Theory of Generalized Cottrellian Current at Rough ... - ACS Publications

10066

J. Phys. Chem. C 2010, 114, 10066–10076

Theory of Generalized Cottrellian Current at Rough Electrode with Solution Resistance Effects Shruti Srivastav and Rama Kant* Department of Chemistry, UniVersity of Delhi, Delhi-110007 ReceiVed: March 25, 2010; ReVised Manuscript ReceiVed: April 10, 2010

A theory for effect of uncompensated solution resistance on Nernstian (reversible) charge transfer at an arbitrary rough electrode is developed. The significant deviation from the Cottrellian behavior is explained, as arising from the resistivity of the solution and geometric irregularity of the interface. Results are obtained for various electrode roughness models, viz., (i) deterministic surface profile, (ii) roughness as random surface profile with known statistical properties, and (iii) random functions with limited self-affine fractal properties. Expressions for concentration, current density, and total current transients have a systematic operator structure in Fourier transformed (deterministic) surface profile function. For a randomly rough electrode, the statistically averaged response is obtained by ensemble averaging over all possible surface configurations. An elegant mathematical formula between the average electrochemical current transient and surface structure factor of roughness is obtained. Realistic fractal roughness is characterized as self-affine scaling property over limited length scales. Limiting behavior in short time, is attributed to resistive effects of the electrolyte, roughness factor and surface curvatures. In the intermediate time, the anomalous power law behavior is attributed to the diffusion length weighted spatial frequency features of roughness. Our results help with the quantitative understanding of generalized Cottrellian response of moderately supported electrolytic solution at rough electrode/electrolyte interface. Introduction Transient response in potential controlled experiments is a standard procedure which can yield information about many electrochemical processes and several kinetic parameters. However, solution resistance can have a serious effect on electrochemical measurements as in polarographic measurements,1,2 potential step technique,3,4 rotating disk electrode,5,6 and PEMFC.7,8 Many users underestimate the ohmic drop, regarding it often merely as a marginal problem, which only has to be taken into account in experiments with very high currents or low conductivity of the electrolyte. However, in many applications involving complex interfacial systems, it is desirable or essential to use reduced or zero levels of added electrolyte, posing a challenging fundamental problem. If the ohmic drop in the electrolyte between the working electrode and the reference electrode cannot be neglected, the current-electrode potential relationship differs significantly, sometimes qualitatively, from the measured current-voltage relationship.9 It has been established that geometry of the electrodes and electrode assembly plays an important role in the ohmic drops.10-12 Experimentalists have been trying for a long time to minimize the ohmic (iR) drop using various methods. In general, the inert electrolytes and highly conducting supporting media are widely used. Other precautions include minimizing the distance between the working electrode and Luggins’ capillary.13 These precautions suppress the migration content; however, they do not eliminate the resistance losses and there is still uncompensated resistance even in a supported medium. Studies have been carried out to understand the effect of ohmic drops on chronoamperograms and voltammograms.14,15 The delicate link lost in these studies was the effect of surface roughness. Disordered or rough * To whom correspondence should be addressed. E-mail: rkant@ chemistry.du.ac.in.

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 assuming it as fractals,16-20 using scaling argument,21-23 fractional diffusion,24,25 ab initio derivation methods,26-32 and numerical approach.33-36 There are two features at the disordered rough electrode one is diffusion to the rough interface and second is resistive effects at the rough interface. There are two phenomenological length scales, i.e., (i) diffusion layer thickness LD ) (Dt)1/2 and (ii) phenomenological diffusion-resistance length scale (LΩ) arising out of solution resistivity LΩ ∝ Fel e (Fe is the solution resistivity and l e is the distance between the WE and RE). The magnitudes of the two phenomenological lengths control the response of the system. It is worth mentioning here that LΩ is dependent on the experimental quantities for solution resistance. This problem has been addressed for planar electrodes numerically by Wein,37 Richtering and Doblhofer,38 and Compton and co-workers39,40 and analytically by Delahay.41-44 However, the theoretical problem of ohmic losses has not been addressed for the rough electrodes. These losses are significant for the experimental studies, and theory must provide for capturing the essence of real processes at rough electrode/ electrolyte interface. Our methodology for such class of problems is an elegant and implementable to several important electrochemical problems to account for various types of surface irregularities both with known geometries and stochastic geometries of the interface. Formulation Our purpose is to formulate the effect of ohmic drop on the potentiostatic current transient at a realistic rough electrode. The mass transfer is assumed to be largely due to diffusion; hence,

10.1021/jp102708w  2010 American Chemical Society Published on Web 05/07/2010

Theory of Generalized Cottrellian Current

J. Phys. Chem. C, Vol. 114, No. 21, 2010 10067

the contributions due to migration and convection are neglected. b) to the gradient (∇) of Fick’s first law relates the flux (J concentration difference (δC(r b, t)) through a proportionality constant as diffusion coefficient (D)

δCO ) -

CO0 - CR0θenfjRΩ 1 + θenfjRΩ

(8)

where θ is given by

b J ) -D∇δC(b, r t)

(1) θ ) e-nf(E-E

0′)

(9)

where δC(r b, t) is defined as

δC(b, r t) ) Ci(b, r t) -

Ci0

(2)

C0i is the bulk concentration (i ) O, R); b r is the threedimensional vector and t being time. Initial and bulk boundary conditions are

LΩ∂nδCO(z ) ζ(b r |), t) - δCO(z ) ζ(b r |), t) ) CS

δCi(b, r t ) 0) ) 0

(3)

δCi(b r |, z f ∞, t) ) 0

(4)

For a single step charge transfer process, O + ne- h R, at a rough electrode (z ) ζ(r b|)) cathodic current density (j) in the normal direction is given by

j(z ) ζ(b r ||), t) ) nFD∂nδC(b, r t)

(5)

The observable, i.e., total, current (I) at the arbitrary interface is obtained by integrating the current density (j) over the whole surface (z ) ζ(r b|)).

