# Transport of ions in hydrophobic nanotubes

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

###### Abstract

The theory of electrokinetic ion transport in cylindrical channels of a fixed
surface charge density is revisited. Attention is focused on impact of the
hydrophobic slippage and mobility of adsorbed surface charges. We formulate
generalised Onsager relations for a cylinder of an arbitrary radius and then
derive exact expressions for the mean electro-osmotic mobility and
conductivity. To employ these expressions we perform additional electrostatic
calculations, with the special focus on the non-linear electrostatic effects.
Our theory provides a simple explanation of a giant enhancement of the
electrokinetic mobility and conductivity of hydrophobic nanotubes by
highlighting the role of appropriate electrostatic and hydrodynamic length
scales and their ratios. We also propose a novel interpretation of zeta
potentials of cylindrical channels.

##  I Introduction

Nanopores are ubiquitous to nature, and nanoporous materials found
applications in various technologies. Electrokinetic transport phenomena in
nanotubes include an electro-osmotic flow in response to an applied electric
field, a conductance, emerging due to a convective ion transfer by this flow
in addition to a conventional migration of ions (electrophoresis), and also a
streaming current that is generated by pressure gradient if any [1, 2]. The
transport of ions (i.e. streaming and conductivity currents) in these
nanoscale systems is of especial interest being important for physiological
phenomena, energy harvesting, biosensing and more [3, 4, 5, 6, 7]. Besides,
from the measured streaming current the zeta potential of surfaces can be
inferred, which is important in colloid and interface science [8, 9, 10].
Finally, with the advent of nanofluidics there has been considerable interest
in an unusually high conductance of electrolyte solutions in confined systems.

Extensive efforts have gone into investigating conductivity in nanotubes and
flat-parallel slits experimentally. Stein  _et al._ [11] found a remarkably
high conductivity of dilute solutions in nanoslits that is independent on the
salt concentration. The corresponding to a saturation plateau conductivity
augments with the surface charge density [12]. Measurements in nanotubes have
shown that the conductivity plateau decreases with the radius [13]. Balme  _et
al._ [14] reported that the height of conductance plateaus is augmented in
hydrophobic nanopores. The body of experimental work investigating the
streaming current (including inferring the zeta potential) is much less than
that for the conductivity current, although there is some literature in this
area [15, 16, 17, 18]. However, despite a rapid rise of an experimental
activity, ion transport has been given insufficient attention compared with
the transport of water. For example, we are unaware of any previous work that
has addressed the impact of hydrodynamic (hydrophobic) slippage [19, 20, 21,
22, 23, 24] on the streaming current, although the amplification of an
electro-osmotic flow in slippery channels was a subject of active experimental
research [25, 26, 27, 28].

There is some literature describing attempts to provide a satisfactory theory
of electric currents emerging in confined electrolytes. The majority of
previous work considered planar geometries (where the exact solution to a non-
linear electrostatic problem exists, although can only be expressed in terms
of an elliptic integral) and assumed the classical no-slip boundary conditions
at the charged walls [29]. The conductivity plateau at low salt has been
attributed to the surfaces of a constant charge density [11]. Bocquet and
Charlaix [2] have briefly discussed the expected shift of the conductance
plateau due to a hydrodynamic slip, assuming immobile surface charges.
Applying some simple scaling arguments these authors propose that this should
be $\propto b/\ell_{GC}$, where $b$ is the slip lenth and $\ell_{GC}$ is the
Gouy-Chapman length, which is inversely proportional to the surface charge
density [30]. At slippery surfaces adsorbed charges could be mobile, and,
therefore, responding to the external electric field as discussed by Maduar
_et al._ [31] and supported by _ab initio_ simulations [32]. Mouterde and
Bocquet [33] derived scaling expressions describing a contribution of a
hydrodynamic slip taking into account the mobility of adsorbed ions that later
allowed one to interpret the computer simulation data on channel conductivity
with physisorbed surface charges [34]. Their results, however, apply only for
a situation of channels of a thickness that significantly exceeds the Debye
length $\lambda_{D}$ of the bulk electrolyte. Recently Vinogradova  _et al._
[35] proposed an analytical theory of ion conductivity in slippery parallel-
plate channels of an arbitrary thickness. These authors derived simple
expressions for the mean conductivity of the channel in two regimes, of thick
and thin slits, highlighting the role of hydrodynamic slip and mobility of
adsorbed surface charges.

Cylinders constitute a more realistic model for artificial nanotubes and real
porous materials, but remain much less theoretically understood. The main
challenge is related to electrostatic calculations in the cylindrical
geometry. Since the exact solution to non-linear electrostatic problem is
still unknown, one needs to rely on numerical solutions or approximations. The
analytical theory of an electro-osmotic flow in no-slip nanotubes developed by
Rice and Whitehead [36] is restricted to small surface potentials. Later
Levine  _et al._ [37] extended this model for a no-slip cylinder to higher
potentials and obtained expressions for streaming and conductivity currents,
but only in the integral form. During last few years some approximate
analytical expressions for a conductance in cylindrical nanopores have been
obtained by postulating either the overall electroneutrality (‘Donnan
equilibrium’) [38, 39], and also for the ‘counter-ions only’ case (so-called
co-ion exclusion approximation) [40]. A slippery nanotube has been considered
in [41]. These authors proposed an expression relating the conductivity to the
integral of an electrostatic potential, assuming the surface charges are
immobile. Numerical integration showed that at a given (high) surface charge
the conductivity increases with the slip length, but a physical interpretation
of this result has not been proposed. Green [42] also studied hydrophobic
nanotubes with immobile surface charged and derived some useful expressions
for transport coefficients. He showed that the hydrodynamic slip boundary
conditions are consistent with the Onsager relations. Silkina  _et al._ [43]
performed calculations of an electro-osmotic flow in nanotubes by applying
electro-hydrodynamic slip boundary conditions [31], i.e. allowing for a
mobility of adsorbed charges in response to electric field. They obtained a
rigorous asymptotic result for a situation of strongly overlapping diffuse
layers at high surface potential and predicted a parabolic velocity (and
potential) profile. This solution, however, is accurate only when
$\lambda_{D}/\ell_{GC}\leq 1$, i.e. when the surface charge is sufficiently
small. The authors interpreted the shift (amplification) of the electro-
osmotic velocity profile due to slippage and concluded that the mobility of
adsorbed charges reduces the effect, but made no attempt to investigate the
transport of ions.

In this paper we present the results of a study of ion transport in the
hydrophobic nanotubes with the fixed surface charge density. The
hydrophobicity implies that the electro-hydrodynamic boundary condition [31]
is imposed, i.e. besides hydrophobic slippage we deal with a migration of
adsorbed ions in response to an applied electric field. The theory has the
merit of yielding useful analytical results as well as being very well suited
to numerical work. Our work provides new insight into streaming and
conductivity currents in the nanotubes, as well as gives an interpretation of
their zeta potential.

Our paper is arranged as follows. In Sec. II, we define our system and present
some general considerations concerning the Onsager relations and transport
coefficients. Section II.1 describes the derivation of the Onsager relations
for a hydrophobic nanotube. A detailed calculation of coefficients for the
mobility matrix is presented in Sec. II.2. The concept of a zeta potential of
hydrophobic nanotubes is introduced. The approximate equations for zeta
potentials and mean conductivity are derived in Sec. III. Numerical results
for zeta-potentials and conductivity are presented in Sec. IV and compared
with approximate theoretical expressions. We conclude in Sec. V. In Appendix A
we briefly discuss the possible reduction of ion mobilities with salt
concentration to argue that this effect can safely be neglected in our system.
Appendix B contains a derivation of electrostatic relationships required for
calculations of streaming and conductivity currents.

