Honerkamp - A Note on Estimating Mastercurves

8/18/2019 Honerkamp - A Note on Estimating Mastercurves

1/8

Rheologica cta R h e o l A c t a 3 2 : 5 7 - 6 4 1 99 3 )

A n o t e o n e s t i m a t i n g m a s t e r c u r v e s

J . H o n e r k a m p a n d J . W e e s e

Universit~t Freiburg, Germany

A b s t ra c t : Th ere are several quan tities in rheo log y which show a scaling behavior.

One well know n example is the t ime- tem perature superposi t ion pr inciple of ma-

ter ia l funct ion s character iz ing the l inear viscoelas tic proper t ies o f po lyme r melts .

We propose a mathe matical sh if t procedure for the calculat ion o f mastercurves

and th e cor respond ing scal ing factors f rom exper imen tal data wh ich show such

a scal ing behavior . In order to demonstra te the applicabi l i ty of the shif t pro-

cedure mastercurves and scal ing factors are determined fo r mater ia l funct ions o f

several polys tyrene melts and for the specif ic viscosity of polyisobuty lene in

cyclohexane.

K ey w o rd s : Scaling behavior - mastercurve - t ime- temp erature superposi t ion

princip le - specific viscosity

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

I t is w e ll k n o w n t h a t m a t e r i a l f u n c t i o n s c h a r a c t e r i z -

i n g t h e l i n e a r v i s c o e la s t ic p r o p e r t i e s o f p o l y m e r m e l t s

o r s o l u t i o n s i n t h e t e r m i n a l a n d p l a t e a u r e g i o n s h o w

a s c a l i n g b e h a v i o r : b y s c a l i n g t h e e x p e r i m e n t a l d a t a

f o r a m a t e r i a l f u n c t i o n m e a s u r e d a t d i f f e r e n t t e m p e r -

a t u r e s a u n i q u e m a s t e r c u r v e r e f e r r i n g t o a n a r b i t r a r y

r e f e r e n c e t e m p e r a t u r e T o c a n b e o b t a i n e d F e r r y ,

1 9 8 0 ). F o r t h e s t o r a g e m o d u l u s G ~ r c o ) t h i s s o c a l l e d

t i m e - t e m p e r a t u r e s u p e r p o s i t i o n p r i n c i p l e c a n b e e x -

p r e ss e d b y t h e e q u a t i o n

G r (o ) ) : C To T ) G ro (a ro ( T )co ) .

1 )

T h e r e b y t h e s c a l in g f a c t o r s a r 0 T ) a n d C T o (T ) h a v e

b e e n i n t r o d u c e d .

T h e s e m a t e r i a l f u n c t i o n s a r e n o t t h e o n l y q u a n t i t ie s

s h o w i n g s u c h a s c a l i n g b e h a v i o r . T h e s p e c i f i c v i s c o s i -

t y r / ~ t c ) o f p o l y m e r s o l u t i o n s i n d e p e n d e n c e o n t h e

c o n c e n t r a t i o n c is a n o t h e r e x a m p l e . W i t h a p p r o p r i a t e

s c a l i n g f a c t o r s a M o ( M ) a n d C M o ( M ) t h e s p e c i f i c

v i s c o s i t y t / ~ t c ) i s i n d e p e n d e n t o f t h e m o l a r m a s s M

o f t h e p o l y m e r sa m p l e U t r a c k i a n d S im h a , 1 9 63 ;

B a l o c h 1 9 8 8 a , 1 9 8 8 b ) :

r/~ c) = CMo M ) rl~o aMo M ) c)

2)

T h e r e b y , M 0 d e n o t e s a n a r b i t r a ry r e f e r e n c e m o l a r

m a s s .

I n p r a c t i c e t h e r e a re t w o r e a s o n s f o r t h e d e t e r m i n a -

t i o n o f s c a li n g f a c t o r s f r o m e x p e r i m e n t a l d a t a o b t a i n -

e d a t d i f f e r e n t t e m p e r a t u r e s o r f o r d i f f e r e n t m o l a r

m a s s e s . F i r st , t h e m e a s u r i n g r a n g e o f a n e x p e r i m e n t a l

d e v i c e c a n b e e x t e n d e d b y t h e c a l c u l a t i o n o f m a s t e r -

c u r v e s . F u r t h e r m o r e , t h e s c a l i n g f a c t o r s p r o v i d e i n -

f o r m a t i o n a b o u t p h y s i c a l p r o p e r t i e s o f th e s y s t e m .

U n f o r t u n a t e l y , t h e r e is n o w e l l - k n o w n m a t h e m a t i -

c a l m e t h o d f o r t h e c a l c u l a t i o n o f s c a l i n g f a c t o r s a n d

g r a p h i c a l m e t h o d s a r e u s ed t o e s t im a t e t h e m . F o r t h a t

r e a s o n w e p r o p o s e a n d d i s c u s s a m a t h e m a t i c a l s h i f -

t i n g p r o c e d u r e i n t h i s p u b l i c a t i o n .

