X-ray meridional scattering in drawn polyethylene. Account of matching orientation of crystallites along the fibril

Polymer Science U.S.S.R. Vol. 31, No. 11, pp. 2646-2652, 1989 Printed m Poland

0032-3950/89 $10.00 + .00 0 1991 Pergamon Press pie

X-RAY MERIDIONAL SCATTERING IN D R A W N POLYETHYLENE. ACCOUNT OF MATCHING ORIENTATION OF CRYSTALLITES ALONG THE FIBRIL* A. YE.

AZRIEL', V. A. VASIL'EV and L. G. KAZARYAN

R e s e a r c h and P r o d u c t i o n A s s o c i a t i o n " P l a s t m a s s y "

(Received26 May 1988) From an experimentally determined dependence of the half-width of meridional reflections on the reflection order, the crvstallite dimensions as calculated from the half-width of the 002 reflection are considerably larger than the long period, while the dimensions as calculated from the half-widths of the 004 and 006 reflections are shorter than the value of the long period. An explanation of this dependence-is proposed, assuming coherent scattering by the whole polymeric fibril. A calculation was carried out of diffraction on a model fibril consisting of alternating crystalline and paracrystalline domains. IT has been known [1-5] that at suiticently high draw ratios with some polymers, the crystallite dimension d~ along the chain axis, as calculated from the half-width of the meridional reflection 001, exceeds the value of the long period L. For simplicity we shall designate this as the the d,>L effect. An explanation of this effect is important for our understanding of the structure of oriented pc, lymets and of their mechanical properties [2, 4]. In [1-4], the d~>L effect was observed for PE. In these studies, CuK~ radiation was used, therefore only one meridional reflection 002 could be observed. The possibility of a manifestation of this effect at other reflection orders was not discussed. In the meantime it was discovered [5] that for some polymers, with the period along the chain axis c ~ 3 0 A, the d~>L effect occurs only for the lowest order I of the 001 reflections, while with growing I the half-widths of fll reflections sharply increase, so that the crystallite dimensions calculated from these fll values already are considerably smaller than L. It was shown that this kind of change in fli with growing ! cannot be caused merely by defects in the crystallites, and cannot be explained within the structural models proposed in [1-4]. The authors explained the obtained results by cooperative scattering on the crystallites in the fibril (the determination of crystallite dimensions from half-widths of reflections assumes that the crystallites scatter independently). Cooperativity appears as a consequence of the equal nature of the tie chains in cryst~llites a~d amorphous domains of the fibril. * Vysokomol. soyed. A31: No. 11, 2412-2417, 1989. 2646

X-ray meridional scattering in drawn polyethylene

2647

At high draw ratios, sections of the tie molecules in amorphous domains of the fibril are straightened out, assuming a conformation near to that of the same chains in the crystallites. Each such nearly periodic section of the tie chain in an amorphous domain of the fibril continuously passes into strictly periodic sections of the same tie chain in the neighbouring crystallites of the fibril, and therefore the waves scattered by these crystallites have similar phases, reinforcing each other. Intensity in the peak maximum is enhanced (as compared to the intensity of scattering by independent crystallites), and in order to maintain constant integral intensity, the half-width is reduced. The mean crystallite size as calculated from this half-width exceeds the true crystallite size, and at sufficient straightening of the tie chains it may even exceed L (in the limiting case of fully extended sections of the tie chains in amorphous domains, all crystallites in the fibril scatter strictly in phase and the "dimension of the crystallite" is equal to the length of the whole fibril). With growing reflection order 1, the half-widths increase and the d~>L effect disappears, even when there are no defects in the crystallites. This is because the phase differences grow in proportion to the growing 1, and their distribution between 0 and 2n becomes more uniform. Thus while for the nearest reflection order, the waves scattered by neighbouring crystallites have similar phases and the d~>L effect occurs, for sufficiently high 1 the phase differences will be uniformly distributed from 0 to 2n. But this means that the crystaUites scatter independently and the d~>L effect disappears. For polymers with c ~ 3 0 A, with a length of amorphous domains D ~ 2 c (as described in [5]), such an explanation appears natural. However, for polymers with c ~ 2 A and D,-~30 c (as in the case of PE), a different approach is necessary, as periodicity defects in monomeric units of the tie chain in the amorphous domain can accumulate in a large number of units and result in complete independence of scattering by the crystallites. The present paper is devoted to the solution of this problem on the example of PE. Oriented samples were obtained by drawing (at 65* and rate of clamp motion 21 mm/min) of dumbbells cut from plates pressed at 150° from PE powder of M,=4 x 104. Meridional reflections were recorded with the diffractometer DRON-3 by the transmission method, with MoK~ radiation (2=0-71 A, tube power 2.4 kW, focus 1 x 10 ram); for a sample with strain ratio 30, the reflections 002, 004 and 006 could be recorded. Corrections for primary beam width and sample thickness were made by a computational procedure. The relatively large splitting of the K~K,z doublet for Mo made possible its separation in the diffractogram by the method of Ratchinger [6]. Crystallite dimensions were calculated by the Scherrer method. Small-angle reflections were recorded with the diffractometer KRM-I with slit collimation (CuK~ radiation). In the studied range of strain ratios K from 10 to 30, the value of the long period L is independent of K and is equal to 200 A (Fig. 1). For the sample with K = 30, the dependence of the half-width fll of the 001 reflections on the reflection order 1 is shown in Fig. 2 (curve 1) (flz is measured in the units 4n s = ~-sin 0, where 2 is the wavelength and 20 the scattering angle). The sharp growth

