Download Multiphysics Simulation of Ion Concentration Polarization Induced

Supplementary Information:
Multiphysics Simulation of Ion Concentration Polarization
Induced By Nanoporous Membranes in Dual Channel Devices
Mingjie Jia1 and Taesung Kim1,2*
*CORRESPONDENCE: Taesung Kim ([email protected])
1
Department of Mechanical Engineering, Ulsan National Institute of Science and Technology
(UNIST), 50 UNIST-gil, Eonyang-eup, Ulsan 689-798, Republic of Korea.
2
Department of Biomedical Engineering, Ulsan National Institute of Science and Technology
(UNIST), 50 UNIST-gil, Eonyang-eup, Ulsan 689-798, Republic of Korea.
Contents
Supplementary Results and Discussion
Governing Equations
Modeling
Numerical Method
Ion Concentration Distributions
Boundary Conditions
Simulation Result of Flow Fields
Simulation Result of Electric Fields
Effect of Different EPMs and Charge Densities on ICP
Supplementary Table
Supplementary Figures
References
S-1
Supplementary Results and Discussion
Governing Equations
Ionic concentration of species i (ci), fluid velocity ( u ), and electric potential (ϕ) are the main
variables in the governing equations, respectively. The generation and development of ICP
phenomena depend on the complex effect of those variables on each other.
First, the Nernst-Planck equation is in charge of the concentration distribution of different
species of ions and trace molecules. The equation is shown as below:
ci
F


 ci  u    D i ci  zi ci Di
 
t
RT


(1)
where ci, Di, and zi are the concentration of an ion species i, its corresponding diffusion coefficient,
and ionic valence in an electrolyte solution, respectively. T represents the temperature of 25°C and
F and R are the Faraday and ideal gas constant, respectively.
Second, the Navier-Stokes equation describes the motion of incompressible fluid flow with
the continuity equation:

u
  (u )u  p  2 u   E 
t
 u  0
(2)
(3)
where  is the density of the solution,  is the dynamic viscosity, and  represents the net charge
density in the solution.
E  F i zi ci
(4)
Third, the Poisson equation relates the ionic concentration of a buffer solution to the electric
potential:
2  
E

(5)
where is the permittivity of the medium. Since the governing equations are coupled with each
other, we performed numerical multiphysics simulations to find out CEFs, ionic concentration
distributions, ionic currents, electric fields, flow fields including vortex flows, and so on.
S-2
Modeling
Each channel of the model was 2000 μm long and 100 μm wide so that the configuration of
an electric potential (VH = 4 V and VL = 2 V) produced an electric field of 103 V/cm in the anodic
channel. Since all the channels were connected with reservoirs, the concentration of ionic species
at the ends of the channels was ci = c0, which was the concentration of a buffer solution. The
concentration of trace molecules at the high potential end was ctr = 1 nM while ctr = 0 at the other
ends. The trace molecules were not permitted to pass through the nanoporous membrane due to
size exclusion. The initial concentration of the buffer solution in the channel and the membrane
was c(t = 0) = c0 while ctr(t = 0) = 0. The material properties used in the simulations are shown in
Table S1 and were obtained from many literatures. H+ and Cl- were used as cation and anion,
respectively, without considering the influence of H+ on deprotonation/protonation inside the
nanoporous membrane.
The diffusivity of ions in the membrane depends on the molecular dimension, pore
dimensions, porosity, etc. The value used in the simulations is based on the estimation from other
literatures. For example, Shen et al. used one tenth of the bulk value for both cations and anions
because the nanoporous membrane used in their work is Nafion membrane whose pore size is
around 3~5 nm, leading to a low diffusivity of ions in it[1]. On the other hand, Dhopeshwarkar et
al. used one half of the bulk value because the pore size of the membrane is about 100 nm[2]. The
values mentioned above were all estimated mainly based on the properties of the relevant
nanoporous membrane, so it appears that one tenth of the value used in our work would be
reasonable for a small pore-size membrane. As for other membranes, we can adjust the value case
by case, but it would not be easy to estimate an exact value without accurate experimental data.
Usually, it is believed that the diffusion of ions is not related to their charges and the surface charge
density in the membrane, but it indirectly influences its electrophoretic mobility according to the
Nernst-Einstein equation (µ i ~ Di/RT). In the Nernst-Planck equation, we can see that the charge
of ions influences the electrophoresis in the bulk solution or in the membrane, while the influence
of the surface charge density of the membrane on the EPM of the ions in the membrane seems not
to be formulated to our best knowledge. The factor of 4 was obtained by measuring current and
then calculating conductance to get an equivalent value for the electrophoretic mobility of proton
in a 2-nm wide nanochannel. It is certain that the factor varies, depending on the kind of ions and
the properties of the membrane, such as surface charge density, pore size, etc. It could be larger or
S-3
smaller according to different conditions. That is another reason why we used different EPMs in
the membrane for simulations.
