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A stochastic phase-field model determinedfrom molecular dynamics

Published online by Cambridge University Press:  17 March 2010

Erik von Schwerin  and
Anders Szepessy
Erik von Schwerin
Affiliation:
Applied Mathematics and Computational Sciences, KAUST, Thuwal, Saudi Arabia.
Anders Szepessy
Affiliation:
Department of Mathematics, Royal Institute of Technology (KTH), Stockholm, Sweden. szepessy@kth.se
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Abstract

The dynamics of dendritic growth of a crystal in an undercooled melt isdetermined by macroscopic diffusion-convection of heat and by capillary forcesacting 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 ona continuum level; it couples the energy equation to a phenomenological Allen-Cahn/Ginzburg-Landauequation modelling the dynamics of an order parameter determining the solid and liquid phases,including also stochastic fluctuations to obtain the qualitatively correctresult of dendritic side branching.This work presents a method to determine stochastic phase-field models from atomisticformulations by coarse-graining molecular dynamics. It has three steps: (1) a precisequantitative atomistic definition of the phase-field variable, based on the localpotential energy;(2) derivation of its coarse-graineddynamics 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, bychoosing coarse-grained drift and diffusion functions that minimize the approximation error of observables in thisensemble average.

Keywords

Phase-fieldmolecular dynamicscoarse grainingSmoluchowski dynamicsstochastic differential equation
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 4 , July 2010 , pp. 627 - 646
Copyright
© EDP Sciences, SMAI, 2010

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