Anomalous Localization of Electrochemical Activity in Reversible

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Anomalous Localization of Electrochemical Activity in Reversible Charge Transfer at Weierstrass Fractal Electrode: Local Electrochemical Impedance Spectroscopy Rama Kant, Shweta Dhillon, and Rajesh Kumar J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/jp512297f • Publication Date (Web): 11 Feb 2015 Downloaded from http://pubs.acs.org on February 17, 2015

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The Journal of Physical Chemistry

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Anomalous Localization of Electrochemical Activity in Reversible Charge Transfer at Weierstrass Fractal Electrode: Local Electrochemical Impedance Spectroscopy

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Rama Kant∗ and Shweta Dhillon and Rajesh Kumar Department of Chemistry, University of Delhi, Delhi 110007, India E-mail: [email protected]

Abstract

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The knowledge of local electrochemical activity over an electrode surface is of

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utmost importance in understanding several processes like electrocatalysis, dendrite

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growth and corrosion. For redox system with unequal diffusivities, we have devel-

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oped second order perturbation theory in surface profile for the local electrochemical

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impedance spectroscopy (LEIS) for a reversible charge transfer system at an electrode

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with arbitrary surface profile corrugations. Detailed analysis of LEIS is performed for

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simple roughness model of sinusoidal surface and realistic roughness model of finite

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fractal Weierstrass function as surface corrugation. At low roughness surface, local

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electrochemical activity is localized at peaks and depleted at valleys. But at moder-

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ate and high roughness surfaces, there is anomalous bifurcation beside peak positions,

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making local electrochemical activity localized at cliff (high slopes) locations. Max-

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imum depletion is observed at valleys, remains more depleted at valley as compared ∗

To whom correspondence should be addressed

1 ACS Paragon Plus Environment

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to peak positions of the surface even at high aspect ratio of roughness and frequency

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of external signal. Complex multi-furcations behavior of local activity and inactivity

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occur in Weierstrass corrugation for moderate and high roughness. The localization

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phenomenon of impedance on Weierstrass fractal surface depends upon the fractal

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dimension size of finest feature and mean square width of roughness. Influence of vari-

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ous feature of Weierstrass surface is also studied through local impedance distribution

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functions.

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Keywords

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Sinusoidal roughness, Weierstrass Function, Local electrochemical impedance spectroscopy

24

(LEIS), Dendrites, Diffusion limited process, Activity localization

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Introduction

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Various complexities present in electrochemical systems originate from ubiquitous surface

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disorders caused by roughness and electrochemical heterogeneity. Recent spurt of activity in

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nanostructured electrodes is an attempt to control these surface disorders and its properties.

29

Designing such nanostructured electrodes usually helps in modifying electrochemical proper-

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ties and enhancing interfacial activities which control overall electrochemical (e.g. catalytic)

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behavior of the system. 1–3 Other interesting class of problems arise in electrochemical growth

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and dissolution which shows temporal and spatial scaling behavior and understanding their

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local behavior is of utmost importance. 4,5 Designing and understanding surfaces with bet-

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ter electrochemical activity is of primary interest in the applied electrochemistry. Dendrite

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growth during electrodeposition 6–11 can occasionally cause severe problems such as short

36

circuiting, local heating and energy loss in batteries . 12–15 Rechargeable lithium, sodium and

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aluminum metal-based batteries are among the most versatile platforms for high-energy, cost-

38

effective electrochemical energy storage. Dendrite formation on the negative electrode during



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repeated cycles of charge and discharge are major hurdles to commercialization of energy-

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storage devices. 16–18 Experiments shows that the dendrite formation is affected by current

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density, time, temperature and concentration. 12–15,19 But effect of local shape or disorder of

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electrode surface over which dendrite formation take place and its theoretical understanding

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is still has to be done. An understanding of the shape evolution is possible, through an

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understanding of the localization of growth processes that take place during electrodeposi-

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tion/thin film deposition under diffusion controlled fast charge transport phenomenon. Local

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electrochemical impedance spectroscopy (LEIS) can offer critical understanding of complex

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electrochemical surface processes and their distribution over the surface. Local reactivity is

48

inversely proportional to local impedance hence high local impedance sites offers low reac-

49

tivity zones and vice-versa.

