E

- - F T " 7Γ a

line

to D = 0 g i v e s

s

W

E q u a t i o n 12 has been d e r i v e d p r e v i o u s l y (11) S u b s t i t u t i o n o f E q u a t i o n (12) i n E q u a t i o n (11) Dn

(1 + 2.5 Φ )

s

τ

Β

B|

gives

kT 1 +4 n a"D

(13)


s

which i s the e q u a t i o n we would expect to d e s c r i b e Ostwald f l o w . τ i s determined by e x t r a p o l a t i o n o f the e x p e r i m e n t a l data and φ i s c a l c u l a t e d as a hydrodynamic volume f r a c t i o n , φ „ , from the measured d i f f e r e n t i a l v i s c o s i t y at h i g h shear r a t e , u s i n g the E i n s t e i n r e l a t i o n : β

n

= n

P L

s

(1 + 2.5

φ )

(14)

Η

The i n t e r p r e t a t i o n o f Φ i n terms o f the volume f r a c t i o n w i l l now be d e a l t w i t h . Η

expected

(ii) The Hydrodynamic Volume F r a c t i o n . The e f f e c t i v e volume o f the p a r t i c l e i s augmented by the presence o f the adsorbed polymer and the hydrodynamic volume f r a c t i o n , φ^, i s r e l a t e d to the volume f r a c t i o n o f core p a r t i c l e s , φρ, by the simple r e l a t i o n

V*P

=

C ( a

o

+

where δ i s t h e ^ e x t e n s i o n o f the polymer loops i n t o the s o l u t i o n ; 6 depends on the b a l a n c e between the heat o f d i l u t i o n (κ ) and the entropy o f d i l u t i o n (ψ^) f a c t o r s f o r the polymer - s o l v e n t i n t e r a c t i o n . For the s o l u t i o n to behave i d e a l l y these parameters must be equal and at t h i s p o i n t the change i n f r e e energy f o r i n t e r p é n é t r a t i o n and compression o f the polymer i s equal to z e r o , and hence, so too i s the r e p u l s i o n energy V . These are c a l l e d the t h e t a c o n d i t i o n s f o r the polymer i n t h a t s o l v e n t and i t then assumes i t s u n p e r t u r b e d r o o t - m e a n - s q u a r e e n d - t o end d i s t a n c e h i n f r e e s o l u t i o n . The dimensions under o t h e r c o n d i t i o n s may be r e p r e s e n t e d thus R

2

< r >

where a ,

h

the

2

= α **

i n t r a m o l e c u l a r expansion

(16)

factor,

is


a

198


BEHAVIOR

measure o f the s o l v e n c y o f the d i s p e r s i o n medium f o r the s t a b i l i z i n g m a c r o m o l e c u l e s . The PEO polymer i s under t h e t a c o n d i t i o n s i n 0.39 M MgSO. at a temper a t u r e o f 320 Κ and i n 0.45M K S 0 at 308 Κ and these are the d i s p e r s i o n media used i n the p r e s e n t studies. A d s o r p t i o n o f the polymer onto an i m p e n e t r a b l e s u r f a c e r e s t r i c t s the segments i n such a way as to i n c r e a s e the mean e x t e n s i o n . We may w r i t e 2 > % (17) δ α < 6l δ < r ο 2

4


r

where δ i s a d i m e n s i o n l e s s parameter which element ary s t a t i s t i c a l n o t i o n s would suggest s h o u l d be equal to about 2. (iii)

The Bingham Y i e l d V a l u e ,

τ .

