# A theoretical understanding of ionic current through a nanochannel driven by
a viscosity gradient

Amer Alizadeh  Corresponding author<EMAIL_ADDRESS>Department of
Chemical and Petroleum Engineering, Schulich School of Engineering,University
of Calgary, AB,Canada.  Hirofumi Daiguji  Department of Mechanical
Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656,
Japan  Anne M. Benneker  Corresponding author<EMAIL_ADDRESS>Department of Chemical and Petroleum Engineering, Schulich School of
Engineering,University of Calgary, AB,Canada.

###### Abstract

It has been recently shown that a viscosity gradient could drive electrical
current through a negatively charged nanochannel (Wiener and Stein, arXiv:
1807.09106). To understand the physics underlying this phenomenon, we employed
the Maxwell-Stefan equation to obtain a relation between the flux of solvent
species and the driving forces. Our 1D model, which was derived for both ideal
and non-ideal solvents, shows that the ionic current depends on the ideality
of the solvent, though both scenarios demonstrated good agreement with
experimental data. We employed the model to understand the impact of solution
bulk ionic strength and pH on the drift of ionic species with same reservoirs
solution properties. Our modeling results unveiled the significant impact of
bulk solution properties on the drift of ions which is in agreement with the
experiments. Moreover, we have shown that the diffusion gradient along the
nanochannel contributes significantly into driving ionic species if we even
apply a small ionic concentration gradient to both reservoirs. Our modeling
results may pave the way for finding novel applications for drift of ions
toward a diffusion gradient.

††preprint: APS/123-QED

##  I Introduction

Migration of ionic species is triggered by asymmetrical system conditions such
as gradients of concentration [1, 2], pressure [3], temperature [4, 5, 6], and
electrical potential [7]. These methods of ion transport phenomena have found
numerous applications in energy harvesting [8, 9, 10, 11], water treatment
[12, 13, 14, 15], or logical part of nanofluidic chips [16, 17, 18, 4, 19].
Recently, Wiener and Stein [20] have shown that by applying a viscosity
gradient along a nanochannel one can drive an ionic current, similar to the
other asymmetrical conditions. They deduced that this ionic current is a
result of ions drifting toward the lower viscosity solvent, in which ions
encounter a higher diffusivity.

Employing a viscosity gradient to drive particles has been studied for long
time. For instance, Qiu and Mao [21] employed a viscosity gradient to separate
different size nanoparticles. In their experiments, a pure viscosity gradient
has been established by aqueous ployvinylpyrrolidone (PVP) solutions which
could have considerable viscosity differences but have almost the same
densities. They have shown that the viscosity gradient can be employed to
effectively separate nanoparticles as an alternative to the density gradient-
based methods. Hendrick de Haan and Salter [22] have theoretically shown that
a polymer can translocate into a nanopore separating solvents of different
viscosity. Their theory predicted that a pumping effect arises when the
viscosity along the nanopore changes. Their modeling results stated that the
direction of polymer translocation could be toward higher viscosity based on
Brownian dynamics or, in contrast, toward lower viscosity by Langevin
dynamics. Liebchen et al. [23] investigated the swimming of micro-particles
such as bacteria toward favorable viscosity, so-called visco-taxis. They have
theoretically shown that the body shape of the biological or synthetic
particles will create visco-taxis. Their model predicted that the shape of
these micro-particles may prevent them to drift towards the lower viscosity
region where they loose their swimming ability. Later, Datt and Elfring [24]
developed a theory for the visco-taxis of active particles. They suggested
that an effect, namely the impact of viscosity gradient on the thrust force
generated by the micro-swimmers, which was not considered by Liebchen et al.
[23] could determine the visco-taxis direction of these particles. Recently,
Qiu et al. [25] found that a viscosity gradient applied to a nanopore can
manipulate ionic rectification behavior. They reported the unexpected finding
that by increasing the viscosity gradient, the rectification ratio increases
for the forward bias applied electric field. Furthermore, Kurup and Basu [26]
studied the migration and sorting of droplets in a viscosity gradient, so-
called visco-phoresis which is introduced to a PDMS microchannel with
different inlets and outlets. They have shown that if we have a parallel
laminar stream of low and high viscosity, a droplet will migrate toward the
lower viscosity by passing through the interface of the phases. They
demonstrated that the viscosity gradient could separate the droplet by their
sizes.

Considering the above mentioned works, it is clear that different terminology
was proposed (visco-taxis, visco-phoresis) for describing a particle movement
inside a viscosity gradient. In this paper, we aim at a mechanistic
understanding of the Wiener and Stein [20] method in driving ionic species by
employing a viscosity gradient. Since we are dealing with ionic species, we
suggest visco-migration as the terminology for drifting ions in a viscosity
gradient, analogous to electro-migration in which ions migrate under the
influence of an external electric field. We will provide a theory based on the
Maxwell-Stefan diffusion equations for ideal and non-ideal fluids.
Successively, a simple equation will be developed to describe the relation of
ionic drift velocity and the ionic current. For ideal fluids, Wiener and Stein
[20] developed a simple relation to obtain the diffusion and as a result the
viscosity along the nanochannel by employing the Maxwell-Stefan equations.
Their modeling results are in good agreement with their experimental
measurements, but we extend the framework to a mechanistic approach which
compared the ideal and non-ideal fluids with increasing the deviation of the
fluid from the ideal scenario. After evaluating our model with the
experimental data, we study the impact of bulk solution properties (i.e. pH
and ionic strength) on the visco-migration ionic current. We obtained the zeta
potential of the nanochannel by utilizing an EDL model, so-called electrical
quad-layer model [27]. Last but not least, we developed a simple equation to
obtain the total ionic current when we not only apply a small ionic
concentration gradient but also we have diffusion gradient owing to the
viscosity gradient.

##  II Problem definition

Recently, Wiener and Stein [20] have experimentally shown that a nanochannel
bridging two microchannels filled with different viscosity solutions
containing dissolved ionic species will drive an ionic current, even when the
bulk ionic concentration in both fluids is equal. Fig. 1 presents a 2D
schematic illustration of the system which consists of two parallel
microchannels and a nanochannel connecting them. It is assumed that the
microchannels are continuously refreshed with glycerol and formamide solution
allowing for the assumption that the viscosity gradient over the nanochannel
is kept constant as a function of time.

