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우종표

명지대학교 공과대학 화학공학과

Analysis of Cystalline Oientation of Semicrystalline Polymers Via Representation of Orientation Ellipsoids

Jong-Pyo Wu

Department of Chemical Engineering, College of Engineering

Myong Ji University

INTRODUCTION

Most polymer products which experience deformation during processing inevitably result in molecular orientation. It is known that this molecular orientation significantly affects the characteristics of products, such as their mechanical, optical, and electrical properties. For this reason there has been much effort to describe the state of orientation of polymer molecules.[1-5] Many different approaches possibly exist to represent the state of orientation of polymer  chains such as a complete probability distribution function for the orientation of polymer chain segments and a more compact form of orientation factor. Though either approach has its own advantages, they usually rely on assumptions about the directions of principal symmetry axes of orientation. These assumptions are obvious in the case of melt spun fibers and biaxially stretched or tubular blown films. The fiber axis in melt spun filaments and the machine, transverse, and normal directions in tubular blown or stretched films correspond to the symmetry axes of orientation. However, in general, the symmetry axes do not coincide with the experimental axes and one does not know them in the complex orientation state. Orientation ellipsoid can be applied in the experimental determination of these symmetry axes of orientation. In this manuscript, we will develop orientation ellipsoid using wide angle x-ray diffraction for the crystalline phase orientation in the semicrystalline polymers.

THEORETICAL ANALYSIS

Biaxial orientation factors based on two polar angles coordinate system have been developed by White and Spruiell. They described second moment orientation factors based on the polarizability tensor. Due to the nature of the second order tensor character of polarizability □ij, it can always be transformed to their symmetry axes, Ⅰ, Ⅱ, and Ⅲ.

□11 □12 □13     □Ⅰ  0   0

□ij =	□21 □22 □23  =   0  □Ⅱ  0	(1)

□31 □32 □33      0   0  □Ⅲ

This concept was applied by White and Cakmak[6] for the representation of orientation with unknown symmetry axes. Here the only and prerequisite problem in the characterization of orientation is the determination of the transformation angles for a unified views of chain orientation in fibers, films, and molded products.

The representation of ellipsoid is particularly useful in interpretation of the symmetry axes of second order tensors including those which are useful to specify orientation. The polarizability tensor leads to a dielectric tensor and a refractive index tensor[7]. Refractive index can be written in the form of Fresnel's ellipsoid.

On the other hand the pole figure from wide angle x-ray diffraction represents a 3-dimensional distribution of plane normals of specific crystallographic planes. The pole figures inherently reveal the detailed texture of orientation of the crystalline phase, i.e., they imply the orientation information of the high orders such as the second moment, fourth moment, etc.. If the second order tensorial quantity, which is related to specification of crystalline orientation, can be developed from the pole figures of wide angle x-ray diffraction, it will play an important role in the experimental determination of symmetry axes of crystalline orientation.

One can apply the concept of analogy between the plane normal vector in wide angle x-ray diffraction and the transition moment vector in infrared absorption[8] to extract a new second order tensorial quantity from x-ray diffraction data. Consider two coordinate systems. In one of them the axes (a, b, c) are adapted to a unit cell of the polymer crystal, while the axes (x, y, z) are the principal coordinate system. Suppose a plane normal vector N has the three component Na, Nb, Nc in the chain coordinate system, then the components Nx, Ny, Nz are given by

Nx,y,z = T Na,b,c	(2)

where T is the second order tensor which correlates the transformation between the polymer chain coordinates and the experimental coordinate system. When one performs the pole figure experiment, where the arbitrary machine direction in the pole figure is maintaining specific polar angles (□,□,□) with respect to coordinate system (x, y, z), each component Nx, Ny, and Nz of the individual plane normal is involved in the determination of averaged cosine square and its value is proportional to N2□,□,□. Since the distribution function I(□,□) obtained from the pole figure experiment corresponds to the number of plane normals pointing to the specific segment, the averaged cosine square value, X□,□,□, is given by

∬I(□,□) sin□ N2□,□,□d□d□

X□,□,□ =	(3)

