Analytical solution of the convection-diffusion equation for uniformly accessible rotating disk electrodes via the homotopy perturbation method

Journal of Electroanalytical Chemistry 799 (2017) 175–180

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Short communication

Analytical solution of the convection-diffusion equation for uniformly accessible rotating disk electrodes via the homotopy perturbation method

MARK

P.G. Jansi Rania, M. Kirthigaa, Angela Molinab, E. Labordab, L. Rajendrana,⁎ a b

Sethu Institute of Technology, Department of Mathematics, Kariapatti 626115, India Department of Physical Chemistry, University of Murcia, 30100 Murcia, Spain

A R T I C L E I N F O

A B S T R A C T

Keywords: Rotating disk electrode Mathematical modeling Homotopy perturbation method Convection-diffusion equation

The mathematical problem corresponding to a one-electron reversible electron transfer at a rotating disk electrode is solved under transient and steady state conditions by using the homotopy perturbation method. Analytical solutions for the time-dependent and stationary concentration profiles, current response and diffusion layer are deduced for finite values of the Schmidt number. The solutions enable us to obtain the response in chronoamperometry, normal pulse voltammetry and steady state voltammetry. The analytical results are assessed by comparison with previous analytical solutions for limiting cases as well as with numerical simulations, finding a satisfactory agreement.

1. Introduction Controlled enhancement of the mass transport rate in electrochemical experiments can be achieved by using hydrodynamic methods or microelectrodes. In the former case, mass transport conditions can be varied conveniently to resolve (electro)chemical phenomena of different kinetics: electron transfers, adsorption/desorption processes, coupled chemical reactions, etc. Different hydrodynamic methods have been developed over years (rotating disc/ring, channel, wall-jet and dropping mercury electrodes) and applied to the study of the most frequent reaction mechanisms: EC, EC′, ECE/DISP, etc. [1–9]. The rotating disc electrode (RDE) is still the most popular method [10] to which much theoretical work has been devoted as evidenced in Table S1. This includes both the derivation of analytical solutions and the use of numerical methods (see SI), the present work developing within the former context. Since Levich's seminal work [11], different analytical solutions for the current response at RDEs have been deduced. For steady state conditions, Levich [11] obtained his well-known expression for the RDE limiting current under the assumption of infinite Schmidt numbers (Sc); this limitation was overcome later by several authors that reported solutions valid for finite Sc-values (Newman [12], Gregory et al. [13], Montella et al. [8], Rajendran et al. [14]). The (semi)analytical theoretical treatment of several reactions mechanisms at an RDE under steady state conditions are also found in the literature: ECE [15], EC2 [16], DISP [17], … Theoretical works under non-steady state conditions are more scarce, although transient measurements are helpful for

⁎

mechanistic deductions and quantitative kinetic studies. Thus, for infinite Sc-values, several expressions for the transient limiting current are available (Bruckenstein [18], Siver [19], Newman [20], Kontturi [5], …). In this communication analytical expressions for the concentration profiles, diffusion layer and current-potential response of simple electron transfers at RDEs are derived using the homotopy perturbation method (HPM) [21]. The theoretical solutions cover both transient and steady state conditions as well as finite Sc-numbers (Sc > 100). 2. Mathematical formulation of the boundary value problem In general, the convection-diffusion equation can be used to describe the transfer of many physical quantities, such as particles and energy, as long as the transfer occurs only due to two processes: convection and diffusion. The general form of the convection-diffusion equation is

∂c = D∇2 c − v. ∇c ∂t

O+e↔R Eq. (1) in one dimensional form can be simplified to [1]

Corresponding author. E-mail addresses: [email protected] (A. Molina), [email protected] (E. Laborda), raj_sms@rediffmail.com (L. Rajendran).

http://dx.doi.org/10.1016/j.jelechem.2017.05.053 Received 6 April 2017; Received in revised form 9 May 2017; Accepted 30 May 2017 Available online 31 May 2017 1572-6657/ © 2017 Elsevier B.V. All rights reserved.

