Birefringence and stress in polymer melts under shear and extension

BIREFRINGENCE AND STRESS

IN POLYMER MELTS

UNDER SHEAR AND EXTENSION

J.A. van Aken

BIREFRINGENCE AND STRESS

IN POLYMER MELTS

UNDER SHEAR AND EXTENSION

B i b l i o t h e e k TU D e l f t

BIREFRINGENCE AND STRESS

IN POLYMER MELTS

UNDER SHEAR AND EXTENSION

PROEFSCHRIFT ter verkrijging van

de graad van doctor in de

technische wetenschappen

aan de Technische Hogeschool Delft,

op gezag van de rector magnificus,

prof. ir. B.P.Th. Veltman

voor een commissie aangewezen

door het college van dekanen

te verdedigen op

donderdag 10 september 1981

te 16.00 uur door

J A C O B U S A R I E V A N AKEN

natuurkundig ingenieur

geboren te Kloetinge

Dit proefschrift is goedgekeurd door de promotor

PROF. DR. H.R.K.N. JANESCHITZ-KRIEGL

H i e r b i j w i l i k g r a a g a l l e n d i e hebben b i j g e d r a g e n aan h e t t o t s t a n d komen v a n d i t p r o e f s c h r i f t h a r t e l i j k bedanken.

Met name w i l i k bedanken

- De medewerkers v a n de C o n s t r u c t i e Tekenkamer en de I n s t r u m e n t m a k e r i j v a n de a f d e l i n g S c h e i k u n d e v o o r hun d e s k u n d i g e s t e u n .

- De h e r e n W.J. de Haas, H.C. N i e u w p o o r t , J . V e r b e e k en G. de Vos v o o r h e t ontwerp en de bouw van h e t r e k s t r o m i n g s a p p a r a a t .

- D r . i r . F.H. Gortemaker en Dr. H.M. L a u n v o o r hun b i j d r a g e aan h o o f d s t u k 2 en h e t b e s c h i k b a a r s t e l l e n van gegevens en g r a f i e k e n .

- De medewerkers v a n de R e p r o g r a f i s c h e D i e n s t v a n de a f d e l i n g S c h e i k u n d e v o o r h e t v e r z o r g e n v a n h e t f o t o w e r k v o o r de p u b l i c a t i e s en v o o r h e t p r o e f s c h r i f t .

Mevr. E. W a r f f e m i u s w i l i k b i j z o n d e r h a r t e l i j k danken v o o r h e t z o r g v u l d i g t y p e n van h e t p r o e f s c h r i f t .

B i j z o n d e r v e e l dank ben i k v e r s c h u l d i g d aan i n g , R. v a n D o n s e l a a r v o o r z i j n h u l p b i j h e t r e a l i s e r e n v a n de v e r s c h i l l e n d e meetsystemen en v o o r de w i j z e waarop h i j m i j b i j h e t o n d e r z o e k met r a a d en daad t e r z i j d e h e e f t g e s t a a n .

De i n d i t p r o e f s c h r i f t b e s c h r e v e n o n d e r z o e k e n z i j n u i t g e v o e r d met s t e u n v a n de N e d e r l a n d s e O r g a n i s a t i e v o o r Z u i v e r W e t e n s c h a p p e l i j k Onderzoek (Z.W.O.).

CONTENTS page 1 GENERAL OUTLINE 7 1.1 I n t r o d u c t i o n 7 1.2 T e n s i l e S t r a i n 9 1.3 S i m p l e Shear S t r a i n 9 1.4 The S t r e s s T e n s o r 10 1.5 R h e o l o g i c a l C o n s t i t u t i v e E q u a t i o n s 12 1.6 The F l o w B i r e f r i n g e n c e T e c h n i q u e 18 1.6.1 E x t i n c t i o n A n g l e and B i r e f r i n g e n c e 18 1.6.2 The E h r i n g h a u s Compensator 19 R e f e r e n c e s 20

2 QUASI-LINEAR RHEOLOGICAL BEHAVIOUR OF POLYMER MELTS. COMPARISON

BETWEEN MECHANICAL AND IMPROVED FLOW BIREFRINGENCE MEASUREMENTS 21

2.1 I n t r o d u c t i o n 21 2.2 D e s c r i p t i o n o f t h e R e v i s e d R o t o r U n i t 22 2.3 New Measurements 24 2.4 D i s c u s s i o n 29 2.5 A p p e n d i x 34 R e f e r e n c e s 36

3 NEW APPARATUS FOR THE SIMULTANEOUS MEASUREMENT OF STRESSES AND

FLOW BIREFRINGENCE IN BIAXIAL EXTENSION OF POLYMER MELTS 37

3.1 I n t r o d u c t i o n 37 3.2 K i n e m a t i c s 38 3.3 The F l o w B i r e f r i n g e n c e T e c h n i q u e 39 3.4 The A p p a r a t u s 40 3.4.1 The C y l i n d e r 40 3.4.2 The D r i v e System 41 3.4.3 The F o r c e Measurements 42 3.4.4 The O p t i c a l Measurements 42 3.4.5 Temperature Contrôle 43 3.5 L u b r i c a t i o n 44 3.6 P r e p a r a t i o n o f t h e Sample 44 3.7 The F o r c e Measurement 45 3.8 R e s u l t s and D i s c u s s i o n 47 3.9 C o n c l u s i o n 52 52 R e f e r e n c e s

4 SIMULTANEOUS MEASUREMENT OF TRANSIENT STRESS AND FLOW BIREFRINGENCE

IN ONE-SIDED COMPRESSION (BIAXIAL EXTENSION) OF A POLYMER MELT 53

4.1 I n t r o d u c t i o n 53 4.2 T h e o r y 54

4.2.1 The R u b b e r l i k e L i q u i d Model 54 4.2.2 Non L i n e a r V i s c o e l a s t i c Model 55 4.3 Some Comments C o n c e r n i n g F u r t h e r Improvement

o f t h e E x p e r i m e n t a l T e c h n i q u e 57 4.3.1 L u b r i c a t i o n 57 4.3.2 C a l c u l a t i o n o f t h e Normal S t r e s s f r o m the F o r c e Measurements 61 4.3.3 The T e m p e r a t u r e Measurement 62 4.4 C h a r a c t e r i z a t i o n o f t h e Sample 62 4.5 P e r f o r m a n c e o f t h e Measurements and R e s u l t s 64 4.6 C o n c l u s i o n s 77 R e f e r e n c e s 78 SUMMARY 79 SAMENVATTING 81 L I S T OF SYMBOLS S3

CHAPTER 1 GENERAL OUTLINE I n t r o d u c t i o n I n o r d e r t o u n d e r s t a n d t h e f l o w b e h a v i o u r o f m a t e r i a l s p o s s e s s i n g v i s c o u s and e l a s t i c p r o p e r t i e s w h i c h c a n n o t be d e s c r i b e d by t h e c l a s s i c a l t h e o r i e s o f h y d r o d y n a m i c s ( f o r Newtonian f l u i d s ) o r o f e l a s t i c i t y ( f o r p u r e l y e l a s t i c s o l i d s ) , a number o f r h e o l o g i c a l c o n s t i t u t i v e e q u a t i o n s have b e e n d e v e l o p e d . A good s t a r t i n g p o i n t f o r the s t u d y o f the u s e f u l n e s s o f c o n s t i t u t i v e e q u a t i o n s f o r p o l y m e r m e l t s i s the r u b b e r l i k e l i q u i d t h e o r y o f Lodge ( 1 ) . T h i s i n t e g r a l model has been m o d i f i e d by d i f f e r e n t a u t h o r s . These m o d i f i c a t i o n s a r e p a r t l y b a s e d on m o l e c u l a r c o n s i d e r a t i o n s , p a r t l y on e x p e r i m e n t a l o b s e r v a t i o n s ( s e e

s e c t i o n 1.5). Anyhow, t h e o r i g i n a l model o f Lodge c a n s t i l l be v e r y u s e f u l f o r the d e s c r i p t i o n o f e x p e r i m e n t a l r e s u l t s , i f t h e t o t a l s t r a i n a p p l i e d on t h e m a t e r i a l i s l e s s t h e n u n i t y . The p r e d i c t i o n s o f t h e r h e o l o g i c a l e q u a t i o n o f s t a t e s h o u l d be compared w i t h the r e s u l t s o f e x p e r i m e n t s f o r a l l t y p e s o f d e f o r -m a t i o n s . I f one exa-mines t h e r e s p o n s e o f poly-mer -m e l t s on d i f f e r e n t t y p e s o f d e f o r m a t i o n s e p a r a t e l y , one o b t a i n e s r e s u l t s a l s o h e l p f u l f o r t h e u n d e r s t a n d i n g o f what happens d u r i n g i n d u s t r i a l p r o c e s s i n g . F o r t h e s e e x a m i n a t i o n s d i f f e r e n t t y p e s o f r o t a t i o n a l , tube and e x t e n s i o n rheometers have been d e s i g n e d . In a d d i -t i o n , -the f l o w b i r e f r i n g e n c e -t e c h n i q u e has p r o v e n -t o be a h e l p f u l me-thod i n -the s e a r c h f o r a s u i t a b l e c o n s t i t u t i v e e q u a t i o n . The a d v a n t a g e s o f t h i s i n d i r e c t m e a s u r i n g method a r e :

a) t h e a p p a r a t u s c a n be b u i l d as s t i f f as n e c e s s a r y , w h i c h means t h a t t h e f l o w w i l l n o t be d i s t u r b e d n o t i c e a b l y by the f l e x i b i l i t y o f the a p p a r a t u s needed f o r the m e c h a n i c a l f o r c e measurement.

b) f o r the d e t e c t i o n o f s m a l l s t r e s s e s t h e o p t i c a l method i s h i g h l y a c c u r a t e . c) an i n a d m i s s i b l e t e m p e r a t u r e r i s e i n t h e m e l t , t h e e f f e c t o f d e g r a d a t i o n o f t h e polymer d u r i n g the d e f o r m a t i o n o r a d i s t u r b a n c e o f the f l o w f i e l d a r e i m m e d i a t e l y d e t e c t e d .

