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J. Phys. Chem. C 2010, 114, 19357–19364

19357

Theory of Absorbance Transients of an Optically Transparent Rough Electrode Rama Kant* and Md. Merajul Islam Department of Chemistry, UniVersity of Delhi, Delhi-110007, India ReceiVed: June 21, 2010; ReVised Manuscript ReceiVed: August 10, 2010

Theory for the potentiostatic absorbance transient for a species generated at the rough electrode/electrolyte interface is developed. Absorbance transients are strongly affected by electrode geometries, particularly the effect of electrode roughness which is the least understood area, is dealt here. A general operator structure between concentration, surface absorbance density, absorbance transients and arbitrary roughness profile of electrode is emphasized. The statistically averaged absorbance transients is obtained by ensemble averaging over all possible surface roughness configurations. An elegant mathematical formula between the average spectroelectrochemical absorbance transient and surface structure factor or power spectrum of roughness is obtained. This formula is used to obtain an explicit equation for the absorbance on an approximately selfaffine (or realistic) fractal and nonfractal electrode. Our results are also applicable to chronocoulometric response as its directly related to chronoabsorptometric response. Limiting behavior in short time, is attributed to the electrode roughness factor and surface curvatures. Limiting behavior in long time is like a smooth electrode response but the transition region is attributed to the mean square width of roughness. In the intermediate time, the absorbance has anomalous power law behavior which is attributed to the diffusion length weighted spatial frequency features of roughness and becomes almost time independent for large roughness electrodes. Finally, this theory provides a quantitative description of the role of roughness on potential step transmission chronoabsorptometry and chronocoulometric measurements, therefore opening up the way to explore and understand various anomalies in their response. Introduction Spectroelectrochemistry (SEC) is an important technique to identify reaction intermediates, products, mechanisms, and other electron transfer processes taking place at an electrode/ electrolyte interface.1–4 The main motivation of this technique has been to provide the means for obtaining information about spectroelectrochemical systems that could not be gathered by either purely electrochemical or spectrochemical methods. Kuwana et al.5 was the first group who monitored the course of an electrochemical reaction by transmission spectroscopy for diffusion controlled reaction in 1964. His group was the main contributor toward the development of this important research field.6–13 This technique is used to analyze various systems like optically transparent minigrid electrodes,14 current step spectrochemical relations for diffusion,15 potential step transmission chronoabsorptometry,16 absorbance measurements by the reflected light,17 microelectrode,18 electrochromic effects and devices,19–25 carbon nanostructures,26 graphene,27 metal clusters,28–32 and biomolecules.33–36 Optically transparent electrodes (OTEs) are valuable platforms for a broad range of applications stemming from fundamental spectroelectrochemical investigations of electron transfer mechanisms to applied technological applications.27 For the redox reaction O + ne- h R taking place at an electrode/electrolyte interface, by using Beer-Lambert and Faraday’s laws we can find the concentration and charge (chronocoulometrically), respectively. The combination of these two laws gives the relationship between charge and absorbance; that is, the absorbance is proportional to the amount of charge transferred.37 If the wavelength is such that one of the species (reduced or oxidized) absorbs light in the UV-visible region, * Corresponding author. E-mail: [email protected].

then the change in absorbance with time can be monitored spectroelectrochemically. In general, the chronocoulometric charge and the absorbance transient of species are related as

A(t) )

εQ(t) nFA0

(1)

where A is the absorbance, ε is the molar absorptivity of reduced (photoactive) species, n is the number of electron(s) involved in the reaction, F is the Faraday constant, A0 is the projected area of the electrode, Q(t) is the charge, and t is the time. Equation 1 interrelates two important techniques that are experimentally different. The absorbance for the diffusionlimited processes at the optically transparent planar electrode is given by2

AP(t) )

2εCO0√Dt

√π

(2)

where AP(t) is the absorbance for the smooth optically transparent planar electrode, CO0 is the concentration of the oxidized species (if reduced species is absorbing) and D is the diffusion coefficient. The quantity nFA0AP(t)/ε is the total amount of charge required to produce the (reduced) photoactive species per unit projected area. The total charge used for reduction on the planar electrode under diffusion-limited phenomena is given by the Anson equation as38

10.1021/jp1057226  2010 American Chemical Society Published on Web 10/22/2010

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J. Phys. Chem. C, Vol. 114, No. 45, 2010

QP(t) )

