A Molecular View of Bulk Deformation - American Chemical Society

Chapter 11 A

Molecular

View

of B u l k

Deformation

Robert C. Cook

Downloaded by RUTGERS UNIV on January 12, 2018 | http://pubs.acs.org Publication Date: August 29, 1989 | doi: 10.1021/bk-1989-0404.ch011

Lawrence Livermore National Laboratory, University of California, P.O. Box 5508 (L-482), Livermore, CA 94550

The l o c a l conformation and storage of energy in i n d i v i d u a l polymer chains during a deformation of a bulk polymer sample are examined by the computer simulation of a r e l a t i v e l y simple model. It i s shown that as the i n t e r a c t i o n between the chain atoms and surrounding medium increases, r o t a t i o n a l angle motion i s suppressed during the deformation, and large amounts of energy are stored i n backbone bond angle and bond length distortions. The r e l a t i o n s h i p of t h i s phenomena to T and the i m p l i c a t i o n s for chain r e l a x a t i o n are discussed. g

In r e c e n t y e a r s o u r u n d e r s t a n d i n g o f p o l y m e r c h a i n c o n f o r m a t i o n a l t r a n s i t i o n s f o r polymers i n d i l u t e s o l u t i o n has i n c r e a s e d m a r k e d l y t h r o u g h a c o m b i n a t i o n o f t h e o r e t i c a l and computer s i m u l a t i o n s t u d i e s . (JLz£.) I t i s a f a c t , however, t h a t modern t e c h n o l o g y i s more i n t e r e s t e d i n b u l k p o l y m e r s t h a n d i l u t e s o l u t i o n systems, and a l t h o u g h i t has l o n g been r e a l i z e d t h a t c o n f o r m a t i o n a l t r a n s i t i o n s and c h a i n m o t i o n i n g e n e r a l a r e key t o u n d e r s t a n d i n g many p r o p e r t i e s o f b u l k polymer systems (7.) , p r o g r e s s i n u n d e r s t a n d i n g t h e s e m o t i o n s i n b u l k systems (jjLi 12.) has l a g g e d f a r b e h i n d d i l u t e s o l u t i o n s y s t e m s . The r e a s o n s need h a r d l y be expanded upon, we o n l y n o t e t h a t a fundamental d i f f e r e n c e i s t h a t i n d i l u t e s o l u t i o n t h e Brownian a c t i o n o f t h e s o l v e n t promotes and m e d i a t e s c o n f o r m a t i o n a l motion, w h i l e i n t h e b u l k s t a t e , p a r t i c u l a r l y below T , t h e s u r r o u n d i n g c h a i n s i n h i b i t o r prevent s i g n i f i c a n t c o n f o r m a t i o n a l motion. For glassy b u l k p o l y m e r s , s i g n i f i c a n t c o n f o r m a t i o n a l m o t i o n can be e x p e c t e d o n l y when t h e sample i s s u b j e c t e d t o d e f o r m a t i o n , and i n f a c t t h e ease o f e x e c u t i n g t h e s e d e f o r m a t i o n s i s i n t i m a t e l y r e l a t e d t o t h e freedom i n d i v i d u a l c h a i n s have i n e x e c u t i n g c o n f o r m a t i o n a l motions. g

0097-6156/89/0404-0107$06.00/0 © 1989 American Chemical Society

Provder; Computer Applications in Applied Polymer Science II ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

