Here I report an electrochemical simulation work that compares voltammetric current and resistance of a complex electrochemical reaction over a potential scan. For this work, the finite element method is employed which are frequently used for voltammetry but rarely for impedance spectroscopy. Specifically, this method is used for simulation of a complex reaction where a heterogeneous faradaic reaction is followed by a homogeneous chemical reaction. By tracing the current and its polarization resistance, I learn that their relationship can be explained in terms of rate constants of charge transfer and chemical change. An unexpected observation is that even though the resistance is increased by the rate of the following chemical reaction, the current can be increased due to the potential shift of the resistance made by the proceeding faradaic reaction. This report envisions a possibility of the FEMbased resistance simulation to be applied to understand a complex electrochemical reaction. Until now, resistance simulations are mostly based on equivalent circuits or complete mathematical equations and have limitations to find proper models. However, this method is based on the firstprinciples, and is expected to be complementary to the other simulation methods.
Introduction
Cyclic voltammetry (CV) is the best known electrochemical technique which can show complex electrochemical information such as faradaic and nonfaradaic, chargetransfer controlled and masstransfer controlled processes, reversible and irreversible reactions and so on. Even though CV is the best as a diagnostic tool for electrochemistry, it is strongly recommended to resort to other techniques if precise evaluation of electrochemical parameters is required.
[1

2]
Various kinds of pulse voltammetry, currentcontrolled techniques, and hydrodynamic methods may be used for specific purposes. Among them, electrochemical impedance spectroscopy (EIS) has attracted many attentions in the recent decades
[3]
because it resolves the electrode/electrolyte interface by frequencies to which individual electrochemical processes differently respond. Nevertheless, EIS is not so frequently used as CV is because it is hard to understand and explain the measured impedance data.
[2
,
4]
At that situation, computeraided simulation can help.
[5

6]
However, there still remain difficulties such as finding an proper equivalent circuit or mathematical expressions for theoretically modelled electrochemical processes.
[7

8]
Especially, when the electrochemical reaction is complex
[9]
or the electrode is modified with functional materials
[10]
, it would be harder or could be almost impossible to find a good electrochemical model to carry out the computeraided simulation.
In this report, I adapt the finite element method (FEM) to conducting the digital simulation,
[11

