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Submitted on 1 Jan 1985
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SHEAR ELASTICITY AND VISCOSITY IN COLLOIDAL CRYSTALS AND LIQUIDS
H. Lindsay, P. Chaikin
To cite this version:
H. Lindsay, P. Chaikin. SHEAR ELASTICITY AND VISCOSITY IN COLLOIDAL CRYS- TALS AND LIQUIDS. Journal de Physique Colloques, 1985, 46 (C3), pp.C3-269-C3-280.
�10.1051/jphyscol:1985321�. �jpa-00224638�
JOURNAL DE PHYSIQUE
Colloque C3, supplkment au n03, Tome 46, mars 1985 page C3-269
SHEAR ELASTICITY AND VISCOSITY IN COLLOIDAL CRYSTALS AND LIQUIDS
H.M. L i n d s a y and P.M. ~ h a i k i n *
Department o f P h y s i c s , U n i v e r s i t y o f C a Z i f o r n i a , Los A n g e l e s , CA 90024, U.S.A.
* ~ x x o n R e s e a r c h and E n g i n e e r i n g Co., Annandale, NJ 08801, U.S.A.
and Department o f P h y s i c s , U n i v e r s i t y o f PennsyZvania, PhiZadeZphia, PA 19104-3859, U.S.A.
~ 6 s u m 6
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Nous a v o n s mesur6 les c o n s t a n t e s B l a s t i q u e s de c i s a i l - l e m e n t d e s c r i s t a u x c o l l o i d a u x e n u t i l i s a n t u n e t e c h n i q u e d e r d s o n a n c e e n g 6 o m e t r i e c y l i n d r i q u e a i n s i que l e u r v i s c o s i t 6 a l ' a i d e d ' u n e s 6 r i e d e v i s c o s i m i t r e s . L e s c o n s t a n t e s 6 l a s t i q u e s s o n t b i e n d g c r i t e s 2 l ' a i d e d e l a t h 6 o r i e d e D e b y e - ~ u c k e l S c o n d i t i o n d e p r e n d r e u n e c h a r g e r e n o r m a l i s Q e . L e s c r i s t a u x f o n d e n t p a r a d d i t i o n d ' h l e c t r o l y t e e t l e l i q u i d e r i s u l t a n t a une v i s c o s i t 6 d e p e n d a n t f o r t e m e n t d e s i n t e r a c t i o n s h l e c t r o s t a - t i q u e s e n t r e c o l l o ' i d e s c h a r g e s . L ' e x t r a p o l a t i o n d e s c o n s t a n t e s B l a s t i q u e s d a n s l 1 & t a t l i q u i d e e t l ' i n t r o d u c t i o n d ' u n t e m p s d e r e l a x a t i o n ph6nom6nologique p e r m e t d e p r 6 v o i r l a v i s c o s i t 6 du l i q u i d e . Nous o b s e r v o n s Bgalement l a f u s i o n d e c i s a i l l e m e n t p a r a p p l i c a t i o n d'une c o n t r a i n t e s u f f i s a n t e . C e t t e t r a n s i t i o n e s t accompagnee d ' u n e Q l e v a t i o n d e l a v i s c o s i t d l o r s q u e l ' o n p a s s e du s o l i d e a u l i q u i d e .A b s t r a c t
-
We h a v e m e a s u r e d t h e s h e a r e l a s t i c c o n s t a n t s o f c o l l o i d a l c r y s t a l s u s i n g a s t a n d i n g c y l i n d r i c a l r e s o n a n c e t e c h n i q u e and t h e v i s c o s i t y w i t h a v a r i e t y of r h e o m e t e r s . The e l a s t i c c o n s t a n t s a r e w e l l d e s c r i b e d b y D e b y e - H u c k e l t h e o r y w i t h a r e n o r m a l i z e d c h a r g e . The c r y s t a l s m e l t upon a d d i t i o n o f e l e c t r o l y t e and t h e r e s u l t i n g l i q u i d h a s a v i s c o s i t y w h i c h i s s t r o n g l y d e p e n d e n t upon t h e e l e c t r o s t a t i c i n t e r a c t i o n s b e t w e e n t h e c h a r g e d c o l l o i d s . E x t r a p o l a t i o n o f t h e e l a s t i c c o n s t a n t s i n t o t h e l i q u i d r e g i m e , t o g e t h e r w i t h a p h e n o m e n o l o g i c a l r e l a x - a t i o n t i m e , a l l o w us t o p r e d i c t t h e v i s c o s i t y o f t h e l i q u i d . We a l s o o b s e r v e a t r a n s i t i o n f r o m s o l i d t o l i q u i d , " s h e a r m e l t i n g " , upon a p p l i c a t i o n o f s u f f i c i e n t l y h i g h stress. T h i s t r a n s i t i o n i s a c c o m p a n i e d b y a n i n c r e a s e i n v i s c o s i t y i n g o i n g f r o m s o l i d t o l i q u i d .I . I n t r o d u c t i o n
-
C o l l o i d a l c r y s t a l s h a v e a t t r a c t e d much i n t e r e s t i n t h e p a s t s e v e r a l y e a r s . I n t h e c o l l o i d a l c r y s t a l we h a v e a s i m p l e , known p o t e n t i a l , a s p h e r i c a l l y s y m m e t r i c s c r e e n e d Coulomb p o t e n t i a l b e t w e e n p a r t i c l e s . N o t o n l y i s t h e p o t e n t i a l s i m p l e , b u t i t a l s o c a n b e v a r i e d o v e r a wide r a n g e , b y a l t e r i n g p a r t i c l e c h a r g e , i n t e r p a r t i c l e s p a c i n g , a n d i o n i c s c r e e n i n g . T h e r e f o r e we c a n c h e c k o u r u n d e r - s t a n d i n g o f c r y s t a l phenomena, m e l t i n g , s t r u c t u r e , etc. i n ways n o t p o s s i b l e i n o r d i n a r y a t o m i c s o l i d s . I n a d d i t i o n , t h e c r y s t a l l i n e s t r u c t u r e o f t h e c o l l o i d a l c r y s t a l i s t h e e l a s t i c m o d u l i of t h e s e s y s t e m s a r e o n t h e o r d e r o f 1B1"t",*;es l o w e r t h a n o r d i n a r y s o l i d s . T h i s m a k e s it p o s s i b l e t o e x a m i n e t h e f l o w b e h a v i o r a t stresses f a r l o w e r and t i m e s f a r s h o r t e r t h a n t h o s e r e q u i r e d t o make o r d i n a r y m a t e r i a l s f l o w .
