A phase-field approach to non-isothermal transitions

The purpose of the paper is to set up a phase-field model for first-order transitions. The phase field is identified with the concentration of a phase and hence is subject to the continuity equation with a mass growth due to the progression of the transition. The continuity equation is viewed as a differential constraint. The body, in the transition layer, is regarded as a viscous material with time- and space-dependent fields for the mass density, the (absolute) temperature, and the phase-field. Consistency of the model suggests that gradients up to third order are considered. The thermodynamic restrictions are derived by letting the second law be expressed by the Clausius-Duhem inequality and allowing for an extra-entropy flux which, eventually, proves essential to the whole thermodynamic scheme. Results are obtained by using the Helmholtz and Gibbs free energies. As a check of the model, Clapeyron's equation is derived by means of the Gibbs free energy. A maximum theorem is proved which shows that the phase field takes values between 0 and 1 if it so does at an initial time.

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