6 X-ray Structural Analysis of Crystalline Polymers RICHARD J. CELLA* and ROBERT E. HUGHES Department of Chemistry, Cornell University, Ithaca, Ν. Y. 14853
The d e t e r m i n a t i o n of the atomic s t r u c t u r e of c r y s talline polymers by wide a n g l e X - r a y diffraction tech n i q u e s p r e s e n t s f o r m i d a b l e problems w h i c h cannot c o n v e n i e n t l y be t r e a t e d by the c o n v e n t i o n a l methods of structure analysis. Consequently, existing methods need t o be m o d i f i e d and new approaches t o the p r o b l e m d e v e l o p e d t o e x t r a c t the maximum amount of s t r u c t u r a l i n f o r m a t i o n f r o m the a v a i l a b l e e x p e r i m e n t a l d a t a . The n a t u r e of the problems e n c o u n t e r e d will be d i s c u s s e d i n this p a p e r , t o g e t h e r w i t h a description of methods t h a t have been utilized t o overcome them. A subsequent p a p e r will d e a l w i t h the a p p l i c a t i o n s o f t h e s e methods to the d e t e r m i n a t i o n of the m o l e c u l a r s t r u c t u r e of several crystalline polyethers. Characteristically, c r y s t a l s composed of l o n g c h a i n m o l e c u l e s p r o v i d e much l e s s diffraction d a t a than do s i n g l e c r y s t a l s of low m o l e c u l a r weight s u b s t a n c e s , and o f t e n t h i s d a t a i s of v e r y p o o r quality. This i s a d i r e c t consequence of the limited l o n g range o r d e r e x h i b i t e d by crystalline polymers; small crystallite size, imperfect crystallite orientation, structural d e f e c t s , and the p r e s e n c e of a non-crystalline compo nent in the sample all c o n t r i b u t e t o the r a p i d a t t e n u a tion of the a v a i l a b l e d a t a . The first of these deficiencies, the l i m i t e d l o n g range t h r e e d i m e n s i o n a l o r d e r , i s the s i n g l e most i m p o r t a n t f a c t o r i n v o l v e d in c r e a t i n g the difficulties e n c o u n t e r e d i n the structural a n a l y s i s of macromolecular m a t e r i a l s . The g r e a t l e n g t h of the polymer c h a i n s and the concomitant possibilities f o r chain entangle ments prohibit the m o l e c u l e s from o r d e r i n g themselves *E. I . du Pont de Nemours and Company, ton, Delaware 19898 86
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over an extended region w i t h a h i g h degree o f r e g u l a r i t y . Furthermore, the i n d i v i d u a l molecules themselves are not p e r f e c t l y r e g u l a r , and often c o n t a i n i n t r i n s i c s t r u c t u r a l defects such a s s t e r i c inversions o f asym metric sites, c h a i n branches, and c r o s s l i n k s . Such d e f e c t s d on o t n e c e s s a r i l y p r e v e n t t h e c h a i n f r o m becoming i n c o r p o r a t e d into a c r y s t a l l i n e domain; often times a molecule c o n t a i n i n g a minor s t r u c t u r a l imper f e c t i o n w i l l manage t opack reasonably w e l l into the s t r u c t u r e , thereby e n l a r g i n g the c r y s t a l l i n e domain but at the same time d e s t r o y i n g the p e r f e c t r e g u l a r i t y o f the assemblage. Another type o fp a c k i n g defect occurs when a "multi-state s t r u c t u r e i s p o s s i b l e . This happens when each successive c h a i n may enter the c r y s t a l l i t e i n any of several s p a t i a l l y d i f f e r e n t