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985321
JOURNAL DE PHYSIQUE
11. S h e a r Modulus
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W h i l e t h e c r y s t a l s t r u c t u r e r e v e a l s t h e p a r t i c l e c o n f i g u r a t i o n s and d e n s i t y , e l a s t i c c o n s t a n t m e a s u r e m e n t s a r e needed t o d e t e r m i n e t h e i n t e r p a r t i c l e f o r c e s . The f i r s t measurement o f a n e l a s t i c modulus i n t h e c o l l o i d a l c r y s t a l was made i n 1977 by C r a n d a l l and w i l l i a m s l . They made u s e o f t h e f a c t t h a t t h e p o l y b a l l s a r e d e n s e r t h a n w a t e r
( P o = 1.05 g / c m 3 ) , s o t h a t t h e r e was a g r a v i t a t i o n a l c o m p r e s s i o n o f t h e c r y s t a l which c o u l d b e q u a n t i f i e d b y m e a s u r i n g t h e c h a n g e i n t h e l a t t i c e c o n s t a n t a s a f u n c t i o n o f h e i g h t i n t h e i r c e l l . From t h i s c o m p r e s s i o n o f t h e l a t t i c e t h e y d e r i v e d a Y o u n g ' s m o d u l u s o f
-
1~ ~ n e / c m ~ .
To m e a s u r e t h e s h e a r modulus r e q u i r e s a d i f f e r e n t t e c h n i q u e . One s t a n d a r d method i s t o m e a s u r e t h e v e l o c i t y o f t r a n s v e r s e sound waves i n t h e sample.
From ~ o a n n y ~ , t h e e q u a t i o n o f m o t i o n f o r t h e p o l y b a l l s i s
T h e m o d i f i e d N a v i e r - S t o k e s e q u a t i o n f o r t h e s o l v e n t i n c l u d e s a v i s c o u s t e r m b e t w e e n t h e s o l v e n t and t h e p o l y b a l l s , and i s w r i t t e n
w h e r e c = p o l y b a l l c o n c e n t r a t i o n , f i s t h e S t o k e s d r a g c o e f f i c i e n t 6 nrl a , i s t h e s o l v e n t v i s c o s i t y , E i s t h e s h e a r modulus, V i s t h e s o l v e n t v e l o c i t y f i e l d , U i s t h e p o l y b a l l d i s p l a c e m e n t f i e l d f r o m t h e l a t t i c e s i t e s , a n d p i s t h e s o l v e n t d e n s i t y . I n t h e l o n g w a v e l e n g t h l i m i t f o r t r a n s v e r s e waves, we g e t a s i m p l e wave e q u a t i o n w i t h v i s c o u s damping:
T h i s y i e l d s p r o p a g a t i n g t r a n s v e r s e waves when w r / ~ < 1. We t h e r e - f o r e c h o o s e o u r e x p e r i m e n t a l f r e q u e n c y r a n g e t o m e e t t h i s c o n d i t i o n . L o n g i t u d i n a l modes d o n o t p r o p a g a t e , d u e t o v i s c o u s damping. I n t r a n s v e r s e waves, t h e p o l y b a l l s and t h e s o l v e n t move t o g e t h e r , i n l o n g i t u d i n g a l waves, t h e s o l v e n t h a s t o f l o w back a r o u n d t h e p o l y - b a l l s , l e a d i n g t o h i g h f r i c t i o n . L o n g i t u d i n a l waves must b e s t u d i e d by o t h e r means, s u c h a s t h e t h e r m a l d i f f u s e s c a t t e r i n g o f ~ u r d ~ . A t v e r y h i g h f r e q u e n c i e s , when t h e v i s c o u s p e n e t r a t i o n d e p t h i s l e s s t h a n t h e i n t e r p a r t i c l e s p a c i n g w e would a g a i n e x p e c t l o n g i t u d i n a l p r o p a g a t i n g modes. But t h e s e w i l l o c c u r f o r w > 1~~ hz.