I n t h e f o l l o w i n g s e c t i o n t h e s h i f t p r o c e d u r e i s i n -

t r o d u c e d . I n S e c t . 3 t h i s p r o c e d u r e i s u s e d t o c a l c u l a t e

t h e s c a l i n g f a c t o r s f o r m a t e r i a l f u n c t i o n s o f s e v e r a l

p o l y s t y r e n e s a m p l e s. A s a n o t h e r a p p l i c a t i o n e x a m p l e

t h e s c a l i n g b e h a v i o r o f t h e s p e c i f i c v i s c o s i t y o f a

p o l y m e r s o l u t i o n i s s t u d i e d i n S e c t . 4 .

2 . S h i f t p r o c e d u r e

T h e f i r s t p a r t o f t h i s s e c t i o n i n c l u d e s a m a t h e -

m a t i c a l f o r m u l a t i o n o f t h e p r o b l e m w h i c h m u s t b e

s o l v e d in o r d e r t o c a l c u l a t e s h i f t f a c t o r s f r o m e x p e r i -


2/8

58 Rheologica Acta , Vol . 32, No. 1 (1993)

m e n t a l d a t a . T h e s h i f t p r o c e d u r e i t se l f i s p r e s e n t e d i n

t h e s e c o n d p a r t . I n t h e l a s t p a r t o f t h i s s e c t i o n

s i m u l a t e d d a t a a r e u s e d t o t e s t t h e s h i f t p r o c e d u r e .

2 .1 F o r m u l a t i o n o f t h e p r o b l e m

F o r t h e d e f i n i t i o n o f t h e s h i f t p r o c e d u r e i t i s c o n v e -

n i e n t t o c o n s i d e r t h e s c a l i n g r e l a t i o n ( E q s . ( 1 ) o r ( 2 ) )

o n a d o u b l e l o g a r i t h m i c s c a l e . O n t h i s s c a l e t h e s c a l -

i n g b e c o m e s a s h i f t a n d t h e e q u i v a l e n t t o t h e s c a l i n g

r e l a t i o n c a n b e w r i t t e n a s

gx (t ) = g~0 (t + rXo (2 ) ) + Y*0 2 ) .

3)

I n t h i s n o t a t i o n g a ( t ) r e p r e s e n t s t h e l o g a r i t h m o f t h e

s t o r a g e m o d u l u s G ~ ( c o ) ( s p e c i f i c v i s c o s i t y t / ~ t ( c ) ) , t

r e p r e s e n t s t h e l o g a r i t h m o f t h e f r e q u e n c y co ( c o n c e n -

t r a t i o n c ) a n d 2 d e n o t e s t h e t e m p e r a t u r e T ( m o l a r

m a s s M ) . r a 0( 2 ) a n d 7 ~ 0( 2) c o r r e s p o n d w i t h t h e

l o g a r i t h m o f t h e s c a l i n g f a c t o r s a r 0 ( T )

( a M o ( M ) )

a n d

C T o ( T ) ( C M o ( M ) )

a n d s a t i sf y th e e q u a t i o n s

r z o ( 2 o ) = O ( 4 a )

yao(20) = 0 . (4b )

I n p r a c t i c e , t h e f u n c t i o n

g ~ ( t )

i s m e a s u r e d f o r

s e v e r a l p a r a m e t e r s 2 d e n o t e d b y 2 0

. . . . A m

i n t h e

f o l l o w i n g t e x t . F o r e v e r y p a r a m e t e r 2 j t h e r e is

t h e r e f o r e a d a t a s e t

[g~, i;

i = 1 . . . . n j.} f o r t h e f u n c -

t i o n

g a j ( t )

a t t h e p o i n t s { tj i ; i = 1 . . . .

n j }

a v a i l a b l e .

B e c a u s e o f t h e e x p e r i m e n t a l ' n o i s e , th e s e d a t a a r e c o n -

s i d e r e d a s r e a l i z a t i o n s o f t h e r a n d o m v a r i a b l e s

Ga

, , = g 2 j ( t j , i ) + f f f fj , i g j , i ,

i = l , . . . , n j , j = O , . . . , m .