(1)

where c denotes the concentration of the diffusing species, D is the diffusion coefficient, v is the velocity of the electrolyte and ∇2 is the Laplacian operator. For a heterogeneous electron transfer at an RDE, (2)

Journal of Electroanalytical Chemistry 799 (2017) 175–180

P.G. Jansi Rani et al.

∂ci ∂c ∂ 2c + vz i = Di 2i ∂t ∂z ∂z

and the initial and boundary condition into:

(i ≡ O, R)

(3)

θO (z , 0) = 1, θR (z , 0) = 0 θO (∞, t ) = 1, θR (∞, t ) = 0

where ci is the concentration profiles of the oxidized and reduced forms and Di is the corresponding diffusion coefficient and vz the component of the fluid velocity normal to the RDE surface that can be described by the Cochran series solution of von Kármán equations [1,22,23]

vz = −0.51023

v−1 2Ω3 2z 2

1 + v−1Ω2z 3 + … 3

θO (0) =

ψ (τ ) =

(i ≡ O, R)

a 1 δ ⎛, ⎞ ⎝ D⎠

( )

∂c R ∂z z = 0

(6)

F ′ (E − E 0 ) RT

i (τ ) nFAcb D 2 3a1

(7) 0′

where E is the applied potential, E the formal potential and F, R and T have their usual meanings [1–3]. Once the above problem is solved and the concentrations profiles are known, the current response (i(t)) is calculated from

3

=

c b eη , 1 + eη

(8)

cb 1 + eη

a 13 ζ = z ⎛, ⎞ , ⎝ D⎠ − 1/2

θi =

ψlim (τ ) =

(9)

ci cb

(i ≡ O, R)

⎜

a 1 δ ⎛, ⎞ D ⎝ ⎠

(10)

where a = 0.51023 ν Ω , τ is the dimensionless time, ζ the dimensionless distance and θi the dimensionless concentration of the electroactive species i. Taking into account the definitions in Eq. (10), now Eq. (5) becomes into:

(i ≡ O, R)

⎟⎜

⎟

⎟

(16)

1 1 kτ 3 2 0.016666 5 + τ− + τ πτ 4 2 π π 1 + 0.0107142 k 2 τ 7 2 π

2

−

0.09375 3 kτ 6 (17)

that reduces to the Cottrell equation in the limit τ → 0 (i.e., t → 0 and/or 1 Ω → 0): ψlim (τ ) = πτ . Also, the thickness of the linear ‘diffusion’ layer can be calculated from Eq. (17) as follows:

3/2

∂2θi (ζ , τ ) ∂θ (ζ , τ ) ⎞ ∂θi (ζ , τ ) = (ζ 2 − kζ 3) ⎛⎜ i ⎟ + ∂ζ 2 ∂τ ⎝ ∂ζ ⎠

(15)

where θ2(ζ, τ) is the term obtained in the third iteration (given in SI3). Note that the concentration profile of the reduced species can be immediately calculated from Eq. (16) taking into account that: θO(ζ, τ) + θR(ζ, τ) = 1. From Eqs.(14) and (16), the following expression for the transient current response under limiting current conditions (ψlim) is deduced:

Note that this also enables us to de-couple the mathematical problems corresponding to species O and R (see below). To proceed with the resolution of the problem, the following dimensionless parameters are introduced:

τ = (Da2)1 2t ,

⎟

⎜

c R (0) =

(14)

ζ ⎞⎛ ζ 3 τζ ζ ⎞ + ⎞ θO (ζ , τ ) = erf ⎛ + erfc ⎛ τ 2 τ 2 24 4⎠ ⎝ ⎠ ⎝ ⎠⎝ 2 4τ ) 2 3 ( − ζ kζ kτζ ⎞ e τ ⎛ζ + − − + θ2 (ζ , τ ) π 4 2 ⎠ ⎝4

