Journal of Electroanalytical Chemistry 726 (2014) 27–35
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Journal of Electroanalytical Chemistry
journal homepage: www.elsevier.com/locate/jelechem
Highly disparate activity regions due to non-uniform potential
distribution in microfluidic devices: Simulations and experiments
Adrian Bîrzu a,b,⇑, Jasmine Coleman b, István Z. Kiss b
a
b
Department of Chemistry, Al. I. Cuza University, Carol I 11 Blvd., 700506 Ias!i, Romania
Department of Chemistry, Saint Louis University, 3501 Laclede Ave., St. Louis, MO 63103, United States
a r t i c l e
i n f o
Article history:
Received 25 December 2013
Received in revised form 18 April 2014
Accepted 2 May 2014
Available online 14 May 2014
Keywords:
Flow cell
Edge effect
Segmented electrodes
Laplace equation
Butler–Volmer kinetics
a b s t r a c t
In a microfluidic flow cell activity pattern can occur along a thin band electrode due to the potential distribution in the cell. For quantitative characterization of the pattern formation exclusively due to electric
effect and for elimination of interactions of reaction sites from concentration distribution along the flow
channel, a partial differential equation model is formulated for the spatiotemporal variation of electrode
potential with Butler–Volmer kinetics limited by mass transfer. At constant applied circuit voltage, with
increase of the electrode size a limiting current is achieved because of the spatial pattern formation. The
limiting current arises due to the formation of high activity at the downstream edge, and low (nearly
open circuit potential) activity at the upstream edge. The spatial pattern (e.g., ratio of active vs. inactive
region) depends on the electrode size, the applied voltage, the conductivity of the electrolyte, and the
distance from the downstream electrode edge to the reservoir. It is also shown that by placing equally
spaced insulating stripes on the electrode much of the activity can be retained and the current does
not decrease significantly due to the lessened surface area (as long as the surface area of the insulating
stripes is less than about 50% of the entire electrode area). The model simulations are interpreted with
a coupled ordinary differential equation model of segmented electrodes and the occurrence of the strong
edge effects is confirmed with experiments of a four-electrode electrode array in a microfluidic flow cell
with ferrocyanide oxidation.
! 2014 Elsevier B.V. All rights reserved.
1. Introduction
Microfluidic devices that incorporate electrochemical flowthrough cells gained considerable interest in electroanalytical
chemistry [1–3], where one electrode can be used as a detector,
or multiple electrodes can be integrated in a generator–collector
detector system [4–6]. In addition, the devices can be used as
kinetic probes [7–10], in miniaturized biological and electrochemical fuel cells as portable, renewable energy generators [11,12], for
electrosynthesis [13,14], or as template device in materials science
[15]. The mathematical modeling and theoretical analysis of electrochemical systems, which are often required for the description
and optimization of the microfluidic devices, are challenging
tasks because of the presence of multiscale, integrated processes
[16,17]. Detailed descriptions for mass transfer, reactions kinetics,
and potential drop in the electrolyte are often required for
characterization of the spatiotemporal response of the cell in terms
of concentration and current density profiles [18,19].
⇑ Corresponding author at: Department of Chemistry, Al. I. Cuza University, Carol
I 11 Blvd., 700506 Ias!i, Romania. Tel.: +40 728 148736.
E-mail address: abirzu@uaic.ro (A. Bîrzu).
http://dx.doi.org/10.1016/j.jelechem.2014.05.002
1572-6657/! 2014 Elsevier B.V. All rights reserved.
Modeling efforts have been often focused on considering one or
two dominant types of processes (e.g., mass transport and chemical reaction); even with such simplifications, a wide variety of
responses are possible. For example, modeling of convective flow,
diffusion, and mass-transfer limited (quasistationary) chargetransfer chemical reactions led to identification of six different
zones of different mass transfer characteristics of a single electrode
[20], and three major (sequential, coupling, and crosstalk) operational regions in dual electrode configurations [21]. Flow profiles
have been simulated computationally for various experimental
techniques such as chronoamperometry, linear and cyclic sweep
voltammetry [22–25] and, with similar approach, limitations on
experimentally measurable kinetic rate constants were obtained
as a function of cell geometry [7].
In addition to mass transfer, potential drop in the electrolyte
also has a great impact on the behavior of electrochemical cells.