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Local electrochemical impedance spectroscopy (LEIS) technique is used for mapping the

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impedance distribution, as a function of frequency, of an electrode. 20,21 LEIS is used to pre-

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cisely locate site of chemical and electrochemical activity at the substrate surface as well

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as the substrate topography. 20–23 LEIS also provides a powerful tool for exploration of elec-

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trode heterogeneity. 23 Chemical reactions at surfaces are of great fundamental and applied

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interest, therefore it is of paramount importance to evaluate surface reactivity and appli-

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cability at electrode surface. 24 LEIS offers a very broad range of important applications in

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corrosion processes for various metals and alloys, 22,25,26 fuel cell optimizations where cat-

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alytic reactions proceed at noble metal clusters. 27 LEIS enables mapping of the membrane

59

resistance at various electrode positions. 28 Chemically-selective and spatially-localized redox

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activity at Ta/Ta2 O5 electrodes is studied, it has importance in microelectronics, catalysis,

61

and corrosion resistance. 29

62

The analytical intricacies in formulating useful models for the topography of surfaces

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have made rough surface characterization a popular area of investigation in the applied

64

chemistry. These surface complexities are often understood through fractal models which

65

includes the concepts of dilational symmetry (associated fractal dimension) and length scales


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corresponding to surface morphologies. 30 Majumdar and his collaborators used Weierstrass

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functions to model rough surface profile in aid of analyzing rough surface topographies and

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surface contact mechanics. 31–33 The modeling of rough surfaces via profiles greatly simplifies

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the problem, since a profile measures the surface height along a single axis or direction. Of

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course, if the surface is isotropic, a surface profile is sufficient to characterize its topography.

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Weierstrass functions are apt for modeling surface roughness profiles, and besides providing

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fractal characterization of roughness, they can also be used to extract statistical informa-

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tion that can be compared with classical random measures of the features of rough surface

74

topography. This interesting and promising approach is applicable to study the pattern and

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disorder in fractal growth processes, 34 contact mechanics 35 and ramified fractal growth. 36,37

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The degree of the non-uniformity of the current density distribution across the electrode

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systems is important in the study of rechargeable lithium ion battery (capability to store

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2 to 3 times than the nickel-metal hydride battery). 38–41 Recently, we have successfully ex-

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plained potentiostatic current transient on realistic fractal roughness and their experimental

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validation for nanometers to micrometer scales of roughness. 42,43 Similarly, the role of surface

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irregularities on diffusion controlled charge transfer process (Warburg impedance) for realis-

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tic self-affine isotropic fractal surfaces 44,45 and diffusion and homogeneous kinetics coupled

83

with a fast heterogeneous charge transfer reaction (Gerischer admittance) 46 have also been

84

explained. Here, similar ab initio methodology is used to understand the admittance density

85

localization for Weierstrass surface.

86

In this paper, we are presenting the approach to find out the LEIS for an arbitrary surface

87

profile electrode in presence of electroactive species of unequal diffusivities (DO 6= DR ).

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We are showing LEIS results for reversible charge transfer system. Firstly, idealized and

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realistic surface models are discussed. Then, expressions for LEIS are obtained for idealized

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and realistic surface. Later in results and discussion section, LEIS behavior at sinusoidal

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and Weierstrass surface are discussed in details, under reversible charge transfer system

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with unequal diffusion coefficients. Probability distribution of local impedance density at



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weierstrass surface are analyzed numerically from simulated LEIS.

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Idealized and realistic roughness model

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To investigate the spatially resolved electrochemistry of rough electrode, the simplest rough-

96

ness model is sinusoidal surface corrugation. However, this model considers the rough surface

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as a single sinusoidal, and cannot describe the real random rough surface completely. But

98

this approach is useful to understand the behavior of local impedance at various regions of

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rough surface and helps in building the understanding of local impedance behavior at rough

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surfaces. The sinusoidal surface profile is given as, 2πb ζ(x) = h cos x λ0

(1)

101

where h is the width of interface, b is the frequency multiplier and λ0 is fundamental wave-

102

length in the roughness profile. In figure 1, lower contour plot shows the waveform of

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sinusoidal surface profile. This will act as an input for surface roughness in the model of

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local admittance density or local impedance density.