In this contribution, we assumed that the mixtures of the glycerol & water and
formamide & water are produced by considering the same volume fraction of
water. We can obtain the water fraction for the mixtures by

$\displaystyle\phi_{w,i}=\frac{\ln\left(\eta_{mix,i}\right)-\ln\left(\eta_{i}\right)}{\ln\left(\eta_{w}\right)-\ln\left(\eta_{i}\right)}$
(1)

where $\mathit{\phi_{w,i}}$ denotes the water fraction in the mixture which is
defined as $\mathit{\phi_{w,i}=c_{w,i}/\left(c_{i}+c_{w,i}\right)}$ and
$\mathit{c_{w,i}}$ represents the concentration of water in this mixture. The
volume fraction of glycerol/formamide for the left solvent mixture can be
obtained as $\phi_{i}=1-\phi_{w,i}$. Given the volume fractions of water and
glycerol/formamide, one can obtain the molar concentration as

$\displaystyle C_{i}=\phi_{i}\times 10^{3}\times\frac{\nu_{i}}{M_{i}}$ (2)
$\displaystyle C_{w,i}=\phi_{w,i}\times 10^{3}\times\frac{1}{18.015}.$ (3)

In all above equations, the subscript $i$ represents the glycerol ($g$) or
formamide ($f$), $w$ stands for water, and $w,i$ for mixture of water &
glycerol/formamide. Finally, we can obtain the concentration (number of
molecules/$\text{m}^{3}$) of each component as $c_{i}=N_{A}\times 10^{3}\times
C_{i}$ where $N_{A}$ is the Avogadro number. Table 1 summarizes the parameters
that we used for this system.

Figure 1: Schematic illustration of the nanochannel under viscosity gradients
at both ends. It is assumed that the both ends of the nanochannel always
subjected to fresh solutions with identical solution properties (i.e, bulk
ionic strength and pH). The mixture of water & glycerol and water & formamide
have different viscosity, $\eta_{1}$ and $\eta_{2}$, respectively, which
results in different diffusion coefficients. The imposed diffusion gradient
will drive ionic species toward higher diffusivity with velocity
$\mathcal{V}$. Because of the nanochannel’s counter-ion selectivity
(overlapped electrical double layer), an ionic current will be measured.

##  III Theory

Generally, a system can be in thermodynamic or mechanical equilibrium. At
thermodynamic equilibrium, the entropy production of the entire system is zero
[28]. For a system at mechanical equilibrium, there is no acceleration of the
center of mass while entropy production is non-zero [29]. For our problem, we
assumed that the mechanical equilibrium is always accessible while the
thermodynamic equilibrium is not reached. Thus, the entropy production can be
defined as the production of flux and driving forces [30, 29, 31]

$\displaystyle T\sigma=J_{q}X_{q}+\sum_{i=1}^{n}\left(J_{i}X_{i}\right)$ (4)

where $\sigma$, T, $\textit{X}_{q}$, $\textit{X}_{i}$ and $J$ denote the
entropy production strength, the temperature, thermodynamic force due to a
temperature gradient, the species-driven thermodynamic forces and fluxes,
respectively. The last term on the right-hand side of Eq. (4) is responsible
for considering the total thermodynamic force owing to the flux of ionic
species as well as the solvents, wherein n denotes the number of solution’s
components (i.e., n=4 indicates two ionic species and two solvents). If we
assume isothermal conditions, we can ignore the first term on the right hand
side of Eq. (4) ($J_{q}X_{q}$). Regarding the thermodynamic force for driving
the miscible liquids and ionic species we can write [31]

$\displaystyle X_{i}=F_{i}-T\nabla\left(\frac{\mu_{i}}{T}\right)$ (5)

where $\mathit{F_{i}}$ denotes the external body force (i.e. electrical body
force) on the ionic and solvent species and $\mathit{\mu}_{i}$ represents the
chemical potential of the $\textit{i}^{th}$ species. If we extend the
thermodynamic force for iso-thermal mass transfer (Eq. 5) we will have

$\displaystyle
X_{i}=F_{i}-\left(\nabla{\mu_{i}}-\mu_{i}\frac{\nabla{T}}{T}\right).$ (6)

Here, we should point out that we assume a zero effective external body force
on the solvent molecules and ionic species. This assumption can be validated
as the hydraulic resistance of the nanochannel is sufficiently high and
balances the possible external pressure gradient. Consequently, the
thermodynamic force which drives the solvent molecules and ionic species will
be solely a function of the chemical potential gradient and could be written
as

$\displaystyle X_{i}=-\nabla{\mu_{i}}.$ (7)

In the present work, we focus on the flux of solvents resulting from the
chemical potential gradient at both ends of the nanochannel and we will
investigate the related flux of ionic species which is a result of drifting
toward diffusion gradient as well as an applied ionic concentration difference
at both ends of nanochannel (c.f. section IV). Therefore, Eq. 7 can be
expanded for solvent molecules as

$\displaystyle X_{i}=-k_{B}T\nabla{\ln{c_{i}}},$ (8)

where the index i changes from 1 to 2 to represent the glycerol & water (i=1)
and formamide & water (i=2) mixture.

In order to relate the species flux to the driving force, we start with the
Maxwell-Stefan (MS) equation. The MS equation simply relates the friction
forces between different components of a solution. In the mixture of the
electrolyte solutions with different viscosity and in isothermal and isobaric
conditions, the generalized MS equation is [30]

$\displaystyle c_{t}k_{B}T\mathbb{d}_{i}=c_{i}\nabla_{T,p}{\mu_{i}},$ (9)

where $\mathit{c_{t}k_{B}T\mathbb{d}_{i}}$ denotes the driving force on the
solvent molecules which is expressed by [30]

$\displaystyle\mathbb{d}_{i}=-\sum_{j=1,i\neq{j}}^{n}\frac{\phi_{i}\phi_{j}\left(\mathbb{u}_{i}-\mathbb{u}_{j}\right)}{D_{ij}},$
(10)

and n is the number of solvents, $\mathit{\phi_{i}}$ and $\mathit{\phi_{j}}$
are the volume fraction of the species, defined as the ratio of the component
concentration to the total concentration of the components in our system;
$\mathit{\phi=c/c_{t}}$. $\mathit{D_{ij}}$ denotes the Maxwell-Stefan
diffusion coefficient of species $\mathit{i}$ with respect to species
$\mathit{j}$. We know that the species fluxes with respect to the solvent are
defined as $\mathit{\mathbb{J}_{i}=c_{i}\mathbb{u}_{i}}$. If we introduce this
relation into Eq. 10, then we have

$\displaystyle\mathbb{d}_{i}=\sum_{j=1,i\neq{j}}^{n}\frac{\phi_{i}\mathbb{J}_{j}-\phi_{j}\mathbb{J}_{i}}{c_{t}D_{ij}}.$
(11)

where $\mathit{c_{t}}$ is the total number of solvent molecules per unit
volume. Here it is worth pointing out that $\mathit{c_{t}}$ is constant along
the nanochannel which is an essential assumption for an isobaric system. We
can non-dimensionalize Eq. 11 by introducing
$\mathit{\bar{x}=x/L},\mathit{\bar{c}=c/c_{t}},\mathit{\bar{D}_{ij}=D_{ij}/(\DJ_{i}+\DJ_{j})}$,
and $\mathit{\bar{\mathbb{J}}=\mathbb{J}/((\DJ_{i}+\DJ_{j})c_{t}/L)}$, where
$\DJ$ denotes the self diffusion coefficient, into this equation which gives
rise to

