Density functional theory for dense nematic liquid crystals with steric interactions

The celebrated work of Onsager on hard particle systems, based on the truncated second order virial expansion, is valid at relatively low volume fractions for large aspect ratio particles. While it predicts the isotropic-nematic phase transition, it does not provide a realistic equation of state in that the pressure remains finite for arbitrarily high densities. In this work, we derive a mean field density functional form of the Helmholtz free energy for nematics with hard core repulsion. In addition to predicting the isotropic-nematic transition, the model provides a more realistic equation of state. The energy landscape is much richer, and the orientational probability distribution function in the nematic phase possesses a unique feature—it vanishes on a nonzero measure set in orientation space.

Figure 1

Equilibrium order parameter $S$ versus $\varphi$ with uniaxial solutions only, superimposed on the energy per particle contours. The black curve represents the stable branch of the bifurcation, the green (light gray) curves correspond to the metastable branches, and the red (dark gray) curves to the unstable branches.

Figure 2

Density plot of the orientational distribution function for selected equilibrium uniaxial solutions. Top row, stable configuration with order parameter $S>0$; second row, unstable configuration with $S>0$; third row, unstable saddle point configuration with $S<0$; bottom row, metastable configuration with $S<0$.

Figure 3

(a) Lower integration limit $x_0$ versus $\varphi$ for uniaxial equilibrium solutions with $S>0$; the top horizontal line indicates the upper limit of the integration. (b) Upper limit $x_0$ vs $\varphi$ for equilibrium solutions with $S<0$; the bottom horizontal line indicates the lower limit of the integration. Black, green (light gray), and red (dark gray) colors correspond to stable, metastable, and unstable solutions in Fig. 1.

Figure 4

These diagrams illustrate two-phase coexistence: (a) the double tangent construction and (b) the Maxwell, equal area construction. The dependence of pressure on $\varphi$ in the isotropic and nematic phases is also shown in (b).

Figure 5

Ternary diagram for representing $\mathbf{Q}$.

Figure 6

Energy contours for two representative volume fractions: (a) $\varphi =-0.1$ and (b) $\varphi =0.1$.

Figure 7

The graph of ${\eta }_{\text{NI}}$ as in Eq. (77) against the reciprocal ${\kappa }^{-1}$ of the aspect ratio for $\lambda$ as in Eq. (78). Simulation data are represented by red circles. Data have been collected from [2] and [40] for both prolate and oblate ellipsoids, and reported here for the effective aspect ratio $\kappa >1$ (which is the real aspect ratio for prolate ellipsoids and its reciprocal for oblate ellipsoids). Data taken from [2] (see the first two columns of their Table 7) and from [39] (see the fifth and eighth columns of their Table VI, where ${\rho }_{\mathrm{cp}}=\sqrt{2}$) include both ends of the coexistence range (and their $\eta$ values have been multiplied by $\pi /6$ to account for the fact that the data of [2] and [39] are scaled to the volume $L{W}^2=6v_0/\pi$). Data taken from [40] are all those in the first column of Table I, but the two referring to the cases 1.3:1-prolate and 1:1.3-oblate, as these latter fall inside the crystal region of the phase diagram (as also shown in Fig. 5 of [40]).

Figure 8

The excluded volume of a pair of congruent ellipsoids of revolution $V_{\text{exc}}$ (normalized to its isotropic average $B_0$) is plotted against the angle $\theta$ between their symmetry axes according to four different approximate formulas: (long-dashed red lines) the expansion in Eq. (A1) truncated at $k=20$ with the coefficients computed according to the theory of [43]; (solid blue lines) our approximate formula in Eq. (A4); (dash-dotted black lines) Sheng's formula from [21]; and (dashed green lines) the HGO formula in Eq. (A11). Values of the ellipsoids' aspect ratio: (a) $\kappa =3$, (b) $\kappa =5$, and (c) $\kappa =10$.