2nFCO0A0√Dt

√π

Kant and Islam

(3)

where QP(t) is the charge for the planar electrode. Most of the real surfaces are not smooth. So, the study of geometrical (roughness) aspects of the electrode (here OTE) is of practical importance since roughness affects the spectroelectrochemical response, like current, charge, impedance, absorbance, etc. The recent interest in realistic fractal geometry and methodology used for its characterization has opened a new window to understand the influence of surface disorder. This includes potential sweep methods on an arbitrary topography,39 Cottrellian current at a rough electrode with the solution resistance effect,40 partial diffusion-limited interfacial transfer/ reaction,41,42 Gerischer admittance at a rough electrode,43 anomalous Warburg impedance at a rough electrode,44–46 and anomalous diffusion reaction rates or diffusion controlled potentiostatic current transients.47–52 In this paper also, we wish to use same approach to characterize the rough OTE in relation to the diffusion-limited reversible redox reactions taking place at the electrode/electrolyte interface through absorbance transients. Figure 1 depicts the rough OTE with emphasis on morphological and phenomenological scales that control the spectroelectrochemical response of such electrode/electrolyte interface. The paper is organized as follows: we formulate the problem and solve it for the arbitrary profile of the electrode by using a perturbation method. The absorbance transient is obtained for the surfaces with a known profile as well as with random roughness profiles. Results for the absorbance transient are analyzed for self-affine fractal surfaces and nonfractal surfaces. Finally, results, discussion, and conclusions are reported. Formulation The total absorbance A(t) for diffusion controlled processes under a single potential step experiment (δE(t) ) E - E0′; where E and E0′ are the applied potential and formal potential, respectively), for an interfacial redox reaction, O + ne- h R, can be obtained by solving appropriate diffusion equation. The b,t), satisfies the diffusion equation as concentration profile Ci(r follows:

∂ r ) Di∇2Ci(b,t) r C (b,t) ∂t i

(4)

Figure 1. Schematic diagram of diffusion controlled charge transfer at rough optically tranparent electrode (OTE) where the charge transfer reaction O(sol) + ne- h R(sol) under Nerstian conditions is coupled to the bulk diffusion of oxidized or reduced species. The rate determining step is diffusion. Here the detector is a spectrophotometer. Various morphological characteristics control the spectroelectrochemical responses of electrode/electrolyte interface, such as lower (l) and upper (L) cutoff length scales of fractality, width (h), and diffusion length [(Dt)1/2].

b),t) ) -(CO0)/(1 + θ) and the above eq 5 becomes δCO(z)ζ(r ) -Cs. For backward reaction, if the oxidized species is light absorbing, under this condition initially CO0 ) 0 then the above b),t) ) (CR0θ)/(1 + θ) ) -Cs. If the equation will be δCO(z)ζ(r potential is large enough (diffusion-limited case), then θ ) 0 b),t) ) -CO0 ) -Cs. We wish to and it is reduced to δCO(z)ζ(r develop a perturbation solution in boundary profile of this boundary-value problem for a deterministic as well as random roughness geometry. Here, the quantity of interest is current since we can calculate charge and absorbance if we know the current. Using Fick’s first law, the important and observable quantity that is, current density is given as

i(z ) ζ(b r |),t) ) nFD∂nCO

(6)

The charge density (σ), is time integration of current-density (i) and can be written as

σ(t) )

∫0t i(z ) ζ(br |),t′) dt′

(7)

where i ≡ O, R representing the oxidized or reduced species, δCi(ζ) is the difference between surface and bulk concentration, Di is diffusion coefficient (for simplicity we assume DO ) DR ) D), and b r is the three-dimensional vector, b r ≡ (x, y, z). There is a local transfer kinetics limitation at the interface (ζ) that in general is represented by the Nerstian boundary condition

The local absorbance density R(t), of an optically transparent electrode is given as

CO0 - CR0θ δCO(z ) ζ(b),t) r )) -Cs 1+θ

where σ(t) is the charge density. Therefore, the local absorbance R, up to time t is given by

(5)

where θ ) exp(-nf(E - E0′)) and δCO(t) is a function of the impressed potential, through Nernstian assumption and f ) F/RT where F, R, and T have the usual meanings. At initial time, and far from the interface a uniform initial and bulk concentration b,t)0) ) Ci(Zf∞,t) ) C0i . Since at (C0i ), is maintained, viz., Ci(r initial times, only the oxidized species is present so CR0 ) 0

R(t) )

R(z ) ζ(b r |),t) )

εD A0

( )

ε σ(t) nFA0

∫0t ∂nδCO(z ) ζ(br |),t′) dt′

(8)

(9)

and the total absorbance over surface ζ(r b|) at the interface up to time t is given by

Theory of Absorbance Transients

εD A (t) ) A 0

J. Phys. Chem. C, Vol. 114, No. 45, 2010 19359

∫0t ∫S dS0 β∂nδCO(z ) ζ(br |),t′) dt′ 0

∫S dx dy βR(z ) ζ(br |),t)

)

0

(10)

nFCO0 nFCO0 aˆ0 ) and aˆ1 ) (q - q) qp(1 + θ) qp(1 + θ) | 0 nFCO 1 ˆ2 ) d2K|′ q|q|,|′ - (q2 + qq|) a′ 2 2 (2π) qp(1 + θ) b| - K b|′ | 2 - 1 K b| - K b|′ ) b′ · (K |K 2 | (13)

∫

[

]

where nˆ ) (1/β)(-∇|ζ(r|),1), ∇| ) (∂/∂x,∂/∂y), and ∂n ) nˆ · ∇ signifies the outward drawn normal derivative at the surface. For calculating the absorbance at an electrode/electrolyte interface one needs to evaluate the concentration profile around an arbitrary rough electrode/electrolyte interface, as indicated through eq 10.

