Ohmic resistance and constant phase element effects on cyclic voltammograms using a combined model of mass transport and equivalent circuits

Accepted Manuscript Ohmic resistance and constant phase element effects on cyclic voltammograms using a combined model of mass transport and equivalent circuits Patcharawat Charoen-amornkitt, Takahiro Suzuki, Shohji Tsushima PII:

S0013-4686(17)32443-X

DOI:

10.1016/j.electacta.2017.11.079

Reference:

EA 30669

To appear in:

Electrochimica Acta

Received Date: 23 June 2017 Revised Date:

31 October 2017

Accepted Date: 12 November 2017

Please cite this article as: P. Charoen-amornkitt, T. Suzuki, S. Tsushima, Ohmic resistance and constant phase element effects on cyclic voltammograms using a combined model of mass transport and equivalent circuits, Electrochimica Acta (2017), doi: 10.1016/j.electacta.2017.11.079. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Ohmic Resistance and Constant Phase Element Effects on Cyclic Voltammograms Using a Combined Model of Mass Transport and

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Equivalent Circuits

Patcharawat Charoen-amornkitta*, Takahiro Suzukia, Shohji Tsushimaa

Department of Mechanical Engineering, Graduate School of Engineering, Osaka

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University, Suita-shi, Osaka 565-0871, Japan

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* Corresponding author. Fax: +81 6 6879 4491

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E-mail address: [email protected] (P.Charoen-amornkitt)

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ACCEPTED MANUSCRIPT Abstract Cyclic voltammetry is a very useful tool for estimating several parameters such as electron transfer kinetics, diffusivity of active species, and effective surface area in a redox system. In cyclic voltammetry modeling, a simulated cyclic

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voltammogram is usually treated by neglecting the nonlinear effects of an electrical double layer and ohmic resistance. However, this approach leads to inaccurate prediction of such parameters. In this study, numerical modeling, including the

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combined effects of ohmic resistance, constant phase element, mass transfer, and faradaic processes, of cyclic voltammetry was carried out to show that, using this

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approach, nonlinear behaviors of the time-domain response of cyclic voltammetry can be encompassed. The model showed a good agreement with the experimental measurements. Furthermore, the numerical investigation of ohmic resistance and

Keywords

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constant phase element effects on a cyclic voltammogram were performed.

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CPE, Electrical Double Layer, Cyclic Voltammetry, Modeling, Non-Faradaic Current

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ACCEPTED MANUSCRIPT 1. Introduction Cyclic voltammetry, one of the most powerful electroanalytical methods used for studying the properties of an analyte or the electrochemical reactivity of an electrode, has been widely explored for several decades [1-4]. This method possesses

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several advantages, e.g., sensitive response to a reaction, easy implementation, and few parameters to be adjusted. In cyclic voltammetry, an immobile electrode is submerged in an unstirred analyte, to which a linear potential scan is applied. The

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analyte is intentionally left at rest to ensure that mass transfer occurs via the diffusion process alone. While the potential is changing, electrochemical reactions occur, and

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thus, the concentration of the species around the electrode surface changes. Cyclic voltammetry is, therefore, considered a transient method that involves mass transfer and electrochemistry. Moreover, besides the faradaic current obtained from the electrochemical reactions of interest, a cyclic voltammogram also involves a non-

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faradaic current generated from an electrical double layer. However, software packages available in the market, such as Digisim® [5-7], Digielch [8,9], and Comsol Multiphysics® [10], concentrate on the faradaic current alone and lack the practical

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non-faradaic current.

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The electrical characteristic of an electrical double layer, which produces the non-faradaic current, does not behave as an ideal capacitor but rather acts as a constant phase element (CPE) [11,12]. The time-domain responses of electrochemical systems with interfacial CPE behavior have received very less attention, although CPE has been widely accepted and used in the electrical equivalent circuits of electrochemical impedance spectroscopy (EIS) analysis [13-15]. Sagüés et al. [16] performed a numerical study of the time-domain response of reinforcing steel in concrete during a polarization resistance test based on the electrical equivalent circuit

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ACCEPTED MANUSCRIPT of the system. Sadkowski [17] conducted modeling for predicting the sweep voltammogram of corrosion systems in the presence of a CPE using fractional calculus. Feliu et al. [18] employed numerical simulation for estimating the corrosion rate based on the discretized approximation of a fractional derivative operator.

