Experimental Validation of Roughness Power Spectrum-Based Theory

Article pubs.acs.org/JPCC

Experimental Validation of Roughness Power Spectrum-Based Theory of Anomalous Cottrell Response Shruti Srivastav, Shweta Dhillon, Ratnesh Kumar, and Rama Kant* Department of Chemistry, University of Delhi, Delhi 110007, India ABSTRACT: We experimentally validate theoretical relation between the roughness power spectrum (PS) and electrochemical current transient for a reversible charge transfer system under a single potential step. Roughness features at the electrochemically roughened electrode are characterized using standard measurements such as scanning electron microscopy (SEM), atomic force microscopy (AFM) and cyclic voltammetry (CV). The PS obtained from AFM shows composite finite fractal and nonfractal nature in roughness, whereas the PS from SEM shows only a finite fractal nature. AFM or SEM measurements provide knowledge of fractal dimension (DH) and two fractal cutoff lengths (S and L ). Topothesy length (Sτ ) or the related proportionality factor ( μ ≡ S2τ DH − 3) from PS data of AFM requires extrapolation of data for unit wavenumber, but this method usually provides unphysical values of μ. We provide a novel method to determine the topothesy of electrodes from CV measurements of electroactive area in conjunction with SEM or AFM measurements. Chronoamperometric measurements were made on morphologically characterized Pt-electrodes for a solution of K4[Fe(CN)6] and K3[Fe(CN)6] in 3 M NaNO3. The transient response observed experimentally is validated using the measured PS in the theoretical equation for the current. The transient response does not show contributions from Gaussian PS in the low wavenumber region; this is due to the fact that the effective lower cutoff wavenumber is usually limited up to the inverse of the width of roughness (or topothesy length). Fractal dimensions obtained through chronoamperometric measurement on electrodes using Pajkossy’s approach do not correspond to the one obtained from AFM and SEM measurements. Finally, the anomalous response in the Cottrell measurements can be understood through PS-based theory.

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electrodes, several theories have popped up.1 Among these, scaling laws,19,20 fractional diffusion,21 deterministic profiles, and perturbation approach have enjoyed quite a bit of attention. However, some of these approaches do not take into account the complete topographical features at the electrode surface. The most popular parameter characterizing the morphology of surfaces is the root-mean-square (rms) roughness, which represents the variance of surface height fluctuation and roughness factor (R*), which is the ratio of active area to its geometric area. However, this statistical description, though simple, does not account for the lateral distribution of various surface features. A more complete description of electrode morphology is provided by the power spectrum (PS) of the surface topography of electrode, which performs a decomposition of the surface profile into its spatial wavelengths and expresses the roughness power per unit spatial frequency or wavenumbers over the sampling length. This allows comparison of roughness features over different spatial frequency ranges. Transient response in potentiostatic experiments is a standard procedure that can yield information about many electrochemical processes and several kinetic parameters such as rate

INTRODUCTION Surface disorder and random roughness can play a significant role in many diffusion limited physical and chemical processes at interfaces, for instance in membrane transport, enzyme catalysis, spin relaxation, and heterogeneous catalysis. Surface disorder and random roughness can play a determining role in many physical and chemical properties of surfaces and interfaces. Highly dispersed metals used in heterogeneous catalysis and electrocatalysis exhibit extremely rough and irregular surfaces. Microstructured and nanostructured materials are currently of interest for devices such as electrochemical sensors,1,2 supercapacitors,3 photovoltaics and fuel cells4 because of their large surface area, novel size effects,5 significantly enhanced kinetics, and so on. For this reason, the objective characterization of rough surface topography and its response has been an important investigation subject for quite some time now. Much attention has been paid to the characterization of the real surfaces made by fracture,6−9 electrodeposition,10−13 vapor deposition14−16 using fractal geometry, and comparing fractal geometry employing electrochemical impedance spectroscopy (EIS) data and scanning electron microscopy (SEM) images.17 The real surfaces are usually believed to show a self-affine scaling property, i.e., asymmetric scaling behavior perpendicular to the surface18 rather than a self-similar scaling property, i.e., isotropic scaling behavior in all directions. In connection to this aspect of disordered © 2013 American Chemical Society

Received: September 14, 2012 Revised: March 26, 2013 Published: April 1, 2013 8594

dx.doi.org/10.1021/jp4015965 | J. Phys. Chem. C 2013, 117, 8594−8603

The Journal of Physical Chemistry C

Article

knowledge of PS, along with other features such as mean square gradient and mean square curvature.23,24

constant, diffusion coefficient, and number of electrons transferred. In more than a decade long work, Kant and co-workers have established a series of relationships between observables such as current, admittance, absorbance, and charge transients with the morphology of the electrodes via PS of the rough interface.22−35 These include anomalous diffusion reaction rates or diffusion-controlled potentiostatic current transients,22,24−26 anomalous Warburg impedance at a rough electrode,32 solution resistance effect on anomalous Cottrellian current31 and anomalous Warburg impedance34 at a rough electrode, partial diffusion-limited interfacial transfer/reaction at rough electrode,28 and Gerischer admittance at a rough electrode.33 Other important problems are absorbance transient at a rough optically transparent electrode (OTE),29 chronocoulometric response at the rough electrode,30 arbitrary potential sweep methods on an arbitrary topography electrode,35 and electrical double layer potential capacitance.28,36,37 Despite these theoretical developments in the direction of relating PS and electrochemical response, the need to unravel the connection between the experiments still remains unanswered. In this regard, this paper attempts for the first time to validate one of the theoretical results22−25,27 through experiments. We wish to address in these preliminary investigations the PS characterization of electrode morphology using standard techniques to correlate and validate the roughness signature in the potentiostatic current transient response. We compare the results with de Gennes38 and Pajkossy’s19 scaling result on rough fractal electrodes.

