Kinetic Monte Carlo Method to Model Diffusion Controlled Phase Transformations in the Solid State

  • Georges Martin
  • Frédéric Soisson


The classical theories of diffusion-controlled transformations in the solid state (precipitate-nucleation, -growth, -coarsening, order-disorder transformation, domain growth) imply several kinetic coefficients: diffusion coefficients (for the solute to cluster into nuclei, or to move from smaller to larger precipitates…), transfer coefficients (for the solute to cross the interface in the case of interface-reaction controlled kinetics) and ordering kinetic coefficients. If we restrict to coherent phase transformations, i.e., transformations, which occur keeping the underlying lattice the same, all such events (diffusion, transfer, ordering) are nothing but jumps of atoms from site to site on the lattice. Recent progresses have made it possible to model, by various techniques, diffusion controlled phase transformations, in the solid state, starting from the jumps of atoms on the lattice. The purpose of the present chapter is to introduce one of the techniques, the Kinetic Monte Carlo method (KMC).


Solute Atom Vacancy Concentration Face Center Cubic Kinetic Monte Carlo Jump Frequency 
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  1. [1]
    A.R. Allnatt and A.B. Lidiard, “Atomic transport in solids”, Cambridge University Press, Cambridge, 1994.Google Scholar
  2. [2]
    T. Morita, M. Suzuki, K. Wada, and M. Kaburagi, “Foundations and Applications of Cluster Variation Method and Path Probability Method”, Prog. Theor. Phys. Supp., 115, 1994.Google Scholar
  3. [3]
    K. Binder, “Applications of Monte Carlo methods to statistical physics”, Rep. Prog. Phys., 60, 1997.Google Scholar
  4. [4]
    Y. Limoge and J.-L. Bocquet, “Monte Carlo simulation in diffusion studies: time scale problems”, Acta Met., 36, 1717, 1988.CrossRefGoogle Scholar
  5. [5]
    G.E. Murch and L. Zhang, “Monte Carlo simulations of diffusion in solids: some recent developments”, In: A.L. Laskar et al. (eds.), Diffusion in Materials, Kluwer Academic Publishers, Dordrecht, 1990.Google Scholar
  6. [6]
    C.P. Flynn, “Point defects and diffusion”, Clarendon Press, Oxford, 1972.Google Scholar
  7. [7]
    J. Philibert, “Atom movements, diffusion and mass transport in solids”, Les Editions de Physique, Les Ulis, 1991.Google Scholar
  8. [8]
    D.N. Seidman and R.W. Balluffi, “Dislocations as sources and sinks for point defects in metals”, In: R.R. Hasiguti (ed.), Lattice Defects and their Interactions, Gordon-Breach, New York, 1968.Google Scholar
  9. [9]
    J.-L. Bocquet, G. Brebec, and Y. Limoge, “Diffusion in metals and alloys”, In: R.W. Cahn and P. Haasen (eds.), Physical Metallurgy, North-Holland, Amsterdam, 1996.Google Scholar
  10. [10]
    M. Nastar, V.Y Dobretsov, and G. Martin, “Self consistent formulation of configurational kinetics close to the equilibrium: the phenomenological coefficients for diffusion in crystalline solids”, Philos. Mag. A, 80, 155, 2000.CrossRefADSGoogle Scholar
  11. [11]
    G. Martin, “The theories of unmixing kinetics of solids solutions”, In: Solid State Phase Transformation in Metals and Alloys, pp. 337–406. Les Editions de Physique, Orsay, 1978.Google Scholar
  12. [12]
    A. Perini, G. Jacucci, and G. Martin, “Interfacial contribution to cluster free energy”, Surf. Sci., 144, 53, 1984.CrossRefADSGoogle Scholar
  13. [13]
    T.R. Waite, “Theoretical treatment of the kinetics of diffusion-limited reactions”, Phys. Rev., 107, 463–470, 1957.CrossRefADSGoogle Scholar
  14. [14]
    I.M. Lifshitz and V.V. Slyosov, “The kinetics of precipitation from supersaturated solid solutions”, Phys. Chem. Solids, 19, 35, 1961.CrossRefADSGoogle Scholar
  15. [15]
    C.J. Kuehmann and P.W. Voorhees, “Ostwald ripening in ternary alloys”, Metall. Mater Trans., 27A, 937–943, 1996.CrossRefGoogle Scholar
  16. [16]
    J.W. Cahn, W. Craig Carter, and W.C. Johnson (eds.), The selected works of J.W. Cahn., TMS, Warrendale, 1998.Google Scholar
  17. [17]
    G. Martin, “Atomic mobility in Cahn’s diffusion model”, Phys. Rev. B, 41, 2279–2283, 1990.CrossRefADSGoogle Scholar
  18. [18]
    C. Desgranges, F. Defoort, S. Poissonnet, and G. Martin, “Interdiffusion in concentrated quartenary Ag-In-Cd-Sn alloys: modelling and measurements”, Defect Diffus. For., 143, 603–608, 1997.CrossRefGoogle Scholar
  19. [19]
    S.M. Allen and J.W. Cahn, “A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening”, Acta Metal., 27, 1085–1095, 1979.CrossRefGoogle Scholar
