Institute for Computational Physics, University of Stuttgart, 70569 Stuttgart,
Germany Univ. Grenoble Alpes, CNRS, LIPhy, 38000 Grenoble, France Institut
Laue Langevin, F-38042 Grenoble, France schlaich24b_twist-a_supplement.pdf

# Bridging Microscopic Dynamics and Hydraulic Permeability in Mechanically-
Deformed Nanoporous Materials

Alexander Schlaich<EMAIL_ADDRESS>Stuttgart
Center for Simulation Science (SC SimTech), University of Stuttgart, 70569
Stuttgart, Germany  Matthieu Vandamme  Navier, Ecole des Ponts, Univ. Gustave
Eiffel, CNRS, Marne-la-Vallée, France  Marie Plazanet  Univ. Grenoble Alpes,
CNRS, LIPhy, 38000 Grenoble, France  Benoit Coasne  benoit.coasne@univ-
grenoble-alpes.fr Univ. Grenoble Alpes, CNRS, LIPhy, 38000 Grenoble, France

###### Abstract

In the field of nanoconfined fluids, there are striking examples of
deformation/transport coupling in which mechanical solicitation of the
confining host and dynamics of the confined fluid impact each other. While
this intriguing behavior can be potentially used for practical applications
(e.g. energy storage, phase separation, catalysis), the underlying mechanisms
remain to be understood as they challenge existing frameworks. Here, using
molecular simulations analyzed through concepts inherent to interfacial
fluids, we investigate fluid flow in compliant nanoporous materials subjected
to external mechanical stresses. We show that the pore mechanical properties
significantly affect fluid flow as they lead to significant pore deformations
and different density layering at the interface accounted for by invoking
interfacial viscous effects. Despite such poromechanical effects, we show that
the thermodynamic properties (i.e. adsorption) can be linked consistently to
Darcy’s law for the permeability by invoking a pore size definition based on
the concept of Gibbs’ dividing surface. In particular, regardless of the pore
stiffness and applied external stress, all data can be rationalized by
accounting for the fluid viscosity and slippage at the interface independent
of a specific pore size definition. Using such a formalism, we establish that
the intimate relation — derived using the linear response theory — between
collective diffusivity and hydraulic permeability remains valid. This allows
for linking consistently microscopic dynamics experiments and permeability
experiments on fluid flow in compliant nanoporous materials.

Fluid/solid interfaces, which are inherent to vicinal and confined liquids as
encountered in nanofluidic devices and nanoporous materials, are host to a
wealth of molecular mechanisms such as adsorption and chemical reactions, but
also to electrokinetic aspects (electrical double layer, crowding, etc.).1, 2,
3, 4, 5, 6 With the boost in nanosciences and nanotechnologies, these specific
surface phenomena are already implemented in important applications such as
energy storage, catalysis, lubrication or depollution.7, 8, 9, 10, 11 Yet,
despite significant progress in our understanding of surface forces and
confinement effects on the thermodynamics and dynamics of fluids, the behavior
of nanoconfined systems still challenges existing frameworks even when simple
liquids are considered. Among important aspects that remain to be understood,
there are now important experimental and numerical reports on the role of
mechanical deformation on transport of gases and liquids in nanoconfined
environments.12, 13, 14, 15, 16 Formally, the coupling between mechanics of
the confining host and dynamics of the confined fluid can manifest itself at
various scales and in different fashions. Locally, i.e. at the fluid/solid
interface, momentum transfer between phonons in the solid phase and molecules
in the fluid phase are expected to give rise to complex surface thermodynamic
and dynamical aspects.16, 13, 15 At the pore scale and beyond, mechanical
deformation of the pore network drastically impacts the fluid microscopic
dynamics and, in turn, its macroscopic permeability.17, 18, 19, 20, 21

Originally, despite the complexity of biological channels and biological
objects in general, biology is an important field in which fascinating
mechano-transport mechanisms were identified. Detailed investigation on ionic
channels and lipid membranes have unraveled complex phenomena such as the
mechanotransduction response of ionic transfer upon mechanical or pressure
solicitation.22 First systematic efforts to stimulate such coupling employed
man-made nanochannels (carbon nanotubes).12 Recently, using single digit
experiments on carbon nanotubes, Marcotte et al. were able to reproduce
mechanically activated ionic transport.15 Using 2D nanoporous membranes, Noh
and Aluru used molecular dynamics simulations displaying a striking impact of
membrane mechanics on water desalination.16 Depending on the deformation
amplitude and frequency, these authors observed a drastic effect on water
permeability accompanied by a small decrease in salt rejection. By analyzing
in detail the dynamics of the deformable membrane and vicinal water, the
vibrational matching between the membrane and water molecules was found to be
the key factor governing the resulting flow. Despite the important works cited
above, several key questions remain left unanswered. In particular, while
available data point to the phonon–fluid coupling as the origin of flow
modifications, the exact role of mechanical deformation — including its
quantitative impact — remains to be assessed. In particular, the validity of
the intricate connection between permeability — as defined in macroscopic
experiments such as in Darcy’s law — and the collective microscopic
diffusivity remains to be established when mechanical solicitation is applied.
Extending such formalism to porous materials and, in particular, to nanoporous
solids is an important step to design mechanical control and stimulation of
fluid adsorption and flow in confined geometries.

Darcy’s law for fluid flow through a porous medium has originally been
proposed for macroscopic permeability measurements.23 Yet, it can be
rigorously derived from fundamental conservation laws, i.e. the Navier–Stokes
equations.24, 25 In detail, in the case of a single-phase fluid considered
here (assumed to be confined in the $z$-direction), one considers the
molecular flux

$J=\int_{-L_{z}/2}^{L_{z}/2}\rho(z)v(z)\,\mathrm{d}z$ (1)

induced by a pressure gradient $\nabla P$, where $\rho(z)$ is the local fluid
molecular density and $v(z)$ its corresponding velocity. The integration
boundaries denote a length-scale $L_{z}$, which corresponds to the domain
size. If the density is taken to be homogeneous, this molecular flux
$J=\rho\bar{v}$ corresponds to a flow rate (mean velocity)
$\bar{v}=-k/\eta\nabla P=-K\nabla P$, where $\eta$ is the fluid viscosity, $k$
the permeability and $K=k/\eta$ the (hydraulic) permeance.26, 25 Direct
assessment of $K$ is difficult in general in molecular simulations since a
constant pressure gradient simulation needs to be set-up,27, 28, 29 requiring
for explicit treatment of reservoirs and therefore also introducing surface
effects.30 On the other hand, Onsager’s relation
$J=-D_{0}/\left(k_{\mathrm{B}}T\right)\times\nabla\mu$ relates the flux to the
chemical potential gradient $\nabla\mu=-f_{x}$ in the direction $x$ of the
flow ($k_{\mathrm{B}}$ and $T$ are the Boltzmann constant and temperature,
respectively). The latter situation can be directly considered in molecular
simulations through the measurement of the collective diffusivity $D_{0}$ as a
response to a constant force $f_{x}$. A general approach to relate the
transport coefficients due to different driving forces was developed by
Onsager.31 To relate the above Darcy and Onsager laws, an incompressible
liquid is usually assumed for which we can use straightforwardly the
Gibbs–Duhem equation: $\rho\,\mathrm{d}\mu=\mathrm{d}P$ (which therefore
relates the pressure and chemical potential gradients as driving forces). In
so doing, one obtains the following classical result for an incompressible
fluid:32

$K=D_{0}/\left(\rho k_{\mathrm{B}}T\right).$ (2)

Previous studies have considered density-dependence of diffusion, 33, 34, 35,
36 spatial density and viscosity heterogeneity,29, 37 and the the influence of
pore size38, 39 or flexibility on transport.16, 14 Here, we report on a
molecular simulation and theoretical approach to investigate the coupling
between mechanical load and flow permeability. Using a simple yet
representative model of nanoporous materials, we design a set-up in which the
flow of water induced by a driving force (pressure gradient) is monitored as a
function of the mechanical stress applied in a direction perpendicular to the
pore surface. To assess the validity of our results, both equilibrium and non-
equilibrium molecular dynamics are employed; using a fluctuation-dissipation
approach based on the Green–Kubo formalism allows verifying that the data and
analysis are not biased due to unphysical coupling between the flow and
deformation when driving forces are applied in non-equilibrium molecular
dynamics. For different pore sizes and mechanical loadings, we analyze the
role of the external mechanical stimulus (stress) and pore mechanical property
(stiffness) on the system’s response — both the resulting flow and mechanical
deformation are analyzed simultaneously to unravel their interplay. The
advective transport of confined water under mechanical solicitation is then
analyzed through the prism of the fundamentals of interfacial fluids as
developed in the field of nanofluidics and fluids confined in nanoporous
materials. This allows us to probe the origin of the impact of mechanical
deformation on flow through its effect on the interfacial viscous layer. Using
the linear response theory and the underlying fluctuation-dissipation theorem,
we show that — akin to non-compliant porous materials — the macroscopic
permeability can be linked to the microscopic collective diffusivity as probed
using simple molecular simulation. Beyond verifying its validity for compliant
systems, this fundamental relation provides a means to define the pore size
that consistently describes both the thermodynamics and dynamics of the
confined fluid. In particular, using excess quantities as defined in the Gibbs
dividing surface formalism, our data for the collective diffusivity and
permeability are found to quantitatively match.

Figure 1: Adsorption in compliant nanoporous materials. (a) Set-up of the
compliant slit-pore considered in this study. The positions of the left pore
wall atoms are kept fixed whereas the right pore wall is allowed to move as a
rigid body. The two pores are connected via a Hookean spring of stiffness
$\kappa_{zz}$ acting between their respective centers of mass (COM) and
several $\kappa_{zz}$ are considered to tune the mechanical pore properties.
An external stress $\sigma_{zz}$ is applied as a body force acting on the
fluctuating surface. For a given spring constant $k$, depending on the
chemical potential of the external reservoir, the pore gets filled as
indicated in (b). The resulting surface-to-surface separation fluctuates with
an average position $\langle H\rangle$ defined as the distance from the planes
passing through the atom centers at the wall external layer as indicated in
(a). (c) and (d) show water density profiles for a slit pore with $\langle
H\rangle\sim 1.7\,\mathrm{nm}$ for a stiff (c) and soft (d) compliant material
for different applied external stresses $\sigma_{zz}$. The chemical potential
$\mu$ is fixed to a value corresponding to a pressure of about 215 bar to
ensure all pores are filled. The vertical solid bars, which indicate the
positions of the lower and upper walls, define the pore spacing $H$ (the left
wall is fixed so that its position does not fluctuate). For
$\sigma_{zz}=1000\,\mathrm{bar}$ in (d) one water layer gets expelled, leading
to a reduction of the pore size from $1.6\,\mathrm{nm}$
($\sigma_{zz}=500\,\mathrm{bar}$) to $1.25\,\mathrm{nm}$.