5 )

I n t h i s e q u a t i o n t h e

ej , i

a r e in d e p e n d e n t , s t a n d a r d

n o r m a l l y d i s t r i b u t e d , r a n d o m v a r i a b l e s a n d t h e

a j , i

s p e c i f y t h e e r r o r o f e a c h d a t a p o i n t u p t o a n u s u a l l y

u n k n o w n s c a l i n g f a c t o r a . ( I f e . g . a j . i = 1 , t h e d a t a

a r e a f f e c t e d b y a b s o l u t e e r r o r s o f s i z e a . I f e . g .

a j , i = g 2 j ( t j , i )

t h e d a t a a r e a f f e c t e d b y r e l a t i v e e r r o r s

o f t h e s i z e a . ) F r o m t h e s e d a t a t h e s h i f t f a c t o r s

y ,~o(21) . . . . y ,~o(2m ) an d r~0(2 i ) . . . . . l'~ to (2m) m u s t be

c a l c u l a t e d .

2 . 2 E s t i m a t i n g m a s t e r e u r v e s b y o p t i m i z a t i o n

I f t h e f u n c t i o n g ~ o ( t ) w o u l d b e k n o w n , t h e d e t e r -

m i n a t i o n o f t h e s h if t f a c t o r s y a o (2 1 ) . . . . 7 ~o (2 m) a n d

1 2 0 ( 2 1 ) . . . . . ~ ' ) . 0 ( 2 m )

w o u l d b e n o p r o b l e m : a s i m p l e

l e a s t - s q u a r e s m e t h o d ( P r e s s e t a l. , 1 9 8 6 ) c o u l d b e u s e d

a n d e s t i m a t es y { 0 ( 2 1 ) , . . . , y ~ o ( 2 m ) a n d r a o (2 1 ) . . . . ,

r { 0 ( 2 m ) f o r t h e s h i f t f a c t o r s w o u l d b e o b t a i n e d b y

m i n i m i z i n g

Z 2 = ~

- T f - (g ~ , i - (g a o ( t j , i + r ( 2 j ) ) + Y ~ o ( 2 j ) ) ) 2

j = O i = 1 • j , i

6 )

wi th re spe c t to Yao 21) . . . . . 7ao

A m )

a n d rZ o ( 2 1 ) . . . . .

r a o ( 2 m ) . T h e f u n c t i o n

g , o ( t )

i s , h o w e v e r , n o t k n o w n .

W e p r o p o s e t h e r e f o r e to a p p r o x i m a t e t h e fu n c t i o n

g ~ o ( t )

b y a p o l y n o m i a l

np

g ~ o ( t ) -~ ~ c k t k

(7)

k = 0

a n d t o d e f i n e t h e e s t i m a t e s y ~ o ( 2 1 )

. . . . ~ , ' ) ~ o ( 2 m )

a n d

G ¢7

~ ') ~0 2 1 ) ' . . , 2 2 0 ( 2 m) b y t h e m i n i m u m o f

m ~ 1

z =

j = O i = l T j , i

X g j , i - - , i + 72) .0

e 8)

w i t h r e s p e c t t o y ~ 0 ( 2 1 )

. . . . ) , '2 0 ( 2 m ) , T ~ 0 ( 2 1 ) . . . . .

Z 'X 0(2 m) a n d c 0 . . . . % : i n a d d i t i o n t o t h e s h i f t f a c -

t o r s 7 z o ( 2 0 , . . . , 7 ~o (2p) an d rzo(21) . . . , r~o(2m ) the

f u n c t i o n g z o ( t ) i t s e l f h a s t o b e f i t t e d t o t h e d a t a .

C o n c e r n i n g t h e s h i f t p r o c e d u r e d e s c r i b e d s o f a r

s o m e r e m a r k s a r e n e c e s s a r y :

F o r t h e a p p r o x i m a t i o n o f th e f u n c t i o n g ~ o ( t )

s p l i n e - f u n c t i o n s se e m t o b e m o r e a p p r o p r i a t e . I f ,

h o w e v e r , l i n e a r o r c u b i c s p l i n e s ( S t o e r , 1 9 8 9 ) a r e

u s e d t h e q u a n t i t y ~ 2 h a s d i s c o n t i n u i t i e s w h i c h

l e a d t o s e r i o u s p r o b l e m s i n p r a c t ic e . T h e s e p r o -

b l e m s c a n b e a v o i d e d b y t h e u s e o f a p o l y n o m i a l .

T h e

r a n g e i n w h i c h t h e d e g r e e

n p

o f t h e

p o l y n o m i a l c a n b e c h o s e n i s l i m i t e d b y

m

l < n p < ~ ]

n j - 2 m .

(9)

j = O

F o r o t h e r c h o i c e s t h e q u a n t i t y Z 2 h a s n o l o c a l

m i n i m u m .

