Simple model for biaxial smectic-$A$ liquid-crystal phases

We have generalized the McMillan theory of liquid crystalline smectic order in uniaxial particle fluids to biaxial particles. Upon varying the control parameter, a uniaxial nematic phase may: (i) order biaxially first, then smectically; (ii) order smectically first, then biaxially; and (iii) simultaneously order biaxially and smectically. We investigate, in the limit of complete orientational order of the molecular major axes, which of these scenarios are realized for a simple model of particles with the symmetry of rectangular parallelepipeds. We also present a generic variational derivation of the theory based on the identification of the dominant order parameters for the most ordered phase.

Figure 1

Schematic representation of the biaxial smectic phases of (a) block-shaped molecules $\left(\mathrm{Sm}{A}_{\mathrm{B}}\right)$, and (b) V-shaped molecules $\left(\mathrm{Sm}{A}_{\mathrm{BP}}\right)$.

Figure 2

Symmetry of the molecular model: biaxial parallelepipeds of edge lengths $l$, $b$, and $w$. $\mathbf{l}$, $\mathbf{m}$, and $\mathbf{n}$ are the three directors for the biaxial phase, with $\mathbf{n}$ the usual uniaxial director.

Figure 3

Diagram showing the regions of instability, on increasing ${\lambda }^{\left(0\right)}$, of the uniaxial nematic $\left(N_{\mathrm{U}}\right)$ phase relative to the biaxial nematic $\left(N_{\mathrm{B}}\right)$, uniaxial smectic $\left(\mathrm{Sm}{A}_{\mathrm{U}}\right)$ and biaxial smectic $\left(\mathrm{Sm}{A}_{\mathrm{B}}\right)$ phases, as a function of the potential parameters $I_0$ and $J_4^{*}$. We do not concern ourselves with the hatched region, where $J_4^{*}>0$ and the molecular minor axes tend to align perpendicular, rather than parallel, to one another, hence no biaxial phases (in the usual sense) are expected to exist.

Figure 4

Order parameters $B$, $\tau$, and ${\sigma }_B$ versus the control parameter $\lambda$. (a) $I_0=1$, $J_4^{*}=-1$: $N_{\mathrm{U}}\text{−}{N}_{\mathrm{B}}$ transition at $\lambda \approx 1.33$, and $N_{\mathrm{B}}\text{−}\mathrm{Sm}{A}_{\mathrm{B}}$ transition at $\lambda \approx 7.11$. (b) $I_0=2.5$, $J_4^{*}=-2$: $N_{\mathrm{U}}\text{−}\mathrm{Sm}{A}_{\mathrm{B}}$ transition at $\lambda \approx 0.54$. (c) $I_0=-1$, $J_4^{*}=-1$: $N_{\mathrm{U}}\text{−}{N}_{\mathrm{B}}$ transition at $\lambda \approx 1.33$. (d) $I_0=-1$, $J_4^{*}=-0.5$: $N_{\mathrm{U}}\text{−}\mathrm{Sm}{A}_{\mathrm{U}}$ transition at $\lambda \approx 2.0$. All of the transitions from the perfectly aligned $N_{\mathrm{U}}$ phase appear to be continuous and occur at $\lambda \simeq {\lambda }_c^{\left(0\right)}$ found from Eqs. (45, 46, 47). The $N_{\mathrm{B}}\text{−}\mathrm{Sm}{A}_{\mathrm{B}}$ transition in (a) also appears continuous, but cannot be located using the bifurcation analysis performed in this paper: One would need to study the stability of the $N_{\mathrm{B}}$ phase with respect to a perturbation with $\mathrm{Sm}{A}_{\mathrm{B}}$ symmetry.

Figure 5

(Color online) Diagram showing the regions of instability, on increasing the control parameter $\lambda =\rho w^3∕k_{\mathrm{B}}T$, of the uniaxial nematic $\left(N_{\mathrm{U}}\right)$ phase relative to the biaxial nematic $\left(N_{\mathrm{B}}\right)$, uniaxial smectic $\left(\mathrm{Sm}{A}_{\mathrm{U}}\right)$, and biaxial smectic $\left(\mathrm{Sm}{A}_{\mathrm{B}}\right)$ phases, as a function of particle length $l$ and breadth $b$, for the layer spacing $d=1.5l$. We do not concern ourselves with the region $l∕w,b∕w<1$, where ordering occurs at very high densities/low temperatures and is likely preempted by transitions into solid phases. This diagram should be interpreted as follows: Take a system characterized by a certain pair of values $(l∕w,b∕w)$ and compress it/cool it from its $N_{\mathrm{U}}$ phase; if it lies in the top region of the diagram, then its first instability is relative to the $\mathrm{Sm}{A}_{\mathrm{U}}$ phase: If it lies in the bottom right region of the diagram, its first instability is relative to the $N_{\mathrm{B}}$ phase; finally, if it lies along the boundary between the two regions, its first instability is relative to the $\mathrm{Sm}{A}_{\mathrm{B}}$ phase. Further instabilities may occur at higher densities/lower temperatures, but these are not captured by the present diagram. The white circles are the locations of the four systems whose order parameters are plotted in Fig. 6.

Figure 6

Order parameters $B$, $\tau$, and ${\sigma }_B$ versus the control parameter $\lambda$. (a) $l∕w=3$, $b∕w=1.5:N_{\mathrm{U}}\text{−}\mathrm{Sm}{A}_{\mathrm{U}}$ transition at $\lambda \approx 0.08$, and $\mathrm{Sm}{A}_{\mathrm{U}}\text{−}\mathrm{Sm}{A}_{\mathrm{B}}$ transition at $\lambda \approx 4.04$. (b) $l∕w=1.5$, $b∕w=1.5$: $N_{\mathrm{U}}\text{−}\mathrm{Sm}{A}_{\mathrm{U}}$ transition at $\lambda \approx 0.86$. (c) $l∕w=1.5$, $b∕w=2.63$: $N_{\mathrm{U}}\text{−}\mathrm{Sm}{A}_{\mathrm{B}}$ transition at $\lambda \approx 0.77$. (d) $l∕w=1.25$, $b∕w=2.63$: $N_{\mathrm{U}}\text{−}{N}_{\mathrm{B}}$ transition at $\lambda \approx 0.77$, and $N_{\mathrm{B}}\text{−}\mathrm{Sm}{A}_{\mathrm{B}}$ transition at $\lambda \approx 2.96$. All of the transitions from the perfectly aligned $N_{\mathrm{U}}$ phase appear to be continuous and occur at $\lambda \simeq {\lambda }_c^{\left(0\right)}$ found from Eqs. (62, 63, 64). The $\mathrm{Sm}{A}_{\mathrm{U}}\text{−}\mathrm{Sm}{A}_{\mathrm{B}}$ transition in (a) and the $N_{\mathrm{B}}\text{−}\mathrm{Sm}{A}_{\mathrm{B}}$ transition in (d) also appear continuous, but cannot be located using the bifurcation analysis performed in this paper: One would need to study the stability of the $\mathrm{Sm}{A}_{\mathrm{U}}$ or $N_{\mathrm{B}}$ phases with respect to a perturbation with $\mathrm{Sm}{A}_{\mathrm{B}}$ symmetry.