Skip to main content
SpringerLink
Log in
Book cover

Mathematics Applied to Engineering, Modelling, and Social Issues pp 569–598Cite as

New Phase-Field Models with Applications to Materials Genome Initiative

  • Chapter
  • First Online:

  • 844 Accesses

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 200))

Abstract

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.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Chen, L.: Phase-field models for microstructure evolution. Annu. Rev. Mater. Res. 32, 113–140 (2002)

    Google Scholar 

  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. 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),83–90

    Google Scholar 

  4. Qin, R.S., Bhadeshia, H.K.: Phase field method. Mater. Sci. Technol. 26(7), 803–811 (2010)

    Google Scholar 

  5. Steinbach, I.: Phase-field models in materials science. Modeling Simul. Mater. Sci. Eng. 17(7):073001-1–073001-31 (2009)

    Google Scholar 

  6. Materials Genome Initiative for Global Competitiveness: Executive Office of the President National Science and Technology Council. USA, Washington (2011)

    Google Scholar 

  7. Materials Genome Initiative (MGI) Strategic Plan. Executive Office of the President National Science and Technology Council, Washington, USA (2014)

    Google Scholar 

  8. Nishizawa, T.: Thermodynamics of Microstructures. ASM International, Ohio (2008)

    Google Scholar 

  9. Feynman, R., Leyton, R., Sands, M.: The Feynman Lectures on Physics, vol. 1. Addison-Wesley Publishing Company (1964)

    Google Scholar 

  10. Cooper, T., et al.: (the ASM Handbook Committee), ASM Handbook, vol. 9. Metallography and Microstructures, ASM International, Ohio (1985)

    Google Scholar 

  11. Allen, S., Bever, M.: Structure of materials, in Encyclopedia of Materials Science and Engineering at MIT Press., Cambridge MA (1986)

    Google Scholar 

  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. 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. Cahn, R., Haasen, P. (eds.): Physical metallurgy Three, vol. Set, 4th edn. North-Holland, Amsterdam (1996)

    Google Scholar 

  15. Eshelby, J.: Collected works of J. D. Eshelby, The Mechanics of Defects and Inhomogeneities. Springer-Verlag (2006)

    Google Scholar 

  16. Maugin, G.: Material Inhomogeneities in Elasticity, vol. 3. Appl Math and Math Comput. Chapman & Hall (1993)

    Google Scholar 

  17. Kienzler, R., Herrmann, G.: Mechanics in Material Space: With Applications to Defect and Fracture Mechanics. Springer-Verlag, Berlin, Heidelberg (2000)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  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. Hornbogen, E., Warlimont, H.: Metallkunde, vol. 4. Springer-Verlag, Auage (2001)

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  23. Buratti, G., Huo, Y., Müller, I.: Eshelby tensor as a tensor of free enthalpy. J. Elasticity 72, 31–42 (2003)

    MathSciNet  MATH  Google Scholar 

  24. Müller, R., Gross, D.: 3D simulation of equilibrium morphologies of precipitates. Computational Materials Sci. 11, 35–44 (1998)

    Google Scholar 

  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. 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)

    MathSciNet  MATH  Google Scholar 

  27. Alber, H.-D.: Evolving microstructure and homogenization. Continum. Mech. Thermodyn 12, 235–287 (2000)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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. Ball, J., James, R.: Fine phase mixtures as minimizers of energy. Arch. Rati. Mech. Anal. 100, 13–52 (1987)

    MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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). https://doi.org/10.3934/dcdsb.2011.15.309

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  38. Alber, H.-D., Zhu, P.: Solutions to a model for interface motion by interface diffusion. Proc. Royal Soc. Edinburgh 138A, 923–955 (2008)

    MathSciNet  MATH  Google Scholar 

  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. 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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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. Cahn, J.: On spinodal decomposition. Acta. Meta. 9, 795–801 (1961)

    Google Scholar 

  45. Cahn, J.: Free Energy of a Nonuniform System. II. Thermodynamic Basis. J. Chem. Phys. 30, 1121–1124 (1959)

    Google Scholar 

  46. Cahn, J., Hilliard, J.: Free energy of a nonuniform system. I interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)

