Characterization of Highly Cross-linked Polymers - American

5 Light Scattering of Randomly Cross-linked Polystyrene 1

KANJI KAJIWARA and WALTHER BURCHARD

Downloaded by TUFTS UNIV on October 27, 2017 | http://pubs.acs.org Publication Date: February 16, 1984 | doi: 10.1021/bk-1984-0243.ch005

Institute of Macromolecular Chemistry, University of Freiburg Stefan-Meier-Str. 31, 7800 Freiburg i.Br., Federal Republic of Germany

The dynamic and static light scattering behavior of crosslinked polystyrene chains has been studied in a good and a theta solvent. Two series of samples, based on two different primary chain lengths, have been prepared by γ-irridiation.The mean square ra dius of gyration as function of the molecular weight of the different samples were found to f a l l on a com mon line, and the same behavior was obtained for the diffusion coefficient.The ratio ρ of the geometric to the hydrodynamic radii increases slightly with the molecular weight, i.e. with increasing extent of crosslinking, but i t then decreases sharply when the gelpoint is approached. The independence of ρ of the primary chain length and also the virtual constancy of this parameter in a wide range of mole cular weights is in agreement with theory. The sharp decrease near the gelpoint is unexpected and indi cates either a certain heterogeneity in the crosslinking density or a hydrodynamic behavior which up to date cannot be described by the KirkwoodOseen approach for the hydrodynamic interaction. Recent development i n photon c o r r e l a t i o n spectroscopy has p r o v i d e d a new way o f c h a r a c t e r i z i n g a polymer molecule i n s o l u t i o n i n terms o f i t s hydrodynamic r a d i u s . The hydrodynamic r a d i u s i s d e f i n e d v i a the S t o k e s - E i n s t e i n r e l a t i o n s h i p R

h

= k T/(6irn D) B

(1)

o

with D and η » being the t r a n s l a t i o n a l d i f f u s i o n c o e f f i c i e n t and the s o l v e n t v i s c o s i t y respectively.We r e l y on Kirkwood (I) f o r the c a l c u l a t i o n o f the hydrodynamic r a d i u s o f models f o r polymer molecules. Then a dimensionless q u a n t i t y ρ can be d e f i n e d as a r a t i o o f two r a d i i , i . e . the r a d i u s o f g y r a t i o n ^ ^

1 /

^

2

and the

1

Current address: Institute for Chemical Research, Kyoto University, Uji, Kyoto, 611 Japan.

0097-6156/84/0243-0055S06.00/0 © 1984 American Chemical Society Labana and Dickie; Characterization of Highly Cross-linked Polymers ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

HIGHLY CROSS-LINKED POLYMERS

hydrodynamic r a d i u s ^2%l/2 .


For p o l y d i s p e r s e systems t h i s parameter i s d e f i n e d by the

(2) z-avera-

Here ρ can be c a l c u l a t e d f o r various model polymers, and i t depends on molecular s t r u c t u r e as w e l l as molecular weight d i s t r i b u t i o n (2). Thus ρ provides a u s e f u l information concerning molecular s t r u c t u r e and p o l y d i s p e r s i t y , when the s t r u c t u r e i s known. For example, p o l y d i s p e r s i t y causes an i n c r e a s e and bran ching a decrease o f p. Excluded volume i n c r e a s e s t h i s value. Fur t h e r examples are d i s c u s s e d i n d e t a i l i n (2). Here we mention i n p a r t i c u l a r the s p e c i a l case o f the f - f u n c t i o n a l random polycondensates where according t o theory the decrease o f ρ i s e x a c t l y ba lanced by the i n c r e a s e as the r e s u l t o f the very pronounced p o l y d i s p e r s i t y ; the ρ-parameter remains constant i n the whole p r e - g e l r e g i o n up t o the g e l p o i n t . The t r a n s l a t i o n a l d i f f u s i o n c o e f f i c i e n t D i s obtained from the f i r s t cumulant Γ of the e l e c t r i c f i e l d time c o r r e l a t i o n func t i o n g ^ ( t ) , which i s d i r e c t l y measured by photon c o r r e l a t i o n technique (3) where (4)

The f i r s t cumulant Γ i s g e n e r a l l y c a l c u l a t e d by assuming the hydrodynamic i n t e r a c t i o n as d e s c r i b e d by Oseen (_3) where no know ledge of the space-time c o r r e l a t i o n f u n c t i o n i s needed (4-6). The purpose of the present c o n t r i b u t i o n i s an experimental t e s t o f t h e o r e t i c a l r e l a t i o n s h i p s which are based on the Flory-Stockmayer (FS) branching theory (7) o f the s o l u t i o n p r o p e r t i e s from ran domly c r o s s l i n k e d monodisperse primary chains. Most o f the theo r e t i c a l work and p a r t o f the experimental work has been p u b l i s h e d p r e v i o u s l y (8-13). We, t h e r e f o r e , b r i n g here only a short o u t l i n e o f the theory and confine ourselves mainly t o the d i s c u s s i o n o f the dynamic p r o p e r t i e s . T h e o r e t i c a l Background According t o Kirkwood (1) the t r a n s l a t i o n a l d i f f u s i o n c o e f f i c i e n t D o f an x-mer i s given as

(5)

Labana and Dickie; Characterization of Highly Cross-linked Polymers ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

5.

