Generalized van der Waals theory for phase behavior of two-dimensional nematic liquid crystals: Phase ordering and the equation of state

Liquid crystalline ordering of anisotropic particles in two dimensions is important in many physical and biological systems and their phase behavior is still a topic of interest. A generalized van der Waals theory is formulated, accounting for repulsive excluded volume and attractive van der Waals and Maier-Saupe interactions, for rectangles confined to two dimensions. The phase ordering transitions and equation of state are analyzed as a function of the model parameters (aspect $\mathrm{ratio}L/B$ and isotropic and anisotropic interaction parameters $\chi$ and $\nu$). Different phase transitions are observed: continuous isotropic-nematic (high $L/B$ and $\nu$), first-order isotropic-nematic (intermediate $L/B$ and small $\nu$), and continuous isotropic-tetratic (small $L/B$ and $\nu$) followed by a continuous tetratic-nematic transition at higher densities. Increasing $L/B$ decreases the pressure, and this effect is more pronounced in the nematic than in the isotropic phase. Increasing both interaction parameters decreases pressure and can lead to phase separation.

Figure 1

Order-disorder transition for an athermal system. (a) Order parameter as a function of the scaled area density, for different values of $L/B=1.5$, 2.5, 4, 6, 9, 14, 20, 40, 100, and 10 000, increasing as indicated by the arrow. The nematic order parameter $S_2$ is indicated with a full line and, for $L/B=1.5$, the tetratic order parameter $S_4$ is indicated with a dashed line. (b) Area density at the order-disorder transition, ${\varphi }_{\mathrm{IN}}$, for small values of $L/B$. Different ordering transitions (full line) are indicated as follows: in region 1, isotropic-tetratic; in region 2, discontinuous isotropic-nematic; in region 3, continuous isotropic-nematic. Tetratic-nematic transition is indicated with the dashed line. In the inset, the scaled area density ${\varphi }_{\mathrm{IN}}^{*}$ is shown in the whole range of $L/B$.

Figure 2

Order parameter as a function of the scaled density, for $L/B=10000$ (a) and 1.5 (b), and the following values of the quadrupolar order parameter (increasing as indicated by the arrow): (a) 0, 0.0625, 0.175, 0.45, 1.25, 3.75, and 16 and (b) 0, 0.5, 1.2, 2.3, 4.65, 10, and 50.

Figure 3

Density at the order-disorder transition as a function of the quadrupolar order parameter, for the same values of $L/B$ as in Fig. 1. The inset shows the inverse of the quadrupolar interaction parameter vs scaled density at the transition, for the same values of $L/B$ as in Fig. 1. Regions 1, 2, and 3 indicate different phase transitions as in Fig. 1.

Figure 4

Pressure of the isotropic phase as a function of area density, for a system with no interaction and the following values of $L/B$ (increasing in the direction of the arrow): 1, 2, 5, 15, 60, 200, 800, 2500, and 10 000. The inset shows the region of small pressure, in linear scale.

Figure 5

Surface pressure of the isotropic phase as a function of area density. The insets show the behavior for small pressures, in linear scale, where ideal behavior is plotted as a dotted line. (a)$L/B=1.5$; $\chi =0$, 0.75, 1.9, 3.2, 3.9, and 4.3. (b) $L/B=6$; $\chi =0$,0.5, 0.975, 1.2, 1.33, and 1.47. (c) $L/B=14$; $\chi =0$, 0.26, 0.52, 0.7, 0.82, and 0.91. (d) $L/B=100$; $\chi =0$, 0.1, 0.2, 0.335, 0.425, and 0.485. For a given surface density, increasing $\chi$ decreases the pressure. The last $\chi$, corresponding to red curves (in the online color version) with the horizontal inflection point, is the critical $\chi$ for each $L/B$.

Figure 6

Surface pressure a function of area density, for a system with no interaction and the following values of $L/B$ (increasing in the direction of the arrow): 1.5, 6, 14, and 100. The inset shows the region of small pressure, in linear scale. The dotted line represents the metastable isotropic phase in conditions where the order phase is the equilibrium one.

Figure 7

Surface pressure as a function of area density for systems without isotropic interactions ($\chi =0$). The insets show the behavior for small pressures, in linear scale. (a) $L/B=1.5$; $\nu =0$, 0.65, 1.1, 2, 3, 3.9, and 4.65 (note that the first few values are difficult to distinguish due to the proximity of the different curves, and some are not shown in the inset). (b) $L/B=6$; $\nu =0$, 0.125, 0.25, 0.45, 0.76, 1, and 1.22. (c) $L/B=14$; $\nu =0$, 0.06, 0.12, 0.185, 0.3, 0.45, and 0.575. (d) $L/B=100$; $\nu =0$, 0.02, 0.033, 0.05, 0.0675, 0.083, and 0.093. Increasing $\nu$ decreases the pressure at constant surface density.

Figure 8

Surface pressure as a function of area density for systems without nematic interactions ($\nu =0$). The insets show the behavior for small pressures, in linear scale. (a) $L/B=6$; $\chi =0$, 0.5, 0.825, 1.2, and 1.47. (b) $L/B=14$; $\chi =0$, 0.13, 0.26, 0.4, and 0.52. (c) $L/B=100$; $\chi =0$, 0.03, 0.06, 0.08, and 0.096. Increasing $\nu$ decreases the pressure at constant surface density. Results for $L/B=1.5$ are not shown as the curves are barely distinguishable from the isotropic case shown in Fig. 5.