A TEST OF THE REFINEMENT PROCEDURE FOR DETERMINING THE CRYSTALLITE ORIENTATION DISTRIBUTION: POLYETHYLENE TEREPHTHALATE

A test of the refinement procedure for improving the crystallite orientation distribution function is presented for a fiber texture sample of polyethylene terephthalate. This is a particularly difficult example because the triclinic unit cell offers no simplification due to symmetry, and the pole figures are sharply peaked. The analysis employed 17 observed pole figures and an additional 29 unobserved pole figures reconstructed from the crystallite orientation distribution function. After three cycles of refinement, in which the maximum value of the coefficient was increased from 6 to 16, the standard deviations, cr and a,, of the plane-normal and crystallite orientation distributions were reduced by about a factor of 3. The refined crystallite orientation distribution function indicates that the c-axis tends to align along the fiber axis for this polyethylene terephthalate sample,


INTRODUCTION
In an ideal crystal, the vectors representing the three unit cell translations have quite definite directions. Real single crystals generally have a mosaic structure which has a significant effect on the measured integrated intensities, but the angular divergence is only of the order of minutes of arc. By contrast, one must consider a broad distribution of orientations for a polycrystalline specimen. Operations such as drawing and rolling influence the mechanical properties through changes in the crystallite orientation distribution, hence it is a matter of some practical importance to develop procedures for characterizing this distribution quantitively.
Although the distribution of crystallite orientations may be investigated using a variety of techniques, such as sonic velocity, optical birefringence, infrared dichroism, wide-line NMR, and fluorescence depolarization, the most complete description is furnished by x-ray diffraction. For example, birefringence and infrared dichroism yield only the second moment of the distribution, and the fourth moment can be obtained from NMR and fluorescence depolarization, but only diffraction permits evaluation of the higher moments.
One does not obtain the crystallite orientation distribution directly from the diffraction measurements. The information gained is the angular distribution of the normal to each observed Bragg plane. This may be conveniently represented by the plane-normal distribution, or the pole figure, for each plane. One is then faced with the problem of deducing the crystallite orientation distribution from the diffraction data. Jetter, McHargue and Williams introduced the concept of the inverse pole figure, and proposed a graphical method for affecting the transformation to this function. Quantitative analytical procedures for deducing the crystallite orientation distribution, or the equivalent inverse pole figure, for samples having fiber symmetry were given by Bunge, 2 and by Roe and Krigbaum, 3 and Roe ' has subsequently generalized the latter treatment for application to samples of arbitrary texture. Experimental applications to samples having fiber texture have been presented for aluminum, polyethylene, and polyethylene terephthalate, 6 while Roe's generalized treatment has been successfully applied to biaxially oriented polyethylene samples. 7 One unfortunate feature of these procedures is  (2), hexagonal (4), trigonal (8), orthorhombic (12), monoclinic (23), and triclinic (45). Moreover, it is often found in. practice that the polynomial series must be truncated prior to this theoretical limit because the simultaneous equations become ill-conditioned in the higher coefficients, presumably due to experimental errors in the pole figures. The resolution of the crystallite orientation distribution is, of course, a direct function of the number of terms retained. On the other side, the resolution required in any particular case depends upon the character of the preferred orientation. A small number of terms may suffice if the planenormal distributions are gradual and smooth, whereas a considerably larger number may be required to obtain a satisfactory representation if the plane-normal distributions are sharply peaked. One of us a has proposed to alleviate this problem through a refinement procedure based upon the analogy between the method used to deduce the crystallite orientation distribution function from pole figure data and that used to solve crystal structures from the measured structure amplitudes. Application of the refinement procedure was illustrated using four pole figures for an isotactic polystyrene sample, which was treated as pseudohexagonal. It was found possible to extend the series beyond the theoretical limit, l 22, to 1 36 by making use of the reconstructed pole figures. The representation achieved, as judged by several criteria, was significantly improved by the refinement. That example concerned a case of high crystal symmetry, and it was recognized that a sample of low crystal symmetry would be a more severe test of the refinement procedure. A test of this type is the subject of the present paper.

