Method of processing of small angle X-ray paterns of oriented amorphous-crystalline polymers

Oriented amorphous-crystalline polymers

1009

4. V. V. KORSHAK, S. V. VINOGRADOVA, V. A. VASNEV, E. B. MUSAYEVA, A. P. GORSHKOV, G. K. SEI~E[N and L. N. GVOZDEVA, Dold. AN" SSSR 226: 350, 1976 5. V. HANOUSEK, Czech. Pat. 88272, 1959, Chem. Abstrs. 54: 8855, 1960 6. C. A. BUEHLER, R. L. BRAWN, G. M. HOLBERT, J. M. FULMER and G. W. PARKER, J. Organ. Chem. 6: 902, 1941 7. K. KROB, Czech. Pat. 139788, 1971; Chem. Abstrs. 76: 85549, 1972 8. H. E. FAITH, J. Amer. Chem. Soc. 72: 837, 1950 9. A. MOSHFEGH, S. FALLAB and H. ERLEMMEYER, Holy. chim. acta 40: I157, 1957; Chem. Abstrs. 52: 3744, 1958 10. P. MARCH, M. BUTLER and B. CLARK, :Industr. and Engng Chem. 41: 2176, 1946 l 1. V. V. KORSHAK, S. V. VINOGRADOVA and V. A. VASNEV, Vysokomol. soyed. AI0: 1329, 1968 (Translated in Polymer Sci. U.S.S.R. 10: 6, 1543, 1968) 12. A. WAISBERGER, E. PROSKAUER, J. RIDDICK and E. TUPPS, Organic Solvents, 1958

Polymer ScienceU.S.S.R. Vol. 20, pp. 1009-1016. C) PergamonPress Lid. 1979. Prit~tedin Poland

0032-3950/78/0401-1009507.5010

METHOD OF PROCESSING OF SMALL ANGLE X-RAY PATERNS OF ORIENTED AMORPHOUS-CRYSTALLINE POLYMERS* B. A. ASHEROV and B. M. GrsZBURG High Polymer Institute, U.S.S.R. Academy of Sciences

(Received 5 July 1977) A new technique has been developed for processing the intensity distribution curves _,re(s) along the meridian of small angle X-ray patterns of highly oriented amorphous-crystalline polymers. The technique is based on a model approach whereby I (s) curves calculated for different one dimensional models of the supermolecular organization (SMO) of polymers are compared with all or most of the experimental curves. By means of this teelmique it is possible to determine the statistics of the one dimensional lattice modelling SMO.

THE supermolecular organization (SMO) of oriented amorphous-crystalline polymers is normally modelled by a one dimensional lattice with paracrystalline distortions of periodicity [ 1-8]; the lattice is made up of crystalline and amorphous regions alternating along the direction of orientation, and lattice distortions stem from polydispersity of crystallites and amorphous regions in respect to their dimensions in one or other direction. * Vysokomol. soyed. A20: No. 4, 894-899, 1978.

