Liquid Crystals. Iam-Choon Khoo

Liquid Crystals - Iam-Choon Khoo


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

      Above Tc, 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.

      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)upper F equals upper F 0 plus one half upper A left-parenthesis upper T right-parenthesis sigma-summation Underscript rho comma alpha Endscripts upper Q Subscript italic alpha beta Baseline upper Q Subscript italic beta alpha Baseline one third upper B left-parenthesis upper T right-parenthesis sigma-summation Underscript alpha comma beta comma gamma Endscripts upper Q Subscript italic alpha beta Baseline upper Q Subscript italic alpha gamma Baseline upper Q Subscript italic gamma alpha Baseline plus upper O left-parenthesis upper Q Baseline 4 right-parenthesis comma

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

      (2.23)upper A left-parenthesis upper T right-parenthesis equals alpha left-parenthesis upper T minus upper T Superscript asterisk Baseline right-parenthesis comma

      where T * is very close to, but lower than, Tc. Typically, upper T Subscript c Baseline minus upper T Subscript c Superscript asterisk Baseline equals 1 normal upper K.

      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)upper Q 1 equals Start 3 By 3 Matrix 1st Row 1st Column negative xi 2nd Column 0 3rd Column 0 2nd Row 1st Column 0 2nd Column negative xi 3rd Column 0 3rd Row 1st Column 0 2nd Column 0 3rd Column 2 xi EndMatrix

Schematic illustration of free energies F(Q) for different temperatures T.

      (2.24b)upper Q 2 equals Start 3 By 3 Matrix 1st Row 1st Column xi 2nd Column 0 3rd Column 0 2nd Row 1st Column 0 2nd Column xi 3rd Column 0 3rd Row 1st Column 0 2nd Column 0 3rd Column minus 2 xi EndMatrix equals minus upper Q 1

      (which signifies some alignment of the molecules in the xy plane).

      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)upper F Subscript i n t Baseline equals minus integral Subscript 0 Superscript upper H Baseline bold upper M dot d bold upper H comma

      (2.26)upper M Subscript alpha Baseline equals sigma-summation Underscript beta Endscripts chi Subscript italic alpha beta Baseline upper H Subscript beta Baseline period

      Thus

      (2.27)upper F Subscript i n t Baseline equals minus one half sigma-summation Underscript alpha Endscripts chi Subscript italic alpha beta Superscript m Baseline upper H Subscript beta Baseline upper H Subscript alpha Baseline period

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