Liquid Crystals. Iam-Choon Khoo

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 by the induced dipoles surrounding the molecules. For systems where the molecules are not correlated, the effective field can be fairly accurately approximated by some local field correction factor [3]; these correction factors are much less accurate in liquid crystalline systems. For a more reliable determination of the order parameter, one usually employs non‐electric‐field‐related parameters, such as the magnetic susceptibility anisotropy:

(2.10)

      2.1.3. Long‐ and Short‐range Order

The order parameter, defined by Eq. (2.2) and its variants such as Eqs. (2.4) and (2.8), is an average over the whole system and therefore provides a measure of the long‐range orientation order. The smaller the fluctuation of the molecular axis from the director axis orientation direction, the closer the magnitude of S is to unity. In a perfectly aligned liquid crystal, as in other crystalline materials, 〈cos² θ〉 = 1 and S = 1; on the other hand, in a perfectly random system, such as ordinary liquids or the isotropic phase of liquid crystals, 〈cos² θ〉 = and S = 0.

      An important distinction between liquid crystals and ordinary anisotropic or isotropic liquids is that, in the isotropic phase, there could exist a so‐called short‐range order [1, 2]; that is, molecules within a short distance of one another are correlated by intermolecular interactions [4]. These molecular interactions may be viewed as remnants of those existing in the nematic phase. Clearly, the closer the isotropic liquid crystal is to the phase transition temperature, the more pronounced the short‐range order and its manifestations in many physical parameters will be. Short‐range order in the isotropic phase gives rise to interesting critical behavior in the response of the liquid crystals to externally applied fields (electric, magnetic, and optical) (see Section 2.3.2).

      As pointed out at the beginning of this chapter, the physical and optical properties of liquid crystals may be roughly classified into two types: one pertaining to the ordered phase, characterized by long‐range order and crystalline‐like physical properties; the other pertaining to the so‐called disordered phase, where a short‐range order exists. All these order parameters show critical dependences as the temperature approaches the phase transition temperature T_c from the respective directions.

2.2. MOLECULAR INTERACTIONS AND PHASE TRANSITIONS

      In principle, if the electronic structure of a liquid crystal molecule is known, one can deduce the various thermodynamical properties. This is a monumental task in quantum statistical chemistry that has seldom, if ever, been attempted in a quantitative or conclusive way. There are some fairly reliable guidelines, usually obtained empirically, that relate molecular structures with the existence of the liquid crystal mesophases and, less reliably, the corresponding transition temperatures.

      One simple observation is that to generate liquid crystals, one should use elongated molecules. This is best illustrated by the nCB homolog [5] (n = 1, 2, 3,…). For n