##  II General theory

We consider an 1:1 aqueous electrolyte solution of a dynamic viscosity $\eta$
and permittivity $\varepsilon$ in contact with a charged nanotube of radius
$R$ as sketched in Fig. 1. The local number density (per unit volume) of
cations and anions are $n_{+}$ and $n_{-}$, respectively, and, therefore, the
charge densities at each point are $\rho_{\pm}=en_{\pm}$, where $e$ is the
elementary positive charge. Defining $n_{\infty}$ as the number density in the
bulk one can write for the Boltzmann distribution of density
$n_{\pm}(r)=n_{\infty}\exp(\mp\phi(r))$, where $\phi(r)=e\Phi(r)/(k_{B}T)$ is
the dimensionless electrostatic potential, where $k_{B}$ is the Boltzmann
constant, $T$ is a temperature of the system.

The cylindrical system $(z,r)$ of coordinates is defined, such that $z$
coincides with the axis of the nanotube, and $r$ is the radial axis. The
surface of nanotube is thus located at $r=R$ and we denote its potential as
$\Phi_{s}$ and a charge density as $\sigma$. Without loss of generality, the
surface charges are taken as cations ($\sigma$ is positive).

###  II.1 Onsager relations

Figure 1: Sketch of the charged nanotube of radius $R$ that is coupled with a
bulk electrolyte reservoir characterized by the Debye length $\lambda_{D}$. An
applied pressure gradient $-\partial_{z}p$ induces a hydrodynamic flow and
streaming current. An applied electric field $E$ induces an electro-osmotic
velocity and a conductivity current. The confined electrolyte flows with the
mean velocity $U$ and has the mean current density $J$.

In the general case, the nanotube subject to a pressure gradient
$\partial_{z}p$ and an electric field $E$ in the $z$ direction. Thus the flow
inside the nanotube satisfies the Stokes equation, which includes an
electrostatic body force $E\rho$

$\eta\nabla^{2}u=\partial_{z}p+E\rho,$ (1)

where $\nabla^{2}=\partial_{r}^{2}+r^{-1}\partial_{r}$ for the cylindrical
geometry, $u(r)$ is the fluid velocity and $\rho$ is the total charge density
at each point $r$. It follows from the Poisson equation that

$E\rho=-E\dfrac{\varepsilon\nabla^{2}\Phi}{4\pi}.$ (2)

Note that CGS (Gaussian) electrostatic units are used throughout our paper.
However, by expressing all characteristic lengths in terms of the introduced
below Bjerrum and Debye lengths we will obtain the results that are
independent of any specific system of units.

If the bulk reservoir represents a 1:1 salt solution of concentration
$c_{\infty}$, then

$E\rho=-E\dfrac{\varepsilon k_{B}T\nabla^{2}\phi}{4\pi
e}=-\dfrac{eE\nabla^{2}\phi}{4\pi\ell_{B}},$ (3)

where $\ell_{B}=\dfrac{e^{2}}{\varepsilon k_{B}T}$ is the Bjerrum length. Note
that $\ell_{B}$ of water is equal to about $0.7$ nm for room temperature.

Consequently, Eq.(1) can be rewritten as

$\eta\nabla^{2}u=\partial_{z}p+\dfrac{eE}{4\pi\ell_{B}}\nabla^{2}\phi$ (4)

We assume weak field, so that in steady state $\phi(r)$ is independent on the
fluid flow and satisfies the nonlinear Poisson-Boltzmann equation

$\nabla^{2}\phi=\dfrac{1}{r}\dfrac{d}{dr}\left(r\dfrac{d\phi}{dr}\right)=\lambda_{D}^{-2}\sinh\phi$
(5)

where $\lambda_{D}=\left(8\pi\ell_{B}n_{\infty}\right)^{-1/2}$ is the Debye
screening length of an electrolyte solution. Note that by analysing the
experimental data it is more convenient to use the concentration
$c_{\infty}[\rm{mol/L}]$, which is related to $n_{\infty}[\rm{m^{-3}}]$ as
$n_{\infty}=N_{A}\times 10^{3}\times c_{\infty}$, where $N_{A}$ is Avogadro’s
number. We also recall that a useful formula for 1:1 electrolyte is [44]

$\lambda_{D}[\rm{nm}]=\frac{0.305[\rm{nm}]}{\sqrt{c_{\infty}}[\rm{mol/L}]},$
(6)

so that upon increasing $c_{\infty}$ from $10^{-7}$ (in pure water, where the
ionic strength is due to the dissociating H+ and OH- ions) to $10^{-1}$ mol/L
the screening length is reduced from about 1 $\mu$m down to ca. 1 nm.

To integrate Eq.(5) we impose two electrostatic boundary conditions. The first
condition always reflects the symmetry of the channel
$\phi^{\prime}|_{r=0}=0$, where ′ denotes $d/dr$. The second condition is
applied at the walls and can be either that of a constant surface potential
(conductors) or of a constant surface charge density (insulators). Here we
limit ourselves by the constant charge condition, which can be formulated as

$\phi^{\prime}|_{r=R}=\dfrac{2}{\ell_{GC}},$ (7)

where $\ell_{GC}=\dfrac{e}{2\pi\sigma\ell_{B}}$ is the Gouy-Chapman length.
For high surface charges, say $\sigma\simeq 36$ mC/m2, the Gouy-Chapman length
is small, $\ell_{GC}\simeq 1$ nm. However, $\sigma\simeq 0.73$ mC/m2 gives
$\ell_{GC}\simeq 50$ nm.

The linearity of Eq.(4) implies that its solution represents the decoupled and
superimposed contribution of the pressure-driven and electro-osmotic flows

$u=-m_{h}\partial_{z}p+m_{e}E,$ (8)

where $m_{h}$ and $m_{e}$ are the functions of $r$ that depend on
electrostatic and electro-hydrodynamic boundary conditions at the solid wall.
Below we refer these functions to as hydrodynamic and electro-osmotic
mobilities. Note that the first term in (8) is taken with minus to provide
positive $m_{h}$ when $\partial_{z}p<0$. As we will see below, a sign of
$m_{e}$ depends on the sign of the surface charge (potential).

It is convenient to use the average values of variables, $U=\overline{u}$,
$M_{h}=\overline{m_{h}}$, and $M_{e}=\overline{m_{e}}$. For any variable $f$
its average value across the cross-section of the cylindrical nanotube is
given by

$\overline{f}\,=\frac{2}{R^{2}}\int\limits_{0}^{R}\,rfdr.$ (9)

Formally the above considerations are equivalent to the statement that in the
linear response regime, the transport of water and ions through the nanotube
can be then expressed in terms of mobility matrix $\mathcal{M}$:

$\begin{pmatrix}U\\\ J\end{pmatrix}=\begin{pmatrix}M_{h}&M_{e}\\\
M_{e}&K\end{pmatrix}\begin{pmatrix}-\partial_{z}p\\\
E\end{pmatrix}=\mathcal{M}\begin{pmatrix}-\partial_{z}p\\\ E\end{pmatrix}$
(10)

where $J$ [A/m2] is the mean current density

$J=\overline{j}+J_{\sigma},$ (11)

and $K$ [S/m] is the mean conductivity of the channel. The first term in (11)
is associated with the transport of the diffuse cations and anions inside the
nanotube that generates the local density of current $j=j_{+}+j_{-}$. The
second term reflects the contribution of the surface current $j_{\sigma}$
emerging due to adsorbed mobile ions that are located at the ring of perimeter
$2\pi R$. Therefore, this surface averaged contribution to the mean current
density is given by $J_{\sigma}=2j_{\sigma}/R$.

