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{{Short description|Mathematical model}}
A '''phase-field model''' is a mathematical model for solving interfacial problems. It has mainly been applied to solidification dynamics,<ref>{{Cite journal|doi=10.1146/annurev.matsci.32.101901.155803|title=Phase-Field Simulation of Solidification|journal=[[Annual Review of Materials Research]]|volume=32|pages=163–194|year=2002|last1=Boettinger|first1=W. J.|last2=Warren|first2=J. A.|author3-link=Christoph Beckermann|last3=Beckermann|first3=C.|last4=Karma|first4=A.}}</ref> but it has also been applied to other situations such as [[viscous fingering]],<ref name="viscous">{{Cite journal|doi=10.1103/PhysRevE.60.1734|pmid=11969955|title=Phase-field model for Hele-Shaw flows with arbitrary viscosity contrast. II. Numerical study|journal=Physical Review E |volume=60|issue=2|pages=1734–40|year=1999|last1=Folch|first1=R.| last2=Casademunt|first2=J.|last3=Hernández-Machado|first3=A.|last4=Ramírez-Piscina| first4=L.|bibcode=1999PhRvE..60.1734F|arxiv=cond-mat/9903173|s2cid=8488585}}</ref> [[fracture]] mechanics,<ref name=":0">{{Cite journal|last1=Bourdin|first1=B.|last2=Francfort|first2=G.A.|last3=Marigo|first3=J-J.|date=April 2000|title=Numerical experiments in revisited brittle fracture|journal=Journal of the Mechanics and Physics of Solids|volume=48|issue=4|pages=797–826|doi=10.1016/S0022-5096(99)00028-9|bibcode=2000JMPSo..48..797B }}</ref><ref name=":1">{{Cite journal|last=Bourdin|first=Blaise|date=2007|title=Numerical implementation of the variational formulation for quasi-static brittle fracture|journal=Interfaces and Free Boundaries|volume=9 |issue=3 |pages=411–430|doi=10.4171/IFB/171|issn=1463-9963|doi-access=free}}</ref><ref name=":2">{{Cite journal|last1=Bourdin|first1=Blaise|last2=Francfort|first2=Gilles A.|last3=Marigo|first3=Jean-Jacques|date=April 2008|title=The Variational Approach to Fracture|journal=Journal of Elasticity|volume=91|issue=1–3|pages=5–148|doi=10.1007/s10659-007-9107-3|s2cid=120498253|issn=0374-3535}}</ref><ref>{{Cite journal|doi=10.1103/PhysRevLett.87.045501|pmid=11461627|title=Phase-Field Model of Mode III Dynamic Fracture|journal=Physical Review Letters|volume=87|issue=4|pages=045501|year=2001|last1=Karma|first1=Alain|last2=Kessler|first2=David|last3=Levine|first3=Herbert|bibcode=2001PhRvL..87d5501K|arxiv=cond-mat/0105034|s2cid=42931658}}</ref> [[hydrogen embrittlement]],<ref>{{Cite journal|doi=10.1016/j.cma.2018.07.021|title=A phase field formulation for hydrogen assisted cracking|journal=Computer Methods in Applied Mechanics and Engineering|volume=342|pages=742–761|year=2018|last1=Martinez-Paneda|first1=Emilio|last2=Golahmar|first2=Alireza|last3=Niordson|first3=Christian|arxiv=1808.03264|bibcode=2018CMAME.342..742M |s2cid=52360579}}</ref> and [[Vesicle (biology and chemistry)|vesicle]] dynamics.<ref>{{Cite journal|doi=10.1103/PhysRevE.72.041921|pmid=16383434|title=Phase-field approach to three-dimensional vesicle dynamics|journal=Physical Review E|volume=72|issue=4|pages=041921|year=2005|last1=Biben|first1=Thierry|last2=Kassner|first2=Klaus|last3=Misbah|first3=Chaouqi|bibcode=2005PhRvE..72d1921B}}</ref><ref name="Ashour Valizadeh Rabczuk 2021 p=113669">{{cite journal | last1=Ashour | first1=Mohammed | last2=Valizadeh | first2=Navid | last3=Rabczuk | first3=Timon | title=Isogeometric analysis for a phase-field constrained optimization problem of morphological evolution of vesicles in electrical fields | journal=Computer Methods in Applied Mechanics and Engineering | publisher=Elsevier BV | volume=377 | year=2021 | issn=0045-7825 | doi=10.1016/j.cma.2021.113669 | page=113669| bibcode=2021CMAME.377k3669A | s2cid=233580102 }}</ref><ref name="Valizadeh Rabczuk 2022 p=114191">{{cite journal | last1=Valizadeh | first1=Navid | last2=Rabczuk | first2=Timon | title=Isogeometric analysis of hydrodynamics of vesicles using a monolithic phase-field approach | journal=Computer Methods in Applied Mechanics and Engineering | publisher=Elsevier BV | volume=388 | year=2022 | issn=0045-7825 | doi=10.1016/j.cma.2021.114191 | page=114191| bibcode=2022CMAME.388k4191V | s2cid=240657318 | doi-access= }}</ref><ref name="Valizadeh Rabczuk 2019 pp. 599–642">{{cite journal | last1=Valizadeh | first1=Navid | last2=Rabczuk | first2=Timon | title=Isogeometric analysis for phase-field models of geometric PDEs and high-order PDEs on stationary and evolving surfaces | journal=Computer Methods in Applied Mechanics and Engineering | publisher=Elsevier BV | volume=351 | year=2019 | issn=0045-7825 | doi=10.1016/j.cma.2019.03.043 | pages=599–642| bibcode=2019CMAME.351..599V | s2cid=145903238 }}</ref>
The method substitutes boundary conditions at the interface by a partial differential equation for the evolution of an auxiliary field (the phase field) that takes the role of an [[order parameter]]. This phase field takes two distinct values (for instance +1 and −1) in each of the phases, with a smooth change between both values in the zone around the interface, which is then diffuse with a finite width. A discrete location of the interface may be defined as the collection of all points where the phase field takes a certain value (e.g., 0).
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A phase-field model is usually constructed in such a way that in the limit of an infinitesimal interface width (the so-called sharp interface limit) the correct interfacial dynamics are recovered. This approach permits to solve the problem by integrating a set of partial differential equations for the whole system, thus avoiding the explicit treatment of the boundary conditions at the interface.
Phase-field models were first introduced by Fix<ref>G.J. Fix, in Free Boundary Problems: Theory and Applications, Ed. A. Fasano and M. Primicerio, p. 580, Pitman (Boston, 1983).</ref> and Langer,<ref>{{cite book| doi=10.1142/9789814415309_0005|chapter=Models of Pattern Formation in First-Order Phase Transitions|title=
publisher=World Scientific|location= Singapore
}}</ref> and have experienced a growing interest in solidification and other areas.
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: <math>F[e,\varphi]=\int d{\mathbf r} \left[ K|{\mathbf\nabla}\varphi|^2 + h_0f(\varphi) + e_0u(\varphi)^2 \right]</math>
where <math>\varphi</math> is the phase field, <math>u(\varphi)=e/e_0 + h(\varphi)/2</math>, <math>e</math> is the local [[enthalpy]] per unit volume, <math>h</math> is a certain polynomial function of <math>\varphi</math>, and <math>e_0={L^2}/{T_M c_p}</math> (where <math>L</math> is the [[latent heat]], <math>T_M</math> is the melting temperature, and <math>c_{p}</math> is the specific heat). The term with <math>\nabla\varphi</math> corresponds to the interfacial energy. The function <math>f(\varphi)</math> is usually taken as a double-well potential describing the free energy density of the bulk of each phase, which themselves correspond to the two minima of the function <math>f(\varphi)</math>. The constants <math>K</math> and <math>h_{0}</math> have respectively dimensions of energy per unit length and energy per unit volume. The interface width is then given by <math>W=\sqrt{K/h_0}</math>.
