# Crystal orientation and twinning in cold-rolled ultrahigh molecular weight polypropylene.

INTRODUCTIONIn a preceding paper (1), we sought to understand the relationship between crystal orientation and mechanical anisotropy of cold-rolled ultrahigh molecular weight polypropylene (UHMWPP) films. It was found that all reciprocal lattice vectors of (110), (040), and (130) planes aligned perpendicular to the direction of rolling with preferential orientation in the film normal direction. Such peculiar orientation behavior is very different from the so-called [Alpha] and [Beta] type orientation of monoclinic crystals originally proposed for biaxially drawn or cold-rolled conventional polypropylene (PP) (2, 3). According to Takahara et al. (2) the a type orientation is generally characterized by a preferential orientation of crystal b-axis toward the film normal accompanied with the random orientation of crystal [a.sup.*] and c-axis around it. This process would give rise to a preferential planar orientation of (040) crystal plane within the film surface. The physical interpretation of the [Alpha] type orientation may be the lamellar orientation in the film surface, i.e., the planar lamellar orientation. The [Beta] type orientation is characterized by the preferential orientation of (130) plane normal perpendicular to the film surface associated with the (110) twinning due to an increased biaxial strain. When the strain is removed, the restoring force would allow the (110) twin to rotate such that the (110) plane normal would orient perpendicular to the film surface. This process is known as the y type orientation. The authors (2) pointed out that the appearance of the [Gamma] type orientation resembled the (110) twinning of the cold-rolled polyethylene proposed by Frank et al. (4).

In the present study, we further analyze the deformation behavior of the unidirectionally and orthogonally rolled UHMWPP films by a model simulation based on the orientation distribution function approach. We adopt the three types of orientations here which were originally proposed by Takahara et al. (2). A combination of [Alpha] and [Gamma] type orientations and slippage mechanisms have been incorporated in the model calculation under the assumption that the volume remains constant during deformation. The calculated pole figures are compared with the observed pole figures of (110), (040), and (130) planes.

A MODEL SIMULATION

Introduction of Slippage Angle Around the Rolled Direction

To introduce the slippage angle around the rolled direction, it is convenient to redefine the Euler coordinate system by transforming the Cartesian coordinate 0 - [a.sup.*] bc of a monoclinic PP crystal with respect to the reference Cartesian coordinate 0 - [x.sub.1][x.sub.2][x.sub.3] such that the polar angle [Theta] is now defined as the angle between the crystal c axis with respect to the rolled direction. Now the transformation matrix [a.sub.im] may be described in terms of three Euler angles [Theta], [Phi], and [Eta];

[Mathematical Expression Omitted],

where [a.sub.im] is the direction cosine of the crystal c-axis and [x.sub.i] axis [ILLUSTRATION FOR FIGURE 1 OMITTED]. The components of this matrix correspond to that of the coordinate system in Fig. A-1 by the following equation, i.e.,

[Mathematical Expression Omitted].

and

cos [Theta] = sin [[Theta].sup.*] sin [[Phi].sup.*]

sin [Theta] sin [Phi] = sin [[Theta].sup.*] cos [[Phi].sup.*]

sin [Theta] cos [Phi] = cos [[Theta].sup.*] (3)

Squaring both sides of the second and third relations of Eq 3 and adding them, one obtains the following relation;

[sin.sup.2] [Theta] = [sin.sup.2] [[Theta].sup.*] [cos.sup.2][[Phi].sup.*] + [cos.sup.2][[Theta].sup.*]

[Theta] = [sin.sup.-1]([+ or -][[[sin.sup.2][[Theta].sup.*] [cos.sup.2][[Phi].sup.*] + [cos.sup.2][[Theta].sup.*]].sup.1/2]) (4)

The third relation of Eq 3 may be rearranged as

[Phi] = [cos.sup.-1](cos [[Theta].sup.*] / sin [Theta])

Since [Mathematical Expression Omitted]

sin [Theta] sin [Eta] = -sin [[Phi].sup.*] cos [[Theta].sup.*] sin [[Eta].sup.*] + cos [[Phi].sup.*] cos [[Eta].sup.*]

[Eta] = [sin.sup.-1] [(-sin [[Phi].sup.*] cos [[Theta].sup.*] sin [[Eta].sup.*]

+ cos [[Phi].sup.*] cos [[Eta].sup.*])/sin [Theta]] (5)

Eq A-22 may be rewritten using the Euler angles [Theta], [Phi], and [Eta] as follows;

[Mathematical Expression Omitted]

where w(cos [Theta], [Phi], [Eta]) represents the orientation distribution function in an undeformed state with respect to the deformed state w[prime](cos [Theta][prime], [Phi][prime], [Eta][prime]) [ILLUSTRATION FOR FIGURE 1B OMITTED].

