On the relation between the standard phase-field model and a "thermodynamically consistent" phase-field model

A further comparison is made between the standard phase-field equations af_t=?²f+( 1 ?²)[g(f)-u], u_t=? ²u+ 1 2lf_t, and the relevant "thermodynamically consistent model of phase transitions" proposed by the authors [Physica D 43 (1990) 44-62]. Here we concentrate on the usual case where g(f)=f-f³, and for comparison purposes retain this expression for the analogous nonlinear functions in the latter model. It is brought out, among other things, that the standard model is thermodynamically consistent in the sense of being derivable from a free-energy functional. However, this free-energy functional is of a somewhat unusual kind: it implies that, at constant temperature, the energy density varies linearly with the order parameter f and the entropy density is a non-concave function of f. The example of a hard sphere system indicates that such behaviour is not impossible, but in most other models the energy density and the entropy are both strictly concave in f. © 1993.