Theory for Cyclic Staircase Voltammetry of Two Step Charge Transfer

Article pubs.acs.org/JPCC

Theory for Cyclic Staircase Voltammetry of Two Step Charge Transfer Mechanism at Rough Electrodes Parveen and Rama Kant* Complex Systems Group, Department of Chemistry, University of Delhi, Delhi 110007, India ABSTRACT: We have developed theory for cyclic (staircase) voltammetry (CSCV) of a two step reversible charge transfer (EE) mechanism for redox species with unequal diffusion coefficients at rough electrodes. The various surface microscopies provide details of random morphology of the electrode which is characterized through the power spectrum in our theory. For the finite fractal electrode model, anomalous enhancement and scan rate dependence of the CSCV or CV response is caused by fractal dimension (DH), lower length scale of fractality (S ), and topothesy length (S τ). The peak current corresponding to both charge transfer steps is enhanced with an increase in roughness of the fractal electrode (through increase in DH or S τ or decrease in S ). The onset and offset of the anomalous regime is controlled by S τ and S , respectively. Results show that the electrode roughness has the potential to enhance the sensitivity of CSCV as an analytical technique. This is due to a decrease in the difference between the peak currents of two charge transfer steps and an increase in the ratio of the peak to the valley (minimum existing between the two peaks) current in the presence of roughness. Finally, we show that not accounting for roughness in data analysis may cause errors in estimation of composition, diffusion coefficient, improper assignment of electrode mechanism, and so forth.

■

INTRODUCTION The presence of stepwise electron transfer mechanisms is proclaimed in numerous applied systems like dyes, dendrimers, polymers, enzymes,1−3 and so forth. Many of these systems frequently encounter the double step charge transfer processes, popularly known as EE mechanisms. Commonly used solid electrodes are ubiquitously rough, e.g., screen printed electrodes, electrodes modified with nanoparticles, films, and dendritic growth. These added complexities are not accounted for in the theoretical models developed earlier for the EE mechanism.5,6 In order to develop further physical insights and make these models more compatible with the applied and realistic systems, electrode roughness or the ruggedness of the electrode boundary is a crucial aspect to be included. A rough or morphologically complex surface exhibits different characteristics in its response compared to that of a smooth surface. Apart from high surface area effects, various other anomalies have been observed in the rough electrode response of single-step charge transfer mechanisms such as deviation of chronoamperometric response from square root dependence on time7,14,15 and that of peak current of cyclic voltammetry (CV) from square root dependence on scan rate.8,9 Not only are the peak currents found to be altered in CV of single-step charge transfer reactions, but the separation between the peaks is also.8,15 All these observations prompt us to further extend the rough electrodes model to the conventional technique of cyclic voltammetry for the EE processes. Cyclic voltammetry is a prevalent technique to investigate multielectron transfer systems. It provides quantitative mecha© 2016 American Chemical Society

nistic and kinetic information through well-defined peak-shaped responses where undesirable effects are greatly minimized. In CV, due to the application of forward and reverse potential scans, both anodic as well as cathodic processes can be studied in a single potential cycle. The problem of modeling the voltammetric response of a general uncomplicated multistep charge transfer process was attempted numerically by Gokhshtein et al.4 at stationary plane electrode for LSV. The specific case of EE mechanism was analyzed for cyclic voltammetry in 1966 by Shain.5 Only numerical and approximate series solutions could be obtained for LSV and CV attributing to the continuous time dependence of the potential scan in these techniques. The analytical solution of the problem was presented for the digital analogue of CV, i.e., cyclic staircase voltammetry,6 for which the stationary plane electrode response is expressed as follows: N−1

Ip(t ) = IC(t )CsEE(0) +

∑ [CsEE(m) − CsEE(m − 1)]IC(t − mτ) m=1

(1)

where IC = nFA 0D1C10/ πD1t denotes the Cottrell current, C01, D1 denotes bulk concentration and diffusion coefficient of first species, A0 denotes the macroscopic area of the electrode, and F (2) (2) is Faraday’s constant. CEE s (m) = (2+θ (m))/(1+θ (m) + θ(1)(m) θ(2)(m)), θ(k)(m) = exp{(−nk f(Ei−Em−E0k ′)}, with Em representing the amplitude of the mth pulse in the cyclic staircase Received: January 25, 2016 Published: January 27, 2016 4306

DOI: 10.1021/acs.jpcc.6b00810 J. Phys. Chem. C 2016, 120, 4306−4321

Article

The Journal of Physical Chemistry C potential sequence, E0k ′ the formal potential of the kth reduction step, nk denoting the number of electrons transferred in the kth charge transfer step and f = F/RT, and R and T denoting the gas constant and temperature, respectively. Though the influence of other geometries of the electrodes on CV response of the EE mechanism has been discussed,6 the question of the morphological aspect of the electrode surface is yet to be addressed. To explain the anomalous electrochemical response observed at randomly rough electrodes, fractal electrode models are often employed.10−15,17−23 Various approaches are used to model the response of fractal electrodes, viz., equivalent circuit,12,13 fractional diffusion,11,20−22 scaling argument,18,61,62 ab initio derivation,7,23,24,28,30−39 and numerical methods.14−16 Equivalent circuit, fractional diffusion, and scaling argument are oneparameter theories which characterize the morphology through fractal dimension (DH) only. Realistic surfaces show fractal behavior under limited cutoff length8,24−27 scales which are important for proper characterization of the electrode surface. Electrode morphology can be statistically characterized through the power spectrum of roughness. The power spectral density (PSD) of the surface can be easily extracted by processing AFM or SEM images of the electrode surface or by employing the hybrid CV-SEM method which has recently been developed by Kant and co-workers.25,27 Using PSD as morphological information parameter, Kant et al. have developed an ab initio approach to model the electrochemical response of various techniques like chronoamperometry,23,28−31 chronocoulometry,32 absorptometry,33 voltammetry,8,9,34 impedance,35,36 and admittance37−40 at rough electrodes. The electrochemical response is found to show strong dependence on characteristic length scales, viz., lower length scale cutoff (S ) and the topothesy length (S τ), along with fractal dimension (DH). Redox couples in various systems, including ionic liquids, are proving these models to be successful in predicting the influence of electrode roughness on the electrochemical response.8,24,26 In the present work, we develop a theoretical model for cyclic staircase voltammetry at arbitrary roughness as well as a selfaffine finite fractal electrode for a double-step charge transfer system. The formalism is developed for the general case of unequal diffusion coefficients of all the electroactive species. The paper is organized in various sections. The model is formulated for an arbitrary potential perturbation applied at an arbitrary topography electrode in the following section. SCV and CSCV current expressions are then deduced as special cases. The explicit expressions for finite fractal roughness model are obtained in the subsequent subsection. The results are then graphically explored and elaborately discussed in the Results and Discussion section. Finally, the paper is closed with concluding remarks.

absence of any other chemical reaction when the diffusion coefficient of all the electroactive species are equal. Recently, Compton et al.41,42 have shown in their numerical and experimental work that the compropotination/disproportination effects will only be seen when the difference between the diffusion coefficients of the electroactive species and separations between formal potentials of the two charge transfer steps are large (D2/D1 ≈ D3/D1 ≥ 1.5). Usually these effects on the reversible charge transfer reactions are shown for the large formal potential separations, e.g., E02′−E01′ ≈ 500 mV. We develop the formulation for an arbitrary time dependent potential perturbation (E(t)) applied at the rough electrode interface from which SCV and CSCV expressions can be obtained as special cases for EE mechanism. A general time dependent form of E(t) can be written as E(t) = Ei+ νg(t). Ei is the initial step over which further perturbation is applied. ν is a constant, whereas g(t) denotes the time dependent part.34 For LSV and CV, the function g(t) = t and thus the potential perturbation of these techniques have linear dependence on time. At the rough interface the total current (I(z,t)) due to any arbitrary potential perturbation is related to the current density, i, as follows I(z = ζ( r ⃗), t ) =

0

i(z = ζ( r ⃗), t ) = i1(z = ζ( r ⃗), t ) + i2(z = ζ( r ⃗), t )

O2 + n2e− ⇌ O3

E20′

(4)

Using Fick’s first law, the relation between current densities and flux at the interface can be expressed as follows i1 = D1∂nδC1(ζ , t ) n1F

(5)

i2 i − 1 = D2∂nδC2(ζ , t ) n 2F n1F

(6)

i2 = −D3∂nδC3(ζ , t ) n 2F

(7)

where n1 and n2 are the number of electrons transferred in the first and second steps, respectively; Cα and Dα are the concentration profile and diffusion coefficient of αth species, with α representing 1, 2, or 3 which corresponds to species first, second, and third, respectively. The difference concentration, δCα (=Cα − C0α), is the deviation of concentration profile from bulk concentration (C0α) of the species α. ∂n is the outward drawn normal at the surface ζ. To obtain the explicit form of current densities, the following diffusion equation which is satisfied by the concentration profiles of the electroactive species needs to be solved

MATHEMATICAL FORMULATION The reaction scheme for a two step charge transfer (EE) mechanism can be represented as follows: E10′

(3)

where S0 is the projection of the rough surface S at the reference (z = 0) plane, ζ(r∥⃗ ) is the surface profile of the electrode, r∥⃗ represents the 2-D vector, r∥⃗ ≡ (x,y), β = [1+(∇∥ζ(r∥⃗ ))2]1/2, and ∇∥ (= i∂̂ x+ j∂̂ y) represents the 2-D Laplacian operator. The current density for an EE mechanism is the sum of current densities of both the charge transfer steps (i1 and i2)

■

O1 + n1e− ⇌ O2

∫S dS0βi(z = ζ( r ⃗), t )

