Frank elastic constant

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)).

We have omitted discussing such interesting properties of liquid-crystal solutions as the Frank elastic constants, the Leslie viscosity coefficients, cholesteric pitch, textured structure (or defects), and rheo-optics. Some of them are reviewed in recent literature [8,167], but the level of their experimental and theoretical studies still remains largely qualitative. 

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). 

FIG. 15.52 Elastic responses due to the deformation of the director field Frank elastic constants. Kindly provided by Prof. SJ. Picken (2003). 

TABLE 15.10 Some values for the Miesowicz viscosities and Frank elastic constants... 

In microscale models the explicit chain nature has generally been integrated out completely. Polymers are often described by variants of models, which were primarily developed for small molecular weight materials. Examples include the Avrami model of crystallization,- and the director model for liquid crystal polymer texture. Polymeric characteristics appear via the values of certain constants, i.e. different Frank elastic constant for liquid crystal polymers rather than via explicit chain simulations. While models such as the liquid crystal director model are based on continuum theory, they typically capture spatiotemporal interactions, which demand modelling on a very fine scale to capture the essential effects. It is not always clearly defined over which range of scales this approach can be applied. 

Only the most essential terms are taken into account, where k and kg are temperature-independent elastic constants. The quantity estimates the average nematic Frank elastic constant."... 

Here nd are elastic constants. The first, is associated with a splay deformation, K2 is associated with a twist deformation and with bend (figure C2.2.11). These three elastic constants are termed the Frank elastic constants of a nematic phase. Since they control the variation of the director orientation, they influence the scattering of light by a nematic and so can be determined from light-scattering experiments. Other techniques exploit electric or magnetic field-induced transitions in well-defined geometries (Freedericksz transitions, see section (C2.2.4.1I [20, M]. 

Fig. 4.12. (a) Temperature dependence of azimuthal Wtp dots) and zenithal Wo (squares) anchoring coefficients for nematic 5CB on rubbed Nylon with Tni being the transition temperature into the isotropic phase (b) A comparison of the ratios of the two anchoring coefficients W jW p (circles) with the ratio of the corresponding Frank elastic constants KijK2 [Q5](solid line) [64]. 

The general mean-field results, presented in this section, enable us to clarify this problem. It should be noted that Straley s theory was developed for a system of rigid rods and thus it takes into consideration only a short-range steric repulsion between molecules. On the other hand, in the theory of Helfrich and Petrov and Derzhanski the flexocoefficients are expressed in terms of Frank elastic constants, which, in turn, are determined by both the intermolecular attraction and repulsion. The relation between the two contributions can be clarified using Eqs (1.31) and (1.32), which can be used to obtain the following estimate of the flexoelectric coefficients ... 

Several molecular theories have been developed for correlating the Frank elastic constants with molecular constiments. The commonly employed one is mean-field theory [53,54]. In the mean-field theory, the three elastic constants are expressed as... 

Here, 0(z,t) deflnes the orientation of the director n=(cosd,0, sind), and Vx(z,t) is the x component of the flow velocity v=(u,0,0) pi is the moment of inertia per unit volume, which will be taken as zero in our calculation because its effect can be signifleant only during a subnanosecond pulse excitation Ku and K33 are the Frank elastic constants, and y and Y2 Leslie... 

Frank elastic constants were introduced giving the energy cost of individual deformation modes for splayed, K22 for twisted, and K33 for bent di-... 

The director takes on a specified orientation at the boundary. The strength of the nematic effect in flow is determined by a dimensionless Ericksen number E = ( 3 — a2)L Y/K, where L is the characteristic length scale and K is a representative Frank elastic constant. Orientation boundary layers will develop because of competition between the alignment induced by the preferred boundary orientation and the nematic potential in the bulk. 

Measured values of the Frank elastic constants Ku, 1 22. and K33 and their temperature dependence are available in the literature. " The common order of magnitude far from Tni is around 10 N. K n and K33 are close (they are equal in the Landau-de Gennes analysis). For the 5CB, K33 1.3/Tii, while K22 0.5/fn. Similar trends... 

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