In traditional macrocells potential drops can cause ohmic losses
resulting in bistability and oscillations [26], can impact explosive
growth of metastable pitting corrosion [27] and can induce
coupling resulting in stationary patterns [28,29], traveling and
standing waves [18], and synchronization patterns [30]. In
traditional configurations one of the simplest manifestation of
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A. Bîrzu et al. / Journal of Electroanalytical Chemistry 726 (2014) 27–35
pattern formation is the ‘edge effect’ when the shape of the electrodes imposes a pattern because of the high activity of the edges
[31]. The presence of potential drop distribution in the flow channel proves to be a great challenge in the modeling of microfluidic
devices. On one hand, ohmic drops can be considered as a necessary nuisance that could be avoided by non-traditional design of
counter [32] and reference [33] electrodes. In some examples,
modeling studies can help designing of cell geometry as it was
done with microfluidic electrokinetics [34] or with interdigitated
electrode arrays [35]. On the other hand, nonuniform potential
field can be advantageous, e.g., for bipolar electrode focusing [36]
or microelectrochemical gates and integrated circuits [37]. In
microfluidic settings the potential drop in the flow channel can
be very large because of the small volume of the conducting electrolyte. For example, this ohmic potential drop can result in nonuniform conducting polymer growth on a single electrode [15] or
in strong electrical coupling that can drive synchronous variations
of currents with electrocatalytic [38] and corrosion [39] systems.
In this paper, we explore the effect of potential distribution on
total current and current density distribution in a flow channel
for electrochemical reactions with Butler–Volmer kinetics with
mass transfer limitation on both single and segmented electrodes.
The primary goal of the model is to shed light on contributions of
non-uniform potential distribution to the unusually disparate
activity of the electrode edges. To eliminate interactions among
reacting sites due to concentration distribution, we consider uniform concentrations of analytes along the flow channel, and a thin
Nernst diffusion layer across the height of the flow channel. Numerical simulations of the model equations are carried out with a single
electrode to explore the effect of electrode size and electrolyte conductivity on quasistationary linear sweep voltammograms, in particular, to the onset potential and current density of limiting
current regions with large electrode widths. The role of edge effects
is clarified for electrode configurations with single (one-sided) and
double (two-sided) counter/reference electrode placements as well
as for segmented working electrodes. The numerical findings of
separation of high and low activity regions in the partial differential
equation model are interpreted with a coupled ordinary differential
equation model of segmented electrodes. Finally, the findings of
model equations are compared to an experimental measurement
with four segmented electrodes in microfluidic flow channel with
the ferrocyanide oxidation reaction on Pt.
Fig. 1. Electrochemical cell geometries. (a) 3-electrode flow cell, (b) Two counter
electrode (CE) cell and (c) Segmented working electrode (WE) cell. e.p. marks the
equipotential plane(s).
The migration current density (imig) is the sum of capacitive
(icap) and Faradaic (iF) terms. As a consequence:
C DL
!
@/DL
@ u!
¼ %r !! % iF ð/DL Þ
@t
@y WE
ð2Þ
where u is the electrostatic potential in the electrolyte, t the time, y
the vertical coordinate, /DL is the potential drop across the electric
double layer in front of the WE, r is the electrical conductivity, and
CDL is the double layer capacitance per unit area of the WE [40]. We
define dimensionless quantities:
u0 ¼
F
u
RT
ð3Þ
where F = 96,500 C/mol, R ¼ 8:314 molJ K and T is the absolute
temperature. The /DL is rescaled in the same manner to give /0DL .
The dimensionless Faradaic current density is:
0
iF
j0
2. The distributed system: model
iF ¼
The cell geometries used in this model are depicted in Fig. 1. All
of them represent two-dimensional approximation of a threedimensional microfluidic flow cell geometry, where the lateral
extension of the microcell was neglected. In the simplest geometry
from Fig. 1a, the working electrode (WE) of length W is embedded
in insulator, on the bottom side of the cell of length D and height H,
while L denotes the distance from the WE to the Reference Electrode (RE). Both RE and the Counter Electrode (CE) are modeled
as an equipotential plane on the right side of the cell. (The reference and counter electrode are typically placed at the reservoir,
which has minimal ohmic potential drop because of the large size.)
Fig. 1b represents a more symmetrical situation, when both left
and right walls of the cell are described by equipotential planes
kept at the applied voltage; this would correspond to a cell geometry with two CEs. Here L1 and L2 represent the distances from the
edges of the WE to the equipotential planes. Fig. 1c describes a segmented WE, where active electrode stripes alternate with insulating regions.
At the WE, the charge balance equation reads:
where j0 is the exchange current density. The dimensionless time is:
imig ¼ icap þ iF
ð1Þ
t0 ¼
ð4Þ
Fj0
t
C DL RT
ð5Þ
and the dimensionless coordinates are:
x0 ¼
x
;
L0
y0 ¼
y
L0
ð6Þ
where L0 = 0.1 cm. Replacing Eqs. (3)–(6) in (2), we obtain the
dimensionless charge balance equation:
!
0!
@/0DL
0
0@u !