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Recently, Allagui et al. [19] provided derived-based expressions, showing that the nonlinear effect of electrical double layer capacitors on potential–time and current– potential plots can be described through the CPE behavior, which is not possible in

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the classical capacitor model. However, such studies, which concentrated on the timedomain responses of CPE, were mainly based on the corrosion systems and electrical

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equivalent circuit calculation, which therefore lacked mass transfer involvement. Montella [20] proposed a model for predicting the dynamic response of cyclic voltammetry in the presence of ohmic potential drop and interfacial CPE behavior using the generalized Mittag-Leffler function. To calculate the faradaic processes,

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they used the semianalytic approach and the so-called master equation proposed by Mahon et al. [21]. This approach calculated the mass transport using Laplace transform with some manipulations without spatial discretization on the transport field.

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The results of Mahon et al. [21] showed a good agreement with that of Digisim®,

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which required less than 1 s for calculating a 1000-point voltammogram on a typical personal computer. However, this method can only deal with uniformly accessible geometries, particularly semi-infinite geometries. Since they were mainly interested in attaining cyclic voltammograms, the proposed semianalytic approach obtained the results without achieving the time variation of the mass distribution, which is essential in certain specific cases. Therefore, to the best of our knowledge, there has been no study on the combined effects of ohmic resistance, CPE behavior, and faradaic processes on a

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ACCEPTED MANUSCRIPT cyclic voltammogram, which concerns the time variation of the mass distribution, validation of their results against the experimental results, and application to more complex geometries. Furthermore, these effects have not been highlighted by a previous study [20]. In this study, a conventional model for predicting the time-

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domain responses of cyclic voltammetry was combined with the electrical equivalent circuit concept to include the effects of ohmic loss and current flow over CPE, for simulating the total current flowing through the electrode. The results of the proposed

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model, which combines spatial discretization on the transport field and the equivalent circuit, were compared against those of a conventional model, which concentrated on

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faradaic current alone, to show that could the proposed model provide a better prediction of the system responses. Also, this study attempted to study the effects of the parameters obtained from EIS analysis, namely the pseudo-capacitance, CPE exponent, and ohmic resistance, on the responses of cyclic voltammetry by comparing

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the cyclic voltammograms. The results provided useful information about the interpretation of different shapes of the cyclic voltammogram that were affected by

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ohmic resistance and CPE parameters.

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2. Model Development

2.1 Modeling the faradaic current In the present study, a simple electrode reaction that occurs at a stationary

cylindrical electrode was considered. The cylindrical electrode was used since it is one of the simplest electrode assemblies [22] and can be conveniently constructed [23,24]. The reaction can then be described as follows:

R

O + ze −

5

(1)

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The electroactive species R lost their z electrons at the electrode surface, and subsequently, produced the electroactive species O . The cyclic voltammetry model was developed in FreeMat software using finite difference techniques, which included

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a set of several equations. Assuming that the mass transport of both species was held

∂c j

= ∇ ⋅ ( D j ∇c j )

(2)

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∂t

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by Fick’s second law, the diffusion equations can be expressed in a general form as

where the subscript j indicates the electroactive species R or O , c j is the species concentration, D j is the diffusivity of the species, and t is the time.

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The electrode kinetics can be described by the Butler–Volmer equation as

(3)

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j  (1 − α ) zFη   −α zFη  = k0 (cR ) s exp   − k0 (cO ) s exp   nF RT    RT 

where j is the current density generated via the faradaic processes, F is the Faraday

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constant, k0 is the reaction rate constant, subscript s indicates the electrode surface,

α is the charge-transfer coefficient, R is the universal gas constant, T is the temperature, and η is the overpotential, which is the difference between the actual potential applied to the electrode Eact and the formal potential E o ' of the redox couple. The current density at the electrode surface is related to the gradient of species concentration at the electrode surface, which can be expressed as 6

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j = DR ∇ ( cR ) s = − DO ∇ ( cO ) s nF

(4)

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Since the potential is swept linearly with time, the applied potential at any time can be given by

Eapp = Ei + ν t

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(5)

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where Eapp is the applied potential on the system, Ei is the initial potential, and ν is the sweep rate. In cyclic voltammetry modeling, the total faradaic current responds to the actual applied potential on the electrode surface, and is expressed as

(6)

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I f = j f Aeff

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where Aeff is the effective surface area at which the reactions occur. The initial

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conditions are given by

cR = cR* , cO = cO* at t = 0

(7)

cR (t ) + cO (t ) = cR* + cO*

(8)

with

where c*R and cO* are the initial concentrations of the electroactive species R and O , respectively.