To m e a s u r e t h e v e l o c i t y o f t h e s h e a r w a v e s , we m e a s u r e s t a n d i n g s h e a r modes i n a c y l i n d r i c a l g e o m e t r y . T h i s t e c h n i q u e w a s f i r s t u s e d by p i e r a n s k i 3 , who o b s e r v e d t h e v i b r a t i o n o f K o s s e l l i n e s t o f i n d t h e r e s o n a n t f r e q u e n c i e s . I n o u r a p p a r a t u s 5 we u s e m e c h a n i c a l methods t o d e t e c t t h e modes, a l l o w i n g u s t o m e a s u r e m a t e r i a l s i n which K o s s e l l i n e s a r e d i f f i c u l t t o d e t e c t , o r a l t o g e t h e r a b s e n t . I n o u r s y s t e m , a g l a s s v i a l i s f i l l e d w i t h s a m p l e t o a f i x e d l e v e l . A g l a s s b o b b i n e x t e n d s i n t o t h e s a m p l e , s u p p o r t e d by a c e n t r a l p o s t w i t h a s u p p o r t i n g s a p h i r e b e a r i n g . A s m a l l p e r m a n e n t magnet and m i r r o r a r e g l u e d o n t o t h e bobbin. The v i a l i s p l a c e d b e t w e e n a p a i r o f H e l m h o l t z c o i l s , w h i c h p r o d u c e a n a c m a g n e t i c f i e l d a n d o s c i l l a t e t h e bobbin. The a m p l i t u d e o f o s c i l l a t i o n of t h e b o b b i n i s d e t e r m i n e d by r e f l e c t i n g t h e beam f r o m a l a s e r o f f t h e m i r r o r t o a p o s i t i o n s e n s i t i v e d e t e c t o r . By s w e e p i n g t h e f r e q u e n c y o f t h e m a g n e t i c f i e l d a t c o n s t a n t a m p l i t u d e w h i l e d e t e c t i n g t h e a m p l i t u d e
o f o s c i l l a t i o n of t h e b o b b i n u s i n g a l o c k i n a m p l i f i e r , a r e s o n a n c e c u r v e c a n b e f o u n d . T h e p e a k s c a n b e i d e n t i f i e d w i t h t h e n o r m a l modes o f a c y l i n d e r :
- / "ln 2 + ( 2 p + 1) 2 2 n
= J E / p (?)
n P 0 4 2
where a a r e t h e z e r o s o f t h e B e s s e l f u n c t i o n J1, ro i s t h e r a d i u s o f t h e c y l i n d e r , a n d R i s t h e h e i g h t o f t h e c y l i n d e r . T y p i c a l l y o n l y t h e f i r s t peak i s u s e d t o d e t e r m i n e t h e s h e a r modulus.
F i g u r e 1 shows e x p e r i m e n t a l d a t a f o r a c r y s t a l made o f ,109 m i c r o n d i a m e t e r s p h e r e s , w i t h a volume f r a c t i o n o f 2%. The s h e a r modulus i s p l o t t e d a g a i n s t added H C 1 , w h i c h i n c r e a s e s t h e s c r e e n i n g b e t w e e n p a r t i c l e s . The s h e a r modulus f a l l s s m o o t h l y w i t h i n c r e a s i n g s c r e e n - i n g (and d e c r e a s i n g p a r t i c l e i n t e r a c t i o n ) a s one would e x p e c t . A t a v a l u e o f 11 m o l e s / l added H C 1 , t h e s h e a r modulus d r o p s a b r u b t l y t o z e r o . T h i s i s n o t a n i n s t r u m e n t a l e f e c t , a s we c a n m e a s u r e t h e
5
s h e a r m o d u l u s t o l e s s t h a n 1 d y n e / c m
.
T h i s d r o p o c c u r s a s t h e c r y s t a l m e l t s , a s o b s e r v e d by t h e l o s s o f o p a l e s c e n c e i n t h e c o l - l o i d . S i m i l a r d r o p s i n t h e s h e a r modulus a r e s e e n a t t h e m e l t i n g t r a n s i t i o n i n 4 % and 1 0 % volume f r a c t i o n s a m p l e s . T h i s d r o p i n d i - c a t e s t h a t t h e t r a n s i t i o n i s f i r s t o r d e r a s f o u n d i n t h e s e l f - d i f f u s i o n m e a s u r e m e n t s o f ~ o z i e r ~ .We s t u d y t h e s h e a r modulus t o u n d e r s t a n d t h e i n t e r p a r t i c l e f o r c e s i n t h e c r y s t a l . A s i m p l e model c o n n e c t i n g t h e s h e a r modulus w i t h t h e p o t e n t i a l was found by Joanny:
w h e r e d i s t h e n e a r e s t n e i g h b o r d i s t a n c e , V i s t h e Debye-Huckel
~ o t e n t i a l :
a n d K i s t h e s c r e e n i n g l e n g t h
K2 - 4ne2 (cZ
*
+c
n . 2 . ) ( 6 b )~k sT 1 I * 1
w i t h E = d i e l e c t r i c c o n s t a n t , Z = e f f e c t i v e c h a r g e o f p o l y b a l l s , and ni and z i t h e number and c h a r g e of any a d d i t i o n a l i o n s .
