Research Article
Electrokinetic transport through
nanochannels
This article presents a numerical study of the electrokinetic transport phenomena
(electroosmosis and electrophoresis) in a three-dimensional nanochannel with a circular
cross-section. Due to the nanometer dimensions, the Boltzmann distribution of the ions
is not valid in the nanochannels. Therefore, the conventional theories of electrokinetic
flow through the microchannels such as Poisson–Boltzmann equation and Helm-
holtz–Smoluchowski slip velocity approach are no longer applicable. In the current
study, a set of coupled partial differential equations including Poisson–Nernst–Plank
equation, Navier–Stokes, and continuity equations is solved to find the electric potential
field, ionic concentration field, and the velocity field in the three-dimensional nano-
channel. The effects of surface electric charge and the radius of nanochannel on the
electric potential, liquid flow, and ionic transport are investigated. Unlike the micro-
channels, the electric potential field, ionic concentration field, and velocity field are
strongly size-dependent in nanochannels. The electric potential gradient along the
nanochannel also depends on the surface electric charge of the nanochannel. More
counter ions than the coions are transported through the nanochannel. The ionic
concentration enrichment at the entrance and the exit of the nanochannel is completely
evident from the simulation results. The study also shows that the flow velocity in the
nanochannel is higher when the surface electric charge is stronger or the radius of the
nanochannel is larger.
Keywords:
Electrokinetics / Electroosmosis / Nanochannels
DOI 10.1002/elps.201000564
1 Introduction
In many microfluidic instruments, electrokinetic effects are
exploited to perform various applications ranging from
biological process such as cell culture [1] to cooling
microelectronic devices [2]. Extensive studies have been
conducted on electrokinetic transport phenomena in the
microscale channels [3, 4]. Sophisticated theories have been
proposed to model the electrokinetic effects in the micro-
channels. Usually, Poisson–Boltzmann equation is used to
find the electric potential in the microchannels; Helm-
holtz–Smoluchowski theorem is commonly utilized to
model the electroosmotic flow through the microchannels.
With the advancement of nano-fabrication technology [5],
more and more attention has been paid to transport
phenomena in devices involving nanochannels [6–9]. By
reducing the dimensions of the channels to the submicron
and nanoscales, these theories may not be applicable
anymore. Mostly, this is because the ion distribution in
the nanochannel cannot be described by Boltzmann
distribution, and hence the electric field generated by the
nanochannel’s surface charge does not obey the Poisson–
Boltzmann equation. The current understanding of the
electrokinetic effects in the nanochannels is very limited.
In modeling the electrokinetic flow in microchannels,
Boltzmann distribution is one of the fundamental equa-
tions. Boltzmann distribution requires a semi-infinite large
liquid phase, the equal number of coions, and the counter
ions in positions sufficiently far away from the charged
surface, and no significantly overlapped electric double layer
fields. However, all these key assumptions are not valid for
smaller nanochannels. For example, the concentrations of
co- and counter ions are not equal in the nanochannels [10].
Since the Boltzmann distribution is not valid, the Pois-
son–Boltzmann equation cannot be used to describe the
electric field in small nanochannels. Furthermore, there is
an essential difference in defining the electrical boundary
condition between the microchannels and the nanochan-
nels. In microchannels, Poisson–Boltzmann equation can
Saeid Movahed
Dongqing Li
Department of Mechanical and
Mechatronics Engineering,
University of Waterloo, Waterloo,
Ontario, Canada
Received October 25, 2010
Revised February 3, 2011
Accepted February 4, 2011
Colour Online: See the article online to view Figs. 1-9 in colour.
Correspondence: Professor Dongqing Li, Department of
Mechanical and Mechatronics Engineering, University of Water-
loo, Waterloo, Ontario, N2L 3G1 Canada
E-mail: dongqing@mme.uwaterloo.ca
Fax: 11-519-885-5862
& 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.electrophoresis-journal.com
Electrophoresis 2011, 32, 1259–1267 1259

be utilized to obtain a relationship between the surface
electric charge and the surface electric potential, approxi-
mately the z-potential. That is, fixing the surface electric
charge is equivalent to fix the z-potential [3]. Therefore, a
constant z-potential is usually used as the boundary condi-
tion to solve for the electric double layer field in the
microchannels. However, in small nanochannels, because
the Poisson–Boltzmann equation is not valid, the surface
electric charge, via the use of the Poisson equation, can
relate only to the electric potential gradient at the charge
surface, not the z-potential. Thus, the z-potential cannot be
used as the boundary condition for the electric field in the
small nanochannels. Instead, the surface electric charge
should be utilized as the electrical boundary condition at the
walls of the nanochannels. By considering these facts, the
conventional theories of the electrokinetic flow are no longer
valid in the small nanochannels.
