Orientation in crystalline polymers related to deformation

Orientation & Crystalline Polymers Related to Deformation Z . W. WILCHINSKY A mechanism is proposed to account for the preferred orientation of the crystal c axis in de]ormed polymers, the c axis being parallel to the molecule backbone. It is assumed that tie molecules provide mechanical linkages between the c axes of neighbouring crystals, so that preferred c axis orientation takes place in the direction o[ extension. The orientation is expressed as the weight average
THE MOST commonly observed type of orientation produced by deformation of a polymer is one in which the molecule backbone, or a crystal axis parallel to the molecule backbone, is preferentially oriented in the direction of extension, this orientation being an increasing function of the extension. A mechanism relating the molecular orientation to a uniform deformation of a sample was proposed by Bailey1 in his studies of polystyrene. The nature of this orientation was described as one that would be experienced by slender rigid rods in a viscous uniformly deformed matrix. Although physically the orienting units cannot be regarded as slender rigid rods, Bailey's approach nevertheless may be quite useful. For example, this type of deformation proposed would transform a spherical region in a polymer into one of ellipsoidal shape. In crystalline polymers containing spherulites, one might expect an initially spherical spherulite to be deformed into an ellipsoidal one. Such deformations have indeed been observed 2'2. However, if this mechanism is operative in microscopic regions, it would require the crystals to undergo a similar deformation and every crystallographic direction would be unselectively affected. As this does not happen, this mechanism cannot be used for crystalline polymers without further qualifications. In the present paper, the mechanism considered by Bailey is modified to take into account the presence of undeformed crystals in a crystalline polymer. Relationships between the extension ratio and crystal orientation are then derived for several simple cases. MODEL

OF THE

POLYMER

In order to formulate a relationship between the orientation and the deformation producing it, one needs a physical model providing the appropriate constraints on the movement of the crystals during the deformation of the sample. Also, a chosen model should be consistent with known structural features of polymers. The following model of the polymer was chosen in accordance with the above considerations. Within a crystal, the molecule backbone is parallel to a principal crystal axis, taken by convention as the c axis. In this discussion, the term 271

Z. W. WILCHINSKY orientation will refer to the orientation of the c axis, which, of course, is equivalent to the orientation of the molecule backbone in the crystallized polymer. It will be assumed that the crystals themselves are not deformed when a sample of polymer is deformed; therefore, there must be mechanical interaction between a crystal and its neighbours. It will be further assumed that crosslinking among molecules is absent and that the interaction is in effect limited to the ends of the crystals, i.e. to the extremities of the crystal c axes. Such interaction is inherent in the older fringed miceUe concept of polymer structure. In the more recent folded chain concept, the interaction can be obtained by postulating the presence of 'tie molecules4,6. The treatment will not be restricted to either of these structural concepts; however, it will be assumed that the interaction is such that the c axis tends to orient in the direction of the deforming strain. To complete the picture, the interacting crystallites may constitute a larger structural unit, i.e. the spherulite. However, this is not a necessary requirement for the simple version of the mechanism.

Ca)

(c)

(b)

(d)

Figure /--Orientation induced by deformation.

(a) Spherical region I:efore deformation; three parallel lines in region shown. (b) Orientation of lines in direction Q is increased after a uniform deformation. (c) Crystals in spherical region, linked by tie molecules. (d) Crystal orientation is increased in direction (} after orientation

DERIVATION OF ORIENTATION

RELATIONSHIPS AND EXTENSION

BETWEEN RATIO

The nature of the orientation being considered is illustrated schematically

inFigure 1. A spherical region of a polymer is shown in (a) and an arbitrary direction in the sphere is represented by one of the lines. Other lines parallel 272