REFINEMENT PROCEDURE
We have selected for this test the pole figure data for one of the four samples of polyethylene terephthalate studied by Krigbaum and Balta. 6 This is an especially difficult case because it couples the low symmetry of the triclinic class with extremely sharp plane-normal distributions. The particular sample chosen, S-2, was prepared by melting and quenching, drawing at the glass transition temperature (70C) to an elongation ratio of 3.0, and crystallizing at fixed extension for 45 minutes at 159C. The crystallinity as estimated from the measured density was 27.5 . Pole figure data were obtained for 17 reflections. Analysis according to the Roe-Krigbaum procedure could include, as an upper limit, the 45 coefficients A,,, and Bl,,, through 8. In actual fact, Krigbaum and Balta 6 found the best representation to be obtained by retaining only the 28 coefficients through l 6. During the course of the refinement procedure, unobserved pole figures were calculated for an additional 29 planes. The pertinent crystallographic data for all 46 planes appear in Table I. The observed planes are numbered 1-17 in column one. Coltimn three lists the 20 values, where 0 is the Bragg angle, while columns four and five list the polar and azimuthal angles, (R) and , which locate each reciprocal lattice vector, r, in the coordinate frame, XYZ, of the unit cell (see Figure 1).
These values are based upon the unit cell parameters given for polyethylene terephthalate by Bunn and co-workers, 9  We begin our analysis by solving the crystallite orientation distribution function, w(, ), for the coefficients A t,,, and B through lm,x 6. We have fiber oxis   (003) and (111) planes. In each case the circles represent the observed plane-normal distribution, while the full and dashed curves show the series approximations obtained upon truncating at 2 16 and 6, respectively. As can be seen in Figure 2, the (003) planenormal distribution is fairly well represented with 2 6, althougli its maximum has been considerably broadened. On the other hand, truncation at the same level produces a very poor result for the (1ll) plane-normal distribution, as shown in 00:5 plane  In the first refinement cycle, which utilized the 17 observed and 20 smoothed, unobserved pole figures, the maximum value of retained was increased to 14. Hence, this solution for the crystallite orientation distribution function, w(, b), makes much more effective use of the measured pole figures. This first refined version of w(, b) was employed to recalculate the 20 unobserved planenormal distributions to a better approximation, 2 16. These were again smoothed, although the changes required in the second smoothing were more modest. These 37 pole figures were used in the second refinement cycle to solve w(, b) with 2 16. At this stage the number of unmeasured pole figures which were reconstructed from the crystallite orientation distribution function was increased to 29, the additional planes appearing in Table I

EVALUATION OF THE REFINEMENT PROCEDURE
One can see from Figures 2 and 3 that, as lm-x is increased during the refinement procedure, there is a noticeable improvement in the extent to which the information contained in the measured pole figures can be utilized in solving for the crystallite orientation distribution function. Correspondingly, when the pole figures for the observed planes are reconstructed from the crystallite orientation distribution function, as illustrated in Figure 4, these resemble the observed pole figures more closely as 2 is increased. This clearly reflects an improvement in the crystallite distribution function during refinement. We presume that the reconstructed pole figures for the unobserved planes are similarly improved, although in this case we can only observe a closer resemblance to the smoothed profiles as 2 is increased, as indicated by comparing the dotted and full curves in Figures 5 and 6. A more quantitative assessment is possible through the standard deviation, %, of the planenormal distribution, defined by .f_ (4) where the QI are calculated from the pole figures using Eq. (2). In order to estimate % for the unobserved planes, we have taken for q(r) the smoothed profiles from the second cycle (2 16), since these were used as input data for the final cycle. This standard deviation furnishes a measure of series termination effects. If the QI are calculated from the crystallite orientation distribution function using Eq. (4), the resulting standard deviation, designated 02, reflects errors in the crystallite    Table II. Comparison of the average values shows that both aq and tr2 are significantly reduced by the refinement process. The two columns labeled 2 18 indicate that the crystallite orientation distribution was not significantly improved by deleting the five planes which were judged to be poorly behaved. This means either that the selection of poorly behaved planes was unsuccessful, or that there was sufficient redundancy in the collection of 47 planes so that errors in the individual pole figures could be minimized by the least-squares procedure in solving for 2 18. The crystallite orientation distribution is expressed in terms of the angles 0 and tk shown in   Figure 7, where In (Q27 and the natural logarithm of the righthand side of Eq. (7)are separetely plotted as a function of 12. There appears to be a tendency for the circles to fall on a curve having upward curvature. Hence, if we use the approximation (Q2) ae-bt2, and replace the sum in Eq. (8) by an integral, we will obtain a lower bound for From the intercept and slope of the line drawn in Figure 7, a=0.246 and b= 9.3x10 -3 which gives for 2 6 1.30, for ;t 6. We conclude that the refinement procedure has reduced a, by a factor of between three and four, which is a significant improvement.