In o r d e r

β

to

1

(a„ + δ ) use E q u a t i o n (12) we now s u b s t i t u t e a where a i s the r a d i u s o f the core p a r t i c l e and φ= φ „ . The energy o f s e p a r a t i o n o f a p a i r o f p a r t i c l e s i s equal to the d i f f e r e n c e between the van der Waals and the r e p u l s i o n energy: (18) A a N

Q

V

12H

R ο and the a n a l y s i s o f Evans and Napper (1_) p r o v i d e s an e x p r e s s i o n f o r V_ which when combined w i t h E q u a t i o n s (12) and (18) g i V e s :

3ΦH

A a

o 12H

2 3 π a

2

-(2ir) / > 27 5

2

3 / 2

2

2

5

3

u> a N ( a - a ) k T S ο A J

1

(19)

M

where ω= c o n c e n t r a t i o n o f s t a b i l i s e r s u r f a c e (assuming 100% c o v e r a g e ) ;

i n g cm

-2

of

m o l e c u l a r weight o f PEO (here 96000). segment d e n s i t y f u n c t i o n .

= A v o g a d r o s number; M f

= a dimensionless

H , the d i s t a n c e o f c l o s e s t approach can be expressed as a m u l t i p l e o f the u n p e r t u r b e d r . m . s . l e n g t h : Q

Η

o

= δ

o

α < r

2

ο

>h = δ

ο

< τ

2

>h

since i s c a l c u l a t e d f o r v a r i o u s v a l u e s o f the d i s t a n c e p a r a m e t e r ^ , and δ i s the minimum v a l u e o f δ c o r r e s p o n d i n g to c l o s e s ? a p p r o a c h .


13.


HUNTER E T A L .

199

At the (iv) The van der Waals c o n s t a n t , A. t h ê t a - t e m p e r a t u r e , θ , the v a l u e o f α i s u n i t y and so the r e p u l s i o n energy i s z e r o . The y i e l d s t r e n g t h at t h i s temperature c a n , t h e r e f o r e , be used to e s t i m a t e a v a l u e f o r the van der Waals c o n s t a n t : A =

, π2a3 4

τ Β (0) (21) *H o p r o v i d e d t h a t [ can be e s t i m a t e d . A l t e r n a t i v e l y , the v a l u e o f A / H may be o b t a i n e d from the v a l u e o f τ D at the t h e t a temperature ( T ( 8 ) ) and t h i s used __-,_..·,_ — δ — We c o u l d then f o r c a l c u l a t i o n s at o t h e r temperatures, write ^3/2 2 5/2 < r > ω a N^(a - & ) kT S. o A τ (β) - Η (2ir) 3 27 Χ π a


a

Ό

r

5

3 φ

K

2

Β

(22)

= τ ( θ ) - Β (Τ) f (α) Β

where Β ( Τ ) i s a s l o w l y v a r y i n g f u n c t i o n o f a t u r e and f(o)-

1

temper-

h ^ T i L " - " Jv a r i e s

r a p i d l y w i t h temper / Vi a t u r e , c h i e f l y because o f the second t e r m . For Τ > θ , α < 1 and so f ( a ) i s n e g a t i v e . The y i e l d s t r e n g t h s h o u l d , t h e r e f o r e , always be g r e a t e r than i t s v a l u e at the t h e t a t e m p e r a t u r e , but s i n c e f ( a ) goes through a maximum ( i n a b s o l u t e v a l u e ) , so too does T g . I t o c c u r s at about the same p o i n t as the minimum i n α - α ( i . e . at a s ( 3 / 5 ) ^ ) . It should be p o i n t e d o u t , however, t h a t the t h e o r y from which the v a l u e o f V i s c a l c u l a t e d i s u n l i k e l y to be v e r y r e l i a b l e above the θ - t e m p e r a t u r e . Polymer s c i e n t i s t s show l i t t l e i n t e r e s t i n the r e g i o n where α< 1 s i n c e i n f r e e s o l u t i o n t h i s corresponds to such poor s o l v e n c y t h a t the polymer i s thrown out as a s e p a r a t e phase. Here i t merely s e t t l e s down onto the p a r t i c l e s u r f a c e . The p h y s i c a l reason f o r the maximum i n τ i s not d i f f i c u l t to f i n d . I f T i s w r i t t e n i n t e form Β k