Our setup consists of a compliant slit pore composed out of a realistic dense
and hydrophobic carbon material.40 The two pore walls separated by a distance
$H$ between the outermost carbon atoms are treated as a rigid body and
connected by a Hookean spring of stiffness $\kappa_{zz}$ acting between the
center of mass (COM) of the pore walls, see Fig. 1(a) and Section II of the
Supplementary Information. While the left wall in Fig. 1(a) is fixed in space,
the right wall is allowed to fluctuate freely and to respond to an external
stress in $\sigma_{zz}=f_{z}N_{\mathrm{s}}/(L_{x}L_{y})$ in the $z$-direction
normal to the surface (such an external constraint is applied via a force
$f_{z}$ that acts on all $N_{\mathrm{s}}$ surface atoms with
$L_{x}=L_{y}=2.5\,\mathrm{nm}$ the lateral dimensions of the periodic
simulation box). The harmonic potential $V(l)=\kappa_{zz}(l-l_{0})^{2}$, which
depends on the spring extension $l$, can be related to the slit pore’s young
modulus, $E_{zz}=\kappa_{zz}l_{0}/(L_{x}L_{y})$, where $l_{0}$ is the
equilibrium length between the wall’s COM resulting in the pore width $H$, see
Figure S1 in the Supporting Information. To mimic water exchange with the bulk
external environment as indicated in Fig. 1(b), we perform Grand-Canonical
Monte-Carlo (GCMC) and Molecular Dynamics (MD) simulations using the LAMMPS
simulation package, all details are given in Methods. In line with intrusion
experiments in such hydrophobic materials, the water chemical potential is
chosen such that its bulk pressure corresponds to $P_{0}=215\,\mathrm{bar}$,
see Section III of the Supplementary Information.

We study two systems: (i) a non-compliant pore ($E_{zz}=\infty$) and (ii) a
compliant system, which can swell upon water adsorption. For the compliant
case, we consider two values for the pore modulus, $E_{zz}=0.2$ and 2 GPa.
Then, in a second step, for the compliant systems we explicitly address the
impact of pore size fluctuations on water transport through these pores. In
detail, we let the pore relax to its equilibrium size in the presence of
water, but then fix it to its average value $\langle H\rangle$. In so doing,
we keep the thermodynamic state of the confined water and can study the
influence of fluctuating and non-fluctuating pore walls (by fixing the pore
size $H$ to $\langle H\rangle$ or by letting $H$ fluctuate around $\langle
H\rangle$).

###  0.1 Adsorption and swelling in compliant nanoporous materials

The density profiles shown in Fig. 1(c) and (d) reveal significant layering at
the surface fixed in space (left walls).41, 42, 43 Similarly, layering is also
observed — albeit less pronounced — for the stiff pore
($E_{zz}=2\,\mathrm{GPa}$) shown on the right side of Fig. 1(c), where the
data shows that the application of an external stress leads to expulsion of
confined water and stronger layering effects. For the softer pore
($E_{zz}=0.2\,\mathrm{GPa}$), the data shown on the right side of Fig. 1(d)
show that surface fluctuations smear out the layering at the fluctuating
surface. Upon increasing the stress $\sigma_{zz}$ in the range from $1$ to
$1000\,\mathrm{bar}$, the water density profiles change significantly since
the average pore separation is able to adjust to the water structure. At
$\sigma_{zz}=1000\,\mathrm{bar}$, in the case of the soft pore, we observe a
transition in the profiles corresponding to the expulsion of one water layer,
corresponding to a jump in pore size from about $1.6\,\mathrm{nm}$ at
$500\,\mathrm{bar}$ to roughly $1.25\,\mathrm{nm}$ at $1000\,\mathrm{bar}$.
The data shown for that case correspond to an equilibrium length of the spring
of $l_{0}=2.84\,\mathrm{nm}$, which is the largest value considered in this
work.

Figure 2: Water confinement and pore swelling in compliant nanoporous
materials. (a) Water surface density in molecules/$\mathrm{nm}^{2}$ as
obtained using GCMC simulations for the soft ($E_{zz}=0.2\,\mathrm{GPa}$,
circles) and stiff ($E_{zz}=2\,\mathrm{GPa}$, squares) compliant nanoporous
material at different external stresses $\sigma_{zz}$. Black triangles denote
the simulation data for a non-compliant, i.e. infinitely stiff, nanopore. The
black line corresponds to the expected variation based on the bulk water
density taking into account excess surface density $n_{\sigma}$. The inset
shows the same data versus the equilibrium pore size $H_{0}$ as obtained in
the absence of any fluid. The dashed line shows the slope expected from the
bulk water density. (b) Stress-strain relation for soft pores,
$E_{zz}=0.2\,\mathrm{GPa}$, with different nominal pore size $H_{0}$ as
indicated by the legend on the right. Dashed lines denote fits of the
effective modulus according to $\sigma_{zz}=\varepsilon
E_{zz}^{\mathrm{eff}}$. The solid line denotes the empty pore mechanical
response without any effect due to adsorbed water, $\sigma_{zz}=\varepsilon
E_{zz}$. The inset shows the equivalent circuit model as discussed in the
text. (c) Effective modulus of the soft system for different equilibrium pore
sizes obtained from the fits in (b). The vertical lines denote the different
regimes, where in the density profiles 1-4 water layers can be observed. Above
$\sim 1.4\,\mathrm{nm}$ the water in the center of the slab is bulk-like, cf.
Supplementary Figures 3 and 4.

To further quantify water adsorption in the compliant material, we show in
Fig. 2(a) the mean number of water molecules in the slit nanopore normalized
to the lateral area of the pore,
$n_{\mathrm{w}}=N_{\mathrm{w}}/\left(L_{x}L_{y}\right)$. The dashed line
corresponds to the bulk water molecular volume, $n_{\mathrm{w}}/\langle
H\rangle=33.2\,\mathrm{nm^{-3}}$.37 The inset of Fig. 2(a) shows the areal
water number as a function of the equilibrium pore size in the absence of any
fluid, which follows from the initial choice of the length of the harmonic
spring, $H_{0}=l_{0}-0.84\,\mathrm{nm}$ that connects the walls’ COM. For a
non-compliant, i.e. frozen, pore (black triangles), the slope corresponding to
the bulk density is well reproduced, but with a shift $z_{\mathrm{G}}$ in the
surface position corresponding to the Gibbs Dividing Surface.44 In detail, the
latter is defined for a single interface via the water surface excess with
respect to its bulk phase,

$z_{\mathrm{G}}=z_{a}+\int_{z_{a}}^{z_{b}}\frac{\rho(z_{b})-\rho(z)}{\rho(z_{b})-\rho(z_{a})}\,\mathrm{d}z,$
(3)

where $\rho(z_{a})\sim 0$ and $\rho(z_{b})=\rho_{\mathrm{b}}$ are the
corresponding fluid densities far from both sides of the interface, i.e. no
water inside the solid and water bulk density in the fluid phase. For the
compliant pores (squares and circles), either swelling or compression is
observed depending on the mechanical load applied, cf. inset of Fig. 2(a). All
simulation data collapse to a master curve in Fig. 2(a) when the average
effective pore separation $\langle H\rangle$ — defined as the distance between
the outermost surface atoms — is employed. The excellent agreement in the
slope observed for different surfaces stresses when large separations are
considered reveals that nanoconfined water is nearly incompressible with a
density that sufficiently away from the interface corresponds to the bulk
density. According to Gibbs adsorption theory, we determine the surface excess
defined for a single interface as

$n_{\sigma}=\int_{-\infty}^{z_{\mathrm{G}}}\left[\rho(z)-\rho(z_{a})\right]\,\mathrm{d}z+\int_{z_{\mathrm{G}}}^{\infty}\left[\rho(z)-\rho(z_{b})\right]\,\mathrm{d}z.$
(4)

This can be conveniently done for confined system by fitting fitting the
offset of the line corresponding to bulkd density to all simulation data for
which $\langle H\rangle>1\,\mathrm{nm}$ in Fig. 2(a) and yields
$2n_{\sigma}=-8.6\,\mathrm{nm^{-2}}$. Such a negative value indicates that
water is depleted from the hydrophobic surface. This result was expected for
the hydrophobic surface chemistry studied here which leads to non-favorable
fluid/solid interactions. The associated surface separations $\langle
H\rangle<\sigma$ at which the slit pore accommodates no water molecules,
$\sigma=2.65\,\mathrm{\AA}$, perfectly coincides with the water kinetic
diameter (which is related to the mean free path of molecules in a fluid
phase).45

The change of pore size upon external mechanical stimulation can be further
analyzed by determining the stress-strain relation in Fig. 2(b) For all
systems considered, we find the strain $\varepsilon=(H_{0}-\langle
H\rangle)/H_{0}$ to be roughly related linearly to the normal stress
$\sigma_{zz}$, where here, per convention, a shrinkage corresponds to a
positive strain. This is not necessarily the expected behavior since the
interfacial structure and compressibility of water are non-negligible for the
small pores considered. However, the strain-stress linearity allows us to
extract the apparent (effective) modulus of the slit pore under drained
conditions, $E_{zz}^{\mathrm{eff}}=\sigma_{zz}/\varepsilon$, in analogy to the
apparent tangent drained bulk modulus derived from a generalized poromechanics
framework.46 The corresponding equivalent circuit depicted in the inset of
Fig. 2(b) consists of the mechanical properties of the empty pore,
characterized through the modulus $E_{zz}$ acting in parallel with the water
mechanical response under drained conditions. In fact, the latter can be
further dissected into the water bulk and surface contributions,
$E_{\mathrm{b}}$ and $E_{\mathrm{s}}$, respectively, acting in series (as the
total strain upon application of a normal stress $\sigma_{zz}$ must be equal
to the sum of the strains of the individual springs). The corresponding
effective modulus follows as

${E_{zz}^{\mathrm{eff}}}={E_{zz}}+\frac{1}{E_{\mathrm{b}}^{-1}+2E_{\mathrm{s}}^{-1}}.$
(5)

