Thermomechanics of Evolving Phase Boundaries in the Plane

This is one of the few books on the subject of mathematical materials science. It discusses the dynamics of two-phase systems within the framework of modern continuum thermodynamics, stressing fundamentals. Two general theories are discussed: a mechanical theory that leads to a generalization of the classical curve-shortening equation and a theory of heat conduction that broadly generalizes the classical Stefan theory. This original survey includes simple solutions that demonstrate the instabilities inherent in two-phase problems. The free-boundary problems that form the basis of the subject should be of great interest to mathematicians and physical scientists.

This book is one of the very first on the subject of mathematical materials science and presents a view different from that prevalent in the physical literature. Issues foundational in nature are stressed with the emphasis on the interplay between mathematics and physics. It discusses the dynamics of two-phase systems within the framework of modern continuum thermodynamics. Two general theories are considered and the resulting equations exhibit unstable growth patterns. The free boundary problems that form the basis of the subject should be of great interest to mathematicians and physical scientists.

1 - Curves
1.1 - Preliminary definitions
1.2 - Convex curves
1.3 - Integrals
1.4 - Piecewise-smooth curves
1.5 - Infinitesimally wrinkled curves
2 - Evolving curves
2.1 - Definitions
2.2 - Transport identities
2.3 - Integral identities
2.4 - Steadily evolving interfaces
2.5 - Piecewise-smooth evolving curves
2.6 - Variational lemmas
3 - Phase regions, control volumes, and inflows
3.1 - Phase regions and control volumes
3.2 - Inflows, the pillbox lemma, and infinitesimally thin evolving control volumes
4 - Balance of forces
4.1 - Balances of forces
4.2 - The power identity
5 - Energetics and the dissipation inequality
6 - Constitutive theory
6.1 - Constitutive equations and the compatibility theorem
6.2 - Balance of capillary forces revisited; corners
7 - Digression: Statistical theory of interfacial stability; convexity, the Frank diagram, and corners; Wulff regions
7.1 - Preliminaries; Polar diagrams
7.2 - Convexity; the extended and convexified energies, and the Frank diagram
7.3 - Stability
7.4 - Instability of the total energy
7.5 - Equilibria of the total energy; Wulff regions
7.6 - Wulff's theorem
8 - Evolution equations for the interface: basic assumptions
8.1 - Isotropic interface
8.2 - Anisotropic interface
8.2.1 - Basic equations
8.2.2. - Equations when the interface is the graph of a function
8.2.3 - Equations when the interface is a level set
8.3 - Plan of the next few chapters
9 - Stationary interfaces and steadily evolving interfaces
9.1 - Stationary interfaces
9.2 - Steadily evolving facets
9.3 - Steadily evolving interfaces that are not flat
10 - Global behaviour for an interface with stable energy
10.1 - Existence of evolving interfaces from a prescribed initial curve
10.2 - Growth and decay of the interface
10.3 - Evolution of curvature; fingers
11 - Unstable interfacial energies and interfaces with corners
11.1 - Admissibility; corner conditions
11.2 - The initial-value problem
11.3 - Facets and wrinklings that connect evolving curves
11.4 - Equations near a corner when the curve is a graph
11.5 - Interfaces with arbitrary angle-set; infinitesimal wrinklings
11.6 - Stationary interfaces and steadily evolving interfaces with corners
12 - Non smooth interfacial energies: crystalline energies
12.1 - Crystalline energies
12.2 - The Wulff region
12.3 - The capillary force at preferred orientations
12.4 - Corners between preferred facets
12.5 - Crystalline motions
12.6 - Interfaces of arbitrary orientation, infinitesimal wrinklings, and generalized motions
12.7 - Evolution of a rectangular crystal
13 - Regularized theory for smooth unstable energies; dependence of interfacial energy on curvature
13.1 - Balance of forces and moments; power
13.2 - Energetics and the dissipation inequality
13.3 - Constitutive equations
13.4 - Evolution equations for the interface
13.5 - Linearized equations; spinodal decomposition on the interface
14 - Review of single-phase thermodynamics
14.1 - Basic equations and the first two laws
14.2 - Constitutive equations and thermodynamic restrictions
14.3 - The heat equation
15 - Thermodynamics of two-phase systems
15.1 - Basic quantities and the first two laws
15.2 - Local forms of the interfacial laws
16 - Constitutive theory
16.1 - Constituive equations for the bulk material
16.2 - The transition temperature
16.3 - Constitutive equations for the interface
17 - Free-boundary problems
17.1 - Bulk equations and interface conditions
17.2 - Initial conditions and boundary conditions
17.3 - Free-boundary problems near the transition temperature for weak surfaces
17.3.1 - Approximate interface conditions
17.3.2 - Approximate free-boundary problems
17.3.3 - The first two laws for the approximate theories
17.3.4 - Growth theorems
17.3.5 - Perfect conductors
18 - Instabilities induced by supercooling the liquid phase
18.1 - The one-dimensional problem: growth of the solid phase
18.2 - Instability of a flat interface