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Fractal geometry, 33 pioneered by Mandelbrot, can be observed in various natural phe-

106

nomena, such as precipitation, turbulence, and surface topography. These approaches have

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been applied in various media such as molten media, 47 non-aqueous media of lithium sec-

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ondary batteries, 48,49 etc. For fractally corrugated surfaces, surface profile can be generated

109

using Weierstrass function. The Weierstrass function is the best model to generate rough

110

surface profile but its properties like non-differentiability and infinite length scales of rough-

111

ness put limitation on it as a useful model for realistic surfaces. But band limited form

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of Weierstrass function with finite number of sinusoids in summation can circumvent these

113

difficulties. Hence, we confine our theory for the band limited Weierstrass model 50–54 and


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represent it as: N2 X 2(1 − b−2H ) b−mH ζ(x) = h −2HN −2H(N +1) 1 2 b −b m=N1 x cos 2πbm + φm λ0

r

(2)

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where H = 2−DH is Hurst’s exponent. Here, DH is Hausdorff-Besicovitch dimension (simply

116

called as fractal dimension) and b > 1 is the frequency multiplier. ζ(x) is the surface profile

117

along the x direction, h is a prefactor representing length, φm is the random phase angle in

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m-th sinusoids (varies in between 0 to 2 π). The random phase φm is used to prevent the

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coincidence of different frequencies at any point of the surface profile. N1 and N2 are the

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lower and upper harmonics in the spectrum. Variation in the lower and upper harmonics of

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the surface causes variation in the roughness features of the surface. In case of pure fractal

122

processes, N1 = −∞ and N2 = ∞ and hence the frequency varies between 0 and ∞. Decrease

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in N1 causes the increase in low wavenumber sinusoids. So the surface shows persistence

124

behavior. Whereas increase in the value of N2 leads to increase in high wavenumber features

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where the anti persistent behavior of fractal surface dominates. Similarly, for fixed N1 and

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N2 , but increase in fractal dimension (DH = 2 − H), will cause increase in weights of

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high wavenumber sinusoids and hence anti-persistent behavior of surface will increase. This

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equation can be used as model for simulating fractal surfaces. Any real surface has fractal

129

nature within certain length scales. Equation 2 function has been used for generating the

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self-affine fractal surface profiles.

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The local slope (ζx (x)) of the rough surface is obtained as the first derivative of surface

132

profile. Derivative of the surface profile can be used to obtain the information of surface slope.

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These are important in understanding diffusion problem on rough interfaces. Weierstrass

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surface profile behaves differently, under the influence of roughness due to different fractal

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dimension, DH , cutoff length scales (L = λ0 b−N1 and ` = λ0 b−N2 ) and width of interface, (h).

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The roughness has remarkable effect on nature of interfacial phenomena and show different 6 ACS Paragon Plus Environment


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behavior at different roughness factor. A surface having large fractal dimension, DH , large

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roughness factor have more pronounced peaks as compared to low fractal dimension small

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roughness factor as shown in figure 3. DH < 1.5 shows super-Brownian nature of fractals

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while DH > 1.5 shows sub-Brownian nature of fractals. DH < 1.5 corresponds to the

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case where more weight is given to the upper cutoff length scale (L). Such a surface has a

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greater mean radial distance between consecutive zero crossing level compared to a Brownian

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fractal, which means that the sub-Brownian fractal has a persistent surface profile. Similarly,

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DH > 1.5 corresponds to a case where the lower cutoff length scale (`) has higher weight and

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such a surface has a smaller mean radial distance between consecutive zero level crossing

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than a Brownian fractal does, which means that the super-Brownian fractal has antipersistent

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nature in its surface profile. 52,55,56

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For insight of various interfacial phenomena, we require not only surface profile but also

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its other geometric properties like slope etc. Figure 1 shows the relationship between surface

150

profile and its slope. This figure offers a relationship among the peaks, valleys and cliffs of

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the surface profile with the local slope and curvature. The maxima of slope profile (figure 1)

