Theory of Anomalous Dynamics of Electric Double Layer at

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Theory of Anomalous Dynamics of Electric Double Layer at Heterogeneous and Rough Electrodes Maibam Birla Singh and Rama Kant* Department of Chemistry, University of Delhi, Delhi 110007, India S Supporting Information *

ABSTRACT: A generalized model for dynamics of the electric double layer (EDL) at a heterogeneous and rough electrode is developed using the Debye−Falkenhagen equation for the potential. The influence of surface heterogeneities which causes the distribution in relaxation time in the compact layer is included through the current balance boundary constraint at the outer Helmholtz layer. The results for the admittance response are obtained for deterministic and stochastic roughness. The response for the deterministic surface is expressed as a functional of an arbitrary surface profile and the stochastic roughness as a functional of an arbitrary power spectrum of roughness. The dynamics is understood in terms of phenomenological (viz., dynamic diffuse layer and polarization) lengths and various relaxation (viz., compact layer, diffuse layer, and mixed) frequencies resulting from the interaction of compact and diffuse double layer. A strong influence of heterogeneity, finite fractal roughness, electrolyte concentration, and their diffusion coefficient is found. Our model unravels anomalous roughness-dependent pseudoGerischer behavior at high frequency, classical Helmholtz behavior at intermediate frequencies, and emergence of CPE at low frequency due to heterogeneity of the surface. Comparison of the theory with experimental data shows good agreement.

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addressed by equivalent circuit,44−46 scaling,44,45,47−51 fractional diffusion,52,53 and the ab initio approach.6−13,21−24 The equivalent circuit and scaling methods treat the CPE exponent as a measure of roughness (or heterogeneity)42,46,52,54 and use it to explain impedance data of heterogeneous,29,35,38 rough,55 porous56 electrodes and the corroding system.57 But the CPE exponent is found to be a model-dependent quantity42 with no specific relation to the fractal dimension of the surface.58,59 The analyses using equivalent circuits or scaling are misleading as they neglect the influence of physical length scales of roughness and heterogeneity seen in experiments.14,42,43 The ab initio approach which employs fundamental equations describing various electrochemical phenomena with appropriate boundary conditions on disordered boundary provides an elegant methodology. Kant and co-workers have developed this formalism successfully to understand the influence of roughness on various electrochemical phenomena at the electrode.9−12,15−20,60−62 The methodology has been applied to understand the diffusion-limited (1) charge transfer process (Warburg15 and Anson12,18), (2) quasireversible charge transfer,9−11 (3) charge transfer coupled to bulk reaction (Gerischer,16 catalytic reaction17) (4) pseudo quasi-reversibility (anomalous Warburg admittance19 due to ohmic loss) and (5) diffusion-limited adsorption20 on the rough electrode. All these theories characterized random electrode roughness through the

INTRODUCTION The influence of electrode morphology and heterogeneity on the electrochemical response is ubiquitous, and these are considered theoretically difficult problems1−24 in electrochemistry. One important problem still unsolved is the anomalous capacitive behavior25 and constant phase element (CPE) associated with the electric double layer formed at the electrode−electrolyte interface.26−33 The problem of CPE remains controversial as there is no universal theory describing varieties of causes which are responsible for this phenomenon.34 Usually, CPE behavior is attributed to interfacial heterogeneity (inhomogeneities)14,35−37 and/or surface disorder/ roughness.29,35,38 The surface features of an electrode are classified as atomic scale (0.1 mm) according to sizes of geometrical features.35 The atomic scale roughness is an energetically nonuniform surface and may consist of steps, kinks, and dislocations.14 The capacitance dispersion observed in a single crystalline electrode due to residual roughness induces energetic inhomogeneities in the electric double layer.29,35,36,38 The charge accumulation at different crystal faces is different and hence causes differences in compact and diffuse layers.14,39−41 The influence of surface heterogeneity is perceptible when the size of the crystal faces in the polycrystalline surface and the correlation length of roughness is comparable to Debye length.6−8,14,42,43 The effect of roughness and the physical origin of CPE or capacitance dispersion in the electrochemical interface are © 2014 American Chemical Society