$\displaystyle\mathbb{\bar{d}}_{i}=\sum_{j=1,i\neq{j}}^{n}\frac{\phi_{i}\mathbb{\bar{J}}_{j}-\phi_{j}\mathbb{\bar{J}_{i}}}{\bar{D}_{ij}}.$
(12)

where by considering Eq. 9, we can relate $\mathit{\mathbb{\bar{d}}_{i}}$ to
gradient of volume fraction of species $\mathit{i}$ as
$\mathit{\mathbb{\bar{d}}_{i}=\sum_{j\text{=1}}^{\text{n-1}}\Gamma_{ij}\nabla\phi_{i}}$,
where $\Gamma_{ij}$ represents the activity coefficient of species i in the
the mixture and defined as [30]

$\displaystyle\Gamma_{ij}=\delta_{ij}+\phi_{i}\frac{\partial{\ln\gamma_{i}}}{\partial{\phi_{j}}}.$
(13)

For dilute gas mixtures, it is reasonable to assume the mixture as an ideal
fluid where $\gamma_{i}=1$, however for higher density fluids or dense gases,
$\gamma_{i}$ will be a function of the volume fractions. For instance, for a
binary solution, we have

$\displaystyle\ln\gamma_{1}=A\phi_{2}^{2},$ (14)

which we will use for a non-ideal mixture scenario. Hereinafter, for sake of
avoiding more symbols, we drop all over-bars. Therefore, all parameters are
non-dimensional, unless otherwise noted.

###  III.1 Binary ideal mixture

Considering an ideal binary system, Eq. 12 can be re-written as

$\displaystyle\mathcal{X}_{1}=\frac{\phi_{1}\mathbb{J}_{2}-\phi_{2}\mathbb{J}_{1}}{D_{12}},$
(15)
$\displaystyle\mathcal{X}_{2}=\frac{\phi_{2}\mathbb{J}_{1}-\phi_{1}\mathbb{J}_{2}}{D_{21}},$
(16)

where $\mathcal{X}_{i}=\nabla{\phi_{i}}$ when $\Gamma_{ij}=1$. Eqs. 15 and 16
have four unknowns ($\phi_{1},\phi_{2},\mathbb{J}_{1}$, and $\mathbb{J}_{2}$)
indicating we need two more equations to close the system of equations.
Because of the Gibbs-Duhem restriction we have
$\sum_{i=1}^{n}\mathcal{X}_{i}=0$. This gives rise to the symmetrical behavior
of MS diffusion coefficient $D_{12}=D_{21}$ according to Eqs. 15 and Eq. 16.

The isobaric condition enforces the total number of molecules along the
nanochannel to be constant which requires that the sum of the volume fractions
must be

$\displaystyle\phi_{1}(x)+\phi_{2}(x)=1.$ (17)

When introducing Eq. 17 into Eqs. 15 and 16 we have

$\displaystyle\mathcal{X}_{1}=\frac{\phi_{1}}{D_{12}}\left(\mathbb{J}_{1}+\mathbb{J}_{2}\right)-\frac{\mathbb{J}_{1}}{D_{12}},$
(18)
$\displaystyle\mathcal{X}_{2}=\frac{\phi_{2}}{D_{12}}\left(\mathbb{J}_{1}+\mathbb{J}_{2}\right)-\frac{\mathbb{J}_{2}}{D_{12}}.$
(19)

Solving Eqs. 18 and 19 for $\mathbb{J}_{1}+\mathbb{J}_{2}$ and substituting
$\mathcal{X}_{2}$ with $-\mathcal{X}_{1}$ (Eq. 20) and $\mathcal{X}_{1}$ with
$-\mathcal{X}_{2}$ (Eq. 21), we finally have

$\displaystyle\frac{1}{\phi_{1}}\left(D_{12}\mathcal{X}_{1}+\mathbb{J}_{1}\right)=\mathbb{J}_{1}+\mathbb{J}_{2}.$
(20)
$\displaystyle\frac{1}{\phi_{2}}\left(D_{12}\mathcal{X}_{2}+\mathbb{J}_{2}\right)=\mathbb{J}_{1}+\mathbb{J}_{2}.$
(21)

The final equation required to close our system can be obtained by considering
the fact that the right-hand side of Eqs. 20 and 21 is a function of both
component 1 and 2 while the left-hand side is only function of component 1 and
2, respectively. As a result, the only solution to Eq. 20 and 21 is
$\mathbb{J}_{1}+\mathbb{J}_{2}=0$, which justifies the fact that in an
$\mathit{n}$ component system, only $\mathit{n}-1$ fluxes are independent
[30]. Considering the left-hand side of Eq. 20, we have
$\mathbb{J}_{1}=-D_{12}\mathcal{X}_{1}$ which denotes the Fick’s diffusion
law. The same procedure can be applied to Eq. 21 to obtain
$\mathbb{J}_{2}=-D_{12}\mathcal{X}_{2}$.

To obtain the flux of solvents ($\mathbb{J}_{1}$ and $\mathbb{J}_{2}$) we need
to find the MS diffusion coefficient ($D_{12}$). For binary systems which do
not present a large deviation from ideality, the Maxwell-Stefan diffusion of a
mixed solution can be directly related to the intradiffusion coefficients
(Fick’s diffusion) of the both components through the relation [32]

$\displaystyle D_{12}=\phi_{1}\DJ_{2}+\phi_{2}\DJ_{1}$ (22)

where $\mathit{\DJ_{\text{1}}}$ and $\mathit{\DJ_{\text{2}}}$ denote the
diffusion of glycerol and formamide in de-ionized water and are equal to
$\DJ_{g}$ and $\DJ_{f}$ in Table 1.

According to the literature [33], Eq. 22 is suitable if no viscosity data is
available for the mixture of solutions. Another model which was proposed by
Vignes [34] is recommended in general for concentrated binary liquids as

$\displaystyle D_{12}=\DJ_{1}^{\phi_{1}}\DJ_{2}^{\phi_{2}}.$ (23)

As Eq. 22 and 23 show, the diffusion of the mixed solution is a function of
the solvents volume fractions
($\mathit{D_{12}=f(\phi_{\text{1}}(x),\phi_{\text{2}}(x))}$). In the present
study, we employ Eq. 23 to obtain the MS diffusion coefficient.

