Phase Separation in Binary Alloys - Modeling Approaches

  • Peter Fratzl
  • Richard Weinkamer
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 453)


The physical principles and modern modeling approaches for phase separation in binary alloys are reviewed. A fundamental distinction between the different simulation models is their description of the moving interface during the separation process. The interface between the phases is either atomically rough, or a diffuse interface with a smooth transition between the phases, or finally a sharp geometric interface with a discontinuous jump of the bulk properties of one phase to the other. Advantages and disadvantages of the these microscopic, mesoscopic and macroscopic modeling approaches are presented. Concerning examples given some emphasis is put on the influence of elastic interactions caused by a misfit between the phases on the transformation process.


Phase Separation Binary Alloy Phase Field Near Neighbor Spinodal Decomposition 
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6. References

  1. Abinandanan, T. A., Haider, F. and Martin, G. (1998). Computer simulations of diffusional phase transformations: Monte Carlo algorithm and application to precipitation of ordered phases. Acta Materialia 46: 4243–4255.CrossRefGoogle Scholar
  2. Aichmayer, B., Fratzl, P., Puri, S. and Sailer, G. (2003). Surface-directed spinodal decomposition on a macroscopic scale in a nitrogen and carbon alloyed steel. Physical Review Letters 91: 015701/1–4.CrossRefGoogle Scholar
  3. Akaiwa, N., Thornton, K. and Voorhees, P. W. (2001). Large-scale simulations of microstructural evolution in elastically stressed solids. Journal of Computational Physics 173:61–86.MATHCrossRefGoogle Scholar
  4. Apel, M., Boettger, B., Diepers, H. J. and Steinbach, I. (2002). 2D and 3D phase-field simulations of lamella and fibrous eutectic growth. Journal of Crystal Growth 237: 154–158.CrossRefGoogle Scholar
  5. Asta, M. and Foiles, S. M. (1996). Embedded-atom-method effective-pair-interaction study of the structural and thermodynamic properties of Cu-Ni, Cu-Ag, and Au-Ni solid solutions. Physical Review B 53: 2389–2404.CrossRefGoogle Scholar
  6. Athenes, M., Bellon, P. and Martin, G. (1997). Identification of novel diffusion cycles in B2 ordered phases by Monte Carlo simulation. Philosophical Magazine a-Physics of Condensed Matter Structure Defects and Mechanical Properties 76: 565–585.Google Scholar
  7. Athenes, M., Bellon, P. and Martin, G. (2000). Effects of atomic mobilities on phase separation kinetics: A Monte-Carlo study. Acta Materialia 48: 2675–2688.CrossRefGoogle Scholar
  8. Athenes, M., Bellon, P., Martin, G. and Haider, F. (1996). A Monte-Carlo study of B2 ordering and precipitation via vacancy mechanism in bcc lattices. Acta Materialia 44: 4739–4748.CrossRefGoogle Scholar
  9. Becker, R. (1938). Die Keimbildung bei der Ausscheidung in metallischen Mischkristallen. Annalen der Physik 32: 128–140.Google Scholar
  10. Bellon, P. (2003). Kinetic Monte Carlo simulations in crystalline alloys: principles and selected applications. In Finel, A. e. a., Eds. NATO Sci. Ser. II Math., Thermodynamics, Microstructures and Plasticity, Kluwer Academic. 395–409.Google Scholar