ORIENTATION IN CRYSTALLINE POLYMERS to the first are also shown. Let the polymer be subjected to a deformation such that the deformation ratios in three mutually orthogonal directions do not change from ono representative region of the polymer to another. Such a deformation will be referred to as a uniform deformation. After a uniform deformation by an extension in a direction Q, the sphere is deformed to an ellipsoid, and the lines have a greater degree of orientation in the Q direction. However, as indicated in (b), the orientation of each line has been changed by the same amount. Figure l(c) represents crystallites within a spherical region, the numbered members having the same orientation as the lines in (a). The region after a deformation similar to that of the preceding case is shown in (d). By virtue of the interactions via the tie molecules, the numbered crystallites have the same orientation as the lines in (b). For this simple model, it will be shown later that the final orientation depends only on the initial orientation (with respect to a chosen direction) and the deformation ratios in three orthogonal directions. The final orientation does not depend, for example, on whether a crystal in a spherulite is in a tangential, radial, or any other special position. Although the deformation on a macroscopic scale may be uniform, it will not be strictly uniform on a microscopic scale, since the crystals are not deformed. This factor will be neglected for the time being, but will be discussed later. Therefore, the change of orientation, according to the foregoing simple concept, will be described in terms of the transformation of a vector in a homogeneous material undergoing a uniform deformation s. Let a unit vector R have the direction of the c axis of a crystaUite. In terms of rectilinear coordinates, one may write

R = ( x ~ - X l ) i + (y~-yl) j+ (z~-zl) k = x i + Yj + Z'k

(1)

where xl, y~, Zl and x~, Y2, z~ are the coordinates of the ends of the vector and X = x2 - Xx, etc. If a reference direction Q is taken along the x axis, then the direction cosine along this direction may be written cos ,p = X

(2)

A measure of the degree of orientation in the direction Q can be quantitatively expressed in terms of the parameter 7 (cos 2 ~p)~ where the brackets denote a 'weight' average. In principle, this average can be obtained in the usual way:

(cos ~ ~> =~ cos ~ ~,AM,/I~/XM, t

(3)

i

where AM~ is the weight of material oriented at angle ¢i. As a result of a uniform deformation, the orientation of the material AM~ will have changed from 9i to 9~,d, and the orientation parameter after deformation is = ~ cos 2 ¢,.aAM,/2~AM, s

(4)

i

If the summations are replaced by integrals expressed in spherical coordinates, one may rewrite equations (3) and (4), respectively, as 273

Z. W. WILCHINSKY 2n


o ~° 2~.

(5)

f f l(9'~b)sin~d~bd¢ 0

o

and ~r 2~

o

o

(cos' 9a) = • 2.

(6)

f f l (¢, ~) sin f~d~ df~ 0

0

where the axis of the coordinate sphere of unit radius is along the x axis, ~b is the angle of longitude a n d I (~, ~b) is the appropriate distribution function. It may be noted that the orientation after deformation is expressable in terms of the initial distribution. Let the polymer undergo a deformation characterized by the deformation ratios X,/~ and ~ along the x, y and z axes, such that X > 1 and 2,/~ = 1. In accordance with the assumed orientation mechanism, the c axis having an initial orientation in the direction of R will have an orientation after deformation in the direction of Rx,,,,7 given by Rx,..,~ =X (x2 - x , ) i + ~ (Y2 -Y,) J+~/(z.~ -z~) k =XXi + eYj + ,,TZk

(7)

Also, after the deformation, the direction cosine of the crystal c axis along x is cos ~, =

xX/(x2X ' + #~Y~+

~2Z~)~/~

(8)

Thus cos 9d can be expressed in terms of the deformation ratios and the initial direction cosines along the coordinate axes. The orientation parameter after the deformation is then obtained by substituting (8) into the integrand of equation (6). In general, the solution is rather difficult. However, solutions were obtained for two important cases which will be referred to as the ideally stretched fibre and the sheet. Ideally stretched fibre The x coordinate and the direction of extension will be taken along the fibre axis. For a fibre to maintain a circular cross section during a uniform deformation, the deformation ratios will be 2~, and/~ =~/=)t -1/2. Designating the direction cosine along the fibre axis by cos ~ one then obtains by substituting into (8) : cos ~ = cos ¢4 =XX/{~2X~ + X-1 (Y2 + Z2)} 1/2

(9)

In terms of the spherical coordinates ¢ and ¢ cos ~ = cos ~/(cos 2 ~ + X-s sin 2 ~)1/2 274

(10)

ORIENTATION IN CRYSTALLINE POLYMERS since the transformation relationships can be expressed by X = c o s 9, Y = sin 9 sin ~ and Z = sin ~ cos q. The orientation parameter (cos ~ 9x> can be calculated by substituting cos 9x for cos Ca in equation (6). If the orientation is initially random, the distribution function l(9,~b) is a constant, and the integral is readily evaluated to give (cos ~,x> = ~

{1 - tan-1 (x~ - 1)1/5t ~a- -- -1)-iT~ j

(11)

