Читать книгу Liquid Crystals - Iam-Choon Khoo - Страница 39

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).