The phase-field model can then be obtained from the following variational relations:<ref>{{Cite journal|doi=10.1103/RevModPhys.49.435|title=Theory of dynamic critical phenomena|journal=Reviews of Modern Physics|volume=49|issue=3|pages=435|year=1977|last1=Hohenberg|first1=P.|last2=Halperin|first2=B.|bibcode=1977RvMP...49..435H|s2cid=122636335 }}</ref>
: <math>
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===Alternative energy-density functions===
The choice of free energy function, <math>f(\varphi)</math>, can have a significant effect on the physical behaviour of the interface, and should be selected with care. The double-well function represents an approximation of the [[Van der Waals equation|Van der Waals equation of state]] near the critical point, and has historically been used for its simplicity of implementation when the phase-field model is employed solely for interface tracking purposes. But this has led to the frequently observed spontaneous drop shrinkage phenomenon, whereby the high phase miscibility predicted by an Equation of State near the critical point allows significant interpenetration of the phases and can eventually lead to the complete disappearance of a droplet whose radius is below some critical value.<ref>{{Cite journal|doi=10.1016/j.jcp.2006.11.020|title=Spontaneous shrinkage of drops and mass conservation in phase-field simulations|journal=Journal of Computational Physics|volume=223|issue=1|pages=1–9|year=2007|last1=Yue|first1=Pengtao|last2=Zhou|first2=Chunfeng|last3=Feng|first3=James J.|bibcode=2007JCoPh.223....1Y|citeseerx=10.1.1.583.2109}}</ref> Minimizing perceived continuity losses over the duration of a simulation requires limits on the Mobility parameter, resulting in a delicate balance between interfacial smearing due to convection, interfacial reconstruction due to free energy minimization (i.e. mobility-based diffusion), and phase interpenetration, also dependent on the mobility. A recent review of alternative energy density functions for interface tracking applications has proposed a modified form of the double-obstacle function which avoids the spontaneous drop shrinkage phenomena and limits on mobility,<ref>{{Cite journal|doi=10.1016/j.ijmultiphaseflow.2011.02.002|title=Diffuse interface tracking of immiscible fluids: Improving phase continuity through free energy density selection|journal=International Journal of Multiphase Flow|volume=37|issue=7|pages=777|year=2011|last1=Donaldson|first1=A.A.|last2=Kirpalani|first2=D.M.|last3=MacChi|first3=A.|bibcode=2011IJMF...37..777D |url=https://nrc-publications.canada.ca/eng/view/accepted/?id=43fdc6f4-70c8-42df-9f15-caf46c4e6c1a}}</ref> with comparative results provide for a number of benchmark simulations using the double-well function and the [[Volume_of_fluid_method|volume-of-fluid]] sharp interface technique. The proposed implementation has a computational complexity only slightly greater than that of the double-well function, and may prove useful for interface tracking applications of the phase-field model where the duration/nature of the simulated phenomena introduces phase continuity concerns (i.e. small droplets, extended simulations, multiple interfaces, etc.).
===Sharp interface limit of the phase-field equations===
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A phase-field model can be constructed to purposely reproduce a given interfacial dynamics as represented by a sharp interface model. In such a case the sharp interface limit (i.e. the limit when the interface width goes to zero) of the proposed set of phase-field equations should be performed. This limit is usually taken by asymptotic expansions of the fields of the model in powers of the interface width <math>\varepsilon</math>. These expansions are performed both in the interfacial region (inner expansion) and in the bulk (outer expansion), and then are asymptotically matched order by order. The result gives a partial differential equation for the diffusive field and a series of boundary conditions at the interface, which should correspond to the sharp interface model and whose comparison with it provides the values of the parameters of the phase-field model.