- I n p r a c t i c e , p o l y n o m i a l s w i t h


3/8

Hone rka mp and Weese, A note on estimatin g mastercurves 59

m m

nj/20


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60 Rhe olog ica Acta, Vo l. 32, No. 1 (1993)

t h e d e g r e e o f t h e p o l y n o m i a l i n t h i s r a n g e . F o r ~o~

p o l y n o m i a l s w i t h a h i g h e r d e g r e e t h e e s t i m a t e s c o u l d

n o t b e c a l c u l a t e d b e c a u s e o f n u m e r i c a l i n s ta b i li t ie s . I n

p r a c t i c e t h i s l i m i t a t i o n i s n o d i s a d v a n t a g e b e c a u s e t h e ~o~

e s t im a t e s a r e n o t e x p e c t e d t o b e c o m e b e t t e r. O n l y t h e

c a l c u l a t i o n t i m e i n c r ea s e s w i t h a n i n c r e as i n g d e g r e e o f

t h e p o l y n o m i a l .

3 App l ica t io n to ma ter ia l funct io ns

~ 1 0 ~

®

lO ~

I n t h i s s e c t i o n t h e s h i f t p r o c e d u r e i s u s e d t o

c a l c u l a t e m a s t e r c u r v e s f o r m a t e r i a l f u n c t i o n s o f s i x

n e a r l y m o n o d i s p e r s e P S s a m p l e s a n d a c o m m e r c i a l

~ o ~

P S s a m p l e . I n e v e r y e x a m p l e t h e s h i f t p r o c e d u r e a

s h o u l d l e a d t o t h e s a m e r e s u l t f o r t h e s c a li n g f a c t o r s .

T h e a g r e e m e n t o f t h e r e s u l t s c a n t h e r e f o r e b e c o n -

s i d e r e d a s a t e s t o f t h e s h i f t p r o c e d u r e . ~ o %

3 . 1 R e s u l t s f o r s i x n e a rl y m o n o d i s p e r s e P S s a m p l e s

T h e d a t a u s e d i n t h i s s e c t i o n w e r e p u b l i s h e d b y

S c h a u s b e r g e r e t a l . (1 9 8 5 ) . T h e y c h a r a c t e r i z e th e d y -

n a m i c m o d u l i o f s i x n e a r l y m o n o d i s p e r s e P S s a m p l e s

w i t h m o l a r m a s s e s b e t w e e n 3 9 - 1 0 3 a n d 3 0 0 0 . 1 0 3

( T a b l e 2 ) a n d w e r e m e a s u r e d a t t e m p e r a t u r e s b e t w e e n

1 5 0 ° a n d 2 7 0 ° C . T h e r e s u lt s f o r t h e m a s t e r c u r v e s a n d

t h e c o r r e s p o n d i n g s c a l i n g f a c t o r s a z o T ) a n d CTo T)

a r e s h o w n i n F ig s . 4 a n d 5 a n d T a b l e 3 . A s r e f e r e n c e

t e m p e r a t u r e T o = 1 5 0 ° C w a s c h o s e n .

F o r a l l s a m p l e s t h e s h i f t p r o c e d u r e l e d t o s m o o t h

m a s t e r c u r v e s ( F ig . 4 ) . T h e v a l u e s f o r t h e h o r i z o n t a l

s c a l i n g f a c t o r s a r o T ) ( F i g . 5 a ) o b t a i n e d f o r t h e s i x

s a m p l e s a r e i n g o o d a g r e e m e n t w i t h e a c h o t h e r . T h e

ver t i c a l s ca l ing f a c to r s CT0 T) ( F ig . 5 b ) a r e w i th in the

e r r o r g i v e n b y 1 .

T h e h o r i z o n t a l s c a l i n g f a c t o r s

a r o T )

a r e we l l r ep-

r e s e n t e d ( F i g . 5 a , s o l id l i n e ) b y t h e W L F - r e l a t i o n

( F e r r y , 1 9 8 0 )

c T o ) T - T o )

l o g

aro T ) -

(11)

T - To~

Table 2. Molar mass and polydispers i ty index of the near ly

monodisperse PS samples (Schausberger e t a l . , 1985)

S am ple M 10- 3

)~lw/MN

PS 1 39 1.05

PS 2 70 1.06

PS 3 128 1.05

PS 4 275 1.07

PS 5 770 1.07

PS 6 3000 1.05

u

@

D

@

u

@

u

@

o

k ~ 2

k

+

k ~ 2

+

2

o

o

i , u . ~ I i i n . . I i i , m , I i , m , u j l l u m l t i l u m l i t u r n 1 1 i i n , , I i J . ~ , . , .