Provided that the diffusion coefficients of the two electroactive species are equal (DO = DR = D), it can be demonstrated that the total concentration of electroactive species remains constant at any time of the experiment and in any region of the solution, that is: cO(z, t) + cR(z, t) = cb. Combining this result with the Nernstian condition in Eq. (6), the surface concentrations of the electroactive species are immediately obtained:

cO (0) =

∂θ = −⎜⎛ O ⎟⎞ ⎝ ∂ζ ⎠ζ = 0

θO (ζ → ∞, τ ) − θO (ζ → 0, τ ) (∂θO ∂ζ )ζ = 0

⎜

i (t ) ∂c = −DO ⎛ O ⎞ FA ⎝ ∂z ⎠ z = 0

3

The homotopy perturbation method (HPM) has been proven very powerful in a variety of problems in physics and engineering [24–26]. This method is a combination of homotopy in topology and perturbation techniques in functional analysis and in a new approach of the HPM [27] only a few iterations are needed to find an asymptotic solution. The problem given by Eqs. (11)–(12) is solved using the HPM (details given in the Appendix) for the application of a potential pulse ′ under limiting current conditions (i.e., E < < E0 so that θO(0) = 0). The following analytical expression for the dimensionless concentration profile of the oxidized species is obtained:

with:

η=

(13)

2.1. Analytical expressions for the transient concentration profile and the current response using the homotopy perturbation method (HPM)

cO (z , 0) = c b, c R (z , 0) = 0 cO (∞, t ) = c b, c R (∞, t ) = 0 cO (0, t ) = e ηc R (0, t ) = − DR

3

and the dimensionless thickness of the linear ‘diffusion’ layer by:

subject to the following boundary conditions where it is assumed that only the oxidized species is initially present, the electron transfer is reversible and the diffusion coefficient of the two electroactive species is equal [1]:

∂cO ∂z z = 0

(12)

Given that typical Sc values are larger than 100, then typical k-values are below 0.18. The dimensionless current is given by:

(5)

( )

1 1 + eη

v −1 k = 0.8175 × Sc −1 3 = 0.8175 × ⎛, ⎞ D ⎝ ⎠

with v being the kinematic viscosity of the electrolyte, and Ω the angular velocity of the electrode. By considering the first two terms in Eq. (4), an accurate description is achieved for most solvents (Schmidt number ≥ 100 [8,9]) (see below). Thus, taking the first two terms in the Cochran expansion into Eq. (3) yields,

DO

θR (0) =

where k is a function of the Schmidt number (Sc) defined as:

(4)

∂ci 1 ∂c ∂ 2c + ⎛−0.51023 v−1 2Ω3 2z 2 + v−1Ω2z 3⎞ i = Di 2i ∂t 3 ∂z ⎝ ⎠ ∂z

eη , 1 + eη

3

=

θO (ζ → ∞, τ ) − θO (ζ → 0, τ ) = [ψlim (t )]−1 (∂θO ∂ζ )ζ = 0

(18)

When the diffusion coefficients of the electroactive species are equal and there are no chemical or electrochemical kinetic limitations, the current response of any electron transfer mechanism upon the application of a potential pulse only differs from the limiting current value by a potential-dependent factor, the form of which depends on the reaction scheme. In the case of one-electron transfers, this factor is 1 η

(11)

1+e

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τ

1.0

=

θO (ζ ) = 1 − 0.3733 Γ

0.01

0.8

steady state

0.4

⎡ + k 2 ⎢−0.0387285 Γ ⎣ +

0.2

0.0

(

)

1 1 3 , ζ 3 3

2 1 ⎤ + 0.269198 k Γ ⎜⎛ 3 , 3 ζ 3) ⎥ ⎝ ⎦

0.3

θ

)