The p r e s e n t i n v e s t i g a t i o n s o f the f l o w b e h a v i o u r o f polymer m e l t , as c a r -r i e d o u t w i t h the a i d o f t h e f l o w b i -r e f -r i n g e n c e t e c h n i q u e , f o -r m a c o n t i n u a t i o n o f the e x p e r i m e n t s c a r r i e d out by Wales (2) and Gortemaker ( 3 ) . Wales showed i n h i s e x p e r i m e n t s w i t h a c l o s e d c o n e - a n d - p l a t e a p p a r a t u s t h a t i n t h i s f l o w

t r y t h e c h e c k o f t h e v a l i d i t y o f t h e l i n e a r s t r e s s o p t i c a l r e l a t i o n was r e s t r i c -ted t o modest v a l u e s o f t h e s h e a r r a t e . However, h i s o p t i c a l measurements i n the 1,3-plane, as c a r r i e d o u t i n a s l i t geometry, p r o v e d t h e v a l i d i t y o f t h e l i n e a r s t r e s s o p t i c a l r e l a t i o n f o r a wide range o f s h e a r r a t e s up t o t h e r e g i o n of m e l t f r a c t u r e . The d i s c r e p a n c y between t h e s e r e s u l t s was c a u s e d by t h e p a r a -s i t i c b i r e f r i n g e n c e w h i c h o c c u r -s a t t h e r i m o f t h e c o n e - a n d - p l a t e a p p a r a t u -s b e i n g c l o s e d f o r t h e o p t i c a l measurements. L a t e r Govtemaker s u c c e e d e d i n r e d u -c i n g t h e p a r a s i t i -c b i r e f r i n g e n -c e by i n t e r -c h a n g i n g t h e -cone and t h e p l a t e . T h i s m o d i f i c a t i o n c a u s e d an i n c r e a s e o f t h e r a n g e o f s h e a r r a t e s o v e r w h i c h t h e l i n e a r s t r e s s o p t i c a l r e l a t i o n c o u l d be p r o v e n , b u t t h e r e g i o n o f t h e m e l t f r a c t u r e was n o t r e a c h e d y e t . From h i s t i m e dependent e x p e r i m e n t s Govtemaker has b e e n a b l e t o draw t h e c o n c l u s i o n t h a t t h e r u b b e r l i k e l i q u i d model o f Lodge

i s v a l i d f o r polymer m e l t s a t e x t r e m e l y low r a t e s o f s h e a r , a t any r a t e o f s h e a r , i f the t o t a l s h e a r does n o t exceed u n i t y . As a c o n t i n u a t i o n o f t h e men-t i o n e d measuremenmen-ts men-t h e p r e s e n men-t i n v e s men-t i g a men-t i o n e x i s men-t s o f men-the f o l l o w i n g i men-t e m s : a) A new m o d i f i c a t i o n o f t h e c o n e - a n d - p l a t e a p p a r a t u s f o r f l o w b i r e f r i n g e n c e measurements. T h i s m o d i f i c a t i o n e n a b l e s us t o e x t e n d t h e r e g i o n o f s h e a r r a t e s , i n w h i c h t h e s t r e s s o p t i c a l c o e f f i c i e n t c a n be p r o v e n , up t o and i n -c l u d i n g t h e r e g i o n o f m e l t f r a -c t u r e . T h i s m o d i f i -c a t i o n and t h e p e r t i n e n t e x p e r i m e n t s a r e d e s c r i b e d i n C h a p t e r 2. b) An a t t e m p t t o use t h e same k i n d o f m o d i f i c a t i o n i n a c l o s e d v e r s i o n o f an a p p a r a t u s c o n s i s t i n g o f two c o n c e n t r i c c y l i n d e r s , i n o r d e r t o e x t e n d t h e range o f s h e a r r a t e s i n t h i s t y p e o f a p p a r a t u s f o r polymer m e l t s . F o r t h e low s h e a r r a t e s t h e measurements w i t h t h i s a p p a r a t u s were v e r y a c c u r a t e . A t h i g h e r s h e a r r a t e s , however, t h e measurements f a i l e d b e c a u s e o f t h e f a c t t h a t i n t h e a r e a s where t h e end s u r f a c e s o f t h e r o t a t i n g c y l i n d e r and t h e f i x e d p l a t e s f o r m i n g b o t t o m and c o v e r meet, v e r y h i g h l o c a l s h e a r r a t e s seemed i n a v o i d a b l e , r e s u l t i n g i n the d e g r a d a t i o n o f t h e m e l t . A l t h o u g h t h e l i g h t beam d i d n o t p a s s t h r o u g h t h i s r e g i o n where t h e d e g r a d a t i o n o c c u r r e d no measurement was p o s s i b l e b e c a u s e o f t h e d e g r a d e d m e l t b e i n g s p r e a d t h r o u g h t h e whole a p p a r a t u s .

c ) A new a p p a r a t u s was c o n s t r u c t e d f o r t h e measurement o f t h e time dependent b e h a v i o u r o f t h e s e c o n d n o r m a l s t r e s s d i f f e r e n c e . I n a c l o s e d s l i t a s h e a r f l o w was c r e a t e d by moving one o f t h e w a l l s p a r a l l e l t o t h e e x t e n s i o n o f t h e s l i t . The m e n t i o n e d s t r e s s d i f f e r e n c e c a n be d e t e c t e d by m e a s u r i n g t h e b i r e -f r i n g e n c e e -f -f e c t o b s e r v a b l e w i t h a l i g h t beam p a s s i n g t h r o u g h t h e m e l t p a r a l l e l t o t h e s t r e a m l i n e s . To e l i m i n a t e t h e e n d - e f f e c t s , u s e must be made of s l i t s o f d i f f e r e n t l e n g t h s . I t i s hoped t h a t t h e measurements o f t h e t i m e dependent second normal s t r e s s d i f f e r e n c e w i l l be c a r r i e d o u t a t L i n z U n i v e r -s i t y .

d) Not o n l y a p p a r a t u s s e s f o r t h e measurement o f polymer m e l t s i n s h e a r f l o w were i n v e s t i g a t e d . A l s o an a p p a r a t u s f o r t h e measurement o f t h e e x t e n s i o n a l v i s c o s i t y was d e v e l o p e d . W i t h t h i s a p p a r a t u s i t i s p o s s i b l e t o c a r r y o u t m e c h a n i c a l and o p t i c a l measurements s i m u l t a n e o u s l y . A d e s c r i p t i o n o f t h i s a p p a r a t u s and t h e r e s u l t s o f s t e a d y s t a t e and t r a n s i e n t measurements a r e g i v e n i n C h a p t e r 3 and 4.

I n t h e f o l l o w i n g s e c t i o n s o f t h i s f i r s t c h a p t e r a s h o r t d e s c r i p t i o n o f t h e used T h e o l o g i c a l e q u a t i o n s i s g i v e n . C h a p t e r s 2, 3 and 4 o f t h i s work have a l -r e a d y been p u b l i s h e d as s e p a -r a t e p a p e -r s i n R h e o l o g i c a A c t a . 1.2 T e n s i l e S t r a i n I f a r e c t a n g u l a r element o f m a t e r i a l o f a l e n g t h 1 i s e x t e n d e d t o a l e n g t h 1 + 61, t h e Cauchy s t r a i n i s d e f i n e d as The r a t e o f s t r a i n i s g i v e n by O n l y i f a m a t e r i a l r e t u r n s t o i t s o r i g i n a l l e n g t h 1 when t h e s t r e s s a c t i n g o n i t i s removed, t h e Cauchy s t r a i n and r a t e o f s t r a i n has a s i g n i f i c a n t meaning. F o r m a t e r i a l s l i k e polymer m e l t s and s o l u t i o n s , however, t h e o r i g i n a l l e n g t h 1 has no meaning from t h e moment the m a t e r i a l s t a r t s t o f l o w . I n such c a s e s t h e Hencky s t r a i n i s t o be p r e f e r r e d . T h i s Hencky s t r a i n i s t h e summation o f s u c -c e s s i v e Cau-chy s t r a i n s : o