12]
where molecular species and their concentrations are digitized as finite elements of time and space at molecular levels and the firstprinciple calculation based on charge transfer, mass transfer and chemical reaction rate are carried out to find values of each finite element. Using this simulation method, I study an electrochemical reaction coupled with a chemical reaction by finding the relationship between the current of CV and the resistance of EIS.
 Theories and discussion
An electrochemical reaction can be described as O + e → R, and the electrode potential (
E
) and the resulting current (
i
) can be measured by voltammetry. When the potential is scanned along time, it can be expressed as a function of current and concentrations of redox molecules changing with the scanning time. Hence, the timederivative is described as below.
[1]
The first term on the right side is known as the polarization resistance,
R_{p}
, which tells the rate of change transfer at the electrified interface as described by eq(2)
[13]
, while the rest part tells the rate of mass transfer through the diffusion layer.
k_{f}
and
k_{r}
are rate constants for reduction and oxidation reactions,
α
transfer coefficient,
n
the number of electron transferred,
F
faradaic constant,
R
gas constant and
T
temperature. From the above equation, we can guess that 1/
R_{p}
= 1/
R_{p,f}
+ 1/
R_{p,r}
where each is respectively referred to the following equations.
Here, we can learn that ln
R
_{p,f}
and ln
R
_{p,r}
are linear with the electrode potential and the slopes are the same as those of the Tafel plots
[1]
and the total resistance,
R
_{p}
, is sum of them. By the way, the intercepts are different due to the concentrations of O and R at the electrified interface.
When an electrochemical reaction becomes complex by the proceeding or following chemical reactions,
R
_{p}
should be changed because
R
_{p,f}
and
R
_{p,r}
are dependent on concentrations of O and R species. When a 1order homogeneous reaction follows the heterogeneous faradaic reaction,
[14]
the changing concentrations are taken into account at eq(3). Simulations with changing
k
^{0}
and
k_{chem}
leads us to a short conclusion that
R_{p,f}
and
R_{p,r}
can be almost separately modulated to manipulate
R_{p}
. Decrease of
k
^{0}
increases the intercept of
R_{p,f}
while increase of
k_{chem}
increases that of
R_{p,r}
. In other words, the decreased
k
^{0}
slows down the electron transfer rate of the forward reaction of which the resistance is increased, and the increased
k_{chem}
decreases
C_{R}
(
t
) so that the resistance of eq (32) is increased. As results of properly controlling
R_{p,f}
and
R_{p,r}
, we observe different behavior of
R_{p}
. First, even when
R
_{p,f}
increases by decreased
k
^{0}
,
R_{p,r}
can be almost unchanged due to the facile consumption of the R species by the following chemical reaction (R → Y). Consequently, the unbalanced change of
R_{p,f}
and
R_{p,r}
will make a shift of the lowest
R_{p}
to the positive potential direction. Eventually at a very small
k
^{0}
, huge value of
R_{p,f}
overwhelms
R_{p,r}
such that the electrochemical reaction is totally controlled by the slow forward reaction. This situation is described as an irreversible faradaic reaction because only the forward reaction is dominantly observable and confirmed by simulation of cyclic voltammetry as shown in
Fig. 1(a)
. Also, the total
R_{p}
is seen to be mostly determined by the overwhelming
R_{p,f}
no matter
R_{p,r}
is controlled by
k_{chem}
as shown in
Fig. 1(b)
.
The cyclic voltammogram (a) and the polarization resistance (b) are obtained by the FEMbased simulation. Simulation parameters are following: the scan rate = 100 mV/s, k^{0} = 0.001 cm/s, α = 0.5 and varying k_{chem} = 0, 0.5, 1, 2, 5 and 10 mM/s. The arrow shows the increasing direction of k_{chem}.
On the other hand, when
k_{chem}
is gradually increased with high
k
^{0}
, the concentration of R,
C_{R}
(
t
), is decreased by eq(4) and
R_{p,r}
is increased by eq(3) subsequently while little change of
R_{p,f}
is expected from the change of
k_{chem}
.
Fig. 2(b)
shows how the total
R_{p}
changes according to the change of
k_{chem}
. The value of the lowest
R_{p}
increases and its position shifts to the positive direction as
k_{chem}
increases. These results can be clearly shown by simulations of cyclic voltammograms in
Fig. 2(a)
. It is not surprising that the position shifts are in accordance with the shifts of the current curves because the lowest
R_{p}
goes with the current peaks. However, it is surprising to observe that the current peaks increase with increase of
R_{p}
values.
The cyclic voltammogram (a) and the polarization resistance (b) are obtained by the FEMbased simulation. Simulation parameters are following: the scan rate = 100 mV/s, k^{0} = 1 cm/s, α = 0.5 and varying k_{chem} = 0, 0.5, 1, 2, 5 and 10 mM/s. The arrow shows the increasing direction of k_{chem}.
This ironic observation is explained from the aspect of overpotential. When the
R_{p}
curves or the current curves of a reduction reaction shift to the positive direction of potential, less overpotentials are required to activate such faradaic reactions. The reduced overpotential generates extra potential at a certain electrode potential. Such superfluous potential can facilitate the charge transfer process, and when the facilitation effect is large enough, it can beat the resistance of charge transfer,
R_{p}
. The reduced overpotentials (∆
E_{p}
) are estimated from the potentials of the lowest
R_{p}
, and plotted with the increased peak currents (∆
i_{p}
) in
Fig. 3
. The linear relationships is explained by the BulterVolmer equation below,
The potential shifts and peak currents obtained by varying values of k_{chem} are plotted. The plots are fitted to the ButlerVolmer equation as down the red line. Deviation is shown when the chemical reaction rate is too high because it severely decrease C_{R}(t) in eq(32). As a result of that, R_{p,r} is increased more than exp[−αFΔE/RT] so that the current becomes less than the expected value.
This is very similar to the Frumkin effect but works oppositely; the Frumkin effect corrects the current with decreased overpotential,
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16]
however this effect does with increased over potential.
Nevertheless, the accelerated charge transfer by the redundant potential is not always observable. When the extrapotential is not enough to overcome the resistance of transferring charges at the electrified interface, we can only observe potential shift without current increase. Such conditions can be made with moderate values of
k
^{0}
and
k_{chem}
. In
Fig. 4
, we can see that increases of
k
_{0}
and
k_{chem}
increase both
R_{p,f}
and
R_{p,r}
, which lead to increase of total
R_{p}
with small potentials shifts. At this situation, decrease of overpotential is canceled out by increase of
R_{p}
, so that effective acceleration of charge transfer is hampered by the stronger resistance. Eventually, no current increase is observed even with potential is shifted.
Fig. 5
shows the overall effects of the combination of
k
^{0}
and
k_{chem}
on the peak current. When
k
^{0}
is low,
k_{chem}
has little effect on the electrochemistry. However, when
k
^{0}
is high enough,
k_{chem}
makes significant effects onto the electrochemical reaction.
The cyclic voltammogram (a) and the polarization resistance (b) are obtained by the FEMbased simulation. Simulation parameters are following: the scan rate = 100 mV/s, k^{0} = 0.1 cm/s, α = 0.5 and varying k_{chem} = 0, 0.5, 1, 2, 5 and 10 mM/s.
The effects of the coplay between k_{chem} and k_{0} on change of the cathodic current peak (i_{cp}). The simulation conditions are the same in Fig. 1, 2 and 4.
 Experimental of the FEMbased simulation
All the simulations are done using Matlab. The concentrations of O and R species in the linear transfer space and time are defined by
C_{O}
(
x
,
t
) and
C_{R}
(
x
,
t
), and redefined by
C_{O}
(
n
,
m
) and
C_{R}
(
n
,
m
) as finite elements and framed in a 2Dmatrix. Here,
n
and
m
are dimensionless parameters determined by
n
=
x
/Δ
x
, and
m
=
t
/Δ
t
, respectively, as used in electrochemistry simulations.
[11
,
17