I(z ) ζ(b r |), t) )

∫S

0

dx dy βj(z ) ζ(b r |), t)

) nFD

∫S

0

RΩ is the solution resistance, E0′ is the formal potential, and E is the applied potential. In the absence of solution resistance contribution, viz., RΩ f 0, boundary constraint simplifies to b|), t) ) -CS ) -(CO0 - CR0θ)/(1 + θ).27 When eq δCO(z ) ζ(r 8 is linearized and the resultant equation is rearranged, we obtain electrode surface boundary constraint (Appendix A)

(6)

dx dy β∂nδC

∂n ) nˆ · ∇ signifies the outward drawn normal derivative at b|), 1), ∇| ) the electrode surface (z ) ζ(r b|)), nˆ ) 1/β(-∇|ζ(r b, t) refers to the (∂/∂x, ∂/∂y), β ) [1 + (∇|ζ)2]1/2 and δC(r difference in concentration at a given point and initial or bulk concentrations. The diffusive transfer at a random electrode for a Nernstian charge transfer reaction satisfies the diffusion equation for the concentration difference as

∂ δC ) D∇2δCi ∂t i The flux balance condition, initial and bulk boundary constraints along with simplifying assumption of DO ) DR ) D, the concentrations for O and R are relateable through δCO(r b,t) b,t) ) δC(r b,t). Hence, also at the surface, the following ) -δCR(r identity must be satisfied

(10) where

LΩ )

n2F2DFel eθ(CR0 + CS) RT(1 + θ)

The phenomenological diffusion-resistance length (LΩ), directly related to the experimental such as resistivity of the solution (Fe) and distance between the working and reference electrode (l e). Mathematically problem of Cottrellian problem with small solution resistance and quasi-reversible charge transfer28 are identical, and hence, the length scale (LΩ) can be related to the apparent rate of charge transfer imposed by the solution resistance. This apparent heterogeneous rate constant can be defined as kΩ ) D/LΩ which signifies the ohmic control of reversible charge transfer through the resistive parameters (viz., Fe and l e). kΩ is dependent on the applied potential through θ/(1 + θ) which varies between 0 and 1 and CS. kΩ imply that one has to be careful in calculating the kinetic parameters in presence of the sizable solution resistance. The concentration field is to be defined at the electrode/ electrolyte interface. This concentration field is the function of boundary profile, therefore we can say that roughness is sensed by the concentration field in an electrochemical measurements. To write this concentration profile we use the perturbation of boundary profile. This is done by using Taylor expansion of surface boundary condition about a mean reference plane (z ) 0). So, we can write

∂zδCO(z ) 0, t) - L-1 ˜ (b r |, z ) 0, t) Ω δCO(z ) 0, t) ) χ0 + χ (12) χ˜ (b r |, z ) 0, t) ) [χˆ 1 + χˆ 2 + ...]δCO(b, r t)

whatever be the surface profile or time. The surface constraint under reversible (Nernstian) charge transfer in the presence of ohmic losses has the following form:

(13)

χ0 is given by the following relation

δCO(z ) ζ(b r |), t) ) -δCR(z ) ζ(b r |), t) ) δC(z ) ζ(b r |), t)

(7)

(11)

χ0 )

CS LΩ

χˆ 1, χˆ 2, ... are the operators which operate on concentration and generate effective source terms arising from Taylor expan-

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J. Phys. Chem. C, Vol. 114, No. 21, 2010

Srivastav and Kant

sion27 of boundary condition. The detailed form of operators χˆ 1 and χˆ 2 are defined as follows

{ {

(

(

1 ∂2 ∂3 1 2 r |)P0 ζ (b 2 LΩ ∂z2 ∂z3

)]

(

)}

1 ∂ ∂2 χˆ 1 ≡ ∇|ζ(b r |) · P0∇|| + ζ(b r |)P0 - 2 LΩ ∂z ∂z 1 ∂ ∂ χˆ 2 ≡ (∇|ζ)2P0 + ∇|ζ(b r |)ζ(b r |)P0 · ∇| + 2 ∂z ∂z

)}

(14)

[ (

)

1 (z - z′)2 exp + 4Dτ 2(πDτ)1/2 z + z′ (z + z′)2 1 (Dτ)1/2 erfc + × exp 4Dτ LΩ LΩ 2(Dτ)1/2 (z + z′) Dτ exp exp 2 (18) LΩ L

Z (b, z z', t, t') )

[

[

Ω

] ]

The formal solution for the effect of mixed boundary condition prescribed at z ) 0 plane, between t ) 0 and t, is propagated by the integral27

The projection operator (P0) in above equation is defined as

P0 f(z) ≡ f(0);

P 20 f(z)

≡ f(0)

(15)

The projection operator projects the value of an arbitrary function of the spatial coordinate z to that at z ) 0 and the further use of this projection operator on this function does not affect it. In deriving the above χˆ i’s, we have used binomial expansion of [1 + (∇|ζ)2]1/2 and boundary surface profile around reference plane (z ) 0) valid for gently fluctuating (sloping) class of surface, i.e., (∇|ζ)2 < 1. it is important to stress here that the boundary itself is fluctuating in two spatial dimension, the surface profile for a two-dimensional rough surface can be expressed as z ) ζ(r b|) and location on it by the surface vector b|)). Mathematically problem of Cottrellian problem with (r b|, ζ(r small solution resistance and quasi-reversible charge transfer28 are identical and, hence their derivational details are similar. This ab initio derivational method is discussed in following sections. Green’s Function and Formulation of Integrodifferential Equation. The precise knowledge of the Green’s function is sufficient to find the solution of the problem. In the absence of exact knowledge of Green’s function on an arbitrary surface profile, Green’s function on reference surface also provides an approach to solution of the diffusion equation with an arbitrary inhomogenity and boundary profile. This formulation provide an integrodifferential equation which is amenable to systematic perturbation analysis. Moreover, these analysis can be interpreted physically. Therein lies the attractiveness of the Green’s function approach. The Green’s function is a solution to the adjoint (partial differential) operator equation with delta function inhomogeneities in space and time and depends on the form of boundary conditions applied.45,46 The Green’s function tells us how point source at an arbitrary position and time propagate through concentration field which evolves in time and space. The Green’s function G (r b,t | b,t′) r for diffusion problem follow the partial differential equation

