Global orientation dynamics for liquid crystalline polymers

We study global orientation dynamics of the Doi–Smoluchowski equation with the Maier–Saupe potential on the sphere, which arises in the modeling of rigid rod-like molecules of polymers. Using the orientation tensor we first reconfirm the structure and number of equilibrium solutions established in [H.L. Liu, H. Zhang, P.W. Zhang, Axial symmetry and classification of stationary solutions of Doi–Onsager equation on the sphere with Maier–Saupe potential, Comm. Math. Sci. 3 (2) (2005) 201–218]. We then examine global orientation dynamics in terms of eigenvalues of the orientation tensor via the Doi closure approximation. It is shown that for small intensity 0<α<40<α<4, all states will evolve into the isotropic phase; for large intensity α>4.5α>4.5, all states will evolve into the nematic prolate phase; and for the intermediate intensity 4<α<4.54<α<4.5, an initial state will evolve into either the isotropic phase or the stable phase of two nematic prolate phases, depending on whether such an initial configuration crosses a critical threshold. Moreover, the uniaxial symmetry structure is shown to be preserved in time.