Conductivity tensor properties

X is the rank-two conductivity tensor for a particular material. In Eq. 1.24, x is the material property that relates both the magnitude of effect Jq to the cause E and their directions—Jq is not necessarily parallel to E. 

It should be noted that because [TCNQ-TTF] crystallizes in monoclinic form and because conductivity is a tensor property, four independent pieces of conductivity data are required to completely define the conductivity in the principal directions. However,... 

The components of a symmetrical second-rank tensor, referred to its principal axes, transform like the three coefficients of the general equation of a second-degree surface (a quadric) referred to its principal axes (Nye, 1957). Hence, if all three of the quadric s coefficients are positive, an ellipsoid becomes the geometrical representation of a symmetrical second-rank tensor property (e.g., electrical and thermal conductivity, permittivity, permeability, dielectric and magnetic susceptibility). The ellipsoid has inherent symmetry mmm. The relevant features are that (1) it is centrosymmetric, (2) it has three mirror planes perpendicular to the... 

There has been a number of attempts [116,117] to analyze galvanomagnetic properties of inhomogeneous media. The effective conductivity tensor S was... 

It was supposed, that each of the phases is characterized by two parameters the ohmic conductivity <7o(r) and the Hall factor p(r). However each of properties CTo(r) and p(r) from the conductivity tensor (285) admit of only two values Co = cii and p [5, in the first phase, cto = ct2 and p = p2 in the second phase. The essence of ideas described in [118,119] consists in linear transformations from the old fields (j,E) to new fields (j,E ) such that the macroscopic properties of the new system are equivalent to those of the original system. These transformations can be applied only to a two-dimensional system, since they do not then change the laws governing a direct current ... 

Voloshinskaya and Fedorov (1973) and Voloshinskaya and Bolotin (1974) investigated the possibility of a correlation between magneto-optical properties and the anomalous Hall effect. The conductivity tensor and the dielectric tensor are related by... 

For the HM base case the mean mechanical properties have been used to calculate the hydraulic aperture distributions over the depth of the model. The continuum model and the applied methodology for the HM coupling in fractured rock does not allow the modelling of a fully HM-coupled system, hence the HM-modified hydraulic conductivity tensors were calculated at the mid point values of several depth ranges (Table 1). The results were assigned uniformly to the formation within each depth range (25m=>0m-50 m, 75 m => 50 m - 100 m, 175 m => 100 m -250 m, 375 m => 250 m - 500 m and 750 m 500 m - 1000 m). The variation in the calculated aperture values decreases as depth increases, which allows for the larger depth bands at the base of the model. 

As a consequence of the uniaxial symmetry all material properties of nematics have to be represented by tensors. For instance, the dielectric displacement D and E are connected by the dielectric susceptibility tensor e as ) = eoeE = tole E + i —e ) n-E)n]. Thus e depends in general on the local director orientation and is specified by two dielectric constants, ey and e (for E parallel and perpendicular to n, respectively). An analogous representation applies to the electric conductivity tensor [Pg.102]

Another example of a second rank tensor property is electrical conductivity which relates the current flow j in a particular direction to the electric field ... 

Liquid crystals as anisotropic fluids exhibit a wide range of complex physical phenomena that can only be understood if the appropriate macroscopic tensor properties are fully characterized. This involves a determination of the number of independent components of the property tensor, and their measurement. Thus a knowledge of refractive indices, electric permittivity, electrical conductivity, magnetic susceptibilities, elastic and viscosity tensors are necessary to describe the switching of liquid crystal films by electric and magnetic fields. Development of new and improved materials relies on the design of liquid crystals having particular macroscopic tensor properties, and the optimum performance of liquid crystal devices is often only possible for materials with carefully specified optical and electrical properties. 