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ACCEPTED MANUSCRIPT 2.2 Modeling the non-faradaic current

The impedance of the CPE in the Laplace domain can be simply expressed as

1 Y0 sγ

(9)

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Z CPE (s) =

where Y0 is the pseudo-capacitance, s is the Laplace variable, and γ is the CPE

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thus, Eq. (9) reduces to the following simpler form:

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exponent, which satisfies 0 < γ ≤ 1 . In an ideal capacitor case, γ is equal to 1, and

Z C (s) =

1 Cs

(10)

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where Y0 becomes C , which is the ideal capacitance. However, this equation cannot describe the nonlinear behavior of the current–potential relation, as stated by Allagui et al. [19]. Therefore, Eq. (9) was used for calculating the non-faradaic current, which

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can be expressed in the Laplace domain as

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I nf (s) =

E act ( s ) = Y0 s γ E act (s) Z CPE ( s )

(11)

where the subscript nf indicates the non-faradaic condition and E is the potential deviation from the initial condition. Taking the inverse Laplace transformation, the non-faradaic current in the time domain can be obtained as

I nf (t) =

d E act (t ) ∗ A(t ) + E act (0) A(t ) dt 8

(12)

ACCEPTED MANUSCRIPT with

Y0t −γ A(t ) = Γ (1 − γ )

(13)

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where the symbol ∗ denotes the convolution operation and Γ represents the gamma function. Note that Eq. (13) is identical to the expression derived by Sagüés et al. [16] and Sadkowski [17]. Hence, taking the convolution, Eq. (12) can be expressed as the

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sum of the integration of the response to the infinitesimal potential applied across the element and that to a finite potential deviated from the initial potential when the time

I nf (t ) = ∫

t

d E act (τ ) A(t − τ )dτ + E act (0) A(t ) dτ

(14)

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0

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is equal to zero.

By uniformly discretizing the time interval into n subintervals, the first term

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of Eq. (14) can be approximated as follows:

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k E act (t j +1 ) − E act (t j ) (j +1) ∆t Y0 d E act (τ ) Y0 (t − τ ) −γ d τ ≈ (tk +1 − τ )− γ dτ ∑ ∫0 dτ Γ (1 − γ ) ∫ j ∆ t Γ (1 − γ ) j = 0 ∆t t

k E act (t j +1 ) − E act (t j ) (k − j) ∆t Y0 ≈ τ −γ dτ ∑ ∫ (k − j + 1) ∆ t Γ (1 − γ ) j =0 ∆t

≈

k E act (t k +1− j ) − E act (t k − j ) (j +1) ∆t − γ Y0 ∑ ∫j∆t τ dτ Γ (1 − γ ) j =0 ∆t

k E ( j + 1)1−γ − j1−γ  ∆t 1−γ act (tk +1− j ) − E act (t k − j )  Y0  ≈ ∑ Γ (1 − γ ) j =0 ∆t (1 − γ )

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Y0 ∆t 1−γ k E act (tk +1− j ) − E act (tk − j )  ( j + 1)1−γ − j1−γ  ∑ Γ ( 2 − γ ) j =0 ∆t

A(∆t ) k E act (tk +1− j ) − E act (tk − j ) ( j + 1)1−γ − j1−γ  + E act (0) A(t ) (15) ∑ (1 − γ ) j =0

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I nf (t ) =

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Finally, Eq. (14) can be expressed as

divided into n subintervals.

