  This paper is concerned with a diffusion model of phase-field type,
consisting of a parabolic system of two partial differential equations,
interpreted as balances of microforces and microenergy, for two unknowns: the
problem's order parameter and the chemical potential; each equation includes a
viscosity term and the field equations are complemented by Neumann homogeneous
boundary conditions and suitable initial conditions. In the recent paper
arXiv:1103.4585v1 [math.AP] we proved that this problem is well posed and
investigated the long-time behavior of its solutions. Here we discuss the
asymptotic limit of the system as the viscosity coefficient of the order
parameter equation tends to 0. We prove convergence of solutions to the
corresponding solutions for the limit problem, whose long-time behavior we
characterize; in the proofs, we employ compactness and monotonicity arguments.
