New Phase-Field Models with Applications to Materials Genome Initiative

  • Peicheng ZhuEmail author
  • Yangxin Tang
  • Yeping Li
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 200)


Advanced materials are crucial to economic security and human well-being. American then-President Obama launched in 2011 the Materials Genome Initiative (MGI) that is a novel and multi-stakeholder effort so that discovery and deployment of advanced materials can be significantly accelerated while the cost can be considerably reduced. Integrated computation is a key tool of MGI. Phase-field approach is a young, however, has now emerged as a powerful tool in theoretical and numerical analysis of phenomena at the meso-scale, therefore it has important applications to MGI. We shall mainly review two types of phase-field models, formulated recently by Alber and the first author of this article, for solid-solid phase transitions driven by configurational forces, with applications to martensitic phase transitions in, e.g. smart materials like shape memory alloys, and to sintering which is a process in, for instance, powder metallurgy. Mathematical and numerical investigations of these models will be presented and open problems related to the models are listed. Finally we shall also introduce phase-field crystal method which can be regarded as an extension of phase-field approach.


Phase-field models Material Genome Initiative Materials science Martensitic phase transitions Interface motion by interface diffusion Configurational force Partial differential equations Elasticity system Weak solutions Simulations Shape memory alloys Sintering 



The authors would like to express their sincere thanks for the anonymous reviewer(s) for his/her useful comments. Zhu and Tang are supported in part by the Start-up grant of 1000-plan Scholar Program from Shanghai University, and by Key grant (Grant No. 2017YFB0701502) from the Ministry of Science and Technology of P. R. China, and Li is supported in part by the National Science Foundation of China (Grant No. 11671134) and the Ph.D. Program Foundation of Ministry of Education of China (Grant No. 20133127110007).


