Paul C. Fife

Thermodynamically consistent models of phase-field type for the kinetics of phase transitions

Abstract A general framework is given for the phenomenological kinetics of phase transitions in which not only the order parameter but also the temperature may vary in time and space. Instead of a Ginzburg-Landau free energy functional, as used in formulating the Cahn-Hilliard equation, we use the analogous entropy functional. Model entropy functionals, and the kinetic equations resulting from them, are constructed for various cases: phase transitions with and without a critical point, and (in the former case) with or without a latent heat. The class considered is general enough to include the entropy functionals for the Ising model in mean-field approximation, the van der Waals fluid, and a simplified version of the density-functional theory of freezing. A case without critical point, for which the energy is conserved but the order parameter is not, provides a thermodynamically consistent derivation of the phase-field equations studied by Caginalp, Fix and others, and also leads in a natural way to the Lyapunov functional given by Langer for these equations; but the treatment also suggests that a modified version of the phase-field equations might provide a more realistic model of freezing.

Target patterns in a realistic model of the Belousov–Zhabotinskii reaction

Periodic expanding target patterns of chemical activity are observed in thin layers of solution containing bromate, malonic acid and ferroin in dilute sulfuric acid. Commonly these patterns appear as thin blue (oxidized) rings propagating out from a central point into red (reduced) bulk medium. Recently, the opposite pattern has been observed: red waves of reduction propagating through an oxidized bulk medium. We discuss both of these patterns under the assumption that there is a heterogeneity at the center of the pattern—most likely a dust particle or a scratch on the glass—which changes the kinetics locally from a stable excitable steady state to a stable periodic oscillatory state. The temporal oscillation at the origin triggers waves of chemical activity which propagate radially into the excitable medium. Our approach is to combine recent advances in the mathematical description of traveling wave front solutions of reaction–diffusion equations with a realistic model of the kinetics of the reaction med...

The approach of solutions of nonlinear diffusion equations to travelling wave solutions

The paper is concerned with the asymptotic behavior as t → ∞ of solutions u(x, t) of the equation ut—uxx—∞;(u)=O, x∈(—∞, ∞) , in the case ∞(0)=∞(1)=0, ∞′(0)<0, ∞′(1)<0. Commonly, a travelling front solution u=U(x-ct), U(-∞)=0, U(∞)=1, exists. The following types of global stability results for fronts and various combinations of them will be given. 1. Let u(x, 0)=u0(x) satisfy 0≦u0≦1. Let \(a\_ = \mathop {\lim \sup u0}\limits_{x \to - \infty } {\text{(}}x{\text{), }}\mathop {\lim \inf u0}\limits_{x \to \infty } {\text{(}}x{\text{)}}\). Then u approaches a translate of U uniformly in x and exponentially in time, if a− is not too far from 0, and a+ not too far from 1. 2. Suppose \(\int\limits_{\text{0}}^{\text{1}} {f{\text{(}}u{\text{)}}du} > {\text{0}}\). If a− and a+ are not too far from 0, but u0 exceeds a certain threshold level for a sufficiently large x-interval, then u approaches a pair of diverging travelling fronts. 3. Under certain circumstances, u approaches a “stacked” combination of wave fronts, with differing ranges.

Some Nonclassical Trends in Parabolic and Parabolic-like Evolutions

An overview will be given of some nonlinear parabolic-like evolution problems which are off the classical beaten track, but have increased in importance during the past decade. The emphasis is on problems which are nonlocal, pattern-forming (including exhibiting propagative phenomena), and/or lead in some singular limit to free boundary problems. In all cases they have been proposed as models for phenomena in the natural sciences. Also emphasized are the relationships among these various trends.

Dynamics of Layered Interfaces Arising from Phase Boundaries

The dynamics of a material in two phases is studied in the context of phase-field models based on a Landau–Ginzburg free energy functional. They consist of a system of two nonlinear diffusion equat...

The dynamics of nucleation for the Cahn-Hilliard equation

When a constant metastable solution of the Cahn–Hilliard equation is subjected to a spatially localized large-amplitude perturbation, a transition process may be triggered leading to a globally stable stationary solution. In one space dimension, the existence and instability of a third stationary solution with the same mass is proved: A spike-like solution called a canonical nucleus. Within the class of solutions which are even with respect to the center of the spike, it has a one-dimensional unstable manifold. In addition, the process of nucleation by formal arguments using two space scales and two timescales is described. The last stage in the process can be approximated by a nonlinear Stefan free boundary problem.

Pattern formation in reacting and diffusing systems

A mechanism is described, whereby stable sharply differentiated (dissipative) structures can evolve naturally within a mixture of reacting and diffusing substances. Our model has two reacting components, with one diffusion coefficient much smaller than the other. Unlike patterned states obtained by small amplitude analysis near uniform states, our structures have large amplitude and serve to divide the reactor into subregions, each corresponding to a distinct phase for the system. The evolution of the structured stationary state from an arbitrary initial distribution occurs in two stages. The first involves differentiation into subregions, and the second involves the migration of the boundaries of the subregions into a stable final configuration. A singular perturbation analysis and the theory of motion of wavefronts is used to deduce these qualitative properties.

On the relation between the standard phase-field model and a “thermodynamically consistent” phase-field model

Abstract A further comparison is made between the standard phase-field equations αφ t =▽ 2 φ+( 1 ξ 2 )[g(φ)−u] , u t =▽ 2 u+ 1 2 lφ t , and the relevant “thermodynamically consistent model of phase transitions” proposed by the authors [Physica D 43 (1990) 44–62]. Here we concentrate on the usual case where g(φ)=φ−φ3, and for comparison purposes retain this expression for the analogous nonlinear functions in the latter model. It is brought out, among other things, that the standard model is thermodynamically consistent in the sense of being derivable from a free-energy functional. However, this free-energy functional is of a somewhat unusual kind: it implies that, at constant temperature, the energy density varies linearly with the order parameter φ and the entropy density is a non-concave function of φ. The example of a hard sphere system indicates that such behaviour is not impossible, but in most other models the energy density and the entropy are both strictly concave in φ.

A phase plane discussion of convergence to travelling fronts for nonlinear diffusion

AbstractThe paper is concerned with the asymptotic behavior as t → ∞ of solutions u(x,t) of the equation