Let us now return to the problem arising from non-deformed crystals. Specifically, let us consider these crystals in a spherulite which has been deformed from a spherical to an eUipsoidal shape. In a completely uniform deformation, all regions in the spherulite would be characterized by the same set of deformation ratios. Consider the spherulite divided up initially into laminar regions as indicated in Figure 2(a). These shells are not

(a)

(b)

(c)

Figure 2--Deformation modes within spherulite. (a) Initial spherulite subdivided into spherical shells. (b) Uniform deformation; shapes of all ellipsoidal surfaces are similar. (c) Spherulite deformed to same external shape as (b), but. thickness of an elemental shell is uniform intended to imply a laminar growth habit of the spherulite, but serve here only as a 'bookkeeping' device to follow the movement of the crystals within the spherulite. Each region constituting a spherical shell would be deformed to an ellipsoidal shell as indicated in Figure 2, (a) and (b). The wall thickness of any ellipsoidal shell is not uniform, but is proportional to the distance from the centre of the shell. To fulfil the geometric requirements of the deformation, the crystals in a spherical shell must be accommodated in the corresponding ellipsoidal shell. This might require a considerable flow of material and a considerable expenditure of energy, accompanied perhaps by disruption of crystals or local recrystallization of the polymer. Furthermore, the situation is aggravated progressively from the inner to the outer shells, since the displacements of crystals become more extreme and the amount of material involved becomes greater. To reduce the severity of this situation, the following mode of deformation was proposed ~, which in general greatly reduces the displacement of the crystals relative to their neighbours. Let a spherical shell be deformed into an ellipsoidal shell of approximately uniform thickness as indicated in Figure 2(c). The deforming strain 275

Z. W. WILCHINSKY thus tends to be equalized within a shell. Furthermore, in such a deformation carried out with a minimum of rearrangement, the displacements of the crystals relative to each other are smaller in the outer shells, which contain the bulk of the material, and greater in the more highly deformed core, which contains a smaller fraction of the material. The shape of the outermost shell, being the same as that of the spherulite, can be characterized by the extension ratio for the entire specimen. Similarly, the shapes of the inner shells can be characterized by higher extension ratios. It will be assumed that the orientation parameter for any shell is given by equation (11) provided that the effective extension ratio for that shell is used. Then, for the entire deformed spherulite, the orientation parameter (cos s 9x>, corrected for the enhanced orientation of the inner shells, can be expressed formally by the following equation v

1

= V l (cos2 ~x,v) dV

(12)

0

where V is the volume of the spherulite and dV is the volume of an elemental shell of uniform thickness having an orientation parameter
o-~

@

O'E

0-4' F

I

0-2

I

I

0'4

I

I

0"6

t

t

0.8

)

1.0

l/A, [extension ratio] -1

Figure 3 - - O r i e n t a t i o n p ~ r a m e t e r c a l c u l a t e d f o r a n ideally s t r e t c h e d fibre. (A) U n i f o r m d e f o r m a t i o n . (B) D e f o r m a t i o n b y m o d e o f Figure 2(e)

Curves for the orientation parameter according to equations (11) and (12) are shown as functions of 1/A in Figure 3. As can be noted, the values differ by approximately ten per cent over a considerable range of 1/A.

Ideally stretched sheet This case is characterized by the deformation ratios A,/x = 1 and ,/= 1 [A, i.e. the width remains constant while the thickness changes by the factor 1/Z. The expression for cos ~9x is then 276

O R I E N T A T I O N IN CRYSTALLINE POLYMERS

cos 2 9x = ?d cos ~ ~ / (sin ~ ~pcos s ~p+ As cos ~ 9 + X-s sin s ~psin 2 ,~)

(13)

Making the substitution of cos s 9x for cos 2 9d in equation (6) and integrating with respect to 4/, one obtains ~r

= ½f [As cos ~ ~ sin ~p/ {(A~ cos ~ ~ + •-= sin s 9) (x~ cos ~ ~ + sin s 9)} 1/2] d9 0

(14) A method for integrating the above expression analytically was not found; however, the integral was evaluated numerically by an electronic computer. The values thus obtained are listed in the Appendix. 1"0

08

0.6

0"~ 0

,,, I

I

0"2

t

I

I

I

L

04 0"6 1/)., [extension ratio] q

1

0'8

I

10

Figure 4--<)rientation parameter calculated for a stretched sheet whose width is held constant. (A) Uniform deformation. (B) Deformation by mode of Figure 2(c) Spherulitic deformation by the mode indicated in Figure 2(c) was also considered for the case of the sheet. Corrections were approximated for the enhanced deformation o f the inner laminae of the spherulite by the same general procedure that was ased for the fibre case. Further details are given in the Appendix. The orientation parameter, with and without the correction, is shown as a function of 1/X in Figure 4. Approximately the same difference between the two curves was obtained as for the fibre case.