Received: November 8, 2013 Revised: February 11, 2014 Published: February 18, 2014 5122

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dimension, lowest length of roughness, and topothesy length (through strength of fractality). Finally, we present the results and conclusions of our work.

power spectrum, which is usually obtained from AFM or SEM measurement.60,61 The experimental validation of roughness power-spectrum-based theory62 of the anomalous Cottrell response has also been obtained.61 The EDL behavior can be understood at both the static and dynamic levels. The static behavior of the double layer is understood by employing the Poisson−Boltzmann equation or the Debye−Hückel equation for potential,6−9,13,21−24 and the dynamic behavior is understood by employing the Laplace equation for potential2−4 and the Debye−Falkenhagen equation for charge or potential.26,63,64 Recently, we have developed a theory for dynamics of EDL in presence of electrode heterogeneities, and the admittance can be expressed as26 YDF(ω) =

A0 z H(ω) + zG(ω)

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MODEL FOR EDL FORMED AT ROUGH ELECTRODE WITH SURFACE HETEROGENEITY We follow the treatment by Falkenhagen26,64 for relaxation of ionic atmosphere but treat a diffuse double layer at a rough heterogeneous electrode. This methodology provides the dynamics of diffuse layer ions near electrode surface and is called the Debye−Falkenhagen (DF) equation.26,63,64 This equation provides momentary excess charge density due to inequality in the diffuse double layer region and is given as ∂ρ = D(∇2 − κ 2)ρ (4) ∂t −1 where ρ is the charge density and κ is the Debye screening length

(1)

where z H is the compact layer impedance including heterogeneity and sluggish charge transfer26 z H(ω) =

κ −1 =

RH 1 − [1 +

RHR ct−1

+

(ιωτH0)γ (1

+

RHR ct−1)1 − γ ]−1

c i = c i∞exp( ± z ieϕ/kBT )

1/τD + iω

(6)

Nonlinear contributions in concentration can be ignored for electrochemical techniques with small external potential perturbations. Now for symmetrical electrolytes z+ = z− = z ∞ and assuming c∞ + = c− = c0, linearizing eq 6 under condition of small applied potential, ϕ ≪ kBT/ze and substituting in eq 4, we obtained the linearized Debye−Falkenhagen (LDF) equation for potential ϕ as

Z0 D

(5)

where I = is the ionic strength, kB is the Boltzmann’s constant, NA is the Avogadro number, T is the temperature, ε0 is the permittivity of free space, and εr is dielectric constant of the solvent. The DF eq 4 relates the dynamic reorganization of charges resulting in homogenization of charge in the ionic sphere. It consists of two process: (1) the rate of change of charge density due to diffusional motion of charges and (2) kinetic hopping of excess charge out of the diffuse layer. Thus, the DF eq 4 is the diffusion equation in local excess charge density. Now we assume that near equilibrium, the local concentration of ions ci is given by the Boltzmann distribution

where RH is the Helmholtz layer (HL) resistance, A0 is the geometric area of electrode, Rct is the charge transfer resistance, γ is the measure of extent of heterogeneity of the surface, τ0H is the surface dependent charging or relaxation time of HL, and τ0H = RHcHav where cHav is the average capacitance of HL. Equation 2 shows that the admittance depends on the heterogeneity coupled to the Helmholtz layer resistance, surface-dependent charging or relaxation time of HL, and the charge transfer resistance. When Rct is large compared to RH, eq 2 reduces to the case of the blocking electrode where zH (ω) = RH[1 − (1 + 1/(ιωτ0H)γ)]−1. Thus, the admittance given by eq 2 is the generalized HL admittance of the partially blocking spatially heterogeneous electrode. The diffuse layer impedance ZG is due to fast diffusion and kinetic hopping/relaxation in the diffuse layer 1/2

