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Research Article
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Optimizing Water Transport through Graphene-Based Membranes:
Insights from Nonequilibrium Molecular Dynamics
Jordan Muscatello,*,† Frederike Jaeger,‡ Omar K. Matar,† and Erich A. Müller†
†
Department of Chemical Engineering, and ‡Department of Physics, Imperial College London, SW7 2AZ, London, U.K.
S Supporting Information
*
ABSTRACT: Recent experimental results suggest that
stacked layers of graphene oxide exhibit strong selective
permeability to water. To construe this observation, the
transport mechanism of water permeating through a
membrane consisting of layered graphene sheets is investigated
via nonequilibrium and equilibrium molecular dynamics
simulations. The effect of sheet geometry is studied by
changing the offset between the entrance and exit slits of the
membrane. The simulation results reveal that the permeability
is not solely dominated by entrance effects; the path traversed
by water molecules has a considerable impact on the
permeability. We show that contrary to speculation in the literature, water molecules do not pass through the membrane as
a hydrogen-bonded chain; instead, they form well-mixed fluid regions confined between the graphene sheets. The results of the
present work are used to provide guidelines for the development of graphene and graphene oxide membranes for desalination
and solvent separation.
KEYWORDS: graphene, membranes, separation, permeation, molecular dynamics, water, confined fluids
1. INTRODUCTION
further insight into the mechanisms underlying the observed
high water permeance.
Previous studies of water transport in confined nanogeometries have presented surprising results. Most notable is
the observation that water molecules can flow within carbon
nanotubes at rates that surpass by orders of magnitude
predictions based on Hagen−Poiseuille flow.10,11 Such transport behavior can be attributed to the occurrence of slip flow at
the hydrophobic carbon walls as well as confinement effects on
the density distribution of water molecules. This phenomenon
has since been observed experimentally. However, a microscopic understanding has been predominantly provided by
simulation studies.12
A further pertinent element of discussion is the quantification
of the energetic barriers required for a water molecule to
remove itself from the bulk state and enter into a nanoconfined
system. For round entrances to nanotubes or perforated
graphene sheets, these energy barriers can be significant and
have to be considered when accounting for the use of
nanoporous carbons as membrane materials.13 Modification
of these barriers is fundamental to maximizing the efficiency of
transport, and this mechanism is key to the ion selectivity of
nanochannels in biological membranes.14 In contrast to the
observations made of water permeation, the MD studies of gas
diffusion through GO layers do not seem to show any
As the recent discovery of lower dimensional allotropes of
carbon (fullerenes and nanotubes) generated an abundance of
research around their applications, the isolation of graphene has
sparked an interest in using these materials as the basis for thin
membranes capable of remarkable separations. Most relevant to
this manuscript are the recent papers by Geim et al.1 that claim
that thin film graphene oxide (GO) sheets are capable of a high
water permeance while presenting an extremely restricted
porosity. Subsequent papers have confirmed these experimental
observations,2 and the field is now open for new technologydriven applications of this phenomenon, particularly in the area
of membrane separations. Here one can envisage membranes
with an active layer made entirely from GO3,4 or otherwise the
incorporation of GO flakes within a membrane material5,6 with
the aim of water desalination and/or solvent purification.
A proposed explanation of the apparent paradox raised by
the above-mentioned helium-tight but water-permeable membrane is that water forms a linear “conga line” consisting of a
hydrogen bonded network that stretches from the entrance to
the exit of the membrane,7 resulting in a collective motion of a
single monolayer of water molecules. This view, presented in a
schematic way in the seminal papers,1 has been reproduced in
further publications8 and is becoming the de facto most
plausible explanation. However, other works question this
mechanism.9 The present work revisits some aspects of this
view and employs molecular dynamics (MD) simulations of
water permeation through GO-like sheets in order to gain
© 2016 American Chemical Society
Received: December 11, 2015
Accepted: April 28, 2016
Published: April 28, 2016
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particularly remarkable results,15 and it seems understandable
that the low porosity of typical GO membranes allows only for
Knudsen-type diffusion, which is extremely limited.