Therefore, the Laplace transformed absorbance at an arbitrary profile electrode is given by

A(p) ) Perturbation Solution for an Arbitrary Profile Electrode The perturbative solution for the concentration profile can be written in Fourier and Laplace transform domains for a weakly fluctuating rough surface. Therefore, the concentration profile for a weakly fluctuating two-dimensional rough surface in Fourier and Laplace transform domains is written as50,51

b|,z,p) ) δC(K

(

)

-CO0 b|) + exp(-q|z){(2π)2δ(K p(1 + θ) b|) + χˆ ζˆ (K b| - K b |′ )ζˆ (K b |′ )} (11) qζˆ (K

( )∫ ε nFA0

S0

b|) + aˆ1ζˆ (K b |) + d2r| [aˆ0(2π)2δ(K b|′ ) ζˆ (K b| - K b|′ ); K b| f b r |] (14) aˆ2ζˆ (K

b|); where notation for the inverse Fourier transform is {f(K b| · b 2 2 jK r| b b r|} ) (1/(2π) )∫d K| e f(K|) with j ) √-1. aˆ0, aˆ1, and K| f b aˆ2 are the operators for the smooth surface and the first- and second-order terms in Laplace transformed domain, respectively. Operator aˆ2 in eq 14 and aˆ2′ in eq 13 differ only in coefficient of last term. The operator aˆ2 is defined as follows

aˆ2 )

nFCO0

1 d2K|′ [q|q|,|′ - (q2 + qq|) ∫ 2 (2π) qp(1 + θ) 2

]

b|′ | 2 - K b|′ · (K b|′ ) b| - K b| - K |K b|) is the two-dimensional Dirac delta function in where δ(K b|, ζˆ (K b|) is the two-dimensional Fourier transform wave vector K Fourier transform of the rough surface profile ζ(r b|), and χˆ is an operator that depends on phenomenological diffusion length (1/q b|). ) (D/p)1/2) and wave vector of surface roughness profile (K The operator χˆ is defined as χˆ ) (q/2(2π)2)∫d2K|′ [2q|,|′ - q], b| - K b|′|2]1/2, where q ) (p/D)1/2, q| ) [q2 + K|2]1/2, q|,|′ ) [q2 + |K p is Laplace transform variable and the magnitude of wave b||. Equation 11 is useful to predict the concentravector, K| ) |K tion distribution around an arbitrary surface roughness profile under the influence of a single potential step function through θ(E) ) exp(-nf(E - E0′)). This can be achieved by taking inverse Laplace transform and inverse Fourier transform of eq 11 for a given roughness profile. The local absorbance density (in eq 9), R, in Laplace and Fourier transformed domain for weakly and gently fluctuating two-dimensional rough surface is obtained as

R(ζˆ ,p) )

( )

ε b|) + aˆ1ζˆ (K b |) + [aˆ (2π)2δ(K nFA0 0 b|′ )ζˆ (K b| - K b|′ )] (12) aˆ′2ζˆ (K

where aˆ0, aˆ1, and aˆ2′ are the operators for the smooth surface and the first- and second-order terms in Laplace and Fourier transformed domains, respectively. These operators are functional of phenomenological diffusion length (1/q ) (D/p)1/2), b|) and external wave vector of surface roughness profile (K perturbation potential (E) and are defined as follows

(15)

Equation 14 can be used to predict absorbance after taking the inverse Laplace transform and inverse Fourier transform for a known surface profile. This expression is very important to provide insights for the response of a nanometer to micrometer scales of a structured electrode, which represents an arbitrary geometric profile or a rough electrode with known profile. Absorbance for Surfaces with Random Roughness The irregularity of surface (or interface) can be satisfactorily described by a statistically homogeneous random surface ζ(r b|). The statistics for such centered Gaussian fields (surfaces) may b|)|2〉,which is the be written as the surface structure factor, 〈|ζˆ (K Fourier transform of the surface correlation function.47,50 The surface structure factor or power spectrum of roughness is the ensemble average of all the possible configurations. The ensemble averaged Fourier transformed surface profile and its correlation is assumed to follow the equation below

b|)〉 ) 0 〈ζˆ (K b|) ζˆ (K b|′ )〉 ) (2π)2δ(K b| + K b|′ )〈|ζˆ (K b|)| 2〉 〈ζˆ (K

(16)