Whereas such expansions were in early phase-field models performed up to the lower order in <math>\varepsilon</math> only, more recent models use higher order asymptotics (thin interface limits) in order to cancel undesired spurious effects or to include new physics in the model. For example, this technique has permitted to cancel kinetic effects,<ref name=thin/> to treat cases with unequal diffusivities in the phases,<ref>{{Cite journal|doi=10.1016/S0167-2789(00)00064-6|title=Thin interface asymptotics for an energy/entropy approach to phase-field models with unequal conductivities|journal=Physica D: Nonlinear Phenomena|volume=144|issue=1–2|pages=154–168|year=2000|last1=McFadden|first1=G.B.|last2=Wheeler|first2=A.A.|last3=Anderson|first3=D.M.|bibcode=2000PhyD..144..154M|hdl=2060/20000014455|s2cid=119641692 |hdl-access=free}}</ref> to model viscous fingering<ref name="viscous" /> and two-phase Navier–Stokes flows,<ref>{{Cite journal|doi=10.1006/jcph.1999.6332|title=Calculation of Two-Phase Navier–Stokes Flows Using Phase-Field Modeling|journal=Journal of Computational Physics|volume=155|issue=1|pages=96–127|year=1999|last1=Jacqmin|first1=David|bibcode=1999JCoPh.155...96J}}</ref> to include fluctuations in the model,<ref>{{Cite journal|doi=10.1103/PhysRevE.71.061603 |pmid=16089744 |title=Sharp-interface projection of a fluctuating phase-field model |journal=Physical Review E |volume=71 |issue=6 |pages=061603 |year=2005 |last1=Benítez |first1=R. |last2=Ramírez-Piscina |first2=L. |bibcode=2005PhRvE..71f1603B |arxiv=cond-mat/0409707 |s2cid=28956874 }}</ref> etc.
== Multiphase-field models==
[[image:Multi Phase Field Order Parameters.jpg|280px|right|thumb|Multiple-order parameters describe a polycrystalline material microstructure.]]
In multiphase-field models, microstructure is described by set of order parameters, each of which is related to a specific phase or crystallographic orientation. This model is mostly used for solid-state phase transformations where multiple grains evolve (e.g.
== Phase-field models on graphs ==
{{main|Phase
Many of the results for continuum phase-field models have discrete analogues for graphs, just replacing calculus with [[Calculus on finite weighted graphs|calculus on graphs]].
==Phase Field Modeling in Fracture Mechanics==
Fracture in solids is often numerically analyzed within a finite element context using either discrete or diffuse crack representations. Approaches using a finite element representation often make use of strong discontinuities embedded at the intra-element level and often require additional criteria based on, e.g., stresses, strain energy densities or energy release rates or other special treatments such as virtual crack closure techniques and remeshing to determine crack paths. In contrast, approaches using a diffuse crack representation retain the continuity of the displacement field, such as continuum damage models and phase-field fracture theories. The latter traces back to the reformulation of Griffith’s principle in a variational form and has similarities to gradient-enhanced damage-type models. Perhaps the most attractive characteristic of phase-field approaches to fracture is that crack initiation and crack paths are automatically obtained from a minimization problem that couples the elastic and fracture energies. In many situations, crack nucleation can be properly accounted for by following branches of critical points associated with elastic solutions until they lose stability. In particular, phase-field models of fracture can allow nucleation even when the elastic strain energy density is spatially constant.<ref>{{cite journal|last1=Tanné|first1=E.|last2=Li|first2=T.|last3=Bourdin|first3=B.|last4=Marigo|first4=J.-J.|last5=Maurini|first5=C.|date=2018|title=Crack nucleation in variational phase-field models of brittle fracture|journal=Journal of the Mechanics and Physics of Solids|volume=110|pages=80–99|doi=10.1016/j.jmps.2017.09.006|bibcode=2018JMPSo.110...80T |s2cid=20139734 |url=https://hal.sorbonne-universite.fr/hal-01568702/file/paper.pdf }}</ref>