If we assume a linear relation to hold, linear fits to the simulation data in
Fig. 2(b) allow for the determination of the effective Young modulus in
presence of water, $E_{zz}^{\mathrm{eff}}=\sigma_{zz}/\varepsilon$.47, 48 Such
data are shown in Fig. 2(c) for the soft compliant pore with
$E_{zz}=0.2\,\mathrm{GPa}$. For sufficiently large separations $H_{0}\gtrsim
1.5\,\mathrm{nm}$, interfacial effects (at sufficiently large strain, see Fig.
2(b)) become negligible if the water is allowed to exchange with the
reservoir. In that case, $E_{zz}^{\mathrm{eff}}\sim E_{zz}$ is recovered, i.e.
the response to an applied stress of the filled pore system corresponds to
that of the confining material. Since the water bulk modulus is about
$25\,\mathrm{kbar}$,49 this implies that the term $2E_{\mathrm{s}}^{-1}$
dominates the denominator in Eq. 5, i.e. the surface layer modulus must be
small such that the total fraction vanishes. Further analysis of the data
shown in Fig. 2(c) for the soft pore and in Figure S2 in the Supplementary
Information for the stiff pore reveals that $E_{zz}^{\mathrm{eff}}\lesssim
E_{zz}$. Since $E_{\mathrm{b}}$ is positive and independent of the pore width,
this indicates that $E_{\mathrm{s}}$ depends on $H_{0}$ and $E_{zz}$, and is
negative. Although this is a counterintuitive result at first (since it
indicates a negative apparent surface compressibility), it is in line with
generalized poromechanics stating that the apparent tangent bulk modulus can
be smaller than its counterpart in bulk depending on the adsorption
properties.46 Physically, it corresponds to the concept of a disjoining
pressure,50 which depending on the wetting properties either is attractive or
repulsive.51 Note, that the typical signature of hydration forces — the
oscillatory stress vs. pore size curve — is encapsulated in the effective
modulus since the pore can mechanically adapt to the molecular water
structure.

Interestingly, deviations for the smaller pores with
$E_{zz}^{\mathrm{eff}}\neq E_{zz}$ can be correlated with the density layering
shown in Fig. 1 (Figures S3 and S4 in the Supplementary Information show the
same data for all systems). The vertical blue lines in Fig. 2(c) indicate the
corresponding number of water layers. For $H_{0}\gtrsim 1.5\,\mathrm{nm}$, the
water density profile in the slit pore center becomes homogeneous and equal to
its bulk value (so that density heterogeneities correspond to two distinct
layers at the solid/fluid interface). Depending on the pore separation $H$, in
the case of the smaller pores, water organizes into one, two, three or four
layers. Moreover, significant deviations from the prescribed modulus can be
observed. The exact influence of the interfacial contribution depends
sensitively on the equilibrium surface separation $\langle H\rangle$ of the
system in contact with the reservoir. It also depends on the applied
mechanical stress $\sigma_{zz}$ which strongly affects the thermodynamic state
of nanoconfined water. Indeed, even if confined water in each system is at the
same chemical potential $\mu$ and temperature $T$, its thermodynamic state
also depends on the prescribed mechanical condition $\sigma_{zz}$.

###  0.2 Transport in compliant nanoporous materials

Collective diffusivity. Having assessed the thermodynamic and adsorption
behavior of water in compliant slit nanopores, we now turn to transport in the
presence of a fluctuating interface. We first focus on Onsager flow, where the
universal thermodynamic driving force is a gradient in the chemical potential,
$\nabla\mu$. Solving Eq. 1 for the mean velocity $\bar{v}$ while taking into
account the heterogeneous density distribution $\rho(z)$ yields

$\bar{v}=\frac{\int\rho(z)v(z)\,\mathrm{d}z^{\prime}}{\int\rho(z)\,\mathrm{d}z^{\prime}}=-\frac{D_{0}}{k_{\mathrm{B}}T}\nabla\mu$
(6)

where Onsager’s relation defined in Eq. 2 was used. Eq. 6 rigorously defines
$D_{0}$ from the velocity $v(z)$ and density $\rho(z)$ profiles without any
need of defining a pore size $H$ or homogeneous density $\rho$. This is a
significant advantage in the definition of $D_{0}$, which can be probed
experimentally — e.g. using quasi-elastic coherent neutron scattering. In
contrast, the permeance $K$ defined in Darcy’s equation relies on the
ambiguous definition of a pore size as discussed below.

Figure 3: Probing dynamics in equilibrium and non-equilibrium conditions. (a)
Typical fits for the mean water velocity $\langle v\rangle$ vs. driving force
$\nabla\mu$. Color code corresponds to the stiffness as shown in legend. Data
are shown for three characteristic pore sizes $H\approx 0.75\,\mathrm{nm}$
(solid lines), 1.25 nm (dashed lines) and 1.75 nm (dashed-dotted lines). The
inset shows the collective diffusion coefficient determined from the linear
response regime of the mean water velocity (non-equilibrium molecular
dynamics) vs. the collective diffusion coefficient determined from the mean-
squared displacement (equilibrium molecular dynamics). All values for the
flexible pores are shown for $\sigma_{zz}=1\,\mathrm{bar}$. (b) Diffusion
coefficient in flexible pores vs. frozen pores (same mean pore size and water
number but non-fluctuating walls). The black line indicates the bisector for
clarity. The inset shows the collective diffusion constant $D_{0}$ vs. the
self-diffusion coefficient $D_{\mathrm{s}}$ to display the role of collective
contributions (cross-terms) in diffusion. The vertical dashed line corresponds
to the bulk value $D_{\mathrm{s}}^{\mathrm{(b)}}=3.03\,\mathrm{m^{2}/s}$
determined from independent simulations with finite size scaling, see Section
IV of the Supplementary Information for details.

In the following, we unravel the effect of pore elasticity by limiting the
discussion to the specific case where the external stress is set to
$\sigma_{zz}=1\,\mathrm{bar}$. Equation 6 provides a means to directly
determine $D_{0}$ from non-equilibrium simulations, where a chemical potential
gradient in the $x$-direction parallel to the pore is applied via a
homogeneous force field acting on each water molecule, $\nabla\mu=-f_{x}$, see
Methods for details. Figure 3 shows the resulting mean velocity which can be
obtained directly from the density and velocity profiles according to Eq. 6.
In the case of point particles, the flow rate can be further simplified to
$\bar{v}=1/\tau\int_{0}^{\tau}\mathrm{d}t\,1/N_{\mathrm{w}}\sum_{i=0}^{N_{\mathrm{w}}}v_{i}(t)$,
where $v_{i}(t)$ denotes the instantaneous velocity of a water molecule’s
center of mass in the direction of the flow at time $t$ and the integral
averages over the simulation time $\tau$.

Figure 3(a) shows $\bar{v}$ as a function of the driving force
$f_{x}=-\nabla\mu$ for different pore widths $\langle H\rangle$ and
stiffnesses $E_{zz}$. Whereas Eq. 6 is intrinsically a linear relation, the
linear response regime is not achieved in the molecular simulations (as well
as in experiments) when too large driving forces are used. However, we find
that for sufficiently small driving forces the linear response limit can be
probed [lines in Fig. 3(a)]. In particular, the interfacial friction between
water and the surface governs the onset of nonlinear effects. As a result, the
critical driving force at the crossover between the linear and nonlinear
regimes sensitively depends on the surface separation as well as the pore
flexibility.37 The frictional stress at the interface increases with
$\bar{v}^{3}$ within the classical Darcy-Weisbach theory. This explains the
emergence of a critical value $\bar{v}_{\mathrm{crit}}=20\,\mathrm{m/s}$ which
is rather independent of the pore size and stiffness. Such a velocity cutoff
for the linear response regime can be rationalized with an activated diffusion
of water molecules over barriers at the interface.52 Note that, however, the
corresponding driving forces are highly dependent on the pore size, which is
in line with the expectation from classical Hagen-Poiseuille theory.

In practice, we fit $D_{0}^{\mathrm{(NE)}}$ in Eq. 6 to the non-equilibrium
simulation data in Fig. 3(a) for velocities $\bar{v}<\bar{v}^{\mathrm{crit}}$.
The fits are indicated as black and color lines, whereas the horizontal dashed
gray line in Fig. 3(a) denotes the employed value
$\bar{v}^{\mathrm{crit}}=20\,\mathrm{m/s}$. As shown in the inset of Fig.
3(a), the obtained values are in perfect agreement with the equilibrium
molecular dynamics results relying on the mean-squared displacement and the
fluctuation-dissipation theorem, see Methods and Figures S7 and S8 in the
Supplementary Supplementary Information. As can be observed from the data in
Fig. 3(a), the collective diffusion $D_{0}$ increases with pore size in line
with a Poiseuille flow, but also with pore flexibility (black via red to blue
data). To shed further light on the dependence $D_{0}(E_{zz})$, we performed
an additional set of simulations at constant (non-fluctuating) pore size with
a value fixed at $H=\langle H\rangle$, where the brackets denote the time-
average of the fluctuating pore size for the specific stiffness $E_{zz}$. The
resulting diffusion coefficients in Fig. 3(b) perfectly agree between the
fluctuating and non-fluctuating pores with the same fixed water number
$N_{\mathrm{w}}$. This result points to the fact that $D_{0}$ depends on
adsorption/layering rather than on pore size fluctuations. However, we note
that the rather large pore size employed here together with the
incompressibility of water and the employment of a stiff surface can explain
the negligible impact of pore size fluctuations. In other words, for smaller
pores (or, equivalently, larger surface to volume ratios), pore size
fluctuations and, in general, mechanical deformations are expected to affect
liquid flow in nanoconfinement. This point will be addressed further below.

