Hybrid molecular dynamics and Navier-Stokes method in complexnanoflow geometries

Matthew K. Borg*, William D. Nicholls and Jason M. Reese

Department of Mechanical Engineering,University of Strathclyde, Glasgow G1 1XJ, UK

* matthew.borg@strath.ac.uk

ABSTRACT

We present a new hybrid molecular dynamics andcontinuum solver for modelling nanoflows in arbitrarydomain geometries. The method makes use of a domaindecomposition, in which the molecular sub-domain isapplied adjacent to interfaces or walls, and is coupledto a continuum sub-domain in the rest of the flow fieldin order to reduce the computational cost of full MDsimulations. Coupling between disparate sub-domainsemploys an alternating time framework, and consists ofmutual two-direction (i.e. continuum-to-molecular andmolecular-to-continuum) boundary conditions. We demon-strate the hybrid technique in shear flow configurations.

Keywords: Nanofluidics, Molecular Dynamics, Hy-brid Continuum-Molecular, Boundary Conditions

1 INTRODUCTION

Nanofluidics is an emerging technology of the 21st

century, and molecular dynamics (MD) is perhaps themost important numerical method available to captureliquid phenomena at small scales. MD is, however, fartoo computationally intensive, limiting it to only smallsystems and relatively short time-scales. Hybrid meth-ods for liquids [1]–[4] have recently been proposed inorder to alleviate the overwhelming computational costof full MD simulations. This is achieved by applying MDsolely in small regions of the domain where it is required,and coupling it with a computationally-fast continuumformulation for the bulk of the flow field. In the liter-ature, only relatively simple systems in cubic domainshave been treated using hybrid methods, and periodicboundary conditions in MD still pose many limitationsto the development of new coupling strategies. In orderto study nanoflows in more engineering-related problemsboth MD and hybrid MD-continuum models need tobe applicable in arbitrary geometries, treat boundariesas non-periodic entities, and handle parallel processing[5]–[7].

2 METHOD

We consider a domain of arbitrary geometry, definedby an unstructured polyhedral mesh. The domain is

segmented into sub-meshes assigned to ComputationalFluid Dynamics (CFD) and MD solvers1, allowing anoverlap region at all CFD-MD interfaces, in which cou-pling between disparate sub-domains can occur at regu-lar time intervals of the simulation. An example of thehybrid domain decomposition methodology for a shearflow with a complex coupling region is portrayed in Fig-ure 1. The MD fluid occupies its sub-domain and con-

MD

CFD

coupling

region

continuum

mesh

molecular

mesh

buffer

region

(a) Hybrid domain

(b) Common coupling regions on sub-domains

10nmx

y

z

(C) (M)

C M

M C

NPBCs

(N-S)

uwall

Figure 1: (a) Hybrid domain decomposition of a shearflow with a complex fixed-wall topology. (b) Schematicof the coupling region between continuum and molecularsub-domain meshes.

sists of “molecules” that interact through a pair-wisepotential U(rij), where rij = |rij | and rij = ri − rj isthe separation vector between a pair of molecules (i, j).

1The continuum and molecular solvers are both implementedin OpenFOAM, an open-source toolbox of C++ libraries for com-putational physics. It may be downloaded freely from [8].

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The standard shifted Lennard-Jones (LJ) 12-6 potential[9] is used in this paper. The positions ri and veloci-ties vi of molecules evolve according to standard Newto-nian dynamics, using a molecular time-integration stepΔtm = 10 fs.

The continuum sub-domain uses a finite-volume dis-cretisation of the compressible, isothermal Navier-Stokesequations, ∂φ/∂t = −∇ · Jφ, where φ = {ρ, ρu} are thedensities, while Jφ = {ρu, ρu+Π} are the flux-densities;Π = p1 + τ is the stress tensor, including the normalpressure and the viscous shear-stress. The continuumtime-integration step is given by Δtc = 0.2 ps.

