Coupling Microstructure Characterization with Microstructure Evolution

  • Chen Shen
  • Ning Ma
  • Yuwen Cui
  • Ning Zhou
  • Yunzhi Wang


Microstructure reconstruction in 3D and quantitative digital representation are enabling consideration of polycrystalline and multi-phase microstructures in mechanics codes in a realistic way. To take full advantage of these advances, we discuss in this chapter the synergy of coupling quantitative microstructure characterization by experimental imaging techniques with quantitative microstructural evolution modeling by image-based computer simulation techniques such as the phase field method. Specific attention will be paid to the fundamentals of the phase field method for microstructure representation and description of microstructure evolution, and procedures of using experimental images as model inputs. Through individual examples we show how to use the phase field method at different length scales to: explore mechanisms of microstructural evolution, extract important materials parameters, carry out physics-based repairs of experimentally reconstructed microstructures, and evolve existing microstructures or generate new microstructures to populate digital microstructural database for different time, temperature, stress, and other service conditions for mechanical property explorations.


Phase Field Phase Field Model Orientation Imaging Microscopy Phase Field Method Phase Field Simulation 
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.



We gratefully acknowledge financial supports by the Office of Naval Research through the D 3D program (Grant No. N00014-05-1-0504), U.S. Air Force Office of Scientific Research through the Metals Affordability Initiative Program on Durable High Temperature Disks and the STW21 Program on Multi-Materials System with Adaptive Microstructures for Aerospace Applications (Grant No. FA9550-09-1-0014), and the National Science Foundation (Grant No. CMMI-0728069). The simulations were performed on supercomputers at the Arctic Region Supercomputing Center and the Ohio Supercomputing Center.


  1. Allen SM, Cahn JW. A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metallurgica 1979;27:1085.CrossRefGoogle Scholar
  2. Almgren RF. Second-order phase field asymptotics for unequal conductivities. SIAM Journal on Applied Mathematics 1999;59:2086.MATHMathSciNetCrossRefGoogle Scholar
  3. Andersson JO, Helander T, Hoglund L, Shi PF, Sundman B. THERMO-CALC & DICTRA, computational tools for materials science. CALPHAD 2002;26:273.CrossRefGoogle Scholar
  4. Ansara I, Dupin N, Lukas HL, Sundman B. Thermodynamic assessment of the Al-Ni system. Journal of Alloys and Compounds 1997;247:20.CrossRefGoogle Scholar
  5. Bhandari Y, Sarkar S, Groeber M, Uchic M, Dimiduk D, Ghosh S. 3D polycrystalline microstructure reconstruction from FIB generated serial sections for FE Analysis. Computational Materials Science 2007;41:222.CrossRefGoogle Scholar
  6. Boettinger WJ, Warren JA, Beckermann C, Karma A. Phase-field simulation of solidification. Annual Review of Materials Research 2002;32:163.CrossRefGoogle Scholar
  7. Braun RJ. Adaptive finite-difference computations of dendritic growth using a phase-field model. Modelling Simul. Materials Science and Engineering 1997;5:365.CrossRefGoogle Scholar
  8. Cahn JW, Hilliard JE. Free energy of a nonuniform system. I. Interfacial free energy. Journal of Chemical Physics 1958;28:258.Google Scholar
  9. Cahn JW, Hilliard JE. Free energy of a nonuniform system. III. Nucleation in a two-component incompressible fluid. Journal of Chemical Physics 1959;31:688.Google Scholar