(- ∂t∂ - D∇ )G (b,r t|b,r t') ) δ(t - t')δ(r - r') 2

(

where Z is defined below

(-| b r| - b r |′|/4Dτ) 4πDτ

)

(17)

∫0t dt' ∫S

0

dS0′ ∂n′G(b, r t| b′, r t')[χ0 + r t')] (19) (χˆ 1 + χˆ 2 + ...)δC(b′,

G is the Green’s function, in half plane z g 0 and can be obtained by using the method of images satisfying mixed or radiation boundary condition, ∂n′ is outward normal on planar surface, viz., ∂n′ ) -∂/∂z′ and χˆ i are the operators generating the source function at planar surface given by eq 14. The above boundary value problem has been rewritten into a compact integrodifferential equation (eq 19) by using the Green’s function for average surface profile z ) 0 and rewriting the boundary condition accordingly. This enables us to use the techniques developed for studying problems in wave propagation in stochastic media, though the equation having an infinite number of source terms in it. In the first flush a perturbative approach seems to be useful for the solution of this equation. Perturbative Solution in Surface Profile. The perturabtive solution of concentration profile in Laplace-Fourier domain (up to second order term) is (see the Supporting Information for derivation details)

b|, p) ) δC(K

-CS b|) + Cˆ 1ζˆ (K b |) + ˆ 0(2π)2δ(K [C p ˆ 2ζˆ (K b|)(K b| - K b |′)] (20) C

b|) is the Fourier transform47 of the surface profile (ζ(r b|)) ζˆ (K and is defined as

∫ d2r| exp(-iKb| · br |)ζ(br |)

b |) ) ζˆ (K

where i ) -1. The operators C0, C1, and C2 are defined below

Cˆ 0 ≡

(16)

and its solution for the mixed boundary value problem is given by the following equation:46

G (b, r t| b, r t') ) Z (b, z z', t, t') exp

δC(b, r t) ) D

{

}

e-qz ; 1 + qLΩ

{

ˆ1 ≡ C

{

qe-q|z 1 + q|LΩ

}

(21)

′ qq|,| q2 + + 2(q|LΩ + 1) q|LΩ + 1 b|′)LΩ b|′) b|′ · (K b| - K b|′ · (K b| - K K L Ωq K q (LΩq| + 1) q|,|′LΩ + 1 2(qLΩ + 1) q|LΩ + 1 (22)

e-qz Cˆ 2 ≡ (2π)2

where

∫ d2K|′

-

}

Theory of Generalized Cottrellian Current q ) (p/D)1/2 ;

q| ) [q2 + K|2]1/2 ;


b| - K b|′ | 2)1/2 q|,|′ ) (q2 + |K

The structure of the problem is mathematically isomorphic to the problem of diffusion to rough interfaces with finite charge transfer rate addressed by Kant-Rangarajan28 and Jha-Kant.31 It is important to stress here that the two different physical problems resolve to same mathematical structure, implying the importance of understanding each of these processes in a comprehensive manner. The expression for current density for the arbitrary surface profile up to second order perturbation term is given by

b|, p) ) nFDCS[Iˆ0(2π)2δ(K b|) + Iˆ1ζˆ (K b |) + j(K b| - K b|′)] b|)(K Iˆ2′ζˆ (K

(23)

The operators Iˆ0, Iˆ1, and Iˆ2′ are defined below

{

}

1 q(qLΩ + 1) q| 1 q Iˆ1 ≡ q q|LΩ + 1 qLΩ + 1 Iˆ0 ≡

{(

Iˆ2′ ≡

1 (2π)2

∫ d2K|′

{(

)}

(24)

)

q| 1 q + 2 qLΩ + 1 q|LΩ + 1

q|q|,|′ q q(q|LΩ + 1) q|,|′LΩ + 1 b|′) b|′| 2 b| - K b|′ · (K b| - K K |K 1 + LΩq(q|LΩ + 1) q|,|′LΩ + 1 q(q|,|′LΩ + 1) b|′) b| - K b|′ · (K K 1 2LΩq(qLΩ + 1) q|LΩ + 1

}

(25)

This current density expression presents two structures in the equation; first term is the planar response to the ohmic losses and subsequent terms represent the first and second perturbation corrections in surface profile. The above equation can be expanded in jK|/q, and taking the inverse Fourier transform,47 we obtain curvature expansion for current density in Laplace domain

j ≈ nFDCs

[

1

Hq (qLΩ + 1)2 (3qLΩ + 1) 2 1 K H + 3 3 3 2q (qLΩ + 1) 2q (qLΩ + 1)

q(qLΩ + 1)

2

]

(26)

where H is the mean curvature and K is the Gaussian curvature described for small curvature surface as

(

2

2

)