¼
%
r
% iF ð/0DL Þ
@t0
@y0 !WE
ð7Þ
where the dimensionless conductivity is:
r0 ¼ r
RT
FL0 J 0
ð8Þ
For the simplicity of notation, we drop the prime, and all quantities referred from this point on, related to the spatially distributed system modeling, are dimensionless. In order to model the
A. Bîrzu et al. / Journal of Electroanalytical Chemistry 726 (2014) 27–35
WE kinetics, we used a simple Butler–Volmer kinetics with mass
transfer contribution [41]. The dimensionless reaction current density is:
iR ¼ e
b/DL
%e
%b/DL
ð9Þ
where b = 0.5 is the transfer coefficient. The dimensionless Faradaic
current density iF (assuming a renormalized mass transfer limiting
current of 1) is given by:
1 1
¼ þ1
iF iR
ð10Þ
which leads to
iF ¼
eb/DL % e%b/DL
% e%b/DL þ 1
eb/DL
ð11Þ
The potential of the metal of the WE is taken as reference and
thus the following relationship is valid between the drop of potential across the electrical double layer in front of WE and the electrostatic potential in front of WE:
/DL ¼ %ujWE
ð12Þ
In stationary state, according to Eq. (7) the capacitive current
density vanishes and the migration current density equals the Faradaic current density. The total current at the WE will be:
Itot ¼
Z
W
0
iF dx
ð13Þ
The potential at the right wall of the cell – in case of Fig. 1a and
c geometries – or at both lateral walls for Fig. 1b is equal to the
applied voltage (Vappl) in a classical three electrode setup with
potentiostatic control:
ujREþCE ¼ V appl
ð14Þ
We assume that the electrolyte is concentrated enough such
that at any point in the electrolyte electroneutrality condition is
valid and thus Laplace equation is satisfied:
Du ¼ 0
ð15Þ
On all the walls, excepting the electrode areas, a no-flux boundary condition was used, which describes an insulating area:
rujR ¼ 0
ð16Þ
Because the local current density in Eq. (11) is a unique function
of the drop of potential across the electrical double layer, the concentration distribution in the flow channel does not affect the solutions. This approximation is equivalent to the assumption that
there is no change of concentration along the flow channel (in
the x direction); the only concentration change occurs across the
height of the flow channel (in the y direction) in the form of a
Nernst diffusion layer that gives an overall kinetic expression
shown in Eq. (11). An investigation on the impact of concentration
distribution along the flow channel on the amperometric response
will be the subject of future work.
29
3. The distributed system: numerical simulation results
Fig. 2 shows the current density–voltage characteristics for
three different cases. The continuous line corresponds to the
point-like system and was obtained plotting iF from Eq. (11) as a
function of /DL. For low /DL values, b/DL ' 1 and, from Eq. (11), iF ( 2b/DL, i.e., the Faradaic current will increase linearly with /DL,
while for large potential drops the current saturates, iF ? 1. The
dots correspond to a spatially distributed system, where the WE
is short – the electrode length W = 0.1. (The other parameter values
are D = 12, L = 1, and r = 10; these parameters correspond to a
fairly conducting electrolyte with a relatively close placement of
WE to the reservoir). The applied voltage was fixed at different values, and the stationary value of the WE current density was determined. As it can be noticed from Fig. 2, the current–voltage
characteristic for a short WE is identical to that for a point-like system and saturation is reached for applied voltages around 10. This
result implies that a small electrode has uniform current distribution that represents a point-like electrode.
In contrast, when the WE is long enough (here, W = 8, triangles
on the plot), the potential drop in the microchannel has significant
impact; the current increases with the applied voltage only gradually and current plateau is achieved at very large applied voltage
(here, over 40).
Fig. 3 illustrates the dependence of the total current as a function of the WE length W, for different values of the operational
parameters: dimensionless conductivity r, lateral position of the
WE, L, and applied voltage, Vappl.
The cell geometry with the WE embedded in insulator prevents
the onset of uniform electrode states [16]. However, if the WE is
short enough, the potential distribution on the inner part of the
WE will be approximately uniform and Eq. (13) reduces to:
Itot ¼ WiF
ð17Þ
i.e., total current at the WE will be proportional to the electrode
length. This is visible in Fig. 3a for a cell with far WE placement
(L = 8). For small W values, Itot increases linearly with W. Fig. 4a1
shows the vertical component of the potential gradient, @@yu, in the
vicinity of the WE. In these plots, lower part of the colorbar means
large values of the migration current (active), while upper part
means inactive (close to zero potential gradient), and the lateral
extension of the cell is marked by the line at the bottom of the plot.
When the WE is short (W = 0.1), the vertical potential gradient is
nearly uniform at the WE.
As it is shown in Fig. 3a, beyond a critical electrode length, Wc,
the current saturates at a value Imax
tot . The origin of this saturation is
2.1. Numerical aspects
Laplace Eq. (15) was discretized using space centered finite difference schemes for considered cell geometries with a fixed grid
size of 0.01 & 0.01. The resulted algebraic systems were solved
using the HSL 2002 library [42]. The stability of the numerical solutions were confirmed with comparisons of potential distributions
with smaller grid sizes of 0.005 & 0.005 and comparisons of cell
resistance to a variable grid size (with 5825 nodes and 11,264
triangles) solution obtained previously [43]. The time dependent
boundary condition (7) was integrated using a fourth order
Runge–Kutta algorithm.