  11. Binder, K. (1997). Applications of Monte Carlo methods to statistical physics. Reports on Progress in Physics 60: 487–559.CrossRefGoogle Scholar
  12. Binder, K. (1998). Spinodal decomposition in confined geometry. Journal of Non-Equilibrium Thermodynamics 23: 1–44.MATHCrossRefGoogle Scholar
  13. Binder, K., Billotet, C. and Mirold, P. (1978). Theory of Spinodal Decomposition in Solid and Liquid Binary-Mixtures. Zeitschrift Fur Physik B-Condensed Matter 30: 183–195.Google Scholar
  14. Binder, K. and Fratzl, P. (2001). Spinodal decomposition. In Kostorz, G., Eds. Phase Transformations in Materials, Wiley-VCH. Chapter 6, 409–480.Google Scholar
  15. Binder, K. and Heermann, D. W. (2002). Monte Carlo simulation in statistical physics: an introduction. Berlin; New York: Springer.MATHGoogle Scholar
  16. Binder, K. and Stauffer, D. (1974). Theory for Slowing Down of Relaxation and Spinodal Decomposition of Binary-Mixtures. Physical Review Letters 33: 1006–1009.CrossRefGoogle Scholar
  17. Blaschko, O. and Fratzl, P. (1983). Experimental-Observation of a Time-Scaling Characteristic in Alloy Decomposition in the Alznmg System. Physical Review Letters 51: 288–291.CrossRefGoogle Scholar
  18. Boettinger, W. J., Warren, J. A., Beckermann, C. and Karma, A. (2002). Phase-field simulation of solidification. Annual Review of Materials Research 32: 163–194.CrossRefGoogle Scholar
  19. Bortz, A. B., Kalos, M. H. and Lebowitz, J. L. (1975). New Algorithm for Monte-Carlo Simulation of Ising Spin Systems. Journal of Computational Physics 17: 10–18.CrossRefGoogle Scholar
  20. Bortz, A. B., Kalos, M. H., Lebowitz, J. L. and Zendejas, M. A. (1974). Time Evolution of a Quenched Binary Alloy-Computer-Simulation of a 2-Dimensional Model System. Physical Review B 10: 535–541.CrossRefGoogle Scholar
  21. Cahn, J. W. (1961). On Spinodal Decomposition. Acta Metallurgica 9: 795–801.CrossRefGoogle Scholar
  22. Cahn, J. W. (1968). 1967 Institute of Metals Lecture Spinodal Decomposition. Transactions of the Metallurgical Society of Aime 242: 166-&.Google Scholar
  23. Cahn, J. W. and Hilliard, J. E. (1958). Free Energy of a Nonuniform System. 1. Interfacial Free Energy. Journal of Chemical Physics 28: 258–267.CrossRefGoogle Scholar
  24. Cahn, J. W. and Hilliard, J. E. (1959). Free Energy of a Nonuniform System. 3. Nucleation in a 2-Component Incompressible Fluid. Journal of Chemical Physics 31: 688–699.CrossRefGoogle Scholar
  25. Cahn, R. W. and Haasen, P. (1983). Physical metallurgy. Amsterdam: North-Holland Physics Publishing.Google Scholar
  26. Chandler, D. (1987). Introduction to modern statistical mechanics. New York: Oxford University Press.Google Scholar
  27. Chen, L. Q. (2002). Phase-field models for microstructure evolution. Annual Review of Materials Research 32: 113–140.CrossRefGoogle Scholar
  28. Chen, L. Q. and Khachaturyan, A. G. (1991). Computer-Simulation of Structural Transformations During Precipitation of an Ordered Intermetallic Phase. Acta Metallurgica Et Materialia 39: 2533–2551.CrossRefGoogle Scholar
  29. Chen, L. Q., Wolverton, C., Vaithyanathan, V. and Liu, Z. K. (2001). Modeling solid-state phase transformations and microstructure evolution. Mrs Bulletin 26: 197–202.Google Scholar
