Microfluidics and Nanofluidics    

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Supplementary Material:

Combining Molecular Dynamics and Lattice Boltzmann Simulations: A

Hierarchical Computational Protocol for Microfluidics

Aline O. Pereira · Lucas S. Lara · Caetano R. Miranda

Benchmark Example: Montmorillonite Channel filled with API Brine

In this benchmark example we simulate a flux of an API brine (8% NaCl and 2% CaCl2)

solution inside a Montmorillonite (MMT) channel using the proposed hierarchical

computational protocol combining Molecular Dynamics (MD) and Lattice Boltzmann

Method (LBM) simulations.

The computational details for the MD simulations are similar to those employed in the

manuscript paper. However, in this case the dimensions of the computational box are

12.0x10.0x12.5 nm3. It has been considered 32000 H2O molecules and the salt is randomly

inserted in solution according with the concentrations of 8.0% NaCl and 2.0% CaCl2. At the

end of the system equilibration, the channel width is of about 6 nm. The MD simulation box

is schematically represented in Figure S1(a). After the equilibration a non-equilibrium MD

simulation is performed. In this case an external force of Fx = 5.16 x 10-12 kg/m2s2 is applied

to the system in the x-direction (parallel to the MMT layers). The velocities for the H2O

molecules and ions are collected every 0.1 ps for 8 ns, in bins along y-direction of width

0.1 Å, after an equilibration period of 2 ns.

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Figure S1. (a) Molecular dynamics snapshot of the computational box after equilibrium is reached. This is composed

of a Montmorillonite channel filled with an API brine solution. The Na, Cl and Ca ions are represented by blue, green

and red dots, respectively. (b) The 2D compatible Lattice Boltzmann simulation box representing the Montmorillonite

channel with 6 nm of empty space. The black lines represent the Montmorillonite walls and the blue region the brine

solution. The computational box dimensions are 80x160 l02 (lattice units).

The equilibrium MD density and viscosity were mapped into LBM simulation parameters

and a LBM flux simulation of brine solution inside a MMT channel is performed. The LBM

simulation box is schematically represented in Figure S1(b). The MD physical parameters

and the correspondent LBM simulation parameters are show in Table S1. In the LBM

simulations the external force is include as an external gravitational force in the x-direction

(Fx = ρ g = 8.0 x 10-8).  

Table S1: Molecular dynamics equilibrium physical parameters and the corresponding Lattice Boltzmann simulation

parameters. The characteristic scale are: l0 = 7.5x10-11 m, t0 = 1.09x10-15 s and m0 = 4.29x10-28 kg.

Molecular Dynamics Lattice Boltzmann

Density 1017 kg/m3 1.000

Kinematic Viscosity 0.860 x 10-6 m2/s 0.167 (τ = 1.0)

External Force (x-direction) 5.160 x 10-12 kg/m2s2 8.000 x 10-8

(a) Molecular Dynamics simulation box (b) Lattice Boltzmann simulation box

x

y

y

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The comparison of the velocity profiles obtained via MD and LBM simulations is

shown in Figure S2. As expected, the maximum velocity is found in the center of the

channel. The maximum velocity intensity for MD and LBM are 25.48 m/s and 25.76 m/s,

respectively. These results show that, even though the LBM velocities are 0.28 m/s higher

than the MD velocities, the proposed hierarchical computational model is consistent, i. e.,

after the mapping of the MD physical parameters into LBM simulation parameters the

velocity profiles are in agreement for both simulation methods.

The difference found between the velocity profiles of MD and LBM may be related to

the width of the channel and the interaction between the channel walls and the fluid. Since

the MMT surface is not totally flat, the width varies around 6nm along the channel. In the

LBM simulations the channel width is set to 6nm for the whole channel, and this may cause

some discrepancies in the velocity profiles. Also, in the LBM simulations the wall is

considered as a barrier, where the fluid particles that reach the wall are bounced-back to the

fluid domain (gMMTb = 0). These approximations may be the origin of such slight deviation

between these velocity profiles.

Figure S2: Velocity profiles for Molecular Dynamics and Lattice Boltzmann simulations of an API brine solution in a

montmorillonite channel with 6.0 nm of width.

Fx#

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Error Estimation

The error bars for the Molecular Dynamic simulation parameters have been already

discussed in previous papers from Lara et al. [J. Chem. Phys. 136, 164702 (2012); J. Phys.

Chem. 116, 14667 (2012)]. For the systems studied in our manuscript paper, the most

important error bar to consider is for the interfacial tension, which is determined in MD

simulations with an error bar of 0.001kg/s2.

In order to investigate how these errors in the interfacial tension would affect the LBM

results we explore the oil displacement process by brine and brine+NP-PEG2 in the pore

network model shown in Fig.2(e) of the manuscript, under an injection rate of 0.003 l0/t0. To

do so, we consider the error in the interfacial tension δγ = 0.001 kg/s2, and evaluate the areal

sweep efficiency number (Se) for γ+δγ and γ-δγ. These results are also compared to the

results presented in the manuscript (γ). This way, we can determine the changes in the areal

sweep efficiency number due to the error bar of the interfacial tension (±δγ).

In such calculations, the LBM simulation parameters for viscosity (relaxation time)

and wetting parameters are the same as in Tab. 2 of the manuscript. The parameters for

interfacial tension (gbo) are:

System MD Interfacial

Tension (kg/s2)

LBM Interfacial

Tension

(dimensionless)

LBM

parameter gbo

NoNP: γ 0.043 0.100 0.190

NoNP: γ+δγ 0.044 0.103 0.192

NoNP: γ-δγ 0.042 0.098 0.188

NP-PEG2: γ 0.029 0.066 0.164

NP-PEG2: γ+δγ 0.030 0.070 0.166

NP-PEG2: γ-δγ 0.028 0.065 0.162

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In Figure S3 we present the final fluid configurations for the

Montmorillonite/Oil/Brine and Montmorillonite/Oil/Brine+PEG2 (Pegylated nanoparticles

dispersed in brine solution) systems, at 11.32 µs. From these results we can see that a

variation of approximately ±0.003 on the Se number as we change the interfacial tension

from γ to γ±δγ. Therefore, an error of δγ = ±0.001 kg/s2 in the interfacial tension value

(originated from MD simulations) will lead to an error of δSe = ±0.003 in the areal sweep

efficiency number.

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Figure S3: Extrapolation tests on the effect of interfacial tension errors with the fluid final configurations (at 11.32

µs). Black, blue, and red lines represent the fluid interfaces, for γ, γ+δγ, and γ-δγ, respectively for the

Montmorillonite/Oil/Brine and Montmorillonite/Oil/Brine+PEG2 (Pegylated nanoparticles dispersed in brine

solution) systems.

ROCK%

OIL%

BRINE&

Without%Nanopar3cles:%OIL/BRINE%�: Se = 0.628� + ��: Se = 0.625� � ��: Se = 0.631

ROCK%

OIL%

BRINE+PEG2*

With%PEG2%Nanopar6cles:%OIL/BRINE+PEG2%�: Se = 0.669� + ��: Se = 0.666� � ��: Se = 0.671