As a side note, in experiment one usually measure the conductance $G$ [S]. For
a nanotube of length $L$

$G=K\dfrac{\pi R^{2}}{L}$ (12)

A $2\times 2$ mobility matrix $\mathcal{M}$ is expected to be positive
definite and symmetric (with equal off-diagonal coefficients), as assumed in
(10), by analogy with Onsager’s relations in (bulk) non-equilibrium
thermodynamics. The consequence of the symmetry is so-called
electrohydrodynamic coupling

$\dfrac{J}{-\partial_{z}p}|_{E=0}=\dfrac{U}{E}|_{-\partial_{z}p=0}.$ (13)

The mobility matrix $\mathcal{M}$ fully characterizes electrokinetic phenomena
in the nanotubes and, once its elements are known, Eq.(10) can be used to
find, without tedious calculations, the liquid flows and currents that are
generated by any combination of two applied forces.

However, it is by no means not obvious that by imposing appropriate for
hydrophobic surfaces electro-hydrodynamic boundary conditions, Eq.(10) will
necessarily hold. More precisely, the equality of off-diagonal elements
$M_{e}$ is still an assumption that should be proven for a situation when the
fluid velocity at $r=R$ satisfies [31]

$u|_{r=R}=b\left(-\partial_{r}u|_{r=R}+\dfrac{(1-\mu)eE}{2\pi\eta\ell_{B}\ell_{GC}}\right)$
(14)

The parameter $\mu$ is the fraction of immobile surface charges that can vary
from 0 for fully mobile charges to 1 in the case when all adsorbed ions are
fixed. The hydrodynamic slip length $b$ in (14) can be of the order of several
tens of nanometers [21, 22, 23, 24], but not much more. In our calculations we
use $b=10$ nm. However, our discussion will also include a larger value of
$b=100$ nm reported in experiment [24].

To prove the consistency of Eq.(14) with the Onsager relations we have to
calculate the coefficients of the mobility matrix. In doing so it is
convenient to set one of two possible driving forces to zero.

###  II.2 Coefficients of the mobility matrix $\mathcal{M}$

####  II.2.1 Pressure-driven flow

If $E=0$, Eq. (10) reduces to

$\begin{pmatrix}U\\\ J\end{pmatrix}=\begin{pmatrix}M_{h}&M_{e}\\\
M_{e}&K\end{pmatrix}\begin{pmatrix}-\partial_{z}p\\\
0\end{pmatrix}=-\partial_{z}p\begin{pmatrix}M_{h}\\\ M_{e},\end{pmatrix}$ (15)

i.e. $U=-M_{h}\partial_{z}p$ (Poiseuille’s law), and $J=-M_{e}\partial_{z}p$.
In other words, an applied pressure gradient induces not only a hydrodynamic
flow, but also an electric current, which is traditionally termed a streaming
current.

Local mobilities can be obtained by integrating Eq. (4) without an electric
body force. The boundary condition (14) reduces to a classical hydrodynamic
slip boundary condition, and second condition is naturally
$\partial_{r}u(r)|_{r=0}=0.$ This gives for a velocity of the pressure-driven
flow

$u=-\dfrac{R^{2}}{4\eta}\left[1-\dfrac{r^{2}}{R^{2}}+\dfrac{2b}{R}\right]\partial_{z}p$
(16)

The local and mean hydrodynamic mobilities are, therefore,

$m_{h}=\dfrac{R^{2}}{4\eta}\left[1-\dfrac{r^{2}}{R^{2}}+\dfrac{2b}{R}\right],\,M_{h}=\dfrac{R^{2}}{8\eta}\left(1+\dfrac{4b}{R}\right)$
(17)

The local densities of current due to diffuse ions are $j_{\pm}=\pm en_{\pm}u$
with $u$ given by (16). Consequently,

$\overline{j_{\pm}}=\pm\dfrac{2en_{\infty}}{R^{2}}\int_{0}^{R}re^{\mp\phi}udr$
(18)

Summing up these functions and using the Poisson-Boltzmann equation (5), we
conclude that the contribution of the diffuse ions to the density of current
can be obtained by taking the integral

$\overline{j}=-\dfrac{4en_{\infty}\lambda_{D}^{2}}{R^{2}}\int_{0}^{R}ru\nabla^{2}\phi
dr$ (19)

Performing (twice) standard integration by parts we then obtain

$\overline{j}=-\dfrac{e}{4\pi\eta\ell_{B}}\partial_{z}p\left[\overline{\phi}-\phi_{s}-\dfrac{2b}{\ell_{GC}}\right]$
(20)

If the adsorbed mobile ions are transferred along the wall with the slip
velocity of liquid $u(R)=-\dfrac{bR}{2\eta}\partial_{z}p$, then they generate
the surface current density

$j_{\sigma}=\sigma(1-\mu)u(R)=-\dfrac{e(1-\mu)bR}{4\pi\eta\ell_{GC}\ell_{B}}\partial_{z}p,$
(21)

whence

$J_{\sigma}=-\dfrac{e(1-\mu)b}{2\pi\eta\ell_{GC}\ell_{B}}\partial_{z}p$ (22)

Substituting Eqs.(20) and (22) into (11) we derive for the mean streaming
current density

$J=-\dfrac{e}{4\pi\eta\ell_{B}}\left(\overline{\phi}-\phi_{s}-\dfrac{2\mu
b}{\ell_{GC}}\right)\partial_{z}p$ (23)

The corresponding mean electro-osmotic mobility is

$M_{e}=\dfrac{e}{4\pi\eta\ell_{B}}\left(\overline{\phi}-\phi_{s}-\dfrac{2\mu
b}{\ell_{GC}}\right)=-\dfrac{e}{4\pi\eta\ell_{B}}\zeta$ (24)

The new dimensionless parameter $\zeta$ introduced above can be termed a
(dimensionless) electro-hydrodynamic or zeta potential

$\zeta=\zeta_{0}+\Delta\zeta,$ (25)

where

$\zeta_{0}=\phi_{s}-\overline{\phi}$ (26)

is expected for a hydrophilic nanotube and associated with the (sensitive to
amount of added salt) electrostatic contribution, and

$\Delta\zeta=\dfrac{2\mu b}{\ell_{GC}}$ (27)

is associated with the slip-driven contribution that does not depend on the
salt concentration.

Note that if we postulate the no-slip boundary condition ($b=0$) and assume
that the capillary is infinitely thick (which is equivalent to
$\overline{\phi}\to 0$), Eq. (25) predicts that $\zeta$ must be equal to the
surface potential. However, in the case of slippery nanotubes the situation is
more complicated and $\zeta$ does not solely reflect $\phi_{s}$, but also
depends on the finite average potential in the nanotube and on the charged
wall slippage properties. We shall return to this issue later.

####  II.2.2 Electro-osmotic flow

If $\partial_{z}p=0$, one can reduce Eq. (10) to

$\begin{pmatrix}U\\\ J\end{pmatrix}=\begin{pmatrix}M_{h}&M_{e}\\\
M_{e}&K\end{pmatrix}\begin{pmatrix}0\\\
E\end{pmatrix}=E\begin{pmatrix}M_{e}\\\ K\end{pmatrix},$ (28)

which indicates that an applied electric field $E$ induces both an electro-
osmotic flow with the average velocity $U=M_{e}E$ and an electric current of
mean density $J=KE$ (Ohm’s law). This current is referred below to as a
conductivity current.

The electro-osmotic velocity is obtained by integrating Eq. (4) with
prescribed boundary conditions (of symmetry and Eq.(14))

$u=\dfrac{eE}{4\pi\eta\ell_{B}}\left[\phi-\phi_{s}-\dfrac{2\mu
b}{\ell_{GC}}\right],$ (29)

whence the local electro-osmotic mobility is

$m_{e}=\dfrac{e}{4\pi\eta\ell_{B}}\left[\phi-\phi_{s}-\dfrac{2\mu
b}{\ell_{GC}}\right]$ (30)

and consequently, $M_{e}$ is again expressed by Eq.(24). This result provides
a proof of Onsager relations (10) if boundary condition (14) are applied. We
stress, however, that to satisfy the Onsager relations the condition (21) for
the surface current density should necessary supplement Eq.(14).

The mean density of the emerging conductivity current is given by Eq.(11), but
besides a convective transfer with the fluid velocity that is now described by
Eq.(29), ions migrate under an applied field, i.e. they also move relative to
a fluid.

The velocity of diffuse ions is thus $\mathcal{U}_{\pm}=u\pm m_{\pm}E$, where
$u$ is the velocity of the electro-osmotic flow given by (29) and $m_{\pm}$ is
the (electrophoretic) mobility of ions. To keep calculations as transparent as
possible we set $m_{\pm}=m=e/(6\pi\eta\mathcal{R})$, where $\mathcal{R}$ is
the hydrodynamic radius of both cations and anions (in calculations below we
will use $\mathcal{R}=0.3$ nm). In such a definition $m$ is treated as equal
to the ion mobility at zero ionic strength and we neglect its the possible
(weak with our parameters) reduction on increasing $n_{\infty}$ (see Appendix
A). If so,

$\overline{j_{\pm}}=\pm\dfrac{2en_{\infty}}{R^{2}}\int_{0}^{R}e^{\mp\phi}(u\pm
m_{\pm}E)rdr$ (31)

and consequently

$\overline{j}=\dfrac{4en_{\infty}}{R^{2}}\int_{0}^{R}r\left(-u\sinh{\phi}+\dfrac{eE}{6\pi\eta\mathcal{R}}\cosh\phi\right)dr,$
(32)

which is equivalent to

$\overline{j}=K_{\infty}E\left[-\dfrac{3\lambda_{D}^{2}\mathcal{R}}{\ell_{B}R^{2}}\int_{0}^{R}m_{e}\nabla^{2}\phi
rdr+\overline{\cosh\phi}\right],$ (33)

where

$K_{\infty}=\dfrac{e^{2}n_{\infty}}{3\pi\eta\mathcal{R}},$ (34)

is the conductivity of the bulk electrolyte solution.

Substituting expression (30) for $m_{e}$ and integrating twice by parts we
obtain

$\overline{j}=K_{\infty}E\left[\overline{\cosh{\phi}}+\dfrac{3\lambda_{D}^{2}\mathcal{R}\overline{(\phi^{\prime})^{2}}}{2\ell_{B}}+\dfrac{12\mu
b\lambda_{D}^{2}\mathcal{R}}{R\ell_{GC}^{2}\ell_{B}}\right]$ (35)

The adsorbed mobile ions generate the density of current

$\displaystyle j_{\sigma}$ $\displaystyle=$
$\displaystyle\sigma(1-\mu)[u(R)+mE]$ (36) $\displaystyle=$
$\displaystyle\dfrac{(1-\mu)e}{2\pi\ell_{GC}\ell_{B}}\left(-\dfrac{eE}{2\pi\eta\ell_{B}}\dfrac{\mu
b}{\ell_{GC}}+\frac{eE}{6\pi\eta\mathcal{R}}\right),$

where we have substituted $u(R)=-\dfrac{eE\mu b}{2\pi\eta\ell_{B}\ell_{GC}}$.
Standard manipulations lead to

$J_{\sigma}=K_{\infty}E\left(\dfrac{4\lambda_{D}^{2}(1-\mu)}{\ell_{GC}}-\dfrac{12\mathcal{R}(1-\mu)\lambda_{D}^{2}\mu
b}{\ell_{GC}^{2}\ell_{B}}\right)$ (37)

Summing up Eqs.(35) and (37) we find the mean density of the conductivity
current $J$, and then by dividing it by $E$ obtain the expression for the mean
conductivity

$K=K_{\infty}\left[\overline{\cosh{\phi}}+\dfrac{3\lambda_{D}^{2}\mathcal{R}\overline{(\phi^{\prime})^{2}}}{2\ell_{B}}+\dfrac{4\ell_{Du}}{R}\left(\dfrac{3\mathcal{R}\mu^{2}b}{\ell_{GC}\ell_{B}}+1-\mu\right)\right],$
(38)

where we use another electrostatic length scale

$\ell_{Du}=\frac{\lambda_{D}^{2}}{\ell_{GC}}$ (39)

termed the Dukhin length. Note that $\ell_{Du}$ can be larger than any
conceivable Debye length. For example, the values of $\lambda_{D}=50$ nm and
$\ell_{GC}=2$ nm lead to $\ell_{Du}=1.25$ $\mu$m, which for a nanotube of
$R=10$ nm gives $\ell_{Du}/R=125$.

It is convenient to divide the mean conductivity of the channel given by (38)
into a “no-slip” conductivity $K_{0}$ expected for hydrophilic channels, and
slip-driven contribution $\Delta K$:

$K=K_{0}+\Delta K,$ (40)

where

$K_{0}=K_{\infty}\displaystyle\left[\dfrac{3\lambda_{D}^{2}\mathcal{R}\overline{(\phi^{\prime})^{2}}}{2\ell_{B}}+\overline{\cosh{\phi}}\right],$
(41)

and

$\Delta
K=K_{\infty}\frac{4\ell_{Du}}{R}\left[\frac{3\mu^{2}b}{\ell_{GC}}\frac{\mathcal{R}}{\ell_{B}}+1-\mu\right].$
(42)

Equation (41) is equivalent to the formula for $K_{0}$ of a plate-parallel
channel [35]. However, now $\overline{(\phi^{\prime})^{2}}$ and
$\overline{\cosh\phi}$ should be calculated for a cylindrical geometry. The
form of Eq.(42) is identical to that for $\Delta K$ of a plate-parallel
channel [35]. The only difference is that Eq.(42) includes the nanotube radius
$R$ instead of the channel thickness.

##  III Approximate solutions

In the previous section we have proven the Onsager relations and obtained the
general equations for the mean electro-osmotic mobility and the mean
conductivity of the nanotube. In order to employ them a detailed information
concerning mean electrostatic parameters, such as $\overline{\phi}$,
$\overline{\cosh\phi}$, etc, is required. The derivation of equations that
determine them is given in Appendix B.

###  III.1 Electro-osmotic mobility and zeta potential

From Eq. (24) it follows that to calculate the electro-osmotic mobility it is
enough to find the zeta potential, which incorporates all geometry and
electrostatic parameters of the problem.

For a thick cylinder, Eq.(58) for $\phi_{s}$ of isolated surface remains
accurate. Substituting this equation into (26) and making the additional
assumption that $\overline{\phi}\simeq 0$ one obtains

$\zeta_{0}\simeq 2\operatorname{arsinh}(\frac{\lambda_{D}}{\ell_{GC}})$ (43)

An approximate $\zeta_{0}$ given by Eq.(43) may slightly overestimate the
exact value since in the cylinder of finite $R$ the average potential does not
vanish in the general case (see Appendix B). For weakly charged surface
Eq.(43) reduces to

$\zeta_{0}\simeq\dfrac{2\lambda_{D}}{\ell_{GC}}$ (44)

All together, $\zeta_{0}$ of a thick cylinder always depends on salt
concentration, and is approximately equal to $\phi_{s}$.

In the thin-channel regime, substituting (67) and (68) into Eq.(26) we derive

$\zeta_{0}\simeq\dfrac{R}{2\ell_{GC}}$ (45)

Thus, $\zeta_{0}$ of a thin nanotube is not equal to $\phi_{s}$ given by (67)
and depends only on the surface charge and the nanotube radius. Importantly,
the zeta potential of a thin hydrophilic cylinder does not depend on the salt
concentration, although $\phi_{s}$ given by (67) reduces on increasing
$c_{\infty}$.

The additional slip-driven contribution $\Delta\zeta$ in Eq.(25) does not
depend on salt and is of the same form for a cylinder of any thickness and is
given by Eq.(27). It is instructive to calculate the relative enhancement of
zeta potential $\zeta/\zeta_{0}=1+\Delta\zeta/\zeta_{0}$ due to slippage. Say,
straightforward calculations give for a thick cylinder

$\dfrac{\zeta}{\zeta_{0}}\simeq 1+\dfrac{\mu
b}{\ell_{GC}\operatorname{arsinh}(\dfrac{\lambda_{D}}{\ell_{GC}})}$ (46)

When the nanotube is weakly charged, Eq. (46) reduces to

$\dfrac{\zeta}{\zeta_{0}}\simeq 1+\dfrac{\mu b}{\lambda_{D}},$ (47)

which indicates that an enhancement of zeta potential can become very large at
high salt.

For a thin cylinder

$\dfrac{\zeta}{\zeta_{0}}\simeq 1+\dfrac{4\mu b}{R}$ (48)

is defined only by the ratio $\mu b/R$ and does not depend on salt. To give an
idea on possible zeta potential enhancement, with $\mu b=100$ nm and $R=10$
nm, we get ca. 40 times amplification!

###  III.2 Conductivity

The substitution of Eqs.(63) and (64) into Eq. (41) allows one to immediately
obtain an expression for electrical conductivity in the thick hydrophilic
cylinder

$K_{0}\simeq
K_{\infty}\left[1+\dfrac{4\lambda_{D}}{R}\left(\sqrt{1+\dfrac{\ell_{Du}}{\ell_{GC}}}-1\right)\left(1+\dfrac{3\mathcal{R}}{\ell_{B}}\right)\right].$
(49)

In the thin-channel regime, $\overline{\cosh{\phi}}$ and
$\overline{(\phi^{\prime})^{2}}$ are given by Eqs.(70) and (69), respectively.
Substituting them to (41) we derive

$K_{0}\simeq
K_{\infty}\left[\sqrt{1+\left(\dfrac{4\ell_{Du}}{R}\right)^{2}}-\dfrac{2\ell_{Du}}{\ell_{GC}}\left(1-\dfrac{3\mathcal{R}}{2\ell_{B}}\right)\right],$
(50)

where the first term is of the leading-order. The significant deviations from
the bulk conductivity are expected when $\ell_{Du}/R\geq 1$. In this case
$K_{0}$ does not depend on $c_{\infty}$:

$K_{0}\simeq K_{\infty}\dfrac{4\ell_{Du}}{R}$ (51)

In other words, the salt dependence of $K_{0}$ should demonstrate a
conductivity plateau in dilute solutions.

In the case of hydrophobic channels the conductivity can be obtained from Eq.
(40), i.e. by summing up the corresponding $K_{0}$ and $\Delta K$ given by
(42). The conductivity amplification due to slippage, $K/K_{0}=1+\Delta
K/K_{0}$, can then be easily found. For example, if $\ell_{Du}/R\geq 1$ and
$\lambda_{D}/\ell_{GC}$ is small enough, i.e. in the thin channel regime, it
follows from Eq. (51) that

$\dfrac{K}{K_{0}}\simeq\frac{3\mu^{2}b}{\ell_{GC}}\frac{\mathcal{R}}{\ell_{B}}+2-\mu,$
(52)

indicating that the enhancement compared to a hydrophilic nanotube can be very
large, a few tens of times, provided $\ell_{GC}$ is small (highly charged
surfaces) and adsorbed charges are immobile ($\mu=1$). Interestingly,
$K/K_{0}=2$ when $\mu=0$, i.e. with fully mobile surface charges the mean
conductivity remains enhanced compared to that in hydrophilic nanotube. We
recall that in this situation the electro-osmotic flow is the same as in a
hydrophilic nanotube.

##  IV Results and discussion

In order to assess the validity of the above analysis, we perform a numerical
resolution of Eq.(5) complemented by the boundary condition (7) following the
numerical approach developed by Bader and Asche. Once $\phi$ and corresponding
average functions are found numerically (see Appendix A), the streaming and
conductivity currents, zeta potential and mean conductivity can be obtained
using the expressions derived in Sec. II and Sec. III. All results in this
Section are obtained for nanotubes of $R=10$ nm.

###  IV.1 Streaming current

Figure 2 shows the streaming current as a function of applied pressure
gradient. The calculations are made for several hydrophobic nanotubes of the
same radius $R=10$ nm, surface charge and slip length, but different $\mu$,
computed for two concentrations of salt, $c_{\infty}=10^{-5}$ mol/L and
$c_{\infty}=10^{-1}$ mol/L which corresponds to $\lambda_{D}\simeq 100$ nm and
1 nm, i.e. to the regimes of thin and thick cylinder. The fixed $\ell_{GC}=5$
nm corresponds to a surface charge density $\sigma\simeq 7.3$ mC$/$m2. The
response is always linear, and the slope of the straight lines is invoked to
find the electro-osmotic mobility $M_{e}$ that depends on $c_{\infty}$. The
computed data show that for nanotubes of the same $\mu$ a larger $J$ is
observed at lower concentration, but a four orders of magnitude increase in
$c_{\infty}$ reduce the streaming current only slightly. However, on reducing
$\mu$ from 0.75 down to $0.25$ the streaming current is significantly
suppressed. Once $M_{e}$ is known, we can determine the value of $\zeta$.
Calculations from Eq. (24) show that the numerical results (from top to
bottom) in Fig. 2 correspond to $\zeta\simeq 4.7,4.2$ and 2.2. Also included
in Fig. 2 are the streaming currents calculated analytically using $\zeta$
given by (25) with $\zeta_{0}$ obtained for the thin and thick cylinder limits
(Eqs.(43) and (45)). We see that with these parameters Eq. (43) provides an
excellent fit to numerical results, but Eq. (45) slightly overestimates the
results. It must be remembered that this is a first-order calculation only,
and given the simplifications made we do not expect it to be very accurate.

Figure 2: Streaming current density $J$ vs. pressure drop $\partial_{z}p$
computed for hydrophobic cylinders of $R=10$ nm, $b=10$ nm and $\ell_{GC}=5$
nm. The solid line shows numerical results obtained using $\mu=0.75$ and
$c_{\infty}=10^{-5}$ mol/L. The dashed line is computed for $\mu=0.75$ and
$c_{\infty}=10^{-1}$ mol/L. The dash-dotted line is computed for
$c_{\infty}=10^{-5}$ mol/L and $\mu=0.25$. Circles and squares show
calculations using $\zeta_{0}$ given by Eqs.(43) and (45), respectively.
Figure 3: Zeta potential as a function of $c_{\infty}$ computed for
cylindrical channels of $R=10$ nm, $\ell_{GC}=5$ nm, $b=10$ nm using $\mu=0.5$
(solid curve) and 0 (dashed curve). The surface potential $\phi_{s}$ is shown
by dash-dotted curve. Filled circles show $\zeta$ calculated using $\zeta_{0}$
predicted by Eq.(43), open circles are calculated using $\zeta_{0}$ given by
Eq.(62). Squares are obtained using $\zeta_{0}$ calculated from Eq.(45).