  30. Cook, H. E. (1970). Brownian Motion in Spinodal Decomposition. Acta Metallurgica 18: 297-&.CrossRefGoogle Scholar
  31. Cook, H. E. and De Fontaine, D. (1969). On Elastic Free Energy of Solid Solutions. I. Microscopic Theory. Acta Metallurgica 17: 915–924.CrossRefGoogle Scholar
  32. Eshelby, J. D. (1957). The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems. Proceedings of the Royal Society of London Series a-Mathematical and Physical Sciences 241: 376–396.MATHMathSciNetGoogle Scholar
  33. Eshelby, J. D. (1959). The Elastic Field Outside an Ellipsoidal Inclusion. Proceedings of the Royal Society of London Series a-Mathematical and Physical Sciences 252: 561–569.MATHMathSciNetGoogle Scholar
  34. Fan, D. and Chen, L. Q. (1997). Computer simulation of grain growth using a continuum field model. Acta Materialia 45: 611–622.CrossRefGoogle Scholar
  35. Fischer, D. and Nielaba, P. (2000). Phase diagram of a model alloy with lattice misfit. Physica a-Statistical Mechanics and Its Applications 279: 287–295.CrossRefGoogle Scholar
  36. Fischer, F. D. and Simha, N. K. (2004). Thermodynamics and kinetics of phase and twin boundaries. In Eds. This book ##, Udine: CISM.Google Scholar
  37. Fratzl, P. (2003). Small-angle scattering in materials science-a short review of applications in alloys, ceramics and composite materials. Journal of Applied Crystallography 36: 397–404.CrossRefGoogle Scholar
  38. Fratzl, P., Lebowitz, J. L., Penrose, O. and Amar, J. (1991). Scaling Functions, Self-Similarity, and the Morphology of Phase-Separating Systems. Physical Review B 44: 4794–4811.CrossRefGoogle Scholar
  39. Fratzl, P. and Penrose, O. (1994). Kinetics of Spinodal Decomposition in the Ising-Model with Vacancy Diffusion. Physical Review B 50: 3477–3480.CrossRefGoogle Scholar
  40. Fratzl, P. and Penrose, O. (1995). Ising-Model for Phase-Separation in Alloys with Anisotropic Elastic Interaction. 1. Theory. Acta Metallurgica Et Materialia 43: 2921–2930.CrossRefGoogle Scholar
  41. Fratzl, P. and Penrose, O. (1996). Ising model for phase separation in alloys with anisotropic elastic interaction. 2. A computer experiment. Acta Materialia 44: 3227–3239.CrossRefGoogle Scholar
  42. Fratzl, P. and Penrose, O. (1997). Competing mechanisms for precipitate coarsening in phase separation with vacancy dynamics. Physical Review B 55: R6101–R6104.CrossRefGoogle Scholar
  43. Fratzl, P., Penrose, O. and Lebowitz, J. L. (1999). Modeling of phase separation in alloys with coherent elastic misfit. Journal of Statistical Physics 95: 1429–1503.MATHMathSciNetCrossRefGoogle Scholar
  44. Fratzl, P., Penrose, O., Weinkamer, R. and Zizak, I. (2000). Coarsening in the Ising model with vacancy dynamics. Physica A 279: 100–109.CrossRefGoogle Scholar
  45. Gerold, V. and Kern, J. (1987). The Determination of Atomic Interaction Energies in Solid-Solutions from Short-Range Order Coefficients-an Inverse Monte-Carlo Method. Acta Metallurgica 35: 393–399.CrossRefGoogle Scholar
  46. Gunton, J. D. (1999). Homogeneous nucleation. Journal of Statistical Physics 95: 903–923.MATHCrossRefGoogle Scholar
  47. Gupta, H., Weinkamer, R., Fratzl, P. and Lebowitz, J. L. (2001). Microscopic computer simulations of directional coarsening in face-centered cubic alloys. Acta Materialia 49: 53–63.CrossRefGoogle Scholar