2648

A. YE. AZnmL' er aL

in #~ on transition from 002 to 004 and the considerably smaller change on transition f r o m 004 to 006 cannot be explained by crystallite defects, as all known types of defects (and their combination) lead to a completely different kind of d e p e n d e n c e curve 1 (Fig. 2) would be concave instead of convex [7]. "Crystallite dimensions" dl calculated for this sample are equal to 300, 175 and 125 A for the 002, 004 and 006 reflections, respectively, i.e. d i > L for 1=2, and dm
c,,, 30 A [51. I

fit" ~0-z'4-1

I

10

I

30

50 20'

4

I

I

2

#

l

FIO. 2 I~o. 1 FIo. 1. Meridional small-angle reflections for PE samples of different strain ratio. Here and in Fig. 2 K = 3 0 (1), 15 (2), 10 (3). FIo. 2. Dependence of half-width #~ of 001 reflections on reflection order ! for PE samples of various strain ratio. Dependences of fl~ on I for samples with K = 10 and 15 are shown in the same Fig. 2. In these curves, the change in fl~ on transition from 002 to 004 is not so sharp as in curve 1, and in curve 3 this change is weaker than in curve 2. The intensity of reflection 006 is very weak, and therefore its half-width was not measured for samples with K = 10 and 15. With these samples, the d~>L effect is not observed.

Fro. 3. Fragment of a model fibril in an oriented polymer. Black and white points represent scattering eentres in crystallites and in amorphous sections, respectively. The experimental data obtained are in agreement with the theoretical concepts presented at the beginning of this communication. Nevertheless, these considerations are of a qualitative nature; for a more rigorous quantitative interpretation of experimental data it is necessary to introduce a model which would respect the basic features of fibril structure that are essential -for the problem considered, and which at the same time would be suitable for a rigorous calculation of scattering intensities. The authors assume that the dL>L effect is due to the similarity of conformations of the tie chains

X-ray meridionai scattering in drawn polyethylene

2649

in amorphous domains and in the crystallites in a fibril, and to the continuity of transition between these domains. In the Appendix it will be shown that in such a model, the intensity of meridional scattering per crystallite in a fibril is

J (s) = Jc(s) + Jo(s) + 4 Re {
(1)

Here Yc (s) is the scattering intensity from a collection of crystallites in a fibril