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EXPERIMENTAL SECTION Electrode Pretreatment. Platinum wire of diameter 0.5 mm purchased from Arora Matthey and Company was used as the working electrode for electrochemical measurements. Two platinum working electrodes (R1 and R2) were carefully polished with emery paper of grade NLT and then polished with alumina of size 10 μm on a microcloth for 10 min. After polishing, each electrode was sonicated in piranha solution for 20 min followed by sonication in triply distilled water for 10 min. Electrodes were rinsed with distilled water. R3 was polished with emery paper of grade NLT and then treated with boiling nitric acid, followed with flame annealing. The electrodes were then electrochemically cleaned and activated in 0.5 M H2SO4. These electrodes were then subject to a electrochemical roughening procedure for different times. R1 was electrochemically pretreated in 0.5 M H2SO4 and used as such for the electrochemical experiments. The R2 and R3 electrodes were roughened as per the procedure suggested by Reiner and co-workers.40 R2 was cycled between −0.2 and 1.4 V at 50 V/s in 0.5 M H2SO4 for 2500 cycles. After every 50 000 cycles, R2 was subject to stabilization at 150 mV/s for 10 cycles. Similarly, R3 was cycled between −0.2 to 1.4 V at 50 V/s in 0.5 M H2SO4 for 2000 cycles. After every 50 000 cycles, R3 was subject to stabilization at 150 mV/s for 2500 cycles. Electrolyte Preparation. Electrolyte solutions were prepared using “analytical grade reagents” in triply distilled water. All solutions were degassed with purified nitrogen. Electrochemical cleaning and roughening was done in 0.5 M Merck Suprapure sulphuric acid, and electrochemical measurements were taken in a mixture of 15 mM solution of AR-grade potassium ferrocyanide and potassium ferricyanide, prepared in 3 M NaNO3. Electrochemical Experiments. Experiments were carried out using μAutolab III. Prior to each experiment, nitrogen purging had been done for about 15 min. The electrochemical experiments were conducted potentiostatically in a typical threeelectrode cell with a platinum rod counter electrode and a 3 M Ag/AgCl electrode reference (E0′ = 0.210 V vs standard hydrogen electrode (SHE)) for R1, R2 electrodes and saturated calomel electrode (SCE) (E0′ = 0.250 V vs SHE) for the R3 electrode at room temperature. In this cell, the geometrical area of the working electrodes R1, R2, and R3 exposed to electrolyte was 0.080, 0.099, and 0.11 cm2, respectively. The pretreated electrodes were used to examine 15 mM ferrocyanide and ferricyanide in 3 M NaNO3 background electrolyte so as to suppress the migration effects. After this, chronoamperometric measurements were carried out for the 1:1 v/v mixture at the electrodes for various time durations at the potential bias of 0.305 V vs Ag/AgCl electrode reference for R1 and R2 electrode and 0.341 V vs SCE reference for R3 electrode. These chronoamperograms were then superimposed to get the complete recording ranging from milliseconds to tens of seconds. No iR compensation was applied while carrying out these measurements.

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THEORETICAL FRAMEWORK The electrochemical observable, the total current, is obtained by the perturbation theory for the diffusion to the rough electrode electrolyte interface for a Nernstian charge transfer process. The total (averaged) current at the stationary, Gaussian random electrode surface (see height distribution curves in the Appendix, section A) is given by22−27 ⎡ 1 ⟨I(t )⟩ = Ip(t )⎢1 + ⎣ 2(2π )2 Dt

2

⎤

∫ d2K (1 − e−K Dt )⟨|ζ (̂ K⃗ )|2 ⟩⎥⎦ (1)

where ⟨I(t)⟩ is the ensemble averaged current at the rough electrode, Ip(t) is the Cottrell current given by (nFA0Cs√D)/ (πt)1/2, A0 is the geometric area, and t is time. Cs is the potentialdependent change in surface concentration given by Cs = (C0O −C0Rθ)/(1 + θ); θ is e−nf(E−E0′). C0O and C0R are the bulk concentration of oxidized and reduced species, respectively; E0′ is the formal potential, E is the step potential, f = F/RT, F is Faraday’s constant, T is the temperature, D is the diffusion coefficient, ⟨|ζ̂(K⃗ ∥)|2⟩ is the PS of the surface, and |K⃗ ∥| is the magnitude of two-dimensional (2-D) wavevector. This expression relates the surface topography to the current transient through the PS of roughness. Equation 1 for the response of an irregular electrode is applicable for random surfaces that are statistically described in terms of the first two correlation functions, viz. centered Gaussian fields, and are statistically homogeneous surfaces (i.e., two point correlation function does not depend on the absolute position of points but only on their separation distance).39 Furthermore, the two-point correlation function is related to the PS of roughness by its Fourier transform. Hence, the electrode’s statistical topography can be described through the PS function. Most popular parameters characterizing the morphology of surfaces are the rms roughness and roughness factor, which can be obtained through the

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RESULTS AND DISCUSSION

PS characterization can be easily done with atomic force microscopy (AFM) as it produces high contrast on relatively flat surfaces, and produces three-dimensional, digital data by default. AFM can directly supply data suitable to measure 8595



