Phase-Field Models
Phase-field models have become popular in recent years to describe a host of free-boundary problems in various areas of research. The key point of the phase-field approach is that surfaces and interfaces are implicitly described by continuous scalar field
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ave rapidly gained popularity over the last two decades in various fields. Some examples for their applications are discussed in recent reviews on the formation of microstructures during solidification by Boettinger et al. (2002) and Plapp (2007), on solid-state transformations R. Mauri (ed.), Multiphase Microfluidics: The Diffuse Interface Model © CISM, Udine 2012
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M. Plapp
by Chen (2002),Steinbach (2009), and Wang and Li (2010), and on multiphase flows by Anderson et al. (1998), but it is almost impossible to give an exhaustive list of the topics treated with the help of phase-field methods since the development of phase-field models for ever new applications is a rapidly advancing field. All the topics mentioned above have in common that they involve the motion of interfaces or boundaries in response to a coupling of the boundary with one or several transport fields (such as diffusion, flow, stress or temperature fields). This interaction generates morphological instabilities and leads to the spontaneous emergence of complex structures. All these different models have in common that they describe the geometry of the boundaries through one or several phase fields. This name was originally coined in solidification, where the phase field indicates in which thermodynamic phase (solid or liquid) each point of space is located. These fields have fixed pretedermined values in each of the domains occupied by a bulk phase, and vary continuously from one bulk value to the other through an interface that has a well-defined width. In other words, in all phase-field models the interfaces are diffuse. In fact, in the literature, the terms “diffuse-interface model” and “phasefield model” are often used as synonyms. In contrast, in the present contribution I will make a distinction between these two classes of models that is based on the two different and complementary viewpoints that can be taken to derive them. In the first perspective, which could be called “bottom-up”, one starts from a microscopic picture of a physical system and performs a coarse-graining. While this operation can rarely be carried out explicitly, conceptually it is well defined. As an example, consider a liquid-vapor interface. The relevant quantity that characterizes the difference between the two phases is the number density of molecules, which is high in the liquid but low in the vapor. However, to define a smooth density, the local number of atoms has to be averaged over a volume that is large enough to contain a significant number of molecules, but small enough to remain “local”, that is, smaller than any geometric scale of the two-phase pattern to be described. Then, the total free energy of the system may be expressed as a functional that is obtained by averaging all quantities on the coarse-graining scale, and all the coefficients of the functional can in principle be calculated from the elementary intermolecular interactions. The first example for such a model was the van der Waals theory of the liquid-vapor interface, see Rowlinson (1979), but since then many other sim
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