A Note on the Onsager Model of Nematic Phase Transitions

We study Onsager’s free energy functional for nematic liquid crystals with orientation parameter on a unit circle. For a class of interaction potentials we obtain explicit expressions for all critical points, analyze their stability, and construct the corresponding bifurcation diagram. We also derive asymptotic expansions of the equilibrium density of orientations near the critical and zero temperatures. In his seminal paper [7] Onsager introduced a simple model describing nematic phase transitions of rod-like polymers. His theory is based on the second-order virial approximation of the free energy as a functional of density of rod orientations. Under a suitable variational ansatz, he was able to demonstrate that at sufficiently high concentrations, rod-like polymers form a nematic phase. In this note we discuss Onsager’s model with orientation parameter on a unit circle. The corresponding free energy functional reads,