2.3 Modeling the total current

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where t k is defined as k ∆t and ∆t is the time step by which the time interval is

In this study, we considered the RΩ + ( R f + Z w ) / Z CPE equivalent circuit

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(Randles circuit), where the symbols + and / represent series and parallel connections, respectively, RΩ is the sum of the ohmic resistances, R f is the charge-transfer

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resistance, and Z w is Warburg impedance. Thus, the total current flowing through the electrode is the sum of the faradaic current flowing through ( R f + Z w ) and the non-

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faradaic current flowing through Z CPE . While the set of equations in section 2.1 was used for calculating the faradaic current, the non-faradaic current was calculated using Eq. (15). The total current can be expressed as follows.

I total = I nf + I f

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(16)

ACCEPTED MANUSCRIPT Because of the presence of ohmic drop, the actual potential applied to the electrode can be modeled as

Eact = Eapp − I total RΩ

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(17)

Hence, combining the definition of the overpotential with Eq. (16), the

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overpotential is expressed as

(18)

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η = Ei + ν t − I total RΩ − E ° '

Considering the above equation, the faradaic and non-faradaic currents are then coupled through ohmic drop effects, and hence, cannot be solved separately. Using the set of Eqs. (3), (4), (5), (15), (16), (17), and (18), the actual potential

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applied to the electrode and the non-faradaic current at a given time can be evaluated. Inserting the actual applied potential back into the Butler–Volmer equation and the

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surface current equation yields the surface concentration. Solving the Fick’s second law by setting the obtained surface concentration as a boundary condition at one end

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and the concentration gradient at the other end gives the concentration profile at a time instant. Finally, the faradaic current is computed using Eq. (4) together with Eq. (6), and combined with the non-faradaic current to obtain the total current. The flowchart of the numerical algorithm for calculating the cyclic voltammogram is summarized in Fig. 1. The following assumptions were made on the model: •

The mass transport of both reducing and oxidizing species is governed by Fick’s second law. 11

ACCEPTED MANUSCRIPT •

The mass transport occurs only in the radial direction and does not vary in other directions.

•

The diffusivity of both reducing and oxidizing species has a common value of D obtained from Konopka and McDuffie [25].

Isothermal condition is assumed.

•

The Butler–Volmer equation describes the electrode reaction.

•

The electroactive species do not interact with each other in the bulk fluid and

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•

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there is no other reaction, except the reaction of interest.

Based on the finite difference method, both time and space domains need to be

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uniformly discretized into small grids. The accuracy of the results is directly proportional to the size of the grids. Therefore, the basic analysis, i.e., gridindependent analysis, was conducted prior to the calculation to ensure that the results did not significantly change with grid spacing. The computational nodes of 21, 51,

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101, and 501 in the space domain were used to investigate the grid independence in the faradaic current calculation. It was found that the cyclic voltammogram changed insignificantly if computational nodes beyond 101 were used. Thus, the computational

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nodes of 101, which corresponds to a grid size of 5 µm, were selected for discretizing

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the space domain. For the time-domain investigation, the grid-independent analysis was separately held in the faradaic and non-faradaic current calculations. The results revealed that, in the faradaic current calculation, 250 time steps were sufficient for obtaining a result that did not change significantly with grid spacing. In the nonfaradaic calculation, a very fine time step size was required for accurately capturing the rapid change in the initial region of potential sweeping. Nevertheless, after a few time steps, the results did not show a significant change with different numbers of time steps. Hence, the number of time steps of 450, which corresponds to the time

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ACCEPTED MANUSCRIPT steps of 0.1 s at a scan rate of 10 mV/s, was selected for discretizing the time domain since it was an acceptable trade-off between computational time and accuracy.

3. Experimental

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0.56 mm cylindrical carbon materials was prepared by heat treating at 400°C for 30 min to enhance the electrical double layer effects and remove the unexpected contamination, so that only the carbon surface was exposed to the electroactive

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species while being submerged in the electrolyte. A conventional three-electrode system was employed using a platinum counter electrode and a Ag/AgCl reference

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electrode. The Ferri/Ferrocyanide redox couple was selected for the experiment since it undergoes a standard single-electron transfer process and exhibits reversible behavior. Potassium Ferricyanide (K3[Fe(CN)6]), supplied by KISHIDA Chemical, was used together with deionized water to fabricate the initial electrolyte solution. 1.0

