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## Über dieses Buch

This IMA Volume in Mathematics and its Applications ON THE EVOLUTION OF PHASE BOUNDARIES is based on the proceedings of a workshop which was an integral part of the 1990- 91 IMA program on "Phase Transitions and Free Boundaries". The purpose of the workshop was to bring together mathematicians and other scientists working on the Stefan problem and related theories for modeling physical phenomena that occurs in two phase systems. We thank M.E. Gurtin and G. McFadden for editing the proceedings. We also take this opportunity to thank the National Science Foundation, whose financial support made the workshop possible. A vner Friedman Willard Miller, Jr. PREFACE A primary goal of the IMA workshop on the Evolution of Phase Boundaries from September 17-21, 1990 was to emphasize the interdisciplinary nature of contempo­ rary research in this field, research which combines ideas from nonlinear partial dif­ ferential equations, asymptotic analysis, numerical computation, and experimental science. The workshop brought together researchers from several disciplines, includ­ ing mathematics, physics, and both experimental and theoretical materials science.

## Inhaltsverzeichnis

### Phase Field Equations in the Singular Limit of Sharp Interface Problems

Abstract
In one of the singular limits as interface thickness approaches zero, solutions to the phase field equations formally approach those of a sharp interface model which incorporates surface tension. Here, we use a modification of the original phase field equations and prove this convergence rigorously in the one-dimensional and radially symmetric cases. Convergence to motion by mean curvature in another distinguished limit is also proved.
Gunduz Caginalp, Xinfu Chen

### A Phase Fluid Model: Derivation and New Interface Relation

Abstract
We develop a very general model of phase boundaries in which the fluid properties play a role. The variables pressure, fluid velocity and specific volume are considered in conjunction with temperature and order parameter using the phase field approach. Upon making some choices and approximations one obtains a system of parabolic differential equations. An asymptotic analysis leads to a new interface relation (generalizing Gibbs-Thomson) which indicates that the front velocity in the kinetic undercooling term should be replaced by the front velocity minus the normal fluid velocity. The temperature term involves a quadratic term due to thermal expansion and isothermal compressibility terms.
G. Caginalp, J. Jones

### Geometric Evolution of Phase-Boundaries

In the memory of Professor Kôsaku Yosida
Abstract
This paper continues our study [CGG], [GG] of a motion of phase-boundaries whose speed locally depends on the normal vector field and curvature tensors.
Yoshikazu Giga, Shun’ichi Goto

### The Approach to Equilibrium: Scaling, Universality and the Renormalisation Group

Abstract
Evidence is accumulating that the long-time behaviour of certain non-equilibrium systems shows scaling behaviour. This assertion is demonstrated in the cases of spinodal decomposition, block copolymer phase separation, crystal growth, and a particular diffusion process which can occur in porous media. These, and hopefully other non-equilibrium problems may be studied by computationally efficient numerical methods, which are based not upon discretising partial differential equations, but upon a coarse-grained description of the dynamics.
In the case of non-linear diffusion in a porous medium, it is shown that the renormalisation group can be used to study the long-time behaviour, and to calculate perturbatively the exponents characterising the anomalous diffusion.
Nigel Goldenfeld

### Evolving Phase Boundaries in the Presence of Deformation and Surface Stress

Abstract
Recent papers of Gurtin [1986, 1988a, 1988b], Angenent and Gurtin [1989], and Gurtin and Struthers [1990] form an investigation whose goal is a nonequilibrium thermomechanics of two-phase continua based on Gibb’s notion of a sharp phase-interface endowed with energy, entropy and superficial force. In all of these studies except the last the crystal is rigid, an assumption that forms the basis of a large class of problems discussed by material scientists, but there are situations in which deformation is the paramount concern, examples being shock-induced transformations and mechanical twinning. In this note I discuss the results of Gurtin and Struthers [1990]1, who consider deformable crystal-crystal systems with coherent interface.
Morton E. Gurtin

### Effect of Modulated Taylor-Couette Flows on Crystal-Melt Interfaces: Theory and Initial Experiments

Abstract
An important problem in the process of crystal growth from the melt phase is to understand the interaction of the crystal-melt interface with fluid flow in the melt. This area combines the complexities of the Navier-Stokes equations for fluid flow with the nonlinear behavior of the free boundary representing the crystal-melt interface. Some progress has been made by studying explicit flows that allow a base state corresponding to a one-dimensional crystal-melt interface with solute and/or temperature fields that depend only on the distance from the interface. This allows the strength of the interaction between the flow and the interface to be assessed by a linear stability analysis of the simple base state. The case of a time-periodic Taylor-Couette flow interacting with a cylindrical crystalline interface is currently being investigated both experimentally and theoretically; some preliminary results are given here. The results indicate that the effect of the crystal-melt interface in the two-phase system is to destabilize the system by an order of magnitude relative to the single-phase system with rigid walls.
G. B. McFadden, B. T. Murray, S. R. Coriell, M. E. Glicksman, M. E. Selleck

### A One Dimensional Stochastic Model of Coarsening

Abstract
A one dimensional model of the coarsening of intervals on a line is considered in which the boundary points between adjacent intervals execute independent random walk with a common diffusion coefficient D/2; when two boundary points meet, they coalesce into a single point that continues to execute random walk. We calculate the following quantities in the asymptotic limit of long times: 1) The average interval length (i.e., $$\;\left\langle l \right\rangle = \sqrt {\pi Dt}$$ , 2) the time-independent probability density for the reduced length $$\sigma = 1/\left\langle l \right\rangle$$ , and 3) the expected value of dl/dt for a given l, which is positive for and negative for $$l > {l_c} = \sqrt {2/\pi } \left\langle l \right\rangle$$ and negative for l < l c . The model is similar to one proposed by Louat for grain growth. Although it is not a good representation of the details of most physical processes of coarsening, it is of theoretical interest since it is one of the few cases for which analytic results can be obtained.
W. W. Mullins

### Algorithms for Computing Crystal Growth and Dendritic Solidification

Abstract
We report on a numerical method for computing the motion of complex solid/liquid boundaries in crystal growth. The model we solve includes physical effects such as crystalline anisotropy, surface tension, molecular kinetics and undercooling. The method is based on a single history-dependent boundary integral equation on the solid/liquid boundary, which is solved by means of a fast algorithm coupled to a level set approach for tracking the evolving boundary. Numerical experiments show the evolution of complex crystalline shapes, development of large spikes and corners, dendrite formation and side-branching, and pieces of solid merging and breaking off freely.
James A. Sethian, John Strain

### Towards a Phase Field Model for Phase Transitions in Binary Alloys

Abstract
To date phase field models have only been used to model non-isothermal phase transitions in a pure material. Here we describe recent steps which aim to extend phase field models to deal with binary alloys; a situation of metallurgical and industrial importance. To this end we present a new phase field model for isothermal phase transitions in a binary alloy and discuss the results of an asymptotic analysis. Finally we suggest ways in which these models may be further developed to achieve our aim of a non-isothermal phase field model of a binary alloy.
A. A. Wheeler, W. J. Boettinger
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