(2)

to first and second charge transfer steps, respectively. Often, the stepwise reversible electron transfer reactions are coupled with the comproportionation/disproportionation reactions. The influence of the latter on the voltammetric response can be neglected if diffusion is the only means of mass transport, in the where E01′ and E02′ are the formal potentials corresponding

∂ δCα( r ⃗ , t ) = Dα ∇2 δCα( r ⃗ , t ) ∂t

(8)

∇2 = (∂2/∂ x2 + ∂2/∂ y2 + ∂2/∂ z2). A uniform initial and bulk c o n c e n t r a t i o n C α ( r⃗ , t ) i s a s s u m e d , v i z . , Cα( r ⃗ , t = 0) = Cα( r ⃗ , z → ∞ , t ) = Cα0 for all three species. 4307


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The Journal of Physical Chemistry C

perturbing potential signal, defined as χ01(t) = θ(1)(t) θ(2)(t) C̃ s − C01, χ02(t) = θ(2)(t) C̃ s − C02, and χ03(t) = C̃ s − C03; C̃ s = (ξ1,3C01 + ξ2,3C02 + C03)/(1 + ξ2,3θ(2)(t) + ξ1,3θ(1)(t) θ(2)(t)). The operators, Ô 0α , Ô1α and 6̂ ′2α , are functions of the phenomenological diffusion length which are defined for a smooth surface and the first- and second-order terms in Fourier transformed surface profile, respectively, as follows

Being second order in space and first order in time, eq 8 needs three constraint conditions to be solved. Apart from initial and bulk conditions, the third constraint is imposed by the reversibility of the redox couples in the form of Nernstian boundary condition at the electrode surface: 0′ C1(z , t ) |z = ζ( r ⃗ ) = e−n1f (E(t ) − E1 ) = θ (1)(t ) C2(z , t ) 0′ C2(z , t ) |z = ζ( r ⃗ ) = e−n2f (E(t ) − E2 ) = θ (2)(t ) C3(z , t )

nF Ô 0α ≡ qα

(9)

where E01′ and E02′ represent the formal potentials of first and second charge transfer steps. In order to decouple concentrations in the reversibility constraints described above, we further need an expression which can relate the three concentration profiles. Such a relation is supplied by following identity for the general case of unequal diffusion coefficients of all the species involved (see Appendix): D1 δC1(z = ζ( r ⃗), t ) +

D2 δC2(z = ζ( r ⃗), t ) +

C1(z = ζ( r ⃗), t )

I(p) = − pχ0̅ (p) 1

⎤ ⎡ ξ1,3C10 + ξ2,3C20 + C30 ⎥ = θ (1)(t )θ (2)(t )⎢ ⎢⎣ 1 + θ (2)(t )ξ2,3 + θ (1)(t )θ (2)(t )ξ1,3 ⎥⎦

∫S

C2(z = ζ( r ⃗), t )

1 2 (q + q α q α ) 2 α (15)

∫S d2r [Ô 0 (2π )2 δ(K⃗ ) + Ô1 ζ(̂ K⃗ ) 1

1

0

2

d r [Ô 02(2π )2 δ(K⃗ ) + Ô12ζ (̂ K⃗ ) + Ô 22ζ (̂ K⃗ ′)ζ ̂ 2

0

(16)

The symbol K⃗ ∥ is the Fourier transform variable corresponding to the space variable, r∥⃗ . The operators Ô 0α and Ô1α are same given in eq 15, whereas Ô 2 is defined as follows:

(12)

ξ1,3C10 + ξ2,3C20 + C30

α

1 + ξ2,3θ (2)(t ) + ξ1,3θ (1)(t )θ (2)(t )

Ô 2α ≡

(13)

the symbol ξk , j = Dk /Dj . Using the decoupled forms of the surface concentrations of electroactive species, one can obtain the current density expression for a weakly and gently varying rough surface in the Fourier and Laplace domain by following the steps mentioned in Appendix B.

nF qα(2π )2

⎡

∫ d2K ′ ⎢⎣qα qα

− , ′

⎤ |K⃗ − K⃗ ′|2 − K⃗ ′·(K⃗ − K⃗ ′)⎥ ⎦

1 2 (q + q α q α ) − 2 α (17)

Here, the Laplace and Fourier transformed total current expression contains the morphological information on an arbitrary profile. For deterministic roughness profiles, results can be easily obtained from eq 16. To deal with the more realistic aspect, we can simplify the expression for the case of random roughness which is ubiquitous in the electrode systems. Voltammetric Current at Random Surface Topography. Random roughness topographies can only be statistically described. To characterize a statistically homogeneous random surface (ζ(r∥⃗ )), the Fourier transformed two-point correlation function (|ζ̂(K⃗ ∥)|2) is employed. The statistics for such centered Gaussian fields may be written as

i(ζ ̂, p) = −pχ0̅ (p)[Ô 01(2π )2 δ(K⃗ ) + Ô11ζ (̂ K⃗ ) 1 + 6̂ ′21 ζ (̂ K⃗ ′)ζ (̂ K⃗ − K⃗ ′)] − (n2p /n)χ0̅ (p) 2 [Ô (2π )2 δ(K⃗ ) + Ô ζ (̂ K⃗ ) + 6̂ ′ ζ (̂ K⃗ ′)ζ ̂ (K⃗ − K⃗ ′)]

− , ′

(K⃗ − K⃗ ′)]

(2)

12

⎡


+ Ô 21ζ (̂ K⃗ ′)ζ (̂ K⃗ − K⃗ ′)] − p(n2 /n)χ0̅

(11)

02

qα

Equation 14 can be used to predict the current density after the inversion from Laplace transform and Fourier domains for a known surface profile. The current density is an important quantity to provide insight into local phenomenon like nonuniform deposition and dissolution on arbitrary profile electrodes. Our main goal is to obtain the expression for total current which is the most easily accessible quantity in the electrochemical measurements. Equation 3 expresses total current as current density integrated over the average area of the electrode. Total current for EE reaction occurring at a rough interface is obtained by substituting eq 14 into the Laplace and Fourier transformed eq 3 and retaining terms up to second order in ζ. This yields

Equation 10 is an exact identity for the planar electrode as well as for the situation of equal diffusion coefficients (D1 = D2 = D3) at the rough electrode. This assumption holds good for electrodes of gently fluctuating morphology and is also applicable for the cases of small and large diffusion length in general (Appendix A). The decoupled forms of surface concentrations under the Nernst reversible charge transfer constraints obtained using the above identity are as follows:

C3(z = ζ( r ⃗), t ) =

nF(qα − qα)

⎤ 2 1 − K⃗ − K⃗ ′ − K⃗ ′·(K⃗ − K⃗ ′)⎥ ⎦ 2

(10)

⎤ ⎡ ξ1,3C10 + ξ2,3C20 + C30 ⎥ ⎢ = θ (t ) ⎢⎣ 1 + ξ2,3θ (2)(t ) + ξ1,3θ (1)(t )θ (2)(t ) ⎥⎦

nF qα(2π )2

6̂ ′2α ≡

D3 δ

C3(z = ζ( r ⃗), t ) = 0

O1̂ α ≡

22

(14)

where χ0α(p) is the Laplace transform of the nongeometric function χ0α(t) which consists of all information about the 4308


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The Journal of Physical Chemistry C ⟨ζ (̂ K⃗ )⟩

=

The quantity dχ̃0α(t)/dt = −(ξα,1/Cs) dχ0α(t)/dt where the differential of χ0α(t) is defined by eqs B.6, B.7, and B.7 for α assumed to be first, second, and third species, respectively. In eq 23, the term outside the integral represents the potentiostatic current transient for an EE mechanism. The LSV current expression reduces to potentiostatic current under the limit ν = 0. Similarly, one can obtain the expression for cyclic voltammetric current. The form of g(t) is the same as that for LSV when time is between 0 and λ, whereas g(t) = 2λ − t when t > λ. In the reverse cycle, one should also notice that the change in the concentrations of third and second species will contribute to the flux. Thus, the CV current expression for the EE mechanism will be given by

0

⟨ζ (̂ K⃗ )ζ (̂ K⃗ ′)⟩ = (2π )2 δ(K⃗ + K⃗ ′)⟨|ζ (̂ K⃗ )|2 ⟩

(18)

The angular brackets denote the ensemble average over various possible configurations of the electrode. The power spectrum, ⟨|ζ̂(K⃗ ∥) |2⟩, is also known as the surface structure factor. It contains as much morphological information as the two-point correlation function, as they are related in the Fourier domain. Equation 16 can be averaged over various possible configurations of the electrode to obtain the total current expression for the random surface profile electrode in the Laplace domain. The term containing the averaged random surface will vanish and the resultant expression is simplified as follows ⎡⎛ nFA 0 ⎞ nFA 0 ⎟+ ⟨I(p)⟩ = − pχ0̅ (p)⎢⎜⎜ 1 ⎢⎣⎝ q1 ⎟⎠ (2π )2

∫

2

⎤ d2K (q1 − q1)⟨|ζ (̂ K⃗ )|2 ⟩⎥ ⎥⎦

⎡⎛ nFA 0 ⎞ nFA 0 ⎟+ − (n2p /n)χ0̅ (p)⎢⎜⎜ 2 ⎢⎣⎝ q2 ⎟⎠ (2π )2

∫ d2K (q2

⟨I(t )⟩ =

α=1 k=α

nk χ ̃ (0)⟨IgC(t , Dα )⟩ + n 0α

∫0

λ

dt ′

⎡ 2 2 n dχ ̃ (t − t ′) ⎤ ⎢ ∑ ∑ k ⟨IgC(t ′, Dα )⟩ 0α ⎥ ⎢⎣ α = 1 k = α n ⎥⎦ dt

− q2)

