Anomalous Warburg Impedance: Influence of ... - ACS Publications

ARTICLE pubs.acs.org/JPCC

Anomalous Warburg Impedance: Influence of Uncompensated Solution Resistance Shruti Srivastav and Rama Kant* Department of Chemistry, University of Delhi, Delhi 110007, India ABSTRACT: We present the theoretical results elucidating the influence of uncompensated solution resistance on anomalous Warburg’s impedance. Here, we obtain the mathematical expression which incorporates the diffusion at the rough electrode/electrolyte interface and bulk solution resistance via phenomenological length scales—diffusion length ((D/ω)1/2) and pseudoreaction penetration length (LΩ). The roughness at the interface is contained in the expression through the surface structure factor, which can be used to describe statistically any random surface morphology. Detailed analysis for realistic fractal electrodes, characterized as a finite self-affine scaling property with two lateral cutoff lengths, is presented. Limiting behavior in higher frequency is attributed to resistive effects of the electrolyte, and surface roughness is seen through the roughness factor. In the intermediate frequency regime, the anomalous power law behavior is attributed to the diffusion length weighted spatial frequency features of roughness (marking impedance loss) along with the interplay of LΩ and at lower frequencies the response crossover to classical Warburg’s behavior. This crossover frequency is dependent on the smallest of root-mean-square width (h) or lateral correlation length (L), while high-frequency crossover is dependent largely on LΩ. Phase response also shows the maximum phase gain for intermediate frequencies owing to the roughness features and is a signature to the fractal or nonfractal roughness at the interface. Our results show that owing to solution resistance the impedance response can mimic pseudo-quasireversibilty inducing a delay in the onset of diffusion-controlled regime and can hack the anomalous response due to roughness partially or completely.

’ INTRODUCTION Impedance spectroscopy has been proven to be a powerful tool for analysis of several phenomena occurring inside an electrolytic cell.1 In contrast to other electrochemical techniques, it is noninvasive and can be used for investigating bulk as well as interfacial processes connected with time constants ranging from minutes down to microseconds. It has been widely used to investigate the electrode kinetics, characterizing the reaction within the film, corrosion studies, and double layer kinetics. In the case of a metal/solution interface, the nature of both the metal and the solution exerts an influence on the charge transfer processes occurring there. As a convention, to study a metal/ solution interface, one first investigates the simplest case which corresponds to a simple Nernstian charge transfer process. A severe limitation of the non-steady-state techniques comes from the electrolyte resistance located between the equipotential surface defined by the reference electrode and the working electrode. The finite conductivity of the electrolyte limits the use of these techniques by adding a parasitic ohmic drop in series with the Faradaic overvoltage. When this ohmic drop is large compared with the latter, it is difficult to obtain accurate results of the Faradaic processes. Potentiostatic modes are basically affected. The HaberLuggin capillary technique could be used to diminish ohmic drop but not to cancel it completely. The problem of the ohmic drop in electrochemical polarization measurements seems to have only minor significance, but it r 2011 American Chemical Society

has actually troubled electrochemists for a long time.2,3 In general, the inert electrolytes and highly conducting supporting media are widely used. Other precautions include minimizing the distance between the working electrode and Luggin’s capillary. The solution resistance cannot be obtained by fitting the measurement model to the impedance. The solution resistance is treated as an arbitrarily adjustable parameter when fitting to the circuit analysis of the impedance in most studies. The ohmic contribution is usually represented by a pure resistor in the circuit analysis. One serious handicap of using circuit analysis for estimating solution resistance is the extraction of kinetic parameters, diffusion coefficient, and other information tractable from impedance measurements. In the circuit analysis, nevertheless, the dynamic interplay of the two effects, i.e., surface roughness and solution resistance, cannot be depicted. At the same time, these get erroneously included for kinetic measurement. However, in many applications it is desirable or essential to use reduced or zero levels of added electrolyte, posing a challenging fundamental problem. It has long been observed that the impedance of an electrode depends on the morphological nonuniformities of the surface, viz., roughness and porosity. In spite of very significant experimental work carried out worldwide during recent decades, it was Received: March 15, 2011 Revised: May 1, 2011 Published: May 25, 2011 12232

dx.doi.org/10.1021/jp2024632 | J. Phys. Chem. C 2011, 115, 12232–12242

The Journal of Physical Chemistry C

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et al.3538Numerical results have been obtained for Nernst Planck Poisson (NPP) type of equations for a finite system using random walk19,20 and fractional diffusion.17,21 We describe anomalous Warburg behavior at fractal interfaces,35,37 homogeneous kinetics coupled to it,38 and quasireversible charge transfer36,45 using an ab initio approach.

’ FORMULATION Our purpose is to formulate the effect of ohmic drop on the total admittance at a rough electrode. For a single-step charge transfer process, O(Sol) þ ne h R(Sol), at a randomly rough electrode, the diffusion equation can be written as Figure 1. Schematic representation of the phenomenon at rough working electrode. The diagram depicts the uncompensated solution resistance between the average plane of a rough electrode and Luggin’s capillary. LΩ is the length scale quantifying the uncompensated solution resistance. (D/ω)1/2 is the dynamic diffusion length. DH is the fractal dimension of the surface, l , and L is the lateral cutoff which takes into account the finite fractality of the rough working electrode.