The coupling procedure applies M→C and C→Mboundary conditions via an arbitrary pair of coupledcells within the overlapping region, as shown in thetwo highlighted regions of Figure 1(b). C→M couplingis achieved through non-periodic boundary conditions(NPBCs): mainly the imposition of feedback loop con-trollers [7] that operate to converge the continuum-derivedstate properties (e.g., density, velocity and tempera-ture). M→C coupling is achieved by coarse-graining ofproperties from cell-occupant molecules, and replacingthe corresponding boundary values on the continuumfields. Properties transferred between continuum andmolecular subdomains employ a relaxation technique inorder to improve stability of the coupling scheme. Forexample, the target MD fields for an arbitrary stateproperty φ is:

φ(n)m = θφ(n−1)

m + (1 − θ)φ(n)c , (1)

where θ (0 < θ < 1) is the relaxation parameter, sub-scripts (m, c) denote molecular and continuum fields re-spectively, and n denotes the time index of coupling.

A coupling time-framework is used to advance theMD and CFD in a sequential manner by a common cou-pling time-interval Δtcoupling = τCΔtc = τMΔtm = 40ps:

1. Advance continuum solution t → t+Δtcoupling byτC = 200 time-steps. MD waits.

2. Apply C→M BCs.

3. Advance MD t → t + Δtcoupling by τM = 4000time-steps. CFD waits.

4. Apply M→C BCs.

5. Repeat the dual time-marching scheme until theend of the simulation.

3 RESULTS

In order to test the hybrid method we investigateflows of liquid argon molecules, which have a character-istic length scale of σ = 0.34 nm, a characteristic energy

of ε = 120kb = 1.65678×10−21 J, where kb is the Boltz-mann constant, and a mass m = 6.69×10−26 kg. A cut-off radius of rcut = 2.5σ = 0.85 nm is used. We presentour results in reduced units: time, t∗ = t

√ε/mσ2, num-

ber density ρ∗ = ρσ3, temperature T ∗ = T (kb/ε) andvelocity u

∗ = u√

m/ε.

3.1 Simple shear flows

We validate the hybrid method by simulating a start-up Couette flow configuration and a simple oscillatingflow. For start-up Couette flow, we consider a domainof dimensions (Lx = 188σ, Ly = 20σ, Lz = 20σ) withtwo moving molecular walls at the extremities of thedomain. We apply MD at the two wall regions (50σthickness), and the compressible Navier-Stokes equa-tions in the central part of the domain. Two inde-pendent coupling regions are therefore required at thecontinuum-molecular interfaces, while periodic bound-ary conditions are applied in the y− and z−directions.The shear viscosity in the entire continuum field is cho-sen to match closely that in the coupling region of theMD domain, i.e., η = 0.899(

√εm/σ2), where the bulk

state of the fluid is ρ∗ = 0.6, and T ∗ = 2.4 for num-ber density and temperature, respectively. The walls(A, B) are accelerated from rest using a ramp velocityfunction over a time-period Δt∗ = 20, until a maximumvelocity u

∗

A = (0.0,−2.5, 0.0) and u∗

B = (0.0, 2.5, 0.0)are reached. The walls are then maintained at uniformvelocity. The transient velocity profiles for both hybridand full MD simulations are shown in Figure 2, wheregenerally good agreement is observed.

-2

-1

0

1

2

3

100 120 140 160 180 200

VE

LO

CIT

Y,

u

y*(x

*)

CHANNEL HEIGHT, x*

MD (B)

N-S

FULL MD

HYBRID: N-S

HYBRID: MD

wall

Figure 2: Transient half-channel velocity profiles forthe start-up Couette flow problem, with relaxationθ = 0.6. Profiles are shown for times: t∗ ={100, 500, 1000, 4000.}. Hybrid solutions are comparedwith the full molecular solution.

