Algebraic equations of state for the liquid crystalline phase behavior of hard rods

Based on simplifications of previous numerical calculations [H. Graf and H. Löwen, Phys. Rev. E 59, 1932 (1999)], we propose algebraic free energy expressions for the smectic-A liquid crystal phase and the crystal phases of hard spherocylinders. Quantitative agreement with simulations is found for the resulting equations of state. The free energy expressions can be used to straightforwardly compute the full phase behavior for all aspect ratios and to provide a suitable benchmark for exploring how attractive interrod interactions mediate the phase stability through perturbation approaches such as free-volume or van der Waals theory.

Figure 1

Schematic images of the isotropic (I), nematic (N), smectic-A (SmA), AAA crystal, and ABC crystal phase states of hard spherocylinder suspensions.

Figure 2

Two types of free area in a 2D hexagonal crystal are indicated for the central particle. The yellow area denotes the free area the particle has when confined within the Wigner–Seitz cell, which is indicated by the solid lines. The red and yellow area combined denote the free area the particle has within the confinement of the neigboring particles.

Figure 3

Volume fraction dependence of the excess free energy of the smectic-A phase of Eq. (15), $f_{\mathrm{or}}+f_{\mathrm{pack}}$. Free energy obtained using numerical minimization (data points) is compared to approximate analytical results (solid curves) for aspect ratios $L/D=5$, 50, and 500, where both use the correction parameter $A=0.41\eta h$ based on fits to the simulation data [6, 7].

Figure 4

Predicted equations of state (curves) for hard spherocylinders of aspect ratio $L/D=$ (a) 3.4, (b) 4, (c) 5, and (d) 40 as indicated. The symbols are simulation results from McGrother et al. [6] (filled) and from Bolhuis and Frenkel [7] (open).

Figure 5

Phase diagram of hard spherocylinders in terms of the volume fraction $\eta$ and inverse aspect ratio $D/L$. Curves are our predictions based on algebraic expressions for all phase states. The data points are simulation results from Bolhuis and Frenkel [7].