Tensor order

An aligned monodomain of a nematic liquid crystal is characterized by a single director n. However, in imperfectly aligned or unaligned samples the director varies tlirough space. The appropriate tensor order parameter to describe the director field is then... 

Every free index represents an increase in the tensor order one free index for vectors, uu two free indices for matrices (second order tensors), r -, three free indices for third order tensors,... 

The hypothesis that the constituents of the mixture have a Lagrangian microstructure (in the sense of Capriz [3]) means that each material element of a single body reveals a microscopic geometric order at a closer look then it is there assigned a measure Vi(x) of the peculiar microstructure, read on a manifold Mi of finite dimension rnp e.g., the space of symmetric tensor in the theory of solids with large pores or the interval [0, v) of real number, with v immiscible mixture (see [5, 9]). We do not fix the rank of the tensor order parameter u%. 

The notation used in the generic equation is strictly only valid for scalar properties. In the particular case when a vector property is considered the tensor order of the corresponding variables is understood to be adjusted accordingly. Hence, the quantities ipk, 4>k and second order tensors. 

The normalization constant G is often conveniently defined by setting Qzz equal to unity in perfectly oriented system. By definition, Q is real, symmetric, and traceless. Q vanishes in the isotropic phase as per the requirement of its suitability for an order parameter to describe the I-N transition. The macroscopic tensor order parameter Q can always be diagonalized ... 

We write the free energy expression (2.5.1) in a more general form in terms of the tensor order parameter (2.3.5), with i=j = 3 corresponding to the long molecular axis ... 

Optical second-harmonic generation experiments give a more detailed description of the anchoring at a microscopic scale [68,69 see also Chapter 5]. The molecule/surface interaction determines the orientational distribution in a thin surface layer extending up to 1 nm. The bulk uniaxial order develops on top of this layer via a transition layer of thickness which is well described by the usual mean-field theory, possibly including non-uniaxial components of the tensor order parameter. 

In conclusions, the elastic theory is adequate as far as the thickness exceeds the coherence length In future SFA experiments at a smaller length scale (down to about 1 nm) a more complete mean-field description of the spatial variations of the tensor order has to be developed in the firamework of the mean-field theory. 

Here W2 is the strength of the interaction and Qs is the preferred value of the tensor order parameter at the substrate, located at z = zs- In the case of uniaxial nematic order the anchoring strength W2 can be related to the anchoring strength W for the bare director description as W = 3iC2 S. ... 

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. 

As well known, the tensor order parameter in the bulk introduced by de Gennes is... 

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

A Landau theory for blue phase was proposed by Brazovskii, Dmitriev, Homreich, and Shtrik-man [7-10]. In this theory, the free energy of the blue phase is expressed in terms of a tensor order parameter which is expanded in Fourier components. The free energy is then minimized with respect to the order parameter with the wave vector in various cubic symmetries. In a narrow temperature region below the isotropic transition temperature, the stmctures with certain cubic symmetries have free energy lower than both the isotroic and cholesteric phases. 

This tensor order parameter is traceless and symmetric and vanishes in the isotropic phase. The anisotropic physical properties of the liquid crystal are closely related to the tensor order parameter. For example, the dielectric tensor of the hquid crystal is... 

For a cholesteric hquid crystal with the chirality q and helical axis along the z direction, the tensor order parameter is... 

As defined, <2 is a 3 x 3 symmetric traceless tensor, and therefore Q (rr) is a 3 x 3 symmetric traceless tensor. Q (independent parameters. It is convenient to expand the tensor order parameter in terms of the second-order spherical harmonics ... 

In order to achieve these goals, we have adopted a multi-scale approach that comprises molecular and mesoscopic models for the liquid crystal. The molecular description is carried out in terms of Monte Carlo simulations of repulsive ellipsoids (truncated and shifted Gay-Berne particles), while the mesoscopic description is based on a dynamic field theory[5] for the orientational tensor order parameter, Q. ... 

This chapter is organized as follows. In section 1.1, we introduce our notation and present the details of the molecular and mesoscale simulations the expanded ensemble-density of states Monte Carlo method,and the evolution equation for the tensor order parameter [5]. The results of both approaches are presented and compared in section 1.2 for the cases of one or two nanoscopic colloids immersed in a confined liquid crystal. Here the emphasis is on the calculation of the effective interaction (i.e. potential of mean force) for the nanoparticles, and also in assessing the agreement between the defect structures found by the two approaches. In section 1.3 we apply the mesoscopic theory to a model LC-based sensor and analyze the domain coarsening process by monitoring the equal-time correlation function for the tensor order parameter, as a function of the concentration of adsorbed nanocolloids. We present our conclusions in Section 1.4. 

To extract local static properties of the system, such as the density p and the tensor order parameter Q, a separate series of NVT runs were performed with spheres fixed at a few selected values of the reaction coordinate. The corresponding simulation configurations were analyzed by binning the LC molecules in a rectangular grid and computing the quantities of interest for each bin. In particular, for each bin with a volume Vb the density was found as p = Nb/Vb, where Nb is the number of molecules within this bin the tensor order parameter was computed as... 

As an alternative to the continuum theory, the mesoscopic approach can be based in a dynamic field theory for the tensor order parameter, Q. This tensor can be viewed as a coarse-graining of the microscopic probability distribution function 0(u, r, t) for the molecular orientation u. In this sense, Q corresponds to the symmetric, traceless part of the tensor of second moments of ip at the point r and time t ... 

---