2021

# Fluids at the Nanoscale: from continuum to sub-continuum transport

Nikita Kavokine  1 Roland R. Netz  2 and Lydéric Bocquet1 1Laboratoire de
Physique de l’École Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne
Université, Université Paris-Diderot, Sorbonne Paris Cité, Paris, France;
email<EMAIL_ADDRESS><EMAIL_ADDRESS>2Fachbereich Physik,
Freie Universitat Berlin, Berlin 14195, Germany

###### Abstract

Nanofluidics has firmly established itself as a new field in fluid mechanics,
as novel properties have been shown to emerge in fluids at the nanometric
scale. Thanks to recent developments in fabrication technology, artificial
nanofluidic systems are now being designed at the scale of biological
nanopores. This ultimate step in scale reduction has pushed the development of
new experimental techniques and new theoretical tools, bridging fluid
mechanics, statistical mechanics and condensed matter physics. This review is
intended as a toolbox for fluids at the nanometre scale. After presenting the
basic equations that govern fluid behaviour in the continuum limit, we will
show how these equations break down and new properties emerge in molecular
scale confinement.

###### keywords:

nanofluidics, slippage, ion transport, sub-continuum, non-mean-field,
statistical mechanics

##  1 INTRODUCTION

Fluid flows at the nanometre scale have been studied indirectly in various
disciplines for the last fifty years (Eijkel & van den Berg (2005)). However,
it is only fifteen years ago that nanofluidics has emerged as a field on its
own, first as a natural extension of microfluidics towards smaller scales.
Back then, it was an issue in itself to establish that nanofluidics deserves
its own name, meaning that there are specific effects at the nano-scale that
are not present at the micro-scale.

Indeed, the ”ultimate scale” for observing specific effects is set by the
molecular size of the fluid ; more precisely, a critical confinement
$\ell_{c}=1~{}\rm nm$ has been generally accepted as the limit of validity for
the Navier-Stokes equation (Bocquet & Charlaix (2010), Sparreboom et al.
(2010)). Moreover, it is at the molecular scale that the fluidic functions of
biological systems emerge: from the giant permeability and perfect selectivity
of the aquaporin (Murata et al. (2000)), to the ion specificity of KcsA
channels (MacKinnon (2004)), to the mechano-sensitivity of Piezo channels (Wu
et al. (2017)), to name a few. However, ten years ago, the exploration of this
ultimate scale was hindered by technical challenges, as molecular scale
channels could not be fabricated artificially.

A decade later, nanofluidics has firmly established itself as a field (Bocquet
(2020)). Indeed, lengthscales associated with the electrostatics and the
fluctuations of surfaces may reach up to several tens of nanometres, and their
effects may be probed specifically in systems without molecular scale
confinement (Schoch et al. (2008), Bocquet & Charlaix (2010), Sparreboom et
al. (2010)). These lengthscales govern the key nanofluidic phenomena that have
been demonstrated over the last ten years, such as, for instance,  fast flows
in carbon nanotubes (Holt et al. (2006)), diffusio-osmotic energy conversion
(Siria et al. (2013)) or diode-type effects (Vlassiouk & Siwy (2007)).

However, the progress in fabrication technology has now allowed to overcome
the challenges that have hindered the development of nanofluidics at the
ultimate scales, and artificial devices with confinement down to about one
water molecule size (3 $\overset{\circ}{\rm A}$) have been achieved, in 0D, 1D
or 2D geometry (Feng et al. (2016a), Tunuguntla et al. (2017), Gopinadhan et
al. (2019)). It is therefore an exciting time for nanofluidics, since it now
has the potential to reverse-engineer biological functions: minimal artificial
systems that mimic biological processes may be designed and studied.
Furthermore, nanofluidics is known for its short path from fundamental science
to applications and innovation, and developments in single channel fabrication
are likely to have direct implications for filtration and membrane science
(Bocquet (2020)).

Figure 1: An overview of nanofluidic lengthscales. Main ingredients of the
physics above the continuum limit, and below.

At this time of accelerated development  of the field, one should realize that
the  nanoscales under study require new tools for understanding the physics at
play. The need for ”new physics” is particularly emphasised in a recent review
by Faucher et al. (2019), which identifies ”critical knowledge gaps in mass
transport through single-digit nanopores” – nanopores that are less than 10 nm
in size. Those knowledge gaps  exist largely because the tools applied for
understanding the experiments  are built on macroscopic fluid mechanics and
continuum electrostatics. But consider now a typical nanochannel of radius 1
nm. At physiological salt concentration, it contains only a single ion per 50
nm channel length. Similarly, the carbon nanotube porins studied by Tunuguntla
et al. (2017) each contain about 30 water molecules. These numbers strongly
suggest that discrete particle effects may be important, and call for a
statistical mechanics description. Furthermore, below 1 nm confinement, the
length scales associated with the fluid dynamics become comparable to
characteristic length scales of the electrons in the confining solids, such as
the Thomas-Fermi length (Mahan (1990)). This points to the necessity of
describing the confining solids at the level of condensed-matter physics, and
not simply as a space impenetrable to the fluid molecules. Overall,
understanding fluidic phenomena at the nanometer scales requires bridging the
gap between fluid mechanics, statistical mechanics and condensed matter
physics.

In the past years, the breakdown of continuum equations has often set a hard
limit for fluid mechanics: below the continuum limit was the realm of
molecular simulations. However, the need for understanding experiments has
pushed for the development (or rediscovery) of analytical tools that have
allowed to identify some specific phenomena and associated length scales. The
description of these phenomena is a key part of this review, as summarised in
figure 1. The review is organised as follows. In section 2, we give a brief
overview of available nanofluidic systems and fabrication methods. In section
3, we focus on continuum modelling of nanofluidic systems, with particular
emphasis on the precautions that should be taken when applying it to the
smallest scales. Finally, in section 4, we go below the continuum limit, and
highlight the specific phenomena that emerge along with the theoretical tools
to describe them.

##  2 THE TOOLBOX OF EXPERIMENTAL SYSTEMS

Nanofluidics generally follows a bottom-up approach. Elementary phenomena are
understood at the well-controlled scale of the individual channel, before
eventually being applied to more complex systems. Hence, the design of these
well-controlled systems is paramount to the development of the field. We start
this review by going through the systems that have so far been achieved, in
order of dimensionality.

Figure 2: State-of-the-art nanofluidic systems of various dimensions. a. TEM-
drilled nanopore in single layer $\rm MoS_{2}$ (Feng et al. (2016a)). b. Boron
nitride nanotube inserted into a SiN membrane (Siria et al. (2013)). c.
Schematic and TEM image of the nanoslit device fabricated by Radha et al.
(2016).

###  2.1 Nanopores

Nanopores are channels whose length $L$ and diameter $d$ are both in the
nanometre range (figure 2a).  Initial studies focused on solid-state nanopores
drilled through membranes made out of ceramics such as SiN of SiC (Keyser et
al. (2006), Dekker (2007)). More recently the advent of 2D materials, such as
graphene, hexagonal boron nitride (hBN) or $\rm MoS_{2}$, allowed for the
exploration of nanopores in atomically thin membranes (Garaj et al. (2010),
Sahu & Zwolak (2019)). Essentially three types of fabrication pathways have
been reported for well-controlled nanopores in 2D materials.

Drilling with an electron or a focused ion beam (FIB). A single 5 nm pore
drilled in monolayer graphene was first reported by Garaj et al. (2010).
Celebi et al. (2014) reported FIB drilling of arrays of nanopores in bilayer
graphene, ranging from 7.6 nm to 1 $\mu\rm m$ in diameter.

Electrochemical etching. Feng et al. (2015) reported the opening of pores in
monolayer $\rm MoS_{2}$ when placed in a salt solution in between two
electrodes. Applying a potential above the oxydation potential of $\rm
MoS_{2}$ resulted in the gradual removal of single $\rm MoS_{2}$ units, thus
creating an opening of controlled size. Nanopores down to 0.6 nm in diameter
have been reported (Feng et al. (2016b)).

The use of intrinsic defects in 2D materials. Large area membranes made of
graphene or hBN are known to exhibit defects in the form of pores, ranging in
size from a few angstroms to 15 nm depending on conditions (Walker et al.
(2017), O’Hern et al. (2012)), or such defects may be generated on purpose in
smaller membranes using ultraviolet-induced oxidative etching (Koenig et al.
(2012)). The chemical vapour deposition (CVD) graphene membranes produced by
Jain et al. (2015) exhibited pores in the sub-2 nm range spaced by 70 to 100
nm. Placing the membrane on top of a 30-40 nm pore of a SiN membrane allowed
to statistically isolate and study a single pore.

###  2.2 Nanotubes

Nanotubes are cylindrical channels of diameter $d$ in the nanometre range, and
length $L\gg d$ (figure 2b). They are typically made out of carbon, or the
isoelectronic boron nitride. The nanotubes themselves, as a product of self-
assembly, are readily available, but interfacing a nanotube to a fluidic
system and avoiding leakage is still an experimental challenge. Three distinct
strategies for addressing this challenge have been reported.

Building a microfluidic system on top of CVD-grown single-walled carbon
nanotubes. The systems may comprise one or several carbon nanotubes, typically
of diameter 1 to 2 nm. The tubes generally have very high aspect ratio with
lengths up to 0.5 mm (Lee et al. (2010), Choi et al. (2013), Yazda et al.
(2017)), although similarly built systems with 20 $\mu\rm m$ long nanotubes
have also been reported (Pang et al. (2011)).

Insertion of a multiwalled nanotube into a solid state membrane. Siria et al.
(2013) reported the fabrication of nanofluidic devices comprising a single
boron nitride nanotube, inserted into a hole milled in a SiN membrane by
direct nanomanipulation under SEM. The hole could be sealed _in situ_ by
cracking of naphtalene induced by the electron beam. The method was later
extended to carbon nanotubes (Secchi et al. (2016a)), of 30 to 100 nm inner
diameter and about 1 $\mu\rm m$ in length, and recently to smaller, 2 nm inner
diameter, multiwall carbon nanotubes.

Insertion of nanotubes into a lipid membrane. Liu et al. (2013) reported the
insertion of very short (5 to 10 nm) and very narrow (0.8 to 2 nm in diameter)
nanotubes into a supported lipid membrane. Nanotubes were brought in contact
with the lipid bilayer thanks to a microinjection probe. Recently, Tunuguntla
et al. (2017) reported the self-assembly of similar nanotubes, which they term
carbon nanotube porins, into phospholipid vesicles. A single patch of membrane
could also be isolated in order to study a single porin.

###  2.3 Nanoslits down to angström confinements

Slit-like channels with one dimension below tens of nanometers were first made
using micro- and nano- fabrication techniques. But recently, Radha et al.
(2016) reported the manufacturing of two-dimensional channels by van der Waals
assembly of 2D materials (figure 2c). A few layers of graphene were used as
spacers between two crystals of graphite, hBN or $\rm MoS_{2}$, allowing for
atomically smooth channels of a few $\mu\rm m$ in length, 100 nm in width and
down to 7 Å in height, that is the thickness of two graphene layers. Very
recently (Gopinadhan et al. (2019)), water transport through one-graphene-
layer thick (3.4 Å) channels was reported.

This brief overview highlights that nanofluidics at the molecular scale is now
a reality. Not only molecular scale confinement is possible, but the geometry
of the confinement and the nature of the confining materials can also be
tuned.