Fig. 2. Current density–voltage characteristics. Continuous line corresponds to
point-like system, points to a short WE and triangles to a long WE, for a spatially
distributed system with cell geometry 1.
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A. Bîrzu et al. / Journal of Electroanalytical Chemistry 726 (2014) 27–35
closer to the reservoir, both the critical width and the limiting current increase (see Fig. 3b for L = 1); higher portion of active region
can be achieved with close WE placement. Therefore, we see that
by closer placement of the WE, the spatial profile of the potential
distribution can be affected, perhaps because of the stronger coupling in the system due to the decrease in total solution resistance
[18]. Note also that for small electrode widths the slope of total
current vs. electrode width is the same (one) as expected for the
mass transfer limited activity. From Fig. 3a–c, the lateral position
of the WE, L, is kept constant, while the conductivity, r, is
decreased ten times. Consequently the cell resistance, Rcell,
increases with one order of magnitude and both the saturation current and the critical length strongly decrease.
According to Christoph and Eiswirth [44], the local charge balance equation at the WE, (5), can be written using the coupling
function method as:
V appl % /DL
@/DL
¼ %iF ð/DL Þ þ
þ const: )
@t
q
0
) ð/DL ðx0 ; tÞ % /DL ðx; tÞÞ dx
Z
0
W
H0 ðjx % x0 jÞ
ð18Þ
where q(x) is the local resistance felt at the position x on the WE, H0
is the coupling function and the constant in front of the integral
depends on the electrolyte conductivity r and cell geometry. Taking
a spatially uniform initial condition at the WE, the integral vanishes
in the r.h.s. of Eq. (12), allowing the calculation of the local resistance, q(x):
q¼
Fig. 3. Total current as function of electrode length for geometry 1, D = 12. (a) L = 8,
r = 10, the arrows correspond to Fig. 4 and (b) L = 1, r = 10; (c) L = 8, r = 1.
shown in Fig. 4a2 with W = 3.98 > Wc. Compared to the small
electrode in Fig. 4a1, the stationary state changes dramatically.
The vertical component of the potential gradient at upstream edge
(3 < x < 4) has similar values as for the uniform electrode. However,
for x < 3 the magnitude of the gradient drops and at the upstream
electrode parts the gradient (that produces the migration current)
diminishes. We thus see that the electrode surface is strongly
nonuniform with coexisting active (downstream) and inactive
(upstream) regions.
Because of the dependence of the Faradaic current on /DL, the
saturation current, Imax
tot , will depend on the applied voltage, Vappl.
When Vappl is decreased to 10 (Fig. 3a) the saturation current and
the critical width Wc decrease; the lower applied voltage is capable
of maintaining smaller active region. When the electrode is placed
V appl % /DL
limt!0 d/dtDL þ iF ð/DL Þ
ð19Þ
Then logarithm of the local resistance q is plotted function of
the horizontal coordinate along WE (see Fig. 4b1 and b2).
For a small electrode width (Fig. 4b1) the downstream edge of
the electrode has the lowest resistance. Although the vertical component of the potential gradient has a maximum value at the center of the WE, the maximum of q is closer to the upstream edge of
the WE because of the geometrical asymmetry of the cell. The values of q decrease at the WE edges, in a similar way to that
described by Newman for a disk WE embedded in insulator [16].
However, in our case, the geometry of the microcell leads to a more
complex local resistance distribution.
In Fig. 4b2, we kept constant all the parameter values from
Fig. 4b1 (including the lateral WE position) and the WE length,
W, was increased to 3.98. Compared to the previous case, the stationary state changes dramatically. Due to the microchannel properties, the local resistance q increases from low values at the
downstream WE edge to extremely high values for the remaining
part of the WE, including the upstream edge. These high q values
render most of the WE in an inactive state, where the values of
@u
are very low. Thus we see that the reason for the strongly non@y
uniform current distribution is the quickly (exponentially) growing
local resistance into the flow channel from the downstream edge.
We investigated the effect of a change in the cell geometry,
from Fig. 1a, to a more symmetrical one, with 2 CEs, depicted in
Fig. 1b on the features of the spatial pattern formation. The parameter values for these simulations are the same as in Fig. 3b, and
Vappl = 10. Under these circumstances, curve 1 from Fig. 5a coincides with the curve from Fig. 3b, for Vappl = 10, presenting the total
WE current, Itot, as a function of the WE length, W.
