Fiber Diffraction Methods - ACS Publications - American Chemical

9

Downloaded by UNIV MASSACHUSETTS AMHERST on September 29, 2014 | http://pubs.acs.org Publication Date: November 17, 1980 | doi: 10.1021/bk-1980-0141.ch009

Computational Methods for Profile Resolution and Crystallite Size Evaluation in Fibrous Polymers 1

A. M. H I N D E L E H , D. J. JOHNSON, and P. E. M O N T A G U E Textile Physics Laboratory, Department of Textile Industries, University of Leeds, Leeds, LS2 9JT U.K.

X-ray diffraction patterns from fibres generally contain a few closely overlapping peaks, each broadened by the contributions of crystallite size, crystallite-size distribution, and lattice distortion. In order to achieve complete characterisation of a fibre by X-ray methods, it is f i r s t necessary to separate the individual peaks, and then to separate the various profile-broadening contributions. Subsequently, we can obtain measures of crystallite size, lattice distortion and peak area crystallinity, to add to estimates of other characteristics obtained i n complementary experiments. Four major computational steps are necessary to separate the individual peaks and the different profile-broadening components: (i) correction and normalisation of the diffraction data, (ii) resolution of the total peak scattering from the so-called background scatter, and resolution of crystallographic, paracrystalline, and amorphous peaks from each other, (iii) correction of the resolved profiles for instrumental broadening, (iv) separation of the corrected profiles into size and distortion components. In this paper we w i l l discuss these steps i n turn, but most attention w i l l be paid to the hitherto largely neglected step of profile resolution. Experimental A modified Hilger and Watts Y115 diffractometer mounted on a Hilger and Watts Y90 constant output X-ray generator, u t i l i z i n g either CuKa or MoKa radiation was used throughout. A l l specimens were mounted i n special holders at the centre of the specimen table; bundles of parallel fibres were examined i n the symmetrical transmission mode, powder specimens were examined i n the symmetrical reflection mode. Counting was usually carried out at 15 steps per degree, occasionally at 30 steps per degree 'Current address:

Department of Physics, University of Jordan, P.O. Box 13093> Amman, Jordan.

0-8412-0589-2/80/47-141-149$08.50/0 © 1980 American Chemical Society In Fiber Diffraction Methods; French, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1980.

150

FIBER

DIFFRACTION

METHODS

(two t h e t a ) . The d i f f r a c t o m e t e r t r a c e s were r e c o r d e d o n p u n c h e d paper tape v i a a s c i n t i l l a t i o n counter, p u l s e - h e i g h t a n a l y s e r , and t i m e r - s c a l e r s y s t e m . F o r s e t t i n g up p u r p o s e s , o u t p u t c a n b e d i r e c t e d t o a chart recorder v i a a ratemeter. A l l c o m p u t a t i o n was c a r r i e d o u t o n t h e U n i v e r s i t y o f L e e d s C o m p u t i n g S e r v i c e ' s I C L 1906A c o m p u t e r . The p r o g r a m s w e r e w r i t t e n i n ALGOL 6 0 .


Computational

Methods

( i ) C o r r e c t i o n and N o r m a l i s a t i o n The s t a n d a r d method f o r n o r m a l i s a t i o n o f d i f f r a c t e d i n t e n s i t y d a t a i n t o e l e c t r o n u n i t s , i s t o compute b o t h t h e mean s q u a r e a t o m i c s c a t t e r i n g f a c t o r a n d t h e mean i n c o h e r e n t s c a t t e r f o r t h e p a r t i c u l a r m o l e c u l a r r e p e a t o v e r a r a n g e o f h i g h two t h e t a v a l u e s ( s a y 40°-60°) w h e r e t h e i r t o t a l v a l u e c a n b e c o n s i d e r e d t o b e e q u i v a l e n t t o t h e a c t u a l d i f f r a c t i o n from t h e m o l e c u l a r system c o n c e r n e d . An a p p r o p r i a t e n o r m a l i s a t i o n f a c t o r i s t h e n a p p l i e d to the experimental i n t e n s i t y data a f t e r geometrical c o r r e c t i o n and, f i n a l l y , i n c o h e r e n t s c a t t e r i s s u b t r a c t e d ( i j . T h i s method, a l t h o u g h w e l l e s t a b l i s h e d , i s n o t e n t i r e l y s a t i s f a c t o r y , since the experimental s c a t t e r i s d i f f i c u l t t o m e a s u r e a c c u r a t e l y o v e r a s m a l l r a n g e o f t h e two t h e t a s c a l e . We p r e f e r t o u s e a n o r m a l i s a t i o n procedure based on V a i n s h t e i n ' s l a w of conservation o f intensity. This law states that t o t a l scatter over i d e n t i c a l r e g i o n s o f r e c i p r o c a l space w i l l be equal d e s p i t e d i f f e r e n t d e g r e e s o f l a t t i c e o r d e r (2). I n t e r m s o f a r a n d o m l y o r i e n t e d s p e c i m e n , f o r example a powder o r f i n e l y d i v i d e d f i b r e , we u s e t h e r e l a t i o n / 2 4 7 r l l ( s ) s ds

=

4irj

Jo

/

2 2 f s ds

Jo

and f o r e q u a t o r i a l t r a c e s f r o m a f i b r e b u n d l e w i t h symmetry we u s e t h e r e l a t i o n

2wl

l ( s ) s ds

=

27r/ f

2

cylindrical

s ds

where

ZN. i s t h e mean s q u a r e a t o m i c s c a t t e r i n g f a c t o r f o r X - r a y s o f t h e m o l e c u l e c o n c e r n e d . N i i s t h e number o f m o l e c u l e s o f t y p e i i n the m o l e c u l a r r e p e a t , and s i s t h e r e c i p r o c a l space v e c t o r , s = s i n 6 / X , where X i s t h e w a v e l e n g t h o f t h e r a d i a t i o n , a n d I i s the corrected i n t e n s i t y .

In Fiber Diffraction Methods; French, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1980.

9.

HINDELEH

The a t o m i c expression

scattering factors

f (x) = A ±


151

Profile Resolution and Crystallite Size

ET AL.