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M potassium chloride (KCl) was used as the supporting electrolyte. The initial electrolyte solution was injected into the supporting electrolyte to obtain the desired concentration of the Ferricyanide solution. Finally, the Ferricyanide solution was

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deaerated by nitrogen purging into the solution for 20 min before starting the

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experiment, and then tank blanketed with nitrogen during the experiment. Cyclic voltammetry measurements were carried out using a computer-

controlled electrochemical test system and run at several scan rates of 2, 5, 10, and 20 mV/s. Between each scan, the solution was stirred to restore the initial condition. The initial potential was 0.45 V, while the switching potential was 0 V. EIS measurements were performed prior to each cyclic voltammetry experiment with an AC signal of 10 mV from 20 kHz to 0.1 Hz. The obtained data were fitted using the equivalent circuit

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ACCEPTED MANUSCRIPT to evaluate the parameters used in the model, i.e., CPE exponent, pseudo-capacitance, and ohmic resistance. One of the main issues in modeling is finding the correct and realistic value of the parameters to be input into a model. In this study, therefore, the properties and

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parameters used in the modeling were obtained from the experiments, i.e., cyclic voltammetry and EIS measurements, and previous studies. Table 1 summarizes the properties and parameters. Note that the outer diameter of the computational domain

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should be equal to the inner diameter of the solution-containing vial. However, the investigation of Pérez-Brokate et al. [26], which had been conducted in the cellular

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automata, as well as ours, suggested that there was no significant change, even though other distances beyond the diffusion layer were used. The computational domain can, thus, be reduced to decrease the computational time.

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4. Results and Discussion

Fig. 2a presents a careful validation of the model used for predicting the cyclic voltammogram. The figure clearly shows a reasonable agreement between the

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numerical results of the model, including both faradaic and non-faradaic currents, and

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the experimental results. The model did not only respond accurately to the scan rate variation but also provided satisfactory responses to the different properties of CPE, as shown in Table 2. Fig. 2b displays the calculated non-faradaic current of the electrodes treated at 400°C for 30 min, while Fig. 2c depicts the calculated faradaic current. It can also be observed that these faradaic currents did not contain any round form around the potential switch, which proved that the non-faradaic current was not involved. The use of this model consequently led to a more convenient situation for evaluating the peak currents and potentials by the conventional approach of

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ACCEPTED MANUSCRIPT extrapolating baselines since the baselines of the faradaic current can be simply defined (Fig. 2c). On the other hand, when the non-faradaic current was included in Fig. 2a, the slope of the baselines was difficult to be defined since the non-faradaic current was nonlinear with respect to the potential sweeping, as shown in Fig. 2b. This

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led to possibly inaccurate peak current and potential evaluation.

The small discrepancy between these results around the reverse potential was apparently caused by the parameters obtained from EIS, especially the CPE

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parameters, as well as the side reaction. Nevertheless, overall, it is adequate to use the developed model since it can provide a good fit as compared to the conventional

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model [22,27], which concentrates on the faradaic current alone (Fig. 2c). Since the current depicted on cyclic voltammograms is the summation of the faradaic and nonfaradaic currents, using the model that concentrates on the faradaic current alone would overestimate the peak of the faradaic current, and hence, the effective surface

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area of the electrode. Although the conventional model can be conveniently and accurately used for studying the characteristics of a specific cyclic voltammogram, the

also involved.

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current model can be used in a more general case where the non-faradaic current is

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Although, in principle, the CPE exponent in the range 0 < γ ≤ 1 can be considered, in practice, for a flat rough electrode, it is usually higher than 0.5 and typically higher than 0.8 [28]. Hence, the CPE exponent of 0.5 was selected for investigating the pseudo-capacitance and ohmic resistance effects since it is considered an extreme case of a flat rough electrode. The numerical investigation of the pseudo-capacitance effects on the cyclic voltammogram is presented in Fig. 3a-c. Using the selected parameters, the total current in Fig. 3a, which increased asymmetrically with the larger increasing current, was found to be negative. This

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ACCEPTED MANUSCRIPT phenomenon was mainly caused by the non-faradaic current, as shown in Fig. 3b. The asymmetric non-faradaic current occurred due to the CPE behavior and potential sweeping, which started with a negative scan rate. When the electrical double layer did not behave as a pure capacitor, the non-faradaic current became asymmetric.