difficult to establish a precise relationship between geometrical characteristics of the roughness and the resulting impedance. The theoretical studies, which tried to establish an analytical relation between the roughness and the impedance, faced great mathematical difficulties. Disordered or rough electrodes are ubiquitous in electrochemistry, but their theoretical aspects are still less understood. There have been several studies wherein the roughness of the interface is modeled as fractals,411 using scaling argument,1214 fractional and anomalous diffusion,1521 ab initio derivations2224 and numerical approach.25 There are two features at the disordered electrode: one is diffusion to the interface and the second is resistive effects at the rough interface. There are two phenomenological length scales: (i) diffusion layer thickness LD = (D/ω)1/2 and (ii) phenomenological length scale pseudo reaction penetration length (LΩ RΩ Fel e ) where l e is the distance between the reference and working electrode and Fe is the solution resistivity. The magnitude of the two phenomenological lengths controls the response of the system. It is worth mentioning here that LΩ is dependent only on the experimental quantities, while LD is dynamic in nature (see Figure 1). These systems belong to the difficult class of problems, hence electrochemists have avoided ab initio approach. The most common technique which analyzes the impedance response is the equivalent circuit analysis.26,28 While representation of the impedance by an equivalent electric circuit can be convenient, this simplified representation may also cause a misunderstanding. It is definitely wrong to analyze experimental impedance data by just fitting to an equivalent circuit roughly corresponding to a network chosen by trial and error.29 The reason for this is that the impedance response of several equivalent circuits can follow exactly the same function of frequency, only with different meanings of the corresponding elements. In addition, a fit will always be successful if an unlimited number of parameters is admitted. Without having an a priori model, the meaning of these parameters is undefined. This calls for a need for concrete theoretical treatment for the impedance response, particularly the origin of the anomalous Warburg impedance and the influence of other processes to it. Several attempts have been made to describe and generalize the anomalous Warburg impedance by Macdonald et al.,18,19 Bisquert et al.,21 and Kant

DδCR ð B r , tÞ ¼ DR r2 δCR ð B r , tÞ ð1Þ Dt The Nernstian boundary condition is valid for systems with a large value of exchange current density. The Nernstian boundary constraint is linearized under assumption of small external perturbation potential and can be written as δCO δCR 0 ¼ nf ηðtÞ C0O CR

ð2Þ

Incorporating the small ohmic losses in overpotential, the linearized Nernstian boundary constraint becomes δCO δCR 0 ¼ nf ðηðtÞ jRΩ Þ C0O CR

ð3Þ

where η(t) is the electrode potential possessing the form η0eiωt; RΩ is the uncompensated solution resistance between the working electrode (taken from the average plane (z = 0) of rough working) and the reference electrode; j is the current density at the electrode surface; n is the number of electrons transferred in redox reaction; η0 is the amplitude of input sinusoidal potential; and f = F/RT (here F is Faraday’s constant, R is the gas constant, and T is the absolute temperature). In formulating the above boundary condition, i.e., eq 3, we have assumed that there is no position-dependent effect in RΩ due to height fluctuations between the working and reference electrodes (as the fluctuations are much smaller than the two electrodes separation). For sinusoidal perturbation of the form eiωt, the diffusion equation changes as follows r Þ ¼ DR r2 δCR ð B rÞ iωδCR ð B

ð4Þ

where R = O, R, ω is the angular frequency and i is (1)1/2. Along with the above boundary condition and initial and bulk condition, viz., δCR(r B,0) = δCR(r B f ¥,t). Using flux-balance and equal diffusion coefficient conditions (DO = DR = D), we get δCO(r B,t) þ δCR(r B,t) = 0. Substituting this identity in eq 4, one can rewrite the boundary condition for δCO in the form: LΩ∂nδCO = δCO þ Γη0/nF. This comprises a phenomenological length scale, i.e., pseudoreaction penetration length (LΩ), which represents the thickness of a charge transfer layer due to uncompensated resistance through which the electrons apparently would be transferred. LΩ ¼ ΓDRΩ

ð5Þ

þ has a dimension of where Γ = n F capacitance per unit volume (specific diffusion capacitance); RΩ is uncompensated electrolyte resistance; ∂n is the outward drawn normal derivative; and D is the diffusion coefficient. This 2 2

12233

/RT(1/C0O

1/C0R)

dx.doi.org/10.1021/jp2024632 |J. Phys. Chem. C 2011, 115, 12232–12242


ARTICLE )

)

)

)

ω20 ζð K BÞ ½^y ð2πÞ2 δð K B Þ þ ^y1 ^ RΩ 0 )

)

yðω, ^ ζð K B ÞÞ ¼

LΩ ω ω0 ω0 ω LΩ þ 1 ω0 LΩ þ 1

ð9Þ

)

)

0

LΩ ω0 ω 2 ω0 LΩ þ 1 ω LΩ þ 1 )

d2 K

)

ω0 ðω LΩ þ 1Þ

LΩ ω0 ω , 0 LΩ þ 1 ) )

LΩ ω ω , 0 )

þ

Z

)

1 ð2πÞ2

)

ð8Þ

) ) )

^y02

ð7Þ

LΩ ω0 ðω0 LΩ þ 1Þ

^y0

^y1

)

)

0 0 ζð K B Þ^ þ ^y2 ^ ζð K B K B Þ

0

0

0

)

)

) )

)

)

)

)

) )

)

)

)

)

B K K B 3ð K B Þ LΩ j K LΩ B K B j2 ω0 ðω LΩ þ 1Þ ω , 0 LΩ þ 1 ω0 ðω , 0 LΩ þ 1Þ # 0 0 K B 3ð K BÞ B K LΩ þ ð10Þ 2ω0 ðω0 LΩ þ 1Þ ω LΩ þ 1

)

)

) )

)

)

where ω0 is (iω/D)1/2; ω = (ω20 þ K2)1/2; ω , 0 = (ω20 þ (K B K B 0 )2)1/2. The expression for the total admittance is given by taking the surface integral of the admittance density and retaining terms up to second order in the Fourier transformed surface

)

)

)

)

)

)

)

)

0 B Þ þ ^y2 ^ζð K B Þ^ζð K B K B Þ; K B f B r g þ ^y1 ^ζð K

ð11Þ

)

) )

)

) ) )

)

)

)

) )

)

)

where the bracket notation Rfor inverse Fourier transform is B f Br } 1/(2π)2 d2K exp(iK B Br )f(K B ). ^y0 and ^y1 {f(K B );K are operators the same as in eq 7, and ^y2 is Z 1 ω0 ω 2 0 1 d ^y2 K 2 ω0 LΩ þ 1 ω LΩ þ 1 ð2πÞ2 ω ω, 0 ω0 þ ω0 ðω LΩ þ 1Þ ω , 0 LΩ þ 1 0

0

)

) )

)

)

)

K B 3ð K BÞ B K 1 ω0 ðω LΩ þ 1Þ ω , 0 LΩ þ 1 0

) )

)

jK B K B j2 ω0 ðω , 0 LΩ þ 1Þ )

1/2

0

0

)

#

)

)

)