  1. 1.
    Chen, L.: Phase-field models for microstructure evolution. Annu. Rev. Mater. Res. 32, 113–140 (2002)Google Scholar
  2. 2.
    Moelans, N., Blanpain, B., Wollants, P.: An introduction to phase-field modeling of microstructure evolution. Calphad. Comp. Coupl. Phase Diag. Thermochemi. 32(2), 268–294 (2008)Google Scholar
  3. 3.
    Provatas, N., Dantzig, J., Athreya, B., Chan, P., Stefanovic, P., Goldenfeld, N., Elder, K. (2007) Using the phase-field crystal method in the multi-scale modeling of microstructure evolution. J. Minerals, Metals and Materials Soc., JOM 59(7),83–90Google Scholar
  4. 4.
    Qin, R.S., Bhadeshia, H.K.: Phase field method. Mater. Sci. Technol. 26(7), 803–811 (2010)Google Scholar
  5. 5.
    Steinbach, I.: Phase-field models in materials science. Modeling Simul. Mater. Sci. Eng. 17(7):073001-1–073001-31 (2009)Google Scholar
  6. 6.
    Materials Genome Initiative for Global Competitiveness: Executive Office of the President National Science and Technology Council. USA, Washington (2011)Google Scholar
  7. 7.
    Materials Genome Initiative (MGI) Strategic Plan. Executive Office of the President National Science and Technology Council, Washington, USA (2014)Google Scholar
  8. 8.
    Nishizawa, T.: Thermodynamics of Microstructures. ASM International, Ohio (2008)Google Scholar
  9. 9.
    Feynman, R., Leyton, R., Sands, M.: The Feynman Lectures on Physics, vol. 1. Addison-Wesley Publishing Company (1964)Google Scholar
  10. 10.
    Cooper, T., et al.: (the ASM Handbook Committee), ASM Handbook, vol. 9. Metallography and Microstructures, ASM International, Ohio (1985)Google Scholar
  11. 11.
    Allen, S., Bever, M.: Structure of materials, in Encyclopedia of Materials Science and Engineering at MIT Press., Cambridge MA (1986)Google Scholar
  12. 12.
    Fix, G.J.: Phase field methods for free boundary problems. In: Fasano, A., Primicerio, M. (eds.) Free Boundary Problems: Theory and Applications, vol. II, pp. 580–589. Pitman, Boston (1983)Google Scholar
  13. 13.
    Langer, J.S.: Models of pattern formation in first-order phase transitions. In: Grinstein, G., Mazenko, G. (eds.) Directions in Condensed Matter Physics, pp. 165–186. World Scientific, Singapore (1986)Google Scholar
  14. 14.
    Cahn, R., Haasen, P. (eds.): Physical metallurgy Three, vol. Set, 4th edn. North-Holland, Amsterdam (1996)Google Scholar
  15. 15.
    Eshelby, J.: Collected works of J. D. Eshelby, The Mechanics of Defects and Inhomogeneities. Springer-Verlag (2006)Google Scholar
  16. 16.
    Maugin, G.: Material Inhomogeneities in Elasticity, vol. 3. Appl Math and Math Comput. Chapman & Hall (1993)Google Scholar
  17. 17.
    Kienzler, R., Herrmann, G.: Mechanics in Material Space: With Applications to Defect and Fracture Mechanics. Springer-Verlag, Berlin, Heidelberg (2000)zbMATHGoogle Scholar
  18. 18.
    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)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Alber, H.-D., Zhu, P.: Evolution of phase boundaries by configurational forces. Arch. Rational Mech. Anal. 185, 235–286 (2007)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Zhu, P.: Solid-Solid Phase Transitions Driven by Configurational Forces: A phase-field Model and its Validity. Lambert Academy Publishing (LAP), Germany (2011)Google Scholar
  21. 21.
    Hornbogen, E., Warlimont, H.: Metallkunde, vol. 4. Springer-Verlag, Auage (2001)Google Scholar
  22. 22.
    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)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Buratti, G., Huo, Y., Müller, I.: Eshelby tensor as a tensor of free enthalpy. J. Elasticity 72, 31–42 (2003)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Müller, R., Gross, D.: 3D simulation of equilibrium morphologies of precipitates. Computational Materials Sci. 11, 35–44 (1998)Google Scholar
  25. 25.
    Socrate, S., Parks, D.: Numerical determination of the elastic driving force for directional coarsening in Ni-superalloys. Acta Metall. Mater. 41(7), 2185–2209 (1993)Google Scholar
  26. 26.