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correspond to cliff positions of surface (figure 1), while zero crossing correspond to peak or

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valley positions. Figure 2 shows the relationship between surface profile and slope. Figure 2

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shows the plot of band-limited Weierstrass function as surface profile (ζ(x)) and its slope

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(ζx (x)) with distance at particular fractal dimension. This figure offers a relationship among

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the peaks, valleys and cliffs of the surface profile with the local slope. The maxima of slope

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profile plots correspond to cliff positions of surface profile, while zero crossing correspond

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to peak or valley positions. Figure 3 shows corrugated profiles of surface of various fractal

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dimensions. It is observed from figure that as the fractal dimension increases ruggedness

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in surface increases, making it more closer to many real rough surfaces. High roughness

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or ruggedness in surface can be interpreted from the lines in contour plot, as number of

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divisions becomes larger as fractal dimension increases.

163

The moments of power spectrum (m2α ) are important physical measures of geometric


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properties of roughness. The mean square α − th derivative for the band-limited Weierstrass

165

surface profile is given by 57 r m2α =

h

2(1 − b−2H ) b−2HN1 − b−2H(N2 +1)

!2

N2 X

b−mH (2πbm )α

2

(3)

m=N1

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m0 is the mean square height, when α is 0. m0 is a measure of the width (h) of fluctuation √ in surface profile. It is related to the zeroth moment (m0 ) of power spectrum as, h = m0 .

168

When α is 1, then m2 is the mean square slope of the interface. Surfaces with small m2

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values have small roughness and with large values correspond to large roughness. The low

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roughness surface have small roughness factor (R∗ ) and the large roughness surfaces have

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large R∗ . The exact expression for the roughness factor of a random surface is given by 57,58

166

1 1 1 U , 2, R =√ 2 2m2 2m2 ∗

(4)

where U(a, b, c) is the confluent hypergeometric functions. 59 For the small roughness surfaces: 57,58 R∗ = 1 +

m2 2

and for large roughness surfaces: 57,58

R∗ ∼

p

2m2 /π

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Theory of localized impedance

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The majority of the models appearing in the electrochemical literature apply either to sys-

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tems in which electrode geometry does not play a role or to those in which the electrode

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geometry is important, but still relatively simple, i.e., planar, cylindrical or spherical. But,

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it has been long observed that impedance of an electrode depends on the roughness of its



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surface. 60 Some significant work has been done on theoretical impedance of rough electrodes

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with smooth shapes of roughness 61–64 and it was found that the influence of surface roughness

179

of solid electrodes on electrochemical measurements is significant.

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The physical quantity of interest is the local and global admittance/impedance. The

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current density would vary with the position vector (~ rk ) on the electrode surface. 65–68 So the

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local admittance function y(~ rk ) is not only the function of frequency but also the position

183

as well. 65–67 Its relation with (Laplace transformed) local current density (i(~ rk , jω)) under

184

potentiostatic condition is: y(~ rk , ω) = R i(~ rk , jω)jω/η0

(5)

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from the knowledge of y(~ rk , ω), we are able to predict localization behavior of admit-

186

tance/impedance density.

187

The admittance density for simplest model of rough, sinusoidal profile, can be obtained

188

using general result for the current density for weakly and gently fluctuating corrugated

189

rough surface, it is given by 69 (see equation 24 therein). The equation for the current den-

190

sity profile is expressed in terms of spatial Fourier and temporal Laplace transformed domain.

191

Using Fourier and Laplace transformation techniques one can write the admittance density

192

expression for corrugated surface which will be used in constructing the formulation for arbi-

193

trary roughness. This work achieves second order perturbation solution for the admittance

194

density for an arbitrary corrugated surface profile and is given as (Appendix for derivation

195

details): √ 2 2 jω n F ˆ 0) ˆ x )+ O ˆ x −K 0 )ζ(K ˆ ω) = ˆ 2α ζ(K ˆ 1α ζ(K √ √ 2πδ(K )+ O y(Kx , ζ, x x x RT 1/( DO CO0 ) + 1/( DR CR0 ) (6)

196

where α = O or R, CO0 and CR0 are the bulk concentrations of oxidized and reduced species,


