Theory for Anomalous Response in Cyclic Staircase Voltammetry

Article pubs.acs.org/JPCC

Theory for Anomalous Response in Cyclic Staircase Voltammetry: Electrode Roughness and Unequal Diffusivities Parveen and Rama Kant* Department of Chemistry, University of Delhi, Delhi 110007, India ABSTRACT: We develop a theory for cyclic staircase voltammetry (CSCV) of a reversible charge transfer process with unequal diffusivities on a randomly rough electrode. The roughness power spectrum based approach is developed and detailed analysis is performed for a finite fractal model. An elegant expression for the statistically averaged CSCV current is obtained in terms of single potential step current at a rough electrode. The extent of anomalous response due to finite fractal roughness is determined by scan rate. The characteristic peak current and peak to peak separation in CV are dependent on finite fractal features, namely, fractal dimension (DH), topothesy length (Sτ ), and finest length scale of fractality (S ). Peak to peak separation is reduced while peak current is enhanced with increase in DH and Sτ and decrease in S . The roughness induced anomalous regime often introduces errors in estimating diffusion coefficient from classical Randles−Sec̆vik equation. Our theory suggests that this error can be reduced using low scan rate regime data. Enhanced effects of roughness are seen in case of slowly diffusing electroactive species. Unequal diffusivities of redox species introduce shape asymmetry in CV of rough electrode. CV experiments have been carried out on Fc/Fc+ redox couple in room-temperature ionic liquid (BmimMF4) medium. Effects of morphology of mechanically and chemically roughened gold electrode and unequal diffusivities of Fc/Fc+ redox species in BmimMF4 are well captured by theoretical CV plots.

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Randles9 and Sevc̆ik10 were among the early investigators of CV who gave the famous Randles−Sevc̆ik relation that relates peak current (Ipeak) with scan rate (ν) at a planar electrode:

INTRODUCTION

For the investigation of an electrochemical reaction scheme, cyclic voltammetry (CV) is almost always chosen as the first option. The popularity of this technique originates from its ability to extract information about the chemical kinetics, for example, checking the reversibility of the redox couple, number of electrons involved in a redox process, rate constants, formal potentials, reaction mechanisms, formation constants, and diffusion coefficients.1−3 Apart from investigating the chemical kinetics of redox couples, this technique is also exploited in various fields like study of electrode materials,4,5 characterization of surface morphology of the electrodes,6 electrochemical energy devices, electrochemical sensing of various chemicals and biochemicals,8 etc. Although all transient techniques can, in principle, explore the same i−E−t (current−potential−time) space to obtain the needed data, CV, in which applied potential is a function of time, allows one to see easily the effects of E and t on the current in single experiment. However, the digital instruments that are currently in use, apply the cyclic potential and voltage in the form of a number of small steps or small duration pulses so the technique is better known as cyclic staircase voltammetry (CSCV). Such pulse based techniques, which were recently reviewed by Molina et al.7 for smooth electrodes of various geometries, plane, cylinder, hemispherical, spherical, disc, etc., additionally enable further reduction of background current by adjusting the value of the sampling time. © 2014 American Chemical Society

Ipeak ∝ ν1/2

(1)

Perone and co-workers11 extended the analytical solution presented by Christie and Lingane12 for staircase voltammetry for reversible systems to CSCV, which better mimics the CV response of the digital instruments used these days. The landmark paper of Nicholson and Shain13 categorized the CV response of various reaction schemes and provided numerical solution of the integrals obtained from boundary value problems. Some authors described CV as “electrochemical spectroscopy”.14 But the characteristic shapes and position of voltammetric waves not only fingerprint the individual properties of the redox couple but also are affected by a number of factors like electrode roughness and surface heterogeneity,15 solution resistance, adsorption phenomenon occurring at the electrode, viscosity of the solution, etc. Electrode roughness, because it influences the mass transport to the surface of the electrode, can significantly affect CV response, which is a diffusion dependent technique. Various approaches used to model rough electrode surfaces mathematically consider the topography of the surface as either deterministic or random. Deterministic roughness profiles are described by exact mathematical functions, while random Received: October 17, 2014 Published: October 24, 2014 26599

dx.doi.org/10.1021/jp510469b | J. Phys. Chem. C 2014, 118, 26599−26612

The Journal of Physical Chemistry C

Article

technique, for example, CV, through diffusion layer thickness,26 which acts as a “yardstick”. The system used by the authors has roughness amplitude smaller than the diffusion layer thickness. Hence, roughness is not sensed in the CV response. Such morphological scales can still affect the response if diffusion coefficients are lowered using viscous or ionic liquid medium27 or if the scan rates used are very high. Also, in contrast to the above-mentioned model, realistic electrodes have fractal roughness,22,23,27,37 which is found to have significant effect on linear scan voltammetry (LSV)36 even at lower scan rates. So realistic fractal roughness is expected to influence CV to a good extent. Fractal electrode roughness shows a self-affine scaling property, that is, asymmetric scaling behavior perpendicular to the surface, rather than a self-similar scaling property, that is, isotropic scaling behavior in all directions. An initial foundation for the study of CV at fractal electrodes was given by Pajkossy et al.19 and Stromme and co-workers,20 who used a scaling argument approach to study the effect of fractal roughness on CV. The Randles−Sevc̆ik relation was generalized by Stromme et al. for fractal electrodes, which they characterized only through fractal dimension DH:

surface profiles are described by their statistical properties.21−25,27 Whether the influence of roughness will be sensed in the CV is decided by the diffusion layer thickness. In the scan rate window where diffusion layer thickness is comparable to the morphological features of the electrode, response will be affected by roughness. For the system with linear scan voltammetry, diffusion layer thickness (δe(E)) for reversible charge transfer is given by26 δe(E) =

δe(E) =

πDRT ; nFν

πD|E − Ep| ν

E ≤ Ep

;

E > Ep

(2)

(3)

Ep is peak potential, ν is scan rate, D is diffusion coefficient of electroactive species, and R, T, F, and n are gas constant, standard temperature, Faraday’s constant, and the number of electrons involved in redox reaction, respectively. The current in CV can be estimated from eqs 2 and 3. The current (through Fick’s first law) is directly proportional to number of electrons transferred (n), diffusion coefficient (D), concentration difference across depletion or diffusion layer thickness (Cs), and area of the electrode (A0) and inversely proportional to effective diffusion layer thickness (δe(E)): nFA 0Cs(E)D I (E ) ≈ δe(E)

Ipeak ∝ ν(DH − 1)/2

Another approach found in the literature is the fractional diffusion equation.47−49 It relies on the assumption that the electrode is smooth, whereas diffusion toward it follows a fractional diffusion equation. As its impact, the fractional exponent that accounts for roughness induced anomalous response in this approach remains as a fitting parameter and does not relate directly to any morphological features of the electrode. Most of the clean electrodes surfaces generated by mechanical, chemical, and electrochemical etching or nanoparticle deposition have band-limited fractal and bifractal natures.6,37 In order to account for the electrochemical response of such electrodes, Kant and co-workers have used the concept of band-limited (finite) fractals to model rough electrodes during the investigations of various electroanalytical techniques like chronoamperometry,21−25,27 chronocoulometry,28 absorbance transient,29 impedance,30−32 admittance,33,34 voltammetry,35,36 etc. Realistic roughness (fractals) can be well characterized through a band limited power-law power spectrum obtained from AFM and SEM images. The power spectrum is basically the Fourier transform of the two point correlation function of the surface profile (ζ(r∥⃗ )), which gives a measure of the rapidity of variation of a given surface. The power spectral density (PSD) of the surface topography of an electrode performs a decomposition of the surface profile into its spatial wavelengths and expresses the roughness power per unit spatial frequency or wavenumbers over the sampling length. Recently, the power spectrum based theory was experimentally verified by Kant and co-workers37 for the generalized Cottrell current expression derived in 1993:21,38

