Phase Diagram of Nematic Polymer Monolayers with the Onsager Interaction Potential
Abstract
We present an asymptotic study on the phase diagram of two-dimensional nematic liquid crystal
polymer monolayers with the Onsager intermolcular potential. In contrast to the case of Maier-
Saupe interaction potential where there is only one nematic branch, our analysis reveals that there
are infinite many nematic branches in the case of the Onsager interaction potential. An asymptotic
expression is derived for each nematic branch. For small polymer concentration the isotropic branch
is the only equilibrium state. As the polymer concentration is increased, nematic branches appear
one by one, starting with the first nematic branch. The polymer orientation distribution of the first
nematic branch has a two fold rotational symmetry, the second branch has a four fold rotational
symmetry, the third branch has a six fold rotational symmetry, and so on. To determine the stability
of these nematic branches, we derive an asymptotic expression of free energy for each nematic
branch. We find that free energies of all nematic branches are lower than that of the isotropic
state, and the first nematic branch has the lowest free energy among all branches. To further
investigate the stability and meta-stability, we carry out asymptotic analysis of the free energy when
each nematic state is perturbed. We conclude that (1) the isotropic branch is stable until the first
nematic branch appears, (2) the first nematic branch is stable, and (3) the isotropic branch (after
the appearance of the first nematic branch) and all other nematic branches are unstable when
perturbed by the leading Fourier mode in the first nematic branch. We also present a spectrum
numerical method for calculating nematic branches and free energies. The spectrum method yields
results that are accurate up to the computer precision. All of asymptotic results are confirmed by
numerical results obtained with the spectrum method.
Description
The article of record as published may be located at http://dx.doi.org/10.1166/jctn.2010.1417
Rights
This publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States.Collections
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