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J. Phys. Chem. C 2009, 113, 19558–19567
Theory of Generalized Gerischer Admittance of Realistic Fractal Electrode Rajesh Kumar and Rama Kant* Department of Chemistry, UniVersity of Delhi, Delhi 110007, India ReceiVed: April 26, 2009; ReVised Manuscript ReceiVed: September 15, 2009
We developed a theory for the generalized Gerischer admittance for an irregular interface, operating under diffusion and homogeneous kinetics coupled with a fast heterogeneous charge transfer reaction. The generalized admittance expressions are obtained for a deterministic surface (viz, the exact mathematical function of roughness profile is known) as well as for a stochastic surface (viz, the statistical properties of roughness profiles are known). A roughness power spectrum or structure factor is sufficient to characterize the statistical properties of stochastic geometrical irregularities. An elegant expression for the generalized Gerischer admittance is obtained as a functional of the roughness power spectrum. This equation is applicable for fractal as well as nonfractal stochastic roughness. Realistic fractal electrodes have a fractal nature over limited length scales and possess an approximate self-affine property, which is characterized in terms of a band-limited power law function for the power spectrum. The generalized Gerischer impedance shows three frequency regimes, viz, (i) low frequency region, which has a kinetic-controlled frequency (ω) independent impedance and phase angle that follows constraint, 0 e φ(ω) < 45°, (ii) anomalous power law behavior for intermediate frequency and their phase angle φ(ω) > 45° and shows an approximately constant phase angle region, and (iii) high frequency limiting Warburg impedance behavior. Introduction The alternating current (ac) impedance technique is extensively applied in the field of electrochemistry and solid state electrochemistry because of its sensitivity and ability to separate different processes in complex systems. It can be used for investigation of a system such as nanostructured photoelectrodes,1,2 basic electrode kinetics,3 bioelectrochemistry, and mechanism of interfacial processes.3,4 The method of impedance is also used comprehensively in applied systems like batteries, corrosion, and photoelectrochemical solar cells. Most of these measurements are strongly influenced because of surface irregularities, which are reflected in their response like an anomalous frequency dependence of magnitude of impedance and constant phase angle behavior. These surface irregularities are best described as fractals over a limited range of length scales, which often possess a self-affine scaling property.5-9 Recently, many attempts have been made to understand the problem of simultaneous occurrence of diffusion and homogeneous first-order reaction, which is important in understanding the response for an oxide electrode,10 mixed conducting solid electrolyte systems,11 solid oxide fuel cells,12 catalytic electrode process application in organic syntheses,13 electrocatalytic properties of hydrogen evolution reaction,14 polymer electrolyte membrane fuel cells,15 and thin film (finite-length) type Gerischer impedance.16 Most of these works and references cited therein addressed the diffusion reaction problem on smooth electrodes, but they do not address issues related to interfacial geometric disorders. Interfacial geometric disorder can be a deterministic surface, viz, the exact mathematical function of a roughness profile is known, as well as for a stochastic surface, viz, the statistical properties of roughness are known. The roughness power spectrum or structure factor is sufficient to characterize the statistical properties of stochastic geometrical irregularities. Deterministic as well as stochastic geometrical irregularities can * E-mail:
[email protected].
be fractal and nonfractal in nature. The surface disorder is effectively modeled as fractal with limited length scales of fractality, which is the basis of our “realistic fractal morphological models”.7,8,17,18 Our analytical methodology constitutes not only an implementable model for a large class of electrochemical systems with surface roughness but is also an elegant approach.6,19-21 It is well-understood that diffusion plays an important role in a number of chemical transformations, in single-phase and multiphase systems. When one or several of the reactants are inhomogeneously distributed in space (r b), then it is characterized by spatial and temporal dependence of the concentrations, C(r b, t), with an effective rate constant k and a diffusion coefficient D. Diffusion kinetic-controlled processes are systems where the diffusion and homogeneous first-order kinetics proceed simultaneously and control the concentration of electroactive species. The partial differential equation characterizing such processes in a single phase with diffusion and first-order kinetics can be expressed as
∂C(b, r t) r t) - kC(b, r t) ) D∇2C(b, ∂t
(1)
where C(r b, t), D, k are the concentration, diffusion coefficient, and rate constant, respectively. Equation 1 can be transformed into a standard diffusion equation by including a reaction term b, t) in the form of an exponential kinetic factor as C*(r b, t) ) ektC(r and simplified to a standard diffusion equation as
∂C*(b, r t) r t) ) D∇2C*(b, ∂t
(2)
There is a need to develop a method for understanding the overall kinetics for such systems, which is influenced by roughness. Here, at initial time and far off from the interface,
10.1021/jp903827w CCC: $40.75 2009 American Chemical Society Published on Web 10/21/2009
Theory of Generalized Gerischer Admittance
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Figure 1. Schematic diagram of a randomly rough interface of width, h, where the charge transfer reaction is coupled to a homogeneous chemical reaction. Two phenomenological length scales are diffusion layer thickness (D/ω)1/2 and reaction layer thickness (D/k)1/2. The random roughness profile is characterized by a power spectrum. Inset shows a power spectrum of realistic self-affine fractal roughness with four morphological characteristics, viz, (i) fractal dimension (DH) is obtained from the slope, (ii) strength of fractality (µ) is obtained from the intercept, (iii) lower cutoff (l ) and upper cutoff length scale (L) are obtained from two crossover points.
