Macroscopic differential model for hysteresis and butterfly-shaped behavior in ferroelectric materials

In the current paper, a macroscopic differential model is constructed oil the basis of the Landau theory of the first order phase transformation. Hysteresis loops and butterfly-shaped behaviors are modeled as a consequence of polarizations and orientation switchings. A non-convex free energy function is constructed to characterize different polarization orientations in the materials. Polarizations and orientation switchings are modeled by formulating the system state switching from one equilibrium state to another, as differential equations. The hysteresis loops and butterfly-shaped behaviors are successfully modeled. Comparison of the model results with the experimental counterpart is also presented.