1 1 3 , ζ 3 3

⎡ − k ⎢0.13607 Γ ⎣

0.1

0.6

(

(

) + 0.098123 Γ ⎛⎝

1 1 3 , ζ 3 3

⎜

2 1 3⎞ , ζ 3 3 ⎟

6 3 ⎛⎜ 0.0242656 ζ − 0.00970645 ζ + 2 ⎝ 0.070759 ζ − 0.0291193

0.0

0.5

1.0

1.5

2.0

2.5

⎠

ζ3 e− 3

⎞⎟ ⎤ ⎥ ⎠⎦

where Γ is the upper incomplete gamma function. By making k → 0, Eq. (21) yields the solution for infinite Schmidt number (Eq. (8) in [8]). From Eq. (21), the following expression for the stationary limiting current (ψlim , SS) is obtained:

ζ Fig. 1. Dimensionless concentration profile of the oxidized species (Eq. (16)) in a limiting-current chronoamperometry experiment for a finite Schmidt number (Sc = 100).

ψlim, SS = ψlim (τ → ∞) = 0.77645 − 0.283037 k − 0.0805585 k 2 such that the transient current-potential response at an RDE can be expressed as follows:

ψ (τ ) = =

+

{

1 π

1 πτ

ψSS = ψ (τ → ∞) = 1

+ 4τ −

kτ 3 2 2 π

0. 0107142 k 2 τ 7

+

0 . 016666 π

τ5

2

−

0 . 09375 6

}

(22)

as well as for the steady state voltammetry:

1 ψ (τ ) 1 + e η lim

1 1 + eη

(21)

1 (0.77645 − 0.283037 k − 0.0805585 k 2) 1 + eη

(23)

Note that the first term in Eq. (22) correspond to the known value of the dimensionless steady state current at an RDE for very large Schmidt numbers (i.e., k → 0) [8].

kτ 3

2

(19)

3. Discussion The expressions deduced in previous sections enable a comprehensive description (concentration profiles, diffusion layer and current response) of the RDE system from transient to steady state conditions and for finite Sc-values, as shown in Figs. 1–4. The latter contrasts with previous analytical treatments of the transient RDE problem where infinite Sc numbers were assumed as only the first term of the Cochran expansion for the fluid velocity was considered (Eq. (4)). The analytical solutions for the current response (Eqs. (17) and (22)) have been validated by comparison with numerical simulations. Fig. 2 shows the comparison between the dimensionless current values obtained from the analytical solutions here deduced (Eqs. (17) and (22) of the main text) and finite-difference numerical simulations [28] where the component of the fluid velocity normal to the RDE surface is accurately described by [29,30]:

2.2. Analytical expressions for the steady state concentration profile and the current response using the HPM Under steady state conditions, the dimensionless form of the convection-diffusion equation concentration (Eq. (11)) is as follows:

dθ (ζ , τ ) ⎞ d 2θi (ζ , τ ) (ζ 2 − kζ 3) ⎜⎛ i ⎟ + =0 dζ dζ 2 ⎝ ⎠

(i ≡ O, R)

(20)

that is subject to the boundary conditions given in Eqs. (12). The analytical expression for the dimensionless steady state concentration profile under limiting current conditions using the HPM after 3 iterations is given by:

Fig. 2. Comparison of the current response at an RDE under transient (short τ-values) and steady state conditions as obtained from the analytical solutions (Eqs. (17) and (22)) and from numerical simulations [28].

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P.G. Jansi Rani et al.

Fig. 3. (A) Dimensionless current-time response (Eq. (17)) and thickness of the diffusion layer (inset, Eq. (18)) in a limiting-current chronoamperometry experiment with an RDE for different Sc-values. The case corresponding to diffusion-only mass transport is also included (grey line) for comparison. (B) Schmidt number correction in the determination of diffusion coefficients from the steady-state limiting current at an RDE: theoretical Levich plots (Eq. (22)) for infinite Sc-numbers and for Sc = 189, cb = 10(− 6) mol/cm3; (inset) error in the determination of diffusion coefficients when neglecting the Schmidt number effect.