a uniform concentration of oxidized species C0 is maintained, b, t ) 0) ) CO(zf∞, t) ) C0, for absence of reduced viz, CO(r b, t ) 0) ) CR(zf∞, t) ) 0. The linearized species, viz, CR(r boundary condition under Nernstian constraint for oxidized species can be written as
δC(z ) ζ(x, y), t) ) -nfη0C0 ) -CS
(3)
where δC(z ) ζ(x, y), t) is the difference in concentration of oxidized species at surface, CS is the excess surface concentration, n is number of electrons transferred, f ) F/RT, F is the Faraday constant, R is the gas constant, T is the absolute temperature, and η0 is the magnitude of applied sinusoidal potential. Equation 3 is obtained for DO ) DR ) D, along with flux balance constraint to show δCO(x, y) ) -δCR(x, y). Here, we are not accounting any contribution from solution resistance (assuming presence of excess inert electrolyte) and double layer in our proposed model. The exact theoretical analysis of electrochemical impedance response for irregular electrodes and their explicit interrelation with various morphological characteristics is a difficult problem. The qualitative understanding of this phenomenon in the case of realistic surface roughness with a limited length scale of irregularities7,8,17,18 can be illustrated through a schematic diagram of a rough working electrode as shown in Figure 1. This figure indicates two morphological cutoff length scales in a realistic fractal surface, i.e., a lower cutoff length (l ), the length above which surface shows fractal behavior and a upper cutoff length (L), the length below which surface shows fractal behavior (Figure 1, inset). The other two characteristics of a power spectrum are fractal dimension (DH) or the HausdroffBesicovitch dimension, which is the global characteristics of morphology of surface and strength of fractality (µ), which is equal to the value of a power spectrum at a unit wavenumber. There exist two phenomenological length scales, i.e., (i) a diffusion layer thickness [LD ) (D/ω)1/2], i.e., a measure of size of the diffusion limited region or depletion layer near electrode and (ii) a reaction layer thickness [LR ) (D/k)1/2], i.e., a measure of distance traveled by reacting species during homogeneous
reaction time. Here, ω is the angular frequency of a perturbing signal. The ratio of these two parameters determines the relative importance of two processes, i.e., the LD/LR < 1 system is diffusion controlled and the LD/LR J 1 system is kineticcontrolled. The intermediate regime is LD < LR and h > LD > l (h is the width of interface), where the system is dynamically sensitive to morphological features of the surface. The relative importance of these surface morphological characteristics, phenomenological length scales, and their implication on Gerischer impedance is analyzed in this paper. The admittance Y(ω) or its reciprocal quantity impedance Z(ω) is an important method in extracting the electrochemical information about the interface and interfacial processes. The total admittance, Y(ω), of an interfacial redox reaction coupled with a homogeneous first-order reaction, driven by sinusoidal interfacial potential (δE ) η0eiωt, where η0 is the magnitude of potential) can be obtained by a solving diffusion equation with an appropriate boundary condition. A specific type of impedance of a first-order homogeneous chemical reaction in series with a charge transfer was first derived theoretically by Gerischer22 in 1951. Gerischer’s results were obtained for a smooth mercury solution interface under a semi-infinite linear diffusion condition. The “Gerischer admittance” [YG(ω)] for a planar electrode-electrolyte interface is given as22
YG(ω) ) A0ΓD1/2(k + iω)1/2
(4)
where A0 is the projected area of the electrode, Γ is the capacitance per unit volume (specific diffusion capacitance), which is equal to n2F2C0/RT, i ) (- 1)1/2, and ω is the angular frequency. Equation 4 implies that the Gerischer admittance for diffusion coupled with a first-order homogeneous chemical reaction (CE process) is related to the area of electrode, specific diffusion capacitance, and the complex kinetic diffusion layer thickness [D/(k + iω)]1/2. The Gerischer admittance has been observed in the frequency response of mixed conducting homogeneous solid electrolyte systems for a CE type reaction. This work has been extended further by Sluyters, Smith, and Schuhmann23-25 for ac polarography and a uniformly accessible rotating disk electrode. Canas et al.26 developed a numerical simulation technique to obtain the impedance diagrams for various reactions exhibiting EC, CE, and CEC mechanisms. The characteristic features and importance of Gerischer for smooth surfaces are discussed in several reviews.23,27 Wider applicability of this coupled system makes it an important class of problem to analyze in the presence of geometric disorder (roughness) at the interface. If the charge transfer reaction is only governed by diffusion, then the Gerischer admittance response becomes a Warburg admittance [YW(ω)] response (i.e., kf0 limit) as
YW(ω) ) A0Γ(iωD)1/2
(5)
Phase angle is an important and useful characteristic of interfacial processes. For a Warburg admittance, it is invariably equal to 45°, whereas for a Gerischer admittance phase angle 0 e φ(ω) e 45° holds for a smooth surface. This condition is not true for the processes involving a rough or fractal interface.17,28 Many authors used the scaling concept29-32 to explain intermediate frequency and time behavior of diffusion-limited reactions, which occur in fractal spaces (porous) or on fractal boundaries
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Kumar and Kant
(rough). Experimental studies29-32 have shown that reaction rates follow approximately a power law relation in the intermediate frequency domain. The intermediate scaling form for the admittance, Y(ω), is expressed as a power law relation given as
Y(ω) ∼ (iω)γ
(6)
where the exponent, γ, depends upon the interfacial roughness. De Gennes33 scaling result justifies the interrelation of γ with DH in eq 6 as γ ) (DH - 1)/2 for the problem of diffusioncontrolled nuclear magnetic relaxation in porous media with fractal dimension (DH), and a further generalized form is described in ref 34. Admittance or impedance measurements are extremely sensitive because one looks for the response at a particular known frequency. Impedance and admittance are readily interconvertible electrochemical observables. Complex nonlinear least-squares analysis offers a fractal modification of the Gerischer impedance equation approximated as11
Z(ω) ∼ [k + iω]-γ
(7)
when γ ) 0.5, it gives the classical Gerischer impedance. In other applications, a double fractal modification presented to account for the dispersion in data11
Z(ω) ∼ [k + (iω)ε]-γ
(8)