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COMPUTER APPLICATIONS IN APPLIED POLYMER SCIENCE II

To f u r t h e r c l a r i f y t h i s p o i n t and t o s e t t h e s t a g e f o r t h e c a l c u l a t i o n s d e s c r i b e d i n t h i s paper, c o n s i d e r a p o r t i o n o f a s i n g l e c h a i n imbedded i n a b u l k p o l y m e r i c s o l i d w h i c h i s e l o n g a t i o n a l l y deformed by a f a c t o r o f two. The c h a i n b e g i n s i n some c o n f o r m a t i o n a p p r o p r i a t e f o r i t s end-to-end length, but a f t e r the deformation the end-toend l e n g t h has d o u b l e d , and t h e o r i g i n a l c o n f o r m a t i o n i s no l o n g e r a p p r o p r i a t e . I f the c o n s t r a i n i n g f o r c e s a r e not t o o g r e a t , a s i t u a t i o n one might e n c o u n t e r i n a m a t e r i a l s i g n i f i c a n t l y above T t h e c h a i n w i l l r e l a x by c h a n g i n g c o n f o r m a t i o n t o a sequence o f r o t a t i o n a l a n g l e s c o n s i s t e n t w i t h i t s new e n d - t o - e n d l e n g t h . These k i n d s o f m o t i o n r e q u i r e t h e t r a n s l a t i o n o f c h a i n segments and t h u s s i g n i f i c a n t l o c a l f r e e volume and m o b i l i t y o f t h e neighboring chains. I f we a r e s u f f i c i e n t l y below T however, t h e s i t u a t i o n w i l l be m a r k e d l y d i f f e r e n t . During the d e f o r m a t i o n t h e c h a i n c o n f o r m a t i o n a l m o t i o n w i l l be p a r t i a l l y s u p p r e s s e d by n e i g h b o r i n g c h a i n s , and t h o s e m o t i o n s n e c e s s a r y t o r e l a x t h e c h a i n w i l l be b l o c k e d . S i n c e t h e c h a i n i s s t i l l b e i n g e l o n g a t e d , a good d e a l o f t h e d e f o r m a t i o n e n e r g y may be t a k e n up i n t h e backbone bond a n g l e s and bond l e n g t h s . In t h e work t h a t f o l l o w s , we e x p l o r e t h i s a r e a o f e n e r g y s t o r a g e d u r i n g deformation by m o d e l i n g a p o l y e t h y l e n e - l i k e c h a i n imbedded i n a b u l k medium w h i c h i s m e c h a n i c a l l y s t r a i n e d .


gr

gr

Model Our model i s b a s e d on a p o l y e t h y l e n e - l i k e c h a i n s i m i l a r t o t h a t u s e d by H e l f a n d , e t . a l . Q ) The p o t e n t i a l f u n c t i o n can be e x p r e s s e d as X>T = \>b + l>0 + ity + where t h e t o t a l energy, i s e x p r e s s e d as a sum o f c o n t r i b u t i o n s from t h e bond l e n g t h s , 0)^, backbone bond a n g l e s , VQ, bond r o t a t i o n a l a n g l e s , tty, and t h e c o n s t r a i n t s o f n e i g h b o r i n g chains, V . No i n t r a c h a i n e x c l u d e d volume terms a r e i n c l u d e d s i n c e t h e segments o f c h a i n we w i l l be e x a m i n i n g are r e l a t i v e l y short. We n o t e t h a t u n l i k e t h e H e l f a n d s t u d y we have u s e d r e a l i s t i c v a l u e s f o r t h e bond l e n g t h and bond a n g l e d e f o r m a t i o n f o r c e c o n s t a n t s . (13.) The m o d e l i n g o f t h e c o n s t r a i n i n g p o t e n t i a l due t o n e i g h b o r i n g c h a i n s was a c c o m p l i s h e d as f o l l o w s . To s i m p l i f y m a t t e r s we w i l l c o n s i d e r o n l y t h e c a s e where a s e c t i o n o f c h a i n i s s i t u a t e d s u c h t h a t t h e l i n e between t h e ends o f t h i s s e c t i o n o f c h a i n i s a l i g n e d w i t h t h e d i r e c t i o n o f an e l o n g a t i o n a l d e f o r m a t i o n . As t h e medium i s e l o n g a t e d t h e n e i g h b o r i n g c o n s t r a i n i n g medium i s a l s o elongated. Thus c h a i n m o t i o n i n t h e d i r e c t i o n o f t h e d e f o r m a t i o n i s l i k e l y t o be l e s s c o n s t r a i n e d t h a n m o t i o n in a d i r e c t i o n perpendicular t o the elongation d i r e c t i o n . In o r d e r t o model t h i s s i t u a t i o n , we a t t a c h t o e a c h atom i n t h e undeformed c h a i n segment Hookean c o n s t r a i n i n g s p r i n g s , which a c t o n l y i n d i r e c t i o n s normal t o t h e deformation d i r e c t i o n . Thus i f we t a k e t h e d e f o r m a t i o n t o C


11.