##  3 NANOFLUIDICS IN THE CONTINUUM LIMIT

###  3.1 Liquid transport

####  3.1.1 Basic equations

The two-centuries-old Navier-Stokes equation is remarkably robust at
describing fluid flow down to the smallest scale, typically $\ell_{c}=1~{}\rm
nm$ for water in normal pressure and temperature conditions (Bocquet &
Charlaix (2010)). This length scale is essentially a lower bound for defining
a fluid viscosity $\eta$. Indeed, in macroscopic fluid mechanics, the
kinematic viscosity $\nu=\eta/\rho$, where $\rho$ is the mass density, plays
the role of a diffusion coefficient for the fluid momentum. For such a
diffusion coefficient to be defined, the time required for momentum to diffuse
across the system, $\ell_{c}^{2}/\nu$, must be larger than the timescale of
molecular motion, which is the microscopic origin of diffusion. A water
molecule at a thermal agitation speed of $300~{}\rm m\cdot s^{-1}$ moves by
its own size in $\tau_{c}=10^{-12}~{}\rm s$, which defines a molecular time
scale. Therefore, viscosity may be defined down to a system size

$\ell_{c}\sim\sqrt{\nu\tau_{c}}\sim 1~{}\rm nm.$ (1)

Below this length scale, water structuring due to surfaces, memory effects and
other sub-continuum phenomena come into play: these will be discussed in
section 4. For water flow at $10~{}\rm nm$ length scales, the Reynolds number
remains smaller than 0.1 up to fluid velocities of $10~{}\rm m\cdot s^{-1}$.
Hence, in nanofluidic systems, inertial effects may be safely neglected, and
the fluid flow is described by the Stokes equation:

$\eta\Delta v+f=\nabla p,$ (2)

where $p$ is the pressure and $f$ a body force, which may be due, for example,
to the application of an electric field (see section 3.3).

####  3.1.2 Boundary conditions

Stokes flow is often solved with no-slip boundary conditions: the velocity of
the liquid is assumed to vanish at a solid-liquid interface. This is, however,
a limiting case of the more general Navier partial slip boundary condition,
which enforces that the viscous stress at the interface should be balancing
the solid liquid friction force. Within linear response theory, the friction
force is proportional to the liquid velocity. For a fluid flowing in the
direction $x$ along a surface of normal $z$, the force balance per unit area
writes $\sigma_{xz}=\lambda v_{x},$ with $[\sigma]$ the stress tensor and
$\lambda$ the friction coefficient per unit area (expressed in $\rm N\cdot
s\cdot m^{-3}$). For a Newtonian fluid, $\sigma_{xz}=\eta\partial_{z}v_{x}$,
which allows to rewrite the Navier boundary condition as

$v_{x}=\left.b\frac{\partial v_{x}}{\partial z}\right|_{\rm wall},$ (3)

introducing the slip length $b=\eta/\lambda$. The slip length can be
geometrically interpreted as the depth inside the solid where the linearly
extrapolated fluid velocity profile vanishes. Accordingly, the no-slip
boundary condition corresponds to $\lambda\to\infty$ or $b\to 0$. The effect
of the partial slip condition is to simply shift the no-slip velocity profile
by the slip velocity, which is not negligible roughly within a slip length
from the wall. Since slip lengths  up to tens of nanometers have been measured
on atomically flat  (and hydrophobic) surfaces, slippage is expected to play a
crucial role in nanofluidics, and some of its effects will be discussed in the
following sections. In the smallest channels, of size $R\ll b$, a perfect slip
boundary condition may even be appropriate: the flow is then controlled by
entrance effects.

####  3.1.3 Geometry and entrance effects

Experimentally, the flow profile inside a nanofluidic channel can hardly be
resolved, and one typically measures the total flow rate $Q$. Under a pressure
drop $\Delta P$ and no-slip boundary conditions, the flow rate through a
cylindrical channel of radius $R$ is given by the Hagen-Poiseuille formula:

$Q_{c}=\frac{\pi R^{4}}{8\eta L}\Delta P.$ (4)

This formula assumes a channel length $L\gg R$, and thereby neglects the
effect of channel mouths on the flow rate. But the transition from a
macroscopic reservoir to a nanoscale channel is a source of viscous
dissipation, as the streamlines need to be bent in order for the fluid to
enter the channel. These entrance effects may be examined by considering the
flow through an infinitely thin nanopore, which is of interest in itself,
given the geometry of certain nanofluidic devices (see section 2.1). This
problem was addressed by Sampson (1891). For a nanopore of radius $R$ (and
vanishing length) under pressure drop $\Delta P$, Sampson obtained the
expression of the flow rate as

$Q_{p}=\frac{R^{3}}{3\eta}\Delta P.$ (5)

The scaling in Sampson’s formula naturally emerges from a Stokes equation
where the only lengthscale is $R$: $\eta\Delta v=\nabla p\Rightarrow\eta
v/R^{2}\sim\Delta P/R$, and the typical fluid velocity is $v\sim Q/R^{2}$. In
order to estimate the flow rate through a channel taking into account entrance
effects, one may simply add the hydrodynamic resistances of the pore
(${\cal{R}}_{p}$) and the channel (${\cal{R}}_{c}$).  If one writes
$Q_{c}=(\Delta P)_{c}/{\cal{R}}_{c}$ and $Q_{p}=(\Delta P)_{p}/{\cal{R}}_{p}$,
then the entrance-corrected flow rate $Q_{pc}$ is obtained by imposing
$Q_{p}=Q_{c}=Q_{pc}$ and $\Delta P=(\Delta P)_{c}+(\Delta P)_{p}$, so that

$Q_{pc}=\frac{\Delta P}{{\cal{R}}_{h}+{\cal{R}}_{p}}=\frac{\pi R^{4}}{8\eta
L}\frac{\Delta P}{1+\frac{3\pi}{8}\frac{R}{L}}.$ (6)

An exact computation (Dagan et al. (1982)) shows that the error made by this a
priori crude approximation is less than 1%. Equation 6 makes a continuous
transition between the nanopore and nanochannel regimes, and shows that
entrance effects are apparently negligible for channel lengths that exceed a
few channel radii.

Figure 3: Entrance effects in nanofluidics. a. Viscous dissipation rate, and
streamlines, for the pressure-driven flow of water across a nanopore, as
obtained from a finite elements solution of the Stokes equation (COMSOL). The
colour scale, from blue to red, encodes the viscous dissipation. b. Geometric
model of the aquaporin as considered by Gravelle et al. (2013). c.
Permeability of the model aquaporin as a function of the cone angle $\alpha$.

However, the above discussion has crucially not taken into account slippage,
which, as we have highlighted in the previous section, is a strong effect at
the nanoscale. Introducing a non-zero slip length $b$, the flow rate though a
channel becomes

$Q_{c}=\frac{\pi R^{4}}{8\eta L}\left(1+\frac{4b}{R}\right)\Delta P,$ (7)

while the flow rate through a pore is not significantly affected (Gravelle et
al. (2013)), since the source of dissipation in that case is mostly geometric.
A full expression can be obtained by gathering previous results, but in the
limit where $b\gg R$, the entrance-corrected flow rate becomes

$Q_{pc}=\frac{R^{3}}{3\eta}\frac{\Delta P}{1+\frac{2L}{3\pi b}}.$ (8)

Thus, the hydrodynamic resistance is actually dominated by entrance effects as
long as the channel is shorter than the slip length, rather than the channel
radius (see figure 3a). In the presence of significant slippage, one should
check whether the low Reynolds number assumption still holds. The average
velocity through a channel of radius $R=5~{}\rm nm$ and length $L=1~{}\rm\mu
m$, with slip length $b=30~{}\rm nm$, under a pressure drop $\Delta P=1~{}\rm
bar$ is $v=8~{}\rm mm\cdot s^{-1}$, which is 25 times faster than the no-slip
result, but still well below the $1~{}\rm m\cdot s^{-1}$ threshold established
in 3.1.1.

Entrance-effect-dominated transport is particularly relevant for biological
nanochannels, due to their relatively small aspect ratio. A striking example
is aquaporin, which was recently studied from a hydrodynamic point of view by
Gravelle et al. (2013). Aquaporins are channel proteins that selectively
transport water across the cell membrane. A simplified geometrical model for
the aquaporin consists of two conical vestibules connected by a subnanometric
channel where water flows in single file (figure 3b). The single-file
transport is expected to be nearly frictionless, and therefore the limit to
the aquaporin’s permeability is set by the entrance effects in the conical
vestibules. Moreover, since the channel in question is less than 1 nm wide and
slip lengths of the order of 10 nm are expected, perfect slip boundary
conditions are relevant for the flow, and the viscous dissipation has a purely
geometric origin in the curvature of the stream lines. With the notations of
figure 3b, Gravelle _et al._ express the total hydrodynamic resistance of the
aquaporin as

$R_{\rm AQP}=R_{1}+R_{2}=C_{\infty}\frac{\eta}{a^{\prime
3}}+C_{\infty}\sin\alpha\frac{\eta}{a^{3}}.$ (9)

Both terms are the analogue of Sampson’s formula (eq. 5). $C_{\infty}=3.75$
replaces the factor 3 in the case of perfect slip boundary conditions, and the
factor $\sin\alpha$ appears because at the cone-cylinder transition, the
stream lines turn by an angle $\alpha$, as opposed to $\pi/2$ when the
transition is from an infinite reservoir to a pore. With that, the
permeability $K=R_{\rm AQP}^{-1}$ turns out to be a non-monotonous function on
$\alpha$ (figure 3c), with the maximal permeability reached for cone angles in
the range measured on aquaporin structures obtained by X-ray crystallography.
It appears, therefore, that the geometry of the aquaporin is an optimum for
hydrodynamic entrance effects, highlighting their particular relevance for
nanoscale fluid transport.

###  3.2 Gas transport

Figure 4: a. Schematic of gas transport through a nanopore in the Knudsen
regime $\ell\gg R$. A diffuse reflection at the pore wall is represented. b.
Schematic of osmotic transport of water mediated by a nanobubble, reproduced
from Lee et al. (2014b).

In a manner similar to liquid flow, gas flow can also be used to probe
nanofluidic systems. Gas flow may display two limiting regimes, depending on
the value of the Knudsen number, defined as

$Kn=\frac{\ell}{R},$ (10)

where $\ell$ is the mean free path and $R$ is the typical system size. The
mean-free path scales as the inverse density according to
$\ell\sim(\rho\sigma^{2})^{-1}$, with $\sigma$ the molecular diameter. For
$Kn\ll 1$, the transport is dominated by intermolecular collisions and is
therefore described by hydrodynamics; for example, by the Poiseuille or
Sampson formulae introduced above. For $Kn\gg 1$, the transport is dominated
by collisions with the walls and is described by molecular diffusion: this is
the so-called Knudsen regime (Lei et al. (2016)). Consider a cylindrical
channel of radius $R$ and length $L\gg R$, connecting two gas reservoirs at
pressures $P_{1}$ and $P_{2}$ and same temperature $T$ (see figure 4a). One
may then define a diffusion coefficient that relates the molecular flow rate
$Q_{K}$ to the density gradient $\Delta n/L$ across the channel through a
Fick-type law:

$\frac{Q_{K}}{\pi R^{2}}=D_{K}\frac{\Delta
n}{L}=\frac{D_{K}}{k_{B}T}\frac{\Delta P}{L}.$ (11)

The last equality uses that $n=P/k_{B}T$ through the ideal gas law, and
$\Delta P=P_{1}-P_{2}$. The only parameters involved in the diffusion are the
channel radius $R$ and the average thermal velocity $v^{*}=\sqrt{8k_{B}T/\pi
m}$, where $m$ is the molecular mass of the gas. On dimensional grounds, the
Knudsen diffusion coefficient should scale as $D_{K}\sim Rv^{*}$. A kinetic
theory computation (Knudsen (1909), Steckelmacher (1966)) yields
$D_{K}=(2\pi/3)Rv^{*}$, and the Knudsen formula:

$Q_{K}=\frac{8}{3}\frac{\pi R^{3}}{\sqrt{2\pi mk_{B}T}}\frac{\Delta P}{L}.$
(12)

Now the Knudsen formula relies on the crucial approximation of diffuse
reflexion on the channel walls: a molecule that hits a wall has a new velocity
randomly picked out of a Boltzmann distribution. It was proposed by Maxwell
that only a fraction $f$ of reflections should be diffuse, and the rest should
be specular, that is correspond to elastic collisions. Smoluchowski then
derived the corresponding correction to the Knudsen formula (Lei et al.
(2016), v. Smoluchowski (1910)):

$Q_{K}=\frac{2-f}{f}\cdot\frac{8}{3}\frac{\pi R^{3}}{\sqrt{2\pi
mk_{B}T}}\frac{\Delta P}{L}.$ (13)

There is a divergence of the flow rate in the limit $f\to 0$, that is the
channel opposes no more resistance to the gas flow when all the wall
reflections are specular. This is to be expected, since in that case a
molecule that enters the channel necessarily exits at the other end. The flow
resistance is then dominated by entrance effects; in other words, the flow
rate is given by the rate of molecules hitting the channel apertures. Gas
molecules at a density $n$ hit an aperture of area $\pi R^{2}$ at a rate
$(1/4)nv^{*}\pi R^{2}$. Hence the effusion flow rate through an opening of
radius $R$ is

$Q_{e}=\pi R^{2}\Delta n\sqrt{\frac{k_{B}T}{2\pi m}}=\frac{\pi
R^{2}}{\sqrt{2\pi mk_{B}T}}\Delta P,$ (14)

which also sets the flow rate in a long channel with specular reflection at
the walls. One notes there is a close analogy between the Poiseuille and
Sampson formulae for liquids, on the one hand, and the Knudsen and effusion
formulae for gases, on the other hand. The fraction of specular reflections
$1-f$ plays a role similar to the slip length: if it is large, the transport
in a long channel may still be dominated by entrance effects.