This form has an additional exponent ε to describe data. The analytical approach to the diffusion problem over arbitrary or random surface profile interfaces are difficult and solved under varying degrees of approximations. There are some general results which emphasize the effect of roughness on interfacial transport,5-8,19-21 viz, potentiostatic current transients5,7,8,19-21,35 and impedance.6,17,18 These ab initio theories based on surface roughness (disorder), characterized through a power spectrum of random roughness, have been most successful in explaining several experimental observations.7,8 Recently, we have successfully explained potentiostatic current transient on realistic fractal roughness and their experimental validation for nanometers to micrometer scales of roughness.7,8 Similarly, the role of surface irregularities on Warburg impedance for realistic self-affine (isotropic) fractal surfaces are discussed in refs 17 and 18. We wish to develop a similar ab initio methodology and formalism for an admittance problem where the diffusion coupled to homogeneous first-order kinetics proceed simultaneously on realistic (isotropic) self-affine fractal roughness. The aim of the present paper is to solve this problem by our ab initio approach and develop a theoretical understanding of admittance that describes the interplay between a diffusion and homogeneous first-order kinetic reaction over an arbitrary or a random surface profile. The generalized admittance expressions are obtained for a deterministic surface as well as for a stochastic surface. Realistic fractal electrode surfaces have a self-affine fractal property over a limited length scales and are characterized in terms of a band-limited power law function for the roughness power spectrum. A roughness power spectrum or structure factor is assumed to be sufficient to characterize the statistical properties of stochastic geometrical irregularities. Our analytical model for such a realistic fractal allow us to probe the simultaneous consequences of fractal dimension (DH), strength
of fractality (µ), and limited length scales, i.e., lower cutoff length (l ) and upper cutoff length scale (L) of roughness (Figure 1). Diffusion-Reaction Admittance for an Arbitrary Roughness Profile. Our aim is to evaluate the expressions for the Gerischer admittance for an arbitrary surface profile. The total admittance, Y(ω), of a rough interface is related to the Laplace transform (with respect to t) of the total interfacial current, I(t), under potentiostatic conditions as6,19
Y(ω) )
I(iω) iω ) δE(iω) η0
∫0∞ dte-iωtI(t)
(9)
where I(iω) and δE(iω) are the Laplace transformed current and potential, respectively. η0 is the magnitude of applied potential, and we use iω for the Laplace transform variable instead of usual notation p used in our earlier works. Using Fick’s first law, one can write a current density (j) expression3,19 in terms of normal derivative of concentration at the surface (S)
j(z ) ζ(b r |), t) ) nFD∂nC
(10)
where b r| represents a two-dimensional surface vector (in x and y directions), n is the number of electrons transferred, F is the Faraday constant, D is the diffusion coefficient, and ∂n ) nˆ.∇ signifies the outward drawn normal derivative at the surface S b|), 1), ∇| ) (∂/∂x,∂/∂y), β and a normal vector: nˆ )(1/β)(-∇|ζ(r ) [1 + (∇|ζ(r b|))2]1/2. The total current, I(t), of the molecules or ions at the interface in terms of a normal derivative of the concentration at the interface is given as3,19
I(z ) ζ(b r |), t) ) nFD
∫S dS∂nC ) ∫S dxdyβj(z ) ζ(br |), t) 0
(11) S0 is the projection of the surface S on the mean (reference) surface z ) 0. Therefore, for calculating the current density and current at an interface (eqs 10 and 11), we need to evaluate the concentration field. The concentration field is a nonlinear functional of the boundary profile. To solve this problem analytically, we use the method of perturbation with respect to the boundary profile, which is obtained by Taylor expansion of the concentration profile about a smooth surface. Taylor expansion of a concentration difference profile about z ) 0 can be expressed as
δC*(z ) 0, b r |, t) ) χ*(S, t) ∞
∑ χ*j
χ*(S, t)
)
χ*0
) -ektCS
χ*m
r |) ) -ζm(b
m)0
1 ∂m δC*(z ) 0, t) m! ∂zm
(12) for m > 0. This procedure is similar to the one used in solving a diffusion limited charge transfer problem in ref 19. The effect of an inhomogeneous Dirichlet boundary condition at the z )
Theory of Generalized Gerischer Admittance
J. Phys. Chem. C, Vol. 113, No. 45, 2009 19561
0 plane between t ) 0 and t is propagated by the integral19,36,37 and can be achieved by Green’s function approach.36,37
δC*(b, r t) ) D
∫0t dt' ∫S dS0′∂nG(b,r t|r'f, t')χ*(S', t') 0
(13) G is Green’s function in half plane z ) 0 and can be obtained by using the method of images,36 ∂′n is outward normal on a planar surface, viz, ∂′n ) -∂′/∂z′, and χ*(S′, t′) are the source functions at a planar surface given by eq 12. An integrodifferential equation has been obtained by substituting eq 12 into eq 13
δC*(b, r t) ) D
{
∫0t dt' ∫S dS0′∂nG(br |, z, t|br |′, z', t') × 0
∂ 1 ∂2 -ekt'CS - ζ(b r |′) δC* - ζ2(b r |′) δC* - · · · ∂z' 2! ∂z2
}
z')0
(14)
where ζ(r b|) is taken for a two-dimensional rough surface profile. The Green’s function for the Dirichlet boundary value problem about the z ) 0 plane37 for the three-dimensional Laplacian is
e-| br |-br |′ | /4Dτ G(b r |, z, t| b′ r |, z', t') ) × 4πDτ 2 2 e-(z-z') /4Dτ e-(z+z') /4Dτ 2(πDτ)1/2 2(πDτ)1/2
in formulating admittance density and total admittance. This is obtained by expanding eq 10 up to second order in ζ and taking the Fourier and Laplace transform of this expanded equation. The expression for current density for a weakly and gently fluctuating two-dimensional (2-D) rough surface in frequency (ω) domain can be written as nFCSD b|)ωk + ωk(ω b|) + {(2π)2δ(K ˜ | - ωk)ζˆ (K iω ˜| ωkω f )ω f ˆ b d2K|ζˆ (K' | ˜ | | ζ(K| - K'|) (2π)2 ˜ |) ωk(ωk2 + ωkω f )ζˆ (K b| - f d2K|ζˆ (K' K'|) | 2 2(2π) ωk f )|K b| - f b| - f d2K|ζˆ (K' K'||2ζˆ (K K'|) | (2π)2
b|, ζˆ , iω) ) j(K
∫
∫
∫
ωk
∫ d K K'f ζˆ (K'f )(Kb 2
2(2π)2
|
|
|
|
b| - f -f K'|)ζˆ (K K'|)} + O(ζˆ 3)
(17)
Similarly, the total interfacial current in the Laplace transform domain at a two-dimensional rough interface is obtained by taking the surface integral (eq 11) of the current density expression. The measured current at the electrode surface can be expressed as
2
{
}
(15)
τ ) t - t′. Using the Laplace transform technique and shift theorem of Laplace transform,38 we obtain a perturbative solution for the concentration profile in Fourier and Laplace transform domains for a weakly fluctuating two-dimensional rough surface to obtain admittance eq 9 (detailed description is shown in Appendix) and can be expressed as
( )
-CS -ω˜ |z b|, z, iω) ) b|) + ωkζˆ (K b |) + δC(K e {(2π)2δ(K iω ω2k f )×ζˆ (K' f )[2γ - 1]} + O(ζˆ 3) b| - K' d2K'|ζˆ (K | | |,| ′ 2(2π)2 (16)
∫
b|| is the magnitude of wave vector (K b|) or simply where K| ) |K b|) is the Fourier transform (performed over wavenumber, ζˆ (K two spatial coordinates, i.e., x and y) of surface roughness profile b|) is the twob|) ) ∫d2r|e-iKb|.rb|ζ(r b|)], and δ(K ζ(x, y) [ζˆ (K b |. dimensional Dirac delta function in wave vector K
( iω D+ k )
1/2
ωk
)
ω ˜ | ) ωkγ|
2 b |2]1/2 ) [ωk + K
f |2]1/2 ω ˜ |,| ′ ) ωkγ |,| ′ ) [ω2k + |K b| - K' | Equation 16 simplifies to diffusion limited concentration19 profile under kf 0 limit. The current density and total current at the interface in Fourier and Laplace transform domain are useful quantities and used
I(iω) )
nFCSD iω
{[
∫S d2r| 0
{
1 (2π)2
∫ d2K|eiK br b|
|
×
b|)ωk + ωk(ω b|) + (2π)2δ(K ˜ | - ωk)ζˆ (K
˜| ωkω
∫
f )ω f ˆ b d2K|ζˆ (K' | ˜ | | ζ(K| - K'|) (2π)2 ωk(ω2k + ωkω ˜ |) f )ζˆ (K f)b| - K' d2K|ζˆ (K' | | 2 2(2π) ωk f )|K f |2ζˆ (K f)b| - K' b| - K' d2K|ζˆ (K' | | | (2π)2 ωk f ζˆ (K' f )(K f )ζˆ (K f) ; b| - K' b| - K' d2K|K' | | | | 2 2(2π)
∫
∫
∫
}
] }
b| f b r | + O(ζˆ 3) K
(18)