Jc(s) = <]Fc[2) + 2 Re {
Jo(s) = <[Fo]2) + 2 Re {~0-3, the half-widths of all

A. YE. AZI~L' et aL

2650

reflections are equal to the constant quantity ft. T h e crystallite dimension calculated with this half-width (by the Scherrer formula) coincides with the value preset in the model, 137 A, i.e. at 6>~0.3, the crystallites scatter independently. At smaller & (higher K) first of all the half-width of the reflection 002 is reduced m o r e than any other. T h e dependences of the half-width fl, of 00l reflections on reflection order !, calculated by means of f o r m u l a (1) for various values of ~ are shown in Fig. 6. In full accord with experimental data, at 6 = 0 . 0 9 (curve 1), a shaxp increase in fit occurs on transition f r o m 002 to 004, and a considerably smaller one on transition from 004 to 006. With increase in ~ (smaller K) this trend becomes less pronounced, and it disappears at 6 = 0 - 3 . W h e n "crystallite dimensions" dl are calculated (by the Scherrer formula) f r o m the half-widths determined at &=0.09, then the values 270, 164 and 147 A are obtained for 002, 004 and 006, respectively, i.e. d , > L for 1=2 and d~L gradually disappears (although the dependence of fli on 1 is preserved), so that at 6 = 0 . 3 , flL no longer depends on 1 (the crystallites scatter independently).

9(,=)

Pz:10,'A-'

C-A

C

C+ A x

~. IoN-'

I

0.1

I

0.2

~"

2

#

l

F[o. 4 F[o. 5 FIG. 6 FIo. 4. Probability density ~ (x) of spacings x between neighbouring scattering centres in amorphous sections, used in modelling the scattering on PE. FIG. 5. Dependence of the half-widths fl, of 00l reflections, as calculated by means of formula (1), on parameter & for l = 2 (1), 4 (2), 6 (3). F[o. 6. Dependence of the half-widths fl~ of 00l reflections, as calculated by means of formula (1), on reflection order l, 1or the following values of the parameter & 0.09 (1); 0.12 (2); 0.15 (3) and 0'3 (4). T h e theoretical (Figs. 5, 6) and experimental dependences obtained for PE (Fig. 4) are of complex appearance. Analogous dependences have been observed [5] for polymers with a period along the chain axis ,,-30 A. This enables us to conclude that the p r o p o s e d m e c h a n i s m of the d~>L effect is correct, and that its nature is the same for all polymers. APPENDIX

Derivation of formula (1). Axis X of a Cartesian coordinate system will be selected so as to coincide with the fibril axis (positive X direction will be to the right). A section of the fibril containing the centres of the crystallite and of the right neighbouring amorphous section will be called a unit. An axbitrary unit will be assigned the number 0; units to th© risht and to the left will be assigned th© numbers _+.1, + 2, + 3, ... etc., in sequential order.

X-ray meridional scattering in drawn polyethylene

2651

The centres in every amorphous section will be numbered from left to right. The distance tt from the ith centre of the amorphous section to the crystalline of the same unit will be called the coordinate of that centre in the unit (evidently fi+ ~ > t~). For amorphous sections containing n centres, the probability density of the coordinates of these centres in the unit is equal to

v.(tt, t2 ..... t,)=g(ta)g(t2-ta) ... g(t,-t~_a)

(2)

The set of all n such coordinates in the unit will be designated by one letter, for example t = (tt, t2, ..., tn). Similarly, dt=(dtt ..... dr.) and

o

[1

tm-t

When the unit contains a crystallite with m centres and an amorphous section with n centres, with coordiua.es in the unit within (t, t+dt), then we shall speak of this unit as belonging to class (m, n, t). The intensity I of meridional scattering on the fibril can be represented [8] by a sum of two members I= I~ + I=. where It=

~lFjl 2,

12=2Re

J

{,Y., ~ ~)Fj+~,e ''(='+'-*')} Jr'-1

Here Fj is the scattering amplitude of the jth unit, calculated for the case that its left end coincides with the origin of coordinates; xj is the coordinate of the left end of the jth unit; Fj is the complex conjugate of Fj. Let us first evaluate J.

12(N)---2Re { ~ ~

?jFj+te"("+'-*"),

J-Jtk~l

where N is a large, but final number of units with the numbers Jr, ]2 . . . . . is. Let x~ +k be the distance between the left ends of the jth unit of class (m, n, t) and of the ( j + k ) t h unit, and ~0t (m, n, t, x) be the probability that xJ +~ E (x, x+dx). Then among N selected units we find

NQm P, v,(t) dtQ~ P, u,(r) drq~k(m, n, t, x) (Ix units of class (m, n, t) with numbers j E (it, J2 ..... iN) such that the units with numbers j + k belong to class (p, v, 3) and ~+t ~ (x, x+dx). Replacing in I2 (N) summation over j by summation over m, n, p, v and integration (as N is very large!) over t, ~, x, we obtain I2(N)=Re[<~F) Y.QmP.

2N

[

'~."

