A stochastic phase-field model determined from molecular dynamics

von Schwerin, Erik; Szepessy, Anders

ESAIM: Modélisation mathématique et analyse numérique, Tome 44 (2010) no. 4, pp. 627-646.

The dynamics of dendritic growth of a crystal in an undercooled melt is determined by macroscopic diffusion-convection of heat and by capillary forces acting on the nanometer scale of the solid-liquid interface width. Its modelling is useful for instance in processing techniques based on casting. The phase-field method is widely used to study evolution of such microstructural phase transformations on a continuum level; it couples the energy equation to a phenomenological Allen-Cahn/Ginzburg-Landau equation modelling the dynamics of an order parameter determining the solid and liquid phases, including also stochastic fluctuations to obtain the qualitatively correct result of dendritic side branching. This work presents a method to determine stochastic phase-field models from atomistic formulations by coarse-graining molecular dynamics. It has three steps: (1) a precise quantitative atomistic definition of the phase-field variable, based on the local potential energy; (2) derivation of its coarse-grained dynamics model, from microscopic Smoluchowski molecular dynamics (that is brownian or over damped Langevin dynamics); and (3) numerical computation of the coarse-grained model functions. The coarse-grained model approximates Gibbs ensemble averages of the atomistic phase-field, by choosing coarse-grained drift and diffusion functions that minimize the approximation error of observables in this ensemble average.

Classification : 65C30, 65C35, 82B26
Mots clés : phase-field, molecular dynamics, coarse graining, Smoluchowski dynamics, stochastic differential equation

[1] G. Amberg, Semi sharp phase-field method for quantitative phase change simulations. Phys. Rev. Lett. 91 (2003) 265505-265509.

[2] M. Asta, C. Beckermann, A. Karma, W. Kurz, R. Napolitano, M. Plapp, G. Purdy, M. Rappaz and R. Trivedi, Solidification microstructures and solid-state parallels: Recent developments, future directions. Acta Mater. 57 (2009) 941-971.

[3] J.T. Beale and A.J. Majda, Vortex methods. I. Convergence in three dimensions. Math. Comp. 39 (1982) 1-27. | Zbl

[4] W.J. Boettinger, J.A. Warren, C. Beckermann and A. Karma, Phase field simulation of solidification. Ann. Rev. Mater. Res. 32 (2002) 163-194.

[5] E. Burman and J. Rappaz, Existence of solutions to an anisotropic phase-field model. Math. Methods Appl. Sci. 26 (2003) 1137-1160. | Zbl

[6] E. Cances, F. Legoll and G. Stolz, Theoretical and numerical comparison of some sampling methods for molecular dynamics. ESAIM: M2AN 41 (2007) 351-389. | Numdam | Zbl

[7] A. De Masi, E. Orlandi, E. Presutti and L. Triolo, Glauber evolution with the Kac potentials. I. Mesoscopic and macroscopic limits, interface dynamics. Nonlinearity 7 (1994) 633-696. | Zbl

[8] W. E and W. Ren, Heterogeneous multiscale method for the modeling of complex fluids and micro-fluidics. J. Comput. Phys. 204 (2005) 1-26. | Zbl

[9] D. Frenkel and B. Smit, Understanding Molecular Simulation. Academic Press (2002). | Zbl

[10] J. Gooodman, K.-S. Moon, A. Szepessy, R. Tempone and G. Zouraris, Stochastic Differential Equations: Models and Numerics. http://www.math.kth.se/~szepessy/sdepde.pdf.

[11] R.J. Hardy, Formulas for determining local properties in molecular dynamics: shock waves. J. Chem. Phys. 76 (1982) 622-628.

[12] J.J. Hoyt, M. Asta and A. Karma, Atomistic and continuum modeling of dendritic solidification. Mat. Sci. Eng. R 41 (2003) 121-163.

[13] J.H. Irving and J.G. Kirkwood, The statistical mechanical theory of transport processes. IV. The equations of hydrodynamics. J. Chem. Phys. 18 (1950) 817-829.

[14] L.P. Kadanoff, Statistical physics: statics, dynamics and renormalization. World Scientific (2000). | Zbl

[15] A. Karma and W.J. Rappel, Phase-field model of dendritic side branching with thermal noise. Phys Rev. E 60 (1999) 3614-3625.

[16] M. Katsoulakis and A. Szepessy, Stochastic hydrodynamical limits of particle systems. Comm. Math. Sciences 4 (2006) 513-549. | Zbl

[17] P.R. Kramer and A.J. Majda, Stochastic mode reduction for particle-based simulation methods for complex microfluid systems. SIAM J. Appl. Math. 64 (2003) 401-422. | Zbl

[18] H. Kroemer and C. Kittel, Thermal Physics. W.H. Freeman Company (1980).

[19] L.D. Landau and E.M. Lifshitz, Statistical Physics Part 1. Pergamon Press (1980). | JFM | Zbl

[20] T.M. Liggett, Interacting particle systems. Springer-Verlag, Berlin (2005). | Zbl

[21] S. Mas-Gallic and P.-A. Raviart, A particle method for first-order symmetric systems. Numer. Math. 51 (1987) 323-352. | Zbl

[22] G. Morandi, F. Napoli and E. Ercolessi, Statistical Mechanics: An Intermediate Course. World Scientific Publishing (2001). | Zbl

[23] E. Nelson, Dynamical Theories of Brownian Motion. Princeton University Press (1967). | Zbl

[24] S.G. Pavlik and R.J. Sekerka, Forces due to fluctuations in the anisotropic phase-field model of solidification. Physica A 268 (1999) 283-290.

[25] T. Schlick, Molecular modeling and simulation. Springer-Verlag (2002). | Zbl

[26] H.M. Soner, Convergence of the phase-field equations to the Mullins-Sekerka Problem with kinetic undercooling. Arch. Rat. Mech. Anal. 131 (1995) 139-197. | Zbl

[27] A. Szepessy, R. Tempone and G. Zouraris, Adaptive weak approximation of stochastic differential equations. Comm. Pure Appl. Math. 54 (2001) 1169-1214. | Zbl

[28] N.G. Van Kampen, Stochastic Processes in Physics and Chemistry. North-Holland (1981). | Zbl

[29] E. Von Schwerin, A Stochastic Phase-Field Model Computed From Coarse-Grained Molecular Dynamics. arXiv:0908.1367, included in [30].

[30] E. Von Schwerin, Adaptivity for Stochastic and Partial Differential Equations with Applications th Phase Transformations. Ph.D. Thesis, KTH, Royal Institute of Technology, Stockholm, Sweden (2007).