# Atomistic-Continuum Couplings for Liquids

Nanofluidics deals with the study of fluid flows inside and outside nanostructures. Nanoscale flows are often embedded in larger scale systems, when for example nanofluidic channels interface microfluidic domains. Despite the success of atomistic simulation models (like Molecular Dynamics (MD)), their limitations in accessible length and time scales are stringent and allow only the analysis of elementary systems and for short times. As fully atomistic simulations are prohibitively expensive, purely continuum approaches are not possible due to the lack of the correct boundary conditions for the continuum solver (no-slip boundary condition may be not valid at the nanoscale)

Hybrid atomistic-continuum simulations are necessary to study large systems for reasonable times. We develop novel computational concepts based on dynamic control theory for the exchange of information between atomistic and continuum descriptions

Continuum simulation:
• Solve for the continuum velocity field subject to appropriate boundary conditions.
MD simulation:
• Compute the interaction between the atoms including the control boundary force.
• Impose on the MD system the velocity boundary conditions. Move the atoms.
• Move the interface. Bounce the atoms that have hit it, reset it to its initial position to keep a constant frame of reference.
• Reinsert the particles that have left the domain.
• Measure the velocities inside the whole MD subdomain and provide them to the continuum solver.

### Coupling Molecular Dynamics and Navier Stokes solver

We implement a domain decomposition algorithm, based on the Schwarz alternating method, to couple the MD description of water with a Finite Volume and a Lattice Boltzmann model solving the Navier-Stokes equations.

The simulations using the hybrid solvers are (L/l)**3 times faster than the full MD simulations , where L is the size of the continuum and l the size of the atomistic domain in the hybrid solvers [1].
[1] T. Werder, J. H. Walther, and P. Koumoutsakos, “Hybrid atomistic–continuum method for the simulation of dense fluid flows,” J. Comput. Phys., vol. 205, iss. 1, p. 373–390, 2005.
[Bibtex] [Abstract] [PDF] [DOI]

We present a hybrid atomistic-continuum method for multiscale simulations of dense fluids. In this method, the atomistic part is described using a molecular dynamics description, while the continuum flow is described by a finite volume discretization of the incompressible Navier-Stokes equations. The two descriptions are combined in a domain decomposition formulation using the Schwarz alternating method. A novel method has been proposed in order to impose non-periodic velocity boundary conditions from the continuum to the atomistic domain, based on an effective boundary potential, consistent body forces, a particle insertion algorithm and specular walls. The extraction of velocity boundary conditions for the continuum from the atomistic domain is formulated by taking into account the associated statistical errors. The advantages and drawbacks of the proposed Schwarz decomposition method as compared to related flux-based schemes are discussed. The efficiency and applicability of the method is demonstrated by considering hybrid and full molecular dynamics simulations of the flow of a Lennard-Jones fluid past a carbon nanotube.

``````@article{werder2005a,
author = {Thomas Werder and Jens H. Walther and Petros Koumoutsakos},
doi = {10.1016/j.jcp.2004.11.019},
journal = {{J. Comput. Phys.}},
month = {may},
number = {1},
pages = {373--390},
publisher = {Elsevier {BV}},
title = {Hybrid atomistic{\textendash}continuum method for the simulation of dense fluid flows},
url = {http://www.cse-lab.ethz.ch/wp-content/papercite-data/pdf/werder2005a.pdf},
volume = {205},
year = {2005}
}``````

People: Evangelos Kotsalis, Alvaro Foletti, Jens Walther
Funding: ETH Zurich

### 2007

• A. Dupuis, E. M. Kotsalis, and P. Koumoutsakos, “Coupling lattice boltzmann and molecular dynamics models for dense fluids,” Phys. Rev. E, vol. 75, iss. 4, 2007.
[BibTeX] [Abstract] [PDF] [DOI]

We propose a hybrid model, coupling lattice Boltzmann (LB) and molecular dynamics (MD) models, for the simulation of dense fluids. Time and length scales are decoupled by using an iterative Schwarz domain decomposition algorithm. The MD and LB formulations communicate via the exchange of velocities and velocity gradients at the interface. We validate the present LB-MD model in simulations of two- and three-dimensional flows of liquid argon past and through a carbon nanotube. Comparisons with existing hybrid algorithms and with reference MD solutions demonstrate the validity of the present approach.