~. (~p~(m, n , t , x ) ~ U d x l , J

(3)

~'' d o

where ( y ) - - - ~ Q . P . y v.(t)y(m, n, t)dt is the mean value o f y . m , fl k-I

XJ+k

As j

: z-)-, . + t. + x + ~ D j+,, where D , , : ( m - 1 ) c and t. are the fixed length of a crystallite rml

and the coordinate of the nth centre of the amorphous section in the jth unit of class (m, n, t), x =xJ+k-t., Dj+, is the length of the ( j + r ) t h unit, then ek(m, n. t, x ) = ~ (x-DIn) * O ( x - t . ) *g(x) * [ ( k - 1) • ~p(x)],

(4)

where ~ (x) is the delta function, • is the convolution symbol, ¢ (x) is the probability density of unit lengths, [(k - 1)*~ (x)] = ¢~(x)*¢ ( x ) . . . * ¢ (x) - (k - 1) times. But w (x)=~¢ (x)*¢° (x) [8], where q~(x):~,Q.,6(x-D.,) [9]. ~nd ~ . ( x ) = ~ P . [ ( n + l ) * o ( x ) ] . m

n

as the fraction of amorphous sections with n centres is equal to P., and the length of each section is equal to the sum of n + 1 random distances between centres. Substituting ~ (x) into expression (4), we find Ct (m, n, t, x).

A. YE. AzRn~I.' et aL

2652 It is easy to show that

F(m, n, t)

=

Fc(m) + Fo(n) e r'°',

(5)

n

where Fc(m)=fee

axe the scattering amplitudes of a crys,ffii tallite and of a~ amorphous section for a unit of class (m, n, t). Substituting relations (2), (4) and (5) into expression (3) we obtain 2 sin - ~

in ~

, F,(n)=f°

I2(N)/N= 2 Re {[(Fc)20° + (,F.)20c +
(6)

Similaxly we can calculate

It(N)/N= ([F[') = <[F~[') +
(7)

Jfs)= lira {11(~r)/N+ I,(N)/N}

(8)

It is evident that N~m

Substituting relations (6) and (7) into expression (8) we obtain relation (1). For information we present expressions useo in our calculations.

( F c ) = Y.Q~F¢(m),

(IF=l=>=Y.Q,,[Fc(m)I 2,

m

m

c= ~a(x)e'=dx, 0

(6")= I2P,6", II

(n)= Y.nP,, n

(I+-G-)--2]'I-GI 0 , = Z Q , e IsDm, •

O,=G(G"),

O=OcOo.

m

T h e a u t h o r s are i n d e b t e d to D. Ya. T s v a n k i n for reviewing the results of this work.

Translated by D. DOSKO(ZILOVA REFERENCES I. L. A. MANRIQUE, Jr. and R. S. PORTER, Polymer Preprints 16: 471, 1975 2. A. G. GIBSON, G. R. DAVIES and L M. WARD, Polymer 19: 683, 1978 3; A. N. OZERIN and Yu. A. ZUBOV, Vysokomol. soyed. A26: 394, 1984 (Translated in Polymer Sci. U.S.S.R. 26: 2, 440, 1984) 4. Yu. A. ZUBOV, S. N. CHVALUN, A. N. OZERIN, V. S. SHIRETS, V. I. SELIKHOVA, L. A. OZERINA, A. V. CHICHAGOV, V. A. AULOV and N. F. BAKEYEV, Vysokomol. soyed. A26: 1766, 1984 (Translated in Polymer Sci. U.S.S.R. 26: 8, 1979, 1984) 5. A. Ye. AZRIEL, V. A. VASIL'EV and L. G. KAZARYAN, Vysokomol. soyed. A28: 810, 1986 (Translated in Polymer Sci. U.S.S.R. 28: 4, 906, 1986) 6. A. GUINIER, Rentgenografiya kristallov, p. 41 I, Moscow, 1961 7. B. WUNDERLICH, Fizika makromolekul, p. 510, Moscow, 1976 8. D. Ya. TSVANKIN, Thesis... Doctor of phys.-mat, sciences, 342 pp, IVS AN SSSR, Leningrad, 1970 9. T. A. AGEKYAN, Teoriya veroyatnostei dlya astronomov i fizikov (Probability Theory for Astronomers and Physicists). p 74, Moscow, 1974