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1

A thermostat-consistent fully coupled molecular dynamics – generalised fluctuating

hydrodynamics model for non-equilibrium flows

Xinjian Liu, Ivan Korotkin, Zhonghao Rao, Sergey Karabasov*

X. Liu, Z. Rao.

School of Electrical and Power Engineering, China University of Mining and Technology,

Xuzhou, 221116, China

X. Liu, S. Karabasov

The School of Engineering and Materials Science, Queen Mary University of London, Mile

End Road, E1 4NS London, United Kingdom

E-mail: s.karabasov@qmul.ac.uk

I. Korotkin

Mathematical Sciences, University of Southampton, University Rd., SO17 1BJ, United

Kingdom

Keywords: hybrid molecular dynamics-continuum hydrodynamics methods, molecular

dynamics, multiscale modeling, linearization, non-equilibrium

The thermostat-consistent fully coupled molecular dynamics – generalised fluctuating

hydrodynamics method is developed for non-equilibrium water flow simulations. The model

allows for strong coupling between the atomistic and the continuum hydrodynamics

representations of water and shows an improved stability in comparison with the previous

formulations of similar multiscale methods. Numerical results are demonstrated for a periodic

nano-scale Poiseuille flow problem with SPC/E water. The computed time-averaged velocity

profiles are compared with the analytical solution, and the thermal velocity fluctuations are well

reproduced in comparison with the Equilibrium Molecular Dynamics (EMD) simulation.

Several options to account for the long-range electrostatics interactions available in

GROMACS are incorporated in the model and compared. It is demonstrated that the suggested

non-equilibrium multiscale model is a factor of 4 to 18 faster in comparison with the standard

all-atom equilibrium molecular dynamics model for the same computational domain size.

1. Introduction

One of the popular categories of multiscale methods for atomistic-scale resolving

simulations of non-equilibrium liquid flows is based on the idea of dividing the computational

domain into an overlapping region of the continuum Navier-Stokes equations and the molecular

dynamics region following the pioneering work of O’Connell and Thompson [1]. In further

models, more accurate continuum models of dense liquids were used such as the Landau-

Lifshitz Fluctuating Hydrodynamics (LL-FH) equations [2], which also preserve thermal

fluctuations thereby enabling more consistent coupling with molecular dynamics [3-5]. A finite

overlap (buffer) region between the models of different resolutions is beneficial for a smooth

2

transition between the continuum and atomistic parts of the model to avoid sharp oscillations

[6]. In multi-particle methods [7-8], such buffer region also contains a multi-resolution region

where discrete particles transition from the fully atomistic to a coarse-grained representation.

Consequently, the coarse-grained particles are coupled with the continuum flow models which

lead to a micro-meso-macro scale formulation[8-9]. For multi-resolution fluid dynamics

modelling, the coupling between the models of different resolution should respect conservation

of both the mass and the linear momentum. To automatically satisfy these conservation laws,

the two-phase analogy multiscale method was suggested in [10-11], which considered the

continuum and atomistic representations of a liquid as phases of a nominally two-phase flow.

The phases are allowed to transition one to another in accordance with a concentration field, a

user-defined function that determines which parts of the computational domain need to be

modelled at molecular dynamics resolution, and which do not. The continuum phase is

governed by the LL-FH-type equations integrated over control volumes, which are smoothly

replaced by the averaged fields obtained from the molecular dynamics equations in the region

of atomistic resolution of the multiscale model. To avoid separation of the two phases in the

buffer region, forcing functions are introduced in the molecular dynamics equations, which

correspond to the equivalent source and sink terms in the control volume averaged mass and

momentum equations. The latter equations are more complex in comparison with the LL-FH

model. Hence, in a number of cases, where the effect of the molecular phase on the continuum

part of the model could be included in the calibration of the constitutive relations of the

continuum model, such as the equation of state, a one-way coupled model case was used. Such

one-way coupled model accounted for the continuum flow effect on microscopic particles while

ignoring the feedback [12-14]. Despite the relative simplicity, the one-way-coupled approach

performs well for a range of problems including the diffusion of peptides in cross flow [15-16],

oscillations of a PCV2 virus capsid in water [17], and the nano-confined water effects in a High-

Speed Atomic Force Microscope experiment [18]. However, to achieve the full potential of the

3

hybrid multiscale method in terms of accuracy and computational performance, the two-way

coupling of the phases is needed. To do so efficiently, the governing equations of the two-phase

flow analogy model [19] were rearranged to the single set of Generalised Landau-Lifshitz

