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Abstract
Phase-field modelling is still a young discipline in condensed-matter physics, which established itself for the class of systems that can be characterised by domains of different phases separated by a distinct interface. Driven out of equilibrium, their dynamics result in the evolution of those interfaces which might develop into well defined-structures with characteristic length scales at the nano-, micro- or meso-scale. Since the material properties of such systems are to a large extent determined by those small-scale structures, acquiring a precise understanding of the mechanisms that drive the interfacial dynamics is a great challenge for scientists in this field. Phase-field modelling is an approach that allows this challenge to be tackled in a simulation-based manner. This review provides a critical overview of the conceptual background of the phase-field method, the most relevant fields of condensed-matter physics that have been approached using phase-field modelling, as well as the respective model formulations and the insight gained so far via their simulation and analysis. Moreover, we discuss directions of further development and the quality of the scientific contributions to be expected from those.
Acknowledgements
I am grateful for all of the discussions at different conferences and visits, which allowed me to see more and more connections between different systems of condensed-matter physics when approaching them in a mechanism-oriented manner and anticipating how to model them in a phase-field approach. In this context I am particularly grateful for the scientific exchange with the colleagues helping to set up the DFG priority program 1296 (http://www.spp1296.rwth-aachen.de), namely K. Binder, D. Herlach, P. Leiderer, H. Löwen, B. Nestler, Th. Palberg and R. Schmid-Fetzer, but also the international colleagues supporting our first initiatives, in particular L. Gránásy, A. Karma, M. Plapp and D. Weitz. Moreover, I would like to thank the members of my own group for letting me experience phase-field modelling as a fully lively and dynamic challenge day by day. A special thanks applies to S. Cottenier, J. Hubert, A. Klein, R. Prieler and R. Siquieri for their careful proof-reading of the manuscript helping to improve its readability at several points, as well as a critical referee for continuing their efforts.
Notes
Notes
1 For three reviews focusing solely on the recent developments of phase-field modelling in material science the reader is referred to Citation3–5.
2 Such as the capillary length (i.e. k≪1).
3 Again the phase-field model in Citation38 provides an example for a concrete phase-field model based on this assumption.
4 Note that to derive the transport equations of a thermodynamic field in the context of a moving boundary system, there is an underlying assumption of instantaneous relaxation of fields to the respective interface position. In that sense the assumption of being close to local thermodynamic equilibrium that was necessary for the calculations of the previous section still holds for those fields. It is the moving interface itself which turns the system into a system which is driven out of equilibrium globally.
5 We refrain from calling Φ itself an order parameter to point out the difference between equilibrium phase transitions and the conditions of interfacial growth, that is, a non-equilibrium conditions in which the introduction of Φ has to be motivated slightly differently than in the context of the former. This is discussed in some more detail in the remainder of this section.
6 An earlier approach in a similar direction can be found in Citation13.
7 Here eventually refers to the limit t→∞.
8 For the sake of simplicity we stick to the case of one transport field in the following. Extension to two or more is straightforward.
9 There are articles on phase-field modelling where φ might take other values such as − 1 on one side and 1 on the other. Only the smooth variation on a scale much smaller the physical length scale of the structuring of the substrate is important.
10 This occurs because there is no intersection of the liquidus and solidus curves.
11 In the context of solidification, latent heat refers to thermal energy which is released at the interface owing to the phase change. Thus, a transformation of the inner energy of the system naturally involves a transformation of the latent heat as well.
12 Note that within this context the dimensionless latent heat is normalised to 1.
13 This is done by proceeding as in (Equation144) and (Equation145).
14 The above formalism can usually be extended to cover systems described by a reduced phase diagram as in .
15 This is despite it being proven to be beyond all orders of perturbation theory.
16 A schematic phase diagram giving rise to eutectic solidification in binary systems is depicted in .
17 For details see http://www.computherm.com.
18 For details see http://www.thermocalc.com/Products/Dictra.html.
19 The irreversible part is provided by the standard viscous stress term of a Newtonian liquid.
20 Normal with respect to the interface.
21 A thorough review discussing velocity selection i the case of diffusion limited dendritic growth is given in Citation118.
22 Note that in this context the width over which the phase-field variable varies essentially gains a physical interpretation.
23 These equations were obtained following the standard variational procedure of phase-field theory (see, e.g. Citation6,Citation225,Citation226).
24 Appropriate functionals involving the distribution of mean and Gaussian curvature have been proposed by Canham Citation287 and Helfrich Citation288 and reviewed by Seifert Citation289.