⎤ ⟨|ζ (̂ K⃗ )|2 ⟩⎥ ⎥⎦

2

∑∑

−

∫λ

t

⎡ 3 dt ′⎢ ∑ ⎢⎣ α = 2

α−1

∑ k=1

dχ0̃ (2λ − (t − t ′)) ⎤ nk ⎥ ⟨IgC(t ′, Dα )⟩ α ⎥⎦ n dt

(19)

(24)

In order to report the current in time domain, we need to invert the Laplace transformed current equation as

For the case of equal diffusion coefficients of all the electroactive species under the limit n1 = n2 = 1, the expression for Cs[dχ̃01(t)/ dt + (n2/n) dχ̃02(t)/dt] has the same form as described by Camacho et al.45 for the quantity dm(t)/dt at the planar electrode. As is apparent from eqs 23 and 24, for these continuous time techniques, only numerical solutions can be obtained for the total current functional. However, for the discretized techniques, SCV and CSCV, an analytical route can be opened. Staircase and Cyclic Staircase Voltammetries. The techniques where potential, rather than being continuous, is a discrete function of time can be conveniently represented using Heaviside unit step function (u(t)) as follows:

⟨I(t )⟩ =

t

⎡ dt ′⎢χ0̃ (t − t ′)⟨IgC(t ′, D1)⟩ ⎣ 1

d dt

∫0

+

⎤ n2 χ0̃ (t − t ′)⟨IgC(t ′, D2)⟩⎥ 2 ⎦ n

(20)

where χ̃0i(t) = −ξi,1χ0i(t)/Cs, Cs = (C01 − θ(1)1 C3)/(1 + θ(1)1), ⟨IgC⟩ is the generalized Cottrellian current at random roughness profile electrode described as follows:8 ⟨IgC(t , Di)⟩ = ICR(t , Di)

(21)

E(t ) = E1(u(t ) − u(t − τ )) + E2(u(t − τ ) − u(t − 2τ ))

where R(t , Di) = 1 +

1 2(2π )2 Dit

+ E3(u(t − 2τ ) − u(t − 3τ )) + ···

2

∫ d2K (1 − e−K D t )⟨|ζ(̂ K )|2 ⟩ i

E1, E2, and so forth are heights or amplitude of pulses applied in sequence after every constant time duration, τ. For SCV, the applied potential is represented as follows

(22)

Equation 20 expresses the EE reaction current for an arbitrary potential perturbation at a randomly rough electrode in the form of differential of convolution integrals between the generalized Cottrellian current and the potential functions representing surface concentrations of the redox species. Equation 20 provides a general result from which, using the form of potential applied in a particular technique, one can obtain the current expression for any potential dependent technique like LSV, CV, DPV, or SWV. For linear sweep (LSV) and cyclic voltammetric (CV) techniques, the potential perturbation E(t) is a linear function of time. For LSV, the time dependent part in the potential perturbation is g(t) = t. Thus, E(t) = Ei + ν g(t) and the LSV current can be written as 2

⟨I(t )⟩ =

2

∑∑ α=1 k=α

nk χ ̃ (0)⟨IgC(t , Dα)⟩ + n 0α

∫0

N−1

E SCV (t ) = Ei +

∑ ΔEu(t − nτ) (26)

n=1

where ΔE is the potential step size of the staircase and N is the total number of steps applied over Ei. Due to the discrete nature of the potential, χ̃0α can also be simply written into the discretized form as follows: χ0̃ (t ) = f1 (0)[u(t ) − u(t − τ )] + f1 (1) 1

[u(t − τ ) − u(t − 2τ )] + ··· + f1 (N )u(t − (N − 1)τ )

(27)

t

dt ′

⎡ dχ ̃ (t − t ′) ⎤ n ⎢ ∑ ∑ k ⟨IgC(t ′, Dα)⟩ 0α ⎥ dt ⎢⎣ α = 1 k = α n ⎥⎦ 2

(25)

χ0̃ (t ) = f2 (0)[u(t ) − u(t − τ )] + f2 (1) 2

2

[u(t − τ ) − u(t − 2τ )] + ··· + f2 (N )u(t − (N − 1)τ )

(23) 4309

(28)


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The above equation, in the present form, describes the cathodic current. To obtain the SCV current for an anodic bias potential, similar methodology will be applicable, but the difference will be that for the anodic cycle, the current density as well as the current will be composed of concentration profiles of species second and third. In the current density expression described by eq B.16, concentration profile of species first will be replaced by the concentration profile of species third along with the replacement of D1 by D3. Thus, for an anodic potential bias, the current expression will be represented as follows:

χ0̃ (t ) = f3 (0)[u(t ) − u(t − τ )] + f3 (1) 3

[u(t − τ ) − u(t − 2τ )] + ··· + f3 (N )u(t − (N − 1)τ )

(29)

Here, the potential dependent surface concentration, which contains exponential dependence over time, is simplified into a set of discrete functions ( fα(m)) which are constant over a particular time interval. f1 (m) = (C10 − θ (1)(m)θ (2)(m)Cs̃ (m))/Cs

f2 (m) =

ξ2,1(C20

(2)

− θ (m)Cs̃ (m))/Cs

Cs̃ (m) =

a ⟨ISCV (t ,

α=2 k=1 N−1

+

(32)

1 + ξ2,3θ (2)(m) + ξ1,3θ (1)(m)θ (2)(m)

(33)

0′

θ (1)(m) = e−nf (Ei + mΔE − E1 )

(34)

−nf (Ei + mΔE − E 20 ′)

M

ECSCV (t ) = Ei +

(35)

∫0

t

⎡ 2 2 ⎢∑ ∑ ⎢⎣ α = 1 k = α

N−1

∑ m=1

∫0

where N is the total number of steps and M is the number of steps after which the potential ramp is reversed. Using the above expression for potential perturbation, the form of χα0 will remain the same as described by eqs 27, 28, and 29; the only difference will be in the definitions of θ(1) and θ(2) stated as follows: 0′

θ (1)(m) = e−nf [Ei + (M−| m − M |)ΔE − E1 ] 0′

θ (2)(m) = e−nf [Ei + (M−| m − M |)ΔE − E2 ]

2

⟨ICSCV (t , N )⟩ =

t

2

α=1 k=α N−1

+

∑

(fα (m) − fα (m − 1))

m=1

nk (f (m) − fα (m − 1))δ n α

⎤ ⟨IgC((N − m)τ , Dα )⟩⎥ + ⎥⎦

3

α−1

N−1

∑∑ ∑ α=2 k=1 m=M+1

nk n

(fα (m − 1) − fα (m))⟨IgC((N − m)τ , Dα )⟩

(41)

Equations 37 and 41 represent the SCV and CSCV current at a random roughness electrode, respectively. The contribution of roughness is interpreted by the power spectrum containing the term in the generalized Cottrellian current. By replacing ⟨IgC(t)⟩ by the Cottrell current, the expressions for SCV and CSCV current at the rough electrode reduce to that presented by Molina6 at the planar electrode under the assumption of equal diffusion coefficients of all the species when the concentration of only the first is present initially. Also, the results of the spherical and disc electrodes6 can be obtained if ⟨IgC(t)⟩ is replaced by the potentiostatic current transient of the sphere and disc electrodes, respectively (for E process). Pulse Voltammetric Current at Finite Self-Affine Fractal Electrode. The power spectrum of roughness, which

nk [f (0)⟨IgC(Nτ , Dα )⟩ n α

∑ (fα (m) − fα (m − 1)) m=1

⟨IgC((N − m)τ , Dα )⟩]

nk ⎡ ⎢f (0)⟨IgC(Nτ , Dα )⟩ n ⎢⎣ α

(N − 1) ≤ M

+

where δij denotes the Kronecker delta function. Now the integrals can be easily performed using properties of Dirac delta functions. The SCV current expression obtained for a random roughness profile is as follows 2

2

∑∑ α=1 k=α

dt ′

(36)

∑∑

(40)

In cyclic voltammetry, as the anodic current starts after the vertex potential, i.e., the Mth step, the cyclic voltammetric current will be expressed as

2

⎤ (t − t ′ − mτ )⟨IgC(t − t ′ − mτ , Dα )⟩⎥ ⎥⎦

⟨ISCV (t , N )⟩ =

ΔEu(t − nτ )

n=M+1

(39)

⎡ n dt ′δ(t − t ′)⎢ ∑ ∑ k fα (0) ⎢⎣ α = 1 k = α n

⎤ ⟨IgC(t ′, Dα )⟩⎥ + ⎥⎦

N−1

∑ ΔEu(t − nτ) − ∑ n=1

Interestingly, the form of the surface concentrations obtained is similar to that encountered during the use of the superposition principle. This is a direct consequence of the discrete potential applied to the electrode which is described in terms of unit step functions. It should be noted here that it is the form of the dependence of the boundary condition on the potential that decides whether the step representation of the boundary is possible or not. Using the Leibniz theorem,49 the differential of the integrals can be simplified into the integrals to be carried over the functions which contain Dirac delta functions. 2

(38)

Now we can proceed to obtaining the current expression for cyclic staircase voltammetry for which the potential can be described as follows

(2)

⟨ISCV (t , N )⟩ =

∑ (fα (m) − fα (m − 1))

⟨IgC((N − m)τ , Dα )⟩]

For SCV, θ (m) and θ (m) are defined as

θ (2)(m) = e

nk [f (0)⟨IgC(Nτ , Dα )⟩ n α

m=1

ξ1,3C10 + ξ2,3C20 + C30

(1)

∑∑

N )⟩ =

(31)

f3 (m) = ξ3,1(C30 − Cs̃ (m))/Cs

α−1

3

(30)