  10. Campbell CE, Boettinger WJ, Kattner UR. Development of a diffusion mobility database for Ni-base superalloys. Acta Materialia 2002;50:775.CrossRefGoogle Scholar
  11. Cha PR, Kim SG, Yeon DH, Yoon JK. A phase field model for the solute drag on moving grain boundaries. Acta Materialia 2002;50:3817.CrossRefGoogle Scholar
  12. Chen LQ. A novel computer-simulation technique for modeling grain-growth. Scripta Metallurgica Et Materialia 1995;32:115.CrossRefGoogle Scholar
  13. Chen LQ. Phase field models for microstructure evolution. Annual Review of Materials Research 2002;32:113.CrossRefGoogle Scholar
  14. Chen LQ, Shen J. Applications of semi-implicit Fourier-spectral method to phase field equations. Computer Physics Communications 1998;108:147.MATHCrossRefGoogle Scholar
  15. Chen Q, Ma N, Wu K, Wang Y. Quantitative phase field modeling of diffusion-controlled precipitate growth and dissolution in Ti-6Al-4V. Scripta Materialia 2004;50:471.CrossRefGoogle Scholar
  16. Collings EW. Materials Properties Handbook: Titanium Alloys. Materials Park, OH: ASM International, 1994.Google Scholar
  17. Dobrich K, Rau C, Krill CE. Quantitative characterization of the three-dimensional microstructure of polycrystalline Al-Sn using X-ray microtomography. Metallugical and Materials Transaction A 2004;35:1953.CrossRefGoogle Scholar
  18. Elder KR, Grant M, Provatas N, Kosterlitz JM. Sharp interface limits of phase-field models. Physical Review E 2001;64:021604.CrossRefGoogle Scholar
  19. Eshelby JD. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proceedings of the Royal Society of London. Series A 1957;241.Google Scholar
  20. Eshelby JD. The elastic field outside an ellipsoidal inclusion. Proceedings of the Royal Society of London. Series A 1959;252:561.MATHMathSciNetCrossRefGoogle Scholar
  21. Fahrmann M, Hermann W, Fahrmann E, Boegli A, Pollock TM, Sockel HG. Determination of matrix and precipitate elastic constants in (gamma-gamma ’) Ni-base model alloys, and their relevance to rafting. Materials Science and Engineering A 1999;260:212.CrossRefGoogle Scholar
  22. Fan DN, Chen LQ. Diffuse-interface description of grain boundary motion. Philosophical Magazine Letters 1997a;75:187.Google Scholar
  23. Fan D, Chen LQ. Computer simulation of grain growth and ostwald ripening in alumina-zirconia two-phase composites. Journal of the American Ceramic Society 1997b;80:1773.CrossRefGoogle Scholar
  24. Fan D, Chen SP, Chen LQ. Computer simulation of grain growth kinetics with solute drag. Journal of Materials Research 1999;14:1113.MathSciNetCrossRefGoogle Scholar
  25. Feng WM, Yu P, Hu SY, Liu ZK, Du Q, Chen LQ. Spectral implementation of an adaptive moving mesh method for phase-field equations. Journal of Computational Physics 2006;220:498.MATHMathSciNetCrossRefGoogle Scholar
  26. Foiles SM, Hoyt JJ. Computation of grain boundary stiffness and mobility from boundary fluctuations. Acta Materialia 2006;54:3351.CrossRefGoogle Scholar
  27. Gabb TP, Backman DG, Wei DY, Mourer DP, Furrer D, Garg A, Ellis DL. γ formation in a nickel-base disk superalloy. In: Pollock TM, Kissinger RD, Bowman RR, Green KA, McLean M, Olson S, Schirra JJ, editors. Superalloys 2000. Warrendale, PA: TMS, 2000. p. 405.Google Scholar
  28. Ghosh S, Bhandari Y, Groeber M. CAD based Reconstruction of three dimensional polycrystalline microstructures from FIB generated serial sections, Journal of Computer Aided Design, Vol. 40/3 pp 293–310, 2008.Google Scholar