conformations. Thus, i f a c h a i n may b eaccommodated i n t o the c r y s t a l l a t t i c e i n e i t h e r a n ' u p " o ra " d o w n " p o s i t i o n a t w o - s t a t e s t r u c ture w i l l r e s u l t ; i fa side group may assume any o f t h r e e d i f f e r e n t a n g u l a r p o s i t i o n s i nt h e s o l i d phase, then a three-state s t r u c t u r e i sp o s s i b l e . These defects may occur s p o r a d i c a l l y w i t h low p r o b a b i l i t y o r they may occur randomly each time the o p p o r t u n i t y p r e s ents i t s e l f . I nt h e l a t t e r c a s e i ti sl i k e l y t h a t a s t a t i s t i c a l d i s t r i b u t i o n o f several d i f f e r e n t models w i l l h a v e t o b e i n v o k e d i no r d e r t o o b t a i n s a t i s f a c t o r y agreement t othe experimental data. Each o fthe s t r u c t u r a l defects described w i l l tend to attenuate the i n t e n s i t y o f the d i f f r a c t e d beam a t higher s c a t t e r i n g angles, and a s i t u a t i o n i s soon r e a c h e d i n w h i c h t h e s c a t t e r e d b e a m i s o fl o w e r i n t e n s i t y than the background s c a t t e r i n g . This f a l l - o f f o f the d a t a a t h i g h e r s c a t t e r i n g a n g l e s i ss i m i l a r t o t h a t c a u s e d b yt h e r m a l m o t i o n a n d a t o m i c s c a t t e r i n g , but these phenomena also occur i n non-polymeric systems and can b ec o r r e c t e d f o r rather e a s i l y . The c y l i n d r i c a l l y symmetric d i s t r i b u t i o n o f c r y s t a l l i t e s about the f i b e r axis also leads t oa decrease i n the q u a n t i t y o fdata obtainable, although for a n e n t i r e l y d i f f e r e n t reason. The r o t a t i o n a l character o f the d i f f r a c t i o n p a t t e r n causes a l l l a t t i c e plans whose r e c i p r o c a l l a t t i c e points have the same ξ and ζ c o o r d i n a t e s t o d i f f r a c t i n t o t h e s a m e p o i n t i ns p a c e , i r r e s p e c t i v e o ft h e v a l u e o ft h e i r a n g u l a r r e c i p r o c a l l a t t i c e c o o r d i n a t e (l). T h i s l e a d s t o a no v e r l a p p i n g o f r e f l e c t i o n s which would otherwise b e d i s t i n g u i s h a b l e were i tnot f o r the c y l i n d r i c a l symmetry o f the p o l y meric f i b e r . Thus, i n many instances two or more independent r e f l e c t i o n s produce o n l y one observable d i f f r a c t i o n maximum. 1
1
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I s o t a c t i c poly(propylene oxide), PPO, provides a n u m e r i c a l example of the above c o n s i d e r a t i o n s . U s i n g copper r a d i a t i o n , this polymer should i n p r i n c i p l e g e n e r a t e a p p r o x i m a t e l y 400 u n i q u e r e f l e c t i o n s (regard l e s s of whether or not t h e i r i n t e n s i t i e s are large enough to be measurable above background scattering) o u t t o a r e s o l u t i o n o f Ο.77 A. In p r a c t i c e ^ an a c t u a l f i b e r o f P P O y i e l d e d d a t a o u t t o o n l y Ο.97 A, thereby r e d u c i n g t h e p o t e n t i a l n u m b e r o f d a t a t o I78 r e f l e c tions; for this purpose, each allowable r e f l e c t i o n con t r i b u t i n g to an o v e r l a p p i n g group of r e f l e c t i o n s is counted separately, and a r e f l e c t i o n is counted whether or not i t is of s u f f i c i e n t i n t e n s i t y to be detectable. When