A stochastic phase-field model determined from molecular dynamics

Type
Article

Authors
von Schwerin, Erik
Szepessy, Anders

KAUST Department
Applied Mathematics and Computational Science Program
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Online Publication Date
2010-03-17

Print Publication Date
2010-07

Date
2010-03-17

Abstract
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. © EDP Sciences, SMAI, 2010.

Citation
Von Schwerin, E., & Szepessy, A. (2010). A stochastic phase-field model determined from molecular dynamics. ESAIM: Mathematical Modelling and Numerical Analysis, 44(4), 627–646. doi:10.1051/m2an/2010022

Publisher
EDP Sciences

Journal
ESAIM: Mathematical Modelling and Numerical Analysis

DOI
10.1051/m2an/2010022

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