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Phase-Field Models

  • Mathis Plapp
Chapter
Part of the CISM Courses and Lectures book series (CISM, volume 538)

Abstract

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 fields that take constant values in the bulk phases and vary continuously but steeply across a diffuse front. In the present contribution, a distinction is made between models in which the phase field can be identified with a physical quantity (coarse-grained on a mesoscopic scale), and models in which the phase field can only be interpreted as a smoothed indicator function. Simple diffuse-interface models for the motion of magnetic domain walls, the growth of precipitates in binary alloys, and for solidification are reviewed, and it is pointed out that is such models the free energy function determines both the bulk behavior of the dynamic variable and the properties of the interface. Next, a phenomenological phase-field model for solidification is introduced, and it is shown that with a proper choice of some interpolation functions, surface and bulk properties can be adjusted independently in this model. The link between this phase-field model and the classic free-boundary formulation of solidification is established by the use of matched asymptotic analysis. The results of this analysis can then be exploited to design new phase-field models that cannot be derived by the standard variational procedure from simple free energy functionals within the thermodynamic framework. As examples for applications of this approach, the solidification of alloys and the advected field model for two-phase flow are briefly discussed.

Keywords

Free Energy Domain Wall Surface Free Energy Free Boundary Problem Free Energy Density 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Bibliography