(37) 4310


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⟨IaC(t)⟩ is replaced by the potentiostatic current transient of sphere and disc electrodes, respectively (for E process).6 In the limits of E01′ = E02′ (=E0′) and C2 = 0, θ(2)(t) (=C2/C3) 0 goes to zero, whereas the product θ(1)(t) θ(2)(t) (=e−nf(E(t)−E ′) = C1/C3) gives the ratio of oxidized (CO) and reduced (CR) species concentrations and eqs 44 and 45 reduce to the single-step charge transfer result for SCV and CSCV, respectively. The voltammetric response expressions deduced for EE mechanism at rough electrode interface are exactly applicable for the coupled comproportionation/disproportionation reaction scheme in the presence of equal diffusion coefficients of all the electroactive species (see Appendix C). For the planar case, this is discussed by Saveant at al.46,47 considering single electron transfer in each electrochemical step. On similar grounds, the validity of the rough electrode results for the scheme where EE mechanism is coupled with comproportionation/disproportionation reaction in the case of equal diffusivities of redox species is discussed in Appendix C without any constraint on the number of electrons transferred in any of the electrochemical steps.

is a morphological information parameter in the current expressions deduced in the previous section, can assume any form depending on the type of electrode randomness considered. An important class of morphological randomness found in electrodes is of band limited self-affine isotropic random fractals.51−62 The power spectrum of such morphologies shows power-law behavior under limited length scales. ⎯→ ⎯ ⟨|ζ (̂ K⃗ )|2 ⟩ = S2τ DH − 3| K |2DH − 7

⎯→ ⎯ for 1/L ≤ | K | ⩽ 1/S (42)

The length scales, S and L, are basically the lower and upper statistical cutoff lengths between which fractal behavior is observed. DH is fractal dimension which is a scale invariant property of roughness; S τ is topothesy length and it is related to width of the interface. S τ → 0 represents the smooth electrode. These four parameters, viz., DH, S τ, S , and L are the characteristics of the finite fractal morphology. Substitution of the value of the power spectrum in the generalized Cottrell current expression yields the potentiostatic current expression for a realistic fractal electrode. The current transient of these electrodes shows anomalous power law behavior in the intermediate time regime, and for this reason, it is known as anomalous Cottrell current (⟨IaC(t)⟩). The expression for ⟨IaC(t)⟩, derived by Kant,23,29 is represented as follows:

■

RESULTS AND DISCUSSION Here we explore graphically how the surface roughness influences the CSCV current response of an EE mechanism. Effects of roughness have been analyzed through a statistical model developed for fractal roughness. Current response of a fractal electrode is dominantly controlled by three morphological characteristics, viz., fractal dimension (DH), lower length scale cutoff (S ), and topothesy length (S τ) (refer to Figure 1 for schematic of the problem). All the plots are generated using Mathematica software.

⎛ S2DH − 3 ⎛ S −2δ − L−2δ ⎜⎜ ⟨IaC(t , Dα )⟩ = IC(t )⎜⎜1 + τ δDα t 8π ⎝ ⎝ Γ(δ , Dα t /S2, Dα t /L2) ⎞⎞ ⎟⎟⎟⎟ (Dα t )1 + δ ⎠⎠

+

(43)

where IC(t) is Cottrell current. Replacing ⟨IgC(t)⟩ in eq 37 by ⟨IaC(t)⟩, the expression for SCV current at finite self-affine fractal electrode can be obtained as follows 2

2

∑∑

⟨ISCV (t , N )⟩ =

α=1 k=α N−1

+

nk [f (0)⟨IaC(Nτ , Dα )⟩ n α

∑ (fα (m) − fα (m − 1)) m=1

⟨IaC((N − m)τ , Dα )⟩]

(44)

Similarly, we can obtain the CSCV current transient at realistic electrode 2

⟨ICSCV (t , N )⟩ =

2

∑∑

nk [f (0)⟨IaC(Nτ , Dα )⟩ n α

α=1 k=α (N − 1) ≤ M

+

∑

(fα (m) − fα (m − 1))

Figure 1. Schematic of the problem for a two step reversible charge transfer under cyclic staircase potential modulation applied at the rough electrode. Finite fractal roughness characteristics which dominantly control the response are fractal dimension (DH), lower cutoff length scale of fractality (S ), and topothesy length (S τ) which is related to h (width of interface).

m=1 3

⟨IaC((N − m)τ , Dα )⟩] +

α−1

∑∑ α=2 k=1

nk n

N−1

∑

(fα (m − 1) − fα (m))⟨IaC((N − m)τ , Dα )⟩

m=M+1

(45)

Effect of Dominant Fractal Characteristics on Cyclic Staircase Voltammograms. Figure 2a shows the variation in cyclic staircase voltammograms with varying fractal dimension (DH) which is a global property that describes the scale invariance property of roughness. The black curve shows the response of the planar electrode and the subsequent upper plots

In eqs 44 and 45, the additional contribution of roughness is buried into ⟨IaC(t)⟩ at each potential step. Under the limit S τ → 0 the expressions for SCV and CSCV current at the rough electrode reduce to the planar electrode response.6 Also, the results of the spherical and disc electrodes can be obtained if 4311


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Figure 2. Influence of fractal characteristics on voltammograms: (a) Fractal dimension (DH). (b) Lowest length scale of fractality (S ). (c) Strength of fractality (μ). Fractal characteristics used are (a) DH = 2.25, 2.3, 2.32, L = 10 μm, S = 200 nm, μ = 5 × 10−6 (a.u.); (b) DH = 2.25, L = 10 μm, S = (500, 150, 30) nm, μ = 5 × 10−6 (a.u.); (c) DH = 2.25, L = 10 μm, S = 10 nm, μ = 2, 4, 6 × 10−6 (a.u.). Other parameters used in the above calculations are diffusion coefficient (D1 = D2 = D3 = 5 × 10−6 cm2/s, electrode area (A0 = 0.01 cm2), scan rate = 10 V/s, concentration (C01 = 4 × 10−6 mol/cm3, C02 = C03 = 0); the difference between the formal potentials of two steps is 150 mV. The black line represents the response of the smooth electrode.

Figure 3. Double logarithmic plots of peak current vs scan rate with varying (a) fractal dimension (DH), (b) lowest length scale of fractality (S ), (c) strength of fractality (S τ). Fractal characteristics used are (a) DH = 2.25, 2.4, L = 10 μm, S = 200 nm, μ = 5 × 10−6 (a.u.); (b) DH = 2.3, L = 10 μm, S = 100, 300 nm, μ = 5 × 10−6 (a.u.); (c) DH = 2.3, L = 10 μm, S = 200 nm, μ = 1, 5 × 10−6 (a.u.). Other parameters used in the above calculations are diffusion coefficient (D = 5 × 10−6 cm2/s, electrode area (A0 = 0.01 cm2), concentration (C01 = 4 × 10−6 mol/cm3, C02 = C03 = 0); the difference between the formal potentials of two steps is 150 mV. The solid lines correspond to first cathodic peak whereas dashed lines correspond to the second cathodic peak. Black lines represent the response of the smooth electrode.

decrease with an increase in S . The electrode becomes more rough with a decrease in S and peak currents increase. Effect of Fractal Characteristics on Peak Current vs Scan Rate Plots. Figure 3 shows the variation of double logarithmic plots of peak current vs scan rate. Dashed lines represent the current corresponding to the second peak whereas solid lines represent the current of the first peak. Black plots are for smooth electrodes which show the typical Randles-Sevčik behavior, whereas other colored plots correspond to rough electrodes. In the low scan rate regime, classical Randles-Sevčik behavior (Ipeak ∝ ν1/2) is followed at rough electrodes, whereas in the very high scan rate regime, peak current follows the roughness factor times the classical behavior for both redox peaks. In the intermediate regime, peak current vs scan rate plots in the presence of electrode roughness show anomalous behavior. The dependence of peak current on scan rate increases from half for both the peaks. Also, in this regime, the difference between the currents of the two peaks decreases with an increase in roughness. Figure 3a shows variation of peak current vs scan rate plots with variation in fractal dimension. With an increase in DH the electrode roughness increases and, therefore, an increase in the slopes of the Randles-Sevčik lines and decrease in the difference between the lines corresponding to two redox peaks of the same voltammogram is observed in the intermediate regime. Similarly, enhancement in values of the topothesy length also increases roughness and thus, in the anomalous regime, the

show rough electrode response of the EE mechanism with increasing fractal dimension. In each half cycle of the voltammograms, two current peaks are observed, one corresponding to each electron transfer step. The influence of roughness is more pronounced in and around the peaks region in cathodic as well as anodic cycles. As fractal dimension is increased, there is an increase in the peak heights. This behavior can be ascribed to the enhancement in surface irregularity with increase in fractal dimension of the electrode since DH is a direct measure of roughness. After the peaks, all the voltammograms converge toward the planar response. Figure 2c unravels the influence of variation in topothesy length (S τ), which is related to the width of roughness, on the cyclic staircase voltammograms of EE mechanism. With increase in topothesy length (S τ), peak heights corresponding to both charge transfer steps increase (see Figure 2c) in cathodic as well as anodic cycles. Thus, with increase in S τ, there is an increase in roughness of the electrode. This leads to a higher magnitude of peak currents. The lower cutoff length scale (S ) is, basically, a measure of the finest fractal length scale of the electrode roughness. Decrease in S means an increase in the roughness containing finer features. Thus, the effect of increase in lower length scale cutoff (S ) is the reverse of that of DH and S τ on the cyclic voltammograms (Figure 2b). Peak currents corresponding to both charge transfer steps 4312


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Figure 4. Ratio of peak to valley current with variation in fractal characteristics: (a) Fractal dimension (DH). (b) Lowest length scale of fractality (S ). (c) Strength of fractality (μ). Fractal characteristics used are (a) L = 10 μm, S = 100 nm, μ = 5 × 10−6 (a.u.); (b) DH = 2.3, L = 10 μm, μ = 5 × 10−6 (a.u.); (c) DH = 2.3, L = 10 μm, S = 200 nm. Other parameters used in the above calculations are diffusion coefficient (D = 5 × 10−6cm2/s, electrode area (A0 = 0.01 cm2), scan rate = 5 V/s, concentration (C01 = 4 × 10−6 mol/cm3, C02 = C03 = 0); the difference between the formal potentials of two steps is 150 mV. The black line represents the response of the smooth electrode.The inset in (c) shows the zoomed plot of the peak to valley current ratio for the second peak.

