Liquid Crystals. Iam-Choon Khoo

and switching devices based on laser‐induced order parameter changes in liquid crystals are given in Chapters 9 and 12.

2.4. ISOTROPIC PHASE OF LIQUID CRYSTALS

      Above T_c, liquid crystals lose their directional order and behave in many respects like liquids. All bulk physical parameters also assume an isotropic form, although the molecules are anisotropic.

The isotropic phase is, nevertheless, a very interesting and important phase for both fundamental and applied studies. It is fundamentally interesting because of the existence of short‐range order, which gives rise to the critical temperature dependence of various physical parameters just above the phase transition temperature. These critical behaviors provide a good testing ground for liquid crystal physics.

      On the other hand, recent studies have also shown that isotropic liquid crystals may be superior in many ways for constructing practical nonlinear optical devices (see Chapter 12), in comparison to the other liquid crystalline phases (see Chapter 8). In general, the scattering loss is less and thus allows longer interaction lengths, and relaxation times are on a much faster scale. These properties easily make up for the smaller optical nonlinearity for practical applications.

      2.4.1. Free Energy and Phase Transition

      We begin our discussion of the isotropic phase of liquid crystals with the free energy of the system, following deGennes’ pioneering theoretical development [1, 2]. The starting point is the order parameter, which we denote by Q.

      In the absence of an external field, the isotropic phase is characterized by Q = 0; the minimum of the free energy also corresponds to Q = 0. This means that, in the Landau expansion of the free energy in terms of the order parameter Q, there is no linear term in Q; that is,

      (2.22)

      where F₀ is a constant and A(T) and B(T) are temperature‐dependent expansion coefficients:

      (2.23)

      where T * is very close to, but lower than, T_c. Typically, .

      Note that F contains a nonzero term of order Q [3]. This odd function of Q ensures that states with some nonvanishing value of Q (e.g. due to some alignment of molecules) will have different free‐energy values depending on the direction of the alignment. For example, the free energy for a state with an order parameter Q of the form

      (2.24a)

Figure 2.6. Free energies F(Q) for different temperatures T. At T = T_c, ∂F/∂Q = 0 at two values of Q, where F has two stable minima. On the other hand, at , there is only one stable minimum where ∂F/∂Q = 0.

(i.e. with some alignment of the molecule in the z direction) is not the same as the state with a negative Q parameter

      (2.24b)

      (which signifies some alignment of the molecules in the x‐y plane).

The cubic term in F is also important in that it dictates that the phase transition at T = T_c is of the first order (i.e. the first‐order derivative of F, ∂F/∂θ, is vanishing at T = T_c, as shown in Figure 2.6). The system has two stable minima, corresponding to Q = 0 or Q ≠ 0 (i.e. the coexistence of the isotropic and nematic phases). On the other hand, for , there is only one stable minimum at Q ≠ 0; this translates into the existence of a single liquid crystalline phase (e.g. nematic or smectic).

      2.4.2. Free Energy in the Presence of an Applied Field

      In the presence of an externally applied field (e.g. dc or low‐frequency electric, magnetic, or optical electric field), a corresponding interaction term should be added to the free energy.

      For an applied magnetic field H, the energy associated with it is

      (2.25)

where M is the magnetization given by

      (2.26)

      Thus

      (2.27)

      Using Eq. (2.9), we can rewrite F_int as

(2.28)