    James, R.: Configurational forces in magnetism with application to the dynamics of a small-scale ferromagnetic shape memory cantilever. Contin. Mech. Thermodyn. 14(1), 55–86 (2002)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Alber, H.-D.: Evolving microstructure and homogenization. Continum. Mech. Thermodyn 12, 235–287 (2000)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Alt, H.W., Pawlow, I.: On the entropy principle of phase transition models with a conserved order parameter. Adv. Math. Sci. Appl. 6(1), 291–376 (1996)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Wechsler, M., Lieberman, D., Read, T.: On the theory of the formation of martensite. Trans. AIMS. J. Met. 197, 1503–1515 (1953)Google Scholar
  30. 30.
    Ball, J., James, R.: Fine phase mixtures as minimizers of energy. Arch. Rati. Mech. Anal. 100, 13–52 (1987)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Ball, J., James, R.: Proposed experimental tests of a theory of fine microstructure, and the two-well problem. Phil. Trans. Roy Soc. Lond. A 388, 389–450 (1992)zbMATHGoogle Scholar
  32. 32.
    Alber, H.-D., Zhu, P.: Comparison of a rapidly converging phase field model for interfaces in solids with the Allen-Cahn model. J. Elast. 111, 153–221 (2013)zbMATHGoogle Scholar
  33. 33.
    Kawashima, S., Zhu, P.: Traveling waves for models of phase transitions of solids driven by configurational forces, Discrete Continuous Dynamical Systems - Seri. B 15(1), 309–323 (2011).
  34. 34.
    Alber, H.-D., Zhu, P.: Solutions to a model with Neumann boundary conditions for phase transitions driven by configurational forces. Nonlinear Anal. Real World Appl. 12(3), 1797–1809 (2011)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Zhu, P.: Solvability via viscosity solutions for a model of phase transitions driven by configurational forces. J. Diff. Eqn. 251(10), 2833–2852 (2011)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Zhu, P.: Regularity of solutions to a model for solid-solid phase transitions driven by configurational forces. J. Math. Anal. Appl. 389(2), 1159–1172 (2012)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Ou, Y., Zhu, P.: Spherically symmetric solutions to a model for phase transitions driven by configurational forces. J. Math. Phys. 52(093708), 21 (2011)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Alber, H.-D., Zhu, P.: Solutions to a model for interface motion by interface diffusion. Proc. Royal Soc. Edinburgh 138A, 923–955 (2008)MathSciNetzbMATHGoogle Scholar
  39. 39.
    H.-D. Alber and Zhu, P.: Spherically symmetric solutions to a model for phase transitions driven by configurational forces. In: preparation (2018)Google Scholar
  40. 40.
    Alber, H.-D., Zhu, P.: Interface motion by interface diffusion driven by bulk energy: justification of a diffusive interface model. Continuum Mech. Thermodyn 23(2), 139–176 (2011)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Sheng, W., Zhu, P.: Viscosity solutions to a model for solid-solid phase transitions driven by material forces. Nonliner Anl. RWA 39, 14–32 (2018)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Alikakos, N., Bates, P., Chen, X.: Convergence of the Cahn-Hilliard equation to the Hele-Shaw model. Arch. Rational Mech. Anal. 128, 165–205 (1994)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Allen, S., Cahn, J.: A microscopic theory for anti-phase boundary motion and its application to anti-phase domain coarsening. Acta Met. 27, 1084–1095 (1979)Google Scholar
  44. 44.
    Cahn, J.: On spinodal decomposition. Acta. Meta. 9, 795–801 (1961)Google Scholar
  45. 45.
    Cahn, J.: Free Energy of a Nonuniform System. II. Thermodynamic Basis. J. Chem. Phys. 30, 1121–1124 (1959)Google Scholar
  46. 46.
    Cahn, J., Hilliard, J.: Free energy of a nonuniform system. I interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)Google Scholar
  47. 47.
    Cahn, J., Taylor, J.: Surface motion by surface diffusion. Acta Metall. Mater 42(4), 1045–1063 (1994)Google Scholar
  48. 48.
    Carrive, M., Miranville, A., Pierus, A.: The Cahn-Hilliard equation for deformable elastic continua. Adv. Math. Sci. Appl. 10(2), 539–569 (2000)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Chen, X.: Generation and propagation of interfaces for reaction-diffusion equations. J. Diff. Equa. 96(1), 116–141 (1992)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Chen, X.: Spectrums for the Allen-Cahn, Cahn-Hilliard, and phase-field equations for generic interface. Comm. Partial Diff. Eqns 19(7/8), 1371–1395 (1994)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Chen, X.: Global asymptotic limit of solutions of the Cahn-Hilliard equation. J. Diff. Geom. 44, 262–311 (1996)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Elliott, C., Zheng, S.: On the Cahn-Hilliard equation. Arch. Rat. Mech. Anal. 96, 339–357 (1986)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Emmerich, H.: The Diffuse Interface Approach in Materials Science. Lecture Notes in Physics, Springer, Heidelberg (2003)zbMATHGoogle Scholar