Fig. 4. Dimensionless (A) NPV (Eq. (19)) and (B) steady state responses (Eq. (23)) at an RDE for different Sc-values (shown on the curves). Insets correspond to the derivative voltammograms.

10

∑ aj j=2

vz = − ν Ω 1+

1 0.88447

Ω ν

10

j

j=2

satisfactory agreement (< 0.25% deviations for Sc ≥ 10) with numerical simulations (which include up to z10-order terms in vz) and previous steady state solutions [8,12–14] as shown in Table 1. Hence, the analytical results here reported enable a satisfactory description of the current response at RDEs in all time regimes for most typical Sc-values: Sc > 100. The Schmidt number effect on the current response is shown in Fig. 3A. The thickness of the diffusion layer (δ) has also been evaluated (inset in Fig. 3A) given the importance of this magnitude that accounts for the mass transfer rate of electroactive species and the extent of the perturbed region in solution, with important implications with respect to, for example, the possible incidence of non-uniform accessibility to the RDE surface. As can be inferred from Fig. 3A, the Schmidt number correction becomes more significant as τ increases; for example, the current and δ-values for Sc = 100 differ by > 5% from the value predicted for Sc → ∞ when τ > 0.8 and by 6.7% under steady state conditions (not shown). The above can have an effect on the accuracy of the determination of diffusion coefficients with the RDE method. Thus, Fig. 3B shows theoretical Levich plots calculated from Eq. (22) for a hypothetical system with infinite Sc-value (black line) and for Sc = 189 (red line), the latter corresponding to ferrocene in acetonitrile at 25 °C [31,32]. As expected from Eq. (22), finite Sc-values do not distort the linearity of the current vs Ω1/2 plot (Fig. 3B) but it does affect the value of the slope: the smaller the Sc-value, the smaller the slope. Hence, underestimated values of the diffusion coefficient are obtained from the analysis of the Levich plot of a system with finite Sc-value with a theoretical model for infinite Sc numbers. The magnitude of such underestimation is plotted in the inset of Fig. 3B and it is larger than 5% for Sc < 800. Finally, the transient and steady state current-potential responses of a reversible electron transfer at an RDE are displayed in Fig. 4 as calculated from Eqs. (19) and (23), respectively. As can be inferred from

j

( z) z) ∑a( Ω ν

j

(C1)

where the values for the coefficients aj are given in [29]. As can be observed, the transient HPM solution (Eq. (17)) describes accurately (error < 3%) the current response for infinite and finite Scnumbers when τ < 1. In the limit of large τ-values, steady state conditions are practically achieved (within 3%) where the current is accurately described by the steady state HPM solution (dotted line, Eq. (22)); indeed, the steady state HPM solution (Eq. (22)) shows a very Table 1 Comparison of the steady state current (Eq. (22)) with previous analytical solutions [8,12–14]. Sc

This work Eq. (22) (3 terms)

Diard and Montella [8] (5 terms)

Newman [12]

Gregory [13]

Rajendran [14]

10,000 5000 2000 1000 800 500 200 100 50 20 10

0.7656 0.7627 0.7577 0.7528 0.7509 0.7464 0.7353 0.7241 0.7097 0.6839 0.6574

0.7656 0.7628 0.7578 0.7529 0.7510 0.7466 0.7356 0.7244 0.7103 0.6848 0.6588

0.7657 0.7628 0.7579 0.7530 0.7511 0.7467 0.7359 0.7251 0.7113 0.6874 0.6639

0.7662 0.7634 0.7586 0.7538 0.7520 0.7478 0.7373 0.7270 0.7142 0.6927 0.6722

0.7657 0.7628 0.7579 0.7529 0.7511 0.7467 0.7357 0.7247 0.7106 0.6850 0.6584

178

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P.G. Jansi Rani et al.

them, finite Schmidt numbers affect the magnitude of the current (the smaller the Sc-value, the smaller the normalized current Ψ) though not the shape and position of the voltammetric wave from which the reversibility of the electron transfer can be confirmed and the corresponding formal potential can be determined. For this, the derivative voltammograms are shown in the insets of Fig. 4 where, regardless of the value of τ and Sc, the peak potential coincides with the formal potential and the half-peak width takes the well-known value for oneelectron reversible electrode reactions: 90 mV at 25 °C.

compared with numerical solutions for chronoamperometry, normal pulse voltammetry and steady state voltammetry and they are in very good agreement with the numerical simulation data.

