Phase behavior of shape-changing spheroids

We introduce a simple model for a biaxial nematic liquid crystal. This consists of hard spheroids that can switch shape between prolate (rodlike) and oblate (platelike) subject to an energy penalty $\mathrm{\Delta }\epsilon$. The spheroids are approximated as hard Gaussian overlap particles and are treated at the level of Onsager's second-virial description. We use both bifurcation analysis and a numerical minimization of the free energy to show that, for additive particle shapes, (i) there is no stable biaxial phase even for $\mathrm{\Delta }\epsilon =0$ (although there is a metastable biaxial phase in the same density range as the stable uniaxial phase) and (ii) the isotropic-to-nematic transition is into either one of two degenerate uniaxial phases, rod rich or plate rich. We confirm that even a small amount of shape nonadditivity may stabilize the biaxial nematic phase.

Figure 1

(a) Expansion coefficients $B_{ij}^{\left(l\right)}$ for $l=0,2$. (b) Bifurcation density ${\rho }^{*}$ of the I phase with respect to the N phase vs rod elongation ${\kappa }_R$ for different $\mathrm{\Delta }\epsilon ={\epsilon }_R-{\epsilon }_P$. Here $\nu =1$ (additive shapes).

Figure 2

(a) Order parameters and (b) composition for the $\mathrm{I}\text{−}{\mathrm{N}}_U^{+}$ phase transition for ${\kappa }_R=1/{\kappa }_P=5$. $\mathrm{\Delta }\epsilon =0$ and $\nu =1$. Lines without symbols: L2 approximation; lines with symbols: full numerical solution.

Figure 3

(a) Order parameters and (b) composition for the $\mathrm{I}\text{−}{\mathrm{N}}_U^{-}$ phase transition for ${\kappa }_R=1/{\kappa }_P=5,\mathrm{\Delta }\epsilon =0$ and $\nu =1$. Lines without symbols: L2 approximation; lines with symbols: full numerical solution.

Figure 4

(a) Order parameters and (b) composition for the $\mathrm{I}\text{−}{\mathrm{N}}_B$ phase transition for ${\kappa }_R=1/{\kappa }_P=5,\mathrm{\Delta }\epsilon =0$, and $\nu =1$. Lines without symbols: L2 approximation; lines with symbols: full numerical solution.

Figure 5

Chemical potential, or Gibbs free energy per particle, of ${\mathrm{N}}_U,{\mathrm{N}}_B$, and I phases vs pressure for $\mathrm{\Delta }\epsilon =0,\nu =1$ and (a) ${\kappa }_R=1/{\kappa }_P=5$ and (b) ${\kappa }_R=1/{\kappa }_P=20$. Lines without symbols: L2 approximation; lines with symbols: full numerical solution.

Figure 6

(a) Order parameters and (b) composition for the $\mathrm{I}\text{−}{\mathrm{N}}_U^{+}$ phase transition in the L2 approximation for ${\kappa }_R=1/{\kappa }_P=5,\mathrm{\Delta }\epsilon =-1$, and $\nu =1$.

Figure 7

(a) Order parameters and (b) composition for the $\mathrm{I}\text{−}{\mathrm{N}}_U^{-}$ phase transition in the L2 approximation for ${\kappa }_R=1/{\kappa }_P=5,\mathrm{\Delta }\epsilon =1$, and $\nu =1$.

Figure 8

(a) Order parameters and (b) composition for the $\mathrm{I}\text{−}{\mathrm{N}}_U^{+}$ phase transition in the L2 approximation for ${\kappa }_R=1/{\kappa }_P=5,\mathrm{\Delta }\epsilon =0$, and $\nu =3$.

Figure 9

(a) Order parameters and (b) composition for the $\mathrm{I}\text{−}{\mathrm{N}}_U^{-}$ phase transition in the L2 approximation for ${\kappa }_R=1/{\kappa }_P=5,\mathrm{\Delta }\epsilon =0$, and $\nu =3$.

Figure 10

(a) Order parameters and (b) composition for the $\mathrm{I}\text{−}{\mathrm{N}}_B$ phase transition in the L2 approximation for ${\kappa }_R=1/{\kappa }_P=5,\mathrm{\Delta }\epsilon =0$, and $\nu =3$.

Figure 11

Chemical potential, or Gibbs free energy per particle, of ${\mathrm{N}}_U,{\mathrm{N}}_B$, and I phases vs pressure in the L2 approximation for ${\kappa }_R=1/{\kappa }_P=5,\mathrm{\Delta }\epsilon =0$. and $\nu =3$.