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(Received 8 December 1971) A method is proposed for determining the non-square angle of disorientation of fibrils of micron dimensions in orientated crystallizable polymers. The method is based on comparison of the azimuthal distribution of the intensity of scattered light between crossed and parallel polaroids. Correlation has been found between the mean-square angles of disorientation of fibrils and crystaUites in samples of orientated, medium-pressure polyethylene. OBSERVATIO~ of orientated, crystallizable polymers Under the electron microscope has shown that they contain fibril-like elements of dimensions in the region of a micron [1]. In many instances one of the crystallographic axes of the crystallites making up a fibril-like formation lies parallel to the axis of the latter and the other two are orientated at random about the first. Because of difference in polarizabilit y along the different polarographic axes these formations are optically anisotropic. The scattering of polarized light at low angles has been used successfully for s t u d y of these formations [2]. In the present paper a method for determination of the disorientation of fibrils of micron dimensions is proposed. The method is based on comparison of the azimuthal distribution of the intensity of scattered light between crossed and parallel polaroids. Calculation of the scattering of light by an assembly of disorientated fibrils. We shall model a polymeric fibril b y a cylinder of height H, radius R and polarizabilities slong the axis of the fibril of gh and in the perpendicular direction of ~r. The orientation of the fibril will be characterized b y a single vector parallel to the axis of the fibril. I f all the individual vectors have a common origin their ends will be distributed on the surface of a hemisphere of unit radius. The density of the distribution of the ends of the vectors we shall write as the function

f(c) = K e x p - - (c~/2J~),

(1)

of cylindrical symmetry in relation to the texture axis. Here c is the angle between the texture axis and the axis of the fibril, K the normalizing factor and J a parameter characterizing the degree of disorientation. When J ~<0.2 the section * Vysokomol. soyed. A15: No. 9, 1953-1958, 1973. 2200

Determination of disorientation offibrillar structural elements in orientated polymers

2201

of the hemisphere on which the greater part of ends of the vectors are distributed m a y be considered to be a quasi-plane. Then K=½u5 ~,

c~=a~+b ~

(2)

and the mean-square angle of disorientation is given by +co +co

2uJ ~ --CO

o

r-t-

-p

/

(3)

--~0

where a is the angle of inclination of the fibril to the primary beam when the direction of the latter is perpendicular to the texture axis, and to the angle between the texture axis and the projection of the axis of the fibril on the plane perpendicular to the primary beam (Fig. 1).

',1//

/// FIG. 1. Coordinate system for calculating scattering by fibrils: z--texture axis; and/~--scattering angles; a, b and c--angles of orientation of the fibril; So and S'-the directions of the incident and scattered beams respectively. Assuming that the fibrils are non-cohering formations the intensity of scattering at the angles 0 and I~from an assembly of N fibrils can be defined as

+ +~ ["a2d_b~'~. I(O, lu)=N_co~ -~I I ' ( O , l ~ , a , b ) × K e x p - - ~ ) a a d b ,

(4)

where I ' (0, #, a, b) is the intensity of scattering by a single fibril orientated at the radial and azimuthal scattering angles of a and b, and 0 and # respectively. According to reference [3] the intensity of scattering from an anisotropic cylindrical formation is given b y

I' (O, lu, a, b ) ~ ~ I ll (z---~)sin z

UT

(5)

2202

Yu. V. BRESTKINand D. RASHIDOV

where 11 (z) is a first-order Bessel function; 2uR z---{[cos a (l--cos {))~-sin a sin 0 sin (p--b)]2-~-sin 2 0 cos2 (]/--b)} 1I-~

U-= ~H [sin a(1--cos 0)--sin 0 cos a sin (]/--b)]

(6)

(7)

where ~l is the wavelength. The value of • i~ dependent on the anisotropy of the cylinder and on the mutual orientation of the cylinder axis and the planes of polarization of the polaroids. When the direction corresponding to t h e angles a----0 and b = 0 is vertical, for Hv and Vh polarizations ~bHv=~vh=(~h--~r)sin b cos b cos 2 a,

(8)

and for V~ and H h polarizations

Cy=o~ r cos2b+~r sin 2 a sin2b+ah cos 2 a sin2b

(9)

~bHh=~h cOS2 a cos2b+~r sin 2 a cos2b+~r sin2b

(10)

Restricting observation to small angles 0 and ~, and assuming that for low degrees of disorientation the angles a and b are small, equations (6)-(10) can be written in the form z z 2~R {a~02(/t--4~)2+02 [ 1 _ (tt__bZ]},l,

(11)

as U--

~. b2

q~H:~Va=(O~tt--~r) b e x p - - ( ~ - + a 2)

(13)

~ r ~ = a r exp--(b ~)

(14)

~Hh----~h exp-- (b2-~-a2

(15)

Making use of the approximation

[ 2I (z) 7 3

oxp-k/ = V ) ,

we obtain the following expression for the distribution of scattering intensities from an isolated fibril with H~ and Vh polarizations t

I~, (0, p ) ~ I r h (0, p),,~ b ~ exp--(b*--F-2a2+~-~-t)

(16)

Determination of disorientation of fibrillar structural elements in orientated polymers

2203

and for V,, and H a polarizations i~o (0, g)~exp--(2b2-~-e+t) 1'i-I, (0, p ) ~ e x p - - ( 2 b 2 + 2 a 2 + e + t ) ,

(17) (18)

where (19)

(l-a? ( -bV

(20)

2A 2

By substituting expressions (16)-(18) in equation (4), integrating and neglecring terms containing fourth and higher powers of fi, we find the distribution of the intensities of scattering from a fibril in the case of crossed polaroids

f

A,fl +AI~

{~A~_~fl~)} exp-- (e)