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Since the sweeping direction started with a negative sweep, the non-faradaic current deviated toward the negative region. As the pseudo-capacitance increased, the baselines for obtaining the peak current became difficult to be defined since the non-

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faradaic current during the forward sweep became more nonlinear. Using this proposed model, the faradaic current could be simply distinguished and shown in Fig.

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3c. The results revealed that although the non-faradaic current increased dramatically due to an increase in pseudo-capacitance and the faradaic current seemed to be larger when considering the total current in Fig. 3a alone, the faradaic current did not change. Hence, the peak current could be conveniently and accurately estimated in Fig. 3c.

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Fig. 4a-c shows the numerical results of the CPE exponent effects on the responses of cyclic voltammetry. As expected, the total current increased more toward the negative region as the CPE exponent deviated further away from 1 (Fig. 4a).

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When the CPE exponent equaled 1, it behaved as a pure capacitor, as mentioned

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earlier, and became a resistor when the exponent equaled 0. The results revealed that the non-faradaic current took an ideal rectangular shape as the CPE exponent equaled 1 and became sharper around the potential switch region as the value of the CPE exponent decreased. Although the pseudo-capacitance was set to be constant in this calculation, its magnitude when the CPE exponent equaled 0.3 seemed to be larger than that when the pure capacitor was considered. However, the magnitude of the pseudo-capacitance did not change and this was a behavior of the CPE that was affected by the exponent. Since the non-faradaic current shifted more toward the

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ACCEPTED MANUSCRIPT negative region as the exponent decreased in Fig. 4b, the peak current and potential of the total current in Fig. 4a when the anodic current dominated shifted toward the irreversible regime and slightly decreased in magnitude. In addition, the peak current when the cathodic current dominated increased significantly. Obviously, this might

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lead to a misunderstanding that the parameters, e.g., the rate constant, effective surface area, and initial concentration of the species, changed and the side reaction occurred when the CPE exponent equaled 0.3. Nonetheless, our model suggested that

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such changes can be caused by the CPE exponent alone and the faradaic current was not affected by the variation in the exponent (Fig. 4c). Note that a CPE exponent of

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0.3 was included in this investigation to show how a low CPE exponent can affect the cyclic voltammogram in principle, although the CPE exponent is usually found to be higher than 0.5 for a flat rough electrode [28].

Since ohmic resistance was always involved in the system, Fig. 5a-c displays

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the effects of ohmic resistance on cyclic voltammograms. The selected resistances were varied between 10 Ω and 2,000 Ω. This high resistance could be found in the cyclic voltammetry and it could be as high as 5,430 Ω [29], which was twice as high

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as the maximum magnitude used in this investigation. The ohmic resistance

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extraordinarily affected the cyclic voltammogram, as seen in Fig. 5a. The results revealed that, as the resistance increased, the peak current significantly shifted toward the irreversible regime, and when the ohmic resistance equaled 2,000 Ω, the peak current could not be observed. Again, this could lead to a misunderstanding that the reaction rate constants used in these cases not only differed between each case, but also differed when performing the forward and reverse sweep. The ohmic resistance seemed to provide a much larger effect on the peak current when the cathodic current was dominant, since it extremely shifted toward the irreversible regime, as compared

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ACCEPTED MANUSCRIPT to the case in which the anodic current had a higher influence. This was because of our selected parameters, which provided the magnitude of the negative current larger than that of the positive current. In addition, we observed that the ohmic resistance affected not only the peak current but also the non-faradaic current in Fig. 5b. When

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the ohmic resistance increased, the shape of the non-faradaic current was different from regular, which led to a more difficult situation for estimating the peak current. The conventional approach of extrapolating the baselines led to an inaccurately

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estimated peak current. This phenomenon occurred since the faradaic and nonfaradaic currents are coupled through ohmic drop effects.