b b| f b b|) is a twob|); K r|} ) [1/(2π)2]∫d2K| e- iK| ·br|f(K where {f(K 39 dimensional inverse Fourier transform, and the outer integral is performed over the surface of a plane z ) 0. Using eq 18 in eq 9, one can write the general admittance expression for a homogeneous chemical reaction coupled with a diffusioncontrolled charge transfer process, occurring at an arbitrary interface with profile ζ(r b|) and can be represented as a sum of various-order terms in surface roughness using eq 9
Y(ω) ) ΓD
{
∫S d2r| 0
1 (2π)2
∫ d2K|eiK br {y0(ω, Kb|) + b|
|
}
b|) + y2(ω, K b|)} y1(ω, K where y0, y1, and y2 can be expressed as
(19)
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b|) ) (2π)2δ(K b|)ωk y0(ω, K b|) ) ωk(ω b |) y1(ω, K ˜ | - ωk)ζˆ (K ωk f )ζˆ (K f )[ω b |) ) b| - K' y2(ω, K d2K'|ζˆ (K' ˜ | | | | ˜ |ω 2 (2π) 1 2 f |2 - K' f (K f b| - K' b ˜ |) - |K (ω + ωkω | | | - K'|) 2 k
∫
b|) is the Fourier transform of surface roughness profile where ζˆ (K ζ(r b|). Equation 19 generalizes the Gerischer admittance for an arbitrary profiled electrode. The first term in eq 19 corresponds to a smooth electrode response, and the effect of roughness is manifested through the second and third terms. Under the limit of kf0, eq 19 will become a Warburg admittance expression for an arbitrary surface profile.6 Equation 19 can be used as a generalized expression for obtaining the admittance expression of some known rough profile functions, i.e., the array of bellshaped roughness over the surface is obtained from ζ(r b|) ) h cos 2(λ1x) cos 2(λ2y). Fourier transformation of this function is: b|) ) hπ2({[δ(Kx + 2λ1) + δ(Kx - 2λ1)]/2 + δ(Kx)} {[δ(Ky -ζˆ (K + 2λ2) + δ(Ky - 2λ2)]/2 + δ(Ky)});39 h is the width of interface, and λ1 and λ2 are wavenumbers corresponding to the x and y direction, respectively. Using the technique of Fourier transform and Laplace transform, one can find out the concentration, current density, total current, admittance density, and impedance for such surfaces or any other known surface profiles. Diffusion-Reaction Admittance for a Random Surface Model. The observables of interest for the random-surface model are not quantities like total admittance, but rather the statistical average (indicated as 〈.〉) of this quantity over various surface configurations. These random surface profiles, ζ(r b|), and b|), are characterized by a their Fourier transformed form, ζˆ (K b|)|2〉. The surface roughness structure factor,7,8,17,40 i.e., 〈|ζˆ (K power spectrum can be measured from AFM studies and are useful in describing fractal and nonfractal random roughness. The ensemble averaged (over random surface configurations) admittance of a randomly rough electrode under diffusion and homogeneous kinetics coupled with fast heterogeneous charge transfer reaction (i.e., generalized Gerischer admittance) using eq 19 gives
iω + k ∞ b| × 〈Y(ω)〉 ) YG(ω){1 + 1 dK|K 0 2π D iω + k iω + k ˆ b 2 b |2 〈|ζ(K|)| 〉} +K D D (20)
[
∫
]
where 〈Y(ω)〉 is the ensemble averaged admittance for the rough b|)|2〉 is the surface structure factor, which is the electrode. 〈|ζˆ (K Fourier transform of the two-point surface correlation function.41 YG(ω) is the Gerischer admittance of a smooth surface shown in eq 4. The generalized admittance equation (eq 20) is proportional to the Gerischer admittance on a projected surface and a frequency dependent complex dynamic roughness factor. The dynamic roughness factor contains all of the information about the roughness of the surface through the power spectrum or structure factor. The kernel inside the integral depends on the complex kinetic diffusion length and picks value of a power spectral density at appropriate wavenumbers. Equation 20 generalizes the conventional Gerischer admittance on a smooth electrode to randomly rough electrode. Equation 20 under a k
Kumar and Kant f 0 limit is same as the expression shown in ref 6 for a generalized Warburg equation (eq 13). Theoretical Model for Realistic Self-Affine Fractal Surfaces. A surface model is employed here for rough electrodes, and the electrode-electrolyte interface is considered a random surface with roughness over several length scales. Roughness of these surfaces can be modeled approximately as self-affine fractals. This kind of roughness has a structure factor of the form which is a band-limited power law function7,8,17,40 and exhibits statistical self-similarity over a limited range of length scales. b|)|2〉, for such a The power spectrum of roughness, 〈|ζˆ (K realistic self-affine fractal surface, is defined in terms of following band-limited functions as
{
b| |1/l, |K 0,
(21)
This surface structure factor represents the statistically isotropic surfaces of a realistic fractal. There are four fractal morphological characteristics of roughness in a power spectrum: fractal dimension (DH), which describes the scale invariance property; lower cutoff length (l ); upper cutoff length (L); and strength of fractality (µ) (see Figure 1 for a graphical illustration of these four characteristics). As mentioned earlier, µ is the strength of fractality, which is related to the topothesy of fractals.17,40 Its unit is [Length]2DH - 3, and µf0 means there is no roughness. The lower length scale is the length above which the surface shows fractal behavior. Similarly, the upper length scale is the length below which the surface shows the fractal behavior. L can also be looked upon as a correlation length above which the height fluctuation is considered as uncorrelated. It is appropriate to recapitulate that the power spectrum or surface structure factor is (two-dimensional) Fourier transform of a two point height-height correlation function of the rough surface. The power spectrum has units of [Length]4. The ensemble averaged admittance expression for an approximately self-affine random fractal surface roughness is b|)|2〉) obtained by substituting eq 21 (surface structure factor 〈|ζˆ (K in eq 20. Solving the resultant integral, we obtained the following equation for total admittance for a band-limited (isotropic) fractal power spectrum as follows
[Y(ω)] ) YG(ω)[1 + RI(ω)]
(22)
the function RI(ω) is the frequency dependent contribution from roughness in the total admittance. Here subscript “I” expresses the intermediate region of a power spectrum. This region shows the dominant effect on the electrochemical response. The low wavenumber contribution of a power spectrum in total admittance can be ignored because the admittance is a weak function of L. So admittance response is no longer influenced by L, whereas high wavenumber contribution (which signifies a sharp cutoff) will be zero.7,17 The function RI(ω)is
RI(ω) )
[
]
(iω + k) µL-2δ µ -2δ - l + ψ(ω) D 4πδ 4πδ
and ψ(ω) is give as
(23)
Theory of Generalized Gerischer Admittance
ψ(ω) )
[ [
J. Phys. Chem. C, Vol. 113, No. 45, 2009 19563
]
µl -2δ -1 -D F δ, , δ + 1, 2 4πδ 2 1 2 l (iω + k) µL -2δ -1 -D F δ, , δ + 1, 2 4πδ 2 1 2 L (iω + k)
]
(24)