Figure 5. Variation in anodic−cathodic paek separation with varying (a) fractal dimension (DH), (b) lowest length scale of fractality (S ), (c) strength of fractality (S τ). Fractal characteristics used are (a) L = 10 μm, S = 200 nm, μ = 5 × 10−6 (a.u.); (b) DH = 2.3, L = 10 μm, μ = 5 × 10−6 (a.u.); (c) DH = 2.3, L = 10 μm, S = 200 nm. Other parameters used in the above calculations are diffusion coefficient (D = 5 × 10−6cm2/s, electrode area (A0 = 0.01 cm2), scan rate = 2.5 V/s, concentration (C01 = 4 × 10−6 mol/cm3, C02 = C03 = 0); the difference between the formal potentials of two steps is 150 mV. The solid lines correspond to the first cathodic peak whereas dashed lines correspond to the second cathodic peak. Black lines represent the response of the smooth electrode.

from the fact that SCV and CV are composed of a number of potentiostatic steps. When potentiostatic chronoamperometry is performed, two characteristic times arise that determine the crossovers from classical Cottrell to anomalous and anomalous to roughness factor controlled regimes of chronoamperometry.28 Corresponding to those crossover times of potentiostatic chronoamperometry, now we have crossover scan rates in SCV and CV. Effect of Fractal Characteristics on Sensitivity of Cyclic Voltammetry toward EE Mechanism. The first thing on which the sensitivity of CV for the double-step charge transfer mechanism depends is the difference between the formal potentials of the two steps. The more separated the two charge transfer steps are, the more easily they can be sensed in CV. Another factor that impacts the sensitivity is the ratio of the peaks to the minimum (valley) present between them. The lower the minimum compared to the peaks, the higher the sensitivity of the two steps would be. Though with roughness and the separation between the two steps is not altered, the second factor adding to the sensitivity, i.e., the ratio of peak currents to the minimum current, is found to be changed. As roughness is increased by increasing DH or S τ or by decreasing S , the ratio of peak currents to the minimum present between the two peaks increases. Figure 4a,b,c shows the variation in ratios of both peaks to the minimum. The increase in the ratio for the first peak is higher than the second. This observation is in accordance with the behavior seen

difference between the currents of two peak is lower for electrodes having higher S τ (cf. Figure 3c). The trend is reversed with S (Figure 3b). Roughness increases with a decrease in S . Thus, the difference between the currents of two peaks in the anomalous regime decreases with a decrease in S . The onset and ending of the anomalous regime is dependent on statistical morphological characteristics. The quantity, scan rate, is dependent on the potential step size (ΔE) and the step duration (τ). In an experiment ΔE is preferably kept constant. This is done in order to avoid the shift in peak position which may give rise to a pseudovariation in kinetics. Thus, for a given experiment, the varying parameter in scan rate is τ. Depending on the electrode morphology, we can find characteristic step intervals which decide at a particular step size which window of scan rate will show the anomalous behavior. The inner characteristic time interval (τi = S 2τ /πD) is controlled by the larger morphological length scale, S τ, which gives the inner crossover scan rate (νi = π DRT/(nFS 2τ )). νi basically represents the crossing point of the low scan rate regime of classical Randles-Sevčik behavior to the intermediate anomalous regime. Similarly, at higher scan rates also corresponding to the point of crossing of anomalous response to the microscopic area modified Randles-Sevčik response, we have a characteristic size of step interval (τo) that decides the outer crossover scan rate (νo). The outer characteristic time, τo (=S 2/πD) is controlled by S , and thus νo is given by πDRT/(nFS 2). The characteristic times have basically evolved 4313


Article

The Journal of Physical Chemistry C in the Randles-Sevčik plots. In the anomalous regime, roughness tries to bring the smaller peak closer to the higher (second) peak. The comparative increase in the ratio of the lower peak than the higher one also adds to the sensitivity which can be explained as follows. For an uncomplicated EE process at the smooth electrode, the ratio of first peak to the valley current is always smaller than that of the second peak. Sometimes it is so small that the first peak seems to be a shoulder rather than a peak and, in such cases, the identification of the location of the first peak is a tough job. However, even in such extreme cases most of the time the second peak is easily identifiable. So, anything that adds to the ratio of the first peak current to the minima current will enhance the ease of the location of the first peak and thus amplify the sensitivity. Here we want to emphasize that the enhancement in the comparative ratio of smaller peak to valley compared to that of the higher peak is subjected to the anomalous region of the scan rate. Effect of Fractal Characteristics on Anodic−Cathodic Peak Separation. With variation in roughness characteristics, there is a shift in positions of anodic as well as cathodic peaks corresponding to both charge transfer steps. Though the difference between the peaks positions of the two charge transfer steps remains unaltered, the separation between anodic and cathodic peaks of the individual E processes changes. With increase in roughness, the separation between the anodic and cathodic peaks decreases. An increase in the values of DH leads to an increase in roughness and thus the separation between anodic and cathodic peaks decreases (Figure 5). In other words, the cathodic peaks slide toward the more negative potentials whereas the anodic peaks shift toward the more positive potentials. The shift in peak positions of both steps is same in cathodic as well as anodic arms. As a consequence, the formal potential values of the individual charge transfers remain unaltered. A similar observation is there with an increase in the values of S τ. As the topothesy length increases, the interfacial width increases and thus the large features of roughness become larger and the electrode becomes rougher. Due to this, the separation between the anodic and cathodic peaks decreases individually for both E steps. With variation in S , the trend is reversed. As a decrease in S means introducing fractal behavior to smaller length scales, the roughness of the electrode is enhanced with a decrease in S . Thus, with declining S , the anodic−cathodic peak-to-peak separation decreases. This decrease in anodic−cathodic peak splitting with increase in roughness in the case of the EE process is similar to that observed for the single-step charge transfer process.8 The variation in the peak separation of the anodic−cathodic peaks due to electrode roughness can give rise to varying values of the heterogeneous rate constant, which is a characteristic of the electrode mechanism. Influence of Variation of Difference Between the Formal Potentials on the Voltammograms. Figure 6 contrasts the influence of variation in the difference between the formal potentials of the two charge transfer steps on the planar (Figure 6a) and rough electrode (R* = 14.1) responses (Figure 6b). The difference between the formal potentials is mentioned in the figures. As the difference between the formal potentials is increased, the separability of the two steps through cyclic staircase voltammetry increases for planar as well as rough electrodes. Figure 6 suggests that the separability is enhanced to a grater extent in the case of rough electrode as compared to the planar counterpart. This is due to the greater enhancement in the current of the first peak compared to the second at the rough

Figure 6. Influence of variation in the difference between formal potentials of two steps: (a) smooth electrode response, (b) rough electrode response. Fractal characteristics used for (b) are DH = 2.4, S = 200 nm, μ = 5 × 10−6 (a.u.), L = 10 μm. Other parameters used in the above calculations are diffusion coefficient (D = 5 × 10−6 cm2/s), electrode area (A0 = 0.01 cm2), scan rate = 12 V/s, concentration (C01 = 4 × 10−6 mol/cm3, C02 = C03 = 0). The difference between the formal potentials for each plot is mentioned in the figure legend.

electrode. A very good comparison can be made through the plot having a formal potential difference of 150 mV (brown curve). In the case of the planar electrode, only a shoulder is present corresponding to the first peak, whereas, in the rough electrode response, two clear-cut peaks are visible. Thus, the enhanced sensitivity of the rough electrode toward the EE mechanism is again established through this observation. The results emphasize the accounting of electrode roughness in calculating various kinetic parameters from solid electrode response data. Table 1 shows probable error while calculating Table 1. Error in Estimating Diffusion Coefficient, Concentration and Number of Electrons Transferred Using Classical Randles-Sevčik Equation at Rough Electrode R*

ν (V s−1)

h(μm)

% error in D

% error in C0

napp

2.3

0.1 1.0 10.0 0.1 1.0 10.0

2.28 2.28 2.28 5.07 5.07 5.07

1.94 13.70 67.60 9.93 78.50 560.20

1.0 6.8 29.9 1.9 14.9 67.6

1.01 1.04 1.20 1.04 1.21 1.87

5.1

diffusion coefficient, concentration, and number of electrons transferred from the first peak current data of the rough electrode. Depending on the scan rate and roughness factor, the apparent value of the number of electrons transferred in the first step calculated using the classical Randles-Sevčik equation may come out as twice the actual number of electrons transferred. Table 1 suggests that a huge error may also get introduced into the diffusion coefficient values estimated by fitting rough electrode data in the classical Randles-Sevčik equation. Similarly, the error may also get buried into the concentration of the electroactive species calculated in the electroanalytical experiments by employing the classical theory without accounting for the morphological aspect of the electrode. The effect of large roughness (viz., electrodes with large roughness factor and width of roughness) is detectable in aqueous as well as ionic media, while the effect of low roughness (R* ∼ 2) is detectable in ionic liquid (see Table 1 as well as Table 1 in ref 26).