±

a r e e v a l u a t e d from t h e

2 expC-a^x ) + B

2 expt-b.jX ) + C

±

±

where x = s i n Q / X a n d t h e c o n s t a n t s A i , B ^ , C i , a i , b i , a r e g i v e n byL e e a n d P a k e s (4.). I t i s w o r t h n o t i n g t h a t A i + + = Z±, t h e a t o m i c number o f t h e atom o f t y p e i . The mean i n c o h e r e n t s c a t t e r C i s g i v e n b y C = 2 N.C. 1

1

2 N. 1

where C i i s t h e i n c o h e r e n t s c a t t e r f o r atoms o f t y p e i a n d N i i s a g a i n t h e number o f atoms o f t y p e i i n t h e m o l e c u l a r r e p e a t . V a l u e s o f C were o b t a i n e d from t a b l e s and b e s t - f i t t h i r d degree p o l y n o m i a l s computed f o r s u b s e q u e n t u s e i n t h e n o r m a l i s a t i o n program. I n p r a c t i c e we e v a l u a t e t h e t o t a l s c a t t e r i n t e r m s o f e i t h e r t h e a r e a A R 1 ( 1 ) o r t h e a r e a A R 1 ( 2 ) , where

_

r z— 2$

AR1(1) =/

2

(f

2e

r 2r—

+ C) s d s =/

(f

2

+ C) s i n e c o s 0 d ( 2 O ) X

f o r a p a r a l l e l bundle

A R 1 ( 2 ) =f

2

%

o ffibres,

2

+ C) s d s =f

specimen type ( 1 ) , and

(f

2

2

+ C) s i n 0 c o s 0 d ( 2 0 )

f o r powders o r f i n e l y d i v i d e d f i b r e s , specimen t y p e ( 2 ) . The a r r a y o f e x p e r i m e n t a l l y o b s e r v e d i n t e n s i t i e s I-j(20) must f i r s t be corrected f o r p o l a r i z a t i o n t o give

I

2

( 2 0 ) = I., ( 2 9 )

and f o r t h e L o r e n t z g e o m e t r i c 1^ or

factor to give

(20) = I ( 2 0 ) 2

2

2/ (1 + c o s 2 0 )

s i n 20

1^ (20) = I ( 2 0 ) 2 s i n 0cos0 2

specimen type ( 1 ) specimen type ( 2 )

C e l l a , L e e a n d Hughes have s t u d i e d t h e L o r e n t z c o r r e c t i o n f o r f i b r e s i n some d e t a i l ; they consider that the c o r r e c t i o n 2 s i n ^ 0 c o s 0 i s most a p p r o p r i a t e f o r t h e e q u a t o r i a l s c a t t e r f r o m a p a r a l l e l bundle o f f i b r e s .


152

FIBER DIFFRACTION

a r e a A R 2 ( l ) o r AR2(2) i s t h e n e v a l u a t e d ,

The

-I

METHODS

where

2d

AR2(1)

0

°l

20


AR2(2)

= /

U

d(20)

specimen type ( 1 )

s i n 9cos6 d(29)

specimen type ( 2 )

I-. (20) sin0cos0

2

5

X

2

o

I , (29) 5

x

3

so t h a t t h e n o r m a l i s a t i o n f a c t o r i s t h e n

AR2(1) AR1(1)

or

AR2 AR1

and i s a p p l i e d t o t h e a r r a y o f i n t e n s i t i e s 1^(20) t o g i v e t h e _ i n t e n s i t i e s 14(20) i n e l e c t r o n u n i t s . The i n c o h e r e n t s c a t t e r C computed e a r l i e r i s t h e n s u b t r a c t e d t o g i v e t h e a r r a y 1^(20). The s t a n d a r d v e r s i o n o f t h i s p r o g r a m i s i d e n t i f i e d a s STEPSCAN; v a r i a t i o n s a r e a v a i l a b l e f o r u s e w i t h d a t a o b t a i n e d from X-ray o r e l e c t r o n d i f f r a c t i o n photographs v i a a Joyce-Loebl a u t o d e n s i d a t e r , and c o n t a i n a p p r o p r i a t e c o r r e c t i o n s f o r o p t i c a l d e n s i t y . The i n p u t d a t a a r e u s u a l l y a v e r a g e d i n g r o u p s o f 3 o r 5 steps, s i n g l e step data being used l e s s f r e q u e n t l y . A c o r r e c t i o n f o r a i r s c a t t e r may b e c a r r i e d o u t , a n d t h e t r u e b a c k ground r a d i a t i o n , a s measured s e p a r a t e l y , i s s u b t r a c t e d . Because the c o r r e c t i o n f o r a b s o r p t i o n w i t h i n the specimen i s n e g l i g i b l e o v e r t h e r a n g e o f two t h e t a n o r m a l l y c o v e r e d , t h i s h a s b e e n omitted. E x p e c t e d peak p o s i t i o n s are a l s o i n p u t ( f o r peakr e s o l u t i o n p u r p o s e s ) t o g e t h e r w i t h g r a p h p l o t t i n g p a r a m e t e r s . The o u t p u t i s i n t h e f o r m o f a r r a y s f , C, I i ( 2 0 ) t o 1 5 ( 2 0 ) , t h e a r e a c a l c u l a t i o n s , e s t i m a t e s o f peak h e i g h t and w i d t h , and a graph o f t h e t y p e shown i n F i g u r e 1. A p p r o p r i a t e d a t a a r e h e l d i n a f i l e f o r subsequent p r o f i l e r e s o l u t i o n . 2

Results. T y p i c a l normalised e q u a t o r i a l X-ray d i f f r a c t i o n t r a c e s f r o m t h e same p o l y ( e t h y l e n e t e r e p h t h a l a t e ) ( P E T ) s p e c i m e n axe shown i n F i g u r e s 1 a n d 2. The_mean s q u a r e a t o m i c s c a t t e r i n g f a c t o r f 2 , t h e i n c o h e r e n t s c a t t e r C, a n d t h e t o t a l s c a t t e r t , a x e a l s o i n c l u d e d . The c o r r e c t e d i n t e n s i t y d a t a a x e e v i d e n t l y d e p e n d e n t o n t h e method o f n o r m a l i s a t i o n e m p l o y e d . F i g u r e 1 was o b t a i n e d w i t h two t h e t a l i m i t s o f 5° a n d 40°, F i g u r e 2 h a s l i m i t s 30° a n d 40°; i t i s e v i d e n t i n t h i s c a s e t h a t f i t t i n g t o t h e c o h e r e n t s c a t t e r o v e r a h i g h two t h e t a r a n g e g i v e s a l m o s t d o u b l e t h e i n t e n s i t y compared w i t h f i t t i n g t o t h e t o t a l s c a t t e r w i t h i n the range covered. The e f f e c t o f t h e L o r e n t z c o r r e c t i o n i s i l l u s t r a t e d i n F i g u r e 3. Here the e q u a t o r i a l t r a c e o f a v i s c o s e rayon specimen i s shown u n c o r r e c t e d (LOR 0 ) , c o r r e c t e d i n t h e n o r m a l way f o r f i b r e s (LOR 1 ) , a n d f o r p o w d e r s (LOR 2 ) . T a b l e I shows t h e r e s u l t s o f p r o f i l e a n a l y s i s on the v i s c o s e r a y o n specimen u s i n g the d i f f e r e n t



9.

HINDELEH

ET

AL.


153

Figure 1. Corrected equatorial trace in electron units for a PET specimen normalized over the full two theta range (f mean-square atomic scattering factor; C incoherent scatter; t total scatter) 2

Figure 2. Corrected equatorial trace in electron units for the same PET specimen as Figure 1, normalized over the two theta range 30°^40° (J mean-square atomic scattering factor; U incoherent scatter; t total scatter) 2



154

FIBER

TWO

DIFFRACTION

METHODS

THETA

Figure 3. Equatorial trace for viscose rayon, normalized over full two theta range (uncorrected (LOR 0); corrected as normal forfibers(LOR 1); and corrected as normal for powders (LOR 2))


HINDELEH ET AL.

