Liquid Crystals. Iam-Choon Khoo

Liquid Crystals

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.

Schematic illustration of sheer flow in the presence of an applied magnetic field.

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

3.5.3. Flows with Fixed Director Axis Orientation

Consider here the simplest case of flows in which the director axis orientation is held fixed. This may be achieved by a strong externally applied magnetic field (see Figure 3.11), where the magnetic field is along the direction ModifyingAbove n With ampersand c period circ semicolon . Consider the case of shear flow, where the velocity is in the z‐direction, and the velocity gradient is along the x‐direction. This process could occur, for example, in liquid crystals confined by two parallel plates in the y‐z plane.

In terms of the orientation of the director axis, there are three distinct possibilities involving three corresponding viscosity coefficients:

1 η1: is parallel to the velocity gradient, that is, along the x‐axis (θ = 90°, ϕ = 0°).

2 η2: is parallel to the flow velocity, that is, along the z‐axis and lies in the shear plane x‐z (θ = 0°, ϕ = 0°).

3 η3: is perpendicular to the shear plane, that is, along the y‐axis (θ = 0°, ϕ = 90°).

These three configurations have been investigated by Miesowicz [19], and the ηs are known as Miesowicz coefficients. In the original paper, as well as in the treatment by deGennes [3], the definitions of η₁ and η₃ are interchanged. In deGennes notation, in terms of η_a, η_b, and η_c, we have η_a = η₁, η_b = η₂, and η_c = η₃. The notation used here is attributed to Helfrich [6], which is now the conventional one.

To obtain the relations between η_1,2,3 and the Leslie coefficients α_1,2,…,6, one could evaluate the stress tensor σ_αβ and the shear rate A_αβ for various director orientations and flow and velocity gradient directions. From these considerations, the following relationships are obtained [3]:

(3.62) StartLayout 1st Row eta 1 equals one half left-parenthesis alpha 4 plus alpha 5 minus alpha 2 right-parenthesis comma 2nd Row eta 2 equals one half left-parenthesis alpha 3 plus alpha 4 plus alpha 6 right-parenthesis comma 3rd Row eta 3 equals one half alpha 4 period EndLayout

In the shear plane x‐z, the general effective viscosity coefficient is actually more correctly expressed in the form [20]

(3.63) eta Subscript e f f Baseline equals eta 1 plus eta 2 cosine squared theta plus eta 2

in order to account for angular velocity gradients. The coefficient η_1,2 is related to the Leslie coefficient α₁ by

(3.64) eta Subscript 1 comma 2 Baseline equals alpha 1 period

3.5.4. Flows with Director Axis Reorientation

The preceding section deals with the case where the director axis is fixed during fluid flow. In more general situations, director axis reorientation often accompanies fluid flows and vice versa. Taking into account the moment of inertia I and the torque ModifyingAbove normal upper Gamma With right harpoon with barb up equals ModifyingAbove n With ampersand c period circ semicolon times ModifyingAbove f With right harpoon with barb up , where is the molecular internal elastic field defined in Eq. (3.11), is the torque associated with an externally applied field, and ModifyingAbove normal upper Gamma With right harpoon with barb up Subscript v i s is the viscous torque associated with the viscous forces, the equation of motion describing the angular acceleration dΩ/dt as the director axis may be written as

(3.65) upper I StartFraction d normal upper Omega Over italic d t EndFraction equals left-parenthesis n times ModifyingAbove f With right harpoon with barb up plus ModifyingAbove normal upper Gamma With right harpoon with barb up Subscript e x t Baseline right-parenthesis minus ModifyingAbove normal upper Gamma With right harpoon with barb up Subscript v i s Baseline period

The viscous torque ModifyingAbove normal upper Gamma With right harpoon with barb up Subscript v i s consists of two components [3]: one arising from pure rotational effect (i.e. no coupling to the fluid flow) given by gamma 1 ModifyingAbove n With ampersand c period circ semicolon times ModifyingAbove upper N With right harpoon with barb up and another arising from coupling to the fluid motion given by gamma 2 ModifyingAbove n With ampersand c period circ semicolon times ModifyingAbove upper A With ampersand c period circ semicolon ModifyingAbove n With ampersand c period circ semicolon . Therefore, we have

(3.66) ModifyingAbove normal upper Gamma With right harpoon with barb up Subscript v i s Baseline equals n times left-bracket gamma 1 ModifyingAbove upper N With right harpoon with barb up plus gamma 2 italic An right-bracket period

Here ModifyingAbove upper N With right harpoon with barb up is the rate of change of the director with respect to the immobile background fluid, given by

(3.67) ModifyingAbove upper N With right harpoon with barb up equals StartFraction italic d n Over italic d t EndFraction minus ModifyingAbove omega With ampersand c period circ semicolon times n comma

where ModifyingAbove omega With ampersand c period circ semicolon is the angular velocity of the liquid. In Eq. (3.66) Скачать книгу