The Structure of the Amorphous Phase in Synthetic Polymers

12 The Structure of the Amorphous Phase in Synthetic Polymers An X-ray Approach

Downloaded by UNIV OF BATH on July 1, 2016 | http://pubs.acs.org Publication Date: November 17, 1980 | doi: 10.1021/bk-1980-0141.ch012

GEOFFREY R. MITCHELL, RICHARD L O V E L L , and A L A N H. WINDLE Department of Metallurgy and Materials Science, University of Cambridge, Pembroke Street, Cambridge CB2 3QZ U.K.

Previous studies of the local conformation and packing of polymer chains based on measurements of wide-angle scattering (X-ray or electron) and radial distribution function analysis have apparently justified widely different models, albeit from similar data (for example 1,2). We present a method which enables detailed and consistent conclusions to be made through systematic comparison of the experimental scattering with the scattering calculated for a wide range of models. This method may be applied to polymeric glasses, melts and rubber and we also show in principle how it may be applied to the non-crystalline component of semi-crystalline polymers. The determination of the structure of the crystalline component of semi-crystalline polymers is simplified if fibre diffraction patterns may be obtained, enabling the chain repeat distance to be calculated from the layer line spacing. For a polymer glass, however, the degree of orientation that may be introduced is limited : Figure 1 shows the scattering from extruded polycarbonate (extension ratio, X = 3), and although there is an anisotropic distribution of intensity there are no fibre type features. Attempts to improve the apparent alignment by azimuthal deconvolution techniques have not been wholly successful (3,4) . For a semi-crystalline polymer, in general, the degree of orientation that may be induced into the amorphous component in a fibre is very low (Figure 2), and for a polymer melt the introduction of any orientation at all for the duration of an X-ray experiment would require particular ingenuity! These problems indicate that any general method for evaluating the structure must be based on the isotropic scattering from unoriented samples. Method The information from the oriented glasses is not redundant since it suggests in broad terms the different origins of the diffuse peaks in the scattering curve. Those which are weighted 0-8412-0589-2/80/47-141-215$05.00/0 © 1980 American Chemical Society French and Gardner; Fiber Diffraction Methods ACS Symposium Series; American Chemical Society: Washington, DC, 1980.


216

FIBER DIFFRACTION

Figure 1.

METHODS

Contour map of the uncorrected scattering (CuKa radiation) from extruded polycarbonate (\ = 3)

Figure 2.

Fiber pattern for isotactic polystyrene

French and Gardner; Fiber Diffraction Methods ACS Symposium Series; American Chemical Society: Washington, DC, 1980.


12.

MITCHELL

ET AL.

Amorphous Phase in Synthetic Polymers

217

towards the equator must a r i s e from the i n t e r f e r e n c e between the a l i g n i n g segments and those weighted towards the meridian r e s u l t from c o r r e l a t i o n s along such segments. In general these segments w i l l be p o r t i o n s o f the chains rather than sidegroups and so we may d i s t i n g u i s h between i n t e r c h a i n d i s t a n c e s and i n t r a c h a i n distances. Of course i f the side groups form a s i g n i f i c a n t p r o p o r t i o n o f the chain (for example polystyrene) then the i n t e r p r e t a t i o n may not be so s t r a i g h t f o r w a r d . Similar information may be d e r i v e d from e v a l u a t i n g peak s h i f t s with temperature (3/5) and by comparing, f o r c r y s t a l l i s a b l e polymers, the p o s i t i o n s o f the c r y s t a l l i n e and amorphous peaks i n the s c a t t e r i n g curve. The separation o f the s c a t t e r i n g curve i s f a c i l i t a t e d by the d i s t r i b u t i o n o f the peaks. T y p i c a l l y the i n t e r c h a i n peak i s a t low s c a t t e r i n g v e c t o r (s = 1 . 0 - 1 , where s = 4irsin0/X) while the i n t r a c h a i n components are a t higher s c a t t e r i n g v e c t o r s (>2.o8~ ). The d i v i s i o n o f the s c a t t e r i n g i n t h i s way enables the s t r u c t u r e determination t o proceed i n two stages: f i r s t , using the i n t r a c h a i n s c a t t e r i n g t o r e s o l v e the p e r s i s t e n t chain conformation, and then the i n t e r c h a i n s c a t t e r i n g t o explore the p o s s i b l e packing arrangements o f t h i s conformation. We have shown (6^,7_)that when the width o f the f i r s t i n t e r c h a i n peak i s c o r r e c t l y modelled using e i t h e r p a r a l l e l o r randomly o r i e n t e d ( i . e . non-aligned) chains, the p r e d i c t e d i n t e n s i t y o f the second and subsequent orders i s so low as t o be i n s i g n i f i c a n t . Hence the separation i n t o i n t e r c h a i n and i n t r a c h a i n e f f e c t s as o u t l i n e d appears j u s t i f i e d ; It i s however d i f f i c u l t t o achieve i n r e a l space since the F o u r i e r transformation o f s p h e r i c a l l y averaged data causes some superp o s i t i o n o f i n t e r c h a i n and i n t r a c h a i n i n f o r m a t i o n , and f o r o r i e n t e d data c y l i n d r i c a l d i s t r i b u t i o n functions are necessary which have l i m i t e d a p p l i c a b i l i t y t o the s c a t t e r i n g from p a r t i a l l y aligned materials. For these reasons modelling i n r e c i p r o c a l space i s much p r e f e r r e d . The ' i s o l a t e d i n t r a c h a i n s c a t t e r i n g i s used t o determine the p e r s i s t e n t l o c a l conformation by comparison between the s c a t t e r i n g from s i n g l e chains and the experimental data. The experimental data needs t o be c o r r e c t e d f o r a b s o r p t i o n , p o l a r i s a t i o n , m u l t i p l e and incoherent s c a t t e r i n g and i n a d d i t i o n the i n t e n s i t i e s normalised t o e l e c t r o n u n i t s , before q u a n t i t a t i v e i n t e n s i t y comparisons are p o s s i b l e . The s-weighted reduced i n t e n s i t y f u n c t i o n s i ( s ) i s a b e t t e r choice f o r such comparisons where: 1