The collective diffusivity as defined in Onsager’s law, i.e. Eqs. 1 and 6,
differs strongly from the self-diffusion coefficient $D_{\mathrm{s}}$ that
follows from tracking a tagged molecule.53 The collective correlations in a
fluid are usually positive and thus strongly enhance $D_{0}$ compared to
$D_{\mathrm{s}}$ (and in the dilute limit $D_{0}\to D_{\mathrm{s}}$). The
inset of Fig. 3(b) shows the corresponding relation $D_{0}$ vs.
$D_{\mathrm{s}}$ for a confined system, for details on the calculation see
Methods. Interestingly, due to interfacial depletion/adsorption effects,
$D_{\mathrm{s}}$ in a confined system can also become larger/smaller than its
counterpart in bulk at the same chemical potential/pressure,
$D_{\mathrm{s}}^{(0)}=3.03\,\mathrm{m^{2}/s}$, which we determined in
independent simulations for our water model, see Section IV of the
Supplementary Information. A detailed discussion of such effects is out of
scope of this work and well represented in text books.54

Velocity profiles. Further insight can be obtained from analyzing the velocity
and density profiles. To this end, we perform a coordinate transformation from
the lab frame [Fig. 1(a)] into the pore frame, i.e. $z\to
z-z_{\mathrm{COM}}(t)$, where
$z_{\mathrm{COM}}=\sum_{i}[z_{i}(t)m_{i}]/\sum_{i}m_{i}$ is the fluctuating
center of mass of the pore and the sum runs over all $i$ pore wall atoms. The
resulting density profiles, which are illustrated for the stiffer system and
an average pore size $\langle H\rangle=1.4\,\mathrm{nm}$ in Fig. 4(a), reveal
that in this frame the density profiles are symmetric as expected.
Furthermore, as already noted above, the first peak of the density profile is
strongly depleted from the position of the interface (defined by the outermost
position of the wall atoms). The velocity profile [purple data in Fig. 4(a)]
is parabolic in the slab center and decays nearly linearly to zero at the pore
wall, $z=H/2$. Importantly, the velocity profiles are perfectly in line with
the solution of the Stokes flow in the presence of an interfacial fluid film
of thickness $w$. The modified Poiseuille profile accounting for continuity of
the velocity and stress at the interface between the two zones is given by

$\displaystyle v_{x}(z)$ $\displaystyle=\frac{\nabla
P}{2}\left[\frac{1}{\eta_{\mathrm{b}}}\left(a^{2}-z^{2}\right)+\frac{1}{\eta_{\mathrm{s}}}\left(\frac{H^{2}}{4}-a^{2}\right)\right]$
$\displaystyle\mathrm{for}$ $\displaystyle\quad 0\leq|z|\leq\frac{a}{2}$
$\displaystyle v_{x}(z)$ $\displaystyle=\frac{\nabla
P}{2\eta_{\mathrm{s}}}\left[\frac{H^{2}}{4}-z^{2}\right]$
$\displaystyle\mathrm{for}$ $\displaystyle\quad\frac{a}{2}\leq|z|\leq H,$ (7)

where we introduced the width of the bulk-like region, $2a=H-2w$ to simplify
notation and $\eta_{\mathrm{b}}$ and $\eta_{\mathrm{s}}$ denote the bulk and
surface layer effective viscosities, respectively. To apply Eq. 7 to our
simulations involving a gradient in the chemical potential instead of the
pressure, we make use of the Gibbs–Duhem relation for incompressible water
($\textrm{d}P=\rho\textrm{d}\mu$) to obtain $\nabla
P=-f_{x}N_{\mathrm{w}}/\left(L_{x}L_{y}H\right)$.

Figure 4: Velocity profiles and interfacial properties. (a) Velocity (purple
line) and density (green line) profiles in the pore-center coordinate frame
(see text) obtained in the linear regime for the slit pore with $\langle
H\rangle=1.4\,\mathrm{nm}$ (indicated by the dash-dotted vertical black lines)
and pore stiffness $E_{zz}=2\,\mathrm{GPa}$. The red line denotes the fit
against a two-zone Poiseuille flow accounting for an interfacial viscous
layer, c.f. Eq. 7. The yellow shaded area denotes the fitted interfacial width
$w$, the shaded area of the velocity profile the standard error of the mean.
(b) Fitted values of the interfacial width and (c) viscosity for the different
pore stiffnesses and sizes. The dashed horizontal line in (b) denotes the
value $\sigma=0.26\,\mathrm{nm}$ obtained via the surface excess in Fig. 2(a).
The dashed horizontal line in (c) denotes the bulk value of the water model
used in the simulations, $\eta_{\mathrm{b}}=0.695\,\mathrm{mPa\cdot s}$,
whereas the solid line and shaded area denote the averaged viscosity in the
interfacial layer, $\eta_{\mathrm{s}}=0.16\pm 0.04\,\mathrm{mPa\cdot s}$. The
filled symbols in (b) and (c) correspond to fluctuating pores whereas the
empty symbols are obtained for frozen pores, which nearly perfectly overlap
with the data for the fluctuating pores. For $\langle H\rangle\leq 2\sigma$,
the viscous behavior is purely interfacial as indicated by the shaded areas in
(b) and (c).

Fits of Eq. 7 to the simulation data shown in Fig. 4 reveal excellent
agreement with the two-zone Poiseuille flow (see also Figures S3 and S4 in the
Supplementary Information for all systems). These data show a rather universal
value for the interfacial width $w\approx\sigma=0.26\,\mathrm{nm}$, c.f. Fig.
4(b). To improve numerical stability during the fitting procedure, we fix
$w=\sigma$ in the following discussion. This approach is justified since it
leads to comparable root-mean-square error in the fits to the velocity
profiles. The interfacial layer has a strictly reduced viscosity as shown in
Fig. 4(c). As can be seen in Fig. 4(a), water is strongly depleted from the
surface within the width $w$. This is believed to result in a reduced value of
$\eta_{\mathrm{s}}$ that is quasi independent of pore stiffness and size for
the compliant pores with a value $\bar{\eta_{\mathrm{s}}}=0.16\pm
0.04\,\mathrm{mPa\,s}$ [solid black line and corresponding shaded area in Fig.
4(c)]. Importantly, the results of the fitting procedure are independent of
whether we consider fluctuating or non-fluctuating pores at the same pore size
and water content [empty symbols in Fig. 4(b,c)]. Noteworthy, in the non-
compliant nanopores, $E_{zz}=\infty$, i.e. when the pore cannot adapt to the
water structure by any modification of the pore size, water layering is
enhanced (Figure S3 in the Supplementary Information) and the interfacial
viscosity converges only slowly for comparatively large pores
$>2\,\mathrm{nm}$ towards $\bar{\eta_{\mathrm{s}}}$.

As observed from Fig. 4(a), both the density and velocity profiles approach
zero at the steric definition of the pore size $H$ and the usual no-slip
boundary conditions perfectly holds. In the pore center, at a distance larger
than $\sigma=0.26\,\mathrm{nm}$ from the surface, the velocity profile is
parabolic by construction from Eq. 7. It is noteworthy that this parabolic
region of width $H-2\sigma$ is different from the thermodynamic pore size
$H-\sigma$ as defined above from the Gibbs dividing plane. Since both the
density and velocity decay quickly to zero at distances smaller than $\sigma$
from the surface, this region barely contributes to the flux according to Eq.
1. Thus, it is tempting to introduce a coarse-grained model of a simple planar
Poiseuille flow in the region $[-(H/2-\sigma),(H/2-\sigma)]$. Two questions
naturally arise at this stage. First, what is the corresponding boundary
condition at the pore surface, $v\left(\pm(H/2-\sigma)\right)$? Second, what
is the contribution of the deviations from the parabolic profile in the
vicinity of the surface to the overall flow? Since the fluid velocity does not
vanish at the interface to the bulk-like flow region, the no-slip boundary
condition does not hold. In this case, a slip boundary condition is introduced
instead with a characteristic slip length

$b=\pm
v_{x}(z_{\mathrm{surf}})/\left(\mathrm{d}v_{x}/\mathrm{d}z\right)_{z_{\mathrm{surf}}}.$
(8)

Defining $b$ requires the knowledge of the position of the corresponding
surface $z_{\mathrm{surf}}$, where the hydrodynamic boundary condition is
applied. There is no absolute natural choice for $z_{\mathrm{surf}}$ in the
case of an atomistically resolved surface.55 This has led to different choices
in the literature ranging from definitions based on the shear-stress
correlation function,56, 57 the use of the Gibbs’ dividing surface, the
repulsive contribution of the surface interaction,58 or simply based on the
position of the surface atoms.59 All these definitions sensitively influence
the numerical value inferred for the slip length $b$.

Based on the excellent description obtained above with the two-zone Poiseuille
flow accounting for interfacial viscosity for the simulated velocity profiles,
we here propose the effective hydrodynamic diameter $H-2\sigma$. In so doing,
accounting for slippage and an effective viscosity reproduces the bulk-like
region of the simulation results. Such a coarse-graining approach — despite
the question about its validity to be addressed below — significantly
simplifies the analytical treatment of the flow profiles. In detail, the
Poiseuille flow with slip is given by

$v_{x}(z)=\nabla
P\frac{{H^{\star}}^{2}}{8\eta}\left[1+\frac{4b}{H^{\star}}-\left(\frac{2z}{H^{\star}}\right)^{2}\right],$
(9)

where the asterisk denotes the — generally unknown — effective pore size. We
now identify $H^{\star}=H-2\sigma$ and consider only the velocities in the
region $[-(H/2-\sigma),(H/2-\sigma)]$ — where the profiles shown in Fig. 4 and
for all systems in Figures S3 and S4 in the Supplementary Information are
parabolic. Concerning the width $w$ of the interfacial layer, Eqs. 8 and 9
define the slip length
$b=\eta_{\mathrm{b}}/\left(2\eta_{\mathrm{i}}\right)\times\left(Hw-w^{2}\right)/\left(H/2-w\right)$.
With this definition, using the values $w=\sigma$ and
$\bar{\eta_{\mathrm{s}}}$ for large $H$ results in $b=1.2\pm
0.3\,\mathrm{nm}$, cf. Figure S5 in the Supplementary Information.

Importantly, to obtain the above expression for $b$, we have redefined the
pore size as the effective hydrodynamic pore size according to
$H^{\star}=H-2\sigma$. It is worth mentioning, that if the pore size
$H^{\star}=H$ would be employed and only the parabolic part of the velocity
profile were considered, both the slip length and the apparent viscosity
$\eta_{\mathrm{app}}=\eta_{\mathrm{b}}+2w/H\times\left(\eta_{\mathrm{i}}-\eta_{\mathrm{b}}\right)$
become dependent on $H$.60 Depending on the interfacial properties,
$\eta_{\mathrm{app}}$ can be orders of magnitude larger than the bulk
viscosity,61, 43, 37, 62 or significantly decrease,63, 64 as observed also for
the hydrophobic surfaces considered here. In summary, our observations from
the density and velocity profiles reveal the two following points. (1) The
thermodynamic pore size as defined from Gibbs’s dividing surface differs from
a steric definition $H$ by a the characteristic size of a water molecule,
$\sigma=2.65\,\mathrm{\AA}$. (2) At the length of a molecular size, one
observes interfacial viscous effects, i.e. the pore size where bulk
hydrodynamic behavior is found is $H-2\sigma$. Note that this is only
partially equivalent to claiming that the hydrodynamic pore size is
$H-2\sigma$ because one needs to take into account slippage. In such a coarse-
grained model — besides the fact that in general the pore size cannot be
rigorously defined such that the apparent viscosity needs to be fitted — one
misses a part in the flux due to neglecting the interface layer. This
contribution to the total flux is in general rather small as will be discussed
below.