SI

2- d i r e c t i o n o f t h e v e l o c i t y g r a d i e n t 3- n e u t r a l d i r e c t i o n . S i m p l e s h e a r i s r e p r e s e n t e d b y : 1 x] + Y x2 [1.3.1] x3 x3 i n w h i c h ( x j , x^, x^) a r e t h e c o o r d i n a t e s o f a " p a r t i c l e " a t p r e v i o u s time t ' and ( X j , X £ , x^) a r e t h e c o o r d i n a t e s o f t h e same p a r t i c l e a t p r e s e n t time t . I n the p e r t i n e n t time i n t e r v a l s h e a r y i s a p p l i e d i n some way. The s h e a r r a t e y i s

d j d t Y = I f t h e s h e a r r a t e i s time i n d e p e n d e n t t h e s h e a r c a n be w r i t t e n a s : Y( t , t ' ) = ( t - t ' ) Y [1-3.3] 1.4 The S t r e s s T e n s o r I n a r i g h t handed C a r t e s i a n c o o r d i n a t e s y s t e m t h e s t r e s s t e n s o r i s g i v e n by: CT11 0) 2 al 3 g = °21 a2 2 °23 [1.4.1] 0-3, a3 2 CT33

The d i a g o n a l e l e m e n t s 0 j j , a n <* ° 3 3a re t'l e n o r m a^ - s t r e s s components and t h e

o f f - d i a g o n a l e l e m e n t s a r e t h e s h e a r s t r e s s components. I n g e n e r a l one c a n choose a c o o r d i n a t e s y s t e m by w h i c h a l l t h e s h e a r components v a n i s h . The axes o f t h i s p a r t i c u l a r c o o r d i n a t e s y s t e m a r e c a l l e d t h e p r i n c i p a l a x e s . When an i n c o m p r e s -s i b l e f l u i d i -s i n a q u i e -s c e n t -s t a t e t h e r e i -s o n l y an i -s o t r o p i c h y d r o -s t a t i c p r e s s u r e whereas t h e s h e a r s t r e s s components a r e z e r o . I n a f l o w i n g s y s t e m some o f t h e s h e a r s t r e s s e s a r e f i n i t e , whereas t h e normal s t r e s s components may d i f f e r f r o m each o t h e r . The normal s t r e s s components 0j j » 022 a n c* °33a re c^an_

ged by e q u a l amounts, i f t h e h y d r o s t a t i c p r e s s u r e i s changed b e c a u s e t h i s p r e s -s u r e doe-s n o t i n f l u e n c e t h e -shape o f t h e m a t e r i a l . F o r t h i -s r e a -s o n -s e p a r a t e normal s t r e s s e s have no r h e o l o g i c a l meaning. The d i f f e r e n c e s o f t h e normal s t r e s s e s a . j - <^22' a2 2 ~ a3 3 a n t^a l 1 _ °33a r e n ot i -nf luence<^ byt ne h y d r o s t a

t i c p r e s s u r e . They g i v e , t o g e t h e r w i t h the s h e a r s t r e s s e s , t h e r h e o l o g i c a l r e s -ponse t o a change o f the shape o f t h e m a t e r i a l . F o r the m a t e r i a l s c o n s i d e r e d i n t h i s t h e s i s one may assume t h a t the s t r e s s t e n s o r i s symmetric, a t l e a s t f o r n o t too h i g h r a t e s o f d e f o r m a t i o n , v i z . :

a = a

As a c o n s e q u e n c e t h e number o f i n d e p e n d e n t components o f the s t r e s s t e n s o r i s r e d u c e d f r o m n i n e t o s i x . I f a s i m p l e s h e a r d e f o r m a t i o n i s c o n s i d e r e d , one can c o n c l u d e f r o m the symmetry o f the f l o w f i e l d t h a t a^ = = 0 and = aj j=0 .

The g e n e r a l s t a t e o f s t r e s s f o r an o r i g i n a l l y i s o t r o p i c m a t e r i a l i s p r e s e n -ted d u r i n g a s h e a r d e f o r m a t i o n by t h e f o l l o w i n g s t r e s s t e n s o r : ai l CT12 a2 1 °22 0 0 J33 [1.4.3]

The t h r e e i n d e p e n d e n t " d e v i a t o r i c " s t r e s s components a r e the s h e a r s t r e s s 0j2 = ° 2 1 ' t'l e ^•"•r s t n or ma l s t r e s s d i f f e r e n c e 0 - a^^ an<l tne second n o r m a l

s t r e s s d i f f e r e n c e ~ I f the s t r e s s t e n s o r has o n l y d i a g o n a l e l e m e n t s (and a l l o f f d i a g o n a l components a r e z e r o ) , the t h r e e p r i n c i p a l a x e s o f the s t r e s s e l l i p s o i d c o i n c i d e w i t h the axes o f the C a r t e s i a n c o o r d i n a t e s y s t e m . F o r a s i m p l e s h e a r d e f o r m a t i o n , however, o n l y the 3 - a x i s o r n e u t r a l d i r e c t i o n o f the l a b o r a t o r y c o o r d i n a t e system i s a p r i n c i p a l a x i s Cthe I l l a x i s ) . The two r e m a i -ming p r i n c i p a l axes must l i e i n the 1,2 p l a n e , the " p l a n e o f f l o w " . By d e f i n i t i o n ,

the f i r s t p r i n c i p a l a x i s i s the a x i s making an a n g l e s m a l l e r t h a n f o r t y f i v e d e g r e e s w i t h the d i r e c t i o n o f the s t r e a m l i n e s ( s e e f i g u r e 1.1). I f one t r a n s -forms t h e p r i n c i p a l s t r e s s e s 0 , a and 0J-J-J i n t o the components o f the s t r e s s t e n s o r i n t h e l a b o r a t o r y s y s t e m one o b t a i n e s w i t h 0^ — o" = A 0. A

a

0

1

Fig. 1.1. Laboratory coordinate system:

x: direction of flow (1-direction); y : direction of velocity gradient (2-direc-tion); J and II : directions of stress; xm •' orientation angle of stress e l l i p

-soid; : velocity; q : shear rate.

1.5 R h e o l o g i c a l C o n s t i t u t i v e E q u a t i o n s I n t h i s s e c t i o n o n l y a few o f t h e r h e o l o g i c a l c o n s t i t u t i v e e q u a t i o n s and m o d i f i c a t i o n s o f t h e s e e q u a t i o n s a r e d i s c u s s e d b r i e f l y . An e x t e n s i v e d i s c u s s i o n o f t h e c o n s t i t u t i v e e q u a t i o n s o f t h e i n t e g r a l and o f t h e d i f f e r e n t i a l t y p e f o r l i n e a r and n o n - l i n e a r b e h a v i o u r i s i n p r e p a r a t i o n ( J a n e s c h i t z - K r i e g l ( 4 ) ) . As a s t a r t i n g p o i n t f o r t h e d i s c u s s i o n o f c o n s t i t u t i v e e q u a t i o n s w h i c h p r o p e r l y d e s c r i b e t h e b e h a v i o u r o f a polymer m e l t , an e x a m i n a t i o n i s g i v e n o f the equa-t i o n d e r i v e d f o r r u b b e r , b e c a u s e equa-t h e b e h a v i o u r o f a polymer m e l equa-t has much i n common w i t h t h a t o f r u b b e r . As w e l l known, an i d e a l r u b b e r shows "neo-Hookean" b e h a v i o u r i n s h e a r . I t i s p o s s i b l e t o d e r i v e a c o n s t i t u t i v e e q u a t i o n f o r such systems from t h e s t a t i s t i c a l t h e o r y . I f the d e f o r m a t i o n o c c u r s a t c o n s t a n t volume and w i t h no change o f i n t e r n a l e n e r g y , one h a s :

O + p I = G B [1.5.1]

where 0 i s the s t r e s s t e n s o r , p the h y d r o s t a t i c p r e s s u r e , G the shear modulus and B t h e F i n g e r s t r a i n t e n s o r . The components o f t h e F i n g e r s t r a i n t e n s o r B a r e d e f i n e d i n C a r t e s i a n c o o r d i n a t e s a s :

3 5x. óx. r l 1

[1.5.2]

where a r e t h e components o f the p o s i t i o n v e c t o r x' o f a p a r t i c l e b e f o r e t h e d e f o r m a t i o n and x t h e components o f t h e c o r r e s p o n d i n g v e c t o r x a f t e r t h e d e f o r -m a t i o n . The Cauchy d e f o r -m a t i o n t e n s o r C i s o f t e n used i n r h e o l o g i c a l e q u a t i o n s . I t i s t h e i n v e r s e o f t h e F i n g e r s t r a i n t e n s o r :

C = -1 [1.5.3]

The components o f t h e Cauchy d e f o r m a t i o n t e n s o r a r e thus g i v e n i n C a r t e s i a n c o o r d i n a t e s by: 3

I

[1.5.9] °>2 = G Y a , , + p - 6(1 + Y2) a3 3 + P = G By a p p r o p r i a t e s u b t r a t i o n one g e t s r i d o f t h e u n d e t e r m i n e d h y d r o s t a t i c p r e s s u r e P = o-j, - a2 2 = G Y2 [1.5.10] a2 2 - a3 3 = 0 [1.5.11] One n o t i c e s t h a t t h e c o n s t i t u t i v e e q u a t i o n f o r an i d e a l r u b b e r [1.5.1] r e s u l t s i n an u n r e s t r i c t e d l y l i n e a r r e l a t i o n between the s h e a r y and the s h e a r s t r e s s

UP to h i g h v a l u e s o f s h e a r (neo—Hookean b e h a v i o u r ) , a f i r s t normal s t r e s s

d i f f e r e n c e w h i c h i s p r o p o r t i o n a l t o t h e s q u a r e o f t h e s h e a r Y>and a second

normal s t r e s s d i f f e r e n c e o f t h e v a l u e z e r o .