∂ζ 1 ∂ζ + 2 2 ∂x2 ∂y ∂2ζ ∂2ζ ∂2ζ K≈ 2 2 ∂x∂y ∂x ∂y

H≈

I(p) ) nFDCS

∫S

0

b|) + d2r| {Iˆ0 + Iˆ1ζˆ (K b|)ζˆ (K b| - K b|′);K b| f b r |} (27) Iˆ2ζˆ (K

where the braket notation for the inverse Fourier transform is b|); K b| f b b| · b b|). Iˆ0 and Iˆ1 {f(K r|} ≡ (1/(2π)2)∫ d2K| exp(iK r|)f(K are operators same as in eq 24 and Iˆ2 is

Iˆ2 ≡

1 (2π)2

∫ d2K|′

{(

)

q| 1 q + 2 qLΩ + 1 q|LΩ + 1

q|q|,|′ q q(q|LΩ + 1) q|,|′LΩ + 1 b|′) b|′| 2 b|′ · (K b| - K b| - K K |K 1 LΩq(q|LΩ + 1) q|,|′LΩ + 1 q(q|,|′LΩ + 1) b|′) b| - K b|′ · (K K q| 2q(qLΩ + 1) q|LΩ + 1

where the integral is performed over the surface S0 (i.e., z ) 0 plane). It is necessary to mention here that Iˆ2′ and Iˆ2 are similar except for numerical coefficient of the last term. These operators consist of diffusion characteristic (q) and ohmic loss characteristic (LΩ) while eqs 23, 26, and 27 emphasize the roughness profile of the interface to highlight the mathematics involved in an amenable manner. Explicitly, the first operator defines the effect of diffusion and ohmic 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. These equation are useful in predicting local concentration, current density profile and total current. This includes known deterministic profiles like sine, conical, triangular profiles, etc. Surface profiles which can be represented in term of cosine or sine functions are easy to write final expressions for concentration and current as their Fourier representation have only Dirac delta functions. Random Surface Electrode

〈ζ(b r |)〉 ) 0

( )

(28)

and the two point correlation function is

〈ζ(b r |)ζ(b r |′)〉 ) h2 W(| b r| - b r |′|)

2

}

For a random surface electrode, the surface profile is characterized by a centered Gaussian field with the statistical properties: the ensemble averaged value of the random surface and the two point correlation function.48 The ensemble averaged value of the random surface is

1

-

The expression for the total current is given by taking the surface integral of the current density and retaining terms up to b|)). second order in surface profile (ζˆ (K

(29)

where 〈 · · · 〉 denotes ensemble average over various possible surface configurations. The normalized correlation function (W(|r b| - b r′|)) gives the measure of rapidity of variation of surface | and h2 is mean square height fluctutions or width of interface.

10070


Srivastav and Kant

For slowly varying surface, the correlation between two positions on the surface is nonzero for even large values of their r|′|. The correlation function vanishes on separation |r b| - b r′|. This correlation function increasing the relative distance |r b| - b | contains information about the surface morphological characterstics as area, mean square height, slope, curvature, and correlation length. Averages in the Fourier plane are

b|)〉 ) 0 〈ζˆ (K

(30)

b|)ζˆ (K b|)〉 ) (2π)2δ(K b| + K b|′)〈|ζˆ (K b|)| 2〉 〈ζˆ (K

(31)

It is to be brought to notice that at a random surface it is not the local quantities but rather average quantity represented by ensmble average over surface configuration space, which are quantities of interest. The average current transient is given by

[

[

]

∫

1 b|)| 2〉 + d2K| K2| 〈|ζˆ (K 2 2(2π) 1 1 [1 - F (√Dt/LΩ)] × d2K| [1 - DtK2| 2 LΩ (2π)2

〈I(t)〉 ) I†p(t) F (√Dt/LΩ) 1 +

∫

]

b|)| 2〉 exp(-K2| Dt)]〈|ζˆ (K

(32)

where I†p(t) is the Cottrellian current at a smooth electrode without ohmic losses

I†p(t) )

nFA0√DCS

√πt

(33)

The contribution of solution resistance is contained in function F [(Dt)1/2/LΩ]

( ) ( )

√πDt √Dt Dt exp 2 erfc F (√Dt/LΩ) ) LΩ LΩ LΩ

(34)

(35)

) I†p(t)F (√Dt/LΩ) I*(t) p

(36)

the roughness factor of the electrode, defined as the microscopic area and projected area. The of solution resistance for rough electrode is function F *[(Dt)1/2/LΩ] and is defined as

F *(√Dt/LΩ) )

- F (√Dt/LΩ)]

F (√Dt/LΩ)

∫0∞ dK| K|[1 - DtK|2 b|)| 2〉 exp(-K|2Dt)]〈|ζˆ (K

(38)

The short time limit (t f 0) of eq 33 is

〈I(t)〉 ≈

0 0 nFA0DCSR* RT (1 + θ)(CO - θCR) A0R* ) LΩ nF RS θ(CO0 + CR0) (39)

Equation 39 stresses the effect of the ohmic losses contained in LΩ or RS at the short times. It is important to realize that while using eq 39 for estimating solution resistance (RS) one has to account for the correct area through roughness factor. The long time asymptotic limit shows the Cottrellian current transient

f I†p ) I*(t) p

nFA0√D

√πt

CS

(40)

Equation 40 shows that at long time transient response is independent of the ohmic parameters and is simply Cottrellian current. However, in the intermediate time domain roughness features dynamically come into picture. The resistive effects are represented in the time dependent function F *. The function R† is the generalized time dependent roughness factor. We can further extend the limiting behavior of eq 35 beyond the eq 39 level to see the contribution of various morphological features of rough surface on the measured current. We expand eq 38 for small K|2Dt. The roughness integral R† for isotropic statistics can also be written as the moments of the surface profile ∞

m

(Dt)m+1〈(∇||mζ)2〉 ∑ 2(m(-1) + 2)!