Pore size fluctuations. To address the underlying effects of pore size
fluctuations in a flexible nanopore, we assume in the following that the pore
size of a pore with spring equilibrium length $H_{0}$ filled with water can be
described using a Gaussian distribution around the mean spring extension $e$
given by
$f(e;E_{zz})={1}/{\left(\varsigma\sqrt{2\pi}\right)}\exp{\left[-{1}/{2}\left(\left(e-\bar{e}\right)/\varsigma\right)^{2}\right]}$
(the extension $e$ is due to swelling and the variance is $\varsigma$). Note
that we dropped the dependence on the external stress since in the following
we limit our discussion to $\sigma_{zz}=1\,\mathrm{bar}$. For all systems, we
find that the fluctuations are well described by this mathematical form for
$f(e)$, cf. inset of Fig. 5(a). Dropping the dependence on the modulus, the
average pore size follows as $\langle
H\rangle=H_{0}+\int_{-\infty}^{\infty}ef(e)\,\mathrm{d}e$.

Figure 5: Influence of fluctuating vs. non-fluctuating pore surfaces. (a)
Variance of the pore size fluctuations in equilibrium (empty symbols) and non-
equilibrium simulations (full symbols). The inset shows an example of the pore
extension distribution function $f(e)$ for $E_{zz}=0.2\,\mathrm{GPa}$ and
$H_{0}=1\,\mathrm{nm}$, red line denotes a fit of a Gaussian to the data. (b)
Permeance of the compliant, fluctuating pores vs. the respective value in a
rigid pore at the same average pore size and particle number. The inset shows
the expected relative permeability enhancement $s_{K}$ due to pore size
fluctuations (see text).

To ensure a compact presentation of the following discussion, we here employ
the coarse-grained model presented above. In other words, the water velocity
profiles in compliant pores can — neglecting interfacial effects — be modelled
by a Hagen-Poiseuille law when accounting for slip and effective viscosity,
see Eq. 9. An extension to the case where the interfacial viscous effects are
treated explicitly is straightforward. A further simplification that is often
made in the analysis of Darcy flows is to treat the water density
$\rho=N_{\mathrm{w}}/V_{\mathrm{w}}=N_{\mathrm{w}}/(A\langle H\rangle)$ as
homogeneous ($V_{\mathrm{w}}=A\langle H\rangle$ is the volume accessible to
the water and $A=L_{x}L_{y}$ is the lateral area of the slit pore). Noting
that the density profiles presented in Fig. 4 and for all systems in Figures
S3 and S4 in the Supplementary Information are not homogeneous, we will
discuss the limitations and impact of this simplification below. The permeance
of a fluctuating nanopore with a Poiseuille velocity profile and slip follows
from Eq. 9 as

$K=\frac{\langle H^{\star}\rangle^{2}}{4\eta}\left(\frac{2b}{\langle
H^{\star}\rangle}+\frac{1}{3}\right).$ (10)

As discussed above, our NEMD simulations reveal that the slip length quickly
approaches its asymptotic constant value $b=1.2\,\mathrm{nm}$ for sufficiently
large nanopores. Moreover, using a consistent definition of the pore size, the
viscosity $\eta=\eta_{\mathrm{b}}$ is found to be independent of $\langle
H\rangle$ and pore fluctuations. Furthermore, employing $f(e)$ to express the
mean pore size, we combine Eqs. 6 and 10 and perform the Gaussian integral,
such that the collective diffusion coefficient can be expressed as

$D_{0}=k_{\mathrm{B}}T\langle\rho
K\rangle=\frac{k_{\mathrm{B}}T}{6\eta}\frac{N_{\mathrm{w}}}{A\left(1+\frac{2\sigma}{\langle
H^{\star}\rangle}\right)}\langle H^{\star}\rangle\left(1+\frac{3b}{\langle
H^{\star}\rangle}\right),$ (11)

which yields the important result of $D_{0}$ being independent of pore size
fluctuations, in good agreement with the results presented in Fig. 3(b).
Although being a first order approximation neglecting any fluid density
heterogeneities or changes in the effective viscosity and slip, Eq. 11
captures remarkably well the qualitative behavior observed in the present work
(see gray dashed line in Figure S6 in the Supplementary Information).

Plugging the pore size distribution using $f(e)$ into Eq. 1 allows us further
to define the average flux according to $J=-\langle\rho
v\rangle=-N_{\mathrm{w}}/\left(A\langle v/H\rangle\right)$. This directly
yields the permeance for a fluctuating slit pore,

$K=-\frac{A}{N\nabla\mu}\langle vH\rangle.$ (12)

For a non-fluctuating pore $\langle vH\rangle=\langle v\rangle\langle
H\rangle$ which yields the classical result
$K^{\mathrm{(frozen)}}=D_{0}/\left(k_{\mathrm{B}}T\right)\times A\langle
H\rangle/N$. The latter expression relates the permeance and collective
diffusion via the thermal energy and fluid density $\rho=N/\left(A\langle
H\rangle\right)$. On the contrary, Eq. 12 reveals that a coupling between pore
size fluctuations and the fluid velocity can impact transport in compliant
pores (in contrast to the collective diffusivity which is expected to be
independent of fluctuations as shown above). For the Poiseuille flow, the
coupling can be made quantitative by calculating the second moment of the
Gaussian integral appearing in Eq. 10, resulting to leading order in $K\sim
H_{0}^{2}+\int_{-\infty}^{\infty}e^{2}f(e)\,\mathrm{d}e=\langle
H\rangle^{2}+\varsigma^{2}$, i.e. under these assumptions the fluctuations
enhance transport in a fluctuating slit pore.

We show in Fig. 4(a) the variance $\varsigma$ of the pore size fluctuations
determined both from equilibrium simulations, i.e. in the absence of flow, as
well as from simulations with applied driving force $\nabla\mu$. Notably, EQMD
and NEMD simulation results agree perfectly, revealing that flow does not
affect the fluctuations in the linear response regime. Furthermore,
fluctuations are in the range of about $(1-3)\times 10^{-2}\,\mathrm{nm}$ for
the pore sizes $H\approx(0.5-2.5)\,\mathrm{nm}$ studied here, i.e.
fluctuations are about two orders of magnitude smaller than the average pore
size, which we attribute to the near-incompressibility of water under these
thermodynamic and transport conditions. The resulting permeance
$K^{\mathrm{(flexible)}}$ obtained from NEMD simulations with fluctuating
pores is plotted in Fig. 4(b) against the values $K^{\mathrm{(frozen)}}$
obtained from simulations with the pore size fixed to the mean pore size
$H=\langle H\rangle$ and same water number $N_{\mathrm{w}}$ (i.e. the
positions of the surfaces were fixed). As expected from the small values of
$\varsigma$, the permeance $K$ is quasi independent of the fluctuations. To
quantify the expected flow enhancement due to pore fluctuations, we define the
coefficient
$s_{K}=\left(K^{\mathrm{(flexible)}}-K^{\mathrm{(frozen)}}\right)/K^{\mathrm{(frozen)}}=\left(\langle
H\rangle^{2}+\varsigma^{2}\right)/\langle H\rangle^{2}$ shown in the inset of
Fig. 5(b). We find values $s_{K}$ as small as $10^{-3}-10^{-4}$ for water in
the studied slit pores, revealing that the impact of pore fluctuations on
permeance are negligible in that case. However, for stronger fluctuating pores
— and, hence, more compressible liquids — the discussed effects are expected
to be more pronounced.

Flow enhancement in compliant nanopores. Our simulation data allow us to
quantify the effect of the compliance of a nanoporous material on the
resulting permeance according to Eq. 12. In Fig. 6(a), we plot the permeance
$K$ as a function of the resulting average pore size $\langle H\rangle$.
Comparison of the fluctuating compliant pores with simulations at the same
mean pore size and water numbers [filled vs. empty symbols in Fig. 6(a)] shows
that fluctuations in the slit nanopore do not enhance transport, also see Fig.
5(b) and in line with the findings discussed in the previous paragraph. Yet,
the molecular simulation data show a strong increase of $K$ with decreasing
stiffness at the same average pore size $\langle H\rangle$. In other words,
softer pores lead to transport enhancement due to the pore compliance itself.
This is due to the fact that at the same bulk pressure (i.e. chemical
potential), due to the different elastic properties of the slit pores, water
can accommodate in different density profiles, and thus a larger net amount of
water is present that can be transported, c.f. Figs. 1 and 2. These
thermodynamic effects that change the water density profile and viscosity
close to the surface seem to be the main driving parameter leading to flow
properties (Fig. 4).

Figure 6: Permeance. The permeance $K$ is shown as a function of pore size
$\langle H\rangle$ for the soft (blue circles), stiff (red squares), and rigid
pores (black triangles). Filled and empty symbols denote simulation data
evaluated according to Eq. 12 for the fluctuating and frozen pores,
respectively. The gray dashed-dotted line denotes the expected permeance from
Poiseuille equation Eq. 10 if a homogeneous density profile, bulk viscosity,
and a slip length of $1.2\pm 0.3\,\mathrm{nm}$ are assumed (see Figure S5 in
the Supplementary Information). Shaded area denotes the corresponding
uncertainty in the extrapolated slip length. Colored dashed lines further take
into account the fitted slip lengths for each pore height and stiffness, data
$H<1\,\mathrm{nm}$ are excluded due to their large uncertainty, see Figure S5
in the Supplementary Information. Solid lines are constructed from the fitted
Poiseuille profiles taking into account the density weighted velocity
profiles, Eq. 13. (b) Permeance $K^{\star}$ calculated according to Eq. 14 vs.
the permeance measured directly from the flux in (a). Solid symbols correspond
to the classical result of Eq. 2, neglecting the surface excess $n_{\sigma}$,
whereas empty symbols take into account the fitted surface excess
$n_{\sigma}=-8.6\,\mathrm{nm^{-2}}$ obtained in Fig. 2. The dashed line serves
as guide to the eye.

The solid lines in Fig. 6(a) show the permeance predicted from the two-zone
Poiseuille flow weighted by the water density profile,

$K=-\frac{\int\rho(z)v(z)\,\mathrm{d}z}{\int\rho(z)\,\mathrm{d}z}\frac{\langle{H}\rangle
A}{N\nabla\mu}.$ (13)

In this equation, the term $\left(\langle H\rangle
A\right)/\left(N\nabla\mu\right)=1/\nabla P$ again relates chemical potential
and pressure gradients as driving force according to a Gibbs–Duhem relation.
The simulation data are excellently described by Eq. 13 with a systematic
underestimation of the permeance that increases with increasing softness and
pore width. This is due to the limited ability of the two-zone Poiseuille flow
model to appropriately describe the profiles, see Figures S3 and S4 in the
Supplementary Information. While more sophisticated models for the interfacial
viscosity — like e.g. an exponential decay to the interfacial value 37 — can
be established, their discussion lies out of scope of the present work and the
observed deviations do not change our main conclusions.