Polymer m e l t s and s o l u t i o n s a r e t h o u g h t t o c o n s i s t o f temporary n e t w o r k s . I n s u c h a network e n t a n g l e m e n t s a r e supposed t o be d i s r u p t e d and r e f o r m e d c o n -t i n u o u s l y under -t h e i n f l u e n c e o f -t h e -t h e r m a l m o -t i o n . To d e s c r i b e -t h i s sys-tem

Lodge (1) p r o p o s e d h i s r u b b e r l i k e l i q u i d m o d e l . I n t h i s t h e o r y each component

o f t h e s t r e s s t e n s o r o b s e r v e d a t p r e s e n t time t i s r e l a t e d t o t h e c o r r e s p o n -d i n g component o f the F i n g e r s t r a i n t e n s o r a t p r e v i o u s time t ' by t h e same

o memory f u n c t i o n y ( t - t ' ) . C o n s e q u e n t l y , Lodge's c o n s t i t u t i v e e q u a t i o n f o r r u b b e r l i k e l i q u i d s i s t (g + P l )t = p ( t - t ' ) B ( t , t ' ) d t ' [1.5.12] t ' = 00

The memory, as e x p r e s s e d by the f u n c t i o n p ( t - t ' ) f a d e s f o r an i n c r e a s i n g t i m e i n t e r v a l ( t - t ' ) . T h i s memory f u n c t i o n i s r e l a t e d t o t h e r e l a x a t i o n s h e a r

o

modulus G ( t - t ' ) , as known from t h e l i n e a r t h e o r y o f v i s c o e l a s t i c i t y , by

g (t -1- ) = " s ^ ; , 0 t i . 5 . 1 3 ]

The time dependent b e h a v i o u r o f r u b b e r l i k e l i q u i d s c a n be c a l c u l a t e d from equat i o n [1.5.12] and [ 1 . 5 . 1 3 ] , i f equat h e r e l a x a equat i o n s h e a r modulus G ( equat equat ' ) i s w r i equat -t e n as a s e r i e s o f e x p o n e n -t i a l s . - - ( t - t » ) / t . G ( t - t ' ) = I g e [1.5.14] i = l 1 A f t e r s u b s t i t u t i o n i n e q u a t i o n [1.5.13] one o b t a i n e s f o r t h e memory f u n c t i o n : 14

» g - ( t - t ' ) / T U ( t - f ) = I ^ e [1.5.15] i = l i I n s h e a r e x p e r i m e n t s one o b t a i n e s f o r t h e s t r e s s i n g e x p e r i m e n t , w h i c h i s t h e b u i l d - u p o f t h e s h e a r s t r e s s and t h e f i r s t normal s t r e s s d i f f e r e n c e a t c o n s t a n t r a t e o f s h e a r Y : a2 ]( t ) = Y I g£ T . ( l - e x) [1.5.16] i = l - a22)t = 2 Y2 I fe T . a - a Ti) [1.5.17] i = l A s i m i l a r c a l c u l a t i o n c a n be made f o r e x t e n t i o n a l f l o w . T h i s c a l c u l a t i o n i s g i v e n i n C h a p t e r 4 where t h e time dependent measurements, w h i c h have been made w i t h t h e new a p p a r a t u s f o r t h e e x t e n s i o n a l f l o w s , a r e d i s c u s s e d .

F o r r a t h e r s m a l l t o t a l d e f o r m a t i o n s t h e r e s u l t s o f s h e a r and e x t e n s i o n a l e x p e r i m e n t s a r e i n agreement w i t h t h e r u b b e r l i k e l i q u i d m o d e l . On t h e o t h e r hand, f o r t h e s t e a d y s t a t e s i t u a t i o n o f f l o w t h e r e s u l t s o f Lodge's model do n o t always a g r e e w i t h t h e e x p e r i m e n t a l f a c t s . The s t e a d y s t a t e s h e a r v i s c o s i t y , d e f i n e d a s : ns = o2]/y [1.5.18] and t h e f i r s t normal s t r e s s d i f f e r e n c e c o e f f i c i e n t °11 " ° 2 2 8 „ - . 2 [1.5.19] Y

a r e s h e a r r a t e dependent. The second normal s t r e s s d i f f e r e n c e i s n o t z e r o . I n f a c t , e x p e r i m e n t s show f i n i t e second normal s t r e s s d i f f e r e n c e s w h i c h have always a s i g n r e v e r s e t o t h a t o f t h e f i r s t normal s t r e s s d i f f e r e n c e and a r e o f t h e o r d e r o f one t e n t h o f t h e l a t t e r q u a n t i t y . T h i s p r o b l e m c a n f o r m a l l y be s o l v e d by t h e r e p l a c e m e n t o f t h e F i n g e r t e n s o r B by a l i n e a r c o m b i n a t i o n o f t h e F i n g e r t e n s o r and t h e Cauchy t e n s o r (1 - a) B - a C [1.5.20] where a i s a p o s i t i v e c o n s t a n t o f t h e o r d e r o f one t e n t h . T h i s m o d i f i c a t i o n goes b a c k t o Mooney and Rivlin ( 5 ) , who a d a p t e d i n t h i s way e q u a t i o n [1.5.1] f o r n o n - i d e a l r u b b e r s . U n f o r t u n a t e l y t h e r e i s no m o l e c u l a r i n t e r p r e t a t i o n f o r

t h i s m o d i f i c a t i o n u n t i l now. I n o r d e r t o d e s c r i b e t h e dependency o f t h e v i s c o -15

s i t y on t h e r a t e o f s h e a r , s e v e r a l m o d i f i c a t i o n s o f t h e memory f u n c t i o n were p r o p o s e d . The a s s u m p t i o n i s made t h a t t h e memory f u n c t i o n i s n o t o n l y a f u n c t i o n o f t h e e l a p s e d t i m e ( t - t ' ) b u t a l s o o f t h e f i r s t and second i n v a r i a n t s o f t h e F i n g e r s t r a i n t e n s o r B ( t h e t h i r d i n v a r i a n t i s 1^ = D e t ( B ^ jc) = 1 b e c a u s e o f t h e c o n s t a n t volume a s s u m p t i o n ) . One h a s : u ( t - t ' ; I , , I2) [1.5.21] I n t h e most g e n e r a l f o r m u l a t i o n t h e memory f u n c t i o n s h o u l d be a f u n c t i o n o f t h e r a t e o f s t r a i n t e n s o r as w e l l . I n o r d e r t o o b t a i n u s e f u l r e s u l t s t h i s model c a n be s i m p l i f i e d by a f a c t o r i s a t i o n o f t h e memory f u n c t i o n , as g i v e n by eq. [ 1 . 5 . 2 1 ] , v i z . : u ( t - f ; I]; I2) = y ( t - t ' ) h ( I , , I2) [1.5.22] o , where U ( t - t ' ) i s t h e memory f u n c t i o n o f t h e l i n e a r v i s c o e l a s t i c t h e o r y and

h ( I j , I2) i s a f u n c t i o n o f t h e s t r a i n , c a l l e d t h e "damping f u n c t i o n " . T h i s

damping f u n c t i o n i s e x t e n s i v e l y e v a l u a t e d by Wagner ( 6 , 7) and Laun ( 8 , 9 ) . I n t h e I n d e p e n d e n t A l i g n m e n t model o f Doi and Edwards (1013) t h i s damping f u n c -t i o n i s i n c o r p o r a -t e d i n -t h e -t e n s o r Q w h i c h r e p l a c e s -t h e F i n g e r -t e n s o r B i n

Lodge's e q u a t i o n [1.5.12]

t (a + p p c =

t1 =—CO

I n t h i s l a t t e r model t h e memory f u n c t i o n i s t h e same as i n e q u a t i o n [ 1 . 5 . 1 3 ] : i t i s o n l y dependent on t h e e l a p s e d t i m e ( t - t ' ) . Doi and Edwards have worked out t h e i r t h e o r y f o r s h e a r f l o w , s t r e t c h i n g f l o w (A > 1) and c o m p r e s s i o n f l o w

o

(A < 1 ) . I f one wants t o use t h e memory f u n c t i o n p ( t - t ' ) as o b t a i n e d f r o m l i n e a r v i s c o e l a s t i c p r o p e r t i e s , t h e t e n s o r Q i n t h e o r i g i n a l p a p e r s must be m u l t i p l i e d by a f a c t o r 5 as e x p l a i n e d b y Janesehitz-Kriegl ( 4 ) .