Numerical Method
COMSOL Multiphysics (ver 4.3b), which is based on a finite element method, was utilized
to solve the governing equations. Correspondingly, the microfluidics module, chemical reaction
engineering module, and AC/DC module were utilized. High and low electric potentials were
applied at the right (VH) and the left (VL) end of the anodic channel, respectively, whereas the
electrodes at the cathodic channel were connected to an electric ground (VG). Both the anodic ends
were set up as an open boundary without external pressure. As a result, the electroosmotic slip
boundary condition significantly reduced the computational load for calculating EOF. That is, the
flow velocity (U) at the channel walls was approximated by the Helmholtz-Smoluchowski
equation as shown below:
U 
 Et

(6)
where 𝐸𝑡 is the tangential electric field and  is the zeta-potential at the charged wall surface. Since
the velocity profile of the EOF is fairly uniform across the channel and the electric double layer
(EDL) is as thin as 10 nm, the error caused by the electroosmotic slip boundary condition must be
negligible. The boundary conditions at the channel walls near the membrane were changed to nonslip ones by repeatedly validating the flow field, because the velocity profile of the EOF would be
locally broken due to the existence of extremely fast vortex flows. The reason will be described
later.
Ion Concentration Distributions
Once an electric potential is applied across a nanoporous membrane, ICP starts to develop
and an IDZ is generated near the membrane in the anodic channel. The extension of the IDZ is
shown in Fig. S1. Initially, the ion concentration in the entire channel is 1 mM. After the
application of an electric potential (VH = 4 V and VL = 2 V), a small IDZ (dark blue area) is induced
near the anodic interface in a second. As time goes on, the IDZ extends along the channel toward
the left end. As reported by Rubinstein and Zaltzman[7], the distribution of ion concentrations
S-4
depends on diffusion and convection while the influence of electrophoresis is negligible. Since the
EOF flows from the right end (VH) to the left end (VL), the extension of the IDZ is inhibited toward
the right end but accelerated toward the left end so that the IDZ continuously propagates only
toward the left end. The ion concentrations at both the ends of the anodic channel are assumed to
be kept at 1 mM because of the boundary conditions; for experiments huge reservoirs are used so
that the assumption is very common in numerical simulations. When the IDZ extends, the ICP
relatively reaches a steady state, although the ion concentrations in the cathodic channel
continuously increase. The sharp decrease of the ion concentrations in the anodic channel
significantly reduces the conductance of the device, while the mild increase of the ion
concentrations in the cathodic channel continuously increases the conductance. Therefore, the
ionic current shown in Fig. 6c and f in the main text sharply decreases initially and then increases
gradually. The pre-concentrated area of the trace molecules is also related to the IDZ. The sharp
change of the electric field near the IDZ generates an electrical barrier that prevents the trace
molecules from moving forward but helps them to be further accumulated.
Boundary Conditions
We used a combined boundary condition that divides the anodic channel walls into two regions:
near the membrane and far from the membrane (the rest parts) as shown in Fig. 1 in the main text.
It is straightforward that the electroosmotic slip boundary condition for the Navier-Stokes equation
is applied to the channel walls far from the membrane because it can induce a very similar bulk
flow as an EOF across the channel and the error is negligible (EDL ~ 10 nm). However, the
electroosmotic slip boundary condition for the entire channel walls can possibly interfere with the
generation of vortex flows and its additional flows (e.g. slow-flow zone); according to
experimental results, slow-flow zones exist. Therefore, the electroosmotic slip boundary condition
for the entire channel walls potentially holds a critical weakness and in turn it produces less
accurate simulation results especially for pre-concentration than the combined boundary condition.
Fig. S2 shows the flow field when a slip boundary condition (constant velocity calculated based
on the average electric field across the channel) was used near the membrane instead of the nonslip one; here, the elelctroosmotic slip boundary condition is not suitable because the extremely
strong, local electric field strength may cause an unrealistic velocity of EOF. Compared with the
flow field shown in Fig. 4a, the streamline adjacent to the wall boundary in the rectangle in Fig.
