## The Diffuse Interface Approach in Materials Science: Thermodynamic Concepts and Applications of Phase-Field Models, Volume 73Many inhomogeneous systems involve domains of well-de?ned phases se- rated by a distinct interface. If they are driven out of equilibrium one phase will grow at the cost of the other. Examples are phase separation by sp- odal decomposition or nucleation and subsequent growth of the nucleus in the nourishing phase [139]. Another example which has often been discussed as a paradigmatic problem is that of dendritic solidi?cation [29, 64, 79, 199]. The phenomenological description of these phenomena involves the de?- tion of a precisely located interfacial surface on which boundary conditions are imposed. One of those boundary conditions typically yields a normal - locity at which the interface is moving. This is the so-calledsharp interface approach, adopted both in analytical and numerical studies for a variety of contexts involving a moving boundary. The origin of such a description is - ten transparent, being obtained by symmetry arguments and common sense. |

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### Contents

I | 1 |

II | 4 |

III | 7 |

IV | 19 |

V | 21 |

VI | 23 |

VII | 26 |

VIII | 28 |

XXVII | 81 |

XXVIII | 92 |

XXIX | 93 |

XXX | 97 |

XXXI | 99 |

XXXII | 104 |

XXXIII | 119 |

XXXIV | 127 |

### Common terms and phrases

anisotropy applied assumed asymptotic analysis boundary conditions bulk Cahn Cahn-Hilliard equation Chap chemical potential coefficient computational conservation context curvature curve cusp defined denotes density change flow derivation diffuse interface approach diffuse interface modeling diffusion limited dendritic dimensionless discussion dynamics element employed energy functional epitaxial growth Euler-Lagrange equation evolution free energy density G.B. McFadden Gibbs free energy given gradient grid growth problem hydrodynamic inner expansion interfacial growth interfacial region kinetic leading-order limited dendritic growth liquid phase matching method Moreover morphology non-conserved numerical obtained order parameter outer expansion phase diagram phase-field equations Phys physical quadtree relation respect Sect sharp interface equations sharp interface formulation sharp interface limit sharp interface model simulation solid phase solidification solidus surface tension temperature term theory thermodynamic consistency thermodynamic equilibrium thermodynamic potential thin interface limit transport equations underlying variational vector velocity y-direction of space yields

### References to this book

Crystal Growth - From Fundamentals to Technology Georg Müller,Jean-Jacques Métois,Peter Rudolph Limited preview - 2004 |