Fluctuating Hydrodynamics equations (GLL-FH), implemented in GROMACS [20], and applied

for liquid argon simulations [21]. In comparison with the standard LL-FH model, the GLL-FH

equations are mathematically equivalent to the control-volume averaged molecular dynamics

equations and reduce to the standard LL-FH equations for the pure continuum hydrodynamics

phase. To extend the original model to more complex liquids such as water, the two-way

coupled GLL-FH model was further extended to the local thermostat equations in Ref.[22],

where the thermostat-consistent fully coupled molecular dynamics – generalised fluctuating

hydrodynamics model of SPC/E water was developed. Because of the simple Langevin

thermostat used, which does not preserve the linear momentum of macroscopic flow, the

implementation was limited to water fluctuations in equilibrium conditions. Hence, the present

work is devoted to extending the model to non-uniform flow by using a new linearisation

approach, which also allows one to couple the suggested method with an external macroscopic

flow model (such as Fluid-Structure-Interaction in the future). To achieve this goal, a more

stable approximation of the source terms of the GLL-FH equations is developed and several

formulations for simulating long-range electrostatic interactions available in GROMACS are

implemented.

2. Method

2.1 Two-phase flow analogy equations and the Generalised Landau-Lifshitz Fluctuating

Hydrodynamics model

Following Ref.[10], the equations of conservation of mass (1),(2) and momentum (3),(4)

of the two-phase flow representative of SPC/E water are considered, where a user-defined

concentration function s is introduced. The latter function enables the smooth transition of

model resolution from the pure molecular dynamics zone (s = 0) to the pure continuum zone (s

4

= 1). In the intermediate buffer region (0 < s < 1), the two phases co-exist, which gives rise to

the exchange (source/sink) terms on the right-hand-side of the phase equations:

6

1

1

tsm s d t J t

un

(1)

6

1

1 1 1

11

N

N

t p p p p p

pp

s m s d t J t

un

(2)

6

2

1

t i i i

smu s u d t sF V t J t

un

(3)

6

2

1 1 1 1

1 1 1

N

NN

MD

t p p ip p p ip p p ip

p p p

s m u s u d t s F V t J t

un

(4)

Here i=1, 2, 3 denotes x, y and z components, variables with and without sub-index p correspond

to molecular dynamics phase and the continuum cell-volume/flux averaged values,

respectively;

corresponds to one of the six faces of the hexahedral control volume of the

computational grid,

V

;

m

and

are the local mass and density of the continuum phase per

given control volume;

p

m

and

pp

mV

are the particle mass and its effective density per

control volume, respectively;

p

u

and

u

correspond to particle velocity and velocity of the two

phase ‘mixture’, which is given by

1

1

N

i i p p ip

p

u s u s u

. The conventional

hexahedral finite-volume representation of control-volume-averaged gradients,

f

, which are

computed in accordance with the Gauss-Ostrogradski (Divergence) theorem,

1,6 fd

V

n

is

noted. The mixture density is defined as

1

1N

p

p

ss

.

N

is the number of particles per

cell volume, and

N

denotes the number of particles crossing the cell face in the direction of

the area normal

d

n

,

t

represents the change of each quantity over one time step.

MD

ip

F

refers

to the MD particle force exerted on each particle. The continuum force,

()

i j ij ij

F

5

includes both the deterministic and stochastic continuum Reynolds stress forces in accordance

with the Landau-Lifshitz Fluctuating Hydrodynamic (LL-FH) model[2], where the amplitude of

thermal fluctuations is inversely proportional to the square root of the continuum

hydrodynamics integration time step

FH

t

and control volume size,

V

and also linearly

proportional to the square root of the thermodynamic temperature,

T

, which is equal to the

target temperature of the MD ensemble,

0

T

. For isothermal processes of interest in this work,

an isothermal Equation of State (EoS),

0

( , )p p T

is assumed, which relates thermodynamic

pressure and density of the continuum phase, and which is calibrated from a separate all-atom

equilibrium MD simulation of SPC/E water. In the current work, the calibrated EoS can be

expressed as

( ) cp a b

, and the simulation parameters including MD details are

summarised in Table 1.

The source and sink terms,

1

J

and

2

J

, which determine the phase interaction are defined

so that the residuals corresponding to the differences of the cell-averaged particle density and

momenta from the same of the two-phase mixture,

1

N

p

p

and

1

N

i i p ip

p

q u u

, are

forced to decay to zero (or, at least, stay bounded) to avoid the unwanted phase separation, in

accordance with the additional relations:

6

11

11N

ti i p

p

u dn s s

tV

, (5)

6

11

11N

ti i j j i p ip

p

qq u dn s s u u

tV

, i, j = 1, 2, 3 (6)

Accordingly, to satisfy the conservation of particle mass and momentum equations,

6

1

11

0

N

tp N

pp

p

p

mdd

t dt

xn

and (7)