(4)

where Cs(E)/δe(E) is the potential dependent concentration gradient. Equation 4 is simplified by replacing E by peak potential (Ep), and substituting δe(E) from eq 2, we get the Randles−Sevc̆ik expression for peak current. ⎛ F 3 ⎞1/2 3/2 1/2 Ipeak ≈ ⎜ ⎟ n A 0 D Cs(Ep)ν ⎝ πRT ⎠

(5)

The minimum length scale of roughness (S ) that can be sensed by diffusion layer thickness is also estimated by eq 2, that is, δe(E) ≈ S . The smallest length scale of roughness (S ) and corresponding (high) scan rate (νS ) (which will sense this roughness) are related as

νS =

πDRT nF S2

(6)

Equation 3 decides the dynamic contribution from the presence of multiple length scales of roughness. The largest length scale of roughness, Sτ , decides the lowest scan rate above which the anomalous regime due to fractal roughness will be visible. ντ =

πD|E − Ep| Sτ 2

(8)

(7)

Usually |E − Ep| is on the order of 0.1−0.2 V. After |E − Ep| exceeds the value of 0.2 V, diffusion layer thickness becomes so large that the diffusion front assumes planarity, no longer sensing the roughness features of the electrode. Recently, Compton and co-workers presented a computational approach for CV on a rough electrode where the roughness is modeled as a deterministic surface with regular array of peak shaped protrusions.16 They concluded that rough electrodes are remarkably insensitive to purely diffusional effects when studied with CV. The effect of roughness is sensed by the diffusional

⎛ 1 ⟨IgC(t )⟩ = IC(t )⎜1 + 2(2π )2 Dt ⎝

2

∫ d2K (1 − e−K Dt )

⎞ ⟨|ζ (̂ K⃗ )|2 ⟩⎟ ⎠

(9)

where IC(t) is the Cottrell current given by 26600


The Journal of Physical Chemistry C IC(t ) =

nFA 0DCs ; πDt

Cs =

CO0 − θC R0 1+θ

Article

(10)

0

where θ = e−nf(Ei−E ′), f = F/(RT), Ei and E0′ represents applied potential and formal potential, respectively, C0O and C0R are the bulk concentrations of oxidized and reduced species, A0 is geometrical area of the electrode, n is number of electrons transferred, F is Faraday’s constant, R is the gas constant, T is the standard temperature, and D represents the diffusion coefficient (for simplicity, it is assumed that diffusion coefficients of reduced and oxidized species are equal, that is, DO = DR = D). In 2010, Kant proposed a general theory for an arbitrary potential sweep voltammetry on any arbitrary roughness electrode under reversible charge transfer conditions.35 This work beautifully related total current transient for any arbitrary type of perturbing potential to the convolution of single potential step current transient with impressed transient potential function for any arbitrary roughness electrode profile: ⟨I(t )⟩ =

d dt

∫0

Figure 1. Schematic of the problem for a reversible charge transfer under cyclic staircase potential modulation applied at the rough electrode. Finite fractal model of roughness characterized with a fractal dimension DH and two cutoff lengths, S and L, of fractality, h being the width of interface.

t

χ0̃ (t − t ′)⟨IgC(t ′)⟩ dt ′

coefficients of oxidized and reduced species are unequal. Here we have highlighted the impact of unequal diffusion coefficients on CV. The paper is organized in several sections. First, the problem of cyclic staircase voltammetry at any arbitrary profile electrode is mathematically formulated for the case of unequal diffusion coefficients of the redox species. The explicit expression for current transient is obtained for a finite fractal electrode. Further, various aspects of variations in morphological characteristics on CSCV current response are explored. Finally, the Fc/Fc+ system in a room-temperature ionic liquid (RTIL; BmimBF4) is used to experimentally validate the theory on a rough gold electrode before we summarize our results and conclusions.

(11)

where, ⟨IgC(t)⟩ represents the potentiostatic current response of any arbitrary random roughness electrode represented by eq 9 and χ̃0 is expressed as follows: 0

χ0̃ (t ) =

χ0 (t ) −Cs

CRθ (e−σg(t ) − 1) CO0 − C R0 θ ; θ + 1 + θ (e−σg(t ) − 1)

1− = 1

σ = nfν (12)

χ0(t) is the difference between surface and bulk concentrations, that is, CO(0, t) − C0O, obtained from Nernst constraint with the assumption, DO = DR = D. In our last publication extending the work to the analytical solution for the LSV current transient36 at a finite self-affine fractal electrode, a significant effect of roughness was seen on both peak current and peak position, which are the characteristics of LSV, and important conclusions are drawn on its basis. This prompted us to further extend this approach to the more popular and frequently used technique of CV. In rest of the work carried out by our group,21−25,27−36 it is assumed that diffusion coefficients of oxidized and reduced species are equal. It is a good approximation for systems like K3Fe(CN)6−K4Fe(CN)6 in aqueous solvents. In higher viscosity solvents like ionic liquids, which are extensively used these days, Unwin et al.15 have revealed that even subtle variations in the electrode surface heterogeneity affect the rate of the reduction process. In such systems, effects of roughness are expected to be more exaggerated attributed to their low diffusion coefficients that often vary significantly for oxidized and reduced species due to their different charge and thus difference in the hydration sphere or, in general, due to difference in their environment. Moreover, in CV, both anodic and cathodic cycles are analyzed simultaneously so the individual diffusion coefficients become more important to account for in this technique. In the present manuscript, we will formulate an analytical model for cyclic staircase voltammetry at a rough electrode for the case when DO ≠ DR (see Figure 1). It is a fundamentally important problem, and apart from other applied systems, this work has distinguished importance in ionic liquids electrochemistry. Most of the time, in ionic liquids, the diffusion

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MATHEMATICAL FORMULATION Semi-infinite mass transport for a single charge transfer step, O + ne− ⇌ R, is expressed by following partial differential equation ∂ Cα( r ⃗ , t ) = Dα ∇2 Cα( r ⃗ , t ) ∂t

(13)

where Cα(r,⃗ t), the concentration profile with α, represents oxidized (O) or reduced species (R), Dα is the diffusion coefficient, ▽2 ≡ (∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2) and r ⃗ represents the 3-D vector, r ⃗ ≡ (x, y, z). A uniform initial and bulk concentration Cα(r,⃗ t) is assumed, namely, Cα(r,⃗ t = 0) = Cα(r,⃗ z → ∞, t) = C0α where C0α represents the bulk concentration of the species. We assume that the charge transfer process considered is reversible and is not complicated by double layer charging, migration, or solution resistance. For fast charge transfer processes, the Nernstian constraint relates the concentration of species involved in the redox couple to the applied potential E(t) at the interface: CO(z = ζ( r ⃗), t ) = C R (z = ζ( r ⃗), t ) exp( −nf (E(t ) − E 0 ′))

(14)

To solve this boundary value problem, we want to use the Green’s function approach. The Green’s function method is simplified using the decoupling approximation for a Nernstian system with unequal diffusivities. The decoupling approxima26601



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analytical route for solving the integral present in eq 11. The Heaviside step function assumes only two values.40 When this property is used, the time dependency in the form of unit step functions can be taken out of the exponential, and the above expression further simplifies to

tion for oxidized and reduced species relates their concentrations, δCO and δCR, at the interface as (see Appendix): DO δCO = − DR δC R

(15) 2

Equation 15 is an exact identity for a planar electrode, as well as for the situation when DO = DR at a rough electrode.39 This assumption will be applicable for gently fluctuating surfaces and, in general for small and large diffusion length situations. With the above decoupling relation for the Nernst equation, anodic (IagC(t)) and cathodic (IcgC(t)) current responses at a rough electrode for a single potential step impulse applied to the system of unequal diffusivities of redox couples are given by c ⟨IgC (t )⟩ a ⟨IgC (t )⟩

= IC(t )R(t , DO)

χ0̃ (t ) = u(t ) − u(t − τ ) + f (1)[u(t − τ ) − u(t − 2τ )] + ... + f (N )u(t − Nτ )

where f(m) is given by 1−

(16)