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roughness parameters. Therefore, an AFM of the electrochemically roughened wire suits our purpose to obtain a PS for the surface that acts as an input to our model equations. Conventionally, PS functions obtained from AFM measurements have roughness values in a limited range of spatial frequencies. The range depends on the scan length and sampling distance, and it also can be additionally restricted or constrained by the effect of measurement artifacts.41 These limitations, however, can be overcome when the topographic measurements performed on different scales are appropriately combined. Recordings were performed on 5 × 5 μm2 scan size. In order to condense the acquired information and facilitate the comparison between the samples, 2-D PS is calculated.42 Scans made by rotating the samples R1 and R2 in different directions were practically identical, so it could be assumed that the sample is isotropic, whereas R3 shows slight anisotropic surface. For R3, averaging was done by rotating the sample in the x and y directions. For a large statistically homogeneous surface, the ensemble average PS in eq 1 can be related to surface average PS, as the large surface may possess all possible configurations of roughness. The PS is defined as follows: 2 1 ⟨ ζ (̂ K⃗ ) ⟩ = lim 2 | L →∞ L

+L /2

It is evident from eqs 4 and 5 that the most important quantities to completely characterize the fractal surface are DH, Sτ , and S . The nonfractal nature of random roughness can be captured through several PS functions. The most prominent one is the Gaussian PS (⟨|ζ̂(K⃗ ∥)|2⟩G) and is given by22,23 ⟨|ζ (̂ K⃗ )|2 ⟩G = πa 2h2 exp( −a 2K 2/4)

(6)

For the Gaussian PS there are two features: correlation length (a) and rms width (h). The Gaussian PS is usually more significant in the lower wavenumbers. Its mean square height is m0 = h2, and the mean square gradient is m2 = 4h2/a2.24 However, most surfaces may require a combination of these two or more PSs as the random roughness cannot be controlled in a particular fashion. Depending upon the technique employed, one may get a single or a combination spectrum. The PS obtained through AFM is generally a combination of several PSs41 as seen in our measurements. AFM has the vertical resolution of 0.1 nm and hence the underlying Gaussian PSs with very low root-meansquare widths (h1,h2) also get resolved in the AFM recordings. However, this may not be the case in low vertical resolution techniques, e.g., SEM. Figures 1, 2, and 3 show the experimental

+L /2

∫−L/2 ∫−L/2

dx dyζ(x , y)

× exp( −ι(xKx + yK y))|2

(2)

where ⟨|ζ̂(K⃗ ∥)|2⟩ is the 2-D PS. L is the sample length, K∥ is the wavenumber in two dimensions, Kx and Ky are the spatial wavenumber in the x and y directions, ι is (−1)1/2, and ζ(x,y) is the 2-D surface profile. The PS can be used to characterize the fractal and nonfractal nature of randomly rough surfaces. The fractal surfaces exhibit statistical self-resemblance over a range of length scales and can be described using power-law PS.16 The PS of a band-limited fractal surface (|ζ̂(K⃗ ∥)|2⟩F) is given by16,24,25,27 ⟨|ζ (̂ K⃗ )|2 ⟩F = μ |K⃗ |2DH − 7 ;

1/L ≤ |K⃗ | ≤ 1/S

(3) Figure 1. PS of R1 rough electrode obtained from AFM and SEM. Solid violet line is the total calculated PS, colored circles are the recorded PS from SEM image analysis, closed black circles are the PS from AFM. Dotted pink and green lines are the Gaussian part of the PS; dotted blue line is the power law region with the lower length scale cutoff from AFM, and dotted black line is the cutoff length from SEM.

where DH is the Hausdroff dimension, which is a global property that describes the scale invariance property of the roughness, μ is the strength of fractality or the proportionality factor related to the topothesy length (Sτ ) of the electrode given by S2τ DH − 3 . The sub-Brownian fractal surface (DH < 2.5) has a persistent behavior in roughness profile. These statistically isotropic finite fractals have a lower (S ) and a upper length (L) scale cutoff. Usually the band-limited fractal PS is more evident at the higher wavenumber of the PS. The sub-Brownian fractal surface (DH < 2.5) has a persistent behavior in roughness profile. Similarly, superBrownian fractal surface (DH > 2.5) corresponds to a case indicating antipersistence behavior in its roughness profile.43,44 Another interesting quantity of physical relevance that can be incepted are moments of PS, rms width (m0), and rms gradient (m2) given by the expression.24−27 For the cases where L or the outer cutoff length scale (mostly the sample length) is usually much larger than all other length scales in the system, these can be approximated as25,26 m0 =

μ (S −(2DH − 5) − L−(2DH − 5)) 2π (2DH − 5)

(4)

m2 =

μ (S −(2DH − 3) − L−(2DH − 3)) 2π (2DH − 3)

(5)

PS from AFM and SEM analysis along with the SEM images in the inset for R1, R2, and R3 rough electrodes obtained, respectively. It should be mentioned that the SEM image electrode for R2 indicates a multivalued surface, while that of R3 shows slight anisotropy. For the AFM measurements (open black circles), the experimental PS can be treated as a combination of several PS functions. The presence of two shoulder shapes in low wavenumber of recorded (double log plot of) PS is an indication of two Gaussian PSs (green dashed line and violet dashed line) and the presence of a high wavenumbers linear regime is an indication of a power law PS (overlapping with power law of AFM). Hence, in the lower wavenumber we use two Gaussian PSs with different correlation lengths followed by the power law PS with a sharp cutoff in the high wavenumber. The results here are analyzed in detail using the composite PSs. Hence, the overall PS is given by the following equation: 8596