(41)

m)0

where the ensemble averaged square mth order profile derivative, 〈(∇|mζ)2〉, is related to 2 mth moments of the power spectrum as

+ F *(√Dt/LΩ)R†(t)] 〈I(t)〉 ) I*(t)[R* p

2Dt/L2Ω[1

1 4πDt

R†(t) ) -

The average current 〈I(t)〉 can be written as

where R* is the ratio of contribution contained in

R†(t) )

〈(∇||mζ)2〉 )

1 (2π)2

∫ d2K| K|2m〈|ζˆ (Kb|)|2〉

(42)

When this expansion is substituted into eq 35 and leading order terms are retained, we obtain the average current transient as follows: 〈I(t)〉 ) I*p (t)[R* - F *(√Dt/LΩ)Dt〈H2〉 + ...]

(43)

where the second term consists of an ensemble averaged square of mean curvature

(37)

The roughness integral R†(t) contains structure sensitive effects as a function of time. This is characterstic of the surface statistics or the surface structure factor. The roughness integral for isotropic surface statistics is given by28

1 〈H2〉 ≈ 〈(∇|2ζ)2〉 4 An arbitrary power spectrum can have band-limited fractal or single correlation length dominated nonfractal random profile.27-29,32,49-53 The results are therefore analyzed for two



types of surface profiles characterized using (a) fractal surface with limited self-affine scaling property and (b) nonfractal surface with fixed transversal correlation length. Random Fractal Electrodes. A random surface can often characterized by concept of fractals. Fractals were first introduced by Mandelbrot54 to describe disordered objects, using fractional dimensions. The concept of fractals is well-known in electrochemistry. In order to capture the complexity arising from the irregular interfaces (i.e., rough, porous and partially active interfaces) one often uses the concept of fractals.55,58 The fractal irregularities are usually understood in particular in terms of self-similar54,59,60 or in general as self-affine26,29,32,53-57,59,60 fractals. These fractal boundaries exhibit statistical selfresemblance over all length scales and can be described using power law power spectrum.61 For realistic electrode surface we use band limited power spectrum to circumvent the mathematical difficulty of nondifferentiability and nonstationarity. These band limited fractals have a lower and a upper length scale cutoff. The quantity we seek for an isotropic fractal surface is the power spectrum as the input for eq 32. This power spectrum of a band limited fractal surface is given by61

b||)| 2〉 ) µ|K b|| | 2DH-7, 〈|ζˆ (K

b|| | e 1/l 1/L e |K

(44) where DH is the Hausdroff dimension, µ is the strength of fractality, and l and L are lower and upper length scale cutoffs of fractality, respectively. The fractal with finite L and l f 0 is called a finite fractal and one with L f ∞ and l f 0 is called an ideal fractal. The finite fractal will show a nonfractal behavior under the limit: l f L. The moments of above power spectrum are related to various morphological features of rough surface, viz., root-mean-square (rms) width (m0), rms gradient (m2), rms curvature (m4), etc. The general moments of power spectrum (i.e. 2kth moments, m2k) are easily obtained for the above-mentioned power spectrum and amount to important morphological characteristic of surface roughness. m2k is given by the expression

µ (l -2(δ+k) - L-2(δ+k)) 4π(δ + k)

m2k )

(45)

δ ) DH - 5/2 is the deviation from Brownian fractal dimension. Average current at an isotropic self-affine fractal surface electrode is obtained using eq 37 and the power spectrum in eq 44 as

[

〈I(t)〉 ) I*p (t) 1 +

m2 (1 - F *(√Dt/LΩ) + F *(√Dt/LΩ)RF(t)) 2

(46)

where RF(t) is the fractal roughness as defined below

RF(t) )

]

m0 µ + δ+1 2Dt 8π(Dt) Γ(δ, Dt/l2, Dt/L2)

where Γ(R, x0, x1)62 denotes incomplete Gamma function.

Γ(R, x0, x1) ) Γ(R, x0) - Γ(R, x1) ) γ(R, x1) - γ(R, x0)

This equation includes the fractal features dependent powerlaw as well as contribution from the resistive losses at the electrode. The resulting current-time behavior is an interplay between the interfacial potential, the resistance of the medium and the fractal roughness features of the interface. The total current is the summation of smooth surface response and an anomalous excess flux due to fractal roughness. The exact form of eq 46 under the diffusion-limited constraint explains and capture large quantity of experimental data available for the intermediate time53 without taking into account the ohmic parameters. The RF(t) is dynamic roughness factor for an isotropic fractal. The resistive effects are represented in the function F *[(Dt)1/2/LΩ]. The function F *[(Dt)1/2/LΩ] contains the generalized roughness factor. At short time limit (t , L2Ω/ D), it behaves as a process dominated by resistive effects of the solution. Random Nonfractal Electrodes. A random isotropic surface is characterized as a Gaussian function, as a simplest case for non fractal morphology. A Gaussian surface has two parameters that can describe roughness of the surface, they are mean square width (h2) and correlation length (a). Gaussian correlation function is given by

〈ζ(b r |)ζ(b r |)〉 ) h2 exp(-| b r| - b r | | 2 /a2)

(47)

Its Fourier transform is the surface structure factor and is given by

b|)| 2〉 ) πa2h2 exp(-|K b| | 2a2 /4) 〈|ζˆ (K

(48)

Roughness factor for isotropic random surface modeled as a Gaussian function is

R* ) 1 + 2h2 /a2

(49)

The transient response for such surface statistics is calculated using eqs 32 and 48. The current transient so obtained are

[

〈I(t)〉 ) I*(t) 1+ p

(

2h2 4Dt 1 - F *(√Dt/LΩ) 2 2 a a + 4Dt

)]

(50)

