Characterization of stable kinetic equilibria of rigid, dipolar rod ensembles for coupled dipole–dipole and Maier–Saupe potentials

We study equilibria of the Smoluchowski equation for rigid, dipolar rod ensembles where the intermolecular potential couples the dipole–dipole interaction and the Maier–Saupe interaction. We thereby extend previous analytical results for the decoupled case of the dipolar potential only (Fatkullin and Slastikov 2005 Nonlinearity 18 2565–80; Ji et al Phys. Fluids at press; Wang et al 2005 Commun. Math. Sci. 3 605–20) or the Maier–Saupe potential only (Constantin et al 2004 Arch. Ration. Mech. Anal. 174 365–84; Constantin et al 2004 Discrete Contin. Dyn. Syst. 11 101–12; Constantin and Vukadinovic 2005 Nonlinearity 18 441–3; Constantin 2005 Commun. Math. Sci. 3 531–44; Fatkullin and Slastikov 2005 Commun. Math. Sci. 3 21–6; Liu et al 2005 Commun. Math. Sci. 3 201–18; Luo et al 2005 Nonlinearity 18 379–89; Zhou et al 2005 Nonlinearity 18 2815–25; Zhou and Wang Commun. Math. Sci. at press), and prove certain numerical observations for equilibria of coupled potentials (Ji et al Phys. Fluids at press). We first derive stability conditions, on the magnitude of the polarity vector (the first moment of the orientational probability distribution function) and on the direction of the polarity. We then prove that all stable equilibria of rigid, dipolar rod dispersions are either isotropic or prolate uniaxial. In particular, all stable anisotropic equilibrium distributions admit the following remarkable symmetry: the peak axis of orientation is aligned with both the polarity vector (first moment) and the distinguished director of the uniaxial second moment tensor. The stability is essential in establishing the axisymmetry. To demonstrate that the 5 Also at: School of Mathematics, Nankai University, Tianjin, 300071, People’s Republic of China. 0951-7715/07/020277+21$30.00 © 2007 IOP Publishing Ltd and London Mathematical Society Printed in the UK 277