To calculate the absorbance for a random electrode/electrolyte interface, we have to find the mean absorbance that senses the surface morphology through the single surface roughness characteristics, the surface structure factor.50 The spectroelectrochemically observed quantity, the total absorbance for electrode/electrolyte interface in Laplace domain for the twodimensional homogeneous random rough surface is given by

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J. Phys. Chem. C, Vol. 114, No. 45, 2010

〈A(p)〉 )

{

εCO0 q 1+ qp(1 + θ) (2π)2

∫ d K [q 2

|

|

Kant and Islam

b|)| 2〉 - q]〈|ζˆ (K

}

(17)

1 〈H2〉 ≈ 〈(∇|2ζ(r|))2〉 4

The inverse Laplace transform of it is written as

〈A(t)〉 )

2εCO0√Dt

√π(1 + θ)

ˆ 〈|ζˆ (K b|)| 2〉) (1 + R

(18)

ˆ brings in the dynamic effect of surface where operator R roughness on the absorbance through its action on the power spectrum of the roughness of the electrode, given by

ˆ )R

1 2(2π)2Dt

√πK| Dt erf(√K| Dt)) 2

(19)

Equation 18 is an elegant equation for the ensembled averaged absorbance transient in terms of surface structure factor. This equation possess information of various morphological features of the rough electrode surface (or interface) and provides physical insights into anomalies and possible interpretation for the spectroelectrochemical responses. To correlate the average absorbance with the various morphological features of the surface, we consider first the short time behavior of eq 18. Noting that the 2kth moments of power spectrum is related to mean square kth derivatives of surface profile,50 we have

1 (2π)2

∫ d2K| K|2k〈|ζˆ (Kb|)|2〉 ) 〈(∇|kζ(r|))2〉 (20)

Expansion of eq 18 for small t results in an equation that clearly interrelates the absorbance to several morphological features of roughness. The equation of absorbance transient of electrogenerated species in terms of moments of power spectrum or mean square derivatives of roughness is as follows

2εCO0√Dt

(

)

∞

(-1)km2k+2 〈A(t)〉 ) 1+ (Dt)k 2(k + 1)(2k + 1)k! √π(1 + θ) k)0 (21) )

∑

2εCO0√Dt

1 1 (1 + 〈(∇|ζ)2〉 - 〈(∇|2ζ)2〉Dt + 2 12 √π(1 + θ) 1 〈(∇ 3ζ)2〉(Dt)2 - ...) (22) 60 |

The above equation may be recast in terms of explicit physical features of roughness, viz., the excess surface area and mean square mean curvature (〈H2〉) as

〈A(t)〉 )

2εRCO0√Dt

√π(1 + θ)

(

1+

)

∆A 〈H2〉 Dt + ... A0 3

(23)

(24)

Equation 23 suggests that at very early times, the absorbance transient behaves as that of a smooth surface electrode (with excess area), and after some time it starts sensing local shape of the electrode and as time progresses it sees the contribution of higher and higher derivatives of the surface profile. Next, we consider the long time asymptotic expansion of eq 18, which is obtained by using an asymptotic formula for -erf[(K|2Dt)1/2] ∼ 1 - exp(-K|2Dt)/(πK|2Dt)1/2.60 We find

∫ d2K| (1 - exp(-K|2Dt) 2

m2k )

where the excess area is given by ∆A/A0 ) 〈(∇|ζ)2〉/2, the mean curvature27 is given approximately by H ≈ 1/2∇|2ζ(r|), and the ensemble averaged mean curvature for random surface is

〈A(t)〉 )

2εRCO0√Dt

√π(1 + θ)

(

1-

)

m0 + ... 2Dt

(25)

Thus, in the long time limit, the mean absorbance is sensitive to geometrical characteristics like the average roughness or mean square width of roughness, i.e., m0 ) h2. Equations 23 and 25 clearly demonstrate that absorbance transients strongly depend on the morphological characteristics like mean square (MS) width, MS gradient, MS curvature, etc. Absorbance Properties of Isotropic Self-Affine Fractal Surface Fractal geometry is very useful tool for describing many natural and artificial surface topographies. Most of these surfaces are described in terms of idealized fractals for the sake of simplicity. Idealized fractal models take into account the roughness over infinite length scales. However, these idealized fractal roughness profiles cannot be handled rigorously for practical purposes because of their peculiar mathematical properties, among which nondifferentiability and nonstationarity restrict their applicability in physical systems. Therefore, for practical purposes and realistic modeling of roughness the bandlimited self-affine fractal are often taken into consideration.54 Fractal surfaces, which exhibit statistical self-resemblence over all length scales, can be described by the complete power law power spectrum. However, any real fractal surface is characterized by the power law spectrum over a few decades in wavenumber with a high and a low wavenumber cutoff. The band-limited power law spectrum in several cases shows a gradual flattening for low wavenumber and a sharp decrease for the high wavenumbers. Such a power spectrum can be estimated by a function with a sharp cutoff at low wavenumbers and a sharp cutoff at high wavenumbers (i.e., power at high wavenumber is very small and low wavenumber features do not contribute significantly to flux heterogeneity in diffusion controlled mass transfer) which is given by53,54

b|)| 2〉 ) µ|K| | 2DH-7 〈|ζˆ (K

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

(26)

