Frank elasticity

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 a flowing liquid crystal, both the viscous stresses and Frank elastic stresses are normally important. Thus, the Ericksen theory for the viscous stresses, must somehow be combined with the Frank theory for the elastic stresses. This was accomplished by Leslie, who... 

The scaling law in Eq. (10-33) was predicted by Marrucci (1984) by assuming that the disclination density at steady state is set by a balance between the viscous energy density r]Y and the Frank elastic energy density K ja. Since the areal density pa is proportional to Pvh a h/a, this balance is... 

We start with Eq. (10-3) for the time evolution of the director n in the absence of Frank elastic... 

Problem 10.3(b) (Worked Example) Derive Eq. (10-31), the time- or strain-dependent shear viscosity in the absence of Frank elastic stresses. 

To evaluate the second term of Eq. (10-10), we need to obtain N. Using the definition of N in Eq. (10-12) along with Eq. (10-3), which is valid in the absence of Frank elastic stresses, we get... 

The radius a of the onions in the intermediate shear-rate regime of lyotropic smectics depends on shear rate, scaling roughly as a A similar texture size scaling rule is found in nematics (see Section 10.2.7) there it reflects a balance of shear stress r y against Frank elastic stress. In smectics, the two important elastic constants B and Ki have differing... 

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. 

Their proof relies on the fact that in the limit of small spatial distortions. The Marrucci-Greco potential of Eq. (18) reduces to the Frank elastic energy of Eq. (2) with... 

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]. 

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