  We are concerned with a phase field system consisting of two partial
differential equations in terms of the variables thermal displacement, that is
basically the time integration of temperature, and phase parameter. The system
is a generalization of the well-known Caginalp model for phase transitions,
when including a diffusive term for the thermal displacement in the balance
equation and when dealing with an arbitrary maximal monotone graph, along with
a smooth anti-monotone function, in the phase equation. A Cauchy-Neumann
problem has been studied for such a system in arXiv:1107.3950v2 [math.AP], by
proving well-posedness and regularity results, as well as convergence of the
problem as the coefficient of the diffusive term for the thermal displacement
tends to zero. The aim of this contribution is rather to investigate the
asymptotic behaviour of the problem as the coefficient in front of the
Laplacian of the temperature goes to 0: this analysis is motivated by the types
III and II cases in the thermomechanical theory of Green and Naghdi. Under
minimal assumptions on the data of the problems, we show a convergence result.
Then, with the help of uniform regularity estimates, we discuss the rate of
convergence for the difference of the solutions in suitable norms.