U n t i l h e r e o n l y c o n s t i t u t i v e e q u a t i o n s o f t h e i n t e g r a l t y p e were p r e s e n t e d . The f o l l o w i n g n o n - l i n e a r c o n s t i t u t i v e e q u a t i o n i s o f t h e d i f f e r e n t i a l t y p e . T h i s e q u a t i o n was p r o p o s e d by Acierno, Marruoai, La Montia, Rizzo and Titomanlio i n a s e r i e s o f p a p e r s ( 1 4 1 6 ) . They f o r m u l a t e d t h e i r t h e o r y i n terms o f an a p p r o -x i m a t e l i n e s p e c t r u m . The c o n t r i b u t i o n s o f t h e s e p a r a t e r e l a -x a t i o n p r o c e s s e s a r e s i m p l y summed up. The p e r t i n e n t e q u a t i o n s a r e :

a = I 0 [1.5.24]

V(t - t ' ) Q d t [1.5.23]

I i s 1 CT + T i ï t (G 7 2I> - 2 T i 2 [1-5.25] 1 G. = G . x. ; T. = T . x^,'4 [1.5.26] l o i l l o i 2 L J dx 1 , E _ = _ (, _ x. ) _ a x . - / - 11.5.27] i i i The d i f f e r e n t i a l e q u a t i o n [1.5.25] c o n t a i n s a c o n t r a v a r i a n t t i m e d e v i a t i v e and i s s i m i l a r t o t h e w e l l - k n o w n Maxwell e q u a t i o n . G ^ i s t h e r e l a x a t i o n s t r e n g t h , T ^ t h e r e l a x a t i o n time o f the i - t h r e l a x a t i o n p r o c e s s a c c o r d i n g t o t h e l i n e a r v i s c o e l a s t i c b e h a v i o u r o f t h e m a t e r i a l , t h e p a r a m e t e r x. w i t h 1 > x. > 0 i s l — l — an i n t e r n a l s t r u c t u r a l p a r a m e t e r . P a r a m e t e r a i s a d j u s t a b l e , p r e f e r a b l y t o t h e non-Newtonian s t e a d y s h e a r v i s c o s i t y , b u t i s e s s e n t i a l l y t h e same f o r a l l t y p e s o f p o l y m e r s . The symbol D s t a n d s f o r t h e r a t e o f s t r a i n t e n s o r D = | ( V v + V vt) . [1.5.28]

and i s t h e c o n t r i b u t i o n o f the i - t h r e l a x a t i o n mechanism t o t h e e l a s t i c e n e r g y , v i z . : Ei = I t r g... [1.5.29] As c a n be seen f r o m e q s . [ 1 . 5 . 2 6 ] , t h e n o n l i n e a r i t y i s i n t r o d u c e d by t h e i n t e r -n a l s t r u c t u r a l p a r a m e t e r x^. I -n eq. [1.5.27] t h e r a t e o f c r e a t i o -n o f -n e t w o r k j u n c t i o n s o f t h e t y p e " i " (due t o t h e B r o w n i a n M o t i o n ) i s g i v e n by (1 - x ^ ) / x ^ , x. E.

whereas the t e r m a — / d e s c r i b e s t h e r a t e o f d e s t r u c t i o n o f t h e network. T. G.

l l

I f the e l a s t i c e n e r g y E ^ i n c r e a s e s the d e s t r u c t i o n o f t h e network i s enhanced. T h i s model h a s s u c c e s s f u l l y been checked by De Cindio (17) w i t h t h e a i d o f s h e a r e x p e r i m e n t s c a r r i e d o u t by Gortemaker ( 3 ) . F o r t h e same polymer t h i s t h e o r y i s e v a l u a t e d i n C h a p t e r 4 f o r s t e a d y s t a t e e x t e n s i o n a l e x p e r i m e n t s .

1.6 The F l o w B i r e f r i n g e n c e T e c h n i q u e

In t h i s s e c t i o n o n l y the e x p r e s s i o n s needed i n the f o l l o w i n g c h a p t e r s a r e d i s c u s s e d . The e x p e r i m e n t a l t e c h n i q u e f o r the d e t e c t i o n o f the components o f the r e f r a c t i v e i n d e x t e n s o r i s b r i e f l y o u t l i n e d . F u r t h e r , the l i n e a r s t r e s s - o p t i c a l r e l a t i o n i s i n t r o d u c e d . I t c l a i m s a s i m p l e p r o p o r t i o n a l i t y between the d e v i a -t o r i c componen-ts o f -the s -t r e s s -t e n s o r and o f -the r e f r a c -t i v e i n d e x -t e n s o r , v i z . :

n = C g, [1.6.1]

where C i s the s t r e s s - o p t i c a l c o e f f i c i e n t . From t h i s r e l a t i o n i t i s a l s o o b v i o u s t h a t t h e r e s p e c t i v e t e n s o r e l l i p s o i d s a r e c o a x i a l . D e v i a t i o n s from t h i s s i m p l e r u l e have o n l y been found a t t e n s i l e s t r e s s e s h i g h e r t h a n 10^ Pa (18) and a t a t i m e - t e m p e r a t u r e - s c a l e c l o s e t o the g l a s s t r a n s i t i o n ( 1 9 ) . Owing t o non—New-t o n i a n v i s c o s i non—New-t y , s p u r non—New-t (20) and m e l non—New-t - f r a c non—New-t i o n (21) s h e a r s non—New-t r e s s e s o f 10 Pa have n e v e r been r e a c h e d . A l s o d e v i a t i o n s from eq. [1.6.1] have n e v e r been f o u n d i n s h e a r e x p e r i m e n t s . I n a d d i t i o n , C has been found to be i n d e p e n d e n t o f tempe-r a t u tempe-r e i n the s p e c i a l c a s e o f p o l y s t y tempe-r e n e ( 2 ) .

1.6.1 E x t i n c t i o n A n g l e and B i r e f r i n g e n c e

I f a l i n e a r l y p o l a r i z e d l i g h t beam p a s s e s t h r o u g h an o p t i c a l l y i s o t r o p i c medium ( l e t us say a polymer m e l t i n i t s s t a t i o n a r y s t a t e ) and the emerging

l i g h t beam i s o b s e r v e d t h r o u g h an a n a l y s e r , w h i c h i s p l a c e d i n a c r o s s p o s i t i o n w i t h r e s p e c t to the p o l a r i z e r , e x t i n c t i o n w i l l be o b s e r v e d . However, i f t h i s polymer m e l t i s f o r c e d to f l o w , i t becomes o p t i c a l l y b i a x i a l . T h i s means t h a t the r e f r a c t i v e i n d e x t e n s o r has got t h r e e d i f f e r e n t p r i n c i p a l a x e s . In the s h e a r f l o w d i s c u s s e d i n C h a p t e r 2 the l i g h t beam p a s s e s t h r o u g h the m e l t i n the 3 - d i r e c t i o n w h i c h i s - by symmetry - one o f the p r i n c i p l e a x e s , say the a x i s I I I . I n o t h e r words, the p l a n e o f o b s e r v a t i o n i s the 1,2 p l a n e . The o t h e r two p r i n c i p a l axes l i e i n the l a t t e r p l a n e . By d e f i n i t i o n , the p r i n c i p a l a x i s I makes an a n g l e o f XQ < 45° w i t h the 1 - a x i s o f the l a b o r a t o r y c o o r d i n a t e system

( f l o w d i r e c t i o n ) . T h i s a n g l e XQ i s c a l l e d the e x t i n c t i o n a n g l e . I f c r o s s e d

p o l a r i z e r and a n a l y s e r a r e t u r n e d s i m u l t a n e o u s l y , u n t i l the p o s i t i o n o f one o f them c o i n c i d e s w i t h the a n g l e XD> no l i g h t w i l l p a s s t h r o u g h the system b e c a u s e

the p l a n e o f v i b r a t i o n o f the l i n e a r l y p o l a r i z e d l i g h t o f w a v e l e n g t h \ makes an a n g l e o f 4 5 ° w i t h one o f the p r i n c i p a l axes i n the p l a n e o f o b s e r v a t i o n , the r e s p e c t i v e d i f f e r e n c e o f the r e f r a c t i v e i n d i c e s c a n be d e t e r m i n e d . T h i s i s a c c o m p l i s h e d by the measurement o f the phase d i f f e r e n c e 6 between the two mu-t u a l l y p e r p e n d i c u l a r l y p o l a r i z e d componenmu-ts o f mu-the l i g h mu-t , w h i c h emerge f r o m mu-the 18

m e l t . The o p t i c a l p a t h d i f f e r e n c e as " r e t a r d a t i o n " T i s t h e n g i v e n by

[ 1 . 6 . 2 ]

and the b i r e f r i n g e n c e An i s

An = ~ [ 1 . 6 . 3 ]

where L i s the o p t i c a l p a t h l e n g t h t h r o u g h the m e l t . F o r the components o f the s t r e s s t e n s o r i n s h e a r f l o w one o b t a i n e s w i t h the a i d o f the s t r e s s o p t i c a l r e l a t i o n :

° 1 2 = Tc Sin 2 Xo t1-6-4]

ai l " ° 2 2 = H I C0S 2 Xo [ 1 . 6 . 5 ]

w i t h An >= ij - n ^ .

I n t h e e x t e n s i o n e x p e r i m e n t ( C h a p t e r s 3 and 4 ) the d i r e c t i o n s of the p r i n -c i p a l axes a r e e q u a l t o t h o s e o f the l a b o r a t o r y system. The e x t i n -c t i o n a n g l e becomes zero o r n i n e t y d e g r e e s and t r i v i a l . The o p t i c a l and m e c h a n i c a l measure-ments a r e s i m p l y r e l a t e d by

An = C a j j [ 1 . 6 . 6 ]

whereaj j (= ° j ) i s the t e n s i l e s t r e s s ( w i t h = ^-J-J = 0 ) * \ The experiments d i s c u s s e d i n t h i s t h e s i s were c a r r i e d o u t on a p o l y s t y r e n e , w h i c h has p r e v i o u s -l y b e e n used f o r a s e r i e s o f r h e o -l o g i c a -l i n v e s t i g a t i o n s ( 3 ) ( 1 7 ) .

