The director equation

The macroscopic nematodynamic equations describe the dynamics of the slowly relaxing variables, which usually are either connected with conservation laws or with the Goldstone modes of the spontaneously broken symmetries. To formulate them we wUl follow the traditional approach [65-67] rather than the one based more directly on the principles of hydrodynamics and irreversible thermodynamics [68]. In the nematic state isotropy is spontaneously broken and the averaged molecular alignment singles out an axis whose orientation defines the director n, i. e. an object that has the properties of a unit vector with n = -n. The static properties are conveniently expressed in terms of a free energy density whose orientational elastic part is given by [69] 

The elastic constants kn, 22, and 33 pertain to the three basic deformations splay, twist, and bend, respectively. For typical nematics with prolate molecules one has hi hi k22 and hi 10 N. 

Electric and magnetic fields E and H exert torques on n that can be derived from an additional contribution to the free energy 

Here Xa = X - Xx and 6a = e - Sx are the relative diamagnetic and dielectric anisotropies, so that the uniaxial susceptibility and dielectric tensors can be written in Cartesian coordinates in the form 

Static director configurations are obtained by minimizing the total free energy F = JAV feiast+fend Under the condition njnj = 1 with suitable boimdary conditions, which results in equating to zero the torque density F on the director. The generalization to dynamic situations is usually done by defining a vector 

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There are several major simplifications that can be used to tackle these equations. First of all, fluid flow is almost always neglected altogether. Then the Navier-Stokes equation is not needed at all, v is no longer a variable and we only need to solve the director equation (7) which will now be... 

The quantity f 6)d0 is the fraction of molecules that have their symmetry axes at an angle between 0 and 0- -d6 with respect to the director. Equation (3.14) now involves Dqq (0) and D (0)) = (Pl cos0)). Thus,... 

In Eq. (291) we derived an expression for the director equation of motion with dielectric and ferroelectric torques included. If the origin of the dielectric torque is in the dielectric biaxiality, the equation (with the elastic term skipped) will be the closely analogous one... 

Figure C2.2.11. (a) Splay, (b) twist and (c) bend defonnations in a nematic liquid crystal. The director is indicated by a dot, when nonnal to the page. The corresponding Frank elastic constants are indicated (equation(C2.2.9)).

The angular momentum conservation equation couples the viscous and the elastic effects. The angular profiles of the director and the effective viscosity data are computed for one set of material parameters based on published data in literature. The velocity profiles are also attained from the same dataset. The results show that the alignment of molecules has a strong influence on the lubrication properties. 

This equation describes the orientation around a common axis called the director of the domain. For perfectly parallel orientation (P2) equals 1. The orientation of the directors in the solution is described by the order parameter PD. The overall orientational order of the anisotropic solution is given by... 

This equation indicates that the average anisotropic part of A along the magnetic field is now scaled by a reduction factor P2(cos ft). As a consequence of the director tilt, the total Hamiltonian becomes... 

Consider a lamellar mesophase, being macroscopically aligned so that the symmetry axis, referred to as the director, has the same direction throughout the sample. If the transformation from the molecular coordinate system to the laboratory system is performed via the director coordinate system (D), Equation 2 reads... 

Here % specify the transformation from coordinate system j to system i. In Equation 3 only Dq q (Qdm) varies with the molecular motion. Since amphiphilic liquid crystalline systems generally are cylindrically symmetrical around the director Dq q (nDM) = 0 if qf 0. If it also is assumed that a nucleus stays within a domain of a given orientation of the director over a time that is long compared with the inverse of the quadrupole interaction, one now obtains for the static quadrupole hamiltonian... 

Comparison of experimental data with Equation 7 makes it possible to determine how the director is oriented with respect to the constraint (14) responsible for macroscopic alignment. 

Throughout this chapter we focus on the extended hydrodynamic description for smectic A-type systems presented in [42,43], We discuss the possibility of an undulation instability of the layers under shear flow keeping the layer thickness and the total number of layers constant. In contrast to previous approaches, Auernhammer et al. derived the set of macroscopic dynamic equations within the framework of irreversible thermodynamics (which allows the inclusion of dissipative as well as reversible effects) and performed a linear stability analysis of these equations. The key point in this model is to take into account both the layer displacement u and the director field ft. The director ft is coupled elastically to the layer normal p = in such a way that ft and p are parallel in equilibrium z is the coordinate perpendicular to the plates. 

To solve these equations we need suitable boundary conditions. In the following we will assume that the boundaries have no orienting effect on the director (the homeotropic alignment of the director is only due to the layering and the coupling between the layer normal p and the director h). Any variation of the layer displacement must vanish at the boundaries ... 