We present here a study of the flow of water through a GOlike membrane with an appropriate geometry, i.e., a net water
flow normal to the plane of graphene flakes. The results of the
permeance for different geometries, calculations of the potential
of mean force and of the average mean number of hydrogen
bonds paint a picture of the transport that is at odds with the
current explanations available. Specifically, we show that energy
barriers exist which coincide with the breaking and forming of
hydrogen bonds during the passage of water through the
membrane and that significant mixing of water molecules
occurs within the membrane.
The rest of this paper is organized as follows. The
methodology is outlined in section 2, a discussion of the
results is presented in section 3, and concluding remarks are
provided in section 4.
conditions applied in all directions. The membrane was formed
of graphene sheets with a slitlike opening placed in the center
of the cell in the xy plane. Membranes consisting of both a
single and double layer of graphene were simulated. As
indicated in Figure 1, the center-to-center separation of carbon
2. METHODOLOGY
2.1. Boundary-Driven Nonequilibrium Molecular Dynamics. In this work we use a boundary-driven nonequilibrium
molecular dynamics (NEMD) method to establish a constant
pressure difference, and hence maintain a steady state mass flux,
across a graphene membrane. For this purpose a force along the
z-axis is applied to all molecules located in a thin slab of width
dext = 10.0 Å approximately half the system cell length away
from the membrane. In doing so a density gradient is set up in
this layer, resulting in a net mass flux in the simulation cell and
a corresponding pressure drop across the membrane. In these
simulations the force is applied only to the oxygen sites of each
SPC/E water molecule. Increasing the applied force results in a
proportional increase in pressure drop; thus the linearity
between the pressure drop and the resultant mass flux can be
investigated. This method has been thoroughly described in
previous studies of mass transport across porous media.16−18
The externally applied force is specified as a multiple of the
ratio between the ϵ and σ Lennard-Jones parameters for the
oxygen atom of an SPC/E water molecule19 and will be
denoted in such units, i.e.,
Fext = A ϵ/σ
Figure 1. Geometry of the double-sheet system (a) and resultant
pressures profiles for various applied forces (b). The bottom panel was
generated from the c = 0.0 simulation. The intersheet distance and
carbon−carbon slit opening is fixed at 6 Å. The dashed red lines in the
bottom figure indicate the positions of the atoms comprising the
graphene sheets.
atoms between adjacent sheets was fixed at 6 Å. This value is
commensurate with other computational studies of hydrated
graphene oxide using reactive force fields.24,25 The carbon−
carbon opening of the slit, dslit, was initially fixed at 6 Å. The
free edges of the graphene sheet were functionalized with
hydrogen atoms, using the parameters for aromatic groups from
Mooney et al.22 Water molecules were represented using the
SPC/E model.19 The interactions between water and carbon
were modeled using oxygen−carbon Lennard-Jones parameters
fit to give the macroscopically observed wetting behavior on
graphitic surfaces.26 A time step δt = 2 fs was used for all
simulations. The SHAKE algorithm was used to constrain the
bonds and angle of the SPC/E molecule.27
Electrostatic interactions were evaluated using the modified
Wolf method proposed by Fennel and Gezelter whereby the kspace part of the Ewald summation is not computed but instead
replaced by a damping function.28 This has been observed to
yield equation of state behavior in fluid phases comparable to
the full Ewald sum.29
Results obtained from nonequilibrium simulations were
produced from averages over runs of 10−16 ns with a 2 ns
transient period where the system was allowed to reach a
steady-state mass flux.
For the case of a double-sheet membrane the geometry of
the system can be specified by changing the offset doff between
the openings of the graphene sheets as a fraction of the cell
length along the y-axis. In this work we consider three offsets
doff = cLy, where c = {0.0, 0.25, 0.5}, to study the effect of
increasing the relative distance between the openings in the
sheet. Each carbon atom in the membrane is tethered to a fixed
(1)
where we have chosen A to vary from 0.2 to 1.0.
