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Phase segregation dynamics in particle systems with long range interactions II: interface motion
, 1996
"... We present and discuss the derivation of a nonlinear nonlocal integrodifferential 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 ddimensional lattice gas evolving via Kawasaki exchange dynamics, i ..."
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Cited by 66 (6 self)
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We present and discuss the derivation of a nonlinear nonlocal integrodifferential 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 ddimensional 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(γx − 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 CahnHilliard equation.
Convergence of nonlocal threshold dynamics approximations to front propagation, Preprint arXiv:0805.2618
"... Abstract. In this note we prove that appropriately scaled threshold dynamicstype algorithms corresponding to the fractional Laplacian of order α∈(0,2) converge to moving fronts. When α ≧ 1 the resulting interface moves by weighted mean curvature, while for α < 1 the normal velocity is nonlocal o ..."
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Cited by 19 (0 self)
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Abstract. In this note we prove that appropriately scaled threshold dynamicstype algorithms corresponding to the fractional Laplacian of order α∈(0,2) converge to moving fronts. When α ≧ 1 the resulting interface moves by weighted mean curvature, while for α < 1 the normal velocity is nonlocal of “fractionaltype. ” The results easily extend to general nonlocal anisotropic threshold dynamics schemes.
Binary Fluids with Long Range Segregating Interaction I: Derivation of Kinetic and Hydrodynamic Equations.
, 2000
"... 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 phas ..."
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Cited by 17 (7 self)
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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 spacetime 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 VlasovEuler and incompressible VlasovNavierStokes equations.
Phase segregation and interface dynamics in kinetic systems, Nonlinearity 19
, 2006
"... Abstract. We consider a kinetic model of two species of particles interacting with a reservoir at fixed temperature, described by two coupled VlasovFokkerPlank equations. We prove that in the diffusive limit the evolution is described by a macroscopic equation in the form of the gradient flux of t ..."
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Cited by 5 (4 self)
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Abstract. We consider a kinetic model of two species of particles interacting with a reservoir at fixed temperature, described by two coupled VlasovFokkerPlank 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 MullinsSekerka motion for the difference of the chemical potentials and the velocity of a HeleShaw 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: VlasovFokkerPlank equation; phase segregation; sharp interface limit; interface motion. 1.
Triple junction motion for an AllenCahn/CahnHilliard system
 Physica D
, 1996
"... In honor of John Cahn's 69 th birthday Long time asymptotics are developed here for an AllenCahn/CahnHilliard system derived recently by Cahn & NovickCohen [11] as a di use interface model for simultaneous orderdisorder and phase separation. Proximity to a deep quench limit is assumed, a ..."
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Cited by 3 (1 self)
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In honor of John Cahn's 69 th birthday Long time asymptotics are developed here for an AllenCahn/CahnHilliard system derived recently by Cahn & NovickCohen [11] as a di use 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. Quasistatic surface di usion 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 lms.
Marra R.: Sharp Interface Motion of a Binary Fluid
 Mixture, Jour. Stat. Phys
, 2006
"... Abstract. We derive hydrodynamic equations describing the evolution of a binary fluid segregated into two regions, each rich in one species, which are separated (on the macroscopic scale) by a sharp interface. Our starting point is a VlasovBoltzmann (VB) equation describing the evolution of the one ..."
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Abstract. We derive hydrodynamic equations describing the evolution of a binary fluid segregated into two regions, each rich in one species, which are separated (on the macroscopic scale) by a sharp interface. Our starting point is a VlasovBoltzmann (VB) equation describing the evolution of the one particle position and velocity distributions, fi(x, v, t), i = 1, 2. The solution of the VB equation is developed in a Hilbert expansion appropriate for this system. This yields incompressible NavierStokes equations for the velocity field u and a jump boundary condition for the pressure across the interface. The interface, in turn, moves with a velocity given by the normal component of u. 1.
Breakup of Universality in the Generalized Spinodal Nucleation Theory
, 2003
"... The problem of nucleation near spinodal is revisited. It is shown that the standard scaling argument due to Unger and Klein [Phys. Rev. B 29:2698–2708 (1984)] based on neglecting all but the first two terms of the Taylor expansion of the potential in the free energy functional is only valid below c ..."
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The problem of nucleation near spinodal is revisited. It is shown that the standard scaling argument due to Unger and Klein [Phys. Rev. B 29:2698–2708 (1984)] based on neglecting all but the first two terms of the Taylor expansion of the potential in the free energy functional is only valid below critical dimension. At critical dimension, the nucleating droplet has a bigger amplitude and a smaller spatial extent than predicted by the standard scaling argument. In this case the structure of the droplet is determined in a nontrivial fashion by the next order term in the expansion of the potential. Above critical dimension, the amplitude of the nucleating droplet turns out to be too big to justify expanding the potential in Taylor series, and no universality is to be expected in the shape and size of the droplet. Both at and above critical dimension, however, the free energy barrier remains finite, which indicates that the nucleation rate does not vanish at spinodal as predicted by the standard scaling argument. KEY WORDS: Nonclassical nucleation; spinodal; critical droplet; matched asymptotics; scaling. 1.
Online at stacks.iop.org/Non/19/115
, 2005
"... doi:10.1088/09517715/19/1/007 Phase segregation and interface dynamics in kinetic systems ..."
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doi:10.1088/09517715/19/1/007 Phase segregation and interface dynamics in kinetic systems
MultiSpike States Of The CahnHilliard Model For Phase Transitions
"... this article we choose to carefully examine the existence of stationary states for the CahnHilliard equation having many `nuclei'. That is, states with several localized areas of higher concentration of one of the two species in a background of almost constant concentration. These states turn ..."
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this article we choose to carefully examine the existence of stationary states for the CahnHilliard equation having many `nuclei'. That is, states with several localized areas of higher concentration of one of the two species in a background of almost constant concentration. These states turn out to be unstable but nevertheless have important implications for the evolution of patterns. For results related to spinodal decomposition, evolution of interfaces, and Ostwald ripening, the reader may consult [G], [MPW], [WS], [Mo], [LM], [ABF], [BX I & II], [AF I & II], [ABC], [ABCF], and the references contained in those articles. The bulk of the article will be quite technical but first we take some time to develop the model and give some background material to place it into context. We then to give a heuristic description of the mathematical approach to the problem of finding stationary states with many nuclei. MULTISPIKE STATES 5 1.1 Energy. We take this as a fundamental principle: A material structure evolves in such a way that its FREE ENERGY decreases as quickly as possible