The free effusion prediction has been verified in systems of graphene
nanopores (Celebi et al. (2014)), with a transition to the hydrodynamic
behaviour (Sampson formula) observed upon reducing the Knudsen number.
Specular reflections have also been evidenced in longer channels. Gas flow
exceeding the Knudsen prediction was measured in carbon nanotubes (Holt et al.
(2006), Majumder et al. (2008)), and, recently, nearly ballistic transport was
evidenced in angstrom-scale slits (Keerthi et al. (2018)). These results point
out that the tendency to anomalously fast transport in nanoscale confinement
exists not only for liquids, but also for gases.

A striking example of gas-mediated osmotic flow was demonstrated by the group
of Karnik. Lee et al. (2014b) fabricated a nanoporous (70 nm pore size)
membrane with partially hydrophobic pores, so that a nanobubble is trapped in
each pore when the membrane is immersed in water (see figure 4b). The
nanobubbles are impermeable to salt, but permeable to water through its vapour
phase transport. They show that the membrane reaches an ion rejection of
$99.9\%$, while competing with the permeability of state-of-the-art polyamide-
based membranes. Such high permeability might seem counterintuitive since
vapour phase transport is expected to scale with the density of water vapour,
1000 times lower than that of liquid water. However, using gas phase transport
as the ion rejection mechanism allows for much larger pores (70 nm) then what
would be required in the liquid phase ($\sim 1~{}\rm nm$) to ensure steric
rejection of ions. The scaling of the water transport rate with the pore size
cubed (Bocquet (2014)) then explains the high transport efficiency in the gas
phase. As such, nanoscale gas transport shows promise for addressing the
permeability-selectivity tradeoff in membrane science.

###  3.3 Ion transport

The behaviour of ions in nanofluidic systems is of great practical interest
with applications ranging from biological ion channels (MacKinnon (2004)) to
ionic liquids inside nanoporous electrodes (Chmiola et al. (2006), Merlet et
al. (2012)). Ion transport also provides an indirect way of probing fluid
transport, which is often useful, since electric currents are much easier to
measure than fluid flow. However, due to their long range Coulomb interactions
and diffusive dynamics, ions in nanochannels give rise to physics of great
complexity. In this section, we only sketch the main results with particular
emphasis on how well-known equations apply to the smallest channels, and we
refer refer the reader to other reviews on the subject (Bocquet & Charlaix
(2010), Schoch et al. (2008), Sparreboom et al. (2010)) for more details.

####  3.3.1 Basic equations

Consider an aqueous solution of monovalent salt. Let $\rho_{+},\rho_{-}$ be
the densities of positive and negative ions, respectively; $D$ the diffusion
coefficient, here assumed to be the same for ions of either sign, and $\phi$
the electrostatic potential. In a mean-field treatment, the convective-
diffusive dynamics of ions are described by a Smoluchowski equation:

$\frac{\partial\rho_{\pm}}{\partial
t}=\nabla\cdot\left(D\nabla\rho_{\pm}\mp\frac{eD}{k_{B}T}(\nabla\phi)\rho_{\pm}+v\rho_{\pm}\right),$
(15)

where $e$ is the unit charge, $\phi$ the electrostatic potential and $v$ is
the fluid velocity field. The mean-field assumption implies in particular that
correlations between the ions can be neglected: the potential importance of
such correlations in nanofluidics will be discussed in section 4. Until then,
we proceed by specifying the electrostatic potential through Poisson’s
equation,

$\Delta\phi=-e\frac{\rho_{+}-\rho_{-}}{\epsilon},$ (16)

where $\epsilon$ is the dielectric permittivity of water. For now we assume it
to be isotropic, though this assumption may break down for nano-confined
water, as we discuss in section 4. Lastly, we specify the flow velocity
through the Stokes equation, which now includes an electrostatic term:

$\eta\Delta v-e(\rho_{+}-\rho_{-})\nabla\phi=\nabla p.$ (17)

We now apply these three coupled equations to a specific geometry, though the
discussion that follows could be generalised to channels of any shpae. For
simplicity, we consider a slit-like channel of height $h$, width $w$ and
length $L$, with $w,L\gg h$, connecting two reservoirs of salt solution at
concentration $\rho_{s}$, extending along the direction $x$, between $z=0$ and
$z=h$. When considering ion transport, it is important to note that most
surfaces are charged in water, due either to the dissociation of surface
groups or to the adsorption of ions (Perram et al. (1973), Grosjean et al.
(2019), Mouhat et al. (2020)). We hence assume the channel wall carries a
surface charge density $-\Sigma e$ ($\Sigma$ is expressed in elementary
charges per unit area, and we assume here the surface charge to be negative).

####  3.3.2 Ionic conductance

We first neglect the coupling of ion transport to water transport, and
consider the electrophoretic (EP) contribution to the ionic current under an
applied electric field $E$: this means that we start by setting the fluid
velocity $v=0$. In the steady state, the Smoluchowski equation (15) reduces to
the so-called Nernst-Planck equations for the constant ionic fluxes (along the
$x$ direction):

$j_{\pm}=D\nabla_{x}\rho_{\pm}\mp\frac{eD}{k_{B}T}(\nabla_{x}\phi)\rho_{\pm}.$
(18)

Together with the Poisson equation (16), these constitute the widely used
Poisson-Nernst-Planck (PNP) framework. In our geometry, the condition $L\gg h$
ensures that in the middle of the channel $\nabla_{x}\rho_{\pm}=0$; moreover
to first order in $E$, $\nabla_{x}\phi=-E$ and the densities reduce to their
equilibrium values. Hence, the EP contribution to the ionic current writes

$I_{\rm
ep}=w\int_{0}^{h}\mathrm{d}z(j_{+}-j_{-})=w\frac{e^{2}D}{k_{B}T}\int_{0}^{h}\mathrm{d}z(\rho_{+}+\rho_{-})E.$
(19)

In order to compute $I_{\rm ep}$, one needs to find the equilibrium solution
of the coupled PNP equations for $\rho_{+}$ and $\rho_{-}$. At equilibrium
$j_{\pm}=0$ and the Nernst-Planck equations (18) can be integrated, imposing
that in the reservoirs $\phi=0$ and $\rho_{\pm}=\rho_{s}$. This yields a
Boltzmann distribution for the ions in the electrostatic potential

$\rho_{\pm}=\rho_{s}\exp\left(\mp\frac{e\phi}{k_{B}T}\right)\equiv\rho_{s}e^{\mp\psi},$
(20)

introducing a dimensionless potential $\psi$. Combining this with the Poisson
equation (16) yields the Poisson-Boltzmann (PB) equation:

$\Delta\psi-\lambda_{D}^{-2}\sinh(\psi)=0,$ (21)

which introduces the Debye length $\lambda_{D}=(8\pi\rho_{s}\ell_{B})^{-1/2}$,
with $\ell_{B}=e^{2}/(4\pi\epsilon k_{B}T)$ the Bjerrum length. In our
geometry, the PB equation has an implicit solution in terms of an elliptic
integral (Levine et al. (1975), Andelman (1995)). We will not exploit it here,
however, and we will instead recover the relevant limiting behaviours from
qualitative considerations.

It is well known that, roughly speaking, the Debye length sets the extension
of the diffuse layer of counterions next to a charged surface (Israelachvili
(2011)). Hence, if the channel height $h\gg\lambda_{D}$, its two opposing
walls do not ’see’ each other. We expect the conductance to be the sum of a
bulk term, and a surface term originating in the two Debye layers:

$I_{\rm ep}=2w\frac{e^{2}D}{k_{B}T}E(\rho_{s}h+\Sigma).$ (22)

where $\rho_{s}h$ and $\Sigma$ account for the number of charge carriers in
the bulk and at surfaces, respectively.  In the opposite limit where
$h\ll\lambda_{D}$, there is no more distinction between surface and bulk. All
the quantities may be considered uniform across the channel: this is called
the Debye overlap regime. However, one may not assume that the channel
contains only counterions, and one should go back to the thermodynamic
equilibrium with the reservoirs, which in this case bears the name of Donnan
equilibrium (Bocquet & Charlaix (2010)). One has
$\rho_{\pm}=\rho_{s}e^{\mp\psi}$, which implies a chemical equilibrium
$\rho_{+}\rho_{-}=\rho_{s}^{2}$ in the channel. Going further, in the limit of
long channel length, there should be local electroneutrality:
$h(\rho_{+}-\rho_{-})=2\Sigma$. This yields

$\rho_{\pm}=\sqrt{\rho_{s}^{2}+(\Sigma/h)^{2}}\pm\Sigma/h,$ (23)

and the current-voltage relation in the Debye overlap is

$I_{\rm ep}=2w\frac{e^{2}D}{k_{B}T}E\sqrt{(\rho_{s}h)^{2}+\Sigma^{2}}.$ (24)

Equation 24 displays the first peculiarity of small channels: one may not
simply add the surface and bulk contributions. Table 1 (see appendix) lists
the values of Debye length for different electrolyte concentrations, showing
that the Debye overlap regime is indeed relevant for experimentally accessible
nanofluidic systems (see section 2). Qualitatively, eqs. (22) and (24) both
predict saturation of the conductance at low salt concentrations at a value
determined by the surface charge. The saturation occurs when
$\rho_{s}\sim\Sigma/h$, which can be recast in the form
$h\sim\rho_{s}/\Sigma\equiv\ell_{Du}$. $\ell_{Du}$ is called the Dukhin length
and quantifies the competition between bulk and surface contributions to the
conductance. For a channel narrower than $\ell_{Du}$, surface contributions
dominate, and vice versa. The Dukhin length is going to be important in our
upcoming discussion of entrance effects.

[] Bjerrum length$\ell_{B}=\frac{e^{2}}{4\pi\epsilon_{0}k_{B}T}$. Distance
between two unit charges at which their interaction energy is $k_{B}T$. Debye
length$\lambda_{D}=(8\pi\rho_{s}\ell_{B})^{-1/2}$. Thickness of the diffuse
layer of counterions next to a charged surface. Dukhin
length$\ell_{Du}=\Sigma/\rho_{s}$. Channel width below which surface
conductance dominates over bulk conductance. Gouy-Chapman
length$\ell_{GC}=(2\pi\Sigma\ell_{B})^{-1}$. Distance a unit charge must
travel from a charged surface so that its electrostatic energy is reduced by
$k_{B}T$.