■

SUMMARY AND CONCLUSIONS The formalism developed enables us to understand the voltammetric response of the two step charge transfer 4314


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the enhanced sensitivity of this technique for the EE process in the presence of electrode roughness. The formalism presented here can be extended to the general multistep charge transfer mechanism. Also, with a small modification, the model can be adapted to understand the response in the more complex intercalation electrode systems. These systems have strong interactions and require accounting for activity coefficients of intercalating particles which are usually accounted43,44 by the use of an additional exponent “z” in the right-hand side of the boundary constraints in eq 9.

mechanism in the presence of unequal diffusion coefficients of electroactive species at the rough electrode. The explicit expression for current transient is obtained for any arbitrary roughness where one can directly insert the electrode surface data from various microscopic techniques, viz., AFM, SEM, or STM, to predict the voltammetric response. The detailed analysis is performed for random roughness with finite fractal model. The fractal electrode current response is dominantly controlled by three fractal characteristics obtained from the measured power spectrum, viz., fractal dimension (DH), finest length scale of fractality (S ), and topothesy length (S τ), which is related to the interfacial width. The following conclusions can be drawn on the basis of the theoretical model developed in the present paper: 1. Roughness induces the current enhancement at and around the peaks of the voltammograms. This enhancement in current is seen with increase in DH or S τ or decrease in S . 2. The ratio of peak current to valley (minimum present between two peaks) current increases for both peaks with an increase in roughness. This is an indication of enhanced sensitivity of cyclic voltammetry toward the EE mechanism at rough electrodes. 3. The classical Randles-Sevčik equation is followed in the lower scan rate regime (ν < rate regime, viz., ν >

πDα S2

■

APPENDIX

Appendix A

Diffusion equations followed by the concentration profiles of first, second, and third species can be written as ∂δCα = Dα ∇2 δCα ∂t

(A.1)

Along with the initial and bulk conditions, we have one surface boundary condition to solve these diffusion equations for the case of single potential step applied at the electrode:

πDα

RT ). Similarly, in the high scan nF S2τ RT the classical Randles-Sevčik nF

( ) ( )

behavior, viz., Ip ∝ ν1/2, is observed but with a multiplicative roughness factor (R*). 4. The Randels-Sevčik plots show a deviation from linearity in the intermediate scan rate (anomalous) regime where the slope of the double logarithmic plots of peak current and scan rate increases from half in the presence of roughness for both the charge transfers. The exponent of the scan rate, along with DH, is controlled by S and S τ as well. Also, in this anomalous regime, the difference between the peak currents corresponding to the two charge transfers decreases with an increase in roughness. 5. Rough electrodes displays enhanced contrast for the EE process due to a decrease in the difference between the peak currents of two charge transfer steps and greater enhancement in the current at peak positions compared to the valley position. This observation is of particular importance in the systems where separations (E01′ − E02′) between two charge transfer steps is lower and, hence, the influence of comproportionation/ disproportionation reactions is suppressed. So the present theory can be used for more number of systems. 6. Electrode roughness is found to influence the positions of the peaks also. The separation between the anodic and cathodic peaks corresponding to each charge transfer step decreases with an increase in roughness without any alteration in the difference between the cathodic−cathodic and anodic−anodic peak positions. The decrease in the anodic−cathodic peak separations due to roughness may give rise to varying values of the heterogeneous rate constant which is a characteristic of the electrode mechanism. 7. The observations emphasize the accounting of electrode roughness in the classical theory; ignorance of which may cause errors in the estimation of composition, diffusion coefficients, and number of electrons transferred in the mechanism from rough electrodes data. Conclusively, we can say that our theory offers an elegant route to study the influence of electrode roughness on cyclic staircase voltammetry of the two step charge transfer mechanism and many crucial points are highlighted in this work which emphasize

C1(z , t ) |z = ζ( r ⃗ ) = θ (1)(t ) C2(z , t )

(A.2)

C2(z , t ) |z = ζ( r ⃗ ) = θ (2)(t ) C3(z , t )

(A.3)

The diffusion equations can be solved simultaneously by using the method of Green’s function only if we can decouple the Nernstian boundary condition using some relation between the surface concentration profiles of electroactive species. Such an identity under the condition of equal as well as unequal diffusion coefficients has been derived by us previously for the case of the single charge transfer mechanism.8 To further generalize the method for the case of double step charge transfer processes under unequal diffusion coefficients (D1 ≠ D2 ≠ D3) condition, we can proceed as follows: Flux balance condition is j = nFD1∂nδC1 + n2FD2∂nδC2 = −(nFD3∂nδC3 + n1FD2∂nδC2)

(A.4)

Simplifying, D1∂nδC1 + D2∂nδC2 = −D3∂nδC3

(A.5)

We can use flux balance condition to find the relation between δC1, δC2, and δC3. Expanding ∂n δC1 around z = 0 plane, ∂δC1( r ⃗ , z , t ) ∂z −

|z = 0 = −

D2 ∂nδC2(z = ζ( r ⃗), t ) D1

D3 ∂ 2δC1(z , t ) ∂nδC3(z = ζ( r ⃗), t ) − ζ( r ⃗) |z = 0 D1 ∂z 2

+ O(ζ 2)

(A.6)

Similarly, flux for C2 and C3 species can also be projected at the z = 0 plane 4315


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The Journal of Physical Chemistry C ∂δC2( r ⃗ , z , t ) ∂z −

D = − 1 ∂nδC1(z = ζ( r ⃗), t ) D2

|z = 0

δC3( r ⃗ , z , t ) = −D3

(A.7)

∂δC3( r ⃗ , z , t ) ∂z

+ ζ( r ⃗ )

D = − 1 ∂nδC1(z = ζ( r ⃗), t ) D3

z=0

+ O(ζ )

{e

Now we can write the concentration using Green’s function for Neumann boundary condition at the z = 0 plane:

∫0

t

∫S dS′0 G1( r ⃗ , t| r ′⃗ , t′)

dt ′ ×

0

∂δC1(S′, t ′) ∂z

z=ζ

∫0

dt ′ ×

z=0

0

∂δC2(S′, t ′) ∂z

∫0

t

∫S dS′0 G3( r ⃗ , t| r ′⃗ , t′)

dt ′ ×

0

z=0 z=ζ

∂δC3(S′, t ′) ∂z

z=0

−(z + z ′)2 /4Diτ ⎫ ⎪

+

e e e ⎨ ⎬ + 1/2 ⎪ 1/2 2(πDiτ ) 2(πDiτ )1/2 ⎪ ⎭ ⎩ 2(πDiτ )

z=ζ

e

−( r ⃗ − r ⃗ ′)2 /4D1τ

2(πD1τ )1/2 + ζ( r ⃗ )

∫0

2

∫ d2r′

2 z=0

⎫ ⎪ + O(ζ 2)⎬ ⎪ ⎭

+

δC 2( r ⃗ , z , t ) = −D2 e

−( r ⃗ − r ⃗ ′)2 /4D2τ 1/2

2(πD2τ ) + ζ( r ⃗ )

∫0

+

∫ d r′ 2

⎧ ⎪ D D × ⎨ 1 ∂nδC1(z = ζ( r ⃗ ), t )+ 3 ∂nδC3(z = ζ( r ⃗ ), t ) ⎪D D 2 2 ⎩

∂ 2δC 2(z , t ) ∂z

2

e−z /4D2τ dt ′ × (πD2τ )1/2

2 z=0

⎫ ⎪ + O(ζ 2)⎬ ⎪ ⎭

⎪

⎪

⎪

∫ d2r′

+ ζ( r ⃗ )

∂ 2δC1(z , t ) ∂z

z=ζ

2 z=0

⎫ + O(ζ 2)⎬ ⎭

∫0

t

⎫ ⎧ e−q2z dt ′⎨ ; p → τ⎬ × ⎭ ⎩ q2 ⎪

⎪

⎪

⎪

∫ d2r′

⎧D ; K → | r ⃗ − r ′⃗ | × ⎨ 1 ∂nδC1(S′, t ′) ⎩ D2

}

+ ζ( r ⃗ )

∂ 2δC 2(z , t ) ∂z 2

z=ζ

z=0

⎫ + O(ζ 2)⎬ ⎭

∫0

t

⎧ −q3z ⎫ ⎪e ⎪ × dt ′⎨ ; p → τ⎬ ⎪ ⎪ ⎩ q3 ⎭

∫ d2r′

⎧D ; K → | r ⃗ − r ′⃗ | × ⎨ 1 ∂nδC1(S′, t ′) ⎩ D3

D2 ∂nδC 2(S′, t ′) D3

}

+ ζ( r ⃗ ) z=ζ

∂ 2δC3(z , t ) ∂z

2 z=0

⎫ + O(ζ 2)⎬ ⎭

⎤ D3 ⟨∂nδC3(z , p , K⃗ ) z = ζ ⟩⎥ D1 ⎦

(A.19)

⎯→ 1 ⎡ D1 ⟨δC2(z = ζ , p , K )⟩ ≈ ⎢ ⟨∂nδC1(z , p , K⃗ ) z = ζ ⟩ q2 ⎣ D2

(A.13)

Similarly, we can write for δC2 and C3 t

⎪

⎯→ 1 ⎡ D2 ⟨δC1(z = ζ , p , K )⟩ ≈ ⎢ ⟨∂nδC2(z , p , K⃗ ) z = ζ ⟩ q1 ⎣ D1

⎧ ⎪D D × ⎨ 2 ∂nδC 2(z = ζ( r ⃗ ), t )+ 3 ∂nδC3(z = ζ( r ⃗ ), t ) ⎪ D D1 ⎩ 1

∂ 2δC1(z , t ) ∂z

e−z /4D1τ × (πD1τ )1/2

⎫ ⎧ e−q1z dt ′⎨ ; p → τ⎬ × ⎭ ⎩ q1

Expanding the difference concentrations at surface for small ζ and after ensemble averaging over surface randomness, the inhomogeneity term arising from the first order perturbation of flux of oxidized species vanishes (contains first moment of ζ). Writing total Fourier and Laplace transformed expression,