From this statistical framework we get four fractal morphological characteristics of power-spectrum of roughness, namely, fractal dimension (DH), strength of fractality (µ), and lower and upper cutoff length scales (l) and (L). The fractal dimension (DH) is the global property that describes the scale invariance property of the roughness and is largely associated with

Theory of Absorbance Transients

J. Phys. Chem. C, Vol. 114, No. 45, 2010 19361

anomalous behavior in diffusion controlled response; hence its time exponent is usually assumed to be a function of roughness. The strength of fractality (µ) is the measure of width of the interface and is related to topothesy. The geometrical interpretation of the topothesy is the length scale over which the profile has a mean slope of 1.55 The lower cutoff length scale (l) is the length above which we observe roughness and the upper cutoff length scale (L) is the length above which we do not observe the roughness. As mentioned earlier, the moments of the power spectrum of roughness have important morphological information. In general, m2k are the 2kth moments of the power spectrum of band-limited fractal and its expression is given as48

(

µ l-2(δ+k) L-2(δ+k) m2k ) 4π δ + k δ+k

)

(27)

〈A(t)〉 ) AP(t)(1 - RF1(t) + RF2(t,l) - RF2(t,L)) m0 µ + Γ(DH - 5/2,Dt/l2,Dt/L2) RF1(t) ) 2Dt 8π(Dt)DH-3/2 µ RF2(t,λ) ) × 8π(DH - 2)λ2DH-3√t*

)

(28) where dimensionless time t* ) Dt/λ2, Γ(R,x0,x1) ) Γ(R,x0) Γ(R,x1) ) γ(R,x1) - γ(R,x0), Γ(R,xi), and γ(R,xi) are the incomplete Gamma functions60 and the minimum wavenumber (1/L) and maximum wavenumber (1/l) are two cutoff wavenumbers in the power spectrum. An alternative method for derivation of the leading asymptotic form of eq 28 is from the leading asymptotic form of the current transient equation of refs 47 and 48. The approximate form of the current for an isotropic self-affine fractal surface can be written as47,48

nFDA0CO0

√πDt(1 + θ)

(

1+

(

µ l-2δ Γ(δ) 8π δDt (Dt)δ+1

))

t > ti

2εCO0√Dt

√π(1 + θ)

(

1-

In most of the cases electrode roughness is modeled as realistic fractals but in some cases, roughness clearly has no self-similarity property and has a dominant length scale of randomness. Such roughness can be modeled with a power spectrum function with a single lateral correlation length. One of the statistical models for nonfractal surfaces can be described in terms of a Gaussian correlation function (W(r b|)) with a mean MS width (h2) and a correlation length, a. The Gaussian correlation function is given as50,56

W(b r |) ) h2e-r| /a 2

2

(32)

Its Fourier transform is the surface structure factor that possesses various morphological features for the nonfractal surfaces and is given by

b|)| 2〉 ) πh2a2e-K|2a2/4 〈|ζˆ (K

(33)

(29)

where δ ) DH - 5/2. Equation 29 is applicable for time greater than the inner fractal crossover time, ti. Integrating eq 29 between limits ti and t, we obtained an asymptotic expression for the charge transient. Substituting the charge transient in eq 1 we obtained the expression for the absorbance transient. The absorbance for t . ti for isotropic self-affine fractal electrode/ electrolyte interface is given by

〈A(t)〉 ≈

ti is obtained by the solution of the above equation, and the region near ti can be observed if the observation time window starts at a time smaller than ti. The inner transition is a strong function of fractal dimension and lower cutoff length (l) through m2 ≈ (µ/4π)[l-2(δ+1)/(δ + 1)] and m4 ≈ (µ/4π)[l-2(δ+2))/(δ + 2)]. Similarly, the outer transition time, to, is the time at which anomalous time behavior of the absorbance ends. Time to cannot be obtained by equating (25) and (30), as the intermediate and long time asymptotically merge with each other at very long times. Outer crossover time (to) can be estimated using method discussed in ref 51. Absorbance for Nonfractal Electrode Surfaces

(√π erf(√t*) - t*2-DHγ(DH - 3/2,t*))