The e f f e c t i v e c h a r g e o f t h e p o l y b a l l i s l e s s t h a n t h e r e a l c h a r g a s e x p l a i n e d i n t h e c h a r g e r e n o r m a l i z a t i o n w o r k o f A l e x a n d e r
9. .
F u r t h e r t e r m s beyond t h e n e a r e s t n e i g h b o r s y i e l d c o r r e c t i o n s o f a t most 2 5 % , and a r e i n c l u d e d i n t h e t h e o r e t i c a l c u r v e s shown.
I n F i g u r e 1, t h e s o l i d c u r v e r e p r e s e n t s t h e p r e d i c t i o n s o f t h e t h e o r y , u s i n g t h e v a l u e o f t h s h e a r modulus a t , z e r o added e l e c t r o - ~ l y t e t o s e t t h e v a l u e o f z
.
I n t h i s p l o t z = 3%5 e l e c t r o n i c c h a r g e s . The t h e o r y f i t s t h e d a t a w e l l f o r t h e 2% and 4 % s a m p l e s , e x c e p t a t t h e m e l t i n g t r a n s i t i o n where we c a n n o t e x p e c t t h e t h e o r y t o h o l d . A t h i g h volume f r a c i o n (la%), t h e r e i s some d i s c r e p a n c y b e t w e e n t h e o r y and observation'.,
P a r t o f t h i s d i s c r e p a n c y i s c a u s e d by t h e a d o p t i o n o f a c o n s t a n t z.
A t t h e h i g h e l e c t r o l y t e concen- t r a t i o n s u s e d i n t h i s e x p e r i m e t , t h e r e n o r m a l i z e d c h a r g e s h o u l d.r
grow c l o s e r t o t h e r e a l c h a r g e
,
r a k s i n g t h e s h e a r modulus v a l u e s above t h o s e c a l c u l a t e d f o r c o n s t a n t z.
I n F i g u r e .,2, t h e c o n c e n t r a t i o n dependence o f t h e s h e a r modulus i s shown w i t h a r a n g e o f volume f r a c t i o n s o f 1% t o 10%. The modulus i s s t r o n g l y d e p e n d e n t on t h e c o n c e n t r a t i o n , and i s i n good a g r e e m e n t w i t h t h e o r y .
JOURNAL DE PHYSIQUE
.
109MICRON 2%
Figure 1
-
Shear modulus as a function of added electrolyte (HC1 in m i c r o e q u i v a l e n t s / l i t e r ) f o r .109 m i c r o n p o l y s t y r e n e spheres, 2%v o l u m e fraction, in water. S o l i d c u r v e i s calculated d e p e n d e n c e from eq. 4.
0 0.02 0.04 0.06 0.08 0.1
volume fraction
F i g u r e 2
-
S h e a r m o d u l u s vs. v o l u m e f r a c t i o n for .109 m i c r o n spheres. Solid curve from eq. 4.I f we mix s t r o n g l y i n t e r a c t i n g m o n o d i s p e r s e c o l l o i d s w i t h t w o d i f f e r e n t p a r t i c l e s s i z e s (and more i m p o r t a n t l y , c h a r g e s ) t o g e t h e r , we may e x p e c t one o r more o f t h e f o l l o w i n g r e s u l t s :
1 ) A c o l l o i d a l l i q u i d w i t h n o l o n g r a n g e o r d e r a n d z e r o s h e a r modulus;
2) a phase s e p a r a t i o n i n t o c r y s t a l l i n e r e g i o n s c o n t a i n i n g o n l y one s i z e o f p a r t i c l e ;
3 ) a compound s t r u c t u r e , g i v i n g a f i n i t e s h e a r modulus w i t h Bragg s c a t t e r i n g ;
4 ) an a l l o y , a l s o w i t h f i n i t e s h e a r modulus and Bragg s c a t t e r i n g ; and
5 ) a c o l l o i d a l g l a s s , w i t h f i n i t e s h e a r m o d u l u s b u t no B r a g g s c a t t e r i n g .
E x p e r i m e n t a l l y , i f we mix .109 p ( z *
-
3 0 0 ) a n d .220 ~ ( z *-
600-7 0 0 ) p o l y b a l l s , a t p a r t i c l e a n d e l e c t r o l y t e c o n c e n t r a t i o n s h i g h enough f o r t h e p u r e s a m p l e s t o c r y s t a l l i z e , we f i n d a c o l l o i d a l g l a s s . The g l a s s h a s a s h e a r m o d u l u s s i m i l a r i n m a g n i t u d e t o c o l l o i d a l c r y s t a l s . The c o l l o i d a l g l a s s s a m p l e s , i f a l l o w e d t o s t a n d f o r s e v e r a l months, w i l l sometimes form r e g i o n s i n which Bragg s c a t t e r i n g i s seen, p r o b a b l y i n d i c a t i n g compound f o r m a t i o n .