``````@article{dupuis2007b,
author = {A. Dupuis and E. M. Kotsalis and P. Koumoutsakos},
doi = {10.1103/physreve.75.046704},
journal = {{Phys. Rev. E}},
month = {apr},
number = {4},
publisher = {American Physical Society ({APS})},
title = {Coupling lattice Boltzmann and molecular dynamics models for dense fluids},
url = {http://www.cse-lab.ethz.ch/wp-content/papercite-data/pdf/dupuis2007b.pdf},
volume = {75},
year = {2007}
}``````

• E. M. Kotsalis, J. H. Walther, and P. Koumoutsakos, “Control of density fluctuations in atomistic-continuum simulations of dense liquids,” Phys. Rev. E, vol. 76, iss. 1, 2007.
[BibTeX] [Abstract] [PDF] [DOI]

We present a control algorithm to eliminate spurious density fluctuations associated with the coupling of atomistic and continuum descriptions for dense liquids. A Schwartz domain decomposition algorithm is employed to couple molecular dynamics for the simulation of the atomistic system with a continuum solver for the simulation of the Navier-Stokes equations. The lack of periodic boundary conditions in the molecular dynamics simulations hinders the proper accounting for the virial pressure leading to spurious density fluctuations at the continuum-atomistic interface. An ad hoc boundary force is usually employed to remedy this situation. We propose the calculation of this boundary force using a control algorithm that explicitly cancels the density fluctuations. The results demonstrate that the present approach outperforms state-of-the-art algorithms. The conceptual and algorithmic simplicity of the method makes it suitable for any type of coupling between atomistic and continuum descriptions of dense fluids.

``````@article{kotsalis2007a,
author = {E. M. Kotsalis and J. H. Walther and P. Koumoutsakos},
doi = {10.1103/physreve.76.016709},
journal = {{Phys. Rev. E}},
month = {jul},
number = {1},
publisher = {American Physical Society ({APS})},
title = {Control of density fluctuations in atomistic-continuum simulations of dense liquids},
url = {http://www.cse-lab.ethz.ch/wp-content/papercite-data/pdf/kotsalis2007a.pdf},
volume = {76},
year = {2007}
}``````

### 2005

• T. Werder, J. H. Walther, and P. Koumoutsakos, “Hybrid atomistic–continuum method for the simulation of dense fluid flows,” J. Comput. Phys., vol. 205, iss. 1, p. 373–390, 2005.
[BibTeX] [Abstract] [PDF] [DOI]

We present a hybrid atomistic-continuum method for multiscale simulations of dense fluids. In this method, the atomistic part is described using a molecular dynamics description, while the continuum flow is described by a finite volume discretization of the incompressible Navier-Stokes equations. The two descriptions are combined in a domain decomposition formulation using the Schwarz alternating method. A novel method has been proposed in order to impose non-periodic velocity boundary conditions from the continuum to the atomistic domain, based on an effective boundary potential, consistent body forces, a particle insertion algorithm and specular walls. The extraction of velocity boundary conditions for the continuum from the atomistic domain is formulated by taking into account the associated statistical errors. The advantages and drawbacks of the proposed Schwarz decomposition method as compared to related flux-based schemes are discussed. The efficiency and applicability of the method is demonstrated by considering hybrid and full molecular dynamics simulations of the flow of a Lennard-Jones fluid past a carbon nanotube.

``````@article{werder2005a,
author = {Thomas Werder and Jens H. Walther and Petros Koumoutsakos},
doi = {10.1016/j.jcp.2004.11.019},
journal = {{J. Comput. Phys.}},
month = {may},
number = {1},
pages = {373--390},
publisher = {Elsevier {BV}},
title = {Hybrid atomistic{\textendash}continuum method for the simulation of dense fluid flows},
url = {http://www.cse-lab.ethz.ch/wp-content/papercite-data/pdf/werder2005a.pdf},
volume = {205},
year = {2005}
}``````