3

d

T

2

r

- Ε ( Φ ) E
0.20 a p p r o x i m a t e l y (13) and w i l l not be considired further here. A model f o r the secondary e l e c t r o v i s c o u s e f f e c t has been proposed by Chan, B l a c h f o r d and G o r i n g (14, 15). I t a t t r i b u t e s the enhanced v i s c o s i t y to an i n c r e a s e i n the s i z e o f the r o t a t i n g d o u b l e t due to the r e p u l s i o n between the particles. T h i s term would be c o n t a i n e d w i t h i n the parameter Ε which we e v a l u a t e from the e x p e r i m e n t a l Tg. E q u a t i o n (10) and (24) combine to y i e l d 2

ν = 3B 2 - f x π a Dn

f,

K

1

c

i

A +

kT

1

c E

1 S 4 ru a D /

Experimental. The PMMA l a t e x p a r t i c l e s were p r e p a r e d by a m o d i f i e d K o t e r a p r o c e d u r e (16) and monodisperse s p h e r i c a l p a r t i c l e s w i t h s i z e s i n the range 0.09 - 0.2μιη were o b t a i n e d . The t r i b l o c k s t a b i l i s i n g polymer was made by the method o f Napper(3) u s i n g p o l y e t h y l e n e oxide o f v i s c o s i t y average m o l e c u l a r weight 96,000 (polyox WSRN10, Union Carbide), The copolymer was adsorbed on to the PMMA p a r t i c l e s by adding a p p r o x i m a t e l y 100 times the amount o f PEO r e q u i r e d f o r 100% coverage to a c o l d d i s p e r s i o n o f PMMA. A f t e r a 24-hour e q u i l i b r a t i o n the excess was removed by c e n t r i f u g a t i o n . C r i t i c a l f l o c c u l a t i o n temperatures o f the s u s p e n s i o n s were determined by t u r b i d i m e t r i c measure ments at 655 nm and were found to be v e r y c l o s e to


13.

HUNTER


E T AL.

201

the t h e t a temperatures o f the f r e e polymer i n the corresponding s o l u t i o n . The expansion f a c t o r , α , r e q u i r e d f o r E q u a t i o n (19) was c a l c u l a t e d from the approximate F l o r y (17) equation : α

5

- α

3

= 2 C

ψ

M

χ

(26)

(1 - Θ / Τ ) yfe


for temperatues below the θ temperature; is a constant. Above the t h e t a temperatureoCwas d e t e r mined from the r e l a t i o n s h i p (18)

is

where

the

l i m i t i n g v i s c o s i t y number.

Results (i) Ostwald Flow B e h a v i o u r . F i g u r e 2 shows the b e h a v i o u r e x h i b i t e d by a s t a b i l i s e d s u s p e n s i o n i n 0.39 M MgS0 f o r which the θ - t e m p e r a t u r e i s 320K. At lower temperatures these samples showed a s m a l l y i e l d v a l u e and an Ostwald Type flow c u r v e . The f u l l l i n e s i n F i g u r e 2, are the t h e o r e t i c a l curves from E q u a t i o n (13) and they o b v i o u s l y g i v e a f a i r r e p r e s e n t a t i o n o f the e x p e r i m e n t a l d a t a , c o n s i d e r i n g the p o s s i b l e e r r o r s i n v o l v e d . F i g u r e 2 shows the same s u s p e n s i o n at two d i f f e r e n t temperatures below the t h e t a t e m p e r a t u r e , w h i l s t F i g u r e 3 shows a comparison o f two d i f f e r e n t suspensions w i t h d i f f e r e n t v a l u e s o f Ν and a at the same t e m p e r a t u r e . 4