Substituting Green’s function and inhomogeneity term from the flux expression: dt ′

t

(A.18) (A.12)

t

∫0

D3 ∂nδC3(S′, t ′) D2

−K 2 /4D3τ

Gi( r ⃗ , z , t | r ′⃗ , z′, t ′)

δC1( r ⃗ , z , t ) = −D1

(A.15)

}

δC3( r ⃗ , z , t ) =

{e =

⎫ ⎪ + O(ζ 2)⎬ ⎪ ⎭

(A.17)

(A.11)

−( r ⃗ − r ′⃗ )2 /4Diτ ⎧ −(z − z ′)2 /4Diτ ⎪

z=0

D3 ∂nδC3(S′, t ′) D1

−K 2 /4D2τ

+

(A.10)

δC3( r ⃗ , t ) = D3

2

⎧D ; K → | r ⃗ − r ′⃗ | × ⎨ 2 ∂nδC 2(S′, t ′) ⎩ D1

δC 2( r ⃗ , z , t ) =

{e

∫S dS′0 G2( r ⃗ , t| r ′⃗ , t′)

∫ d2r′

(A.16)

Similarly, δC2 and δC3 can be written as δC 2( r ⃗ , t ) = D2

∂z

−K 2 /4D1τ

+

(A.9)

t

∂ 2δC3(z , t )

δC1( r ⃗ , z , t ) =

(A.8)

δC1( r ⃗ , t ) = D1

e−z /4D3τ × (πD3τ )1/2

Taking local Fourier and Laplace transforms,

∂ 2δC3(z , t ) D − 2 ∂nδC2(z = ζ( r ⃗), t ) − ζ( r ⃗) |z = 0 D3 ∂z 2 2

dt ′

2 ⎧ D e−( r ⃗ − r ⃗ ′) /4D3τ ⎪ D1 ⎨ ∂nδC1(z = ζ( r ⃗ ), t )+ 2 ∂nδC 2(z = ζ( r ⃗ ), t ) × ⎪D D3 2(πD3τ )1/2 ⎩ 3

D3 ∂ 2δC2(z , t ) ∂nδC3(z = ζ( r ⃗), t ) − ζ( r ⃗) |z = 0 D2 ∂z 2

+ O(ζ 2)

∫0

2

t

⎤ D3 ⟨∂nδC3(z , p , K⃗ ) z = ζ ⟩⎥ D2 ⎦

(A.20)

⎯→ 1 ⎡ D1 ⎢ ⟨∂nδC1(z , p , K⃗ ) z = ζ ⟩ ⟨δC3(z = ζ , p , K )⟩ ≈ q3 ⎣ D3 +

(A.14) 4316

⎤ D2 ⟨∂nδC 2(z , p , K⃗ ) z = ζ ⟩⎥ D3 ⎦

(A.21) DOI: 10.1021/acs.jpcc.6b00810 J. Phys. Chem. C 2016, 120, 4306−4321

Article

The Journal of Physical Chemistry C Using flux balance condition described by eqs A.5, A.20 and A.21 can be simplified as ⎯→ 1 ⟨δC2(z = ζ , p , K )⟩ ≈ ⟨∂nδC2(z , p , K⃗ ) z = ζ ⟩ q2 ⎯→ 1 ⟨δC3(z = ζ , p , K )⟩ ≈ ⟨∂nδC3(z , p , K⃗ ) z = ζ ⟩ q3

is a nonlinear functional of the boundary profile. Using Taylor expansion in roughness profile, we can transfer the surface boundary conditions to z = 0 plane. Now, the inhomogeneity, which previously was only a function of time, contains space dependent terms also.

(A.22)

(A.23)

Now, the expressions for fluxes of second and third species can be extracted from eqs A.22 and A.23 and can be substituted in eq A.19 as follows ⎯→ ⟨δC1(ζ , p , K )⟩ ≈

⎯→ 1 D3 ⟨δC3(z , p , K ) ⟩ q1 /q3 D1 z=ζ

2

(B.2)

δC3(z = 0, r , t ) = χ0 (t ) + χ3̃ ( r ⃗ , z = 0, t )

(B.3)

∞

q1 =

p + K2 = D1

⎞ K 2D1 p ⎛ ⎜⎜1 + + ···⎟⎟ D1 ⎝ 2p ⎠

q2 =

p + K2 = D2

p D2

⎛ ⎞ K 2D2 ⎜⎜1 + + ···⎟⎟ 2p ⎝ ⎠

(A.26)

p + K2 = D3

⎞ K 2D3 p ⎛ ⎜⎜1 + + ···⎟⎟ D3 ⎝ 2p ⎠

(A.27)

(A.25)

χm̂ (ζ , t ) = −

dχ0 (t ) 1

dt (A.28)

⎡ = ⎢θ (1)(0)θ (2)(0)e−σ12g(t )((σ2 − σ12)ξ2,3θ (2)(0) ⎢⎣

/⎡⎣1 + ξ2,3θ (2)(0)e−σ2 g(t ) + ξ1,3θ (1)(0)θ (2)(0)

(A.29)

e−σ12 g(t )⎤⎦ 2 ×

Thus, the identity for relation between ensemble averaged (over the surface configurations) concentrations of oxidized and reduced species given by above equation is applicable at the interface of a weakly and gently fluctuating surface. For our further calculations, we will assume that

dχ0 (t ) 2

dt

dg(t ) dt

(B.6)

⎡ = ⎢θ (2)(0)e−σ2 g(t )((σ12 − σ2)ξ1,3θ (1)(0)θ (2)(0) ⎢⎣ ⎤ e−σ12 g(t ) − σ2)(ξ1,3C10 + ξ2,3C20 + C30)⎥ ⎥⎦

D1 δC1(z = (ζ , r ⃗), t ) ≈ − D2 δC2(z = (ζ , r ⃗), t ) D3 δC3(z = (ζ , r ⃗), t )

(B.5)

⎤ e−σ2 g(t ) − σ12)(ξ1,3C10 + ξ2,3C20 + C30)⎥ ⎥⎦

D1 ⟨δC1(z = (ζ , r ⃗), t )⟩ = −( D2 ⟨δC2(z = (ζ , r ⃗), t )⟩

−

1 m ∂m ζ ( r ⃗)70 m m! ∂z

for m > 1, and the projection operator (70 ) has the property 70f (z) = f (0). These terms consist of products of various order terms and derivatives of unknown concentration at the average plane, that is, z = 0. The validity regime of this expansion is decided by the convergence of the Taylor expansion, which relies on the extent of smoothness that can be associated with the surface profile. Here, we can define other important quantities which are nothing but the differentials of the potential functions, χ01(t), χ02(t), and χ03(t) as follows:

Taking inverse Fourier and inverse Laplace,

D3 ⟨δC3(z = (ζ , r ⃗), t )⟩)

(B.4)

χ̂1, χ̂2, ... are operators which operate on concentration and generate effective source terms arising from Taylor expansion of boundary condition. The detailed form of operators χ̂m are expressed as follows

⎛ D 2 ⟨δC2(ζ (̂ K⃗ ), p)⟩ ⟨δC1(ζ (̂ K⃗ ), p)⟩ = −⎜⎜ ⎝ D1

+

∑ χm̂ δCi( r ⃗ , t ) m=1

Under the small diffusion length limit or small wavenumber, the contribution of terms containing K∥ can be neglected. Thus, under the small diffusion length and small ζ conditions eq A.24 can be simplified as

+

δC2(z = 0, r , t ) = χ0 (t ) + χ2̃ ( r ⃗ , z = 0, t )

χĩ ( r ⃗ , z = 0, t ) = χ ̂ δCi( r ⃗ , t ) =

qO∥ and qR∥ can be expanded for small diffusion length (DO/p) in comparison to K∥ as

⎞ D3 ⟨δC3(ζ (̂ K⃗ ), p)⟩⎟⎟ D1 ⎠

(B.1)

where χ01(t) = θ(1)θ(2) C̃ s−C01, χ02(t) = θ(2) C̃ s−C02, and χ03(t) = C̃ s−C03; C̃ s = (ξ1,3C01+ ξ2,3C02+C03)/(1+ξ2,3θ(2) + ξ1,3θ(1)θ(2)). Here, θ(1) and θ(2) both are the functions of applied potential (E(t)) and hence that of time. (A.24)

q3 =

1

3

⎯→ 1 D2 ⟨δC2(z , p , K ) ⟩ q1 /q2 D1 z=ζ +

δC1(z = 0, r , t ) = χ0 (t ) + χ1̃ ( r ⃗ , z = 0, t )

(A.30)

/⎡⎣1 + ξ2,3θ (2)(0)e−σ2 g(t ) + ξ1,3θ (1)(0)θ (2)(0)

Appendix B

The surface boundary constraints prescribed by eqs 11, 12, and 13 have only time dependent inhomogeneity. But the concentration field itself, as defined over a fluctuating interface,

e−σ12 g(t )⎤⎦ 2 × 4317

dg(t ) dt

(B.7) DOI: 10.1021/acs.jpcc.6b00810 J. Phys. Chem. C 2016, 120, 4306−4321

Article

The Journal of Physical Chemistry C dχ0 (t ) 3

dt

⎡ = ⎢(σ2ξ2,3θ (2)(0)e−σ2 g(t ) + σ12ξ1,3θ (1)(0)θ (2)(0) ⎢⎣

method allows us to satisfy order by order the right-hand side and left-hand side of eqs B.11 and B.12. The perturbative solution of concentration profile in LaplaceFourier domain (up to second order term) is ⎯→ δC1( K , z , p) = −χ0̅ (p) exp( −q1 z)[(2π )2 1

