Academic literature on the topic 'Liquid crystals Calculus of tensors'

When different portions of a solid are at different temperatures, thermal energy is transported from the warmer to the cooler regions. The thermal conductivity coefficient provides a quantitative measure of the rate at which thermal energy is transported along the thermal gradient. Thermal conductivity coefficients k relate the heat flux h [W/m2] to temperature gradient dT/dZ. In tensor form, The minus sign appears because heat flows from hot to cold. Thermal conductivity is measured in units of W/m K. Four contributions to thermal conductivity are illustrated in Fig. 18.1. The two principal mechanisms are from conduction electrons and from lattice vibration phonons. In transparent solids, especially at high temperature, photon transport can also be important. In porous media, convection currents from gas or liquid molecules can contribute to the thermal conductivity. Thermal conductivity is a polar second rank tensor like electric permittivity, magnetic susceptibility, and electrical resistivity but there is a basic question regarding the symmetry of transport properties such as electrical and thermal conductivity. The symmetry of tensors is partly dictated by geometrical considerations through Neumann’s Principle, and partly through thermodynamic arguments. For triclinic crystals there are nine nonzero conductivity coefficients kij . If the tensor is symmetric then kij = kji, and there are only six independent coefficients to be determined. For the dielectric constant it was shown that Kij = Kji, based on thermostatic energy arguments (Section 9.2). This argument does not hold for transport properties, but there is another principle based on irreversible thermodynamics. Onsager’s Theorem states that for transport properties, involving the flow of charge, heat, or atomic species, then . . . kij = kji. . . . The proof of Onsager’s Theorem depends on statistical mechanics and is beyond the scope of this book. From a practical point of view, Onsager’s Theorem is not very important because most transport experiments are performed on high symmetry metals, semiconductors, and insulators.