B K K B 3ð K BÞ ω LΩ 2ω0 ðω0 LΩ þ 1Þ ω LΩ þ 1 )

where j(t) is the current density and i is (1) . The local admittance density can be evaluated with the knowledge of local interfacial current density. The analytical treatment to obtain the local interfacial current density expression under potentiostatic condition for the mathematically isomorphic problem of the quasi-reversible charge transport problem to arbitrary surface is described in refs 36 and 42. With the knowledge of the Laplace transform technique and using eq 6, one can evaluate the local admittance density. The local admittance density expression (Y(ω,ζ(r B ), B ))) for an arbitrary roughness surface profile, ζ(r can be represented as the sum of various order terms in surface roughness, i.e., ^y0, ^y1, and ^y0 2.36,42,44 One can write the admittance density to a rough electrode as the following

)

Z ω20 d2 r f^y0 ð2πÞ2 δð K BÞ Y ðω, ζð B r ÞÞ ¼ RΩ z ¼ 0

)

’ PERTURBATION SOLUTION IN SURFACE PROFILE For calculating the admittance, we need to define the concentration field at the electrode/electrolyte interface. This concentration field is the function of the boundary profile; therefore, we can say that roughness is sensed by the concentration field in electrochemical measurements. We use the methodology given in refs 36,42, and 44. Using this methodology, one can define the current density and hence the admittance at the electrode. Admittance is related to the current density through the following relation Z iω ¥ dt eiωt jðtÞ ð6Þ yðωÞ ¼ η0 0

profile (ζ^(K B ))

)

pseudoreaction penetration length depends upon experimental quantities such as the diffusion coefficient, resistivity of the solution, and separation between the working and reference electrode.

where the integral is performed over the surface, i.e., z = 0 plane. It is necessary to mention here that ^y20 and ^y2 are similar except for the numerical coefficient of the last term. These operators consist of diffusion length (ω1 0 ) and ohmic loss characteristic length (LΩ), while in eq 11 the roughness profile of the interface is emphasized to highlight the mathematics involved in an amenable manner. Explicitly, the first operator defines the effect of diffusion and solution resistance phenomenon for a smooth surface, whereas the first-order and the second-order terms take into account the surface modulation or the fluctuations around the reference plane. It is worth realizing that these expressions are valid for an arbitrary surface profile.

’ RANDOM ELECTRODE MORPHOLOGY Any real surface in general does not have deterministic surface profiles. By most means of synthesis, it is not achievable to have deterministic profiles. Some workers, however, target the problem at such surfaces, ignoring the fact that real surfaces always have some randomness. An idealized rough surface composed of a regular “array” of bell-shaped protrusions conferring a significant surface roughness, is claimed by Compton and co-workers.47,48 However, the experimental procedure in ref 47 does not result in idealized structures, but rather a very pronounced randomness is seen in the AFM image of the rough GC electrode. It is emphatic that randomness is a ubiquitous property at the electrode surface, and the modeling must provide for that. Random surfaces have long been observed in electrochemistry. They are the result of several processes like electrodeposition,49,50 electropolishing, and electrocleaning. These random surfaces can be characterized as fractal and nonfractal. Often fractured, corroded, and electrodeposited surfaces are characterized as fractals.7,5053 Other surfaces on polishing, on scratching, and on treatment gain greater degrees of nonfractal randomness. For a random electrode, the surface profile is characterized by a centered Gaussian field with the statistical properties given by 12234



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)

mean and correlation. The ensemble averaged value of the 54 random surface (ζ(r B )) is ð12Þ

)

Æζð B r Þæ ¼ 0 and the two point correlation function is 0

0

ð13Þ

)

)

)

)

Æζð B r Þζð B r Þæ ¼ h2 Wð B r B r Þ

Æ^ ζð K B Þæ ¼ 0 )

ð14Þ

0

0

)

)

)

)

)

Æ^ ζð K B Þ^ ζð K B Þæ ¼ ð2πÞ2 δð K B þ K B ÞÆj^ ζð K B Þj2 æ

ð15Þ )

)

)

)

)

)

)

where δ(K B ) is a Dirac delta function in wave-vector K B and R Æζ^(|K B )|2)æ = h2 d2r eiKB 3 Br W(rB ) . It is to be brought to notice that at a random surface it is not the local quantities but rather average quantities represented by the ensemble average over the surface configuration space which are quantities of interest. The average admittance is given by

ð18Þ

The statistically isotropic surfaces of the finite fractal form have four surface morphological features of roughness on which the admittance response is analyzed here. The four morphological features are: the surface roughness amplitude (μ), fractal dimension (DH), lower cutoff length (l ) and upper cutoff length L. μ is related to the topothesy of fractals;66 its units are cm2DH3; and μ f 0 means there is no roughness. The lower length scale cutoff (l ) is the length above which the surface shows fractal behavior. The ensemblaveraged admittance expression for a finite self-affine isotropic fractal surface for problems involving sizable ohmic losses for solution resistance can be obtained by substituting eq 18 in eq 16 and solving resultant the integral (see Appendix A). The ensembleaveraged admittance expression for a finite (isotropic) fractal power spectrum can be expressed as ÆY ðωÞæ ¼ Yp ðωÞ½1 þ ψ1 ðωÞ þ ψ2 ðωÞ

ð19Þ

where ψ1(ω) and ψ2(ω) are defined below ¥

dK K 0

ψ1 ðωÞ ¼ ½ðAðl Þ AðLÞÞ ðH1 ðl Þ H1 ðLÞÞ " μω0 LΩ l 2ðδ þ 1Þ L2ðδ þ 1Þ ψ2 ðωÞ ¼ 4πðδ þ 1Þð1 þ LΩ ω0 Þ 2 H2 ðl Þ H2 ðLÞ ð20Þ 1 LΩ ω0

K 2 LΩ ω ω0 Æj^ζð K B Þj2 æ þ 1 þ ω LΩ 2ð1 þ ω0 LΩ Þ )

Z

#

)

ω0 2π

#

)

¼ Yp ðωÞ 1 þ

"

)

"

) )

ÆY ðωÞæ

Æj^ζðjKjÞj2 æ ¼ μjKj2DH 7 , 1=L e jKj e 1=l

)

)

)

)

)