Our system can be considered as a 1D problem since we do not have any flux
perpendicular to the nanochannel walls. Therefore, according to mass
conservation for a steady state condition without any chemical reactions, we
have

$\displaystyle\frac{dJ_{1}}{dx}=0\rightarrow-\ln\left(\frac{\DJ_{1}}{\DJ_{2}}\right)\mathcal{X}_{1}^{2}=\frac{d\mathcal{X}_{1}}{dx}$
(24)

Eq. 24 can be simplified by introducing
$d\mathcal{X}_{1}/dx=\mathcal{X}_{1}d\mathcal{X}_{1}/d\phi_{1}$ and
$\lambda=\ln(\DJ_{1}/\DJ_{2})$ which gives rise to

$\displaystyle-\lambda\mathcal{X}_{1}=\frac{d\mathcal{X}_{1}}{d\phi_{1}},\mathcal{X}_{1}\neq
0.$ (25)

Eq. 25 can be solved for $\phi_{1}(x)$ as

$\displaystyle\phi_{1}(x)=\frac{1}{\lambda}\ln\left[\lambda\left(C_{1}x+C_{2}\right)\right],\lambda\neq
0$ (26) $\displaystyle BC:\phi_{1}(0)=1,\phi_{1}(1)=0,$ (27)

where the constant parameters ($C_{1}$ and $C_{2}$) can be obtained by
introducing the above boundary conditions into Eq. 26 as

$\displaystyle\begin{split}C_{1}=\frac{1-\exp(\lambda)}{\lambda},\\\
C_{2}=\frac{\exp(\lambda)}{\lambda},\end{split}$ (28)

Re-calling Eq. 17, we have $\phi_{2}(x)=1-\phi_{1}(x)$. Once we obtained
$\phi_{\text{1}}(x)$ and $\phi_{\text{2}}(x)$, we already have
$D_{\text{12}}(x)$ and, as a result, the flux of the solvents.

###  III.2 Binary non-ideal mixture

For a binary non-ideal mixture, we can write the MS equations as [30]

$\displaystyle\Gamma\mathcal{X}_{1}=\frac{\phi_{1}\mathbb{J}_{2}-\phi_{2}\mathbb{J}_{1}}{D_{12}},$
(29)
$\displaystyle\Gamma\mathcal{X}_{2}=\frac{\phi_{2}\mathbb{J}_{1}-\phi_{1}\mathbb{J}_{2}}{D_{21}},$
(30)

where based on the Gibbs-Duhem restriction we have
$\Gamma(\mathcal{X}_{1}+\mathcal{X}_{2})=0$ and isobaric conditions imposes
$\phi_{1}(x)+\phi_{2}(x)=1$. The activity coefficient for a binary mixture can
be found by considering the Gibbs excess energy as
$\mathcal{Q}=A\phi_{1}\phi_{2}$ which gives rise to [30]

$\displaystyle\Gamma=1-2A\phi_{1}(x)\phi_{2}(x),$ (31)

where $A$ is an arbitrary parameter to determine the deviation of the mixture
from ideality.

If we introduce Eq. 31 into Eqs. 29 and 30 and follow the same approach as for
the ideal mixture, we finally have

$\displaystyle\frac{1}{\phi_{1}}\left(\Gamma
D_{12}\mathcal{X}_{1}+\mathbb{J}_{1}\right)=\mathbb{J}_{1}+\mathbb{J}_{2},$
(32)

which gives rise to $\mathbb{J}_{1}=-\Gamma D_{12}\mathcal{X}_{1}$. Likewise
the approach we employed to ideal mixture, we have

$\displaystyle\frac{dJ_{1}}{d\phi_{1}}=0\rightarrow\left(\Gamma\ln(\DJ_{1})-2A(\phi_{2})\right)\mathcal{X}_{1}=-\Gamma\frac{d\mathcal{X}_{1}}{d\phi_{1}},$
(33)

where for sake of having simpler form of ordinary differential equation, we
employed the chain rule to the derivative of flux as

$\displaystyle\frac{dJ_{1}}{dx}=\frac{dJ_{1}}{d\phi_{1}}\mathcal{X}_{1}=0$
(34)

that for $\mathcal{X}_{1}\neq 0$, we have $dJ_{1}/d\phi_{1}=0$.

Eq. 33 can be solved by separation of variables and taking integral of both
sides which gives

$\displaystyle\begin{aligned} &\ln(\mathcal{X}_{1})=\\\
&-\lambda\phi_{1}-2A\int\frac{1-2\phi_{1}}{1-2A\phi_{1}\left(1-\phi_{1}\right)}d\phi_{1}+C_{1}^{\prime},\end{aligned}$
(35)

where by knowing that the integral has analytical solution as
$\int\frac{1-2\phi_{1}}{1-2A\phi_{1}\left(1-\phi_{1}\right)}d\phi_{1}=\frac{\ln\left(2A\phi_{1}^{2}-2A\phi_{1}+1\right)}{2A}$,
we re-arrange Eq. 35 by introducing $\mathcal{X}_{1}=d\phi_{1}/dx$ which gives
rise to

$\displaystyle\left(2A\phi_{1}^{2}-2A\phi_{1}+1\right)\exp\left(\lambda\phi_{1}\right)d\phi_{1}=C_{1}^{\prime}dx.$
(36)

Integrating both sides of Eq. 36, we finally have

$\displaystyle\begin{aligned}
&\frac{\mathrm{e}^{\lambda\phi_{1}(x)}}{\lambda}\left[2A(\phi_{1}(x)^{2}-\phi_{1}(x))+\frac{2A}{\lambda}(1-2\phi_{1}(x))+\right.\\\
&\left.\frac{4A}{\lambda^{2}}+1\right]=C_{1}^{\prime}x+C_{2}^{\prime},\lambda\neq
0.\end{aligned}$ (37) $\displaystyle BC:\phi_{1}(0)=1,\phi_{1}(1)=0$ (38)

Eq. 37 is a transcendental equation which has no closed-form solution for
$\phi_{1}(x)$. Therefore, we can solve it by approximating the left hand-side
by a Taylor expansion that for $\mathcal{O}(\phi_{1}(x)^{11})$ the maximum
error will be on the of order $10^{-6}$. Consequently, we can safely utilize
the Taylor expansion (around $x=0$) and solve Eq. 37 for $\phi_{1}(x)$ with
the proper boundary conditions (Eq. 38). It is worth pointing out that for
$A=0$ (ideal fluid), Eq. 37 re-covers Eq. 26.

