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Supporting Information for “Ion Transport and Selection through
DCGC Based Electroosmosis in a Conducting Nanofluidic Channel”
Cunlu Zhao, Chun Yang*
School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang
Avenue, Singapore 639798, Republic of Singapore
*Address correspondence to: [email protected]
S1 Details of boundary conditions
For completion of the mathematical formulation of the problem, boundary conditions are necessary for
the Poisson equation (eq 3), the Nernst-Planck equation (eq 5) and the Navier-Stokes equation (eq
7).
Since AH is a symmetric boundary (see Figure 2), a zero potential gradient condition is used for the
Poisson equation
n φ = 0 on the boundary AH
(s1)
where n represents the unit outward normal to the corresponding planes.
Along the boundary AB, the electric potential is specified as
φ = φ0 on the boundary AB
(s2)
and the potential, φ0, is known a priori and can be adjusted by the power source shown in Figure 1.
The boundary GH is taken as the potential reference (i.e., grounded as shown in Figure 1)
φ = 0 on the boundary GH
(s3)
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The boundaries BC and FG in the reservoirs are not charged since they are assumed in the bulk
electroneutral solution. Besides, the walls of the reservoirs (planes CD and EF) are assumed not to
carry any fixed electric charge, and hence no charge boundary conditions for the potential are used
n φ = 0 on the boundaries BC, CD, EF and FG
(s4)
On the conducting walls of the nanochannel, two kinds of boundary conditions are considered
in this study: (i) if the switch in Figure 1 is in off-state, the conducting wall floats in the electric
field set up by the two driving electrodes inside two reservoirs. Then the floating boundary
condition for the potential is used (Stratton 1941)
  n φ  ds
2
= Q on the boundary DE
(s5)
s2
where Q is the overall dimensionless free surface charge acquired by the conducting wall before the
application of the external electric field. Specifically, Q is normalized with respect to ε0 εr φref . (Here
s2 denotes the surface element for the boundary DE and is normalized with respect to Lref). Eq s5
constrains the total charge on the conducting surface. In the present work, the conducting wall of
the nanochannel is uncharged such that Q =0 is used. (ii) When the switch is in on-state, the electric
potential on the conducting wall is supplied by the electric power source, and a fixed potential
boundary applies
φ = φ1 on the boundary DE
(s6)
As for the Nernst-Planck equation, the concentrations of two ionic species on the boundaries of AB
and GH are equal to the bulk electrolyte concentration in two reservoirs, namely
ci = 1 on the boundaries AB and GH
(s7)
2
Since ions cannot pass through the solid walls of two reservoirs and the nanochannel (represented
by the boundary segments CD, DE, and EF in Figure 2), ionic fluxes (electric current) normal to
these walls vanish, namely
n  uci - Di ci / Pe - Di zi ci φ / Pe  = 0 on the boundaries CD, DE, and EF
(s8)
Boundaries BC and FG are assumed to be far from the nanochannel inlet (outlet) are in the bulk
electrolyte solution. Consequently, ionic fluxes normal to these boundaries should be zero,
n  uci - Di ci / Pe - Di zi ci φ / Pe  = 0 on the boundaries BC and FG
(s9)
On the symmetric boundary AH, a symmetric boundary condition applies to the Nernst–Planck
equations
n  uci - Di ci / Pe - Di zi ci φ / Pe  = 0 on the boundary AH
(s10)
Finally, boundary conditions for the hydrodynamic problem governed by eqs 6 and 7 are prescribed.
On the solid walls of the nanochannel and the reservoirs (refer to boundary segments CD, DE, and
EF in Figure 2), no-slip boundary conditions (i.e., u = v = 0) are specified. Since there is no pressure
difference imposed between two reservoirs, we set p = 0 on the boundaries AB and GH. A
symmetric boundary condition is used along the boundary of symmetry, AH. Finally, because
boundaries BC and FG are in the bulk electrolyte solution and are not affected by the inlet and
outlet of the nanochannel, viscous effects at these boundaries disappear (e.g., a perfect slip
boundary conditions).
S2 Numerical method and validation
The presented model is different from most of the previous relevant studies on induced-charge
electrokinetic phenomena around microscale conducting surfaces (Squires and Bazant 2004; Bazant
and Squires 2004). In those studies the ionic concentrations are described by the Boltzmann
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distribution. Then the electric potential is governed by the Poisson–Boltzmann equation, and thus
the electrostatic and hydrodynamic problems are decoupled. However, the Boltzmann distribution is
only valid to describe the ionic concentration in the semi-infinite stationary space over a charged
surface. This is not the case in the present study because the channel is in nanometric scale and the
EDLs around the walls of the nanochannel overlap. Besides, the ionic species are also convectively
transported by electroosmotic flow. Evidently, the distributions of ionic species do not exactly
follow the Boltzmann distribution, and hence the Poisson-Boltzmann equation cannot give good
description of the electric potential. Overall, the electrostatic problem, the hydrodynamic problem
and the ionic transport in this study are strongly coupled, and then the numerical method is used to
tackle this coupled system. A similar model was utilized for describing the transport of ions in
insulating nanofluidic channels (Daiguji et al. 2003; Qian et al. 2007).
The strongly coupled system presented in this work is solved by using the commercial finite
element software package COMSOL Multiphysics 4.3. The simulated domain in Figure 2 is fairly
regular in shape, and triangular elements are used to mesh the domain with dense mesh elements
within the nanochannel. Furthermore, to capture the details inside the EDL on the wall (segment DE)
of the nanochannel, the region near the wall is finely meshed with more than 20 mesh elements
inside the EDL. In addition, the solutions obtained for different numbers of mesh elements are
compared to exclude the possibility of mesh-size dependence of numerical solutions. The results
presented in this work are all independent of the size of mesh when the simulation domain is
meshed with 1,631,488 triangular elements. The direct solver MUMPS is used for solving all the
governing equations with a relative tolerance set to be 10-8. Reservoirs with different sizes are also
considered in the simulation and the results do not noticeably depend on the reservoir size. Hence,
numerical simulations for a reservoir size of 100h×100h are presented in this work. For testing the
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above numerical model, we simulate a conventional electroosmosis with fixed surface charge
density on the channel walls, and compare the numerical results with the benchmark analytical
solutions to validate the numerical model.
We use an analytical solution as a benchmark for the conventional electroosmosis in a straight
insulating microchannel with a fixed surface charge density. The solution in terms of the
dimensionless axial velocity takes the form (Keh and Tseng 2001; Stein et al. 2004; Zhao et al.
2008)
u  
Ex cosh  h   cosh  hy  
1 