)

where Æ 3 3 3 æ denotes ensemble average over various possible random surface configurations; h2 is the surface roughness amplitude; and W is the normalized correlation function. The correlation function gives the measure of rapidity of variation of surface and varies between 0 and 1. For slowly varying surfaces, the correlation between two positions on the surface ζ(r B ) and 0 0 ) is nonzero for even large values of separation |r ζ(r B Br |. B The correlation function vanishes on increasing the relative 0 distance |r B Br |. This correlation function contains information about the surface morphological characteristics such as area, mean square height, slope, curvature, and correlation length. Averages in the Fourier plane are

experimentally realized fractured surfaces, 60 rough film electrodes, 15 porous electrodes,61 especially for hydrogen storage, 15,62 and electrodeposited surfaces,49,63,64 columnar structured arrays,51,52 and roughened or deformed surfaces 53 as examples of random fractal surfaces. These fractal boundaries exhibit statistical self-resemblance over all length scales and can be described using the power law power spectrum.22,23,59,65 For a realistic electrode surface, we use the bandlimited power spectrum to circumvent the mathematical difficulty of nondifferentiability and nonstationarity. These band-limited fractals have a lower and an upper length scale cutoff. The quantity we seek for an isotropic fractal surface is the power spectrum as the input for equation eq 16. This power spectrum is given by

ð16Þ Yp(ω) is the admittance at a smooth electrode Yp ðωÞ ¼ A0 Γ

26,27

ðiωDÞ1=2 1 þ ΓðiωDÞ1=2 RΩ

ð17Þ

Γ denotes the diffusion capacitance per unit volume and is 01 written as: Γ = (n2F2)/(RT [C01 O þ CR ]). An arbitrary power spectrum can have a band-limited fractal. A narrow band-limited fractal is practically a nonfractal or one with a single correlation length dominated nonfractal random profile.30,32,33,35,42,55 The results here are analyzed in detail for the finite fractal electrodes.

’ FINITE FRACTAL SURFACE Fractals were introduced by Mandelbrot56to describe disordered objects, using fractional dimensions. The concept of fractals is wellknown in electrochemistry. A rough electrode surface can often be characterized by the concept of fractals. To capture the complexity arising from the irregular interfaces (i.e., rough, porous, and partially active interfaces), one often uses the concept of fractals.56,57 The fractal irregularities are usually understood, in particular, in terms of self-similar5658 or in general as self-affine2224,31,32,35,5659 fractals. In electrochemistry, the concept of the band-limited fractal electrodes has been well established and accepted. Many workers have

where A(), H1(), and H2() are given as AðuÞ

" # μω20 1 D DL2Ω 2δ ¼ u F1 δ; , 1; δ þ 1; 2 , 2 2 u iω u ðD iωL2Ω Þ 4πδð1 LΩ ω0 Þ

ð21Þ " # μω20 DL2Ω 2δ u 2 F1 1, δ; δ þ 1; 2 H1 ðuÞ ¼ 4πδð1 LΩ ω0 Þ u ðD iωL2Ω Þ

ð22Þ "

H2 ðuÞ ¼ u

2ðδ þ 1Þ

DL2Ω 2 F1 1, δ þ 1; δ þ 2; 2 u ðD iωL2Ω Þ

#

ð23Þ where δ = DH 5/2; F1 is the first Appell function; and 2F1 is the hypergeometric function. Analytic continuations of hypergeometric functions are wellknown; however, in the case of Appell’s (F1), they are not as 12235



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simple as their one-variable partners.68 There are a variety of transformations of the F1 function, ranging from the simplest cases, where one of the variables or parameters is zero, to the analytic continuations and transformations to series of hypergeometric functions67 for the wider range of arguments. So, one can get the intermediate frequency expansion yielding Y ðωÞ ¼ Yp ðωÞ½1 þ ψðl Þ ψðLÞ ðH1 ðl Þ H1 ðLÞÞ þ ψ2 ðl Þ ψ2 ðLÞ

ð24Þ where ψ(l ) and ψ(L) are ψðl Þ μω0 Γð3=2Þ ðω2 l 2 Þr l ð2δ þ 1Þ ¼ Γð3=2 rÞΓðr þ 1Þ 0 4πð1 LΩ ω0 Þ " # 1 2δ þ 1 2δ þ 3 L2Ω r; r; 2 2 F1 1, 2δ þ 1 2 2 l ð1 ω20 L2Ω Þ r 2 ð25Þ

∑

ψðLÞ ¼

μω20 Γð3=2Þ 1 L2δ Γð3=2 rÞΓðr þ 1Þ ðω20 L2 Þr 4πð1 LΩ ω0 Þ " # 1 L2Ω ð26Þ 2 F1 1, δ þ r; δ þ r þ 1; 2 ðδ þ rÞ L ð1 ω20 L2Ω Þ

∑

0 B 2ðδ þ 1Þ Bu μω0 LΩ B ψ2 ðuÞ ¼ 4πðδ þ 1Þð1 þ ω0 LΩ ÞB 2 @ " 2ðδ þ 1Þ u 2 F1

#1 L2Ω 1, δ þ 1, δ þ 2, 2 u ð1 ω20 L2Ω Þ C C C C ð1 ω0 LΩ Þ A ð27Þ

This representation has been adopted to circumvent the evaluation of Appell’s function, by using the hypergeometric function which ensures good convergence at most arguments. This equation includes the fractal features dependent power law as well as the contribution from the resistive losses at the electrode. The resulting admittance behavior is an interplay between the interfacial potential, the solution resistance, and the roughness features of the interface. The total admittance is the summation of smooth surface response and an anomalous excess flux due to fractal roughness. Here δ = DH 5/2; F1 is Appell’s function;68 and 2F1 is the hypergeometric function69 which can be numerically evaluated with the help of Mathematica software. The admittance expression shown in eq 24 is a function of four fractal morphological characteristics of roughness, say (μ, DH, l , L), and two phenomenological lengths, say (LD, LΩ). Equation 24 extends the conventional result of the admittance on the smooth surface electrode to the fractally rough electrode with solution resistance effects, providing the insight into the frequency regimes which each of these phenomenological length scales would affect. It becomes important to mark the isomorphism

