Liquid Crystals. Iam-Choon Khoo

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)

The first term on the right‐hand side of Eq. (2.28) is independent of the orientation of the (anisotropic) molecules, and it can therefore be included in the constant F₀.

On the other hand, the second term is dependent on the orientation of the molecules. Using Eq. (2.10) for the order parameter Q_αβ we can write it as

      (2.29)

      Therefore, the total free energy of a liquid crystal in the isotropic phase, under the action of an externally applied magnetic field, is given by

      (2.30)

      Without solving the problem explicitly, we can infer from the magnetic interaction term that a lower energy state corresponds to some alignment of the molecules in the direction of the magnetic field (for Δχ ^m > 0).

      Using a similar approach, we can also deduce that the electric interaction contribution to the free energy is given by (in inks units)

      (2.31)

The orientation‐dependent term is therefore

      (2.32) Скачать книгу