Q

(ii) Hydrodynamic Volume F r a c t i o n . The s l o p e o f these curves at h i g h shear r a t e g i v e s an e x p e r i m e n t a l v a l u e f o r η . which when s u b s t i t u t e d i n E q u a t i o n (14) g i v e s an e x p e r i m e n t a l v a l u e f o r φ„. T h i s v a l u e was used i n a l l o t h e r e q u a t i o n s i n p l a c e o f the p a r t i c l e volume fTSLCtion^^. When s u b s t i t u t e d i n E q u a t i o n (15) i t corresponded to a δ v a l u e o f 55 nm which g i v e s a value o f =2.1 from E q u a t i o n (17) . T h i s i s i n very s a t i s f a c t o r y agreement w i t h the expected v a l u e o f about 2. ρ

1

(iii) The Bingham Y i e l d V a l u e , τ . A c c o r d i n g to E q u a t i o n (19) τ s h o u l d be p r o p o r t i o n a l to φ^, and F i g u r e 4 shows t h a t t h i s i s a p p r o x i m a t e l y t r u e even f o r the τ - φ r e l a t i o n . F i g u r e s 4,5 and 6 show how i m p o r t a n t t h e ^ e f f e c t o f temperature i s on τ i n the neighbourhood o f the t h e t a t e m p e r a t u r e . The r e s u l t s f o r MgS04 were l e s s r e l i a b l e because £

β

2

β

β




BEHAVIOR

Figure 3. Comparison of Ostwaldflowcurves for particles of different radii (116 nm (a) and 164 nm (b), respectively) at the same temperature (35°C). Note that the shear rate at which the linear region begins de creases with increase in particle size.

Figure 4. The effect of volume fraction on yield value for two systems of the same particle radius (167 nm); φ = 0.072, φ = 0.055. The dotted line shows the square of the ratio of the gravi metric volume fractions which should be equiva lent to the τΒ ratios at that particular temperature. The points (O) give the experimental values for this ratio. ί

2


13.

1.5


203


HUNTER ET A L .

I-

30

40 Temperature PC)

50

Figure 5. n— temperature curves for a series of particles in K SO (0.45M). The effect of particle radius on the position of the maximum in the curve is described in the text. The particle radii were (a )i = 164 nm; (a ) = 116 nm; (a ) = 103 nm. T

2

0

fl

0

2

0

40 Temperature

3

SO (°C)

Figure 6. Comparison of theoretical (1) and experimental (2) τ —temperature data for a particular size (116 nm) par ticle using a Hamaker constant of 1 X 10~ J n

20


204


BEHAVIOR

the h i g h e r temperatures i n v o l v e d made measurements more d i f f i c u l t but they showed the same q u a l i t i v e b e h a v i o u r p a t t e r n as the K S 0 systems i l l u s t r a t e d in Figures 4 - 6 . The r i s e i n the v a l u e o f τ at about the t h e t a temperature and i t s r e a c h i n g a maximum v a l u e at s l i g h t l y h i g h e r temperatures are i n q u a l i t a t i v e a c c o r d w i t h E q u a t i o n s (19) and ( 2 2 ) . Note t h a t f o r most o f the systems the Tg v a l u e never f a l l s below i t s v a l u e at the t h e t a temperature ( c f . E q u a t i o n 2 2 ) . Note p a r t i c u l a r l y t h a t the e x t e n t o f the r i s e and f a l l in f o r Τ > θ i s c r i t i c a l l y dependent on the core p a r t i c l e r a d i u s ( F i g u r e 5 ) . Figure 6 gives a comparison between the e x p e r i m e n t a l and t h e o r e t i c a l curves. 2