9.

155


TABLE I


Resolved parameters f o r an u n t r e a t e d viscose with different Lorentz corrections

Parameters

LOR 0 No c o r r e c t i o n

LOR 2

LOR 1 (sin20)""

rayon

1

(sin20sin9)""

f 101

0.2

0.6

0.7

A 101

14.5

8.0

4.4

2.3

2.1

2.1

12.0

12.1

12.1

-0.2

0.0

0.2

32.1

30.0

26.1

2.4

2.3

2.2

20.0

20.0

20.1

0.7

0.5

0.3

25.6

30.6

33.8

1.8

1.9

1.9

21.7

21.8

21.8

%k

48.6

47.1

46.1

%B

51.4

52.9

53.9

I

A

W

101

r

101

I

101

A

-

w

ioT

p

ioT

f

002

A

002

W

002

P 002

f - p r o f i l e f u n c t i o n parameter, A - peak h e i g h t ( e u ) , W - peak w i d t h (two t h e t a ) , P - peak p o s i t i o n (two t h e t a ) %k - p e r c e n t a g e a r e a u n d e r t h e p e a k s , %B - p e r c e n t a g e a r e a under t h e background.


1

156

FIBER DIFFRACTION

METHODS

Lorentz corrections. As c a n be seen, t h e r e a r e s i g n i f i c a n t changes i n t h e r e s o l v e d p a r a m e t e r s , p a r t i c u l a r l y t h e peak p o s i t i o n s and peak h e i g h t s ; t h e e f f e c t on peak w i d t h i s l e s s m a r k e d . The s t a n d a r d f i b r e c o r r e c t i o n (LOR 1) g i v e s t h e most r e a s o n a b l e r e s u l t s , t h e c o r r e c t i o n (LOR 2 ) , s u g g e s t e d f o r t h e e q u a t o r i a l t r a c e o f a f i b r e specimen, i s n o t c o n s i d e r e d t o be realistic.


(ii)

Resolution

Our s t a n d a r d r e s o l u t i o n p r o g r a m RESOLVE f i n d s t h e optimum f i t o f t s e p a r a t e peaks and a p o l y n o m i a l background t o t h e c o r r e c t e d a n d n o r m a l i s e d i n t e n s i t y t r a c e (6). E a c h p e a k p r o f i l e i s c o n s i d e r e d t o have t h e form f G t

t

(1 - f C )

+

t

t

where G, i s t h e G a u s s i a n f u n c t i o n

A

t

exp ) - I n 2 \

and Cj. i s t h e C a u c h y

2(X - P ) t

L

—w;

function

A

t

/ | l

+[2(X-P )/W ]'} t

t

The p e a k s a r e d e f i n e d b y t h e p a r a m e t e r s A-t t h e p e a k h e i g h t , W-fc t h e w i d t h o f t h e p e a k a t h a l f h e i g h t , a n d P-^ t h e p e a k p o s i t i o n ; f-t i s t h e p r o f i l e f u n c t i o n p a r a m e t e r , w h i c h c a n v a r y between - 0 . 5 and 1.0 f o r s e n s i b l e p r o f i l e s and e f f e c t i v e l y d e s c r i b e s t h e t a i l r e g i o n o f t h e p r o f i l e (j). The s c a t t e r f r o m d i s o r d e r e d m o l e c u l e s i n t h e f i b r e i s c o n s i d e r e d t o have t h e p o l y n o m i a l form a + bX + c X

2

+ dX

5

and c o r r e s p o n d s t o t h e s o - c a l l e d b a c k g r o u n d s c a t t e r . X may b e e i t h e r t h e two t h e t a o r t h e s s c a l e . The f u n c t i o n t o b e minimised i n terms o f these parameters i s S =

E

^

2

v

(1/ - If \.) (calc)i (norm;I 1

v

7

I n p u t i s i n t h e f o r m o f a r r a y s o f two t h e t a a n d I c , e x p e c t e d p e a k p o s i t i o n s , p e a k h e i g h t s a n d w i d t h s e s t i m a t e d b y STEPSCAN, and a p p r o p r i a t e b a c k g r o u n d p a r a m e t e r s . The l a t t e r may a l l s t a r t a t z e r o , a l t e r n a t i v e l y t h e y may b e f o u n d t o a f i r s t a p p r o x i m a t i o n b y p u t t i n g a s m a l l number o f p o i n t s ( f r o m a n assumed b a c k g r o u n d ) i n


9.

HINDELEH

ET AL.