the r o t a t i o n a l symmetry of the f i b e r specimen is taken into c o n s i d e r a t i o n and o v e r l a p p i n g r e f l e c t i o n s a r e c o u n t e d a s o n e o b s e r v a t i o n o n l y 94 o b s e r v a t i o n s are to be expected. O f t h e s e , o n l y 60 a r e o f s u f f i c i e n t i n t e n s i t y to be experimentally measurable above the background s c a t t e r i n g . This example d r a m a t i c a l l y i l l u s t r a t e s t h a t a t b e s t l e s s t h a n 25$ of the t h e o r e t i c a l l y a l l o w a b l e data may be observed, and of these less than two-thirds a c t u a l l y are observed. Furthermore, i t should be emphasized that i s o t a c t i c PPO is a polymer w h i c h y i e l d s a r e l a t i v e l y large amount of data, and many examples could be quoted i n which an even smaller f r a c t i o n of the data is a c t u a l l y obtained. The poor q u a l i t y of the data obtained is a conse quence of the imperfect alignment of the c r y s t a l l i n e domains w i t h respect to the f i b e r axis of the sample. The c r y s t a l l i t e s are not p e r f e c t l y u n i a x l a l l y oriented, and this m i s o r i e n t a t i o n causes the r e f l e c t i o n s from polymeric substances to appear as elongated arcs rather than as sharp, w e l l defined spots. Measuring the total, or integrated, i n t e n s i t y of these arcs proves to be extremely d i f f i c u l t , and even the simpler task of determining the peak i n t e n s i t y is subject to large e r r o r (2). F u r t h e r m o r e , a l l c r y s t a l l i n e p o l y m e r s c o n t a i n a c e r t a i n amount of n o n - c r y s t a l l i n e m a t e r i a l which produces only diffuse s c a t t e r i n g and has the e f f e c t of i n c r e a s i n g the background l e v e l , thereby r e n d e r i n g even more d i f f i c u l t the process of measuring i n t e n s i t i e s . These two f a c t o r s t a k e n t o g e t h e r , therefore, are r e sponsible f o r the poor q u a l i t y of the data when com pared w i t h the accuracy achieved i n single c r y s t a l analyses. The c h a r a c t e r i s t i c features of polymer structures d e s c r i b e d above were d i s c u s s e d at some l e n g t h because they are almost wholly responsible f o r the d i f f i c u l t i e s encountered i n the s t r u c t u r a l analysis of these m a t e r i a l s . Many of the powerful methods existent f o r single
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t a l studies are inadequate when d e a l i n g w i t h such m i t e d amount o fdata, and the poor q u a l i t y o f the merely serves t o i n t e n s i f y the problem. Thus, niques such a s d i r e c t methods, P a t t e r s o n synthesis, somorphous replacement are not a p p l i c a b l e t o p o l y c r y s t a l s t r u c t u r e analyses, and r e s o r t must b e made r i a l a n d e r r o r m e t h o d s t os e l e c t a r e a s o n a b l e t i n g s t r u c t u r e w h i c h must subsequently b e r e f i n e d . The standard technique for r e f i n i n g a proposed s t r u c t u r e i sa l e a s t s q u a r e s a n a l y s i s i n w h i c h the s t r u c t u r a l parameters are s y s t e m a t i c a l l y v a r i e d s o a s to produce the best agreement between the c a l c u l a t e d d i f f r a c t i o n p a t t e r n and the experimental data. When a p p l i e d t o the determination o fpolymer s t r u c t u r e