6

6

1 1 1 1

N

NN

p ip

t p ip p ip p

p p p

d du

m u u d t m t

dt dt

xn

, (8)

the molecular dynamics equations are modified to

1

(1 ) (1 ) N

ip

p ip p ip p p p

p

i

dx s u s u s s x

dt

(9)

and

1

11

1

1

1

(1 )

1

N

p p jp p

NN

p

jp MD

p jp p p jp p p

N

pp

ip

pp

N

jp

p

i

p

s s u

du x

s F s F x

dt

s s q

(10)

where i, j = 1,2,3 are Cartesian coordinate components, p is the particle sub-index which refers

to the point/ field value defined/interpolated to the particle location, and

,0

are adjustable

coupling constants, which are obtained from the model calibration by comparison with the all-

atom MD simulations in the equilibrium case.

For numerical solution of equations (1) – (4), following Ref.[21], new dependent variables

*

and

i i i

q u q

are introduced to re-arrange the equations to the so-called

Generalised Landau-Lifshitz Fluctuating Hydrodynamics (GLL-FH) form, which is a single

phase formulation of the original two-phase-like equations (1)-(4),

6

1

*1* (1 s)

tii

s u dn Q s

tV

(11)

6

1

1(1 s)

ti i j j i i i

qsqu dn Q sF s q

tV

i, j=1, 2, 3. (12)

where the right-hand-side terms account for the control-volume-averaged mass and momentum

terms sources corresponding to the feedback from the MD particles to the continuum

hydrodynamics phase,

6

11

11

N

p ip p i

p

Q s u dn

V

(13)

7

and

6

1 1 1

1

11

N

NMD

i p ip p jp p ip j

pp

Q s F s u u dn

V

i, j=1, 2, 3. (14)

In the actual MD implementation, an additional Langevin dissipative term is included in

the MD velocity equation

*

ip i

dx u

dt

(15)

*

ip

ip i

F

du u

dt

,

1

MD

t

, (16)

where

is the rescaling parameter of the Berendsen thermostat [23],

11

ref

MD T

t

T

,

is its characteristic relaxation time,

ref

T

is the target MD temperature, and

T

is the

instantaneous temperature of MD particles for the relevant ensemble averaging. Furthermore,

to include the continuum hydrodynamics effect in the multi-resolution domain, the s-dependent

local thermostat was suggested[22] to partly account for the particle inertia effect,

0

1

ref

s

TT

fs

, where

=(1 2)f s s

. (17)

By integrating the MD particle equations, the modified Leapfrog scheme is obtained [22]

*e1

t

ip ip i

t t tx x u

(18)

/2 3

*ee

e

22

tt

t

ip ip i

tt

ut Fut

, (19)

1

*(1 ) (1 ) N

ip

p ip p ip p p p

p

ii

dx

u s u s u s s x

dt

,

where

8

1

11

1

1

*

1

(1 )

1

N

p p jp p

NN

p

MD

p jp p p jp p p

N

pp

ip

p

N

jp

p

i

j

F

s s u x

s F s F x

s s m

(20)

This will lead to the corresponding modification of the momentum source term of the GLL-FH

equations in comparison with (14),

6

1 1 1 1

1

11

N

NN

MD

i p ip p jp p ip j ip p

p p p

Q s F s u u dn u

V

(14a)

2.2 Application to the Poiseuille flow: the linearisation approach and the computational

method

To enforce a non-uniform background incompressible flow,

0,

0

ux

, a linearization

algorithm is introduced as follows.

The GLL-FH solution is considered as a linear superposition of the steady solution and the

unsteady fluctuation,

0

( , ) ( ( , ), ( ) ( , ))tt

0

u x u x u x

, where the double prime denotes

the thermal fluctuation. Such linearization reduces the problem to (i) the solution of the

governing GLL-FH equations (11) and (12) for density and thermal velocity fluctuation,

,), ( t

ux

under the simplified periodic boundary conditions and (ii) the determination of the

background flow solution

0,

0

ux

. Notably, the latter can be obtained from a separate

continuum flow calculation or specified analytically, as in the case of the Poiseuille flow case

considered in the numerical example section. The MD equations (18)-(20) are solved with the

full reconstructed solution

( , )

u

. Hence, for consistency, the MD velocities appearing in the

source term

i

Q

(14a) are redefined to subtract the mean values,

1

1N

ip ip ip

p

u u u

N

.

It can be noted that the GLL-FH equations are solved for the velocity fluctuation variable

consistently with MD equations. In accordance with this formulation, the non-uniform

9

background flow is included separately and, thereby, is not affected by the Langevin dissipation.