2NAe 2I (1/2)∑icizi2

(2)

zG(ω) =

εr ε0 kBT

(3)

where Z0 = (D/σ) and τD = 1/κ2D, where D is the diffusion coefficient of ions, σ is the conductivity, and κ is the inverse Debye screening length of electrolyte. Equations 1−3 govern the dynamics of the diffuse and compact layer on the heterogeneous but geometrically smooth electrode. In this work, we wish to further develop a theory for the Debye−Falkenhagen (DF) admittance of rough electrode in the presence of heterogeneity. The paper is organized as follows: We begin by formulating the problem of ion transport and relaxation at the rough and heterogeneous electrode analogous to the Debye−Falkenhagen model of ionic relaxation. We solved the DF equation for potential perturbatively employing the inhomogeneous boundary conditions valid at the outer Helmholtz surface. The admittance/ impedance is obtained as a function of roughness features of the surface. We employ a finite fractal model of surface roughness and analyze impedance response of (i) homogeneous and (ii) heterogeneous rough electrodes (obtained from the general result). The responses are compared to understand the effects of electrolyte concentration, diffusion coefficient of ions, and various morphological parameters of surface: fractal

∂ϕ = D(∇2 − κ 2)ϕ ∂t

(7)

Thus, we have recast the DF eq 4 expressed in terms of charge density to a linearized DF equation in terms of potential in eq 7. For a sinusoidal applied potential at a planar electrode, ϕ (z,t) = ϕ (z) eiωt, the complex potential in LDF equation is simplified to (∇2 − κe2)ϕ = 0

(8)

where κe = κ((1 + iωτD)) is the dynamic inverse screening length. The LDF eq 8 reduces to the linearized Poisson− Boltzmann equation or the Debye−Hückel equation (∇ − κ2)ϕ = 0 when the ω ≪ ωD (= 1/τD). Now let us consider a rough boundary at the electrode− electrolyte interface, as shown in Figure 1, described by a random function z = ζ (r∥⃗ ) of the in-plane position vector r∥⃗ in xy-plane with the average flat interface at z = 0 or ensemble averaged surface function, ⟨ζ (r∥⃗ )⟩ = 0. The EDL formed at the interface consists of a compact Helmholtz layer (HL) whose 1/2

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This length characterizes the effective thickness of the polarized interfacial layer and is the measure of distance up to which reorganization of charges in solution takes place. The length |ΛC(ω)−1| contains the influence of the compact layer heterogeneity (γ), capacitance (cHav), resistance (RH), and the interfacial conductivity of the electrolyte σ. At low frequency, the |ΛC(ω)−1| is usually larger than κ−1 e , and at high frequency, both are comparable except for very dilute electrolyte and low diffusion coefficient.26 Thus, the two dynamic phenomenological lengths viz., |ΛC(ω)−1| and κ−1 e , essentially influences the dynamic behavior of the interface, and the response is controlled by the smaller of the two. The parameter γ in eq 2 accounts for the heterogeneity of the surface and γ = 1 for a smooth planar or nonheterogeneous electrode surface (like mercury), where γ < 1 for a heterogeneous or rough electrode (single crystal or polycrystalline gold or platinum) surface. For a homogeneous planar electrode where γ = 1 and when the charge transfer resistance is large (Rct ∼ 108Ω cm2, corresponding to blocking case), the inverse complex polarization length may be rewritten for simplicity as ΛC(ω) = (σ/i ωcH + W)−1, where W = σRH is a characteristic resistive length which may be called as modified Wagner number,66,68,69 and the averaged capacitance of the Helmholtz layer cHav is written as cH for simplicity. The quantity σ/iωcH is a natural length of the problem which also arise in impedance analysis of rough (quasi-fractals) and porous electrodes44,70 and can be interpreted as length of a cube of electrolyte limited by planar electrode face whose volume resistance |Λc(ω)|/σ is equal to the capacitive impedance of that face 1/(ωcH|ΛC(ω)|−2).44,70