Table 1:  Parameters employed to model the experimental measurements. Note that subscript $g$ and $f$ refer to glycerol and formamide, respectively Parameters | Amounts
---|---
$\rho$ | 999.99 (kg/$m^{3}$)
$\eta_{mix,g}$ | 3.4$\times 10^{-3}$ (Pa.s)
$\eta_{mix,f}$ |  $10^{-3}$ to $5\times 10^{-3}$ (Pa.s)
$\eta_{w}$ |  $8.9\times 10^{-4}$ (Pa.s)
$\eta_{g}$ | 0.934 (Pa.s)
$\eta_{f}$ |  $3.34\times 10^{-3}$ (Pa.s)
$M_{g}$ | 92.094 (g/mol)
$M_{f}$ | 45.04 (g/mol)
$\nu_{g}$ | 1.26 (g/$cm^{3}$)
$\nu_{f}$ | 1.13 (g/$cm^{3}$)
$\DJ_{g}=\DJ_{1}$ | 2.2$\times 10^{-12}$ ($m^{2}$/s)
$\DJ_{f}=\DJ_{2}$ | 5.5$\times 10^{-10}$ ($m^{2}$/s)
Channel length$\equiv$L | 200 ($\mu$m)
Channel Width$\equiv$W | 150 ($\mu$m)
Channel Height$\equiv$H | 50 ($\mu$m)
Vacuum permittivity$\equiv$$\epsilon_{0}$ | 8.854$\times$10-12 (Fm-1)
Water electrical permittivity$\equiv$ $\epsilon_{r}$ | 78.54
Boltzmann constant$\equiv$ $k_{b}$ | 1.38$\times$10-23 (JK-1)
Solution temperature$\equiv$T | 298.15 K
Electron charge$\equiv$ $e$ | 1.602$\times$10-19 (C)

##  IV Results and Discussion

So far, we found the volume fraction of the species along the nanochannel
(Eqs. 26 and 37) which provides the MS diffusion coefficient and species flux.
In the following sections we will first benchmark our model and, second,
investigate the impact of bulk solution pH and ionic strength on the
nanochannel’s surface charge by solving the available electrical double layer
models such as electrical-quad layer model [27]. Finally, we will compare
visco-migration and reverse electrodialysis ionic current by not only applying
a viscosity gradient but also an ionic concentration gradient.

###  IV.1 Model benchmark

We have solved Eqs. 26 and 37 to obtain the MS diffusion coefficient
($D_{12}(x)$) and successively the flux of species. As Wiener and Stein [20]
mentioned, their experimental data are consistent with the isothermal rule
[35] in stochastic displacement models. According to the Hanggi model [35], a
particle drifts toward the lower viscosity (higher diffusion) with a speed

$\displaystyle\mathcal{V}=\frac{dD_{12}(x)}{dx}$ (39)

where, as we stated above, $D_{12}$ and $x$ are non-dimensional parameters.

Non-dimensionalizing the equations in section III allowed us to figure out
that for the ideal Eq. 40 and non-ideal Eq. 41 scenarios we have

$\displaystyle\text{ideal:}\>{{J}_{1}}=\frac{-1}{\lambda}{\mathcal{V}}.$ (40)
$\displaystyle\text{non-
ideal:}\>{J}_{1}=\frac{-\Gamma}{\lambda}{\mathcal{V}}.$ (41)

Studying Eq. 41 reveals that since $\Gamma$ is a function of $x$ (Eq. 31),
therefore ${\mathcal{V}}$ for non-ideal fluids cannot be a constant amount in
contrast to what we had for ideal fluids. In other words, since the left-hand
side of Eq. 41 is constant amount (flux of species), as a result, the
multiplication of $\Gamma\>{\mathcal{V}}$ must be a constant amount. This fact
suggests that the Hanggi model (Eq. 39) can be modified for ideal and non-
ideal fluids as

$\displaystyle{\mathcal{V}}=\Gamma\frac{d{D}_{12}(x)}{d{x}}.$ (42)

Therefore, we can simply derive a general equation for the flux of ideal and
non-ideal species as function of drift velocity

$\displaystyle J_{1}=\frac{-1}{\lambda}\mathcal{V}.$ (43)

Hereinafter, we will present the non-dimensional parameters with over-bar.
Ionic species are typically moving in an isothermal domain owing to the three
transport phenomena: advection, diffusion, and electromigration [7, 36]. Since
in Wiener and Stein’s experiments [20] there is no applied external electric
field, pressure gradient, and ionic concentration gradient, we can simply
relate the drift velocity to the advection of ionic species which gives rise
to

$\displaystyle
I_{visco}=\left(\frac{\DJ_{1}+\DJ_{2}}{L}\right)\bar{\mathcal{V}}\>\rho_{ave}\>\mathcal{A}$
(44)

where $\mathcal{A}=W\>H$ represents the cross sectional area of the
nanochannel and the average net electric charge density $\rho_{ave}$ (Cm-3)
defined as

$\displaystyle\rho_{ave}=\frac{\int_{0}^{H/2}\rho_{e}dy}{H/2}.$ (45)

In Eq. 45, $\rho_{e}$ denotes the local net electric charge density which is
obtained as

$\displaystyle\rho_{e}=-\epsilon_{0}\epsilon_{r}\frac{d\psi(y)}{dy}$ (46)

where $\psi(y)$ is the internal electric potential [V] owing to the electrical
double layer effect which for a channel with similar charge on the walls can
be obtained as [36]

$\displaystyle\psi(y)=\frac{\sigma}{\epsilon_{0}\epsilon_{r}\sinh{\left(\kappa
H/2\right)}}\cosh\left(\kappa y\right).$ (47)

It is worth noting that in Eq. 47, $\sigma$ represents the measured surface
charge on the nanochannel’s wall which Wiener and Stein [20] measured for
their setup (100 mM NaCl) is $\sigma=-200$ (mC/m2). Furthermore,
$\kappa=\sqrt{\epsilon_{0}\epsilon_{r}k_{b}T/2e^{2}n_{b}}$ is the inverse of
Debye length (m-1).

Now, let us re-write Eq. 44 by substituting the drift velocity with the
species flux from Eq. 43 which reads

$\displaystyle{I}_{visco}=-\left(\frac{\DJ_{1}+\DJ_{2}}{L}\right)\>\lambda\>{\bar{J}_{1}}\>{\rho}_{ave}\>{\mathcal{A}}.$
(48)

Eq. 48 shows that the ionic current because of the visco-migration phenomenon
can be obtained by multiplying the species fluxes by $\lambda$, average net
charge density, and the cross-section of the nanochannel.

Figure 2 illustrates our ideal and non-ideal models’ prediction against the
experimental data for glycerol & water with $\eta_{1}=3.4$ [mPas] and 1.5
[mPas]. Our modeling results show that the ionic current owing to visco-
migration phenomenon is a function of the applied viscosity gradient as well
as the deviation from ideal fluid. It has been unveiled that by increasing the
$\eta_{\text{2}}$, the impact of deviation from ideality increases.