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Moreover, there was a difference between the effects of CPE parameters and ohmic resistance since the faradaic current changed significantly with ohmic resistance variation, as depicted in Fig. 5c, while there was no observable change found with the variation in CPE parameters, as displayed in Figs. 3c and 4c. Such

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cyclic voltammograms could lead to inaccurate interpretation, as mentioned earlier. Nonetheless, the ohmic potential drop increased due to an increase in either the total current or ohmic resistance due to Ohm’s law. Although, in this case, only the ohmic

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resistance variation was found to affect the cyclic voltammograms, other parameters

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such as effective surface area, pseudo-capacitance, CPE exponent, and diffusivity of the electroactive species, which affected the total current by their variation, could also affect the cyclic voltammograms. The cyclic voltammograms, thus, had to be considered and analyzed carefully. The reason behind the shifting of the peak current and potential was that the actual potential applied to the electrode did not increase linearly as expected due to the ohmic potential drop. The potential actually applied to the electrode was the ohmic potential drop subtracted from the potential applied to the system, which has

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ACCEPTED MANUSCRIPT been already presented in Eq. (17). Fig. 6 shows the variation in the actual potential applied to the electrode during the potential sweeping. As the ohmic resistance increased, the actual scan rate became slower than that applied to the system, resulting in a higher applied potential at the potential switching point. Furthermore, since the

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actual scan rate was slow, the final potential applied to the electrode could not reach the expected potential, which was 0.45 V. Consequently, the peak current seemed to shift to the irreversible regime, as seen in Fig. 5c. However, this shifting resulted from

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plotting the currents against the potential applied to the system.

Alternatively, Fig. 7a-c plots the total, faradaic, and non-faradaic currents

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against the actual potential applied to the electrode. These graphs clearly show that the switching potentials differed in each case. Interestingly, Fig. 7c shows that the peak potential was same in each case, except for the case of 2,000 Ω ohmic resistance, in which the actual applied potential did not travel beyond the peak potential during

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the backward sweep. Nevertheless, we observed that the peak current decreased as the ohmic resistance increased. This decrease was the effect of the scan rate, which differed in each case. Therefore, the shifting of the peak potential and decrease in

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peak current in Fig. 5c can be apparently explained through Figs. 6 and 7c, and they

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were not the effects of the reaction rate constant variation. Owing to the problem that the effects of the electric double layer and ohmic

resistance occasionally could not be avoided, the results of this study provided an idea that they should be treated carefully, and using the present model, the important parameters, e.g., the peak current and potential, could be obtained conveniently through the faradaic current–potential plot. Since the faradaic and non-faradaic currents were coupled through ohmic resistance, the peak current and potential should be based on extrapolating the baselines in the faradaic current–potential plot rather

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ACCEPTED MANUSCRIPT than the total current–potential plot. Furthermore, this model can be conveniently extended to other more complex geometries and can be used in some other specific cases in which the solution is not left at rest to investigate the mass transport. A

the analyte solution flows is ongoing in our research group.

5. Conclusion

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further study of the time-domain responses of the electroanalytical methods in which

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A model for predicting the time-domain responses of cyclic voltammetry, which combined the effects of ohmic resistance, CPE behavior, mass transfer, and

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faradaic processes, was successfully developed in FreeMat software using finite difference techniques. The model validation was carried out and a good agreement was reached between the results of the developed model and the experimental data. The current model was also compared with the conventional model, which

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concentrates on the faradaic current alone. It was found that the conventional model overestimates the peak current and the effective surface area since it considers only the faradaic current, while in a practical situation, the non-faradaic current is also

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involved. In addition, we numerically investigated the effects of CPE parameters and

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ohmic resistance on the cyclic voltammograms. Based on our selected parameters, the faradaic current, which was the main parameter of cyclic voltammetry, did not change despite the total current dramatically changing with variation in CPE parameters. On the other hand, the ohmic resistance significantly affected the faradaic current via the ohmic potential drop. Due to the ohmic resistance, the actual potential applied to the electrode and the scan rate remarkably differed from those applied to the system. The effects of CPE parameters and ohmic resistance can lead to the misinterpretation of cyclic voltammograms. On the other hand, our model can provide a better insight for

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ACCEPTED MANUSCRIPT cyclic voltammogram interpretation and conveniently obtain the important parameters, e.g., peak current and potential, by separating the faradaic and non-

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faradaic currents.

