Free-energy power expansion for orientationally ordered phases: Energy and entropy

We propose an approach to the description of orientational phase transitions that utilizes the specific features of orientational energy and entropy. The approach is applied to building a model for nematic phases in materials with nonpolar parallelepiped-type molecules with symmetry $D_{2h}$. The model operates with complex order parameters, generalizes the Landau–de Gennes theory, and predicts the existence of a biaxial nematic phase for the fourth-order expansion of free energy.

Figure 1

Chains of arrows represent the solutions [Eq. (9)]; each arrow $\{M_{n-1},M_n\}$ corresponds to $\langle M_{n-1}{\tilde{m}}_{n-1}2m_n|M_n{\tilde{m}}_n\rangle$.

Figure 2

Temperature dependencies of normalized OP amplitudes $\widehat{s},\widehat{p}$ (solid) and OP phases $\alpha$, $\beta$ (dashed) in $N_u$ (thin) and $N_b$ (thick) phases for $\widehat{b}=0.5$, ${\widehat{c}}_2=0.2$, $v=0.09$, and $\theta =\pi /18$. The vertical line at ${\widehat{u}}_{iu}=T^{*}/T_{iu}$ corresponds to the phase transition between isotropic and $N_u$ phases at the temperature $T_{iu}$. Note that $\widehat{s}=0.37$ at ${\widehat{u}}_{iu}$, thus the normalization factor for $\widehat{s}$ and $\widehat{p}$ is close to unity.

Figure 3

Phase diagrams in $\{\widehat{u},v\}$ plane for $\theta =0$ and different sets of $\widehat{b}$ and ${\widehat{c}}_2$: (1) $\widehat{b}=0.5$, ${\widehat{c}}_2=0.0$; (2) $\widehat{b}=0.5$, ${\widehat{c}}_2=0.2$; (3) $\widehat{b}=0.6$, ${\widehat{c}}_2=0.2$. Curves correspond to phase transitions: $I\stackrel{1}{\leftrightarrow }{N}_u$ (thin solid), $I\stackrel{1}{\leftrightarrow }{N}_b$ (thick solid), $N_u\stackrel{2}{\leftrightarrow }{N}_b$ (dotted), $N_u\stackrel{1}{\leftrightarrow }{N}_b$ (dashed).