Liquid Crystals. Iam-Choon Khoo

Figure 2.1. Coordinate system defining the microscopic order parameter of a nematic liquid crystal molecule. Here, i, j, and k are the molecular axes, whereas is the laboratory axis that denotes the average direction of liquid crystal alignment.

where , , and are unit vectors along the molecular axes. With reference to Figure 2.1, the three diagonal components S_ii, S_jj, and S_kk are given by

      (2.3a)

      (2.3b)

      (2.3c)

      Note that S_ii+ S_jj+ S_kk= 0. Put another way, S is a traceless tensor because its diagonal elements add up to zero.

For a complete description of the statistical properties of the liquid crystal orientation, functions involving higher powers of cos² θ are needed. The most natural functions to use are the Legendre polynomials P₁(cos θ) (l = 0, 1, 2,…), in terms of which we can write Eq. (2.1) as S = 〈P₂〉, which measures the average of cos² θ. The next nonvanishing term is 〈P₄〉, which provides a measure of the dispersion of 〈cos² θ〉.

      The order parameters defined previously in terms of the directional averages can be translated into expressions in terms of the anisotropies in the physical parameters such as magnetic, electric, and optical susceptibilities. For example, in terms of the optical dielectric anisotropies Δε = ε_‖ − ε_⊥, one can define a so‐called macroscopic order parameter that characterizes the bulk response:

(2.4)

      It is called macroscopic because it describes the bulk property of the material. To be more explicit, consider a uniaxial nematic liquid crystal such that in the molecular axis system ε_αβ is of the form

      (2.5)

      Writing Q_αβ explicitly in terms of their diagonal components, we thus have

      (2.6)

and

      (2.7)

      It is useful to note here that, in tensor form, ε_αβ can be expressed as

(2.8)

      Note that this form shows that ε = ε_|| for an optical field parallel to and ε = ε_⊥ for an optical field perpendicular to .

      Similarly, other parameters such as the magnetic (χ ^m ) and electric (χ) susceptibilities may be expressed as

      (2.9a)

      and

      (2.9b)

      respectively, in terms of their respective anisotropies Δχ ^m and Δχ.

      In general, however, optical dielectric anisotropy and its dc or low‐frequency counterpart (the dielectric anisotropy) provide a less reliable measure of the order parameter because they involve electric fields. This is because of the so‐called local field effect: the effective electric field acting on a molecule is a superposition of the electric field from the externally applied source and the field created

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