At this point, a remark should be made concerning the range of validity of
equations (22) and (24). Indeed, they have been derived from qualitative
considerations, without reference to the exact solution of the PB equation.
Now, from eqs. (19) and (20), one obtains more generally

$I_{\rm
ep}=2w\frac{e^{2}D}{k_{B}T}\left(\rho_{s}h\cosh(\psi(h/2))+\frac{\mathcal{E}}{2k_{B}T}\right),$
(25)

where $\mathcal{E}=(\epsilon/2)\int_{0}^{h}(\partial_{z}\phi)^{2}\mathrm{d}z$
is the electrostatic energy per unit area. The electric double layer can be
pictured as a capacitor with charge $\Sigma$, hence one would expect its
electrostatic energy to scale as $\Sigma^{2}$;  moreover, this is the
prediction of the linearised PB equation, i.e. eq. (21) with the approximation
$\sinh\psi\approx\psi$. This is in contrast to, eq. (22), which predicts a
linear scaling of the conductance with $\Sigma$: this scaling must therefore
come from the non-linearities of the PB equation. The PB equation may be
linearised if the potential varies by less than $k_{B}T$ across the Debye
layer (or across the channel if there is Debye overlap). This is the case in
the high concentration/low surface charge limit and specifically when the
Debye length is smaller than the so-called Gouy-Chapman length:
$\ell_{GC}=(2\pi\Sigma\ell_{B})^{-1}$. On the other hand, Eq. (24), valid for
the Debye overlap regime, is safe from a condition on $\ell_{GC}$, as it
predicts both quadratic and linear scalings depending on the value of
$\Sigma$, as long as there is Debye overlap.

####  3.3.3 Entrance effects

Similarly to what we have discussed for liquid and gas transport, we may now
ask, for ion transport, what is the additional electrical resistance due to
the channel - reservoir interface. Equivalently, we may want to estimate the
ionic conductance of a nanopore of small aspect ratio, say of radius $R$ and
length $L\sim R$. The problem has first been considered in the context of
biological channels by Hille (1968) and Hall (1975). Hall solved the
electrostatic problem with an electrode at infinity and an equipotential disk
accounting for the entrance of the pore. Translating the solution into scaling
arguments, the current through the pore entrance is $I\sim\pi
R^{2}\kappa_{b}\Delta V_{\rm out}/R$, where $\Delta V_{\rm out}$ is the
voltage drop at the entrance of the pore, which is expected to occur over a
distance $R$, and not over the macroscopic distance between the electrodes.
This defines an electrical access resistance as the ratio $\Delta V_{\rm
out}/I$. We now specialise to the thin Debye layer regime $\lambda_{D}\ll R$.
If one simply sums the access resistance and the channel resistance as given
by eq. (22) (adapted to cylindrical geometry), the current-voltage relation
becomes

$I_{\rm ep}=\kappa_{b}\left(\frac{L}{\pi
R^{2}}\frac{1}{1+\ell_{Du}/R}+\frac{1}{\alpha R}\right)^{-1}\Delta V,$ (26)

with $\alpha$ a geometric factor which is 2 in Hall’s computation, and
introducing the bulk conductivity $\kappa_{b}=2e^{2}D\rho_{s}/(k_{B}T)$. The
above equation predicts vanishing conductance as $\rho_{s}\to 0$, since the
access resistance becomes infinite in this limit;  however, this is not what
is observed experimentally in short nanopores (Lee et al. (2012), Feng et al.
(2016a)). This inconsistency arises because, for a surface-charged pore, the
access current has a surface contribution, in addition to the bulk
contribution. This surface contribution may be evaluated starting from charge
conservation at the surface, which imposes a relation between the axial and
radial components of the electric field, as pointed out by Khair & Squires
(2008):

$\kappa_{b}E_{r}=\partial_{x}[\kappa_{s}\theta(x)E_{x}],$ (27)

with $\kappa_{s}$ the surface conductance and $\theta$ the Heaviside function,
accounting for the discontinuity of surface charge at the pore boundary, which
leads to subtle consequences.  Indeed, it reveals that the Dukhin length,
$\ell_{Du}=\Sigma/\rho_{s}\sim\kappa_{s}/\kappa_{b}$, is the relevant
lengthscale for the surface contribution to the electric field outside the
pore, instead of the channel radius  or Debye length. The Dukhin length
appears here as an electrostatic healing length:  feeding the surface
conduction at the nanopore mouth disturbs the electric field lines in the bulk
over a length $\ell_{Du}$. This interpretation is supported by the numerical
results of Lee et al. (2012) as shown in figure 5a.  The bending of the
electric field lines can be estimated by a perturbative approach (Lee et al.
(2012)) that leads to an analytical expression for the corrected current-
voltage relation:

$I_{\rm ep}=\kappa_{b}\left(\frac{L}{\pi
R^{2}}\frac{1}{1+\ell_{Du}/R}+\frac{1}{\alpha
R+\beta\ell_{Du}}\right)^{-1}\Delta V.$ (28)

The surface-charged pore therefore appears, from the perspective of entrance
effects, as an uncharged pore of effective size $R+\ell_{Du}$, since the
geometrical prefactor $\beta\approx 2$. In the limit of vanishing salt
concentration, $\rho_{s}\to 0$, the conductance indeed saturates, and, as
expected, the entrance correction disappears in the limit of large aspect
ratio pores. Lee _et al._ successfully compared the prediction of eq. (28) to
experimental measurements in SiN nanopores in the range $100-500~{}\rm nm$.
More recently, it has been used to describe conductance measurements in $\rm
MoS_{2}$ nanopores down to 2 nm in diameter (Feng et al. (2016a)).

Figure 5: a. Colour plot of the electrostatic potential around a nanopore
immersed in a salt solution and subject to a voltage drop, in the charged and
in the uncharged case. Around a charged pore, the lengthscale for variation of
the potential is set by the Dukhin length. Adapted from Lee et al. (2012). b.
and c. Schematic representation of coupled ion and fluid transport in a
nanochannel under electric field driving: in the thin Debye layer limit (b.)
and in Debye overlap in the presence of slippage (c.). The electro-osmotic
contribution dominates in the latter case. d. Peclet number (defined in the
text), as a function of channel width and Gouy-Chapman length. The region in
parameter space where $Pe>1$ is coloured in dark red in the no-slip case, and
in light red when slippage is present.

####  3.3.4 Coupling with fluid flow

So far, we have neglected any coupling of ion transport to fluid transport.
However, in the presence of charged surfaces, an external electric field
exerts a net force on the charged Debye layer, which sets the fluid in motion.
This interfacially-driven flow is termed electro-osmotic (EO) flow. The EO
flow in turn drags along the ions in the Debye layer, which makes a
supplementary contribution to the ionic current, that we denote $I_{\rm eo}$.
This contribution has a convenient exact expression in the slit-like geometry
considered in section 3.3.2. In analogy to eq. (19),

$I_{\rm eo}=2w\int_{0}^{h/2}e(\rho_{+}-\rho_{-})v(z)\mathrm{d}z.$ (29)

From the Poisson equation (16), we may replace
$e(\rho_{+}-\rho_{-})=-\epsilon\partial^{2}_{z}\phi$. Then, integrating by
parts,

$I_{\rm
eo}=2w\epsilon\left[\int_{0}^{h/2}\partial_{z}\phi\partial_{z}v\,\mathrm{d}z+v(0)\partial_{z}\phi|_{0}\right].$
(30)

Now, we may use the partial slip boundary condition
$v(0)=b\partial_{z}v|_{0}$, as well as the electrostatic boundary condition
$\partial_{z}\phi|_{0}=e\Sigma/\epsilon$. This stems from the Gauss theorem
applied to the surface, and the assumption that the medium outside the channel
has much lower dielectric permittivity than water (Andelman (1995)). Moreover,
an integration of the Stokes equation (17) allows the replacement
$\partial_{z}v=-(\epsilon E/\eta)\partial_{z}\phi$. Altogether, we obtain

$I_{\rm
eo}=w\left[\frac{2\epsilon}{\eta}\mathcal{E}+\frac{2be^{2}\Sigma^{2}}{\eta}\right]E,$
(31)

where $\mathcal{E}=(\epsilon/2)\int_{0}^{h}(\partial_{z}\phi)^{2}\mathrm{d}z$
is the electrostatic energy per unit surface introduced in 3.3.1. It appears
here that slippage has an additive contribution to the EO current, which
strikingly does not depend on any electrolyte properties.

We now consider the result in two limiting cases. First, in the thin Debye
layer limit, by comparing with eq. (25), one may identify

$I_{\rm eo}=\frac{k_{B}T}{2\pi\ell_{B}\eta D}I_{\rm ep}^{\rm
surf}+2w\frac{be^{2}\Sigma^{2}}{\eta}E,$ (32)

where $I_{\rm ep}^{\rm surf}$ is the surface contribution to the
electrophoretic current. To quantify the importance of the EO contribution,
one may compute the analogue of a Peclet number:

$Pe=\frac{I_{\rm eo}}{I_{\rm ep}^{\rm
surf}}=\frac{3}{2}\frac{d_{i}}{\ell_{B}}\left(1+\frac{b}{\ell_{GC}}\right).$
(33)

We have introduced here the ion diameter $d_{i}$ by using the Einstein
relation $D=k_{B}T/(3\pi\eta d_{i})$ and we recall that
$\ell_{GC}=(2\pi\Sigma\ell_{B})^{-1}$. In water at room temperature,
$d_{i}\sim\ell_{B}\sim 0.7~{}\rm nm$, therefore, in the absence of slippage,
the EO contribution is of the same order as the surface EP contribution,
$Pe\sim 1$. Table in the supplemental appendix lists, for reference, values of
the Gouy-Chapman length for typical surface charge values. As these are
generally in the nanometre range, even in the case of moderate slippage
($b\sim 10~{}\rm nm$), there is a strong enhancement of the EO contribution.
The threshold confinement below which the resulting surface contribution
dominates over the bulk contribution is given by a rescaled Dukhin length:
$\ell_{Du}^{*}=(b/\ell_{GC})\ell_{Du}$.

Second, in the Debye overlap regime, the EO current is readily determined from
eq. (29), since the ion densities may then be considered uniform across the
channel and are given by eq. (23). Uniform ionic densities also imply that the
flow has no longer a surface, but rather a volume driving. It is simply a
Poiseuille flow, with the pressure gradient $\Delta P/L$ replaced by the
electric driving force $e(\rho_{+}-\rho_{-})E$. Altogether one obtains

$I_{\rm eo}\simeq\frac{wh}{3\eta}e^{2}\Sigma^{2}\left(1+\frac{6b}{h}\right)E.$
(34)

Since in Debye overlap it makes no more sense to distinguish a surface and a
bulk contribution, we define the Peclet number as the ratio of the EO current
to the total EP current:

$Pe=\frac{I_{\rm eo}}{I_{\rm
ep}}=\frac{d_{i}}{\ell_{B}}\frac{h}{4\ell_{GC}}\frac{1+6b/h}{\sqrt{1+(h/\ell_{Du})^{2}}}.$
(35)

Figure 5d illustrates the dependence of this Peclet number on $h$ and
$\ell_{GC}$, with $\rho_{s}=10^{-2}~{}\rm M$ so that there is Debye overlap.
In the absence of slippage, the EO contribution dominates only for high
surface charges. However, if a small slip length $b=5~{}\rm nm$ is introduced,
the Peclet number exceeds 1 for all reasonable surface charges. Indeed, the
Peclet number is then essentially determined by the ratio $b/\ell_{GC}$. This
highlights that in the Debye overlap regime, ionic conduction should be mostly
driven by electro-osmosis.

If strong EO flows are expected, then the reciprocal effect, streaming
current, is expected to be large as well. The streaming current results from
the application of a pressure gradient $\Delta P/L$: in Debye overlap, the
charge density $2\Sigma/h$ is simply dragged along by the Poiseuille flow. The
current reads

$I_{\rm str}\simeq
hw\frac{eh\Sigma}{6\eta}\left(1+\frac{6b}{h}\right)\frac{\Delta P}{L}\equiv
hw\frac{-\epsilon\zeta}{\eta}\frac{\Delta P}{L},$ (36)

where we have phenomenologically defined the zeta potential ($\zeta$) via the
streaming mobility. Independent measurements of the surface charge through the
voltage-driven current and of the zeta potential from the pressure-driven
current may allow in principle to estimate the slip length $b$. Such an
estimate does not replace a direct measurement, however, in particular because
the slip length may directly depend on surface charge (Xie et al. (2020)); and
additional charge-surface coupling effects may also occur, as we discuss
below.

####  3.3.5 Surface modifications to PNP theory

A remark is in order at this point concerning the nature of surface charge,
that we have so far assumed to be constant, whatever the conditions. Usually,
surface charge is considered to result from the acid base reactivity of the
surface when dipped into water, of the type $\rm[AH]_{s}\to[A^{-}]_{s}+H^{+}$,
where the negative group $\rm[A^{-}]_{s}$ remains fixed on the surface, while
the proton diffuses in solution.  In the case of the air-water interface,
surface charge may result from the adsorption of hydronium ions (Mamatkulov et
al. (2017)), or charged impurities. The adsorption of surfactant impurities
was also proposed as a charging mechanism for general hydrophobic surfaces
(Uematsu et al. (2019)). Graphite and hBN, which are of particular relevance
in nanofluidics, have a priori no obvious acid-base reactivity in water, but
they may develop a surface charge, through chemisorption or physisorption of
hydroxyde ions, as inferred from experiments (Secchi et al. (2016b), Siria et
al. (2013)), and recently confirmed by _ab initio_ simulations (Grosjean et
al. (2019)). Such a picture of surface charge implies that it may actually
depend on electrolyte concentration, which is an example of charge regulation.
Qualitatively, the salt concentration affects the surface potential, and
therefore the concentration of $\rm H^{+}$ and $\rm OH^{-}$ ions at the
surface, which in turn affects the chemical (or the adsorption) equilibrium
which governs the surface charge. Such charge regulation has been invoked to
explain several experiments on carbon nanotubes, where a scaling
$I\propto\rho_{s}^{1/2}$ or $I\propto\rho_{s}^{1/3}$ was observed (Secchi et
al. (2016b), Liu et al. (2010), Pang et al. (2011)). Various models have been
developed (Secchi et al. (2016b), Biesheuvel & Bazant (2016), Manghi et al.
(2018), Uematsu et al. (2018)), predicting  a rich panorama of different
sublinear scalings of conductance with salt concentration, depending on the
conditions.

A further step that may be required to accurately describe the surface charge
is to take into account its mobility. Surfaces charges may be mobile when
embedded in lipid bilayers, or with more relevance to nanofluidics, when
resulting from adsorbed ions. For instance, the simulations of Grosjean et al.
showed that physisorbed hydroxyde ions on graphene surfaces retain a high
lateral mobility (Grosjean et al. (2019)). The effect of a mobile surface
charge on ion and fluid transport coefficients (in the thin Debye layer
regime) was  the subject of several investigations (Maduar et al. (2015),
Silkina et al. (2019), Mouterde & Bocquet (2018)).  Using the framework of
Mouterde & Bocquet (2018), one may introduce two new friction coefficients:
$\lambda_{-}$ between the adsorbed (negative) ions and the wall, and $\xi_{-}$
between the adsorbed ions and the fluid, in addition to the water-wall
friction coefficient $\lambda$. The force balance on the interfacial fluid
layer results in a modified partial slip boundary condition  involving the
tangential electric field at the surface:

$b_{\rm
eff}\partial_{z}v|_{z=0}=v|_{z=0}-e\frac{\xi_{-}}{\xi_{-}+\lambda_{-}}\frac{b_{\rm
eff}}{\eta}\Sigma(-\partial_{x}\phi)|_{z=0},$ (37)

where the effective slip length is

$b_{\rm
eff}=\frac{b}{1+\frac{\lambda_{-}\xi_{-}}{\lambda(\lambda_{-}+\xi_{-})}\Sigma}.$
(38)

The slip length is therefore reduced by the surface charge mobility, and the
boundary condition involves an extra electric term. This has in general a
moderating effect on the transport coefficients. For example, in the case of
streaming current, the fluid brings along surfaces charges in addition to the
counterions, so that the total current is reduced. Notable exceptions are
diffusio-osmotic mobility and conductance: the overall ionic conductance
increases when surface charges are able to move in response to the electric
field.

We have just introduced ion-wall and ion-water friction for adsorbed ions in
the interfacial layer. Now, for the smallest accessible channels, which are
comparable to the ion size (Radha et al. (2016), Lee et al. (2010)), such
friction needs to be taken into account for all the ions, and should appear at
the level of the transport equations. This was proposed, for instance, to
rationalise the non-linear voltage-pressure couplings observed in angstrom-
scale slits by Mouterde et al. (2019). They observed that the streaming
mobility $\mu$, defined by $I_{\rm str}=hw\mu\Delta P/L$, depends on the
applied voltage in a qualitatively different way for graphite and BN slits. In
the proposed ’surface PNP’ model, the ion-wall ($\lambda_{\pm}$) and ion-water
($\xi_{\pm}$) friction coefficients are introduced in the same way as above,
except they no longer apply to adsorbed ’surface charge’ ions, but to all the
regular salt ions. The equivalent of the Smoluchowski and Stokes equations
need then to be rederived by considering the force balance on a single ion and
on an element of fluid, respectively, as shown in the Appendix.

There is a rich phenomenology to be harnessed from the new couplings that
appear when all components of the nanofluidic system interact with the
surfaces. Nevertheless, such surface PNP theory is still in its infancy, and
it requires input from more microscopic modelling in order to estimate the
various friction coefficients it introduces. It even raises mathematical
difficulties in the writing of a one-dimensional Poisson equation, whose
Green’s function has infinite range. Finally, the relevance of this
phenomenological theory should be assessed in light of the correlation and
structuring effects – specific to the smallest scales – that will be described
in the next section.

##  4 BEYOND THE CONTINUUM DESCRIPTION

In this section, we focus on the specific nanoscale effects that are not
described in the framework of continuum hydro- and electro-dynamics. These
effects mainly originate from thermal fluctuations, interparticle correlations
and structuring effects, and, as such, they require theoretical tools that
bridge the gap between physics of continuous media and statistical mechanics.

###  4.1 Fluctuations

####  4.1.1 Fluctuating hydrodynamics

The relative particle number fluctuations in an open system of $N$ particles
scale as $1/\sqrt{N}$. In a nanopore of radius $1~{}\rm nm$ and length
$10~{}\rm nm$, there are on average about 1000 water molecules and these
fluctuations  are of a few percent and already not negligible. Fundamentally,
the description of such fluctuations requires a statistical mechanics
framework, such as density functional theory (Barrat & Hansen (2003)).
However, these approaches remain quite formal  and difficult to implement,
particularly for  finite length pores which are non translationally-invariant
systems.

A simpler coarse-grained approach to fluctuations – the so-called fluctuating
hydrodynamics (FH) – has been introduced by Landau & Lifshitz (1977). In
fluctuating hydrodynamics, a random stress tensor $S$ is added to the Navier-
Stokes equation:

$\rho\partial_{t}\mathbf{v}+\rho(\mathbf{v}\cdot\nabla)\mathbf{v}=-\nabla
p+\eta\Delta\mathbf{v}+\nabla\cdot S,$ (39)

where $S$ satisfies a fluctuation-dissipation theorem: $\langle
S_{ij}(\mathbf{r},t)S_{kl}(\mathbf{r}^{\prime},t^{\prime})\rangle=2\eta
k_{B}T(\delta_{ij}\delta_{kl}+\delta_{il}\delta_{jk})\delta(\mathbf{r}-\mathbf{r}^{\prime})\delta(t-t^{\prime})$,
similarly to the fluctuating force in a Langevin equation;  $\rho$ is the mass
density of the fluid. This description introduces thermal fluctuations in a
setting where, otherwise, a continuum hydrodynamic description holds. FH has
been extensively discussed in the literature (Hauge & Martin-Lof (1973), Fox &
Uhlenbeck (1970), Mashiyama & Mori (1978)) and we only consider here its basic
implications for fluid transport in a nanopore.

In standard hydrodynamics, the fluid inside a nanopore is at rest when subject
to no external force. The effect of fluctuations is to induce a stochastic
flow through the nanopore, which can be pictured as a stochastic motion of the
fluid centre of mass  (CM). Detcheverry & Bocquet (2012) solved the FH
equations for the velocity correlation function, and found that the CM motion
was described by a non-Markovian Langevin equation of the type

$m\frac{\mathrm{d}v}{\mathrm{d}t}=-\int_{-\infty}^{t}\mathrm{d}t^{\prime}\xi(t-t^{\prime})v(t^{\prime})+F(t),$
(40)

where the random force $F(t)$ and the memory kernel $\xi$ are related by
$\langle F(t)F(0)\rangle=2k_{B}T\xi(t)$. The memory kernel is indeed non-
trivial (i.e. not reduced to a $\delta$ function), as confirmed by molecular
dynamics (MD) simulations (Detcheverry & Bocquet (2013)). Memory effects in
the diffusion of fluid inside a nanopore are indeed expected from the analysis
of relevant timescales. A nanopore of radius $R$ and length $L$ contains a
mass $m=\rho\pi R^{2}L$ of fluid, and its friction coefficient on the wall
can be calculated from the Navier-Stokes equation as $\xi_{s}\approx 8\pi\eta
L$. The velocity of the fluid CM then relaxes on a timescale $\tau\sim
m/\xi_{s}\sim R^{2}/\nu$. But $R^{2}/\nu$ is the time required for fluid
momentum to diffuse across the nanopore, that is the time for the wall
friction force to establish itself. Therefore, the friction force may not
adapt instantaneously to the CM velocity, and memory effects are to be
expected.

Properties that are determined by short timescale dynamics may be affected by
those memory effects as it was shown to be the case for solute mobility in
confinement (Daldrop et al. (2017)). More generally, memory effects play an
important role in barrier-crossing processes (Kappler et al. (2018)). But
other properties only depend on the long timescale dynamics. In particular,
the fluid centre of mass diffusion coefficient, given by
$\mathcal{D}=k_{B}T/(2\int_{0}^{\infty}\xi(t)\mathrm{d}t)$, reduces to the
Einstein expression $\mathcal{D}=k_{B}T/\xi_{s}$, that would be expected if
the diffusion was Markovian. Similarly, the hydrodynamic slip length $b$ is
found to satisfy a Green-Kubo relation whatever the form of the memory kernel
$\xi(t)$ (Bocquet & Barrat (2013)).

From a practical perspective, one may evaluate the diffusion coefficient $\cal
D$. From the Einstein relation, and the assumption of Poiseuille flow with
slip length $b$, one obtains

$\mathcal{D}=\frac{k_{B}T}{\xi}=\frac{k_{B}T}{\frac{8\pi\eta
L}{1+4b/R}+3\pi^{2}\eta R},$ (41)

where, in the most general case, the friction coefficient $\xi$ has two
contributions, one from the channel interior and one from the entrance effects
(Detcheverry & Bocquet (2012)). Overall, the fluid CM diffusion $\mathcal{D}$
can be seen as a supplementary contribution to the diffusion coefficient of a
particle inside the nanopore. If the size of the particle is comparable to
that of the nanopore, its self-diffusion is strongly hindered (Renkin (1954))
and the fluid contribution may actually dominate the particle diffusion
(Detcheverry & Bocquet (2012)).

####  4.1.2 Noise

Thermal fluctuations are at the origin of the noise in ionic current
measurements through a nanopore. At zero applied voltage, the total noise
amplitude satisfies the Nyquist relation $\langle I^{2}\rangle=2k_{B}TG$,
where $G$ is the nanopore conductance. The electro-osmotic term in the
conductance makes a supplementary contribution to the noise, which can
actually be traced back to the hydrodynamic fluctuations of the fluid CM
described in the previous paragraph (Detcheverry & Bocquet (2012)).

When resolved in frequency, numerous experiments on artificial nanochannels
(Smeets et al. (2008), Siwy & Fuliński (2002), Hoogerheide et al. (2009),
Tasserit et al. (2010), Secchi et al. (2016b)) and biological pores (Wohnsland
& Benz (1997), Bezrukov & Winterhalter (2000)), have shown that the current
spectrum exhibits ’pink noise’, that is noise that scales with frequency $f$
as $f^{-\alpha}$, with $\alpha$ in the range $0.5-1.5$. Such a spectrum is
traditionally described by the empirical Hooge’s law (Hooge (1969)):

$\langle\delta I^{2}\rangle(f)=A\frac{\langle I^{2}\rangle}{f^{\alpha}},$ (42)

with $A$ a coefficient inversely proportional to the number of charge
carriers. Notably, in experiments, the $f^{-\alpha}$ scaling is observed down
to frequencies below 1 Hz, showing that some correlations in the system under
study exist even at such low frequencies. Correlations between ions have been
put forward as a possible origin for the pink noise (Zorkot et al. (2016)). On
the other hand, Gravelle et al. (2019) demonstrated in MD simulations that
pink noise persisted even with non-interacting ions, if reversible ion
adsorption was allowed on the channel wall. For simple diffusive dynamics, the
longest correlation time that may be expected for adsorption-desorption
processes is $R^{2}/D$, with $R$ the pore radius and $D$ the ion diffusion
coefficient. For a 10 nm pore, this would correspond to a frequency  cutoff of
100 MHz, below which the noise spectrum should be flat.  Such a high cutoff
frequency is in clear contradiction with experiment. However, Gravelle et al.
showed that in the presence of reservoirs, excursions of ions outside the pore
coupled to adsorption-desorption processes on the pore surface result in ionic
correlations on much longer time scales,  hence predicting pink noise down to
very low frequencies.

####  4.1.3 Wall fluctuations

So far, we have only considered thermal fluctuations of the fluid itself.
However, the confining walls might also be subject to fluctuations. Wall
fluctuations are known to enhance fluid self-diffusion by inducing flows,
through the Taylor-Aris mechanism (Taylor (1953), Aris (1956)). On the other
hand, static constrictions slow down diffusion in a channel through Fick-
Jacobs entropic trapping (Malgaretti et al. (2014), Reguera & Rubí (2001)).
Recently, MD simulations have shown diffusion enhancement due to the
propagation of phonons in carbon nanotubes (Ma et al. (2015), Cruz-Chú et al.
(2017)).

On the theoretical side, a very general framework for evaluating the self-
diffusion coefficient  for a fluid confined in a fluctuating channel was
developed by Marbach et al. (2018). In a slit like-channel of height $h$, the
fluctuation-corrected diffusion coefficient $D$ is formally related  to the
spectrum of wall fluctuation $S(k,\omega)$ by

$D=D_{0}\left(1-\frac{1}{h^{2}}\int\frac{\mathrm{d}k\mathrm{d}\omega}{(2\pi)^{2}}\frac{(D_{0}k^{2})^{2}-3\omega^{2}}{(D_{0}k^{2})^{2}+\omega^{2}}S(k,\omega)\right),$
(43)

where $D_{0}$ is the bare diffusion coefficient. Equation (43) has two
limiting regimes, that are governed by a ’fluctuation-related’ Peclet number,
which may be defined as $Pe=\ell^{2}/(D_{0}\tau)$, where $\ell$ and $\tau$ are
the characteristic length and time scales of the fluctuations. The diffusion
coefficient is enhanced for $Pe>1$ ($D=D_{0}(1+3\langle\delta
h^{2}\rangle/h^{2})$), and reduced for $Pe<1$ ($D=D_{0}(1-\langle\delta
h^{2}\rangle/h^{2})$)  and both tendencies are observed experimentally
(Marbach et al. (2018)). Equation (43) thus bridges the limiting Taylor-Aris
and Fick-Jacobs results.

The permeability of a fluctuating channel is also expected to be impacted by
wall fluctuations. However, permeability is related to the diffusion of the
fluid centre of mass (see eq. (41)), which is different from the fluid self-
diffusion; and extending the above framework to centre of mass diffusion
remains to be done.

###  4.2 Fluid structuring

####  4.2.1 Phenomenology

While a fluid appears  disordered at hydrodynamic lengthscales, its molecular
nature manifests itself under confinement in the form of structuring effects.
Even at a non-confined solid-liquid interface, the attractive interactions
between the solid and the liquid result in molecular layering, as verified
experimentally by x-ray spectroscopy (Cheng et al. (2001)). In between two
surfaces, a liquid adopts a layered structure even in the absence of
interactions, simply due to geometrical constraints (Israelachvili & Pashley
(1983)). The onset of structuring clearly represents the transition from
continuum to sub-continuum transport.

The threshold confinement at which this transition occurs has been studied for
both planar and cylindrical geometries using MD simulations. For water in
carbon nanotubes, Thomas & McGaughey (2009) have found that it retains a bulk-
like disordered structure down to a tube diameter of 1.39 nm ((10,10) tube).
In a (9,9) tube (1.25 nm) water was found to structure in stacked hexagons
(see figure 7), and in a single file chain in a 0.83 nm (6,6) tube. For water
between two graphene sheets, four distinct layers could be observed at a 1.35
nm distance between sheets (Neek-Amal et al. (2016)), and a single monolayer
of water was observed for confinement below 0.8 nm (Neek-Amal et al. (2018)).
In should further be noted that the observation of an exotic square ice phase
for few-layer water between two graphene sheets has been claimed (Algara-
Siller et al. (2015)).

In the sub continuum-regime, transport properties show strong qualitative
deviations from bulk expectations. The MD simulations of Thomas & McGaughey
(2009) showed non-monotonous permeability for carbon nanotubes as a function
of diameter, and similar observations were made for the capillary filling
velocity of carbon nanotubes, studied by Gravelle et al. (2016).  On the
experimental side, this behaviour can be put in perspective with the recent
results by Radha et al. (2016) for capillary-driven flow through slit-like
graphene channels  where a peak in flow rate at around 1.3 nm confinement was
observed.

Figure 6: MD simulation snapshots of water structure inside carbon nanotubes
of different radii, adapted from Thomas & McGaughey (2009).

####  4.2.2 Disjoining pressure

Thermodynamically, the effect of  molecular fluid structuring may be described
by a supplementary contribution to the pressure, the so-called disjoining
pressure. For a fluid confined between two walls of area $\mathcal{A}$
separated by a distance $h$, in equilibrium with a reservoir at pressure
$P_{0}$,  it is defined in full generality as

$\Pi_{d}(h)=-\frac{1}{\mathcal{A}}\left(\frac{\partial G}{\partial h}\right),$
(44)

where $G$ is the Gibbs free energy of the whole system (Israelachvili (2011),
Barrat & Hansen (2003)).

In principle, $G$ varies with $h$ because of distance-dependent interactions
between the surfaces; the disjoining pressure can then be seen as the force
the fluid needs to exert in order to oppose those interactions. These are
generally of two types. On the one hand, there are van der Waals interactions
that scale algebraically as $h^{-2}$, yielding a contribution
$\Pi_{vdW}=-{A_{H}\over 6\pi h^{3}}$ to the disjoining pressure, with $A_{H}$
the Hamaker constant (although more complex dependencies may occur in specific
geometries). On the other hand, electrostatic interactions lead to
contributions that scale exponentially with the distance $h$ due to ionic
screening. The combination of these two interaction terms is the basis for the
DLVO (Derjaguin-Landau-Verwey-Overbeek) theory.

Now, if there is fluid structuring that depends on confinement, it makes a
supplementary contribution to the variation of $G$. This contribution exists
even for a hard sphere fluid  due to excluded volume effects, and is purely
entropic in nature. It quantifies essentially how much the structure of the
fluid in confinement differs from the one in the bulk. For hard spheres of
diameter $\sigma$,  the disjoining pressure is usually described by the
expression (Israelachvili (2011), Kralchevsky & Denkov (1995)):

$\Pi_{d}(h)=-\rho_{\infty}k_{B}T\cos(2\pi h/\sigma)e^{-h/\sigma},$ (45)

where $\rho_{\infty}$ is the bulk density. This is an oscillating function of
$h$ that decays to 0 at large $h$, where there is no more fluid structuring.
It is bounded by $\Pi_{d}(0)=\rho_{\infty}k_{B}T$, which evaluates to about
1000 bar for water at room temperature. In practice,  for fluid flow
measurements where a pressure drop is imposed between two reservoirs,  this (
huge) contribution to the pressure does not intervene directly. However, in a
capillary flow geometry,  which involves liquid-vapour interfaces, the
disjoining pressure may  be much larger than the bare capillary pressure and
act as the main driving force. Indeed, the capillary pressure drop across a
meniscus of radius $R=1~{}\rm nm$ is typically $2\gamma/R=140~{}\rm bar$ (for
water with surface tension $\gamma=72~{}\rm mN\cdot m^{-1}$).  Driving by
disjoining pressure was observed in simulations of carbon nanotube capillary
filling (Gravelle et al. (2016)), where the subtle dependence of the water
molecular structure – and therefore of the disjoining pressure – on nanotube
radius resulted in the imbibition velocity displaying oscillatory behaviour
versus the confinement. Experimentally, a similar phenomenology was observed
by Radha et al. (2016), who measured the capillary flow of water across
angstrom scale slits and found a non-monotonous dependence of the evaporation
rate on confinement. These measurements could be interpreted in terms of
disjoining pressure, which was indeed found to be of the order of 1000 bar in
MD simulations of the experimental system (Neek-Amal et al. (2018)).

####  4.2.3 Descriptions of fluid transport

A first understanding of sub-continuum transport may be provided by extending
the continuum description with effective, confinement-dependent, values for
density, viscosity and slip length. In a layered fluid, the density in each
layer is generally higher than in the bulk, and a general trend is that
viscosity increases (Schlaich et al. (2017)). More exotically, an oscillating
viscosity as a function of confinement was observed in simulations of water
between two graphite slabs (Neek-Amal et al. (2016)). Neek-Amal et al. (2018)
could reproduce the experimental results of Radha et al. (2016) by using
simulated values of disjoining pressure, along with a Poiseuille formula with
effective density and viscosity; however, they assumed a viscosity-independent
slip length. Now, in general, care should be taken in defining a viscosity for
a structured fluid: as its density is non-uniform, a position-dependent
viscosity should be introduced in order to describe the details of the flow
profile. For confinement below five molecular diameters (in the case of a
Lennard-Jones fluid) Travis et al. (1997) have shown that even a position-
dependent viscosity is not sufficient and a non-local viscosity kernel should
be used (Zhang et al. (2004)).  Ultimately, for confinement below a few
molecular layers (typically 1 nm for water), the notion of viscosity itself –
which is intrinsically a continuum quantity – looses its fundamental meaning.

Nevertheless, a simple picture of sub-continuum fluid transport is possible in
the case of large slippage,  since surface friction then becomes the main
mechanism resisting fluid transport. This  appears indeed when looking at the
limit $b\gg R$ in the Poiseuille formula (7), which yields for the average
flow velocity:

$v=\frac{Rb}{2\eta L}\Delta P=\frac{R}{2\lambda L}\Delta P,$ (46)

where we have obtained the second equality by relating the slip length to the
liquid-wall friction coefficient $\lambda$: $b=\eta/\lambda$;  note that, for
simplicity, we forget here about entrance. The viscosity does not enter the
permeability, as expected for surface dominated friction. The fluid moves
indeed in the channel as a single block,  and dissipation only intervenes at
the surface. The crucial parameter governing the transport is then the solid-
liquid friction coefficient $\lambda$. A model for evaluating $\lambda$,
tested against MD simulations of water in carbon nanotubes, was proposed by
Falk et al. (2010). They obtain

$\lambda\approx\frac{\tau}{k_{B}T}f_{q_{0}}^{2}S(q_{0}),$ (47)

where $q_{0}$ is the wave-vector corresponding to the solid lattice spacing,
$f_{q_{0}}$ is the Fourier component of the fluid-solid interaction potential
at that wave-vector, and $S(q)$ is the liquid structure factor; $\tau$ is the
correlation time of the force between the solid and the liquid, which is found
to not depend on confinement. In this way, $\lambda$ is found to be directly
related to the liquid structure and to the liquid wall interaction. Such a
friction-dominated approach was applied successfully,  for example, to predict
the permeability of ultra-confined alcanes in a nanoporous matrix, showing a
scaling dependence of the permeance on the alcane length (Falk et al. (2015)).
However, predicting the solid-liquid friction coefficient remains to large
extents an open problem. As we discuss in the following (see 4.2.6), its
determination in MD simulations still remains ambiguous quantitatively, and
the role of underlying quantum effects should be assessed.