The s h e a r modulus of t h e c o l l o i d a l g l a s s i s s i m i l a r i n magnitude t o t h a t o f t h e c r y s t a l . A s t h e r a t i o o f .109 t o .220 p a r t i c l e s i s v a r i e d af c o n s t a n t o v e r a l l volume f r a c t i o n , t h e s h e a r modulus v a r i e s
smoothly
.
111. V i s c o s i t y
-
We b e g a n s t u d y i n g t h e v i s c o s i t y of c o l l o i d a l c r y s t a l s i n t h e s i m p l e s t p o s s i b l e e x p e r i m e n t a l s e t up
-
measuring t h e f l o w o f f l u i d down a c a p i l l a r y t u b e , u s i n g an Ostwald v i s c o s i m e t e r . We measured t h e e f f e c t i v e v i s c o s i t y o f t h e sample a s a f u n c t i o n o f added e l e c - t r o l y t e . We soon d i s c o v e r e d a c u r i o u s r e s u l t : A s t h e i n t e r p a r t i c l e p o t e n t i a l d e c r e a s e d w i t h added e l e c t r o l y t e , we would s e e a sudden r i s e i n t h e e f f e c t i v e v i s c o s i t y , f o l l o w e d by a c o n t i n u a l d e c r e a s e i n v i s c o s i t y . T h i s r i s e i n v i s c o s i t y d i d n o t o c c u r when t h e sample was a l r e a d y m e l t e d w i t h no a p p l i e d s h e a r .One i n d i c a t i o n of t h e n a t u r e o f t h i s r i s e i s t h a t w h i l e t h e m a t e r i a l i n t h e r e s e r v o i r s was c r y s t a l l i n e , t h e sample i n t h e c a p i l l a r y t u b e o f t e n seemed t o l o s e i t s o p a l e s c e n c e , a l t h o u g h t h e d i f f e r e n c e was s o m e t i m e s h a r d t o s e e . T h i s i n d i c a t e d t h a t w e p i g h t b e s e e i n g a s h e a r m e l t i n g t r a n s i t i o n , a s r e p o r t e d by Hoffman and. Ackerson and c l a r k 9 . The c o n c e p t i s t h a t below t h i s m e l t i n g t r a n s i t i o n you have o r d e r e d f l o w o f p l a n e s , and t h a t above t h e s h e a r m e l t i n g t r a n s i t i o n t h e p a r t i c l e s a r e d i s o r d e r e d , y i e l d i n g an i n c r e a s e i n v i s c o s i t y . TO t e s t t h i s s h e a r m e l t i n g h y p o t h e s i s , we compared t h e v i s c o s i t y of a c r y s t a l l i n e sample w i t h a c o l l o i d a l g l a s s . I n t h e g l a s s we e x p e c t t o s e e no such t r a n s i t i o n , s i n c e t h e flow i s a l w a y s d i s o r d e r e d . I n F i g u r e 3 t h e v i s c o s i t i e s o f g l a s s a n d c r y s t a l l i n e s a m p l e s w i t h comparable s h e a r moduli a r e shown. The c o l l o i d a l g l a s s , a s expec- t e d , shows no jump i n v i s c o s i t y . I n a d d i t i o n , t h e v i s c o s i t y of t h e g l a s s i s much h i g h e r t h a n t h e v i s c o s i t y o f c r y s t a l b e f o r e t h e s h e a r m e l t i n g t r a n s i t i o n , w h i l e a f t e r t h e t r a n s i t i o n t h e v i s c o s i t i e s a r e c o m p a r a b l e . T h i s i n d i c a t e s t h a t t h e o r d e r i n g i n t h e f l o w d o e s r e d u c e t h e v i s c o s i t y .
JOURNAL DE PHYSIQUE
microeq/l HCI
o glass + crystal
F i g u r e 3
-
E f f e c t i v e v i s c o s i t y f o r a c o l l o i d a l " g l a s s " and a c o l l o i d a l c r y s t a l with approximately equal s h e a r modulus a s a func- t i o n of added e l e c t r o l y t e .-109
MICRON
4% 0 HCLSHEAR
RATEHZ
x 1 0 2F i g u r e 4
-
S h e a r s t r e s s vs. s h e a r r a t e f o r a 4% volume f r a c t i o n c o l l o i d a l c r y s t a l of .lo9 micron spheres.Flow down a tube i s not a good way t o s y s t e m a t i c a l l y study a non- Newtonian f l u i d , s i n c e t h e s h e a r s t r e s s v a r i e s d r a m a t i c a l l y over t h e c r o s s s e c t i o n of t h e t u b e . We t h e r e f o r e d e c i d e d t o move t o a c o u e t t e geometry, where v a r i a t i o n s i n t h e s h e a r can be reduced. We have c o n s t r u c t e d two types of viscometers. I n one viscometer we r o t a t e t h e o u t e r c y l i n d e r , s e t t i n g a known s h e a r r a t e , and t h e n measure t h e torque on t h e i n n e r c y l i n d e r , thereby determining t h e s h e a r s t r e s s . The o t h e r design we used was a Z i m m viscometer. I n t h i s design, an i n n e r g l a s s c y l i n d e r , s e t t o n e u t r a l buoyancy, i s supported by t h e sample between t h e two c y l i n d e r s . The i n n e r c y l i n - d e r i s c e n t e r e d by s u r f a c e t e n s i o n . Within t h e i n n e r c y l i n d e r i s an aluminum block, w h i l e o u t s i d e t h e o u t e r c y l i n d e r i s a magnet. The magnet i s r o t a t e d v i a a motor, c r e a t i n g a r o t a t i n g magnetic f i e l d , which produces a torque on t h e aluminum block. I n t h i s s e t u p we s e t t h e torque, which determines t h e shear s t r e s s , and then measure t h e r o t a t i o n r a t e of t h e i n n e r c y l i n d e r , thereby f i n d i n g t h e s h e a r r a t e . One fundamental d i f f e r e n c e between t h e two a p p a r a t i i s t h a t i n t h e f i r s t one described, t h e o u t e r c y l i n d e r r o t a t e s , w h i l e i n t h e Z i m m viscometer t h e i n n e r c y l i n d e r r o t a t e s . This means t h a t i n t h e Z i m m viscometer Taylor i n s t a b i l i t i e s may be found, b u t not i n t h e o t h e r viscometer.
