Nematic phase constitutive equation

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). 

Equations (10.75) and (10.78) determine the stress for a given velodty gradient and can be regarded as a constitutive equation. It should be emphasized that this constitutive equation holds in both the isotropic and the nematic phases since no presumption has been required about the equilibrium state. 

We shall now study the flow properties of the nematic phase. From the microscopic viewpoint, no spedal consideration may seem necessary sinoe the constitutive equation (10.75) and eqn (10.78) applies both for the isotropic and the nematic states, llns is not tte case. The rheological properties of solutions of todlike polymers are dian d entirely when the tem becomes nematic. 

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 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. 

Under isothermal conditions the constitutive equations for the description of flow phenomena in nematic and cholesteric liquid crystals are identical [48]. Nevertheless, a series of novel effects are caused by the helical structure of cholesteric phases. They arise firstly because of the inhomogeneous director orientation in the undistorted helix and secondly because of the winding or unwinding of the helix due to viscous torques. 

The foundations of continuum theory were first established by Oseen [61] and Zocher [107] and significantly developed by Frank [65], who introduced the concept of curvature elasticity. Erickson [17, 18] and Leslie [15, 16] then formulated the general laws and constitutive equations describing the mechanical behavior of the nematic and chiral nematic phases. 

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. 

---