Molecular model for nematic, smectic-$A$, and smectic-$C$ liquid crystals

We analyze a molecular model to describe the phase transitions between the isotropic, nematic, smectic-A, and smectic-C phases. The smectic phases are described by the use of a pair potential, which lacks the full rotational symmetry because of the cylindrical symmetry around the smectic axis. The tilt of the long molecules inside the smectic layers is favored by a biquadratic pair potential, which compete with the pair potential of the McMillan model. The part of the phase diagram showing the first three phases is similar to that of the McMillan molecular model. The smectic-C phase is separated from the nematic by a continuous phase transition line along which the tilt angle is nonzero. The tilt angle vanishes continuously when one reaches the line separating the smectic-C and the smectic-A line.

Figure 1

Phase diagram of the McMillan model showing the isotropic (I), the nematic (N), and the smectic-A phases, in the plane temperature $T$ versus the parameter $\alpha$. The dashed and solid lines represent discontinuous and continuous transitions, respectively. The triangle and the full circle represent triple point and a tricritical point, respectively. The phase diagram corresponds also to the anisotropic model when $c=0$.

Figure 2

Ground state or zero temperature phase diagram showing the nematic (N) and the two smectic phases. The smectic-C occurs between the two solid lines. The dashed lines are lines of constant tilt angle $\omega$. In the region between the dot-dashed line and the solid line the tilt angle is $\omega =\pi /2$.

Figure 3

Phase diagram of the anisotropic model showing the isotropic (I), the nematic (N), the smectic-A, and the smectic C phases, in the plane temperature $T$ versus the parameter $\alpha$ for $c/a=2$. The heavy dashed line and the solid line represent discontinuous and continuous transitions, respectively. The triangle and the full circle represent a triple point and a tricritical point, respectively, whereas the square represents a Lifshitz point. The light dashed lines are lines of constant tilt angle $\omega$. Along the line separating the smectic-C and the smectic-A, $\omega =0$. Along the critical line separating the N and ${\mathrm{S}}_{\mathrm{C}}$ phases the tilt angle $\omega$ decreases from $\omega =\pi /2$ at the empty circle to $\omega =0$ at the Lifshitz point. In the smectic-C domain, the triangle region with the open circle has tilt angle $\omega =\pi /2$.