4


β

(iv) The van der Waals c o n s t a n t , A . Values o f A c a l c u l a t e d from τ ( θ ) u s i n g E q u a t i o n (21) depend c r i t i c a l l y on the c h o i c e o f the d i s t a n c e o f c l o s e s t approach. The s m a l l e s t p o s s i b l e v a l u e i s set by the c o n d i t i o n t h a t the i n t e r v e n i n g polymer chains are compressed to t h e i r normal d e n s i t y i n the s o l i d . T h i s would o c c u r at Η = 4 nm which i s o b v i o u s l y too s m a l l f o r i t w o u l â r e q u i r e the e x p u l s i o n o f a l l s o l v e n t from around the polymer c h a i n s . For avalue o f 5 nm the van der Waals c o n s t a n t comes out to be 1 χ 10"20j which i s v e r y r e a s o n a b l e . The much l a r g e r v a l u e s o f H which are c a l c u l a t e d f o r s t a b l e systems by e q u a t i n g the hydrodynamic compressive f o r c e w i t h the e l e c t r o s t a t i c r e p u l s i o n (14) , would require correspondingly higher values for A. That model, however, does not a t t r i b u t e the e x t r a energy d i s s i p a t i o n to van der Waals a t t r a c t i o n (see b e l o w ) . β

Discussion. The s e p a r a t i o n energy which appears i n E q u a t i o n (10) f o r the Ostwald flow curves does not a r i s e from van der Waals a t t r a c t i o n energy between core p a r t i c l e s . These systems are s t a b l e and so the r e p u l s i o n s h o u l d i n any case dominate over a t t r a c t i o n . Also, F i g u r e s 2 and 3 and T a b l e I show t h a t the s e p a r a t i o n energy i s independent o f core r a d i u s but dependent on temperature which i s the o p p o s i t e of the b e h a v i o u r expected o f a van der Waals energy. We b e l i e v e t h a t the s e p a r a t i o n energy i s due to the n e c e s s i t y f o r the shear f i e l d to t e a r p a r t i c l e s apart at a r a t e which i s f a s t e r than the r e l a x a t i o n time o f the i n t e r a c t i n g polymer c h a i n s . The r e l a x a t i o n time f o r the PEO chains can be expected to be (19, 20) about 5 χ 10" s e c . The l i f e - t i m e o f a 4



δ

*

ΕΡ*

3

suspension

SP*

2

SP:

SP*

3

polymerisation

35

43

35

25

0

160

116

116

116

r

Latex P r e p a r - Temperature a a t i o n Method °C ° . (nm)

SP*

No.

190

150

160

170

^

* ΕΡ:

a = a + Hδ* inm9 (nm? « - 2ι

0.05

0.06

0.05

-1

polymerisation.

6

8

6

4

c

Field. E /kT

—§

0.03-.01

N m

Β

τ„

by Shear

Emulsion

0.20

0.12

0.14

0. 15

Ρ

φ

Energy R e q u i r e d f o r S e p a r a t i o n o f Doublets

2

Fig.

—

Table 1.


206


BEHAVIOR

shear d o u b l e t , t g , i s g i v e n by E q u a t i o n (6) and even f o r D = 10^ sec i t i s about 7 χ 10-3 c; i t w i l l be even l o n g e r f o r lower shear r a t e s , so t h e r e i s ample time f o r the chains to f i n d t h e i r time averaged c o n f i g u r a t i o n s as the p a r t i c l e s approach and r o t a t e around one another i n a d o u b l e t . Suppose t h a t the p a r t i c l e s are b e i n g s e p a r a t e d by a simple l a m i n a r shear f i e l d w i t h a v e l o c i t y g r a d i e n t o f 300 s e c " . I f the p a r t i c l e c e n t r e s are d i s p l a c e d by one diameter i n the d i r e c t i o n o f the v e l o c i t y g r a d i e n t , then the v e l o c i t y o f one p a r t i c l e w i t h r e s p e c t to the o t h e r i s 3 χ 10~3 χ 300 = 1 0 ' cm s e c " . I f the d i s t a n c e o f polymer i n t e r p é n é t r a t i o n i s 5 nm then the time f o r p u l l i n g the d o u b l e t a p a r t i s 5 χ 1 0 " / 1 0 - 2 = 5 χ 10~ sec or an o r d e r o f magnitude f a s t e r than the polymer c h a i n s can r e l a x . Hence the shear f i e l d has to "tear the p a r t i c l e s a p a r t . On the o t h e r hand, when s e p a r a t i o n o c c u r s by Brownian m o t i o n , the p r o c e s s can o c c u r s l o w l y enough to a l l o w the polymer c h a i n s to r e l a x (tg > 10" sec). In our a n a l y s i s o n l y the energy r e q u i r e d f o r shear s e p a r a t i o n i s i m p o r t ant and T a b l e I shows t h a t i t has a v a l u e o f a few times k T . As a f u r t h e r p i e c e o f c i r c u m s t a n t i a l evidence f o r the h y p o t h e s i s t h a t c h a i n : c h a i n i n t e r a c t i o n s are r e s p o n s i b l e f o r the Bingham y i e l d v a l u e , we show i n F i g u r e 7 a p l o t o f E a g a i n s t (1 - Θ / Τ ) . A c c o r d i n g to F l o r y s a n a l y s i s (17), the f r e e energy change which o c c u r s on m i x i n g polymer chains i s -

