A thermodynamic approach to ferromagnetism and phase transitions

The paper provides a modelling of the magnetization curve and of the ferromagnetic–paramagnetic transition within a continuum thermodynamic setting. The general model of the nonlinear, time dependent behaviour of ferromagnetic materials is accomplished by regarding the magnetization vector as an internal variable, namely as a vector field whose time evolution is a constitutive equation subject to the requirements of the second law of thermodynamics. The exchange interaction of the magnetization is modelled through a dependence of the free energy on the magnetization gradient. Consistent with the non-simple character of the material, the second law allows for a non-zero extra-entropy flux. A general three-dimensional scheme is elaborated which seems to be new in the literature. The three-dimensional setting is then established for stationary and homogeneous fields thus finding the collinearity and the corresponding form of the magnetic susceptibility. The whole evolution problem for the temperature and the magnetization is provided so that temperature-induced transition processes are allowed. The model accounts also for the dependence of the saturation magnetization on the temperature. Also for the sake of comparison with the existing literature, the evolution equations for the direction and the intensity of magnetization are derived. Known models, such as those of Landau–Lifshitz and Gilbert, are recovered as particular cases of saturated bodies. Next, the model is made more specific so as to account in detail for the saturation, the residual or spontaneous magnetization and the coercive field. First, the classical potential, which traces back to Ginzburg, and the Weiss model are revisited. The corresponding lack of the saturation effect or the description via implicit relations are emphasized. Hence, a new potential, with a logarithmic dependence on the magnetization, is investigated which provides the residual magnetization and the coercive field in an explicit way and satisfies expected properties of the residual magnetization as a function of the temperature.