    MATH  Google Scholar 

  47. Cahn, J., Taylor, J.: Surface motion by surface diffusion. Acta Metall. Mater 42(4), 1045–1063 (1994)

    Google Scholar 

  48. Carrive, M., Miranville, A., Pierus, A.: The Cahn-Hilliard equation for deformable elastic continua. Adv. Math. Sci. Appl. 10(2), 539–569 (2000)

    MathSciNet  MATH  Google Scholar 

  49. Chen, X.: Generation and propagation of interfaces for reaction-diffusion equations. J. Diff. Equa. 96(1), 116–141 (1992)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  51. Chen, X.: Global asymptotic limit of solutions of the Cahn-Hilliard equation. J. Diff. Geom. 44, 262–311 (1996)

    MathSciNet  MATH  Google Scholar 

  52. Elliott, C., Zheng, S.: On the Cahn-Hilliard equation. Arch. Rat. Mech. Anal. 96, 339–357 (1986)

    MathSciNet  MATH  Google Scholar 

  53. Emmerich, H.: The Diffuse Interface Approach in Materials Science. Lecture Notes in Physics, Springer, Heidelberg (2003)

    MATH  Google Scholar 

  54. Emmerich, H.: Advances of and by phase-field modeling in condensed-matter physics. Adv. Phys. 57(1), 1–87 (2008)

    Google Scholar 

  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)

    MATH  Google Scholar 

  56. Fratzl, P., Penrose, O., Lebowitz, J.: Modeling of phase separation in alloys with coherent elastic misfit. J. Statist. Phys. 95, 1429–1503 (1999)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  58. Garcke, H.: On Cahn-Hilliard systems with elasticity. Proc. R. Soc. Edinb., Sect. A, Math. 133(2), 307–331 (2003)

    MathSciNet  MATH  Google Scholar 

  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. 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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  62. Pego, R.: Front Migration in the Nonlinear Cahn-Hilliard equation. Proc. R. Soc. Lond. 422A(1863), 261–278 (1989)

    MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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. Goldenfeld, N., Athreya, B., Dantzig, J.: Renormalization group approach to multiscale modeling in materials science. J. Stat. Phys. 125(5/6), 1019–1027 (2006)

    MATH  Google Scholar 

  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. Kobayashi, R., Warren, J., Craig Carter, W.: A continuum model of grain boundaries. Phys. D 140, 141–150 (2000)

    MathSciNet  MATH  Google Scholar 

  69. Crandall, M., Lions, P.: Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277, 1–42 (1983)

    MathSciNet  MATH  Google Scholar 

  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. 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)

    MathSciNet  MATH  Google Scholar 

  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. Gurtin, M.: Configurational Forces as Basic Concepts of Continuum Physics, vol. 137. Springer-Verlag, Applied Math Sci (2000)

    Google Scholar 

Download references

Acknowledgements

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).

Author information

Author notes

  1. Yangxin Tang

    Present address: Institute of Statistics and Applied Mathematics, Anhui University of Finance and Economics, No. 962 Caoshan Road, Bengbu, 233030, Anhui Province, People’s Republic of China

Authors and Affiliations

  1. Materials Genome Institute and Department of Mathematics, Shanghai University, Shangda Road 99, Shanghai, 200444, People’s Republic of China

    Peicheng Zhu

  2. Department of Mathematics, Shanghai University, Shangda Road 99, Shanghai, 200444, People’s Republic of China

    Yangxin Tang

  3. Department of Mathematics, East China University of Science and Technology, Shanghai, 200237, People’s Republic of China

    Yeping Li

Authors
  1. Peicheng Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yangxin Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yeping Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peicheng Zhu .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University College London, London, UK

    Frank T. Smith

  2. Department of Mathematics, Gauhati University, Guwahati, Assam, India

    Hemen Dutta

  3. Department of Mathematics, Center for Mathematics of Uncertainty, Creighton University, Omaha, NE, USA

    John N. Mordeson

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Zhu, P., Tang, Y., Li, Y. (2019). New Phase-Field Models with Applications to Materials Genome Initiative. In: Smith, F.T., Dutta, H., Mordeson, J.N. (eds) Mathematics Applied to Engineering, Modelling, and Social Issues. Studies in Systems, Decision and Control, vol 200. Springer, Cham. https://doi.org/10.1007/978-3-030-12232-4_18

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-12232-4_18

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-12231-7

  • Online ISBN: 978-3-030-12232-4

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation