# Slippery and mobile hydrophobic electrokinetics: from single walls to
nanochannels

Olga I. Vinogradova<EMAIL_ADDRESS>Elena F. Silkina  Evgeny S. Asmolov
Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy
of Sciences, 31 Leninsky Prospect, 119071 Moscow, Russia

###### Abstract

We discuss how the wettability of solid walls impacts electrokinetic
properties, from large systems to a nanoscale. We show in particular how could
the hydrophobic slippage, coupled to confinement effects, be exploited to
induce novel electrokinetic properties, such as a salt-dependent giant
amplification of zeta potential and conductivity, and a much more efficient
energy conversion. However, the impact of slippage is dramatically reduced if
some surface charges migrate along the hydrophobic wall under an applied
field.

##  1 Introduction

Controlling the wettability of solid materials is a key issue in surface
engineering, and wetting phenomena have been studied for at least two
centuries. In many common situations the equilibrium contact angle lies
between $0$ and $90^{\circ}$ (i.e., a hydrophilic case), but some solids can
have a contact angle greater than $90^{\circ}$ (a hydrophobic case). This can
generate some special properties of practical interest, based on water
repellency, such as, for example, an extremely long-ranged hydrophobic
attraction. Over past few decades, the cutting edge in the research on smooth
hydrophobic surfaces has shifted from equilibrium wetting and surface forces
towards properties that can impact the dynamics of liquids due to hydrophobic
(hydrodynamic) slippage. In the last years research on hydrophobic slippage is
rapidly advanced being strongly motivated by potential applications, such as
drag reduction and more. These hydrodynamic studies recently raised a question
of a possible influence of hydrophobic slippage on the electrokinetic
transport phenomena in micro- and nanofluidic channels, which include an
electro-osmotic flow in response to an applied electric field, a conductance,
emerging due to this flow and a migration of ions, and also a streaming
current generated by pressure gradient (see Fig.1). This was also motivated by
an awareness of many experimental puzzles, such as an enormous conductivity of
dilute electrolyte solutions confined in nanochannels [1, 2] enhanced by
hydrophobization of the walls [3], the saturation of the conductivity of
nanometric foam films [4], and more.

Figure 1: (a) Nanopore in single layer MoS2 [5] and boron nitride nanotube
inserted into a SiN membrane [6], (b) Salt dependence of conductivity for
membranes made of holey-graphene-like network of different thicknesses [7],
(c) Streaming current as a function of pressure for a 140 nm thick fused
silica slit at low and high salt concentrations and schematics of this
experiment [8].

In this article we review recent developments in the theory of electrokinetic
transport near a hydrophobic wall and in hydrophobic channels by concentrating
on the fundamental understanding and expectations. Our main focus will be on
channels of micro- and nanometric size, where transport phenomena can be very
strongly modified by tuning the electrostatic and wetting properties of the
confining surfaces. We attempt to give the flavour of some of the recent work
in this field, but our review is not intended to be comprehensive. Emphasis is
placed to a continuum description of electrokinetic phenomena; the choice
reflects the authors own interests. Thus, our discussion applies for channels
down to a few nanometers and we limit ourselves by 1:1 electrolyte solutions
of the concentration range from $10^{-6}$ to $10^{-1}$ mol/l, which provides a
very accurate description of the ionic distributions within the Poisson-
Boltzmann theory [9]. Correlations and various nonidealities, such as hydrated
ion volume effects [10], dielectric mismatch [11], dispersion forces between
ions [12], ion-specifity [13], which would be important for channels of one or
two nanometers or in molecular-scale confinement, warrant a separate
discussion. We, therefore, refer the readers to the recent reviews on sub-
continuum electrokinetic effects [14] and relevant simulation techniques [15,
16, 17].

##  2 Electrokinetic transport: electro-osmosis, conductivity currents and
beyond

One imagines a bulk electrolyte solution of a dynamic viscosity $\eta$ and
permittivity $\varepsilon$ in contact with a symmetric channel. In the general
case, the channel subject to a pressure gradient $\partial_{x}p$ and an
electric field $E$ in the $x$ direction as sketched in Fig. 2. The axis $z$ is
defined normal to the surfaces of potential $\Phi_{s}$ and charge density
$\sigma$ (without loss of generality, the surface charges are taken here as
cations) located at $z=\mathcal{L}$. Although channels of variable thickness
have been considered as a model for certain pores [18, 19], two main
geometries are still favoured: slits and cylinders. In the former the
electrolyte is confined between two parallel walls of (infinite) area,
separated by a finite distance $H$. For such a channel it is enough to
consider $\mathcal{L}=H/2$ because of the $z\leftrightarrow-z$ symmetry. For
the cylinder geometry the interior radius $\mathcal{L}=R$ is finite but the
length is infinite. Let the bulk reservoir represent a 1:1 salt solution of
number density of ions $n_{\infty}[\rm{m^{-3}}]$. Clearly, by analysing the
experimental data it is more convenient to use the concentration
$c_{\infty}[\rm{mol/l}]$, which is related to a number density as
$n_{\infty}=N_{A}\times 10^{3}\times c_{\infty}$, where $N_{A}$ is Avogadro’s
number. Ions are assumed to obey the Boltzmann distribution
$n_{\pm}(z)=n_{\infty}\exp(\mp\phi(z))$, where $\phi(z)=e\Phi(z)/(k_{B}T)$ is
the dimensionless local electrostatic potential, $e$ is the elementary
positive charge, $k_{B}$ is the Boltzmann constant, $T$ is a temperature of
the system, and the upper (lower) sign corresponds to the positive (negative)
ions.

Figure 2: (a) Sketch of the nanotube and slit. An applied electric field $E$
and/or a pressure gradient $\partial_{x}p$ generate a mean fluid velocity
$\mathcal{U}$ and electric current of the mean density $\mathcal{J}$. (b)
Profiles of a dimensionless electrostatic potential $\phi(z)$ in thick and
thin channels.

An electrolyte solution builds up a so-called electrostatic diffuse layer
(EDL) close to the channel walls, where the surface charge is balanced by the
cloud of counterions. In thick channels EDLs do not overlap and at their
central part contains the (electro-neutral) bulk electrolyte, so $\Phi(0)=0$.
However, when the channel is sufficiently thin, the EDLs overlap. There exists
no bulk electrolyte solution inside and $\Phi(0)\neq 0$ \- see Fig. 2.

When an electric field $E$ is applied tangent to a charged wall, an electro-
osmotic flow of a velocity $U(z)$ is induced. The electroosmosis takes its
origin in the EDL, where a tangential electric field $E$ generates a force
that, in turns, sets the fluid in motion. The successful understanding of
electro-osmosis is due to Smoluchowski [20], who argued that the velocity of a
plug flow in the bulk is given by

$U=-\dfrac{\varepsilon E}{4\pi\eta}\Phi_{s}$ (1)

Eq.(1) implies that in the case of the positively charged surface, the
direction of fluid flow is opposite to that of the electric field (see Fig.
3). Note that Smoluchowski considered the single wall ($\mathcal{L}\to\infty$)
and postulated the no-slip boundary condition. Only if so, the electrokinetic
potential, later termed zeta potential $Z$, that as a matter of fact should
appear in (1) via the Stokes equation, coincides with the surface potential
$\Phi_{s}$. However, the conclusion that $Z=\Phi_{s}$, which became a dogma in
colloidal science and long time invoked in the interpretation of the
electrokinetic data, is by no means obvious for a sufficiently thin channel
and/or for a situation where the no-slip boundary condition is violated.

Figure 3: Sketch of the electroosmotic velocity profiles $U(z)$ near
positively charged hydrophilic (a solid curve) and hydrophobic (a dash-dotted
curve) single walls of a potential $\Phi_{s}$. A finite slip velocity $U_{s}$
at the hydrophobic wall causes the difference between $\Phi_{s}$ and $Z$.

An applied field also generates an electric (so-called conductivity) current
that is both due to their convective transfer by an emerging electroosmotic
flow and a migration of ions relative to a solvent. The bulk conductivity can
be found as $K_{\infty}=2en_{\infty}m$, where $m$ is the mobility of ions. In
the simplest case, the electric force $eE$, causing the migration of ions, is
balanced by the Stokes force, which for ions of hydrodynamic radius
$\mathcal{R}$ (of the order of a few tenths of nm) gives
$m=e/(6\pi\eta\mathcal{R})$. Consequently,

$K_{\infty}=\frac{e^{2}n_{\infty}}{3\pi\eta\mathcal{R}}$ (2)

Clearly, if $\mathcal{L}$ is large enough, an average ionic conductivity of a
confined solution turns to $K_{\infty}$.

Similarly, the pressure-driven flow induces a convective transfer (but no
migration) of ions and thus generates a current, referred to as streaming
current. Since the streaming current can only be generated within the EDLs,
but not in the electro-neutral bulk electrolyte, its depth-averaged value
becomes negligible in the large systems.

##  3 Spectrum of electrostatic lengths

There are a number of electrostatic length scales that control electrokinetic
transport of water and ions. These length scales are related to properties of
either electrolyte solution or surfaces, or both. Specific micro- and
nanofluidic phenomena will show up when the channel size $\mathcal{L}$ becomes
of the order of or smaller than these characteristic lengths. Another
advantage of using a dimensionless potential and characteristic lengths is
that the results become independent on the choice of any specific system of
electrostatic units.

  * 1.

The so-called Bjerrum length is a length for which the thermal energy is equal
to the Coulombic energy between two unit charges

$\ell_{B}=\dfrac{e^{2}}{\varepsilon k_{B}T}$ (3)

Defined by such a way the Bjerrum length is a property of solvent and does not
depend on the electrolyte concentration. For water at a room temperature
$\ell_{B}\simeq 0.7$ nm. We remark that $\ell_{B}/\mathcal{R}=O(1)$.

  * 2.

The extension of the EDLs is defined by the Debye (screening) length

$\lambda_{D}=\left(8\pi\ell_{B}n_{\infty}\right)^{-1/2}$ (4)

of a bulk solution. For a monovalent salt in water at room temperature
$\lambda_{D}[\rm{nm}]\simeq\dfrac{0.3[\rm{nm}]}{\sqrt{c_{\infty}}[\rm{mol/l}]}$
Therefore, by increasing $c_{\infty}$ from $10^{-6}$ to $10^{-1}$ mol/l, we
reduce the screening length ca. from 300 down to 1 nm. The channel of
$\mathcal{L}\gg\lambda_{D}$ is traditionally referred to as thick and of
$\mathcal{L}\ll\lambda_{D}$ is termed thin.

  * 3.

The Gouy-Chapman length is inversely proportional to the surface charge
density $\sigma$

$\ell_{GC}=\dfrac{e}{2\pi\sigma\ell_{B}}$ (5)

It is often convenient to define $\ell_{GC}$ of the same sign with $\sigma$,
although some researchers use only positive definite $\ell_{GC}$. For
monovalent ions in water at room temperature
$\ell_{GC}[\mathrm{nm}]\simeq\dfrac{36[\mathrm{nm}]}{\sigma[\mathrm{mC/m}^{2}]}$.
A typical (high) surface charge density $\sigma\simeq 36$ mC/m2 gives
$\ell_{GC}\simeq 1$ nm, but small $\sigma$ provides much larger $\ell_{GC}$.
The important ratio is $\lambda_{D}/\ell_{GC}\propto\sigma n_{\infty}^{-1/2}$,
which reflects the _effective surface charge_. The surfaces are referred to as
weakly charged when $\lambda_{D}/\ell_{GC}\leq 1$, and to as strongly charged
if $\lambda_{D}/\ell_{GC}\gg 1$. Say, a surface of $\ell_{GC}=3$ nm is weakly
charged when $c_{\infty}\geq 10^{-2}$ mol/l and strongly charged when
$c_{\infty}\leq 10^{-4}$ mol/l.

  * 4.

The Dukhin length is defined as

$\ell_{Du}=\frac{\lambda_{D}^{2}}{\ell_{GC}}=\dfrac{\sigma}{4en_{\infty}}$ (6)

For a surface of $\sigma\simeq 18$ mC/m2 on increasing $c_{\infty}$ from
$10^{-6}$ to $10^{-1}$ mol/l, one can reduce $\ell_{Du}$ from ca. 45 $\mu$m
down to 0.5 nm. In very dilute solutions, therefore, $\ell_{Du}$ can be much
larger than any conceivable Debye length.

##  4 Governing equations and boundary conditions

The flow inside the channel satisfies the _linear_ Stokes equation with an
electrostatic body force

$\eta\nabla^{2}U=\partial_{x}p+\dfrac{eE}{4\pi\ell_{B}}\nabla^{2}\phi,$ (7)

where $U$ is the fluid velocity. One can also define a dimensionless velocity
as $u=\dfrac{4\pi\eta\ell_{B}}{eE}U$.

In steady state $\phi(z)$ is independent on the fluid flow and satisfies the
_nonlinear_ Poisson-Boltzmann equation (NLPB) [21, 22]:

$\nabla^{2}\phi=\lambda_{D}^{-2}\sinh\phi,$ (8)

which can be linearized only if $\phi_{s}\leq 1$ (or $\Phi_{s}\leq 25$ mV).

When modeling electrokinetics in nanochannels and nanopores, special care
should be given to the selection of boundary conditions. Clearly, the symmetry
conditions $\partial_{z}u|_{z=0}=\partial_{z}\phi|_{z=0}=0$ are always hold,
but the conditions at the walls might be different depending on their
material.

###  4.1 Electrostatic boundary conditions and the contact theorem

To solve NLPB it is convenient to assume either a constant surface charge
density (insulators)

$\phi^{\prime}|_{z=\mathcal{L}}=\dfrac{2}{\ell_{GC}},$ (9)

or a constant surface potential (conductors)

$\phi|_{z=\mathcal{L}}=\phi_{s}$ (10)

These situations, referred below to as CC and CP cases, provide rigorous
bounds on any solutions obtained by imposing a so-called charge regulation
(CR) - another commonly used boundary condition.

The first integration of the NLPB equation leads to a relation between
$\ell_{GC}$ and $\phi_{s}$, known as the _contact theorem_. Once this relation
is found, the NLPB solution obtained for the CC case can be immediately
transformed into a solution for the CP case and vice versa.

Throughout this article we present analytic results for the following two
experimentally relevant modes:

  * 1.

In the limit of the thick channel, $\mathcal{L}/\lambda_{D}\gg 1$, the contact
theorem coincides with the (exact) Grahame equation for a single wall [23]

$\phi_{s}=2\operatorname{arsinh}\left(\frac{\lambda_{D}}{\ell_{GC}}\right)$
(11)

For weakly charged surfaces, $\lambda_{D}/\ell_{GC}\leq 1$, this equation can
be linearized, so is the NLPB. In this case, $\phi_{s}\propto
c_{\infty}^{-1/2}$. Does the Graham equation applies for a channel of
$\mathcal{L}/\lambda_{D}=O(1)$? The answer is yes, but only if surfaces are
strongly charged [24]. Such channels are termed quasi-thick and are said to be
in the _thick channel mode_.

  * 2.

In the so-called _thin channel mode_ defined below, the contact theorem reads
[25]

$\phi_{s}\simeq\operatorname{arsinh}\left(\dfrac{2\alpha\ell_{Du}}{\mathcal{L}}\right),$
(12)

where $\alpha$ reflects the geometry being equal to $1$ for a slit, and to $2$
for a cylinder. Clearly, the linearization of (12) and, consequently, of (8)
cannot be justified when $\ell_{Du}/\mathcal{L}\geq 1$, even provided the
effective charge is small. In essence, the derivation of Eq.(12) does not
require a thin channel limit, $\mathcal{L}/\lambda_{D}\ll 1$, but only imposes
that $\phi_{s}>\mathcal{L}/\ell_{GC}$. This is equivalent to saying that the
effective charge should be below $\phi_{s}\lambda_{D}/\mathcal{L}$ or that
$\mathcal{L}/\lambda_{D}<\ell_{GC}\phi_{s}/\lambda_{D}$. This condition can be
satisfied even if $\mathcal{L}/\lambda_{D}=O(1)$. To fix the idea, set
$\mathcal{L}=\lambda_{D}=10$ nm and $\alpha=2$. Choose $\ell_{GC}=5$ nm. For
such a nanotube Eq.(12) gives $\phi_{s}\simeq 2.8$, which exceeds
$\mathcal{L}/\ell_{GC}=2$. Thus, this nanotube is quasi-thin, i.e. does fall
into a thin channel mode, but one can easily verify that the nanotube of
$\ell_{GC}=1$ nm, does not.

The derived below electrokinetic equations can be used both for CC and CP
cases. For a CC channel $\ell_{GC}$ does depend on neither $c_{\infty}$ nor
$\mathcal{L}$, and $\ell_{Du}\propto c_{\infty}^{-1}$. The expressions for
$\ell_{Du}$ and $\ell_{GC}$ of a CP channel, depending on the mode, follow
directly from Eqs.(11) or (12) and are summarised in Table 1.

Table 1: The Dukhin and Gouy-Chapman lengths for a CP channel. Thick channel mode | Thin channel mode
---|---
$\ell_{Du}=\lambda_{D}\sinh\dfrac{\phi_{s}}{2}\propto c_{\infty}^{-1/2}$ | $\ell_{Du}\simeq\dfrac{\mathcal{L}\sinh\phi_{s}}{2\alpha}\propto c_{\infty}^{0}$
$\ell_{GC}=\dfrac{\lambda_{D}}{\sinh\dfrac{\phi_{s}}{2}}\propto c_{\infty}^{-1/2}$ | $\ell_{GC}\simeq\dfrac{2\alpha\lambda_{D}^{2}}{\mathcal{L}\sinh\phi_{s}}\propto c_{\infty}^{-1}$

Note that some approximate expressions for a conductance in thin nanopores
have been also obtained by either postulating the uniform potential across the
pore (“Donnan equilibrium”) [26, 27], or for the “counter-ions only” case (so-
called co-ion exclusion approximation) [28, 29]. To what extent and when can
these primitive models be employed is discussed in a recent review [22].

###  4.2 Hydrodynamic boundary conditions and the slip length

The fluid flow at interfaces introduces an additional, hydrodynamic length
scale of the problem. Namely, the so-called slip length, $b$, that is defined
as

$U|_{z=\mathcal{L}}=-b\partial_{z}U|_{z=\mathcal{L}}$ (13)

Equation (13) represents a boundary condition for a pressure driven flow.
Physically, the (scalar) slip length characterizes the friction of the fluid
at the interface, and large $b$, indicating low friction, is associated with
the wettability of the surface [30]. For poorly wetted hydrophilic surfaces
$b=0$, i.e. the no-slip boundary condition is hold. The slip length of
hydrophobic surfaces can be of the order of tens of nanometers [31, 32, 33,
34], but not much more.

Theoretical estimates of the intrinsic and apparent slip length have been
discussed in review articles [35, 36], and it now seems certain that there is
not one, but many mechanisms of hydrophobic slippage. They depend on
substrate, contact angle and dissolved gas. However, the slip length of water
has defied complete understanding thus far, with accumulating experimental
evidence for surface charge- and curvature-dependent hydrodynamic slippage:

  * 1.

The very high surface charge is known to reduce the slip length [37, 38]. For
a homogeneous charge distribution this effect can be observed when $\sigma\geq
200$ mC/m2 or $\ell_{GC}\leq 0.18$ nm [37], which is beyond a scope of the
current review.

  * 2.

Some authors revealed large and radius-dependent surface slippage in carbon
nanotubes. Thus, it has been reported that $b$ shows an order of magnitude
increase on reducing the carbon nanotube radius from 50 down to 15 nm [39]. In
other words, when they become narrower, water flows faster. A concept of a
radius-dependent slippage is not yet widely accepted and requires more
experimental confirmation.

Here we keep the analysis at the simplest level by assuming that $b$ is
independent on $\sigma$ and $\mathcal{L}$, but this is immaterial to our main
thesis. Since the values of $b$ are comparable with the Debye lengths
$\lambda_{D}$ of electrolyte solutions, and also with $\ell_{GC}$, it is
natural to expect that hydrophobic slippage can significantly affect the whole
spectrum of electrokinetic phenomena.

###  4.3 Electro-hydrodynamic boundary conditions

For more than a hundred years, colloid scientists have assumed that the
potential-determining surface ions are immobile. However, recently it has been
recognized that this cannot be justified for a liquid-gas interface (bubbles
or drops, foams) as well as for slippery hydrophobic solids [40]. The charges
associated with such surfaces can migrate relative to liquid in response to
$E$. Such an adsorbed ion layer reacts to an electric field and drags the
fluid in the direction opposite to the main electroosmotic flow inside the
channel. This enhances the shear stress at the interface.

To describe the fluid velocity $u_{s}=u|_{z=\mathcal{L}}$ at the hydrophobic
surfaces that is generated by an applied electric field, an electro-
hydrodynamic boundary condition has been formulated [40]

$u_{s}=b\left[-\partial_{z}u|_{z=\mathcal{L}}+\frac{2(1-\mu)}{\ell_{GC}}\right],$
(14)

where $\mu$ can vary from 0 to 1. Integrating Eq.(7) and using (9) one can
reduce (14) to [25]

$u_{s}=-\dfrac{2\mu b}{\ell_{GC}}$ (15)

Thus, in the former case $u_{s}$ vanishes even when $b$ is large. In the
latter case $u_{s}$ attains its maximal value. Clearly, $u_{s}$ is constant
for a CC channel, but it increases with $c_{\infty}$ in the CP case (see Table
1).

Figure 4: Schematic representation of surface cations migrating under an
applied electric field due to a low physisorption potential (a), and snapshots
(adapted from [41]) of chemisorbed (b) and physisorbed (c) hydroxide ions on
hexagonal boron nitride surfaces.

Returning to $\mu$, one mechanism [42] has to do with a momentum portion that
(physisorbed) surface charges transfer to the wall under an applied electric
field and pressure gradient (see Fig. 4). Another [40] involves the “gas
cushion model” [43]: the mobile surface ions (of portion $1-\mu$) are located
at the interface of a thin gas coating, but immobile ions are fixed to a solid
surface itself (e.g. chemisorption). This model implies that in a pressure-
driven flow, the mobile surface charges translate with the velocity of a
hydrodynamic slip given by Eq.(13), but do not migrate relative to liquid.
Both mechanisms that are a corollary of hydrophobicity lead to boundary
condition (14), but in essence, the parameter $\mu$ is still awaiting for a
more detailed interpretation. The point is that such a mobility exists, is
supported by simulation data [40, 44], but has immense variability depending
on substrate material [41]. This subject is becoming more into focus.

Finally, it should be emphasized that Eq.(14) becomes singular if $b=\infty$
and $\mu=0$, which is appropriate for an interface between bulk liquid and
gas. In this situation, the boundary condition for an electro-osmotic flow
takes the form [40]

$\partial_{z}u|_{z=\mathcal{L}}=-\frac{2}{\ell_{GC}}$ (16)

The implications of condition (16) for electrokinetic transport remain largely
unexplored and warrant more investigations.

##  5 Mobility matrix

The linearity of Eq.(7) implies that the transport of water and ions through
the channel can be expressed in terms of a mobility matrix $\mathcal{M}$

$\begin{pmatrix}\mathcal{U}\\\
\mathcal{J}\end{pmatrix}=\begin{pmatrix}M_{h}&M_{e}\\\
M_{e}&K\end{pmatrix}\begin{pmatrix}-\partial_{x}p\\\
E\end{pmatrix}=\mathcal{M}\begin{pmatrix}-\partial_{x}p\\\ E\end{pmatrix}$
(17)

Here $\mathcal{U}$ [m/s] is the mean fluid velocity and
$\mathcal{J}=\overline{J}+\alpha J_{\sigma}/\mathcal{L}$ [A/m2] is the mean
current density, where $\overline{J}$ denotes a volume-averaged value and the
second term reflects the contribution of adsorbed mobile ions. The elements of
matrix $\mathcal{M}$ represent so-called transport coefficients. Namely,
$M_{h}$ [m2/(s$\times$Pa)] is the hydrodynamic mobility, $M_{e}$
[m2/(s$\times$V)] is the electroosmotic mobility, and $K$ [S/m] is the mean
conductivity.

A $2\times 2$ mobility matrix $\mathcal{M}$ is positive definite and symmetric
(with equal off-diagonal coefficients), as assumed in (17), by analogy with
Onsager’s relations in (bulk) non-equilibrium thermodynamics [45]. The
equality of off-diagonal elements of matrix $\mathcal{M}$ has been confirmed
for hydrophobic surfaces with immobile surface ions ($\mu=1,J_{\sigma}=0$)
[46], and later for the case of mobile surface charges, providing the
additional proof of validity of electro-dynamic boundary condition (14) [47].
However, for a pressure induced ionic current Eq.(14) should necessarily be
supplemented by a condition [47]

$J_{\sigma}=(1-\mu)\sigma U|_{z=\mathcal{L}}$ (18)

In other words, whatever the physical mechanism of adsorbed ion mobility is,
boundary condition (14) would be consistent with (17), if and only if, their
contribution to a streaming current density is given by (18). A corollary of a
symmetry is the so-called electro-hydrodynamic coupling

$M_{e}=\dfrac{\mathcal{J}}{-\partial_{x}p}|_{E=0}\equiv\dfrac{\mathcal{U}}{E}|_{-\partial_{z}p=0}$
(19)

Eq.(19) indicates that the magnitude of the streaming current induced by a
pressure gradient is directly related to that of the electro-osmotic flow
generated by the applied electric field.

Figure 5: Sketch of $U(z)$ and $J(z)$ for channels with $\phi$ profiles as in
Fig.2(b) for situations of $-\partial_{x}p=0$ (a) and $E=0$ (b). The thick
channel (left) is hydrophilic, whilst the thin (right) is hydrophobic.

The mobility matrix fully characterizes electrokinetic phenomena in the
channel and, once its elements are known, Eq.(17) can be used to find, without
tedious calculations, the liquid flows and currents that are generated by any
combination of two applied forces. We return to that in Sec. 6. The
coefficients of $\mathcal{M}$ can be obtained by setting one of two possible
driving forces to zero. Figure 5 shows a sketch of $U$ and $J$ profiles for
these two situations discussed in detail below.

###  5.1 Electro-osmotic flow and conductivity current

If $\partial_{z}p=0$, one can reduce Eq. (17) to

$\begin{pmatrix}\mathcal{U}\\\
\mathcal{J}\end{pmatrix}=E\begin{pmatrix}M_{e}\\\ K\end{pmatrix}$ (20)

Electro-osmotic mobility $M_{e}$ is given by

$M_{e}=-\dfrac{e}{4\pi\eta\ell_{B}}\zeta,$ (21)

where

$\zeta=\dfrac{eZ}{k_{B}T}=\phi_{s}-\overline{\phi}+\dfrac{2\mu
b}{\ell_{GC}}=\phi_{s}-\overline{\phi}-u_{s}$ (22)

is the dimensionless zeta (or electrokinetic) potential. Thus, zeta potential
of hydrophobic channels includes a contribution of an electrostatic potential
as well as a slip velocity at the wall, which in turn depends both on
electrostatic (surface charge) and wetting (hydrophobic slip length and
surface charge mobility) properties of the walls.

When $\mathcal{L}\to\infty$, which is the case of an isolated surface,
$\overline{\phi}\to 0$, a relation between $\phi_{s}$ and surface charge
density is given by Eq.(11), and $\zeta$ becomes a property of the surface
itself. However, zeta potential of hydrophobic surfaces no longer reflects the
sole surface potential, except the situation of $\mu=0$, since it is enhanced
due to a finite fluid velocity at the walls

$\zeta=\phi_{s}-u_{s}$ (23)

(or $Z=\Phi_{s}+\dfrac{4\pi}{\varepsilon}\mu b\sigma$). A sketch of $U(z)$
near a slippery wall is included in Fig. 3 along with the Smoluchowski
profile. As a side note, similar mechanism of zeta potential enhancement is
observed for hydrophilic surfaces coated by porous nanofilms that are
permeable for water and ions [48, 49]. For a homogeneous hydrophilic surface
($b=0$), however, $\zeta=\phi_{s}$, which is the classical Smoluchowski result
[20].

It is instructive now to clarify how to use Eq.(23) in the CC and CP cases.
Substituting (11) we obtain the CC expression

$\zeta=2\operatorname{arsinh}\left(\frac{\lambda_{D}}{\ell_{GC}}\right)+\dfrac{2\mu
b}{\ell_{GC}},$ (24)

which, say, for a low effective surface charge, $\lambda_{D}/\ell_{GC}\leq 1$,
gives $\phi_{s}\simeq 2\lambda_{D}/\ell_{GC}$ and $\zeta/\phi_{s}\simeq 1+\mu
b/\lambda_{D}$. Thus, although $\phi_{s}$ itself decreases with salt, in
concentrated solutions $\zeta\gg\phi_{s}$, provided $\mu b/\lambda_{D}$ is
large. By expressing $\ell_{GC}$ in (24) through $\phi_{s}$ (see Table 1) one
can immediately obtain a solution for the CP case

$\zeta=\phi_{s}+\dfrac{2\mu b}{\lambda_{D}}\sinh\dfrac{\phi_{s}}{2}$ (25)

For $\phi_{s}\leq 1$ this gives the same $\zeta/\phi_{s}$ as in the CC case,
but since the surface potential is constant, $\zeta$ becomes very large at
high salt. Thus, if the hydrophobic slippage is ignored, one can erroneously
infer a huge $\phi_{s}$ from the measurements.

Figure 6: Zeta potential as a function of salt concentration computed for a
channel of $\mathcal{L}=10$ nm and $b=30$ nm using fixed $\ell_{GC}=5$ (a) or
$\phi_{s}=4$ (b). Solid curves represent $\zeta$ of a slit, dashed ones - of
the cylinder. The surface potential computed for a cylinder is shown by dash-
dotted curves. Squares and circles show calculations from Eqs.(25) and (26)
with $\alpha=2$.

In the general case, the zeta potential is a global electrodynamic property of
the channel. Finite $\overline{\phi}$ reduces its value, but the hydrophobic
slippage if any tends to augment $\zeta$. The competition between these
effects defines a magnitude of $\zeta$, which can be larger or smaller than
$\phi_{s}$, and provides a diversity of its behavior. This is illustrated in
Fig. 6, where $\phi_{s}$ is compared with the values of $\zeta$ computed using
$\mu=0$ and 1 (both for a slit and a cylinder of $\mathcal{L}=10$ nm and
$b=30$ nm). It can be seen that there exists some quantitative difference
between cylinders and slits, but the qualitative features of the
$\zeta$-curves are the same. The CC calculations shown in Fig. 6(a) are made
with the value of $\ell_{GC}=5$ nm. In the case of $\mu=0$, identical to a no-
slip channel, $\zeta=\phi_{s}-\overline{\phi}\leq\phi_{s}$. Zeta potential
remains constant at $c_{\infty}\leq 10^{-3}$ mol/l (which is the thin channel
mode, as discussed in Sec. 4.1) and begins to converge slowly to $\phi_{s}$
only at a higher concentration. The emergence of the plateau, which points out
that $\mathcal{U}$ becomes independent on $c_{\infty}$, is an entirely
unexpected result that is impossible for a single hydrophilic wall. When
$\mu=1$, $\zeta$ is large, exceeds $\phi_{s}$, and depends neither on amount
of salt nor geometry. In the thin channel mode the difference between
$\phi_{s}$ and $\overline{\phi}$ can be calculated analytically, so is $\zeta$

$\phi_{s}-\overline{\phi}\simeq\dfrac{2}{2+\alpha}\dfrac{\mathcal{L}}{\ell_{GC}},\,\zeta\simeq-
u_{s}+\dfrac{2}{2+\alpha}\dfrac{\mathcal{L}}{\ell_{GC}}$ (26)

Consequently, when $\mu b/\mathcal{L}\ll 1$, one can derive
$\zeta\propto\mathcal{L}/\ell_{GC}$. This is the case of $\mu=0$ in Fig. 6(a):
the value of $\zeta$ at saturation reflects only (constant) $\sigma$. However,
if $\mu b/\mathcal{L}\gg 1$, then $\zeta\simeq-u_{s}\propto\mu b/\ell_{GC}$,
so that when $\mu=1$, the plateau appears due to a constant $b\sigma$ and can
be used to infer its value. In the thick channel mode it is generally assumed
that $\overline{\phi}\simeq 0$, so the Eq.(23) could be sensible
approximation. The same calculations, but for the CP channel of $\phi_{s}=4$
are shown in Fig. 6(b). One important conclusion is that $\zeta$ now grows
with $c_{\infty}$. If $\mu=1$, the zeta potential significantly exceeds
$\phi_{s}$ in the thick channel mode, and for this branch of the curve
$\zeta\propto c_{\infty}^{1/2}$. Also included in Fig. 6 some analytical
results obtained for CC and CP cylinders (that constitute a more realistic
model for artificial nanotubes and real porous materials). It can be seen that
the fit is quite good.

These results imply that if $\mu b=0$ any CR model should lead to a power-law
scaling $\zeta\propto c_{\infty}^{\gamma}$) with an exponent between 0 and 1
in the thin channel mode, or from -1/2 to 0 in the thick channel mode. A
finite $\mu b$ not only increases $\zeta$, but also leads to a different
scaling: in concentrated solutions $\gamma$ is now bounded by 0 and 1/2.

Some further comments should be made. That $\overline{\phi}$ augments on
dilution is apparent since EDLs begin to occupy a large portion of the
channel. In essence, the emergence of the plateau branch only indicates that
$\phi_{s}-\overline{\phi}\propto c_{\infty}^{0}$, but it does not require a
thin channel mode or even an overlap of EDLs. The analytical calculations of
$\phi_{s}-\overline{\phi}$ beyond the thin channel mode appear to be missing
and remain a challenge.

Figure 7: Electrical conductivity as a function of salt concentration computed
for the same CC and CP channels as in Fig. 6. Dash-dotted lines show
conductivity of the bulk electrolyte.

Finally, we turn to the mean conductivity of the channel $K$. Numerical
results are shown in Fig. 7 and compared with $K_{\infty}$. The calculations
are made for the same channels as in Fig. 6. One can see an obvious
correlation with zeta potential data: in the CC case the conductivity plateau
appears, and exactly in the same concentration range. The conductivity
amplification at the plateau branch compared to the bulk is huge, several
order of magnitudes, and depends on $\mu$ (and geometry).

It is conventional to divide $K$ into two contributions: $K_{0}$ arising for
hydrophilic channel, and slip-driven $\Delta K$ that is associated with
hydrophobicity

$K=K_{0}+\Delta K$ (27)

For a channel of any thickness

$K_{0}=K_{\infty}\displaystyle\left[\dfrac{3\lambda_{D}^{2}\mathcal{R}\overline{(\phi^{\prime})^{2}}}{2\ell_{B}}+\overline{\cosh{\phi}}\right],$
(28)

where $K_{\infty}$ is given by Eq.(2) The first and second terms in (28) are
associated with the convective and migration contributions, correspondingly.
To use (28) the mean square derivative of the electrostatic potential
$\overline{(\phi^{\prime})^{2}}$, which is the measure of the electrostatic
field energy (per unit area), and the mean osmotic pressure
$\overline{\cosh\phi}$ should be substituted. Their detailed calculations for
slits and cylinders can be found in [24, 47] and the results for $K_{0}$ can
be summarized as follows. The surface conductivity, i.e. the conductivity
associated with the EDL, dominates over the bulk one when
$\ell_{Du}/\mathcal{L}=O(1)$ or larger. If so,

$K_{0}\simeq K_{\infty}\dfrac{2\alpha\ell_{Du}}{\mathcal{L}}\beta,$ (29)

with $\beta=O(1)$. Namely, it is equal to $1+3\mathcal{R}/\ell_{B}\simeq 2$ in
the thick and $\simeq 1$ in the thin channel mode. This implies that in CC
channels $K_{0}\propto e\alpha\beta\sigma/\mathcal{L}$ does not depend on the
salt concentration (since $K_{\infty}\ell_{Du}\propto c_{\infty}^{0}$). Thus,
the surface contribution to the conductivity shows up as a saturation of the
conductivity in dilute solutions, where the bulk contribution is practically
absent [1, 2]. The height of the conductivity plateau augments on increasing
surface charge density and reduce on decreasing $\mathcal{L}$ (but the
conductance $\propto K_{0}\mathcal{L}$ does not depend on the channel
thickness). By expressing $\ell_{Du}$ in (29) through $\phi_{s}$ one can
obtain $K_{0}$ in the CP case. In the thick channel mode

$K_{0}\simeq
K_{\infty}\dfrac{2\alpha\lambda_{D}}{\mathcal{L}}\sinh\dfrac{\phi_{s}}{2}\propto
c_{\infty}^{1/2},$ (30)

but for a thin channel mode $K_{0}\propto K_{\infty}\sinh\phi_{s}\propto
c_{\infty}$.

The slip-driven contribution $\Delta K$ is given by

$\Delta K=K_{\infty}\frac{2\alpha\ell_{Du}}{\mathcal{L}}\left(1-\mu-\frac{3\mu
u_{s}}{2}\dfrac{\mathcal{R}}{\ell_{B}}\right),$ (31)

where the proportional to $(1-\mu)$ term is associated with a migration
contribution of adsorbed ions and the last term accounts for an additional
convective conductivity due to a finite slip velocity. Eq. (31) was first
derived for a thick slit [42], but a later study has proven that it is valid
for any channel [24].

It follows then from (27) that

$K\simeq
K_{0}\left(1+\dfrac{1-\mu}{\beta}+\frac{3\mu^{2}b}{\beta\ell_{GC}}\dfrac{\mathcal{R}}{\ell_{B}}\right),$
(32)

where the expression in the brackets represents the amplification of
conductivity due to hydrophobization. It is nonlinear function of $\mu$, which
can exhibit a minimum [24]. For the sake of brevity here we only mention the
cases of $\mu=0$, where only a migration of surface ions generates a
complimentary current, and of $\mu=1$, where only a convective ionic current
enhances the conductivity. In the former case $K/K_{0}\simeq 1+1/\beta$, i.e.
the conductivity increases, and can be twice larger than $K_{0}$ (in thin
channel mode). Importantly, in any modes, and independently of electrostatic
boundary conditions, $K$ at $\mu=0$ scales with $c_{\infty}$ exactly as
$K_{0}$ does. In the case of $\mu=1$

$K\simeq
K_{0}\left(1+\frac{3b}{\beta\ell_{GC}}\dfrac{\mathcal{R}}{\ell_{B}}\right)$
(33)

The amplification of conductivity, $K/K_{0}\propto b/\ell_{GC}$, can be huge,
provided $b/\ell_{GC}$ is large. If so, in the CC case $K$ does not depend on
salt and the effect of slippage shows up simply in the shift of the saturation
plateau towards a much larger value. If we deal with the CP channel of
$\mu=1$, the scaling of $K$ with salt becomes different. It is straightforward
to show that in the thick channel mode

$K\simeq
K_{0}\left(1+\frac{b}{\lambda_{D}}\dfrac{3\mathcal{R}}{\beta\ell_{B}}\sinh\dfrac{\phi_{s}}{2}\right)$
(34)

If the second term dominates, $K\propto c_{\infty}$, and slippage appears as a
shift of the bulk conductivity curve. In the thin channel mode

$K\simeq
K_{0}\left(1+\dfrac{b\mathcal{L}}{2\alpha\lambda_{D}^{2}}\dfrac{3\mathcal{R}}{\beta\ell_{B}}\sinh\phi_{s}\right),$
(35)

indicating that the conductivity increment does exist and is salt-dependent,
but quite small. Thus, we might argue that a sensible scaling should be
$K\propto c_{\infty}$, but note that conductivity can become smaller than at
$\mu=0$. At first sight this is surprising, but we recall that in very dilute
solutions $K$ is twice larger than $K_{0}$. Some of these theoretical results
are included in Fig. 7. It can be seen that the approximate formulas are very
reliable.

Summarising the scaling properties of the CC and CP channels, one can expect
that any CR model should lead to $K_{0}\propto c_{\infty}^{\gamma}$ with
$\gamma$ bounded by 0 and 1 in the thin channel mode or 0 and 1/2 in the thick
channel mode. The values of $\gamma=1/3$ [50] and 1/2 [26] obtained for dilute
solutions lie in between these attainable bounds. These two results have been
actively discussed in the literature [28, 51, 29], but in essence, in the thin
channel mode, $\gamma$ can vary smoothly from 0 to 1 depending on parameters
of the CR model. Hydrophobic slippage has the effect of allowing augmented
conductivity, $K\propto c_{\infty}^{\gamma}$, but if $\mu=0$ the scaling
behavior does not change. For $\mu=1$ in the thin channel mode
$0\leq\gamma\lesssim 1$, and in the thick channel mode $1/2\leq\gamma\leq 1$.
Thus, the hydrophobicity not just increases the conductivity, but augment
$\gamma$, although in concentrated solutions only. To the best of our
knowledge such predictions have not been tested in experiment and simulations
yet.

###  5.2 Pressure-driven flow and streaming current

If $E=0$, Eq.(17) reduces to

$\begin{pmatrix}\mathcal{U}\\\
\mathcal{J}\end{pmatrix}=-\partial_{z}p\begin{pmatrix}M_{h}\\\
M_{e}\end{pmatrix}$ (36)

Thus, the streaming current measured as a function of applied pressure
gradient allows one to determine $M_{e}$, and then to use Eq.(21) to infer
$\zeta$. From a pragmatic view, the choice of this experiment is dictated by
the complexity (or impossibility) of the fluid velocity measurements in narrow
channels. In any event, the streaming current studies are always much easier
to perform than any velocimetry experiment.

The hydrodynamic mobility, or a coefficient which relates applied pressure
gradient with the mean velocity is given by

$M_{h}=\dfrac{\mathcal{L}^{2}}{\alpha(2+\alpha)\eta}\left(1+\dfrac{(2+\alpha)b}{\mathcal{L}}\right)$
(37)

The value of $M_{h}$ thus depends on the size and geometry of the channel and
the wetting properties of its walls, but does not depend on electrostatics.

##  6 Applications

The main purpose of Sec. 5 has been to show that electrokinetic phenomena in a
narrow channel can be significantly different from those near a single wall,
and also why and when. This may open many novel applications. We list below
what we believe are the most relevant.

###  6.1 Probing surface properties

By measuring (streaming or conductivity) currents and plotting $\zeta$ and/or
$K$ against $c_{\infty}$, the surface properties can be tested. That the
surface is of the constant $\sigma$ is signalled by an emergence in dilute
solutions of the zeta potential and conductivity plateau. The height of the
conductivity plateau was long time used to infer $\sigma$ of hydrophilic
silica, but the amount of its shift due to hydrophobicity reflects the values
of $\mu$ and $b$ that can also be determined. Such a procedure is beset with
difficulties. The point is that the electrostatic and hydrodynamic effects are
strongly coupled. To disentangle them, a series of several different (multi-
step) experiments is required. Again, evidences of large $\zeta$ and $K$ in
concentrated solutions, coupled with some particular exponents of power-law
scaling are signalling that surfaces keep $\phi_{s}$ constant (or obeying CR
conditions) and point to the absence or existence of slip and surface charge
mobility. A properly designed experiment would allow to infer their magnitude.

###  6.2 Energy conversion

The amplification by hydrophobic slippage of all elements of the mobility
matrix $\mathcal{M}$ opens very interesting perspectives in the context of
electrokinetic energy conversion.

  * 1.

The emergence of streaming current represents a tool for conversion
hydrostatic energy into electrical power. The maximum efficiency, i.e. the
ratio of the output to the input power is [52]

$\mathcal{E}=\frac{\Theta}{\Theta+2\left(\sqrt{1-\Theta}+1-\Theta\right)},$
(38)

where $\Theta=M_{e}^{2}/\left(KM_{h}\right).$ Slip lengths of a few tens of
nanometers are predicted to increase the efficiency of the energy conversion
up to 40% [53]. However, the mobility of surface charges significantly reduces
$E$ [54]. These results would require thorough experimental validation with
various hydrophobic materials.

  * 2.

The electrokinetic energy can be converted to a mechanical energy. For
example, one can generate a high pressure gradient at the nanoscale. Suppose
we close one end of the channel and apply $E$. Substituting $\mathcal{U}=0$ to
(17) leads to $-\partial_{z}p/E=-M_{e}/M_{h}$. Could the hydrophobicity
augment $-\partial_{z}p/E$? The answer to this question is by no means obvious
since both $M_{e}$ and $M_{h}$ increase with $b$. Using (26) and (37) one can
be readily demonstrated that for the nanotube of $\mathcal{L}=10$ nm and
$b=30$ nm in the thin channel regime $-\partial_{z}p/E$ “increases” in $\mu$
times. Thus, a hydrophobicity has detrimental or no effect. However, in the
thick channel regime $\partial_{z}p/E$ for a given hydrophobic cylinder
augments in ca. $\mu\mathcal{L}/4\lambda_{D}$ times. This implies that for
$\mu=1$ and $c_{\infty}=10^{-1}$ mol/l the amplification of $\partial_{z}p/E$
in 2.5 times can be expected. To our knowledge such predictions have not been
made before, nor tested in experiment.

##  7 Conclusion

This review was motivated by an awareness of many unusual experimental
results, in part on giant conductivities and zeta potentials measured in
micro- and nanochannels. These observations have posed serious issues for
classical theories of electrokinetic phenomena, and also about the
manifestations of hydrophobicity. The effects of hydrophobic slippage until
recently have not been considered, and the new “ingredient” is the existence
of a migration of adsorbed surface ions with respect to liquid under an
electric field. During the last several years theory has made striking
advances leading to interpretation of electrokinetic experiments in
hydrophilic channels, as well as predictions of novel effects that might be
expected to occur when channels are hydrophobic. The time is probably right
for more detailed simulations and experimental studies. If these effects were
better clarified and tested, the implications are large.

## Acknowledgements

This work was supported by the Ministry of Science and Higher Education of the
Russian Federation.

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