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Asymptotics and numerical efficiency of the Allen–Cahn model for phase interfaces with low energy in solids

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Abstract

We study how the propagation speed of interfaces in the Allen–Cahn phase field model for phase transformations in solids consisting of the elasticity equations and the Allen–Cahn equation depends on two parameters of the model. The two parameters control the interface energy and the interface width, but change also the interface speed. To this end, we derive an asymptotic expansion of second order for the interface speed, called the kinetic relation, and prove that it is uniformly valid in both parameters. As a consequence, we show that the model error is proportional to the interface width divided by the interface energy. We conclude that simulations of interfaces with low interface energy based on this model require a very small interface width, implying a large numerical effort. Effective simulations thus need adaptive mesh refinement or other advanced techniques.

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References

  1. Abels, H., Schaubeck, S.: Sharp interface limit for the Cahn–Larché system. Asymptot. Anal. 91(3–4), 283–340 (2015)

    MathSciNet  MATH  Google Scholar 

  2. Abeyaratne, R., Knowles, J.K.: On the driving traction acting on a surface of strain discontinuity in a continuum. J. Mech. Phys. Solids 38(3), 345–360 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  3. Alber, H.-D.: Evolving microstructure and homogenization. Continum. Mech. Thermodyn. 12, 235–286 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  4. Alber, H.-D.: Evolution of phase interfaces by configurational forces: a phase field model. Thermodynamische Materialtheorien, Oberwolfach 12.2.2004–18.12.2004. Oberwolfach Rep. 1(4), 2981–2985 (2004)

    Google Scholar 

  5. Alber, H.-D., Zhu, P.: Solutions to a model with nonuniformly parabolic terms for phase evolution driven by configurational forces. SIAM J. Appl. Math. 66(2), 680–699 (2006)

    Article  MathSciNet  Google Scholar 

  6. Alber, H.-D., Zhu, P.: Evolution of phase boundaries by configurational forces. Arch. Ration. Mech. Anal. 185(2), 235–286 (2007)

    Article  MathSciNet  Google Scholar 

  7. Alber, H.-D., Zhu, P.: Interface motion by interface diffusion driven by bulk energy: justification of a diffusive interface model. Contin. Mech. Thermodyn. 23(2), 139–176 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  8. Alber, H.-D., Zhu, P.: Comparison of a rapidely converging phase field model for interfaces in solids with the Allen–Cahn model. J. Elast. 111(2), 153–221 (2013)

    Article  MathSciNet  Google Scholar 

  9. Alber, H.-D.: Asymptotics and numerical efficiency of the Allen–Cahn model for phase interfaces with low energy in solids. Version containing full proofs. arXiv:1505.05442 [math-ph]

  10. Almgren, R.F.: Second-order phase field asymptotics for unequal conductivities. SIAM J. Appl. Math. 59(6), 2086–2107 (1999)

    Article  MathSciNet  Google Scholar 

  11. Alikakos, N.D., Bates, P.W., Chen, X.: Convergence of the Cahn–Hilliard equation to the Hele-Shaw model. Arch. Ration. Mech. Anal. 128, 165–205 (1994)

    Article  MathSciNet  Google Scholar 

  12. Bhattachary, K.: Microstructure of Martensite. Oxford University Press, Oxford (2003)

    Google Scholar 

  13. Buratti, G., Huo, Y., Müller, I.: Eshelby tensor as a tensor of free enthalpy. J. Elast. 72, 3142 (2003)

    Article  MathSciNet  Google Scholar 

  14. Caginalp, G.: The role of microscopic anisotropy in the macroscopic behavior of a phase boundary. Ann. Phys. 172, 136–155 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  15. Caginalp, G., Chen, X.: Convergence of the phase field model to its sharp interface limits. Eur. J. Appl. Math. 9, 417–445, 795–801 (1998)

  16. Carlen, E.A., Carvalho, M.C., Orlandi, E.: Approximate solutions of the Cahn–Hilliard equation via corrections to the Mullins–Sekerka motion. Arch. Ration. Mech. Anal. 178(1), 1–55 (2005)

