A constraint-free phase field model for ferromagnetic domain evolution

A continuum constraint-free phase field model is proposed to simulate the magnetic domain evolution in ferromagnetic materials. The model takes the polar and azimuthal angles (ϑ₁,ϑ₂), instead of the magnetization unit vector m(m₁,m₂,m₃), as the order parameters. In this way, the constraint on the magnetization magnitude can be exactly satisfied automatically, and no special numerical treatment on the phase field evolution is needed. The phase field model is developed from a thermodynamic framework which involves a configurational force system for ϑ₁ and ϑ₂. A combination of the configurational force balance and the second law of thermodynamics leads to thermodynamically consistent constitutive relations and a generalized evolution equation for the order parameters (ϑ₁,ϑ₂). Beneficial from the constraint-free model, the three-dimensional finite-element implementation is straightforward, and the degrees of freedom are reduced by one. The model is shown to be capable of reproducing the damping-dependent switching dynamics, and the formation and evolution of domains and vortices in ferromagnetic materials under the external magnetic or mechanical loading. Particularly, the calculated out-of-plane component of magnetization in a vortex is verified by the corresponding experimental results, as well as the motion of the vortex under a magnetic field.