In the application of the externally applied force, the system
undergoes a constant energy input. So as to perturb the
dynamics of the water molecules as little as possible, a
Berendsen thermostat20 was applied to the graphene sheet only
such that the membrane dissipates excess energy. This is
preferred to the case of applying a Nosé−Hoover or Langevin
thermostat to the whole of the bulk fluid, which may artifically
affect the dynamics, especially if the streaming velocity of the
particles is not taken into account.12,21 In order to promote
energy dissipation in the membrane, the carbon atoms of the
graphene sheet are tethered to their nearest neighbors via
harmonic bonds with spring constant kC−C = 478. 4 (kcal/
mol)/Å2 equal to the bond strength for aromatic carbon.22 The
thermostat was set to 300 K. In this work the usual velocitiesVerlet algorithm is used to integrate the equations of motion of
the particles. The simulations were performed with the
LAMMPS parallel molecular dynamics package.23
2.2. Computational Details and System Geometry. In
all simulations an orthogonal system cell with extents (Lx, Ly,
Lz) = (18.4, 23.0, 105.0) Å was used with periodic boundary
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point in space by harmonic bonds with spring constant ktether=
100 (kcal/mol)/Å2 such that the membrane does not change
overall position in the cell when a flux is applied to the water
molecules. This emulates a membrane held in place by an
external support, in that we do not consider the mechanical
properties of the membrane under applied pressure in this
work. The double-sheet system is illustrated in Figure 1 along
with the resultant hydrostatic pressure profiles for varying
applied forces in a system with zero offset, c = 0.0, between the
graphene sheets. The pressure profile was calculated by taking
the average of the diagonal components of the Irving−
Kirkwood expression for the pressure tensor.30 The transmembrane pressure drop was calculated using the difference in
hydrostatic pressure of the bulk fluid on either side of the
membrane.
are clearly visible in Figure 1. This is in accordance with the
definition of the width of a single graphene sheet used by other
authors.33 In doing so, we can extract values for the
permeability of each system geometry from the slope of the
linear fitting to the data. It should be noted that in this case the
linear expression is purely empirical, as hydrodynamic arguments break down at pore sizes of molecular scale.34
The permeability values obtained are shown in Table 1. It
can be seen that the permeability of two sheets decreases with
increasing sheet offset. This dependence shows that the
permeability depends not only on the porosity of the
membrane (which is fixed in this case) but also on the length
of the path taken by the water molecules. The permeability for
a single sheet was found to be approximately 3 times that of the
double-sheet with inline slits.
Care should be taken in comparing the permeabilities for the
single and double slit membranes. Permeability is expressed as a
material property of homogeneous membranes (normalized by
the width of the membrane). However, due to the discrete
number of sheets, the single and double sheets cannot be
considered to have equivalent material properties in this case.
An additional energetic penalty is present in the double sheet
due to the presence of the monolayer of water (see discussion
below). In the limit of a large number of sheets a graphene
based membrane will appear homogeneous at the scale of the
total membrane and the permeability should reach a constant
value. In this study we do not attempt to find this limit.
3.2. Potentials of Mean Force (PMF). To investigate the
energetics of water molecules entering the membrane, the
potential of mean force (PMF) was calculated along a reaction
coordinate which crosses perpendicularly the entrance to the
graphene sheet.
An umbrella sampling method in the canonical ensemble was
used for this purpose.36 The Nosé−Hoover thermostat was
used at T = 300 K with a damping parameter μdamp = 100 fs. A
molecule is chosen at random and attached to a harmonic bias
potential, with equilibrium position at a point along the z-axis,
aligned with the center of the slit opening. The harmonic
1
potential is described by Vspring = 2 k(z − z 0)2 , where k is the
spring constant and z0 the equilibrium position. After an
equilibration period, the position distributions were collected
along the reaction coordinate in steps of 0.2 Å over a period of
6 ns using a spring constant ranging from 5.0 to 12.6 kJ/(mol·
Å2). The data were then analyzed using the weighted histogram
analysis method (WHAM).35
Figure 3 (panels a and b) shows the PMFs for the single
sheet and a zero offset double-sheet membrane with reference
to the bulk with slit width of 6 Å. In both cases free energy
barriers are shown to emerge at the entrance to the membrane.