Due to the linearisation approach, the non-uniform macroscopic flow is preserved and the total

linear momentum conservation property is not violated. A more general approach to enforce

the total momentum conservation would be to use Dissipative Particle Dynamics (DPD)

thermostat, which is based on the relative velocity between a pair of interacting particles. Once

implemented in future work, such approach would not require using the suggested linearisation

method to conserve the bulk flow momentum.

Following Ref.[22], the GLL-FH equations (5),(6), (11)-(13), (18)-(20) are solved by a

predictor-corrector scheme, where the Eulerian part of the model (5),(6), (11)-(13) is solved by

a central finite-volume method on a uniform computational grid of control volumes

V

. The

integration time step of the control-volume-averaged equations,

FH

t

is 10 times larger in

comparison with the MD time step,

MD

t

. In the two-way coupled solution, the hydrodynamic

fields are driven by the collective dynamics of MD particles while coordinates and velocities

of the MD particles are concurrently updated in accordance with (18)-(20).

Most details of the implemented numerical algorithm are identical to the algorithm

published in Ref.[22]. Two modifications to increase the stability of the numerical scheme for

solving the GLL-FH equations are summarised below.

First, the modified version of equations (5) and (6) is solved by re-arranging the source

term to include the evolutionary variable in the dissipation terms thereby enabling an improved

stability,

6

11

1 1 1

11

22

N

ti i p

p

u dn s s s s

tV

(5a)

6

11

1 1 1

11

22

N

ti i j j i p ip

p

qq u dn s s u u s s q

tV

, i, j = 1, 2, 3 (6a)

10

Secondly, for more accurate approximation of the ensemble-averaged MD forces

1

1

NMD

p ip

p

sF

of the momentum source term

i

Q

in (14a), an improved averaging procedure

is implemented. The MD forces are driven by steep MD pair potentials. Without a sufficient

ensemble averaging, the force term has a highly oscillatory behaviour, which may lead to

numerical instability when solving the GLL-FH equations. Hence, to compensate for a limited

time averaging, the MD force term is spatially filtered by averaging over a super-control volume

of 33 cells, where the central cell corresponds to the current control volume of interest (Figure

1).

Figure 1. Schematic of a super-control volume comprising 33 cells for calculation of the cell-

averaged MD force term in the current central cell.

Following Ref.[22], the computational domain includes a large hydrodynamic box domain,

1 2 3

,,L x x x L

that overlaps with a small MD particle domain in the center. Continuum

hydrodynamic equations (5), (6), (11)-(13) are solved in the large box domain, and the modified

MD particle equations (18)-(20) are solved in the particle domain with NVT ensemble. Due to

the improved stability, the current implementation of the GLL-FH model permits the testing of

different long-range electrostatics methods in the MD part of the multiscale model. Specifically,

the implementations of the model based on the Cut-off and Reaction-Field[24] methods in

11

GROMACS are compared. In addition, we have also implemented the same multiscale model

with the Particle-Mesh Ewald (PME) method, which yields similar results to the Reaction-Field,

hence, is not discussed separately.

The background flow corresponding to the planar Poiseuille flow is imposed

2

2

01 01 max 0

,0,0 , 1 x

u u U U

L

0

ux

, (21)

where

2

2

0 max 2

11d

2

L

L

x

U U x

LL

corresponds to the shift applied to subtract the center of

mass velocity in order to simplify the simulation in GROMACS. For the current test,

max 0.05U

nm·ps-1 and L ≡ box domain length with 9×9×9 elementary control volumes are considered.

To investigate the effect of the size of the hydrodynamic computational domains, two

computational domains, which correspond to 9×9×9 and 17×17×17 elementary control volumes

are considered. The MD particle domain in the centre of the hydrodynamic box domain

corresponds to 5×5×5 control volumes of the full grid. On average, each elementary control

volume in the particle domain contains 243 water molecules at the normal atmospheric pressure

(1 bar) and room temperature conditions (298.15 K). Periodic boundary conditions are used for

both the hydrodynamic box and the interior MD particle domains. Together with considering

of the Cut-off and Reaction-Field methods for the long-range MD interactions this amounts to

4 cases (2 domains × 2 models) to be tested.

To complete the hybrid model description, the s-function, which delineates the regions of

the atomistic resolution from the hydrodynamics region is specified following Ref.[21] and

Ref.[22] so that it is spherically distributed with the radial distance from the centre,

2 2 2

1 2 3

r x x x

inside the particle domain

12

max

max

0,

( ) ,

,

MD

MD MD FH

FH MD

FH

rR

rR

s r S R r R

RR

S r R

, (22)

and is set to 1 in the hydrodynamics box domain outside of the MD region.