∬I(□,□) sin□d□d□

where X□,□,□ is the averaged cosine square value obtained from the arbitrarily chosen machine direction in the pole figure experiment. On the other hand, the component of the individual plane normal in arbitrary direction (□,□,□) is given by

N□,□,□ = Nxcos□+Nycos□+Nzcos□	(4)

The Equations (3) and (4) result in

X□,□,□ = Xxcos2□+Xycos2□+Xzcos2□	(5)

The Equation (5) represents the equation of the ellipsoid obtained from the plane normal distributions. The X□,□,□ can be generalized in the form of a second order tensor as

Xij = <cos□icos□j>

∬I(□,□) sin□NiNjd□d□

=	(6)

∬I(□,□) sin□d□d□

The ellipsoid of Xij  can then be expressed in simple form on the appropriate space coordinate system.

¤¤ Xij xixj = 1	(7)

i  j

The schematic representation of the X ellipsoid is shown in Fig. 1.

EXPERIMENTAL

Wide angle x-ray diffraction pole figure experiments were carried out using an x-ray generator equipped with a single crystal orienter by General Electric(Model XRD-6) with a copper target and Ni-□filter(□=1.542Å). The operating voltage and current were 40KV and 50 mA, respectively.

The material used in this study were high density polyethylene, poly(ethylene terephthalate). These samples were mounted on the single crystal orienter which could be tilted up to 90o and rotated up to 360o. Before beginning the pole figure experiments, 2□ scan was performed and diffractometer was set to the maximum 2□ peak of each crystallographic diffraction plane. The planes used in this experiment were (010), (100), and (-105) planes for PET and (110), (200), and (020) planes for HDPE. The data acquisition is automated with step scanning □ from 0o to 90o in 5o intervals and □ from 0o to 360o in every 10o. Total 649 points of diffraction intensity data were obtained for each crystallographic planes.

RESULTS

To apply X ellipsoid from the pole figure data it is necessary to obtain a complete set of data X□,□,□ for all the possible □,□,□ combinations. In this sense it is useful to transform the experimental pole figure intensity distribution data I(□,□), obtained at the specific fixed direction, to the I'(□',□') for the arbitrary □,□,□ direction. The data was transformed to each step of angle □ and □ with the intervals of 5o and 10o, respectively. Then the X values at each step of transformation angle was calculated.

It reveals that the experimental results describe the ellipses well for different crystal unit cell structures. This supports the validity of the X ellipsoid concept developed in this study. Furthermore the experimentally determined symmetry axes of crystalline orientation indicate the unique directions irrespective of the different crystallographic planes. The results of X ellipsoid obtained from the pole figure data are summarized in Table I.

REFERENCES

1.	J.J. Hermans, P.H. Dermans, D. Vermaas, and A. Weidinger, Rec. Trav. Chim.,      65, 427 (1946)

2.	R.J. Roe and W.R. Krigbaum, J. Appl. Phys., 36, 2004 (1965)

3.	S. Nomura, H. Kawai, I. Kimura, and S. Kagiyama, J. Polym. Sci., 5, 479 (1967)

4.	J.L. White, Pure Appl. Chem., 55, 765(1983)

5.	M. Kosc, Colloid & Polym. Sci., 270, 106 (1992)

6.	J.L. White and M. Cakmak, Int. Polym. Proc., 2, 48 (1987)

7.	M. Born and N.E. Wolf, "Principles of Optics", 4th Ed., Pergamon, London         (1970)

8.	R. Zbinden, "Infrared Spectroscopy of High Polymer", Academic Press, N.Y.        (1964)

Table I.  Summarized Results of X Ellipsoid Obtain from the Pole Figure Data

MATERIAL	CRYSTAL STRUCTURE	PLANE	LEAST SQUARES ERROR(%)	SYMMETRY ANGLE(o)
PP	Triclinic	(040) (110)	0.04 0.05	+2 +1
HDPE	Orthorhombic	(020) (110)	0.42 0.63	0 +3
KEVLAR	Orthorhombic	(200) (110)	0.50 0.39	+2 +2
PET	Triclinic	(010) (100)	0.10 0.10	-2 -2

[eqn: rm]

Fig.1  Schematic representation of X ellipsoid