Surface tensor

Making the appropriate transformations from surface to beam coordinates, Koos et al. [122] note that xSjki has the same form as the second-order surface tensor x fk as a result of the applied field being parallel to the surface normal. Because the field is localized at the surface, the xn> effect should mimic the dipolar surface response with an effective susceptibility of... 

Here Wgn are the coefficients of the expansion of the electrostatic free energy, which can be obtained from the free energy Wfl(li), according to Equation (2.269). T2n are the irreducible spherical components of the (second rank) surface tensor, which describe the anisometry of the molecular shape, and can be calculated in the form of integrals over the molecular surface [25]. Given the nematic potential the distribution function... 

Recently, the surface tensor model has been used together with the dielectric continuum model to calculate the orientational order parameters of solutes in nematic solvents [8,9,27], Figure 2.32 shows the theoretical results for anthracene and anthraquinone in nematic solvents with different dielectric anisotropy. Considering only the surface tensor contribution, positive Szz and Sxx and negative are obtained, with Szz > Sxx > Syy. This corresponds to what could be expected on the basis of the molecular shape the long axis (z) is preferentially aligned with the director, and the normal to the... 

Fortunately these terms can be cast into the same form as the surface dipole susceptibility giving effective surface tensor components which are given by... 

The local geometry of a surface is generally characterized by two surface tensors, the metric tensor and curvature tensor. Letting (C, ) = (C C )i the tangent vectors to each of the coordinate curves at a point P can be represented as the basis vectors Uq, = a = 1,2) where the partial derivatives are to be evaluated where the coordinate curves on the curves intersect. It follows that an element of arc length with respect to the surface coordinates is represented by ... 

The molecular surface tensor is a symmetric second-rank tensor, which can be written in the diagonal form ... 

The second-order tensors are characterized by three invariants, that is, it is possible to combine the nine components in three ways to get quantities that are independent of the coordinate systems and express some fundamental properties. For the siuface component there are only two such invariants. The first of these can be written as tr (xj, where tr is short for trace and the operation that sums the diagonal elements of the tensor. The second is l/2 [tr (xj] - XjiXj, where a double dot product has been introduced. Since the trace itself is an invariant, some authors drop this term from the second invariant. In addition, the second invariant of this symmetric siuface tensor is the same as the third invariant in three dimensions, which is the determinant of x (see the remark after Equation 7.E1.8). There is a very important second-order surface tensor in the form of... 

The bulk and the surface energy can be expressed by the power series of an order parameter in this analysis. Therefore the surface tensor order parameter Qij was introduced to obtain the surface energy Q j is the tensor order parameter with the easy axis as its principal axis at the surface. 

The surface deformation energy can be expanded in powers of the gradient of surface tensor order parameter Finally,... 

Surface anisotropy has given rise to the technique of reflectance anisotropy spectroscopy (RAS), in which linearly polarized light is modulated between two principal directions (of the surface tensor) and the difference... 

Fig. 15.21 continued) (b) The corresponding D (l) set of surface tensor harmonics and (right) their application to an octahedral cluster. Note the 90 rotation patterns between corresponding D (l) and D (l) functions. 

Table 15.9 The symmetry properties of the surface tensor harmonics in the octahedral group 0 ...

Using [MoeClg] as an example, apply Stone s surface tensor harmonic model to the species [Ta6Cli2]. To what extent does the topological equivalent orbital model facilitate the application ... 

It is often argued that it is the shape anisotropy which is largely responsible for liquid crystal formation. Two methods have been proposed to introduce this view into the calculation of the interaction tensor for each conformer. In one it is assumed that the tensor is proportional to the moment of inertia tensor which is readily calculated from a knowledge of the molecular geometry [75]. However, it is found that this paramet-rization results in too great a dependence of the N-I transition temperature on the molecular length [76]. This observation was partly responsible for the development of the surface tensor model [77]. In this the interaction tensor is defined in irreducible form as... 

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