The second curve from Fig. 5a corresponds to the same lateral
distance as before (here, with L1 = L2), but for geometry depicted
in Fig. 1b. As a result, the transition from linear dependence to saturation occurs for larger W, and the saturation current doubles its
value compared to the previous case. The reason of these changes
can be noticed in Fig. 5b and c. Compared to the results from Fig. 4,
due to the symmetry of the cell geometry 2, the local resistance q
A. Bîrzu et al. / Journal of Electroanalytical Chemistry 726 (2014) 27–35
31
Fig. 4. (a) Vertical potential gradient and (b) local resistance distribution for L = 8, r = 10, D = 12, Vappl = 20, (1) W = 0.1 and (2) W = 3.98, cell geometry 1.
decreases to low values on both up and downstream edges of the
WE. Thus, the active region on the WE doubles its ‘active’ surface,
as it can be seen on the stationary vertical gradient distribution
from Fig. 5b, where the two active edges are clearly visible.
The consequences of the edge effect are studied further by using
the microcell geometry from Fig. 1c. Here, on the WE, conducting
and insulating regions alternate. The parameter values are the
same as in Fig. 3c, with W = 2, as a low conductivity value enhances
the edge effect. The total WE length, W, as well as its lateral position, L, are kept constant while a number of nins identical stripes of
insulators, each of length 0.09, are placed on the WE with equal
spacing. Assuming a uniform spatial distribution of the migration
current at the WE, the total current Itot would linearly decrease
with nins, reaching its zero value when the entire WE surface would
be covered with insulator. This variation corresponds to the dashed
line in Fig. 6a.
The values obtained from simulations are represented on the
same plot by points. Due to the edge effect, the observed decrease
in Itot when increasing nins is much smaller than the previous one.
In fact, the total current remains almost constant as long as the surface area of the insulators is kept less than 50% of the total surface
area. As a consequence, the total current density, i, plotted on
Fig. 6b, increases with nins. Some of the corresponding vertical gradient distributions in the electrolyte, in the WE area, are plotted in
Fig. 7.
In Fig. 7a, there is no insulating stripe on the WE, the extension
of which is marked with a line at the bottom of the figure. In this
plot is visible the active region located on the downstream edge
of the WE. Fig. 7b and c correspond to one and nine insulating
stripes, respectively. While for nins = 1 the change in the migration
current distribution is small compared to the case when no insulator is present, for nins = 9, the maximum absolute value of @@yu
increases from 0.25 to 0.33, contributing to the raise of the total
current density at the WE. For comparison, Fig. 7d presents the vertical gradient distribution for a continuous WE with the same active
surface and lateral position as in Fig. 7c.
4. Demonstration of pattern formation with electrode arrays:
numerical simulations and experiments
In order to further analyze and characterize the edge induced
pattern formation in the distributed system, we perform an analysis of behavior of segmented electrode arrays in numerical simulations and experiments.
4.1. Numerical simulations
Let us consider a four-electrode array in a flow channel shown
in Fig. 8a. To derive equation for the electrical coupling among
the electrodes, and equivalent circuit was constructed, shown in
Fig. 8b.
Each electrode is represented with a parallel combination of a
capacitive (cd) and a nonlinear Faradaic component. The four electrodes are coupled through the resistive elements of the electrolyte
(four resistors r). The potentiostat sets circuit potential of each
electrode to circuit potential m, therefore:
m ¼ e1 þ rði1 þ i2 þ i3 þ i4 Þ
m ¼ e2 þ rði2 þ i3 þ i4 Þ
m ¼ e3 þ rði3 þ i4 Þ
m ¼ e4 þ ri4
ð20Þ
where e1 ) ) ) e4 and i1 ) ) ) i4 are the electrode potentials and the currents of the four electrodes, respectively. The currents of the four
electrodes are obtained from capacitive, Faradaic, and coupling
terms:
32
A. Bîrzu et al. / Journal of Electroanalytical Chemistry 726 (2014) 27–35
Fig. 6. (a) Total current and (b) current density dependence on number of
insulating stripes on WE, for L = 8, r = 1, Vappl = 10, D = 12, W = 2, geometry 3.
Straight dashed line in (a) corresponds to a linear decrease of total current with nins.
%e1
1
cd dedt1 ¼ m%e
% jF ðe1 Þ þ e2ar
ar
%e2
%e2
cd dedt2 ¼ %jF ðe2 Þ þ e3ar
þ e1ar
%e3
%e3
cd dedt3 ¼ %jF ðe3 Þ þ e2ar
þ e4ar
ð22Þ
%e4
cd dedt4 ¼ %jF ðe4 Þ þ e3ar
Fig. 5. (a) Total current as function of electrode length for L = 1, r = 10, Vappl = 10,
for geometry 1 and geometry 2, respectively, (b) vertical potential gradient and (c)
local resistance distribution, for geometry 2.