Following the idea of a coarse-grained model for the pore flux above, we now
assume that a Poiseuille flow with slip, Eq. 10 describes accurately the
permeance for $\eta=\eta_{\mathrm{b}}$ and $b=1.2\pm 0.3\,\mathrm{nm}$. The
resulting permeance is shown as dashed gray line and corresponding shaded area
in Fig. 6(a), revealing an overall good agreement with the simulation data
(symbols). However, the applicability of this model is limited by the fact
that the extrapolated slip length is subject to rather large uncertainty and
is not necessarily the same for the different pore elasticities considered.
Indeed, we find that good agreement using the coarse-grained model with the
measured permeance for $H\gtrsim 1\,\mathrm{nm}$ when the distance-dependent
values of $b$ are determined according to Eq. 8 from fits of the two-zone
model as shown in Figure S5 in the Supplementary Information. For smaller pore
sizes, the fitting procedure is not robust enough to accurately predict the
ratio $b/H$. Significant deviations in the predicted permeance from this
approach are only observed for the pore with infinite stiffness. This is due
to the fact that density layering is the strongest for this system, which is
included in Eq. 13 and thus shows perfect agreement with the simulation data
[solid lines in Fig. 6(a)]. The observations above suggest that for
sufficiently large pores the assumption of a homogeneous density and parabolic
Poiseuille profile with slip boundary conditions predicts accurately the
permeance for nanopores (even though significant interfacial layering and slip
effects are present). Note that this conclusion still relies on introducing a
hydrodynamic pore size $H^{\star}=H-2\sigma$ as discussed above.

Finally, we now link the permeance — which is a macroscopic quantity amenable
to simple experimental testing — to the microscopic quantity $D_{0}$. For a
homogeneous fluid, one expects Eq. 2 to hold. In Fig. 6(b), we compare the
permeance $K^{\star}$ predicted from Eq. 2 and the collective diffusivity
$D_{0}$ determined from the simulations (Figure S6 in the Supplementary
Information) with the measured permeance $K$. These data are compared for the
different values $E_{zz}$ considered in this work. Eq. 2 strictly
underestimates the permeance due to the neglect of the surface excess when
assuming a homogeneous, bulk-like density. This can be accounted for by a
modified expression,

$K^{\star}=D_{0}H\frac{1}{k_{\textnormal{B}}T\left(n_{\sigma}+\rho_{\mathrm{b}}H\right)},$
(14)

in which we use $n_{\sigma}=-8.6\,\mathrm{nm^{-2}}$ (see discussion above).
The corresponding comparison corresponds to the empty symbols in Fig. 6(b).
The collapse of the data on the bisector reveals that Eq. 14 allows for a
consistent link between $K$ and $D_{0}$ through the concepts of interfacial
thermodynamics (including the significant surface layering which leads to
strong fluctuations with respect to a simple homogeneous density profile). The
latter finding thus paves the way to consistent interpretation of microscopic
dynamics — incorporated in $D_{0}$ and measured experimentally typically in
scattering experiments — to the permeance $K$, that is accessible from
experimental hydraulic flow measurements through a porous material.

In conclusion, whereas the compliance of a nanoporous material strongly
influences the transport in deformable nanopores due to the different density
layering and, thus, different interfacial viscous effects, pore size
fluctuations do not enhance the flow in the system considered here. While
permeance enhancement is expected from a theoretical viewpoint, the absence of
such poromechanical effect is due to the large pores and/or fluid
incompressibility involved in our systems. We note, however, that for smaller
pores and more flexible materials these effects could become highly relevant
and encanced transport due to the compliant material appears. More
importantly, we find that a consistent pore size definition can be employed to
relate accurately the thermodynamics and transport of the nanoconfined fluid.
In particular, when invoking the Gibbs dividing surface, we find that the
classical description through a Poiseuille flow holds perfectly at the
nanoscale. Active driving of fluctuating membranes has been suggested to
increase selectivity and transport,65 whereas additional effects appear
through the phonon-fluid coupling.16 Thus, it is promising to further explore
flow and selectivity manipulation for stronger fluctuating systems in future
work. In this context, by providing a consistent description to account for
transport and thermodynamics of fluids confined in nanoporous materials, we
believe that the present work offers a robust framework for such perspectives.

###  0.3 Methods

####  0.3.1 GCMC/MD Simulations

All molecular simulation results were obtained using the Large-scale
Atomic/Molecular Massively Parallel Simulator (LAMMPS) version 30 Oct 2019.66
Water is treated using the SPC/E model67 kept rigid via the SHAKE algorithm 68
and the slit pore was constructed as described in Section II of the
Supplementary Information. Input files are freely available in the data
repository of the University of Stuttgart (DaRUS).69 In the molecular
simulations, the carbon, oxygen and hydrogen atoms of the host matrix were
treated as Lennard-Jones particles with previously established parameters that
well account for experimental adsorption isotherms,40, 70, 71 see Table S1 in
the Supporting Information. All molecular simulations were run at temperature
$T=300\,\mathrm{K}$ using a Nosé-Hoover thermostat72 with a characteristic
damping time of $0.2\,\mathrm{ps}$. The integration timestep of
$2\,\mathrm{fs}$ was used. Short-range interactions were cutoff and shifted to
zero at $9\,\mathrm{\AA}$ and long-range electrostatics was treated using the
PPPM method73 with an accuracy of $10^{-4}$. For the slab system, we
additionally used the correction by Yeh and Berkowitz74 with a total vacuum
layer of 3 times the slab separation.

During the GCMC/MD simulations, we performed every $1\,\mathrm{ps}$ $2\times
10^{4}$ insertion/deletion steps for water molecules in the slab. To obtain
convergence, typical $10^{6}$ GCMC steps/water molecule were employed, which
corresponds to a total simulation time of about $10\,\mathrm{ns}$ per system
to sample the equilibrium particle number. The water chemical potential was
converted into a corresponding reservoir pressure using independent
simulations of bulk water, as explained in Section III of the Supplementary
Information. The value $\mu=-11.4\,\mathrm{kcal/mol}$ used here was thus found
to correspond to an external pressure of $P_{0}=215\,\mathrm{bar}$. This value
is sufficient for the pores to always be filled within the range of values
$\sigma_{zz},H_{0}$ reported in Fig. 2.

####  0.3.2 Equilibrium MD Simulations

To setup equilibrium MD simulations, slit pores with the average number
$N_{\mathrm{w}}$ as determined above were constructed. All other simulation
parameters were left unchanged. Systems were equilibrated for 1 ns and then
productions runs of 250 ns length were performed during which the unfolded
coordinates were recorded every 1 ps to calculate the mean-squared
displacement. The latter were used to infer the collective and self
diffusivities as
$D_{0}=\lim_{t\to\infty}1/(2N_{\mathrm{w}}dt)\langle\sum_{i,j}\left[\mathbf{r}_{i}(t)-\mathbf{r}_{i}(0)\right]\cdot\left[\mathbf{r}_{j}(t)-\mathbf{r}_{j}(0)\right]\rangle$
and
$D_{\mathrm{s}}=\lim_{t\to\infty}1/(2dt)\langle\left|\mathbf{r}(t)-\mathbf{r}(0)\right|^{2}\rangle$,
respectively. In these equations, $d=2$ is the dimensionality of the slit pore
and the corresponding displacement $\mathbf{r}$ is calculated in the planar
directions only. Typical curves for the mean-squared displacement are shown in
Figure S7 in the Supplementary Information. Care has to be taken in the
determination of the diffusion coefficients from the mean-squared displacement
since on the one hand the limit $t\to\infty$ has to be approximated, but on
the other hand the statistics gets very poor for large $t$ (see Figure S7 in
the Supplementary Information). In practice, to reliably determine a
reasonable fitting range, we determine the time-dependent values
$D_{0}(t)=1/(2N_{\mathrm{w}}dt)\langle\sum_{i,j}\left[\mathbf{r}_{i}(t)-\mathbf{r}_{i}(0)\right]\cdot\left[\mathbf{r}_{j}(t)-\mathbf{r}_{j}(0)\right]\rangle$
and
$D_{\mathrm{s}}(t)=1/(2dt)\langle\left|\mathbf{r}(t)-\mathbf{r}(0)\right|^{2}\rangle$,
shown in Figure S7 in the Supplementary Information, and average the data over
the range where it is roughly constant, i.e. typically between 250-5000 ps to
evaluate $D_{\mathrm{s}}$ and 100-1000 ps to evaluate $D_{0}$ for all systems
except the smallest pores, where the plateau is observed even for smaller time
intervals only due to the small values of the diffusion coefficients.

Alternatively, the diffusion constant can be determined from the velocity
autocorrelation functions,
$D_{0}=1/(N_{\mathrm{w}}d)\int_{0}^{\infty}\langle\sum_{i,j}\mathbf{v}_{i}(t)\cdot\mathbf{v}_{j}(0)\rangle$
and
$D_{\mathrm{s}}=1/d\int_{0}^{\infty}\langle\mathbf{v}(t)\cdot\mathbf{v}(0)\rangle$.
Note that the velocity correlation function of water is dominated by the
femtosecond timescale as shown in Figure S8 in the Supplementary Information,
thus requiring a high dump frequency for the velocities. In practice, we
recorded the velocities every timestep and after 1 ns the correlation function
was evaluated via fast Fourier transformation using the convolution theorem.
The resulting correlation functions are averaged over the total simulation
time and finally integration is performed. The limit $t\to\infty$ needs to be
replaced by some appropriate fitting range to obtain the extrapolation as
shown in Figure S8 in the Supplementary Information. As also shown therein,
both approaches give – as expected – results that are in perfect agreement.
However, $D_{\mathrm{s}}$ involves an average over all $N_{\mathrm{w}}$ water
molecules whereas $D_{0}$ corresponds to the water’s center of mass movement.
Therefore, significantly less statistics is used to measure $D_{0}$.

