Crystal unit cell density

Figure 8.8. The temperature dependence of the crystal unit cell density of polyethylene.

Crystal Structure. The crystal stmcture of PVDC is fairly well estabhshed. Several unit cells have been proposed (63). The unit cell contains four monomer units with two monomer units per repeat distance. The calculated density, 1.96 g/cm, is higher than the experimental values, which are 1.80—1.94 g/cm at 25°C, depending on the sample. This is usually the case with crystalline polymers because samples of 100% crystallinity usually cannot be obtained. A dkect calculation of the polymer density from volume changes during polymerization yields a value of 1.97 g/cm (64). If this value is correct, the unit cell densities may be low. 

Real data is often available only for periodic systems, so only the density in the crystal unit cell need to be considered. Now the X-ray experiment gives structure factors Fh (along with errors at) which are related to the unit cell charge density via a Fourier transform,... 

According to Eq. (1.16), the elastic coherent X-ray scattering amplitude is the Fourier transform of the electron density in the crystal. The crystal is a three-dimensional periodic function described by the convolution of the unit cell density and the periodic translation lattice. For an infinitely extended lattice,... 

Fig. 18. Variation of crystal hardness, measured at room temperature, as a function of unit cell density for the various VF2/F3E compositions...

Fig. 19. Dependence of crystal hardness upon unit cell density through the Curie transition...

Fig. 6. Unit-cell density vs. macroscopic density for bulk-crystallized linear polyethylene62 ...

Samples of PET/PEN copolymers with 10, 20, 30, 50, 80 and 100 mol% PEN have been synthesized. Amorphous films of the samples were obtained by melt pressing above the melting point and quenching in ice-water. The samples were then crystallized by annealing the glassy materials at various temperatures. The degree of crystallinity was calculated from the amorphous density measured on quenched samples and from the crystal density derived from the crystal unit cell. 

The basic principle of the method is to determine by x-ray diffraction the volume of the crystal unit-cell. This, together with the measured crystal density gives directly the mass of the contents of a unit cell, and hence the mass of an integral number of molecules of the material. In most cases of complex organic molecules the number of molecules per cell is small (2-6) and fixed by the crystallographic symmetry and the symmetry of the molecule. 

The concept of percentage crystallinity implies the existence of separate crystalline and amorphous phases of constant structure, whereas there may be defects inside, or folds on the surfaces of, lamellar crystals. Crystallinity is usually measured indirectly, via measurements of density or enthalpy of fusion, although these give slightly different results than X-ray diffraction methods. For polyethylene, the density of the crystal unit cell... 

We can calculate a number of molecules in the crystal unit cell from experimental values of density and lattice parameters. The density of ice can be expressed as... 

Crystallization is characterized by several physical phenomena. With regard to rubbers, a crystal unit cell is of higher density (1.0 g/cm for NR) than is amorphous unfilled rubber (0.91 g/cm for NR). Crystallization is thus accompanied by an increase in density (i.e., decrease in volume) of up to 10%. Small volume changes (less than 3% for NR) have been measured by both hydrostatic balance and dilatometric techniques. Crystallite formation also involves the evolution of heat of fusion, and calorimetry has been used to derive values for the rate and degree of crystallization in NR in broad agreement with volume change data. 

Measurements of the densities of a series of 19 PTT samples with dsc crystallinities from 14 to 35% gave an extrapolated crystal density of 1441 kg/m , in closer agreement to the waxd unit cell density (39). Until more work was done, the author s research group used 1432 kg/m for crystallinity measurement, based on crystal structure parameters (34). Using a group contribution method, an amorphous density of 1295 kg/m was calculated (25) in good agreement with the extrapolated value obtained by Ziabicki (39). 

Hobbs SY, Billmeyer FW Jr. Crystal unit-cell dimensions and densities of linear aUphatic polyesters. J Polym Sci B 1969 7 1119-1121. 

Equation (Bl.8.6) assumes that all unit cells really are identical and that the atoms are fixed hi their equilibrium positions. In real crystals at finite temperatures, however, atoms oscillate about their mean positions and also may be displaced from their average positions because of, for example, chemical inlioniogeneity. The effect of this is, to a first approximation, to modify the atomic scattering factor by a convolution of p(r) with a trivariate Gaussian density function, resulting in the multiplication ofy ([Pg.1366]

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