When compared to the density profile (Figure 3, panels e and
f), it can be seen that these barriers coincide with molecular
3. RESULTS
3.1. Permeability. Figure 2 shows the molar flux of water
molecules versus the transmembrane pressure drop for three
Figure 2. Molar flux of water through membranes with increasing
offset values against transmembrane pressure drop. Symbols are
simulations results. Dashed lines are a linear fitting. Errors bars
represent the standard error.
offset geometries for the double-sheet system as well as for a
single sheet with dslit fixed at 6 Å. We assume that a linear
relationship holds between the molar flux and the pressure drop
for this range of applied pressures. With this assumption, a
relationship resembling Darcy’s law can be applied,31,32
Jz = k
ΔP
lmem
(2)
where Jz is the molar flux through the membrane, ΔP is the
transmembrane pressure drop, k is the coefficient of
permeability, and lmem is considered to be an effective
membrane width. Here we define lmem to be the region over
which the pressure drop occurs, taken from the density profile
as the distance from one bulk region to another. These peaks
Table 1. Table of Effective Membrane Widths (lmem), Permeabilities (k), Mean Residence Times (τres), Mean Hydrogen Bond
Lifetimes (τHB), and Diffusion Coefficients (Dmono) for All Double Slit Simulations and if Applicable Also for a Single Sheeta
c = doff/Ly
lmem (Å)
0.0
0.25
0.5
single sheet
34.0
34.0
34.0
28.0
k (×10−17 m2 s−1 Pa−1)
τres (ps)
τHB (ps)
Dmono (×10−5 cm2 s−1)
±
±
±
±
13.7 ± 1.6
12.0 ± 0.7
17.3 ± 1.5
1.86 ± 0.10
2.28 ± 0.10
2.19 ± 0.10
0.65 ± 0.21
0.72 ± 0.21
0.73 ± 0.21
0.35
0.22
0.08
1.02
0.05
0.03
0.02
0.05
To obtain permeability values in units of m2 s−1 Pa−2, the molar flux was converted to a volumetric flux using the density of water at standard
temperature and pressure.
a
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Figure 3. PMF (panels a and b), hydrogen bonds per molecule (panels c and d), and density (panels e and f) profiles for single sheet (left) and
double-sheet (right, c = 0.0) systems with dslit = 6 Å. The positions of the graphene sheets are indicated by dashed red lines. Free energy barriers
coincide with density peaks due to molecular ordering at the sheet entrances and the breaking of hydrogen bonds as molecules pass through the
membrane. Error bars on the PMFs were obtained via a Monte Carlo bootstrap algorithm.35
to the ring distributions of water molecules observed in carbon
nanotubes.38 Density peaks appear in quantitatively similar
positions.
3.3. Hydrogen Bonding Profile. The degree of hydrogen
bonding was calculated along the z-axis. A hydrogen bond is
considered to exist between two molecules if the following
distance and orientation criteria are met: ROO < 3.4 Å, ROH <
2.425 Å, and the H−O···O angle is less than 30°, where ROO
and ROH are the radial oxygen−oxygen and oxygen−hydrogen
distances, respectively, on neighboring molecules.39,40
The center panels in Figure 3 (panels c and d) show the
average number of hydrogen bonds per molecule calculated in
slabs along the simulation cell for both a single and a doublesheet membrane. The average number of hydrogen bonds in
the bulk is ∼3.5 in both cases. The degree of hydrogen bonding
drops rapidly at the entrance to the membrane, indicating that
there is an energetic penalty incurred as a molecule enters the
membrane; hydrogen bonds must be broken as a molecule
passes through the slit. This provides a physical explanation of
the free energy barriers shown in the top panel. For the case of
the double-sheet the degree of hydrogen bonding in the region
between the sheets approaches that found in the bulk. This
indicates that even though the width of the spacing can only
support a monolayer of water and is therefore too narrow to
support a fully tetrahedrally ordered network, hydrogen
bonding is still present. The relative increase in the height of
the free energy barrier at the slit compared to the single sheet
case can be interpreted as due to an increased disruption and
subsequent rearrangement of the hydrogen bonded network in
order to form the monolayer.