S-5
S2 is fairly straight, indicating that the local flow field is dominated by the slip boundary condition
(constant velocity) and the slow-flow zone is supposed to exist. Since the continuity of EOF near
the vortex is probably broken, it seems that the non-slip boundary condition is capable of better
simulating the mobility balance between electroosmosis and electrophoresis, which is very delicate
for the pre-concentration of the trace molecules near the IDZ, than the electroosmotic slip
boundary condition. In addition, almost no variation in a flow field was found when a slip boundary
condition was applied to the channel walls near the membrane instead of the non-slip one, meaning
that the inertia effect of the EOF is much weaker than that of the vortex flows. Since the slip
boundary condition typically does not generate hydraulic resistances at walls, it seems to be
unrealistic for experiments. For this reason, we applied the non-slip boundary condition to the
channel walls near the membrane. In real experimental conditions, it appears to be very complicate
and difficult to observe the transition from an EOF to vortex flows. Correspondingly, from the
view point of simulations, it seems to be intricate to exactly deal with the local boundary condition
of experiments. The existence of vortex flows could possibly interfere with the local generation of
an EOF so that the electroosmotic slip boundary cannot be valid near the membrane. Thus, the
application of the electroosmotic slip boundary condition to the entire channel walls can not only
be inappropriate but also eliminate slow-flow zones, leading to less accurate results. Consequently,
we validated that the combined boundary condition better compromises with our experimental
results from the view point of the formation of the slow-flow zones and vortex flows. In addition,
the border of the electroosmotic slip and the non-slip boundary conditions should locate as close
as possible to the slow flow zone to minimize the influence of the non-slip boundary condition on
the hydraulic resistance in the channel.
Simulation Result of Flow Fields
Basically, the velocity profile of an EOF across a microchannel is uniform, showing a flat profile
when the EDL is much shorter than the channel dimension. But, in the simulations, we found that
the velocity profile of an EOF can be flat, concave, and even convex when vortex flows exist. To
explore on this, we simulated the same model when no ion-permselective membrane was
employed (open junction) but other numerical parameters and boundary conditions were the same.
The velocity profile of the EOF along A–A’ at a steady state is shown in Fig. S3 (blue, open square).
In addition, the velocity profile of the EOF in a single and straight channel is shown as a reference
S-6
value (red, open circle). The velocity profile of the EOF shows a concave shape because the electric
field in the left half of the anodic channel is locally applied in the opposite direction (from VL to
the junction), which generates an additional EOF against the bulk flow. Without the electric field
which is in the opposite direction, the velocity profile would be flat as the reference value. The
concave profile of the EOF in Fig. 4b and the negative velocity of the flow at the walls in Fig. 4c
at t = 0.1 s are induced by the same reason. Interestingly, the velocity profile for both the cases
cannot develop to a convex shape in the absence of vortex flows.
Simulation Result of Electric Fields
Fig. S4a shows the distribution of the electric potential in the anodic channel at an initial state
and a steady state, respectively. The normalized arrows indicate the direction of electric fields. As
time goes on, the electric potential near the IDZ increases from the initial value of 1.7 V to the
steady state value of 2.6 V. This means that the direction of the electric field in the left half of the
anodic channel changes during the development of ICP and the strength of the electric field in the
right half at the steady state is weaker than the initial value, which could be proved by the velocity
profiles shown in Fig. 4b and c. Due to the electroosmotic slip boundary condition, the velocity of
flows at the channel walls is proportional to the local strength of an electric field. For this reason,
the velocity of flows at the channel walls at the steady state shown in Fig. 4b is smaller than the
initial value (t = 0.1 s), confirming that the strength of the electric field at the steady state is
relatively weaker than that at the initial state. Since the velocity near the walls shown in Fig. 4b
decreases at first and then increases, it could be inferred that the strength of the local electric field
initially decreases and then increases gradually. As mentioned in the main text, the initial sharp
decrease of ion concentrations in the IDZ and afterward the gradual increase of ion concentrations
in the IEZ explains the tendency of the change of electric fields and currents along the channels.
Fig. S4b shows the distribution of the electric potential along a–a’ and b–b’ at the steady state,
respectively. From this result, we can find two interesting facts. First, from the result along b–b’,
the potential drop in the left half is slower than that in the right half, meaning that the strength of
the electric field is rather weaker. This can explain why the velocity of the EOF near the walls is
relatively slower in the former than that in the latter as shown in Fig. 4b and c. Second, from the
result along a–a’, the sharpest potential drop happens near the anodic interface. Therefore, the local
strength of the electric field is extremely high as shown in Fig. 3. The potential drop in the cathodic
S-7
channel is much smaller than that in the anodic channel and the local strength of the electric field
is around 102 V/m.