2 The Ehringhaus Compensator

The o p t i c a l r e t a r d a t i o n T i s d e t e r m i n e d w i t h the a i d o f a compensator a c c o r d i n g t o E h r i n g h a u s ( 2 2 ) . By t u r n i n g a p l a t e , which c o n s i s t s o f two q u a r t z c r y s t a l s , a r e t a r d a t i o n i s p r o d u c e d , w h i c h s e r v e s f o r c o m p e n s a t i o n o f t h e r e t a r -d a t i o n p r o -d u c e -d by the polymer sample. The a n g l e by w h i c h t h e p l a t e i s r o t a t e -d , i s a measure f o r the b i r e f r i n g e n c e . T h i s compensator i s p l a c e d i n f r o n t o f the a n a l y s e r . W i t h t h e E h r i n g h a u s compensator o n l y r e t a r d a t i o n s c o r r e s p o n d i n g w i t h

*) Sometimes i n t h i s work the symbol 0zz i s used i n s t e a d o f O J J , the z - a x i s

b e i n g the a x i s o f the s t r e t c h .

p h a s e - d i f f e r e n c e s s m a l l e r t h a n 2TT can be d e t e c t e d . I f t h e phase d i f f e r e n c e i s more t h a n 2TT, one has t o c o u n t t h e " f r i n g e s " p a s s i n g by f r o m t h e moment when t h e f l o w i s s t a r t e d u n t i l t h e s t e a d y s t a t e i s r e a c h e d .

REFERENCES

Lodge, A.S., " E l a s t i c L i q u i d s " , Academic P r e s s , New Y o r k (1964) Wales, J.L.S.j "The A p p l i c a t i o n o f Flow B i r e f r i n g e n c e to R h e o l o g i c a l

S t u d i e s o f Polymer M e l t s " D e l f t (1976)

Gortemaker, F.H., "A Flow B i r e f r i n g e n c e Study o f S t r e s s e s i n Sheared

Polymer M e l t s , T h e s i s , D e l f t (1976)

Janesehitz-Rriegl, H., "Flow B i r e f r i n g e n c e i n Polymer M e l t Rheometry"

monograph i n p r e p a r a t i o n f o r S p r i n g e r , H e i d e l b e r g

Mooney, M., J . A p p l . P h y s . , _H> 582 C194Q)

Wagner, M.H., R h e o l . A c t a J_8, 33 (1979)

Wagner, M.H., S.E. Stephenson, J . o f R h e o l o g y 23, 489 (1979) Laun, H.M., R h e o l . A c t a J_7, 1 (1978)

Laun, H.M., M.H. Wagner, H. Janesehitz-Kriegl, R h e o l . A c t a J_8, 615 (1979) Doi, M., S.F. Edwards, J.C.S. F a r a d a y I I 74, 1789 (1978)

Doi, M., S.F. Edwards, J.C.S. F a r a d a y I I 74, 1802 (1978) Doi, M., S.F. Edwards, J.C.S. F a r a d a y I I 74, 1818 (1978) Doi, M., S.F. Edwards, J.C.S. F a r a d a y I I 75, 38 (1979)

Aoierno, D., F.F. La Mantia, G. Marruooi, G. Titomanlio, J . Non-Newtonian

F l u i d Mech. 125 (1976)

Aoierno, D., F.F. La Mantia, G. Marruooi, G. Rizzo, G. Titomanlio, J .

Non-Newtonian F l u i d Mech. i_, 147 (1976)

Aoierno, D., F.F. La Mantia, G. Marruooi, J . Non-Newtonian F l u i d Mech.

2, 271 (1977)

De Cindio, B., D. Aoierno, F.H. Gortemaker, H. Janeschitz-Rriegl, R h e o l .

A c t a _T6, 484 (1977)

Matsvmoto, T., D.C. Bogue, J . Polym. S e i . , Polym. Phys. Ed. 1663 (1977) Read, B.E., Polymer 5, 1 (1964)

Vinogradov, G.V., A. Ya. Malkin, Y.G. Yanouskü, E.K. Borisonkova, B.V. Yarlykov, G.V. Berezknaya, J . Polym. S e i . , A2, j_0, 1061 (1972) Den Otter, J.L., P l a s t i c s and P o l y m e r s 38, 155 (1970)

Ehringhaus, A., Z. K r i s t a l l o g r a p h i e 76, 315 ( 1 9 3 1 ) .

CHAPTER 2

QUASI-LINEAR RHEOLOGICAL BEHAVIOUR OF POLYMER MELTS: COMPARISON BETWEEN MECHANICAL AND IMPROVED FLOW BIREFRINGENCE MEASUREMENTS*^

I n t r o d u c t i o n

When the f i r s t a p p a r a t u s (1) f o r c o n t i n u o u s measurements o f t h e f l o w b i r e -f r i n g e n c e o -f polymer m e l t s was p u b l i s h e d i n 1967, i t was n o t c l e a r , whether t h e a p p l i e d c o n e a n d p l a t e geometry was r e a l l y t h e most p r o m i s i n g one f o r e x p e r i -ments i n s h e a r . Some y e a r s l a t e r , Wales (2) a p p l i e d the s l i t geometry i n o r d e r to measure t h e b i r e f r i n g e n c e i n the 1,3-plane. He o b s e r v e d t h a t , i n c o n t r a s t t o t h e e x p e r i e n c e s g a t h e r e d w i t h the c o n e - a n d - p l a t e a p p a r a t u s , measurements c o u l d be e x t e n d e d i n t o the s o - c a l l e d h i g h s h e a r r a t e r a n g e , i . e . t h e r a n g e o f s h e a r r a t e s c h a r a c t e r i s t i c o f polymer p r o c e s s i n g c o n d i t i o n s . As i s well-known, t h i s r a n g e depends on t h e p o l y m e r ( m o l e c u l a r mass and m o l e c u l a r mass d i s t r i b u t i o n ) and i s bound i n e x t r u s i o n e x p e r i m e n t s by the o n s e t o f m e l t f r a c t u r e . O n l y r e -c e n t l y , i t was p r o v e d a t the D e l f t l a b o r a t o r y t h a t a l s o the -c o n e - a n d - p l a t e geo-metry i s c a p a b l e o f c o v e r i n g t h e whole range o f i n t e r e s t i n g s h e a r r a t e s . I n the p r e s e n t c h a p t e r the improvements a r e d e s c r i b e d w h i c h l e a d t o t h i s p r o g r e s s . A p r e r e q u i s i t e f o r s t a b l e f l o w i n a c o n e - a n d - p l a t e a p p a r a t u s i s t h a t t h e r e i s no open r i m where the m e l t c a n t e a r i n . Whereas a c l o s e d c o n s t r u c t i o n c a n n o t e a s i -l y be used f o r the measurement o f t h e norma-l t h r u s t , such a c o n s t r u c t i o n i s p a r t i c u l a r l y u s e f u l f o r o p t i c a l measurements. The d e c i s i v e improvement c o n s i s -t e d i n a r e a r r a n g e m e n -t o f p a r -t s so -t h a -t -t h e l i g h -t beam c a n be d i r e c -t e d -t h r o u g h the "deadwater" r e g i o n s n e a r t h e windows. I n t h i s way the i n f l u e n c e o f p a r a s i -t i c b i r e f r i n g e n c e e f f e c -t s can be m i n i m i z e d . In f a c -t , i n f o r m e r e x -t i n c -t i o n a n g l e measurements (3) t h e s e e f f e c t s c a u s e d enormous t r o u b l e s . Some l a t e s t r e s u l t s a r e p r e s e n t e d . T h e i r i m p l i c a t i o n f o r the i n t e r p r e t a t i o n o f p o l y m e r m e l t f l o w a r e d i s c u s s e d .

*) v a n Aken, J.A., F.H. Gortemaker, H. J a n e s c h i t z - K r i e g l , H.M. Laun R h e o l . A c t a 19, 159 (1980)

.2 D e s c r i p t i o n o f t h e R e v i s e d R o t o r U n i t

The d e s c r i p t i o n o f a r e d e s i g n e d a p p a r a t u s has b e e n p u b l i s h e d q u i t e r e c e n t -l y ( 3 ) . R e s u -l t s , as o b t a i n e d w i t h t h i s a p p a r a t u s , were p u b -l i s h e d and i n t e r p r e t e d i n a s e r i e s o f p a p e r s ( 4 - 6 ) . As a c o n s e q u e n c e , t h e p r e s e n t a u t h o r s c a n r e s t r i c t t h e i r d i s c u s s i o n t o t h e e s s e n t i a l l y new f e a t u r e s i n t r o d u c e d l a t e l y . I n f i g u r e 2.1 t h e v e r y newest c o n s t r u c t i o n o f t h e r o t o r u n i t i s shown. One o f t h e p r e s e n t a u t h o r s (F.H.G.) i s r e s p o n s i b l e f o r t h e u n d e r l y i n g i d e a s . The i m p o r t a n t change w i t h r e s p e c t t o t h e p r e v i o u s a p p a r a t u s c o n s i s t s i n t h e f a c t t h a t , i n s t e a d o f

the r o t o r , t h e s t a t o r (7) i s now formed by t h e " p l a t e " , whereas t h e r o t o r (6) i s m o d e l l e d a s t h e "cone".

