Complex free-energy landscapes in biaxial nematic liquid crystals and the role of repulsive interactions: A Wang-Landau study

General quadratic Hamiltonian models, describing the interaction between liquid-crystal molecules (typically with $D_{2h}$ symmetry), take into account couplings between their uniaxial and biaxial tensors. While the attractive contributions arising from interactions between similar tensors of the participating molecules provide for eventual condensation of the respective orders at suitably low temperatures, the role of cross coupling between unlike tensors is not fully appreciated. Our recent study with an advanced Monte Carlo technique (entropic sampling) showed clearly the increasing relevance of this cross term in determining the phase diagram (contravening in some regions of model parameter space), the predictions of mean-field theory, and standard Monte Carlo simulation results. In this context, we investigated the phase diagrams and the nature of the phases therein on two trajectories in the parameter space: one is a line in the interior region of biaxial stability believed to be representative of the real systems, and the second is the extensively investigated parabolic path resulting from the London dispersion approximation. In both cases, we find the destabilizing effect of increased cross-coupling interactions, which invariably result in the formation of local biaxial organizations inhomogeneously distributed. This manifests as a small, but unmistakable, contribution of biaxial order in the uniaxial phase. The free-energy profiles computed in the present study as a function of the two dominant order parameters indicate complex landscapes. On the one hand, these profiles account for the unusual thermal behavior of the biaxial order parameter under significant destabilizing influence from the cross terms. On the other, they also allude to the possibility that in real systems, these complexities might indeed be inhibiting the formation of a low-temperature biaxial order itself—perhaps reflecting the difficulties in their ready realization in the laboratory.

Figure 1

Essential triangle: Region of biaxial stability. OI and IV are uniaxial torque lines intersecting at the point I. OCT is the dispersion parabola which meets line IV at the Landau point T. Base OV is the limit of biaxial stability for the interaction [33]. IW and OCT are the trajectories along which present simulations have been carried out. Points A (0.048, 0.269), B (0.166, 0.111), ${\mathrm{B}}^{\prime }$ (0.204, 0.061), C (0.215, 0.047) and D ( 0.225, 0.033) are points of particular interest (see text).

Figure 2

The temperature variation of order parameters $R_{00}^2,R_{22}^2$ (superimposed on the specific-heat curves) and $R_{02}^2$(inset) illustrates the phase behavior at the four representative points A, B, C, and D with values of $\lambda {}^{\prime }$ = (a) 0.414 (point A), (b) 0.610 (point B), (c) 0.692 (point C), and (d) 0.709 (point D) ($L=20$).

Figure 3

Phase diagram inside the essential triangle along path IW.

Figure 4

Energy cumulant $V_4$ for certain values of $\lambda {}^{\prime }$ in the range 0.345–0.692 (point C).

Figure 5

Energy cumulant $V_4$ for values of $\lambda {}^{\prime }$ in the range 0.709 (point D) to 0.747 at ($L=15$).

Figure 6

Comparison of results obtained from RW ensembles (hollow black squares) and B ensembles (hollow red circles). Temperature variation of the specific heat (continuous lines) and the order parameter profiles is shown for four values of $\lambda {}^{\prime }$ along the path IW: (a) 0.566, (b) 0.656, (c) 0.692, and (d) 0.709. It is seen that thermal averages of $R_{22}^2$ from RW ensembles differ from the B ensembles in the intermediate $N_U$ phase for values of $\lambda {}^{\prime }>0.566$.

Figure 7

Free energy shown as a function of (a) energy per particle $E$, (b) $R_{00}^2$, and (c) $R_{22}^2$ on cooling from the isotropic phase to the biaxial phase at $\lambda {}^{\prime }=0.610$.

Figure 8

Representative free energy plotted as a function of $R_{22}^2$ for different values of $\lambda {}^{\prime }$: (a) 0.674, (b) 0.692, and (c) 0.709 $[L=20$ for (a) and (b) and $L=15$ for (c)].

Figure 9

Correlation function $G\left(r\right)$ of $x, y$, and $z$ molecular axes plotted as a function of the distance $r$ at the point C ($\lambda {}^{\prime }=0.692$) at two temperatures in the $N_U$ phase.

Figure 10

Uniaxial order ($R_{00}^2$) and biaxial order ($R_{22}^2$) plotted as a function of reduced temperature for values of $\gamma$ ranging from (a) 0.131–0.228 and (b) 0.245–0.333. The inset in (a) shows a magnified version of the $R_{22}^2$ vs $T$ plot where the decrease of $R_{22}^2$ at lower temperatures is seen.

Figure 11

Free energy plotted as a function the biaxial order parameter $R_{22}^2$ for four values of $\gamma$ in the neighborhood of point C (including C, $\gamma =0.212$).

Figure 12

Free energy shown as a function of $R_{22}^2$, on cooling in the uniaxial nematic phase for values of (a) $\gamma =0.228$, (b) $\gamma =0.294$, and (c) $\gamma =0.333$.