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197


ˆ α and O ˆ α are defined as follows: respectively. Various symbols, operators O 1 2 α ˆ 1α = (ωK O − ω0α ) x Z 1 α α α2 α α 0 2 0 0 α ˆ dKx0 2ωK ω − ω − ω ω − 2(K − K ) − K (K − K ) O2 = 0 x x 0 0 Kx x x x x Kx ,Kx 4π p jω/Dα ω0α = q α ωK = ω0α 2 + Kx 2 x q α (7) ω0α 2 + (Kx − Kx0 )2 ωK = 0 x ,Kx

198

Dα is diffusion coefficient of electroactive species, viz. DO for cathodic current and DR

199

for anodic current, n is number of electron transferred, F is Faraday constant, R is gas

200

ˆK ~x ) is the Fourier transform of ζ(x). For constant and T being the absolute temperature. ζ(

201

simplicity, we assumed, DO = DR = D. Using this simplification in equation 6, expression

202

~ k , ω), for corrugated surface profile reduces to 43,69,70 : of admittance density, y(K ~x , ω) = y(K

203

√ n2 F 2 D jω ˆ x) + O ˆ x − K 0 )ζ(K ˆ 0) ~x ) + O ˆ 1 ζ(K ˆ 2 ζ(K 2πδ(K x x 0 0 RT (1/CO + 1/CR )

(8)

ˆ 1 and O ˆ 2 are special case of equation 7 for the case of DO = DR . where operators O

204

Our aim is to find out the band-limited admittance density expression for simplest sinu-

205

ˆ x ), in soidal surface roughness (equation 1). Using Fourier transform 71 of equation 1, ζ(K

206

equation 8, we obtain admittance density expression for sinusoidal surface roughness. The

207

expression for admittance density upto second order in surface profile for a sinusoidal surface



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208

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can be written as:

y(ω) = yw (ω) 1 + h y1 (ω) + h2 y2 (ω) √ n2 F 2 jω √ √ yw (ω) = RT 1/( DO CO0 ) + 1/( DR CR0 ) q y1 (ω) = ω0α 2 + λ2 − ω0α cos(λ x) q q q 1 α2 α 2 α2 α2 α2 2 2 2 y2 (ω) = cos(2λ x) 2 ω0 + 4λ ω0 + λ − ω0 − ω0 ω0 + 4λ − 3λ 4 q 1 2 α α2 α2 2 + 2 ω0 ω0 + λ − 2 ω0 − λ 4 2πb λ = λ0 209

(9)

where yw (ω) is classical Warburg admittance density.

210

Next step is to know the band-limited admittance density expression for Weierstrass sur-

211

face profile, which is sum of several sinusoids and shown in equation 2. For this, using expres-

212

sion of Weierstrass surface profile (equation 2) in local admittance expression (equation 8),

213

we obtain admittance density expression for a Weierstrass surface profile. The expression

214

for admittance density upto second order in surface profile for a Weierstrass surface can be

215

written as: " y(ω) = yw (ω) 1 + h

N2 X

y1m (ω)+ h 2

m=N1 √ 2 2

yw (ω) =

N2 N2 X X

# y2m n (ω)

m=N1 n=N1

n F jω √ √ RT 1/( DO CO0 ) + 1/( DR CR0 )

y1m (ω) = γm (ωλαm − ω0α ) cos(λm x + φm ) α+ y2m n (ω) = γm γn ωm n cos((λm + λn )x + (φm + φn )) + α− γm γn ωm n cos((λn − λm )x + (φn − φm ))


(10)

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216


where

λm =

2πbm λ0

γm = b

−mH

q 2(1 − b−2H )/(b−2HN1 − b−2H(N2 +1) )

1 2ωλαm +λn ωλαm − ω0α 2 − ω0α ωλαm +λn − 2λ2m −λm λn ) 4 1 = 2ωλαm −λn ωλαm − ω0α 2 − ω0α ωλαm −λn − 2λ2m +λm λn ) 4 q

α+ ωm n = α− ωm n

ωλαm = ωλαm ±λn

=

ω0α 2 + λ2m

q

ω0α 2 + (λm ± λn )2

217

λ0 is the fundamental wave number, λm and φm are the wavenumber and phase, respectively.