〈I(t)〉 ≈

m2 m4Dti µl-2δ µΓ(δ) ) δ+1 2 12 8πδDti 8π(2δ + 1)(Dti)

(31)

These moments have physical significance as they are measures of morphological features of rough surface; viz., m0, m2, and m4 are mean square (MS) width, MS gradient, and MS mean curvature of roughness, respectively. The exact representation of absorbance for an isotropic selfaffine fractal surface is obtained by substituting eq 26 in eq 18 and representing various integrals in terms of special functions. The mean absorbance for realistic fractal roughness is as follows:

(

The inner fractal crossover time, ti, is the time after which the anomalous intermediate time regime is observed in the absorbance. Contribution of 〈A(ti)〉 is small for all t . ti; hence it is ignored in above formula. The inner fractal crossover time, ti, is the time after which anomalous intermediate time regime is observed in the absorbance and is used to mark the scaling region of the absorbance. Equating (23) and (30), we obtain for ti

)

µΓ(δ) µl-2δ + 8πδDt 8π(2δ + 1)(Dt)δ+1 t . ti

(30)

In this framework, we get two morphological parameters, viz., width of the interface, h2, and transversal correlation length, a. The transverse correlation length (a) is the length that is a measure of the average distance between consecutive peaks and valleys on the random nonfractal surface, and h is the measure of mean square departure of the surface from flatness. The excess surface area, Areal and average square gradient are related as 1/2〈(∇|ζ)2〉 ) Areal/A0 ) R* - 1 ) 2h2/a2, where A0 is the projected area of the electrode and R* is the roughness factor given by R* ) 1 + 2(h/a)2. The ensemble averaged mean square curvature is 〈H2〉 ≈ 8h2/a4. The ensembled averaged absorbance, 〈A(t)〉, for nonfractal electrode surface (or interface) is given by

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〈A(t)〉 )

2εRCO0√Dt

√π(1 + θ)

(

h2 tan-1 1+

( ) 2√Dt a

a√Dt

)

Kant and Islam

(34)

The second term in parentheses is the contribution due to roughness. If the roughness (h2) is zero, then only the first term survives, giving smooth electrode response. Equation 34 for absorbance is related to the current transient in refs 50 and 51 through an integration with respect to time of the mean current for a Gaussian power spectrum of roughness. Results and Discussion In this section we will analyze absorbance transients in detail for realistic self-affine fractal electrodes as well as for nonfractal electrodes. Figure 1 depicts the rough OTE with fractal morphological and phenomenological scales that control the spectroelectrochemical response. As discussed earlier, the power spectrum of such roughness consists of four morphological characteristics, viz., the fractal dimension (DH), lower (l) and upper (L) cutoff length scales of fractality, and a proportionality factor (µ) or strength of fractality. Out of four only three, i.e., DH, µ, and l, contribute strongly to the spectroelectrochemical response of electrode while L contributes weakly. Figure 2 is the log-log plot of absorbance and time, which illustrates the change in absorbance with time while varying one of the dominant morphological parameters, i.e., DH, µ, and l. This figure illustrates the existence of three time regimes, that is, short, anomalous intermediate, and long time regimes. The anomalous intermediate and long time regimes are seen in the plot. The short time regime with square root time dependence is not visible in the plots because it does not come in the time region shown in the figure. The anomalous intermediate regime for the absorbance can also be characterized by using scaling arguments. The de Gennes scaling law for the diffusion controlled current is given as57–59 I ∝ t-β, and its time integration is proportional to the total charge used in the reaction or absorbance. The scaling form of absorbance transient is as follows

A(t) ∝ t-β+1

(35)

where the exponent, β, depends on the interfacial roughness given by β ) (DH - 1)/2 and it varies from 0.5 (planar surface) to 1; thus, DH varies from 2 (planar surface) to 3 (nearly porous surface), respectively. From this, for DH ) 2 the scaling exponent in absorbance becomes 0.5 while for DH ) 3 it is 0. This means, if the fractal dimension approaches 3, absorbance is very weakly dependent on time and the intermediate region becomes flatter. In other words, if roughness increases initially, absorbance also increases and after a certain value the absorbance shows a weak dependence on time in the intermediate region (see Figure 2) and shows smooth electrode response in the long time regime. It is clear that the scaling result in eq 35 is unable to include a complete set of realistic fractal morphological characteristics of roughness and because of this it cannot capture all aspects of experimental data. Particularly, difficulties arise in the transition region and often mix the data of the scaling region with the transition region, and hence there is an inaccurate prediction of fractal dimension with scaling relations. Figure 2a demonstrates the variation of absorbance transient with fractal dimension. As the fractal dimension increases, the roughness factor also increases; the magnitude of the absorbance