18]
For calculation for the diffusion of
j
species, the Fick’s first and second laws are modified and combined to express a dimensionless concentration,
C_{j}
(
n
,
m
), by the dimensionless parameters as shown below,
Here,
D_{M}
is a dimensionless diffusion coefficient obtained from
D
∆
t
/∆
x
^{2}
and called the model diffusion coefficient.
[17]
For calculation for the flux at the electrified interface, the charge transfer rate and the diffusion rate are balanced to yield the following equation,
For calculation for the change by the homogeneous chemical reaction, the concentrations are obtained by the following equation,
The constant
k_{M}
=
k_{chem}
∆
t
is a dimensionless parameter for the chemical reaction rate. To satisfy the conditions of the semiinfinite diffusion and the explicit finite elements,
D_{M}
< 0.5 and the maximum
n
>6(
D_{M}m
)
^{1/2}
/∆
x
are set.
Conclusion
So far, I have discussed on a complex electrochemical reaction, where a heterogeneous faradaic reaction is accompanied by a homogeneous chemical reaction from viewpoints of voltammetric current and resistance using the digital simulation. The digital simulation is conducted in the finite element method discretizing the dimension, time and concentrations of chemicals. The concentrations are used to theoretically calculate polarization resistances and their flux to obtain voltammograms which can be measured by experiments. By changing the rates of heterogeneous and homogeneous reactions respectively, it is learned how the resistance is reflected on the measurable voltammogram. With decreasing
k
^{0}
, the resistance increases to lead an irreversible faradaic reaction regardless of the chemical reaction. On the other hand, with increasing
k_{chem}
, the resistance increases with lowering overpotential. From this sense, a paradox that current is increased with increased resistance is explained. Also, the coplay of
k
^{0}
and
k_{chem}
studied to predict how the voltammogram appears.
From this report, I present a vision of FEMbased digital simulation study on impedance, more specifically
resistance
. Even though there are many electrochemical measurement techniques such as voltammetry and electrochemical impedance spectroscopy, we still have limitations in understanding the observations especially when the reactions are very complex. As shown in this report, such understandings can be facilitated by computer simulations.
[19]
While theoretical resistance data are mostly simulated based on the equivalent circuit models, ours are done using the finite elements at the molecular levels. Consequently, it is not necessary to find an appropriate equivalent circuit or mathematical equations of potential and current. For the time being, it is only shown that resistance simulation is possible at the molecular level. In the forthcoming reports, it will be reported how an equivalent circuit is modelled from our FEMbased resistance simulation results.
Acknowledgements
This research was supported by a Research Grant of Pukyong National University (2014).
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