An introduction to phase-field modeling of microstructure evolution
01 Jun 2008-Calphad-computer Coupling of Phase Diagrams and Thermochemistry (Elsevier)-Vol. 32, Iss: 2, pp 268-294
TL;DR: In this article, the authors introduce the concept of diffuse interfaces, the phase-field variables, the thermodynamic driving force for microstructure evolution and the kinetic phasefield equations are discussed.
Abstract: The phase-field method has become an important and extremely versatile technique for simulating microstructure evolution at the mesoscale. Thanks to the diffuse-interface approach, it allows us to study the evolution of arbitrary complex grain morphologies without any presumption on their shape or mutual distribution. It is also straightforward to account for different thermodynamic driving forces for microstructure evolution, such as bulk and interfacial energy, elastic energy and electric or magnetic energy, and the effect of different transport processes, such as mass diffusion, heat conduction and convection. The purpose of the paper is to give an introduction to the phase-field modeling technique. The concept of diffuse interfaces, the phase-field variables, the thermodynamic driving force for microstructure evolution and the kinetic phase-field equations are introduced. Furthermore, common techniques for parameter determination and numerical solution of the equations are discussed. To show the variety in phase-field models, different model formulations are exploited, depending on which is most common or most illustrative.
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Dissertation•
01 Oct 1948
TL;DR: In this article, it was shown that a metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.
Abstract: IN two previous notes1, Prof. Max Born and I have shown that one can obtain a theory of superconductivity by taking account of the fact that the interaction of the electrons with the ionic lattice is appreciable only near the boundaries of Brillouin zones, and particularly strong near the corners of these. This leads to the criterion that the metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.
2,042 citations
Book•
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01 Jan 1996
TL;DR: A review of the collected works of John Tate can be found in this paper, where the authors present two volumes of the Abel Prize for number theory, Parts I, II, edited by Barry Mazur and Jean-Pierre Serre.
Abstract: This is a review of Collected Works of John Tate. Parts I, II, edited by Barry Mazur and Jean-Pierre Serre. American Mathematical Society, Providence, Rhode Island, 2016. For several decades it has been clear to the friends and colleagues of John Tate that a “Collected Works” was merited. The award of the Abel Prize to Tate in 2010 added impetus, and finally, in Tate’s ninety-second year we have these two magnificent volumes, edited by Barry Mazur and Jean-Pierre Serre. Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate. For an overview of Tate’s work, the editors refer the reader to [4]. Before discussing the volumes, I describe some of Tate’s work. 1. Hecke L-series and Tate’s thesis Like many budding number theorists, Tate’s favorite theorem when young was Gauss’s law of quadratic reciprocity. When he arrived at Princeton as a graduate student in 1946, he was fortunate to find there the person, Emil Artin, who had discovered the most general reciprocity law, so solving Hilbert’s ninth problem. By 1920, the German school of algebraic number theorists (Hilbert, Weber, . . .) together with its brilliant student Takagi had succeeded in classifying the abelian extensions of a number field K: to each group I of ideal classes in K, there is attached an extension L of K (the class field of I); the group I determines the arithmetic of the extension L/K, and the Galois group of L/K is isomorphic to I. Artin’s contribution was to prove (in 1927) that there is a natural isomorphism from I to the Galois group of L/K. When the base field contains an appropriate root of 1, Artin’s isomorphism gives a reciprocity law, and all possible reciprocity laws arise this way. In the 1930s, Chevalley reworked abelian class field theory. In particular, he replaced “ideals” with his “idèles” which greatly clarified the relation between the local and global aspects of the theory. For his thesis, Artin suggested that Tate do the same for Hecke L-series. When Hecke proved that the abelian L-functions of number fields (generalizations of Dirichlet’s L-functions) have an analytic continuation throughout the plane with a functional equation of the expected type, he saw that his methods applied even to a new kind of L-function, now named after him. Once Tate had developed his harmonic analysis of local fields and of the idèle group, he was able prove analytic continuation and functional equations for all the relevant L-series without Hecke’s complicated theta-formulas. Received by the editors September 5, 2016. 2010 Mathematics Subject Classification. Primary 01A75, 11-06, 14-06. c ©2017 American Mathematical Society
2,014 citations
TL;DR: In this article, the authors reviewed the application of the phase-field method in different fields of materials science, including elastic interactions and fluid flow in multi-grain multi-phase structures in multicomponent materials.
