Liquid Crystals. Iam-Choon Khoo

Liquid Crystals

href="#fb3_img_img_10a945b2-c05c-5bb5-bf80-26c39d959147.png" alt="StartLayout 1st Row ModifyingAbove normal upper Gamma With right harpoon with barb up Subscript m Baseline equals ModifyingAbove upper M With right harpoon with barb up times ModifyingAbove upper H With right harpoon with barb up 2nd Row equals normal upper Delta chi Superscript m Baseline left-parenthesis n dot ModifyingAbove upper H With right harpoon with barb up right-parenthesis left-parenthesis n dot ModifyingAbove upper H With right harpoon with barb up right-parenthesis period EndLayout"/>

These electric and magnetic torques play a central role in various field‐induced effects in liquid crystals.

3.4. OPTICAL DIELECTRIC CONSTANTS AND REFRACTIVE INDICES

3.4.1. Linear Susceptibility and Local Field Effect

In the optical regime, ε_|| > ε_⊥. Typically, ε_|| is on the order of 2.89ε₀ and ε_⊥ is 2.25ε₀. These correspond to refractive indices n_|| = 1.7 and n_⊥ = 1.5. An interesting property of nematic liquid crystals is that such a large birefringence (Δε_⊥ = ε_|| − ε_⊥ ≈ 0.2) is manifested throughout the whole optical spectral regime (from near‐ultraviolet [≈ 400 nm], to visible [≈ 500 nm] and near‐infrared [1–3 μm], to the infrared regime [8–12 μm], i.e. from 400 nm to 12 μm). Figure 3.6 shows the measured birefringence of three typical nematic liquid crystals from the UV to the far‐infrared (λ = 16 μm).

Figure 3.6. Measured birefringence De = ε_|| − ε_⊥ of three nematic liquid crystals.

The optical dielectric constants originate from the linear polarization ModifyingAbove upper P With right harpoon with barb up generated by the incident optical field ModifyingAbove upper E With right harpoon with barb up Subscript o p on the nematic liquid crystal:

(3.30a) ModifyingAbove upper P With right harpoon with barb up equals epsilon 0 ModifyingAbove Above ModifyingAbove chi With right harpoon with barb up With right harpoon with barb up dot ModifyingAbove upper E With right harpoon with barb up period

From the defining equation

(3.30b) ModifyingAbove upper D With right harpoon with barb up equals epsilon 0 ModifyingAbove upper E With right harpoon with barb up plus ModifyingAbove upper P With right harpoon with barb up equals ModifyingAbove Above ModifyingAbove epsilon With right harpoon with barb up With right harpoon with barb up colon ModifyingAbove upper E With right harpoon with barb up comma

we have

(3.30c) ModifyingAbove Above ModifyingAbove epsilon With right harpoon with barb up With right harpoon with barb up equals epsilon 0 left-bracket 1 plus ModifyingAbove Above ModifyingAbove chi With right harpoon with barb up With right harpoon with barb up Superscript left-parenthesis 1 right-parenthesis Baseline right-bracket period

Here ModifyingAbove Above ModifyingAbove chi With right harpoon with barb up With right harpoon with barb up Superscript left-parenthesis 1 right-parenthesis is the linear (sometimes termed “first order”) susceptibility tensor of the nematics. is a macroscopic parameter and is related to the microscopic (molecular) parameter, the molecular polarizabilities tensor α_ij, in the following way:

d Subscript i Baseline equals alpha Subscript italic i j Baseline upper E Subscript j Superscript l o c Baseline comma

(3.31a) ModifyingAbove d With right harpoon with barb up equals ModifyingAbove Above ModifyingAbove alpha With right harpoon with barb up With right harpoon with barb up colon ModifyingAbove upper E With right harpoon with barb up Superscript l o c Baseline comma

(3.31b) ModifyingAbove upper P With right harpoon with barb up equals upper N ModifyingAbove d With right harpoon with barb up comma

where d_i is the ith component of the induced dipole ModifyingAbove d With right harpoon with barb up and N is the number density. In Chapter 8, a rigorous quantum mechanical derivation of α in terms of the dipole matrix elements or oscillator strengths and the energy levels and level populations will be presented. The connection between the microscopic parameter α_ij and the macroscopic parameter χ_ij is the local field correction factor (i.e. the difference between the externally applied field and the actual field as experienced by the molecules). Several theoretical formalisms have been developed to evaluate the field correction factor, ranging from simplified to complex and sophisticated ones.

Most of the approaches used to obtain the local field correction factor are based on the Lorentz results [5], which state that the internal field (i.e. the local field as experienced by a molecule ModifyingAbove upper E With right harpoon with barb up Superscript l o c in a solid) is related to the applied field by

(3.32a) ModifyingAbove upper E With right harpoon with barb up Superscript l o c Baseline equals StartFraction n squared plus 2 Over 3 EndFraction ModifyingAbove upper E With right harpoon with barb up Superscript a p p Baseline period

In particular, Vuks [10] analyzed experimental data and proposed that the local field in an anisotropic crystal may be taken as isotropic and expressed in the form

(3.32b) ModifyingAbove upper E With right harpoon with barb up Superscript l o c Baseline equals StartFraction left pointing angle n squared right pointing angle plus 2 Over 3 EndFraction left pointing angle ModifyingAbove upper E With right harpoon with barb up right pointing angle comma

where left pointing angle n squared right pointing angle equals one third left-parenthesis n Subscript x Superscript 2 Baseline plus n Subscript y Superscript 2 Baseline plus n Subscript z Superscript 2 Baseline right-parenthesis and n_x, n_y, and n_z are the principal refractive indices of the crystal. This approach has been employed in the study of liquid crystals [11]. A more generalized expression for anisotropic crystals is given in

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Liquid Crystals. Iam-Choon Khoo

Liquid Crystals

3.4. OPTICAL DIELECTRIC CONSTANTS AND REFRACTIVE INDICES 3.4.1. Linear Susceptibility and Local Field Effect

3.4. OPTICAL DIELECTRIC CONSTANTS AND REFRACTIVE INDICES

3.4.1. Linear Susceptibility and Local Field Effect