= IC(t )R(t , DR )

f (m) = 1+

(17)

where IC is the Cottrell current given by eq 10. R(t , Di) = 1 +

1 2(2π )2 Dit

i

CO0 − C R0 exp( −nf (E(t ) − E 0 ′)) DO DR

exp( −nf (E(t ) − E 0 ′))

⟨I(t , N )⟩ =

(19)

For the applied cyclic staircase potential perturbation, E(t), can be expressed as M m=1

[

where Ei is the initial potential, ΔE is the potential step applied after every constant time duration τ, and M is the number of steps after which the potential ramp switches to the opposite direction. N is the total number of pulses applied, and u is Heaviside unit step function such that

χ̃0 (t ) = 1+

DO DR

⎛ ⎜1 + ⎝

DO DR

t

c dt ′f (0)δ(t − t ′)⟨IgC (t ′)⟩

∫0

t

N≤M

dt ′[ ∑ (f (m) − f (m − 1)) m=1

∫0

t

dt ′

∑

(f (m) − f (m − 1))δ(t − t ′ − mτ )] (26)

∞

(27)

In the expression for ⟨I(t, N)⟩, f(m) is a dimensionless weight function dependent on the initial composition of electroactive species, their diffusion coefficient, and potential. Using the above-described property of the Dirac delta function, the arguments of IgC are replaced by t − mτ and the rest of the integral of Dirac delta function that is carried out over whole time regime becomes unity. Thus, the CSCV current transient at a randomly rough electrode in the presence of unequal diffusivities of redox species is expressed as

N

M

⎞ θ⎟ ⎠

(25)

∫−∞ dx h(x)δ(x − a) = h(a)

(22)

[e−nf ΔE(∑m = 1 u(t − mτ) −∑m = M + 1 u(t − mτ)) − 1]

θ

[e(M−| m − M |)ΔE * − 1]

IcgC and IagC are the generalized Cottrellian current for a random roughness electrode for cathodic and anodic sweeps, respectively. Explicit expressions for IcgC and IagC in the case of unequal diffusion coefficients of oxidized and reduced species have been derived in the Appendix. The integrals in eq 26 can be solved by using the following property of the Dirac delta function (δ(t)):41

(21)

M

⎞ θ⎟ ⎠

a (t ′)⟩ ⟨IgC

χ̃0 is given by θCR0

DO DR

m=M+1

Using cyclic staircase potential perturbation, the surface boundary condition can be rewritten by segregating the Nernstian constraint for a single potential step, Cs:

CO0 − θCR0

;

θ

N

(20)

1−

∫0

m=M+1

δCO(z = ζ( r ⃗), t ) = −Csχ0̃

⎛ ⎜1 + ⎝

[e(M−| m − M |)ΔE * − 1]

c (t ′)⟩ + δ(t − t ′ − mτ )]⟨IgC

u(t − mτ )ΔE

⎧ 0, t ≤ τ u(t − τ ) = ⎨ ⎩1, t > τ

DO DR

+

N

∑ u(t − mτ)ΔE − ∑

E (t ) = E i +

θC R0

where f(0) = 1. This emerged structure of χ̃0 reveals the superposition of surface concentrations. For asingle potential step experiment, χ̃0 expressed by eq 24 apparently becomes equal to u(t). So, as a consequence of this simplification, the result for chronoamperometry becomes easily recognizable. χ̃0(t) can be substituted from eq 24 into eq 11 to express the cyclic staircase current transient at a random roughness electrode in the presence of unequal diffusivities as follows:

Di is the diffusion coefficient of electroactive species, namely, DO for cathodic current and DR for anodic current. The decoupled Nernstian constraint that applies a local charge transfer limitation at the interface can be written for oxidized species as

1+

θC R0 CO0 −

ΔE* = nf ΔE

2

∫ d2K (1 − e−K D t )⟨|ζ(̂ K )|2 ⟩ (18)

δCO(z = ζ( r ⃗), t ) = −

(24)

N

[e−nf ΔE(∑m = 1 u(t − mτ) −∑m = M + 1 u(t − mτ)) − 1] (23)

Here, we used the unit step function notation for representing cyclic staircase potential. Apart from being more specific, this representation has an additional advantage, which opened the 26602



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N≤M c ⟨I(t , N )⟩ = ⟨IgC (t )⟩ +

∑

(f (m) − f (m − 1))

m=1

∑

(f (m) − f (m − 1))

m=M+1 a ⟨IgC (t

− m τ )⟩

(28)

The formalism developed here provides a general expression for CSCV current transient for any randomly rough electrode. CSCV current is represented in terms of summation of individual generalized Cottrellian current with different potential weight functions. The beauty of the above expression is that with the knowledge of single potential step current at any geometry of electrode, current for the cyclic staircase potential sequence can be written. The generalized result for the continuum version, that is, CV, obtained by Kant35 can be deduced under the limit τ → 0 when DO = DR. Under the limit when potential steps are applied for infinitesimally small time duration, the summation in eq 28 is replaced by the integral ⟨I(t )⟩ = ⟨IgC(t )⟩ +

1 Cs

⟨IgC(t ′)⟩ +

1 Cs

∫λ

t

∫0

λ

CURRENT TRANSIENT AT FINITE SELF-AFFINE FRACTAL ELECTRODE

The fractional dimensionality found in nature can be accounted by using the concept of “fractals”.17 The real surfaces usually show a self-affine scaling property,17,18,50−55 that is, asymmetric scaling behavior perpendicular to the surface, rather than a selfsimilar scaling property, that is, isotropic scaling behavior in all directions. To characterize such geometries, the band limited power-law power spectrum can be used:

N c ⟨IgC (t − m τ )⟩ +

Article

⟨|ζ (̂ K⃗ )|2 ⟩ = μ |K⃗ |2DH − 7 ,

But experimentally, τ cannot be considered too small because double layer charging current interferes near the time regime where the potential step is just started. At each step, charging currents can be minimized if current is sampled in the latter half of the step. A sampling parameter, a, can be incorporated into eq 28, which takes into account the fraction of time after which the current is sampled during the potential step.11,43,44 Substituting t = (N − a)τ into eq 28 gives

i ⟨IaC (t )⟩ =

⎛ −2δ − L−2δ nF DO A 0Cs(E) ⎛ ⎜⎜1 + μ ⎜⎜ l πt δDit 8π ⎝ ⎝

N≤M

(f (m) − f (m − 1))

+

m=1 c ⟨IgC ((N N

+

− a)τ − m τ )⟩

∑

Γ(δ , Dit /l 2 , Dit /L2) ⎞⎞ ⎟⎟⎟⎟ (Dit )1 + δ ⎠⎠

(32)

where, Di = DO for cathodic current (IcaC(t)) and Di = DR for anodic reaction current (IaaC(t)), δ = DH − 5/2, Γ(α, x0, x1) = Γ(α, x0) − Γ(α, x1) = γ(α, x1) − γ(α, x0), and Γ(α, xi) and γ(α, xi) are the incomplete gamma functions.40 The above represented expression for anomalous Cottrell current is composed of two parts. First term is the Cottrell current, that is, the smooth electrode response, whereas the additional part reflects the contribution from roughness. It contains the signature of surface fractal characteristics and can be understood as a smooth electrode response times the dynamic roughness factor. Under the limit μ → 0, the contribution from fractal roughness (shown by second term in the parentheses) vanishes and cathodic as well as anodic anomalous Cottrell current ⟨IiaC(t)⟩ reduce to classical Cottrell current, namely, smooth electrode potentiostatic current response.