Suppose the slippage of (040) and (110) planes occurs in the orthogonal direction during rolling; then Eq 1 may be rewritten as [Mathematical Expression Omitted] where [Omega] is the slippage angle. The prime ([prime]) symbol signifies the deformed state.

The relationship between the orientation distribution of crystallites, w(cos [Theta], [Phi], [Eta]), and that of j-th individual crystal planes, q(cos [[Theta].sub.j], [[Phi].sub.j]), may be obtained through the expansion of Jacobi's polynomials with the expansion coefficient l = 8, i.e.,

w[prime](cos [Theta][prime], [Phi][prime], [Eta][prime])

[Mathematical Expression Omitted]

where the coefficient [A.sub.lmn] and [B.sub.lmn] may be given as

[Mathematical Expression Omitted]

On the other hand, the orientation distribution function of individual diffraction plane may be expanded as

[Mathematical Expression Omitted]

Here, the polar angle [[Theta].sub.j] and the azimuthal angle [[Phi].sub.j] of the [r.sub.j] vectors are defined in Fig. A-1. The coefficients [A.sub.lmn] and [B.sub.lmn] in Eq 9 and [Mathematical Expression Omitted] and [Mathematical Expression Omitted] in Eq 10 may be related in terms of the addition theorem of Legendre polynomials,

[Mathematical Expression Omitted]

[Mathematical Expression Omitted] and [Mathematical Expression Omitted] are the associated Legendre polynomials which may be further related to the Legendre polynomials as follow;

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

The polar angle [[symmetry].sub.j] and the azimuthal angle [[Phi].sub.j] are already defined in Fig. A-1. The angles of (110), (040), and (130) planes of polypropylene are tabulated in Table 1 based on the orthogonality conditions such that [r.sub.040] [multiplied by] [r.sub.c] = 0 ([Alpha] type) for planar lamellar orientation, [r.sub.130] [multiplied by] [r.sub.c] = 0 ([Beta] type) and [r.sub.110] [multiplied by] [r.sub.c] = 0 ([Gamma] type) for the crystal orientation due to twinning.

EXPERIMENTAL

Ultrahigh molecular weight polypropylene (UHMWPP) powders were supplied by Himont Co. The viscosity average molecular weight [M.sub.v] of UHMWPP was approximately 3.9 x [10.sup.6]. The method of cold-rolling of UHMWPP slabs was thoroughly described in [TABULAR DATA FOR TABLE 1 OMITTED] the preceding paper (1). The specimens with rolled ratios 3.6 x 1, 4.0 x 1, 4.5 x 1, 3 x 2, and 4 x 2 were used for wide angle X-ray (WAXD) studies. The WAXD pole figures of (110), (040), and (130) crystal planes were acquired on a 2 kW Rigaku X-ray diffractometer with the aid of a pole figure attachment.

RESULTS AND DISCUSSION

In a preceding paper (1), we reported the observation of the peculiar WAXD pole figures in the unidirectionally rolled UHMWPP in that all plane normals of (110), (040), and (130) were concentrated in the thickness direction. Although the extent of orientation increases with increasing rolled ratio, the general trend is the same for all rolled samples. In the cross-rolled biaxial UHMWPP films, the (110), (040), and (130) plane normals are populated near the thickness direction except that the contour peak broadens toward the transverse direction [ILLUSTRATION FOR FIGURES 2A AND B OMITTED]. This deformation behavior is at variance with that of the conventional cold-rolled PP in that the (110) plane normals are oriented in the transverse direction and that of the (130) is tilted for 40 to 50 [degrees] with respect to the thickness direction (5). The (040) plane normals are concentrated in the thickness direction which is typical for the orientation of the [Alpha] type monoclinic crystals (2). The reason why UHMWPP exhibits a peculiar orientation behavior is presently unclear except that more than one orientation mechanism is required to account for the deformation; the identification of underlying deformation mechanisms is the central issue in this study. It has been known that UHMWPP melt viscosity is extremely high, which implies that there must be substantial chain entanglements in the melt. Such entangled chains may be entrapped during crystallization which probably act like tie chains between inter-crystalline regions. It may be speculated that the entrapped entangled chains, when subjected to severe deformation under cold-rolling, probably cause UHMWPP to exhibit a different type of orientation than normal PP.