VA

B

Fig. 2.1. Cvoaa-aeation through the rotor unit, (1) linearly polarized light beam entering the ring-shaped gap, (2) reflection prism, (3) inner window, (4) outer window, (5) stationary plate, (6) rotor with conical front surface, (7) ring-shaped gap (gap angle exaggerated in the drawing, real gap angle 1°8 ', causing a maximum gap width of ~ 0.4 mm), (8) blind hole for the thermocouple,

(9) sample injection hole, (10) e l l i p t i c a l l y polarized light beam emerging from the gap, (11) analyser.

A s h o r t d e s c r i p t i o n o f t h e o p e r a t i o n o f t h e u n i t i s g i v e n f i r s t . As was p o i n t e d o u t a l r e a d y i n e a r l i e r p a p e r s , n e i t h e r p l a t e n o r cone a r e c o m p l e t e . I n 22

the c e n t r e o f the u n i t a c y l i n d r i c a l chamber i s l o c a t e d , w h i c h c o n t a i n s t h e r e -f l e c t i o n p r i s m ( 2 ) . The r i n g - s h a p e d gap between "cone" and " p l a t e " i s l a t e r a l l y c o n f i n e d by c o n c e n t r i c c y l i n d r i c a l w a l l s n o t shown i n the f i g u r e . The windows (3 and 4) a r e mounted i n t h e s e w a l l s . The p o l a r i z e d l i g h t beam (see the c o u r s e o f t h e arrows f r o m the r i g h t l e t t e r A to t h e upper l e t t e r B) p a s s e s t h r o u g h t h e gap i n a r a d i a l outward d i r e c t i o n . The l i g h t s o u r c e (not shown) and t h e r e f l e c -t i o n p r i s m a r e moun-ted on a h o r i z o n -t a l o p -t i c a l bench h i n g i n g a r o u n g -t h e a x i s

(B-B). Between t h e l i g h t s o u r c e and t h e r e f l e c t i o n p r i s m a p o l a r i z i n g f i l t e r i s p l a c e d on the b e n c h . By t h i s f i l t e r the l i g h t beam i s p o l a r i z e d i n a f i x e d v e r t i c a l d i r e c t i o n . The p l a n e o f p o l a r i z a t i o n i s t h e same b e f o r e and a f t e r r e f l e c -t i o n by -t h e p r i s m . The a n g u l a r p o s i -t i o n o f -the bench d e f i n e s -t h e d i r e c -t i o n o f l i n e a r p o l a r i z a t i o n o f t h e l i g h t e n t e r i n g t h r o u g h t h e i n n e r window ( 3 ) . (The gap a n g l e i s 1 8 ' as i n p r e v i o u s v e r s i o n s . )

A t t h i s s t a g e o f the d e s c r i p t i o n we a r e i n the p o s i t i o n to e x p l a i n t h e m e r i t s o f the l a t e s t improvement. By t h e r e f l e c t i o n p r i s m t h e l i g h t p a t h i s f l e c t e d by e x a c t l y 90 d e g r e e s . T h i s i s not o n l y a c o n s e q u e n c e o f t h e s i m p l e de-s i g n o f t h e p r i de-s m b u t a l de-s o a n e c e de-s de-s i t y . O t h e r w i de-s e , t h e l i g h t beam e m e r g i n g f r o m the p r i s m would n o t p u r s u e the same p a t h i n d e p e n d e n t l y o f t h e a n g u l a r p o s i t i o n o f t h e b e n c h . We a r e f o l l o w i n g t h i s p a t h t h r o u g h t h e gap. I n t h e e a r l i e r c o n -s t r u c t i o n t h i -s l i g h t p a t h wa-s v e r y c l o -s e to the r o t o r , -s i n c e the r o t o r wa-s mo-d e l l e mo-d as t h e p l a t e w i t h i t s s u r f a c e p a r a l l e l to t h e l i g h t p a t h . From an econo-mic p o i n t o f v i e w the o p t i o n was f o r an i n t e r c h a n g e a b l e s t a t i o n a r y c o n e . The main drawback o f such a c o n s t r u c t i o n , however, was t h a t n e a r t h e windows t h e l i g h t beam p a s s e d t h r o u g h zones, where the c o n c e n t r a t i o n o f f l o w l i n e s was ex-t r a o r d i n a r i l y h i g h . I n f a c ex-t , ex-t h e r e a r e ex-two ( c i r c u l a r ) s i n g u l a r l i n e s a ex-t ex-t h e o u t e r edge and a t the i n n e r edge o f t h e r o t o r s u r f a c e , where t h e r o t o r f i t s i n t o the c o n c e n t r i c c y l i n d r i c a l w a l l s m e n t i o n e d above. The s h e a r r a t e i n the f l u i d becomes t h e o r e t i c a l l y i n f i n i t e a t t h e s e s i n g u l a r l i n e s . T h i s means t h a t , w i t h a

l i g h t beam p a s s i n g t h r o u g h t h i s z o n e s , c o n s i d e r a b l e p a r a s i t i c b i r e f r i n g e n c e e f f e c t s can be e x p e c t e d . The r e a s o n f o r the l a t e s t change i s now o b v i o u s . I f , i n c o n t r a s t to t h e e a r l i e r d e s i g n s , the p l a t e i s formed by the s t a t i o n a r y c o u n t e r -p a r t o f the r o t o r - which i s now n e c e s s a r i l y m o d e l l e d as t h e cone - t h e l i g h t beam p a s s e s t h r o u g h the dead-water r e g i o n s n e a r the windows. As an a d d i t i o n a l p r o v i s i o n , t h e c l e a r a n c e s between t h e l a t e r a l c y l i n d r i c a l s u r f a c e s o f t h e r o t o r and t h e c o n f i n i n g c y l i n d r i c a l w a l l s were e n l a r g e d , i n o r d e r to r e d u c e t h e s t r e a m l i n e d e n s i t y n e a r t h e edges o f t h e r o t o r .

.3 New Measurements

I n f i g u r e s 2.2 and 2.3 t h e a c h i e v e d improvements a r e d e m o n s t r a t e d .

9 0 ° r

Fig. 2. 2. Doubled extinction angle 2\ as a function of the nominal shear rate q

for a teohnioal polystyrene (Hostyren N 4000 V) at 170°C according to ref. (3).

The e a r l i e r c o n s t r u c t i o n was used when the e x t i n c t i o n a n g l e c u r v e o f a commer-c i a l p o l y s t y r e n e ( H o s t y r e n N 4000 V, Mw = 240000, Mn = 87000), as d e p i c t e d i n

f i g u r e 2.2, was measured. The i n t e r s e c t i o n s o f t h e dashed v e r t i c a l l i n e s w i t h t h e a b s c i s s a i n d i c a t e t h e s h e a r r a t e s a t w h i c h the main b i r e f r i n g e n c e c a u s e s o p t i c a l p a t h d i f f e r e n c e s o f one, two and more f u l l wave l e n g t h s ( e q u i v a l e n t to f i r s t , second and h i g h e r o r d e r f r i n g e s ) . A t t h e s e p o i n t s t h e main b i r e f r i n g e n c e becomes v i r t u a l l y z e r o so t h a t the p a r a s i t i c b i r e f r i n g e n c e s n e a r t h e windows become p r e d o m i n a n t . An i n t e r p r e t a t i o n o f t h i s phenomenon was t r i e d i n r e f . ( 3 ) . I n any c a s e , a d e d u c t i o n o f t h e c o r r e c t e x t i n c t i o n a n g l e c u r v e becomes r a t h e r d i f f i c u l t on the b a s i s o f t h e e x p e r i m e n t s shown i n f i g u r e 2.2. I n f i g u r e 2.3, t h e c o n s e q u e n c e s o f t h e l a t e s t c o n s t r u c t i o n a r e shown. The s t i l l p e r c e p t i b l e d e v i a t i o n s o f t h e e x t i n c t i o n a n g l e c u r v e f r o m i t s c o r r e c t c o u r s e as o c c u r n e a r

Fig. 2.3. Doubled extinction angle 2x vs. nominal shear rate q for the polysty-rene mentioned in the caption to figure 2.2, as measured with the newest rotor unit at 170°C. The measurements were continued to a sixfold rate of shear of

18 s , The extinction angle remained essentially constant (viz. 2x ~ 37.5°) over this extended range, when taken halfway between the fringes.

the " f r i n g e s " have o b v i o u s l y become r a t h e r s m a l l . F o r p r a c t i c a l p u r p o s e s a smoothed c u r v e i s drawn t h r o u g h t h e p o i n t s l o c a t e d h a l f w a y between t h e f r i n g e s . F u r t h e r i t s h o u l d be m e n t i o n e d t h a t the measurements c o u l d be c o n t i n u e d up t o a s h e a r r a t e o f 18 s ' w i t h o u t d i f f i c u l t i e s , whereas w i t h t h e p r e v i o u s c o n s t r u c -t i o n measuremen-ts became i m p o s s i b l e a l r e a d y h a l f way -t h e s i x -t h f r i n g e (~ 1.8 s ')

W i t h t h e new c o n s t r u c t i o n t h e a c c e s s i b l e r a n g e o f s h e a r r a t e s i s l i m i t e d by the o n s e t o f t o o much f r i c t i o n a l h e a t . As a consequence o f t o o much f r i c t i o n a l h e a t , t e m p e r a t u r e g r a d i e n t s a r e b u i l t up i n the gap, by w h i c h the l i g h t beam i s d e f l e c t e d o u t o f the normal l i g h t p a t h , so t h a t no l i g h t r e a c h e s the eye p i e c e any more. As c a n be shown (7, 8 ) , t h i s i s a v e r y s e n s i t i v e t e s t . T h i s means t h a t f r i c t i o n a l h e a t p r o d u c t i o n i s o f no i m p o r t a n c e under normal o p e r a t i o n c o n d i t i o n , i . e . as l o n g as t h e l i g h t beam i s u n d i s t u r b e d .