s , however, t h i s method s u f f e r s f r o m two d e f i c i e n c i e s . T h e f i r s t o ft h e s e i s t h a t a l e a s t s q u a r e s refinement w i l l c o n v e r g e t ot h e n e a r e s t m i n i m u m i nt h e m u l t i dimensional parameter space, and unless the proposed s t a r t i n g m o d e l i sq u i t e c l o s e t o t h e c o r r e c t s t r u c t u r e this w i l l not b e the g l o b a l minimum. Thus, the l e a s t squares approach w i l l often converge t oan i n c o r r e c t s t r u c t u r e . T h e s e c o n d s h o r t c o m i n g o ft h e l e a s t s q u a r e s method l i e s i nt h e p o o r d a t a t o p a r a m e t e r r a t i o g e n e r a l l y encountered i n polymer s t r u c t u r e s . T y p i c a l l y , the number o fparameters t ob er e f i n e d i n a polymeric s t r u c t u r e i so f t h e same o r d e r o f m a g n i t u d e a s t h a t f o u n d i ns m a l l m o l e c u l e s t r u c t u r e s , y e t t h e r e i s m u c h l e s s data w i t h w h i c h t owork. Consequently, i f one independently refines the atomic p o s i t i o n a l coordinates i n a d d i t i o n t o scale factors and thermal parameters the results w i l l b es t a t i s t i c a l l y meaningless. Thus, f o r these types o f s t r u c t u r e s i twould b eb e n e f i c i a l t o p o s s e s s c o m p u t a t i o n a l m e t h o d s w h i c h d on o t a l l o w a l l o f the atomic coordinates t ovary independently, but r a t h e r which couples them i n a meaningful way t o reduce the number o fparameters i n v o l v e d . The p r e c e d i n g d i s c u s s i o n has d e s c r i b e d the inadequacy o f standard c r y s t a l l o g r a p h i c methods when a p p l i e d to the determination o fpolymer s t r u c t u r e s . The l a c k of an a n a l y t i c a l method o fphase d e t e r m i n a t i o n makes t h e s o l u t i o n o ft h e s e s t r u c t u r e s a m a t t e r o ft r i a l and e r r o r ; t h e r e i sn oa l t e r n a t i v e b u t t o p o s t u l a t e various models and compare the t h e o r e t i c a l l y c a l c u l a t e d i n t e n s i t i e s t ot h o s e a c t u a l l y o b s e r v e d . I tw o u l d b e u s e f u l , therefore, t odevelop computational techniques whereby p h y s i c a l l y r e a l i s t i c models may b es y s t e m a t i c a l l y generated and evaluated i n a s t r a i g h t f o r w a r d , e f f i c i e n t manner. The problem d i v i d e s i t s e l f n a t u r a l l y into two d i s t i n c t parts: (1) t h e c a l c u l a t i o n o f g e o m e t r i c a l l y
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a n d e n e r g e t i c a l l y a l l o w a b l e s t r u c t u r e s , a n d (2) the refinement of these s t r u c t u r e s to produce the b e s t agreement to the a v a i l a b l e X - r a y d a t a . Geometric
C o n s i d e r a t i o n s
It is not p o s s i b l e to r e l y s o l e l y on geometric arguments to determine a polymer s t r u c t u r e ; u l t i m a t e l y , recourse must be made to v a r i a t i o n a l methods. Geomet r i c c o n s t r a i n t s , however, are extremely u s e f u l i n e l i m i n a t i n g from c o n s i d e r a t i o n v a s t numbers of p h y s i c a l l y unreasonable s t r u c t u r e s , thereby r e d u c i n g the number of t r i a l models w h i c h need to be i n v e s t i g a t e d . A survey of the many substances whose m o l e c u l a r s t r u c tures have been e s t