Third Day Poster Session, 17 June 2010 - NanoTR-VI

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P and
Poster Session, Thursday, June 17
Theme F686 - N1123
Analytical Solution of an Electrokinetic Flow in a Nano-Channel using Curvilinear Coordinates
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UMehdi MostofiUP P*, Davood D. GanjiP Mofid Gorji-BandpyP
1
PDepartment of Mechanical Engineering, Noshiravani University of Technology, Babol, Iran
Abstract-In this paper, an electrokinetic flow of an electrolyte in a 15 nm radius nano-channel will be studied. This study will be with existence
of the Electric Double Layer (EDL) and fully analytical. Governing equations for electrokinetic phenomena are Poisson-Boltzmann, Navier-
Stokes, species and mass conservation equations. Induced electric potential force the electrolyte ions and decrease the mass flow rate. In this
paper, it is assumed that, zeta potential has small quantity. In this paper, after getting the equations set from the literature and transforming it
into curvilinear coordinates, the set will be simplified and be solved analytically for small zeta potentials in a nano-channel.
One of the most important subsystems of the micro- and
nano- fluidic devices is their passage or “Micro- and Nano-
Channel”. Nano-channel term is referred to channels with
hydraulic diameter less than 100 nanometers [1]. By decrease
in size and hydraulic diameter some of the physical parameters
such as surface tension will be more significant while they are
negligible in normal sizes.
Concentrating surface loads in liquid – solid interface makes
the EDL to be existed. If the loads are concentrated in the end
of nano-channels, a potential difference will be generated that
forces the ions in the nano-channel. However, induced electric
field is discharged by electric conduction of the electrolyte.
Rice and Whitehead [2], Lu and Chan [3] and Ke and Liu
[4] studied the flow in capillary tube. None of them solved the
problem based on the curvilinear coordinates system. Also, all
of them studied the problem with existence of the pressure
gradient while in the modern applications, the pressure
gradient can be eliminated and consequently, solving the
problem considering this fact is necessary. In this paper, for
small zeta potentials without pressure gradient will be studied
based on the curvilinear coordinates in a capillary tube.
In electrokinetic processes, for the most general form of the
study, seven nonlinear equations govern an electrokinetic
process [5]. In this paper, by some simplifications that will be
mentioned later, this set will be made simpler.
Next in this work, a very long nano- tube will be
investigated. According to the fact that reference length of the
tube according to x direction (L) is very larger than capillary
radius (R) and reference amount for theta ( ), we can neglect
several terms of the equations. In addition, it is assumed that,
electric potential in the x direction is constant. According to
these assumptions, the equations mentioned below will be
available:
1
r
X p
X m
2
(1)
r r
r
1
r
u
r
r r
1 X
r
r r
r
1 X
r
r r
r
p
m
e
E0
RT
2
F U
0
X
p
X
m
(2)
X
p 0
r
(3)
X
m
0
r
(4)
By applying boundary conditions(no slip condition at wall
and free stream velocity in center of nano-channel for velocity
field and zeta potential at wall and finite amount of it at center
of the nano-channel for potential field), we have the following
figures. Figures (a) and (b) show the results for velocity and
potential fields respectively.
In summary, by considering curvilinear coordinates and
using Taylor series, some derivation of Developed Bessel
ODE has been derived and solved for Poisson-Boltzmann
equation. In addition, velocity profile in nano-tube has been
achieved for small amounts of zeta potentials. Results those
are derived by curvilinear coordinates are in good agreement
with those of resulted by rectilinear ones in [5].
Figure 1. Normalized distribution of potential as a function of
normalized radius.
Figure 2. Normalized velocity profile as a function of
normalized radius.
* Corresponding author: HTmehdi_mostofi@yahoo.comT
[1] S. Kandlikar, et. al, Heat Transfer and Fluid Flow in
Minichannels and Microchannels. Elsevier Limited, Oxford (2006).
[2] Rice, C.L. and Whitehead, R. J. Phys. Chem., 69(11), 4017–4023
(1965)
[3] W.Y. Lo, and K. Chan. J. Chem. Phys., 143, 339–353 (1994)
[4] H. Keh, and Y.C. Liu, J. Colloids and Interface Surfaces, 172,
222–229 (1995)
[5] Zheng, Z.: Electrokinetic Flow in Micro- and Nano- Fluidic
Components. Ohio State University, (2003).
6th Nanoscience and Nanotechnology Conference, zmir, 2010 681