Figure 2. Absorbance transients and effect of three dominant fractal morphological characteristics on them: (a) DH, (b) µ, and (c) l. The above plots are generated by using (a) DH ) 2.00, 2.15, 2.20, 2.25, and 2.30 (from bottom), l ) 50 nm, L ) 1 µm, and µ ) 10-5 au, (b) µ ) µ0, 5 × µ0, 10 × µ0, and 20 × µ0 au (from bottom) where, µ0 ) 10-6, DH ) 2.25, l ) 50 nm, and L ) 1 µm, and (c) l ) 50, 100, 200, and 400 nm (from top), DH ) 2.25, µ ) 10-5 au, and L ) 1 µm. Other parameters used in the calculations are as diffusion coefficient (D ) 5 × 10-6 cm2/s), concentration (C ) 10-6 mol/cm3), and ε ) 105 cm2/ mol.

transient increases, but after a certain value of fractal dimension, the anomalous intermediate region shows a weak dependence on time. Figure 2b illustrates the variation of absorbance with width of the interface. If the width of the interface increases, the roughness factor also increases. The absorbance transient increases, but after a certain value of roughness, the anomalous region becomes flat or approximately time independent. Similarly, if the lower cutoff length scale decreases, the roughness factor also increases. Again absorbance increases in the initial time but the anomalous region starts becoming flatter with an increase in roughness due to lowering of the lower cutoff length. Therefore, there is a similar kind of observation: we either

Theory of Absorbance Transients increase the width of the interface (or strength of fractality) in Figure 2b or decrease the lower cutoff length scale in Figure 2c as in the case of an increase of fractal dimension in Figure 2a. The inset figures depict a traditional representation of absorbance transient where the abscissa is t1/2 and the ordinate is A. These figures depict a smooth surface response as a straight line and a rough electrode enhanced response with curved lines. Traditional representation does not explicitly reveal various time regimes while the double logarithmic plot of absorbance vs time can show three distinct time regions in Figure 2c. In the short time region, the diffusion layer thickness is small compared to the surface irregularities. In the intermediate time region, the diffusion layer thickness is comparable to the surface irregularities. Finally, in the long time region, the diffusion layer thickness is much larger than the surface irregularities. Thus, the intermediate time region is the region of transition from short time asymptotic to long time asymptotic behavior. From the above discussion it is clear that absorbance is strongly dependent not only on fractal dimension but also on the other two morphological parameters, µ and l. Thus, rigorous theory based on the power spectrum of roughness characterizes three regimes, viz., an anomalous intermediate region along with short and long time regime, more efficiently and accurately than one based on the scaling approach. The double logarithmic plots of the absorbance and time give more insights into the physical picture in all regimes and explain the transition times as well, which is not possible to observe in nonlogarithmic traditional plots of A(t) vs t1/2 (see insets of Figures 2 and 3). As expected, it is linear for the smooth electrode surface (first plot from bottom). As the roughness increases, it increases absorbance and absorbance starts to deviate from a linear response because of roughness contributions. The deviation is larger with the change in morphological features, which increases the roughness factor especially in the short time and enhancing the absorbance as well. Figure 3 shows the absorbance transients of nonfractal rough electrode and illustrates the effect of two nonfractal morphological charateristics, viz., MS width (h2) and transversal correlation length (a). Figures are plotted as double log plots of absorbance vs time while traditional plots of A(t) vs t1/2 are shown in inset. Common observations in this case are the presence of three time regions, which are the short time t1/2 region, intermediate transition region, and long time t1/2 region. Here, the short time regime falls within experimental time scales in the plots. Figure 3a shows that if the width of the interface increases, absorbance also increases and the outer crossover time (to) also increases. In the short time regime the absorbance transient increases by a roughness factor and follows t1/2 behavior, in the intermediate transition regime there is a slow decrease in the slope of double log plots of the absorbance transient and it eventually becomes weakly dependent on time at higher values of the roughness factor. Finally, it merges with the planar response in the long time regime. Figure 3b shows that if we decrease the transverse correlation length, a, the roughness factor as well as the absorbance increases. Thus, in the short time regime, the absorbance transient increases by a roughness factor while in the intermediate regime of absorbance it increases but shows a weaker dependence; these effects are very similar to what one sees in fractal OTE. In the long time regime, all responses merge with the smooth surface absorbance. Conclusions The theory developed in present paper enables us to discern the effect of the morphology of electrodes on diffusion

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Figure 3. Absorbance transients for a nonfractal OTE and effect of two nonfractal morphological characteristics: (a) root-mean-square (rms) width of the interface, h, at fixed correlation length, and (b) correlation length, a, while rms width of the interface is fixed. The plots are generated by using (a) h ) 3, 5, 7, 10 µm (from bottom), and a ) 6 µm, and (b) a ) 1, 2, 4, 8 µm (from top), and h ) 5 µm. Other parameters are used in the above calculations are as diffusion oefficient (D ) 5 × 10-6 cm2/s), molar absorptivity (ε ) 105 cm2/mol), concentration (C ) 10-6 mol/cm3).