(f (m) − f (m − 1))

m=M+1 a ⟨IgC ((N − a)τ − mτ )⟩

(31)

the power spectrum (Sτ = μ1/(2DH − 3)). Sτ is basically a transversal length that relates to the amplitude of roughness. Sτ → 0 implies no roughness. Two longitudinal lengths, S and L, represent the lower and upper bounds under which the power spectrum follows power-law function. Hence the surface shows fractal behavior between lower cutoff length scale S and the upper cutoff length scale L. The anomalous Cottrell current, that is, the potentiostatic current transient for a finite fractal electrode, is given by38

⎡ (C 0 + C 0 )σθ exp( − σ[2λ − (t − t ′)]) ⎤ R ⎢ O ⎥⟨IgC(t ′)⟩ ⎣ (1 + θ exp( − σ[2λ − (t − t ′)]))2 ⎦ (29)

∑

1/L ≤ |K⃗ | ≤ 1/l

⟨|ζ̂(K⃗ ||)|2⟩ represents the power spectrum of roughness that describes surface roughness as a power-law function of wavenumber (K⃗ ||). From the power spectrum, we get the knowledge of four morphological characteristics, namely, DH, S , L, and Sτ . DH is a fractal dimension that describes the scale invariant property of roughness. But being scale invariant, it does not characterize the surface completely; for example, a mountain and a microscale electrode may have the same fractal dimensions. Here comes the concept of topothesy length, Sτ , which can be extracted from μ, the proportionality constant of

⎡ (C 0 + C 0 )σθ exp(− σ(t − t ′)) ⎤ R ⎢ O ⎥ ⎣ (1 + θ exp(− σ(t − t ′)))2 ⎦

c ⟨I(N , a)⟩ = ⟨IgC ((N − a)τ )⟩ +

for

(30)

where a is defined as a = 1 − τ′/τ and τ′ is the time at which the current is sampled during the potential step, which is always smaller than τ. When IgC is replaced by Cottrellian current, we obtain the result of Perone et al.11 for a planar electrode under the assumption C0R = 0. At a = 0, that is, when sampling of current is done at the end of each potential step, eq 30 reduces to eq 28. If IgC is replaced by a potentiostatic current transient for disc, band, cylinder, and spherical electrodes, we get the corresponding results obtained by Molina et al.42 Results for cyclic multipulse voltammetry can also be obtained. Its just a matter of adjusting the weight factors, that is, ΔE. 26603



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N≤M c ⟨I(N , a)⟩ = ⟨IaC ((N − a)τ )⟩ +

∑

6610LV SEM instrument in secondary electron imaging or morphological mode. This mode provides an excellent topographic image of the surface under observation with very high resolution and depth of field at very high scanning speeds (Figure 2). Cyclic voltammetry, in combination with SEM pictures was used to find the power spectrum, which in turn was exploited to extract additional morphological details.6

(f (m) − f (m − 1))

m=1 c ⟨IaC ((N N

+

− a)τ − m τ )⟩

∑

(f (m) − f (m − 1))

m=M+1 a ⟨IaC ((N − a)τ − mτ )⟩

(33)

In the above expression, the cyclic voltammetric current at a realistic fractal roughness electrode is pictured as the sum of the individual cathodic and anodic anomalous Cottrellian current for cathodic and anodic scans, respectively, each multiplied by the varying difference of consecutive potential weighted functions. Under the limit Sτ → 0, the above expression simplifies to the response for a planar electrode. In the absence of the bulk concentration of the reduced species, the result deduced by Perone et al.11 for a planar electrode CSCV response can be retrieved. If the descending order of staircase, that is, the reverse scan is not applied, the summation carried out from the (M + 1)th step to the Nth step vanishes, and the above expression reduces to the SCV result.36

Figure 2. SEM image of rough gold electrode at 1300× magnification.

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RESULTS AND DISCUSSION Here we graphically explore the effect of morphology of a rough electrode on the current response. Equation 30 contains the statistical information on the morphology of the electrode in the form of the power spectrum. The power spectrum provides four characteristics that describe the surface morphology of the electrode, namely, fractal dimension (DH), which is the measure of the global scale invariance property of roughness, finest length scale of fractality (S ), sample size or upper cutoff length scale of fractality (L), and the topothesy length (Sτ ), which relates to the proportionality constant, μ (=Sτ 2DH − 3). Out of these four, only DH, S , and Sτ are found to dominantly control the electrochemical response of diffusion limited charge transfer systems.22,28,29,34 Before starting the elaborate discussion of the influence of electrode morphology on cyclic voltammetry, we want to analyze the influence of starting potential and initial composition on the current in multiple cycling. Effect of Multiple Cycling on Cyclic Voltammetry. The first cycle or scan in cyclic voltammetry is found to be different from the next or all the subsequent cycles (Figure 3). Figure 3a shows first (green curve) and second (red curve) cycles of CV when the initial concentration of the reduced species is zero. The inset represents the applied potential variation with time (scaled by step duration). If initially, the concentration of the reduced species is zero or the relative concentration of the reduced species is very small, then the first or virgin cycle remains open at the lower vertex. The theory predicts a little higher current in the cathodic scan and complete overlap in subsequent cycles. In Figure 3b, the initial concentration of the reduced species is comparable to that of the oxidized species. Here, the first cycle starts from a high anodic (negative) current value and crosses the anodic arm. In the case of non-zero initial concentration of reduced species, the cathodic (upper) half of the first cycle remains a little lower but very close to the subsequent cycles, which perfectly stack over one another. The first cycle is also affected by the starting or initial potential (Figure 3c). The inset plot shows the applied

EXPERIMENTAL DETAILS Electrode Pretreatment. A 2 mm long gold wire (0.5 mm diameter, 99.5% purity, Arora Matthey Ltd., India) was used as working electrode for electrochemical measurement. One end of the wire was enclosed in glass tube for making an electrical connection. The working electrode was mechanically roughened using 1000 grit emery paper and kept in ethanol for 48 h and was then sonicated in triple distilled water for 10 min. Further, the electrode was kept in piranha solution for 20 min and then rinsed with triple distilled water. In addition, the electrode was electrochemically cleaned and activated in 0.5 M H2SO4 between potential −0.2 to 1.2 V vs SCE at scan rate 100 mV/s for 100 cycles with electrode sheet as counter electrode. Electrochemical Experiments. All chemicals used were of analytical grade and were used as received without any further purification, unless stated. Experiments were carried out using a μAutolab III potentiostat. The geometrical area of the working electrode was 0.033 cm2. The electrolyte solution, 5 mM ferrocene in [Bmim]BF4, was prepared by first dissolving ferrocene in a minimum amount of acetonitrile (0.025 mL in 3 mL of the RTIL) and then adding the RTIL. The actual ratio of bulk concentrations of oxidized (Fc+) and reduced species (Fc) present in the solution was calculated using open circuit potential (OCP) measurement. Cyclic voltammetry was carried out at various scan rates using silver wire as quasi-reference and bright platinum coil as counter electrode in the potential window ranging from 0.1 to 0.5 V. In order to subtract charging current, baseline correction was done in NOVA-1.8 software. Area Measurements. The electrochemically active microscopic area was calculated from the gold oxide region of CV between 0.0 and 1.4 V at 100 mV/s scan rate with SCE as reference electrode and platinum sheet as counter electrode. The oxide reduction peaks were integrated using a charge density associated with reduction of the oxide layer as 390 μC/ cm2 at 25 °C. Using the measured microscopic area (0.1154 cm2), we calculated the roughness factor R*. For further characterization of the electrode, SEM (scanning electron microscopy) imaging was carried out using JEOL-JSM 26604



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Figure 4. Effect of unequal diffusion coefficients with roughness on peak current vs scan rate plots. Dashed lines show cathodic response and solid lines show anodic response (black for planar, red for rough electrode). Fractal parameters used are DH = 2.2, L = 50 μm, S = 0.1 μm, μ = 5 × 10−6 au. Other parameters used in the generation of above plots are diffusion coefficient (DO = 1 × 10−7cm2/s; DR = 5 × 10−7cm2/s), sampling parameter (a = 0), electrode area (A0 = 0.01 cm2), and concentration of oxidized and reduced species (C0O = C0R = 4 × 10−6 mol/cm3).