1 0 6 1 0 s 1 ¢ 4 1 0 3 1 0 2

1 0 1 0

1 d i ~ I C 1 3

1o

o o ~

+ - ~ o o o

O~o

a ×

e

a

i i i i i I i q i L I I I H r I i i i i i i i i I i i H L ] I I I I I I I I I I I I I I I ] ] [ 1 1 I I I I I l l l I I I I I I I L I I I 1 1 1 1 1 1 1 1 I I I I I I I 1 ~

1 0 ~ 1 0 ~ 1 0 s 1 0 ~ 1 0 3 1 0 ~ 1 0 - 1 0 l d 1 0 ~ 0

~ ( s - )

Fig. 4. Mastercurves for the dyn amic moduli G' r0(co and

G}n(co) (b) of the s ix near ly monodisperse PS samples

( < > ' P S I, x P S 2 , + P S 3 , A P S 4 , © P S 5 , [ ] P S 6 )

T h e a p p r o p r i a t e W L F - p a r a m e t e r s h a v e b e e n d e t e r -

m i n e d b y a n o n l i n e a r l e a s t - s q u a r e s f i t ( P r e s s e t a l . ,

1 9 8 6 ) a n d a r e g i v e n b y

c T o ) T o - T o~ ) =

( - 7 6 3 _ 7 8 ) ° C a n d T= = ( 4 4 . 8 0 _ 6 . 3 3 ) ° C . I t s h o ul d

b e m e n t i o n e d t h a t

c T o ) T o - T = )

h a s b e e n c a l c u l a te d

i n s t e a d o f

c T o)

b e c a u s e t h i s q u a n t i t y i s i n d e p e n d e n t

o f t h e r e f e r e n c e t e m p e r a t u r e T o.

3 . 2 R e s u l ts f o r a c o m m e r c i a l P S s a m p l e

F o r t h e c o m m e r c i a l P S s a m p l e d a t a f o r t h e d y n a m -

i c m o d u l i a n d t h e r e l a x a t i o n m o d u l u s w e r e a v a i l a b l e .

T h e d a t a h a v e b e e n m e a s u r e d a t t e m p e r a t u r e s b e -

t w e e n 1 2 2 ° a n d 2 5 0 ° C . T h e r e s ul t s f o r t h e m a s t e r -

c u r v e s a n d t h e c o r r e s p o n d i n g s c a l i n g f a c t o r s a r o T )

a n d

CTo T)

a r e s h o w n i n F i gs . 6 a n d 7 a n d T a b l e 4 .

A s r e f e r e n c e t e m p e r a t u r e T o = 1 8 0 ° C w a s c h o s e n .


5/8

H o n e r k a m p a n d W e e s e, A n o t e o n e s t i m a t i n g m a s t e r c u r v e s 6 1

T a b l e 3 . S c a l i n g f a c t o r s a ~ - o T a n d C ~ . o T c a l c u l a t e d f r o m t h e d a t a f o r t h e d y n a m i c m o d u l i o f t h e n e a r l y m o n o d i s p e r s e

P S s a m p l e s

P S 1 P S 2 P S 3

a)n (160 ) (2 .890 -+ 0 .062) 10- ~ (2 .482 -+ 0 .026) 10-1 (2 .386 -+ 0 .090) 10-

a~ (1 80 ) - (2 .579 ± 0 .047 ) . 10 -z (2 .530 +_ 0 .101) 10 -2

c ~_ (160) 1.038 _+ 0.027 0.99 2 _+ 0.010 1.004 + 0.010

c br~(180) - 1 .063 -+ 0.027 0.993 ± 0.012

P S 4 P S 5 P S 6

a~n (160 ) (2 .399 + 0 .088) 10-1 (2 .314 +_ 0 .040) - 10-1 (1 .970 _ 0 .058) 10-1

a~ro 180 ) (2.20 2 _+ 0.095) 10 -2 (2.286 -+ 0.052) 10 -2 (2.108 + 0.076) 10 -2

a~.,,(200) (4.264 -+ O. 184). 10 - 3 (4.32 0 _+ O. 109). 10 - 3 (5.275 ± 0.250) 10 - 3

a ~ (220) (1.231 _+ 0.05 4). 10 -3 (1.285 + 0.0 33) . 10 -3 (1.583 ± 0.091) 10 -3

a~.~(240) - (4.919 + O. 126). 10 -4 -

a}~(270) - - (1 .594 _+ 0 .095) . 10 -4

c ~-n 160) 0.9 97 -+ 0.00 7 0.9 90 _+ 0.00 7 1.027 _+ 0.0 12

c ~r~(180) 0.99 9 _+ 0.009 0.98 6 + 0.007 1.009 _+ 0.013

c ~_ (200) 0.988 _+ 0.00 9 0.991 +- 0.007 1.029 _+ 0.013

c ~ (220 ) 0.9 43 -+ 0.0 14 0.9 90 -+ 0.0 08 1.02 0 +__0 .014

c ~n (240) - 0 .987 -+ 0.008 -

c ~-0(270) - - 1 .01 7:t :0 .014

l d

lo 0

i d '

8 .