DISCUSSION

A test of the derived equations would require accurate measurements of the c axis orientation for uniformly deformed samples. The orientation measurements can usually be carried out satisfactorily by X-ray diffractometer techniques using pole figure devices 8. However, preparation of suitable samples may often present a more serious problem due to the tendency of crystalline polymers to neck down during extension. Carefully stretched polyethylene films free of neckdown, and measurements of
Z. W. WlLCHINSKY

these films were reported by Stein and Norris r. Although the present theoretical development may not be rigorously applicable to those data, it is nevertheless of interest to compare the theoretical and experimental values for . This is done in Figure 5. A rather good agreement is obtained with the curve for an ideally stretched fibre, corrected for the enhanced orientation of the inner regions of the spherulites. Although this is quite encouraging, a more definitive test is desirable. 1.0

A

0-8

% o ij v

~6

0"4 0

I

I

I

lO0

200 Etongation, %

300

400

Figure 5---Calculated orientation parameters compared with data of Stein and Norris r for stretched film. Curve (A): fibre, uniform deformation. (B) Fibre, deformation by mode of Figure 2(c). (A') Sheet, uniform deformation. (B') Sheet, deformation by mode of Figure 2(c)

In the evaluation of the orientation parameter, the crystals were considered to be randomly oriented initially. If this condition is not fulfilled, an allowance for the initial orientation can be made provided that the orientation parameter is greater than one third. This orientation parameter would correspond to an extension ratio of say ~0, had the orientation been random initially. If the actual specimen is extended by a ratio X', the resulting orientation parameter would then correspond to an extension ratio of ~ = ~0Z' for the polymer with initially random crystal orientation. A final comment will be made regarding the relationship of the spherulite growth habit to the modes of deformation referred to in Figure 2. If the spherulite grows by laminar accretion, then the 'growth rings' can be represented by the concentric laminae in this figure. This type of growth habit has not been observed, to the writer's knowledge. If the spherulite has a radial growth habit, it would be expected to have radial structural regions*-11. One such region is represented schematically in Figure 6(a). After deformation, this region is represented by Figure 6(b) and (c); the former is by the deformation mode of Figure 2(b) and the latter by the mode of Figure 2(c). Further clues regarding spherulite growth habit can sometimes be obtained from a microscopic examination of fractures in polymers. For example, radial cracks in spherulites might be considered 278

O R I E N T A T I O N IN C R Y S T A L L I N E POLYMERS

evidence for a radial fibril structure. Radial cracks have been observed 12-15, but the spherulites in these cases were brittle and were not mechanically deformed. Detailed observations of deformed spherulites might disclose which of the modes of deformation in Figure 6 is the preferred one.

(a)

(b)

(c)

Figure 6--Deformation of a radial structure in a spherulite. (a) Before deformation. (b) After uniform deformation. (c) After deformation by mode of Figure 2(c)

The author is indebted to Mr Howard Oakley for calculating the values o/the integral of equation (14). Esso Research and Engineering Company, Linden, New Jersey (Received July 1963)

APPENDIX CALCULATION

OF

ORIENTATION

PARAMETERS

These calculations are for the fibre and sheet cases discussed for a strictly uniform deformation and also. for the modified mode of spherulite deformation illustrated in Figure 2(c). For a strictly uniform deformation, (cos 2 ~Px) for the fibre case was calculated in a straightforward manner by equation (11). For the sheet case, values of (cos ~~Px) were evaluated by electronic computer for 3. = 1 "25, 1-6, 2, 4, 6 and 10. For ~ = 1 and ~ , (cos ~ ~Px) is ] and 1 respectively. Other values were determined by graphical interpolation from a plot of (cos ~9x) versus 1/)t. Results of the calculations for the fibre and sheet are listed in columns A of Table 1. The evaluation of (cos 2 ~Px) for the modified mode of deformation, which involves an enhanced orientation of the inner region of a spherulite, is expressed formally by equation (12). This equation can be evaluated satisfactorily by the summation N