A limitation of this approach is that nucleation is based on strain energy density and not stress. An alternative view based on introducing a nucleation driving force seeks to address this issue.<ref>{{Cite journal|title=Revisiting nucleation in the phase-field approach to brittle fracture.|journal=Journal of the Mechanics and Physics of Solids|volume=142|pages=104027|doi=10.1016/j.jmps.2020.104027|year=2020|last1= Kumar|first1=A.|last2=Bourdin|first2=B.|last3=Francfort|first3=G. A.|last4=Lopez-Pamies|first4=O.|bibcode=2020JMPSo.14204027K |doi-access=free}}</ref>
== Phase Field Models for Collective Cell Migration ==
A group of biological [[Cell (biology)|cells]] can [[Self-propelled particles|self-propel]] in a complex way due to the consumption of [[Adenosine triphosphate]]. Interactions between cells like [[Cohesion (chemistry)|cohesion]] or several chemical cues can produce movement in a coordinated manner, this phenomenon is called "Collective cell migration". A theoretical model for these phenomena is the phase-field model<ref>{{Cite journal|last1=Najem|first1=Sara|last2=Grant|first2=Martin|date=2016-05-09|title=Phase-field model for collective cell migration|url=https://link.aps.org/doi/10.1103/PhysRevE.93.052405|journal=Physical Review E|volume=93|issue=5|pages=052405|doi=10.1103/PhysRevE.93.052405|pmid=27300922 |bibcode=2016PhRvE..93e2405N }}</ref><ref>{{Cite web|title=Phase-field model for cellular monolayers : a cancer cell migration studyauthors : Benoit Palmieri and Martin Grant {{!}} Perimeter Institute|url=https://www2.perimeterinstitute.ca/videos/phase-field-model-cellular-monolayers-cancer-cell-migration-studyauthors-benoit-palmieri-and|access-date=2021-11-05|website=www2.perimeterinstitute.ca}}</ref> and incorporates a phase field for each cell species and additional field variables like [[Chemotaxis|chemotactic]] agent concentration. Such a model can be used for phenomena like cancer, wound healing, [[morphogenesis]] and [[Ectoplasm (cell biology)|ectoplasm phenomena]].
==Software==
* [https://www.hs-karlsruhe.de/en/research/hska-research-institutions/institute-for-digital-materials-science-idm/pace-3d-software/ PACE3D – Parallel Algorithms for Crystal Evolution in 3D] is a parallelized phase-field simulation package including multi-phase multi-component transformations, large scale grain structures and coupling with fluid flow, elastic, plastic and magnetic interactions. It is developed at the [[Karlsruhe University of Applied Sciences]] and [[Karlsruhe Institute of Technology]].
* [https://github.com/mesoscale/mmsp The Mesoscale Microstructure Simulation Project (MMSP)] is a collection of C++ classes for grid-based microstructure simulation.
* [http://web.micress.de/ The MICRostructure Evolution Simulation Software (MICRESS)] is a multi-component, multiphase-field simulation package coupled to thermodynamic and kinetic databases. It is developed and maintained by [http://www.access-technology.de ACCESS e.V .]
* [[MOOSE (software)|MOOSE]] massively parallel open source C++ [[multiphysics]] finite-element framework with support for phase-field simulations developed at Idaho National Laboratory.
* [http://www.phasepot.com/ PhasePot] is a Windows-based microstructure simulation tool, using a combination of phase-field and Monte Carlo Potts models.
* [http://www.openphase.rub.de/ OpenPhase] is an open source software for the simulation of microstructure formation in systems undergoing first order phase transformation based on the multiphase field model.
*[https://
*[https://microsim.co.in MicroSim] is a software stack that consists of phase-field codes that offer flexibility with discretization, models as well as the high-performance computing hardware(CPU/GPU) that they can execute on.