To illustrate the orientation behavior, we calculate the pole figures of the (110), (040), (130) plane normals and the crystal c-axis based on the ideal, [Alpha], [Beta], and [Gamma] type orientation models. Figure 3a shows the pole figures of (110), (040), and (130) of the typical [Alpha] type orientation in which the plane normal of (040) is concentrated in the film thickness direction while the (110) is oriented in the transverse direction. The (130) is tilted for about 40 to 50 [degrees] between the thickness and transverse direction. In the case of the orientation of the [Beta] type, the plane normal of (130) aligns in the thickness direction [ILLUSTRATION FOR FIGURE 3b OMITTED]. However, the (110) and the (040) show maxima at about 20 to 30 [degrees] and 40 to 50 [degrees] between the thickness and the transverse directions, respectively. On the other hand, the (110) plane normals of the [Gamma] type crystals are populated in the thickness direction while the (130) and (040) plane normals are oriented about 20 to 30 [degrees] and 70 to 75 [degrees] between the thickness and transverse directions, respectively [ILLUSTRATION FOR FIGURE 3c OMITTED]. Obviously, the individual orientation process alone is not adequate to account for the experimentally observed pole figures.

It is natural to consider the combination of various orientation modes to account for the experimentally observed complex orientation behavior of this cold-rolled UHMWPP. According to our SAXS studies in the preceding paper (1), lamellae are oriented preferentially in the film plane with some level of tilting. Such oriented lamellae would give the orientation of (040) plane normal in the thickness direction. Thus the [Alpha] type orientation must be, although by no means sufficient to explain the experimental pole figures, one of the possible deformation mechanisms. We introduce the slippage angle in the deformation to modify the orientation behavior of the [Alpha] type. The distribution of the (040) plane normal broadens in the transverse direction, but the fit is rather poor for other crystal planes. We thought that additional mechanisms such as the [Beta] and [Gamma] types should be incorporated in the simulation. Based on the trend in Fig. 3c, the combination of [Alpha] and [Beta] type would not allow the (110) plane normal to orient perpendicular to the film surface. Thus the combination of the [Alpha] and [Gamma] type orientations appears most reasonable compared with the observed pole figures. Again, a simple addition of the two processes will not give a good fit, therefore a slippage mechanism has been incorporated in the calculation.

Several slippage angles from 20 to 35 [degrees] were employed assuming an equal contribution from the [Alpha] and [Gamma] types. The relative contributions of the [Alpha] and [Gamma] type can be varied in the calculation, but it would become nothing more than an adjustable parameter, and thus this approach was abandoned. As can be seen in Figs. 4a, to d, the calculated pole figures capture the trend of the experimental observation in that all plane normals of (110), (040), and (130) orient in the thickness direction. Among the pole figures, the calculation with a slippage angle of [+ or -]22.5 [degrees] gives the best fit. Moreover, this slippage angle corresponds to the shear plane where the shear stress is maximum. Thus it was utilized for other cases of different rolled ratios. As shown in Figs. 5a to e, this model calculation fits remarkably well with the observed pole figures of unidirectionally rolled and cross-rolled UHMWPP films. It may be concluded that the combination of the crystal orientation and twinning in conjunction with the slippage mechanism is required to account for the observed X-ray pole figures.

CONCLUSIONS

The peculiar WAXD pole figures were observed in cold-rolled UHMWPP in which all plane normals of (110), (040), and (130) were populated in the film thickness direction. This deformation behavior in biaxially cold-rolled UHMWPP is very different from that of conventional PP. It was speculated that chain entanglements entrapped in the intercrystalline regions during solidification probably causes UHMWPP to undergo a different type of deformation during cold-rolling relative to that of ordinary PP. The present pole figures cannot be explained in terms of an individual orientation of the [Alpha], [Beta], or [Gamma] types. We found that the model based on the combined [Alpha] and [Gamma] type orientations in conjunction with the slippage angle of [+ or -]22.5 [degrees] gives satisfactory fit with the observed pole figures. The slippage angle 2 [Omega] = 45 [degrees] corresponds to the shear plane where the maximum shear stress occurs during deformation.