I n F i g u r e 4 i s shown a n e x p e r i m e n t a l s h e a r s t r e s s
-
s h e a r r a t e curve. The sample i s .109 d i a m e t e r p o l y b a l l s w i t h weight f r a c t i o n 4%. No s h e a r m e l t i n g i s seen i n t h i s sample. The sample i s c l e a r l y s h e a r t h i n n i n g . I t might be tempting from t h i s p l o t t o d e s c r i b e t h e sample a s having a y i e l d s t r e s s of-
4 dynes/cm2. But t h i s would n o t be c o r r e c t . I f we p l o t t h e d a t a on a log-log s c a l e , we s e e t h e d a t a forms a smooth curve ( F i g u r e 5 ) , w i t h a power lower r e l a t i o n a t low s h e a r , t h i s type of power law behavior i s seen i n our samples a t low s h e a r , a l t h o u g h t h e e x p o n e n t v a r i e s b e t w e e n 2.5 and 7 . T h i s type of power law creep i s found i n o t h e r m a t e r i a l s , and i s d i s c u s - sed i n t h e paper by Weitz e t a l l 0 .L O G S H E A R R A T E
.I09MICRON
4% 0H C L
Figure 5
-
Log s t r e s s vs. log r a t e f o r t h e d a t a of f i g u r e 4.C3-276 JOURNAL DE PHYSIQUE
F i g u r e 6 shows a s t r e s s - r a t ~ e c u r v e w i t h a s h e a r m e l t i n g t r a n s i t i o n . A t a s h e a r r a t e o f
-
40 Hz t h e r e i s a r a p i d r i s e i n t h e s t r e s s , which c o r r e s p o n d s t o a change i n t h e a p p e a r a n c e of t h e sample from s t r o n g o p a l e s c e n c e t o a m i l k y - w h i t e c o l o r . A t t h e s a m e t i m e t h e i n n e r bobbin i n t h e Z i m m v i s c o m e t e r becomes drawn t o t h e s i d e of t h e o u t e r c y l i n d e r . T h i s i s why t h e r e i s a s m a l l g a p i n t h e d a t a . A t h i g h e r s t r e s s e s t h e bobbin i s a g a i n c e n t e r e d . The i n s t a b i l i t y o f t h e bobbin a t s h e a r m e l t i n g p r o b a b l y i n d i c a t e s a n anomalous d e r i v a - t i v e o f t h e s t r e s s - r a t e curve.shear rate Hz
F i g u r e 6
-
S t r e s s r a t e c u r v e f o r a sample of .91 m i c r o n s p h e r e s , 4%volume f r a c t i o n , 35 m i c r o e q u i v a l e n t s ~ ~ l / ~ i t e r , showing t h e s h e a r m e l t i n g t r a n s i t i o n s .
F i g u r e 7 shows a s e r i e s of s t r e s s - r a t e c u r v e s f o r a .091 d i a m e t e r , 4% sample a s a f u n c t i o n o f added e l e c t r o l y t e . They show a d e c r e a s e i n v i s c o s i t y w i t h i n c r e a s i n g i o n i c s t r e n g t h , and a d e c r e a s e i n t h e s h e a r r a t e s and s t r e s s a t which t h e m e l t i n g t r a n s i t i o n s o c c u r . The d a t a i n d i c a t e d w i t h t r i a n g l e s i s t h e c u r v e t a k e n j u s t a f t e r t h e s a m p l e m e l t e d a t r e s t , and
i s
n e a r l y l i n e a r .I n F i g u r e 8 we p l o t t h e s h e a r s t r e s s a t which s h e a r m e l t i n g o c c u r s v e r s u s t h e s h e a r modulus from t h e added e l e c t r o l y t e .