(22)

and for V v and H h polarizations Ii~(O l~) ~ exp-'

exp-- (e),

(23)

2

where fl~=(3+fi-2) -~ with 62<41 and AI=A[l~-(52/2)]. It follows from equation (23) that the azimuthal intensity distribution for parallel polaroids is a Gaussian distribution with the dispersion a2~-- A~-~ fl 2

(24)

and a half-width (Fig. 2) of PII-a/K', (25) where K'--~ (ln 4) -~ ~ 0.85. By comparing formulae (22) and (23) it is easy to see that for an assembly of disorientated fibrils the azimuthal intensity distribution with crossed polaroids is always broader than with parallel polaroids. This difference is explained by the fact that with parallel polaroids fibrils orientated in the direction parallel to the texture axis make the main contribution to the intensity, whereas when the polaroids are ~rossed these fibrils do not contribute to the scattering (formulae (16)-(18)). We shall now m a k e use of the difference in the breadth of the azimuthal distributions for determining the degree of disorientation. Taking account of equations (24) and (25) we shall write equation (22) in the form

2204

Y u . V. B~ESTKIN a n d I). RASmDOV

Since w h e n / ~ = / ~ , w h e r e / ~ is the half width of the distribution of I± with respect to/~ I ± = 0 - 5 exp--(~) , it follows from equation (26) that

{

'

1-4-1~K,,I~K,2fl 2 exp-- ( 2/z~K,~ j -

(27)

Solving this equation with respect to 6 we find

where

The dependence of/~-//~11 on (c) *, which is found from equation (28), is shown in Fig. 3. I t is seen that/~±/:tl i increases as the mean-square angle of disorientation increases. At a given (c) ~ the ratio/~-//~l~ is greater the lower the value of/~.

~.lj.

I

" 1

b 2 2

J

Radions

/

,.g -20 FIG. 2

04 II

0 ,Uz Pn 20

~.

o

5

O.2 I

I0

I

O'3 I

I

15

Degrees Fio. 3

FIG. 2. V~--i)iffractogram of a polyethylene specimen stretched to 350~o (a) a n d t h e azim u t h a l i n t e n s i t y d i s t r i b u t i o n s of the same specimen w i t h parallel (1) a n d crossed (2) polaroids, w i t h 0 ~ 10 ° (b). FIo. 3. I)ependence of/~jJ~[[ o n (cJ) t for /LI[--10 ° (•); 20 ° (2) a n d 30 ° (3).

Determination of disorientation of fibrillar structural elements in orientated polymers

2205

Equation (28)presents the possibility of finding the mean-square angle of disorientation experimentally. For this it is necessary to find the aximuthal intensity distribution with crossed and parallel polaroids, and to determine their half-widths. The radial angle 0 must be taken such that the measured signal when /~----0 is much stronger than the background in the case of crossed polaroids, and the half-width/~ll should be not more t h a n 10-20 °*. When ]~ii>20 ° the sensitivity of the proposed method may be low (Fig. 3). The po]arizabilities ar and ah of the fibrils do not enter into equation. (28). This is explained by the fact t h a t according to formulae (8), (13) and (15) ar and ah do not affect the distribution of the intensity from a fibril, but affect only the integral scattering intensity. The values of g~ and gh are dependent on the anisotropy and orientation of the structural elements of the fibril, namely of lamellae, crystallites and chain segments. Many methods have been proposed for detrmination of the orientation of these structural elements. The orientation of the crystallites can be measured by wide-angle X-ray scattering [4], the orientation of lamellae by low-angle X-ray scattering [5] and the birefringence method can be used for determining the orientation of segments [6]. The fact t h a t

2206

Y r . V. BRESTKIN and D. RASHIDOV

I t is seen that within the range of degrees of extension studied (200-550~) the mean-square angle of disorientation does not exceed 13.5 °, and it decreases to 7.5 ° as the degree of extension is increased. The above information on the degree of disorientation of fibrils was compared with results on the degree of disorientation of crystallites in the same specimens. The disorientation of the crystallites was determined by large-angle X-ray scattering by the method C H A R A C T E R I S T I C S OF THE P O L Y E T H Y L E N E

Elongation 200 350

/~lJ

L

P±

degree 14 8

18 19

SAMPLES

(c2) ~, deg.

(c~) ~, deg.

Elonga crystaltion fibrils i lities 13.5 8-5

9.5 7

450 550

i

degree 10.5 9.5

15.5 14

fibrils I crystal-lities 7-5 7.5

6 6

of reference [4]. The 110 reflection was used. The mean-square angles of disorientation of the crystallites in the polyethylene specimens are shown in the Table. I t is seen that there is a correlation between the disorientations of the crystallites and fibrils. The authors are indebted to S. Ya. Frenkel' and B. M. Ginsb~rg for their interest in the work and for valuable comments. Translated by E. O. PHILLIPS

REFERENCES 1. F. GEIL, Monokristally polimerov (Polymer Single Crystals). Izd. "Khimiya", 1968 (Russian translation) 2. T. I. VOLKOV and V. (L BARANO~r, Novoye v metodakh issledovaniya polimerov (New Methods for Investigation of Polymers). Izd. "Mir", 1968 3. V. (L BARANOV, Optika i spektroskopiya 21: 610, 1966 4. A. Ye. GROMOV and A. I. SLUTSKER, Fizika tverdogo tela 5: 2185, 1963 5. R. J. SAM~ELS, J. Polymer Sci. A-2, 6: l l 0 l , 1968 6. L. TRELOAR, Fizika uprugosti kauchuka (The Physics of Rubber Elasticity). Foreign Literature Publishing House, 1953, (Russian translation)

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