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differ in their roughness properties vastly, as marked by the roughness factor and other roughness measures. One of the limitations of the AFM technique is that the measured roughness depends on the probe used and its condition, as the horizontal resolution is tip size dependent. This is because less sharp probes will tend to smooth out the data, as they cannot image fine features on the sample surface. However, SEM images show that surfaces formed with large roughness may have several domains of roughness. One general approach to counter this problem is to take AFM for the different domains at different positions and then average over these PS. Usually, for a random surface electrode it is expected that various domains will yield similar PS if the same procedure for their preparation is employed, sometimes only varying in the proportionality factor from one domain to another. Another way to counter the problem of averaging is using a technique that can scan a more comprehensive area of interest in less scans like SEM. For SEM images, however, the PS was obtained through image reconstruction and its statistical analysis.45 SEM as a technique has its own advantages. It is much faster and economical than AFM measurements. Unlike AFM, it can provide a better insight of the both local and global roughness, providing a more representative picture for the surface for different domains. In the raw analogue images obtained from SEM, each pixel on the screen is digitized in a gray scale with 256 levels (.tif format)46 assuming that the gray scale corresponds to the scaled height, for a chemically homogeneous surface (refer to SEM images in the insets of Figures 1, 2, and 3). A word of caution to be mentioned here is that for a chemically inhomogeneous surface, the gray scale conveys different information. These images are used to obtain the relevant PS and statistical values of the surface. SEM data often has high wavenumber noise, which can be seen in their PS near rough to smooth transition. This noise appears as bandlimited white noise. In order to further identify the noise region in PS, the addition of external noise in image data mainly shifts the PS in the higher wavenumbers, signifying the white noise.47 In the case of AFM data, we observe a sharp drop-off at certain wavenumbers in contrast to the SEM analysis where crossover from a power law to white noise PS is identified as a higher cutoff in the wavenumbers. The PS of roughness is a combination of white noise for high wavenumbers and a power law function for the intermediate wavenumber. The PS obtained from each measurement, i.e., AFM and SEM, is normalized with the mean square height or area under each curve.24 The PSs so obtained from AFM and SEM are comparable. S is calculated using the intersection of two lines, i.e., the power law fit and the white noise region or the zero slope region. In Figures 1, 2, and 3, the PS from AFM so obtained is shown in open black circles with a power law fit in the blue dashed line. The regression parameters for fitting the SEM PS and AFM PS are provided in the Appendix, section C. The morphological parameters of roughness that need to be identified in this framework are DH, S , and L for fractal component of PS. The data is fit to a power law whose slope will provide for DH. The rough to noisy crossover in the high wavenumber region gives the value of S , and L is identified as the lowest wavenumber closest to the power law fit on the recorded PS. Vertical and horizontal resolution for SEM can be up to tens of nanometers,48 restricting the information of finer features such as Gaussian PS with low a and h on the surface. All the morphological parameters obtained from each of the measurements is tabulated in Table 1. It can be seen that DH and S from both techniques are in good agreement with each other,

Figure 2. PS of R2 rough electrode obtained from AFM and SEM. Solid violet line is the total calculated PS, colored circles are the recorded PS from SEM image analysis, closed black circles are the PS from AFM. Dotted pink and green lines are the Gaussian part of the PS; dotted blue line is the power law region with the lower length scale cutoff from AFM, and dotted black line is the cutoff length from SEM.

Figure 3. PS of R3 rough electrode obtained from AFM and SEM. Solid violet line is the total calculated PS, colored circles are the recorded PS from SEM image analysis, closed black circles are the PS from AFM. Dotted pink and green lines are the Gaussian part of the PS; dotted blue line is the power law region with the lower length scale cutoff from AFM, and dotted black line is the cutoff length from SEM.

⟨|ζ (̂ K⃗ )|2 ⟩ = ⟨|ζ (̂ K⃗ )|2 ⟩G1 + ⟨|ζ (̂ K⃗ )|2 ⟩G2 + ⟨|ζ (̂ K⃗ )|2 ⟩F = πa12h12 exp(−a12K 2/4) + πa 22h22 × exp(−a 22K 2/4) + μK 2DH − 7 (7)

Equation 7 employed in the wavenumber region 1/L≤|K⃗ | ≤ 1/S . Morphological parameters of roughness needs to be identified in this framework, namely DH, S , and L for the fractal component, and hi and ai for the nonfractal component of PS. The lower cutoff length is evident in the PS and is estimated from the fall-off frequency where the power law sets in a PS curve (vertical blue dashed line). The upper cutoff length is the inverse of the lowest wavenumber in the PS data. In eq 7, a1, a2 and h1, h2 are the correlation lengths of the two Gaussian PSs and the rms width of the surface at different wavenumbers, respectively (violet solid line). R1, R2, and R3 8597



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For the present electrochemical time window of interest, features or mean square height of less than 20 nm do not affect the electrochemical response, as the diffusion layer thickness ((Dt)1/2) is already larger than these features for a typical aqueous system, or, in other words, the electrochemical resolution for aqueous systems is larger than 20 nm. Small values of h1 and h2 for the nonfractal Gaussian PS in the low wavenumber region, for all three electrodes, imply that the resulting current−time behavior is a weak function of these characteristics in the long time response. The overall response therefore has a strong influence on the fractal roughness features of the interface (as shown in the intermediate and larger wavenumber region of PS) and shows dominance in the electrochemical response. Average current at the electrode is obtained using eq 1, and the PS in eq 7 can be represented as the sum of planar and finite fractal component responses:25,26