####  4.2.4 Ultimate structuring: single file transport

The most extreme type of structuring is single-file arrangement of fluid
molecules. In the case of water transport, this is the realm of biological
channels (Horner & Pohl (2018b)), where water is conducted through sub-
nanometric openings in transmembrane proteins. Single file transport may also
occur in artificial channels: it was observed in MD simulations for carbon
(Hummer et al. (2001)) and boron nitride (Won & Aluru (2007)) nanotubes of 0.8
nm in diameter, and experimental observation in carbon nanotubes was recently
claimed (Tunuguntla et al. (2017)). Single-file transport is a field of its
own and there are dedicated reviews on the subject (Horner & Pohl (2018b),
Köfinger et al. (2011)); here we will only sketch the main ideas.

The notion of single file transport and the underlying exclusion transport
models are not restricted to the study of confined fluids. Indeed, a general
feature of particles in one dimension that cannot cross each other is sub-
diffusive behaviour: the mean squared displacement of a particle scales with
time as $\langle\Delta x^{2}\rangle\propto t^{1/2}$, as opposed to linear
scaling for normal diffusion (Levitt (1973)). Such anomalous diffusion is
encountered in a variety of fields (Fouad & Gawlinski (2017)), and may be
understood in terms of the normal diffusion of vacancies in the 1D chain; in
the case of fluid transport, it was studied by Chou (Chou (1999, 1998)).
However, this effect is not expected to play a decisive role in single-file
transport through short channels, where vacancies are unlikely to occur (Kalra
et al. (2003)).

In the case where no vacancies are expected, a simple model for single-file
transport was proposed decades ago by Finkelstein (1987) for describing water
transport in biological channels. It stems from a global force balance on the
water chain, and is as such equivalent to eq. (46), which we have derived as a
strongly confined limit of the Poiseuille formula. In Finkelstein’s model, the
chain of $N$ water molecules (of total length $L$) moving at velocity $v$ is
subject to a friction force $-N\xi v$, and to a pressure driving $Nv_{w}\Delta
P/L$, where $v_{w}$ is the volume of one water molecule. The force balance
then leads to

$v=\frac{v_{w}}{\xi L}\Delta P,$ (48)

which is indeed eq. (46), granted the identification $\xi=(2v_{w}/R)\lambda$.
In the biophysics literature, the preferred quantity for characterising a
channel’s transport properties is the unitary channel permeability $p_{f}$:
$p_{f}$ is the number of water molecules that crosses the channel per unit
time, per unit osmolyte concentration difference applied across the channel,
therefore expressed in $\rm m^{3}\cdot s^{-1}$. For an osmolyte concentration
difference $\Delta\rho_{s}$, the osmotic pressure difference is
$k_{B}T\Delta\rho_{s}$, and hence the unitary permeability is

$p_{f}=\frac{N}{L}\frac{v}{\Delta\rho_{s}}=\frac{v_{w}}{aL}\frac{k_{B}T}{\xi},$
(49)

where we have defined $a=L/N$ the average spacing between water molecules. The
diffusion coefficient of the water chain centre of mass may be expressed
through the Einstein relation as $D=k_{B}T/(N\xi)$, and therefore
$p_{f}=v_{w}D/a^{2}$, hence the diffusion coefficient and the unitary
permeability may be used interchangeably.  A typical value for the
permeability in biological channels such, as Aquaporin or Gramicidin A, is in
the range of $10^{-14}-10^{-13}~{}\rm cm^{3}\cdot s^{-1}$,  which corresponds
to 1-10 water molecules crossing the channel per nanosecond under a 1 bar
pressure drop.

In an alternative phenomenological description, transport through the channel
is viewed as an activated process with activation energy $\Delta G^{\ddagger}$
(Horner & Pohl (2018b)). The permeability is then expressed in the framework
of transition state theory as $p_{f}=v_{w}\,\nu_{0}\,\exp[-\Delta
G^{\ddagger}/k_{B}T]$, with $\nu_{0}\sim 10^{13}$ s-1 a molecular attempt
frequency.  For biological channels, this relation is well verified by
independent measurements of $p_{f}$ and $\Delta G^{\ddagger}$, with  $\Delta
G^{\ddagger}$ of the order of $5~{}\rm kcal\cdot mol^{-1}$.  However, values
of $\Delta G^{\ddagger}$ remains debated for transport measurements in carbon
nanotubes (Horner & Pohl (2018a), Tunuguntla et al. (2018)).

Although it provides a general guiding line, Finkelstein’s formula is
challenged both in artificial and biological channels. In the case of protein
channels, the assumption of uniformly smooth walls breaks down, since there
are discrete hydrogen bonding sites (Horner & Pohl (2018b)). Hence $p_{f}$ is
not found to be inversely proportional to channel length (Saparov et al.
(2006)): it rather has an exponential dependence on the number of hydrogen
bonding sites (Horner et al. (2015)). Such an exponential dependence suggests
a collective transport mechanism of the water chain, with bursts requiring the
breaking of multiple hydrogen bonds at once. There is, however, a notable
disagreement between experiments and simulations, as the simulations of water
transport through polyalanine channels showed no dependence of $p_{f}$ on the
channel length (Portella et al. (2007)).

In simulations of carbon nanotube channels, all the single-file water
molecules were clearly shown to move in a correlated fashion (Hummer et al.
(2001)): water transport occurs when all the molecules simultaneously shift by
one molecular diameter. These dynamics were successfully described by a
continuous time random walk model (Berezhkovskii & Hummer (2002)), or
equivalently by diffusion of a collective coordinate of the water molecules
(Zhu et al. (2004)), which actually echoes our discussion of fluctuations in a
more general setting (eq. 40 and Detcheverry & Bocquet (2012)).

Overall, there is still much to understand about the collective motions and
subtle surface interactions involved in single-file transport, and it is an
active field of research.

####  4.2.5 Structuring and electrostatics: dielectric anomalies

Figure 7: Dielectric anomalies due to fluid structuring. a. Components of the
dielectric permittivity tensor of water in a carbon nanotube, as a function of
nanotube radius, as determined from MD simulations by Loche et al. (2019). b.
Experimental results for the transverse dielectric constant of water in planar
confinement, as a function of confinement width, from Fumagalli et al. (2018).

As confined water becomes structured, its response properties to an external
electric field are accordingly modified. When an electric field $\bf E$ is
applied in water, the individual molecules reorient and polarise, creating an
electric field themselves. The total electric field $\bf E$ is then the sum of
the polarization field of the water molecules and of the externally applied
field $\mathbf{D}/\epsilon_{0}$, where $\bf D$ is called the electric
displacement and $\epsilon_{0}$ is the vacuum permittivity. In bulk water, the
dielectric response, that is the relation between $\bf D$ and $\bf E$, may be
expressed through a single scalar quantity, the relative permittivity
$\epsilon\approx 80$: $\mathbf{D}=\epsilon\epsilon_{0}\mathbf{E}$. However,
the most general (static) linear response may be anisotropic, space-dependent
and non-local:

$D_{\alpha}(\mathbf{r})=\epsilon_{0}\sum_{\beta}\int\mathrm{d}\mathbf{r}^{\prime}\epsilon_{\alpha\beta}(\mathbf{r},\mathbf{r^{\prime}})E_{\beta}(\mathbf{r^{\prime}}).$
(50)

The relative permittivity is then a tensor with components
$\epsilon_{\alpha\beta}(\mathbf{r},\mathbf{r}^{\prime})$. While MD simulations
show that the dielectric response in water may be considered local
($\epsilon_{\alpha\beta}(\mathbf{r},\mathbf{r}^{\prime})=\epsilon_{\alpha\beta}(\mathbf{r})\delta(\mathbf{r}-\mathbf{r^{\prime}})$),
it becomes anisotropic and space-dependent in the vicinity of interfaces, as a
consequence of the water layering (Bonthuis et al. (2012),Bonthuis et al.
(2011)). Qualitatively, the orientations of the water dipoles are anti-
correlated in the direction perpendicular to the interface, resulting in a
reduced permittivity in that direction, while the permittivity is largely
unaffected parallel to the interface. In planar confinement, this behaviour
could be captured by an effective medium model, in which the water is
described by a space-independent, but anisotropic permittivity
$(\epsilon_{\parallel},\epsilon_{\perp})$ (the parallel direction is not
confined). While $\epsilon_{\parallel}$ essentially retains its bulk value,
$\epsilon_{\perp}$ is reduced by up to an order of magnitude for confinements
below 1 nm (Schlaich et al. (2016), Zhang et al. (2013)). Such a reduction of
the perpendicular dielectric response was recently observed experimentally for
water confined between a graphite and a boron nitride crystal (Fumagalli et
al. (2018)). A deviation from the bulk value was measured up to nearly 100 nm
confinement (figure 7). The results were well described by assuming each
interface carried a 7 Å thick layer of very low permittivity ($\epsilon=2.1$)
”electrically dead water”.

An effective medium model based on MD simulations was developed by Loche et
al. (2019) for cylindrical confinement of water in carbon nanotubes. Similarly
to the case planar case, the radial permittivity is found to be reduced by up
to an order of magnitude for tube radii smaller than 3 nm (figure 7). However,
for the smallest tubes (below 1 nm radius), the longitudinal permittivity is
found to increase with respect to its bulk value, and it skyrockets 1 to 3
orders of magnitude in a 0.4 nm radius tube, where water is in a single file
arrangement.

Knowledge of these modified dielectric properties is important when
considering ion transport in strong confinement, as will be discussed in the
following (see 4.3.1).

####  4.2.6 Limits of molecular dynamics

We will close this section on liquid structuring with a word of warning. Many
of the results we have presented so far have been obtained using classical MD
simulations, and like any type of simulation, these come with some underlying
assumptions. In particular, as we are dealing with strongly confined fluids
where all the molecules interact with the confining surface, the modelling of
that surface plays an increasingly important role. Since the solid is
represented by rigid balls interacting with classical force fields, no
electron dynamics may be described by the simulation. However, some coupling
between water and electron dynamics may be expected, based on a lengthscale
argument. The maximum of the dielectric response of a solid indeed occurs at a
lengthscale $k_{\rm TF}^{-1}$, where $k_{\rm TF}$ is the Thomas-Fermi wave-
vector (Mahan (1990)). Typically, $k_{TF}^{-1}\approx 1~{}\rm nm$ in graphite,
hence the electrons in graphite could expected to respond to the dynamics of
individual water molecules.

There is growing evidence that such electron-water couplings indeed occur. A
first consequence of taking into account electron dynamics is that a solid
should appear polarisable: in simulations, the fluctuations of water next to a
graphite surface were found to be strongly affected by the polarisability of
carbon atoms (Misra & Blankschtein (2017)). On the experimental side, the
induction of an electronic current by a water flow inside a carbon nanotube,
as well as the reverse phenomenon, were observed (Ghosh et al. (2003),
Rabinowitz et al. (2020)).

These results call for caution in the interpretation of MD simulations, which,
as we have pointed out, lack electron dynamics. They also call for the
development of new simulation methods that would take such dynamics into
account, while not limited by the very small system size of _ab initio_
simulations.

###  4.3 Electrostatics in extreme confinement

In this section, we discuss a few aspects of the behaviour of ions below the
continuum limit. We will describe how the confinement modifies ionic
interactions, and possible consequences on ion transport.

####  4.3.1 Ionic interactions and self-energy

In bulk water, ions interact via a Coulomb potential
$\phi(r)=e/(4\pi\epsilon_{0}\epsilon_{w}r)$, where $\epsilon_{w}$ is the water
dielectric permittivity. But inside a nanochannel, ions are no longer
surrounded by a homogeneous fluid, and their interaction potential may be
affected by the dielectric properties of the confining medium. In order to
assess the importance of this effect, we consider the simple situation where
an infinitely long cylindrical channel of radius $R$ is filled with water
having isotropic permittivity $\epsilon_{w}$, and the medium outside the
channel is a homogeneous dielectric of permittivity $\epsilon_{s}$, with
typically $\epsilon_{s}\ll\epsilon_{w}$. The electrostatic potential around an
ion placed in the middle of the channel can then be determined by solving
Poisson’s equation in the presence of the dielectric discontinuities, which
has been done analytically by several authors (Parsegian (1969), Levin (2006),
Teber (2005), Kavokine et al. (2019)).

