Supplementary Information for “Induced-charge electrokinetics in a
conducting nanochannel with broken geometric symmetry-towards a flexible
control of ionic transport”
Cunlu Zhaoa, Yongxin Songb and Chun Yanga *
a
School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore
b
Department of Marine Engineering, Dalian Maritime University, Dalian, P. R. China
*E-mail: mcyang@ntu.edu.sg
S1 Numerical method and validation
The strongly coupled governing equations consisting of equations (3), (4) and (5) are solved with
the commercial finite element software package Comsol Multiphysics 4.3. Comsol Multiphysics
features the capability of modelling coupled multiphysical processes in arbitrary geometric
domains. This software is therefore well suited to our nanochannel system that involves three
coupled physical processes, namely electric filed, liquid flow and ion transport. Triangular
elements are employed to mesh the simulated domain shown in Figure 2. Compared to two bulk
reservoirs, the nanochannel domain is more densely meshed to ensure sufficiently accurate
evaluation of large gradients of electric potential, velocity and ion concentration inside the EDL
near the nanochannel wall (boundary DE). The MUMPS direct solver in Comsol with a quite
small tolerance of 10-8 was chosen for solving all three governing equations. The solutions for
different numbers of mesh elements are obtained and compared to ensure that the numerical
results are independent of the number of mesh elements. Particularly, the results presented in this
work are all obtained with a total number of 1,089,344 triangular elements in the chosen
simulation domain. In addition, we also tested the effect of reservoir size on the results of
numerical simulation. It was found that the size of reservoir has insignificant effects on the
numerical results and two reservoirs with size of 100h2×100h2 were used in all simulations.
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As suggested by Daiguji et al.1, 2, three benchmark tests are conducted to verify the numerical
model. The first two tests are designed for the Poisson equation and the Nernst-Planck equation,
and the third test is for the Navier-Stokes equation. At first, for validation purpose, the tapered
channel shown in Figures 1 and 2 needs to be replaced by a straight channel with uniform half
height of h for all the benchmark tests. For a microchannel with large values of  h  h
1 , the
EDLs on two channel walls do not overlap, each wall can be regarded as an isolated surface and
the dimensionless surface potential  s at an isolated surface is related to the dimensionless
charge density q by the Grahame equation 3, 4



 2  h  
 s  2sinh 1 
q
(s1)
The Poisson and Nernst-Planck equations are solved numerically for  h  50 with three different
surface charge densities, -25, -50 and -75. The corresponding surface potentials calculated from
the numerical model are respectively -0.495, -0.962, and -1.386, which are exactly the same as
those calculated from equation (s1) to three decimal places. Such an excellent agreement
suggests high accuracy of our numerical method.
For a channel with uncharged walls, electric potential and ionic species are distributed uniformly
along the y direction, and thus the Poisson and Nernst-Plank equations can be reduced to be one
dimensional along the x direction. In equation (7), the convective transport of ionic species
vanishes because of stationary liquid electrolyte. Furthermore, ionic concentration is uniform in
the x direction, and then there is only the gradient of electric potential left for driving ionic
fluxes. Hence, the ionic flux for species j becomes
2
i j,x  
d 1 2
z j Djc j
dx Pe
(s2)
Under a potential bias of  0 =1000, the ionic currents of K+ and Cl- ions calculated with
equations (s2) and (6) are 19.19 and 19.92, respectively. The corresponding results from
numerical solutions of the Poisson and Nernst-Planck equations are 19.16 and 19.88
respectively, indicating small deviations less than 0.25%.
Finally, to check the numerical model of the Navier-Stokes equation, we consider a purely
pressure-driven flow inside the channel. The resulting flow is Poiseuille in nature and is fully
developed at x=0 (the middle of channel). The fully developed velocity profile at x=0 in the
straight channel is given by
u  y  
1 dp
1 y2 

2 dx
(s3)
For a given pressure gradient of dp / dx =50, the centerline velocities, u(0), calculated from
equation (s3) and obtained from the numerical solution of Navier-Stokes equation are
respectively 25.0 and 24.93, which then ensures the validity and accuracy of our Comsol model
for the numerical solution of Navier-Stokes equation.
It should be noted that in this nanochannel system the electric potential, ionic concentration and
velocity field are strongly coupled via equations (3), (4) and (5). The numerical model presented
in this study is quite general for describing electrokinetic phenomena. In the literature, similar
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numerical models have been successfully used to analyze the ionic transport in nanochannels via
conventional electrokinetics1, 2, 5, 6 and the mixing in microchannels using ICEK phenomena7.
References
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