10 Diffraction from Nonperiodic Structures
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The Molecular Conformation of Polytetrafluoroethylene (Phase II) 1
E D W A R D S. C L A R K , J. J. WEEKS, and R. K. EBY Polymer Science and Standards Division, National Bureau of Standards, Washington, D.C. 20234
The determination of c r y s t a l structure i n synthetic polymers is often made difficult by the lack of resolution i n the diffraction data. The diffuseness of the reflections observed i n most x-ray fiber patterns results from the small size and imperfect l a t t i c e nature of the polymer c r y s t a l l i t e s . Resolution of individual reflections i s also made difficult from misorientation of the c r y s t a l l i t e s about the fiber a x i s . This lack of resolution leads to poor accuracy i n measurement of peak positions. In p a r t i c u l a r , this lack of accuracy makes determination of layer l i n e heights d i f f i c u l t with a corresponding loss of significant figures i n evaluation of the repeat distance for the molecular conformation. In the case of h e l i c a l conformations, the repeat distance may be of considerable length or, as we s h a l l show, indeterminate and, i n effect, nonperiodic. This evaluation requires high accuracy i n measurements of layer l i n e heights. In this paper we examine electron diffraction fiber patterns of the homopolymer polytetrafluoroethylene (-CF2-CF2-)n, PTFE, i n which the resolution is sufficient to y i e l d much more accurate values of layer l i n e heights than were available from the previous x-ray diffraction experiments (1) on the c r y s t a l structure of Phase I I , the phase below the 19°C transition (2). On the basis of x-ray data, the molecule was assigned the conformation 13/6 or thirteen CF2 motifs regularly spaced along s i x turns of the h e l i x . This i s equivalent to a 13 screw a x i s . The r e l a tionship between the molecular conformation and the h e l i c a l symmetry has been studied by Clark and Muus (3) and i s i l l u s trated i n Figure 1. The electron diffraction data of high resol u t i o n enabled us to determine i f this unusual 13-fold symmetry was exact or an approximation of the true symmetry. We have also 2
'Polymer Engineering, The University of Tennessee, Knoxville, TN 37916.
0-8412-0589-2/80/47-141-183$05.00/0 © 1980 American Chemical Society French and Gardner; Fiber Diffraction Methods ACS Symposium Series; American Chemical Society: Washington, DC, 1980.
184
FIBER DIFFRACTION
developed mathematical expressions to s i m p l i f y these data.
METHODS
interpretation
of
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Theoretical A convenient method f o r d e f i n i n g h e l i c a l symmetry and c a l c u l a t i n g the d i s t r i b u t i o n of i n t e n s i t y i n a f i b e r p a t t e r n was devised by Cochran, C r i c k and Vand (CCV) (4). As i n d i c a t e d i n F i g u r e 1, the molecular conformation i s treated as a r e g u l a r s e r i e s of d i f f r a c t i o n u n i t s uniformly spaced along a h e l i x of p i t c h , P, with a x i a l s e p a r a t i o n , s. In PTFE, one h e l i x d e f i n e s the carbon p o s i t i o n s ; two h e l i c e s d e f i n e f l u o r i n e p o s i t i o n s . If there i s a meaningful t r a n s l a t i o n a l i d e n t i t y , c, i t follows that P/s w i l l be the r a t i o of small whole numbers: [1]
r * = u * / t * = P*/s*
f o r a commensurable h e l i x having t turns i n u motif u n i t s . The a s t e r i s k i n d i c a t e s a commensurable h e l i x . The i d e n t i t y p e r i o d i s : [2]
c = u*s* = t*P*.
As shown by CCV, the f i b e r p a t t e r n can be i n t e r p r e t e d i n terms of B e s s e l f u n c t i o n s governing the i n t e n s i t y d i s t r i b u t i o n i n each l a y e r l i n e according to the s e l e c t i o n r u l e : [3]
I = t*n + u*m
where I i s the l a y e r l i n e number, n i s the order of B e s s e l f u n c t i o n and m i s any i n t e g e r (+, -, or 0 ) . With t r a n s l a t i o n a l i d e n t i t y , the l a y e r l i n e s w i l l be confined to h e i g h t s : r
« A . i c u*s*
[4] L
with uniform spacings between the l a y e r l i n e s i n space:
J
reciprocal
The s t r u c t u r e f a c t o r governing the d i s t r i b u t i o n of i n t e n s i t y i n a l a y e r l i n e f o r a commensurable h e l i x may be approximated c l o s e l y 1 the c y l i n d r i c a l l y symmetrical transform of a h e l i c a l molecule with atoms at c y l i n d r i c a l c o o r d i n a t e s , ( r j , $ j , Zj) f o r the asymmetric u n i t repeating along the h e l i x a x i s i n accordance with equation [1]. The c y l i n d r i c a l l y averaged i n t e n s i t y f u n c t i o n has been given i n convenient from by Davies and R i c h (5): 2
F (R,*/c) = Y
L n
(A* + n
B^) n
French and Gardner; Fiber Diffraction Methods ACS Symposium Series; American Chemical Society: Washington, DC, 1980.
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CLARK ET AL.
Diffraction from Nonperiodic Structures
Figure 1. Conformation of PTFE molecule showing 13/6 helical symmetry
m=
ι i * 11
' ι
t
H} ι :
1 n= -ίο
»! Î I ! -5
0
5
t 1 10
Figure 2. Selection rule for 13/6 commensurable helix
French and Gardner; Fiber Diffraction Methods ACS Symposium Series; American Chemical Society: Washington, DC, 1980.
186
FIBER DIFFRACTION
METHODS
where
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[6]
I f t h e r e i s no t r a n s l a t i o n a l i d e n t i t y , t h e r a t i o P/s w i l l be i n c o m m e n s u r a b l e and, i n t h e o r y , t h e d i f f r a c t i o n p a t t e r n w i l l f i l l t h e whole of r e c i p r o c a l space. Layer l i n e s w i l l o c c u r a t h e i g h t s of: 4"f
+
S
[7]
w h e r e P/s i s an i n c o m m e n s u r a b l e f r a c t i o n . However, as e x p l a i n e d by CCV, t h e t r a n s f o r m , i n e f f e c t , w i l l be c o n f i n e d t o t h o s e l a y e r l i n e s o f n o n - u n i f o r m s p a c i n g c l o s e t o t h e v a l u e s f o r t h e commensurable h e l i x which approximates the a c t u a l conformation. This i s b e c a u s e o n l y B e s s e l f u n c t i o n s o f l o w o r d e r , s a y n