218

In our analysis, we have second order perturbation expression for the admittance density. We

219

observe that equation 10 is valid for the small amplitude of fluctuation and for small gradient.

220

Finite radius of convergence limits the behavior and utility of perturbation expansion even if

221

sufficient number of terms are available in the expression. We can circumvent this constraint

222

to some extent using Padè approximants 72 technique and enhance its utility. Using this

223

technique, validity of equation 10 can be extended upto the moderate level of roughness.

224

The [0/2] Padè approximant form of admittance expression (equation 10) can be expressed

225

as: yw (ω) [1 − h f1 (ω) + h 2 (f1 (ω)2 − f2 (ω))] N2 X f1 (ω) = y1m (ω)

ypa (ω) =

(11)

m=N1

f2 (ω) =

N2 N2 X X

y2m n (ω)

m=N1 n=N1

226

Similarly, [0/2] Padè approximant for admittance density at a sinusoidal surface can be



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227

Page 14 of 45

written as,

ypa (ω) =

yw (ω) [1 − hy1 (ω) + h2 (y1 (ω)2 − y2 (ω))]

(12)

228

This simplified expression as shown in equation 11 is suitable to obtain frequency de-

229

pendence of admittance density behavior for the Weierstrass surface profile. The variance

230

in admittance density in frequency region is a function of four fractal morphological param-

231

eters, viz. fractal dimension (DH ), root mean square width of the interface (h) and the

232

lower and upper harmonics in the spectrum (N1 , N2 ). Equations 10 and 11 consist of finite

233

series of unequally spaced sinusoids depending upon the lower cutoff length (` = λ0 b−N2 )

234

and upper cutoff length (L = λ0 b−N1 ), related to upper and lower harmonics of the surface,

235

respectively. More specifically, using equation 11 we are able to predict the localized ad-

236

mittance/impedance density behavior and localization of reaction sites for the band-limited

237

fractal Weierstrass surface profile.

238

Results and Discussion

239

Here, we analyze the theoretical results developed for the diffusion controlled local impedance

240

on a simple sinusoidally varying surface (equation 12) and a complex band-limited Weier-

241

strass surface (equation 11). For simplicity in our calculations, we assumed DO = DR = D 73

242

and used here values of diffusion coefficient is 5×10−8 cm2 /sec, typical value for room tem-

243

perature ionic liquids. Non uniformity in local surface impedance due to localization of the

244

reaction sites and their relation to local geometry is discussed. This helps in identification of

245

enhanced local electrochemical activity and depletion zones. In the following figures, we are

246

analyzing behavior of local impedance density (z(ω)) with scaled distance (with respect to

247

λ0 (fundamental wave number) which is equal to the maximum length scale of roughness).

248

For the greater understanding, we analyze the contour plots and distribution function of

249

local impedance and its interrelation with the geometrical complexity of the surface. 13 ACS Paragon Plus Environment

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250

Localization of impedance over sinusoidally varying surface

251

Figure 4 and 5 show the response of magnitude of local impedance over the single sinusoid

252

surface profile. In figure 4, effect of aspect ratio (h/λ0 ) is studied. In figure 4 at low aspect

253

ratio (h/λ0 ), minima of local impedance are seen at peak positions of surface, while local

254

impedance maxima are observed at valley positions of surface profile. Hence, enhanced lo-

255

cal activity is seen at peak of the surface while valley of surface serve as depletion zones.

256

This is the similar to the results predicted by Louch and Pritzker. 74 Pritzker stated that

257

the surface morphology have an important influence on the local current distribution by

258

causing the localizations of the current maxima and minima at the peak and valley posi-

259

tions, respectively. But on increasing aspect ratio, bifurcation of local impedance minima

260

starts emerging, with their locations besides peak positions of the surface. These minima

261

in local impedance are sites of high electrochemical activity. This anomalous observation of

262

bifurcation in enhanced activity sites to the cliff positions of the surface and also emergence

263

of additional depletion zones were not predicted by earlier authors. 74 This bifurcation shifts

264

enhanced activity regions to cliff positions of the surface. With further increase in aspect

265

ratio, depletion zones at peak positions gets further depleted. Once local impedance maxima

266

at peak and valley positions of surface profile becomes comparable in magnitude, it becomes

267

insensitive to effect of aspect ratio and both maxima of local impedance (at peak and valley

268

of surface profile) simultaneously increase in magnitude with increase in aspect ratio. No

269

further bifurcations were observed with increase in aspect ratio. Figure 5 shows the effect

270

of frequency on magnitude and behavior of local impedance for sinusoidal surface profile.