Acknowledgment

This research was supported by Japan Science and Technology Agency

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(JST), Precursory Research for Embryonic Science and Technology (PRESTO), Grant No. JPMJPR12C6. The authors would also like to express their gratitude to the

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Ministry of Education, Culture, Sports, Science and Technology, Japan, for providing financial support under the scholarship program for foreign students.

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List of figures referred to in the text and their captions Fig. 1 Flowchart of the algorithm used for calculating the cyclic voltammogram including both faradaic and non-faradaic currents. Fig. 2 Model validation of (a) cyclic voltammograms of the electrode heat-treated at 400°C for 30 min, which consists of (b) non-faradaic current and (c) faradaic current, at different scan rates.

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ACCEPTED MANUSCRIPT Fig. 3 Numerical solutions of (a) total current, (b) faradaic current, and (c) nonfaradaic current of the different pseudo-capacitances under the conditions of Aeff = 50.386 mm2, E° ' = 0.241 V vs. Ag/AgCl, RΩ = 2 Ω, γ = 0.5, D = 7×10-6 cm2, k0 =

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7×10-5 m/s, α = 0.5, T = 298 K, ν = 10 mV/s, cO* = 1 mM, and cR* = 0 mM. Fig. 4 Numerical solutions of (a) total current, (b) faradaic current, and (c) nonfaradaic current of the different CPE exponents under the conditions of Aeff = 50.386

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mm2, E° ' = 0.241 V vs. Ag/AgCl, RΩ = 2 Ω, Y0 = 0.0005 Fs γ −1 , D = 7×10-6 cm2, k0 = 7×10-5 m/s, α = 0.5, T = 298 K, ν = 10 mV/s, cO* = 1 mM, and cR* = 0 mM.

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Fig. 5 Numerical solutions of (a) total current, (b) faradaic current, and (c) nonfaradaic current of the different ohmic resistances under the conditions of Aeff = 50.386 mm2, E ° ' = 0.241 V vs. Ag/AgCl, Y0 = 0.002 Fs γ −1 , γ = 0.5, D = 7×10-6 cm2,

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k0 = 7×10-5 m/s, α = 0.5, T = 298 K, ν = 10 mV/s, cO* = 1 mM, and cR* = 0 mM. Fig. 6 Variation in the actual potentials applied to the electrode at different ohmic resistances.

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Fig. 7 Results of (a) total current, (b) faradaic current, and (c) non-faradaic current in Fig. 5 plotted against the actual potentials applied to the electrode instead of those

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applied to the system.

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ACCEPTED MANUSCRIPT Table 1 Properties and parameters used in the simulation HT 400°C 30 min

Parameters

Analysis

Units

(Model validation) 0.560

0.560

mm

Boundary diameter

1.560

1.560

mm

Effective surface area

30.000

50.386

mm2

Formal potential

0.241

0.241

E vs (Ag/AgCl)/V

Ohmic resistance

6.473–7.996

10–2000

Ω

Pseudo-capacitance

2.155–2.462

0.1–2.0

mFs γ −1

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Electrode diameter

0.748–0.767

0.3–1

7×10-6

7×10-6

cm2

7×10-5

7×10-5

m/s

Charge-transfer coefficient

0.5

0.5

Temperature

298

298

K

2-20

10

mV/s

0.45

0.45

V

1

1

mM

0

0

mM

CPE exponent Diffusivity

Initial potential

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Scan rates

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Reaction rate constant

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Oxidized species concentration Reduced species concentration

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Table 2 Parameters of each case obtained from the EIS and used in the simulation 5 mV/s

10 mV/s

20 mV/s

Units

Ohmic resistance

7.996

7.570

6.678

6.473

Ω

Pseudo-capacitance

0.216

0.213

0.221

0.246

mFs γ −1

CPE exponent

0.767

0.766

0.761

0.748

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2 mV/s

Parameters

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ACCEPTED MANUSCRIPT Highlights Effects of CPE and ohmic resistance are encompassed in the proposed model.

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The model conveniently helps separating the faradaic and non-faradaic currents.

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The faradaic and non-faradaic currents are coupled through the ohmic drop effects.

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CPE and ohmic resistance significantly affect cyclic voltammetry responses.

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