1
i1 ¼ cd a dedt1 þ ajF ðe1 Þ þ e2 %e
r
2
2
i2 ¼ cd a dedt2 þ ajF ðe2 Þ þ e3 %e
þ e1 %e
r
r
3
3
i3 ¼ cd a dedt3 þ ajF ðe3 Þ þ e2 %e
þ e4 %e
r
r
ð21Þ
4
i4 ¼ cd a dedt4 þ ajF ðe4 Þ þ e3 %e
r
where a is the surface area of the electrodes, and jF(e) is the Faradaic
current density of the electrodes. By combining Eqs. (20) and (21)
we obtain a set of ordinary differential equations for the electrode
potentials:
These equations show that the electrode segments are coupled
with strength 1/ar; the coupling can be enhanced by using small
electrodes and strongly conducting electrolytes with small r values. The ODE system has two trivial approximate solutions. When
the coupling is strong (small ar values), the coupling terms will
dominate and thus for stationary solutions (de1. . .4/dt = 0) e1 =
e2 = e3 = e4 = m; this solution corresponds to the uniform potential
distribution solution in the distributed system. When the coupling
between the electrodes is weak, (large ar values) the coupling
terms in the equations will be small, therefore the upstream electrode will be inactive exhibiting open circuit potential eoc: e2 =
e3 = e4 = eoc, and the currents of these electrodes are zero
(jF(eoc) = 0). However, for the most downstream electrode, because
1
of the m%e
term, a high current state can be established at potential
ar
1
e1 that corresponds to m%e
¼ jF ðe1 Þ. This solution thus corresponds
ar
to the strong edge effects seen in the distributed system where
the downstream electrode behaves as if it were alone in the flow
channel, while the upstream electrodes are inactive.
The effect of electrode size on the appearance of uniform and
non-uniform current distribution can be demonstrated with
numerical simulations using a general kinetic formula that incorporates Butler–Volmer kinetics and mass transfer limitations:
A. Bîrzu et al. / Journal of Electroanalytical Chemistry 726 (2014) 27–35
33
Fig. 7. Vertical potential gradient for D = 12, L = 8, W = 2, r = 1, Vappl = 10. (a) One segment WE, (b) two segment WE, (c) ten segment WE and (d) A continuous WE with the
same L and active area as (c).
Fig. 8. Numerical simulations with discrete arrays. (a) Schematic of four discrete working electrodes (WE1–WE4) in a flow channel with reference (RE) and counter (CE)
electrodes placed at the end of the flow channel. (b) Equivalent circuit of flow channel. The electrodes, having double layer capacitance cd, are polarized at potential v. The
electrolytes separating the electrodes are represented with resistive elements r. RE/CE denotes the placement of reference/counter electrodes. (c) Linear sweep voltammetry
simulations of Eqs. (22) and (23) showing uniform currents of the four electrodes with Butler–Volmer kinetics and mass transfer limitations using electrodes sizes of
a = 10%5 cm2. Scan rate: 1 mV/s. (d) Simulations of strong edge effects in linear sweep voltammetry curves with electrode sizes a = 10%3 cm2. Scan rate: 1 mV/s.
1 1
1
¼ þ
jF jL 2j0 sinhð0:5ef Þ
ð23Þ
where jL = 10 mA/cm2 is the limiting current, j0 = 10%3 mA/cm2 is
the exchange current density, f = nF/RT = 38.95 1/V, and the symmetry factor is assumed to be 0.5.
With relatively small electrode size of 10%5 cm2, the (quasistationary) linear sweep voltammetry is shown in Fig. 8c (The
coupling resistance r is set to 10 kX). Because of the strong coupling the currents of the electrodes are almost identical at any
potential values. However, when the size of the electrodes is
increased to 10%3 cm2, the edge effects become prevalent, as it
can be seen in Fig. 8d: the downstream electrode reaches limiting
current at about v = 1.0 V; at this circuit potential the most
upstream electrode is largely inactive because it produces only
34% of the limiting current. Note also that the electrodes closer
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A. Bîrzu et al. / Journal of Electroanalytical Chemistry 726 (2014) 27–35
Fig. 9. Experiments: demonstration of edge effects in a four-electrode array in a microfluidic electrochemical cell. (a) Schematic diagram of the four-electrode flow cell during
the oxidation of ferrocyanide. The electrolyte solution is 0.01 M ferrocyanide/0.1 M potassium nitrate pumped through the 100 lm & 200 lm microfluidic channel at flow
rate Q = 1.5 lL/min. Each electrode segment is numbered, with the corresponding color denoting the current level at V = 1.0 V (color code is same as in panel d). RE: Ag/AgCl/
3 M NaCl reference electrode, CE: Pt counter electrode. (b) Optical microscope image of PDMS-based flow channel sealed over the four Pt band electrodes, WE1,2,3,4. (c) Linear
sweep voltammetry scan of ferrocyanide oxidation on Pt band electrodes. Scan rate: 10 mV/s, Q = 1.5 lL/min. Each electrode segment is plotted, denoted by the corresponding
electrode segment number. (d) Colormap of current profile across each individual Pt band WE during ferrocyanide oxidation at V = 1.0 V, corresponding to the scan in panel
(c).
the downstream edge have monotonically larger current values
along the flow channel.