1

(c°s~ ~ ) = V~/~ W,~- v,,_~)

(15)

where V, is the volume within the nth ellipsoid starting with n = 1 for the 279

Z. W. W I L C H I N S K Y Table 1. Calculated values of . Columns A are for a uniform deformation within the spherulite. Values in columns B, calculated by equation (15), take into account the enhanced orientation of the inner regions of the spherulite F/bre

100 50 30 20 15 I0 8 6 5 4 3"5 3 2-5 2.25 2 1'8 1-6 1"5 1.4 1-25 1-18 1-11 1"05 1'01 1

Sheet

A

B

1-0000 0"9984 0"9956 0.9905 0"9827 0"9735 0'9523 0.9343 0.9017 0.8740 0.8309 0"7996 0'7580 0.7010 0"6643 0"6205 0.5789 0"5304 05034 0'4741 0.4281 0.4017 0.3760 O-3530 0"3397 0"3333

1.0000 0"999 0'997 0.994 0"989 0'982 0'968 0-955 0-932 0.911 0.876 0'852 0'818 0.768 0"732 0"691 0-649 0'596 0"564 0'529 0.469 0.435 0.398 0.365 0'342 0'3333

A 20 10 6-6667 6 5 4 3-333 2-857 2.5 2"222 2 1"818 1 "667 1.6 1.539 1.429 1.333 1 "25 1'177 1.111 1 "052 1

I'0000 0.952 0"9076 0"865 0'8516 0.825 0.7852 0.748 0.709 0'673 0"637 0.6029 0"569 0"538 0.5223 0'507 0"479 0"451 0.4242 0.398 0.374 0.352 0"3333

1"000 0.970 0"932 0"898 0-865 0"833 0"801 0"769 0.737 0.704 0-670 0.637 0"603

0"568 0.534 0"499 0'465 0.431 0"396 0-364 0'3333

innermost, and Vo=0; V~ is the volume of the spherulite, and (cos 2 9x..> denotes the orientation parameter for the nth ellipsoidal shell whose volume is V . - V ~ : . The value of
For the sheet, the following relationships among the ellipsoid axes were 280

O R I E N T A T I O N IN C R Y S T A L L I N E POLYMERS

used: A.=A.B., A.=A~C., and A . - C . = a . V,,)

Then the ratio V,,IVu is

x~ / (X,~,- 1) :> ,h~, - ~.~/(~.~,- 0 3

(17)

Results of evaluating equation (15) with the aid of (16) and (17) are listed in columns B of Table 1.

REFERENCES x BAILEY,J. India Rubb. World, 1948, 188, 225 2 HARRIS, P. H. and MAGiLL, J. W. J. Polym. Sci. 1961, 54, $47 a STEIN, R. S., RHODES, M. B., WILSON, F. R. and STEDHAM, S. N. Pure appl. Chem. 1962, 4, 219 4 GEIL, P. H. Meeting of American Chemical Society, Division of Polymer Chemistry, Preprints of papers presented, 1962, Vol. III, No. 2, p 12 BASSET'f, D. C., KELLER, A. and MITSUHASHI,S. J. Polym. Sci. A, 1963, 1, 763 6 WILCHINSKY, Z. W. Meeting of American Chemical Society, Division of Polymer Chemistry, Preprints of papers presented, 1962, Vol. III, No. 2, p 111 STEIN, R. S. and NCRRlS, F. H. J. Polym. Sci. 1956, 21, 381 s WILCHINSKY,Z. W. J. appl. Phys. 1960, 31, 1969 KELLER, A. J. Polym. Sci. 1959, 36, 361 l0 KErrlt, H. D. and PADDEN, F. J., Jr. J. Polym. Sci. 1959, 39, 123 la KHOURY, F. J. Polym. Sci. 1957, 26, 275 1~ INOUE, M. J. Polym. Sci. 1961, 55, 443 ~ REruNS, F. P. and WALTER, F. R. J. Polym. Sci. 1959, 38, 141 a4 REDING, F. P. and BROWN, A. Industr. Engng Chem. (Industr.), 1954, 46, 1962 xz VAN SCHOOTEN,J. J. appl. Polym. Sci. 1960, 4, 122

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