####  0.3.3 Non-equilibrium MD Simulations

Non-equilibrium simulations were performed by applying a constant force
$f_{x}=-\Delta\mu/L_{x}$ on all water oxygen atoms along the $x$-direction.
This driving force corresponds to a chemical potential difference $\Delta\mu$
over the corresponding length $L_{x}$ of the simulation box. We made sure that
the resulting mean velocity and permeance remained within the linear response
regime, cf. Fig. 3. To avoid any coupling between the applied force and the
system thermalization, only the $y$ and $z$ components of the velocity were
considered for calculating the thermal energy. All other simulation parameters
were the same as explained above. After an initial equilibration of 1 ns,
molecular simulations were performed for a total length of 500 ns per system
during which positions and velocities were recorded every 10 ps.

####  0.3.4 Data analysis

All obtained simulation data were analyzed using MDAnalysis75 and our in-house
open source framework MAICoS, https://www.maicos-analysis.org/. Analysis
scripts are freely available on the data repository of the University of
Stuttgart DaRUS, https://doi.org/10.18419/darus-3966.69

##  1 Associated content

###  1.1 Supporting Information

The Supporting Information is available free of charge on the publisher’s
website.

Table S1 and Figures S1-S8, details on the construction of the slit pore,
SPC/E bulk water equation of state for the employed simulation setup,
diffusion coefficient in bulk water and analysis of pore size and particle
number fluctuations.

##  2 Author Information

###  2.1 Author Contributions

AS and BC designed the research project; AS performed the simulations; AS and
BC analyzed the simulation data; all authors were involved in preparing the
manuscript.

###  2.2 Notes

The authors declare no competing financial interest.

##  3 Acknowledgments

We acknowledge funding from the Agence Nationale de la Recherche (Project
TWIST No. ANR-17-CE08- 0003) and computation time on the GRICAD infrastructure
supported by the Equip@Meso project (No. ANR-10-EQPX-29-01). AS acknowledges
funding from the DFG under Germany’s Excellence Strategy - EXC 2075 –
390740016 and SFB 1313 (Project Number 327154368) and support by the Stuttgart
Center for Simulation Science (SimTech).