  29. Ginzburg VL, Landau LD. On the theory of superconductivity. Zhurnal Eksperimentalnoy i Teoreticheskoy Fiziki (USSR), 1950;20:10641082 (in Russian) [English translation: in Men of Physics, vol. 1. 1965. Oxford: Pergamon Press, pp. 138167].Google Scholar
  30. Glavicic MG, Kobryn PA, Bieler TR and Semiatin SL. An automated method to determine the orientation of the high temperature beta phase from measured EBSD data fro the low temperature alpha phase in Ti-6Al-4V, Materials Science and Engineering A, 351, 2003: 258–264.CrossRefGoogle Scholar
  31. Grafe U, Botteger B, Tiaden J, Fries SG. Coupling of multicomponent thermodynamic database to a phase field model: application to solidification and solid state transformations of superalloys. Scripta Materialia 2000;42.Google Scholar
  32. Groeber M, Ghosh S, Uchic M, Dimiduk D. Development of a robust 3D characterization-representation framework for modeling polycrystalline materials. JOM 2007;59:32.CrossRefGoogle Scholar
  33. Gronhagen K, Agren J. Grain-boundary segregation and dynamic solute drag theory – a phase-field approach. Acta Materialia 2007;55:955.CrossRefGoogle Scholar
  34. Gruber J, Ma N, Rollett AD, Rohrer GS. Sparse data structure and algorithm for the phase field method. Modelling and Simulation in Materials Science and Engineering 2006;14:1189.CrossRefGoogle Scholar
  35. Gunton JD, Miguel MS, Sahni PS. The dynamics of first-order phase transitions. In: Domb C, Lebowitz JL, editors. Phase Transitions and Critical Phenomena, vol. 8. New York: Academic Press, 1983.Google Scholar
  36. Hillert M. A Theory of Nucleation of Solid Metallic Solutions. vol. Sc.D. Cambridge, MA: Massachusetts Institute of Technology, 1956.Google Scholar
  37. Hirth JP, Lothe J. Theory of Dislocations. New York: Wiley, 1982.Google Scholar
  38. Hoyt JJ, Asta M, Karma A. Method for computing the anisotropy of the solid-liquid interface free energy. Physical Review Letters 2001;86:5530.CrossRefGoogle Scholar
  39. Hu SY, Chen LQ. A phase-field model for evolving microstructures with strong elastic inhomogeneity. Acta Materialia 2001;49:1897.Google Scholar
  40. Huang Y, Humphreys HJ, Mackenzie JK. Acta Materialia 2000;48:2017.CrossRefGoogle Scholar
  41. Johnson WC. Influence of elastic stress on phase transformations. In: Aaronson HI, editor. Lectures on the Theory of Phase Transformations. Warrendale, PA: The Minerals, Metals & Materials Society, 1999. p. 35.Google Scholar
  42. Jonsson B. Ferromagnetic ordering and diffusion of carbon and nitrogen in BCC CR-FE-NI alloys. Zeitschrift fur MetaIlkunde 1994;85:498.Google Scholar
  43. Karma A. Phase field methods. In: Buschow KHJ, Cahn RW, Flemings MC, Ilschner B, Kramer EJ, Mahajian S, editors. Encyclopedia of Materials: Science and Technology, vol. 7. Oxford: Elsevier, 2001. p. 6873.Google Scholar
  44. Karma A, Rappel W-J. Phase-field method for computationally efficient modeling of solidification with arbitrary interface kinetics. Physical Review E 1996;53:3017.CrossRefGoogle Scholar
  45. Karma A, Rappel W-J. Quantitative phase-field modeling of dendritic growth in two and three dimensions. Physical Review E 1998;57:4323.MATHCrossRefGoogle Scholar
  46. Kaufman L, Bernstein H. Computer Calculation of Phase Diagrams with Special Reference to Refractory Metals. New York: Academic Press, 1970.Google Scholar
  47. Kazaryan A, Wang Y, Jin YMM, Wang YU, Khachaturyan AG, Wang LS, Laughlin DE. Development of magnetic domains in hard ferromagnetic thin films of polytwinned microstructure. Journal of Applied Physics 2002a;92:7408.CrossRefGoogle Scholar