1

s e


1

2

1

7

5

11

g

f

δ(Δ G/kT) -

constant

ψ ( 1 - Θ/Τ)δν 1

and a l t h o u g h the d a t a i s s u b j e c t to a wide margin o f e r r o r i t does appear to c o r r e l a t e w i t h F l o r y s equation. The s u g g e s t i o n t h a t the f a l l o f f i n x at temperatures above θ i s due to a r e d u c t i o n i n the e f f e c t i o n c o l l i s i o n frequency (and hence f(2) i n E q u a t i o n (23)) i s s u p p o r t e d by the d a t a i n F i g u r e 5. For s m a l l p a r t i c l e s , i n c r e a s e s i n temperi have a p r o p o r t i o n a t e l y l a r g e r e f f e c t than fo l a r g e p a r t i c l e s because S*-/a. i s l a r g e r and hence f ( a ) i n E q u a t i o n (22) undergoes l a r g e r changes with t e m p e r a t u r e . The i n t e r p r e t a t i o n o f enhanced v i s c o s i t y i n terms o f the u s u a l d e s c r i p t i o n (14,15) o f the secondary e l e c t r o v i s c o u s e f f e c t ( E q u a t i o n (25)) p r e s e n t s some d i f f i c u l t i e s . A l t h o u g h the e l e c t r o s t a t i c r e p u l s i o n f o r c e s are e s s e n t i a l l y zero i n 1

g

Q



13.

HUNTER

ET

AL.


207

Figure 7. Energy of separation E vs. (1 — θ/T). The straight line suggests that the energy of separation is due to polymer chain interactions. s


