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Continuum Diffuse-Interface Model for Modeling Microstructural Stability

  • L. Q. Chen
  • D. N. Fan
  • Y. Z. Wang
  • A. G. Khachaturyan
Part of the NATO ASI Series book series (NSSB, volume 355)

Abstract

The majority of advanced materials in practical applications are multiphase and/or polycrystalline, and their physical properties depend not only on the atomic structure and properties of each constituent phase, but also crucially on the nanoscale or microscale microstructure. A microstructure is characterized by (1) the number of phases and their volume fractions, and (2) the size, shape, distribution and orientation of each phase. Generally, a microstructure is thermodynamically unstable, or at most, metastable, and at a given temperature, it will evolve as a function of time towards equilibrium. The driving force for the temporal evolution of a microstructure usually consists of one or more of the following: (1) reduction in the bulk chemical free energy; (2) decrease of the total interfacial energy between different phases or between different orientation domains or grains of the same phase; (3) homogeneous and heterogeneous relaxation of the elastic strain energy generated by the lattice mismatch between different phases or different orientation domains; and (4) external fields such as applied stress, electrical, temperature and magnetic fields. The phase changes and resulted microstructural development driven by the decrease of the bulk chemical free energy is usually referred to as phase transformations whereas the microstructural evolution due to reduction in the total excess free energies associated with interfaces is called coarsening (or grain growth in polycrystalline materials). In coherent systems, however, elastic strain energy may play an important, and very often, a controlling role in the microstructural evolution during both phase transformations and coarsening.

Keywords

Tetragonal Phase Spinodal Decomposition Elastic Strain Energy Orientation Domain Antiphase Domain 
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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References