  48. Kazaryan A, Wang Y, Dregia SA, Patton BR. Grain growth in anisotropic systems: comparison of effect of energy and mobility. Acta Materialia 2002b;50:2491.CrossRefGoogle Scholar
  49. Khachaturyan AG. Fizika Tverdogo Tela 1966;8:2710.Google Scholar
  50. Khachaturyan AG. Some questions concerning the theory of phase transformations in solids. Soviet Physics – Solid State 1967;8:2163.Google Scholar
  51. Khachaturyan AG. Theory of Structural Transformations in Solids. New York: Wiley, 1983.Google Scholar
  52. Khachaturyan AG, Shatalov GA. Elastic interaction potential of defects in a crystal. Soviet Physics – Solid State 1969;11:118.Google Scholar
  53. Khachaturyan AG, Semennovskaya S, Tsakalakos T. Elastic strain energy of inhomogeneous solids. Physical Review B 1995;52:15909.CrossRefGoogle Scholar
  54. Kim SG, Park YB. Grain boundary segregation, solute drag and abnormal grain growth. Acta Materialia 2008;56:3739.CrossRefGoogle Scholar
  55. Kim SG, Kim WT, Suzuki T. Phase-field model for binary alloys. Physical Review E 1999;60:7186.CrossRefGoogle Scholar
  56. Kim SG, Kim WT, Suzuki T, Ode M. Phase-field modeling of eutectic solidification. Journal of Crystal Growth 2004;261:135.CrossRefGoogle Scholar
  57. Kitashima T. Coupling of the phase-field and CALPHAD methods for predicting multicomponent, solid-state phase transformations. Philosophical Magazine 2008;88:1615.CrossRefGoogle Scholar
  58. Kitashima T, Harada H. A new phase-field method for simulating gamma’ precipitation in multicomponent nickel-base superalloys. Acta Materialia 2009;57:2020.CrossRefGoogle Scholar
  59. Kobayashi H, Ode M, Kim SG, Kim WT, Suzuki T. Phase-field model for solidification of ternay alloys coupled with thermodynamic database. Scripta Materialia 2003;48:689.CrossRefGoogle Scholar
  60. Krill CE, Chen LQ. Computer simulation of 3-D grain growth using a phase-field model. Acta Materialia 2002;50:3057.Google Scholar
  61. Krill CE, Dobrich K, Michels D, Michels A, Rau C, Weitkamp T, Snigirev A, Birringer R. In: Bonse U, editor. Developments in X-Ray Tomography III, Proc. SPIE, vol. 5335. Bellingham, WA: SPIE Press, 2004. p. 205.Google Scholar
  62. Lan CW, Hsu CM, Liu CC, Chang YC. Adaptive phase field simulation of dendritic growth in a forced flow at various supercoolings. Physical Review E 2002;65:061601.CrossRefGoogle Scholar
  63. Landau L, Lifshitz E. Physikalische Zeit schrift der Sowjetunion 1935;8:153.MATHGoogle Scholar
  64. Langer JS. Statistical theory of the decay of metastable states. Annals of Physics 1969;54:258.CrossRefGoogle Scholar
  65. Langer JS. Theory of spinodal decomposition in alloy. Annals of Physics 1971;65:53.CrossRefGoogle Scholar
  66. Langer JS. Models of pattern formation in first-order phase transitions. In: Grinstein G, Mazenko G, editors. Direction in Condensed Matter Physics. Singapore: World Scientific, 1986. p. 165.Google Scholar
  67. Langer JS. An introduction to the kinetics of first-order phase transitions. In: Godrèche C, editor. Solids Far from Equilibrium. New York: Cambridge University Press, 1992.Google Scholar
  68. Lee YW, Aaronson HI. Anisotropy of coherent interphase boundary energy. Acta Metallurgica 1980;28:539.CrossRefGoogle Scholar
  69. Li DY, Chen LQ. Shape of a rhombohedral coherent Ti11Ni14 precipitate in a cubic matrix and its growth and dissolution during constrained aging. Acta Materialia 1997a;45:2435.CrossRefGoogle Scholar
  70. Li DY, Chen LQ. Computer simulation of morphological evolution and rafting of gamma’ particles in Ni-based superalloys under applied stresses. Scripta Materialia 1997b;37:1271.CrossRefGoogle Scholar
