Solution landscape of the Onsager model identifies non-axisymmetric critical points

Abstract

We investigate critical points of the Onsager free-energy model on a sphere with different potential kernels, including the dipolar potential, the Maier-Saupe potential, the coupled dipolar/Maier-Saupe potential, and the Onsager potential. A uniform sampling method is implemented for the discretization of the Onsager model, and solution landscapes of the Onsager model are constructed using saddle dynamics coupled with downward/upward search algorithms. We first construct the solution landscapes with the dipolar and Maier-Saupe potentials, for which all critical points are axisymmetric. For the coupled dipolar/Maier-Saupe potential, the solution landscape shows a novel non-axisymmetric critical point, named tennis, which exists for a wide range of parameters. We further demonstrate various non-axisymmetric critical points in the Onsager model with the Onsager potential, including square, hexagon, octahedral, cubic, quarter, icosahedral}, and dodecahedral states. The bifurcation diagram is presented to show the primary and secondary bifurcations of the isotropic state and reveal the emergence of the critical points. The solution landscape provides an efficient approach to show the global structure of the model system as well as the bifurcations of critical points, which can not only support the previous theoretical conjectures but also propose new conjectures based on the numerical findings.

References in corpus (8)

• Numerical methods for nonlocal and fractional models
• Transition pathways connecting crystals and quasicrystals
• Searching the solution landscape by generalized high-index saddle dynamics
• Solution landscape of a reduced Landau-de Gennes model on a hexagon
• Constrained high-index saddle dynamics for the solution landscape with equality constraints
• Onsager's missing steps retraced
• Axisymmetry of critical points for the Onsager functional
• Cavity volume and free energy in many-body systems

Cited by in corpus (5)

• Computing solution landscape of nonlinear space-fractional problems via fast approximation algorithm
• Generalization of the Maier-Saupe theory to the ferroelectric nematic phase
• Hierarchies of Critical Points of a Landau-de Gennes Free Energy on Three-Dimensional Cuboids
• Transition pathways in Cylinder-Gyroid interface
• Gentlest ascent dynamics on manifolds defined by adaptively sampled point-clouds