controlled reactions and their influence on chronoabsorptometric and chronocoulometric measurements. The theory developed here allows us to address deterministic as well as stochastic surface roughness in OTE. Stochastic morphology offers realistic models of roughness that may have fractal or nonfractal nature. The randomly rough surface can be characterized by using a power spectrum of roughness. The conclusions drawn are (1) absorbance increases with an increase in roughness of the surface, (2) at very early times the diffusion layer thickness is small compared to the roughness features, and the electrode response is similar to that of a planar electrode (with excess area) (after a certain time it starts to sense the local shape of the electrode and as time progresses it senses the contribution of higher and higher derivatives of the surface), (3) the intermediate regime shows anomalous power-law behavior but at higher value of roughness factor it becomes weakly dependent on time and finally for large roughness it is independent of time, (4) the scaling law for absorbance of fractal OTE based on the de Gennes scaling expression for the flux does capture some features of intermediate times but it is not sufficient for the characterization of all regions in absorbance transients, (5) the long time regime occurs when the diffusion layer thickness is larger than the mean square width of the interface, and (6) our ab initio methodology based on the power spectrum of roughness characterizes all three absorbance regions successfully. Our theoretical model successfully provides the physical picture that relates the response to the phenomenological length (the

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diffusion length) and morphological length scales that characterize the nature of the rough surface. These findings provide insights for spectroelectrochemical responses and techniques that can be critical for assigning the mechanisms, for identification of intermediates and products, and for other electron transfer processes taking place at an electrode/electrolyte interface. Finally, one can say this theory is an indispensable step in the quantitative description of the role of roughness on the single potential step chronoabsorptometric response of an optically transparent electrode. References and Notes (1) Kaim, W.; Klein, A. Spectroelectrochemistry; RSC Publishing: Cambridge, U.K., 2008. (2) Bard, A. J.; Faulkner, L. R. Electrochemical Methodes: Fundamentals and Application; Wiley: New York, NY, U.S., 1980. (3) Kaim, W.; Fiedler, J. Chem. Soc. ReV. 2009, 38, 3373, and references theirin. (4) Plieth, W.; Wilson, G. S.; Delafe, C. G. Pure Appl. Chem. 1998, 70, 1395. (5) Kuwana, T.; Darlington, R. K.; Leedy, D. W. Anal. Chem. 1964, 36, 2023. (6) Strojek, J. W.; Kuwana, T. J. Electroanal. Chem. 1968, 16, 471. (7) Kuwana, T.; Strojek, J. W. Discuss. Faraday Soc. 1968, 45, 134. (8) Strojek, J. W.; Kuwana, T.; Feldberg, S. W. J. Am. Chem. Soc. 1968, 90, 1353. (9) Osa, T.; Kuwana, T. J. Electroanal. Chem. 1969, 22, 389. (10) Strojek, J. W.; Gruver, G. A.; Kuwana, T. Anal. Chem. 1969, 41, 481. (11) Winograd, N.; Blount, H. N.; Kuwana, T. J. Phys. Chem. 1969, 73, 3456. (12) Grant, G. C.; Kuwana, T. J. Electroanal. Chem. 1970, 24, 11. (13) Kuwana, T.; Heineman, W. R. Acc. Chem. Res. 1976, 9, 241. (14) Petek, M.; Neal, T. E.; Murray, R. W. Anal. Chem. 1971, 43, 1069. (15) Petek, M.; Neal, T. E.; McNeely, R. L.; Murray, R. W. Anal. Chem. 1973, 45, 32. (16) Chia-yu, Li; Wilson, G. S. Anal. Chem. 1973, 45, 2370. (17) McCreery, R. L.; Pruiksma, R.; Fagan, R. Anal. Chem. 1979, 51, 749. (18) Robinson, R. S.; McCreery, R. L. Anal. Chem. 1981, 53, 997. (19) Bhandari, S.; Deepa, M.; Pahal, S.; Joshi, A. G.; Srivastava, A. K.; Kant, R. ChemSusChem. 2010, 3, 97. (20) Bhandari, S.; Deepa, M.; Srivastava, A. K.; Kant, R. J. Mater. Chem. 2009, 19, 2236. (21) Bhandari, S.; Deepa, M.; Srivastava, A. K.; Lal, C.; Kant, R. Macromol. Rapid Commun. 2008, 29, 1959. (22) Deepa, M.; Bhandari, S.; Arora, M.; Kant, R. Macromol. Chem. Phys. 2008, 209, 137. (23) Bhandari, S.; Deepa, M.; Srivastava, A. K.; Lakshmikumar, S. T.; Kant, R. J. Nanosci. Nanotechnol. 2008, 9, 1.

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