a b l i s h e d i n d i c a t e s that c e r t a i n s t r u c t u r a l features, namely bond lengths and bond angles, assume values w h i c h are e s s e n t i a l l y constant i n any series of r e l a t e d compounds. On the other extreme, the i n t e r n a l r o t a t i o n angles about single bonds are free to assume a wide range of values, and i n most cases i t is not p o s s i b l e a p r i o r i to a s s i g n values to these parameters w i t h any degree of c e r t a i n t y (the r e l a t i v e i n v a r i a n c e of some m o l e c u l a r parameters i n r e l a t i o n to others is of course a d i r e c t consequence of the energetics i n v o l v e d i n e f f e c t i n g changes i n t h e i r v a lu es , but the problem is b e s t t r e a t e d from a p u r e l y geometric p o i n t of view). Thus, to a good approxima t i o n , i t is p o s s i b l e to assume values f o r the bond lengths and bond angles and to use these values i n the g e o m e t r i c a l d e t e r m i n a t i o n of the i n t e r n a l r o t a t i o n angles. Because of the s m a l l amount of d i f f r a c t i o n data a v a i l a b l e this p r a c t i c e is u n i v e r s a l l y followed i n polymer s t u d i e s , and i t is necessary i n order to reduce the a n a l y s i s to manageable p r o p o r t i o n s . If the data allows, i t w i l l be p o s s i b l e to v a r y these values s l i g h t l y at a l a t e r stage i n the s t r u c t u r a l d e t e r m i n a t i o n . Consider a l i n e a r polymer molecule whose s k e l e t a l a t o m s a r e n u m b e r e d 0 , 1, N, and represented by C , Gi, Cjp s u c c e s s i v e l y f r o m one end of the c h a i n to the other. Let r^ represent the v e c t o r from C i _ to C^ a n d l e t 0j_ r e p r e s e n t t h e s u p p l e m e n t o f t h e angle between r. and "?^ · The angle between the two planes determined by r ^ _ and r^, and r ^ and Γ^ , r e s p e c t i v e l y , is measured from the cis p o s i t i o n i n the d i r e c t i o n of clockwise r o t a t i o n of f a c i n g i n the d i r e c t i o n of r ^ . The c h a i n skeleton of a h e l i x w i t h two backbone atoms i n the r e p e a t i n g u n i t may be completely d e s c r i b e d by s p e c i f y i n g the two types of bond lengths, b = Q
1
+
1
1
+ 1
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= ] Ï 3 - | = . . . = \r \ = ... and b = | r | = |r | = ... = | r j _ | = . .., the two t y p e s o f b o n d a n g l e s 0 i = 0 3 = . . . = 0 j _ = ... and 0 2 = 0 4 = ... = 0 i + i ···* ^ t h e t w o t y p e s o f d i h e d r a l a n g l e s τ.χ = τ = · . . . = τ ι = ... a n d τ = τ = ... T j _ = .... A l t e r n a t i v e l y , the h e l i c a l conformation i s u n i q u e l y determined b y s p e c i f y ing the bond lengths, the bond angles, the r o t a t i o n a l angle about the h e l i x axis, θ (unit twist), and the t r a n s l a t i o n a l distance along the h e l i x axis per repeat u n i t , d ( u n i t t r a n s l a t i o n ) ; t h e r e i so n l y one set o f d i h e d r a l a n g l e s , τχ and τ , c o m p a t i b l e w i t h t h e s e six q u a n t i t i e s . S i m i l a r l y , the s k e l e t a l conformation o fa three a t o m h e l i x i s u n i q u e l y d e t e r m i n e d b ys p e c i f y i n g the f o l l o w i n g q u a n t i t i e s : b i= | r i | = . . |rj_| = 2
±
2
4
+ 1
=
a n
· · · 9
Τ" 1
= |r|
3
0 2 . . .
=
= ... =
3
= .··> =
T" j _
=
0 i + i · · · 9
= ^2.