1010

B. A. AsmcRov and B. M. Gn~BU~O

The concept of a "statistical lattice" has been adduced in literature to characterize these one dimensional lattices, and is a conventional means of denoting t h e sum total of the independent functions o f / / 1 and H2 describing distribution densities in respect to the dimensions of crystalline and amorphous regions* respectively. Without knowledge of the lattice statistics one cannot arrive at any more ~)r less accurate estimation of the lengths of long periods c, since a simple deterruination of c based on Bragg's law m a y result in major errors in the case of paracrystalline lattices [3, 4, 7]. I t was suggested in [3, 4] t h a t estimates of c values (and of crystallite dimensions as well) could be attempted b y using (with a pre-set lattice model, i.e. given its statistics, and with a linear density distribution in the model) the t w o parameters of experimental Ie(s) curves describing the intensity distribution of the small angle scattering of X-rays, namely, the angular position of the m a x i m u m and the reflection width. In papers [7, 8] Cryst suggested the use of a third parameter, the latter being the ratio of angles at which first and second reflection orders appear. According to Cryst [7, 8] even an approximate selection of lattice statistics will remain ambiguous when based on thehe parameters even in the case of small angle reflection of second order, and the determination error for c based on a first reflection, using Bragg's law, m a y be as m u c h as 35%; if on the other hand there is only a single reflection, the error will be still greater (up to 50%), and there will be no question of lattice statistics. However, with highly oriented amorphous-crystalline polymers one fre(tuently finds t h a t there is only a single independent reflection on the meridian o f the small angle X-ray patterns (SXP). The technique proposed b y us for interpretation of SXP of this type, and for the analysis of lattice statistics, is based on a model approach involving use of the entire Ie(s) curve (or most of it), i.e. account is taken of the largest possible amount of experimental information. Attention is drawn in [9] to the importance of using the entire Ie(s) curve in the analysis of SXP.

Formula for calculation of I (s) and for selection of the one dimensional lattic~ Tarameters. The technique is based on a comparison of experimental Ie(s) curves with theoretical I (s) curves calculated with wide variations in the one dimensional model parameters. Calculation of the I(s) curves is carried out using the formula in [10, 11] for a general paracrystalline model with mirror symmetrical transition regions adjoining different ends of a single erystallite

I(8)=l ,1'8 Ir(8)

(1)

* The term "lattice statistics" pertains to a so called general paracrystalli~e model; in the case of a particular paracrystaUine model one uses the s u m total of analogous functions for orystaUites and long periods [7].

Oriented amorphous-crystalline polymers

1011

Here Ir(s) is the Hermans expression [1] for scattering intensit) by an infinite model with effective crystallite sizes a e and dimensions of amorphous regions b~ ~fithout transition regions between the latter

Ir(s )= s~

ll-f°l ~ (l--lhel~)+[l--he]2 (l--[f¢]2) i i_]Ohel ,

(2}.

8 is the projection of vector s on the z axis along which .crystalline regions alternate with amorphous ones; s=2u(sin 20)/A, where 20 is the scattering anglo

i

14

15

2~ Fro. 1

a~-l,]

Groin

u~

FIG. 2

F I e . 1. Appearance of H i ftmctions upon varying p a r a m e t e r m of generalized F - d i s t r i b u tion: m = l (•); 2 (2); 5 (3) and 35 (4). FIG. 2. Schematic representation of the experimental S X P curve, and the separation o f diffuse scattering from the reflection: / - - t a n g e n t whereby the separation is frequently carried out; 2 - - s m o o t h curve under peak intended for the same purpose; 3, 4 - - e n d of the Gaussian curve, and straight line, approximating diffuse scattering in the vicinity of the I(8) minimum; stain--minimal size of anglo in the int&wal used for comparing the experimental curve with theoretical ones.

for X-rays of wavelength ),; re, he, Fourier transforms for functions//~,(a~) and //~,(b~) which are respectively density distributions for effective crystallites and amorphous regions with dimensions; a~-----aj+fl; b~=bj--fl; aj is the length of the j t h erystallite, incorporating transition regions; bj is the length of the amorphous

1012

B . A . Astor.Roy a n d B. M. GIN~UXO

region: fl:(--2zau~t)/s; whereupon

z is the complex independent variable in

v=lw[e

,

i V = ~ [Ptratmition(Z)] nt- - - e x p (iSt) , 8

(3)

where ptransitlon(Z) is the intensity distribution in the transition region; t, the length of the transition region. Functions H~(a~) and H~(b~) are related to the respective functions for true dimensions of cryst~llites and amorphous regions by the expressions I-l~l(a~)=Itl(aj) *J (z--fl) (4)