Figure 2. (a) Effect of electrolyte resistance (RΩ) on the log of impedance magnitude, (b) phase vs log of frequency. (c) Effect of RΩ on the Nyquist plot. RΩ is varied as 10, 20, 50 Ω (lower to upper curves). The plots are generated by using fixed parameters: l = 50 nm, DH = 2.25, projected area (A0 = 1 cm2), diffusion coefficient (D = 5 106 cm2/s), and concentration (CO = CR = 5 mM) are used in our calculations.

with the quasi-reversible charge transfer. It is to be noticed that solution resistance is a bulk phenomena, whereas charge transfer is an interfacial process. However, for the quasi-reversible charge transfer there exists a similar length scale as LΩ, viz., heterogeneous reaction layer (penetration) thickness (LCT). For the quasi-reversible charge transfer process, LCT is dependent on heterogeneous charge transfer kinetics.45 The mathematical parallelism between two physically distinct processes induces the pseudoquasireversibility in the system.

’ RESULTS AND DISCUSSION To demonstrate the effect of ohmic drop on an anomalous impedance response at a rough electrode, we consider the changes in the impedance with respect to different parameters affecting the surface roughness and the solution resistance of the electrolyte. The resulting response resonates with the variation of these parameters to provide a better understanding of both finite fractal roughness as well as uncompensated solution resistance. Figure 2(a) shows the effect of solution resistance on the impedance response in the presence of surface roughness. We find that the impedance response is affected significantly at higher frequencies with the variation of solution resistance (RΩ), thus delaying the onset of the diffusion-controlled process. As the resistance of the solution increases, the impedance increases. 12236



Figure 3. (a) Effect of fractal dimension (DH) on the double logarithm plot of impedance and (b) phase and log of frequency. DH is varied as 2.2, 2.25, 2.3 (lower to upper curves). (c) Effect of DH on the Nyquist plot. The plots are generated by using fixed parameters: l = 30 nm, projected areas (A0 = 1 cm2), diffusion coefficient (D = 5 106 cm2/s), RΩ = 5 Ω, and concentration (CO = CR = 5 mM) are used in our calculations.

The marked presence of plateaus at the higher-frequency regime is attributed to the presence of solution resistance. These plots follow a power law behavior in the intermediate frequency domain. One can see the effect of dynamic morphological features in this intermediate region. It is important to stress here that the mere subtraction of ohmic loss does not suffice, accounting for the response. One can clearly incept the idea that it is a dynamical convolution of both the effects, i.e., morphology and solution resistance, from the above depiction. One process gives way to others at certain frequencies marked by the inner and outer crossover frequencies. The role of solution resistance thus becomes imperative in characterizing the crossover frequency. As one can see in the plots, the higher the value of RΩ, the crossover frequency shifts to the lower values, following a power law in the intermediate region. To demonstrate the effect of ohmic drop on an anomalous impedance response at a rough electrode, we consider the changes in the impedance with respect to different parameters affecting the surface roughness and the solution resistance of the electrolyte. The resulting response resonates with the variation of these parameters to provide a better understanding of both finite fractal roughness as well as uncompensated solution resistance. Figure 2(a) shows the effect of solution resistance on the impedance response in the presence of surface roughness. We find that the impedance response is

ARTICLE

Figure 4. (a) Effect of lower length scale of fractality (l ) on the double logarithm plot of impedance, (b) phase and log of frequency, and (c) effect of l on the Nyquist plot. The plots are generated using fixed parameters: l = 25, 50, 100 nm (from upper to lower curves), DH = 2.3, projected areas (A0 = 1 cm2), diffusion coefficient (D = 5 106 cm2/s), RΩ = 10 Ω, and concentration (CO = CR = 5 mM) are used in our calculations.

affected significantly at higher frequencies with the variation of solution resistance (RΩ), thus delaying the onset of the diffusioncontrolled process. As the resistance of the solution increases, the impedance increases. The marked presence of plateaus at the higher-frequency regime is attributed to the presence of solution resistance. These plots follow a power law behavior in the intermediate frequency domain. Figure 2(c) shows the Nyquist representation for the anomalous Warburg behavior with inclusion of solution resistance contributions. Here, one can see the deviation from the Warburg line (dot-dash line) and planar response with ohmic (dash line) contributions. The increase in RΩ yields the increasing value of Z0 on the real axis. The response shows a knee depicting the crossover between the ohmic-controlled regime and the roughness-controlled regime and then asymptotically merges with the Warburg line. The curves crossover to a resistance-controlled regime to an anomalous regime, i.e., Z0 (ωi) = Z00 (ωi), at the inner crossover frequency ωi ≈ D/(max[l 2 , L2Ω]). The region below this (the higher frequencies) is controlled by the ohmic factors or resistance of the electrolyte. These plots roll down sharply to the real axis marking the value which is a product of the roughness factor and the uncompensated resistance. The region above the crossover knee region shows the roughness control at 12237