157


a s u b s i d i a r y p r o g r a m BASELINE. O u t p u t i s i n t h e f o r m o f p a r a m e t e r l i s t s , t h e a r e a s and c o o r d i n a t e s o f t h e r e s o l v e d p e a k s , t h e t o t a l a r e a u n d e r t h e p e a k s , and t h e a r e a u n d e r t h e b a c k ground. O p t i m i z a t i o n . We h a v e r e c e n t l y t r i e d s e v e r a l methods o f o p t i m i z a t i o n , b u t f o r many y e a r s made s a t i s f a c t o r y u s e o f t h e m i n i m i z a t i o n p r o c e d u r e due t o P o w e l l ( 8 ) , w h i c h e m p l o y s t h e method o f c o n j u g a t e d i r e c t i o n s and e n s u r e s e f f i c i e n t c o n v e r g e n c e i n most c a s e s . A l t h o u g h n o b o u n d s c a n b e p u t o n t h e p a r a m e t e r s , i t i s p o s s i b l e t o c o n s t r a i n any o f t h e p a r a m e t e r s e t s d u r i n g a single optimization cycle by appropriate ordering of the parameter l i s t . T h i s p r o c e d u r e (POWELL 64) h a s r e c e n t l y b e e n w i t h d r a w n f r o m t h e NAG l i b r a r y o f a l g o r i t h m s and we h a v e s e a r c h e d for a s u i t a b l e a l t e r n a t i v e although r e t a i n i n g the procedure i n our own p r o g r a m . The n e a r e s t e q u i v a l e n t (E04CEA) e m p l o y s a quasi-Newton approach w i t h d i f f e r e n c e a p p r o x i m a t i o n s t o t h e gradient. A g a i n n o b o u n d s a r e p o s s i b l e , and c o n s t r a i n t d e p e n d s upon t h e parameter l i s t . A n o t h e r u s e f u l p r o g r a m (E04HAA) p r o v i d e s c o n s t r a i n e d o p t i m i z a t i o n w i t h bounds f o r each parameter u s i n g a s e q u e n t i a l p e n a l t y f u n c t i o n t e c h n i q u e , which e f f e c t i v e l y o p e r a t e s around unconstrained minimization cycles. V e r s i o n s o f RESOLVE w h i c h u s e e i t h e r POWELL 64 (NUSOLVE G ) , E04CEA (NUSOLVE E ) o r E04HAA (NUSOLVE P ) , and g r a p h p l o t t i n g versions are a v a i l a b l e . I t i s also possible to deal with a s y m m e t r i c p r o f i l e s (£). I n t h i s c a s e , two s e p a r a t e p a r a m e t e r s f o r e a c h o f f ^ and W-t a r e e v a l u a t e d , one f o r t h e l e f t - h a n d s i d e o f t h e p r o f i l e , t h e o t h e r f o r t h e r i g h t - h a n d s i d e . We h a v e a l s o t r i e d t h e P e a r s o n V I I f u n c t i o n s u g g e s t e d b y H u e v a l , H u i s m a n and L i n d (10) i n p l a c e o f o u r c o m b i n e d G a u s s i a n - C a u c h y f u n c t i o n . U n f o r t u n a t e l y , because the t a i l - f i t t i n g parameter m tends t o i n f i n i t y f o r a G a u s s i a n f u n c t i o n , we h a v e e x p e r i e n c e d c o m p u t a t i o n a l d i f f i c u l t i e s where p r o f i l e s a p p r o x i m a t e d t o a G a u s s i a n f o r m . However, i n a l l c a s e s w h e r e r e s o l u t i o n was s u c c e s s f u l , t h e f i n a l peak p a r a m e t e r s were i d e n t i c a l i n terms o f b o t h G a u s s i a n - C a u c h y and P e a r s o n V I I f u n c t i o n s . R e s u l t s . Where X - r a y d i f f r a c t i o n p e a k s a r e s h a r p and r e l a t i v e l y w e l l d e f i n e d , and where t h e r e i s no a m b i g u i t y i n t h e number o f p e a k s , t h e r e a r e few p r o b l e m s i n a c h i e v i n g s a t i s f a c t o r y r e s o l u t i o n w i t h any o f t h e t h r e e p r o g r a m s d i s c u s s e d above. T y p i c a l r e s u l t s achieved with the c r y s t a l l i n e c e l l u l o s e I f i b r e , Ramie, and w i t h t h e c r y s t a l l i n e c e l l u l o s e I I f i b r e F o r t i s a n , a r e i l l u s t r a t e d i n F i g u r e s 4 and 5- T h e 101, 101 and 002 p r o f i l e s r e s o l v e d b y NUSOLVE G a r e g i v e n , t o g e t h e r w i t h t h e b e s t - f i t p o l y n o m i a l b a c k g r o u n d . The p a r a m e t e r s a r e l i s t e d i n T a b l e I I , t o g e t h e r w i t h t h e t o t a l peak a r e a e x p r e s s e d a s a p e r c e n t a g e . T h i s l a t t e r parameter can be considered as the peak-area c r y s t a l l i n i t y w i t h i n t h e two t h e t a l i m i t s e m p l o y e d .


158

FIBER

DIFFRACTION

METHODS

t 70A

i

£ 40-

0 0 2

RAnie

so-


•o101

30-

/

101

A A // V A\V

2010-

f

9

11

IS

IS

17

/ J/ 19

21

TWO THETA

Figure 4. Smoothed equatorial trace for Ramie (Cellulose I)fiberswith resolved 101, lOT, and 002 profiles, and best-fit background

TVO THETA

Figure 5.

Smoothed equatorial trace for Fortisan (Cellulose II)fiberswith resolved 101,10T, and 002 profiles, and best-fit background


9.

HINDELEH

ET AL.


159

TABLE I I R e s o l v e d p a r a m e t e r s f o r Ramie ( C e l l u l o s e I ) and F o r t i s a n ( C e l l u l o s e I I ) Parameters

f Downloaded by UNIV MASSACHUSETTS AMHERST on September 29, 2014 | http://pubs.acs.org Publication Date: November 17, 1980 | doi: 10.1021/bk-1980-0141.ch009

Fortisan

0.4

0

20

8.9

101

1.6

1.9

101

14.6

11.6

0.3

0.2

18

40

1.5

1.9

16.3

19.4

0.7

0.6

62

37

101 A

101

W

r

Ramie

x

101

A

ioT

W

101

r

101

f

002

A

002

W

002

1.7

1.8

002

22.4

21.5

r

a b

1.45 -0.5

c

0.068

d

-0.0018

%B

-0.84 0.5 -0.021 0.0003

77

75

23

25

f - p r o f i l e f u n c t i o n parameter, A - peak h e i g h t ( e u ) W - peak w i d t h (two t h e t a ) , P - peak p o s i t i o n (two t h e t a ) , %k - p e r c e n t a g e a r e a u n d e r t h e p e a k s , %B - p e r c e n t a g e area under t h e background.


