Читать книгу Liquid Crystals - Iam-Choon Khoo - Страница 52

The general theoretical framework for describing the hydrodynamics of liquid crystals has been developed principally by Leslie [16] and Ericksen [17]. Their approaches account for the fact that the stress tensor depends not only on the velocity gradients but also on the orientation and rotation of the director. Accordingly, the stress tensor is given by

(3.59)

where the A_αβs are defined by

(3.60)

Note that all the other terms on the right‐hand side of Eq. (3.59) involve the director orientation, except the fourth term, α₄A_αβ.This is the same term as that for an isotropic fluid (cf. Eq. [3.57]), that is, α₄ = 2η.

Therefore, in this formalism, we have six so‐called Leslie coefficients, α₁,α₂,…, α₆, which have the dimension of viscosity coefficients. It was shown by Parodi [18] that

(3.61)

and so there are really five independent coefficients.

In the next few sections, we will study exemplary cases of director axis orientation and deformation, and we will show how these Leslie coefficients are related to other commonly used viscosity coefficients.

Figure 3.11. Sheer flow in the presence of an applied magnetic field.