We now turn to zeta potential of nanotubes. The main issue we address is how
to enhance $\zeta$ by generating a slip velocity at the surface and by tuning
the salt concentration. Figure 3 includes $\zeta$ computed as a function of
$c_{\infty}$. Note that varying $c_{\infty}$ from $10^{-6}$ to $10^{-1}$ mol/L
is equivalent to the range of $R/\lambda_{D}$ from 0.032 to 10.3. In this
calculations we keep $\ell_{GC}=5$ nm fixed, so that $\phi_{s}$ varies with
salt, but the ratio $b/\ell_{GC}=2$ remains constant, i.e. independent on
$c_{\infty}$. The computed $\phi_{s}$ is also included in Fig. 3. It can be
seen that on increasing $c_{\infty}$ it reduces, but never becomes small. The
calculations of $\zeta$ are made using $\mu=0.5$ and 0. As discussed above, in
the latter case of fully mobile surface charges $\zeta=\zeta_{0}$, although
the hydrodynamic slip length is finite. For the thin nanotubes $\zeta$ does
not depend on salt and takes its maximal (constant) value. When nanotubes
become thick, $\zeta$ begins to decrease with $c_{\infty}$. It can be seen
that when $\mu=0$ the surface potential is always larger than
$\zeta=\zeta_{0}$, but at $\mu=0.5$ the zeta potential is significantly
amplified by hydrodynamic slippage and in the sufficiently concentrated
solutions $\zeta$ dramatically exceeds $\phi_{s}$. The numerical data are
compared with the analytical calculations in which $\zeta_{0}$ is evaluated
using Eqs.(43) (thick nanotubes, $c_{\infty}\simeq 10^{-1}$ mol/L) or (45)
(thin nanotubes, $c_{\infty}\leq 10^{-2}$ mol/L). The fits are quite good, but
there is some discrepancy. The analytical equations overestimate the exact
numerical results, which is the consequence of an assumption
$\overline{\phi}\simeq 0$ (see Sec. III.1). Similar fits for thick nanotubes
made using a next order approximation, Eq.(62), would make an improvement to
the fit, but slightly underestimate the numerical results.

The amplification of the zeta potential relative to $\zeta_{0}$ is illustrated
in Fig. 4, where the data are reproduced from Fig. 3. With these parameters
$\zeta$ of the hydrophobic channel is amplified in ca. 3.5 times in dilute
solutions, where $\lambda_{D}\geq R$. In more concentrated solutions (of small
$\lambda_{D}$) $\zeta/\zeta_{0}$ begins to rapidly augment with $c_{\infty}$
to larger values. It is instructive to compare these results with a
calculation of $\zeta/\phi_{s}$, also included in Fig. 4. This ratio increases
with $c_{\infty}$ strictly monotonically and is always lower than
$\zeta/\zeta_{0}$. When $c_{\infty}\leq 10^{-3}$ mol/L, $\zeta/\phi_{s}$ is
smaller than unity, but then $\zeta$ exceeds $\phi_{s}$. This is especially
well seen in Fig. 3. Also included in Fig. 4 are the theoretical results for
$\zeta/\zeta_{0}$ obtained from Eqs.(47) and (48). The theoretical predictions
provide a good qualitative agreement with the numerical data, but show smaller
amplification of the zeta potential. An obvious explanation for this
discrepancy is the neglected average potential of the nanotube. Indeed,
Eq.(62) that takes into account a non-zero $\overline{\phi}$ for the high
concentration branch in Fig. 4 provides a better match to the numerical data.
If similar fits are made to a $\zeta/\phi_{s}$ curve, as shown in Fig. 4, it
is found that the agreement of analytical theory that invokes Eqs.(58) and
(67) for $\phi_{s}$ with numerical results is very good.

Figure 4: $\zeta/\zeta_{0}$ (solid curve) and $\zeta/\phi_{s}$ (dashed curve)
computed as a function of $c_{\infty}$ for cylindrical channels of $R=10$ nm,
$\ell_{GC}=5$ nm, $b=10$ nm, and $\mu=0.5$. Filled circles and squares are
obtained from (47) and (48), respectively. Filled triangles show
$\zeta/\zeta_{0}$ calculated using $\zeta_{0}$ given by (62). Filled circles
and squares are calculations of $\zeta/\phi_{s}$ with $\phi_{s}$ obtained from
(58) and (67).

###  IV.2 Electrical conductivity

Figure 5: Conductivity current density $J$ vs. electric field $E$ computed for
the same nanotubes and $c_{\infty}$ as in Fig. 2. Symbols show $J$ calculated
using Ohm’s law with $K$ given by Eq.(40) and $\Delta K$ determined from (42).
The circles correspond to $K_{0}$ calculated from (49). The squares are
calculations using $K_{0}$ given by (50).

Next, we turn to the mean conductivity of the channel $K$. Figure 5 shows a
current-voltage response ($J-E$) computed for the same nanotubes and
electrolyte concentrations as in Fig. 2. The numerical examples show that $J$
is largest for $c_{\infty}=10^{-1}$ mol/L. The mean electric current density
computed using $c_{\infty}=10^{-5}$ mol/L is ca. an order of magnitude
smaller. The slope of the $J-E$ straight line is invoked to find the
conductivity $K$ at given $c_{\infty}$. Therefore, we conclude that numerical
data at $c_{\infty}=10^{-5}$ mol/L shows conductivity that is an order of
magnitude smaller than this computed using $c_{\infty}=10^{-1}$ mol/L.
Meantime, the bulk conductivity is four order of magnitude smaller as follows
from Eq.(34). These numerical data are compared with the above theoretical
results. The mean conductivity of the nanotube is calculated from Eq.(40). To
calculate $K_{0}$ in this equation in the case of $c_{\infty}=10^{-1}$ mol/L
we use Eq.(49) since $\lambda_{D}/R$ is small. The calculation of $K_{0}$ at
$c_{\infty}=10^{-5}$, where $\lambda_{D}/R$ is large, are performed using
(50). For both concentrations to obtain $\Delta K$ we, of course, employ
Eq.(42). Once $K$ is known, $J$ can be found employing Ohm’s law. Theoretical
results are included in Fig. 5. It can be seen that the numerical data sets
are very well fitted by analytical equations.

Figure 6: $K_{0}/K_{\infty}$ as a function of $c_{\infty}$ computed for
nanotubes of $R=10$ nm, using $\ell_{GC}=1$ and 50 nm (from top to bottom).
Circles show calculations using $K_{0}$ calculated from Eq.(49). Filled and
open squares are obtained using $K_{0}$ from Eqs.(50) and (51). The big
triangles mark the points of $\ell_{Du}/{R}=1$.

To examine the significance of deviations from the bulk conductivity more
closely, in Fig. 6 we plot $K_{0}/K_{\infty}$ as a function of $c_{\infty}$.
The calculations are made using $\ell_{GC}=1$ and 50 nm. The conductivity of
hydrophilic nanotubes, $K_{0}$, can be very large (nearly five orders of
magnitude larger than $K_{\infty}$) if it is highly charged ($\ell_{GC}=1$
nm), but even for $\ell_{GC}=50$ nm we observe ca. three orders of magnitude
enhancement, provided that an electrolyte solution is extremely dilute. On
increasing $c_{\infty}$ (reducing $\lambda_{D}$ and $\ell_{Du}$),
$K_{0}/K_{\infty}$ first decays linearly in this log-log plot indicating that
$K_{0}/K_{\infty}\propto c_{\infty}^{-1}$. However, on increasing $c_{\infty}$
further $K_{0}/K_{\infty}$ begins to reduce weaker (down to 1). We have marked
with triangles the points of $\ell_{Du}/{R}=1$ and one can see that this
branch for both curves occurs in the vicinity of these points. The upper curve
that corresponds to a highly charged cylinder is well fitted using $K_{0}$
given by Eq.(49) indicating that the nanotube effectively behaves as thick
even when $\lambda_{D}/R$ is large. The curve for the weakly charged nanotube
is well described using $K_{0}$ calculated from Eq.(50) when $c_{\infty}$ is
below $10^{-4}$ mol/L. The use of (51) makes practically no difference to the
fit, as expected. At concentrations $c_{\infty}\geq 10^{-3}$ mol/L the weakly
charged nanotube is effectively thick and the numerical data are well fitted
if $K_{0}$ given by (49) is employed.

The mean conductivity of the nanotubes can be amplified by the slippage
effect. This is illustrated in Fig. 7, where the amplification of the mean
conductivity is calculated using $\mu=0.5$ (a half of surface charges is
mobile) and a moderate slip length $b=10$ nm. For these examples we use
$\ell_{GC}=1$ and 50 nm, the same as in Fig. 6. It can be seen, that for low
concentrations ($\ell_{Du}/{R}\geq 1$) the conductivity amplification due to
slip, $K/K_{0}$, is independent on salt and takes its largest value. For a
highly charged nanotube ($\ell_{GC}=1$ nm) the slippage increase the
conductivity by a factor of ca. 3, but for a weakly charged cylinder the
enhancement is low with these papameters. When $\ell_{Du}/{R}$ becomes smaller
than unity, the value of $K/K_{0}$ decreases with salt. It is well seen that
for a curve calculated using $\ell_{GC}=50$ nm $K\simeq K_{0}$ when
$c_{\infty}\geq 10^{-2}$ mol/L. In other words, there is no conductivity
amplification due to slippage. However, a highly charged nanotube
($\ell_{GC}=1$ nm) shows the enhancement of the mean conductivity in ca. 2
times even when $c_{\infty}=10^{-1}$ mol/L. The results of theoretical
calculations are also shown in Fig. 7. The theoretical data for $K_{0}$ of
Fig. 6 are reproduced and summed up with $\Delta K$ given by (42) to obtain
$K$. Calculated values of $K/K_{0}$ fit very well the numerical curves. We
have also verified Eq.(52). As predicted, it represents a sensible
approximation for $K/K_{0}$ if $\ell_{Du}/R\geq 1$ and
$\lambda_{D}/\ell_{GC}\leq 1$. Indeed, the plateau branch of the numerical
curve obtained using $\ell_{GC}=50$ nm is well fitted by Eq.(52).

Figure 7: The conductivity amplification $K/K_{0}$ as a function of
$c_{\infty}$ obtained for the same nanotubes as in Fig. 6, but with $b=10$ nm,
and $\mu=0.5$. Filled symbols are obtained using $K_{0}$ from Fig. 6 and $K$
calculated from (40). Open squares show calculations from Eq.(52). The big
triangles mark the points of $\ell_{Du}/{R}=1$.

##  V Conclusion

We have presented a theory describing the transport of ions in hydrophobic
nanotubes with the constant surface charge density. The electro-hydrodynamic
boundary condition [31] is imposed at the hydrophobic wall, which assumes that
the surface demonstates a hydrodynamic slippage and some portion of the
adsorbed surface charges can migrate relative to fluid by reacting to the
applied electric field (but not to the hydrodynamic tangential stress).
Numerical solutions are presented and fully validate our analysis. These
results are directly relevant for enhanced streaming and conductivity currents
in carbon and boron nitride nanotubes, which are currently the area of very
active research, as well as for conventional nanoporous membranes.

The main results of our work can be summarized as follows. We have proven that
when the electro-hydrodynamic boundary condition is applied the Onsager
relations hold provided the adsorbed surface ions are transferred in a
pressure-driven flow with the slip velocity of liquid. Namely, we have derived
general expressions for elements of the $2\times 2$ mobility matrix, Eq.(10),
and demonstrated that the off-diagonal coefficients are equal if the electro-
hydrodynamic boundary condition, Eq.(14), is imposed and provided Eq.(21) is
valid. These expressions include some mean electrostatic functions, which we
have calculated analytically for two specific regimes (of thin and thick
nanotubes). Importantly, these regimes are defined not by just
$\lambda_{D}/R$, but also controlled by the value of $\lambda_{D}/\ell_{GC}$,
i.e. effective charge of the walls. We have then derived simple analytical
approximations for the electro-osmotic mobility and mean conductivity in these
two regimes. Our results show that qualitative features of the electrosmotic
mobility and conductivity curves for cylinders are the same as for slits, but
there is some important quantitative difference due to different expressions
for electrostatic functions. We have also given a novel interpretation of the
zeta potential of nanotubes.

Our results open strategies to tune the ion transport in nanotubes via a
modification of their walls and, vice versa, to probe surface properties by
measuring the streaming or conductivity currents. Our quantitative results can
be improved by performing more accurate calculations of the central (axis)
potential. This remains a challenging mathematical problem and a subject of
future research. Other fruitful directions would be to extend the results for
so-called charge regulation surfaces and for systems, where both pressure drop
and electric field are applied simultaneously. The latter calculations are now
straightforward and can be done by using the proven Onsager relations.

###### Acknowledgements.

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

## DATA AVAILABILITY

The data that support the findings of this study are available within the
article.

##  Appendix A Mobility of ions in dilute 1:1 electrolyte solutions

In general case, the electrophoretic mobility of ions reduces with the
concentration of salt. If $\mathcal{R}\ll\lambda_{D}$, the mobilities of
monovalent ions can be approximately described by the Hückel formula [45]

$m\simeq\dfrac{e}{6\pi\eta\mathcal{R}}\left(1-\dfrac{\mathcal{R}}{\lambda_{D}}\right)$
(53)

Using (6) this can be rewritten as

$m\simeq\dfrac{e}{6\pi\eta\mathcal{R}}\left(1-\dfrac{\mathcal{R}\sqrt{c_{\infty}}}{0.305}\right)$
(54)

This expression shows that the mobility at infinite dilution represents its
upper possible limit. The deviations from this ideal mobilty in dilute
solutions do not depend on the ion radius and scale with $c_{\infty}^{1/2}$.
Thus, Eq.(54) is functionally equivalent to the Kohlrausch empirical model
[46].

It follows from (54) that at $c_{\infty}=10^{-2}$ mol/L the decrease in the
mobilities of ions of $\mathcal{R}=0.3$ nm compared to those at an infinite
dilution is less than 10%. Concequently for solutions of $c_{\infty}\leq
10^{-2}$ mol/L (where $\mathcal{R}/\lambda_{D}$ is small), this effect can
safely be neglected. At larger concentrations the analysis based on an upper
limit of ion mobilities becomes quite approximate. Nevertheless, it provides
us with some guidance. As a side note, at our largest concentration
$c_{\infty}=10^{-1}$ mol/L the use of Eq.(53) would also represent too rough
an approximation since $\mathcal{R}/\lambda_{D}\simeq 0.3$.

##  Appendix B Derivation of electrostatic equations

Expressions for the mean osmotic pressure $\overline{\cosh{\phi}}$ and for the
mean square derivative of the electrostatic potential
$\overline{(\phi^{\prime})^{2}}$, which is the measure of the electrostatic
field energy (per unit area), can be derived similar to [35]. First
integration of Eq.(5) from the axis ($r=0$) to an arbitrary $r$ gives

$\lambda_{D}^{2}\left(\dfrac{d\phi}{dr}\right)^{2}=2\cosh\phi-\frac{4}{r^{2}}\int_{0}^{r}r\cosh\phi
dr$ (55)

Applying boundary condition (7) then yields

$\overline{\cosh\phi}=\cosh\phi_{s}-\frac{2\ell_{Du}}{\ell_{GC}}.$ (56)

The expression (56) is exact and valid for any channel thickness and surface
charge/potential. In Fig. 8 we present numerical solutions to validate Eq.(56)
and illustrate the variation of $\overline{\cosh\phi}$ in response to
$\lambda_{D}/\ell_{GC}$ and $\lambda_{D}/R$.

Note that from (5) and (7) it follows that

$\overline{\sinh{\phi}}=\dfrac{4\ell_{Du}}{R}$ (57)

Figure 8: $\overline{\cosh\phi}$ as a function of $\lambda_{D}/\ell_{GC}$
computed for cylindrical channels of $R=10$ nm with fixed $\lambda_{D}=100$ nm
(solid curve), $\lambda_{D}=10$ (dashed curve), and $\lambda_{D}=1$ (dash-
dotted curve). The dotted line indicates $\overline{\cosh\phi}=1$. Circles
show predictions of Eq.(56). Squares are obtained using Eqs.(63) and (69).
Figure 9: Surface potential, $\phi_{s}$, as a function of
$\lambda_{D}/\ell_{GC}$ calculated numerically using $R/\lambda_{D}\simeq
0.1,$ 1, and 10 (from top to bottom). Filled and open circles show
calculations from Eqs. (58) and (67).

###  B.1 Thick channel

For a single wall or an infinitely large cylinder the relation between the
surface potential and charge is given by the Grahame equation

$\phi_{s}=2\operatorname{arsinh}\left(\dfrac{\lambda_{D}}{\ell_{GC}}\right)$
(58)

In Fig. 9 we plot $\phi_{s}$ as a function of $\lambda_{D}/\ell_{GC}$. It can
be seen that Eq.(58) remains very accurate for a thick channel and even when
$\lambda_{D}/R=O(1)$ provided $\lambda_{D}/\ell_{GC}$ is large.

Substituting (58) into (56) one obtains $\overline{\cosh\phi}=1$, which is
equivalent to $\overline{\phi}=0$ and $\overline{(\phi^{\prime})^{2}}=0$.
However, this result holds only if $R\to\infty$. The data presented in Fig. 8
show that $\overline{\cosh\phi}$ is well above unity at finite
$\lambda_{D}/\ell_{GC}$, even for a thick channel, indicating that
$\overline{\phi}\neq 0$.

The calculation of $\overline{\phi}$ represents a challenge for a cylindrical
geometry and we leave this for a future research. However, when the nanotube
is weakly charged, $\phi_{s}\leq 1$, one can linearize Eq. (5):

$\dfrac{1}{r}\dfrac{d}{dr}\left(r\dfrac{d\phi}{dr}\right)\simeq\lambda_{D}^{-2}\phi$
(59)

Integrating the above equation (59) we obtain the potential profile:

$\phi\simeq\dfrac{2\lambda_{D}}{\ell_{GC}}\dfrac{I_{0}(r/\lambda_{D})}{I_{1}(R/\lambda_{D})},$
(60)

where $I_{0}$ and $I_{1}$ are the modified Bessel functions. The average
potential is then

$\overline{\phi}\simeq\dfrac{4\ell_{Du}}{R}$ (61)

It follows from (58) that for weakly charged surfaces $\phi_{s}\simeq
2\lambda_{D}/\ell_{GC}$. Using (26) we then obtain

$\zeta_{0}\simeq\dfrac{2\lambda_{D}}{\ell_{GC}}\left(1-\dfrac{2\lambda_{D}}{R}\right)$
(62)

Figure 10: $\overline{\phi}$ as a function of $R/\lambda_{D}$ computed for
cylindrical channels of $R=10$ nm with fixed $\phi_{s}=5$ and 1 (from top to
bottom). Squares are obtained using Eq. (68). Circles show predictions of
linear theory Eq. (61).

The numerical data presented in Fig. 10 confirm the validity of Eq.(61) at
$\lambda_{D}/\ell_{GC}\leq 1$ but show smaller $\overline{\phi}$ than this
theoretical prediction when $\lambda_{D}/\ell_{GC}$ becomes larger. A
calculation of the average potential and its effect on the zeta potential in
the case of a highly charged thick nanotube remains very difficult and well
beyond the scope of this paper. Nevertheless, we see that at large
$\lambda_{D}/\ell_{GC}$ the surface potential (see Fig. 9) is much higher than
$\overline{\phi}$, so that the later can be neglected in the first-order
calculations. Clearly, given this simplification we do not expect the
calculations of the zeta potential to be very accurate.

It is naturally to assume that corrections to $\overline{\cosh\phi}$ and
$\overline{(\phi^{\prime})^{2}}$ are proportional to the ratio of the EDL
length to the channel area. Performing similar to described in [35]
calculations we obtain

$\overline{\cosh\phi}\simeq
1+\dfrac{4\lambda_{D}}{R}\left(\sqrt{1+\dfrac{\ell_{Du}}{\ell_{GC}}}-1\right),$
(63)

and

$\overline{(\phi^{\prime})^{2}}\simeq\dfrac{8}{R\lambda_{D}}\left(\sqrt{1+\dfrac{\ell_{Du}}{\ell_{GC}}}-1\right),$
(64)

Note that it follows then that these functions are related as

$\overline{\cosh\phi}\simeq
1+\dfrac{\lambda_{D}^{2}\overline{(\phi^{\prime})^{2}}}{2}.$ (65)

Calculations from (63) are shown in Fig. 8. It can be seen that Eq.(63) gives
a good match to the numerical data for thick nanotubes, and can even be used
when $R\simeq\lambda_{D}$.

###  B.2 Thin channel

In the thin channel limit, $R/\lambda_{D}\ll 1$, the distribution of a
potential in the channel is approximately given by the parabola [43]

$\phi(r)=\phi_{s}+\dfrac{\sinh\phi_{s}}{4\lambda_{D}^{2}}\left(r^{2}-R^{2}\right).$
(66)

The relation between the surface potential and adsorbed charge reads [43]:

$\phi_{s}=\mathrm{arsinh}\frac{4\ell_{Du}}{R}.$ (67)

The calculations from Eq.(67) are included in Fig. 9. It can be seen that (67)
is accurate only when $\lambda_{D}/\ell_{GC}$ is small enough.

It follows then that the average potential is given by

$\overline{\phi}\simeq\phi_{s}-\dfrac{R^{2}\sinh\phi_{s}}{8\lambda_{D}^{2}}\simeq\mathrm{arsinh}\frac{4\ell_{Du}}{R}-\dfrac{R}{2\ell_{GC}}$
(68)

The calculation from Eq.(68) are compared with numerical results in Fig. 10.
The fit is quite good for $\lambda_{D}/\ell_{GC}\leq 3$, but at larger
$\lambda_{D}/\ell_{GC}$ there is some discrepancy, and the average potential
is higher than predicted by Eq.(68).

By direct integration of Eq. (66) using (56) and (67) we can easily derive

$\overline{\cosh\phi}\simeq\sqrt{1+\left(\dfrac{4\ell_{Du}}{R}\right)^{2}}-\dfrac{2\ell_{Du}}{\ell_{GC}}$
(69)

and

$\overline{(\phi^{\prime})^{2}}\simeq\dfrac{2}{\ell_{GC}^{2}}$ (70)

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