160

FIBER

DIFFRACTION

METHODS


We w i l l now c o n s i d e r t h r e e c a s e s where s a t i s f a c t o r y p e a k r e s o l u t i o n was much more d i f f i c u l t t o a c h i e v e . Case ( a ) , a s p e c i m e n o f PET f i b r e u s e d f o r t e x t u r i s i n g , h a v i n g p o o r l y d e f i n e d c r y s t a l l i n e p e a k s and p o s s i b l y a n a d d i t i o n a l p e a k due t o a n i n t e r m e d i a t e phase (11); case ( b ) , a specimen from a range o f PET f i b r e s w i t h d i f f e r e n t s h r i n k a g e s ( 1 2 ) , a g a i n a n a d d i t i o n a l i n t e r m e d i a t e p h a s e p e a k was a p o s s i b i l i t y ; c a s e ( c ) , c o l d drawn p o l y p r o p y l e n e f i b r e s (15)» a n a d d i t i o n a l p a r a c r y s t a l l i n e p e a k was most l i k e l y h e r e . C a s e ( a ) . When t h e i n p u t p a r a m e t e r s were i l l c h o s e n , s e e Table I I I , i n p u t ( 1 ) , p a r t i c u l a r l y w i t h background parameters z e r o , t h e v e r s i o n s o f NUSOLVE b a s e d o n POWELL 64 a n d E04CEA (NUSOLVE G a n d NUSOLVE E ) , w h i c h h a v e no p a r a m e t e r c o n s t r a i n t s , g a v e i d e n t i c a l r e s u l t s . However, t h e p r o f i l e f u n c t i o n p a r a m e t e r o f t h e 101 p e a k i s - 1 . 3 w h i c h g a v e a d i s t o r t e d p r o f i l e . The c o n s t r a i n e d p a r a m e t e r v e r s i o n NUSOLVE F, b e i n g u n a b l e t o d i s t o r t t h e 010 p e a k , d o e s n o t f i n d a v e r y good s o l u t i o n (S=67.7) w i t h i n t h e l i m i t s i m p o s e d . A r e a s o n a b l e f i t was o n l y f o u n d when t h e b a c k g r o u n d p a r a m e t e r s w e r e i n i t i a l l y w e l l c h o s e n , i . e . t h e y were f o u n d f r o m t h e s u b s i d i a r y p r o g r a m BASELINE. T h e n , i n p u t ( 2 ) , T a b l e I I I , a l l t h r e e v e r s i o n s o f NUSOLVE g a v e s i m i l a r o u t p u t p a r a m e t e r s (S=7.2 t o 7-7) a n d r e a l i s t i c p r o f i l e s , F i g u r e 6. A l t h o u g h t h e r e i s some s a v i n g i n c p u t i m e w i t h E04CEA o v e r POWELL 64 when i l l - c h o s e n p a r a m e t e r s were i n p u t , t h e o u t p u t p a r a m e t e r s were i d e n t i c a l . The c p u t i m e s r e q u i r e d w i t h w e l l c h o s e n i n p u t p a r a m e t e r s were a l m o s t i d e n t i c a l f o r t h e t h r e e programs. Because o f t h e p o s s i b i l i t y t h a t an i n t e r m e d i a t e phase e x i s t s i n PET, s e e C a s e ( b ) , t h e r e s o l u t i o n was c a r r i e d o u t w i t h f o u r peaks and zero background p a r a m e t e r s . E v e n t u a l l y , and o n l y a f t e r s e v e r a l a t t e m p t s , a g o o d f i t was f o u n d w i t h t h r e e c r y s t a l l i n e p e a k s , one v e r y b r o a d n o n c r y s t a l l i n e p e a k , a n d a p o l y n o m i a l b a c k g r o u n d . On a d d i n g t h e b r o a d p e a k t o t h e new b a c k g r o u n d , a n i d e n t i c a l r e s u l t t o t h a t o f F i g u r e 6 ( T a b l e I I I ) was o b t a i n e d , c a s t i n g doubt on t h e e x i s t e n c e o f a t r u e i n t e r m e d i a t e phase. C a s e ( b ) . The p o l y e s t e r s p e c i m e n PET06 was r e s o l v e d i n t o three c r y s t a l l i n e peaks w i t h almost i d e n t i c a l r e s u l t s u s i n g a l l t h r e e NUSOLVE p r o g r a m s , b o t h when i n p u t p a r a m e t e r s f o r t h e b a c k g r o u n d were z e r o , o r when b a c k g r o u n d p a r a m e t e r s f r o m BASELINE w e r e u s e d , s e e F i g u r e 7* However, b e c a u s e o f t h e s u s p e c t e d p r e s e n c e o f a n a d d i t i o n a l p e a k , f u r t h e r a t t e m p t s were made t o r e s o l v e t h e t r a c e i n t o f o u r p e a k s . When t h e a d d i t i o n a l p e a k p a r a m e t e r s w e r e i l l - c h o s e n ( a l l z e r o ) , i t was i m p o s s i b l e t o achieve sensible r e s o l u t i o n without constrained o p t i m i z a t i o n ; h o w e v e r , when t h e i n p u t p a r a m e t e r s f o r t h e a d d i t i o n a l p e a k were w e l l - c h o s e n , a l l programs e v e n t u a l l y gave s i m i l a r r e s u l t s , see F i g u r e 8 and T a b l e I V . The u s e o f c o n s t r a i n e d o p t i m i z a t i o n decreased t h e time r e q u i r e d t o f i n d a s a t i s f a c t o r y s o l u t i o n .



9.

HINDELEH ET AL.


161

Figure 6. Good resolution with well-chosen background parameters. Corrected and smoothed equatorial trace for PET feed yarn resolved into realistic 010 profile, together with 110,100 profiles, and new best-fit background.

Figure 7. Crystalline peak resolution. Corrected and smoothed equatorial trace of a PET fiber specimen resolved into three crystalline peaks, 010, 110, 100, and the best-fit background.


162

FIBER

DIFFRACTION

METHODS

TABLE I I I R e s o l v e d p a r a m e t e r s f o r PET f e e d y a r n w i t h b a c k g r o u n d i n p u t p a r a m e t e r s (1) a l l z e r o , (2) f i t t e d b y BASELINE, f o r t h r e e r e s o l u t i o n p r o g r a m s , NUSOLVE E , NUSOLVE G w i t h o u t c o n s t r a i n t s , NUSOLVE F w i t h c o n s t r a i n t s Input(l)

Output NUSOLVE G

E04CEA E

0

-1.3

-1.3

0.3

19.42

18.2

18.2

8.7

5.0

3.7

3.7

2.2

0

0.2

0.2

0


POWELL64 Parameters f

010

A

010

W

010

f

110

A 110

(1)

25.75

17.1

17.1

E04HAA

F

19.0

A

5.0

2.6

2.6

2.2

0

0.2

0.2

0.6

38.94

39.1

39.1

35.2

5.0

3.1

3.1

2.9

0

33.9

33.9

4.97

b

0

-3.89

-3.89

0.81

c

0

d

0

w

iTo

f

100

A

100

w 100 a w

0.161

0.161

-0.022

-0.0021

-0.0021

17.50

17.82

17.82

17.87

p 110

22.50

22.88

22.88

22.88

p 100 s

25.86

25.89

25.89

25.89

5904

7.56

7.56

67.7

268

183

163

r

010

r

0.000004

r

cpu s f W P cpu

p r o f i l e f u n c t i o n p a r a m e t e r , A - peak h e i g h t ( e u ) , peak w i d t h (two t h e t a ) ,a,b,c,d - background p a r a m e t e r s , p e a k p o s i t i o n ( t w o t h e t a ) , S - l e a s t sum o f s q u a r e s , s - computing time.


HINDELEH ET AL.


TABLE I I I ( c o n t i n u e d )

Output (2)


Input(2) POWELL64 NUSOLVE G

E04CEA E

E04HAA P

0.2

0.5

0.7

0.6

10.0

10.3

9.6

10.0

1.8

1.9

1.9

1.9

0.6

0.2

0.2

0.2

15.0

16.7

16.1

16.4

iTo

2.4

2.6

2.5

2.6

f

100

0.5

0.3

0.3

0.3

A

100

37.0

56.2

36.5

36.5

2.9

2.9

2.9

2.9

Parameters f

010

A

010

W

010

f 110 A -

A

w

110

w 100 a

-14.95

-10.20

-21.09

-16.42

b

3.41

5.08

4.69

4.02

c

-0.125

-0.150

-0.202

-0.173

d

0.00124

0.0016

0.0026

0.0022

P

17.80

17.82

17.82

17.82

110

22.85

22.86

22.86

22.86

p 100

25.89

25.90

25.90

25.90

s

50.26

7.71

7.19

7.28

148

134

151

r

010

P r

-

r

cpu s f W P cpu

p r o f i l e f u n c t i o n p a r a m e t e r , A - peak h e i g h t ( e u ) , peak w i d t h (two t h e t a ) , a,b,c,d - b a c k g r o u n d paramet peak p o s i t i o n ( t w o t h e t a ) , S - l e a s t sum o f s q u a r e s , s - computing time.