Acknowledgements This work was supported by the Department of Science and Technology, SERB-DST (EMR/2015/002279) Government of India. Also the authors thanks to Mr. S. Mohamed Jaleel, the Chairman and Dr. A. Senthil Kumar, the Principal, Sethu Institute of Technology, Kariapatti, India for their constant encouragement. Prof. Molina and Dr. Laborda greatly appreciate the financial support provided by the Fundacion Seneca de la Region de Murcia (Project 19887/GERM/15) as well as by the Ministerio de Economia y Competitividad (Project CTQ2015-65243-P and fellowship “Juan de la Cierva Incorporación 2015”).

4. Conclusion Approximate analytical solutions to the convection diffusion equation are presented by using homotopy perturbation method. This analytical result will be useful to know the behaviour of the reaction system. In addition, we can emphasized that our analytical solutions are Appendix A. Analytical solution of the transient RDE with the HPM

In this Appendix, the HPM is used to solve the RDE problem under transient conditions. Thus, employing the Laplace transformation in Eq. (11) and using the condition given by Eq. (12), the following differential equation in the Laplace space is obtained:

dθ (ζ , τ ) ⎞ d 2θ (ζ , τ ) − sθ + 1 + (ζ 2 − kζ 3) ⎜⎛ ⎟ = 0 dζ 2 ⎝ dζ ⎠

(A.1)

Now the boundary conditions become −

θ (ζ → ∞, s ) =

1 s

(A.2)

θ (ζ = 0, s ) = 0

(A.3)

where s is the Laplace variable and the overbar denotes Laplace-transformed quantities. The set of expressions presented in Eqs. (A.1)–(A.3) defines the initial and boundary value problem in Laplace space. We construct a homotopy to derive the solution of Eq. (11) as follows:

(1 − p) ⎡ ⎢ ⎣

dθ (ζ , τ ) ⎞ ⎤ d 2θ (ζ , τ ) d 2θ (ζ , τ ) =0 − sθ + 1 + (ζ 2 − kζ 3) ⎛⎜ − sθ + 1⎤ + p ⎡ ⎟ 2 2 ⎢ ⎥ dζ ⎠ ⎥ dζ dζ ⎝ ⎦ ⎦ ⎣

(A.4)

where p ∈ [0, 1] is an embedding parameter. The approximate solutions of Eq. (A.4) is:

θ = θ0 + pθ1 + p2 θ2 + p3 θ3 + …

(A.5)

Substituting Eq. (A.5) into Eq. (A.4) and comparing the coefficients of like powers of p, the following differential equations are obtained:

p0 :

d 2θ0 − sθ0 = −1 dζ 2

(A.6)

p1 :

d 2θ1 dθ − sθ1 + (ζ 2 − kζ 3) ⎜⎛ 0 ⎟⎞ = 0 dζ 2 ⎝ dζ ⎠

(A.7)

p2 :

d 2θ2 dθ − sθ2 + (ζ 2 − kζ 3) ⎛⎜ 1 ⎞⎟ = 0 dζ 2 ⎝ dζ ⎠

(A.8)

the boundary conditions of which are:

θ0 (ζ → ∞, s ) =

1 s

and θ0 (ζ = 0, s ) = 0

(A.9)

θ1 (ζ → ∞, s ) = 0 and θ1 (ζ = 0, s ) = 0

(A.10)