Figure 2: The ionic current $\left(I_{visco}\right)$ as a result of an applied
viscosity gradient along the fabricated nanochannel by Wiener and Stein [20]
for two different glycerol & water viscosities $(\eta_{1}=3.4$ and 1.5
[mPas]). Our ideal and non-ideal modeling results are validated with the
experimental data.

After comparing the predicted ionic current $I_{visco}$ with the experimental
measurements, we further scrutinize the distribution of species and the MS
diffusion coefficient along the nanochannel. To this end, we present the
concentration of species for ideal and non-ideal fluids under different
viscosity gradients (Fig. 3) where the symbols represent the ideal and
solid/dashed lines the non-ideal fluids. It is worth mentioning that for the
present system, the red lines and symbols represent the concentration of
glycerol & water ($c_{1}$) while the blue ones stand for formamide &
water($c_{2}$). It can be seen that for $A=-0.1$ (Fig. 3a), the ideal and non-
ideal models predict almost the same distribution of solvent species.

By increasing $A$ from $-$0.1 to $-$1, Fig. 3b demonstrates that the ideal and
non-ideal modeling results deviates, albeit rather small. As was the case for
$A=-0.1$, for $A=-1$ we can see that by increasing the viscosity difference,
an increasingly non-linear behavior of the species concentration as function
of distance is developed. Another parameter that we are interested to see the
behavior of it for different viscosity gradient and fluid ideality, is the MS
diffusion coefficient. Fig. 4 demonstrates $D_{12}$ as function of distance
$x$. It is interesting to note that $D_{12}$ shows linear behavior versus the
distance from the inlet of the nanochannel for both ideal fluid and non-ideal
fluid with $A=-0.1$ (Fig. 4a). On the other hand, more deviation from the
ideal fluid ,$A=-1$, results in non-linear behavior for $D_{12}(x)$.

(a)  (b)

Figure 3: Concentration of species 1 (red lines/symbols, glycerol & water) and
2 (blue lines/symbols, formamide+water) along the nanochannel for different
applied viscosity gradients. Symbols represent the ideal fluid ($\Gamma=1$)
and non-ideal fluids are illustrated by solid and dashed lines for (a)
$A=-0.1$ and (b) $A=-1$.

###  IV.2 Impact of bulk pH and ionic strength

In this section, we investigate the impact of bulk solution properties such as
pH and bulk ionic concentration ($n_{b}$) on the resulting current $I_{visco}$
(Eq. 48). To this end, we employ a recently developed electrical double layer,
so-called electrical-quad layer model (EQL) [27] by considering the following
chemical reactions

$\displaystyle\text{SiOH}^{+}_{2}\rightleftharpoons\text{SiOH}+\text{H}^{+},K^{a1}_{int}$
(49)
$\displaystyle\text{SiOH}\rightleftharpoons\text{SiO}^{-}+\text{H}^{+},K^{a2}_{int}$
(50)
$\displaystyle\text{SiO}^{-}+\text{M}^{+}\rightleftharpoons\text{SiOM},K^{a3}_{int}$
(51)

where M+ denotes the counter-ion (i.e. Na+), and $K^{a1}_{int}$ to
$K^{a3}_{int}$ represent the chemical equilibrium constants. Solving the
equations of the EQL model [27] predicts the $\zeta$-potential of the silica-
aqueous solution interface as function of bulk ionic strength (Fig. 5a) and
the solution pH (Fig. 5b) which were evaluated by the available experimental
data [37, 38]. Since we are dealing with the $\zeta$-potential, the electrical
potential distribution perpendicular to the nanochannel walls can be obtained
by [36]

$\displaystyle\psi\left(y\right)=\frac{\zeta\cosh\left(\kappa
y\right)}{\cosh\left(\kappa H/2\right)},$ (52)

where $\zeta$ represents the zeta potential on the nanochannel walls. Here, we
have to point out that the bulk reservoirs at both ends of the nanochannel
have the same solution properties. Fig. 5a shows the $I_{visco}$ versus the
bulk ionic strength for an NaCl solution with pH=6.5 and for two different
viscosity gradient scenario $\eta_{1}=3.4\text{(mPas)}$,
$\eta_{2}=1\text{(mPas)}$ and $\eta_{1}=3.4\text{(mPas)}$,
$\eta_{2}=5\text{(mPas)}$. Our modeling results show that by increasing the
bulk ionic concentration the absolute value of $I_{visco}$ increases to a
maximum amount and then decreases by further increasing $n_{b}$. This behavior
of the ionic current can be attributed to the competition of the EDL thickness
and $\rho_{ave}$, where by increasing $n_{b}$, the absolute value of
$\rho_{ave}$ increases and then decreases. In other words, when we increase
the bulk ionic concentration, the surface charge density at the nanochannel
walls increases [7, 27, 39]. However, for higher amount of $n_{b}$, leading to
thinner electrical double layers, the major part of the nanochannel is
electro-neutral ($\rho_{e}=0$) which does not contribute into the average net
charge density. Moreover Fig. 5a illustrates that the $I_{visco}$ has higher
sensitivity to the bulk ionic concentration when we have
$\eta_{1}=3.4\text{(mPas)}$, $\eta_{2}=1\text{(mPas)}$ compared to
$\eta_{1}=3.4\text{(mPas)}$, $\eta_{2}=5\text{(mPas)}$.

Regarding the impact of solution pH, Fig. 5b shows that for an NaCl solution
with $n_{b}=0.01\text{M}$, the absolute value of $I_{visco}$ always increases
with pH while similar to Fig. 5a, $I_{visco}$ increases more for viscosity
gradient $\eta_{1}=3.4\text{(mPas)}$, $\eta_{2}=1\text{(mPas)}$. Comparing
Fig. 5a and 5b shows that although higher pH provides a smaller absolute value
of zeta potential in comparison with the bulk ionic strength, the
$I_{\text{visco}}$ for both scenarios are almost in the same order of
magnitude.

(a)  (b)

Figure 4: The MS diffusion coefficient ($D_{12}$) versus the length of the
nanochannel for ideal fluid ($\Gamma=1$) represented by the symbols and non-
ideal fluid which demonstrated by the solid and dashed line for (a) $A=-0.1$
and (b) $A=-1$.

(a)  (b)

Figure 5: The impact of (a) bulk ionic strength ($n_{b}$) with pH=$6.5$ and
(b) solution’s pH with $n_{b}=0.01$ M on $I_{visco}$ for NaCl solution. The
zeta potentials to obtain the average net-charge density is shown by red line
obtained by the EQL model [27] and symbols the experimental measurements [38].