164

FIBER

DIFFRACTION

METHODS

TABLE I V R e s o l u t i o n o f p o l y e s t e r s p e c i m e n PET 06 i n t o t h r e e c r y s t a l l i n e p e a k s , and t h r e e c r y s t a l l i n e peaks and a p a r a c r y s t a l l i n e peak

Peak

f

A

P

010

0.2

28

17.5

1.3

7.3

9.8

110

0.0

37

22.6

1.9

4.9

20.6

100

0.1

84

25.7

1.9

4.7

46.4

W

L

hko

%Area

010

0.3

27

17.5

1.3

7.2

9.2

110

0.3

31

22.6

1.7

5-5

14.5

100

0.2

83

25.7

1.9

4.7

44.8

-1.0

9.2

26.7

5-3

—

14.7

Para

Total area

Background area

76.8

23.2

83.2

16.8

f - p r o f i l e f u n c t i o n p a r a m e t e r , A - peak h e i g h t ( e u ) , P - p e a k p o s i t i o n (two t h e t a ) , W - p e a k w i d t h ( t w o t h e t a ) L , , - c r y s t a l l i t e s i z e normal t o (hk0)(nm). n


9.

HINDELEH E T A L .


165

100

PCT06 4 PEAKS

A 80o 2 |

60C

40-

//*

1 1

/ V/i

010 A

'

1

//•' \

A

/U'v V

20-

15

20

25

30

35 TVO THETA

Figure 8. Paraerystailine peak resolution. Corrected and smoothed equatorial trace of a PET fiber specimen resolved into three crystalline peaks, 010, 110, 100, a paraery stailine peak and the new best-fit background.

INTENSITY


110

«• POLYPROPYLENE 10

«•« *• M-

«• tits-

/

\

\j

c a , c

-

It-

s'

io

is""

20

25

30 TVO THETA

Figure 9. Unrealistic resolution with unconstrained optimization. Corrected equatorial trace of a polypropylene fiber specimen cold drawn X-5-5. Total theoretical trace comprised of unrealistic profiles.



166

FIBER DIFFRACTION

METHODS

Case ( c ) . A t t e m p t s t o r e s o l v e t h e e q u a t o r i a l t r a c e o f a p o l y p r o p y l e n e s p e c i m e n ( c o l d drawn x10) i n t o c r y s t a l l i n e peaks and a b a c k g r o u n d were u n s u c c e s s f u l , d e s p i t e a v a r i e t y o f i n p u t parameters. Without parameter c o n s t r a i n t , nonsensical p r o f i l e s w e r e o b t a i n e d ( F i g u r e 9). The p r o f i l e r e s o l u t i o n c o u l d n o t b e i m p r o v e d e v e n b y t h e u s e o f BASELINE. E v e n w i t h c o n s t r a i n t s , r e s o l u t i o n was c l e a r l y w r o n g ( F i g u r e 10), a l t h o u g h i t p r o d u c e d t h e b e s t - f i t m a t h e m a t i c a l l y p o s s i b l e . Good r e s o l u t i o n was f i n a l l y a c h i e v e d when a n a d d i t i o n a l p e a k was i n t r o d u c e d , b u t o n l y w i t h c o n s t r a i n e d o p t i m i z a t i o n ( F i g u r e 11). With another polyp r o p y l e n e s p e c i m e n ( c o l d d r a w n x5), r e s o l u t i o n i n t o f o u r c r y s t a l l i n e p e a k s a n d a b a c k g r o u n d was r e l a t i v e l y s t r a i g h t f o r w a r d ( F i g u r e 12). E v i d e n t l y t h e f i t i s good except i n t h e r e g i o n 15-16°; as might be a n t i c i p a t e d , t h e a d d i t i o n o f a f i f t h p a r a c r y s t a l l i n e peak gave an e x c e l l e n t f i t throughout ( F i g u r e 13). D e t a i l s o f s a t i s f a c t o r y r e s o l u t i o n o f t h e s e two p o l y p r o p y l e n e s p e c i m e n s a r e g i v e n i n T a b l e V. The t h r e e c a s e s t u d i e s o u t l i n e d h e r e g i v e some i n d i c a t i o n o f the w o r s t d i f f i c u l t i e s t h a t have been e x p e r i e n c e d w i t h peak r e s o l u t i o n , a n d h a v e one f a c t o r i n common; t h e p r e s e n c e o f a n a d d i t i o n a l amorphous o r p a r a c r y s t a l l i n e p e a k was s u s p e c t e d i n e a c h c a s e . A l l t h r e e o p t i m i z a t i o n p r o g r a m s w o r k e d w e l l when t h e i n p u t p a r a m e t e r s w e r e w e l l c h o s e n , b u t c o n s t r a i n t s were n e c e s s a r y i n t h e more d i f f i c u l t c a s e s and when b r o a d p e a k s w e r e p r e s e n t . J u d i c i o u s c h o i c e o f i n p u t parameters always speeds t h e u l t i m a t e s o l u t i o n a n d t h e u s e o f BASELINE i s most h e l p f u l . E r r o r A n a l y s i s . The d i f f i c u l t i e s e n c o u n t e r e d i n r e s o l v i n g p e a k p r o f i l e s f r o m e a c h o t h e r , a n d a t t h e same t i m e d e t e r m i n i n g the background s c a t t e r , c a l l t o question the extent o f t h e e r r o r involved i n t h i s type o f a n a l y s i s , given that the f i n a l r e s o l u t i o n a p p e a r s r e a l i s t i c i n t h e l i g h t o f known i n f o r m a t i o n a b o u t t h e s t r u c t u r e o f t h e m a t e r i a l . The p r o b l e m o f e r r o r i n p r o f i l e r e s o l u t i o n h a s b e e n c o n s i d e r e d (14) i n terms o f t h e e q u a t o r i a l t r a c e o f a v i s c o s e rayon f i b r e specimen which i s s i m i l a r t o F o r t i s a n , F i g u r e 5* When random e r r o r s o f a v e r a g e m a g n i t u d e v a r y i n g b e t w e e n 1 and 8%, a c c o r d i n g t o t h e r e g i o n o f two t h e t a , were added t o 106 d a t a p o i n t s , t h e r e s o l u t i o n g a v e p a r a m e t e r s w i t h e r r o r s a s shown i n Table VI. T e s t s made o n t r a c e s t a k e n o n t h e same s p e c i m e n a t w i d e l y s e p a r a t e d i n t e r v a l s o f t i m e show d i f f e r e n c e s v e r y s i m i l a r t o t h o s e i n t h e t r i a l r e s o l u t i o n w i t h random e r r o r . The a v e r a g e e r r o r margins from t h i s e x e r c i s e a r e g i v e n i n T a b l e V I I and r e p r e s e n t an assessment o f t h e experimental c o n d i t i o n s f o r v i s c o s e r a y o n s p e c i m e n s a t t h e t i m e when t h e w o r k was c a r r i e d o u t . F o r o u r a p p a r a t u s , and f o r s i m i l a r p a r a m e t e r s i n o t h e r s p e c i m e n s , the e r r o r margins w i l l be comparable.


9.

HINDELEH

ETAL.