Table 1. Morphological Parameters Obtained From AFM and SEM AFM electrode

DH

S (nm)

R1 R2 R3

2.07 2.24 2.34

23.2 75.8 38.5

SEM

h1 (nm)

h2 (nm)

a1 (μm)

a2 (μm)

DH

S (nm)

6 55 11

2.5 25 4

0.32 2.20 3.61

1.67 7.78 1.10

2.11 2.24 2.32

22.2 73.2 29.3

establishing the coherence of the measured PS of both techniques. Strength of fractality ( μ ≡ S2τ DH − 3) is related to the topothesy (Sτ ) of the fractal,49 and we elucidate μ from the cyclic voltammetry (CV) of the electrochemically active area of the electrodes. For ideal fractals, μ is often the scale at which the average slope on the surface would equal 1 radian; this can be obtained by extrapolation of the power law to wavenumber 1. However, it is established for real surfaces that the extrapolation method at wavenumber 1 often leads to unphysical results.50 We encounter similar difficulties in the estimation of μ through the extrapolation method using eq 6. We suggest an alternate method of finding μ or topothesy electrochemically, to circumvent the problem associated with the method of extrapolation. The hydrogen adsorption method for calculating electroactive area is often used with transition metals for potential regions prior to massive H2 evolution.51 We used CV for the H-adsorption method to evaluate the roughness factor (R*) of all Pt-electrodes in H2SO4. Roughness factor (R*) is related to the rapidity of surface fluctuations or the second moment of PS, which is the mean square gradient (m2) as defined in eq 5 for the finite fractal electrode surface. Using the probability density function for the gradient of an isotropic random Gaussian surface,52 one can obtain the mean area (⟨A⟩). ⟨A⟩ and R* are related to each other as ⟨A⟩ =

∫0

∞

R* = 1 +

⟨I(t )⟩ = Ip(t ) + IF(t )

where Ip(t) is the Cottrell current, and IF(t) is the current due to the fractal roughness as defined below 2(δ+ 1)⎤ ⎡m Γ(δ , Dt /l 2 , Dt /L2) ⎛ Sτ ⎞ ⎥ ⎟ ⎜ IF(t ) = Ip(t )⎢ 0 + ⎝ Dt ⎠ ⎢⎣ 2Dt ⎥⎦ 8π

(10)

where m0 is given by 4, Γ(α,x0,x1) denotes Γ(α,x0,x1) = Γ(α,x0) − Γ(α,x1) = γ(α,x1) − γ(α,x0) and is the incomplete Gamma function, and δ is DH − 5/2. The total mean current is the summation of smooth surface current and an anomalous excess current due to fractal roughness, or it can also be looked upon as the product of the (1/√t) current and a dynamic roughness factor. The above expression shows an enhancement in current due to additional flux at the rough interface, which can find application in sensorics1,2 where a large signal from small concentration in samples is desirable. Most surfaces after chemical etching and mechanical cleaning yield surfaces with S about 20−50 nm. These surfaces when investigated in aqueous systems have ti far left of the experimental window. Equation 10 has a simple representation for cases where experimental measurement time is larger than inner crossover time (ti = S2τ /D) associated with the response of rough electrodes.26 53

dS0 1 + (∇ ζ( r ⃗))2 = R *A 0 ⎛ 1 ⎛ 1 ⎞ πm 2 exp⎜ ⎟ erfc⎜⎜ 2 ⎝ 2m2 ⎠ ⎝ 2m2

⎞ ⎟⎟ ⎠

(9)

(8)

(δ + 1) ⎡ 2δ 2 ⎤ Γ(δ) ⎛ S2τ ⎞ 1 ⎛ Sτ ⎞ ⎛ S τ ⎞⎥ ⎢ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⟨I(t )⟩ ≈ Ip(t ) 1 − + ⎢ 8π ⎝ Dt ⎠ 8πδ ⎝ S ⎠ ⎝ Dt ⎠⎥⎦ ⎣

where A0 is the geometric area, dS0 is the area element on projected surface, and ∇∥ζ(r∥⃗ ) is the gradient of roughness surface profile. For a gently varying random surface, m2 is small and R* can be approximated as (1+m2). Similarly, for a rapidly varying random surface, m2 is large and R* can be approximated as (πm2/2)1/2. Knowledge of rms gradient ((m2)1/2) from eq 8 can be used to obtain μ in conjunction with eq 5 and other finite fractal quantities (DH, S , L) directly calculated through AFM or SEM analysis. The values so obtained provide reasonable value of μ and are tabulated in Table 1. The features present in the high and the intermediate wavenumbers of the PS influence the current−time response in the small and intermediate time regimes, respectively. The lower wavenumber features such as that of the Gaussian part of the PS should influence the long time regime. This is, however, subject to the magnitude of the relative widths of the Gaussian features: if the width is less than the diffusion layer thickness, then these features fail to influence the electrochemical response significantly. In these cases, mainly the intermediate and high wavenumber region will affect the electrochemical response.