The Journal of Physical Chemistry C the intermediate frequency. These curves then asymptotically merge with the Warburg line, viz., Z(ω) ≈ ZW(ω), at the lower frequencies, i.e., ω < ωo ≈ D/(h2 þ L2Ω). Our plots show deviation from the Warburg behavior characteristic of the rough electrodes in the intermediate frequency. To illustrate this, we investigate graphically the effect of various morphological features of roughness with fractal nature along with solution resistance in the impedance spectra. For a fractal surface, there are three morphological features, which dominantly control the impedance response for a fractally rough surface, namely, the fractal dimension (DH), lower length scale of fractality (l ), and proportionality factor of the power spectrum (μ). Figure 3(a) shows the effect of fractal dimension DH on the impedance plot in the presence of some sizable resistance offered by the electrolyte at a finite self-affine fractal electrode. The quantity DH is the Hausdorff dimension of the surface and is a direct measure of the surface irregularity. In the plots, it can be that as the fractal dimension increases the impedance decreases and becomes parallel at higher frequencies. At lower frequencies, the plots merge with the classical Warburg response. At the intermediate frequencies, the power law impedance response follows, characteristic of the dynamic response to surface roughness. The slope of the power law region changes with the varying fractal dimension, marking the strong dependence of anomalous response on the fractal dimension (DH). The outer cutoff frequency is dependent on the solution resistance, and the inner cutoff frequency is affected by the fractal dimension. In the electrochemical context, the sensitivity toward roughness is affected at higher frequencies in the presence of uncompensated resistance. The phase plot (Figure 3(b)) shows the effect of fractal dimension as φ(ω) > 45 in the intermediate frequency regime. This deviation from 45 increases with the increase in fractal dimension. At the lower frequencies, the phase angle tends toward 45, and at the higher frequency it goes to 0 characteristic of the solution resistance. The surface roughness amplitude (μ) follows the similar behavior as that of the fractal dimension (DH). μ is a function of the topothesy of the fractal. In the physical sense, it can be related directly to amplitude of roughness. Figure 3(c) shows the influence of fractal dimension on the Nyquist representation. The increase in DH yields the overall decrease in the resistive component. The response shows two crossover points: one depicting the high-frequency crossover to dynamic roughness effects and another low-frequency crossover to the Warburg behavior. An increase in DH yields a decrease in the real component in the impedance. The higher-frequency effects are resonated in the lower portion of the curve near the x-axis where solution resistance has a dominant influence. The inner crossover frequency is the point at which Z0 (ωi) = Z00 (ωi) and ωi ≈ D/(max[l 2 ,L2Ω]). This remains the same for all the values of DH. The magnitude for each curve is, however, governed by the fractal dimension at this crossover frequency. The region below the knee (the higher frequencies) is controlled by the ohmic factors or resistance of the electrolyte. The slope changes due to variation in DH in the intermediate frequency owing to values of fractal dimension. All the curves merge ahead to the Warburg response, i.e., Z(ω) ≈ ZW(ω), at the lower frequencies when the ω e D/(h2 þ L2Ω). Figure 4(a) shows the effect of varying the lower length scale cut off (l ) on the impedance response. The lower length scale of fractality (l ) can be identified as the extent of the coarseness of the surface. It is seen that increasing l increases the average impedance. The effect of this morphological parameter is, however, propagated to a larger frequency window in the plots.

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Figure 5. Fractal to nonfractal response transition is obtained by increasing l . The plots are generated using fixed parameters: DH = 2.25, projected areas (A0 = 0.1 cm2), diffusion coefficient (D = 5 106 cm2/s), RΩ = 5 Ω, h = 5.4 μm, L = 1 μm, and concentration (CO = CR = 5 mM) are used in our calculations. The fractal case has small value of l , e.g., l = 30 nm, while a nonfractal case has a large value of l , e.g., l = 67 nm.

It is evident that l not only just affects the higher frequencies but also influences the intermediate power law response to a larger extent showing that the impedance is strongly dependent on the lower length scale cut off. The change in the slope of the curve is, however, enhanced in the presence of electrolytic resistance in the higher-frequency ranges. The phase plot (Figure 4(b)) also shows a higher sensitivity toward the lower cutoff length. The deviation from the planar response is quite significant in the intermediate frequency regime. In the phase response as the lower length scale decreases, it follows that more and more detailing percolates to the finer length scales and a greater deviation from 45, yielding approximately constant phase angle type of response in the intermediate region. 12238


The Journal of Physical Chemistry C Figure 4(c) shows the influence of the lower cutoff length on the Nyquist representation. The increase in l yields the increase in value on the real axis but decrease in the imaginary component. The response shows two crossover ones depicting crossover to dynamic roughness effects and another crossover to classical Warburg’s behavior. Decrease in l yields a decrease in the real part of impedance. Various values of l have a more penetrating effect on the response, marking the importance of finite fractals. The curves crossover from a solution resistance controlled regime to anomalous Warburg’s regime, when Z0 (ωi) = Z00 (ωi) at the inner crossover frequency ωi ≈ (D)/(max[l 2 , L2Ω]). This marks the importance of the relative length scales. Only when l is greater than the phenomenological length scale LΩ, the l affects the crossover frequencies. The region below the knee (higher-frequency region) is controlled by the ohmic factors or resistance of the electrolyte. The slope does not change much due to variation in l in the intermediate frequency; rather, the curves are transcended owing to the greater coarseness in the surface. All the curves merge ahead to the Warburg response, i.e., Z(ω) ≈ ZW(ω), at the lower frequencies when the ω e D/(h2 þ L2Ω). Figure 5(a) presents an interesting situation wherein the crossover from a fractal to a nonfractal surface is achieved using similar morphological features. Using these features, one can achieve a surface where the distinction between the fractal and nonfractal features diminishes. The plots are obtained for the diffusion-controlled regime with low solution resistance. Dotted black lines in the magnitude and phase plot representing the planar behavior with the same magnitude of solution resistance are provided for the reference. The dot-dash (blue) curves represent the nonfractal type behavior, and the solid line represents the typical fractal behavior. For both fractal and nonfractal surfaces, the mean square width of the interface (h2), fractal dimension (DH), and upper cutoff length (L) are kept constant. The phase plot (Figure 5(b)) can give out more appropriate information in the frequency window of interest, where for a fractal phase angle gain happens with the surface of the same mean square width and correlation length (L). However, for a nonfractal electrode the 45 phase response is followed for a higher range of frequency. In the phase response, there is a delay from the crossover of various phenomenological regimes, i.e., from being diffusion controlled to being ohmic controlled. These crossover frequencies are also different for the fractal and nonfractal response despite having the same mean square height and correlation length. The magnitude plot shows that a nonfractal response merges with the planar response faster than the fractal response. The transition between a fractal and a nonfractal can be attained for various parameters. For the Nyquist representation (Figure 5(c)), the contrast between fractal and nonfractal response is marked. The red line is the planar response with ohmic losses, and the black dotted line is the Warburg line. The nonfractal response (dot-dash line) remains close to the Warburg behavior, while the fractal response (solid line) goes farther away from the Warburg marking the greater influence of the fractal roughness. The main players here are the ratio (F) of the upper length scale cutoff (L), the lower length scale cutoff (l ), and the mean square width (h2) of the interface. For the surfaces with F ≈ 10 one can expect a crossover from a nonfractal response to a fractal response for sufficiently low solution resistance. However, this crossover is strongly dependent on the fractal dimension of the surface and curtailed with moderate value of RΩ.