208

C O L L O I D A L DISPERSIONS A N D M I C E L L A E B E H A V I O R

these systems t h e r e are s t i l l r e p u l s i v e f o r c e s between the p a r t i c l e s , as i s e v i d e n t from the s t a b i l i t y o f the s o l . A c c o r d i n g to Chan et a l (14) t h i s would cause an i n c r e a s e i n the hydroïïynamic energy d i s s i p a t i o n because the r o t a t i n g d o u b l e t i s i n c r e a s e d i n s i z e and the e f f e c t s h o u l d i n c r e a s e w i t h i n c r e a s e i n V . Yet we f i n d t h a t r a i s i n g the temperature (which reduces V ) causes an i n c r e a s e i n the v a l u e o f E i n E q u a t i o n (24) (see F i g u r e 7 ) . T h i s i s e s p e c i a l l y t r u e o f the u n s t a b l e s u s p e n s i o n s above the t h e t a temperature where the i n t e r a c t i o n i s wholly a t t r a c t i v e . The c o l l i s i o n doublet would i n t h a t case presumably be s m a l l e r than i t i s at lower t e m p e r a t u r e s . One e f f e c t o f the hydrodynamic f o r c e s i s to l i m i t the approach o f two p a r t i c l e s d u r i n g a c o l l i s i o n because the outflow time f o r the i n t e r v e n i n g f l u i d i s l o n g compared to the c o l l i s i o n t i m e . T h i s i s e s p e c i a l l y t r u e f o r l a r g e r p a r t i c l e s and above a = 5 ym i t becomes the dominant e f f e c t . The b e h a v i o u r o f s m a l l e r p a r t i c l e s ( a < 1 ym) i s not known w i t h such c e r t a i n t y but i t i s c l e a r t h a t i f the hydrodynamic f o r c e s are to be l a r g e enough to s e p a r a t e a c o l l i s i o n d o u b l e t , the p a r t i c l e s i n v o l v e d cannot be p e r m i t t e d to approach to the p o t e n t i a l energy minimum. Perhaps the d i f f e r e n c e i n b e h a v i o u r o f these s t e r i c a l l y s t a b i l i s e d systems can be t r a c e d to the f a c t t h a t the r e p u l s i v e f o r c e does not appear u n t i l polymer c h a i n i n t e r p é n é t r a t i o n i s s i g n i f i c a n t . The d e t a i l e d c a l c u l a t i o n o f the energy d i s s i p a t i o n as the d o u b l e t r o t a t e s and s e p a r a t e s would pose v e r y c o n s i d e r a b l e d i f f i c u l t i e s i n t h i s c a s e , s i n c e the flow b e h a v i o u r o f the i n t e r v e n i n g f l u i d i s undoubted l y i n f l u e n c e d by the l o c a l polymer segments. Some a s p e c t s o f t h i s work are p u b l i s h e d elsewhere {22_, 23) . R

R


g

0

ç

Summary Latex p a r t i c l e s s t a b i l i z e d by adsorbed h y d r o p h i l i c polymers e x h i b i t both p s e u d o p l a s t i c and Ostwald type flow b e h a v i o u r i n the neighbourhood o f the c r i t i c a l f l o c c u l a t i o n temperature ( c f . t . ) , which corresponds c l o s e l y to the t h e t a - t e m p e r a t u r e o f the adsorbed p o l y m e r . Ostwald flow occurs below the t h e t a temperature i n s t a b l e systems w i t h a f i n i t e , though s m a l l , y i e l d v a l u e . I t can be e x p l a i n e d q u a n t i t a t i v e l y by c a l c u l a t i n g the excess energy which the shear f i e l d must s u p p l y to d e s t r o y a l l


13.

HUNTER

209


ET AL.

d o u b l e t s not broken down by Brownian m o t i o n . At and above the c . f . t . a l l systems e x h i b i t a y i e l d v a l u e , τ „ , and the dependence o f τ on temper a t u r e can be q u a l i t a t i v e l y e x p l a i n e d by c o n s i d e r i n g both the i n t e r a c t i o n energy and the p r o b a b i l i t y o f c o l l i s i o n between p a r t i c l e s . The energy o f i n t e r a c t i o n i s independent o f core p a r t i c l e r a d i u s and i s o f the o r d e r o f a few times k T . The p l a s t i c v i s c o s i t y o f the system obeys the E i n s t e i n r e l a t i o n w i t h an e f f e c t i v e volume f r a c t i o n which suggests t h a t the adsorbed polymer chains are s t r e t c h e d to about twice t h e i r normal r . m . s . length.


β

Acknowledgements. T h i s work was supported by g r a n t s from the A u s t r a l i a n Research Grants Committee and by an A u s t r a l i a n Government Post Graduate S t u d e n t s h i p B.A.F. Literature

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M i c h a e l s , A.S., Chem. Fundam.

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Firth, B.A., Neville, J. Colloid Interface

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p . 606

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Colloidal Dispersions and Micellar Behavior

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