  48. Heermann, D. W., Li, Y. X. and Binder, K. (1996). Scaling solutions and finite-size effects in the Lifshitz-Slyozov theory. Physica A 230: 132–148.CrossRefGoogle Scholar
  49. Hu, S. Y. and Chen, L. Q. (2001). Solute segregation and coherent nucleation and growth near a dislocation-A phase-field model integrating defect and phase microstructures. Acta Materialia 49: 463–472.CrossRefGoogle Scholar
  50. Huse, D. A. (1986). Corrections to Late-Stage Behavior in Spinodal Decomposition-Lifshitz-Slyozov Scaling and Monte-Carlo Simulations. Physical Review B 34: 7845–7850.CrossRefGoogle Scholar
  51. Ikeda, H. and Matsuda, H. (1993). Effects of Difference in Elastic-Moduli between Constituents on Spinodal Decomposition Processes. Materials Transactions Jim 34: 651–657.Google Scholar
  52. Inden, G. (2001). Atomic odering. In Kostorz, G., Eds. Phase Transformations in Materials, Wiley-VCH. Chapter 8, 519–581.Google Scholar
  53. Kampmann, R. and Wagner, R. (1984). In Haasen, P., Gerold, V., Wagner, R. and Ashby, M. F., Eds. Decomposition of Alloys: the Early Stages, Oxford: Pergamon Press. 91–103.Google Scholar
  54. Kantelhardt, J. W., Koscielny-Bunde, E., Rego, H. H. A., Havlin, S. and Bunde, A. (2001). Detecting long-range correlations with detrended fluctuation analysis. Physica A 295: 441–454.MATHCrossRefGoogle Scholar
  55. Khachaturyan, A. G. (1965). On Theory of Modulated Structures in Binary Solid Solutions. Soviet Physics Crystallography, Ussr 10: 248-&.Google Scholar
  56. Khachaturyan, A. G. (1983). Theory of structural transformations in solids. New York: Wiley.Google Scholar
  57. Kikuchi, R. and Sato, H. (1972). Diffusion-Coefficient in an Ordered Binary Alloy. Journal of Chemical Physics 57: 4962–4979.CrossRefGoogle Scholar
  58. Laberge, C. A., Fratzl, P. and Lebowitz, J. L. (1995). Elastic Effects on Phase Segregation in Alloys with External Stresses. Physical Review Letters 75: 4448–4451.CrossRefGoogle Scholar
  59. Laberge, C. A., Fratzl, P. and Lebowitz, J. L. (1997). Microscopic model for directional coarsening of precipitates in alloys under external load. Acta Materialia 45: 3949–3962.CrossRefGoogle Scholar
  60. Langer, J. S. and Schwartz, A. J. (1980). Kinetics of Nucleation in near-Critical Fluids. Physical Review A 21: 948–958.MathSciNetCrossRefGoogle Scholar
  61. Lebowitz, J. L., Marro, J. and Kalos, M. H. (1982). Dynamical Scaling of Structure-Function in Quenched Binary-Alloys. Acta Metallurgica 30: 297–310.CrossRefGoogle Scholar
  62. Lee, J. K. (1998). Elastic stress and microstructural evolution. Materials Transactions Jim 39: 114–132.Google Scholar
  63. Lengeler, B. (2001). Coherence in X-ray physics. Naturwissenschaften 88: 249–260.CrossRefGoogle Scholar
  64. Lifshitz, I. M. and Slyozov, V. V. (1961). The Kinetics of Precipitation from Supersaturated Solid Solutions. Journal of Physics and Chemistry of Solids 19: 35–50.CrossRefGoogle Scholar
  65. Lochte, L., Gitt, A., Gottstein, G. and Hurtado, I. (2000). Simulation of the evolution of GP zones in Al-Cu alloys: An extended Cahn-Hilliard approach. Acta Materialia 48: 2969–2984.CrossRefGoogle Scholar
  66. Marro, J., Bortz, A. B., Kalos, M. H. and Lebowitz, J. L. (1975). Time Evolution of a Quenched Binary Alloy. 2. Computer-Simulation of a 3-Dimensional Model System. Physical Review B 12:2000–2011.CrossRefGoogle Scholar