APPENDIX: THEORETICAL BACKGROUND

A transformation matrix [Mathematical Expression Omitted], which characterizes the orientation of a Cartesian coordinate 0 - [u.sub.1] [u.sub.2] [u.sub.3] of a structural unit with respect to a reference Cartesian coordinate 0 - [x.sub.1] [x.sub.2] [x.sub.3] in the bulk specimen (laboratory coordinate), may be described in terms of three Euler angles [[Theta].sup.*], [[Psi].sup.*], and [Eta][prime]:

[Mathematical Expression Omitted]

where [Mathematical Expression Omitted] is the direction cosine of the axis [u.sub.m] of the structural unit with respect to the axis [x.sub.i] of the bulk specimen [ILLUSTRATION FOR FIGURE A-1 OMITTED]. The reference axis [x.sub.1] is chosen to correspond the film normal (thickness) direction, [x.sub.2] to the transverse direction, and [x.sub.3] the rolling direction. The polar angle [[Theta].sup.*] represents the angle between the crystal c-axis and the thickness direction. For monoclinic crystals such as that of polypropylene, Eq A-1 may be rewritten as follows:

[Mathematical Expression Omitted]

where [e.sub.i] and [e.sub.m] are the unit vectors of the 0 - [x.sub.1] [x.sub.2] [x.sub.3] and 0 - [u.sub.1] [u.sub.2] [u.sub.3] coordinates, respectively. The subscripts [a.sup.*], b, and c stand for crystal [a.sup.*], b, and c axes. Note that the [a.sup.*] axis of the reciprocal lattice vector is not identical to the crystal a-axis of a mono-clinic crystal; e.g., it is tilted about 9 [degrees] away from the crystal a-axis of the monoclinic crystal of PP. The orthogonality condition of the components of directional cosine requires that

[Mathematical Expression Omitted]

When an arbitrary vector [r.sub.c] along [u.sub.3] axis is subject to deformation, it changes to [r[prime].sub.c] such that

[Mathematical Expression Omitted]

where [[Lambda].sub.ic] represents the draw ratio in each direction. Rewriting Eq A-4, one obtains

[Mathematical Expression Omitted]

where

[Mathematical Expression Omitted]

Applying Eq A-3 to Eq A-6 leads to

[Mathematical Expression Omitted]

Similarly, one can derive the following expression along the [u.sub.1] axis,

[Mathematical Expression Omitted]

rearranging Eq A-8 gives

[Mathematical Expression Omitted]

Assuming that the orthogonality of vectors [a.sup.*] and c is maintained during deformation, i.e., [Mathematical Expression Omitted], Eq A-9 may be rewritten as,

[Mathematical Expression Omitted]

Subsequently, one obtains the following expression from Eq A-6

[Mathematical Expression Omitted]

then,

[Mathematical Expression Omitted]

Rearranging Eq A-10 gives,

[Mathematical Expression Omitted]

From the second relation of Eq A-8 one gets

[Mathematical Expression Omitted]

Rearranging Eq A-14 gives

[Mathematical Expression Omitted]

From Eq A-9 and A-10,

[Mathematical Expression Omitted]

Differentiating Eq A-15 with respect to [[Eta].sup.*] and [[Eta].sup.*[prime]] and substituting Eq A-16 into Eq A-15, one obtains

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

Substituting Eqs A-12 and A-13 into Eq A-18 leads to

[Mathematical Expression Omitted]

Assuming that volume remains unchanged during deformation,

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

Finally, the orientation distribution function of crystallites after deformation, w[prime](cos [[Theta].sup.*[prime]], [[Psi].sup.*[prime]], [[Eta].sup.*[prime]]), may be expressed as

[Mathematical Expression Omitted]

where [w.sup.*](cos [[Theta].sup.*], [[Psi].sup.*], [[Eta].sup.*]) represents the orientation distribution function in an undeformed state.

ACKNOWLEDGMENT

The authors acknowledge a support from the National Science Foundation, Grant No. MSM 87-13531. S. H. thanks the Ministry of Education, Science, and Culture, Japan for a support of his travel. We are indebted to Dr. Ted Dziemianowicz of Himont Co. for supplying the UHMWPP samples and Mr. T. Sakatani of Rigaku Co. to double check the WAXD pole figures.

REFERENCES

1. S. Hibi, T. Niwa, C. Wang, T. Kyu, and J. S. Lin, this issue.

2. H. Takahara, H. Kawai, Y. Yamaguchi, and A. Fukushima, Sen-I Gakkaishi, 25, 60 (1969).

3. H. Uejo and S. Hoshino, J. Appl. Polym. Sci., 14, 317 (1970).

4. F. C. Frank, A. Keller, and A. O'Connor, Phil. Mag., 3, 64 (1958).

5. S. Hibi, K. Suzuki, T. Hirano, T. Torii, K. Fujita, E. Nakanishi, and M. Maeda, Kobunshi Ronbunsho, 45, 237 (1988).