1 0 2 .

l d

1 0 4

1 4 0 0 1 6 ' 0 0 1 8 ' 0 0 2 0 ' 0 . 0 2 2 1 0 . 0 2 4 1 0 .0 2 6 i 0 . 0 2 8 1 0 . 0

T (oc)

2 . 0

1 . 5

8 , 1 . 0 - I i

o~

0 . 5

0 . 0

1 4 0 0

i ~ 7 8 o

1 6 ' 0 . 0 8 ' 0 . 0 2 0 ' 0 . 0 2 2 1 0 .0 2 4 ' 0 . 0 2 6 ' 0 0 2 8 1 0 0

T ( ° C )

T a b l e 4 . S c a l i n g f a c t o r s a } o T ) a n d c } n ( T ) c a l c u l a t e d f r o m

t h e d a t a f o r t h e d y n a m i c m o d u l i a n d t h e r e l a x a t i o n m o d u l u s

o f t h e c o m m e r c i a l P S s a m p l e

G ( o ) ) a n d G ( c o ) G t )

a~._ (122) (1.52 2_+ 0.03 1). 10 +4

abr~(133) (1 .120 _+0 .021) ' 10 +3

abr~ 143) (1.5 28 _+ 0.0 26 )- 10 +2

a b r~ (1 5 3) ( 2 .7 6 0 ± 0 .0 3 8 ) . 1 0 + 1

abr~(164) (6.6 56 _+ 0.06 1) 10 + 0

a~r° , (200) (1 .958_+ 0 .011) .10 -1

abT~ 220) (6.099 -+ 0.0 37) . 10 -2

a -~ (250) (1 .462 _+ 0 .010 ) . 10 -2

c~ . (122) 0 .527_+0 .003

c~r° , (133) 0 .536_+0 .003

cbr~(143) 0.524_+0.002

c ~ r ~ ( 1 5 3 ) 0 .5 6 7 + _ 0 .0 0 2

c ~° (164) 0.38 5 _+ 0.001

c ~ r ~ ( 2 0 0 ) 0 . 8 3 3 + 0 . 0 0 2

c ~r~ 220) 1.02 9 + 0.00 3

c ~ ( 2 5 0 ) 1 . 1 0 4 - + 0 . 0 0 5

(1.195 ± 0.023) 10 +4

(0.776_+ 0 .00 9) ' 10 + 3

(1.176___ 0.00 8) 10 +2

(2 .428 ± 0 .017) 10 + 1

(5.877 ± 0.026) 10 +o

(1 .858 ± 0 .012) 10-1

(5 .710 ± 0 .068) 10 -2

0.548 _+ 0.003

0 .593 ± 0 .003

0 .572 ± 0 .003

0.591 ± 0.003

0.405 +_ 0.002

0.961 +_ 0.007

1.147 ± 0.017

F i g . 5 . S c a l i n g f a c t o r s a ~ .n( T ) a n d c } n T ) c a l c u l a t e d f r o m

t h e d a t a f o r t h e d y n a m i c r f io d u l i o f t h e ~ ne a rl y m o n o d i s p e r s e

P S s am p le s ( o P S 1 , x P S 2 , + P S 3 , ~ P S 4 , © P S 5 ,

[ ] P S 6 ) . T h e s o l i d li n e ( a) m a r k s t h e t e m p e r a t u r e

d e p e n d e n c e o f t h e s c a li n g fa c t o r a r 0 ( T ) g i v e n b y t h e W L F

r e l a t i o n


6/8

62 Rh eolog ica Acta, Vo l. 32, No. 1 (1993)

F o r t h e d y n a m i c m o d u l i a s w e l l a s f o r t h e r e l a x a - ~0~

t i o n m o d u l u s g o o d r e s u l t s f o r t h e m a s t e r c u r v e s

( F ig . 6 ) h a v e b e e n o b t a i n e d . I n a d d i t i o n , t h e s h i f t p r o -

c e d u r e l e d t o h o r i z o n t a l s c a l i n g f a c t o r s a r 0 ( T ) w h i c h ~o~

a r e i n g o o d a g r e e m e n t w i t h e a c h o t h e r ( F ig . 7 a ) .

S o m e w h a t a m a z i n g is t h e t e m p e r a t u r e d e p e n d e n c e o f ~ . ~0

t h e v e r t i c a l sc a l in g f a c t o r s c r 0 ( T ( F ig . 7 b ) . T h e y

v a r y b e t w e e n 0 . 4 a n d 1 . 2 w i t h o u t a s y s t e m a t i c t e m p e r - ° ~0'

a t u r e d e p e n d e n c e . T h i s f a c t m a y b e a t t r i b u t e d t o i n a c - ~oo

c u r a c ie s i n t h e p r e p a r a t i o n o f t h e s a m p l e s .

