We present and discuss the derivation of a nonlinear non-local integro-differential equation for the macroscopic time evolution of the conserved order parameter ρ(r, t) of a binary alloy undergoing phase segregation. Our model is a d-dimensional lattice gas evolving via Kawasaki exchange dynamics, i.e. a (Poisson) nearest–neighbor exchange process, reversible with respect to the Gibbs measure for a Hamiltonian which includes both short range (local) and long range (nonlocal) interactions. The nonlocal part is given by a pair potential γ d J(γ|x − y|), γ> 0, x and y in Z d, in the limit γ → 0. The macroscopic evolution is observed on the spatial scale γ −1 and time scale γ −2, i.e., the density, ρ(r, t), is the empirical average of the occupation numbers over a small macroscopic volume element centered at r = γx. A rigorous derivation is presented in the case in which there is no local interaction. In a subsequent paper (part II), we discuss the phase segregation phenomena in the model. In particular we argue that the phase boundary evolutions, arising as sharp interface limits of the family of equations derived in this paper, are the same as the ones obtained from the corresponding limits for the Cahn-Hilliard equation.

We study the evolution of a two component fluid consisting of "blue" and "red" particles which interact via strong short range (hard core) and weak long range pair potentials. At low temperatures the equilibrium state of the system is one in which there are two coexisting phases. Under suitable choices of space-time scalings and system parameters we first obtain (formally) a mesoscopic kinetic VlasovBoltzmann equation for the one particle position and velocity distribution functions, appropriate for a description of the phase segregation kinetics in this system. Further scalings then yield Vlasov-Euler and incompressible Vlasov-Navier-Stokes equations.

Long time asymptotics are developed here for an Allen-Cahn/Cahn-Hilliard system derived recently by Cahn & Novick-Cohen [11] as a diffuse interface model for simultaneous orderdisorder and phase separation. Proximity to a deep quench limit is assumed, and spatial scales are chosen to model Krzanowski instabilities in which droplets of a minor disordered phase bounded by interphase boundaries (IPB) of high curvature coagulate along a slowly curved antiphase boundaries (APB) separating two ordered variants. The limiting motion couples motion by mean curvature of the APBs with motion by minus the surface Laplacian of the IPBs on the same time scale. Quasi-static surface diffusion of the chemical potential occurs along APBs. The framework outlined here should also be suitable for describing sintering of small grains and thermal grain boundary grooving in polycrystalline films. KEYWORDS: triple-junction motion, motion by mean curvature, motion by minus the surface Laplacian of mean curvatur...

Abstract. We consider a kinetic model of two species of particles interacting with a reservoir at fixed temperature, described by two coupled Vlasov-Fokker-Plank equations. We prove that in the diffusive limit the evolution is described by a macroscopic equation in the form of the gradient flux of the macroscopic free energy functional. Moreover, we study the sharp interface limit and find by formal Hilbert expansions that the interface motion is given in terms of a quasi stationary problem for the chemical potentials. The velocity of the interface is the sum of two contributions: the velocity of the Mullins-Sekerka motion for the difference of the chemical potentials and the velocity of a Hele-Shaw motion for a linear combination of the two potentials. These equations are identical to the ones in [OE] modelling the motion of a sharp interface for a polymer blend. Keywords: Vlasov-Fokker-Plank equation; phase segregation; sharp interface limit; interface motion. 1.