Abstract
Kinetic Monte-Carlo (KMC) methods are used as an approach to simulate precipitation in Cu-alloyed bcc Fe. In order to characterize the process, transformed fractions, that is, the precipitated atoms, are related to Johnson-Mehl-Avrami-Kolmogorov theory. However, simulated data often deviate from corresponding fit curves and so does the resulting growth exponent when compared to theoretical expectations. Furthermore, some data may suggest the development of a metastable phase. In our study, we show that the characteristics of the transformed fraction and, as a consequence, the derived growth exponents sensitively depend on the number of atoms that are considered to form a particle and hence contribute to the transformed fraction. With a temperature dependence of the critical cluster size and additionally accounting for severe impingement of the particles, we obtain growth exponents which lie close to the expected range between n = 1.5 and n = 2.5 for pre-existing nuclei or continuous nucleation, respectively. From these, we obtain activation energies for nucleation and growth of precipitates. In this way, atomistic KMC simulations yield thermodynamical quantities, which can be valuable input parameters for larger length scale simulation methods, for example, for Phase Field Method simulations.
Similar content being viewed by others
References
Willer, D., Zies, G., Kuppler, D., Föhl, J., Katerbau, K.-H.: Service-Induced Changes of the Properties of Copper-Containing Ferritic Pressure-Vessel and Piping Steels. Final Report, MPA Stuttgart (2001)
Pan, F., Ruoff, H., Willer, D., Katerbau, K.-H.: Investigation on the Microstructure of a Pressure Vessel/Pipe Steel 15 NiCuMoNb 5 (WB36). MPA Report, Stuttgart (1996)
Schick, M., Wiedemann, J., Willer, D.: Untersuchungen zur sicherheitstechnischen Bewertung von geschweissten Komponenten aus Werkstoff 15 NiCuMoNb 5 (WB36) im Hinblick auf die Zähigkeitsabnahme unter Betriebsbeanspruchung, Technischer Bericht, MPA Stuttgart (1997)
Soisson F., Barbu A., Martin G.: Monte Carlo simulations of copper precipitation in dilute iron-copper alloys during thermal ageing and under electron irradiation. Acta Mater. 44, 3789–3800 (1996)
Binkele, P.: Atomistische Modellierung und Computersimulation der Ostwald-Reifung von Ausscheidungen beim Einsatz von kupferhaltigen Stählen. PhD thesis, University of Stuttgart (2006)
Johnson W.A., Mehl R.F.: Reaction kinetics in processes of nucleation and growth. Trans. Am. Inst. Min. Metal. Pet. Eng. 135, 416–442 (1939)
Avrami M.: Kinetics of phase change I—general theory. J. Chem. Phys. 7, 1103–1112 (1939)
Kolmogorov A.E.: Ivz. Akad. Nauk SSR Ser. Fiz. Mat. Nauk 1, 355 (1937)
Mittemeijer, E.J.: Fundamentals of Materials Science. p. 442 Springer, Heidelberg (2011)
Soisson F., Fu C.-C.: Cu-precipitation kinetics in α-Fe from atomistic simulations: vacancy-trapping effects and Cu-cluster mobility. Phys. Rev. B 76, 214102 (2007)
Claudio D., Gonzalez-Hernandez J., Licea O., Laine B., Prokhorov E., Trapaga G.: An analytical model to represent crystallization kinetics in materials with metastable phase formation. J. Non-Cryst. Solids 352, 51–55 (2007)
Robson J.D., Bhadeshia H.K.D.H.: Modelling precipitation sequences in power plant steels Part 1—kinetic theory. Mater. Sci. Technol. 13, 631 (1997)
Pizzini S., Roberts K.J., Phythian W.J., English C.A., Greaves G.N.: A fluorescence EXAFS study of the structure of copper-rich precipitates in Fe-Cu and Fe-Cu-Ni alloys. Philos. Mag. Lett. 61, 223–229 (1990)
Othen P.J., Jenkins M.L., Smith G.D.W.: High-resolution electron-microscopy studies of the structure of Cu precipitates in α-Fe. Philos. Mag. A 70, 1–24 (1994)
Bauer, R., Rheingans, B.F., Mittemeijer, E.J.: The kinetics of the precipitation of Co from supersaturated Cu-Co alloy. Metall. Mater. Trans. A. doi:10.1007/s11661-010-0594-7 (2010)
Vincent E., Becquart C.S., Domain C.: Atomic kinetic Monte Carlo model based on ab initio data: Simulation of microstructural evolution under irradiation of dilute Fe-CuNiMnSi alloys. Nucl. Instrum. Methods Phys. Res. B 255, 78–84 (2007)
Vincent E., Becquart C.S., Domain C.: Solute interaction with point defects in α-Fe during thermal ageing: a combined ab initio and atomic kinetic Monte Carlo approach. J. Nucl. Mater. 351, 88–99 (2006)
Molnar D., Binkele P., Hocker S., Schmauder S.: Atomistic multiscale simulations on the anisotropic tensile behaviour of copper-alloyed alpha-iron at different states of thermal ageing. Philos. Mag. 92, 586–607 (2012)
Soisson F., Fu C.-C.: Energetic landscapes and diffusion properties in FeCu alloys. Solid State Phenom. 129, 31–39 (2007)
Kittel C.: Introduction to Solid State Physics. Wiley, New York (1976)
Mehrer H.: Landolt-Börnstein, numerical data and functional relationships in science and technology, new series III. Springer, Heidelberg (1991)
Bergmann L., Schaefer C., Raith W.: Bergmann, Schaefer, Lehrbuch der experimentalphysik, band 6, Festkörper. Walter de Gruyter, Berlin (1992)
Schmauder S., Binkele P.: Atomistic computer simulation of the formation of Cu-precipitates in steels. Comput. Mater. Sci. 24, 42–53 (2002)
Porter D.A., Easterling K.E.: Phase Transformations in Metals and Alloys, 2nd edn. Taylor and Francis, London (1992)
Liu F., Sommer F., Bos C., Mittemeijer E.J.: Analysis of solid state phase transformation kinetics: models and recipes. Int. Mater. Rev. 52(4), 193–212 (2007)
Starink M.J.: On the meaning of the impingement parameter in kinetic equations for nucleation and growth reactions. J. Mater. Sci. 36, 4433 (2001)
Kooi B.J.: Monte Carlo simulations of phase transformations caused by nucleation and subsequent anisotropic growth: extension of the Johnson-Mehl-Avrami-Kolmogorov theory. Phys. Rev. B 70(22), 224108 (2004)
Wang J. et al.: On discussion of the applicability of local Avrami exponent: errors and solutions. Mater. Lett. 63, 1153–1155 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Oliver Kastner.
Rights and permissions
About this article
Cite this article
Molnar, D., Niedermeier, C., Mora, A. et al. Activation energies for nucleation and growth and critical cluster size dependence in JMAK analyses of kinetic Monte-Carlo simulations of precipitation. Continuum Mech. Thermodyn. 24, 607–617 (2012). https://doi.org/10.1007/s00161-012-0258-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00161-012-0258-5