(11)

In eq 11, the first term belongs to the planar behavior. The second and third terms are of opposite sign but switch their sign as fractal dimension goes from persistent behavior in the surface roughness profile (DH < 2.5) to antipersistent behavior in the surface roughness profile (DH > 2.5). The second term can be understood as source of anomalous behavior as it has topothesy length (Sτ ) of a fractal surface scaled with respect to yardstick (dynamic diffusion) length with exponent containing fractal dimension in it. The third term is related to the rms width of fractal roughness as in eq 4 scaled with respect to yardstick (dynamic diffusion) length. Alternatively, one may look at the contribution of the third term as the ratio of the topothesy length (Sτ ) with lower cutoff length (S ) with fractal dimension in exponent providing the range of fractal roughness seen in the response while dynamics is dictated by the ratio of S2τ /Dt . 8598



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been mixed for various recordings of different time durations to yield the complete response for the diffusion controlled regime. The total current is the summation of the smooth surface response and an anomalous excess flux due to various components of roughness as in eq 9. The values of DH, S , L, and μ are obtained from the AFM, SEM, and CV measurements. These values are used for calculating the theoretical response as shown in solid lines in Figures 4, 5, and 6. The theoretical curves show a good agreement with the experimental graphs at all time regimes. We also show the scaling argument response as a dashed red line. It can be seen from table 1 that the fractal dimension predicted by the linear regression of the double logarithmic current−time data curves differs from the measured fractal dimension obtained from SEM and AFM. The chronoamperometric response of R1 is quite close to the Cottrell response because of small values of R* and rms width (h) as seen from Table 2. Although from SEM images R1 seems rough, it has relative contribution of roughness features such as DH ≈ 2 and h ∼ S . Also, the range of wavenumbers contributing to current− time response is very small, hence no dynamic effects of morphology can be seen in Figure 4. R2 shows significant roughness as per Table 2. It shows anomalous response at intermediate times merging with the Cottrell response at long time, in accordance with the outer crossover time (to), to ≈ min[h , Sτ]2 /D, in the fractal region of the PS for R2. The intermediate power law regime is not just the characteristic of DH but also of the topothesy length (Sτ ) and lower length scale cutoff (S ) as shown by Kant.25,26 All these parameters are obtained directly from measurements such as SEM and CV, which are extensively used by electrochemistry to characterize their interfaces and electrochemical active area. This is in contrast to the scaling approach wherein the determination of I(t) ∝ t−α essentially has two unknown parameters, viz. exponent (α) and a prefactor.19,54 As seen in Table 2, R2 has large R*, which enhances the current and large rms width (h or (m0)1/2) of the interface, which prolongs the perception of roughness features in the response. The chronoamperometric response of R3 shows a signature of ohmic loss below 50 ms in the data due to the different cell configuration used here. The theoretical response is therefore predicted by using theory with ohmic correction (see eq 46 in ref 31). The value of small solution resistance (RΩ ≈ 2 Ω cm2) is accounted in our calculation. R3 shows moderate roughness, and the response goes from early time ohmic controlled to intermediate time anomalous diffusion controlled. The effective lower wavenumber fractal cutoff in PS corresponding to 1/min[h , Sτ] for R3 also marks that the fractal dominant region in the PS will influence the response. As evident from Table 2, for a moderate R*, but large rms width, the enhancement of the current is not high, but the signature of the roughness features are perceived until a longer time in the response, as opposed to the R1 electrode, which has the same order of μ but negligible rms width ((m0)1/2) of the interface.

Figures 4, 5, and 6 show the chronoamperometric response in 15 mM potassium ferrocyanide and ferricyanide 1:1 (v/v) in 3 M

Figure 4. Chronoamperometric response of R1 rough electrode. Solid line is the theoretical plot. Dotted red line is the scaling fit to the data.


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CONTRAST TO SCALING APPROACH Scaling approach has an inherent limitation of identifying crossovers pertaining to the real electrode surfaces. An ideal fractal surface requires two features associated with it: (a) fractal dimension (DH) and (b) topothesy length (Sτ ). Error in estimation of either of these quantities automatically induces the incorrect value of other. Pajkossy’s approach identifies only one of the two parameters from the electrochemical response to yield an effective value of DH. This value of DH, however, does not


NaNO3 as the background electrolyte at R1, R2 pretreated Pt wires as working electrodes19,40 vs Ag/AgCl reference electrode. Similarly, they show that for R3 pretreated Pt wire as the working electrode vs SCE reference electrode for the same aqueous redox system with slightly different cell configuration. The data has 8599



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Table 2. Morphological Parameters Obtained From CV CV electrode R1 R2 R3

A0 (cm2) 0.080 0.099 0.11

R* 1.08 13.27 4.17

(m2)1/2 0.29 10.55 3.16

μ (cm2DH−3) 1.21 × 10 2.74 × 10−5 2.22 × 10−7

correspond to measured DH from SEM and AFM (refer to Table 1). The other parameter gets buried in the prefactor with no definite estimation procedure. In contrast to this, it is to be emphasized here that in the present formulation no unknown or fitting parameters are present; all of the morphological information is obtained through routine measurements such as CV, AFM and SEM. The current time behavior is obtained using these values as an input to the equation, based on rigorous derivation. Another important aspect of the present formulation is that this theory can be generalized to many electrochemical techniques and class of reaction at the interfaces. For instance, here we can directly validate and identify the presence of solution resistance31 effect, which the scaling approach assumes to be absent in order to capture the response.19,20

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Sτ (nm)

−7

composite qty (m0)1/2 (μm)