Parameters of the suggested model such as Smax, RMD and RFH as well as the coupling parameters,

𝛼 and 𝛽 are acquired from a suitable calibration of the model for the equilibrium water

simulation case. Summary of the model parameters is provided in Table 1.

Figure 2. Computational setup for the simulation of water fluctuations: the overlapping

continuum and particle box domains. The insert shows an outline of the spherical pure MD

region (s = 0) inside the MD particle box.

Table 1. Simulation parameters of the GLL-FH model of the SPC/E water flow

Items

value

Number of atoms (molecules)

91125 (30375)

Molecular mass (g·mol−1)

18.015

Temperature (K)

298.15

MD box volume (nm3)

9.686×9.686×9.686

MD time step (ps)

0.001

Continuum solver time step (ps)

0.01

Average density (amu·nm−3)

602.18

Shear viscosity (amu·nm−1·ps−1)

409.496

Bulk viscosity (amu·nm−1·ps−1)

933.41

Maximum concentration of the hydrodynamic

phase in the particle domain

max

S

0.5

Number of control volumes in the MD box domain

5×5×5

Number of control volumes in the continuum box

domain

9×9×9&17×17×17

Dimensionless radius of the pure MD zone,

1/ 3

2MD

RV

0.5

13

Dimensionless radius of the pure MD/FH zone,

1/3

2FH

RV

1

MD/FH coupling parameters, α (ps-1), β (ps-1)

500, 1000

Thermostat relaxation time, (ps)

0.01

Constraints algorithm

SETTLE[25]

Parameters of EoS

()p a b c

a=0.010, b=-10.133, c=2428.920

3 Results and discussion

3.1 Flow profiles

Figure 3 compares the time-averaged velocity profiles of FH solutions and MD solutions

for all four cases, the Cut-off and Reaction-Field electrostatics methods with two domain sizes,

9×9×9 and 17×17×17 cells. The three solutions compared against the analytical Poiseuille flow

solution,

01

u

correspond to the time-averaged velocity profile

1

u

and the time- and cell-

volume-averaged profile of the MD particle velocities,

1

1N

ip

pu

N

in the full particle zone

including the buffer region and in the pure MD zone only. In all cases, the numerical solutions

closely capture the analytical Poiseuille flow profile. It can be seen the MD velocity solutions

of the large computational domain are less sensitive to the choice of the electrostatics interaction

modelling. For the smaller domain, the Reaction-Field method leads to a less noisy and more

accurate solution of the velocity profile. The larger sensitivity of the plain Cut-off method can

be explained by the fact that it cannot accurately describe the long-range dipole-dipole

interactions.

<ΣN

p=1u1p/N>particle zone

<ΣN

p=1u1p/N>pure MD zone

<u1>

uanalytical

-4 -2 0246810 12 14

-0.040

-0.035

-0.030

-0.025

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

Pure MD zone

Particle zone FH zone

u1 (nm·ps-1)

x2 (nm)

FH zone

(a)

<ΣN

p=1u1p/N>particle zone

<ΣN

p=1u1p/N>pure MD zone

<u1>

uanalytical

-15 -10 -5 0 5 10 15 20 25

-0.18

-0.16

-0.14

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

-0.04

-0.03

-0.02

-0.01

0.00

0.01

Pure

MD zone

Particle zone FH zone

u1 (nm·ps-1)

x2 (nm)

FH zone

(b)

14

<ΣN

p=1u1p/N>particle zone

<ΣN

p=1u1p/N>pure MD zone

<u1>

uanalytical

-4 -2 0246810 12 14

-0.040

-0.035

-0.030

-0.025

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

Pure MD zone

Particle zone FH zone

u1 (nm·ps-1)

x2 (nm)

FH zone

(c)

<ΣN

p=1u1p/N>particle zone

<ΣN

p=1u1p/N>pure MD zone

<u1>

uanalytical

-15 -10 -5 0 5 10 15 20 25

-0.18

-0.16

-0.14

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

-0.04

-0.03

-0.02

-0.01

0.00

0.01

Pure

MD zone

Particle zone FH zone

u1 (nm·ps-1)

x2 (nm)

FH zone

(d)

Figure 3. Time-averaged flow velocity profiles of the continuum field and MD particle

solutions for different hydrodynamic box sizes and MD electrostatics methods: (a) Cut-off

method in the 9×9×9 domain, (b) Cut-off method in the 17×17×17 domain, (c) Reaction-field

method in the 9×9×9 domain, and (d) Reaction-field method in the 17×17×17 domain.