Effect of Different EPMs and Charge Densities on ICP
Fig. S5a shows the velocity profiles of flows along A–A’ for the different EPMs of counterions in the membrane at a steady state. For the 0.1x EPM, the velocity profile is concave because
vortex flows are not strong enough to increase the EOF in the channel. That is, the electroosmotic
mobility (eo) is weaker than the EPM (i) of the trace molecules. Thus, the trace molecules cannot
be transported by the EOF, so that pre-concentration does not happen (CEF is nearly zero in Fig.
6a). With the increase of the EPM, the flow rate across the channel increases and causes eo to be
much stronger. Consequently, more trace molecules are delivered in the channel so that a higher
CEF is obtained. The distribution of the electric field across the membrane for the different EPMs
at a steady state is shown in Fig. S5b. As mentioned in the main text, the application of the 0.1x
EPM is inappropriate to simulate experiments. Conversely, when a high EPM (µ i > 0.5x) of
counter-ions in the membrane is assumed, the distributions of the electric field for the different
EPMs are close to each other at the anodic interface. However, the electric field varies significantly
in the membrane. This is because the local conductance is remarkably influenced by the EPMs.
Similarly, the sharp change of the electric field at the cathodic interface is induced by the sudden
change of the EPMs. The velocity profile along A–A’ and the distribution of the electric field
across the membrane for the different charge densities of the membrane are shown in Fig. S5c and
d, respectively. We can hardly tell the difference of these results except some minor difference of
the electric field in the membrane. In other words, the influence of the charge density on ICP is
relatively trivial compared with that of the EPM. Since the velocity profile and the distribution of
the electric field in the channel are similar, the electroosmosis and electrophoresis effect on the
trace molecules are also similar. As a result, the variation of the CEFs shown in Fig. 6d is
insignificant. Lastly, as reported by Yeh et al., the surface charge density in nanopores is related
to local ion concentrations[8], thus the non-uniform distribution of the ion concentrations leads to
the non-uniform distribution of the charge density, making the mechanism of ion transport inside
the membrane more complicated.
We also compared the influence of the charge density of the membrane for a relatively low EPM
(µ i = 0.5x). As shown in Fig S5, the variations of CEFs, ion concentration distributions, currents
S-8
and the velocity profiles of EOF induced by the change of the charge density of the membrane are
relatively insignificant compared with those induced by the change of the EPM of counter-ions in
the membrane. Since the influence of the charge density turned out to be trivial even for a relatively
low EPM condition, we emphasize that the ion-permselectivity of a nanochannel/nanoporous
membrane should be enhanced by a high EPM of counter-ions in it, which generates much more
reliable simulation results than the previous simulation results found in other literature.
S-9
Supplementary Table
Table S1 Properties used in simulations unless otherwise noted[3-6].
Diffusion coefficient of H+
Diffusion coefficient of Cl-
9.36 × 10-9 m2/s
2.032 × 10-9 m2/s
Diffusion coefficient of trace molecule
0.45 × 10-9 m2/s
Valence of trace molecule
-2
Diffusion coefficient of H+ in membrane
9.36 × 10-10 m2/s
Diffusion coefficient of Cl- in membrane
2 × 10-10 m2/s
Ion concentration in electrolyte solution, c0
1 mM
Assumed charge density of the membrane, 𝜌𝑓𝑖𝑥
-0.5 mM
-60 mV
Zeta-potential at the channel walls, 
S-10
Supplementary Figures
VL = 2
t=1s
VH = 4 V
t=5s
t = 10 s
t = 50 s
Fig. S1 The distribution of ion concentrations in the anodic channel when t = 1, 5, 10 and 50 s.
S-11
Fig. S2 The distribution of flow field when constant-velocity slip boundary condition is applied
near the vortex zone.
S-12
60
No membrane
Velocity (m/s)
50
Straight channel
40
30
20
10
0
0
20
40
60
80
100
Distance across channel (m)
Fig. S3 The velocity profile of an EOF in the anodic channel for a DC-ICP device and that of an
EOF in a single and straight channel in a SC-ICP device at a steady state when no ionpermselective membrane is employed. The average strength of an electric field is the same for both
cases.
S-13
a
VL = 2
Initial state
VH = 4 V
VL = 2
Steady state
2.6 V
1.7 V
b’
b
a
Electric potential (V)
b
VH = 4 V
4
3
Left half
Anodic part
2
1
0
a’
a-a'
b-b'
Right half
Near interface
0
1000
2000
Cathodic part
3000
4000
Distance along the line (m)
Fig. S4 (a) The distribution of an electric potential in the anodic channel at an initial (t = 0.1 s)
and a steady state, respectively. The normalized arrows indicate the direction of the electric field.