morphology of fractal electrodes through fractal dimension.19,20 The influence of roughness on peak current vs scan rate plots, assuming equal diffusion coefficients of oxidized and reduced species, have been elaborately explained earlier.36 Here, we are more interested in the case when DO ≠ DR. In Figure 4, dashed lines show cathodic peak current, whereas solid lines show anodic peak currents. Black curves indicate planar response, whereas red lines correspond to rough electrode response. In the low scan rate regime, plots corresponding to anodic and cathodic response of rough electrodes merge with those of planar counterparts. This implies that in the low scan rate regime, the effect of roughness is not visible as diffusion length exceeds Sτ . The difference in anodic and cathodic peak currents is attributed to the difference in the diffusion coefficients of oxidized and reduced species. In the intermediate scan rate regime, anomalous behavior is seen due to roughness;36 the slope lifts from half to a higher value. Here, the influence of roughness and diffusion coefficient couple with each other. The beginning of the anomalous intermediate scan rate regime is controlled by topothesy length (Sτ ) of the electrode:

Figure 3. Comparison of the first two cycles of cyclic voltammetry (a) in the absence of an initial concentration of reduced species, (b) at equal concentration of oxidized and reduced species, and (c) with variation in starting potential. Fractal parameters used are DH = 2.45, L = 12.5 μm, S = 44 nm, μ = 3.8 × 10−9 au. Other parameters used in the generation of above plots are diffusion coefficient (DO = 2.6 × 10−7 cm2/s; DR = 3.4 × 10−7 cm2/s), sampling parameter (a = 0), electrode area (A0 = 0.033 cm2), concentration of oxidized and reduced species (for part a, C0O = 2.5 × 10−6 mol/cm3, C0R = 0; for parts b and c, C0O = C0R = 2.5 × 10−6 mol/cm3).

ντ =

πD(E − Ep) Sτ 2

(34)

The typical magnitude of potential deviation from the peak potential value is |E − Ep| ≈ 0.1−0.2 V. In the high scan rate regime, the influence of diffusion coefficients is seen only in the crossover potential, and the plots with different diffusion coefficients but with same roughness start merging. The crossover scan rate (νS ), above which only the microscopic area effects of roughness control the response and the effect of unequal diffusion coefficients vanishes, is decided by finest length scale of fractality (S ):

potential variation with time for three cyclic voltammograms presented in Figure 3c. When the relative concentration of both the species is significant, the virgin cycle acquires the same initial current value at different potentials, depending upon the value of start potential, whereas no change is noticed in the second cycle. The more anodic is the initial potential, the closer the crossing point of the cathodic arm and anodic half of first cycle comes to the left, lower vertex of the second and subsequent cycles. The second cycle is independent of starting applied potential and relative initial concentrations. Similar observations were made by Molina et al.42,58 for smooth geometries. Effect of Roughness and Diffusion Coefficient on Generalized Randles−Sevcĭ k Plots. Here, we explore the influence of fractal roughness on double logarithmic plots of peak current and scan rate (Figure 4), which are very crucial to understand CV and are also employed to characterize the

πDRT (35) nF S2 Thus, we have observed that in the low scan rate regime, the effect of diffusion coefficient can be segregated from the effect of roughness. Based on this observation, we realize that to calculate diffusion coefficient from CV, low scan rate regime data should be used. Figure 5 shows the effect of scan rate on the value of diffusion coefficient estimated at an electrode νS =

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Figure 7. Effect of unequal diffusion coefficients on voltammograms. Fractal characteristics used to generate the plots are DH = 2.23, L = 50 μm, S = 50 nm, μ = 5 × 10−6 au. Other parameters used in the calculations are diffusion coefficient (for dashed lines, DO = (1, 0.1, and 0.01) × 10−7 cm2/s, DR = (2, 0.2, and 0.02) × 10−7 cm2/s; for solid lines, DO = (1, 0.1, and 0.01) × 10−7 cm2/s, DR = (2.5, 0.25, and 0.025) × 10−7 cm2/s), sampling parameter (a = 0), electrode area (A0 = 0.01 cm2), scan rate = 12.8 mV/s, and concentration of oxidized and reduced species (C0O = C0R = 4 × 10−6 mol/cm3).

Figure 5. Effect of roughness on estimation of diffusion coefficients from peak current of CV at various scan rates. Fractal characteristics used to generate the plots are DH = 2.4, L = 50 μm, S = 200 nm, Sτ = 5 μm. Other parameters used in the calculations are diffusion coefficient (DO = DR = 5 × 10−7cm2/s), sampling parameter (a = 0), electrode area (A0 = 0.1 cm2), and concentration of oxidized and reduced species (C0O = C0R = 4 × 10−6 mol/cm3).

having roughness. The ratio of a diffusion coefficient estimated at a rough electrode of roughness factor 6.7 to that at a smooth electrode comes out to be as high as 16 times the actual value at high scan rates (Figure 5). Effect of Dominant Fractal Characteristics on Peak Current. Figure 6 shows the influence of fractal characteristics (DH, S , and Sτ ) on cyclic voltammograms when diffusion coefficients of both the redox species are equal. Figure 6a,b,c shows the variation in cyclic voltammograms with varying DH, S , and Sτ , respectively. The lower (black line) plot shows the planar response, whereas all subsequent (upper) plots represent the rough electrode response. On increasing roughness, either by increasing DH or Sτ or by decreasing S , the current increases by the same factor for both anodic and cathodic voltammograms. This observation is similar to that observed in the case of SCV,36 which prevails in both the cycles of CV. In the case when DO ≠ DR, an asymmetry in the anodic and cathodic voltammograms is visible (see Figure 7). If the diffusion coefficient of the reduced species is higher, the cathodic voltammogram peak height is larger than the anodic one. This can be explained in terms of diffusion layer thickness. The species having a higher diffusion coefficient forms a larger diffusion layer, so the current due to that particular species is less. In experiments, if one increases the viscosity of a medium, usually both DO and DR decrease, but their ratio remains preserved. This is the case shown in Figure 7, where current decreases but no shift in potentials of anodic and cathodic

peaks is visible. In the case when DO and DR are varied without preserving their ratio, as shown by dashed and solid lines in Figure 7, the alteration in the location of peaks on the potential axis is observed. Effect of Fractal Characteristics on Peak to Peak Separation. In cyclic voltammetry, the magnitude of peak to peak separation is an important parameter that is being used to characterize the extent of reversibility of a redox couple. For the case of equal diffusion coefficients of oxidized and reduced species, we have found that with alteration in roughness, separation between the anodic and cathodic peaks can be altered. This observation raises serious question on “the number”, 59.9 mV, that asserts the reversibility of the redox reaction and supports the simulations carried out earlier.20,45,46 With increase in fractal dimension, the peaks of anodic and cathodic voltammograms come closer to each other, Figure 8a. This behavior of roughness is similar to the one that emerges due to adsorption at the electrode. The trend of peak to peak separation reverses with increase in S (Figure 8b). As we increase S , peak to peak separation increases. Since increase in S means decrease in roughness, this observation is consistent with the fact that roughness induces a decrease in separation between anodic and cathodic peak potentials. Variation in roughness with change in Sτ shows same effect on peak separation as that of DH. As Sτ is increased, peak to peak separation decreases (Figure 8c).

Figure 6. Effect 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 as follows: (a) DH = 2.23, 2.25, 2.26, and 2.27, L = 50 μm, S = 50 nm, μ = 5 × 10−6 au; (b) DH = 2.25, L = 50 μm, S = 500 nm, μ = 5 × 10−6 au; (c) DH = 2.25, L = 50 μm; S = 50 nm, μ = (5, 10, 15, and 20) × 10−6 au. Other parameters used in the above calculations are diffusion coefficient (D = 5 × 10−6 cm2/s sampling parameter (a = 0), electrode area (A0 = 0.01 cm2), scan rate = 7.7 mV/s, and concentration (C0O = C0R = 4 × 10−6 mol/cm3). Black lines in all panels represent the response of a smooth electrode. 26606



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Figure 8. Effect of fractal characteristics on peak separation: (a) fractal dimension; (b) lower cutoff length scale; (c) topothesy length. Fractal characteristics used are as follows: (a) L = 50 μm, S = 50 nm, μ = 5 × 10−6 au; (b) DH = 2.3, L = 50 μm, μ = 5 × 10−6 au; (c) DH = 2.3, L = 50 μm, S = 45 nm. Parameters used in the generation of above plots are diffusion coefficient (D = 5 × 10−6 cm2/s), electrode area (A0 = 0.01 cm2), and concentration (C0O = C0R = 4 × 10−6 mol/cm3).

