Nonparallel solutions of extended nematic polymers under an external field

We continue the study on equilibria of the Smoluchowski equation for dilute solutions of rigid extended (dipolar) nematics and dispersions under an imposed electric or magnetic field [25]. We first provide an alternative proof for the theorem that all equilibria are dipolar with the polarity vector parallel to the external field direction if the strength of the permanent dipole ($\mu$) is larger than or equal to the product of the external field ($E$) and the anisotropy parameter ($\alpha_0$) (i.e. $\mu \ge |\alpha_0| E$). Then, we show that when $\mu < |\alpha_0| E$, there is a critical value $\alpha^$*$\geq 1$ for the intermolecular dipole-dipole interaction strength ($\alpha$) such that all equilibria are either isotropic or parallel to the external field if $\alpha \le \alpha^$*; but nonparallel dipolar equilibria emerge when $\alpha > \alpha^$*. The nonparallel equilibria are analyzed and the asymptotic behavior of $\alpha^$* is studied. Finally, the asymptotic results are validated by direct numerical simulations.