Incompressibility nematics

Six viscosity coefficients required for a description of the dynamics of an incompressible, nematic liquid crystal. 

The equation for mass conservation for an incompressible nematic can still be used in the form of divv = 0. 

The second rank viscous stress tensor found by Leslie for the incompressible nematic phase consists of nine matrix elements, each of them having the form ... 

Equation 4.5 is written in the one-constant approximation Ki = K) for simplicity reasons and the parameter rj is the effective rotational viscosity. In principle, in an incompressible nematic there are five independent viscosity coefficients, three for describing the dissipative stress due to the fluid velocity gradient, one coupling the director with flow and one due to the rotation of the director with respect to the fluid. However, the fluid flow can often be neglected and only an effective rotational viscosity r/ is used. 

Inserting (4.6) into the equation of motion (4.5), linearising for small fluctuations, and solving the dynamical equation for incompressible nematics, the relaxation rates for the two eigenmodes are... 

Continuum theory has also been applied to analyse tire dynamics of flow of nematics [77, 80, 81 and 82]. The equations provide tire time-dependent velocity, director and pressure fields. These can be detennined from equations for tire fluid acceleration (in tenns of tire total stress tensor split into reversible and viscous parts), tire rate of change of director in tenns of tire velocity gradients and tire molecular field and tire incompressibility condition [20]. 

Consider a number n of stiff polymer components (here stiff is used to mean semiflexible ) and define orientation-dependent ideal and interacting response (n x n) matrices X0(Q, u, u ) and X(Q, u, u ) respectively. In this case, orientational correlations have to be included in addition to the usual isotropic ones. Doi et al. [36-38] have developed the theory for solutions of stiff homopolymers. Their formalism is applied in Appendices C and D to multicomponent blend mixtures of stiff polymers without and with the incompressibility condition respectively. The interaction potentials comprise anisotropic (also called nematic) contributions as well as the usual isotropic ones ... 

Another frequently employed simplification is the ID assumption according to which all variables depend only on one coordinate, namely, the one transversal to the plane of the nematic layer, say, the coordinate. This means that the incident fight should be an infinite plane wave (hence this approximation is often called the infinite plane wave approximation), and, by virtue of incompressibility and the boundary conditions, v = Vx z, t), Vy z, t), 0) which... 

The stratified structure of a smectic liquid crystal imposes certain restrictions on the types of deformation that can take place in it. A compression of the layers requires considerable energy - very much more than for a curvature elastic distortion in a nematic - and therefore only those deformations are easily possible that tend to preserve the interlayer spacing. Consider the smectic A structure in which each layer is, in effect, a two-dimensional fluid with the director n normal to its surface. Assuming the layers to be incompressible, the integral... 

A number of important ideas concerning the N, phase have been discussed theoretically - molecular statistical and phenomenological theories, " " continuum theories, " topological theories of de-fects, - " etc. For example, Saupe and Kini " who used different theoretical approaches, have both concluded that the incompressible orthorhombic nematic has 12 curvature elastic constants (excluding three which contribute only to the surface torque) and 12 viscosity coefficients. 

According to this model, the free energy expansion, in the case of uniaxial nematic order and incompressible strain of the sam-... 

An extension of rubber elasticity (i.e. of the description of large, static and incompressible deformations) to nematic elastomers has been given in a large number of papers [52, 61-66]. Abrupt transitions between different orientations of the director under external mechanical stress have been predicted in a model without spatial nonuniformities in the strain field [52,63]. The effect of electric fields on rubber elasticity of nematics has been incorporated [65]. Finally the approach of rubber elasticity was also applied recently to smectic A [67] and to smectic C [68] elastomers. Comparisons with experiments on smectic elastomers do not appear to exist at this time. Recently a rather detailed review of the model of an-... 

Let us assume that a liquid is incompressible, B oo, and discuss orientational (or torsimial) elasticity of a nematic. In a solid, the stress is caused by a change in the distance between neighbor points in a nematic the stress is caused by the curvature of the director field. Now a curvature tensor dnjdxj plays the role of the strain tensor ,y. Here, indices i,j = 1, 2, 3 and Xj correspond to the Cartesian frame axes. The linear relationship between the curvature and the torsional stress (i.e., Hooke s law) is assumed to be valid. The stress can be caused by boundary conditions, electric or magnetic field, shear, mechanical shot, etc. We are going to write the key expression for the distortion fi-ee energy density gji, related to the director field curvature . To discuss a more general case, we assume that gji t depends not only on quadratic combinations of derivatives dnjdxj, but also on their linear combinations ... 

The symmetric part of the strain tensor can be associated with changes in the relative positions of particles within a strained sample. For incompressible materials this is zero, and such an assumption is normally applied to nematic liquid crystals. Howev-... 

Most of the hydrodynamic effects observed in nematic liquid crystals can be explained by the LEP theory. Its generality is sufficient for the following discussion. According to this theory the viscous part of the stress tensor a j for an incompressible uniaxial nematic liquid crystal is... 

A biaxial nematic phase is characterized by two orthogonal directors a and b of length 1. Introduction of a third orthogonal vector c of length 1 simplifies the notation of the following equations. According to Saupe, Covers and Vertogen the viscous part of the stress tensor of an incompressible biaxial nematic phase is the sum of the sym-... 

The chiral nematic is considered incompressible, i.e., of constant density p with a nonpolar unit director n (i.e., i = 1). This implies that the external forces and fields responsible for the elastic deformation, viscous flow, etc., are much weaker than the intermolecular forces giving rise to the local order, i.e., between the chiral molecules. We will consider a volume of material V bounded by a surface 5 v and O) represent linear velocity and local angular velocity, respectively, i.e.. 

The most general expression of the viscous stress tensor,, compatible with the symmetry of a uniaxial and incompressible calamihc nematic liquid crystal, was determined first by Leslie as ... 

A fluid is said to be incompressible whenever its mass density p is constant, resulting in p = 0. Since we shall always suppose that the nematic is incompressible, the result in (4.33) allows us to replace the mass conservation law (4.29) by... 

It is convenient at this point to summarise the Ericksen-Leslie dynamic equations for nematics in the incompressible isothermal theory when the director inertial term is neglected. These are the most frequently used forms of the equations and we state them in the notation introduced in the previous Sections. They consist of the constraints... 

We emphasise again that these rely on the Parodi relation (4.96) being valid. The coefficients a to as can also be considered as a canonical set of five viscosities. This is known to be experimentally viable for a large majority of nematics when they are treated as incompressible. 

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