It~2(b~ ) : H~(bl) *5 (z-{-fl) ,

(4a)

where J is the Dirac delta function; symbol * denotes the coiling operation. If only a single function of some sort characterizing the model (H1, H2 or ptransmon) is varied, then to find I (s) one has to calculate afresh only the appropriate parameters forming par~ of equations (1) and (2), i.e. fe, he or I~]]2 and ft. In selecting the type of functions (H1, H a or ptransmon) we were guided by the following considerations. The crystallization process is to a large extent governed by external conditions [12, 13]. According to the thermodynamic theory of folding [12] the free energy density for a lattice chain (if the crystallization temperature is more than 20-30 ° below the melting point) has a fairly sharp minimum at a definite fold length (i.e. crystallite size); because of this, crystallite distribution according to size is bound to be rather narrow. We accordingly selected, as Hx, Gaussian functions having quite a low degree of dispersion, so that the number of crystallites having a negative length does not exceed 0.15%. As H,* we took the generalized exponential distribution (F denotes distribution)

(vbj)

H2(bj): ~

bj? (rn)

exp (--vbt),

(5)

where F(m) is a gamma function; m, v arc distribution parameters related to the average size b by the relation b : m ] v . Selection of the form of Ha was governed by the consideration that on varying m we may obtain a majority of distributions t h a t have previously been used in literature (Fig. 1). In our calculations we took m = l (exponential distribution) [3, 4]; m----2 [2]; m = 3 5 (given such large m values the distribution scarcely differs from the Gaussian distribution used in [3, 6-8]) and, finally, as the intermediate distribution between "Reinhold" and Gaussian distribution we took distribution with m = 5 . Following the work of Tsvankin [3, 4], who was the first to introduce the matter of transition regions, we assumed that density distribution within the • The idea of u s i n g a generalized e x p o n e n t i a l d i s t r i b u t i o n as Ha comes from S. Ya. Frenkel'.

Oriented amorphous-crystallino polymers

1013

latter to be linear in character. A similar density distribution (pa) is assumed b y t h e authors of [6-8]. Calculation and analysis of distributions I (s). In I (s) calculations done with the computer the following four independent model parameters (see Table) were varied: parameter m for function H2(bj); the degree of crystallinity ale (whero c = a ÷ b is the average size of the long period); the transition region fraction in an average crystallite t/~t; standard A~ for Hl(aj) distribution. Fibrils were assumed to be of infinite length. IN'DEPENDENT PARAMETERS

OF ONE-DIMENSIONAL

FIBRIL MODEL

m

h

1

0.4

t

0.455

2

0.5

0.2~

5 35

0.75 0.9

0.15 0

z/=

0.133 ~--2 3

0

~ o ~ . Values a a n d zl, g i v e n In c.

I(s) calculations were carried out subject to ~ = 1 ; now, the degree of crystallinity is numerically equal to the average crystallite size ~. With fixed values of m and al the dispersion Ab~ for function H~.(b¢) may be unambiguously defined as

2

Ab=mlv

2

.

Calculation of I(s) curves started at values of s = 0 . 1 × 2 ~ . For each curve a scale was selected so that positions of points corresponding to intensity maxima would coincide in the case of superposed coordinate axes. Thus, we obtained a set of 192 curves that could be comparison with experimental ones. This comparison amounts to empirical selection of an I(s) formula containing four independent parameters that will describe the experimental dependence of Ie(s).

Analysis of experimental curves of Ie(s) and comparison of the latter curves with theoretical ones. The Ie(s) curves are to be corrected, if experimental conditions require this (corrections for primary beam width, and for fibril or crystallite orientation as well as for transverse fibril dimensions, etc.), after which the curves have to mapped out on a scale that will make the position of the maximum the same as on theoretical curves. Particular attention must be paid to fibril orientation, as the latter has a marked effect on the S X P width and profile. Initial selection of theoretical curves agreeing with a given experimental curve is carried out visually, after which one has recourse to the following criterion of satisfactory "agreement" between curves [14]: ]I(s)--Ie(s)] moduli must not exceed maximum errors for Ie(s) over the entire range of s values selected for comparison "(Fig. 2).