θ2 (ζ → ∞, s ) = 0 and θ2 (ζ = 0, s ) = 0

(A.11)

Upon solving Eqs. (A.6) and (A.7) and using the boundary conditions (Eqs. (A.9) and (A.10)), we obtain

θ0 (ζ , s ) = θ1 (ζ , s ) =

1 1 − e− s s e− s ζ

⎡ s3 ⎢ ⎣

sζ

(A.12)

s 3 2ζ 2 4

+

sζ − 4

k s 3 2ζ 3 4

−

3k s ζ 2 8

−

3k s ζ + 8

s 2ζ 3 6

−

k s 2ζ 4 8

⎤ ⎥ ⎦

and also

179

(A.13)

Journal of Electroanalytical Chemistry 799 (2017) 175–180

P.G. Jansi Rani et al.

dθ1 (ζ , s ) ⎞ ⎟ dζ

⎠

−

=

e− s ζ s3

⎡s ⎣

3 2ζ

2

3 2 2 ⎡s ζ s5 2 ⎣ 4

e− s ζ

+

+ sζ 4

s 4

−

−

3k s3 2ζ 2 4

k s3 2ζ 3 4

−

−

3k s ζ 4

3k s ζ2 8

−

−

3k s 8

3k s ζ 8

+ +

s 2ζ 2 2 s 2ζ 3 6

k s 2ζ 3 ⎤ 2

−

⎦

k s 2ζ 4 ⎤ 8

−

(A.14)

⎦

Substituting Eq. (A.14) in Eq. (A.8) we get − sζ

⎛e ⎡s s3 ⎣ d 2θ2 ⎜ 2 3 − sθ2 + (ζ − kζ ) dζ 2 ⎜ − e− ⎜ s5 ⎝

3 2ζ

2 sζ 2

+

⎡s ⎣

s 4

−

3 2ζ 2

4

3k s3 2ζ 2 4

+

sζ 4

−

−

3k s ζ 8

k s3 2ζ 3 4

−

−

3k s 8

3k s ζ2 8

s 2ζ 2 2

−

k s 2ζ 3 ⎤ 2

3k s ζ 8

+

s 2ζ 3 6

+

−

⎞ ⎟=0 k s 2ζ 4 − 8 ⎤⎟⎟ ⎦⎠ ⎦

(A.15)

From the solution of the above equation (not included for the sake of shortening), we obtain

45 315k ⎤ 15 45k 3 ⎡ dθ2 (ζ , s ) ⎤ = a⎡ 7 2 − + b⎡ 3 − 7 2⎤ + c⎡ 5 ⎢ dζ ⎥ 16s 4 ⎦ 8s ⎦ ⎣ 8s ⎣ 8s ⎣ 4s ⎣ ⎦ζ = 0

2

−

15k ⎤ 3 3k 1 + d⎡ 2 − 5 2⎤ + l⎡ 3 8s 3 ⎦ 4s ⎦ ⎣ 8s ⎣ 4s

2

−

3k ⎤ 8s 2 ⎦

(A.16)

where

a=

k −1 k 1 3k 1 , b= − , c= − 3 2, d = 3 8 s 6 s 4s 4s 8s 4s

2

−

3k 1 3k , l= 2 − 52 8s 2 4s 8s

(A.17)

Finding the inverse Laplace transform of Eqs. (A.12) and (A.13), the dimensionless concentration profile of the oxidized species (Eq. (16) of the main text) from:

θ (ζ , τ ) = lim θ (ζ , τ ) ≈ θ0 (ζ , τ ) + θ1 (ζ , τ ) + θ2 (ζ , τ )

(A.18)

p→1

From the derivative of Eqs. (A.12) and (A.13) at ζ = 0 together with Eq. (A.16), the expression for the current (Eq. (17) of the main text) is obtained. Appendix B. Supplementary data Supplementary data to this article can be found online at http://dx.doi.org/10.1016/j.jelechem.2017.05.053.

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