We write the solution as the vector X = (6,(j),u,vx,vy,i ,P,) consisting of the angular variables of the director, the layer displacement, the velocity field, the pressure, and the modulus of the (nematic or smectic) order parameter. For a spatially homogeneous situation the equations simplify significantly and the desired solution Xo can directly be found (see Sect. 3.1). To determine the region of stability of Xq we perform a linear stability analysis, i.e., we add a small perturbation Xi to... 

Equation (39) shows that nematic degrees of freedom couple to simple shear, but not the smectic degrees of freedom the modulus of the nematic order parameter has a non-vanishing spatially homogeneous correction (see (39)), whereas the smectic order parameter stays unchanged. The reason for this difference lies in the fact that J3 and /3 include h and p, respectively, which coupled differently to the flow field (see (22) and (23)). Equation (38) gives a well defined relation between the shear rate y and the director tilt angle 9o, which we will use to eliminate y from our further calculations. To lowest order 0O depends linearly on y ... 

Clearly flow aligning behavior of the director is present and do increases linearly with the tilt angle, do. Above a threshold in the Spain rate, y 0.011, undulations in vorticity direction set in. In Fig. 14 the results of simulations for y 0.015 are shown. In Fig. 15 we have plotted the undulation amplitude obtained as a function of the shear rate. The dashed line indicates a square root behavior corresponding to a forward bifurcation near the onset of undulations. This is, indeed, what is expected, when a weakly nonlinear analysis based on the underlying macroscopic equations is performed [54], In Fig. 16 we have plotted an example for the dynamic behavior obtained from molecular dynamics simulations. It shows the time evolution after a step-type start for two shear rates below the onset of undulations. The two solid lines correspond to a fit to the data using the solutions of the averaged linearized form of (27). The shear approaches its stationary value for small tilt angle (implied by the use of the linearized equation) with a characteristic time scale t = fi/Bi. 

The strategy is as follows. We start by rewriting the equations in cylindrical coordinates (r, ,z). The variables we consider are the layer displacement u (now in the radial direction) from the cylindrical state, the director n, and the fluid velocity v. The central part of the cylinder, r < Ri, containing a line defect, is not included. It is not expected to be relevant for the shear-induced instability. We write down linearized equations for layer displacement, director, and velocity perturbations for a multilamellar (smectic) cylinder oriented in the flow direction (z axis). We are interested in perturbations with the wave vector in the z direction as this is the relevant direction for the hypothetical break-up of the cylinder into onions. The unperturbed configuration in the presence of shear flow (the ground state) depends on r and 0 and is determined numerically. The perturbations, of course, depend on all three coordinates. We take into account translational symmetry of the ground state in the z direction and use a plane wave ansatz in that direction. Thus, our ansatze for the perturbed variables are... 

The three elastic constants are the Frank elastic constants, called after Frank, who introduced them already in 1958. They originate from the deformation of the director field as shown in Fig. 15.52. A continuous small deformation of an oriented material can be distinguished into three basis distortions splay, twist and bend distortions They are required to describe the resistance offered by the nematic phase to orientational distortions. As an example, values for Miesowicz viscosities and Frank elastic constants are presented in Table 15.10. It should be mentioned that those material constants are not known for many LCs and LCPs. Nevertheless, they have to be substituted in specific rheological constitutive equations in order to describe the rheological peculiarities of LCPs. Accordingly, the viscosity and the dynamic moduli will be functions of the Miesowicz viscosities and/or the Frank elastic constants. Several theories have been presented that are more or less able to explain the rheological peculiarities. Well-known are the Leslie-Ericksen theory and the Larson-Doi theory. It is far beyond the scope of this book to go into detail of these theories. The reader is referred to, e.g. Aciemo and Collyer (General References, 1996). 

The response of the director of a nematic phase to an applied electric field is dependent upon the magnitude of the dielectric permitivity (dielectric constants) measured parallel and perpendicular, and jl respectively, to the director and to the sign and magnitude of the difference between them, Le. the dielectric anisotropy, Aa, see Equation 9 and Figure 2.10. Since the dielectric permitivity measured along the x-axis is unique and the values of the dielectric permitivity measured parallel to the y- and z-axes are the same. 

Since the director n can be influenced by the flow, an additional dynamic equation must... 

There are four such steady-state solutions in the (n, 3>) plane, as depicted in the Ericksen diagram shown in Fig. 10-5. Since nematics are nonpolar (i.e., the head of the director is indistinguishable from the tail ), two of the four solutions shown in Fig. 10-5 are redundant of the remaining two solutions, one is unstable and the other stable. For A > 1 and a shear rate y that is positive, the stable solution is the one with a positive sign in Eq. (10-4). Equation (10-4) tells us that if A 1, the stable orientation angle 9 approaches 45°, while if A is near unity, 9 approaches zero that is, the director becomes parallel with the flow direction. 

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