Coupling a nano-particle with isothermal fluctuating hydrodynamics

Coarse-graining from microscopic to mesoscopic dynamics

Pep Español, Aleksandar Donev

Research output: Contribution to journalArticle

Abstract

We derive a coarse-grained description of the dynamics of a nanoparticle immersed in an isothermal simple fluid by performing a systematic coarse graining of the underlying microscopic dynamics. As coarse-grained or relevant variables, we select the position of the nanoparticle and the total mass and momentum density field of the fluid, which are locally conserved slow variables because they are defined to include the contribution of the nanoparticle. The theory of coarse graining based on the Zwanzing projection operator leads us to a system of stochastic ordinary differential equations that are closed in the relevant variables. We demonstrate that our discrete coarse-grained equations are consistent with a Petrov-Galerkin finite-element discretization of a system of formal stochastic partial differential equations which resemble previously used phenomenological models based on fluctuating hydrodynamics. Key to this connection between our "bottom-up" and previous "top-down" approaches is the use of the same dual orthogonal set of linear basis functions familiar from finite element methods (FEMs), both as a way to coarse-grain the microscopic degrees of freedom and as a way to discretize the equations of fluctuating hydrodynamics. Another key ingredient is the use of a "linear for spiky" weak approximation which replaces microscopic "fields" with a linear FE interpolant inside expectation values. For the irreversible or dissipative dynamics, we approximate the constrained Green-Kubo expressions for the dissipation coefficients with their equilibrium averages. Under suitable approximations, we obtain closed approximations of the coarse-grained dynamics in a manner which gives them a clear physical interpretation and provides explicit microscopic expressions for all of the coefficients appearing in the closure. Our work leads to a model for dilute nanocolloidal suspensions that can be simulated effectively using feasibly short molecular dynamics simulations as input to a FEM fluctuating hydrodynamic solver.

Original languageEnglish (US)
Article number234104
JournalJournal of Chemical Physics
Volume143
Issue number23
DOIs
StatePublished - Dec 21 2015

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Hydrodynamics
hydrodynamics
Nanoparticles
nanoparticles
finite element method
approximation
Finite element method
Fluids
fluids
coefficients
ingredients
Ordinary differential equations
partial differential equations
closures
Partial differential equations
Molecular dynamics
Mathematical operators
Suspensions
Momentum
differential equations

ASJC Scopus subject areas

  • Physics and Astronomy(all)
  • Physical and Theoretical Chemistry

Cite this

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abstract = "We derive a coarse-grained description of the dynamics of a nanoparticle immersed in an isothermal simple fluid by performing a systematic coarse graining of the underlying microscopic dynamics. As coarse-grained or relevant variables, we select the position of the nanoparticle and the total mass and momentum density field of the fluid, which are locally conserved slow variables because they are defined to include the contribution of the nanoparticle. The theory of coarse graining based on the Zwanzing projection operator leads us to a system of stochastic ordinary differential equations that are closed in the relevant variables. We demonstrate that our discrete coarse-grained equations are consistent with a Petrov-Galerkin finite-element discretization of a system of formal stochastic partial differential equations which resemble previously used phenomenological models based on fluctuating hydrodynamics. Key to this connection between our {"}bottom-up{"} and previous {"}top-down{"} approaches is the use of the same dual orthogonal set of linear basis functions familiar from finite element methods (FEMs), both as a way to coarse-grain the microscopic degrees of freedom and as a way to discretize the equations of fluctuating hydrodynamics. Another key ingredient is the use of a {"}linear for spiky{"} weak approximation which replaces microscopic {"}fields{"} with a linear FE interpolant inside expectation values. For the irreversible or dissipative dynamics, we approximate the constrained Green-Kubo expressions for the dissipation coefficients with their equilibrium averages. Under suitable approximations, we obtain closed approximations of the coarse-grained dynamics in a manner which gives them a clear physical interpretation and provides explicit microscopic expressions for all of the coefficients appearing in the closure. Our work leads to a model for dilute nanocolloidal suspensions that can be simulated effectively using feasibly short molecular dynamics simulations as input to a FEM fluctuating hydrodynamic solver.",
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