=
... = i + i »··^ 3= ... = i + = · · · . this case, however, a knowledge o f the bond lengths, bond angles, θ a n d d i si n s u f f i c i e n t t o d e t e r m i n e a u n i q u e t r i a d [τι, τ , τ } which i s consistent w i t h the given geometric c o n s t r a i n t s . One o f the three d i h e d r a l a n g l e s ( u s u a l l y chosen t ob e τ ) must b e s p e c i f i e d a s an independent v a r i a b l e , and for a given value o f this a n g l e t h e v a l u e s o ft h e o t h e r t w o d i h e d r a l a n g l e s are unique. Various r e l a t i o n s h i p s have been p u b l i s h e d g i v i n g θ and d a s functions o f the bond lengths, bond angles and d i h e d r a l angles for helices c o n t a i n i n g a smany a s s i x b a c k b o n e a t o m s i nt h e r e p e a t i n g u n i t ( 3 - 8 ) . While useful f o r c a l c u l a t i n g the unit twist and u n i t repeat f o r a p a r t i c u l a r h e l i c a l model, these r e l a t i o n s h i p s are o f l i m i t e d u s e i ns t r u c t u r a l a n a l y s i s f o r θ a n d d a r e u s u a l l y k n o w n q u a n t i t i e s , w h i l e l i t t l e o rn o t h i n g i s known about the d i h e d r a l angles. Cognizant o f t h i s , Nagai and Kobayashi have d e r i v e d complex a n a l y t i c a l expressions e x p l i c i t l y g i v i n g the r o t a t i o n a l angles a s functions o f the other h e l i c a l parameters (9). I n the case o fa three atom h e l i x these formulas specify Τ χ a n d τ a s f u n c t i o n s o ft h e b o n d l e n g t h s , b o n d angles, θ, d and τ . A computer program, DIHED, was w r i t t e n w h i c h would solve the necessary matrix equations and c a l c u l a t e a l l allowable h e l i c a l conformations consistent w i t h the imposed stereochemical c o n s t r a i n t s . G i v e n bond lengths, b o n d a n g l e s , θ a n d d., t h e p r o g r a m s y s t e m a t i c a l l y varies τ over any p r e s e l e c t e d range and c a l c u l a t e s the c o r r e s p o n d i n g v a l u e s o f τχ a n d τ ; the coordinates o f the h e l i c a l b a c k b o n e a r e c o m p u t e d i nc y l i n d r i c a l and T
=
a
2
n
d
T
T
3
2
3
2
2
3
2
ΐ
η
92
POLYETHERS
C a r t e s i a n c o o r d i n a t e s . S e v e r a l examples o ft h e program output a r e shown g r a p h i c a l l y i n F i g u r e I . Each curve i s t h e l o c u s o f a l lp o s s i b l e d i h e d r a l a n g l e s , [ τ ι ,τ , τ } , f o r t h e h i g h l y symmetric three atom h e l i x w i t h b i = b = b = 1.54 Â , 0 i = 02 = 0 3 = 109.5° , a n d d = 3.60 Â ; t h e c u r v e s c o r r e s p o n d t o θ = ΐ 8 θ ° , 1 2 0 ° , a n d 90°. T h ev a l u e s o fτχ a n d τ a r e read a st h e i n t e r s e c tions o ft h e curve w i t h a v e r t i c a l l i n e drawn t h r o u g h a g i v e n v a l u e o f τ . T h u s , f o r t h e 4χ h e l i x , [ τ = ΐ4θ° , τ ι ( o r τ ) = 1 β ΐ ° , τ ( o r Τ ι ) = 126°} i s a s o l u t i o n o ft h e geometric problem. T h e r i g h t a n dl e f t hand extremes o ft h e curve ( i . e . , τ = 121° o r 1 6 2 ° ) r e p r e sent those conformations f o rw h i c h τχ = τ . F o r τ g r e a t e r t h a n l62° o r l e s s t h a n 121° n o s o l u t i o n s