We discuss the convergence of the Caginalp-Fix phase field model to the weak formulation of the Stefan problem. In particular, we address two related situations: first, we study the case of a single substance subject to this change of the solidification model; then, we try and repeat the convergence analysis to the case of two different adjoining fluids obeying to transmission conditions at the common boundary both for the temperature and for the phase field. In particular, we assume that the solidification law undergoes a variation only on one side: in this case, the convergence problem is solvable only under additional compatibility conditions, which we try and justify from both the mathematical and the thermodynamical viewpoints.
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