1000

10 15 20

q[s-i

Fig. 2.4. Flow birefringence of the investigated polystyrene (Hostyren N 4000 V) at 170°C as a function of the rate of shear q. At the points of measurements the extinction position was found by interpolation on a smoothed curve obtained from figure 2.3.

I n f i g u r e 2.4 t h e p e r t i n e n t p a t h d i f f e r e n c e , r e c a l c u l a t e d as b i r e f r i n g e n c e i s shown as a f u n c t i o n o f t h e s h e a r r a t e . A s l i g h t d i f f i c u l t y w i t h t h i s r e c a l -c u l a t i o n i s t h a t the e f f e -c t i v e p a t h l e n g t h I i s unknown, i n p r i n -c i p l e , s i n -c e t h e e x a c t f l o w p r o f i l e n e a r t h e edges as w e l l as t h e e x a c t i n f l u e n c e o f t h e d i v e r g e n c e o f the l i g h t beam a r e unknown. As i n the p r e v i o u s p a p e r ( 3 ) , an e s t i mate o f the e f f e c t i v e p a t h l e n g t h I was made ('1 = d 2c, where d i s t h e d i s -t a n c e be-tween -t h e windows and c i s -the c l e a r a n c e be-tween r o -t o r and w a l l ) . The i m p l i c a t i o n s o f t h e s e new r e s u l t s w i l l be d i s c u s s e d i n the n e x t s e c t i o n . I n t h e A p p e n d i x the o b t a i n e d r e s u l t s a r e q u o t e d i n T a b l e 2.1.

The m e c h a n i c a l d a t a were o b t a i n e d by means o f a m o d i f i e d W e i s s e n b e r g R h e o g e n i o -meter (9) under n e a r l y s t e a d y s t a t e c o n d i t i o n s . I n f i g u r e 2.5 t h e t i m e - d e p e n d e n t ' v i s c o s i t i e s " n ( t , y ) , which a r e measured a t c o n s t a n t s h e a r r a t e s y o f 0.1 s ,

I s ' , and 10 s ', s u d d e n l y imposed a t t i m e t = 0, a r e p l o t t e d as f u n c t i o n s o f

3

103 l 1 1 _ i

0 20 40 - 60

Fig. 2.5. Time-dependent "viscosity" r\(t, y) of PS Hostyren N 4000 V versus accumulating shear strain y in "stressing" tests at constant shear rates y and T = 170°C. Mechanical measurements with modified Weisseriberg Rheogoniometer at 24 mm plate diameter and 8° gap angle. Symbols with pip denote repeated measure-ments.

t h e t o t a l a c c u m u l a t i n g s h e a r s t r a i n y = y.t. S h o r t l y a f t e r t h e i m p o s i t i o n o f t h e s h e a r r a t e a pronounced maximum i s o b s e r v e d a t y - 2.5. A f t e r t h e o v e r s h o o t t h e c u r v e l e v e l s o f f . U n f o r t u n a t e l y , a t ] s ' and 10 s ' a c o n t i n u i n g s l i g h t d e -c r e a s e o f t h e v i s -c o s i t y i s f o u n d w i t h i n -c r e a s i n g s h e a r d e f o r m a t i o n . T h i s must be a t t r i b u t e d t o some i n s t a b i l i t y w h i c h may be caused by n e a r l y a d i a b a t i c h e a t i n g o f t h e m e l t o r by a c o n t i n u o u s s l i g h t d e c r e a s e o f t h e e f f e c t i v e r a d i u s o f t h e m e l t i n t h e c o n e a n d p l a t e system. An e x t r a p o l a t i o n p r o c e d u r e was used t o e v a -l u a t e t h e s t e a d y - s t a t e v i s c o s i t y n i n t h i s c a s e . The r a p i d -l y d e c r e a s i n g p a r t o f t h e c u r v e s b e h i n d t h e maximum and t h e s l i g h t l y d e c r e a s i n g p a r t a t h i g h e r d e f o r -m a t i o n s a r e a p p r o x i -m a t e d by s t r a i g h t l i n e s ( b r o k e n l i n e s i n f i g . 2 . 5 ) . The h e i g h t o f t h e i n t e r s e c t i o n p o i n t was assumed t o r e p r e s e n t n • Such a p r o c e d u r e h a s been found t o be r e a s o n a b l e e v e n i n t h o s e c a s e s where t h e m e l t i s p e r c e p t i b l y l e a k i n g

from t h e gap.

0 2 0 ¿ 0 — - 6 0

Fig, 2.8. Time-dependent primary normal-stress coefficient Q(t, y) versus shear

strain y in stressing tests as described in figure 2.5.

The c o r r e s p o n d i n g time—dependent p r i m a r y n o r m a l - s t r e s s c o e f f i c i e n t s • * 2 0 ( t , y ) = N j ( t ) / Y a r e shown i n f i g u r e 2.6. Remarkably, t h e o v e r s h o o t o f t h e normal t h r u s t ( i n t h e s h e a r s t r a i n r a n g e o f y - 12) i s l e s s pronounced t h a n t h a t of t h e s h e a r s t r e s s ( f i g . 2 . 5 ) . A l s o t h e d e c r e a s e a t h i g h d e f o r m a t i o n s i s much l e s s . P o s s i b l y , t h i s p o i n t s t o a d e c r e a s e o f the e f f e c t i v e r a d i u s R' d u r i n g 3

f l o w . I n f a c t , t h e measured t o r q u e i s p r o p o r t i o n a l t o R' whereas t h e normal 2 t h r u s t i s o n l y p r o p o r t i o n a l t o R' , so t h a t t h e f o r m e r e f f e c t i s i n f l u e n c e d more pronouncedly by a d e c r e a s e o f R'. From t h e n e a r l y c o n s t a n t v a l u e s o f 0 a t h i g h s h e a r s t r a i n s the s t e a d y - s t a t e p r i m a r y n o r m a l - s t r e s s c o e f f i c i e n t s 0g (open c i r -c l e s i n f i g . 2.8) were d e t e r m i n e d . I t s h o u l d be emphasized t h a t t h e d e -c r e a s e o f 0s w i t h g r o w i n g s h e a r r a t e i s v e r y s t r o n g . The s t e a d y - s t a t e v a l u e s ng and 0g a r e quoted i n t a b l e 2.2 ( s e e A p p e n d i x ) f o r t h e s h e a r r a t e s a p p l i e d . 28

.4 D i s c u s s i o n

F i r s t o f a l l , we c a n deduce from t h e r e s u l t s o f f i g u r e s 2.3 and 2.4 - and f r o m t a b l e 2.1 - t h a t f l o w i n t h e r i n g shaped gap o f o u r a p p a r a t u s i s s t a b l e up to s h e a r r a t e s comparable w i t h t h o s e a t w h i c h m e l t - f r a c t u r e o c c u r s i n c a p i l l a r y f l o w . A s i m i l a r c o n c l u s i o n c a n be drawn f r o m f l o w b i r e f r i n g e n c e r e s u l t s , a s o b -t a i n e d by Wales and Philippe*ff (10) on a h i g h m o l e c u l a r w e i g h -t s i l i c o n e o i l some y e a r s ago. However, as such an e x p e r i e n c e c o u l d n o t be r e p r o d u c e d w i t h o t h e r p o l y m e r s a t t h a t time, i t was n o t c o n s i d e r e d t o be g e n e r a l l y v a l i d f o r polymer m e l t s . A t t h e moment we know t h e e x p l a n a t i o n f o r t h i s e x c e p t i o n a l p o s i t i o n o f s i l i c o n e s : The o p t i c a l a n i s o t r o p y o f t h e c h a i n backbone o f t h i s m a t e r i a l i s v e r y s m a l l compared w i t h t h a t o f most o t h e r p o l y m e r s , i n p a r t i c u l a r t h a t o f p o l y s t y -r e n e . As a c o n s e q u e n c e , t h e t -r o u b l e s e x p e -r i e n c e d when a f -r i n g e p a s s e d by w i t h i n c r e a s i n g s h e a r r a t e , were n o t undergone w i t h t h i s m a t e r i a l . As a s e c o n d p o i n t t h e v a l i d i t y o f t h e l i n e a r s t r e s s o p t i c a l r u l e s h o u l d be d i s c u s s e d . I n f i g u r e 2.7 t h e s t r e s s - o p t i c a l c o e f f i c i e n t , as o b t a i n e d f r o m t h e e x p e r i m e n t a l d a t a by t h e e q u a t i o n A n s i n 2y 2 Y n [2.1] 10-8r -C[m2/N]l

I

Fig. 2.7. The stvess-optioal eoeffieient of the investigated polystyrene

(Hostyren N 4000 V) at 170°C, plotted against the rate of shear.