where δ ) DH - 5/2 and 2F1 is the hypergeometric function.38 RI(ω) and hypergeometric function in eq 24 can be numerically evaluated easily with the help of standard software like Mathematica. The admittance expression shown in eq 22 is dependent on four fractal morphological characteristics of a power spectrum of roughness and phenomenological (complex) diffusion-reaction length. Equation 22 extends the conventional representation of the reaction-diffusion or Gerischer admittance of a smooth surface electrode to a fractally rough electrode. These equations show that the extent of deviation of admittance from planar electrode admittance is directly related to the fractal nature of the surface as well as coupled homogeneous kinetics. These equations achieve a more realistic characterization of limited scales of roughness which generalizes the Gerischer admittance. The generalized Warburg admittance17 can be obtained as a special case of generalized Gerischer admittance in eq 22 for fractally a rough electrode, under the limit: k f 0. Results and Discussion In this section, we use results developed for theoretical model of the diffusion-kinetic admittance, where diffusion and a homogeneous first-order reaction proceed simultaneously on the band-limited (isotropic) random surface fractal described by eq 22. This equation extends the conventional representation of the Gerischer admittance on the planar electrode to the fractally rough electrode. Our admittance expression shown in eq 22 depends on four fractal morphological characteristics, i.e., fractal dimension (DH), lower cutoff length scale (l ), upper cutoff length scale (L), and strength of fractality (µ). Here, we analyze the impedance response and its deviation from a smooth geometry response under the variation of kinetic (rate constant k) and fractal morphological characteristics of a roughness power spectrum. These results provide clear insight into the influence of roughness and homogeneous kinetics over the electrochemical impedance response. Figure 2 (first column of graphs) illustrates the influence of kinetics (rate constant) over the behavior of impedance plots for the problem when diffusion and the chemical reaction proceed simultaneously. This plot shows that as the rate constant increases from upper curve to lower curve, the magnitude of impedance decreases as well as crossover frequency to anomalous power law region increases. If the reaction is fast, then there is prolonged control of kinetics over diffusion in Gerischer impedance. This plot also shows that the low frequency regime, which is controlled by kinetics, extends toward high frequency as the rate constant increases. Similarly, Figure 2 (second column of graphs) describes the influence of kinetics (with varying rate constant) over the phase angle response. This plot shows that the “approximate” constant phase angle region decreases with an increase in the rate constant from the upper to lower curve. Phase angle response approaches zero at low frequency due to kinetic-controlled regime and anomalous intermediate phase angle response φ (ω) > 45° and high frequency-limiting Warburg behavior with phase angle φ(ω) ) 45°. These graphs show that there are two crossover frequencies, viz, outer crossover frequency (ωo = k) and inner crossover
Figure 2. Effect of kinetics (rate constant k) on the logarithmic of magnitude of impedance log (|Z(ω)|) with the logarithmic of angular frequency log (ω) (first column of the graph) and phase angle [φ(ω)] vs log (ω) (second column of the graph). The graph is generated for varying rate constants; k(s-1) ) 10-2, 10-1, 100, 101, and 102 (from top to bottom and can be seen in the left portion of the graph) and keeping fixed DH ) 2.4, l ) 40 nm, L ) 4 µm, µ ) 2 × 10-6 a.u., A ) 1 cm2, D ) 5 × 10-6 cm2 s-1, and concentration (C0 ) 15 mM).
frequency (ωi = D/l 2). In the low frequency limit (ω e ωo), the Gerischer impedance of a rough electrode is kinetically controlled. Similarly, for a high frequency limit (ω g ωi), it shows classical Warburg impedance behavior. In the intermediate region between these two crossover frequencies, impedance exhibits an approximate power law and constant phase angle behavior. Figure 3 (first column of graphs) illustrates the influence of roughness in terms of fractal morphological characteristics, i.e., fractal dimension (DH), lower cutoff length (l ) and strength of fractality (µ), on generalized Gerischer impedance response. Figure 3A shows the influence of fractal dimension over the behavior of magnitude of impedance for such types of processes. From this plot, we clearly explain how the fractal dimension influences the magnitude of impedance when the fractal morphological characteristics of roughness like lower cutoff length, (l ) upper cutoff length (L), and strength of fractality (µ) are kept constant. The nature of this plot elucidates three different behaviors in the three frequency regimes and two crossover points. The log-log plot of impedance versus frequency shows the crossover from frequency independent response at low frequency (ω < k) followed by anomalous frequency dependent power law behavior at intermediate frequency regime and classical Warburg impedance response at high frequency (ω > D/l 2). These observations can be explained for a low frequency region where ω < k (or LR < LD), which signifies the kinetic-controlled frequency independent impedance (ω < k), and the high frequency region corresponds to LR > LD and ω > D/l 2 which represents the Warburg impedance. There is a constant impedance response for the rough surface and this effect is more prominent in the case of large reaction layer thickness. The anomalous intermediate frequency region can be understood in frequency region; k < ω < D/l 2. Dynamic anomalous response is seen as a power law decay of the impedance response in the intermediate frequency domain. This graph also suggests that when the value of k becomes lower, the impedance response shows delay in the onset of the anomalous region. The magnitude of impedance decreases with an increase in the fractal dimension (DH) from the upper curve to the lower curve, which is similar to smooth geometry but lower in magnitude. Figure 3B shows the influence of the lower cutoff length scale. This plot shows that as the lower cutoff length value increases, the magnitude of impedance increases. Similar behavior is observed in this plot as in Figure 3A, i.e., frequency independent impedance response at low frequency, Warburg impedance type response at higher frequency, and power law behavior in intermediate frequency
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Figure 3. Effect of fractal morphological characteristics of roughness on diffusion-kinetic impedance, i.e., the logarithmic of magnitude of impedance log (|Z(ω)|) vs the logarithmic of angular frequency log (ω) (first column of graphs) and phase angle [φ(ω)] vs log (ω) (second column of graphs). Black line corresponds to planar electrode response. It illustrates the effect of three dominant fractal morphological characteristics of roughness. (A,D) The graph is generated for fixed l ) 40 nm, L ) 4 µm, µ ) 2 × 10-6 a.u., A ) 1 cm2, D ) 5 × 10-6 cm2 s-1, and concentration (C 0 ) 15 mM), while varying fractal dimensions, DH ) 2.2, 2.25, 2.3, 2.35, 2.4, and 2.45 (varying from top to bottom in first column of the graph and bottom to top in second column of the graph). (B,E) Varying lower cutoff length scale, (l), the parameters were taken as DH ) 2.4 and l nm ) 20, 40, 80, 160, 320, and 640 (varying from bottom to top in first column of the graph and top to bottom in second column of the graph) and other parameters are kept constant, i.e., L, µ, A, D, and C 0. (C,F) Varying strength of fractality (µ), the parameters were taken as µ a.u. ) 0.05, 0.2, 0.8, 3.2, and 12.8 × 10-6 (varying from top to bottom in first column of the graph and bottom to top in second column of the graph), DH ) 2.4, l ) 40 nm, A ) 1 cm2, D ) 5 × 10-6 cm2 s-1, L ) 4 µm, and concentration (C0 ) 15 mM).