272

Similar effects are seen here, as seen in the case of increasing aspect ratio (figure 4). This p bifurcation arises when diffusion length ( Dα /ω) becomes greater than the width of the

273

interface (h). Emergence of steady state pattern is seen in figure 5, as increase or decrease in

274

frequency does not introduce bifurcation but only decreases or increases magnitude of local

275

impedance.

271

276

To get the further insight about the behavior of surface profile, its effect on the phase is 14 ACS Paragon Plus Environment


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277

also studied. Figure 6 and 7 shows the effect of aspect ratio and frequency on the phase over

278

the sinusoidally varying surface profile, respectively. In figure 6, at low aspect ratio, phase

279

shows maxima and minima at the valley and peak positions of sinusoidally varying surface

280

profile, respectively. It is observed that phase behaves opposite to the surface profile at low

281

aspect ratio. At peak positions, there is accumulation of charge which lead to the capacitive

282

kind of phase behavior and at valley positions of surface, no such accumulation takes place,

283

which forces surface to follow Warburg kind of behavior. With the increase of aspect ratio,

284

a peak in phase starts appearing out of minima of phase and with further increase of aspect

285

ratio, it dominates other phase maxima (which was present over valley of surface profile).

286

Now, at higher aspect ratios we found, maxima at peak positions of surface profile and nearly

287

flat region at valley positions of the surface profile. Figure 6 shows the effect of frequency

288

on the phase, with increase in frequency phase shifts to higher values. Also, it can be seen

289

that with increase of frequency, from nearly flat phase shifts to modulating phase. At high

290

frequency, maxima can be seen at cliff positions of the surface profile while minima in phase

291

are observed at peak and valley positions of surface profile.

292

Localization of impedance over Weierstrass surface

293

Behavior of local impedance density is studied along the realistic surface profile, generated

294

using band limited Weierstrass function. Enhanced electrochemically active and depletion

295

zones are identified over the corrugated surface. Magnitude and behavior of local impedance

296

density varies along the surface with the change in aspect ratio and frequency in figure 8 and

297

9, respectively. Anomalous inhomogeneous distribution of local impedance density is seen

298

for Weierstrass surface under diffusion limited aggregation/dissolution of the electroactive

299

species.

300

In figure 8, depletion regions are seen at the valley positions of Weierstrass band limited

301

fractal electrode. Also, maxima of local impedance density or depletion zones are seen at the

302

peak positions of Weierstrass surface at high aspect ratio, which are initially absent at low 15 ACS Paragon Plus Environment

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303

aspect ratio. This behavior is similar to local impedance density behavior seen over the single

304

sinusoid surface profile. Minima of local impedance density or enhanced electrochemical

305

activity zones appears to be at cliff positions of Weierstrass surface profile. Gently changing

306

steeps provide higher electrochemical activity zones while steeper cliffs provide comparatively

307

lower activity. Figure 9 shows the effect of frequency on the local impedance density at the

308

Weierstrass surface profile. Similar trends like figure 8 are seen in this study i.e., effects

309

of frequency on local impedance density. Depending upon the frequency and aspect ratio

310

in a Weierstrass surface profile, electrochemical activity which is supposed to be maximum

311

at the peak, 74 shifts to the cliff region of surface profile and minima of electrochemical

312

activity appears at peak and valley positions of the surface profile. This fact is seen in the

313

impedance density profile as a stronger localization at intermediate frequency (or at lower

314

frequency). The impedance density increases due to the depletion of the reacting species

315

near the electrode surface under the diffusion controlled limit at higher frequency which can

316

help in understanding the phenomena of complex electrodeposition (e.g. growth of scattered

317

grains with fractal patterns).