1014

B.A. As~mRov and B. M. GINZBURG

Bringing in the R factor we now have Smax

I lI(s)--Ie(s)[ ds R=..: ~'m,o

Smaz

100%

(6)

I Ie(s)ds 8rain

I t can be shown b y experiments that, with adherence to the foregoing criterio~ of agreement between I(8) and Ie(8), the R factor does not exceed 5-10%; this standard value of R m a y be reduced b y a reduction in the measurement error. Since the density distribution in the infinitely long fibril model examined b y us is fixed statistically, one may assume t h a t by following along a fibril in a n y direction from a selected point we will obtain one and the same density distribution, and the latter m a y be taken to be centrisymmetrical. Now, according to the theorem of uniqueness [1] there is a unique solution of the reciprocal diffraction problem, i.e. there is a single fibril model that has a density distribution satisfying the experimental Ie(s)* curve. In view of the error in measurement results, there will theoretically be an infinite number of theoretical I(s) curves satisfying matching criteria; actually however, steps in variations of fibril parameters (see Table) are not infinitely small, with the result t h a t actually no more than from two to four of the entire number of I (s) curves will coincide with each processed experimental curve. Further differentiation o f theoretical curves coinciding with an experimental one is practicable solely with the aid of additional data, e.g. b y comparing average crystallite dimensions obtained from small and wide angle X-ray patterns. Let us now turn to the problem of the interval of s values selected for comparison of I(s) with Ie(S). In m a n y specimens having a fibrillar structure inhomogeneities are found (e.g. micropores), and the scattering intensity b y t h e latter decreases monotonically as the scattering angle widens (the so-called diffuse scattering). To isolate diffuse scattering from a curve relating to scattering b y intermittent fibrils the method hitherto appearing in literature is one based on a tangent to the Ie(s) curve or alternatively this m a y be done with the aid of a smaoth curve beneath the reflection (Fig. 2). The advantage of the first method lies in the fact that it is unambiguous; the second does involve a considerable degree of randomness, although from a physical standpoint it is more reasonable, since it is less probable t h t t the diffuse scattering intensity will apear as a curve having a more or less a b r u p t discontinuity, which must inevitably be expected where diffuse scattering isola* It must be emphasized, however, that in view of Babine's principle amorphous and crystalline regions may be interchangeable in the case of an infinitely long fibril, in which respect the solution proves to be ambiguous. This ambiguity can only be resolved by using. independont'methods, e.g. by determination of degrees of crystaUinity.

