# A stochastic phase-field model determined from molecular dynamics

Erik von Schwerin; Anders Szepessy

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 44, Issue: 4, page 627-646
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topvon Schwerin, Erik, and Szepessy, Anders. "A stochastic phase-field model determined from molecular dynamics." ESAIM: Mathematical Modelling and Numerical Analysis 44.4 (2010): 627-646. <http://eudml.org/doc/250849>.

@article{vonSchwerin2010,

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.
},

author = {von Schwerin, Erik, Szepessy, Anders},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Phase-field; molecular dynamics; coarse graining;
Smoluchowski dynamics; stochastic differential equation; phase-field; Smoluchowski dynamics},

language = {eng},

month = {6},

number = {4},

pages = {627-646},

publisher = {EDP Sciences},

title = {A stochastic phase-field model determined from molecular dynamics},

url = {http://eudml.org/doc/250849},

volume = {44},

year = {2010},

}

TY - JOUR

AU - von Schwerin, Erik

AU - Szepessy, Anders

TI - A stochastic phase-field model determined from molecular dynamics

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/6//

PB - EDP Sciences

VL - 44

IS - 4

SP - 627

EP - 646

AB -
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.

LA - eng

KW - Phase-field; molecular dynamics; coarse graining;
Smoluchowski dynamics; stochastic differential equation; phase-field; Smoluchowski dynamics

UR - http://eudml.org/doc/250849

ER -

## References

top- G. Amberg, Semi sharp phase-field method for quantitative phase change simulations. Phys. Rev. Lett.91 (2003) 265505–265509.
- 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.
- J.T. Beale and A.J. Majda, Vortex methods. I. Convergence in three dimensions. Math. Comp.39 (1982) 1–27. Zbl0488.76024
- W.J. Boettinger, J.A. Warren, C. Beckermann and A. Karma, Phase field simulation of solidification. Ann. Rev. Mater. Res.32 (2002) 163–194.
- E. Burman and J. Rappaz, Existence of solutions to an anisotropic phase-field model. Math. Methods Appl. Sci.26 (2003) 1137–1160. Zbl1032.35053
- E. Cances, F. Legoll and G. Stolz, Theoretical and numerical comparison of some sampling methods for molecular dynamics. ESAIM: M2AN41 (2007) 351–389.
- A. De Masi, E. Orlandi, E. Presutti and L. Triolo, Glauber evolution with the Kac potentials. I. Mesoscopic and macroscopic limits, interface dynamics. Nonlinearity7 (1994) 633–696. Zbl0797.60088
- W. E and W. Ren, Heterogeneous multiscale method for the modeling of complex fluids and micro-fluidics. J. Comput. Phys.204 (2005) 1–26. Zbl1143.76541
- D. Frenkel and B. Smit, Understanding Molecular Simulation. Academic Press (2002).
- J. Gooodman, K.-S. Moon, A. Szepessy, R. Tempone and G. Zouraris, Stochastic Differential Equations: Models and Numerics. . Zbl0991.65089URIhttp://www.math.kth.se/~szepessy/sdepde.pdf
- R.J. Hardy, Formulas for determining local properties in molecular dynamics: shock waves. J. Chem. Phys.76 (1982) 622–628.
- J.J. Hoyt, M. Asta and A. Karma, Atomistic and continuum modeling of dendritic solidification. Mat. Sci. Eng. R41 (2003) 121–163.
- 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.
- L.P. Kadanoff, Statistical physics: statics, dynamics and renormalization. World Scientific (2000). Zbl0952.82001
- A. Karma and W.J. Rappel, Phase-field model of dendritic side branching with thermal noise. Phys Rev. E60 (1999) 3614–3625.
- M. Katsoulakis and A. Szepessy, Stochastic hydrodynamical limits of particle systems. Comm. Math. Sciences4 (2006) 513–549. Zbl1108.82029
- 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. Zbl1068.60081
- H. Kroemer and C. Kittel, Thermal Physics. W.H. Freeman Company (1980).
- L.D. Landau and E.M. Lifshitz, Statistical Physics Part 1. Pergamon Press (1980).
- T.M. Liggett, Interacting particle systems. Springer-Verlag, Berlin (2005). Zbl1103.82016
- S. Mas-Gallic and P.-A. Raviart, A particle method for first-order symmetric systems. Numer. Math.51 (1987) 323–352. Zbl0625.65084
- G. Morandi, F. Napoli and E. Ercolessi, Statistical Mechanics: An Intermediate Course. World Scientific Publishing (2001). Zbl0968.82002
- E. Nelson, Dynamical Theories of Brownian Motion. Princeton University Press (1967). Zbl0165.58502
- S.G. Pavlik and R.J. Sekerka, Forces due to fluctuations in the anisotropic phase-field model of solidification. Physica A268 (1999) 283–290.
- T. Schlick, Molecular modeling and simulation. Springer-Verlag (2002). Zbl1011.92019
- 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. Zbl0829.73010
- A. Szepessy, R. Tempone and G. Zouraris, Adaptive weak approximation of stochastic differential equations. Comm. Pure Appl. Math. 54 (2001) 1169–1214. Zbl1024.60028
- N.G. van Kampen, Stochastic Processes in Physics and Chemistry. North-Holland (1981). Zbl0511.60038
- E. von Schwerin, A Stochastic Phase-Field Model Computed From Coarse-Grained Molecular Dynamics. arXiv:0908.1367, included in [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).

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.