Considering DO ≠ DR and keeping the ratio of DO and DR fixed, if we increase both the diffusion coefficients, no visible change in the peak separation is observed. If the ratio of the diffusion coefficients is altered, a change in peak separations is also noticed (see Figure 11). Effect of Roughness on Effective Diffusion Layer Thickness. Effective diffusion layer thickness expression for CV experiment at randomly rough electrode is given by56,57 δeα(N , a) =

nFDα A 0Cs(E)f (N ) ⟨I(N , a)⟩

the reduced species increases, the diffusion layer thickness of the reduced species (anodic arm) grows faster than the diffusion layer thickness of the oxidized species. In contrast, the diffusion layer thickness of the oxidized species grows more slowly with increase in DR, that is, with decrease in the ratio DO/DR. Even a difference of ten percent in the diffusion coefficients can be sensed by diffusion layer thickness, shown by the green line in the plot. When DO = DR, shown by black curves, the diffusion layer thickness of both the species grows with approximately same rate but not exactly the same. The asymmetry in the arms with respect to potential axis is attributed to the difference in bulk concentrations of oxidized and reduced species. CV−SEM Surface Characterization. To characterize the rough gold electrode quantitatively, we adopt the method of SEM imaging in combination with cyclic voltammetry.6 First, we denoise the SEM images from “salt and pepper” noise using the Laplacian filter in Mathematica 8.0.1 software. White noise with flat PSD is considered uncorrelated and independently distributed noise, which can be ascertained by varying the regularization parameter of Laplacian filter. For further processing of the SEM image, the 1D power spectrum was obtained by running a discrete Fourier transform consecutively on all lines of image. The PSD of the total SEM image data was extracted by averaging line by line the 1-D PSD. PSD along all lines of a denoised image were averaged to reduce the influence of statistical artifacts.6

(36)

Because the initial part of the current goes mostly into the charging of the double layer, it is better to neglect that part of diffusion layer thickness. Figure 9 shows the effect of roughness

Figure 9. Effect of roughness on diffusion layer thickness. Fractal characteristics used are L = 50 μm, S = 50 nm, μ = 5 × 10−6 au. Parameters used in the generation of above plots are diffusion coefficient (DO = 1 × 10−7 cm2/s, DR = (1, 1.1, 1.4, 1.7, and 2) × 10−7 cm2/s), electrode area (A0 = 0.01 cm2), and concentration of oxidized and reduced species (C0O = 1 × 10−6 mol/cm3, C0R = 2 × 10−6 mol/ cm3).

N

PSD(Kx)s =

Δx |∑ ζ(x)r exp[2πi(r − 1)(s − 1)/N ]| N r=1 (37)

where ζ(x)r are the data points of the surface in gray scale along the x axis having N points, r and s are the running index for ζ(x)r and PSD(Kx)s, respectively. ⟨...⟩ gives the averaging over the surface or ensemble of curves along the other axis. Δx is the pixel size. The magnitude of wavenumbers, Kx and Ky along the x and y axes, respectively, has been calculated using the relation, Kx = Ky = (s − 1)/(ΔxN). Power spectra extracted from various images, resolution ranging from 10000 to 100000, were combined appropriately according to their spatial frequency to give a complete representation of roughness of the whole surface after normalization. Normalization was done basically to eliminate the effects of varying contrast, brightness, instrumental parameters, etc. Now, the actual height information was missing in the PSD because SEM image only gives the relative values of the height profile varying between 0 and 1. To extract

on effective Nernst diffusion layer thickness (δeα). Solid curves show the diffusion layer thickness corresponding to anodic and cathodic cycles at a rough electrode, while dashed curves show that at a planar electrode. The rough electrode current response comes out higher than that of the planar electrode, and effective diffusion layer thickness is inversely related to current, so values of δeα are higher for the planar electrode in comparison to the rough electrode. This means that the effective diffusion layer thickness grows slowly for a rough electrode in comparison to a planar electrode. Diffusion layer thickness is also found to depend on diffusion coefficient. Here we are discussing the case when the diffusion coefficient of the reduced species is higher than that of the oxidized species. As the diffusion coefficient of 26607



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Figure 10. Cyclic voltammograms of 5 mM ferrocene in BmimBF4 ionic liquid performed at a rough electrode (experiment, red circles; theory, solid black line) at various scan rates: (a) 10 V/s; (b) 9 V/s; (c) 8 V/s; (d) 7 V/s; (e) 6 V/s; (f) 5 V/s; (g) 4 V/s; (h) 3 V/s; (i) 2 V/s; (j) 1 V/s; (k) 900 mV/s; (l) 800 mV/s. Fractal characteristics of gold electrode are as follows: (a) DH = 2.45, L = 1.26 μm, S = 44 nm, Sτ = 0.39 μm. Other parameters used in the calculations are diffusion coefficient (DO = 2.6 × 10−7cm2/s, DR = 3.4 × 10−7cm2/s), sampling parameter (a = 0), electrode area (A0 = 0.033 cm2), and concentration (C0O = C0R = 2.5 mM).

this information, we use the R* value obtained from CV to calculate mean square gradient, m2(CV). m2(CV) was then scaled by the mean square gradient obtained from PSD, m2(PSD), to get a scaling factor. Multiplying the morphological PSD with this scale factor gives the actual PSD of morphology of the electrode roughness. Now, the PSD was used to obtain zeroth moment (m0(PSD)), which was further utilized to get the information on mean square width (MSW) of roughness or missing height elevation in the SEM micrograph, h (=(m0)1/2). The size of the finest feature of roughness was calculated from the intersection point of noise PSD and image PSD. Cyclic Voltammetry Experiments at a Rough Gold Electrode. Cyclic voltammetry of 5 mM ferrocene in BmimBF4 at various scan rates was carried out. Throughout the experiment, temperature was maintained at 25 ± 1 °C. Using open circuit potential (OCP) measurement, we estimated the bulk concentrations of oxidized (Fc+) and reduced species (Fc) present in the solution as C0O = C0R = 2.5 mM. Diffusion coefficients of oxidized and reduced species calculated from peak current of CV data at low scan rates are DO = 2.6 × 10−7 cm2 s−1 and DR = 3.4 × 10−7 cm2 s−1. The difference in diffusion coefficients of oxidized and reduced species is attributed to the difference in the hydration spheres of the respective species in ionic liquid medium. Also, the value of the diffusion coefficients is slightly higher than that in pure ionic liquid. This is due to the presence of a small amount of acetonitrile, which was added in order to dissolve ferrocene before adding ionic liquid. CV plots experimentally obtained

were analyzed in NOVA 1.8 software. Because CV is a potential scan technique, it is significantly influenced by alteration in electric double layer due to adsorption and associated charging current. Irrespective of which CV cycle one focuses on, virgin or stabilized, both will require baseline or electric double layer corrections for comparison with diffusion controlled charge transfer theories (which are usually devoid of electric double layer contributions). In Figure 10 experimental data is illustrated by red circles, whereas theoretical plots are shown by solid black lines. The experimental data used is that of the stable CV obtained after a few repeated cycles or scans, and baseline corrections used for electric double layer contributions were applied using NOVA software. Also, most experimental analysis uses stabilized CV after multiple scans; hence we compare the second cycle of our theory with baseline corrected data of the stabilized CV. The potential, scan rate, and roughness factor dependent charging current are expressed as cubic polynomial in potential, which is obtained by fitting data for charging current (Idl): Idl(E) ≈ vR *(a0 + a1E + a 2E2 + a3E3)

