Unified molecular field theory of nematic, smectic-$A$, and smectic-$C$ phases

A unified mean-field molecular theory of nematic ($N_U$), smectic A (Sm$A$), and smectic C (Sm$C$) liquid crystal phases, composed of uniaxial nonpolar molecules, is developed taking into account the variation of all orientational and translational order parameters in these phases. Numerical results, obtained by direct global minimization of the free energy, are presented in the form of three typical phase diagrams of different topology. Temperature variation of the relevant order parameters in different sequences of phases is analyzed for various cross sections of the phase diagrams. The present model enables one to reproduce all possible sequences of phase transitions between the given phases including isotropic (Iso)-$N_U$-Sm$A$-Sm$C$, Iso-$N_U$-Sm$C$, Iso-Sm$A$-Sm$C$, and Iso-Sm$C$. The properties of the NAC point, where the $N_U$, Sm$A$, and Sm$C$ structures coexist, are considered in detail and the shape of the phase diagram in the vicinity of the NAC point is compared with existing experimental data.

Figure 1

Schematic representation of the distribution of long molecular axes in the Sm$C$ phase (a) and the three density waves (b) which correspond to the orientational-translational order parameters ${\Sigma }_k$, ${\Pi }_k$, and $\Lambda$. The average width of the orientational distribution around the layer normal $\mathbf{k}$ in (a) is specified by the nematic order parameter $S_k$, the nonpolar asymmetry of the distribution is characterized by the biaxial order parameter $P_k$ and the tilt order parameter $V$ describes the shift of the distribution away from the layer normal. Panel (b) illustrates the fact that for sufficiently large values of the order parameter ${\Sigma }_k$ local orientational distribution of molecules may depend on the position along the layer normal $\mathbf{k}$ [15] and may even be characterized by negative local order parameters for positions “between the layers.”

Figure 2

Phase diagram for $u=-1,w_0=w_1=-0.04,w_3=w_4=-0.0273+0.7575w_2$. The blue line is the line of transitions from the isotropic phase (Iso), while the red and green lines represent the $N_U$-Sm$A$ and Sm$A$-Sm$C$ phase transitions, respectively.

Figure 3

Temperature variation of the whole set of orientational and translational order parameters defined with respect to the smectic layer normal $\mathbf{k}$ in the case of the direct Iso-Sm$A$ phase transition (a), variation of the orientational (b), and the orientational-translational order parameters (c) defined both in the $\mathbf{k}$ frame and in the director reference frame. The parameters $\Theta$, $S$, $P$, $\theta$, $\Sigma ,$ and $\Pi$, defined in the director frame, have been calculated using Eqs. (24), (25), (26), (27), (28), and (29), respectively. Order parameter profiles have been obtained by minimization of the free energy Eq. (47) for $w_2=-0.95$. The values of other coupling constants are the same as in Fig. 2.

Figure 4

Temperature variation of the whole set of orientational and translational order parameters defined with respect to the smectic layer normal $\mathbf{k}$ in the case of the direct Iso-Sm$C$ phase transition (a), variation of the orientational (b), and the orientational-translational order parameters (c) defined both in the $\mathbf{k}$ frame and in the director reference frame. The order parameter profiles have been calculated for $w_2=-1.6$, while the values of other coupling constants are the same as in Fig. 2.

Figure 5

Phase diagram for $u=-1,w_0=w_1=-0.04,w_3=w_4=-0.0221+0.7725w_2$. The violet line represents the line of $N_U$-Sm$C$ phase transitions, and the colors of the other curves have the same meaning as in Fig. 2. The inset shows the magnified area around the NAC point.

Figure 6

Phase diagram for $u=-1,w_0=w_1=-0.04,w_3=w_4=-0.0393+0.7425w_2$. The colors of the transition lines has the same meaning as in Figs. 2 and 5. The inset shows the magnified behavior in the vicinity of the NAC point.

Figure 7

Temperature variation of the whole set of orientational and translational order parameters in the case of the Iso-${\mathit{N}}_{\mathit{U}}$-Sm$C$ sequence of phase transitions for $w_2=-0.72$; the values of other coupling constants are the same as in Fig. 5.

Figure 8

(a) Variation of the whole set of orientational and translational order parameters for $w_2=-0.45$, which corresponds to the Iso-${\mathit{N}}_{\mathit{U}}$-Sm$A$-Sm$C$ sequence of phase transitions. The values of other coupling constants are the same as in Fig. 5. (b) Temperature variation of the higher-order parameters along the same cross section of the phase diagram.

Figure 9

Temperature variation of the orientational and translational order parameters defined with respect to the smectic layer normal $\mathbf{k}$ along the cross section of the phase diagram in Fig. 6 for $w_2=-0.575$ (a). Variation of the orientational (b) and the orientational-translational (c) order parameters defined both in the $\mathbf{k}$ frame and in the director reference frame along the same cross section of phase diagram.