0.02 21.26 0.48

6.45 × 10 15.13 0.51

−2

scaling approach DH 2.10 2.15 2.21

seen in the low wavenumber side of PS do not affect the response, as they lie outside the effective PS range. • An elusive parameter such as topothesy length (or μ) is obtained by substituting a numerical value of the mean square gradient of roughness (m2) obtained from area measurements from CV in its definition for the finite fractals (along with numerical values for DH, S , and L obtained from SEM or AFM). For diffusion-limited redox reactions at rough electrodes, quantities such as DH, S , and Sτ derived from the PS are sufficient to capture the electrochemical response. • Our theoretical results, which relate the PS to the current− time response, are valid for all times and show time regimes: a short time region and an anomalous intermediate region influenced by the high and intermediate wavenumber fractal PS. This framework can take into account the ohmic loss, which is neglected in the scaling approach. • The small time crossover (ti = S2/D) and long time crossover (to = min[S2τ , h2]/D) identify the anomalous Cottrell regime with higher magnitude of intermediate slope. The region beyond these times is the classical Cottrell response where early time crossover is (usually suppressed by ohmic losses) lying outside the experimental time window. This general approach can be extended to different classes of reaction schemes and random morphologies as it can account for the interplay of both morphology and phenomenological length scales. This first investigation involving detailed characterization of roughness through its PS will open up further experimental and theoretical investigations in complex problems of electrochemistry at rough electrodes.

SUMMARY AND CONCLUSIONS

It has been known for a long time that the electrochemical response of solid electrodes is affected to a significant degree by the ubiquitous geometrical disorder at the electrode/electrolyte interface. Despite early success in the quantitative description of the electrochemical behavior on smooth electrodes, subsequent progress was not satisfactory because of the discouraging mathematical difficulties encountered in solving similar problems on disordered electrodes. Some recent development in rigorous theories of rough electrodes also lack experimental follow up. We have for the first time presented an experimental analysis of the chronoamperometric response at a randomly rough electrode, which is characterized using the PS. • Our analysis clearly shows that the simplified approach of Pajkossy19,20 and de Gennes38 of using one response exponentbased scaling ansatz with an effective fractal dimension from the chronoamperometric data does not correspond to the measured fractal dimension from SEM and AFM. The scaling exponent as perceived by de Gennes Pajkossy scaling ansatz is not just a function of DH but of three characteristic features: DH, S , and Sτ . • The elegance of this formalism lies in the statistical en route connecting the morphology of the interface and the response through the PS of roughness. The idea of characterization of PS in itself resolves or abstains from the point of reproducing a particular morphology. It instead empowers the user to account for and understand the diffusion limited response and various physical phenomenon including loss mechanisms at any kind of rough morphology. • AFM or SEM and CV are sufficient to characterize the roughness PS for the complete characterization of electrode morphology. High and intermediate wavenumber regions of PS influence electrode response most. The lower wavenumber region of PS is usually truncated with an effective wavenumber cutoff, which is an inverse of topothesy length (1/Sτ ) or the inverse of width of roughness, viz. 1/min[Sτ , h]. Hence the effective PS spans wavenumbers between 1/min[Sτ , h] and 1/S . The upper cutoff length scale of fractality (L) usually acts as the sample size without having a significant role in the response. This is the reason that the underlying random nonfractal structures

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APPENDIX

A. Height Distribution

Figures A1, A2, and A3 show height distribution functions of electrode E1, E2, and E3 respectively. These height distribution curves are extracted from SEM images.

Figure A1. Height distribution function of R1 rough electrode obtained from SEM image. 8600



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Figure B2. AFM image of R2 rough electrode.

Figure A2. Height distribution function of R2 rough electrode obtained from SEM image.

Figure B3. AFM Image of R3 rough electrode.

C. Data Analysis for Image Processing

The method of finite fractal characterization from the SEM images in combination with CV measurements of the electroactive area of electrodes is described in ref 45. This analysis has been done in Mathematica 8.0 using the Regress Package in TIFF image format obtained from SEM of three different electrodes. A linear equation of the form y = a + βx

(C1)

was fitted to the PS of SEM images (refer to inset of Figures 1, 2, 3) by using linear regression techniques. The fitted values of α and β are collected in Table C1 along with statistics describing the quality of the regression results at confidence levels of 0.99. R2 is the confidence coefficient of statistical measure in Table C1.

Figure A3. Height distribution function of R3 rough electrode obtained from SEM image.

B. SEM and AFM Characterization

The rough surfaces were characterized by Quanta 200F SEM-FEI and Nanotec-AFM systems in tapping mode. Figure B1 shows

D. Electrochemical Measurements

The pretreated electrodes were used to examine 15 mM ferrocyanide and ferricyanide in 3M NaNO3 background electrolyte so as to suppress the migration effects. After this, chronoamperometric measurements were carried out for the 1:1 v/v mixture at the electrodes for various time durations at the potential bias of 0.305 V vs Ag/AgCl electrode reference for R1 and R2 electrode and 0.341 V vs SCE reference for R3 electrode. These chronoamperograms were then superimposed to get the complete recording ranging from milliseconds to tens of seconds. No iR compensation was applied while carrying out these measurements. The double log data was used for regression analysis for calculating fractal dimension from de Gennes− Pajkossy’s scaling exponent (δ = [DH−1]/2). A linear equation of the form

Figure B1. AFM image of R1 rough electrode.

the AFM micrographs for the first rough electrode (R1), Figure B2 is the AFM image of the second rough electrode (R2), and Figure B3 is the AFM image of third rough electrode (R3).

y = γ + δx

(D1)

was fitted to the recorded CA data by using linear regression techniques, listed in Table D1. 8601



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Table C1. Linear Regression Parameters for Fitting of PS From AFM and SEM AFM

