Hydrodynamic equations, nematics

The hydrodynamic equations of the classical nematic ( 3.1) are applicable to the N, phase as well. There are six viscosity coefficients (or Leslie coefficients) which reduce to five if one assumes Onsager s reciprocal relations. A direct estimate of an effective value of the viscosity of from a director relaxation measurement indicates that its magnitude is much higher than the corresponding value for the usual nematic. 

For low-molecular-weight nematics, it has been known that the constitutive equation of nematics is entirely different from that for isotropic liquids. A phenomenological theory for the hydrodynamics of nematics (of low molecular weight) has been constructed by Ericksen and Leslie. Their equation reads... 

Here the discussion of viscous properties of nematic liquid crystals is based on the approach developed by F. Leslie [7]. For the nematic phase we have the equation for conservation of mass, the modified equations for conservation of momentum and energy E and one additional equation for conservation of the angular momentum of the director [8,9]. Totally there are seven equations two for scalar quantities (p and E), three for momentum m ) and two for director (due to condition = 1), which completely describe the hydrodynamics of nematics. In this case... 

The above is a quick summary of the relevant hydrodynamic equations for a simple fluid. The behaviour of a nematic liquid crystal is more complex in that the stress tensor is now non-symmetric. Another variable that is introduced is the director, , defined by a unit vector, where n —Allied to these constraints, one defines the rate of rotation of the director with respect to the background fluid by... 

It is usual that applied external fields like electric and magnetic fields, gravity, temperature gradients, pressure and concentration, shear and vortex flows carry out the nematic to a new equilibrium state so that these fields must be included in the hydrodynamic equations. 

The constitutive hydrodynamic equations for uniaxial nematic calamitic and nematic discotic liquid crystals are identical. In comparison to nematic phases the hydro-dynamic theory of smectic phases and its experimental verification is by far less elaborated. Martin et al. [17] have developed a hydrodynamic theory (MPP theory) covering all smectic phases but only for small deformations of the director and the smectic layers, respectively. The theories of Schiller [18] and Leslie et al. [19, 20] for SmC-phases are direct continuations of the theory of Leslie and Ericksen for nematic phases. The Leslie theory is still valid in the case of deformations of the smectic layers and the director alignment whereas the theory of Schiller assumes undeformed layers. The discussion of smectic phases will be restricted to some flow phenomena observed in SmA, SmC, and SmC phases. 

Although the constitutive hydrodynamic equations for nematic and polymeric liquid... 

For finite wavelengths, the collective dynamics of bulk nematics can be described within the hydrodynamic equations of motion introduced by Ericksen [4-8] and Leslie [9-11]. A number of alternate formulations of hydrodynamics [12-18] leads essentially to the equivalent results [19]. The spectrum of the eigenmodes is composed of one branch of propagating acoustic waves and of two pairs of overdamped, nonpropagating modes. These can be further separated into a low- and high-frequency branches. The branch of slow modes corresponds to slow collective orientational relaxations of elastically deformed nematic structure, whereas the fast modes correspond to overdamped shear waves, which are similar to the shear wave modes in ordinary liquids. 

On the other hand, in a pure liquid crystal system, liquid crystalline order, such as orientation order in nematic or layer order in smectic, is created under phase transition point, and the symmetry of the system is reduced. At the same time, new hydrodynamic fluctuation motions appear to be associated with new degrees of freedom. The modes of hydrodynamic fluctuations are characterized by a dispersion relation that can be obtained by solving the constitutive hydrodynamic equations of the system, giving the angular frequency wave number q of the fluctuations. It can be said that in a uniform alignment of the pure liquid crystal, the system universally satisfies the dispersion relation from the micrometer scale up to the length of the sample chamber, which means that the material keeps spatial homogeneity for the dynamics in pure system. 

The tendency of LCs to resist and recover from distortion to their orientation field bears clear analogy to the tendency of elastic solids to resist and recover from distortion of their shape (strain). Based on this idea, Oseen, Zocher, and Frank established a linear theory for the distortional elasticity of LCs. Ericksen incorporated this into hydrostatic and hydrodynamic theories for nematics, which were further augmented by Leslie with constitutive equations. The Leslie-Ericksen theory has been the most widely used LC flow theory to date. 

To complete this Section we have to mention the hydrodynamics of a nematic liquid crystal. In the equations described above the macroscopic flow was neglected. Such an approximation is usually justified since we describe only systems with zero total momenum. However, one should bear in mind that due to the coupling between the hydrodynamic and orientational degrees of freedom there are local flows accompanying the orientational changes, the effect known as backflow [38,39]. In some cases the backflow effects can alter substantially the phenomena in question. 

The equations for hydrodynamics of cholesterics are basically the same as for nematics, but there are some specific features related to the helical structure. 

Despite their highly successful record, MD or MC simulations are still hardly extended to the direct interpretation of complex set-ups, typical of most rheological experiments. In such cases it is preferable to employ mean-field or continuum descriptions, based of the numerical solution of the constitutive equations describing hydrodynamic properties. Such techniques were for instance applied to the prediction of transient director patterns of liquid crystalline nematic samples [11-14]. Hydrodynamic treatments are algebraically complex and computationally intensive, and their implementation is limited mostly to nematic phases. 

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]... 

In what follows we sketch the steps of this theoretical formalism for nematics. The first class of hydrodynamic variables is associated with local conservation laws which express the fact that quantities like mass, momentum or energy cannot be locally destroyed or created and can only be transported. If p(r,t), g=pv(r,t) and e(r,t), where v is the hydrodynamic velocity, denote respectively, the density of these quantities, the corresponding conservation equations are ((Landau L.D. and Lifshitz E. 1964). 

Most of the parameters involved in the hydrodynamic and electrodynamic equations for a nematic have been measured for different substances that show a uniaxial nematic phase. Among these one can mention the elastic constants (Blinov L. M and Chigrinov V. G. 1994) specific heat, the flux alignment parameter X and the viscosities Vj, i=l,2...5, the inverse of the diffusion constant yir the thermal conductivity (Ahlers, Cannell, Berge and Sakurai 1994), and the electric conductivity pijE. 

The hydrodynamic continuum theory of nematic liquid crystals was developed by Leslie [1,2] and Ericksen [3, 4] in the late 1960s. The basic equations of this theory are presented in Vol. 1, Chap. VII, Sec. 8. Since then, a great number of methods for the determination of viscosity coefficients have been developed. Unfortunately, the reliability of the results has often suffered from systematic errors leading to large differences between results. However, due to a better understanding of flow phenomena in nematic liquid crystals, most of the errors of earlier investigations can be avoided today. 

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