*[https://prisms-center.github.io/phaseField/ PRISMS-PF] is a massively parallel finite element code for conducting phase-field and other related simulations of microstructure evolution.<ref>{{Cite journal|doi=10.1038/s41524-020-0298-5|title=PRISMS-PF: A general framework for phase-field modeling with a matrix-free finite element method|journal=[[npj Comput Mater]]|volume=6|pages=29|year=2020|last1=DeWitt|first1=S.|last2=Rudraraju|first2=S.|last3=Montiel|first3=D.|last4=Montiel|first4=D.|last5=Andrews|first5=W.B.|last6=Thornton|first6=K.|doi-access=free|bibcode=2020npjCM...6...29D }}</ref> It is based on the [https://www.dealii.org/ deal.II] finite element library and developed and maintained by the [http://prisms-center.org/#/home PRISMS Center] at the University of Michigan.
==References==
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*{{cite journal|doi=10.1146/annurev.matsci.32.112001.132041|title=Phase-Field models For microstructure evolution|journal=[[Annual Review of Materials Research]]|volume=32|pages=113–140|year=2002|last1=Chen|first1=Long-Qing}}
*{{Cite journal|doi=10.1016/j.calphad.2007.11.003|title=An introduction to phase-field modeling of microstructure evolution|journal=Calphad|volume=32|issue=2|pages=268|year=2008|last1=Moelans|first1=Nele|last2=Blanpain|first2=Bart|last3=Wollants|first3=Patrick}}
*{{cite journal|doi=10.1088/0965-0393/17/7/073001|title=Phase-field models in materials science|journal=Modelling and Simulation in Materials Science and Engineering|volume=17|issue=7|pages=073001|year=2009|last1=Steinbach|first1=Ingo|bibcode=2009MSMSE..17g3001S|s2cid=3383625 }}
*{{cite journal|doi=10.3139/146.110013 |title=Upgrading CALPHAD to microstructure simulation: The phase-field method |journal=International Journal of Materials Research |volume=100 |issue=2 |pages=128 |year=2009 |last1=Fries |first1=Suzana G. |last2=Boettger |first2=Bernd |last3=Eiken |first3=Janin |last4=Steinbach |first4=Ingo |bibcode=2009IJMR..100..128F |s2cid=138203262 }}
*{{Cite journal|doi=10.1179/174328409X453190|url=http://www.msm.cam.ac.uk/phase-trans/2010/mst8941.pdf|title=Phase field method|journal=Materials Science and Technology|volume=26|issue=7|pages=803|year=2010|last1=Qin|first1=R. S.|last2=Bhadeshia|first2=H. K.|bibcode=2010MatST..26..803Q |s2cid=136124682}}
*{{cite journal|doi=10.1016/j.ijmultiphaseflow.2011.02.002|title=Diffuse interface tracking of immiscible fluids: Improving phase continuity through free energy density selection|journal=International Journal of Multiphase Flow|volume=37|issue=7|pages=777|year=2011|last1=Donaldson|first1=A.A.|last2=Kirpalani|first2=D.M.|last3=MacChi|first3=A.|bibcode=2011IJMF...37..777D |url=https://nrc-publications.canada.ca/eng/view/accepted/?id=43fdc6f4-70c8-42df-9f15-caf46c4e6c1a}}
*{{Cite journal|arxiv=cond-mat/0305058|last1= Gonzalez-Cinca|first1= R.|title= Phase-field models in interfacial pattern formation out of equilibrium|journal= In Advances in Condensed Matter and Statistical Mechanics,
*Provatas, Nikolas; Elder, Ken (2010). ''Phase-Field Methods in Materials Science and Engineering.'' Weinheim, Germany: Wiley-VCH Verlag GmbH & Co. KGaA. {{doi|10.1002/9783527631520}}. {{ISBN|9783527631520}}
*Steinbach, I.: "Quantum-Phase-Field Concept of Matter: Emergent Gravity in the Dynamic Universe", Zeitschrift für Naturforschung A 72 1 (2017) {{doi|10.1515/zna-2016-0270}}
*Schmitz, G.J.: "A Combined Entropy/Phase-Field Approach to Gravity", ''Entropy'' '''2017''', ''19''(4) 151; {{doi|10.3390/e19040151}}
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