i(s)

= k I(s) - E f j

2

(s)

and I(s) i s the f u l l y c o r r e c t e d data, k i s the n o r m a l i s a t i o n f a c t o r , and E f j (s) i s the independent s c a t t e r i n g from a composition u n i t . The s c a t t e r i n g may be c a l c u l a t e d from atomic coordinates using the Debye equation ( 8 ) : i(s)

= £ £f.f s i n s r . . / s r . . x D i D ID ID


218

FIBER

DIFFRACTION

METHODS

where and f . are the s c a t t e r i n g f a c t o r s f o r the i and j atoms, and r j i s the d i s t a n c e between them. The atomic coordinates a r e generated f o r a chain of p a r t i c u l a r chemical c o n f i g u r a t i o n by s p e c i f y i n g the bond l e n g t h s , bond angles and bond r o t a t i o n angles. For a p a r t i c u l a r polymer we may assume t h a t the bond lengths are e f f e c t i v e l y constant as a r e bond angles, which except f o r some polymers such as PMMA (9_,10) have near t e t r a h e d r a l v a l u e s . Thus the l o c a l conformation o f a chain w i l l be d e f i n e d by a s e t o f r o t a t i o n angles about the s k e l e t a l bonds (plus sidegroup r o t a t i o n s i f a p p l i c a b l e ) . The length o f c h a i n used depends upon the r e g u l a r i t y o f the sequence o f r o t a t i o n angles and i s o f course one o f the parameters t o be determined. To minimise the e n d - e f f e c t s a chain o f more than 6 s k e l e t a l bonds i s r e q u i r e d . I f there i s c o n s i d e r a b l e d i s o r d e r i n the chain then 500-1000 bonds may be r e q u i r e d t o f u l l y represent the d i s t r i b u t i o n o f p e r s i s t e n t conformations. To explore a l l o f the conformational p o s s i b i l i t i e s f o r such chains i s n o t o n l y time consuming b u t unnecessary since many o f the bond r o t a t i o n s may be excluded as they w i l l l e a d t o 'overlapping* atoms.


±

S t e r i c i n f o r m a t i o n i s most e a s i l y summarised i n a conforma t i o n a l energy map. Such c a l c u l a t i o n s have been made by a number o f workers but they r e q u i r e semi-empirical p o t e n t i a l s or the e f f e c t i v e van der Waals r a d i i which are s u b j e c t t o e r r o r and d i s p u t e . We have c a r r i e d out c a l c u l a t i o n s o f t h i s s o r t f o r polyethylene u n i t s using d i f f e r e n t p o t e n t i a l s and f i n d t h a t although the r e l a t i v e energies o f the minima vary c o n s i d e r a b l y , t h e i r l o c a t i o n s a r e not p a r t i c u l a r l y s e n s i t i v e t o the d i f f e r e n t p o t e n t i a l s . Thus the conformations f o r which the s c a t t e r i n g w i l l be c a l c u l a t e d may be r e s t r i c t e d t o those o f low energy without loss of generality. By comparing a l l arrangements o f these r o t a t i o n angles i n v a r y i n g lengths of chain the most l i k e l y p e r s i s t e n t conformation may be determined. We have s u c c e s s f u l l y a p p l i e d t h i s technique to s e v e r a l polymers (6/2/5/12^ ^ i l l u s t r a t e i t here with the s p e c i f i c example o f molten p o l y e t h y l e n e .