Figure 1. Schematic diagram of the electric double layer formed at the heterogeneous rough electrode. The equations and the boundary conditions used to calculate the interfacial impedance of electrode in contact with an electrolyte are also shown. The quantities which characterized the surface morphology of electrodes are S (smallest), L (largest) length of fractality, and S τ the topothesy length relating to the width of interface.

thickness is of molecular dimensions rH (several angstroms in thickness) and diffuse layer whose thickness κ−1 (≈1−10 nm) varies with a change in the concentration of the electrolyte. Now equating the current crossing at outer HL from solution (given by Ohm’s law) to the ratio of potential drop at the outer HL from the electrode (ϕ̃ − ϕ (z = ζ (r∥⃗ ))) to the impedance zH of the Helmholtz layer, we obtain the Robin boundary condition as26,63,65,66 −σ(∂ϕ/∂n)|z = ζ( r ⃗ ) = [ϕ ̃ − ϕ(z = ζ( r ⃗))]/z H

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LOCAL AND OVERALL ADMITTANCE Electrochemical impedance spectroscopy (EIS) is a powerful technique often used to measure the electrochemical response of interfaces in rough, porous and heterogeneous electrodes.71−73 It involves a small perturbation in potential so that all inherently nonlinear aspects of the current−potential curve can be linearized. The small perturbation makes the measurement nondestructive and a broad range of time scale can be measured in a single experiment. In this section we wish to develop an EIS theory for rough electrode with heterogeneity. Our aim here will be to relate the admittance of the interface to the surface feature through the local admittance density at the outer Helmholtz layer. The total electrode admittance Y(ω) of EDL can be obtained by surface integrating the local admittance density y(ω) defined at the outer Helmholtz layer as

(9)

where σ is the conductivity of electrolyte, ϕ (z = ζ (r∥⃗ )) is the potential at the OHP surface at a distance ζ(r∥⃗ ) and beyond, ϕ̃ is the potential of the electrode, zH is the impedance of HL, and |σzH(ω)| is the dynamic polarization (or equilibration) length due to compact layer dynamics. Usually, the geometric separation length between the working and counter electrodes is much larger than the Debye length (κ−1) (there are no overlaps of the two diffuse layers). Hence, it’s appropriate to use (in most cases) the semi-infinite (bulk) boundary condition where ϕ (∞) = 0. The generalized surface boundary condition in eq 9 expresses how the EDL charging current given by the ratio of potential drop at the compact layer to the spatially inhomogeneous impedance of HL (zH) is related to the current in the diffuse layer at the rough outer HL in the electrode/electrolyte interface. This allow us to include in the model compact and diffuse layers independently. The contributions from the diffuse layer are accounted in detail through the (linearized) Debye− Falkenhagen equation (eq 7), but for the influence of surface heterogeneity, we consider that there is heterogeneity in the exponent-dependent distribution of the compact layer relaxation time.26,67 The impedance of HL zH, including the distribution of compact layer relaxation time, can be expressed in terms of the Helmholtz layer resistance RH, charge transfer resistance Rct, and heterogeneity parameter γ as (derived in the Appendix). The boundary condition eq 9 introduces a phenomenological dynamic polarization or equilibration length |ΛC(ω)−1| = |σzH|.