  54. 54.
    Emmerich, H.: Advances of and by phase-field modeling in condensed-matter physics. Adv. Phys. 57(1), 1–87 (2008)Google Scholar
  55. 55.
    Fife, P., McLeod, J.: The approach of solutions of nonlinear diffusion equations to traveling front solutions. Arch. Rati. Mech. Anal. 65, 335–361 (1977)zbMATHGoogle Scholar
  56. 56.
    Fratzl, P., Penrose, O., Lebowitz, J.: Modeling of phase separation in alloys with coherent elastic misfit. J. Statist. Phys. 95, 1429–1503 (1999)MathSciNetzbMATHGoogle Scholar
  57. 57.
    Fried, E., Gurtin, M.: Dynamic solid-solid transitions with phase characterized by an order parameter. Phys. D 72, 287–308 (1994)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Garcke, H.: On Cahn-Hilliard systems with elasticity. Proc. R. Soc. Edinb., Sect. A, Math. 133(2), 307–331 (2003)MathSciNetzbMATHGoogle Scholar
  59. 59.
    Leo, P., Lowengrub, J., Jou, H.: A diffuse interface model for microstructural evolution in elastically stressed solids. Acta Mater. 46(6), 2113–2130 (1998)Google Scholar
  60. 60.
    Ninomiya, H., Taniguchi, M.: Existence and global stability of traveling curved fronts in the Allen-Cahn equations. J. Diff. Eq. 213(1), 204–233 (2005)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Ninomiya, H., Taniguchi, M.: Global stability of traveling curved fronts in the Allen-Cahn equations. Disc. Conti. Dyna. Syst. 15(3), 819–832 (2006)MathSciNetzbMATHGoogle Scholar
  62. 62.
    Pego, R.: Front Migration in the Nonlinear Cahn-Hilliard equation. Proc. R. Soc. Lond. 422A(1863), 261–278 (1989)MathSciNetzbMATHGoogle Scholar
  63. 63.
    Volpert, A., Volpert, V., Volpert, V.: Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, 140. American Mathematical Society, Providence, Rhode Island (1994)zbMATHGoogle Scholar
  64. 64.
    Acharya, A., Matthies, K., Zimmer, J.: Traveling wave solutions for a quasilinear model of field dislocation mechanics. J. Mech. Phys. Solids 58, 2043–2053 (2010)MathSciNetzbMATHGoogle Scholar
  65. 65.
    Hildebrand, F., Miehe, C.: A regularized sharp-interface model for phase transformation accounting for prescribed sharp interface kinetics. Proc. Appl. Math. Mech. 10, 673–676 (2010)Google Scholar
  66. 66.
    Goldenfeld, N., Athreya, B., Dantzig, J.: Renormalization group approach to multiscale modeling in materials science. J. Stat. Phys. 125(5/6), 1019–1027 (2006)zbMATHGoogle Scholar
  67. 67.
    Kobayashi, R., Warren, J., Craig Carter, W.: Vector-valued phase field model for crystallization and grain boundary formation. Physica D 119, 415–423 (1998)Google Scholar
  68. 68.
    Kobayashi, R., Warren, J., Craig Carter, W.: A continuum model of grain boundaries. Phys. D 140, 141–150 (2000)MathSciNetzbMATHGoogle Scholar
  69. 69.
    Crandall, M., Lions, P.: Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277, 1–42 (1983)MathSciNetzbMATHGoogle Scholar
  70. 70.
    Crandall, M., Lions P.: On existence and uniqueness of solutions of Hamilton-Jacobi equations. Nonlinear Anal. T.M.A., 10:353–370 (1986)Google Scholar
  71. 71.
    Juutinen, P., Lindqvist, P., Manfredi, J.: On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation. SIAM J. Math. Anal. 33(3), 699–717 (2001)MathSciNetzbMATHGoogle Scholar
  72. 72.
    Ladyzenskaya, O., Solonnikov, V., Uralceva, N.: Linear and Quasilinear Equations of Parabolic Type, Translations of Math. Monographs 23, AMS, Providence (1968)Google Scholar
  73. 73.
    Gurtin, M.: Configurational Forces as Basic Concepts of Continuum Physics, vol. 137. Springer-Verlag, Applied Math Sci (2000)Google Scholar

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Authors and Affiliations

  1. 1.Materials Genome Institute and Department of MathematicsShanghai UniversityShanghaiPeople’s Republic of China
  2. 2.Department of MathematicsShanghai UniversityShanghaiPeople’s Republic of China
  3. 3.Institute of Statistics and Applied MathematicsAnhui University of Finance and EconomicsBengbuPeople’s Republic of China
  4. 4.Department of MathematicsEast China University of Science and TechnologyShanghaiPeople’s Republic of China

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