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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 000l reflections along the chain axis do not satisfy the extinction rule requested for the P3₂ space group (the appearance of 000l reflections with l = 3, 6, 9, …) [20]. The 2D X‐ray diffraction pattern was measured again using the incident X‐ray beam of shorter wavelength (λ = 0.711 Å) than before, giving 40–50 observed diffraction spots in total [20]. The indexing of the observed peaks was made using the orthorhombic‐type unit cell:

The chain takes the 3/1 helical conformation [17, 18], and the six chains are packed in the unit cell, the same as the model (i). The 18 monomeric units are contained in the unit cell. According to the International Table for Crystallography [53], the orthorhombic unit cell must contain four or eight crystallographically asymmetric units. Therefore, if the orthorhombic system is assumed, the 18 monomeric units (or six chains) cannot be divided into the asymmetric units and are difficult to correlate with each other by the symmetric relation. So, the final space group symmetry selected is P1, which reproduced the observed X‐ray diffraction data well. However, it was impossible to determine the packing structure of many such chains uniquely because the observed diffraction peaks are relatively small in number compared with the total number of the adjustable structure parameters. As the hints to construct the crystal structure model, two important phenomena were observed:

1 The α (or δ) form transforms to the β form by the application of tensile or shear force, wherein the alternate packing structure of the upward and downward chains must be kept between them.

2 The relationships of the unit cell size among the three crystalline forms are a (α) ≈ a (δ) ≈ a (β), b (α) ≈ b (δ) ≈ b/3 (β), and c (α) ≈ c (δ) ≈ 3c (β), suggesting that the positions of the chains in the cell might not change very much before and after the structural transition from the α to β form.

Then, the chain packing structure of the α form was used as an initial model of the β form, which was enlarged three times along the b‐axis. The thus‐constructed model consists of the alternate packing of the upward (U) chains and downward (D) 3/1 chains. The positions and relative orientations of these U and D chains were modified in various ways to obtain the best reproduction of the observed X‐ray diffraction profiles. At present, the two possible models (model 2 and model 3) are considered as the best candidates for the β form (Figure 6.8). The U (and D) chains are surrounded by the U and D chains in a different environment depending on the local position, suggesting the frustrated structure as pointed out by Lotz et al. [18]. The comparison between the observed and calculated 1D‐WAXD profiles is presented in Figure 6.9.

FIGURE 6.8 Crystal structure of PLLA β form (model 2).

Source: Reproduced from Wang et al., Macromolecules 2017, 50, 3285–3300