domain. Figure 3C shows the effect of strength of fractality over the magnitude of impedance. This plot shows that the magnitude of impedance decreases as the strength of fractality increases from the upper curve to the lower curve, and it is controlled by the kinetics of the chemical reaction along the lower frequency and controlled by diffusion toward higher frequencies. In an intermediate frequency domain, these curves follow an approximate power law behavior. The phase angle of a generalized Gerischer impedance shows several important features which are illustrated in plots. Figure 3 (second column of graphs) shows the phase angle plots under the variation of fractal morphological characteristics, i.e., fractal dimension (DH), lower cutoff length (l ), and strength of fractality (µ), with fixed phenomenological length LR (or fixed kinetic rate constant). Figure 3D shows the roughness effect over the phase angle due to the increase in fractal dimension from the lower to upper curves. One can observe from this plot
Kumar and Kant that the phase angle region increases with an increase in fractal dimension (DH). It shows “approximate” constant phase angle behavior in the limited range of frequency, and it also crossovers to the zero phase angle due to the kinetic-controlled regime toward the lower frequency region. The phase angle crosses over to a Warburg value of 45° in the high frequency regime. We also observe that the phase angle region shifts toward a lower frequency region as the fractal dimension increases. This phase angle response, which is independent of frequency and crossover to zero, is due to the fact that at low frequency, phenomenological length LD > LR (ω < k) makes the region kinetic dependent and frequency independent, and at higher frequency, the phenomenological length LD < LR (ω > D/l 2) makes the region diffusion controlled and shows Warburg phase angle behavior. The approximate constant phase angle response is observed in the intermediate frequency region. Figure 3E shows the effect of lower cutoff length (l ) value over the phase angle plot. The phase angle region decreases with an increase in the lower cutoff length value from the upper to lower curve. It also exhibits the “approximate constant phase angle behavior” in the limited frequency range and finally a crossover to zero. The phase angle region shifts toward the higher frequency as we increase the lower cutoff length value. A similar pattern is observed in this graph as was observed in Figure 3D, i.e., frequency independent phase angle response at low frequency and Warburg impedance type phase angle response at high frequency. Figure 3F shows the influence of strength of fractality (µ) over the phase angle behavior for such a type of reaction. The phase angle region increases with an increase in the strength of fractality from bottom to top. It also has an “approximate” constant phase angle behavior for a limited range of frequencies and the phase crossover to zero along low frequency region. The phase angle region shifted toward the lower frequency region as the strength of fractality increases. Finally, all of these results suggest that the interfacial response depends on all three dominant fractal morphological characteristics as well as on rates of coupled reaction on realistic fractals. All graphs reported in this section are generated using Mathematica software. Conclusion The present theory of reaction-diffusion impedance on a realistic fractal electrode offers a new way to understand the combined charge transport under diffusion-controlled and -coupled homogeneous chemical reaction phenomena at the rough electrode. Some of the earlier attempts to study the Gerischer impedance for fractal electrode were mainly based on less precise scaling analysis, where the direct influence of fractal morphological characteristics of roughness like fractal dimension (DH), strength of fractality (µ), and cutoff lengths (lower cutoff length l and upper cutoff length L) were never understood simultaneously. Here, we are analyzing the simultaneous influence of three dominant fractal morphological characteristics of a roughness power spectrum and homogeneous reaction rate effect on the impedance and the phase angle behavior of a rough electrode. From our graphs, we observe how surface irregularities can affect the electrochemical response in comparison to a smooth surface. Graphs show a kinetically controlled region along a lower frequency and diffusioncontrolled region toward higher frequency. An approximate constant phase angle behavior is observed in a limited range of frequency. Our graphs also show two crossover frequencies, viz, outer crossover frequency, ωo = k and inner crossover frequency, ωi = D/l 2. Under a small frequency limit, i.e., ω e
Theory of Generalized Gerischer Admittance ω0, the Gerischer impedance is kinetically controlled. For ω g ωi, it shows diffusion controlled impedance (i.e., Warburg type) behavior. In between these two crossover frequencies, impedance exhibits an approximate power law and constant phase angle behavior. Our results are valid for all frequency regimes. This work unravels the connection between fractal dimension (DH), lower (l ), and upper cutoff (L) length scales and strength of fractality (µ), and support the fact that the fractal dimension is not itself sufficient to understand the problem. Finally, one can say that the present theory offers a quantitative description of the cases where diffusion and coupled homogeneous firstorder kinetics proceed simultaneously across an irregular interface. Nomenclature A0 b, t) CR(r C*(r R b, t)
geometrical or projected area of the surface concentration of Rth species modified concentration including exponential kinetic factor of R -th species bulk concentration of Rth species C0R difference in the concentration at an arbitrary point δCR and the bulk concentration of Rth species excess surface concentration CS diffusion coefficient of Rth species DR Hausdroff-Besicovitch fractal dimension of surface DH E potential of working electrode δE difference in the potential of working electrode F Faraday constant hypergeometric function 2F1[a, b, c, z] G Green’s function I(t) total interfacial current under potentiostatic conditions I(iω) Laplace transform of total interfacial current i ( - 1)1/2 b ˆ j(K|, ζ, iω)Laplace and Fourier transformed current density k homogeneous reaction rate constant b| two-dimensional wave vector, (Kx, Ky) K l lower length cutoff L upper length cutoff diffusion layer thickness LD reaction layer thickness LR n number of electron transferred in redox reaction frequency dependent contribution of roughness from RI intermediate power spectrum t time Y(ω) total admittance 〈Y(ω)〉 ensemble averaged total admittance total Gerischer admittance for planar electrode YG(ω) YW(ω) total Warburg admittance for planar electrode β coefficient of correlation scale δ( · ) Dirac delta function amplitude of input sinusoidal potential η0 γ exponent in formal power law expression for admittance Γ capacitance per unit volume χ*(S′, t′) source function µ normalizing factor of the power law spectra (strength of fractality) ω angular frequency of externally applied potential [(iω + k)/(D)]1/2, inverse of complex reaction-diffusion ωk layer thickness outer crossover frequency ωo inner crossover frequency ωi b|2]1/2 [(iω + k)/(D) + K ω ˜| b|′ |2]1/2 b| - K ω ˜ |,|′ [(iω + k)/(D) + | K
J. Phys. Chem. C, Vol. 113, No. 45, 2009 19565 normal derivative of surface phase angle electrode-electrolyte interface or surface profile of the surface ζˆ Fourier transform of the surface profile b|)|2〉 power spectrum of roughness 〈|ζˆ (K 〈.〉 ensembleaverageoverallrandomsurfaceconfigurations ∂n φ ζ
Acknowledgment. The authors dedicate this work to the memory of their mentor, late Prof. S. K. Rangarajan (formerly Professor, Department of Inorganic and Physical Chemistry, Indian Institute of Science (IISc), Banglore, India). R. Kant thanks University of Delhi for financial support under the Scheme to Strengthen R & D Doctoral Research Programme and R. Kumar is grateful to UGC, New Delhi for the Research Fellowship in Science for Meritorious Students. We also thank the referees for their useful suggestions. Appendix Perturbative Solution for the Concentration Profile for a Weakly Fluctuating Two-Dimensional Rough Surface Substituting eq 15 into eq 14, we obtain the integrodifferential equation for a two-dimensional rough surface as
δC*(b r |, z, t) ) ektδC(b r |, z, t) ) -|b r |-b r |′2/4Dτ ze-z /4Dτ d2r|′ e × 1/2 3/2 4πDτ 2π (Dτ) ∂ 1 ∂2 ekt'CS + ζ(b r |′) δC* + ζ2(b r |′) δC* + · · · ∂z' 2! ∂z2
-D
{
∫
2
t
dt' 0
∫
}
z')0
(25)
The above integro-differential equation up to the second order in surface profile can be written symbolically in the form
δC*(b, r t) ) δC0* + (L˜1 + L˜2)δC*(b′ r |, z', t') + O(ζ3) (26)
b, t). where δC*(r b, t) ) ektδC(r We use (iω) to represent frequency dependence, and it is used in the admittance expression instead of p standard notation in the Laplace domain. δC0* and the operators L˜1 and L˜2 are explained below.
