Liquid Crystals. Iam-Choon Khoo

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      Note that this is similar to the dielectric constant .

      Nematic liquid crystals, in fact, liquid crystals in general, are diamagnetic. Therefore, and are negative of vanishingly small magnitude. As a result of the smallness of these magnetic susceptibilities, the magnetic interactions among the molecules comprising the liquid crystal are small (in comparison with their interaction with the externally applied field). Consequently, the local field acting on the molecules differs very little from the external field, and in general, magnetic measurements are the preferred method to study liquid crystal order parameters and other physical processes.

      3.3.2. Free Energy and Torques by Electric and Magnetic Fields

      In this section, we consider the interactions of nematic liquid crystals with applied fields (electric or magnetic); we will limit our discussion to only dielectric and diamagnetic interactions.

      For a generally applied (dc, low frequency, or optical) electric field , the displacement may be written in the form

      (3.23)

The electric interaction energy density is therefore

(3.24)

Note that the first term on the right‐hand side of Eq. (3.24) is independent of the orientation of the director axis. It can therefore be neglected in the director axis deformation energy. Accordingly, the free‐energy density term associated with the application of an electric field is given by

      (3.25)

      in SI units (in cgs units, ). The molecular torque produced by the electric field is given by

(3.26)

      Similar considerations for the magnetic field yield a magnetic energy density term U_m given by

      (3.27)

      a magnetic free‐energy density (associated with director axis reorientation) F_m given by

      (3.28)

      and a magnetic torque density

(3.29)

      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).

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