A hybrid simulation of an oscillating shear flow prob-lem is then carried out to demonstrate that the hybrid

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algorithm may also be applied to unsteady cases. Thehybrid domain decomposition consists of an MD sub-domain encompassing the oscillating molecular wall andadjacent layer of molecular liquid, coupled to a Navier-Stokes continuum solver with no-slip boundary condi-tions applied at the other end of the domain (i.e. atLx = 100σ). The molecular wall moves with an oscillat-ing motion uwall(t) = umax sin(2πt)/τT , where u

∗

max =(0, 10.0, 0) is the maximum velocity, and τ∗

T = 200 is thetime-period of the oscillatory wave. In Figure 3 we showvelocity profiles at different times in one oscillatory cy-cle, and see that good agreement is achieved betweenthe hybrid and a full-MD case. For both start-up andoscillating shear flow cases, a computational speed-upof approximately two times is achieved using the hybridmethod, although the MD component still takes up themajority of the cost of the entire simulation.

-3

-2

-1

0

1

2

3

10 20 30 40 50 60

VE

LO

CIT

Y, u

y*(x

*)

CHANNEL HEIGHT, x*

COUPLING REGION

Full-MD

t* = 1000

t* = 1050

t* = 1100

t* = 1150

N-S

Figure 3: Velocity profiles shown only in the couplingregion for the simple oscillating shear flow case. Resultsare only displayed for one cycle (τ∗

T = 200) of the oscil-latory motion. Comparisons are made with a full MDsimulation.

3.2 Complex coupling region

We now demonstrate the capability of our hybridmethod by applying it to a Couette flow problem with acomplex-shaped coupling region, as shown in Figure 1.The bottom stationary wall and adjacent layer of liquidis modelled by MD, while the rest of the domain andupper moving wall is simulated by CFD. A moving wallvelocity of u

∗

wall = (0.5, 0, 0), with a no-slip boundarycondition, is applied to the top boundary of the domain,and cyclic boundary conditions are applied in the othertwo directions. At the molecular-continuum interface,a 3D overlap region is present for velocity and densitycoupling between the CFD and MD formulations.

(N-S)

MD

(a) HYBRID SOLUTION

u*

(b) FULL MD SOLUTION

NPBCs

CFD

u*x

y

z

x

y

z

Figure 4: Velocity vector fields: (a) hybrid (top – N-Ssolution; bottom – MD solution) and (b) full-MD simu-lations of the same complex Couette flow case.

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Results are shown in Figures 4 and 5, where generalagreement of the velocity field is observed between thefull-MD and hybrid MD-CFD simulations. However, thehybrid simulation is approximately two times faster thanthe full-MD simulation. Speed-up may also be estimatedfrom the term NFM

mol/NHmol, where NH

mol = 34, 394 andNFM

mol = 68, 944 are the average number of moleculesin the hybrid MD sub-domain and the full-MD domainrespectively.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

20 30 40 50 60 70 80

VE

LO

CIT

Y, u

x*

CHANNEL HEIGHT, y*

NPBCs

M-to-C BCs

C-to-M BCs

Hybrid - CFD solutionHybrid - MD solution

Full-MD solution

Figure 5: Comparisons of the velocity profile in the y-direction starting from the wall protrusion seen in Fig-ure 4.

4 CONCLUSIONS

We have presented a new hybrid MD-CFD methodbased on spatial domain decomposition and an alternat-ing time-coupling framework with relaxation field cou-pling. Our method is tested for shear-flow applications,and shows generally good agreement between the hy-brid and full-MD solutions, with more than twice thespeed-up. Our hybrid method is parallelised and gen-erally applicable to any complex domain configuration,and so may serve as a very useful tool in engineeringdesign and simulation of nano-scale applications.

5 ACKNOWLEDGMENTS

This work is funded in the UK by the James WeirFoundation (MKB), the Institution of Mechanical En-gineers (WDN) and the Royal Society of Edinburgh /Scottish Government (JMR).

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