Reexamination of the mean-field phase diagram of biaxial nematic liquid crystals: Insights from Monte Carlo studies

Investigations of the phase diagram of biaxial liquid-crystal systems through analyses of general Hamiltonian models within the simplifications of mean-field theory (MFT), as well as by computer simulations based on microscopic models, are directed toward an appreciation of the role of the underlying molecular-level interactions to facilitate its spontaneous condensation into a nematic phase with biaxial symmetry. Continuing experimental challenges in realizing such a system unambiguously, despite encouraging predictions from MFT, for example, are requiring more versatile simulational methodologies capable of providing insights into possible hindering barriers within the system, typically gleaned through its free-energy dependences on relevant observables as the system is driven through the transitions. The recent paper from this group [Kamala Latha et al., Phys. Rev. E 89, 050501(R) (2014)], summarizing the outcome of detailed Monte Carlo simulations carried out employing an entropic sampling technique, suggested a qualitative modification of the MFT phase diagram as the Hamiltonian is asymptotically driven toward the so-called partly repulsive regions. It was argued that the degree of (cross) coupling between the uniaxial and biaxial tensor components of neighboring molecules plays a crucial role in facilitating a ready condensation of the biaxial phase, suggesting that this could be a plausible factor in explaining the experimental difficulties. In this paper, we elaborate this point further, providing additional evidence from curious variations of free-energy profiles with respect to the relevant orientational order parameters, at different temperatures bracketing the phase transitions.

Figure 1

Essential triangle: Region of biaxial stability. $OI$ and $IV$ are uniaxial torque lines intersecting at point $I$ ($0,1/3$). $OT$ is the dispersion parabola, which meets the line $IV$ at the Landau point $T$. Point $V (1/2,0)$ is the limit of biaxial stability for the interaction. $C_1 (0,0.2)$ and $C_3 (5/29,19/87)$ are tricritical points, and $C_2 (0.22,0)$ is a triple point [9]. $K (0.2,0.2)$ is a point where $\mu =-1$ (refer to the text).

Figure 2

Temperature variation of the specific heat (in arbitrary units) for different ranges of ${\lambda }^{*}$: (a) 0.1–0.25, (b) 0.26–0.33, (c) 0.33–0.53, and (d) 0.53–0.733. Corresponding variations of the two primary order parameters ($R_{00}^2$ and $R_{22}^2$) are shown in the insets with the same color scheme. Splitting of the $C_v$ peaks in (d) for ${\lambda }^{*}>0.53$ and qualitative changes in the temperature variation of order parameters are clearly observed.

Figure 3

Comparison of the results obtained from $B$ ensembles (hollow red circles) and RW ensembles (hollow black squares): Temperature variation of the specific heat (in arbitrary units) and corresponding variations of the two primary order parameters ($R_{00}^2$ and $R_{22}^2$) are shown for different values of ${\lambda }^{*}$ in regions $OI$ and $IV$: (a) 0.2, (b) 0.33, (c) 0.51, and (d) 0.54. The overlap of the corresponding curves [(a)–(c)] clearly indicates the agreement between the two ensembles up to ${\lambda }^{*}=0.53$ as mentioned in the text. Part (d) shows the qualitative disagreement first noticed at ${\lambda }^{*}=0.54$.

Figure 4

Variation of energy cumulant with temperature, at different ${\lambda }^{*}$ values: (a) 0.18, (b) 0.2, (c) 0.22, and (d) 0.26.

Figure 5

(a) Specific-heat profile with (inset) energy cumulant; (b) order parameters with (inset) susceptibility profiles for ${\lambda }^{*}=0.33$. Point $I$ of the essential triangle is the intersection point of the three uniaxial torque axes [9] and hosts the strongest first order $I-N_B$ transition, as shown by the very sharp features of these physical properties.

Figure 6

Representative free energy (in arbitrary units) as a function of biaxial order parameter $R_{22}^2$ at (a) $\mathit{T}>{\mathit{T}}_{N_{B}I}$, (b) $\mathit{T}={\mathit{T}}_{N_{B}I}$, and (c) $\mathit{T}<{\mathit{T}}_{N_{B}I}$ at ${\lambda }^{*}=0.33$ for a system of size $L=15$.

Figure 7

Specific-heat profile with (inset) energy cumulant $V_4$ and order parameters with (inset) susceptibility profiles at different ${\lambda }^{*}$ values (a) 0.54, (b) 0.58, (c) 0.62, (d) 0.66, (e) 0.69, and (f) 0.72.

Figure 8

Free energy shown as a function of (a) $R_{00}^2$ and (b) $R_{22}^2$, at the transition temperature ${\mathit{T}}_1$ for ${\lambda }^{*}$ values in the region $C_{3}T$ of Fig. 1.

Figure 9

Free energy shown as a function of (a) $R_{00}^2$ and (b) $R_{22}^2$, on cooling from ${\mathit{T}}_1$ to ${\mathit{T}}_2$ for ${\lambda }^{*}$ = 0.65.

Figure 10

Free energy shown as a function of (a) $R_{00}^2$ and (b) $R_{22}^2$, at the transition temperature ${\mathit{T}}_2$ for ${\lambda }^{*}$ values in the region $C_{3}T$ of Fig. 1.

Figure 11

Phase diagram as a function of ${\lambda }^{*}$, derived from RW ensembles. The transition temperature $1/{\beta }^{*}$ is scaled to conform to mean-field values as indicated in the text. Points along $OIV$ in Fig. 1 are mapped onto the ${\lambda }^{*}$ axis for reference. An additional biaxial-biaxial transition is observed in the region $KTV$ in place of a single transition (to the biaxial phase) predicted by the mean-field theory [38].

Figure 12

Comparison of data, as a function of temperature, from the $B$ and RW ensembles, at the Landau point $T$ ($1/3,1/9$): (a) Specific heat $C_v$ and (b) order parameters $R_{00}^2$ and $R_{22}^2$. The insets focus on (a) the energy cumulant $V_4$ and (b) order parameter susceptibilities ($\chi$'s), both derived from RW ensembles (${\lambda }^{*}$ = 0.733).

Figure 13

Contour plots of the distribution of microstates collected in the entropic ensemble at ${\lambda }^{*}\simeq 0.733$: (a) microstate energy vs its uniaxial order and (b) microstate energy vs its biaxial order. The superimposed red (dash-dotted line) and black (dashed) lines are thermal averages from $B$ and RW ensembles, respectively.