Y (w) =

∫S y(ω)dS 0

(10)

where y(ω) = −(σ/ϕ̃ )(∂ϕ/∂n)|z=ζ(r∥⃗ ), and dS is the surface element at outer Helmholtz layer. The local admittance density, y(ω), relates the gradient of potential field at the outer Helmholtz layer to the electrolyte conductivity and potential applied at the electrode. The potential is governed by LDF eq 8 and obeys the surface boundary condition (bc) given in eq 9. On solving eq 8 using surface bc eq 9 along with bulk bc for potential, we obtained the general expression of admittance (the details of the calculation are shown in the Appendix). The overall admittance of an electrode up to the second order term in arbitrary surface profile function ζ is obtained as 5124

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∫S

⎧ 1 d 2r ⎨ ⎩ (2π )2 0

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stationary random surface, the gradient of the random surface profile ζ(r∥⃗ ) and their Fourier transform ζ̂(K⃗ ∥) is related to the various moments of the power spectrum as9

∫ d2K eιK⃗ ·r ⃗ [y0 + y1̂ ζ(̂ K⃗ )

⎫ + y2̂ ζ (̂ K⃗ )ζ (̂ K⃗ − K⃗ ′)]⎬ ⎭

(11)

m2k = ⟨(∇k ζ( r ⃗))2 ⟩ =

⃗ ∞ ⃗ where ∫ d2K⃗ ∥ ≡ ∫ ∞ −∞dKx∫ −∞dKy, κ is the inverse Debye length, 2 2 1/2 ⃗ κ∥ = (κe + K∥) , κ∥,∥ ′ = (κ2e + |K⃗ ∥ − K⃗ ∥′ |2)1/2, |K⃗ ∥| is the magnitude of wave vector K⃗ ∥, δ(K⃗ ∥) is the two-dimensional Dirac delta function in wave vector K⃗ ∥, and ζ̂(K∥) is the Fourier transform of the surface profile ζ (x,y), that is, ζ̂(K⃗ ∥) = ∫ d2r∥e−ιK⃗ ∥·r∥⃗ ζ(r∥⃗ ). The function y0 and operators (ŷ1 and ŷ2) in eq 11 are defined as y0 =

y2̂ =

1 (2π )2 +

⎡ ⎛

∫ d2K⃗ ′⎢⎢ 12 ⎜⎜ κ κκ

⎣ ⎝

e

(13)

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κe(κe + ΛC(ω)) ⎞ κ κe 1⎛ ⎜⎜ ⎟⎟ − κe ⎝ κ + ΛC(ω) κe + ΛC(ω) ⎠

∫ d2K K⃗ 2k⟨|ζ(̂ K⃗ )|2 ⟩

The various orders of moment of the power spectrum are physically observable quantities that provide various morphological information of the rough surface, viz., root-mean-square (rms) width ((m0)1/2), rms gradient ((m2)1/2), rms curvature ((m4)1/2), and so forth, which can be obtained from experiments.61 A convenient hybrid experimental method is developed using a SEM micrograph and microscopic area measurement from electrochemistry.60

(2π )2 δ(K⃗ )

y1̂ =

1 (2π )2

ADMITTANCE FOR ROUGH ELECTRODE WITH HETEROGENEITY Equation 11 gives the local admittance of the electrode/ electrolyte interface various surface profile terms. Due to the electrode roughness, the surface is random in nature. The quantities of interest in the electrochemical response of the interface are not the local quantities but the ensemble averaged over various surface configurations. The ensemble averaged admittance ⟨Y(ω)⟩ of the randomly rough electrode may be obtained from eq 11 as

⎞ κ κe ⎟⎟ − + ΛC(ω) κ + ΛC(ω) ⎠

,

′ κe(κ + ΛC(ω))

K⃗ ′. (K⃗ − K⃗ ′) κ κe − κ , + ΛC(ω) 2κe(κe + ΛC(ω)) κ + ΛC(ω) ′ ⎤ K⃗ ′. (K⃗ − K⃗ ′) |K⃗ − K⃗ ′|2 ΛC(ω) ⎥ − − κe(κ + ΛC(ω)) κ , + ΛC(ω) κe(κ , + ΛC(ω)) ⎥⎦ ′ ′

−

⎡ 1 ⟨Y (ω)⟩ = YDF(ω)⎢⎢1 + 2π ⎣

(12)

The first term in eq 11 inside the bracket corresponds to the operator that represents the admittance response of the heterogeneous electrode surface. The second and third terms in eq 11 represent the first order and second order terms in the surface profile modulation or fluctuation around a reference surface at the outer Helmholtz surface. It is to be noted here that these operators defined in eq 12 are expressed in terms of −1 two phenomenological length scales, κ−1 e and |ΛC(ω) |, and thus the admittance given by eq 11 essentially depends upon the relative importance of these two quantities.