(38)

where the value of a0 is 1.72 μF, a1 = −1.76 μF/V, a2 = 57.07 μF/V2, and a3 = −62.80 μF/V3. If CV data is not corrected for the double layer contributions, it will give rise to an apparent fractal dimension by an electrochemical approach. We have shown amply in our various publications22,27,36 that not only the fractal dimension is responsible for electrochemical 26608



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slowly diffusing species. (6) The effects of the asymmetric diffusivities and the electrode roughness can be recognized from the cyclic voltammograms by varying scan rates. CV measurements in the low scan rate regime, that is, ν < ντ, are weakly influenced by the electrode roughness, and this regime is not influenced by kinetics or pseudokinetics due to ohmic contributions. Hence in the ratio of peak currents of cathodic and anodic voltammograms (at low scan rates) deviation from unity is indication of asymmetric diffusivities of oxidized and reduced species. The measurements in high scan rate (ν > νS ) are mainly influenced by contribution of kinetics or pseudokinetics due to ohmic losses and microscopic area. Similarly, moderate scan rate (ντ ≤ ν ≤ νS ) is influenced by the roughness features while weakly influenced by the charge transfer kinetics or ohmic contributions. (7) In the low scan rate regime, effects of diffusion coefficients can be segregated from the effects of roughness pointing toward the fact that from CV, diffusion coefficients should be calculated in slow scan rate regime. (8) Effective diffusion layer thickness (δe(E)) grows slowly in the case of a rough electrode in comparison to the planar one. In the case of unequal diffusion coefficients, the ratio of DO and DR controls diffusion length, δe(E). δe grows slowly for species with lower diffusion coefficient and will show enhanced influence of roughness in their corresponding voltammogram. Hence it will generate asymmetry in the cathodic and anodic arms of voltammograms. (9) If the bulk concentrations of oxidized and reduced species are not equal, an asymmetry in effective depletion layer arms with respect to potential axis is seen. (10) This theory shows good correlation with CV experiments carried out in Fc/Fc+ system in RTIL (BmimBF4) medium at various scan rates. (11) Thus, our theory is a crucial step toward understanding CV response of rough electrodes especially in high viscosity solvents like ionic liquids. The formulation presented opens the analytical route for other multipulse techniques also.

response exponents, but smallest scale of fractality and topothesy length too are responsible for it. In order to include the effects of using a nonvirgin experimental cycle, theoretical voltammograms used to fit data were taken from the second cycle (which is quite different from the first). Experimental cyclic voltammograms are well in agreement with those theoretically predicted. Figure 11

Figure 11. Comparison of planar (black line) and rough (blue line) theoretical cyclic voltammogram with experimental data (red circles) at scan rate 8 V/s. Parameters are listed in the Figure 10 caption.

illustrates the comparison of a rough electrode CV with a theoretical planar CV plot at 8 V s−1. Rough theoretical and experimental CV are in good agreement. But the theoretical planar cyclic voltammogram underestimates the current, especially around the peak area. This emphasizes that the accounting contribution from the roughness of the electrode in the cyclic voltammetry technique is crucial, especially at high and intermediate scan rates. These effects become more prominent in a high viscosity medium like ionic liquids.

■

SUMMARY AND CONCLUSIONS In the present work we have analyzed the effect of finite fractal roughness on cyclic staircase/CV response. Different diffusion coefficient values for oxidized and reduced species are used while developing the formalism. Following conclusions are drawn on the basis of theory developed: (1) Current of cathodic, as well as anodic, voltammograms increases with increase in fractal dimension and topothesy length but decreases with increase in the size of finest length scale of fractality. Peaks become broader and higher with increase in roughness. (2) Height of cathodic and anodic peak currents depends upon the corresponding diffusion coefficients of the electroactive species. If the diffusion coefficient of the oxidized species is higher than that of the reduced species, the magnitude of the anodic peak current is higher than that of the cathodic peak current for reversible charge transfer. (3) The variation in separation of anodic and cathodic peaks is also observed with alteration in finite fractal features. As the electrode surface becomes more and more rough, due to enhancement in values of DH or Sτ or decrease in value of S , the peak separation is reduced, that is, the anodic and cathodic voltammograms come closer to each other. This observation may have implications in corrosion related studies.59 (4) If the diffusion coefficients are changed in different voltammograms in such a manner that their ratio is kept constant, no shift in peak potentials is noticed. In case of alteration in the ratio of diffusion coefficients, a shift in peak positions is perceptible. (5) In the presence of unequal diffusivities of the redox species, the three regimes shown by double logarithemic plots of anodic or cathodic peak current vs scan rate are found to have difference in their cross over points. The peak current generated due to the fast diffusing species, achieves the crossover at lower values of scan rate in comparison to the peak current generated due to

■

APPENDIX Diffusion equations for oxidized and reduced species can be written as ∂δCO = DO∇2 δCO ∂t

(39)

∂δC R = DR ∇2 δC R ∂t

(40)

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: CO = θC R

(41)

In terms of change in concentration δCO and δCR, Nernst constraint can be rewritten as δCO + CO0 = θ(δC R + C R0 )

(42)

The diffusion equations can be solved simultaneously by using the method of Green’s function only if we can decouple the Nernstian boundary condition. In our previous work, we have used the relation δCO = −δCR, which is valid when diffusion coefficients of oxidized and reduced species are equal. To further generalize the method for the case of unequal diffusion coefficients (DO ≠ DR), we need to find some relation 26609



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between δCO and δCR to decouple the Nernstian boundary

δCO( r ⃗ , z , t ) = −DO

condition. This can be achieved as follows: Flux balance condition is

∫0

∫ (43)

d2r′

e−( r ⃗ − r ′⃗ ) /(4DOτ) 2(πDOτ )1/2

⎧ ⎪D × ⎨ R ∂nδC R (S′, t ′) z = 0 ⎪D ⎩ O

We can use flux balance condition to find relation between δCO and δCR. Expanding ∂nδCO around z = 0 plane,

+ ζ( r ′⃗ )

∂δCO( r ⃗ , z , t )

δC R ( r ⃗ , z , t ) = −DR

z=0

+ O(ζ 2)

∫

plane

∫0

(49)

+ ζ( r ′⃗ ) z=0

D ∂ 2δC R (z , t ) = − O ∂nδCO(z = ζ( r ⃗), t ) − ζ( r ⃗) DR ∂z 2

δCO( r ⃗ , z , t ) =

(45)

∫ d2r′{e−K

Now we can write concentration using Green’s function for

dt ′ ×

∫S dS0′G( r ⃗ , t| r ′⃗ , t′) (46)

z=0

t

dt ′ ×

∫S dS0′G( r ⃗ , t| r ′⃗ , t′) 0

∂δC R (S′, t ′) ∂z

(47)

z=0

(50)

2

∫0

t

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

/(4DOτ )

; K → | r ⃗ − r ′⃗ |}

z=0

(51)

(52)

D 1 ⟨δC R (z = ζ , p , K⃗ )⟩ ≈ ⊗ O ⟨∂nδCO(z , p , K⃗ )|z = ζ ⟩ qR DR

e−( r ⃗ − r ′⃗ ) /(4Dτ) ⎧ e−(z − z ′) /(4Dτ) e−(z + z ′) /(4Dτ) ⎫ ⎨ ⎬ + 1/2 ⎪ 2(πDτ )1/2 ⎪ 2(πDτ )1/2 ⎭ ⎩ 2(πDτ ) 2

⎪

z=0

D 1 ⟨δCO(z = ζ , p , K⃗ )⟩ ≈ ⊗ R ⟨∂nδC R (z , p , K⃗ )|z = ζ ⟩ qO DO

G( r ⃗ , z , t | r ′⃗ , z′, t ′) 2

∂z 2

⎫ ⎪ + O(ζ 2)⎬ ⎪ ⎭

Expanding δCO at the 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,

Similarly, δCR can be written as

∫0

∂ 2δC R

⎫ ⎪ + O(ζ 2)⎬ ⎪ ⎭

0

∂δCO(S′, t ′) ∂z

e−z /(4DR τ) × (πDR τ )1/2

⎧ ⎪D ∂ 2δCO(z , t ) × ⎨ R ∂nδCR(S′, t ′) z = ζ + ζ( r ⃗) ⎪D ∂z 2 ⎩ O

Neumann boundary condition at z = 0 plane: t

dt ′

Taking local Fourier and Laplace transforms, z=0

+ O(ζ 2)