Some studies have been conducted to model and
simulate the electrokinetic effects in the nanoscale channels.
Several of them are based on Poisson–Boltzmann equation
[11–14]. The Boltzmann distribution is derived under the
following boundary conditions: At positions infinitely far
away from the charged solid surface, (i) the electric potential
is 0, and (2) the bulk solution is electrically neutral or has
zero net charge. However, in a small nanochannel with
significantly overlapped electric double layers, the above
conditions do not exist. Therefore, the Boltzmann equation
and hence the Poisson–Boltzmann equation are no longer
valid in small nanoscale channels. A widespread numerical
technique used to model electrokinetic effects in nano-
channels is molecular dynamics simulation [15–17]. Using
this method, Qiao and Aluru modeled ion distribution and
velocity profiles for the electroosmotic flow in the nano-
channel [15]. They proposed electrochemical potential
correction term to modify Poisson–Boltzmann equation and
predict the ion distribution with good accuracy; however,
they considered only the presence of the counter ions in
their simulations. By using this technique, they also studied
the transient response of the electroosmotic flow in the
nanochannels [16]. However, it must mention that mole-
cular dynamics technique needs huge computational effort
and may not be practical for larger computational domains.
In 2000, Freund studied the electroosmotic flow in the
nanoscale channels [18]. In this study, the author considered
only the effects of counter ion on liquid flow; they
also utilized the Poisson–Boltzmann equation to find the
ionic distribution and electric potential. Some studies have
been conducted to investigate EDL overlapping for simple
case of long and slit nanochannels [19–22]; however, the
results of these studies cannot be used for the nanochannels
with sharp changes in geometry such as micro-
channel–nanochannel connection or three-dimensional
nanochannels.
Several new studies utilized Poisson–Nernst–Plank
equations to model the electrokinetics in nanoscale chan-
nels. These studies are usually based on two-dimensional
modeling or did not consider the effect of convection on ion
mass transfer. For example, Choi and Kim used a two-
dimensional model to investigate electrokinetic effects in slit
nanochannel [23]. In 2008, Cheng investigated the electro-
kinetic ion transport in a one-dimensional slit nanochannel
[24]. However, this study did not present any results for the
flow (velocity) field. The more accurate study on the elec-
trokinetic effects in nanoscale channels was performed by
Vlassiouk et al. [25]. However, in that study, the authors did
not consider the effect of electroosmosis on ion transfer;
therefore, the Poisson–Nernst–Plank and Navier–Stokes
equations became decoupled. In this way, it dramatically
decreased the difficulties associated with the numerical
simulations. At the end of mentioned article, the authors
showed that their approximation (neglecting the effect of
electroosmosis on ion mass transfer) can contribute 20% to
the total current (ion mass transfer). In addition, at the exit
of the nanochannel to the microchannel, they could not
model concentration polarization effect that was experi-
mentally shown in the other studies [10].
It is highly desirable to study the electrokinetic effects in
three-dimensional nanoscale channels in order to improve
the current understanding in this field. For small nano-
channels, Poisson–Nernst–Plank equations along with the
modified Navier–Stokes equation and the continuity equa-
tion must be solved in order to find the ionic mass transfer,
electric field, and velocity field. These governing equations
are highly coupled and the results are affected by all these
equations and the corresponding boundary conditions. This
article considers one circular cross-section and three-
dimensional nanochannel connected to two reservoirs at the
ends. The Poisson–Nernst–Plank, Navier–Stokes, and
continuity equations are solved simultaneously to calculate
the electric potential, ionic concentration, and fluid flow in
the nanochannel. The remaining of this article is organized
as follows: Section 2 presents the mathematical model of the
electrokinetic effects in the nanochannels. Section 3
explains the details of numerical method. The results are
described and discussed in Section 4 and the concluding
remarks are provided at the end of the article.
y
x
z
Nanochannel
reservoir 2 reservoir 1
L
+ −
− − − − − − − − − − − − − − − −
− − − − − − − − − − − − − − − −
Figure 1. Schematic diagram of the assumed system of this
study. Two reservoirs are connected to each other by a circular
nanochannel of length L and radius R. Two electrodes located in
the reservoirs apply electric potential to the ends of the
nanochannel.
Electrophoresis 2011, 32, 1259–12671260 S. Movahed and D. Li
& 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.electrophoresis-journal.com