In these equations the angular brackets denote an average over the ensemble of random surface configurations while h2 is the mean-square width of surface from flat surface. Results and Discussion To demonstrate effect of ohmic drop on current transient at a rough electrode, we consider the changes in a chronoamperogram with respect to different parameters affecting the surface roughness and the ohmic drop in the electrolyte. The resulting transient response resonate the variation of these parameters to provide a better understanding of both roughness as well as ohmic factors. Figure 1a shows the effect of resistivity of the electrolyte solution on the current transient. We find that the chronoamperograms are affected significantly at short time by the ohmic parameter thus delaying the onset of diffusion controlled process. As the resistivity of the solution increases, the current transient decreases. The dotted lines in the same color cor-

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Figure 1. Double logarithm plot of the current transient vs time is plotted. (a) Effect of electrolyte’s resistivity (Fe) on the current transient. Fe is varied as 50, 500, 1000 Ω cm (upper to lower curves) and distance between working and counter electrode l e ) 0.73 cm. The dotted lines in the similar color depicts the response at the planar electrodes with same resistivity of the electrolyte. (b) Effect of distance between the working and reference electrode (l e) on the current transient. l e is varied as 0.01, 0.25, 0.75, and 1.5 cm (upper to lower curves) resistivity being; Fe ) 200 Ω cm. The plots are generated by using fixed parameters: l ) 2.50 µm, DH ) 2.46, projected area (A0 ) 0.164 cm2), diffusion coefficient (D ) 5 × 10-6 cm2/s) and concentration (CO ) CR ) 5 mM) are used in our calculations.

respond to the planar response with the same parameters for the ohmic loss at the electrode. The striking feature of this graph is the marked presence of plateaus at short times. In this time domain there is no depletion layer present because of the ohmic control. These plots then follow a power law behavior in the intermediate time domain (viz., (L2Ω + h2)/D g t g L2Ω/D) and around this time (i.e., L2Ω/D) the depletion layer near electrode starts forming. One can see the effect of dynamic morphological features in this intermediate region. Richtering and Doblhofer38 simulated the effect of uncompensated resistance on chronoamperometric measurements at smooth planar electrode. In contrast to their results, an important feature of electrode roughness is included here and we show that presence of the plateau region is followed by an anomalous power law in the intermediate region. This anomalous power law merges into the Cottrellian response later than planar electrode response, marking the effect of roughness parameters in the intermediate time domain. The crossover from Cottrellian to anomalous region is affected in presence of ohmic loss. This change in slope in double logarithim plots of the curves, (i.e., slope g1/ 2) is due to the inherent roughness at the working electrode. Our plots show deviation from Cottrell behavior characterstic to the rough electrodes in the intermediate time. More recently Compton and co-workers39,40 numerically studied the effect of concentration of supporting electrolyte on the transient response in chronoamperometric measurements. In their studies, at moderate ohmic losses they observed a plateau in short time interval following a Cottrellian behavior in the intermediate region while for lower values of supporting electrolyte are attributed as Frumkin effect in the chronoamperometric response. This intermediate time behavior is controlled by the morphological features of the rough electrode which was not accounted in the above-mentioned work. The next plot, Figure 1b, shows the effect of distance between the reference electrode (RE) and working electrode (WE) on the current transient. It is worth mentioning here again that positioning of the probe of reference electrode (Luggin’s capillary) is the key tool for experimentalists to minimize the effect on uncompensated solution resistance in most experiments. It has been noticed that decreasing the distance between the RE and the WE reduces the ohmic drops in the system. The plots provide the similar nature, as when the distance

Srivastav and Kant between the WE and the RE (l e) increases the current transient decreases. This shows that the greater the distance l e greater are the ohmic losses recepted by the system. The nature of this graph mimics the observation that at the early time resistivity affects a diffusion controlled process the most, following a power law behavior in the intermediate time domain. Notice that there is a time delay; when the system switches to diffusion control; as we increase the separation between the electrodes (l e). This intermediate response is attributed to the morphology of the electrode, i.e., the type of roughness present at the interface. Isotropic Self-Affine Fractals. Next we investigate graphically the effect of various morphological feature of roughness with fractal nature alongwith ohmic loss in Cottrell current. For a fractal surface there are three morphological features which dominantly control the response for a fractally rough surface namely fractal dimension (DH), lower length scale of fractality (l ) and proportionality factor of power-spectrum (µ) or mean square width of the interface (h2). Figure 2(a), shows the effect of fractal dimension DH on the current transient in presence of certain resistance offered by the electrolyte at an isotropic selfaffine fractal electrode. The quantity DH is the Hausdorff dimension of the surface and is a direct measure of the surface irregularity. As the fractal dimension increases surface becomes more rough and the current transient increases. This increase in the current transient is however scaled down in presence of ohmic losses in short times and it strongly affects the early response. The effect of ohmic drops is most pronounced at early time and carried until the onset of the diffusion control in the system. Figure 2b shows the effect of varying the lower length scale cut off (l ) on the current transient. Lower length scale of fractality (l ) can be identified as the extent of the coarseness of the surface. It is seen that increasing l decreases the current transient. The effect of this morphological parameter is however propagated to a larger time in the plots. It is evident that l not just effects the short time but influences the intermediate time domain to a larger extent showing that the current transients are strongly dependent on the lower length scale cut off. The change in the slope of the curve is however enhanced in presence of the ohmic drop in the early time domain. Figure 2c shows the effect of root-mean-square height (h) or the width of the interface in presence of ohmic drop. The effect of mean square height is similar to that of DH. It resonates the same influence over the current transient as DH, i.e., when mean square height increases the current transient also increases. As one increases the strength of roughness, it increases the rootmean-square height and so does the electrochemically active area. Hence, increasing root-mean-square height (h) increases the current transient which finally merges with the smooth electrode response when the time is sufficiently larger than the root-mean-square height (h) and all irregularities of the surface is smaller than the yardstick depletion layer thickness after it onset. The time of merger of intermediate anomalous region with classical Cottrellian current increases with increase in the width of interface. Comparison with Experiment. The time scale of the experimental set up for an electrolyte solution is usually ranges from milliseconds to tens of seconds where the diffusion length is much larger than the Gouy layer thickness (region where electroneutrality of system is violated) in the supported system. Hence, presence of the excess supporting electrolyte allows us to neglect the migration contributions, while experiments show that sizable uncompensated solution resistance still persists in



Figure 3. Comparison with experimental data16 for isotropic self-affine fractals. In the inset is the same data using Gaussian power spectrum.