3.4. Residence Times and Water Molecule Dynamics.
The residence time distribution of water molecules in the
membrane for the double-sheet system was calculated for each
sheet offset value. The times were logged only for molecules
contributing to the net mass flux; i.e., the residence time tres =
tout − tin is logged only if a molecule enters the membrane from
the left sheet and subsequently exits through the right sheet
adsorption on the graphene sheets. For the case of the doublesheet the energetic barrier at the slit is significantly higher than
for the single sheet. This can be interpreted as being due to the
presence of a monolayer of water in the intermembrane region
exhibiting a different structure and higher density from the
bulk. Such high density configurations have been frequently
reported in confined systems.37
PMFs were also calculated for single sheets with varying slit
widths. Figure 4 shows the height of the energy barrier at the
Figure 4. Value of potential of mean force at the center of a slit for a
range of slit widths in a single-sheet system. The inset plot shows the
density profile in the y-direction as a function of distance from the slit
center.
slit for dslit = {5, 6, 9, 15} Å. In general, the barrier increases
with decreasing slit width. For larger slit widths confinement
effects become less prevalent indicating a return to bulklike
behavior, as shown in the inset plot illustrating the density
distribution within the slit.
As the slit width is increased, a transition from a single
monolayer to a layer at each edge is observed. This is analogous
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(with respect to the direction of flux). The normalized
residence time distributions are shown in Figure 5. In all
distribitions. This becomes apparent when the mean values are
compared (see Table 1), since the mean residence times are
greater than the hydrogen bond lifetime by a factor of at least 5,
despite the similarity of the distributions at short times.
The hydrogen bond lifetime and associated probability
distribution function were calculated for each offset for NEMD
simulations with A = 1.0. The hydrogen bond probability
distribution was computed for water molecules located in the
membrane in accordance with Han et al.,42 using the same
criteria for the existence of a hydrogen bond as mentioned in
section 3.3. The hydrogen bond lifetime is given by42
τHB =
∞
∫0
tpHB (t ) dt
(3)
where pHB(t) is the probability distribution. Figure 6 shows pHB
for c = 0.25. The values for τHB are given in Table 1. These
values are small compared to the average residence times,
suggesting that during the time that a water molecule remains
in the membrane, the hydrogen bonded network is undergoing
continuous breaking and re-forming.
To further investigate the dynamic behavior of water
molecules within the monolayer, the velocity autocorrelation
function (VACF) was calculated for NEMD runs with A = 1.0
in the xy plane such that only the diffusive motion of water
molecules in the monolayer was taken into account; i.e., only
the vx and vy components of the velocity were used. Figure 6
shows the results for the c = 0.25 case and that of bulk water,
where the normalized VACF is given by
Figure 5. Normalized residence time distributions for water molecules
in the double-sheet membrane with different offset values. The inset
plot shows the same data on log−log axes, as well as the hydrogen
bond lifetime distribution for c = 0.0.
cases the distribution decays approximately as a power law, as
indicated by the inset log−log plot. This shows that although a
water molecule is more likely to spend a short period of time in
the membrane, some molecules can remain diffusing in the
monolayer. The mean residence time (see Table 1) for the c =
0.5 case is approximately 25% longer than the c = 0.25 case.
This may be surprising, as the average path length for the offset
cases, assuming an equal probability of a molecule taking the
most direct route out of the membrane, is the same, since in the
c = 0.25 offset there is also a longer path the molecules can take
(0.75 × Ly).
Analogous to the case of fluid elements in a chemical reactor,
the residence time distribution reveals information regarding
the degree of mixing of water molecules in the membrane.41
For the hypothesized case of a single hydrogen bonded sheet of
water molecules moving collectively through the membrane the
flow would resemble a “plug-flow”-like scenario in which the
residence time distribution is single-valued, i.e., a δ function.
While there is a strong peak for shorter residence times, the
distributions all feature a slowly decaying tail. This indicates
that a significant contribution is made by molecules remaining
in the membrane for longer times. An animated visualization of
this diffusive behavior is higlighted in the Supporting
Information. The inset log−log plot in Figure 5 shows the
residence time distributions compared to the hydrogen bond
lifetime distribution for the c = 0.0 case. At short time scales the
distributions appear similar. However, the hydrogen bond
lifetime does not feature the long decay of the residence time
Ψ(t ) =
⟨v(0) ·v(t )⟩
⟨|v(0)|2 ⟩
(4)
and
⎛v ⎞
v = ⎜⎜ x ⎟⎟
⎝ vy ⎠
for molecules located between the carbon sheets. Integrating
the non-normalized VACF yields the diffusion coefficient for
molecules diffusing in the xy plane, given by
Dmono =
1
d
∫0
∞
⟨v(0) ·v(t )⟩ dt
(5)
where d = 2 for the case of diffusion in a plane.