-kt kt' δC0* ) e L˜e CS -z2/4Dτ
∫0t dt' 2πze1/2(Dτ)3/2 e-k(t-t')CS ∫0t dt'(e-ω z;iω f τ)CS
) -D ) -
(27)
k
where {f(p); pfτ} )1/(2πi)∫dp epτ f(p) is the inverse Laplace b|), an transform of function, τ ) t - t′, L˜1 and L˜2 involve ζ(r
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Kumar and Kant
integral operator L˜, and projection operator P0 and are expressed as
∫ dt'{e ;iω f τ}C e + ∫ dt'{e ;iω f τ} × ∫ d r' {e ;|Kb | f |br - r'f |} × f ;iω f τ′}C e ζˆ (r' )Lˆ ∫ dt''{e ∫ dt'{e ;iω f τ} × ∫ d r' {e ;|Kb | f |br - r'f |} × f f |} × b | f |b ;iω f τ′} × ∫ d r'' {e ;|K r - r' ζˆ (r' )Lˆ ∫ dt''{e f ;iω f τ′′}C e + ζˆ (r″ )Lˆ ∫ dt′′′{e ∫ dt'{e ;iω f τ} × ∫ d r' {e ;|Kb | f |br - r'f |} × f ;iω f τ′}C e (35) ζˆ (r' )Lˆ ∫ dt''{e t
δC*(b, r t) ) δC(b, r t)ekt ) t
-ω0z
∞
-ω0z
t'
(28)
1
|
t
∂ Lˆ1 ) P0 ∂z' 1 ∂2 Lˆ2 ) P0 2! ∂z2
∞
|
-ω0z
t
|
|
|
-ω0z
∫0t dt' 2πze1/2(Dτ)3/2 ∫-∞∞ d2r|′ e
|
kt′′′
∞
-ω0z
-K|2Dτ
2
|
-∞
|
t'
2
|
|
-ω0z
kt″
S
0
dividing eq 35 by ekt. So eq 35 will transform to the expression as t dt'{e-ω z ;iω f τ}CSekt' + ∫ 0 t ∞ b| | f e-kt ∫0 dt'{e-ω z ;iω f τ} × ∫-∞ d2r″|{e-K Dτ ;|K t' f -ω z ˆ f ˆ |b r | - r' ;iω f τ′}CSekt″ | |} × ζ(r'|)L1 ∫0 dt″{e t ∞ b| | f e-kt ∫0 dt'{e-ω z ;iω f τ} × ∫-∞ d2r'|{e-K Dτ ;|K t' f f |b r | - r'| |} × ζˆ (r'|)Lˆ1 ∫0 dt''{e-ω z ;iω f τ′} × ∞ 2 ∫-∞ d r''|{e-K Dτ′;|Kb|| f |br | - r'f||} × t″ f ζˆ (r″|)Lˆ1 ∫0 dt′′′{e-ω z ;iω f τ′′}CSekt′′′+ t ∞ b| | f e-kt ∫0 dt'{e-ω z ;iω f τ} × ∫-∞ d2r'|{e-K Dτ ;|K t' f f |b r | - r'| |} × ζˆ 2(r'|)Lˆ2 ∫0 dt''{e-ω z ;iω f τ′}CSekt″
δC(b, r t) ) -e-kt
0
2 |
0
-(b r |-b′ r |)2/4Dτ
|
S
2
-z2/4Dτ
|
-K|2Dτ′
2
0
0
The projection operation is defined through PRf(z) ) f(R), and the operators for a two dimensional rough surface in eq 29 can be represented as
|
-∞
0
|
L˜ ) -D
-K|2Dτ
2
∞
1
|
kt″
-∞
|
|
S
t″
(29)
|
-ω0z
-ω0z
t'
1
|
0
0
|
-K|2Dτ
2
-∞
0
r |′)Lˆ1 L˜1 ) L˜ζ(b f )Lˆ L˜2 ) L˜ζ2(r' | 2
kt'
S
0
0
4(πDτ)
(30)
2 |
0
0
The rest of the solution of eq 26 is obtained by expanding the surface concentration as a power in the small parameter, viz, the surface “height” h. The method allows us to satisfy order by order the R.H.S. and L.H.S. of eq 26 and obtain the solution. The perturbative solution of the above equation up to second order in ζ is
δC*(b, r t) ) δC0*(z, t) + L˜1δC0*(z', t') + L˜21δC0*(z', t′′) + L˜2δC0*(z', t3) + O(ζ3) (31) b, t). Using Laplace transform38 and where δC*(r b, t) ) ektδC(r Fourier transform,39 eq 30 for operator L˜ can be written as
2 |
0
0
(36) taking the Fourier transform with respect to b r| and using convolution properties of Fourier transform39 of the above equation, we obtain b|, z, t) ) -(2π)2δ(K b|) δC(K
∫
t
0
L˜ ) -D
{
∫0 dt' t
-ω0z
e
D
}∫
;iω f τ
2 |
0
∫ dt'{e t
2
t'
d r|′{e 2
-K|2Dτ
b| | f ;|K
k
s
e-K| Dτ ˆ b ζ(K|) X (2π)2 2
|b r| - b r |′|} (32)
-
∫
t
dt'{e-ωkz ;iω f τ} ×
0
∫ dt''{ω ;iω f τ′}e b )] ∫ dt′′′{ω ;iω f τ′′}C + ∫ dt'{e ζˆ (K
[
t'
-K|2Dτ′
k
0
t″
b| )(Kx, Ky). where ω0 )(iω/D) , K The first brace in the above equation is the inverse Laplace transform, and it is defined as 1/2
{f(p);p f τ} )
1 2πi
∫ dpepτf(p)
(33)
and the second is the inverse Fourier transform in two dimensions defined as
1 b|);K b| f b {f(K r |} ) (2π)2
∫ d K|e 2
b |. b iK r|
b|) f(K
|
Explicitly, eq 31 for a two dimensional (2-D) rough surface can be written as
t
k
0
e-K| Dτ ˆ b b|)] × [ζ(K|) X ζˆ (K (2π)2 2
S
-ωkz
0
∫ dt″ t
0
{
×
;iω f τ} ×
}
ωk2 ;iω f τ' CS (37) 2
b|)] ) ∫d2K′|f(K b′|)g(K b| - K b′|). Using Laplace b|)Xg(K where [f(K transform technique and shift theorem of Laplace transform,38 we obtain the perturbative solution for the concentration profile in Fourier and Laplace transform domains for a weakly fluctuating two-dimensional rough surface
b|, z, iω) ) δC(K ω2k
(34)
;iω f τ}CS -