Abstract: The phase-field method is reviewed against its historical and theoretical background. Starting from Van der Waals considerations on the structure of interfaces in materials the concept of the phase-field method is developed along historical lines. Basic relations are summarized in a comprehensive way. Special emphasis is given to the multi-phase-field method with extension to elastic interactions and fluid flow which allows one to treat multi-grain multi-phase structures in multicomponent materials. Examples are collected demonstrating the applicability of the different variants of the phase-field method in different fields of materials science.
1,004 citations
01 Nov 1992
TL;DR: In this article, a class of phase-field models for crystallization of a pure substance from its melt are presented, which are based on an entropy functional, and are therefore thermodynamically consistent inasmuch as they guarantee spatially local positive entropy production.
Abstract: In an effort to unify the various phase-field models that have been used to study solidification, we have developed a class of phase-field models for crystallization of a pure substance from its melt. These models are based on an entropy functional, as in the treatment of Penrose and Fife, and are therefore thermodynamically consistent inasmuch as they guarantee spatially local positive entropy production. General conditions are developed to ensure that the phase field takes on constant values in the bulk phases. Specific forms of a phase-field function are chosen to produce two models that bear strong resemblances to the models proposed by Langer and Kobayashi. Our models contain additional nonlinear functions of the phase field that are necessary to guarantee thermodynamic consistency.
459 citations
28 Jan 2015
TL;DR: In this article, the state of the art in additive manufacturing and material modelling is presented, focusing on those technologies that have the potential to produce and repair metal parts for the aerospace industry.
Abstract: This paper reviews recent improvements in additive manufacturing technologies, focusing on those which have the potential to produce and repair metal parts for the aerospace industry. Electron beam melting, selective laser melting and other metal deposition processes, such as wire and arc additive manufacturing, are presently regarded as the best candidates to achieve this challenge. For this purpose, it is crucial that these technologies are well characterised and modelled to predict the resultant microstructure and mechanical properties of the part. This paper presents the state of the art in additive manufacturing and material modelling. While these processes present many advantages to the aerospace industry in comparison with traditional manufacturing processes, airworthiness and air transport safety must be guaranteed. The impact of this regulatory framework on the implementation of additive manufacturing for repair and production of parts for the aerospace industry is presented.
337 citations
References
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TL;DR: In this article, it was shown that the thickness of the interface increases with increasing temperature and becomes infinite at the critical temperature Tc, and that at a temperature T just below Tc the interfacial free energy σ is proportional to (T c −T) 3 2.
Abstract: It is shown that the free energy of a volume V of an isotropic system of nonuniform composition or density is given by : NV∫V [f 0(c)+κ(▿c)2]dV, where NV is the number of molecules per unit volume, ▿c the composition or density gradient, f 0 the free energy per molecule of a homogeneous system, and κ a parameter which, in general, may be dependent on c and temperature, but for a regular solution is a constant which can be evaluated. This expression is used to determine the properties of a flat interface between two coexisting phases. In particular, we find that the thickness of the interface increases with increasing temperature and becomes infinite at the critical temperature Tc , and that at a temperature T just below Tc the interfacial free energy σ is proportional to (T c −T) 3 2 . The predicted interfacial free energy and its temperature dependence are found to be in agreement with existing experimental data. The possibility of using optical measurements of the interface thickness to provide an additional check of our treatment is briefly discussed.