⎤ e−σ12 g(t ))(ξ1,3C10 + ξ2,3C20 + C30)⎥ ⎥⎦

δ(K⃗ ) + q1ζ (̂ K⃗ ) + *21ζ (̂ K⃗ − K⃗ ′)ζ (̂ K⃗ ′)]

/[1 + ξ2,3θ (2)(0)e−σ2 g(t ) + ξ1,3θ (1)(0)θ (2)(0) e−σ12 g(t )⎤⎦ 2 ×

dg(t ) dt

(B.13)

⎯→ δC2( K , z , p) = −χ0̅ (p) exp( −q2 z)[(2π )2

(B.8)

where σ2 = n2 fν and σ12 = (n1+n2) fν. The transfer of the surface boundary to the z = 0 plane which leads to the problem with somewhat more complex inhomogeneous boundary condition for the concentration field but on a plane is done in order to use the Green function to solve the diffusion equation.48,50 The Green function provides a powerful tool to solve linear differential equations under given boundary constraints with arbitrary source terms. The Green function is basically defined by a similar problem where all the constraints are homogeneous and the only inhomogeneous term present in the differential equation is a delta function. With the precise knowledge of Green function of a problem one can write down its solution by using magic rule in closed form as linear combinations of integrals involving the Green function and the functions appearing in the inhomogeneities. This formulation provides an integrodifferential equation which is amenable to systematic perturbation analysis. Moreover, these analyses can be interpreted physically. Therein lies the attractiveness of the Green function approach. The Green function tells us how the point source at an arbitrary position and time propagates through the concentration field which evolves in time and space. For the diffusion problem, the Green’s function G(r,⃗ t|r′⃗ , t′) follows the partial differential equation ⎛ ∂ ⎞ ⎜− − Dα ∇2 ⎟Gα( r ⃗ , t | r ′⃗ , t ′) = δ(t − t ′)δ(r − r′) ⎝ ∂t ⎠

2

δ(K⃗ ) + q2ζ (̂ K⃗ ) + *22ζ (̂ K⃗ − K⃗ ′)ζ (̂ K⃗ ′)] (B.14)

χ0α(p) is the Laplace transform of χ0α(t), ζ̂(K⃗ ∥) is the two dimensional Fourier transform of the rough surface profile ζ(r∥⃗ ), δ(K⃗ ∥) is the two- dimensional Dirac delta function in vector K⃗ ∥, *2α is an operator defined as *2α ≡

dt ′

(B.10)

δC2( r ⃗ , t ) = D2

1

∫0

t

dt ′

∫S

2

1

(B.11)

(B.17)

i(ζ ̂, p) = −pχ0̅ (p)[Ô 01(2π )2 δ(K⃗ ) + Ô11ζ (̂ K⃗ )

∂′n G2( r ⃗ , t | r ′⃗ , t ′)

1

+ 6̂ ′21 ζ (̂ K⃗ ′)ζ (̂ K⃗ − K⃗ ′)] − (n2p /n)χ0̅ (p)

0

[χ0 + (χ ̂ + χ2̂ + ···)δC2( r ′⃗ , t ′)]

(B.16)

Performing Laplace and Fourier transformations on the current density expressed by eq B.16 and substituting eqs B.13 and B.14 as the difference concentration profiles of first and the second species, we can obtain the current density expression for a weakly and gently varying rough surface in the Fourier and Laplace domain.

0

1

(B.15)

⎡ 1 7ζ ≡ 70⎢∂z + ζ ∂ 2z + ζ 2∂ 3z − ∇ ζ ·∇ − ζ ∇ ζ ·∇∂z ⎣ 2 ⎤ 1 − (∇ ζ )2 ∂z⎥ + O(ζ 3) ⎦ 2

∫S ∂′n G1( r ⃗ , t| r ′⃗ , t′)

[χ0 + (χ ̂ + χ2̂ + ...)δC1( r ′⃗ , t ′)]

− 1]

where n(=n1+n2) represents the total number of electrons transferred in the EE mechanism. The projection operator, 7ζ , basically projects the normal derivative of the concentration to the arbitrarily fluctuating surface, z = ζ(r∥⃗ ), which is obtained by Taylor expansion of the boundary condition around the z = 0 plane.

The formal solution for the effect of mixed boundary condition prescribed at z = 0 plane, between t = 0 and t, is propagated by the integral48 t

, ′

i = nFD17ζδC1( r ⃗ , t ) + n2FD2 7ζδC2( r ⃗ , t )

(B.9)

2 2 e−| r ⃗ − r ′⃗ | /4Dα(t − t ′) ⎡⎢ e−(z − z ′) /4Dα(t − t ′) = × 1/2 4πDα (t − t ′) ⎣⎢ 2(πDα (t − t ′))

∫0

∫ d2K ′ [2γα

Equations B.13 and B.14 are useful equations to predict the concentration distribution of species first and second around an arbitrary surface roughness profile under the influence of a general input potential function. This can be achieved by taking the inverse Laplace transform and the inverse Fourier transform of eqs B.13 and B.14 for a given roughness profile. The observable physical quantity, the current density, obtained from eqs 4, 5, and 6 can be written in terms of the projection operator (7ζ ) as follows:

Gα( r ⃗ , z , t | r ′⃗ , z′, t ′)

δC1( r ⃗ , t ) = D1

2(2π )2

qα = qαγα = [qα2 + K 2]1/2, K = |K⃗ |

and its solution for Dirichlet boundary problem is given by50

2 e−(z + z ′) /4Dα(t − t ′) ⎤⎥ − 2(πDα (t − t ′))1/2 ⎥⎦

qα2

2

(B.12)

[Ô 02(2π )2 δ(K⃗ ) + Ô12ζ (̂ K⃗ ) + 6̂ ′22 ζ (̂ K⃗ ′)ζ ̂

The solution of the above equation is obtained by expanding the surface concentration as a power in surface height profile, ζ. The

(K⃗ − K⃗ ′)] 4318

(B.18) DOI: 10.1021/acs.jpcc.6b00810 J. Phys. Chem. C 2016, 120, 4306−4321

Article

The Journal of Physical Chemistry C where χα0 (p) consists of all information about the perturbing potential signal, Ô 0α , Ô1α and 6̂ ′2α are operators for a smooth surface and the first- and second-order terms in Fourier transformed surface profile, respectively. These operators are functions of the phenomenological diffusion length and are defined as follows nF Ô 0α ≡ qα 6̂ ′2α ≡

O1̂ α ≡ nF qα(2π )2

K⃗ − K⃗ ′

2

nδC1(z = ζ( r ⃗), t ) + n2δC2(z = ζ( r ⃗), t ) = nχ0 (t ) + n2χ0 (t ) 1

Thus, eq C.5 can be solved for (nδC1(r,⃗ t) + n2δC2(r,⃗ t)) which can be used to predict the current expression from eq B.16 under the assumption of equal diffusion coefficients of the electroactive species. The results comes out to be same as described by eq 45 when D1 = D2 = D3.

nF(qα − qα)

■

qα ⎡


− , ′

1 2 (q + q α q α )− 2 α

⎤ 1 − K⃗ ′·(K⃗ − K⃗ ′)⎥ 2 ⎦

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

(B.19)

ACKNOWLEDGMENTS Parveen is grateful to UGC-India for providing SRF. R. K. is grateful to DST SERB and University of Delhi for financial support.

The scheme of a disproportionation/comproportionation reaction coupled with two charge transfer steps can be represented as follows:

■

O1 + n1e− ⇌ O2 O2 + n2e− ⇌ O3 (C.1)

here, O1, O2, O3 are the electroactive species whereas k1 and k−1 are the forward and backward rate constants for the third (disproportionation) reaction. Under the assumption of equal diffusion coefficients of all the electroactive species (D1 = D2 = D3 = D), the concentration profiles of the species follow the diffusion kinetic equations represented as follows: ∂C1( r ⃗ , t ) n n = D∇2 C1( r ⃗ , t ) + 2 k1C2n( r ⃗ , t ) − 2 k −1 ∂t n n C1n2( r ⃗ , t )C3n1( r ⃗ , t )

(C.2)

∂C2( r ⃗ , t ) = D∇2 C2( r ⃗ , t ) − k1C2n( r ⃗ , t ) + k −1C1n2( r ⃗ , t ) ∂t C3n1( r ⃗ , t )

(C.3)

∂C3( r ⃗ , t ) n n = D∇2 C3( r ⃗ , t ) + 1 k1C2n( r ⃗ , t ) − 1 k −1C1n2 ∂t n n C3n1( r ⃗ , t )

(C.4)

After multiplying the diffusion-kinetic equations of species 1 by n and that of species 2 by n2 and adding them together, one can write the resulting equation for the difference concentration profiles as follows ∂(nδC1( r ⃗ , t ) + n2δC2( r ⃗ , t )) ∂t = D∇2 (nδC1( r ⃗ , t ) + n2δC2( r ⃗ , t ))

(C.5)

Initially and in the bulk solution, the following constraint applies: nδC1( r ⃗ , t = 0) + n2δC2( r ⃗ , t = 0) = nδC1( r ⃗ , z → ∞ , t ) + n2δC2( r ⃗ , z → ∞ , t ) = 0

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k1

k −1

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Corresponding Author

Appendix C

nO2 HooI n2O1 + n1O3

(C.7)

2

(C.6)

Using Nernstian constraint applied at the electrode electrolyte interface, the surface boundary condition can be written as 4319


Article

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Theory for Cyclic Staircase Voltammetry of Two Step Charge Transfer

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