  71. Ma N, Dregia SA, Wang Y. Segregation transition and drag force at grain boundaries. Acta Materialia 2003;51:3687.CrossRefGoogle Scholar
  72. Ma N, Kazaryan A, Dregia SA, Wang Y. Computer simulation of texture development during grain growth: effect of boundary properties and initial microstructure. Acta Materialia 2004;52:3869.CrossRefGoogle Scholar
  73. Ma N, Chen Q, Wang Y. Simulating microstructural evolution with high interfacial energy anisotropy using the phase field method. Scripta Materialia 2006;54:1919.CrossRefGoogle Scholar
  74. Mackenzie JK. Second paper on statistics associated with the random disorientation of cubes. Biometrika 1958;45:229.MATHMathSciNetGoogle Scholar
  75. McFadden GB, Wheeler AA, Anderson DM. Thin interface asymptotics for an energy/entropy approach to phase-field models with unequal conductivities. Physica D 2000;144:154.MATHMathSciNetCrossRefGoogle Scholar
  76. Moelans N, Blanpain B, Wollants P. A phase field model for the simulation of grain growth in materials containing finely dispersed incoherent second-phase particles. Acta Materialia 2005;53:1771.CrossRefGoogle Scholar
  77. Moelans N, Blanpain B, Wollants P. Quantitative analysis of grain boundary properties in a generalized phase field model for grain growth in anisotropic systems. Physical Review B 2008;78:024113.CrossRefGoogle Scholar
  78. Onuki A. Ginzburg-Landau approach to elastic effects in the phase separation of solids. Journal of the Physical Society of Japan 1989;58:3065.CrossRefGoogle Scholar
  79. Poduri R, Chen LQ. Computer simulation of morphological evolution and coarsening kinetics of δ (Al3Li) precipitates in Al-Li alloys. Acta Materialia 1998;46:3915.CrossRefGoogle Scholar
  80. Porter DA, Easterling KE. Phase Transformation in Metals and Alloys. New York: Van Nostrand Reinhold, 1981.Google Scholar
  81. Provatas N, Greenwood M. Multiscale modeling of solidification: phase-field methods to adaptive mesh refinement. International Journal of Modern Physics B 2005;19:4525.CrossRefGoogle Scholar
  82. Provatas N, Goldenfeld N, Dantzig J. Efficient computation of dendritic microstructures using adaptive mesh refinement. Physical Review Letters 1998;80:3308.CrossRefGoogle Scholar
  83. Raabe D. Computational Materials Science: The Simulation of Materials Microstructures and Properties. Weinheim: Wiley-VCH Verlag GmbH, 1998.Google Scholar
  84. Read W, Shockley W. Dislocation models of crystal grain boundaries. Physical Review 1950;78:275.MATHCrossRefGoogle Scholar
  85. Saunders N, Miodownik AP. CALPHAD (Calculation of Phase Diagrams): A Comprehensive Guide. Oxford, New York: Pergamon, 1998.Google Scholar
  86. Searles T, Tiley J, Tanner A. Rapid characterization of titanium microstructural features for specific modelling of mechanical properties. Measurement Science and Technology 2005;16:60.CrossRefGoogle Scholar
  87. Semiatin SL, Scoper JC, Sukonnik IM. Short-time beta grain growth kinetics for a conventional titanium alloy. Acta Materialia 1996;44:1979.CrossRefGoogle Scholar
  88. Semiatin SL, Fagin PN, Glavicic MG, Sukonnik IM, Ivasishin OM. Materials Science and Engineering A 2001;299:225.CrossRefGoogle Scholar
  89. Shen C, Wang Y. Coherent precipitation – phase field method. In: Yip S, editor. Handbook of Materials Modeling, Part B: Models. New York: Springer, 2005. p. 2117.Google Scholar
  90. Shen C, Wang Y. “Phase field microstructure modeling,” in Fundamentals of Modeling for Materials Processing, ASM Handbook, Volume 22A, Eds. D. Furrer and S.L. Semiatin, TMS (2010).Google Scholar