## References

  * Schoch et al. 2008 Schoch, R. B.; Han, J.; Renaud, P. Transport Phenomena in Nanofluidics. _Rev. Mod. Phys._ 2008, _80_ , 839–883
  * Bocquet and Charlaix 2010 Bocquet, L.; Charlaix, E. Nanofluidics, from Bulk to Interfaces. _Chem. Soc. Rev._ 2010, _39_ , 1073–1095
  * Kavokine et al. 2021 Kavokine, N.; Netz, R. R.; Bocquet, L. Fluids at the Nanoscale: From Continuum to Subcontinuum Transport. _Annual Review of Fluid Mechanics_ 2021, _53_ , null
  * Bazant et al. 2011 Bazant, M. Z.; Storey, B. D.; Kornyshev, A. A. Double Layer in Ionic Liquids: Overscreening versus Crowding. _Phys. Rev. Lett._ 2011, _106_ , 046102
  * Lainé et al. 2020 Lainé, A.; Niguès, A.; Bocquet, L.; Siria, A. Nanotribology of Ionic Liquids: Transition to Yielding Response in Nanometric Confinement with Metallic Surfaces. _Phys. Rev. X_ 2020, _10_ , 011068
  * 6 Polster, J. W.; Souna, A. J.; Motevaselian, M. H.; Lucas, R. A.; Tran, J. D.; Siwy, Z. S.; Aluru, N. R.; Fourkas, J. T. The Electrical-Double Layer Revisited. _Natural Sciences_ _n/a_ , e20210099
  * Coppens 2012 Coppens, M.-O. A Nature-Inspired Approach to Reactor and Catalysis Engineering. _Current Opinion in Chemical Engineering_ 2012, _1_ , 281–289
  * Yang et al. 2013 Yang, H. Y.; Han, Z. J.; Yu, S. F.; Pey, K. L.; Ostrikov, K.; Karnik, R. Carbon Nanotube Membranes with Ultrahigh Specific Adsorption Capacity for Water Desalination and Purification. _Nat Commun_ 2013, _4_ , 2220
  * Shao et al. 2020 Shao, H.; Wu, Y.-C.; Lin, Z.; Taberna, P.-L.; Simon, P. Nanoporous Carbon for Electrochemical Capacitive Energy Storage. _Chem. Soc. Rev._ 2020,
  * Eijkel and van den Berg 2005 Eijkel, J. C. T.; van den Berg, A. Nanofluidics: What Is It and What Can We Expect from It? _Microfluid Nanofluid_ 2005, _1_ , 249–267
  * Huber 2015 Huber, P. Soft Matter in Hard Confinement: Phase Transition Thermodynamics, Structure, Texture, Diffusion and Flow in Nanoporous Media. _J. Phys.: Condens. Matter_ 2015, _27_ , 103102
  * Ma et al. 2015 Ma, M.; Grey, F.; Shen, L.; Urbakh, M.; Wu, S.; Liu, J. Z.; Liu, Y.; Zheng, Q. Water Transport inside Carbon Nanotubes Mediated by Phonon-Induced Oscillating Friction. _Nature Nanotech_ 2015, _10_ , 692–695
  * Bocquet and Netz 2015 Bocquet, L.; Netz, R. R. Phonon Modes for Faster Flow. _Nature Nanotech_ 2015, _10_ , 657–658
  * Marbach et al. 2018 Marbach, S.; Dean, D. S.; Bocquet, L. Transport and Dispersion across Wiggling Nanopores. _Nature Phys_ 2018, _14_ , 1108–1113
  * Marcotte et al. 2020 Marcotte, A.; Mouterde, T.; Niguès, A.; Siria, A.; Bocquet, L. Mechanically Activated Ionic Transport across Single-Digit Carbon Nanotubes. _Nat Mater_ 2020, _19_ , 1057–1061
  * Noh and Aluru 2022 Noh, Y.; Aluru, N. R. Phonon-Fluid Coupling Enhanced Water Desalination in Flexible Two-Dimensional Porous Membranes. _Nano Lett._ 2022, _22_ , 419–425
  * Wei et al. 2019 Wei, M.; Liu, Y.; Liu, J.; Elsworth, D.; Zhou, F. Micro-Scale Investigation on Coupling of Gas Diffusion and Mechanical Deformation of Shale. _Journal of Petroleum Science and Engineering_ 2019, _175_ , 961–970
  * Berthonneau et al. 2018 Berthonneau, J.; Obliger, A.; Valdenaire, P.-L.; Grauby, O.; Ferry, D.; Chaudanson, D.; Levitz, P.; Kim, J. J.; Ulm, F.-J.; Pellenq, R. J.-M. Mesoscale Structure, Mechanics, and Transport Properties of Source Rocks’ Organic Pore Networks. _Proceedings of the National Academy of Sciences_ 2018, _115_ , 12365–12370
  * Hoseini et al. 2009 Hoseini, M.; Bindiganavile, V.; Banthia, N. The Effect of Mechanical Stress on Permeability of Concrete: A Review. _Cement and Concrete Composites_ 2009, _31_ , 213–220
  * Zhang et al. 2018 Zhang, Y.; Wan, X.; Hou, D.; Zhao, T.; Cui, Y. The Effect of Mechanical Load on Transport Property and Pore Structure of Alkali-Activated Slag Concrete. _Construction and Building Materials_ 2018, _189_ , 397–408
  * Bai and Elsworth 2000 Bai, M.; Elsworth, D. _Coupled Processes in Subsurface Deformation, Flow, and Transport_ ; American Society of Civil Engineers, 2000
  * Cox et al. 2019 Cox, C. D.; Bavi, N.; Martinac, B. Biophysical Principles of Ion-Channel-Mediated Mechanosensory Transduction. _Cell Reports_ 2019, _29_ , 1–12
  * Darcy 1856 Darcy, H. _Les fontaines publiques de la ville de Dijon: Exposition et application des principes à suivre et des formules à employer dans les questions de distribution d’eau : Ouvrage terminé par un appendice relatif aux fournitures d’eau de plusieurs villes, au filtrage des eaux et à la fabrication des tuyaux de fonte, de plomb, de tôle et de bitume_ ; Victor Dalmont, éditeur, 1856
  * Whitaker 1986 Whitaker, S. Flow in Porous Media I: A Theoretical Derivation of Darcy’s Law. _Transp Porous Med_ 1986, _1_ , 3–25
  * O’Sullivan et al. 2022 O’Sullivan, C.; Arson, C.; Coasne, B. A Perspective on Darcy’s Law across the Scales: From Physical Foundations to Particulate Mechanics. _Journal of Engineering Mechanics_ 2022, _148_ , 04022064
  * Muskat 1937 Muskat, M. _The Flow of Homogeneous Fluids Through Porous Media_ ; McGraw-Hill Book Company, Incorporated, 1937
  * Thompson et al. 1998 Thompson, A. P.; Ford, D. M.; Heffelfinger, G. S. Direct Molecular Simulation of Gradient-Driven Diffusion. _J. Chem. Phys._ 1998, _109_ , 6406–6414
  * Arya et al. 2001 Arya, G.; Chang, H.-C.; Maginn, E. J. A Critical Comparison of Equilibrium, Non-Equilibrium and Boundary-Driven Molecular Dynamics Techniques for Studying Transport in Microporous Materials. _J. Chem. Phys._ 2001, _115_ , 8112–8124
  * Bhatia and Nicholson 2008 Bhatia, S. K.; Nicholson, D. Modeling Mixture Transport at the Nanoscale: Departure from Existing Paradigms. _Phys. Rev. Lett._ 2008, _100_ , 236103
  * Martin et al. 2001 Martin, M. G.; Thompson, A. P.; Nenoff, T. M. Effect of Pressure, Membrane Thickness, and Placement of Control Volumes on the Flux of Methane through Thin Silicalite Membranes: A Dual Control Volume Grand Canonical Molecular Dynamics Study. _J. Chem. Phys._ 2001, _114_ , 7174–7181
  * Onsager 1931 Onsager, L. Reciprocal Relations in Irreversible Processes. I. _Phys. Rev._ 1931, _37_ , 405–426
  * Bhatia et al. 2011 Bhatia, S. K.; Bonilla, M. R.; Nicholson, D. Molecular Transport in Nanopores: A Theoretical Perspective. _Phys. Chem. Chem. Phys._ 2011, _13_ , 15350–15383
  * Falk et al. 2015 Falk, K.; Coasne, B.; Pellenq, R.; Ulm, F.-J.; Bocquet, L. Subcontinuum Mass Transport of Condensed Hydrocarbons in Nanoporous Media. _Nature Communications_ 2015, _6_ , 6949
  * Obliger et al. 2016 Obliger, A.; Pellenq, R.; Ulm, F.-J.; Coasne, B. Free Volume Theory of Hydrocarbon Mixture Transport in Nanoporous Materials. _J. Phys. Chem. Lett._ 2016, _7_ , 3712–3717
  * Sastre et al. 2018 Sastre, G.; Kärger, J.; Ruthven, D. M. Molecular Dynamics Study of Diffusion and Surface Permeation of Benzene in Silicalite. _J. Phys. Chem. C_ 2018, _122_ , 7217–7225
  * Bukowski et al. 2021 Bukowski, B. C.; Keil, F. J.; Ravikovitch, P. I.; Sastre, G.; Snurr, R. Q.; Coppens, M.-O. Connecting Theory and Simulation with Experiment for the Study of Diffusion in Nanoporous Solids. _Adsorption_ 2021, _27_ , 683–760
  * Schlaich et al. 2017 Schlaich, A.; Kappler, J.; Netz, R. R. Hydration Friction in Nanoconfinement: From Bulk via Interfacial to Dry Friction. _Nano Lett._ 2017, _17_ , 5969–5976
  * Chakraborty et al. 2017 Chakraborty, S.; Kumar, H.; Dasgupta, C.; Maiti, P. K. Confined Water: Structure, Dynamics, and Thermodynamics. _Acc. Chem. Res._ 2017, _50_ , 2139–2146
  * Tinti et al. 2021 Tinti, A.; Camisasca, G.; Giacomello, A. Structure and Dynamics of Water Confined in Cylindrical Nanopores with Varying Hydrophobicity. _Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences_ 2021, _379_ , 20200403
  * Bousige et al. 2016 Bousige, C.; Ghimbeu, C. M.; Vix-Guterl, C.; Pomerantz, A. E.; Suleimenova, A.; Vaughan, G.; Garbarino, G.; Feygenson, M.; Wildgruber, C.; Ulm, F.-J.; Pellenq, R. J.-M.; Coasne, B. Realistic Molecular Model of Kerogen’s Nanostructure. _Nature Materials_ 2016, _15_ , 576–582
  * Israelachvili and Pashley 1983 Israelachvili, J. N.; Pashley, R. M. Molecular Layering of Water at Surfaces and Origin of Repulsive Hydration Forces. _Nature_ 1983, _306_ , 249–250
  * Hayashi et al. 2002 Hayashi, T.; Pertsin, A. J.; Grunze, M. Grand Canonical Monte Carlo Simulation of Hydration Forces between Nonorienting and Orienting Structureless Walls. _The Journal of Chemical Physics_ 2002, _117_ , 6271–6280
  * Li et al. 2007 Li, T.-D.; Gao, J.; Szoszkiewicz, R.; Landman, U.; Riedo, E. Structured and Viscous Water in Subnanometer Gaps. _Phys. Rev. B_ 2007, _75_ , 115415
  * Chattoraj and Birdi 1984 Chattoraj, D. K.; Birdi, K. S. _Adsorption and the Gibbs Surface Excess_ ; Plenum Press: New York, 1984
  * Ismail et al. 2015 Ismail, A. F.; Khulbe, K. C.; Matsuura, T. _Gas Separation Membranes: Polymeric and Inorganic_ , 2015th ed.; Springer: Cham Heidelberg New York Dordrecht London, 2015
  * Brochard et al. 2012 Brochard, L.; Vandamme, M.; Pellenq, R. J. M. Poromechanics of Microporous Media. _Journal of the Mechanics and Physics of Solids_ 2012, _60_ , 606–622
  * Gor et al. 2017 Gor, G. Y.; Huber, P.; Bernstein, N. Adsorption-Induced Deformation of Nanoporous Materials—A Review. _Applied Physics Reviews_ 2017, _4_ , 011303
  * Kolesnikov et al. 2021 Kolesnikov, A. L.; Budkov, Y. A.; Gor, G. Y. Models of Adsorption-Induced Deformation: Ordered Materials and Beyond. _J. Phys.: Condens. Matter_ 2021, _34_ , 063002
  * Motakabbir and Berkowitz 1990 Motakabbir, K. A.; Berkowitz, M. Isothermal Compressibility of SPC/E Water. _J. Phys. Chem._ 1990, _94_ , 8359–8362
  * Eskandari-Ghadi and Zhang 2021 Eskandari-Ghadi, M.; Zhang, Y. Mechanics of Shrinkage-Swelling Transition of Microporous Materials at the Initial Stage of Adsorption. _International Journal of Solids and Structures_ 2021, _222–223_ , 111041
  * Kanduč et al. 2016 Kanduč, M.; Schlaich, A.; Schneck, E.; Netz, R. R. Water-Mediated Interactions between Hydrophilic and Hydrophobic Surfaces. _Langmuir_ 2016, _32_ , 8767–8782
  * Erbaş et al. 2012 Erbaş, A.; Horinek, D.; Netz, R. R. Viscous Friction of Hydrogen-Bonded Matter. _J. Am. Chem. Soc._ 2012, _134_ , 623–630
  * Hansen and McDonald 2013 Hansen, J.-P.; McDonald, I. R. _Theory of Simple Liquids: With Applications to Soft Matter_ ; Academic Press, 2013
  * Kärger et al. 2012 Kärger, J.; Ruthven, D. M.; Theodorou, D. N. _Diffusion in Nanoporous Materials_ ; John Wiley & Sons, 2012
  * Boţan et al. 2011 Boţan, A.; Rotenberg, B.; Marry, V.; Turq, P.; Noetinger, B. Hydrodynamics in Clay Nanopores. _J. Phys. Chem. C_ 2011, _115_ , 16109–16115
  * Bocquet and Barrat 1994 Bocquet, L.; Barrat, J.-L. Hydrodynamic Boundary Conditions, Correlation Functions, and Kubo Relations for Confined Fluids. _Phys. Rev. E_ 1994, _49_ , 3079–3092
  * Herrero et al. 2019 Herrero, C.; Omori, T.; Yamaguchi, Y.; Joly, L. Shear Force Measurement of the Hydrodynamic Wall Position in Molecular Dynamics. _J. Chem. Phys._ 2019, _151_ , 041103
  * Sokhan et al. 2002 Sokhan, V. P.; Nicholson, D.; Quirke, N. Fluid Flow in Nanopores: Accurate Boundary Conditions for Carbon Nanotubes. _The Journal of Chemical Physics_ 2002, _117_ , 8531–8539
  * Geng et al. 2019 Geng, X.; Yu, M.; Zhang, W.; Liu, Q.; Yu, X.; Lu, Y. Slip Length and Structure of Liquid Water Flowing Past Atomistic Smooth Charged Walls. _Sci Rep_ 2019, _9_ , 18957
  * Vo and Kim 2016 Vo, T. Q.; Kim, B. Transport Phenomena of Water in Molecular Fluidic Channels. _Sci Rep_ 2016, _6_ , 33881
  * Ma et al. 2015 Ma, L.; Gaisinskaya-Kipnis, A.; Kampf, N.; Klein, J. Origins of Hydration Lubrication. _Nature Communications_ 2015, _6_ , 6060
  * Leng and Cummings 2005 Leng, Y.; Cummings, P. T. Fluidity of Hydration Layers Nanoconfined between Mica Surfaces. _Phys. Rev. Lett._ 2005, _94_ , 026101
  * Shaat et al. 2020 Shaat, M.; Javed, U.; Faroughi, S. Wettability and Confinement Size Effects on Stability of Water Conveying Nanotubes. _Sci Rep_ 2020, _10_ , 17167
  * Köhler and da Silva 2016 Köhler, M. H.; da Silva, L. B. Size Effects and the Role of Density on the Viscosity of Water Confined in Carbon Nanotubes. _Chemical Physics Letters_ 2016, _645_ , 38–41
  * Marbach and Bocquet 2017 Marbach, S.; Bocquet, L. Active Sieving across Driven Nanopores for Tunable Selectivity. _The Journal of Chemical Physics_ 2017, _147_ , 154701
  * Plimpton 1995 Plimpton, S. Fast Parallel Algorithms for Short-Range Molecular Dynamics. _J. Comput. Phys._ 1995, _117_ , 1–19
  * Berendsen et al. 1987 Berendsen, H. J. C.; Grigera, J. R.; Straatsma, T. P. The Missing Term in Effective Pair Potentials. _J. Phys. Chem._ 1987, _91_ , 6269–6271
  * Ryckaert et al. 1977 Ryckaert, J.-P.; Ciccotti, G.; Berendsen, H. J. C. Numerical Integration of the Cartesian Equations of Motion of a System with Constraints: Molecular Dynamics of n-Alkanes. _Journal of Computational Physics_ 1977, _23_ , 327–341
  * Schlaich 2024 Schlaich, A. Replication Data for: Bridging Microscopic Dynamics and Hydraulic Permeability in Mechanically-Deformed Nanoporous Materials. 2024; https://doi.org/10.18419/darus-3966
  * Billemont et al. 2013 Billemont, P.; Coasne, B.; De Weireld, G. Adsorption of Carbon Dioxide, Methane, and Their Mixtures in Porous Carbons: Effect of Surface Chemistry, Water Content, and Pore Disorder. _Langmuir_ 2013, _29_ , 3328–3338
  * Pikunic et al. 2003 Pikunic, J.; Clinard, C.; Cohaut, N.; Gubbins, K. E.; Guet, J.-M.; Pellenq, R. J.-M.; Rannou, I.; Rouzaud, J.-N. Structural Modeling of Porous Carbons: Constrained Reverse Monte Carlo Method. _Langmuir_ 2003, _19_ , 8565–8582
  * Shinoda et al. 2004 Shinoda, W.; Shiga, M.; Mikami, M. Rapid Estimation of Elastic Constants by Molecular Dynamics Simulation under Constant Stress. _Phys. Rev. B_ 2004, _69_ , 134103
  * Plimpton et al. 1997 Plimpton, S.; Pollock, R.; Stevens, M. Particle-Mesh Ewald and rRESPA for Parallel Molecular Dynamics Simulations. In Proceedings of the Eighth SIAM Conference on Parallel Processing for Scientific Computing. 1997
  * Yeh and Berkowitz 1999 Yeh, I.-C.; Berkowitz, M. L. Ewald Summation for Systems with Slab Geometry. _The Journal of Chemical Physics_ 1999, _111_ , 3155–3162
  * Michaud-Agrawal et al. 2011 Michaud-Agrawal, N.; Denning, E. J.; Woolf, T. B.; Beckstein, O. MDAnalysis: A Toolkit for the Analysis of Molecular Dynamics Simulations. _J. Comput. Chem._ 2011, _32_ , 2319–2327

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