Oriented amorphous-crystalline polymers

1015,

tion is based on use of the tangent. In either case, however, by intercepting diffuse scattering one may at the same time "intercept" some of the scattering intensity originating directly from one dimensionally periodic structures, since m a n y theoretical curves feature something resembling diffuse scattering in the form of a plateau or flat curve t h a t does not fall to zero in the region of very small scattering angles. This means t h a t in the isolation of diffuse scattering stemming from nonfibrillar elements oue must proceed with great caution so as to avoid distortion of the true shape of the Ie(8) curve. A suitable means of isolating diffuse scattering from the curve as a whole might be achieved through the introduction of substances having a high electron density into amorphous pai~s of the fibrils (the electron density of the substances used could have values all the way up to the point where the electron densities of crystalline and amorphous regions are equalized [15J). The diffuse scattering intensity by micropores or by other inhomogeneities generally decays markedly as the scattering angle widens. This is borne out by examples of Ie(s) curves relating to some concrete specimens of highly oriented polymers (PVA, PAN, etc.) which for one reason or another give no small angle reflections. :Now, assuming we have a reflection and diffuse scattering the sharply varying portion of the Ie(8) curve pertaining to diffuse scattering m a y be approximated by a portion of the Gaussian curve (Fig. 2); in an approximation of this type diffuse scattering has very little effect on the portion of the Ie(8) curve t h a t follows the minimum under study. A still simpler ~pproximation of the diffuse scattering intensity distribution in the vicinity of the reflection, i.e. approximation by a straight line, produces results practically identical to those obtained by an approximation using a portion of the Gaussian curve. Thus, the value of stain is ascertained from the position of the Io(s) minimum; the value of 8max is determined by the ral)ge of angles where (hc scattered radiation intensity becomes comparable with the background. It appears therefore t h a t by means of the .proposed method one may determine the following parameters of a one dimensional periodic lattice modelling the SMO of highly oriented amorphous-(.rystalline polymers: I) the crystallite fraction in thc long period ~/~; 2) the dispersion of crystallites according to size Aa~ (with an a priori fixed density distribution of crystallites according to size in the form of a Gaussian function); 3) the transition region portion in an average crystallite t/~t; 4) parameter m determining the type of the generalized exponential distribution describing the density distribution of amorphous regions according to lengt, h; 5) the average length of the long period ~; 6) the dispersion of long periods A2c; 7) the average size of an amorphous region b; 8) the dispersion of amorphous regions according to size A~; 9) parameter Z,n relating the long period to the long period length determined by Bragg's law: -c-ZrndBra~. Only four of the parameters listed here are independent ones (the parameters selected t o be independent parameters in the calculations have been stated first).

3016

B . A . ASH]mROVand B. M. GxNZVU]aO

T h e a u t h o r s t h a n k S. ¥ a . F r e n k e l ' , D. Y a . T s v a n k i n , V. I. G e r a s i m o v , Ye. V. K u ' v s h i n s k i i , Yu. V. B r e s t k i n a n d A. M. Y e l ' y a s h e v i c h for p a r t i c i p a t i o n in t h e d i s c u s s i o n o f results a n d for t h e i r critical o b s e r v a t i o n s . Translat~ by R. J. A. HENDRr REFERENCES

1. R. HOSEMANN and S. N. BAGCHI, Direct Analysis of Diffraction by Matte, North-Hol. land Co., 1962 2. C. R. REINHOLD, E. W. FISCHER and A. PETERLIN, J. Appl. Plys. ~5: 71, 1964 3. D. Ya. TSVANKIN, Doktorskaya dissertatsiya, High Polymer Institute, Leningrad, AN SSSR, 1970 4. D. Ya. TSVANKIN, Vysokomol. soyed. 6: 2078, 1964 (Translated in Polymer Sei U.S.S.R. 6: 11, 2304, 1964) 5. D. J. BLUNDELL, Acta crystallogr. A26: 472, 1970 6. D. J. BLUNDELL, Acta crystallogr. A26: 476, 1970 7. B. CRYST, J. Polymer Sci., Polymer Phys. Ed. 11: 1023, 1973 8. B. CRYST and N. MOROSOFF, J. Polymer Phys. Ed. 11: 1023, 1973 9. A. GUINIER and G. FOURNET, Small-Angle Scattering of X-Rays, Wiley, 1955 10. B. A. ASHEROV, B. M. GINZBURG and Sh. TUICHIYEV, Bold. AN Tadzh SSR 19: 18, 1976 11. B. A. ASHEROV, B. M. GINZBURG and T. I. VOLKOV, Dep. VINITI, No. 294-76, 1976 12. F. DZHEIL, Polymeric Monoerystals, Izd. "Khimiya", 1968 13. L. MANDELKERN, Crystallization of Polymers, Izd. "Khimiya", 1966 14. B. M. SHCH]GOLEV, lV~atematieheskaya obrabotka nablyudenii (Mathematical Analysis of Findings). p. 283, Izd. "Nauka", 1969 15. A. I. SLUTSKER, Doktorskaya Dissortatsiya, A. F. Ioffe Physico-technological Institute, Leningrad, 1967