exist and these regions o ft h e space m a y b e excluded from f u r t h e r c o n s i d e r a t i o n . It i sapparent that such a n a n a l y s i s when used i n c o n j u n c t i o n w i t h c h e m i c a l l y reasonable values o f t h e b j / s a n d 0j_ s c a n p r o v e e x t r e m e l y u s e f u l . A l t h o u g h n o unique h e l i c a l conformation r e s u l t s , t h e number o f p o s s i b i l i t i e s t ob e i n v e s t i g a t e d i s d r a s t i c a l l y r e d u c e d . 2
3
2
3
3
2
2
3
3
2
3
2
T
E n e r g e t i c
Considerations
The above geometric arguments a r ebased o n n o n i n t e r a c t i n g p o i n t atoms; b y i n t r o d u c i n g t h e side c h a i n substituents a n dr e p l a c i n g t h ep o i n t atoms b y r e a l a t o m s i tb e c o m e s p o s s i b l e t o u s e e n e r g y considerations to f u r t h e r l i m i t t h e number o fp l a u s i b l e models(10-14). E n e r g y c a l c u l a t i o n s may b e c o n d u c t e d a tv a r i o u s levels of s o p h i s t i c a t i o n ranging from hard sphere atoms w i t h square w e l l p o t e n t i a l s t o" s o f t " atoms using a more r e a l i s t i c p o t e n t i a l f u n c t i o n . Furthermore, o n e m a y i n c l u d e t o r s i o n a l a n dv i b r a t i o n a l e n e r g i e s , p o l a r i n t e r a c t i o n s , h y d r o g e n b o n d i n g a n dv a r i o u s o t h e r terms i n t h e c a l c u l a t i o n . In this study, a n elegant energy m i n i m i z a t i o n com p u t e r program, PACK, w a s employed t oc a l c u l a t e t h e t o t a l p a c k i n g energies o ft h e s t r u c t u r e s under i n v e s t i g a t i o n . T h ep r o g r a m , b e l o n g i n g t oP r o f e s s o r H . Scheraga, c a l c u l a t e s t h e t o t a l energy o fa n assemblage of molecules b y t a k i n g i n t o account t h e f o l l o w i n g con t r i b u t i o n s t ot h e t o t a l e n e r g y o f t h e s y s t e m : fa) non-bonded i n t e r a c t i o n s ho) p o l a r i n t e r a c t i o n s fc) t o r s i o n a l b a r r i e r s id) hydrogen bonding (e) In
bond
a d d i t i o n ,
angle
bending
t h eprogram
and
bond
s t r e t c h i n g
m a y b e used
t og e n e r a t e
a n
CELLA
A N D HUGHES
Crystalline Polymers
94
POLYETHERS
e n e r g y c o n t o u r map a sa f u n c t i o n parameters. Although a comprehen this program i sbeyond the scope (a) a n d (b) l i s t e d a b o v e r e q u i r e T h e p r o g r a m u s e s a "6-12" p o the non-bonded atomic i n t e r a c t i o n p o t e n t i a l energy, U(r), i sgiven i n t e r n u c l e a r distance, r , by:
o fa n y t w o s t r u c t u r sive d i s c u s s i o n o f o fthis paper, item f u r t h e r e x p l a n a t i o n t e n t i a l t o c a l c u l a t s . The p a i r w i s e a sa f u n c t i o n o f th
a l s . e e
"I (m)
0
0
w [l (m)-ffilj(m)]
m
w [l (m)-|lj(m)]«
0
= 0 i f Zlj|(m)£l (m)
=
Observable Reflections
to $
0
c
c
o
0
c
0
c
0
= 0 i f |F (n)|*|F (n)|
Q
f*|F (n)|
= 0 i f |F (n)|i|F (n)|
=
i f |F (n)|>|F (n)|
o
Q
= |F (n)|
Denominator
Index
i f |F (n)|>|F (n)|
c
c
||F (n)|-|P (n)||
= ||F (n)|-f*|F (n)||
=
Numerator
Contribution to
to the Least Squares Residue and Residual
Contribution
Contributions
TABLE I
6.