  91. Shen C, Chen Q, Wen YH, Simmons JP, Wang Y. Increasing length scale of quantitative phase field modeling of growth-dominant or coarsening-dominant process. Scripta Materialia 2004a;50:1023.CrossRefGoogle Scholar
  92. Shen C, Chen Q, Wen YH, Simmons JP, Wang Y. Increasing length scale of quantitative phase field modeling of concurrent growth and coarsening processes. Scripta Materialia 2004b;50:1029.CrossRefGoogle Scholar
  93. Simmons JP, Shen C, Wang Y. Phase field modeling of simultaneous nucleation and growth by explicit incorporating nucleation events. Scripta Materialia 2000;43:935.CrossRefGoogle Scholar
  94. Steinbach I, Pezzolla F. A generalized field method for multiphase transformations using interface fields. Physica D 1999;134:385.MATHMathSciNetCrossRefGoogle Scholar
  95. Steinbach I, Pezzolla F, Nestler B, Seesselberg M, Prieler R, Schmitz GJ, Rezende JLL. A phase field concept for multiphase systems. Physica D 1996;94:135.MATHCrossRefGoogle Scholar
  96. Stogner RH, Carey GF, Murray BT. Approximation of Cahn-Hilliard diffuse interface models using parallel adaptive mesh refinement and coarsening with C1 elements. International Journal for Numerical Methods in Engineering 2006;64:1.Google Scholar
  97. Sutton AP, Balluffi RW. Interfaces in Crystalline Material. New York: Oxford University Press, 1995.Google Scholar
  98. Suwa Y, Saito Y, Onodera H. Phase field simulation of grain growth in three dimensional system containing finely dispersed second-phase particles. Scripta Materialia 2006;55:407.CrossRefGoogle Scholar
  99. Tiaden J, Nestler B, Diepers HJ, Steinbach I. The multiphase-field model with an integrated concept for modelling solute diffusion. Physica D 1998;115:73.MATHCrossRefGoogle Scholar
  100. Uchic MD. 3-D microstructural characterization: Methods, analysis, and applications. JOM 2006;58:24.CrossRefGoogle Scholar
  101. Unocic R, Kovarik L, Shen C, Sarosi P, Wang Y, Li J, Ghosh S, Mills MJ. Deformation mechanisms in Ni-base disk superalloys at higher temperatures. In: Reed RC, Green KA, Caron P, Gabb TP, Fahrmann MG, Huron ES, Woodard SA, editors. Superalloys 2008. Warrendale, PA: TMS, 2008. p. 377.Google Scholar
  102. van der Waals JD. The thermodynamik theory of capillarity under the hypothesis of a continuous variation of density. Konink. Akad. Weten. Amsterdam (Sect. 1) 1893;1:56.(in Dutch) [English translation (with commentary): J. S. Rowlinson, J. Stat. Phys. 20, 197 (1979)].Google Scholar
  103. Vedantam S, Patnaik BS. Efficient numerical algorithm for multiphase field simulations. Physical Review E 2006;73:016703.CrossRefGoogle Scholar
  104. Wang YU. Computer modeling and simulation of solid-state sintering: a phase field approach. Acta Materialia 2006;54:953.CrossRefGoogle Scholar
  105. Wang Y, Chen LQ. Simulation of Microstructural Evolution Using the Field Method. Methods in Material Research. New York: Wiley, 2000. p. 2a.3.1Google Scholar
  106. Wang Y, Chen LQ, Khachaturyan AG. Modeling of dynamical evolution of micro/mesoscopic morphological patterns in coherent phase transformations. In: Kirchner HO, Kubin KP, Pontikis V, editors. Computer Simulation in Materials Science – Nano/Meso/Macroscopic Space and Time Scales. Dordrecht: Kluwer Academic Publishers, 1996. p. 325.Google Scholar
  107. Wang Y, Banerjee D, Su CC, Khachaturyan AG. Field kinetic model and computer simulation of precipitation of L12 ordered intermetallics from fcc solid solution. Acta Materialia 1998;46:2983.CrossRefGoogle Scholar