Figure 2. Logarithm of the current transient vs the logarithm of the time in seconds is plotted. (a) Effect of fractal dimension (DH) on the current transient. The double logarithm plot of the current transient vs time in seconds is plotted. DH is varied as 2.1, 2.2, 2.25, and 2.3 (lower to upper curves), l ) 25 nm. (b) Effect of lower length scale of fractality (l ) on the current transient. The plots are generated using parameters l ) 25, 50, 75, and 100 nm (from upper to lower curves), DH ) 2.3. (c) Effect of h (rms width) on the current transient. h is varied as 19.64, 43.01, 62.10, and 107.55 µm (lower to upper curves), l ) 2.50 µm, DH ) 2.46. The plots are generated by using fixed parameters, projected areas(A0 ) 0.164 cm2), diffusion coefficient (D ) 5 × 10-6 cm2/s), Fe ) 200 Ω cm, distance between the working and reference electrode l e ) 0.73 cm and concentration (CO ) CR ) 5 mM) are used in our calculations.

the system. Our results are, therefore, helpful in accounting the effect for uncompensated resistance at rough electrode in case of fully and moderately supported systems. A comparison between the experimental results of Pajkossy and co-workers for the current transient on rough platinum wire electrode (∆) and theoretical (solid line) curve obtained from eq 46 has shown in Figure 3. They measured current transient over three to four orders of time and over 2-3 orders of magnitude of current. They did not apply iR compensation in short time data so typical effect of the solution resistance can be seen in their data. Using self-affine fractal model Figure 3 clearly depict our theoretical predictions that shows an excellent agreement with experimental data including the data points

showing ohmic losses. An important advantage of our theory is that it predicts various roughness feature of roughness profile like: the roughness factor (R*), the width of roughness (h or [m0]1/2), the gradient ([m2]1/2) and the curvature (1/[m4]1/2) of the surface. Rough surface of wire electrode is imagined as randomly fluctuating surface around a macroscopic plane as the curvature contribution of macroscopic wire geometry is insignificant. Theoretical result is represented by solid line and is generated using fractal morphological parameters as DH ) 2.46, L ) 30 µm, l ) 1.95 µm, A0 ) 0.164 cm2 (projected area), µ ) 1.673 × 10-5 (arbitrary units), h ) 20.26 µm, D ) 5 × 10-6 cm2 s-1 and initial concentrations CO0 and CR0 ) 5 mM, RS ) Fel e/A0 ) 137.2 Ω (solution resistance) is used in our calculations. The roughness factor for isotropic fractal statistics is R* ) 5.49. Major observation here is that our equation provide excellent match for the current transient in short time, intermediate as well as long time regimes. Another very important aspect is that this data can also be captured using nonfractal Gaussian statistics as shown in Figure 3 (inset). The parameters here are correlation length (a) and mean square width of the interface (h2). The value of these two length scales are a ) 9.8 µm, h ) 20.26 µm and the solution resistance is RS ) Fel e/A0 ) 137.2 Ω. The roughness factor using Gaussian statistics is R* ) 5.31. It is to be noticed here that if the correlation length (a) is the order of the lower length scale of fractality (l ) in fractal model, one cannot differentiate through the response between a fractal and nonfractal surface. A word of caution is necessary here that one has to properly analyze the surface statistics through AFM measurements to assert a fractal or a nonfractal surface, since both statistics can yield the similar response. In the case of Pajkossy and co-workers, we know that the electrode is fractal; however, the response can be mimicked through nonfractal statistics also. Hence, it is clear that intermediate anomalous region can originate from random roughness both fractal or nonfractal in nature. Conclusion The formalism developed in present paper enables us to discern the effect of ohmic drop (through the phenomenological diffusion-resistance length, LΩ) and morphology. Formalism developed here allow us to address deterministic as well as