    Article  MathSciNet  Google Scholar 

  17. Chen, X.: Spectrums for the Allen–Cahn, Cahn–Hilliard, and phase-field equations for generic interface. Comm. Partial Diff. Eqns. 19, 1371–1395 (1994)

    Article  MathSciNet  Google Scholar 

  18. Chen, X., Caginalp, G., Eck, Ch.: A rapidely converging phase field model. Discrete Contin. Dyn. Syst. 15(4), 1017–1034 (2006)

    Article  Google Scholar 

  19. Chen, X., Hong, J., Yi, F.: Existence, uniqueness, and regularity of classical solutions of the Mullins–Sekerka problem. Comm. Partial Differ. Equ. 21, 1705–1727 (1996)

    Article  MathSciNet  Google Scholar 

  20. de Mottoni, P., Schatzman, M.: Geometrical evolution of developed interfaces. Trans. Am. Math. Soc. 347, 1533–1589 (1995)

    Article  MathSciNet  Google Scholar 

  21. Escher, J., Simonett, G.: On Hele–Shaw models with surface tension. Math. Res. Lett. 3(4), 467–474 (1996)

    Article  MathSciNet  Google Scholar 

  22. Escher, J., Simonett, G.: Classical solutions for Hele–Shaw models with surface tension. Adv. Differ. Equ. 2(4), 619–642 (1997)

    MathSciNet  MATH  Google Scholar 

  23. Escher, J., Simonett, G.: A center manifold analysis for the Mullins–Sekerka model. J. Differ. Equ. 143(2), 267–292 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  24. Fife, P., Penrose, O.: Interfacial dynamics for thermodynamically consistent phase field models with nonconserved order parameter. Electron. J. Differ. Equ. 1, 1–49 (1995)

    Article  MathSciNet  Google Scholar 

  25. Fried, E., Gurtin, M.: Dynamic solid–solid transitions with phase characterized by an order parameter. Phys. D 72, 287–308 (1994)

    Article  MathSciNet  Google Scholar 

  26. Garcke, H., Stinner, B.: Second order phase field asymptotics for multi-component systems. Interfaces Free Bound 8(2), 131–157 (2006)

    Article  MathSciNet  Google Scholar 

  27. Karma, A., Rappel, W.-J.: Quantitative phase-field modeling of dendritic growth in two and three dimensions. Phys. Rev. E 57, 4323–4349 (1998)

    Article  ADS  Google Scholar 

  28. Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966)

    MATH  Google Scholar 

  29. Luckhaus, S.: Solutions for the two-phase Stefan problem with the Gibbs–Thomson law for the melting temperature. Eur. J. Appl. Math. 1(2), 101–111 (1990)

    Article  MathSciNet  Google Scholar 

  30. Schrade, D., Mueller, R., Xu, B.X., Gross, D.: Domain evolution in ferroelectric materials: a continuum phase field model and finite element implementation. Comput. Methods Appl. Eng. 196, 4365–4374 (2007)

    Article  Google Scholar 

  31. Xu, B.-X., Schrade, D., Müller, R., Gross, D., Granzow, T., Rödel, J.: Phase field simulation and experimental investigation of the electro-mechanical behavior of ferroelectrics. Z. Angw. Mat. Mech. 90(78), 623–632 (2010)

    Article  Google Scholar 

  32. Zhang, W., Bhattacharya, K.: A computational model of ferroelectric domains, part I: model formulation and domain switching. Acta Mater. 53, 182–198 (2005)

    Google Scholar 

  33. Zuo, Y., Genenko, Y.A., Klein, A., Stein, P., Xu, B.: Domain wall stability in ferroelectrics with space charges. J. Appl. Phys. 115, 084110 (2014)

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Authors and Affiliations

  1. Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7, 64289, Darmstadt, Germany

    Hans-Dieter Alber

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  1. Hans-Dieter Alber
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Correspondence to Hans-Dieter Alber.

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Communicated by Andreas Öchsner.

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Alber, HD. Asymptotics and numerical efficiency of the Allen–Cahn model for phase interfaces with low energy in solids. Continuum Mech. Thermodyn. 29, 757–803 (2017). https://doi.org/10.1007/s00161-017-0558-x

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