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’ CONCLUSION An ab initio approach is used to derive the relationship between the impedance response under the influence of the diffusion process and solution resistance effects at rough electrodes. The perturbation analysis up to second order in roughness profile is obtained for detailed analysis of a stationary random surface. The relationship to study the phenomenon of uncompensated solution resistance at deterministic and stochastic electrode geometries is derived. The explicit relationship between the power spectrum of roughness and the impedance response in the presence of typical solution resistance is elucidated. Electrodes with finite fractality in roughness are ubiquitous, hence the analysis for these geometries is presented in detail. The impedance response at a fractal electrode is affected by the morphological features (signified by fractal dimension (DH), surface roughness amplitude (μ), as well as two lateral cutoff lengths in the case of finite fractals, viz, l , L). The following points have been established in the paper: (1) For ideal fractals, it is sufficient to describe the surface using DH and μ. To account for the finiteness of the fractal two length scales: the large (L) and small (l ) lateral cutoffs are also necessary to define surface statistics. These two lateral cutoffs can be related to the coarseness of the surface (l ) and the correlation length (L) for a random surface. These signify that beyond these cutoffs there is no fractality at the surface. The fractal with limits L f ¥ and l f 0 shows an ideal fractal. (2) In the presence of the uncompensated solution resistance (RΩ), the Nernstian charge transfer yields a similar response as for the quasi-reversible charge transfer, due to the mathematical isomorphism in such boundary value problems. The extent of this pseudoquasi-reversibilty is contained in the pseudoreaction penetration length (LΩ) which is a direct measure of RΩ. (3) The high-frequency impedance is dependent on the uncompensated resistance along with gross roughness factor. This observation is in contrast with the earlier results35,37 for purely diffusion-controlled theory where at high frequencies we expected the classical Warburg response (Z(w) (iω)1/2). The anomalous impedance response is dominantly governed by the finite fractal features at the intermediate wavenumbers. As the diffusion layer ((D/ω)1/2) grows out of the pseudoreaction penetration length (LΩ RΩ), the dynamic roughness features come into play and yield the anomalous Warburg response. Therefore, the intermediate frequency regime resonates the effects of DH, l , and μ in convolution with resistive parameters contained in (LΩ). The lowerfrequency regime represents the classical Warburg response due to large diffusion length. (4) The crossover frequency from being a pseudoquasireversible reaction to the roughness-controlled response is dictated by the magnitude of the uncompensated resistance present in an otherwise reversible system as well as axing the anomalous response regime. In the presence of the sufficient amount of solution resistance (LΩ > (D/ω)1/2; for very low ω) there is no anomalous behavior present even for a sufficiently rough electrode. With the increase in RΩ, there is an increase in LΩ, which causes a shift in the outer crossover (ωo) frequency toward lower frequencies. Inner crossover frequency ωi 12239



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’ APPENDIX A. Calculation of Fractal Roughness Admittance. Substitution of the power spectrum shown in eq 18 into eq 16 gives rise to the integral expression. The first part of the integral of eq 16 after substitution of the power spectrum, which is further split in to two limits, can be expressed as

μK 2DH 6 ðω ω0 Þ dK 1 þ ω LΩ

μK 2DH 6 ðω ω0 Þ dK 1 þ ω LΩ )

1=L

μK 2DH 6 ðω ω0 Þ dK 1 þ ω LΩ

)

Z

)

0

ω0 2π

0

ðA.1Þ

)

)

where, ω = (ω20 þ K2)1/2 and ω0 = (iω/D)1/2. To solve the above integral, we substitute y = K l in the first part of eq A.1. )

)

)

)

ðA.2Þ Equation A.2 can be further generalized to make the integral solvable and is given as ω0 5 2DH l 2π Z

0

1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μy2DH 6 ð ðω20 þ y2 =l 2 Þ ω0 Þð1 ðω20 þ y2 =l 2 ÞLΩ Þ dy 1 ðω20 þ y2 =l 2 ÞL2Ω

ðA.3Þ Simplifying eq A.3, we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ω0 5 2DH 1 μy2DH 6 ðω20 þ y2 =l 2 Þ l dy 2π 1 ðω20 þ y2 =l 2 ÞL2Ω 0 Z 1 2DH 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðω20 þ y2 =l 2 ÞLΩ ω2 μy dy þ 0 l 5 2DH 2 2π 1 ðω0 þ y2 =l 2 ÞL2Ω 0 Z ω0 5 2DH 1 μy2DH 6 ðω20 þ y2 =l 2 ÞLΩ l dy 2π 1 ðω20 þ y2 =l 2 ÞL2Ω 0 Z 1 ω2 μy2DH 6 0 l 5 2DH dy ðA.4Þ 2 2 2 2 2π 0 1 ðω0 þ y =l ÞLΩ Rearranging terms into simpler form, we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z μω0 ð1 þ LΩ ω0 Þ 5 2DH 1 y2DH 6 ðω20 þ y2 =l 2 Þ ! dy l 2π 0 y2 L2Ω 2 2 ð1 ω0 LΩ Þ 2 l Z 1 2 2DH 6 μω ð1 þ ω0 LΩ Þ 5 2DH y ! dy l 0 2π 0 y2 L2Ω 2 ð1 ω0 LΩ Þ 2 l Z 1 2DH 4 μω0 LΩ 3 2DH y ! dy ðA.5Þ l 2π 0 y2 L2Ω 2 2 ð1 ω0 LΩ Þ 2 l The solution of the first part of the above equation can be obtained by using the following steps pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z μω0 ð1 þ LΩ ω0 Þ 5 2DH 1 y2DH 6 ðω20 þ y2 =l 2 Þ ! dy l 2π 0 y2 L2Ω 2 2 ð1 ω0 LΩ Þ 2 l

)

1=l

)

Z

)

ω0 2π

)

¼

)

1=L

)

)

1=l

)

Z

)

ω0 2π

The second part of eq A.1 can be similarly calculated, then we have Z 2D 6 ω0 1=l μK H ðω ω0 Þ dK 1 þ ω LΩ 2π 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ω0 5 2DH 1 μy2DH 6 ð ðω20 þ y2 =l 2 Þ ω0 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l ¼ dy 2π 1 þ ðω20 þ y2 =l 2 ÞLΩ 0 )