109

Molecular View of Bulk Deformation

cook

be a l o n g t h e z - d i r e c t i o n , e x p r e s s e d as

the p o t e n t i a l

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CR

be

n

n

v

V

= 2 ^yM-x ) ii0

2

+ 2 ^y (yryi,o)

d)

2

c

where 7 i s t h e Hookean f o r c e c o n s t a n t f o r d i s p l a c e m e n t i n t h e x o r y d i r e c t i o n s , and x± Q and y± o a r e t h e x and y c o o r d i n a t e s o f p a r t i c l e i i n t h e undeformed c h a i n . The magnitude o f y c l e a r l y c o n t r o l s the s t r e n g t h of the c o n s t r a i n i n g f o r c e s and must be r e l a t e d t o t h e i n t e r a c t i o n p o t e n t i a l o f e l e m e n t s on n e i g h b o r i n g c h a i n s , as w e l l as t h e l o c a l f r e e volume. T h i s w i l l c e r t a i n l y v a r y from atom t o atom, and t h u s we a r e m o d e l i n g an a v e r a g e e f f e c t . C

f

f


c

The d e f o r m a t i o n i s e x e c u t e d as f o l l o w s . The undeformed r e l a x e d c h a i n segment, w i t h bond l e n g t h s , bond a n g l e s , and t o r s i o n a l a n g l e s a t t h e i r minima, i s f i r s t a l i g n e d so t h a t t h e f i r s t atom i s a t t h e o r i g i n and t h e l a s t atom on t h e p o s i t i v e z - a x i s . The c h a i n i s t h e n s t r e t c h e d by an amount A , t y p i c a l l y t a k e n t o be 0.001 nm, by m u l t i p l y i n g t h e z - c o o r d i n a t e o f e a c h p a r t i c l e by ( l + A / z ) where z i s t h e z - c o o r d i n a t e o f t h e end p a r t i c l e i n the n - p a r t i c l e c h a i n . T h i s affine d e f o r m a t i o n method p l a c e s most o f t h e i n i t i a l s t r a i n energy i n t h e bond lengths. We t h e n l e t t h e system r e l a x t o i t s l o w e s t energy conformation c o n s i s t e n t w i t h the c u r r e n t end-to-end separation. The c o n f o r m a t i o n a l and e n e r g e t i c v a r i a b l e s for t h i s " r e l a x e d " c o n f i g u r a t i o n are then recorded. The p r o c e s s i s r e p e a t e d u n t i l t h e c h a i n has been s t r a i n e d t h e d e s i r e d amount. We n o t e t h a t our method o f r e l a x a t i o n i s c o n s i s t e n t w i t h a z e r o t e m p e r a t u r e s i m u l a t i o n . Our g o a l a t t h i s p o i n t i s t o u n d e r s t a n d t h e b a s i c m e c h a n i c s o f t h e model and f o r e g o e x p l i c i t t h e r m a l e f f e c t s . F u r t h e r , we b e l i e v e t h i s t r e a t m e n t i s c o n s i s t e n t w i t h t h e mean f i e l d t r e a t m e n t of the s u r r o u n d i n g chains through y . In t h i s c o n t e x t we note t h a t the concept of thermal motion i s i m p l i c i t l y c o n t a i n e d i n t h e magnitude o f y , s i n c e i t c o n t r o l s t h e d e g r e e o f t r a n s l a t i o n a l freedom t h e atoms o f t h e c h a i n enjoy. A p r o p e r t r e a t m e n t o f t h e r m a l e f f e c t s would r e q u i r e an e x p l i c i t m o d e l i n g o f t h e c h a i n e n v i r o n m e n t . Though such s t u d i e s a r e c o m p u t a t i o n a l l y w i t h i n r e a c h , (JL1) our g o a l a t t h i s p o i n t i s t o f o c u s on s i m p l e r models. In t h e work r e p o r t e d h e r e , we have examined t h e c o n f o r m a t i o n a l geometry and e n e r g e t i c s a s s o c i a t e d w i t h the gauche-to-trans t r a n s i t i o n by e x a m i n i n g t h e e x t e n s i o n o f a f o u r - a t o m c h a i n o r i g i n a l l y i n a gauche c o n f i g u r a t i o n , as w e l l as s i x - and e i g h t - a t o m c h a i n s i n tgt and ttgtt configurations respectively. We have examined t h e s e systems w i t h c o n s t r a i n i n g f o r c e c o n s t a n t , y , t h a t v a r i e s from z e r o t o v a l u e s h i g h enough t o s u p p r e s s a l m o s t a l l n

n

c

c

c


110

COMPUTER APPLICATIONS IN APPLIED POLYMER SCIENCE II

t r a n s l a t i o n a l m o t i o n normal t o t h e c h a i n d i r e c t i o n , e f f e c t i v e l y e l i m i n a t i n g the t o r s i o n a l gauche-to-trans motion. Our p a r t i c u l a r i n t e r e s t i s i n e n e r g y s t o r a g e ; t h u s we have f o l l o w e d t h e e n e r g y s t o r e d i n t h e bond l e n g t h s , backbone bond a n g l e s , and t o r s i o n a l a n g l e s as a f u n c t i o n o f c h a i n e x t e n s i o n and t h e magnitude o f t h e constraining force.


