upper T right-parenthesis plus upper K Subscript upper B Baseline upper T integral f left-parenthesis theta comma phi right-parenthesis log 4 italic pi f left-parenthesis theta comma phi right-parenthesis d normal upper Omega plus upper G 1 left-parenthesis p comma upper T comma upper S right-parenthesis comma"/>
where Gi is the free enthalpy of the isotropic phase. Minimizing G(p, T) with respect to the distribution function f, one gets
(2.14)
where
and the partition function z is given by
(2.16)
From the definition of
The coupled Eqs. (2.15) and (2.17) for m and S may be solved graphically for various values of U/KBT, the relative magnitude of the intermolecular interaction to the thermal energies. Figure 2.2 depicts the case for T below a temperature Tc defined by
(2.18)
Figure 2.2 shows that curves 1 and 2 for S intersect at the origin O and two points N and M. Both points O and N correspond to minima of G, whereas M corresponds to a local maximum of G. For T < Tc, the value of G is lower at point N than at point O; that is, S is nonzero and corresponds to the nematic phase. For temperatures above Tc the stable (minimum energy) state corresponds to O; that is, S = O and corresponds to the isotropic phase.
The transition at T = Tc is a first‐order one. The order parameter just below Tc is
(2.19)
It has also been demonstrated that the temperature dependence of the order parameter of most nematics is well approximated by the expression [9]:
Figure 2.2. Schematic depiction of the numerical solution of the two transcendental equations for the order parameter for T ⩽ Tc; there is only one intersection point (at the origin).
where V and Vc are the molar volumes at T and Tc, respectively.
In spite of some of these predictions, which are in good agreement with the experimental results, the Maier–Saupe theory is not without its shortcomings. For example, the universal temperature dependence of S on T/Tc is really not valid [10]; agreement with experimental results requires an improved theory that accounts for the noncylindrical shape of the molecules [8]. The temperature variation given in Eq. (2.20) also cannot account for the critical dependence of the refractive indices. Nevertheless, the Maier–Saupe theory remains an effective, clear, and simple theoretical framework and a starting point for understanding nematic liquid crystal complexities.
2.3.2. Nonequilibrium and Dynamical Dependence of the Order Parameter
While the equilibrium statistical mechanics of the nematic liquid crystal order parameter and related physical properties near Tc are now well understood, the dynamical responses of the order parameter remain relatively unexplored. This is probably due to the fact that most studies of the order parameter near the phase transition point, such as the critical exponent, changes in molar volume, and other physical parameters, are directed at understanding the phase transition processes themselves and are usually performed with temperature changes occurring at very slow rates.
The temperature of the nematics can be abruptly raised by very short laser pulses [11, 12]. The pulse duration of the laser is in the nanosecond or picosecond time scale, which, as we shall see, is much shorter than the response time of the order parameter. As a result, the nematic film under study exhibits delayed signals.
Figure 2.3a and b shows the observed diffraction from a nematic film in a dynamic grating experiment [11]. As explained in more detail in Chapter 9, the diffracted signal is a measure of the dynamical change in the refractive index, Δn(t), following an instantaneous (delta function like) pump pulse. For a nematic liquid crystal, the principal change in the refractive index associated with a rise in temperature is through the density and order parameter [12]; that is,
Figure 2.3. (a) Observed oscilloscope trace of the diffracted signal in a dynamical scattering experiment involving microsecond infrared (CO2 at 10.6 μm) laser pump pulses. The sample