3.2 Thermal fluctuations

The computed standard deviations of density and velocity fluctuations of the MD phase

and the hydrodynamics fields,

,

u

of the four solutions are summarised in Tables 2a and

b. The MD fluctuations are ensemble-averaged over each control volume

V

of the MD particle

part of the computational domains and compared with the reference analytical solutions. The

latter solutions are based on the grand-canonical-ensemble fluctuating hydrodynamics theory,

10

() TB

T

STD c k V

and

0

() B

T

STD u k V

, where

1

T

c

is the isothermal speed of sound and 𝑘𝐵 is

Boltzmann constant.

For all four cases, the standard deviations of the MD particle and continuum fluctuating

hydrodynamics solutions are approximately within 15% for density fluctuations and within 10%

for velocity fluctuations one from the other. This suggests that the artificial phase separation

did not occur, and the phase coupling of two-phase flow analogy method has been consistent.

On the other hand, it can be noted that the thermal fluctuations of density are 30-50%

lower than the analytical solution based on the grand canonical ensemble theory. This is

explained by the effect of the local thermostat implemented in the buffer region. Indeed, it can

be recalled that the s-dependent thermostat (17) is set to decrease the reference temperature

with increase of the hydrodynamic force contribution (s increasing). Hence, when averaged

15

over the entire MD particle domain, the thermal fluctuations of the particle phase of the GLL-

FH are expected to be smaller than the reference analytical solution that also includes the

hydrodynamics contribution. In this respect, the solution of the Reaction-Field method appears

to be more consistent due to the fact it demonstrates moderately lower fluctuations than the

predictions based on the grand canonical ensemble theory, while the Cut-off method tends to

overestimate the fluctuations.

Of particular interest are the thermal velocity and density fluctuations in the pure MD

region, which corresponds to the region of the highest resolution of the GLL-FH model. Hence,

Table 3 shows the standard deviations of effective particle density and particle velocity

fluctuations computed in control volume corresponding to the pure MD zone (s=0), without

considering the buffer region, where the local thermostat is used, i.e. time averaged

2

1

1

N

p

Np

p

pN

N

and

2

11

1

NN

p p p

pp

Ncell cell

p

p

v m m

VV

vN

N

. It can be seen that in all

cases the fluctuations of the hybrid multiscale model are within 0.5% of the solutions of the

reference all-atom equilibrium MD simulation of SPC/E water.

Table 2a. Standard deviations of density fluctuations averaged in the complete particle

domain domain.

_MD

STD

(amu·nm-

3)

_FH

STD

(amu·nm-

3)

Cut-off

(FH cells 9×9×9)

6.834

5.768

Reaction-field

(FH cells 9×9×9)

8.510

7.202

Cut-off

(FH cells 17×17×17)

6.775

5.757

Reaction-field

(FH cells 17×17×17)

8.517

7.206

Analytical solution

10.049

16

Table 2b. Standard deviations of velocity fluctuations averaged in the complete particle

domain.

_x_MD

STD u

(nm·ps-1)

_y_MD

STD u

(nm·ps-1)

_z_MD

STD u

(nm·ps-1)

_x_FH

STD u

(nm·ps-1)

_y_FH

STD u

(nm·ps-1)

_z_FH

STD u

(nm·ps-1)

Cut-off

(FH cells 9×9×9)

0.0235

0.0231

0.0231

0.0254

0.0253

0.0253

Reaction-field

(FH cells 9×9×9)

0.0186

0.0181

0.0181

0.0210

0.0210

0.0210

Cut-off

(FH cells

17×17×17)

0.02312

0.0229

0.02288

0.02515

0.02534

0.02532

Reaction-field

(FH cells

17×17×17)

0.0184

0.0181

0.0181

0.0206

0.0207

0.0207

Analytical solution

0.0238

Table 3a. Standard deviations of particle density fluctuations in the pure MD domain.

_STD

(amu·nm-3)

Cut-off

(FH cells 9×9×9)

0.9721

Reaction-field

(FH cells 9×9×9)

0.9722

Cut-off

(FH cells 17×17×17)

0.9769

Reaction-field

(FH cells 17×17×17)

0.9722

Reference all-atom

MD solution

0.9721

Table 3b. Standard deviations of particle velocity fluctuations in the pure MD domain.

_x

STD u

(nm·ps-1)

_y

STD u

(nm·ps-1)

_z

STD u

(nm·ps-1)

Cut-off

(FH cells 9×9×9)

0.9588

0.9572

0.9609

Reaction-field

(FH cells 9×9×9)

0.9595

0.9601

0.9588

Cut-off

(FH cells 17×17×17)

0.9572

0.9610

0.9584

Reaction-field

(FH cells 17×17×17)

0.9587

0.9591

0.9599

Reference all-atom

MD solution

0.9575

0.9578

0.9576

Figure 4 shows the time history of volume-averaged density and temperature computed in

the pure MD zone for the same four cases. Thanks to the local thermostat implemented in the

GLL-FH model, the temperature in the pure MD zone is well controlled in all cases. No

17

influence of the hydrodynamic domain size on the predicted temperature and density signals

can be noted.