8,720 citations
TL;DR: The renormalization group theory has been applied to a variety of dynamic critical phenomena, such as the phase separation of a symmetric binary fluid as mentioned in this paper, and it has been shown that it can explain available experimental data at the critical point of pure fluids, and binary mixtures, and at many magnetic phase transitions.
Abstract: An introductory review of the central ideas in the modern theory of dynamic critical phenomena is followed by a more detailed account of recent developments in the field. The concepts of the conventional theory, mode-coupling, scaling, universality, and the renormalization group are introduced and are illustrated in the context of a simple example---the phase separation of a symmetric binary fluid. The renormalization group is then developed in some detail, and applied to a variety of systems. The main dynamic universality classes are identified and characterized. It is found that the mode-coupling and renormalization group theories successfully explain available experimental data at the critical point of pure fluids, and binary mixtures, and at many magnetic phase transitions, but that a number of discrepancies exist with data at the superfluid transition of $^{4}\mathrm{He}$.
4,980 citations
"An introduction to phase-field mode..." refers background or methods in this paper
...with ζk(r , t) non-conserved and ψB(r , t) conserved Gaussian noise fields that satisfy the fluctuation–dissipation theorem [18,19]....
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...The stochastic theory of critical dynamics of phase transformations from Hohenberg and Halperin [18] and Gunton et al....
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...The idea was introduced by Langer [37] based on one of the stochastic models of Hohenberg and Halperin [18]....
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...The stochastic theory of critical dynamics of phase transformations from Hohenberg and Halperin [18] and Gunton et al. [19] also results in equations that are very similar to the current phase-field equations....
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01 Jan 1979
TL;DR: In this paper, a microscopic diffusional theory for the motion of a curved antiphase boundary is presented, where the interfacial velocity is linearly proportional to the mean curvature of the boundary, but unlike earlier theories the constant of proportionality does not include the specific surface free energy.
Abstract: Abstract A microscopic diffusional theory for the motion of a curved antiphase boundary is presented. The interfacial velocity is found to be linearly proportional to the mean curvature of the boundary, but unlike earlier theories the constant of proportionality does not include the specific surface free energy, yet the diffusional dissipation of free energy is shown to be equal to the reduction in total boundary free energy. The theory is incorporated into a model for antiphase domain coarsening. Experimental measurements of domain coarsening kinetics in Fe-Al alloys were made over a temperature range where the specific surface free energy was varied by more than two orders of magnitude. The results are consistent with the theory; in particular, the domain coarsening kinetics do not have the temperature dependence of the specific surface free energy.
2,414 citations
"An introduction to phase-field mode..." refers background or methods in this paper
...Approximately 50 years ago, Ginzburg and Landau [16] formulated a model for superconductivity using a complex valued order parameter and its gradients, and Cahn and Hilliard [17] proposed a thermodynamic formulation that accounts for the gradients in thermodynamic properties in heterogeneous systems with diffuse interfaces....
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...It was calculated by Allen and Cahn [74], assuming that l/ρ 1 with 1/ρ the local curvature of the interface and l the interfacial thickness, that the interfacial velocity equals v = Lκ ( 1 ρ ) (64) for curvature driven coarsening of the anti-phase domain structure in Fig....
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...It was calculated by Allen and Cahn [74], assuming that l/ρ 1 with 1/ρ the local curvature of the interface and l the interfacial thickness, that the interfacial velocity equals...
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...3 represented by a single order parameter field, the interfacial energy and width can be calculated analytically [17,74]....
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...The Cahn–Hilliard equation is essentially a diffusion equation of the form 1 Vm ∂xB(−→r , t) ∂t = − −→ ∇ · −→ J B, (67) where the diffusion flux −→ JB is given by −→ J B = −M −→ ∇ δF δxB = −M −→ ∇ [ ∂ f0(xB, ηk) ∂xB − −→ ∇ · −→ ∇ xB(−→r , t) ] ....
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