167

4S-

INT ENS:

Ail

POLYPROPYLENE 10

«oIS-

M-


S'

A lexp.

20IS-

A

J-iJcalc.

/A f

\ X\ \

1

JOS' 10

T5

20

25

30 TVO

THETA

Figure 10. Improved but still unrealistic resolution with constrained optimization. Corrected equatorial trace of a polypropylenefiberspecimen cold drawn Total theoretical trace still comprised of unrealistic profiles. Compare with Figure 9.

Figure 11. Good resolution with constrained optimization and addition of a paracrystailine peak. Corrected equatorial trace of a polypropylenefiberspecimen cold drawn X-5.5. Total theoretical trace now comprised of realistic crystalline peaks 110, 040, 130, 111, 131, 041, and paracrystailine peak at 15.5°. Compare with Figures 9 and 10.


168

FIBER

DIFFRACTION

METHODS

POLYPROPYLENE 5

10

20

-15

25

30 TVO THETA

Figure 12. Good resolution with constrained optimization except in 15° region. Corrected equatorial trace of a polypropylenefiberspecimen cold drawn X2.75. Total theoretical trace comprised of realistic peaks 110, 040, 130, 111, 131, 041, 060, but poorfitin 15° region.

POLYPROPYLENE 5

£40 HENS]


040

040

-30j

110

20-

l \ Jl 10-

10

15

L

060

20

25

30 TVO THETA

Figure 13. Good resolution with constrained optimization and addition of a paracrystalline peak. Corrected equatorial trace of a polypropylenefiberspecimen cold drawn X2.75. Total theoretical trace comprised of realistic crystalline peaks and a paracrystalline peak at 15.5°. Compare with Figure 12.


9.

HINDELEH ET AL.

169


TABLE V


R e s o l u t i o n o f c o l d drawn p o l y p r o p y l e n e specimens i n t o c r y s t a l l i n e and p a r a c r y s t a l l i n e peaks

Peak

f

A

P

W

L, , 1

?6Area

11.9

Total area

Background area

X10

110

0.1

15

14.0

1.2

7.5

040

0.3

11

16.9

1.0

9.3

8.6

130

0.1

14

18.4

6.6

10.2

111,131, 041

0.0

12

21.3

1.4 2.0

3.0

12.8

Para

0.5

7.0

15.5

2.5

-

14.6

110

0.1

18

14.1

0.6

17

7.2

040

0.3

28

16.9

0.6

18

10.4

130

0.0

13

18.6

0.9

11

7.9

111,131, 041

0.2

7

21.4

1.8

5.0

7.5

060

0.4

4.6

25.5

0.7

16

1.7

0.7

4.0

15.6

2.5

-

5.0

Para

39.7

60.3

f - p r o f i l e f u n c t i o n parameter, A - peak h e i g h t ( e u ) , P - p e a k p o s i t i o n (two t h e t a ) , W - p e a k w i d t h ( t w o t h e t a ) , L,, - c r y s t a l l i t e s i z e normal t o ( h k l ) ( n m ) . n


170

FIBER DIFFRACTION

METHODS

TABLE V I Known p a r a m e t e r s a f t e r

Parameters


f

Starting values

Known values

resolution Final values

Error

0.0

0.0

0.05

+0.05

101 A

101

6.5

6.0

5.81

-3.3/0

W

101

4.5

4.0

3.90

-2.596

0.0

0.1

0.04

-0.06

14.0

12.0

11.00

-8%

f

101

A

ioT

w

ioT

f

002

A

002

W

002

4.0

3.25

3.06

-6%

0.0

0.5

0.51

+0.01

17.0

15.0

15.83

+5.5°/o

4.0

3.25

3.30

+1.996

96A

100%

64.9%

62.9%

-5.796

%B

0%

35.196

37.196

+5.7°/o

12.2

12.15

12.145

-0.005

20.1

20.0

19.89

-0.11

21.9

22.0

21.92

-0.08

3.720

3.304

P 101 p _ 101 r

r

p 002 S

442.96

f - p r o f i l e f u n c t i o n parameter, A - peak h e i g h t ( e u ) , W - peak w i d t h (two t h e t a ) , P - peak p o s i t i o n (two t h e t a ) , S - sum o f l e a s t s q u a r e s , %k - p e r c e n t a g e a r e a u n d e r t h e p e a k s , %B - p e r c e n t a g e a r e a u n d e r t h e b a c k g r o u n d .


9.

HINDELEH ET A L .


171

TABLE V I I Error

Margins

Margin

Parameter

±0.15


f A

101,101

A

002

W

101

W

101,002 ±6°/

Pr

- -r P

±0.1

p

- p

r

(iii)

+6%

%A, %B

101

f W %k %B -

±5?6

002

0

101

±0.05

-

101

p r o f i l e f u n c t i o n parameter, A - peak h e i g h t (eu) peak w i d t h (two t h e t a ) , P - peak p o s i t i o n (two t h e t a ) , percentage a r e a under t h e peaks, percentage a r e a under t h e background. Instrumental Broadening C o r r e c t i o n

The two most common m e t h o d s u s e d t o c o r r e c t r e s o l v e d p e a k p r o f i l e s f o r t h e b r o a d e n i n g imposed b y t h e f i n i t e w i d t h o f t h e X - r a y beam i n t h e d i f f r a c t o m e t e r , a r e d u e t o J o n e s (15) a n d S t o k e s (16). B o t h a r e e s s e n t i a l l y u n f o l d i n g o r d e c o n v o l u t i o n methods, b u t t h e J o n e s method d e f i n e s s p e c i f i c f u n c t i o n s f o r b o t h t h e u n c o r r e c t e d and t h e i n s t r u m e n t a l b r o a d e n i n g p r o f i l e . I f t h e uncorrected p r o f i l e i s Gaussian, then

and i f t h e u n c o r r e c t e d p r o f i l e i s C a u c h y ,

then

(3 = B - b where p i s t h e c o r r e c t e d b r e a d t h , B t h e o b s e r v e d b r e a d t h , a n d b the instrumental breadth. The a p p r o x i m a t i o n s due t o J o n e s a r e b a s e d o n i n t e g r a l b r e a d t h s a l t h o u g h many i n v e s t i g a t o r s u s e h a l f widths instead. S i n c e we a r e d e a l i n g w i t h G a u s s i a n / C a u c h y f u n c t i o n s , we h a v e made u s e o f t h e c o r r e c t i o n


172

FIBER

P = f(B - b ) 2

2

2

DIFFRACTION

METHODS

+ (1 - f ) ( B - b )