  67. Marro, J., Lebowitz, J. L. and Kalos, M. H. (1979). Computer-Simulation of the Time Evolution of a Quenched Model Alloy in the Nucleation Region. Physical Review Letters 43: 282–285.CrossRefGoogle Scholar
  68. Mirold, P. and Binder, K. (1977). Theory for Initial-Stages of Grain-Growth and Unmixing Kinetics of Binary-Alloys. Acta Metallurgica 25: 1435–1444.CrossRefGoogle Scholar
  69. Nielaba, P., Fratzl, P. and Lebowitz, J. L. (1999). Growth of ordered domains in a computer model alloy with lattice misfit. Journal of Statistical Physics 95: 23–43.MATHCrossRefGoogle Scholar
  70. Nishimori, H. and Onuki, A. (1990). Pattern-Formation in Phase-Separating Alloys with Cubic Symmetry. Physical Review B 42: 980–983.CrossRefGoogle Scholar
  71. Novotny, M. A. (1995). A new approach to an old algorithm for the simulation of Ising-like systems. Computers in Physics 9: 46–52.CrossRefGoogle Scholar
  72. Onsager, L. (1944). Crystal statistics. I. A two-dimensional model with an order-disorder transition. Physical Review 65: 117–149.MATHMathSciNetCrossRefGoogle Scholar
  73. Onuki, A. and Nishimori, H. (1991). Anomalously Slow Domain Growth Due to a Modulus Inhomogeneity in Phase-Separating Alloys. Physical Review B 43: 13649–13652.CrossRefGoogle Scholar
  74. Orlikowski, D., Sagui, C., Somoza, A. and Roland, C. (1999). Large-scale simulations of phase separation of elastically coherent binary alloy systems. Physical Review B 59: 8646–8659.CrossRefGoogle Scholar
  75. Orlikowski, D., Sagui, C., Somoza, A. M. and Roland, C. (2000). Two-and three-dimensional simulations of the phase separation of elastically coherent binary alloys subject to external stresses. Physical Review B 62: 3160–3168.CrossRefGoogle Scholar
  76. Pareige, C., Soisson, F., Martin, G. and Blavette, D. (1999). Ordering and phase separation in Ni-Cr-Al: Monte Carlo simulations vs three-dimensional atom probe. Acta Materialia 47: 1889–1899.CrossRefGoogle Scholar
  77. Paris, O., Fahrmann, M., Fahrmann, E., Pollock, T. M. and Fratzl, P. (1997). Early stages of precipitate rafting in a single crystal Ni-Al-Mo model alloy investigated by small-angle X-ray scattering and TEM. Acta Materialia 45: 1085–1097.CrossRefGoogle Scholar
  78. Porta, M., Vives, E. and Castan, T. (1999). Vacancy-assisted domain growth in asymmetric binary alloys: A Monte Carlo study. Physical Review B 60: 3920–3927.CrossRefGoogle Scholar
  79. Press, W. H. (1997). Numerical recipes in C: the art of scientific computing. Cambridge Cambridgeshire; New York: Cambridge University Press.Google Scholar
  80. Raabe, D. (1998). Computational materials science-the simulation of materials microstructures and properties. Weinheim: Wiley-VCH.Google Scholar
  81. Rao, M., Kalos, M. H., Lebowitz, J. L. and Marro, J. (1976). Time Evolution of a Quenched Binary Alloy. 3. Computer-Simulation of a 2-Dimensional Model System. Physical Review B 13:4328–4335.CrossRefGoogle Scholar
  82. Rautiainen, T. T. and Sutton, A. P. (1999). Influence of the atomic diffusion mechanism on morphologies, kinetics, and the mechanisms of coarsening during phase separation. Phys Rev B 59: 13681–13692.CrossRefGoogle Scholar