F o r t h e W L F - p a r a m e t e r s t h e v a l u e s c ( T o ) ( T o - T ~ ) = 1 ~

( - 73 8 __+_ 9) °C a n d Too = (4 4. 75 ___3 . 06) °C we r e ob- ::

t a i n e d . T h e s e v a l u e s a r e i n g o o d a g r e e m e n t w i t h t h e ~0-~

v a l u e s i n S e c t. 3 .1 a n d l e a d t o a g o o d a p p r o x i m a t i o n a

o f t h e t e m p e r a t u r e d e p e n d e n c e o f t h e h o r i z o n t a l s c al -

i n g f a c t o r s a r o ( T ). 2 0

1 2 o . o 1 4 1 0 0 1 6 ~ o . o 1 8 ' o .o 2 o ' o . o 2 2 ' 0 . 0 2 4 J o . o 2 6 1 0 0

T ( o c )

1 0 7 ~

o

J

o

o * a B

. o

o •

e

e

o

o

~cf

1 0 z , , , ,, ,, ,1 1 0 - . . . . . 1 0~ ' ' ' ' ' ' ' ' ~ 1 0 0 . . . . . . . 1 0 1 ' ' ' ' i ' ~ ' . . . . . . . 0 l , , ,, , ,, ,~ 1 0~ , , , . q 1 0 5

( s - b

1if=

> , -

& 1 0 : ' ,

' \

l o '

1 0 5 ' ' ' ' 1 0 - ' ' ' l d z ' ' ' ~ 1 0 - ~ ' ' ' 1 0 - ' '' ' ~ 1 0 ° ' ' ' ' 1 0 ' ' ' j ' ~ 2

b t 4

Fig. 6. M astercurves for the dynam ic mod uli G'T0(og) and

G~0(co (a) and the re laxat ion modulus Gro(T) of the com -

mercial PS sample

~ v o 1 . o -

J

0 . 5 -

0 . 0

1 2 0 . 0 1 , $ ' 0 0 1 6 ' 0 . 0 1 8 ' 0 0 2 0 ' 0 . 0 2 2 ~ 0 . 0 2 4 1 0 . 0 2 6 1 0 . 0

b T ( ° C )

Fig. 7. Scaling factors

a~.,~(T)

(a) and

c~,,(T)

(b) calculated

f rom the data for the dy°namic mod uli ~nd the re laxat ion

modulus of the commercial PS sample . The sol id l ine (a)

marks the tem perature depend ence of the scal ing facto r

aro(T)

given by the W LF r e la tion

4 A p p l i c a t io n t o t h e s p e c i f ic v is c o s it y

T h e d a t a f o r t h e s p e c i fi c v i s c o s it y i n d e p e n d e n c e o f

t h e c o n c e n t r a t i o n f o r p o l y i s o b u t y l e n e i n c y c l o h e x a n e

w e r e p u b l i s h e d b y B a l o c h ( 1 9 8 8 ) . T h e s e d a t a

c h a r a c t e r i z e t h e s p e c i f i c v i s c o s i t y o f f i v e p o l y i s o -

b u t y l e n e s a m p le s w i t h m o l a r m a s s es b e t w e e n 1 . 4 .1 0 5

a n d 3 1 . 6 . 1 0 s f o r c o n c e n t r a t i o n s b e t w e e n 0 .0 0 1 g / c m 3

a n d 0 . 4 g / c m 3 a n d r e f e r t o a t e m p e r a t u r e o f 2 5 ° C .

T h e r e s u l ts f o r t h e m a s t e r c u r v e a n d t h e c o r r e s p o n d i n g

s c a l i n g f a c t o r s a M o ( M ) a n d CMo(M a r e s h o w n i n

F i g s. 8 a n d 9 . A s r e f e r e n c e t h e s a m p l e w i t h a m o l a r

m a s s o f M = 1 . 4 . l 0 s w a s c h o s e n .

A s i n t h e e x a m p l e s o f S e c t . 3 t h e d e t e r m i n a t i o n o f

t h e s c a l i n g f a c t o r s w i t h t h e s h i f t p r o c e d u r e f r o m e x -

p e r i m e n t a l d a t a i s n o p r o b l e m a n d a s m o o t h m a s t e r -

c u r v e ( F i g . 8 ) i s o b t a i n e d . T h e h o r i z o n t a l s c a l i n g f a c -


7/8

Ho nerk am p and Weese, A note on es timating mastercurves 63

l d ~

I @

1@

i o

~ . ~ 1 0 6 I

1 0

1 0

I @

i d

i o

1 o

1 o ~

n g °

q ~ o

o o

o

o

o

o

[]

oo

o

[]

t~

[]

[]

o

[]

[]

[]

1 0 - 2 1 0 ~ 1 0 1 0

c ( g / c m ~ )

Fig. 8. M astercu rve f or th e specific visco sity r /~q0 c of

po ly i sobuty lene in cyc lohexan

t o r s

a Mo M

F i g . 9 a ) s h o w a m o l a r m a s s d e p e n -

d e n c e . A g a i n s t t h i s t h e v e r t i c a l s c a l i n g f a c t o r s

Cuo M) F i g . 9 b ) s e e m t o b e i n d e p e n d e n t o f t h e

m o l a r m a s s .

