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# Communications in Mathematical Sciences

## Volume 5 (2007)

### Number 4

### Steady states and their stability of homogeneous, rigid, extended nematic polymers under imposed magnetic fields

Pages: 917 – 950

DOI: http://dx.doi.org/10.4310/CMS.2007.v5.n4.a10

#### Authors

#### Abstract

We study the steady state phase behavior of homogeneous, rigid, extended (polar) nematic polymers or nematic dispersions under imposed magnetic (or electric fields), in which the intermolecular dipole-dipole and excluded volume interaction as well as external field contribution are accounted for. We completely characterize the phase diagram for polar nematics with and without permanent magnetic moments (or dipoles). For nematics without magnetic moments, the steady state is either purely nematic or polar depending on the strength of the excluded volume and the dipole-dipole interaction, in which the nonzero polarity vector (the first moment vector) is coaxial with the second moment tensor; thereby the steady state pdf is determined essentially by up to three scalar order parameters and a rotational group of SO(2) transverse to the imposed field direction. For nematics with permanent magnetic moments (or dipoles), the steady states are polar and the polarity vector is parallel to the external field direction when a necessary condition for parameters is met. When the condition is violated, only stable steady states have their polarity vector parallel to the external field direction and there are thermodynamically unstable nonparallel states. The stability of the steady states is inferred from the minimum of the free energy density.

#### Keywords

nematic liquid crystal; kinetic theory; polymers; magnetic field; electric field; bifurcation; steady state; phase transition; polarity; stability

#### 2010 Mathematics Subject Classification

45K05, 70K50, 82B26, 82B27