  83. Roussel, J. M. and Bellon, P. (2001). Vacancy-assisted phase separation with asymmetric atomic mobility: Coarsening rates, precipitate composition, and morphology. Physical Review B 6318: art. no.-184114.Google Scholar
  84. Rubin, G. and Khachaturyan, A. G. (1999). Three-dimensional model of precipitation of ordered intermetallics. Acta Materialia 47: 1995–2002.CrossRefGoogle Scholar
  85. Sagui, C., Orlikowski, D., Somoza, A. M. and Roland, C. (1998). Three-dimensional simulations of Ostwald ripening with elastic effects. Physical Review E 58: R4092–R4095.CrossRefGoogle Scholar
  86. Schweika, W. (1998). Disordered alloys: diffuse scattering and Monte Carlo simulations. Berlin; New York: Springer.MATHGoogle Scholar
  87. Socrate, S. and Parks, D. M. (1993). Numerical Determination of the Elastic Driving Force for Directional Coarsening in Ni-Superalloys. Acta Metallurgica Et Materialia 41: 2185–2209.CrossRefGoogle Scholar
  88. Soisson, F. and Martin, G. (2000). Monte Carlo simulations of the decomposition of metastable solid solutions: Transient and steady-state nucleation kinetics. Physical Review B 62: 203–214.CrossRefGoogle Scholar
  89. Stadler, L.-M., Sepiol, B., Weinkamer, R., Hartmann, M., Fratzl, P., Kantelhardt, J. W., Zotone, F., Grübel, G. and Vogl, G. (2003). Long-term correlations distinguish coarsening mechanisms in alloys. Physical Review B 68: 180101R-1–4.CrossRefGoogle Scholar
  90. Sur, A., Lebowitz, J. L., Marro, J. and Kalos, M. H. (1977). Time Evolution of a Quenched Binary Alloy. 4. Computer-Simulation of a 3-Dimensional Model System. Physical Review B 15:3014–3026.CrossRefGoogle Scholar
  91. Thornton, K., Agren, J. and Voorhees, P. W. (2003). Modelling the evolution of phase boundaries in solids at the meso-and nanoscales. Acta Materialia 51: 5675–5710.CrossRefGoogle Scholar
  92. Thornton, K., Akaiwa, N. and Voorhees, P. W. (2001). Dynamics of late-stage phase separation in crystalline solids. Phys Rev Lett 86: 1259–1262.CrossRefGoogle Scholar
  93. Vaithyanathan, V. and Chen, L. Q. (2000). Coarsening kinetics of delta’-A13Li precipitates: Phase-field simulation in 2D and 3D. Scripta Materialia 42: 967–973.CrossRefGoogle Scholar
  94. Vaithyanathan, V., Wolverton, C. and Chen, L. Q. (2002). Multiscale modeling of precipitate microstructure evolution. Physical Review Letters 88:-.Google Scholar
  95. Vives, E. and Planes, A. (1993). Ordering Kinetics by Vacancies. International Journal of Modern Physics C-Physics and Computers 4: 701–720.CrossRefGoogle Scholar
  96. Voorhees, P. W. (1985). The Theory of Ostwald Ripening. Journal of Statistical Physics 38: 231–252.CrossRefGoogle Scholar
  97. Voorhees, P. W. (1992). Ostwald Ripening of 2-Phase Mixtures. Annual Review of Materials Science 22: 197–215.CrossRefGoogle Scholar
  98. Wagner, C. (1961). Theorie der Alterung von Niederschlägen durch Umlösen (Ostwald Reifung). Zeitschrift für Elektrochemie 65: 581–591.Google Scholar
  99. Wagner, R., Kampmann, R. and Voorhees, P. W. (2001). Homogeneous second-phase precipitation. In Kostorz, G., Eds. Phase Transformations in Materials, Wiley-VCH. Chapter 5, 309–407.Google Scholar