  1. 1.
    J. D. Gunton, M. S. Miguel and P. S. Sahni, The dynamics of first-order phase transitions, in C. Domb and J. L. Lebowitz(eds), Phase Transitions and Critical Phenomena, Vol. 8., Academic Press, New York, pp. 267-466 (1983). Google Scholar
  2. 2.
    T. M. Rogers, K. R. Elder and R. C. Desai, Numerical study of the late stages of spinodal decomposition, Phys. Rev. B 37:9638 (1988).ADSCrossRefGoogle Scholar
  3. 3.
    Y. Oono and S. Puri, Computationally efficient modeling of ordering of quenched phases, Phys. Rev. Lett. 58:836 (1987).ADSCrossRefGoogle Scholar
  4. 4.
    A. A. Wheeler, W. J. Boettinger and G. B. McFadden, Phys. Rev. A45:7424 (1992).ADSGoogle Scholar
  5. 5.
    G. Caginalp and E. Socolovsky Phase field computations of single-needle crystals, crystal growth, and motion by mean curvature, SIAM J. Sci. Comput. 15:106 (1994).MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    R. Kobayashi, A numerical approach to three–dimensional dendritic solidification, Experim. Math. 3:59 (1994).MATHGoogle Scholar
  7. 7.
    J. A. Warren and W. J. Boettinger, Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method, Acta metall. mater. 43:689 (1995).CrossRefGoogle Scholar
  8. 8.
    A. Onuki , Ginzburg-Landau approach to elastic effects in the phase separation of solids, J. Phy. Soc. Jpn. 58:3065(1989); Long-range interactions through elastic fields in phase-separating solids, J. Phy. Soc. Jpn., 58:3069 (1989); H. Nishimori and A. Onuki, Pattern formation in phase-separating alloys with cubic symmetry, Phys. Rev. B 42:980 (1990).ADSCrossRefGoogle Scholar
  9. 9.
    Y. Wang, Theoretical characterization and modeling of microstructure development during coherent phase transformations in metals and ceramics, Ph.D. dissertation, Rutgers, The State University of New Jersey, NJ, USA (1995); Y. Wang, L. Q. Chen, and A. G. Khachaturyan, Computer simulation of microstructure evolution in coherent solids, in Solid→Solid phase Transformations, W. C.Johnson, J. M. Howe, D. E. Laughlin and W. A. Soffa (eds), The Minerals, Metals & Materials Society, pp245-265 (1994); Y. Wang, H. Y. Wang, L. Q. Chen and A. G. Khachaturyan, Shape evolution of a coherent tetragonal precipitate in partially stabilized cubic ZrO2: a computer simulation, J. Am. Ceram. Soc. 76:3029 (1993); (1995) Microstructural development of coherent tetragonal precipitates in Mg-partially stabilized ZrO2: a computer simulation, J. Am. Ceram. Soc. 78, 657-661. Y. Wang and A. G. Khachaturyan, Effect of antiphase domains on shape and spatial arrangement of coherent ordered intermetallics, Scipta Metall et Mater., 31:1425 (1995); Theoretical characterization of the kinetics of martensitic transformation, to be submitted to Acta metall er mater (1995).Google Scholar
  10. 10.
    D. N. Fan and L. Q. Chen, Computer simulation of twin formation during the dispiaci ve c→t’ phase transformation in the Zirconia-Yttria system, J. Am. Ceram. Soc. 78:769 (1995); Possibility of spinodal decomposition in ZrO2-Y2O3 alloys: a theoretical investigation, ibid 78:1680 (1995). MathSciNetCrossRefGoogle Scholar
  11. 11.
    L. Q. Chen and W. Yang, Computer simulation of the domain dynamics of a quenched system with a large number of nonconserved order parameters: the grain-growth kinetics, Phys. Rev. B50:15752 (1994); L. Q. Chen, A novel computer simulation technique for modeling grain growth, Scripta metall. mater, 32:115 (1995); D. N. Fan, and L. Q. Chen, Computer simulation of grain growth using the diffuse-interface field model, submitted to Acta metall et mater (1995). ADSGoogle Scholar
  12. 12.
    J. W. Cahn and J. E. Hilliard Free energy of a nonuniform system, J. Chem. Phys. 28:258 (1958). ADSCrossRefGoogle Scholar
  13. 13.
    J. W. Cahn, On spinodal decomposition, Acta metall. 9:795 (1961); On spinodal decomposition in cubic crystals, ibid, 10:179 (1962). CrossRefGoogle Scholar
  14. 14.
    E. M. Lifshitz, and Pitaevskii, L. P. Statistical Physics, 3rd edition, Part 1, Landau and Lishitz Course of Theoretical Physics, Vol. 5, Pergamon Press (1980). Google Scholar
  15. 15.
    S. M. Allen and J. W. Cahn A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta metall. 27:1085 (1979). CrossRefGoogle Scholar
  16. 16.
    W. Yang and L. Q. Chen, Computer simulation of the dynamics of 180° ferroelectric domain formation, to be published in the J. Am. Ceram. Soc. (1995).Google Scholar
  17. 17.
    T. Eguchi, K. Oki and S. Matsumura, Kinetics of ordering with phase separation, in MRS Symp. Proc, 21:589 (1984); K. Shiiyama, H. Kanemoto, H. NInomiya and T. Eguchi, Computer simulation of dynamics of the pattern formations in antiphase ordered structure and phase separation induced by ordering in binary alloys, in Proceedings of the International Workshop on Computational Materials Science (IWCMS), 103 (1990). CrossRefGoogle Scholar
  18. 18.
    A. G. Khachaturyan, Some questions concerning the theory of phase transformations in solids, Sov. Phys. Solid State 8:2163 (1967); A. G. Khachaturyan and G. A. Shatalov, Elastic-interaction potential of defects in a crystal, Sov. Phys. Solid State 11:118 (1969). Google Scholar
