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…mehr
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.
* 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
* 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
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