However, depending on the choice of the method for computing long-range interactions,

the mean density calculation shows some variability: the Cut-off method overestimates the cell-

averaged water density by about 7.5-8.7% while the reaction-field overestimates by only about

1%. As water is a highly incompressible substance, the 7-8 fold reduced error in density

suggests that the Reaction-Field method leads to an improved preservation of the local reference

pressure in comparison with the Cut-off method.

0200 400 600 800 1000 1200 1400 1600 1800 2000

270

280

290

300

310

320

330

Temperature (K)

Time (ps)

Cut-off with FH cells 9×9×9

Cut-off with FH cells 17×17×17

Reaction-fieldwith FH cells 9×9×9

Reaction-fieldwith FH cells 17×17×17

Analytical

(a)

0200 400 600 800 1000 1200 1400 1600 1800 2000

540

560

580

600

620

640

660

680

700

720

740

760

ρ (amu·nm-3)

Time(ps)

Cut-off with FH cells 9×9×9

Cut-off with FH cells 17×17×17

Reaction-fieldwith FH cells 9×9×9

Reaction-fieldwith FH cells 17×17×17

Analytical

(b)

Figure 4. Dependence of the temperature (a) and density (b) in the pure MD region of the

computational domain on the domain size and the electrostatics interaction method: Cut-off

method in the 9×9×9 domain, Cut-off method in the 17×17×17 domain, Reaction-Field

method in the 9×9×9 domain, and Reaction-Field method in the 17×17×17 domain.

To complete this sub-section, distributions of the cell-averaged temperature of MD particles

along the flow stream-wise, shear-wise, and span-wise directions, which are centred in the

pure MD region, are shown in Fig.5. The solutions presented correspond to Reaction-Field

method and the 9×9×9 cell computational domain. Other choices of the force field and

computational domain sizes lead to very similar temperature distributions, hence, are not

included. Fig. 5a shows the particle temperature measured computed from the equipartition

theorem. The temperature profiles in all 3 coordinate directions virtually coincide, and the

particle temperature in the central pure MD region is equal to the target value, T0. Fig.5b

shows the reconstructed full temperature including the hydrodynamic phase contribution in

accordance with Eq.(17). Good agreement with the target temperature T0 is evident for all

spatial locations including the hydrodynamics-dominated regions near the boundaries of the

18

MD box. This confirms that the implemented local thermostat works well for the non-

equilibrium flow problem considered.

0.97 2.91 4.84 6.78 8.72

0.80

0.85

0.90

0.95

1.00

1.05

1.10 (a)

Tcell/T0

x1, x2, x3 (nm)

x1

x2

x3

0.97 2.91 4.84 6.78 8.72

0.90

0.95

1.00

1.05

1.10

Tcell/T0

x1, x2, x3 (nm)

x1

x2

x3

(b)

Figure 5. Profiles of the cell-averaged temperature of MD particles along the stream-

wise (x1), shear-wise (x2), and span-wise (x3) directions normalised on the target

temperature T0 and computed from (a) the equipartion theorem and (b) the equipartion

theorem including the hydrodynamic phase contribution in accordance with Eq.(17).

The model corresponds to the Reaction-Field method in the 9×9×9 domain.

3.3 Radial distribution function

The radial distribution function (RDF) is an important indicator showing whether the

interatomic forces in water have been accurately reproduced. Hence, RDF of water atoms has

been computed in the pure MD region for all 4 GLL-FH models and compared with the

reference all-atom equilibrium MD solution. The results for O-O atoms are shown in Figure 6.

Again, no influence of the hydrodynamic domain size on the predicted RDF function can be

noted. However, it can be observed that the solutions of the Cut-off model correctly capture the

first hydration layer and yet smear the subsequent dip associated with repulsion, which is likely

to be due to the water density overestimation. In comparison with this, the Reaction-Field

models capture both the features of the RDF curve and are in excellent agreement with the

reference all-atom MD solution.

19

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

RDF

r (nm)

Cut-off with FH cells 9×9×9

Cut-off with FH cells 17×17×17

pure MD

(a)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

RDF

r (nm)

Reaction-fieldwith FH cells 9×9×9

Reaction-fieldwith FH cells 17×17×17

pure MD

(b)

Figure 6. Radial distribution functions of O-O atoms for different electrostatics methods Cut-

off method (a) and Reaction-field method(b): Cut-off method in the 9×9×9 domain, Cut-off

method in the 17×17×17 domain, Reaction-field method in the 9×9×9 domain and Reaction-

field method in the 17×17×17 domain. The reference pure all-atom MD solution is included

for comparison.