  19. 19.
    Y. Wang, L. Q. Chen and A. G. Khachaturyan, Shape evolution of a precipitate during strain-induced coarsening: a computer simulation, Scripta metall. mater. 25:1387 (1991); Strain-induced modulated structures in two-phase cubic alloys, ibid, 25:1969 (1991); Kinetics of strain-induced morphological transformation in cubic alloys with a miscibility gap, Acta metall. mater. 41:279 (1993); Particle translational motion and reverse coarsening phenomena in multiparticle systems induced by a long-range elastic interaction, Phys. Rev. B 46:11194 (1992). CrossRefGoogle Scholar
  20. 20.
    L. Q. Chen, Y. Wang and A. G. Khachaturyan, Transformation-induced elastic strain effect on the precipitation kinetics of ordered intermetallics, Phil. Mag. Ltr. 64:241 (1992).ADSCrossRefGoogle Scholar
  21. 21.
    Y. S. Yoo, D. Y. Yoon and M. F. Henry, The effect of elastic misfit strain on the morphological evolution of g’-precipitates in a model Ni-base superalloy, in press; Y. S. Yoo, The dendritic growth of g’-precipitates and grain boundary serration in Ni-based superalloys, Ph.D. dissertation, Korea Advanced Institute of Science and Technology, Taejon, Korea (1993).Google Scholar
  22. 22.
    A. Maheshwari and A. J. Ardell, Anomalous Coarsening of Small Volume Fractions of Ni3AI Precipitates, An Explanation of Inhomogeneous Dispersions Observed at Small Undercoolings, Scripta metall mater. 26:347 (1992); Anomalous coarsening behavior of small volume fractions of NÌ3Al precipitates in binary Ni-Al alloys, Acta metall. mater. 40:2661 (1992). CrossRefGoogle Scholar
  23. 23.
    A. G. Khachaturyan, Theory of Structural Transformations in Solids, Wiley, New York, 1983.Google Scholar
  24. 24.
    M. Doi and T. Miyazaki, On the spinodal decomposition in Zirconia-Yttria (Zr02-Y2O3) alloys, Philos Mag. B68:305 (1993).Google Scholar
  25. 25.
    H. V. Atkinson, Theory of normal grain growth in pure single phase systems, Acta metall. 36:469(1988).CrossRefGoogle Scholar
  26. 26.
    D. Weaire and J. A. Glazier, Modeling grain growth and soap froth coarsening: past, present and future, Materials Science Forum, p27, vol. 94–96 (1992) Pt. 1. CrossRefGoogle Scholar
  27. 27.
    D. Weaire and F. Bolton, Rigidity loss transition in a disordered 2D froth, Phys. Rev. Lett. 65:3449(1990).ADSCrossRefGoogle Scholar
  28. 28.
    D. Weaire and J. N. Rivier, Soap, cells and statistics - random patterns in two dimensions, Contemp. Phys. 25:59 (1984).ADSCrossRefGoogle Scholar
  29. 29.
    D. Weaire and H. Lei, A note on the statistics of the mature two-dimensional soap froth, Phil. Mag. Lett 62:427 (1990).ADSCrossRefGoogle Scholar
  30. 30.
    R. L. Fullman, in Metal interfaces, p179, American Society for Metals, Cleveland, 1952.Google Scholar
  31. 31.
    K. Kawasaki, T. Nagai, and K. Nakashima, Vertex Model of Cellular Pattern Growth in Two and Three Dimensions, Phil. Mag. B 60:399 (1989).CrossRefGoogle Scholar
  32. 32.
    M. P. Anderson, D. J. Srolovitz, G. S. Grest and P. S. Sahni, Computer simulation of grain growth -1. kinetics, Acta metall. 32:783 (1984). CrossRefGoogle Scholar
  33. 33.
    D. J. Srolovitz, M. P. Anderson, P. S. Sahni and G. S. Grest, Computer simulation of grain growth - II. grain size distribution, topology, and local dynamics, Acta metall. 32:793 (1984). CrossRefGoogle Scholar
  34. 34.
    M. P. Anderson and G. S. Grest, Computer simulation of normal grain growth in three dimensions, Phil. Mag. B 59:293 (1989).CrossRefGoogle Scholar
  35. 35.
    S. Kumar, S. K. Kurtz, J. R. Banavar, and M. G. Sharma, Properties of a Three-Dimensional Poisson-Voronoi Tesselation, A Monte Carlo Study, J. Statis. Phys. 67:523 (1992).ADSMATHCrossRefGoogle Scholar
  36. 36.
    S. K. Kurtz and F. M. A. Carpay, Microstructural and normal grain growth in metals and ceramics. J. Appl. Phys. 51:5125 (1980).CrossRefGoogle Scholar
  37. 37.
    V. E. Fradkov, L. S. Shvindlerman and D. G. Udler, Computer simulation of grain growth in two dimensions, Scripta. Met. 19:1291 (1985).CrossRefGoogle Scholar
  38. 38.
    C. W. J. Beenakker, Numerical simulation of a coarsening two-dimensional network, Phys. Rev. A 37:1697 (1988).ADSCrossRefGoogle Scholar
  39. 39.
    M. Marder, Soap-bubble growth, Phys. Rev. A 36:438 (1987).ADSCrossRefGoogle Scholar
  40. 40.
    C. V. Thompson, H. J. Frost, and F. Spaepen, The relative rates of secondary and normal grain growth, Acta Metall 35:887 (1987).CrossRefGoogle Scholar
  41. 41.
    G S. Grest, M. P. Anderson and D. J. Srolovitz, Domain-growth kinetics for the Q-state Potts model in two and three dimensions, Phys. Rev. B 38:4752 (1988). ADSGoogle Scholar
  42. 42.
    D. N. Fan and L. Q. Chen, to be submitted to Acta metall et mater.Google Scholar

Copyright information

© Plenum Press, New York 1996

Authors and Affiliations

  • L. Q. Chen
    • 1
  • D. N. Fan
    • 1
  • Y. Z. Wang
    • 2
  • A. G. Khachaturyan
    • 2
  1. 1.Department of Materials Science and EngineeringThe Pennsylvania State UniversityUSA
  2. 2.Department of Materials Science and EngineeringRutgers UniversityPiscatawayUSA

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