FIGURE 6.4 Comparison between the observed (broken line) and calculated (solid line) X‐ray diffraction profiles of PLLA α form for the several layer lines (25°C).

      Source: Reproduced from Wasanasuk et al., Macromolecules 2011, 44, 6441–6552.

The H atom positions, which are important for the theoretical calculation of the mechanical properties, can be determined from the quantitative analysis of the wide‐angle neutron diffraction data, since the neutron beam is scattered almost equally by C, O, and H (and D) atomic species, which is quite in contrast to the X‐ray scattering, in which the scattering amplitude is extremely small for H atoms compared with the C and O atoms [49].

      As mentioned above, the finally‐determined space group is P2₁. This symmetry reduction is needed to generate the 00l diffraction peaks of the odd l values (see Figure 6.5a). Strictly speaking, even this model could not reproduce the observed 00l profile perfectly, although most of the hkl diffraction peaks were reproduced satisfactorily enough (Figure 6.4). This situation requires introducing some structural disorder that was not considered in the structural analyses mentioned above. After many trials, the disordered domain model was found to reproduce the data quite well. Figure 6.5b shows that the domains of a specific size are gathered together with the relative height disorder. The relative‐height shift between the domains is small (0.1–0.2 c), but it gives a good reproduction of the 00l diffraction profile as shown in Figure 6.5a. Since the domain size is large enough, the observed X‐ray diffraction profiles of the general hkl peaks are not seriously affected [14].

      6.2.3 Crystal Structure of the δ Form

The existence of the δ form (or the old name, α′ form) had been controversial for a long time. Historically, the suggestion of the δ form came from the measurement of the melt‐isothermal crystallization rate, which showed an anomalous peak at around 110–118°C [1, 50]. The DSC thermogram showed double melting peaks in the cold‐crystallization process [51]. Simultaneous DSC and WAXD measurement was performed as shown in Figure 6.6. The unoriented δ form sample was heated gradually, during which the 1D X‐ray diffraction profile was measured stepwise. When the temperature increased to 148–165°C, the 203 and 116 diffraction peaks, which were originally single peaks, changed to double peaks and the relative intensity of these two peaks was exchanged, as indicated by the arrows. This indicates that these two phases are coexistent during the transition, implying a thermodynamically first‐order transition. Besides, several peaks (210 and 213) started to appear, which were assigned to the diffraction peaks of the α form. In this way, the δ form is the crystal form thermodynamically independent of the α form.

FIGURE 6.5 (a) Comparison of the observed 00l reflection profile with the calculated curves for both of P2₁2₁2₁ and P12₁1 models and for the disordered domain model.

      Source: Reproduced from Wasanusuk et al., Macromolecules 2011, 44, 6441–6452.

      (b) PLLA domain model (several tens Å size along the a axis).

FIGURE 6.6 Temperature dependence of the 1D WAXD profile measured in the phase transition from the δ (α’) to α form of the unoriented PLLA sample (heating process).

      Source: Reproduced from Zhang et al., Macromolecules 2008, 41, 1352–1357.

FIGURE 6.7 Observed X‐ray 00l reflection profiles of PLLA α and δ forms.

      Source: Reproduced from Wasanasuk and Tashiro, Polymer 2011, 52, 6097–6109.

The crystal structure of the δ form was proposed by analyzing the 2D X‐ray diffraction pattern shown in Figure 6.2b, where the sample was prepared by stretching the melt‐quenched sample followed by annealing at ca. 100°C. In contrast to the α form, the X‐ray diffraction pattern is not very clear and consists of only the 41 broad diffraction spots. In addition, the diffuse streaks are detected relatively strongly along the layer lines, suggesting the existence of the remarkable disorder of the unit cell structure [9]. The orthogonal unit cell was proposed with the parameters a = 10.80 Å, b = 6.20 Å, and c (chain axis) = 28.80 Å. As seen in the 00l profile (Figure 6.7), the 0010 peak is quite strong, but the others are appreciably broad and diffuse, which is different from the sharp 00l peaks observed for the α form. This indicates that the chains takes the 10/3 helical conformation, but they are deformed to more extent than the α form (Figure 6.3). The detailed analysis of the X‐ray diffraction data revealed that not only the conformational disorder but also the chain packing disorder occurs in the crystal lattice of the δ form. In particular, the relative height between the neighboring chains is appreciably random compared with the α form, as seen from the more remarkable streaks along the layer lines of the 2D X‐ray diffraction pattern [9]. Besides, the aggregation of the domains is also more highly disordered than the α form (see Figure 6.5b). The X‐ray‐coherent crystallite sizes, as evaluated from the half width of the various diffraction peaks using Scherrer’s equation [52], are appreciably smaller than those of the α form (see Section 6.3.1).

      6.2.4 Crystal Structure of the β Form

As for the structure of the β form, the two types of the crystal structure were previously proposed by the X‐ray or electron diffraction data analysis: model (i) the orthorhombic type: a = 10.31 Å, b = 18.21 Å, and c (chain axis) = 9.00 Å, in which the six chains of 3/1 helical conformation are packed [17], and model (ii) the trigonal type with a = b = 10.52 Å and c (chain axis) = 8.80 Å, in which the three upward helices of 3/1 conformation are related by the space group symmetry P3₂ [18]. The present authors measured the 2D X‐ray diffraction pattern using a Mo‐Kα beam and analyzed it thoroughly (Figure 6.2d). The several diffraction peaks intrinsic to the β form could not be indexed reasonably by using the model (ii). Besides, the observed

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