Multiple branches of ordered states of polymer ensembles with the Onsager excluded volume potential

We study the branches of equilibrium states of rigid polymer rods with the Onsager excluded volume potential in two-dimensional space. Since the probability density and the potential are related by the Boltzmann relation at equilibrium, we represent an equilibrium state using the Fourier coefficients of the Onsager potential. We derive a non-linear system for the Fourier coefficients of the equilibrium state. We describe a procedure for solving the non-linear system. The procedure yields multiple branches of ordered states. This suggests that the phase diagram of rigid polymer rods with the Onsager potential has a more complex structure than that with the Maier–Saupe potential. A study of free energy indicates that the first branch of ordered states is stable while the subsequent branches are unstable. However, the instability of the subsequent branches does not mean they are not interesting. Each of these unstable branches, under certain external potential, can be made metastable, and thus may be observed. © 2008 Elsevier B.V. All rights reserved. PACS: 61.30.Gd; 61.30.Vx The isotropic-nematic phase transition in rigid rod-like polymers is a classical topic [9]. In 1949, Onsager first examined the isotropic-nematic phase transition theoretically by modeling the steric excluded volume interactions [1]. Onsager’s study was based on a virial expansion, which yields a mean field potential, now bearing his name (the Onsager potential). Using this mean field potential and choosing an appropriate trial function for the orientation distribution function, Onsager was able to argue that when the concentration is high enough, there is a transition from a uniform isotropic state to an orientationally ordered prolate nematic state. Many theories have been formulated after Onsager’s pioneering work. An excellent review on the theoretical advances of liquid crystalline polymers is given in [2]. In this study, we adopt the Doi–Hess model [3,4], which was first developed by Doi and Edwards to describe spatially * Corresponding author. E-mail address: [email protected] (H. Wang). 0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.01.085 homogeneous flows of rodlike liquid crystal polymers and has been used in many studies [5–7]. In the Doi–Hess model, the polymer ensemble is represented by an orientational probability density in a meso-scale much larger than individual polymer rods but much smaller than the macroscopic flow. Each polymer rod undergoes Brownian diffusion and is affected by the interaction with other polymer rods within the meso-scale. The inter-molecular interaction is usually modeled using the Onsager potential or the Maier–Saupe potential, which is an approximation to the Onsager potential. The main mathematical difference between the Onsager potential and the Maier–Saupe potential is that the Onsager potential depends on the whole probability density function while the Maier–Saupe potential depends only on the second moment. As a result, the Doi–Hess model with the Onsager potential is mathematically much more challenging than with the Maier–Saupe potential. The Doi–Hess model with the Onsager potential has been studied numerically and the Onsager theory has been extended to other systems. For example, Larson [8] applied the spherical 3424 H. Wang, H. Zhou / Physics Letters A 372 (2008) 3423–3428 harmonic expansions to solve the Doi–Hess equation with the Onsager potential for the 3D time-dependent orientation distribution function in the presence of shearing flow. Lasher [10] extended Onsager’s work to the nematic ordering of hard rods with a direct application of the scaled particle approach. Vroege and Lekkerkerker [11] provided a comprehensive overview on the theory and experiments in lyotropic colloidal and polymer liquid crystals. In particular, they generalized Onsager’s virial theory to polydisperse solutions and soft interactions. Chrzanowska [12] developed a simple Onsager theory type density functional theory (DFT) of a two-dimensional system of hard needles. Recently, the phase diagram of rigid rod polymers with the Maier–Saupe potential, previously obtained in various numerical studies (for example, see [5] and [13] and references therein), has been studied analytically [14–16]. Thus, it is timely and worthwhile to revisit analytically the phase diagram of rigid rod polymers with the Onsager potential. In studying a phase diagram, it is important to get a full picture of the phase diagram, including all stable and unstable branches. Besides mathematical completeness, there are two more reasons for including all branches when studying a phase diagram: (a) the full structure of the phase diagram helps us understand the dynamics and the stabilities; and (b) under a certain external field, unstable branches may be made meta-stable and thus may be observed. In this Letter, we consider the case of rigid rod polymers in the two-dimensional space with the Onsager intermolecular interaction potential. In the two-dimensional space, orientation is represented by the polar angle θ and the orientational probability density ρ(θ, t) is governed by the Smoluchowski equation