  1. R. F. Almgren. Second-order phase field asymptotics for unequal conductivities. SIAM J. Appl. Math., 59:2086–2107, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  2. D. M. Anderson, G. B. McFadden, and A. A. Wheeler. Diffuse-interface methods in fluid mechanics. Annual Review of Fluid Mechanics, 30:139, 1998.MathSciNetCrossRefGoogle Scholar
  3. D. M. Anderson, G. B. McFadden, and A. A. Wheeler. A phase-field model of solidification with convection. PHYSICA D, 135:175–194, 2000.MathSciNetzbMATHCrossRefGoogle Scholar
  4. M. J. Aziz. Model for the solute redistribution during rapid solidification. J. Appl. Phys., 53(2):1158, 1982.CrossRefGoogle Scholar
  5. J. Beaucourt, T. Biben, and C. Verdier. Elongation and burst of axisymmetric viscoelastic droplets: A numerical study. Phys. Rev. E, 71:066309, 2005.Google Scholar
  6. C. Beckermann, H.-J. Diepers, I. Steinbach, A. Karma, and X. Tong. Modeling melt convection in phase-field simulations of solidification. J. Comput. Phys., 154:468, 1999.zbMATHCrossRefGoogle Scholar
  7. T. Biben, C. Misbah, A. Leyrat, and C. Verdier. An advected-field approach to the dynamics of fluid interfaces. Europhys. Lett., 63:623, 2003.CrossRefGoogle Scholar
  8. T. Biben, K. Kassner, and C. Misbah. Phase-field approach to three-dimensional vesicle dynamics. Phys. Rev. E, 72:041921, 2005.CrossRefGoogle Scholar
  9. W. J. Boettinger, J. A. Warren, C. Beckermann, and A. Karma. Phase-field simulation of solidification. Annu. Rev. Mater. Res., 32:163–194, 2002.CrossRefGoogle Scholar
  10. A. J. Bray. Theory of Phase-ordering kinetics. Adv. Phys., 43:357–459, 1994.MathSciNetCrossRefGoogle Scholar
  11. Q. Bronchart, Y. Le Bouar, and A. Finel. New coarse-grained derivation of a phase field model for precipitation. Phys. Rev. Lett., 100:015702, 2008.CrossRefGoogle Scholar
  12. G. Caginalp. Stefan and hele-shaw type models as asymptotic limits of the phase-field equations. Phys. Rev. A, 39:5887–5896, 1989.MathSciNetzbMATHCrossRefGoogle Scholar
  13. J. W. Cahn and J. E. Hilliard. Free energy of a non-uniform system. 1. interfacial free energy. J. Chem. Phys., 28:258–267, 1958.CrossRefGoogle Scholar
  14. L.-Q. Chen. Phase-field models for microstructure evolution. Annu. Rev. Mater. Res., 32:113, 2002.CrossRefGoogle Scholar
  15. J. B. Collins and H. Levine. Diffuse interface model of diffusion-limited crystal growth. Phys. Rev. B, 31:6119–6122, 1985.CrossRefGoogle Scholar
  16. B. Echebarria, R. Folch, A. Karma, and M. Plapp. Quantitative phase-field model of alloy solidification. Phys. Rev. E, 70(6):061604, 2004.CrossRefGoogle Scholar
  17. K. R. Elder, M. Grant, N. Provatas, and J. M. Kosterlitz. Sharp interface limits of phase-field models. Phys. Rev. E, 64:021604, 2001.CrossRefGoogle Scholar
  18. H. Emmerich. Advances of and by phase-field modelling in condensed-matter physics. Adv. Phys., 57(1):1–87, 2008.CrossRefGoogle Scholar
  19. G. J. Fix. In A. Fasano and M. Primicerio, editors, Free boundary problems: Theory and applications, page 580, Boston, 1983. Piman.Google Scholar
  20. R. Folch, J. Casademunt, A. HernándezMachado, and L. Ramírez Piscina. Phase-field model for hele-shaw flows with arbitrary viscosity contrast. i. theoretical approach. Phys. Rev. E, 60(2):1724, 1999.CrossRefGoogle Scholar
  21. M. E. Glicksman, M. B. Koss, and E. A. Winsa. Dendritic growth velocities in microgravity. Phys. Rev. Lett., 73:573–576, 1994.CrossRefGoogle Scholar
  22. T. Haxhimali, A. Karma, F. Gonzales, and M. Rappaz. Orientation selection in dendritic evolution. Nature Materials, 5:660–664, 2006.CrossRefGoogle Scholar
  23. P. C. Hohenberg and B. I. Halperin. Theory of dynamic critical phenomena. Rev. Mod. Phys., 49:435–479, 1977.CrossRefGoogle Scholar
  24. J. J. Hoyt, M. Asta, and A. Karma. Atomistic and continuum modeling of dendritic solidification. Mat. Sience Eng. R, 41:121, 2003.CrossRefGoogle Scholar
  25. D. Jacqmin. Calculation of two-phase Navier-Stokes flows using phase-field modeling. J. Comput. Phys., 155:96–127, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  26. D. Jamet and C. Misbah. Thermodynamically consistent picture of the phase-field model of vesicles: Elimination of the surface tension. Phys. Rev. E, 78:041903, 2008.CrossRefGoogle Scholar
  27. D. Jamet, O. Lebaigue, N. Coutris, and J. M. Delhaye. The second gradient method for the direct numerical simulation of liquid-vapor flows with phase change. J. Comput. Phys., 169:624–651, 2001.MathSciNetzbMATHCrossRefGoogle Scholar
  28. A. Karma. Phase-field formulation for quantitative modeling of alloy solidification. Phys. Rev. Lett., 87(10):115701, 2001.CrossRefGoogle Scholar
  29. A. Karma and W.J. Rappel. Phase-field method for computationally efficient modeling of solidification with arbitrary interface kinetics. Phys. Rev. E, 53(4):R3017–R3020, 1996.CrossRefGoogle Scholar
  30. A. Karma and W.J. Rappel. Quantitative phase-field modeling of dendritic growth in two and three dimensions. Phys. Rev. E, 57(4):4323–4349, 1998.zbMATHCrossRefGoogle Scholar
  31. A. Karma, Y. H. Lee, and M. Plapp. Three-dimensional dendrite-tip morphology at low undercooling. Phys. Rev. E, 61(4, Part B):3996–4006, APR 2000.CrossRefGoogle Scholar
  32. J. S. Langer. An introduction to the kinetics of first-order phase transitions. In C. Godrèche, editor, Solids far from equilibrium, Edition Aléa Saclay, pages 297–363, Cambridge, UK, 1991. Cambridge University Press.Google Scholar
  33. J. S. Langer. Models of pattern formation in first-order phase transitions. In G. Grinstein and G. Mazenko, editors, Directions in Condensed Matter Physics, pages 165–186, Singapore, 1986. World Scientific.Google Scholar
  34. S. Nguyen, R. Folch, V. K. Verma, H. Henry, and M. Plapp. Phase-field simulations of viscous fingering in shear-thinning fluids. Phys. Fluids, 22:103102, 2010.CrossRefGoogle Scholar
  35. M. Ohno and K. Matsuura. Quantitative phase-field modeling for dilute alloy solidification involving diffusion in the solid. Phys. Rev. E, 79(3): 031603, 2009.CrossRefGoogle Scholar
  36. O. Penrose and P. C. Fife. Thermodynamically consistent models of phase-field type for the kinetics of phase transitions. Physica D, 43:44–62, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  37. M. Plapp. Three-dimensional phase-field simulations of directional solidification. J. Cryst. Growth, 303:49–57, 2007.CrossRefGoogle Scholar
  38. M. Plapp. Remarks on some open problems in phase-field modelling of solidification. Phil. Mag., 91:25–44, 2011.CrossRefGoogle Scholar
  39. J. S. Rowlinson. Translation of J. D. van der Waals, The Thermodynamic Theory of Capillarity under the Hypothesis of a Continuous Variation of Density. J. Stat. Phys., 20:197–244, 1979.MathSciNetzbMATHCrossRefGoogle Scholar
  40. I. Steinbach. Phase-field models in materials science. Model. Simul. Mater. Sci. Eng., 17(7):073001, OCT 2009.MathSciNetCrossRefGoogle Scholar
  41. W. van Saarloos. Front propagation into unstable states. Phys. Reports, 386:29–222, 2003.zbMATHCrossRefGoogle Scholar
  42. Y. Wang and J. Li. Phase field modeling of defects and deformation. Acta Mater., 58:1212–1235, 2010.CrossRefGoogle Scholar

Copyright information

© CISM, Udine 2012

Authors and Affiliations

  • Mathis Plapp
    • 1
  1. 1.Physique de la Matière Condensée, École PolytechniqueCNRSPalaiseauFrance

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