An efficient Monte Carlo Potts method for the grain growth simulation of single-phase systems

Noureddine Maazi1, Balahouane Lezzar1

1 Laboratoire Microstructures et défauts dans les Matériaux (LMDM), Département de Physique, Faculté des Sciences Exactes, Université Mentouri Constantine 1, Algeria.



The choice of the lattice sites to be reoriented in the Monte Carlo Potts algorithm for grain growth simulation is repeated in a non-homogeneous way. Therefore, some grains are favorably growing than others. This fact may seriously affect the simulation results. So a modified MC method is presented. Lattice sites are selected for reorientation one by one following their positions in the matrix in each Monte Carlo step (mcs). This approach ensures that the various selections of one lattice site within every mcs are eliminated, and no favorable growth of grains at the expense of others. The calculation time considerably decreases. The effect of real-time and physical temperature on the grain growth kinetics is discussed.

Cite as:

Maazi, N., & Lezzar, B. (2020). An efficient Monte Carlo Potts method for the grain growth simulation of single-phase systems. Computer Methods in Materials Science, 20(3), 85–94.

Article (PDF):

Key words:

Alloys, Crystal growth, Monte Carlo simulation, Microstructure


Anderson, M.P., Srolovitz, D.J., Grest, G.S., & Sahni, P.S. (1984). Computer simulation of grain growth – I. Kinetics. Acta Metallurgica, 32(5), 783–791.

Atkinson, H.V. (1988). Overview no. 65: Theories of normal grain growth in pure single phase systems. Acta Metallurgica, 36(3) 469–491.

Bortz, A.B., Kalos, M.H., & Lebowitz, J.L. (1975). A new algorithm for Monte Carlo simulation of Ising spin systems. Journal of Computational Physics, 17(1), 10–18.

Geiger, J., Roósz, A., & Barkóczy, P. (2001). Simulation of grain coarsening in two dimensions by cellular-automaton. Acta Materialia, 49(4), 623–629.

Hillert, M. (1965). On the theory of normal and abnormal grain growth. Acta Metallurgica, 13(3), 227–238.

Holm, E.A., & Battaile, C.C. (2001). The computer simulation of microstructural evolution. JOM. The Journal of The Minerals, Metals & Materials Society (TMS), 53(9), 20–23.

Holm, E.A., Glazier, J.A., Srolovitz, D.J., & Grest, G.S. (1991). Effects of lattice anisotropy and temperature on domain growth in the two-dimensional Potts model. Physical Review A, 43(6), 2662–2668.

Kawasaki, K., Nagai, T., & Nakashima, K. (1989). Vertex models for two-dimensional grain growth. Philosophical Magazine B, 60(3), 399–421.

Kim, H.S. (2010). Von Neumann–Mullins equation in the Potts model of two-dimensional grain growth. Computational Materials Science, 50(2), 600–606.

Maazi, N. (2017). Conversion of Monte Carlo steps to real time for grain growth simulation. Advances in Mathematical Physics, 2017, 1–8.

Maazi, N., & Boulechfar, R. (2019). A modified grain growth Monte Carlo algorithm for increased calculation speed in the presence of Zener drag effect. Materials Science and Engineering: B. 242, 52–62.

Maazi, N., & Rouag, N. (2002). Consideration of Zener drag effect by introducing a limiting radius for neighbourhood in grain growth simulation. Journal of Crystal Growth, 243(2), 361–369.

Mason, J.K., Lind, J., Li, S.F., Reed, B.W., & Kumar, M. (2015). Kinetics and anisotropy of the Monte Carlo model of grain growth. Acta Materialia, 82, 155–166.

Messina, R., Soucail, M., & Kubin, L. (2001). Monte Carlo simulation of abnormal grain growth in two dimensions. Materials Science and Engineering: A, 308(1–2), 258–267.

Mullins, W.W. (1956). Two‐Dimensional Motion of Idealized Grain Boundaries. Journal of Applied Physics, 27(8), 900–904.

Neumann, J. von (1952). Discussion – shape of metal grains. In Metal interfaces a seminar on metal interfaces held during the Thirty-third National Metal Congress and Exposition, Detroit, October 13 to 19, 1951 (108–110). American Society for Metals.

Ono, N., Kimura, K., & Watanabe, T. (1999). Monte Carlo simulation of grain growth with the full spectra of grain orientation and grain boundary energy. Acta Materialia, 47(3), 1007–1017.

Phaneesh, K.R., Bhat, A., Mukherjee, P., & Kashyap, K.T. (2012). On the Zener limit of grain growth through 2D Monte Carlo simulation. Computational Materials Science, 58, 188–191.

Radhakrishnan, B., & Zacharia, T. (1995). Simulation of curvature-driven grain growth by using a modified Monte Carlo algorithm. Metallurgical and Materials Transactions A, 26(1), 167–180.

Raghavan, S., & Sahay, S. (2007). Modeling the grain growth kinetics by cellular automaton. Materials Science and Engineering: A, 445–446, 203–209.

Song, X., & Liu, G. (1998). A simple and efficient three-dimensional Monte Carlo simulation of grain growth. Scripta Materialia, 38(11), 1691–1696.

Song, X., Liu, G., & He, Y. (1998). Modified Monte Carlo method for grain growth simulation. Progress in Natural Science, 8, 92–97.

Srolovitz, D.J., Grest, G.S., & Anderson, M.P. (1986). Computer simulation of recrystallization – I. Homogeneous nucleation and growth. Acta Metallurgica, 34(9), 1833–1845.

Suwa, Y., Saito, Y., & Onodera, H. (2006). Phase field simulation of grain growth in three dimensional system containing finely dispersed second-phase particles. Scripta Materialia, 55(4), 407–410.

Wang, Y.U. (2006). Computer modeling and simulation of solid-state sintering: A phase field approach. Acta Materialia, 54(4), 953–961.

Wejrzanowski, T., & Kurzydlowski, K.J. (2005). Modelling of the influence of the grain size distribution on the grain growth in nanocrystals. Solid State Phenomena, 101–102, 315–318.

Weygand, D., Bréchet, Y., & Lépinoux, J. (2000). Inhibition of grain growth by particle distribution: effect of spatial heterogeneities and of particle strength dispersion. Materials Science and Engineering: A, 292(1), 34–39.

Yu, Q., & Esche, S.K. (2003a). A Monte Carlo algorithm for single phase normal grain growth with improved accuracy and efficiency. Computational Materials Science, 27(3), 259–270.

Yu, Q., & Esche, S.K. (2003b). A new perspective on the normal grain growth exponent obtained in two-dimensional Monte Carlo simulations. Modelling and Simulation in Materials Science and Engineering, 11(6), 859–861.