4.2. Experiments
To demonstrate the edge effect in an experiment, the oxidation
of potassium ferrocyanide was studied in a four-electrode array in
a microfluidic flow cell. A schematic of the four-electrode microfluidic flow cell is shown in Fig. 9a.
The electrolyte solution is a 0.01 M ferrocyanide/0.1 M
potassium nitrate solution pumped through a 100 lm & 200 lm
poly-dymethil-siloxane linear microfluidic channel at flow rate
Q = 1.5 lL/min, which corresponds to a linear velocity of
0.125 cm/s. The electrolyte flows over the four Pt band electrodes
to the reservoir, which contains the Ag/AgCl/3 M NaCl reference
electrode (%35 mV vs. standard calomel electrode) and Pt wire
counter electrode (All potentials are given with respect to the
applied reference electrode.) The Pt electrodes were fabricated with
soft lithography technique described in a previous publication [38].
The width of each Pt electrode is 0.5 mm, the distance to the reservoir containing the RE and CE is 1.1 mm, and the spacing between
each electrode segment is 0.1 mm (see Fig. 9b).
The linear sweep voltammogram with four electrodes in the
flow channel is shown in Fig. 9c and d. With all four electrodes
in the flow channel, the more active downstream electrodes reach
the limiting current at V = 1.0 V; at the same time the current of the
upstream electrode, WE4, is suppressed due to the edge effect. At
1.0 V circuit potential where the downstream electrode already
reached the limiting current, the most upstream electrode reached
only 52% of the limiting current. Similar to the numerical predictions, the currents of the electrodes monotonically decrease with
increasing the distance from the most downstream electrode.
The simulations and experiments with segmented electrodes
thus demonstrate the prevalence of the pattern formation effects
with the segmented electrodes. It is expected that the patterns
will become dominant with large electrode sizes and weakly
conducting electrolytes. Note that the numerical simulations and
experiments contradict common expectation that the upstream
electrode would be the most active due to the oxidation of the
fresh electrolyte. In contrast, the downstream electrode was found
to be the most active due to the potential distribution effects.
5. Conclusions
The simulation results predict the coexistence of active and
inactive regions when the length of the electrode in a flow channel
exceeds a critical value. Above the critical length the total current
does not increase anymore with increasing the size of the electrode
because of the extremely high local resistance of the upstream
edge of the electrode. The model was constructed to explore the
potential drop effects and to eliminate contributions from variations of concentrations along the channel due to diffusion and convection. (The model does consider a thin Nernst diffusion layer
across the channel height that has impact on the current obtained
from the electrode through Butler–Volmer kinetics limited by mass
transfer in Eq. (11). Because of assumptions on concentration distribution, quantitative comparison could be possible only in (i)
micropolarization region where the surface concentration is the
same as bulk concentration, (ii) in reactions that are under kinetic
control (e.g., transpassive Ni electrodissolution [43]) where the
flow rate does not affect the behavior and (iii) in flow cells at large
flow rates where there are no concentration variations along the
flow channel. Since the interactions among reacting sites due to
concentration effects were discarded, the contributions from
potential drop effects can be clearly identified. For example, activity of the electrode does not decrease dramatically by incorporating equally placed insulators on the electrode; this feature may
contribute to the effectiveness of multi-electrode based detectors
that can enhance mass transfer to segmented electrodes [4]
without inactivating the electrode surface due to the suppressed
edge effects. Future studies will be directed toward considering
diffusion/convection that could add further complexity and the
A. Bîrzu et al. / Journal of Electroanalytical Chemistry 726 (2014) 27–35
possibility of non-monotonic variations of the current along the
flow channel due to the counteracting potential drop and concentration shielding effects.
Conflict of Interest
The authors declare that there is no conflict of interest with any
financial organization regarding the material discussed in the
manuscript.
Acknowledgements
This material is based upon work supported by the National Science Foundation under Grant Number CHE-0955555. A.B. acknowledges support from the Romania-US Fulbright Commission.
References
[1] J.C. McDonald, G.M. Whitesides, Accounts. Chem. Res. 35 (2002) 491–499.
[2] R.S. Martin, P.D. Root, D.M. Spence, Analyst 131 (2006) 1197–1206.
[3] P.C.H. Li, Fundamentals of Microfluidics and Lab on a Chip for Biological
Analysis and Discovery, CRC Press, Boca Raton, 2010.
[4] C. Amatore et al., Anal. Chem. 80 (2008) 4976–4985.
[5] D.A. Roston, P.T. Kissinger, Anal. Chem. 54 (1982) 429–434.