+

∫0

∞

⎛ κ (κ − κ )Λ (ω) e e C dK K ⎜⎜ ( ) κ + Λ ω C ⎝

⎤ ⎞ ⎟⟨|ζ (̂ K⃗ )|2 ⟩⎥ ⎥ 2(κe + ΛC(ω)) ⎟⎠ ⎦ κeK⃗

2

(14)

where YDF is the DF admittance of EDL formed at a heterogeneous electrode given by eq 1, and ⟨|ζ̂(K⃗ ∥)|2⟩ is the power spectrum of rough surface. Equation 14 relates the ensemble averaged admittance to the morphological features of surface, including roughness and heterogeneity to the phenomenological lengths, screening length of the diffuse −1 layer (κ−1 e ), and the polarization length (|ΛC(ω) |) of the compact layer.

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SURFACE ROUGHNESS AND THEIR STATISTICAL PROPERTIES Most of the electrode surfaces are rough. This may be due to the method of preparation of the electrode or due to various processes (like electrodeposition, electrocleaning, corrosion, etc.) involved at the interface. The roughness is random in nature, and hence, its observable properties are often characterized in the statistical sense. For a random surface, the nature of the surface profile ζ(r∥⃗ ) determines the average statistical properties of the electrode. When ζ(r ∥⃗ ) is homogeneous (stationary), stochastic, and Gaussian, the ensemble averaged value ⟨.⟩ of the random surface configuration is ⟨ζ(r∥⃗ )⟩ = 0. The two-point correlation function for the surface is defined as ⟨ζ(r∥⃗ )ζ(r∥′⃗ )⟩ = h2W(r∥⃗ − r∥′⃗ ), where the normalized correlation function W(r∥⃗ − r′∥⃗ ) varies from zero to unity, and h2 = ⟨ζ2(r∥⃗ )⟩ denotes the mean square departure of the surface from flatness. The averages in the Fourier plane are ⟨ζ̂(K⃗ ∥)⟩ = 0 and ⟨ζ̂(K⃗ ∥)ζ̂(K⃗ ∥′ )⟩ = (2π)2δ(K⃗ ∥ + K⃗ ∥′ )⟨|ζ̂(K⃗ ∥)|2⟩. The Fourier integral representation for ζ(r∥⃗ ) is ζ(r∥⃗ ) = 1/ (2π)2∫ d2K∥eιK⃗ ∥·r∥⃗ ζ̂(K∥). The power spectrum ⟨|ζ̂(K⃗ ∥)|2⟩ is related to the Fourier transform of the correlation function W(r∥⃗ − r∥′⃗ ). It is an experimentally determinable quantity. For a

APPLICATION TO FRACTAL ELECTRODE WITH HETEROGENEITY In order to achieve a better understanding of the effect of roughness, we developed a fractal model of the surface. Fractal geometry74 is a popular model of rough surfaces. The application of the fractal concept in the electrochemistry of a rough and heterogeneous surface is well-known.52,75 The irregularities of rough surfaces are understood in terms of self-similar74 or self-affine74,76 fractals. However, it is found experimentally that the fractal nature of the surface can be observed in finite length scale.61 A fractal surface which exhibits statistical self-resemblance over limited length scales whose statistical properties can be described by the power law power spectrum with structure factor ⟨|ζ (̂ K⃗ )|2 ⟩ = μ|K⃗ |2DH − 7 , 1/L ≤ |K | ≤ 1/S

(15)

