From the volume of the unit cell V = 131 103 Å, the molecular mass M = 1703, and the experimental density of 1.025 g/cm3, the number of molecules per unit cell was calculated using equation (34) as 47.6 and then rounded as 48. As the structure developed from the hexagonal columnar phase we assumed that this basic structure was preserved, implying that there were two columns per unit cell of the P212121 pseudohexagonal phase. Therefore it was concluded that there were twenty four PBI 12-2 molecules per each column and hence twelve PBI 1-2 molecules in an asymmetric unit. 3D Reconstructed Electron Density Maps According to the symmetry of the space group P212121, the structure factor F200 must be real (phase angle 0 or ) while F110 and F201 must be imaginary (phase angle /2). The contribution of F200 and F110 to the electron density map is essentially to produce columns without helicity. Their phases are chosen so that the cores of the columns, formed by the aromatic parts of the molecules, have higher electron density than the alkyl-comprising surroundings. The addition of 201 and 111 reflections makes the columns helical, but the columns are all at the same height (same z) ( c and d). The sense of the helix (left or right) determines the phase angle of 201. In this case the phase angle is chosen arbitrarily to be +/2 corresponding to columns. After the phases of the first three peaks had been determined, F111 was added and its phase angle was varied between 0 and 2 in /6 increments. The resulting 12 electron density maps were examined using isoelectron at different levels. The best phase angle was chosen so as to produce the smoothest helical columns. This procedure is qualitatively equivalent to the maximum entropy method, where the “correct” map is the one containing least structural detail. The next step was the addition of F211, with its phase angle chosen according to the same criterion. As shown in Figures b and b d, the addition of F211 introduces a vertical shift of alternate columns up and down the z-axis. The difference to the maps caused by the inclusion of F211 is further highlighted in . All intensities and phase angles used in the calculation of the maps in b, d are listed in .
chain gives rise to the false meridional streaks, the equivalent to diffuse scattering in single crystals with correlated fluctuations. The molecular chain of poly-9-hydroxynonanoate is determined to possess a planar zigzag conformation. Two of such chains are arranged within an orthorhombic unit cell with lattice parameters, a = 7.47 ± 0.02 Å, b = 5.00 ± 0.02 Å, c = 12.37 ± 0.02 Å, very similar to those of polyethylene, a = 7.40 ± 0.02 Ǻ, b = 4.93 ± 0.02 Ǻ, c = 2.53 ± 0.02 Ǻ. The space group is determined to be Pnam. The angle between the C-C-C... zigzag plane of poly-9-hydroxynonanoate and the b-axis is determined to be 32.7°. Torsion of the carbonyl group at this particular angle between the C-C-C... zigzag plane of and the b-axis is determined to be 0°. The diffraction pattern shows sharp equatorial reflections and a series of reflections on the non-equatorial layer line, although one could confuse it with the 5th layer line due to the presence of noncrystallographic meridional streaks of periodicity of 12 Å. Hitherto this is the first time that a space group for an aliphatic polyester of the type [-(CH2)n-CO-O-]x with an even number of methylene units between successive ester groups is determined because poly-9-hydroxynonanoate is the first to show Bragg reflections on the non-equatorial layer line. In all previous cases studied there were no non-zero layer lines on the diffraction patterns and hence no space group could be determined. In fact, all previous polymers of the type [-(CH2)m-CO-O-]n, with an even number of methylene units between successive ester groups, had no three-dimensional long range order but only two-dimensional and this is why, theoretically, they could be considered as columnar liquid crystals.