###  IV.3 Visco-migration vs Reverse Electrodialysis

So far we have investigated the impact of solution properties on the ionic
current owing to the viscosity gradient. As we have previously alluded to, for
$I_{\text{visco}}$, the sole existence of the ionic current is a result of the
drift of ionic species because of the diffusion gradient resulting from the
viscosity gradient. However, since the viscosity of the solvent is changing
along the nanochannel, based on the Stokes-Einstein equation, the diffusion of
the ionic species must be changed which suggests that if we apply a small
concentration gradient of ionic species, one can take the advantage of this
diffusion gradient to drive an increasing ionic current [40, 10, 41]. In this
section, we aim at proposing a simple analytical solution to the Nernst-Planck
equation when we only have ionic species concentration gradient as

$\displaystyle j_{i}=-D_{i}\nabla c_{i}$ (53)

where $j_{i}$ denotes the flux, $D_{i}$ the diffusion, and $c_{i}$ the
concentration of the $i$th ionic species. It is worth noting that in Eq. 53 we
dropped the electro-migration term since we assumed that the applied
concentration gradient is small enough to ignore the surface charge
heterogeneity and no external electric field is applied [42].

Let us define $n_{s}=c_{1}-c_{2}$ and assume that the diffusivity of both co-
and counter-ions are approximately identical ($D_{1}\simeq D_{2}=D^{\circ}$).
Now we can subtract the flux of the counter-ion ($i$=1) from co-ion ($i$=2)
based on Eq. 53 which gives rise to

$\displaystyle j_{\text{net}}=-D^{\circ}\nabla n_{s}.$ (54)

Employing the mass conservation law to Eq. 54 gives

$\displaystyle B\frac{d\langle
n_{s}\rangle}{dx}+\left(A+Bx\right)\frac{d^{2}\langle n_{s}\rangle}{dx^{2}}=0$
(55)

where $B=dD^{\circ}/dx$. Eq. 55 can be solved analytically (see appendix A for
definition of $A$ and $B$) as

$\displaystyle\langle n_{s}\rangle=a_{1}+a_{2}\ln\left(x+\frac{A}{B}\right)$
(56)

where $\langle n_{s}\rangle=\left(1/H/2\right)\int_{0}^{H/2}n_{s}dy$
represents the average amount of $n_{s}$ perpendicular to the nanochannel’s
walls. Eq. 56 can be solved by having the boundary conditions

$\displaystyle x=0:\langle
n_{s1}\rangle=-2n_{b1}\frac{\int_{0}^{H/2}\sinh\left(\frac{\psi\left(y\right)}{V_{T}}\right)dy}{H/2}$
(57) $\displaystyle x=L:\langle
n_{s2}\rangle=-2n_{b2}\frac{\int_{0}^{H/2}\sinh\left(\frac{\psi\left(y\right)}{V_{T}}\right)dy}{H/2}$
(58)

where $n_{b1}$ and $n_{b2}$ are the bulk ionic concentration at the left and
right reservoirs, respectively, and $V_{T}$ represents the thermal voltage
which is defined as $k_{b}T/e$. Introducing Eqs. 57 and 58 into Eq. 56, we
have

$\displaystyle\begin{split}a_{1}=\langle n_{s1}\rangle-a_{2}\ln\frac{A}{B}\\\
a_{2}=\frac{\langle n_{s2}\rangle-\langle
n_{s1}\rangle}{\ln\left(1+\frac{B}{A}\right)}.\end{split}$ (59)

Given the fact that we have the $j_{\text{net}}$ by introducing Eq. 56 into
Eq. 54, one can simply obtain the ionic current owing to the applied
concentration gradient as

$\displaystyle I_{\text{revED}}=e\>\mathcal{A}\>j_{\text{net}}.$ (60)

Figure 6 shows the relation of reverse electro-dialysis current
$I_{\text{revED}}$ to the visco-migration current $I_{\text{visco}}$ versus
the deviation of the right hand-side bulk ionic concentration
($n_{\text{b2}}$) from the left hand-side bulk concentration
($n_{\text{b1}}$). To study the impact of the viscosity gradient, bulk ionic
concentration, and the solution pH on the aforementioned parameters, we
consider two viscosity gradients (solid
$\eta_{1}=3.4\text{(mPas)},\eta_{2}=1\text{(mPas)}$ and dashed lines
$\eta_{1}=3.4\text{(mPas)},\eta_{2}=3\text{(mPas)}$), two bulk ionic
concentration ($n_{\text{b}}=0.3\text{mM}$ and $n_{\text{b}}=100\text{mM}$)
and two solution pH. On the one hand, our theoretical model shows that
$I_{\text{revED}}\>I_{\text{visco}}^{\text{-1}}$ changes almost linearly with
the $n_{\text{b2}}\>n_{\text{b1}}^{\text{-1}}$ for all scenarios (logarithmic
$y$-axis). Moreover, it is revealed that by decreasing the viscosity gradient,
the impact of diffusion gradient decreases, which results in the domination of
the role of the concentration gradient in ionic current. On the other hand,
Fig. 6 demonstrates that by increasing the bulk ionic concentration or the
solution pH, the contribution of $I_{\text{revED}}$ increases in comparison to
the $I_{\text{visco}}$. It is interesting to note that for certain amounts of
$n_{\text{b2}}\>n_{\text{b1}}^{\text{-1}}$,
$I_{\text{revED}}\>I_{\text{visco}}^{-1}$ falls below unity. This implies that
for very small applied concentration gradient, the visco-migration ionic
current is higher than that from concentration gradient. Generally, when
comparing the two contributions, it can be seen that for sufficiently large
concentration gradients the reverse ED contribution to the current is orders
of magnitude higher than the contribution as a result of the viscosity
gradient.

Figure 6: The relation of ionic current because of concentration gradient
($I_{\text{revED}}$) to ionic current owing to viscosity gradient
($I_{\text{visco}}$) versus the deviation of right reservoir bulk ionic
concentration from the left reservoir. Dashed and solid lines represent
different viscosity gradients for two bulk ionic concentration and solution
pH.

##  V Conclusions

In the present contribution, we theoretically investigated the recently
discovered ionic transport phenomenon (Wiener and Stein [20]) which drives
ionic species owing to a viscosity gradient along a nanochannel, so-called
visco-migration transport phenomenon. Our model has shown that the diffusion
gradient can be linearly changed along the nanochannel when we assume the
fluids are ideal, while for non-ideal fluids, it showed non-linear behavior as
function of the distance from nanochannel entrance. We have shown that the
ionic current owing to visco-migration would be a result of nanochannel’s
surface charge, flux of solvent with higher viscosity, and natural logarithm
of lower to higher diffusion coefficient. After benchmarking our ideal and
non-ideal binary model with the available experimental data from Wiener and
Stein [20], we investigated the impact of bulk solution properties (i.e. pH
and ionic strength) on the visco-migration ionic current by solving the
surface complexation models to obtain the relevant surface charge as function
of solution bulk properties. Our modeling results suggested that:

(I) by increasing the solution’s bulk ionic strength and pH, the absolute
value of visco-migration ionic current is increasing, while the increment
trend is a function of the applied viscosity gradient. Moreover, we have found
that the absolute value of visco-migration ionic current has a maximum point,
which can be attributed to the competition of thinning EDL (in contrast to
thickening/overlapping EDL) and the increment of surface charge;

(II) by applying a small ionic concentration gradient at both reservoirs, the
impact would be twofold; one can exploit the diffusion gradient and
concentration gradient to drive higher amounts of ionic current. For isntance,
for $n_{b2}\>n_{b1}^{-1}=0.99$, $n_{b1}=0.3\>\text{(mM)}$, and
$\eta_{\text{1}}=\text{3.4}\>\text{(mPas)}$ and
$\eta_{\text{2}}=\text{3}\>\text{(mPas)}$ we have greater reverse ED ionic
current (order of $4\times 10^{2}$) than the visco-migration ionic current.