KAJIWARA AND BURCHARD

57

Light Scattering of Polystyrene

where ζ i s the f r i c t i o n c o e f f i c i e n t o f a monomer u n i t . The sum o f the conîigurational average o f the r e c i p r o c a l d i s t a n c e between the i - t h and j - t h u n i t s extends over a l l p a i r s o f u n i t s i n a molecule. The hydrodynamic r a d i u s i s d e f i n e d f o r the non-draining case as 1

V

- (6*VV)/D

=

X

"

2

Σ

R

i j

X

( 6 )


i/j The observable q u a n t i t y , D or i s subject t o v a r i o u s types o f ensemble averages, and the z-average t r a n s l a t i o n a l d i f f u s i o n coef f i c i e n t (or the z-average o f the r e c i p r o c a l e f f e c t i v e hydrodyna mic radius) i s measured by dynamic l i g h t s c a t t e r i n g . Branched p o l y s t y r e n e used i n the present study i s produced by random c r o s s l i n k i n g o f l i n e a r p o l y s t y r e n e o f narrow molecularweight d i s t r i b u t i o n . The a p p l i c a t i o n o f Good's s t o c h a s t i c theory o f cascade processes (8) y i e l d s the ' s t r u c t u r e f a c t o r ' S*(q2) = DP^P^q^) o f randomly cross l i n k e d Gaussian chains (9,10) χ

Z

S

* 2 (1+cOS (q )

(1+αφ)ί(φ)

9

(q ) =

Z—z

= (φ)-1)αφ)

1 - (f

l - o(S*(q ρ

w

(7) )-D

where

1+Φ f

(φ,

2φ

=

W

. 1-φ

D P

WP ^

Z

(l-φ) y

P

2 * ^ > = p S

Z P P

(q

2 }

P

In these equations α denotes the extent o f c r o s s l i n k i n g , ! , e . the f r a c t i o n o f r e p e a t i n g u n i t s b e a r i n g a c r o s s l i n k , and y = DP i s the weight average degree o f p o l y m e r i z a t i o n o f the primary ^ chains, i . e . the chains before c r o s s l i n k i n g . DP i s the weightaverage degree o f p o l y m e r i z a t i o n o f the t o t a l , c r o s s l i n k e d p o l y mer, and Ρ (q^) and Ρ (q ) are the p a r t i c l e s c a t t e r i n g f a c t o r s of the c r o s s l i n k e d ancPof the primary chains r e s p e c t i v e l y . The v a r i a b l e q gives the magnitude o f the s c a t t e r i n g v e c t o r q as q = (4Tr/X)sin(9/2)

(9)

and the v a r i a b l e φ i s d e f i n e d as 2

2

φ = exp(-q b /6)

(10)

2 with b denoting the mean square d i s t a n c e between two adjacent r e p e a t i n g u n i t s . Then the weight-average degree o f p o l y m e r i z a t i o n i s given by DP

w

= S**(0) = y ( l + a ) / ( l - a ( y - l ) )

and the p a r t i c l e s c a t t e r i n g

factor

Labana and Dickie; Characterization of Highly Cross-linked Polymers ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

(11)

58


2 * 2 * P ( q ) = S*

oo c

Ό

> > •z

1

>

64


l y i n c r e a s e s the value o f ρ t o 0.8. A l i s t o f the ρ parameters f o r d i f f e r e n t models i s given i n Table J I . The reduced f i r s t cumulant Γ/q e x h i b i t s f o r the higher a convexed^urve i n the good s o l v e n t when p l o t t e d a g a i n s t q whereas the Γ/q versus q curve i n the t h e t a s o l v e n t i s w e l l a p p r o x i mated by a l i n e a r l i n e . A convexed curve was p r e d i c t e d f o r the s o f t sphere model (17) and was found f o r PVAç .microgels (16) . The molecular weight dependence o f R^ and \ S J £ o f the correspon ding c r o s s l i n k e d p o l y s t y r e n e sampïes i n the good s o l v e n t toluene are shown i n Figure 4. The behavior i s s i m i l a r to that i n cyclohe xane, b u t the p o i n t s o f measurement from the two s e r i e s no longer seem t o form a common curve. The ρ parameters are about 20 t o 45% l a r g e r than i n the t h e t a s o l v e n t , a behavior t h a t i s found a l s o f o r l i n e a r p o l y s t y r e n e and which was p r e d i c t e d by theory (8,18,19). I t should be mentioned, however, t h a t the absolute values o f ρ found by experiment are about 14% lower than p r e d i c t e d from theo ry when the Kirkwood -Oseen approach f o r the hydrodynamic i n t e r a c t i o n (1,3)is taken. 2


2

2 (b) Angular Dependence o f Γ/q . Equation (14) reduces f o r s m a l l q to

D

a

/ P

P

D

-

1

+

«/3)

Characterization of Highly Cross-linked Polymers - American

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