I V . V i s c o s i t y
of
C o l l o i d a l L i q u i d s-
A s w e l l a s measuring t h e v i s c o s i t y of t h e c r y s t a l , we have measured t h e v i s c o s i t y of t h e c o l l o i d a l l i q u i d . W e d e f i n e t h e c o l l o i d a l l i q u i d a s t h e s t a t e o f t h e c o l l o i d i n which t h e i n t e r a c t i o n s a r e t o o weak f o r t h e system t o o r d e r , i-e., t h e c o l l o i d i s "melted". T h i s i s t h e s t a t e i n w h i c h m o s t c o l l o i d a l s u s p e n s i o n s i n n a t u r e a r e found. To d e s c r i b e t h e v i s c o s i t y o f t h e l i q u i d , we u s e a two f l u i d model, i n which we f i n d t h r e e c o n t r i b u t i o n s t o t h e v i s c o s i t y : t h e v i s c o s i t y o f t h e s o l v e n t T i o , t h e v i s c o s i t y o f t h e i n t e r a c t i n g p o l y b a l l f l u i d 0 and a hydrodynamic c o n t r i b u t i o n on t h e o r d e r of
0 HCI 10 HCI
shear rote Hz
o 4 0 HCI 50 HCI
F i g u r e 7
-
S t r e s s r a t e c u r v e s f o r . 9 1 m i c r o n s p h e r e s , 4 % v o l u m e f r a c t i o n a t d i f f e r e n t e l e c t r o l y t e c o n c e n t r a t i o n s . 0 m i c r o e q u i - v a l e n t s H C 1 ,+
1 0 m i m c r o e q u i v a l e n t s H ~ l / l i t e r ,0
4 0 m i c r o e q u i - v a l e n t s H C l / l i t e r , A 50 m i c r o e q u i v a l e n t s ~ C l / l i t e r .shear modulus dynes/cm2
F i g u r e 8
-
S h e a r s t r e s s a t t h e s h e a r m e l t i n g t r a n s i t i o n s a s a func- t i o n o f s h e a r m o d u l u s a t z e r o s t r e s s f o r s a m p l e s o f . 9 1 m i c r o n s p h e r e s , 4% volume f r a c t i o n , w i t h a d i f f e r e n t e l e c t r o l y t e c o n c e n t r a - t i o n s .C3-278 JOURNAL DE PHYSIQUE
t h e volume f r a c t i o n , which i s small. We w r i t e t h e v i s c o s i t y of our s a m p l e
n
a sFrom our measurements of q and t h e known v a l u e s of
no
and $I,
we can determinenl.
Phenomenologically, we may expect t h e v i s c o s i t y of a l i q u i d t o go l i k e
where G i s t h e s h e a r modulus i n t h e l i q u i d , and T i s a "Maxwell"
r e l a x a t i o n t i m e , t h e t i m e t o r e l a x a s t r e s s . To d e t e r m i n e E , we measure t h e shear modulus i n t h e s o l i d , determining t h e e f f e c t i v e charge, and use t h e t h e o r e t i c a l model d i s c u s s e d e a r l i e r (Eq. 5 ) t o f i n d t h e v a l u e o f E i n t h e l i q u i d a t known e l e c t r o l y t e concen- t r a t i o n . We p l o t values of nl, VS. E i n Figure 9, and g e t a l i n e a r r e l a t i o n s h i p . This shows t h a t t h e r e l a x a t i o n t i m e -c i s independent of E.
D e l t a v i s c o s i t y v s C a l c E l a s t i c i t y
- 1 09 micron 5%
1 , I
S h e a r Modulus (dynes/cm )
Figure 9
-
V i s c o s i t y of c o l l o i d a l l i q u i d s from .I09 micron spheres, 5% volume f r a c t i o n a s a f u n c t i o n of t h e s h e a r modulus e x t r a p o l a t e d from measurements i n t h e s o l i d phase.The r e l a x a t i o n t i m e T may be thought of a s t h e time it t a k e s f o r a s i n g l e p a r t i c l e i n t h e c o l l o i d t o d i s t i n g u i s h between t h e s o l i d and l i q u i d phase. I n a s o l i d , v i b r a t i o n s of t h e p a r t i c l e can occur on t h e order of a Lindemann d i s t a n c e , w h i l e i n a l i q u i d t h e p a r t i c l e h a s no such l i m i t . We might expect t h e r e l a x a t i o n t i m e t o be t h e t i m e it t a k e s t h e p a r t i c l e t o d i f f u s e a "Lindemann d i s t a n c e "
XR
w h e r e D i s t h e S t o k e s f r e e d i f f u s i o n constant,-. kT W e u s e t h e s t o k e s - 8 i f f u s i o n c o n s t a n t b e c a u s e o v e r these6'&?ort d i s t a n c e s , p a r t i c l e i n t e r a c t i o n s a r e n o t i m p o r t a n t a n d t h e s e l f - d i f f u s i o n c o n s t a n t i s j u s t e q u a l t o t h e S t o k e s value1'.
The v a l u e which we f i n d f o r T i s
w h e r e Q i s t h e a v e r a g e p a r t i c l e s p a c i n g .
