Phase Diagram of Nematic Polymer Monolayers with the Onsager Interaction Potential

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 MaierSaupe 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.