Results L e t us f i r s t l o o k at a c h a i n w i t h o u t any e x t e r n a l c o n s t r a i n t s e x c e p t t h o s e on t h e f i r s t and l a s t atoms w h i c h c o n t r o l the degree of chain e x t e n s i o n . In our d i s c u s s i o n s we w i l l f o c u s on t h e f o u r atom c h a i n , i n i t i a l l y i n a gauche c o n f i g u r a t i o n . The r e s u l t s f o r t h e s i x - and e i g h t atom c h a i n s a r e q u a l i t a t i v e l y t h e same. As t h e c h a i n i s e x t e n d e d from t h e gauche c o n f i g u r a t i o n , we see a d e c r e a s e i n t h e t o r s i o n a l a n g l e from 120° t o t h e trans s t a t e a t 0°. The r e l a t i v e l y s t i f f backbone bond a n g l e s open s l i g h t l y d u r i n g t h e i n i t i a l s t a g e s o f t h e e x t e n s i o n and t h e n b e g i n t o c l o s e as t h e r o t a t i o n a l a n g l e p a s s e s o v e r i t s e n e r g e t i c maximum a t 60°. D u r i n g t h e e x t e n s i o n most o f t h e e n e r g y i s s t o r e d i n the s o f t e r r o t a t i o n a l angle with very l i t t l e i n t h e bond a n g l e s and even l e s s i n t h e bond l e n g t h s . The d i s t o r t i o n s i n t h e c h a i n geometry d u r i n g d e f o r m a t i o n c a u s e d by t h e c o n s t r a i n i n g f o r c e s a r e shown i n F i g u r e 1, where t h e v a l u e s o f a backbone bond a n g l e and t h e t o r s i o n a l a n g l e a r e shown as a f u n c t i o n c?i4, the d i s t a n c e between atoms one and f o u r , f o r s e v e r a l v a l u e s o f the c o n s t r a i n i n g f o r c e constant, y . I t i s not u n t i l y r e a c h e s a v a l u e o f about 10 k g / m o l - n s t h a t i t s e f f e c t b e g i n s t o become n o t i c e a b l e . At v a l u e s l o w e r t h a n t h i s , t h e c o n s t r a i n i n g f o r c e i s the s o f t e s t f o r c e and t h u s y i e l d s t o t h e t r a n s l a t i o n a l m o t i o n s o f the c h a i n atoms d u r i n g the deformation-induced c o n f o r m a t i o n a l transition. But as t h e c o n s t r a i n i n g f o r c e s become s t r o n g e r , one can see t h a t t h e r o t a t i o n a l a n g l e m o t i o n i s i n c r e a s i n g l y i n h i b i t e d , s i n c e t h i s m o t i o n i n v o l v e s the l a r g e s t t r a n s l a t i o n s f o r the atoms. To accommodate t h e imposed e n d - t o - e n d d i s t a n c e , t h e bond a n g l e s must open ( d e c r e a s e t h e v a l u e o f 0) . T h e r e i s a l s o an i n c r e a s e i n the bond l e n g t h s , a d i s t o r t i o n t h a t r e q u i r e s even l e s s t r a n s l a t i o n a l d i s p l a c e m e n t o f t h e atoms. In F i g u r e 2 we have p l o t t e d the energy s t o r e d i n the bond l e n g t h s , b o n d a n g l e s , and t o r s i o n a l a n g l e as a f u n c t i o n of c h a i n e x t e n s i o n f o r the three v a l u e s of the c o n s t r a i n i n g f o r c e shown i n F i g u r e 1. At a v a l u e o f y = 5 x 10 kg/mol-ns ( s o l i d l i n e s ) , we see t h e e n e r g y p r i m a r i l y s t o r e d i n t h e r o t a t i o n a l a n g l e as i t surmounts the gauche-trans b a r r i e r and t h e n s l i d e s down t h e p o t e n t i a l h i l l t o t h e trans c o n f i g u r a t i o n , a p a t t e r n q u a l i t a t i v e l y s i m i l a r t o the s i t u a t i o n f o r the unconstrained chain. Note t h a t v e r y l i t t l e e n e r g y i s d i s t r i b u t e d t o t h e s t i f f e r bond a n g l e s d u r i n g t h e i n i t i a l c

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Molecular View of Bulk Deformation

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A Molecular View of Bulk Deformation - American Chemical Society

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