  108. Wang YU, Jin YM, Khachaturyan AG. Three-dimensional phase field microelasticity theory and modeling of multiple cracks and voids. Applied Physics Letters 2001;79:3071.CrossRefGoogle Scholar
  109. Wang YU, Jin YM, Khachaturyan AG. Phase field microelasticity theory and modeling of elastically and structurally inhomogeneous solid. Journal of Applied Physics 2002;92:1351.CrossRefGoogle Scholar
  110. Wang YU, Jin YM, Khachaturyan AG. Phase field microelasticity modeling of dislocation dynamics near free surface and in heteroepitaxial thin films. Acta Materialia 2003;51:4209.CrossRefGoogle Scholar
  111. Wang YU, Jin YM, Khachaturyan AG. Dislocation dynamics – phase field. In: Yip S, editor. Handbook of Materials Modeling, Part B: Models. New York: Springer, 2005a. p. 2287.Google Scholar
  112. Wang Y, Ma N, Chen Q, Zhang F, Chen SL, Chang YA. Predicting of phase equilibrium, phase transformation, and microstructure evolution in advanced titanium alloys. JOM 2005b;September:32.Google Scholar
  113. Warren JA, Kobayashi R, Lobkovsky AE, Carter WC, Sutton AP. Acta Materialia 2003;51:6035.CrossRefGoogle Scholar
  114. Wheeler AA, Boettinger WJ, McFadden GB. Phase-field model for isothermal phase transitions in binary alloys. Physical Review A 1992;45:7424.CrossRefGoogle Scholar
  115. Wu K, Zhou N, Pan X, Morral JE, Wang Y. Multiphase Ni-Cr-Al diffusion couple: a comparison of phase field simulations with experimental data. Acta Materialia 2008;56:3854.CrossRefGoogle Scholar
  116. Zhang F, Xie FY, Chen SL, Chang YA, Furrer D, Venkatesh V. Predictions of titanium alloy properties using thermodynamic modeling tools. Journal of Materials Engineering and Performance 2005;14:717.CrossRefGoogle Scholar
  117. Zhang F, Chen SL, Chang YA, Ma N, Wang Y. Development of thermodynamic description of a pseudo-ternary system for multicomponent Ti64 alloy. Journal of Phase Equilibria and Diffusion. 2007;28:115.CrossRefGoogle Scholar
  118. Zhang F, Yang Y, CaoWS, Chen SL,Wu K, Chang YA, Commercial Alloy Phase Diagrams and Their Industrial Applications, in ASM Handbook, Volume 22B, Modeling and Simulation: Processing of Metallic Materials, D.U. Furrer and S.L. Semiatin, editors. ASM International, 2010.Google Scholar
  119. Zhou N, Shen C, Mills MJ, Wang Y. Contributions from elastic inhomogeneity and fro plasticity to γ rafing in single-crystal Ni-Al. Acta Materialia 2008;56:6156.CrossRefGoogle Scholar
  120. Zhou N, Shen C, Mills MJ, Wang Y. to be submitted. 2010a.Google Scholar
  121. Zhou N, Shen C, Mills MJ, Wang Y. Large-scale Three-Dimensional Phase Field Simulation of γ Rafting and Creep Deformation. Philosophical Magazine, 2010b;90:405CrossRefGoogle Scholar
  122. Zhu J, Chen LQ, Shen J, Tikare V. Coarsening kinetics from a variable-mobility Cahn-Hilliard equation: Application of a semi-implicit Fourier spectral method. Physical Review E 1999;60:3564.CrossRefGoogle Scholar
  123. Zhu JZ, Liu ZK, Vaithyanathan V, Chen LQ. Linking phase-field model to CALPHAD: application to precipitate shape evolution in Ni-base alloys. Scripta Materialia 2002;46:401.CrossRefGoogle Scholar
  124. Zhu JZ, Wang T, Ardell AJ, Zhou SH, Liu ZK, Chen LQ. Three-dimensional phase-field simulations of coarsening kinetics of γ particles in binary Ni-Al alloys. Acta Materialia 2004;52:2837.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Chen Shen
  • Ning Ma
  • Yuwen Cui
  • Ning Zhou
  • Yunzhi Wang
    • 1
  1. 1.Department of Materials Science and EngineeringThe Ohio State UniversityColumbusUSA

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