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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