b|) × dt'{e-ωkz ;iω f τ} eK| Dτζˆ (K
∫ dt''{ω ;iω f τ′}C 0
-ωkz
0
2(2π)2
( )
-CS -ω˜ |z b|) + ωkζˆ (K b |) + e [(2π)2δ(K iω
f )×ζˆ (K' f )(2γ ∫ d2K'|ζˆ (Kb| - K' | | |,|
- 1) + O(ζˆ 3)
References and Notes (1) Bisquert, J. J. Phys. Chem. B 2002, 106, 325. (2) Fabregat-Santiago, F.; Garcia-Belmonte, G.; Bisquert, J.; Zaban, A.; Salvador, P. J. Phys. Chem. B 2002, 106, 334.
Theory of Generalized Gerischer Admittance (3) Bard A. J.; Faulkner, L. R. Electrochemical Methods: Fundamentals and Application; Wiley: New York, 1980. (4) Lasia, A. In Modern Aspects of Electrochemistry; Conway, B. E., et al., Eds.; Kluwer Academic: New York, 1999; No. 32, p 143. (5) Kant, R. J. Phys. Chem. B 1997, 101, 3781. (6) Kant, R.; Rangarajan, S. K. J. Electroanal. Chem. 2003, 552, 141. (7) Kant, R.; Jha, S. K. J. Phys. Chem. C 2007, 111, 14040. (8) Jha, S. K.; Sangal, A.; Kant, R. J. Electroanal. Chem. 2008, 615, 180. (9) (a) Mandelbrot, B. B. The Fractal Geometry of Nature; Freeman: San Franscisco, 1997. (b) Feder, J. Fractals; Plenum: New York, 1998. (10) Schiller, R.; Balog, J.; Nagy, G. J. Chem. Phys. 2005, 123, 094704. (11) Boukamp, B. A.; Bouwmeester, H. J. M. Solid State Ionics 2003, 157, 29. (12) Wilson, J. R.; Schwartz, D. T.; Adler, S. B. Electrochim. Acta 2006, 51, 1389. (13) Steckhan, E. Top. Curr. Chem. 1987, 142, 1. (14) Jukic, A.; Metikos-Hukovic, M. Electrochim. Acta 2003, 48, 3929. (15) Meland, A. -K.; Bedeaux, D.; Kjelstrup, S. J. Phys. Chem. B 2005, 109, 21380. (16) Boukamp, B. A.; Verbraeken, M.; Blank, D. H. A.; Holtappels, P. Solid State Ionics 2006, 177, 2539. (17) Kant, R.; Kumar, R.; Yadav, V. K. J. Phys. Chem. C 2008, 112, 4019. (18) Kumar, R.; Kant, R. J. Chem. Sci. 2009, in press. (19) Kant, R.; Rangarajan, S. K. J. Electroanal. Chem. 1994, 368, 1. (20) Kant, R. Phys. ReV. Lett. 1993, 70, 4094. (21) Kant, R.; Rangarajan, S. K. J. Electroanal. Chem. 1995, 396, 285. (22) Gerischer, H. Z. Phys. Chem. 1951, 198, 216. (23) Sluyters-Rehbach, M.; Sluyters, J. H. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1970; Vol. 4, p 65.
J. Phys. Chem. C, Vol. 113, No. 45, 2009 19567 (24) Smith, D. E. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1970; Vol. 1, p 44. (25) Levart, E.; Schuhmann, D. J. Electroanal. Chem. 1974, 53, 77. (26) Canas, P.; Lorenzo, M. S.; Duo, R.; Celdran, R. Z. Phys. Chem 1990, 271, 847. (27) Sluyters-Rehbach, M.; Sluyters, J. H. In ComprehensiVe Treatise of Electrochemistry; Yeager, E.; Bockris, J. O. M.; Conway, B. E.; Sarangapani, S., Eds.; Plenum: New York, 1984; Vol. 9, p 274. (28) Pajkossy, T.; Nyikos, L. New J. Chem. 1990, 14, 233. (29) Kopelman, R. J. Stat. Phys. 1986, 42, 185 Science, 1988, 241, 1620. (30) Chaudhari, A.; Yan, C.-C. S.; Lee, S.-L. Chem. Phys. Lett. 2002, 351, 341. (31) Pajkossy, T.; Nyikos, L. Electrochim. Acta 1989, 34, 171. (32) De Levie, R. J. Electroanal. Chem. 1990, 281, 1. (33) De Gennes, P. G. C. R. Acad. Sci. Paris 1982, 295, 1061. (34) (a) Maritan, A.; Stella, A. L.; Toigo, F. Phys. ReV. B 1989, 40, 9269. (b) Giacometti, D.; Maritan, A. J. Phys. (Paris) 1990, 51, 1387. (35) Oldham, K. B. J. Electroanal. Chem. 1991, 297, 317. (36) Barton, G. Elements of Greens Function and Propagation: Potentials, Diffusion, and WaVes; Clarendon: Oxford, 1991. (37) Carlsaw, H. S.; Jaegar, J. C. Conduction of Heat in Solids; Clarendon; Oxford, 1959. (38) Abramowitz, M., Stegun, I. A. Handbook of Mathematical Functions; Dover Publications, Inc.: New York, 1972. (39) Champaney, D. C. Fourier Transforms and their Physical Applications; Academic Press; London, 1973. (40) Yordanov, O. I.; Atanasov, I. S. Euro Phys. J. B 2002, 29, 211. (41) Kant, R. Phys. ReV. E 1996, 53, 5749.
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