Liquid Crystals. Iam-Choon Khoo

Liquid Crystals

Schematic illustration of stresses acting on opposite planes of an elementary volume of fluid.

Figure 3.10. Stresses acting on opposite planes of an elementary volume of fluid.

Let us ignore the external field for the moment. The formulation of the equation of motion for a fluid element is complete once we identify the viscous forces. Note that, in analogy to the pressure gradient term, the viscous force ModifyingAbove f With right harpoon with barb up Subscript v i s is the space derivation of a quantity that has the unit of pressure (i.e. force per unit area). Such a quantity is termed the stress tensor σ (i.e. the force is caused by the gradient in the stress; see Figure 3.10). For example, the α component of ModifyingAbove f With right harpoon with barb up may be expressed as

(3.55) f Subscript alpha Baseline equals StartFraction partial-differential Over partial-differential x Subscript beta Baseline EndFraction sigma Subscript italic alpha beta Baseline period

Accordingly, we may rewrite Eq. (3.54) as

(3.56)

where summation over repeated indices is implicit.

By consideration of the fact that there is no force acting when the fluid velocity is a constant, the stress tensor is taken to be linear in the gradients of the velocity (see Figure 3.10), that is,

(3.57) sigma Subscript italic alpha beta Baseline equals eta left-parenthesis StartFraction partial-differential v Subscript beta Baseline Over partial-differential x Subscript alpha Baseline EndFraction plus StartFraction partial-differential v Subscript alpha Baseline Over partial-differential x Subscript beta Baseline EndFraction right-parenthesis period

The proportionality constant η in Eq. (3.57) is the viscosity coefficient (in units of g cm⁻¹ S⁻¹). Note that for fluid under uniform rotation ModifyingAbove w With right harpoon with barb up Subscript o (i.e. ModifyingAbove v With right harpoon with barb up equals ModifyingAbove w With right harpoon with barb up Subscript o Baseline times ModifyingAbove r With right harpoon with barb up ), we have ∂v_β/∂x_α = −∂v_α/∂x_β, which means σ_αβ = 0.

Equation (3.56), together with Eq. (3.57), forms the basis for studying the hydrodynamics of an isotropic fluid. Note that since the viscosity for ModifyingAbove f With right harpoon with barb up Subscript v i s is a spatial derivative of the stress tensor, which in turn is a spatial derivative of the velocity, is of the form eta nabla squared ModifyingAbove v With right harpoon with barb up . Equation (3.54), therefore, may be written as

(3.58)

which is usually referred to as the Navier–Stokes equation for an incompressible fluid.

3.5.2. General Stress Tensor for Nematic Liquid Crystals

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) upper A Subscript italic alpha beta Baseline equals one half left-bracket StartFraction partial-differential v Subscript beta Baseline Over partial-differential x Subscript alpha Baseline EndFraction plus StartFraction partial-differential v Subscript alpha Baseline Over partial-differential x Subscript beta Baseline EndFraction right-bracket period

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

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