Amy Novick-Cohen

Nonlinear aspects of the Cahn-Hilliard equation

Abstract This paper treats phase separation within the context of the phenomenological Cahn-Hilliard equation, c t = ∇ · [ M ( c )∇( ∂f / ∂c - K ∇ 2 c )], where c is the concentration, M ( c ) is the mobility, and f ( c ) is homogeneou s free energy, which is assumed here to be a fourth degree polynomial. Natural boundary conditions are introduced. The full set of equilibrium solutions is specified. A comparison theorem for stability criteria which was postulated by Langer is proved here within the framework of the natural boundary conditions. Energy methods are used to define and estimate the limit of monotonic global stability. It is pointed out that within the parameter region where the uniform homogeneous state is the only equilibrium solution, there may still exist some internal “excitable” region in which the homogeneous solution possesses growing fluctuations. Furthermore a periodic instability is shown to exist in the metastable region in addition to the well-known nucleation instability.

The Cahn–Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature

We show by using formal asymptotics that the zero level set of the solution to the Cahn–Hilliard equation with a concentration dependent mobility approximates to lowest order in ɛ. an interface evolving according to the geometric motion, (where V is the normal velocity, Δ 8 is the surface Laplacian and κ is the mean curvature of the interface), both in the deep quench limit and when the temperature θ is where є 2 is the coefficient of gradient energy. Equation (0.1) may be viewed as motion by surface diffusion, and as a higher-order analogue of motion by mean curvature predicted by the bistable reaction-diffusion equation.

Stable patterns in a viscous diffusion equation

We consider a pseudoparabolic regularization of a forward-backward nonlinear diffusion equation ut = A(f(u) + vut), motivated by the problem of phase separation in a viscous binary mixture. The function f is nonmonotone, so there are discontinuous steady state solutions corresponding to arbitrary arrangements of phases. We find that any bounded measurable steady state solution u(x) satisfying f(u) = constant, f(u(x)) > 0 a.e. is dynamically stable to perturbations in the sense of convergence in measure. In particular, smooth solutions may achieve discontinuous asymptotic states. Furthermore, stable states need not correspond to absolute minimizers of free energy, thus violating Gibbs principle of stability for phase mixtures.

The viscous Cahn-Hilliard equation: Morse decomposition and structure of the global attractor

In this paper we establish a Morse decomposition of the stationary solutions of the one-dimensional viscous Cahn–Hilliard equation by explicit energy calculations. Strong non-degeneracy of the stationary solutions is proven away from turning points and points of bifurcation from the homogeneous state and the dimension of the unstable manifold is calculated for all stationary states. In the unstable case, the flow on the global attractor is shown to be semi-conjugate to the flow on the global attractor of the Chaffee-Infante equation, and in the metastable case close to the nonlocal reaction–diffusion limit, a partial description of the structure of the global attractor is obtained by connection matrix arguments, employing a partial energy ordering and the existence of a weak lap number principle.

Evolution equations for phase separation and ordering in binary alloys

We explore two phenomenological approaches leading to systems of coupled Cahn-Hilliard and Cahn-Allen equations for describing the dynamics of systems which can undergo first-order phase separation and order-disorder transitions simultaneously, starting from the same discrete lattice free energy function. In the first approach, a quasicontinuum limit is taken for this discrete energy and the evolution of the system is then assumed to be given by gradient flow. In the second approach, a discrete set of gradient flow evolution equations is derived for the lattice dynamics and a quasicontinuum limit is then taken. We demonstrate in the context of BCC Fe−Al binary alloys that it is important that variables be chosen that accommodate the variations in the average concentration as well as the underlying ordered structure of the possible coexistent phases. Only then will the two approaches lead to roughly the same continuum descriptions. We conjecture that in general the number of variables necessary to describe the dynamics of such systems is equal toN1+N2−1, whereN1 is given by the dimension of the span of the bases of the irreducible representations needed to describe the symmetry groups of the possible equilibrium phases andN2 is the number of chemical components.N1 of these variables are nonconserved, and the remaining are conserved and represent the average concentrations. For the Fe−Al alloys this implies a description of one conserved order parameter and one nonconserved order parameter. The resultant description is given by a Cahn-Hilliard equation coupled to a Cahn-Allen equation via the lower-order nonlinear terms. The rough equivalence of the two phenomenological methods adds credibility to the validity of the resulting evolution equations. A similar description should also be valid for alloy systems in which the structure of the competing phases is more complicated.