'

u

t w

e

w

i

l

1

Polyethylene melt a t 140°C The p u b l i s h e d conformational energy c a l c u l a t i o n s f o r nalkanes (for example see 11,12) have minima a t approximately the f o l l o w i n g p a i r s o f r o t a t i o n angles: 0°,0° trans,trans (tt) 0°,120°

trans,gauche*

Ctg+)

120°, 120°C

gauche+,gauche+

Cg+g+)

120

gauche+,gauche-

(g+g-)

,-120



12.

MITCHELL ET AL.


219

O t h e r p e r m u t a t i o n s may b e g e n e r a t e d b y c h a n g i n g s i g n s . T h e p a i r s are given i n order o f i n c r e a s i n g energy, f o r although the c a l c u l a t i o n s f o r n - a l k a n e s do n o t a g r e e o n t h e e n e r g y d i f f e r e n c e s b e t w e e n s t a t e s t h e y do a g r e e o n t h e o r d e r . The Cgauche+, gauche-) minimum i s s p l i t i n t o two m i n i m a b u t t h i s e f f e c t h a s u s u a l l y b e e n i g n o r e d (11} a s t h e e n e r g y o f t h i s c o n f o r m a t i o n above ( t r a n s , t r a n s ) i s c o n s i d e r a b l e , and s o t h e o c c u r r e n c e o f ( g a u c h e * , gauche-) p a i r s i s l i k e l y t o b e s l i g h t . F i g u r e 3 shows t h e s i ( s ) c u r v e s f o r d i f f e r e n t l e n g t h s o f r e g u l a r s e q u e n c e s o f trie l o w e r e n e r g y p a i r s o f c o n f o r m a t i o n s , compared t o t h e e x p e r i m e n t a l s i ( s ) curve. The a g r e e m e n t b e t w e e n t h e c a l c u l a t e d a n d e x p e r i m e n t a l c u r v e s i s p o o r f o r a n y o f t h e c o n f o r m a t i o n s , however, t h e p e a k s f o r the ( t r a n s , t r a n s ) conformation are a t the c o r r e c t s c a t t e r i n g v e c t o r s although too sharp. These c u r v e s i n d i c a t e t h a t t h e m e l t d o e s n o t c o n t a i n any s i g n i f i c a n t amount o f l o n g r u n s i n s i m p l e regular conformation. The n e x t s t a g e i n t h e p r o p o s e d method i s t o t a k e t h e most p r o m i s i n g r e g u l a r c o n f o r m a t i o n and i n t r o d u c e d e f e c t s i n t h e f o r m of other conformations. This i s achieved by ' b u i l d i n g the chain a c c o r d i n g t o a s e t o f u n c o n d i t i o n a l p r o b a b i l i t i e s . The p r o b a b i l i t y o f the next r o t a t i o n being t r a n s i s p , the p r o b a b i l i t y o f i t b e i n g g a u c h e + ( o r gauche-) i s ( l - p ) / 2 . Figure 4 shows t h e s i ( s ) c u r v e s f o r c h a i n s c o n t a i n i n g 500 s k e l e t a l b o n d s b u i l t using different values of p . Reasonable agreement i s o b t a i n e d f o r p v a l u e s o f 0.5 - 0.6. The s e t o f p r o b a b i l i t i e s may b e v a r i e d i n s e v e r a l w a y s , f o r e x a m p l e b y t h e e x c l u s i o n o f (gauche+,gauche-) p a i r s o r b y a d j u s t i n g t h e p r o b a b i l i t i e s dependent on t h e s t a t e o f t h e p r e c e d i n g bond. T h i s i n t r o d u c e s c o n d i t i o n a l i t y t o t h e p r o b a b i l i t i e s , and t h e s e and o t h e r i m p r o v e m e n t s s u c h a s f l u c t u a t i o n s i n a n g l e s a r e c o n s i d e r e d i n more d e t a i l e l s e w h e r e ( 1 ) . The e n e r g y l e v e l s i n t h e s e m i - e m p i r i c a l c o n f o r m a t i o n a l e n e r g y c a l c u l a t i o n s may b e u s e d t o d e r i v e c o n d i t i o n a l p r o b a b i l i t i e s u s i n g t h e s t a t i s t i c a l w e i g h t s method d e s c r i b e d b y F l o r y ( 1 3 ) . T h i s a p p r o a c h t a k e s i n t o a c c o u n t t h e n e i g h b o u r i n g bonds and t h e i r e f f e c t on the p o p u l a t i o n s of the r o t a t i o n s t a t e s . The s c a t t e r i n g f r o m PE c h a i n s b u i l t a c c o r d i n g t o c o n d i t i o n a l p r o b a b i l i t i e s o b t a i n e d from t h e s t a t i s t i c a l w e i g h t s p r o p o s e d b y Abe, J e r n i g a n and F l o r y (11) i s i n good a g r e e m e n t w i t h t h e e x p e r i m e n t a l d a t a (6). The e n e r g y c a l c u l a t i o n s a l s o i n d i c a t e t h a t t h e e x a c t v a l u e s o f t h e r o t a t i o n a n g l e s a r e dependent on n e i g h b o u r i n g r o t a t i o n s t a t e s , and t h i s r e f i n e m e n t h a s b e e n i n c o r p o r a t e d i n t h e m o d e l chain. Such s o p h i s t i c a t i o n i n t h e c h a i n b u i l d i n g does i n d e e d improve t h e agreement between t h e c a l c u l a t e d s i ( s ) and t h e e x p e r i m e n t a l c u r v e ( F i g u r e 5a,b).Thus o u r a p p r o a c h i s a b l e t o g i v e a good i n d e p e n d e n t i n d i c a t i o n o f t h e b a s i c s t a t i s t i c a l p a r a m e t e r s f o r a p o l y e t h y l e n e c h a i n , and i n a d d i t i o n i t c a n p r o v i d e some d i r e c t experimental j u s t i f i c a t i o n f o r the d e t a i l e d s t a t i s t i c a l p r o p o s a l s ( s u c h a s t h o s e i n 11) o b t a i n e d f r o m m o d e l l i n g t h e molecular t r a j e c t o r y . 1