3.4 Evaluation of computational efficiency

Since hybrid atomistic-scale resolving multiscale methods are considerably more complex

in comparison with the single-scale methods like Molecular Dynamics, computational

efficiency of the suggested multiscale method is of great interest. All current GLL-FH models

have been implemented in GROMACS and run on a workstation computer. The computational

cost of the multiscale solution does not depend on the type of the long-range electrostatics

method, cut-off and reaction-field and only weakly depends on the size of the hydrodynamics

box computational domain. Since the main cost is associated with solving the MD particle

equations in the small interior domain, the equivalent cost of equilibrium all-atom MD

simulation in the entire computational domain is much greater. The difference in the

computational cost of the hybrid multiscale method and the all-atom MD model in the same

computational domains, 9×9×9 and 17×17×17 control volumes are summarised in Table 4.

Notably, for the smaller computational domain, 9×9×9, the suggested multiscale model is a

factor of 4 faster that the all-atom equilibrium MD simulation. For the larger size domain of

17×17×17 volumes, the hybrid method already becomes a factor of 18 faster in comparison

with the all-atom simulation. For large problems, the computational benefits of the hybrid

20

method in comparison with the pure MD method should increase proportionally to the size of

the hydrodynamics box simulation domain.

Table 4. Simulation times of the hybrid multiscale method against the all-atom equilibrium

molecular dynamics for different computational domains, CPU hours for nanosecond

simulated.

Domain size

(9×9×9)

(17×17×17)

CPU hours of the hybrid method

/CPU hours of the all-atom MD

simulation

5.81/21.24=0.274

7.78/138.34=0.056

4. Conclusions

The thermostat-consistent fully coupled molecular dynamics – generalised fluctuating

hydrodynamics model has been extended to non-equilibrium water flow simulations. The key

ingredients of this extension are the increased robustness of the method due to the use of a more

stable approximation of the source terms in the governing fluctuating hydrodynamics equations

and the linearization formulation, which simplifies setting up of the boundary conditions. The

strong coupling and improved numerical stability of the hybrid multiscale method allow using

the advanced algorithms of modelling of long-range electrostatic interactions in water, such as

the Reaction-Field and Particle-Mesh Ewald methods in GROMACS, both of which give

similar results.

For validation, the method is applied for a multi-resolution simulation of the periodic

Poiseuille flow of SPC/E water. It is shown that the time-averaged flow velocity profile

compares well with the analytical solution, the thermal density and velocity fluctuations are

within the expected tolerance from the theoretical predictions of the grand canonical ensemble

fluctuating hydrodynamics theory, and the fluctuations in the pure molecular dynamics region

are within 0.5% from the reference all-atom molecular dynamics simulation. In addition,

distributions of the cell-averaged temperature of MD particles along the flow stream-wise,

shear-wise, and span-wise directions are shown to be in good agreement with the target

temperature value in all spatial locations including those dominated by hydrodynamics.

21

It is shown that the solution based on the Reaction-Field method captures both the mean

temperature and density in the atomistic region of the hybrid model within 1% from the

specified conditions and captures well both the hydration layer peak and the subsequent deep

of the RDF function.

The computational efficiency of the non-equilibrium fully coupled molecular dynamics –

generalised fluctuating hydrodynamics model is similar to the previously reported equilibrium

version of the same model: it is a factor of 4 to 18 faster compared to the all-atom equilibrium

molecular dynamics model depending on the overall domain size. The current implementation

in the popular open-source code such as GROMACS makes the suggested model available to

other researchers working in the area of atomistic scale resolving simulations of non-

equilibrium flows.

Code availability

The multiscale code with all parameter files and README can be downloaded following the

GitHub link: https://github.com/ikorotkin/gromacs_fhmd-langevin_non-equilibrium

Acknowledgements

The work of XL was supported by the China Scholarship Council (CSC) and Postgraduate

Research & Practice Innovation Program of Jiangsu Province, Grant No. KYCX21_2174. IK

and SK gratefully acknowledge the funding under the European Commission Marie

Skłodowska-Curie Individual Fellowship Grant No. H2020-MSCA-IF-2015-700276

(HIPPOGRIFFE). The work was also supported by European Commission in the framework of

the RISE program, Grant No. H2020-MSCA-RISE-2018-824022-ATM2BT and the National

Natural Science Foundation of China, Grant No.51776218.

Received: ((will be filled in by the editorial staff))

Revised: ((will be filled in by the editorial staff))

Published online: ((will be filled in by the editorial staff))

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