CELLA
A N D HUGHES
Crystalline Polymers
99
c a l c u l a t e d i nt h e u s u a l w a y a n d w i l l m o s t l i k e l y b e s l i g h t l y l a r g e r . I tw a s f e l t , h o w e v e r , that the unob s e r v e d r e f l e c t i o n s a r e o fs u f f i c i e n t i m p o r t a n c e t o d e m a n d t h e i r i n c l u s i o n i n t o R-m w h e n t h e c a l c u l a t e d s t r u c t u r e factors are l a r g e r t h a n the threshold values. In other words, any s t r u c t u r a l model w h i c h p r e d i c t s i n t e n s i t y values above the threshold i n t e n s i t y f o r very m a n y o ft h e u n o b s e r v e d d a t a m u s t o fn e c e s s i t y b e i n c o r rect.. I ns i n g l e c r y s t a l w o r k , where t h o u s a n d s o f r e f l e c t i o n s are measurable, this c r i t e r i o n i s unneces sary, since any s t r u c t u r e which p r e d i c t s the observable data w i l l very l i k e l y also p r e d i c t the unobservable data. I na n a l y s i s i n v o l v i n g o n l y a l i m i t e d n u m b e r o f data, however, this may not b e the case and such miss ing data can then give p o s i t i v e s t r u c t u r a l i n f o r m a t i o n . C o n c l u s i o n This paper has described i n very general terms the u n i q u e p r o b l e m s e n c o u n t e r e d i nt h e d e t e r m i n a t i o n o f t h e molecular p a c k i n g i n polymeric c r y s t a l s . Standard c r y s t a l l o g r a p h i c procedures lose t h e i r u t i l i t y when d e a l i n g w i t h such problems, and a d d i t i o n a l information, such a s stereochemical c o n s t r a i n t s and p a c k i n g energies, s h o u l d b e b r o u g h t t o b e a r i nt h e p r o b l e m . F i n a l l y , a method o fs t r u c t u r a l r e f i n e m e n t w h i c h i s e s p e c i a l l y s u i t e d t op o l y m e r i c s t r u c t u r e s s h o u l d b eemployed i n p l a c e o fc l a s s i c a l l e a s t s q u a r e s m e t h o d s . A subsequent paper w i l l deal w i t h the a p p l i c a t i o n o fthese p r i n c i p l e s a n d t e c h n i q u e s t ot h e d e t e r m i n a t i o n o ft h e c r y s t a l structures o f s e v e r a l c r y s t a l l i n e p o l y e t h e r s . Acknowledgment The authors g r a t e f u l l y acknowledge the c o n t r i b u t i o n s o fP . W a r d a n d R . F l e t t e r i c k t o t h e development of the computer programs DIKED and POLYMIN. W e are a l s o p l e a s e d t oa c k n o w l e d g e f i n a n c i a l s u p p o r t through NIH Grant GM-148J2-02 and a d d i t i o n a l support through the M a t e r i a l s S c i e n c e C e n t e r a tC o r n e l l U n i v e r s i t y . Literature Cited 1. Buerger, M., "X-Ray Crystallography," John Wiley and Sons, Inc., New York, N.Y. (1962). 2. Cella, R. J., Lee, B., and Hughes, R. Ε., Acta Cryst., (1970), A26, (6). 3. Miyazawa, T., J. Polym. Sci., (1961), 55, 215. 4. Hughes, R. Ε. and Lauer, J. L., J. Chem. Phys., (1959), 30 (5), 1 1 6 5 . 5. McCullough, R., Polymer
Letters,
(1965),
3,
509.
100
6. Shimanouski, T. and Mizushima, S., J. Chem. Phys., (1955), 23 (4), 707. 7. Sugeta, H. and Miyazawa, T., Biopolymers, (1967), 5, 673. 8. Kijima, H., Sato, T., Tsuboi, M., and Wada, Α., Bull. Chem. Soc. Jap., (1967), 40, 2544. 9. Nagai, K. and Kobayashi, M., J. Chem. Phys., (1962), 36 (5), 1268. 10. Nemethy, G. and Scheraga, H., Biopolymers, (1965), 3, 155. 11. Scott, R. A. and Scheraga, H., J. Chem. Phys., (1966), 45, 2091. 12. Ooi, T., Scott, R., Vanderkooi, G., and Scheraga, H., J. Chem. Phys., (1967), 46, 4410. 13· DeSantis, P., Giglio, E., Liquori, Α., and Ripamonti, Α., J. Polym. Sci., (1963), 1383. 14. Liquori, Α., J. Polym. 12, 209. 15. Del Re, G., Theor. Chim. Acta, (1963), 1, 188. 16. Del Re, G., Rev. Mod.
POLYETHERS
Sci., Phys.,
(1966), (1965),
Part 34,
C, 604.
No.
