A stochastic phase-field model determined from molecular dynamics
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 4, p. 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.

DOI : https://doi.org/10.1051/m2an/2010022
Classification:  65C30,  65C35,  82B26
Keywords: phase-field, molecular dynamics, coarse graining, Smoluchowski dynamics, stochastic differential equation
     author = {von Schwerin, Erik and Szepessy, Anders},
     title = {A stochastic phase-field model determined from molecular dynamics},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {44},
     number = {4},
     year = {2010},
     pages = {627-646},
     doi = {10.1051/m2an/2010022},
     zbl = {1193.82052},
     mrnumber = {2683576},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2010__44_4_627_0}
von Schwerin, Erik; Szepessy, Anders. A stochastic phase-field model determined from molecular dynamics. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 4, pp. 627-646. doi : 10.1051/m2an/2010022. http://www.numdam.org/item/M2AN_2010__44_4_627_0/

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