[6] R.S. Martin et al., Anal. Chem. 72 (2000) 3196–3202.
[7] M. Thompson, E.V. Klymenko, R.G. Compton, J. Electroanal. Chem. 576 (2005)
333–338.
[8] R. Compton et al., J. Appl. Electrochem. 18 (1988) 657–665.
[9] I. Dumitrescu, D.F. Yancey, R.M. Crooks, Lab Chip 12 (2012) 986–993.
[10] K. Aoki, K. Tokuda, H. Matsuda, J. Electroanal. Chem. 79 (1977) 49–78.
[11] J. Yang, Phys. Chem. Chem. Phys. 15 (2013) 14147–14161.
[12] J.W. Lee, M.-A. Goulet, E. Kjeang, Lab Chip 13 (2013) 2504–2507.
[13] R. Ferrigno et al., Electrochim. Acta 44 (1998) 587–595.
[14] C.A. Paddon et al., J. Appl. Electrochem. 36 (2006) 617–634.
[15] A. Power, B. White, A. Morrin, Electrochim. Acta 104 (2013) 236–241.
35
[16] J. Newman, K.E. Thomas-Alyea, Electrochemical Systems, third ed., Wiley,
Hoboken, NJ, 2004.
[17] J. Newman, Ind. Eng. Chem. 60 (1968) 12–27.
[18] K. Krischer, Principles of temporal and spatial pattern formation in
electrochemical systems, in: B.E. Conway, O.M. Bockris, R.E. White (Eds.),
Modern Aspects of Electrochemistry, vol. 32, Kluwer Academic, NY, 1999, pp.
1–142.
[19] F. Plenge et al., Z. Phys. Chem. 217 (2003) 365–381.
[20] C. Amatore et al., Anal. Chem. 79 (2007) 8502–8510.
[21] C. Amatore et al., Anal. Chem. 80 (2008) 9483–9490.
[22] N.V. Rees et al., J. Electroanal. Chem. 542 (2003) 23–32.
[23] R.G. Compton et al., J. Phys. Chem. 97 (1993) 10410–10415.
[24] A.C. Fisher, R.G. Compton, J. Phys. Chem. 95 (1991) 7538–7542.
[25] T.Y. Ou, S. Moldoveanu, J.L. Anderson, J. Electroanal. Chem. 247 (1988) 1–16.
[26] M.T.M Koper, Oscillations and Complex Dynamical Bifurcations in
Electrochemical Systems, in: I. Prigogine, S.A. Rice (Eds.), Adv. Chem. Phys.,
vol. 92, Wiley, NY, 1996, pp. 161–298.
[27] M. Orlik, Self-organization in Electrochemical Systems I, Springer, Berlin, 2012.
[28] M. Orlik, Self-organization in Electrochemical Systems II, Springer, Berlin,
2012.
[29] L. Organ et al., Electrochim. Acta 52 (2007) 6784–6792.
[30] I.Z. Kiss, T. Nagy, V. Gáspár, Dynamical instabilities in electrochemical
processes, in: V.V. Kharton (Ed.), Solid State Electrochemistry II, Wiley-VCH,
Weinheim, 2011, pp. 125–178.
[31] E. Gileadi, Electrode Kinetics for Chemists, Chemical Engineers, and Material
Scientists, Wiley-VCH, NY, 1993.
[32] G.M. Tom, A.T. Hubbard, Anal. Chem. 43 (1971) 671–674.
[33] E.V. Dydek et al., J. Electrochem. Soc. 159 (2012) H853–H856.
[34] A. Persat, M.E. Suss, J.G. Santiago, Lab Chip 9 (2009) 2454–2469.
[35] C. Guajardo, S. Ngamchana, W. Surareungchai, J. Electroanal. Chem. 705 (2013)
19–29.
[36] R.K. Perdue et al., Anal. Chem. 81 (2009) 10149–10155.
[37] B.Y. Chang et al., J. Am. Chem. Soc. 132 (2010) 15404–15409.
[38] I.Z. Kiss, N. Munjal, R.S. Martin, Electrochim. Acta 55 (2009) 395–403.
[39] Y. Jia, I.Z. Kiss, J. Phys. Chem. C 116 (2012) 19290–19299.
[40] A. Birzu, K. Krischer, Phys. Chem. Chem. Phys. 8 (2006) 3659–3668.
[41] A.J. Bard, L.R. Faulkner, Electrochemical Methods, second ed., Wiley, NY, 2001.
[42] HSL2002, A collection of Fortran codes for large scale scientific computation,
<www.hsl.rl.ac.uk>.
[43] A.G. Cioffi, R.S. Martin, I.Z. Kiss, J. Electroanal. Chem. 659 (2011) 92–100.
[44] J. Christoph, M. Eiswirth, Chaos 12 (2002) 215–230.