  100. Wang, Y., Chen, L. Q. and Khachaturyan, A. G. (1993). Kinetics of Strain-Induced Morphological Transformation in Cubic Alloys with a Miscibility Gap. Acta Metallurgica Et Materialia 41: 279–296.CrossRefGoogle Scholar
  101. Wang, Y., Chen, L. Q. and Khachaturyan, A. G. (1996). Modeling of dynamical evolution of micro/mesoscopic morphological patterns in coherent phase transformations. In Kirchner, H. O. and al., e., Eds. Computer Simulation in Materials Science, Kluwer Academic Publishers. 325–371.Google Scholar
  102. Wang, Y. and Khachaturyan, A. G. (1997). Three-dimensional field model and computer modeling of martensitic transformations. Acta Materialia 45: 759–773.CrossRefGoogle Scholar
  103. Wang, Y. U., Jin, Y. M., Cuitino, A. M. and Khachaturyan, A. G. (2001). Nanoscale phase field microelasticity theory of dislocations: Model and 3D simulations. Acta Materialia 49: 1847–1857.CrossRefGoogle Scholar
  104. Wang, Y. U., Jin, Y. M. M. and Khachaturyan, A. G. (2003). Phase field microelasticity modeling of dislocation dynamics near free surface and in heteroepitaxial thin films. Acta Materialia 51: 4209–4223.CrossRefGoogle Scholar
  105. Weinkamer, R. and Fratzl, P. (2003). By which mechanism does coarsening in phase-separating alloys proceed? Europhysics Letters 61:261–267.CrossRefGoogle Scholar
  106. Weinkamer, R., Fratzl, P., Gupta, H., Penrose, O. and Lebowitz, J. L. (2004). Using kinetic Monte Carlo simulations to study phase separation in alloys. Phase Transitions submittedGoogle Scholar
  107. Weinkamer, R., Fratzl, P., Sepiol, B. and Vogl, G. (1998). Monte Carlo simulation of diffusion in a B2-ordered model alloy. Physical Review B 58: 3082–3088.CrossRefGoogle Scholar
  108. Weinkamer, R., Fratzl, P., Sepiol, B. and Vogl, G. (1999). Monte Carlo simulations of Mossbauer spectra in diffusion investigations. Physical Review B 59: 8622–8625.CrossRefGoogle Scholar
  109. Weinkamer, R., Gupta, H., Fratzl, P. and Lebowitz, J. L. (2000). Dynamics of mesoscopic precipitate lattices in phase-separating alloys under external load. Europhysics Letters 52: 224–230.CrossRefGoogle Scholar
  110. Wolverton, C., Zunger, A. and Schonfeld, B. (1997). Invertible and non-invertible alloy Ising problems. Solid State Communications 101: 519–523.CrossRefGoogle Scholar
  111. Yaldram, K. and Binder, K. (1991). Monte-Carlo Simulation of Phase-Separation and Clustering in the Abv Model. Journal of Statistical Physics 62: 161–175.CrossRefGoogle Scholar
  112. Yaldram, K. and Binder, K. (1991). Spinodal Decomposition of a 2-Dimensional Model Alloy with Mobile Vacancies. Acta Metallurgica Et Materialia 39: 707–717.CrossRefGoogle Scholar
  113. Yaldram, K. and Binder, K. (1991). Unmixing of Binary-Alloys by a Vacancy Mechanism of Diffusion-a Computer-Simulation. Zeitschrift Fur Physik B-Condensed Matter 82: 405–418.CrossRefGoogle Scholar
  114. Zhu, J. Z., Liu, Z. K., Vaithyanathan, V. and Chen, L. Q. (2002). Linking phase-field model to CALPHAD: application to precipitate shape evolution in Ni-base alloys. Scripta Materialia 46: 401–406.CrossRefGoogle Scholar

Copyright information

© CISM, Udine 2004

Authors and Affiliations

  • Peter Fratzl
    • 1
  • Richard Weinkamer
    • 1
  1. 1.Department of BiomaterialsMax-Planck Institute of Colloids and InterfacesPotsdamGermany

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