T h i s c o n c l u s i o n a n d t h e a s s u m p t i o n t h a t f o r s m a l l

c o n c e n t r a t i o n s c t h e s p e c i f i c v i s c o s i t y t /~ t c ) is g i v e n

b y

rl~ c) = [~llMC 12)

l e a d s t o t h e r e l a t i o n

aMo M) = [ n l M / [ n l M 0 •

1 3 )

T h e i n t r i n s i c v i s c o s i t y [ r/ ] M a n d t h e s c a l i n g f a c t o r

a Mo M) h a v e t h e r e f o r e t h e s a m e m o l a r m a s s

d e p e n d e n c e . T h i s r e s u l t is in a g r e e m e n t w i t h t h e r e l a -

t i o n s s e v er a l a u t h o r s B a l o c h , 1 9 8 8) p r o p o s e f o r t h e

h o r i z o n t a l s c a l i n g f a c t o r s .

5 C o n c l u s i o n s

A s h i f t p r o c e d u r e f o r t h e c a l c u l a t i o n o f m a s t e r -

c u r v e s a n d t h e c o r r e s p o n d i n g s c a l i n g f a c t o r s f r o m

d a t a s h o w i n g a s c a l i n g b e h a v i o r h a s b e e n p r o p o s e d

a n d d i s c u s s e d . T h e m e t h o d h a s b e e n t e s t e d w i t h e x -

p e r i m e n t a l d a t a f o r m a t e r i a l f u n c t i o n s o f s e v e r a l

p o l y s t y r e n e m e l t s a n d l e d t o c o n s i s t e n t r e s u lt s f o r t h e

ld

I

I

lO

1@ @ 1@

a M

ld

A

o

~ L ~ i t r 7

1@ 1@ lo~

b

Fig. 9. Scaling facto rs

a ~n M

a) and

C~o M

b)

calculated f rom the data for the specif ic vascosi ty of

po ly i sobuty lene in cyc lohexan

s c a l i n g f a c t o r s . I t w a s a l s o p o s s i b l e t o c a l c u l a t e

m a s t e r c u r v e s a n d s c a l i n g f a c t o r s f o r t h e s p e c i f i c

v i s c o s i t y o f p o l y i s o b u t y l e n e i n c y c l o h e x a n e .

T h o u g h t h e s h i f t p r o c e d u r e h a s b e e n a p p l i e d t o e x -

p e r i m e n t a l d a t a w i t h s m a l l s t a t i s t i c a l e r r o r s , i t c a n

a l s o b e a p p l i e d t o e x p e r i m e n t a l d a t a w h i c h a r e a f -

f e c t e d b y l a r g e s t a t i s t i c a l e r r o r s o r b y s y s t e m a t i c e r -

r o r s . H o w e v e r , i t s h o u l d b e k e p t i n m i n d t h a t l a r g e

d a t a e r r o r s le a d t o i n e x a c t re s u l ts . O n t h e o t h e r h a n d ,

t h e r e s u l t s b e c o m e m o r e a n d m o r e a c c u r a t e w i t h

d e c r e a s i n g d a t a e r r o r s .

Acknowledgement

J . W eese grateful ly acknowledges f inancial suppor t by

Rheom et r ic s I nc . , New Je r sey . We thank Rheom et r ic s I nc.

f o r supply ing us wi th the expe r im enta l da ta o f the com m er -

cial polystyrene sample.


8/8

64 Rheologica Acta, Vol. 32, No. 1 (1993)

e ferences

Baloch MK (1988) J Macromol Sci - Chem A 25(4):363

Baloch MK (1988) J Macromol Sci - Phys B 27(2):151

Bard Y (1974) Nonlinear parameter estimation. Academic

Press, New York

Ferry JD (1980) Viscoelastic properties of polymers. J.

Wiley Sons, New York

Press WH, Flannery BP, Teukolsky SA, Vetterling WT

(1986) Numerical recipes. Cambridge University Press

Schausberger A, Schindlauer G, Janeschitz-Kriegl H (1985)

Rheol Acta 24:220

Stoer J (1989) Numerische Mathematik 1. Springer Verlag,

Berlin

Utracki L, Simha R (1963) J Polym Sci A 1:1089

Correspondence to:

Prof. J. Honerkarnp

Fakult~tt for Physik

Universit~tt Freiburg

Hermann Herder Str. 3

W-7800 Freiburg i. Br.

Germany

(Submitted on June 8, 1992;

in revised form on Nov 16, 1992)

Honerkamp - A Note on Estimating Mastercurves

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