This surface structure factor represents the statistically determined isotropic rough surface of a realistic fractal characterized by four fractal morphological parameters of roughness (see Figure 1) in the power spectrum: fractal dimension DH, the lower cutoff length S , upper cutoff length L, 5125

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and strength of fractality μ, which is related to topothesy length as Sτ = μ1/(2DH − 3). Physically, S is the length scale below which we see no fractal behavior, and L is the largest length scale (usually considered as size of sample) above which no fractal behavior is seen. The width of the interface which corresponds to the roughness of the surface is characterized by μ or Sτ . These values may be obtained from experiments through SEM or AFM measurements.61 The power spectrum of the roughness given by eq 15 contains all the fractal morphological information of the RF(ω) =

surface. This is incorporated by introducing eq 15 in eq 14. We obtained (the detailed calculation is shown in the Appendix) the ensemble averaged admittance expression for approximately self-affine random fractal surface roughness with heterogeneity as ⟨Y (w)⟩ = YDF(ω)[1 + RF(ω)]

(16)

where RF is the frequency dependent roughness integral. The integral is solved analytically in the Supporting Information. The value of RF(ω) may be written as

⎫ κ 3Λ (ω) κe3ΛC(ω) ⎧ μ 1 e C ⎨H1(S) − H1(L) + (H2(S) − H2(L))⎬ {A1(S) − A1(L)} 2 2 2 2 2 4πδ (κe − ΛC(ω)) ⎩ 4(δ + 1) (κe − ΛC(ω)) ⎭ (κe − ΛC(ω)) +

κe2 ΛC2 (ω) κe μ μ μ {H1(S) − H1(L)} + + {S −2(δ + 1) − L−2(δ + 1)} 2 4πδ (κe − ΛC2 (ω)) 4πδ 8π(δ + 1) (κe + ΛC(ω))

(17)

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COMPARISON OF ADMITTANCE OF THE SMOOTH, HETEROGENEOUS, AND ROUGH ELECTRODE Figure 2 shows the comparison of DF admittance for the homogeneous smooth (like mercury) electrode with γ = 1, the heterogeneous (like polycrystalline Au or Pt) electrode with γ < 1, and the electrode with heterogeneity and roughness. Figure 2a,b,c show, respectively, the log(|Z|) vs log(ν) plot, the Φ vs log(ν), and the Nyquist plots of impedance. The relation between frequency (ν) and angular frequency ω is ν = ω/2π. The dotted black lines show the admittance of the planar electrode with no heterogeneity (γ = 1). The blue lines show the admittance with heterogeneity and black lines for a rough electrode with heterogeneity. The plots (a) and (c) show that the slope of the impedance plot of the smooth electrode with no heterogeneity is larger than the electrode with heterogeneity and roughness at low frequency. The roughness results in lowering of the magnitude of impedance compared to the planar and heterogeneous electrode. The impedance of the rough electrode with heterogeneity (represented by solid black line) shows four distinct regimes of frequencies: (i) very low frequency, where the CPE behavior is observed and the slope of the magnitude of impedance curve is less than 1, (ii) low frequency, where impedance shows classical Helmholtz (RC) model behavior with magnitude of slope 1, (iii) intermediate frequency, which shows pseudo-Gerischer type behavior, with zero slope in the impedance curve, and (iv) high-frequency pseudo-Warburg behavior with magnitude of slope approaching 1/2. The phase plots for the rough electrode with heterogeneity in Figure 2b also show four regimes: (i) very low frequency, where Φ < 90° shows constant phase element (CPE) behavior, (ii) low frequency, where Φ ≈ 90° (hump region), (iii) intermediate frequency, in which Φ → 0° shows a resistive behavior (valley), and (iv) high frequency, where Φ → 45° shows the diffusive (pseudo-Warburg) behavior. At low frequencies, the homogeneous smooth electrode shows a phase angle of 90°, whereas for both heterogeneous and heterogeneous with roughness, the phase angle Φ < 90°. The inclusion of heterogeneity results in a phase angle value