SEM

electrode

α

β

R2

variance of power law

G1variance

G2variance

α

β

R2

variance

R1 R2 R3

−2.47422 0.203828 −1.08659

−2.8686 −2.51451 −2.42492

0.996234 0.984524 0.983989

0.00245492 0.00487622 0.00322203

0.024395 0.0564913 0.0199408

0.638658 0.539926 0.552533

−2.93 −1.14 −0.15

−2.78 −2.51 −2.37

0.98 0.99 0.99

0.01 0.005 0.007

Electrodeposits. Dimensionality Determination by Scanning Tunneling Microscopy. J. Phys. Chem. 1992, 96, 347−350. (13) Vázquez, L.; Salvarezza, R. C.; Ocón, P.; Herrasti, P.; Arvia, A. J. Self-Affine Fractal Electrodeposited Gold Surfaces: Characterization by Scanning Tunneling Microscopy. Phys. Rev. E 1994, 49, 1507−1511. (14) Gobal, F.; Malek, K.; Mahjani, M. G.; Jafarian, M.; Safarnavadeh, V. A Study of the Fractal Dimensions of the Electrodeposited PolyOrtho-Aminophenol Films in Presence of Different Anions. Synth. Met. 2000, 108, 15−19. (15) Herrasti, P.; Ocón, P.; Salvarezza, R. C.; Vara, J. M. L.; Arvia, A. J. A Comparative Study of Electrodeposited and Vapour Deposited Gold Films: Fractal Surface Characterization Through Scanning Tunnelling Microscopy. Electrochim. Acta 1992, 37, 2209−2214. (16) Mbise, G. W.; Niklasson, G. A.; Granqvist, C. G. Scaling of Surface Roughness in Evaporated Calcium Fluoride Films. Solid Commun. 1996, 97, 965−969. (17) Risović, D.; Poljaček, S. M.; Furić, K.; Gojo, M. Inferring Fractal Dimension of Rough/Porous Surfaces: A Comparison of SEM Image Analysis and Electrochemical Impedance Spectroscopy Methods. Appl. Surf. Sci. 2008, 255, 3063−3070. (18) Duparre, A.; Ferre-Borrull, J.; Gilech, S.; Notni, G.; Steinert, J.; Bennett, J. M. Surface Characterization Techniques for Determining the Root-Mean-Square Roughness and Power Spectral Densities of Optical Components. Appl. Opt. 2002, 41, 154−171. (19) Pajkossy, T. Electrochemistry at Fractal Surfaces. J. Electroanal. Chem. 1991, 300, 1−11. (20) Imre, A.; Nyikos, L.; Pajkossy, T. Electrochemical Determination of the Fractal Dimension of Fractured Surfaces. Acta Metall. 1992, 40 (8), 1819−1826. (21) Bisquert, J.; Compte, A. Theory of the Electrochemical Impedance of Anomalous Diffusion. J. Electroanal. Chem. 2001, 499 (1), 112−120. (22) Kant, R. Can Current Transients be Affected by the Morphology of the Nonfractal Electrode? Phys. Rev. Lett. 1993, 70, 4094−4097. (23) Kant, R. Can One Electrochemically Measure the Statistical Morphology of a Rough Electrode? J. Phys. Chem. 1994, 98, 1663−1667. (24) Kant, R.; Rangarajan, S. K. Effect of Surface Roughness on Diffusion-Limited Charge Transfer. J. Electroanal. Chem. 1994, 368, 1− 21. (25) Kant, R.; Jha, S. K. Theory of Anomalous Diffusive Reaction Rates on Realistic Self-Affine Fractals. J. Phys. Chem. C 2007, 111, 14040− 14044. (26) Jha, S. K.; Sangal, A.; Kant, R. Diffusion Controlled Potentiostatic Current Transients on Realistic Fractal Electrodes. J. Electroanal. Chem. 2008, 615, 180−190. (27) Kant, R. Diffusion-Limited Reaction Rates on Self-Affine Fractals. J. Phys. Chem. B 1997, 101, 3781−3787. (28) Kant, R.; Rangarajan, S. K. Diffusion to Rough Interfaces: Finite Charge Transfer Rates. J. Electroanal. Chem. 1995, 396, 285−301. (29) Islam, Md. M.; Kant, R. Generalization of the Anson Equation for Fractal and Nonfractal Rough Electrodes. Electrochim. Acta 2011, 56, 4467−4474. (30) Kant, R.; Islam, Md. M. Theory of Absorbance Transients of an Optically Transparent Rough Electrode. J. Phys. Chem. C 2010, 114, 19357−19364. (31) Srivastav, S.; Kant, R. Theory of Generalized Cottrellian Current at Rough Electrode with Solution Resistance Effects. J. Phys. Chem. C 2010, 114, 10066−10076.

Table D1. Linear Regression Parameters for Fitting of Chronoamperometric Data for Pajkossy’s Approach CA electrode

γ

δ

R2

variance of CA data

R1 R2 R3

−4.18115 −4.03821 −3.78134

−0.551011 −0.579127 −0.607013

0.997838 0.976617, 0.9997611

0.000868728 0.00493017 0.0000357

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank S. Sampath, IPC, IISc Bangalore, and V. Lakshminarayanan, RRI, Bangalore, for useful suggestions. SS is grateful to CSIR for SRF, SD is grateful to UGC for Non-NET fellowship and RK is grateful to CSIR for JRF. Rama Kant is grateful to DU DST purse grant.

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REFERENCES

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Experimental Validation of Roughness Power Spectrum-Based Theory

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