(b) The distribution of an electric potential along a–a’ and b–b’ at the steady state.
S-14
b
40
30
0.1x
0.5x
1.0x
2.0x
4.0x
20
10
0
Velocity (m/s)
c
Electric Field (V/m)
50
0
20
40
60
80
Distance along A-A' (m)
10
4
10
3
10
2
10
50
40
30
0.10 mM
0.25 mM
0.50 mM
0.75 mM
1.00 mM
10
0
20
40
60
80
Distance along A-A' (m)
Nanoporous
membrane
10
0
100
50
150
6
10
-0.10 mM
-0.25 mM
-0.50 mM
-0.75 mM
-1.00 mM
5
10
4
10
3
10
2
10
Nanoporous
membrane
1
100
100
Distance along the y-axis (m)
d
20
0.1x
0.5x
1.0x
2.0x
4.0x
5
1
60
0
10
Electric Field (V/m)
Velocity (m/s)
a
6
60
10
0
50
100
150
Distance along the y-axis (m)
Fig. S5 Flow fields and electric fields for the different EPMs of counter-ions and the charge
densities in the membrane at a steady state. (a) The velocity profiles along A–A’ at a steady state
when the EPMs are 0.1, 0.5, 1.0, 2.0 and 4.0 folds of the bulk solution. (b) The electric fields across
the membrane from the anodic to the cathodic channel along the y-axis under the same conditions
as (a). (c) The velocity profiles along A–A’ at a steady state when the charge densities are -0.1, 0.25, -0.50, -0.75 and -1.0 mM for the 4x EPM. (d) The electric fields across the membrane along
the y-axis under the same conditions as (c).
S-15
a
b
Ion concentration (mM)
200
CEF
150
100
50
0
Current (mA)
c
-0.10 mM
-0.50 mM
-0.75 mM
-1.00 mM
5
d
3
0
500
1000 1500
Time (s)
mM
mM
mM
mM
2
1
Nanoporous
membrane
0
50
100
Distance along the y-axis (m)
150
60
50
4
2
3
0
Velocity (m/s)
0.0
0.2
0.4
0.6
0.8
1.0
Charge density of membrane (mM)
6
-0.10
-0.50
-0.75
-1.00
4
40
30
20
10
0
2000
-0.10 mM
-0.50 mM
-0.75 mM
-1.00 mM
0
20
40
60
80
Distance along A-A' (m)
100
Fig. S6 Simulation results for a relatively low EPM (0.5x) when the charge densities are -0.1, -0.5,
-0.75 and -1.0 mM. (a) The highest CEF at t = 30 min. (b) Ion concentration distributions along
the y-axis at t = 10 s. (c) The transient ionic currents across the membrane. (d) The velocity profiles
of the EOF along A-A’ at a steady state.
S-16
Reference
1.
2.
3.
4.
5.
6.
7.
8.
Shen, M., et al., Microfluidic Protein Preconcentrator Using a Microchannel-Integrated
Nafion Strip: Experiment and Modeling. Analytical Chemistry, 2010. 82(24): p. 9989-9997.
Dhopeshwarkar, R., et al., Electrokinetics in microfluidic channels containing a floating
electrode. Journal of the American Chemical Society, 2008. 130(32): p. 10480-10481.
Dhopeshwarkar, R., et al., Transient effects on microchannel electrokinetic filtering with
an ion-permselective membrane. Analytical Chemistry, 2008. 80(4): p. 1039-1048.
Kirby, B.J. and E.F. Hasselbrink, Zeta potential of microfluidic substrates: 1. Theory,
experimental techniques, and effects on separations. Electrophoresis, 2004. 25(2): p. 187202.
Kirby, B.J. and E.F. Hasselbrink, Zeta potential of microfluidic substrates: 2. Data for
polymers. Electrophoresis, 2004. 25(2): p. 203-213.
Lide, D.R., Handbook of Chemistry and Physics. 1993.
Rubinstein, I. and B. Zaltzman, Electro-osmotic slip of the second kind and instability in
concentration polarization at electrodialysis membranes. Mathematical Models &
Methods in Applied Sciences, 2001. 11(2): p. 263-300.
Yeh, L.H., M.K. Zhang, and S.Z. Qian, Ion Transport in a pH-Regulated Nanopore.
Analytical Chemistry, 2013. 85(15): p. 7527-7534.
S-17