 h sinh  h   cosh  h  
(s11)
where σ is the dimensionless surface charge density normalized with ε0 εr φref / h ,and E x is the
dimensionless electric field strength along the channel axial direction and is normalized with
φref / h .
The derivation of the analytical formula (s11) is made under the assumptions of the Debye-Hückel
linearization for the Poisson-Boltzmann equation, the linear superposition of external and EDL
electric fields, no axial variation of ionic concentration, and no entrance/exit effects. Since the
present numerical model involves none of these assumptions, it is then not straightforward to make
a comparison between the numerical and analytical results. Nevertheless, if one focuses on the midsection of the channel (corresponding to x = 0, sufficiently far from the inlet and outlet of the
channel), the numerical model would be astonishingly close to the analytical solution available in an
infinitely long channel (Mansouri et al. 2005). Therefore, we compare the axial velocity profile at x
= 0 obtained from the numerical simulation with that obtained using eq s11. It also should be noted
that one needs to take the value of Ex in eq s11 to be equal to the numerical solution at x = 0 when
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comparing the velocity profiles. The numerical simulations for κh = 5 and 10 are conducted with a
constant surface charge of σ=-1 at the channel wall and an applied voltage difference of 1000
between the two driving electrodes (i.e., φ0 = 1000 ). From the steady-state numerical solution, the
value of Ex at x=0 and y=0 is obtained. This value of Ex is then substituted in eq s11 to calculate the
analytical velocity profiles. Figure S1 shows the comparison results between the numerical and
analytical velocity profiles. It is evident that the numerically predicted velocity profiles at the midsection of the channel (x=0) agree extremely well with the velocity profiles obtained from analytical
formula expressed by eq s11.
1.0
h=5
0.8
u
0.6
h=10
0.4
Analytical solutions
Numerical solutions
0.2
0.0
0.0
-0.2
-0.4
y
-0.6
-0.8
-1.0
Figure S1. Comparison of the velocity profiles obtained from the present numerical model and the analytical
formula eq s11 for the electroosmosis with a uniform natural surface charge density of σ = -1 on the channel
walls. In the comparison, the velocity profiles are obtained at the cross section of x=0 and two values of  h
are chosen, viz.,  h  5,10 .
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S3 Supplemental results for negatively biased cases under three operating modes
S3.1 Electroosmosis in conducting nanochannels with floating walls (Mode 1)
When the system is negatively biased ( φ0 = -6 ), it is seen from Figure S2(a) that there are also two
identical potential barriers at the junctions connecting the nanochannel and the two reservoirs. The
ionic distributive characteristics inside the nanochannel can be obtained by simply interchanging the
profiles of K+ and Cl- presented in Figure 3 for the positively biased case ( φ0 = 6 ). The wall
polarization shown in Figure S2(b) reverses as compared to the positively biased case shown in
Figure 5(a). However, as is shown schematically in Figure S2(c), the electrostatic body forces
driving electrokinetic transport inside the conducting nanochannel remain identical to the positively
biased case sketched in Figure 5(b). Therefore, it is expected that the flow characteristics of the
positively and negatively biased cases should be the same, as depicted in Figure 4(b).
0.0
(a)