Triple-junction motion for an Allen-Cahn/Cahn-Hilliard system

Abstract Long time asymptotics are developed here for an Allen–Cahn/Cahn–Hilliard system derived recently by Cahn and Novick-Cohen [J.W. Cahn, A. Novick-Cohen, J. Statist. Phys. 76 (1994) 877–909] as a diffuse interface model for simultaneous order–disorder 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 (IPBs) of high curvature coagulate along a slowly curved antiphase boundaries (APBs) 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 timescale. Quasi-static surface diffusion of the chemical potential occurs along APBs. The framework here yields both sharp interface and diffuse interface modeling of sintering of small grains and thermal grain boundary grooving in polycrystalline films.

Steady states of the one-dimensional Cahn–Hilliard equation

The steady states of the Cahn–Hilliard equation are studied as a function of interval length, L , and average mass, m . We count the number of nontrivial monotone increasing steady state solutions and demonstrate that if m lies within the spinodal region then for a.e. there is an even number of such solutions and for a.e. there is an odd number of such solutions. If m lies within the metastable region, then for a.e. L > 0 there is an even number of solutions. Furthermore, we prove that for all values of m , there are no secondary bifurcations.

The nonlinear Cahn-Hilliard equation: Transition from spinodal decomposition to nucleation behavior

The behavior of the nonlinear Cahn-Hilliard equation for asymmetric systems,ct=∇2(±c+Bc2+c3-∇2c) within the unstable subspinodal region is explored. Energy considerations and amplitude equation methods are employed. Evidence is given for a transition from periodically structured“spinodal” behavior to nucleation behavior somewhere within the traditional spinodal. A mechanism for describing a time-dependent lengthening of the dominant wavelength is explored.

PHASE FIELD EQUATIONS WITH MEMORY: THE HYPERBOLIC CASE*

We present a phenomenological theory for phase transition dynamics with memory which yields a hyperbolic generalization of the classical phase field model when the relaxation kernels are assumed to be exponential. Thereafter, we focus on the implications of our theory in the hyperbolic case, and we derive asymptotically an equation of motion in two dimensions for the interface between two different phases. This equation can be considered as a hyperbolic generalization of the classical flow by mean curvature equation, as well as a generalization of the Born--Infeld equation. We use a crystalline algorithm to study the motion of closed curves for the generalized hyperbolic flow by mean curvature equation our hyperbolic generalization of flow by mean curvature and present some numerical results which indicate that a certain type of two-dimensional relaxation damped oscillation may occur.

A traveling wave solution for coupled surface and grain boundary motion

Abstract We study the coupled motion of a grain boundary in a bicrystal which is attached at a “groove root” to an exterior surface which evolves according to surface diffusion in a “quarter loop” geometry (Dunn et al., Trans. Am. Inst. Min. Engrs. 185 (1949) 708), and prove the existence of a unique traveling wave solution for various partially linearized formulations. Our results complement and complete the earlier analysis by Mullins (Acta metall. 6 (1958) 414) where the groove root and the velocity as a function of groove depth were determined. We demonstrate that the net effect of the coupling to the exterior surface is to reduce the overall velocity relative to a freely moving grain boundary by a factor which is small (≈3.5%) for typical parameter values. For extreme values of the parameters, the coupling may cause an increase in the overall grain boundary velocity. No “jerky” or “stop and go” motion is predicted by our solution, and we conjecture why this is so.