Comparing the VACFs of confined and bulk water, it can be
seen that there is a stronger negative correlation for the
confined molecules than those of the bulk. This is analogous to
the VACF observed in amorphous solids or liquids below the
glass transition.43 The diffusion coefficient was calculated from
Figure 6. (a) Normalized velocity autocorrelation function for water molecules located between carbon sheets for c = 0.25 and for bulk water. (b)
Hydrogen bond lifetime distribution for water molecules located between carbon sheets for c = 0.25.
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■
the VACFs for all offsets, yielding values of ∼0.7 × 10−5 cm2
s−1. This value is lower than that of bulk water by a factor of 5
rather than orders of magnitude lower as would be expected for
glassy system, indicating that liquid-like diffusion is still
prevalent. Furthermore, the time scale associated with diffusion
in the sheet is much shorter than the average residence time.
Therefore, significant diffusive mixing is to be expected for a
molecule remaining in the sheet.
Research Article
ASSOCIATED CONTENT
S Supporting Information
*
The Supporting Information is available free of charge on the
ACS Publications website at DOI: 10.1021/acsami.5b12112.
Video highlighting a single molecule diffusing through
the membrane with c = 0.5 (MPG)
Additional simulation results describing flow through a
membrane only permitting a single path (PDF)
■
4. CONCLUSIONS
In this study we have observed the permeation of water through
stacked graphene layers via nonequilibrium molecular dynamics
simulation. In particular, we investigated the effect of slit widths
and sheet geometries on the transport of water molecules. For
narrow entrances there are energetic barriers on either side of
the membrane due to molecular layering of the adsorbed water
as well as a barrier associated with the passage through the
opening, all of which serve to hinder transport. As the slit width
is increased, bulk behavior is recovered, where no barriers are
present. In addition, there is a further energetic penalty
associated with the formation of a confined phase between
multiple sheets. This is due to molecular rearrangement that
must occur when a molecule passes from the bulk to the
monolayer. Hydrogen bonds are broken and then re-formed as
a molecule enters the membrane.
Results readily reveal that the sheet offset also has a
considerable effect on the resultant permeability of the
membrane, whereby it is favorable for the slits to be as close
as possible in adjacent stacking layers.
The previously proposed mechanism for confined water
transport through graphene-based membranes involves the
continuous pluglike flow of a hydrogen bonded sheet of water.
Other authors have shown alternative mechanisms for fast flow
in confined geometries that support more than one layer of
water.9 In addition we have shown that for highly confined
single monolayer water the previously proposed description is
not accurate. The residence time analysis instead reveals a high
degree of mixing as molecules permeate through the
membrane, supported by analysis of the dynamical properties
of the hydrogen bonded monolayer. Analysis performed on an
additional system permitting only a single route through the
membrane indicates that the above conclusions are not an
effect of the periodicity of the simulations. These results are
presented in the Supporting Information.
The above conclusions suggest clear guiding principles for
the design of graphene-based membranes in terms of
maximizing the transport of water while still maintaining the
potential for use in separation processes. This involves
minimizing the energy barriers upon entrance to the
membrane. We have shown that unimpeded entry to the
sheets can be achieved by using slit widths greater than ∼3
molecular diameters. It has been suggested that tunable sheet
spacings may be achieved by covalently bonded “linkers”.44,45
Rejection properties may also be manipulated in the same way
by ensuring that the spacing between sheets only admits a
single monolayer of water. In summary, by optimization of the
configuration at the molecular scale, the viability of graphene
oxide as a potential material for separations (e.g., desalination,
alcohol dehydration, etc.) can be maximized.
AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected].
Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS
The authors acknowledge funding from the Thomas Young
Centre under Grant TYC-101 and EPSRC Grants EP/E016340
and EP/J014958. F.J. was supported through a studentship in
the Centre for Doctoral Training on Theory and Simulation of
Materials at Imperial College London funded by the Engineering and Physical Sciences Research Council under Grant EP/
G036888/1. F.J. thanks Dr. Carlos Braga for advice regarding
the calculation of PMF profiles. J.M. thanks Prof. Adrian Sutton
and Dr. Arash Mostofi for comments and suggestions.
■
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