The complicated analytical result may be interpreted in the following way (see
figure 8). At short distances (much smaller than the channel radius), only the
dielectric response of the water is visible and the potential is $\phi(x)\sim
1/(\epsilon_{w}x)$. At long distances (much larger than the channel radius),
it is the dielectric response of the confining medium that matters and
$\phi(x)\sim 1/(\epsilon_{s}x)$. At intermediate distances, the electric field
lines are essentially parallel to the channel due to the dielectric contrast
$\epsilon_{w}\gg\epsilon_{s}$, and the potential, which is well described by
an exponential function, resembles a 1D Coulomb potential:

$\phi(x)\approx\frac{e\alpha}{2\pi\epsilon_{0}\epsilon_{w}R}e^{-x/(\alpha
R)},$ (51)

with $\alpha$ a numerical coefficient that depends on the ratio
$\epsilon_{w}/\epsilon_{s}$; $\alpha=6.3$ for $\epsilon_{w}/\epsilon_{s}=40$
(Teber (2005)). It is notable that so far no assumption on the channel radius
was made, so that formally this 1D regime exists for a channel of any size.
However, it is only relevant if it leads to ion-ion interactions stronger than
$k_{B}T$, that is if $e\phi(R)>k_{B}T$, for a monovalent ion. This defines a
limiting channel radius, below which ionic interactions are affected by the
confining medium: $R_{c}\sim 7.5~{}\rm nm$. These modified Coulomb
interactions therefore may have some effect in any single-digit nanopore, and
they have essentially two practical consequences.

Figure 8: a. Schematic representation of the electric field lines around an
ion confined in a narrow channel. b. Potential generated along the axis by an
ion confined in a channel of radius 2 nm. c. Born self-energy of a monovalent
ion as a function of channel radius, computed following Teber (2005).

First, because ions produce a stronger Coulomb potential in confinement than
in the bulk, they have a larger Born self-energy. This phenomenon was pointed
out decades ago by Parsegian (1969) in the case of ions crossing lipid
bilayers, and has since then been commonly considered in the study of
biological ion channels. For small channels where the 1D contribution to the
potential dominates, the additional self-energy due to confinement is ${\cal
E}_{s}=e\phi(0)/2$, with $\phi(x)$ given by eq. (51). An expression valid for
any $R$ was obtained by Teber, and it is plotted in figure 8. It shows that
${\cal E}_{s}$ is larger than $k_{B}T$ up to $R\sim 4~{}\rm nm$. ${\cal
E}_{s}$ represents a supplementary energy barrier for entering the channel.
To illustrate the consequence of this Born self-energy, let us consider a
neutral channel in the absence of correlation effects: a barrier ${\cal
E}_{s}=4k_{B}T$ reduces the ionic concentration by a factor $e^{{\cal
E}_{s}/(k_{B}T)}=100$.

Second, increased Coulomb interactions with respect to the bulk may actually
result in correlation effects. To assess the potential importance of
correlations, one may introduce a coupling parameter $\Gamma=e\phi((\pi
R^{2}\rho)^{-1})/(k_{B}T)$, where $\rho$ is the ion concentration inside the
channel, so that $(\pi R^{2}\rho)^{-1}$ is the average distance between ions
(Bocquet & Charlaix (2010)). Let us take as an example a channel of radius 2
nm, where ${\cal E}_{s}\sim 2.5k_{B}T$ from eq. (51). An effective Coulomb
potential of magnitude $\sim 5k_{B}T$ then extends around an ion over a
distance $\alpha R\sim 10~{}\rm nm$. Therefore, for concentrations above
$\rho=10^{-2}~{}\rm M$, $\Gamma>1$ and correlations are expected to become
important.

Several remarks can be made here. A first point is that we have considered
monovalent ions so far; for ions of valence $z$, the self-energy, for example,
would be multiplied by a factor $z^{2}$. Then, we have gone through the
discussion assuming for simplicity that water has an isotropic permittivity
$\epsilon_{w}$. We have highlighted in the previous section that this
assumption breaks down in cylindrical channels of radius smaller than 5 nm,
and, though it does not affect the qualitative phenomenology, dielectric
anisotropy, as predicted by Loche et al. (2019), should then be taken into
account. It mainly affects the behaviour of the potential at short distances
$x\leq R$. We should also mention that when entering a channel smaller than
its hydrated radius, an ion pays an additional energy penalty due to the
shedding of its hydration shell (Richards et al. (2012), Epsztein et al.
(2019)). For chloride, the total hydration energy is as high as
$155~{}k_{B}T$, but the hydrated radius is 0.4 nm, so that only partial
dehydration arises and only for the smallest pores. Finally, we have assumed
that the confining medium is a dielectric medium with uniform permittivity.
This is a good model for a lipid membrane, but artificial confining materials
may have a variety of electronic properties (see 4.2.6), and their influence
on effective Coulomb interactions has not been thoroughly investigated. In the
case of confinement by a perfect metal, the ionic interactions were found to
be exponentially screened over a distance of order of the channel radius
(Kondrat & Kornyshev (2011), Loche et al. (2019)). Such effective interactions
were found to lead to like-charge attraction in ionic liquids confined in
carbon nanopores (Futamura et al. (2017)).

####  4.3.2 Ion transport beyond the mean-field

We have highlighted in the previous paragraph that correlations may become
important in ion transport through single digit nanopores. If it is the case,
then mean-field theories, such as the PNP framework, may not be directly
applied, and the determination of equilibrium properties, such as ion
concentrations inside the nanopore, requires some form of exact statistical
mechanics. In systems with high aspect ratio (typically nanotubes), the
problem can be reduced to a one-dimensional model of the Ising or 1D Coulomb
gas type, and the partition function may then be exactly computed through a
(functional) transfer matrix formalism (Zhang et al. (2006), Lee et al.
(2014a), Kavokine et al. (2019)). Outside of the 1D geometry, variational
methods may be used (Buyukdagli et al. (2010)), and methods for incorporating
ion pairs into Poisson-Boltzmann theory have been developed (Levin (2002)). A
general feature of these calculations (Zhang et al. (2006), Buyukdagli et al.
(2010)) is that they predict filling transitions: namely, a non-analytic
behaviour of ion concentrations in the nanopore as a function of the salt
concentration in the reservoir, which strongly deviates from the mean-field
Donnan equilibrium result, and may break local electroneutrality (Zhang et al.
(2006)). One should note, however, that breakdown of electroneutrality does
not require ionic correlations and it has recently been shown to arise in a
mean-field setting (Levy et al. (2020)).

If linear response theory applies, then transport properties such as ionic
conductance may be determined directly from the concentrations of charge
carriers at equilibrium. However, this is no longer the case when correlations
are very strong, leading to the formation of tightly bound Bjerrum pairs of
oppositely charged ions. This was first realised by Onsager, who showed that
in a three-dimensional weak electrolyte – an electrolyte that forms Bjerrum
pairs – there is a quadratic current-voltage relation, a phenomenon known as
the second Wien effect (Onsager (1934), Kaiser et al. (2013)). Bjerrum pairing
in one-dimensional confinement was studied by Kavokine et al. (2019), and
stable ion pairs were shown to arise typically for confinement below 2 nm,
similarly to what was observed in MD simulations (Nicholson & Quirke (2003)).
The pairing resulted in some very non-linear behaviour governed by discrete
particle effects,  echoing the phenomenology of Coulomb blockade (Beenakker
(1991), Kaufman et al. (2015)).  As such, ion correlations produced quantised
transport behaviour in a purely classical system.

The coupling of ion transport to fluid transport in the non-mean-field, non
linear regime remains largely unexplored, though it promises a rich
phenomenology. As an example, a recent simulation (Li et al. (2017)) found
that pressure driven water flow through a nanochannel could be blocked by an
ion tightly bound inside. Lastly, we should mention, at the frontier of sub-
continuum ion transport, the Grotthus-like translocation of protons, which has
been observed in a single file (Tunuguntla et al. (2017)) and a single plane
(Gopinadhan et al. (2019)) of water, but still remains poorly understood  in
strongly confined systems.

##  5 CONCLUSION

This review has explored some defining aspects of fluid transport at the
nanoscale. We started from continuum theory and reduced the scale down to its
limit of applicability; we then explored phenomena that occur below the
continuum limit and gave indications of appropriate theoretical descriptions
to tackle them. The main points summarised below were made. Overall, fluids in
molecular scale confinement are largely an uncharted territory for theory, and
recent experiments urge for the development of theoretical tools beyond those
described in this review. [SUMMARY POINTS]

  1. 1.

Experimental systems for studying fluid transport in molecular scale
confinement are today within reach.

  2. 2.

Above 10 nm confinement, fluid transport is governed by continuum hydrodynamic
equations, with coupling to ion transport and surface effects.

  3. 3.

Below 10 nm –the domain of so-called single-digit nanopores – thermal
fluctuations and electrostatic correlations are increasingly important,
challenging continuum and mean-field theory.

  4. 4.

In few nanometre confinement, fluid structuring effects and correlations play
an overwhelming role.

## DISCLOSURE STATEMENT

The authors are not aware of any affiliations, memberships, funding, or
financial holdings that might be perceived as affecting the objectivity of
this review.

## ACKNOWLEDGMENTS

L.B. acknowledges funding from the EU H2020 Framework Programme/ERC Advanced
Grant agreement number 785911-Shadoks. L.B. and R.N acknoweldge support from
ANR-DFG project Neptune. R.N. acknowledges funding from the DFG via SFB1078
and NE810/11. N.K. thanks A. Marcotte for discussions.

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## Appendix

$\rho_{s}$ (M) |  $\lambda_{D}$ (nm)
---|---
$10^{0}$ | 0.3
$10^{-1}$ | 1.0
$10^{-2}$ | 3.1
$10^{-3}$ | 9.6

Material |  $\Sigma$ max ($e\rm\cdot nm^{-2}$) |  $\ell_{GC}$ (nm) |  $\ell_{Du}$ at 0.1 M (nm)
---|---|---|---
Graphene oxidea | 0.06 | 4 | 9
Silicab | 0.3 | 0.8 | 50
Carbon nanotubec | 1 | 0.2 | 170
BN nanotubed | 9 | 0.03 | 1500

aMouhat et al. (2020); bStein et al. (2004);cSecchi et al. (2016b); dSiria et
al. (2013) Left: values of Debye length for different concentrations of
monovalent salt. Right: maximal measured surface charge, Gouy-Chapman length
and Dukhin length at 1 M monovalent salt concentration.

Table 1: Electrostatic lengthscales

## Surface PNP theory

Here we reproduce the main equations of the surface PNP-Stokes framework, that
was introduced by Mouterde et al. (2019) (see 3.3.5). The Nernst-Planck
expression for the ion fluxes is modified according to

$j_{\pm}=\frac{e}{\xi_{\pm}+\lambda_{\pm}}\left[\frac{k_{B}T}{e}(-\partial_{x}\rho_{\pm}\pm\rho_{\pm}(-\partial_{x}\phi)\right]+\alpha_{\pm}\rho_{\pm}v,$
(52)

with $\alpha_{\pm}=\frac{\xi_{\pm}}{\xi_{\pm}+\lambda_{\pm}}$. The
(integrated) Stokes equation becomes

$v=K(\rho_{+},\rho_{-})[(-\partial_{x}p)+e(\alpha_{+}\rho_{+}-\alpha_{-}\rho_{-})(-\partial_{x}\phi)],$
(53)

with $K(\rho_{+},\rho_{-})$ a concentration-dependent permeability, and these
are supplemented by a 1D Poisson equation:

$\partial_{x}[\epsilon h(-\partial_{x}\phi)]=he(\rho_{+}-\rho_{-}).$ (54)