Thermomechanics of Evolving Phase Boundaries in the Plane

Introduction Part I: Kinematics 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 Part II: Mechanical theory of interfacial evolution 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 Part III: Thermodynamical theory of interfacial evolution in the presence of bulk heat conduction 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 References Index