10074


stochastic surface roughness. Stochastic morphology offeres realistic models of roughness. We have analyzed our general model for two types of surface randomness statistics: (i) Roughness with known statistical self-affine fractal properties and power-spectrum of such roughness model has four fractal characteristicssfractal dimension (DH), lower length scale cut off (l ), upper length scale cut off (L), and strength of fractality (µ) which is proportional to mean square height (h2). Three dominant fractal parameters are DH, l , and µ (or h2) which finally control the electrode response. (ii) Statistical behavior of random nonfractal electrode surface and its power spectrum has two characteristic parameters, i.e., mean square height (h2) and the correlation length (a), are sufficient for such Gaussian surface statistics. This paper establishes for such roughness the following points: (1) Chronoamperograms distort under the conditions of Nernstian charge transfer (reversible) due to the ohmic drop across the solution and presence of the surface roughness. The formalism enables us to discern the effect of ohmic drop as well as morphology of an arbitrary rough surface. (2) These results are valid for all times which translate into three time regimes, namely, a short time region (i.e., t < L2Ω/D and l 2/D), an anomalous intermediate time region (i.e., L2Ω/D and l 2/D j t j (L2Ω + h2)/D), and long time region (i.e., t > (L2Ω + h2)/D). These results capture the experimental data for low resitivity electrolyte solutions in all time regimes. (3) The deviation from the classical Cottrellian behavior in the short time domain is found to be dependent primarily on the resistance of the electrolytic solution and the real area of the surface (see eq 39). L2Ω/D is roughly the time up to which a system is affected the most by the resistive parameters. This short time domain is mainly influenced by the external experimental conditions, resistivity of the solution (Fe) and, the distance between the probe and the working electrode (l e). (4) In the absence of the surface roughness, the current crossover to classical Cottrell response as the diffusion length exceeds the diffusion-ohmic length, but in the presence of roughness, there is formation of anomalous intermediate region followed by classical Cottrell region. (5) The intermediate region is marked by the anomalous power law behavior. The presence of an ohmic drop delays the onset anomalous power law region. The anomalous power law region arises due to formation of a depeletion layer which leads to the emergence of dynamic effects of roughness in the response. The depeletion layer start forming as the time exceeds the value L2Ω/D. (6) This intermediate time response contains several morphological details of the surface, viz., mean square width, mean square gradient, mean square curvature, etc. These finer features give shape to the anomalous intermediate region for a rough surface. This intermediate transition region from initial ohmic control to final Cottrellian response (or diffusion control) becomes shorter as the surface becomes smoother. (7) For long time, i.e., when t > (L2Ω + h2)/D, the dynamic roughness response evens out in time due to large depletion layer thickness, and we obtain the classical Cottrellian response. Finally, one can say that the theory presented here succeeds in offering an understanding of a near complete picture of diffusion controlled processes under resistive experimental conditions at irregular interfaces. Acknowledgment. S.S. is grateful to UGC-India for providing financial assistance (Non-NET fellowship). R.K. is grateful to University of Delhi and DST for purse grant.

Srivastav and Kant Appendix A. Derivation of the Nernstian Boundary Condition For a reversible reaction

O(Soln) + ne- h R(Soln)

(A.1)

the surface concentration is given by Nernst equation

CO0 - CR0θ 1+θ

CS ) -δCO )

(A.2)

where θ is given by

θ ) e-nf(E-E

0′)

(A.3)

and CS is the surface concentration difference without the ohmic losses at the electrode. The surface concentration difference obtained from Nernst equation along with ohmic loss is

-δCO )

)

CO0 - CR0θenfjRΩ

[

1 + θenfjRΩ 1+θ

CO0

CR0θ

1-

CR0θ

(enfjRΩ - 1)

θ 1+ (enfjRΩ - 1) 1+θ CO0

CR0θ

(A.4)

]

(A.5)

Linearizing it for small values of ohmic loss

-δCO ) CS

[

1 - nfjRΩ

[

] ]

CR0θ

+ ... CO0 - CR0θ θ 1 + nfjRΩ + ... 1+θ

[

[

] ]

(A.6)

On writing the binomial expansion of the denominator, we obtain

[

-δCO ) CS 1 - nfjRΩ

[

-δCO ) CS -

CR0θ

] ]

+ ... × CO0 - CR0θ θ 1nfjRΩ + ... 1+θ

[ [

]

[[ 1 +θ θ ]nfjR (C Ω

S

]

+ CR0)

]

(A.7)

(A.8)

Substituting for current density (j) as

j(S, t) ) nFD∂nδCO

(A.9)

We obtain

-δCO ) CS -

[

]

2 2 θ n F DRΩ (CS + CR0) ∂nδCO 1 + θ RT (A.10)



(A.11)

ζ(r b|) b|) ζˆ (K ∇| l Γ µ

(A.12)

References and Notes

Rearranging the equation

LΩ∂nδCO - δCO|s ) CS where diffusion-resistance length LΩ is defined below

LΩ )

n2F2DRΩ(CR0 + CS)θ RT(1 + θ)

Mathematically, the problems of the Cottrellian problem with small solution resistance and quasi-reversible charge transfer28 are identical, and hence, their derivational details are similar to those in refs 28 and 63. Derivation of complex problem of Nernstian charge transfer with solution resistance is discussed in Supporting Information. Supporting Information Available: Derivation details of the formal solution of the integrodifferential equation (eq 19), concentration and current density expressions. This material is available free of charge via the Internet at http://pubs.acs.org. List of Symbols A0 C0i

Projected area Bulk concentration of the oxidized species, e.g., i ) O, R Concentration difference obtained using Nernst equaCS tion Change in concentration of oxidized species δCO b|, p) Fourier Laplace transform concentration δC(K Diffusion coefficient of ith species e.g., i ) O, R Di Fractal dimension DH E Potential of the working electrode f F/RT F Faraday’s constant F, F * Dynamic ohmic loss function G Green’s Function for the mixed boundary condition at a plane h Standard deviation of the surface height fluctuation I(p) Laplace transform current I(t) Current transient Smooth electrode response with ohmic losses I*(t) p Cottrellian current Ip† j(t) Current density b|, p) Fourier-Laplace transform current density j(K [K2x + K2y ]1/2 K| b| K Vector (Kx, Ky) Diffusion-resistance length (see eq 11) LΩ n Number of electrons transferred nˆ Normal vector drawn in outward direction p Laplace transform variable Projection operator P0 q (p/D)1/2 q| [(p/D) + K|2]1/2 b r Vector (x, y, z) Vector (x, y) b r| Solution resistance RS R* Roughness factor Dynamic roughness integral R† z Coordinate representing distance away from electrode ∫dx ∫dy ∫d2r| b| b|) Dirac delta function in wave-vector K δ(K

Arbitrary surface profile Fourier transform of the arbitrary surface profile ˆı∂x + ˆj∂y Lower cutoff length scale of a band-limited fractal Gamma function Proportionality constant of power spectrum or strength of fractality

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