≈ D/(L2Ω þ h2) (for L g h) or D/(L2Ω þ L2) (for L e h) is observed toward lower frequency, and outer crossover frequency ωo ≈ D/L2Ω is observed toward high frequency. For the lower-frequency limit, i.e., ω e ωi, the impedance corresponds to classical Warburg behavior. (5) For cases where the anomalous region is observed, phase response can yield crossover from a characteristic fractal to a nonfractal response, and depending upon the degree of interplay of various length scales involved, viz., h/L e 1 and LΩ < LD and F = l /L j 10, one can expect the nonfractal behavior. The anomalous region here is shortened in the magnitude response and leads to loss of broad maxima in the phase response for the physically relevant frequency window. The phase gain is much more prominent for fractal than nonfractal for the electrode with the same width. (6) Nyquist’s representation offers a yet comparable contrast with the anomalous Warburg behavior with and without the ohmic loss phenomena. The influence of roughness and solution resistance can alter the traditional understanding of such plots. For fractal interfaces, they show that the deviation is strongly dependent on the fractal morphological parameters. Plots slowly merge with the smooth electrode response as the (D/ω)1/2 grows out of h. The plots intersect the real axis at the value dictated by the solution resistance and roughness factor. The intermediate section points out the deviation characteristic to the dynamic roughness factor merging with the smooth electrode response. (7) The important thing to note is that impedance measurement often mixes RCT with RΩ, and subtracting RΩ as a usual practice does not suffice in the equivalent circuit analysis for actual calculation of kinetic parameters and elimination of uncompensated resistance. Finally, one can say that the theory presented here succeeds in offering an understanding of impedance response of diffusion-controlled processes under resistive experimental conditions at finite fractal electrodes and their more careful interpretation in terms of surface roughness sensitivity and anomalous response.

Putting y2 = m and rearranging the term, the above integral becomes Z μω0 ð1 þ LΩ ω0 Þ 5 2DH 1 DH 7=2 2 l m ðω0 þ m=l 2 Þ1=2 4π 0 !1 mL2Ω 2 2 ð1 ω0 LΩ Þ 2 dm ðA.6Þ l 12240



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Z 1 ΓðcÞ ma 1 ð1 mÞc a 1 ð1 mz1 Þb1 ð1 mz2 Þb2 dm ΓðaÞΓðc aÞ 0

ðA.7Þ mDH 7=2

0

!1 ð1 mÞ0 dm

Mapping the above integral with the integral shown in eq A.7, we have μω20 5 2DH l F1 DH 5=2; 1=2, 1; 4πð1 ω0 LΩ ÞðDH 5=2Þ 1 L2Ω DH 3=2; 2 2 , ðA.8Þ ω0 l ð1 ω20 L2Ω Þl 2 Solving the second part of eq A.5 using a similar methodology, we have Z

y2DH 6

1 0

y2 L Ω ð1 ω20 L2Ω Þ 2 l

2

! dy

) )

μω20 ð1 þ ω0 LΩ Þ 5 2DH l 2π

solution which possess hypergeometric 2F1 function instead of Appell F1 function, i.e. μω20 l 5 2DH 2 F1 4πðDH 5=2Þð1 LΩ ω0 Þ "

LΩ 2 1, DH 5=2; DH 3=2; 2 l ð1 L2Ω ω20 Þ

# ðA.9Þ

y2 L2 ð1 ω20 L2Ω Þ 2Ω l

! dy

)

0

y2DH 4

1

) )

Z

)

μω0 LΩ 3 2DH l 2π

)

Similarly, with the third part of eq A.5, we have

)

mL2Ω 1 ð1 ω20 L2Ω Þl 2

ð1 þ m=ω20 l 2 Þ1=2

1

)

Z

)

μω0 ð1 þ LΩ ω0 Þ 5 2DH ω0 l 4π ð1 ω20 L2Ω Þ

)

¼

) )

F1 ½a; b1 , b2 ; c; z1 , z2

’ LIST OF SYMBOLS A0 Projected area Bulk concentration of the species, e.g., R = O, R C0R concentration of the reduced species CR Change in concentration of species δCR Diffusion coefficient of the ith species DR Fractal dimension DH f F/RT F Faraday’s constant Appell’s hypergeometric function F1 Hypergeometric function 2F1 h Standard deviation of the surface height fluctuation j(t) Current density [K2x þ K2y ]1/2 K K B Vector (Kx,Ky) Pseudo reaction penetration length LΩ Diffusion length ((D/iω)1/2) LD n Number of electrons transferred ^n Normal vector drawn in outward direction r Vector (x,y,z) B Vector (x,y) r B Uncompensated solution resistance RΩ R Gas constant t Time T Temprature W Normalized correlation function y(ω) Admittance density Y(ω) Admittance Yp(ω) Smooth electrode admittance z Coordinate representing distance away from electrode ∂/∂x RPartial R derivative wrt x R 2 dx dy dr B δ(K B ) Dirac delta function in wave-vector K δ DH 5/2 η(t) Potential perturbation Amplitude of potential perturbation η0 ι (1)1/2 φ(ω) Phase angle F Ratio l /L Resistivity of the electrolyte Fe ω Angular frequency Inner cutoff angular frequency ωi Outer cutoff angular frequency ωo ζ(rB ) Arbitrary surface profile ((iω)/(D))1/2 ω0 ω (ω20 þ K B2)1/2 ω,0 (ω02 þ K B2 K B 0 2)1/2 ζ^(K B ) Fourier transform of the arbitrary surface profile î∂x þ ^j∂y 3 l Lower cutoff length scale of a band-limited fractal le Distance between the WE (reference plane z = 0) and the RE Γ Specific diffusion capacitance Γ() Gamma function μ Proportionality constant of power spectrum or strength of fractality )

Taking out some common terms makes the above integral simple, which can be mapped into the integral of the Appell function shown below

μω0 LΩ l 3 2DH 2 F1 4πðDH 3=2Þð1 L2Ω ω20 Þ "

L2Ω 1, DH 3=2; DH 1=2; 2 l ð1 L2Ω ω20 Þ

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

)

and the solution is

# ðA.10Þ

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