  20. [20]
    P. Bellon and G. Martin, “Coupled relaxation of concentration and order fields in the linear regime”, Phys. Rev. B, 66, 184208, 2002.CrossRefADSGoogle Scholar
  21. [21]
    C. Pareige, R Soisson, G. Martin, and D. Blavette, “Ordering and phase separation in Ni-Cr-Al: Monte Carlo simulations vs Three-Dimensional atom probe”, Acta Mater., 47, 1889–1899, 1999.CrossRefGoogle Scholar
  22. [22]
    Y. Le Bouar and F. Soisson, “Kinetic pathways from EAM potentials: influence of the activation barriers”, Phys. Rev. B, 65, 094103, 2002.CrossRefADSGoogle Scholar
  23. [23]
    E. Clouet and N. Nastar, “Monte Carlo study of the precipitation of Al3Zr in Al-Zr”, Proceedings of the Third International Alloy Conference, Lisbon, in press, 2002.Google Scholar
  24. [24]
    J.-L. Bocquet, “On the fly evaluation of diffusional parameters during a Monte Carlo simulation of diffusion in alloys: a challenge”, Defect Diffus. For., 203–205, 81–112, 2002.Google Scholar
  25. [25]
    R. LeSar, R. Najafabadi, and D.J. Srolovitz, “Finite-temperature defect properties from free-energy minimization”, Phys. Rev. Lett., 63, 624–627, 1989.CrossRefADSGoogle Scholar
  26. [26]
    A.P. Sutton, “Temperature-dependent interatomic forces”, Philos. Mag., 60, 147–159, 1989.ADSGoogle Scholar
  27. [27]
    Y. Mishin, M.R. Sorensen, F. Arthur, and A.F. Voter, “Calculation of point-defect entropy in metals”, Philos. Mag. A, 81, 2591–2612, 2001.ADSGoogle Scholar
  28. [28]
    D. Gendt, Cinétiques de Précipitation du Carbure de Niobium dans la ferrite, CEA Report, 0429–3460, 2001.Google Scholar
  29. [29]
    M. Athènes, P. Bellon, and G. Martin, “Identification of novel diffusion cycles in B2 ordered phases by Monte Carlo simulations”, Philos. Mag. A, 76, 565–585, 1997.CrossRefADSGoogle Scholar
  30. [30]
    M. Athènes and P. Bellon, “Antisite diffusion in the L12 ordered structure studied by Monte Carlo simulations”, Philos. Mag. A, 79, 2243–2257, 1999.CrossRefGoogle Scholar
  31. [31]
    A. Athènes, P. Bellon, and G. Martin, “Effects of atomic mobilities on phase separation kinetics: a Monte Carlo study”, Acta Mater., 48, 2675, 2000.CrossRefGoogle Scholar
  32. [32]
    R. Wagner and R. Kampmann, “Homogeneous second phase precipitation”, In: P. Haasen (ed.), Phase Transformations in Materials, VCH, Weinhem, 1991.Google Scholar
  33. [33]
    F. Soisson, A. Barbu, and G. Martin, “Monte Carlo simulations of copper precipitation in dilute iron-copper alloys during thermal ageing and under electron irradiation”, Acta Mater., 44, 3789, 1996.CrossRefGoogle Scholar
  34. [34]
    P. Auger, P. Pareige, M. Akamatsu, and D. Blavette, “APFIM investigation of clustering in neutron irradiated Fe-Cu alloys and pressure vessel steels”, J. Nucl. Mater., 225, 225–230, 1995.CrossRefADSGoogle Scholar
  35. [35]
    P. Fratzl and O. Penrose, “Kinetics of spinodal decomposition in the Ising model with vacancy diffusion”, Phys. Rev. B, 50, 3477–3480, 1994.CrossRefADSGoogle Scholar
  36. [36]
    J.-M. Roussel and P. Bellon, “Vacancy-assisted phase separation with asymmetric atomic mobility: coarsening rates, precipitate composition and morphology”, Phys. Rev. B, 63, 184114, 2001.CrossRefADSGoogle Scholar
  37. [37]
    F. Soisson and G. Martin, Phys. Rev. B, 62, 203, 2000.CrossRefADSGoogle Scholar
  38. [38]
    E. Clouet, M. Nastar, and C. Sigli, “Nucleation of Al3Zr and Al3Sc in aluminiun alloys: from kinetic Monte Carlo simulations to classical theory”, Phys. Rev. B, 69, 064109, 2004.CrossRefADSGoogle Scholar
  39. [39]
    M. Athènes, P. Bellon, G. Martin, and R Haider, “A Monte Carlo study of B2 ordering and precipitation via vacancy mechanism in BCC lattices”, Acta Mater., 44, 4739–4748, 1996.CrossRefGoogle Scholar
  40. [40]
    G. Martin and P. Bellon, “Driven alloys”, Solid State Phys., 50, 189, 1997.CrossRefGoogle Scholar
  41. [41]
    R.A. Enrique and P. Bellon, “Compositional patterning in immiscible alloys driven by irradiation”, Phys. Rev. B, 63, 134111, 2001.CrossRefADSGoogle Scholar
  42. [42]
    C.H. Lam, C.K. Lee, and L.M. Sander, “Competing roughening mechanisms in strained heteroepitaxy: a fast kinetic Monte Carlo study”, Phys. Rev. Lett., 89, 216102, 2002.CrossRefADSGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Georges Martin
    • 1
  • Frédéric Soisson
    • 2
  1. 1.Commissariat à lÉnergie Atomique, Cab. H.C.Paris Cedex, 15France
  2. 2.CEA Saclay, DMN-SRMPGif-sur-YuetteFrance

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