The relationship between macroscopic properties and molecular properties is a major area of interest, since it is through manipulation of the molecular structure of me-sogens, that the macroscopic liquid crystal properties can be adjusted towards paricu-lar values which optimize performance in applications. The theoretical connection between the tensor properties of molecules and the macroscopic tensor properties of liquid crystal phases provides a considerable challenge to statistical mechanics. A key factor is of course the molecular orientational order, but interactions between molecules are also important especially for elastic and viscoelastic properties. It is possible to divide properties into two categories, those for which molecular contributions are approximately additive (i.e. they are proportional to the number density), and those properties such as elasticity, viscosity, thermal conductivity etc. for which intermolecular forces are responsible, and so have a much more complex dependence on number density. For the former it is possible to develop a fairly simple theory using single particle orientational order parameters. 

We have indicated in Section I that the optical properties of the crystal are characterized by the transverse dielectric tensor ex(k, cu) (1.79). The real and imaginary parts of this tensor being related by the Kramers-Kronig relations resulting from the linearity, ex(k, co) is itself determined by its imaginary part. In what follows, we assume that an eigendirection of c1 is excited, and we consider t"(k, oj) and the optical conductivity cuc (k, a>) under the common denomination of optical absorption . In fact, it is the conductivity that determines the absorption by the crystal of the energy of the plane wave (see Appendix A). 

Equation (2.26) for heat conduction and Eq. (2.3) for momentum transfer are similar, and the flow is proportional to the negative of the gradient of a macroscopic variable the coefficient of proportionality is a physical property characteristic of the medium and dependent on the temperature and pressure. In a three-dimensional transport, Eqs. (2.27) and (2.15) differ because the heat flow is a vector with three components, and the momentum flow t is a second-order tensor with nine components. 

Stress and strain tensors are not matter tensors like susceptibUity or conductivity, which were covered in earlier chapters. They do not represent a crystal property, but are, rather, forces imposed on the crystal, and the response to those forces, which can have any arbitrary direction or orientation. Although the magnitude and direction of strain are influenced by the crystal symmetry, they are also determined by the magnitude... 

The equations governing the steady state, quasi-one-dimensional flow of a reacting gas with negligible transport properties can easily be obtained from equations (l-19)-(l-22). When transport by diffusion is negligible 0 and Dtj 0 for ij = 1,..., N the diffusion velocities, of course, vanish [FJ 0 for / = 1,..., N, see equation (1-14)]. If, in addition, transport by heat conduction is negligible (A 0) and = 0, then the heat flux q vanishes [see equation (1-15)]. Finally, in inviscid flow 0 and K 0), equations (1-16)-(1-18) show that all diagonal elements of the pressure tensor reduce to the hydrostatic pressure, pu = pjj — P33 = P-The steady-state forms of equations (1-20), (l-21a), and (1-22) then become... 

Most engineering materials are isotropic in nature, and thus they have the same properties in all directions. For such materials we do not need to be concerned about the variation of properties with direction. But in anisotropic materials such as (he fibrous or composite materials, (he properties may change with direction. For example, some of the properties of wood along the grain are different than those in (he direction normal to the grain. In such cases the thermal conductivity may need to be expressed as a tensor quantity to account for the variation with direction. I he treatment of such advanced topics is beyond the scope of tlus text, and we will assume the thermal conductivity of a material to be independent of direction. 

It is valid for each continuum independent of the individual material properties and is therefore one of the fundamental equations in fluid mechanics and subsequently also in heat and mass transfer. The movement of a particular substance can only be described by introducing a so-called constitutive equation which links the stress tensor with the movement of a substance. Generally speaking, constitutive equations relate stresses, heat fluxes and diffusion velocities to macroscopic variables such as density, velocity and temperature. These equations also depend on the properties of the substances under consideration. For example, Fourier s law of heat conduction is invoked to relate the heat flux to the temperature gradient using the thermal conductivity. An understanding of the strain tensor is useful for the derivation of the consitutive law for the shear stress. This strain tensor is introduced in the next section. 

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