t

t

t

t


220

FIBER

PE

DIFFRACTION

METHODS

—n=6 fx

V / ;

^

1

n

Mn

" ;

1

2

[tg]n

A expt(l40*C)

^

\ J W Downloaded by UNIV OF BATH on July 1, 2016 | http://pubs.acs.org Publication Date: November 17, 1980 | doi: 10.1021/bk-1980-0141.ch012

1

1

2

1

^ 1

1

4

1

6

1

1

8

1

1

1

10

S(A"*) Figure 3. The experimental s-weighted intensity function si(s) for molten polyethylene at 140°C compared with the calculated si(s) curves for single chains of simple regular conformations

Figure 4.

Calculated s-weighted intensity function si(s) for chains with rotation states randomly distributed, the probability of trans being p t


12.

MITCHELL

ET AL.


221

H a v i n g d e t e r m i n e d t h e t y p e o f p e r s i s t e n t c o n f o r m a t i o n we must consider l i k e l y packing arrangements t h a t a r e c o n s i s t e n t w i t h bulk d e n s i t y , a n d t h e p o s i t i o n , h e i g h t a n d w i d t h o f t h e i n t e r c h a i n peak. S i n c e t h e weighted-average sequence l e n g t h o f t r a n s , f o r t h e c h a i n s p r o v i d i n g t h e b e s t agreement w i t h e x p e r i m e n t a l d a t a , i s a b o u t 3, t h e t y p i c a l 'segment i n a c h a i n i s g l o b u l a r i n n a t u r e . A p o s s i b l e a p p r o a c h t h a t we h a v e p r e s e n t e d i n d e t a i l e l s e w h e r e i s t o c o n s i d e r t h e c e n t r o i d s o f t h e s e g l o b u l a r segments t o be d i s t r i b u t e d a s t h e c e n t r e s o f r a n d o m l y c l o s e - p a c k e d spheres. By a s s u m i n g n o o r i e n t a t i o n a l c o r r e l a t i o n b e t w e e n n e i g h b o u r i n g segments t h e s c a t t e r i n g i n t e n s i t y i s s i m p l y t h e p r o d u c t o f t h e s c a t t e r i n g o f a n a v e r a g e segment a n d t h e s t r u c t u r e f a c t o r f o r a n a s s e m b l a g e o f random s p h e r e s . The s i ( s ) c u r v e f o r s u c h a r e p r e s e n t a t i o n i s shown i n F i g u r e 5 c . T h i s model q u a n t i t a t i v e l y a c c o u n t s f o r a l l t h e f e a t u r e s i n t h e s i ( s ) c u r v e f o r m o l t e n p o l y e t h y l e n e , a n d s u p p o r t s t h e random c o i l concept.


1

S e m i - c r y s t a l l i n e polymers I n v e s t i g a t i o n s o f t h e complete s t r u c t u r e o f s e m i - c r y s t a l l i n e p o l y m e r s u s i n g X - r a y s o r n e u t r o n s have i n g e n e r a l b e e n r e s t r i c t e d t o s m a l l s c a t t e r i n g v e c t o r s (

The Structure of the Amorphous Phase in Synthetic Polymers

Recommend Documents