Finally, we examine how the crystallite orientation distribution function varies during the refinement procedure. The function G(, tk)= 4zw(.,4) has some advantage for graphical presentation, since a unit value of G(, q) indicates that as many crystallites have that particular orientation as would be found in a perfectly random sample. In Figure 8 are compared the G'(., b) function, calculated with 2 6 using the Roe-Krigbaum procedure, and that obtained from the refinement procedure using 46 pole figures. In the latter case 2 16, which we believe offers the best representation, since maxima which we judge to be spurious begin to appear when 2 _> 18. As expected, the G(, 4) function is considerably sharpened by the refinement procedure, the global maximum increasing from 17.1 at 2 6 to 53.8 when ;t 16. Of more significance are changes in the details of the G(, tk) function. We suspect that the contours marked 1, which correspond to G(, tk) 2, are of dubious significance. If these are deleted, the G(, ) functions for 2 6 and 16 are in qualitative agreement. However, a more detailed examination reveals that the maximum moves from 0 6 when 2 6 toward 0 0 as 2 is increased, arid the highest (inner) contours begin to resemble circles about the point 0 0. We therefore believe that for fiber texture samples of polyethylene terephthalate the c-axis tends to align preferentially along the fiber axis. Heffelfinger and Schmidt o also state that the c-axis of polyethylene terephthalate tends to align along the fiber axis, although we are unable to compare their results with ours in any more detailed manner. Daubeny, Bunn and Brown 9 indicated in their study that the c-axis is inclined by approximately 5 to the fiber axis, and Krigbaum and Balta 6 reached a similar conclusion from their earlier analysis of the data utilized here. CONCLUSION This example indicates that the refinement procedure can be successful, even in the difficult case presented by a material having triclinic symmetry and sharply peaked pole figures. The improvement is reflected in the fact that the reconstructed pole figures become more faithful representations of those observed, and through the reductions of trq and r,. An improvement is expected since the information contained in the observed pole figures can be more effectively utilized as higher terms are added to the series during refinement. Although the refinement procedure permits one to exceed the theoretical limit established by the Roe-Krigbaum treatment, which is 8 in the present case, there does appear to be a second limit beyond which the higher coefficients become ill-conditioned. This behaviour probably arises from mutual inconsistencies among the observed pole figures. In this example the best representation of the crystallite orientation 90 k=6 k=16 FIGURE 8 Contour maps of G(, b) in polar stereographic projection. For 2 6 contours 1-5 correspond to probability densities 2, 6, 10, 14 and 16, while for 2 16 contours 1-7 correspond to probability densities 2, 5, 11, 30, 40, 45 and 50.
distribution was obtained with 2 16. Thus, although the theoretical limit can be exceeded, the refinement procedure does not fully overcome the disadvantage arising from low crystal symmetry. This can be seen by comparison of the present results with those obtained by applying the refinement procedure to the more favourable case of isotactic polystyrene. In the latter case, which was treated as pseudohexagonal, it was possible to extend the series to 2 36 using only four observed pole figures, and to obtain much smaller standard deviations, trq 0.037 and aw 0.013.