W e t h e r e f o r e f i n d t h a t t h e v i s c o s i t y o f t h e c o l l o i d a l l i q u i d g o e s a s
s o t h a t by making a s i n g l e d e t e r m i n a t i o n o f t h e s h e a r modulus o f t h e s o l i d , we may p r e d i c t t h e v i s c l ~ s , ~ ~ o f t h e l i q u i d u s i n g a v e r y s i m p l e m o d e l . H e s s a n d K l e i n h a v e made a m o r e i n v o l v e d t h e o r e t i c a l p r e d i c t i o n o f t h e v i s c o s i t y and f i n d t h e r e s u l t
w i t h f o f o r d e r u n i t y w h i c h d i f f e r s f r o m o u r r e s u l t o n l y b y a n u m e r i c a l f a c t o r .
We n o t e t h a t t h e r e l a t i o n s l e a d i n g t o e q u a t i o n 1 0 a r e p r o b a b l y
not
c o r r e c t f o r " s i m p l e " f l u i d s . I n p a r t i c u l a r , t h e r e l a x a t i o n t i m e T
w i l l b e a c t i v a t e d and s t r o n g l y d e p e n d e n t o n E f o r o t h e r f l u i d s . I n t h e c o l l o i d t h e r e i s a n a d d i t i o n a l component, t h e s o l v e n t - w a t e r i n o u r c a s e , w h i c h c a n t a k e momentum a n d s t r e s s o u t o f t h e s y s t e m . Between " c o l l i s i o n s " t h e c o l l o i d a l p a r t i c l e s move by Brownian m o t i o n r a t h e r t h a n b a l l i s t i c a l l y .
V . C o n c l u s i o n s
-
The r h e o l o g y of t h e c o l l o i d a l c r y s t a l s y s t e m p r e s e n t s a f a s c i n a t i n g a n d c o m p l e x v a r i e t y o f b e h a v i o r s e n c o m p a s s i n g r i g i d i t y , p l a s t i c f l o w , a h i g h l y a n i s o t r o p i c a n d n o n l i n e a r f l u i d a n d s h e a r m e l t i n g f r o m s o l i d t o l i q u i d . T h e s o l i d p h a s e i s w e l l c h a r a c t e r i z e d by Debye-Huckel t h e o r y w i t h a r e n o r m a l i z e d c h a r g e . T h i s same r e n o r m a l - i z e d c h a r g e may b e u s e d t o c a l c u l a t e t h e v i s c o s i t y o f t h e l i q u i d s t a t e i f one a s s u m e s a p h e n o m e n o l o g i c a l r e l a x a t i o n t i m e w h i c h i s i n d e p e n d e n t o f t h e i n t e r a c t i o n s t r e n g t h b e t w e e n c o l l o i d p a r t i c l e s b u t r e l a t e s t o t h e Brownian m o t i o n of t h e n o n i n t e r a c t i n g p a r t i c l e s . The s h e a r m e l t i n g t r a n s i t i o n h a s been o b s e r v e d i n t h e s t r e s s - r a t e d e p e n d e n c e of t h e c o l l o i d s and c o r r e s p o n d s t o a hydrodynamic l i k e t r a n s i t i o n f r o m a low v i s c o s i t y o r d e r e d f l o w t o a h i g h v i s c o s i t y d i s o r d e r e d f l o w .
R e f e r e n c e s
1. CRANDALL, R.S. and W i l l i a m s , R . , S c i e n c e 1 9 8 ( 1 9 7 7 ) 293.
2. JOANNY, J.F.
,
J. C o l l o i d , I n t e r f a c e S c i . 71(1979) 622.3 . DUBOIS-VIOLETTE, E., P i e r a d s k i , P., ~ o t h e z F. and S t r z e l e c k i , L., J. P h y s . ( P a r i s )
41
( 1 9 8 0 ) 369.C3-280 JOURNAL DE PHYSIQUE
HURD, A.J., C l a r k , N.A., M o c k l e r , R.C., O ' S u l l i v a n , W.J., P h y s . R e v . A 2 6 ( 1 9 8 2 ) 2 8 6 9 .
L I N D S A Y ~ H.M. a n d C h a i k i n , P.M., J. C h e m . P h y s . 7 6 ( 1 9 8 2 ) 3 7 7 4 . D O Z I E R , W.D., L i n d s a y , H.M., C h a i k i n , P.M., L e s ~ o u c h e s , W i n t e r School ( 1 9 8 4 ) .
ALEXANDER, S., C h a i k i n , P.M., G r a n t , P., M o r a l e s , G . J . , P i n c u s , P., H o n e , D., J . C h e m . P h y s . , t o be p u b l i s h e d .
HOFFMAN, R.L., T r a n s a c t i o n s Soc. R h e o l o g y
16
( 1 9 7 2 ) 155.ACKERSON, B . J . a n d C l a r k , N.A., P h y s . R e v . L e t t .
46
( 1 9 8 1 ) 1 2 3 . W E I T Z , D.A., D o z i e r , W.D. a n d C h a i k i n , P.M., L e s H o u c h e s , W i n t e r S c h o o l .K L E I N , R., L e s H o u c h e s , W i n t e r S c h o o l o n C o l l o i d a l C r y s t a l s ( 1 9 8 4 ) .
K L E I N , R. and H e s s , W., F a r a d a y D i s c . C h e m . S o c .