The theoretical model of this study may pave the way for novel ideas which
combine the visco-migration and reverse electro-dialysis for energy conversion
applications. Furthermore, this contribution would help researchers to
understand how smart selection of solution’s properties (i.e. pH and bulk
ionic strength) can be lead to drive the maximum amount of ionic current from
a viscosity gradient.

## Acknowledgment

AA is very thankful to Prof. Derek Stein and Dr. Benjamin Wiener for fruitful
discussions. This work is financially supported by the Eyes High Postdoctoral
program at the University of Calgary. AB acknowledges financial support from
the Canada First Research Excellence Fund.

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## Appendix A: Deriving equation for reverse electro-dialysis with viscosity
gradient

Considering the ideal fluid which we already have an explicit solution for the
volume fractions of solvents species (Eq. 27), one can simply relate the mixed
fluid viscosity along the nanochannel to the volume fractions by [34]

$\eta_{\text{12}}(x)=\eta_{1}^{\phi_{1}(x)}\eta_{2}^{1-\phi_{1}(x)}.$ (A.1)

Having the viscosity along the nanochannel lets us to obtain the diffusivity
of the ionic species by employing the Stokes-Einstein equation as

$D^{\circ}(x)=\frac{k_{b}T}{6\pi\eta_{\text{12}}(x)r^{\circ}}$ (A.2)

Taking $d/dx$ from both sides of Eq. A.2 gives

$\frac{dD^{\circ}(x)}{dx}=\frac{-k_{b}T\mathcal{X}_{1}\ln\frac{\eta_{1}}{\eta_{2}}}{6\pi\eta_{\text{12}}(x)r^{\circ}}$
(A.3)

where $r^{\circ}$ denotes the Stokes radius of the ionic species that we
assumed approximately identical for both co- and counter-ionic species
(Na+Cl-) equal with 1.94 Å.

The flux of ionic species can be described by the Nerns-Planck equation [44]
by considering the diffusion and electromigration as

$j_{i}=-\left(D_{i}\nabla
c_{i}+\frac{z_{i}}{V_{T}}D_{i}c_{i}\nabla\psi\right).$ (A.4)

According to the mass conservation law for a steady-state condition without
sink or source of ionic species, justifies $\nabla\cdot j_{i}=0$ which gives
rise

$0=\nabla D_{i}\cdot\nabla
c_{i}+D_{i}\nabla^{2}c_{i}+z_{i}\nabla\cdot\left(D_{i}c_{i}\nabla\bar{\psi}\right),$
(A.5)

where $\bar{\psi}=\psi/V_{T}$. If we do subtraction over the ionic species for
Eq. A.5, we have

$0=\nabla D_{i}\cdot\nabla
n_{s}+D_{i}\nabla^{2}n_{s}+\nabla\cdot\left(D_{i}n_{d}\nabla\bar{\psi}\right)$
(A.6)

where $n_{d}=c_{1}+c_{2}$ and $n_{s}=c_{1}-c_{2}$. Since our nanochannel’s
wall are impermeable, therefore the flux of ionic species along the
$y$-direction must be zero which justifies

$j_{i,y}=0\rightarrow\frac{\partial c_{i}}{\partial
y}+z_{i}c_{i}\frac{\partial\bar{\psi}}{\partial y}=0.$ (A.7)

Now $j_{1,y}-j_{2,y}$ can be written as

$0=\frac{\partial n_{s}}{\partial y}+n_{d}\frac{\partial\bar{\psi}}{\partial
y}.$ (A.8)

If we take $\partial/\partial y$ from both sides of Eq. A.8 and introduce the
results into Eq. A.6, we finally have

$\displaystyle\Omega\frac{\partial{n_{s}}}{\partial{x}}+D^{\circ}\frac{{\partial}^{2}{n_{s}}}{\partial{x}^{2}}+n_{d}\Omega\frac{\partial{\bar{\psi}}}{\partial{x}}$
(A.9) $\displaystyle+D^{\circ}\frac{\partial n_{d}}{\partial
x}\frac{\partial\bar{\psi}}{\partial
x}+D^{\circ}n_{d}\frac{\partial^{2}\bar{\psi}}{\partial x^{2}}=0.$

where $\Omega=dD^{\circ}/dx$. As alluded to what previously we noted, since we
are applying very small amounts of concentration gradients and also ignoring
the nanochannel entrance effect, therefore it is reasonable to assume that
$\partial\bar{\psi}/\partial x=0$, which make Eq. A.9 simpler as

$\Omega\frac{\partial n_{s}}{\partial
x}+D^{\circ}\frac{\partial^{2}n_{s}}{\partial x^{2}}=0.$ (A.10)

If we take an average in $y-$direction from both sides of Eq. A.10 and define
$\langle n_{s}\rangle=\int_{0}^{H/2}n_{s}/H/2$, then we have

$\Omega\frac{d\langle n_{s}\rangle}{dx}+D^{\circ}\frac{d^{2}\langle
n_{s}\rangle}{dx^{2}}=0.$ (A.11)

Eq. A.11 cannot be solved analytically since both $\Omega$ (Eq. A.3) and
$D^{\circ}$ (Eq. A.2) are complex functions of $x$. For sake of simplicity, we
can utilize Taylor expansion for $D^{\circ}$ with
$\mathcal{O}\left(D^{\circ}\right)^{2}$ and maximum error on the order of
$10^{-6}$. Consequently, we can re-write Eq. A.11 as

$B\frac{d\langle n_{s}\rangle}{dx}+\left(A+Bx\right)\frac{d^{2}\langle
n_{s}\rangle}{dx^{2}}=0.$ (A.12)

In Eq. A.12, we replace $\Omega$ with the slope of $D^{\circ}$ which
represented by $B$. Eq. A.12 can be analytically solved by a proper boundary
conditions (see Eqs. 56 to 59).