exhibit a nematic phase. For n = 5–7, the material possesses a nematic range. For n > 8, smectic phases begin to appear.
Another reliable observation is that the nematic → isotropic phase transition temperature Tc is a good indicator of the thermal stability of the nematic phase [6]; the higher the Tc, the greater is the thermal stability of the nematic phase. In this respect, the types of chemical groups used as substituents in the terminal groups or side chain play a significant role – an increase in the polarizability of the substituent tends to be accompanied by an increase in Tc.
Such molecular‐structure‐based approaches are clearly extremely complex and often tend to yield contradictory predictions because of the wide variation in the molecular electronic structures and intermolecular interactions present. In order to explain the phase transition and the behavior of the order parameter in the vicinity of the phase transition temperature, some simpler physical models have been employed [6]. For the nematic phase, a simple but quite successful approach was introduced by Maier and Saupe [7]. The liquid crystal molecules are treated as rigid rods, which are correlated (described by a long‐range order parameter) with one another by Coulomb interactions. For the isotropic phase, deGennes introduced a Landau type of phase transition theory [1–3], which is based on a short‐range order parameter.
The theoretical formalism for describing the nematic → isotropic phase transition and some of the results and consequences are given in the next section. This is followed by a summary of some of the basic concepts introduced for the isotropic phase.
2.3. MOLECULAR THEORIES AND RESULTS FOR THE LIQUID CRYSTALLINE PHASE
Among the various theories developed to describe the order parameter and phase transitions in the liquid crystalline phase, the most popular and successful one is the theory first advanced by Maier and Saupe and corroborated in studies by others [8]. In this formalism, Coulombic intermolecular dipole–dipole interactions are assumed. The interaction energy of a molecule with its surroundings is then shown to be of the form [6]:
where V is the molar volume (V = M/p), S is the order parameter, and A is a constant determined by the transition moments of the molecules. Both V and S are functions of temperature. Comparing Eqs. (2.11) and (2.1) for the definition of S, we note that Wint ≈ S 2, so this mean‐field theory by Maier and Saupe is often referred to as the S 2 interaction theory [1]. This interaction energy is included in the free enthalpy per molecule (chemical potential) and is used in conjunction with an angular distribution function f(θ, ϕ) for statistical mechanics calculations.
2.3.1. Maier–Saupe Theory: Order Parameter Near Tc
Following the formalism of deGennes, the interaction energy may be written as
(2.12)
The total free enthalpy per molecule is therefore
(2.13)
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