-1.5
-3.0
-4.5
-6.0
6.0
ci
4.5
-
3.0
Cl
+
K
1.5
0.0
-200
-150
-100
-50
0
50
100
150
200
x
7
6
(b)
4

2
0
-2
-4
-6
-100
-50
0
x
50
100
E
(c)
Fe
Fe
Figure S2. The results for mode 1 when the conducting walls are floating for a negative electric bias of
0  6 . (a) Electric potential  and ion concentration c profiles along the nanochannel axis (y=0), (b)
Induced surface charge density σ distribution along the conducting wall, and (c) A schematic illustration of
electrokinetics in the conducting nanochannel for a reverse electric bias as compared to Figure 5(b).
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S3.2 Electroosmosis in conducting nanochannels with wall potential closer to  0 (Mode 2)
For the negatively biased case ( φ0 = -6 , φ1 = -4 ), Figure S3(a) shows that the electric potential
increases from -6 to 0 along the nanochannel and the potential rise across the left nanochannelreservoir junction is less significant because the conducting wall is biased much closer to φ0 = -6 .
The ionic concentration profiles can be obtained by interchanging the profiles of anions and cations
depicted in Figure 6(a) for the positively based case ( φ0 = 6 , φ1 = 4 ). Contrary to the positively
biased case, the cations predominate inside the longer right portion of the nanochannel while the
anions predominate inside the relatively shorter left portion of the nanochannel. In addition, it can
be concluded from the Figure S3(b) that the wall polarization also reverses as compared to the
positively biased case shown in Figure 6(b).
0.0
(a)

-1.5
-3.0
-4.5
-6.0
6.0
ci
4.5
3.0
-
Cl
+
K
1.5
0.0
-200
-150
-100
-50
0
x
50
100
150
200
9
6
(b)
4

2
0
-2
-4
-6
-100
-50
0
x
50
100
Figure S3. The results for mode 2 when the potential imposed on the conducting walls is set to be closer to
 0 for the negatively biased case of 0  6, 1  4 . (a) Electric potential  and ion concentration c
profiles along the nanochannel axis (y=0) and (b) induced surface charge density σ distribution along the
conducting wall.
S3.3 Electroosmosis in conducting nanochannels with wall potential closer to the ground (Mode 3)
In the negatively biased case ( φ1 = -2 , φ0 = -6 ) depicted in Figure S4(a), the ion distributions can
also be obtained via interchanging profiles of K+ and Cl- presented in Figure 8(a) for the positively
biased case ( φ1 = 2 , φ0 = 6 ). In addition, the wall polarization presented in Figure S4(b) also
reverses as compared to the positively biased case shown in Figure 8(b).
10
0.0

-1.5
(a)
-3.0
-4.5
-6.0
6.0
ci
4.5
3.0
-
Cl
+
K
1.5
0.0
-200
-150
-100
-50
0
x
50
100
150
200
6
4
(b)

2
0
-2
-4
-6
-100
-50
0
x
50
100
Figure S4. The results for mode 3 when the potential imposed on the conducting walls is set to be closer to
the ground for the negatively biased case of 0  6, 1  2 . (a) Electric potential  and ion concentration
c profiles along the nanochannel axis (y=0) and (b) induced surface charge density σ distribution along the
conducting wall.
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References
Bazant MZ, Squires TM (2004) Induced-Charge Electrokinetic Phenomena: Theory and Microfluidic
Applications. Phys Rev Lett 92 (6):066101.
Daiguji H, Yang P, Majumdar A (2003) Ion Transport in Nanofluidic Channels. Nano Lett 4 (1):137-142.
Keh HJ, Tseng HC (2001) Transient Electrokinetic Flow in Fine Capillaries. J Colloid Interface Sci 242
(2):450-459.
Mansouri A, Scheuerman C, Bhattacharjee S, Kwok DY, Kostiuk LW (2005) Transient streaming potential
in a finite length microchannel. J Colloid Interface Sci 292 (2):567-580.
Qian S, Das B, Luo X (2007) Diffusioosmotic flows in slit nanochannels. J Colloid Interface Sci 315
(2):721-730.
Squires TM, Bazant MZ (2004) Induced-charge electro-osmosis. J Fluid Mech 509:217-252.
Stein D, Kruithof M, Dekker C (2004) Surface-Charge-Governed Ion Transport in Nanofluidic Channels.
Phys Rev Lett 93 (3):035901.
Stratton JA (1941) Electromagnetic Theory. 1st edn. McGraw - Hill, New York
Zhao C, Zholkovskij E, Masliyah JH, Yang C (2008) Analysis of electroosmotic flow of power-law fluids in
a slit microchannel. J Colloid Interface Sci 326 (2):503-510.
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