∫0

2

t

2

∂δC R ( r ⃗ , z , t )

=

z=0

2

Similarly, flux for reduced species can also be projected at z = 0

δC R ( r ⃗ , t ) = DR

∂z 2

⎫ ⎪ + O(ζ 2)⎬ ⎪ ⎭

e−( r ⃗ − r ′⃗ ) /(4DR τ) d r′ 2(πDR τ )1/2 ⎧ ⎪D × ⎨ O ∂nδCO(S′, t ′) z = 0 ⎪D ⎩ R

(44)

δCO( r ⃗ , t ) = DO

∂ 2δCO

Similarly, we can write for δCR

z=0

∂ 2δCO(z , t ) D = − R ∂nδC R (z = ζ( r ⃗), t ) − ζ( r ⃗) DO ∂z 2

∂z

2

e−z /(4DOτ) × dt ′ (πDOτ )1/2 2

j = nFDO∂nδCO = −nFDR ∂nδC R

∂z

t

2

(53)

⎪

where ⊗ represents the convolution operator defined as (48)

f (K⃗ ) ⊗ g (K⃗ ) =

Substituting Green’s function and inhomogeneity term from

∫ dK⃗ ′ f (K⃗ ′)g(K⃗

− K⃗ ′)

(54)

By use of the flux balance condition, DR⟨∂nδCR|z=ζ⟩ can be replaced by −DO⟨∂nδCO|z=ζ⟩

flux expression: 26610


The Journal of Physical Chemistry C ⟨δCO(ζ , p , K⃗ )⟩ ≈

Article

D 1 ⊗ R ⟨δC R (z , p , K⃗ )|z = ζ ⟩ qO /qR DO

Voltammetry and SEM Image. Appl. Surf. Sci. 2013, 282, 105−114 and references therein.. (7) Laborda, E.; Gonzalez, J.; Molina, A. Recent Advances on the Theory of Pulse Techniques: A Mini Review. Electrochem. Commun. 2014, 43, 25−30. (8) Huang, X. J.; Aldous, L.; O’Mahony, A. M.; Campo, F. J.; Compton, R. G. Toward Membrane-Free Amperometric Gas Sensors: A Microelectrode Array Approach. Anal. Chem. 2010, 82, 5238−5245. (9) Randles, J. E. B. A Cathode Ray Polarograph. Part II- The Current-Voltage Curves. Trans. Faraday Soc. 1948, 44, 327−338. (10) Sevc̆ik, A. Oscillographic Polarography with Periodical Triangular Voltage. Collect. Czech. Chem. Commun. 1948, 13, 349−377. (11) Miaw, L. H. L.; Boudreau, P. A.; Perone, S. P. Theoretical and Experimental Evaluation of Cyclic Staircase Voltammetry. Anal. Chem. 1978, 50, 1988−1996. (12) Christie, J. H.; Lingane, P. J. Theory of Staircase Voltammetry. J. Electroanal. Chem. 1965, 10, 176−182. (13) Nicholson, R. S.; Shain, I. Theory of Stationary Electrode Polarography. Anal. Chem. 1964, 36, 706−723. (14) Heinze, J. Cyclic Voltammetry-Electrochemical Spectroscopy. New Analytical Methods. Angew. Chem, Int. Ed. Engl. 1984, 23 (11), 831−847. (15) Aaronson, B. D. B.; Lai, S. C. S.; Unwin, P. R. Spatially Resolved Electrochemistry in Ionic Liquids: Surface Structure Effects on Triiodide Reduction at Platinum Electrodes. Langmuir 2014, 30, 1915−1919. (16) Menshykau, D.; Streeter, I.; Compton, R. G. Influence of Electrode Roughness on Cyclic Voltammetry. J. Phys. Chem. C 2008, 112, 14428−14438. (17) Mandelbrot, B. B. The Fractal Geometry of Nature; Freeman: San Francisco, CA, 1977. (18) Feder, J. Fractals; Plenum: New York, 1988. (19) Pajkossy, T.; Nyikos, L. Diffusion to Fractal Surfaces-III Linear Sweep and Cyclic Voltammograms. Electrochim. Acta 1989, 34, 181− 186. (20) Stromme, M.; Niklasson, G. A.; Granqvist, C. G. Determination of Fractal Dimension by Cyclic I-V Studies: The Laplace-Transform Method. Phys. Rev. B 1995, 52, 14192−14197. (21) Kant, R.; Rangarajan, S. K. Effect of Surface Roughness on Diffusion-Limited Charge Transfer. J. Electroanal. Chem. 1994, 368, 1− 21. (22) Jha, S. K.; Sangal, A.; Kant, R. Diffusion-Controlled Potentiostatic Current Transients on Realistic Fractal Electrodes. J. Electroanl. Chem. 2008, 615 (2), 180−190. (23) Kant, R.; Jha, S. K. Theory of Anomalous Diffusive Reaction Rates on Realistic Self-Affine Fractals. J. Phys. Chem. C 2007, 111, 14040−14044. (24) Srivastav, S.; Kant, R. Theory of Generalized Cottrellian Current at Rough Electrode with Solution Resistance Effects. J. Phys. Chem. C 2010, 114, 10066−10076. (25) Kant, R.; Sarathbabu, M.; Srivastav, S. Effect of Uncompensated Solution Resistance on Quasireversible Charge Transfer at Rough and Finite Fractal Electrode. Electrochim. Acta 2013, 95, 237−245. (26) Kant, R.; Kaur, J.; Singh, M. B. Electrochemistry Vol. 12, Nanosystems Electrochemistry; Wadhawan, J. D., Compton, R. G., Eds.; Specialist Periodical Reports; Royal Society of Chemistry: London, 2014; Vol. 12, pp 336−378. (27) Dhillon, S.; Kant, R. Theory of Double Potential Step Chronoamperometry at Rough Electrodes: Reversible Redox Reaction and Ohmic Effects. Electrochim. Acta 2014, 129, 245−258. (28) Islam, M. M.; Kant, R. Generalization of the Anson Equation for Fractal and Non-Fractal Rough Electrodes. Electrochim. Acta 2011, 56, 4467−4474. (29) Kant, R.; Islam, M. M. Theory of Absorbance Transients of an Optically Transparent Rough Electrode. J. Phys. Chem. C 2010, 114, 19357−19364. (30) Kant, R.; Rangarajan, S. K. Effect of Surface Roughness on Interfacial Reaction-Diffusion Admittance. J. Electroanal. Chem. 2003, 552, 141−151.

(55)

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

p +K DO

qR =

p +K DR

2

2

=

=

⎞ K 2DO p ⎛ ⎜⎜1 + + ...⎟⎟ DO ⎝ 2p ⎠

(56)

⎛ ⎞ K 2DR ⎜⎜1 + + ...⎟⎟ 2p ⎝ ⎠

(57)

p DR

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 55 can be simplified as ⟨δCO(ζ (̂ K⃗ ), p)⟩ = −

DR ⟨δC R (ζ (̂ K⃗ ), p)⟩ DO

(58)

Taking inverse Fourier and inverse Laplace, DO ⟨δCO(z = (ζ , r ⃗))⟩ = − DR ⟨δC R (z = (ζ , r ⃗))⟩ (59)

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

■

DO δCO(z = (ζ , r ⃗)) ≈ − DR δC R (z = (ζ , r ⃗))

(60)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS Parveen is grateful to UGC-India for providing fellowship. R.K. is grateful to University of Delhi for R & D grant. REFERENCES

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Theory for Anomalous Response in Cyclic Staircase Voltammetry

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