The S t o k e s method i s e s s e n t i a l l y a F o u r i e r t r a n s f o r m method m a k i n g use o f t h e e n t i r e p r o f i l e , and i s a r e a s o n a b l y s t r a i g h t f o r w a r d c o m p u t a t i o n , a l t h o u g h l i m i t s have t o be a p p l i e d t o t h e p r o f i l e i n t r a n s f o r m s p a c e t o a c h i e v e c o r r e c t r e s u l t s . The m a i n p e a k f r o m h e x a m e t h y l e n e t e t r a m i n e c r y s t a l s c o m p a c t e d a t 85°C (17) h a s b e e n used as o u r s t a n d a r d f o r t h e i n s t r u m e n t a l b r o a d e n i n g peak. R e s u l t s . We h a v e made a d e t a i l e d s t u d y o f t h e e f f e c t o f u s i n g b o t h t h e Jones and t h e Stokes c o r r e c t i o n s on c r y s t a l l i t e s i z e measurements o b t a i n e d f r o m t h e most c r y s t a l l i n e s a m p l e s o f c e l l u l o s e I a n d I I , Ramie a n d F o r t i s a n (.18). Our c o n c l u s i o n was t h a t t h e J o n e s c o r r e c t i o n s were a l w a y s w i t h i n J>% o f t h e S t o k e s c o r r e c t i o n s f o r b o t h h a l f - w i d t h and i n t e g r a l b r e a d t h . C u r r e n t p r a c t i s e i s always t o use a Stokes d e c o n v o l u t i o n procedure f o r t h e c o r r e c t i o n o f a l l r e s o l v e d peak p r o f i l e s , e v a l u a t i n g an apparent c r y s t a l l i t e s i z e I^JJ^J terms o f t h e r e l a t i o n s h i p i

n

L/,, , \ — K — K = K X ^ p ds cos0d(29) n K 1 )

where d s a n d d ( 2 0 ) r e p r e s e n t t h e i n t e g r a l b r e a d t h s ( o r h a l f w i d t h s ) i n u n i t s o f s o r two t h e t a . The S c h e r r e r p a r a m e t e r , K, i s u s u a l l y t a k e n a s 1 f o r i n t e g r a l b r e a d t h s a n d 0.89 f o r h a l f w i d t h s . F i g u r e 14 i l l u s t r a t e s t h e d e c o n v o l u t i o n o p e r a t i o n f o r a PET 100 p e a k ; T a b l e V I I I l i s t s t h e a p p r o p r i a t e p a r a m e t e r s o f t h e o b s e r v e d , i n s t r u m e n t a l , and c o r r e c t e d peaks, w i t h t h e c r y s t a l l i t e s i z e e v a l u a t e d a s above and b y t h e a p p r o p r i a t e J o n e s f a c t o r . A g a i n , t h e v a l u e s a r e w i t h i n y/o. Apparent c r y s t a l l i t e s i z e v a l u e s f o r v a r i o u s s p e c i m e n s , e v a l u a t e d f r o m t h e above e q u a t i o n w i t h K = 1, a r e g i v e n i n T a b l e s I V a n d V. ( i v ) S e p a r a t i o n o f t h e r e s o l v e d and c o r r e c t e d p r o f i l e s

into

s i z e a n d d i s t o r t i o n components The methods i n g e n e r a l u s e f o r s e p a r a t i n g t h e s i z e a n d d i s t o r t i o n b r o a d e n i n g components o f t h e r e s o l v e d a n d c o r r e c t e d p e a k p r o f i l e s c a n b e s e p a r a t e d i n t o two g r o u p s , n o n - t r a n s f o r m and t r a n s f o r m methods. The n o n - t r a n s f o r m methods a r e e s s e n t i a l l y s i m i l a r t o t h e J o n e s method, b e i n g a p p r o x i m a t i o n s t o a c o n v o l u t i o n . The t r a n s f o r m method d i s c u s s e d h e r e makes u s e o f the F o u r i e r c o e f f i c i e n t s found a f t e r t h e Stokes c o r r e c t i o n . N o n - t r a n s f o r m m e t h o d s . The n o n - t r a n s f o r m methods make u s e of Jones type r e l a t i o n s ; t h u s , i f Pg and PD a r e t h e i n t e g r a l b r e a d t h s due t o s i z e a n d d i s t o r t i o n r e s p e c t i v e l y , t h e n , t h e observed p r o f i l e p i s g i v e n by



9.

HINDELEH

Figure 14.


ET AL.

173

Deconvolution correction for instrumental broadening—simulation. Profile b unfolded from Profile a to give Profile c.

TABLE V I I I C o r r e c t i o n o f 100 peak o f PET 06 specimen f o r i n s t r u m e n t a l broadening by Stokes method and b y Jones method Profile

f

A

W

P

IB

P

Uncorrected

0.2

83

1.9

25.7

2.74

0.0478

0.303

-3.3

Instrumental

0.5

89

0.2

25.7

0.30

0.0052

0.033

30

Corrected

0.2

75

1.8

25.7

2.56

0.0447

0.282

3.5

As

L

f - p r o f i l e f u n c t i o n parameter, A - peak h e i g h t ( e u ) , W - peak w i d t h (two t h e t a ) , P - peak p o s i t i o n (two t h e t a ) , IB - i n t e g r a l b r e a d t h ( t w o ^ t h e t a ) , p - i n t e g r a l b r e a d t h ( r a d ) , As - i n t e g r a l b r e a d t h (run" ) , L - c r y s t a l l i t e s i z e (nm) Stokes, L , - c r y s t a l l i t e s i z e (nm) Jones. g


s

L

J

3.6

174

FIBER

2

2

0

h

-

DIFFRACTION

METHODS

2 +

P

D

i f b o t h components h a v e G a u s s i a n p r o f i l e s , a n d b y P

P

=

s

h

+


i f b o t h components h a v e C a u c h y p r o f i l e s . (3g a n d 3^ a r e u s u a l l y d e f i n e d i n u n i t s o f s b y As

=

g

K L

(hkl)

A s ^ = 2 e s = 4e s i n 9

and

X where K i s t h e S c h e r r e r p a r a m e t e r , t y p i c a l l y 1.0 f o r i n t e g r a l b r e a d t h a n d 0.89 f o r h a l f w i d t h , L / ^ i ) i s t h e a p p a r e n t c r y s t a l l i t e s i z e n o r m a l t o t h e p l a n e s ( h k l ) , a n d e = Ad/d i s a m e a s u r e o f t h e maximum d i s t o r t i o n Ad i n t h e l a t t i c e o f s p a c i n g d . I t i s more a p p r o p r i a t e t o t h i n k o f e a s a l a t t i c e s t r a i n . I n o r d e r t o o b t a i n a measure o f t r u e c r y s t a l l i t e s i z e , we c a n make use o f t h e r e l a t i o n K = true c r y s t a l l i t e size apparent c r y s t a l l i t e s i z e p r o v i d e d t h a t we h a v e a u s e f u l v a l u e o f K. The above e q u a t i o n s y i e l d t h e f o l l o w i n g s e r i e s o f r e f l e c t i o n s (001 f o r e x a m p l e ) : (As)

2

As where a

±

= s l and s Q

Q

= (K/L = K/L

=

( 0 Q 1 )

( 0 Q 1 )

)

2

+ 4 e

+ 2 e

S

relationships

2

fora

(1)

2 S l

(2)

l

V

Fiber Diffraction Methods - ACS Publications - American Chemical

Recommend Documents