Macroscopic tensor properties

Above four main features of the ferroics can be summarized as follows (1) the existence of long range order for at least one macroscopic tensor property at r <... 

At this point it appears that three averages, or order parameters, are necessary to relate the molecular tensor property to the macroscopic tensor property of the bulk phase. In fact, only two are necessary, since the three terms in brackets add to zero. Krrowledge of any two of the three terms allows the third to be determirred. Thrrs either before or after performing the average, the brrlk arrisotropy could have been written nsing jrrst two of these terms. The corrverrtion is to rrse the following averages for the two order parameters. 

The second rank-order parameter S can be derived from measurements of the macroscopic tensor properties such as birefringence and diamagnetic susceptibility. It varies typically between 0.4 at the clearing temperature to 0.7 at T ni- r= 20K in nematic phase. The fourth rank-order parameter (P ) may play an important role for a subtle analysis of the orientational distribution function and can be determined using polarized Raman spectroscopy. ... 

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. 

The average over the products of direction cosine matrices contains the orientational order parameters, and in terms of the principal components of the anisotropic part of the macroscopic tensor property becomes ... 

Symmetry restrictions exist for tensors describing macroscopic physical properties of all but triclinic crystals, and for tensors describing the local properties of atoms at sites with point-group symmetries higher than I. 

In order to see how this molecitle contributes to p, the tensor properly of the bulk phase, the components of the molecitlar tensor property in a coordinate system in which the z-axis points along the director must be calculated. To obtain such a coordinate system, the molecular coordinate system must first be rotated by around the z-axis and then this coordinate system must be rotated by 0 about the new y-axis (all counter-clockwise). This is shown in Figure 2.6(b), where the x -, y -, and z -axes denote the macroscopic coordinate system with the z -axis along the director. 

The anisotropy of liquid crystals stems from the orientational order of the constituent molecules, but the macroscopic anisotropy can only be determined through measurement of tensor properties, and macroscopic tensor order parameters can be defined in terms of various physical properties. The anisotropic part of a second rank tensor property can be obtained by subtracting the mean value of its principal components ... 

It has been assumed that molecular properties contribute additively to the macroscopic tensor components, which are consequently proportional to the number density. If intermolecular interactions contribute to the physical property, then deviations from a linear dependence of the property on density are expected. Also the contribution of orientational order will be more complex, since the properties will depend on the degree of order of interacting molecules. Effects of molecular interactions contribute to the dielectric properties of polar mesogens, and are particularly important for elastic and visoelastic properties. Molecular mean field theories of elastic properties predict that elastic constants should be proportional to the square of the order parameter this result highlights the significance of pairwise interactions. 

Several methods have been developed in order to determine the macroscopic optical properties [63], of which the simplest is the oriented gas model due to Chemla et al. [64, 65] In that method, the hnear and nonlinear susceptibilities (Eq. (8.2) are calculated from simple tensor sums of the (hyper)polarizabihties of the molecules constituting the elementary unit cell. Corrective factors can subsequently be added to account for the effects of local electric fields. The relevance of this method is ensured provided the intermolecular interactions are weak, while the macroscopic responses are strongly dependent on the values of local field factors. More sophisticated schemes take into account the intermolecular interactions. They include the supermolecule model [66-69], where an aggregate of... 

Other macroscopic properties can also be related to the tensor order parameter - in fact, any second rank tensor property of the system can be expressed in terms of Q. Using (4.1), the dielectric tensor can be written as [1]... 

The existence of the orientational order of the molecules results in anisotropy of most of the physical properties of liquid-crystalline phases. The magnitude of the observed anisotropy of the macroscopic property under investigation depends on the degree of orientational order, i.e. it depends on 8. However, the order parameter can only be determined experimentally if this property can be related directly to a tensor property of the molecules. Particularly suitable for determination of the order parameter is the anisotropy of the magnetic susceptibility, since the macroscopic volume susceptibility is directly related to the molecular susceptibility = being the... 

The Fisher relation (38) has a structure similar to a fluctuation dissipation relation in statistical mechanics It relates a macroscopic transport coefficient, the hydrodynamic speed, to the diffusion tensor and to the statistical properties of... 

The real and imaginary parts of the refractive index n quantify the scattering and absorption (or amplification) properties of a material- The refractive index is besfl derived from the susceptibility tensor y of the material, defined below, whi j describes the response of a macroscopic "system to incident radiation [212], Spe fically, an incident electric field E(r, t), where r denotes the location in the medium, tends to displace charges, thereby polarizing the medium. The change in dmd(r, the induced dipole moment, from point r to point r + dr is given in terms of th polarization vector P(r, t), defined as... 

Similarly as the trace, the anisotropy of the polarizability tensor of diatomic colli-sional systems can also be related to some macroscopic properties, namely to the refractive properties of atomic gases. The so-called Kerr constant, the anisotropy of the refractive index in the parallel and perpendicular directions to the external static electric field is given by,... 

In addition to the microstructural geometrical features described above, macroscopic, dynamical, rheological properties of the suspensions are derived by Brady and Bossis (1985). Dual calculations are again performed, respectively with and without DLVO-type forces. When such forces are present, an additional contribution (the so-called elastic stress) to the bulk stress tensor exists. In such circumstances, the term (Batchelor, 1977 Brady and Bossis, 1985)... 

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. 

The different symmetry properties considered above (p. 131) for macroscopic susceptibilities apply equally for molecular polarizabilities. The linear polarizability a - w w) is a symmetric second-rank tensor like Therefore, only six of its nine components are independent. It can always be transformed to a main axes system where it has only three independent components, and If the molecule possesses one or more symmetry axes, these coincide with the main axes of the polarizability ellipsoid. Like /J is a third-rank tensor with 27 components. All coefficients of third-rank tensors vanish in centrosymmetric media effects of the molecular polarizability of second order may therefore not be observed in them. Solutions and gases are statistically isotropic and therefore not useful technically. However, local fluctuations in solutions may be used analytically to probe elements of /3 (see p. 163 for hyper-Rayleigh scattering). The number of independent and significant components of /3 is considerably reduced by spatial symmetry. The non-zero components for a few important point groups are shown in (42)-(44). 

The different symmetry properties considered above (p. 131) for macroscopic susceptibilities apply equally for molecular polarizabilities. The linear polarizability a( w w) is a symmetric second-rank tensor hke Therefore, only six of its nine components are independent. It can always be transformed to a main axes system where it has only three independent components,... 

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