318

Local impedance phase studies are also conducted on Weierstrass surface profile. Fig-

319

ure 10 shows the effect of aspect ratio, at constant frequency, on the local impedance phase.

320

At low aspect ratio, maxima of local impedance phase appears at peak and valley positions

321

of Weierstrass surface profile while minima appears at the cliff positions of the surface. On

322

increasing aspect ratio, this order reverse down. Cliff positions of Weierstrass surface are the

323

positions which gives maximum phase. Figure 11 shows the frequency effects on the phase

324

over the rough Weierstrass surface profile. Similar trends as of effect of aspect ratio, are seen

325

in this plot. At high frequency, cliff positions of Weierstrass surface gives maximum phase

326

while peak and valley positions of surface gives minima in phase profile.

327

Local impedance behavior at Weierstrass fractal electrode is more complicated to un-

328

derstand as compared to local impedance behavior at sinusoidal rough surface. Therefore,

329

probability distribution of local impedance density is studied to get the idea of distribution of



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330

local impedance over the rough electrode surface. Figure 12 shows the effect of ` on distribu-

331

tion of local impedance density. Normalized probability distribution function (PDF) of local

332

logarithmic impedance density at Weierstrass surface is plotted. As we can see from figure 12

333

that with increase in `, mean local impedance increases with decrease in `. Also, distribution

334

becomes wider with decrease in `. Mean of local impedance density and standard deviation

335

336

are listed in table 1. With decrease in `, mean and standard deviation increases. Similarly, √ figure 13 shows the effect of width of interface ( m0 ) on the distribution of local impedance

338

density. Table 2 lists the values of mean and standard deviation of local impedance density. √ With increase in m0 mean and standard deviation increases. In figure 14, effect of fractal

339

dimension, DH , is studied.

340

Conclusion

341

We have analyzed the influence of surface morphological complexities (due to roughness)

342

to the local impedance behavior for a reversible redox system having unequal diffusivities

343

(DO 6= DR ). Realistic surface corrugation is modeled through band-limited Weierstrass

344

function. This work unravels the connection between the fractal dimension (DH ) of rough-

345

ness, the lower and upper cutoff length scales (` & L) and width of interface with the local

346

impedance. The analysis of rough surfaces have both obvious and subtle application in ap-

347

plied and fundamental electrochemical systems, therefore a search for the local electrochem-

348

ical impedance of such surfaces is of utmost importance. This is a theoretically challenging

349

problem and hence circumvented using second order perturbation theory in combination

350

with Padè approximant method. The main conclusions drawn from this methodology are as

351

follow:

337

352

• Our theoretical result for LEIS shows non-uniform distribution of surface impedance

353

with frequency for simple reversible charge transfer and its interrelation to morpho-

354

logical features. Strong non-uniform distribution of local impedance is indication of 17 ACS Paragon Plus Environment

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355

localization of reaction sites, viz. high impedance indicated low reactivity while low

356

impedance indicates high reactivity.

357

• This study shows anomalous bifurcation in the magnitude and phase of local impedance

358

even at moderate amplitude sinusoidal surface roughness. For more realistic fractal

359

surface model (of Weierstrass function), even more complex multi-furcations in local

360

impedance are observed which is an indication of complex reactivity pattern on rough

361

electrodes.

362

• At high aspect ratio, bi(/multi)-furcations are observed beside peak positions (i.e., cliff

363

positions) of the surface. It is found that enhanced electrochemical activity sites are

364

localized at the cliff positions of the surface profile whereas depletion zones are found

365

at peak and valley sites of the surface. This could be the cause of dendrite formation

366

on the surfaces.

367

• At low aspect ratio, enhanced electrochemical activity sites are found at peak positions

368

while depletion zones at valley positions of the surface. This is similar to the result

369

predicted by Pritzker, 74 he stated that locations of the current maxima and minima

370

are found at peak and valley positions, respectively.

371

• Peak positions of the surface can have enhanced electrochemical activity sites or de-

372

pletion sites depending on the roughness aspect ratio. Whereas, there will be always

373

activity depletion zones at valley positions of the surface.

374

• For nanostructured surfaces, diffusion length becomes larger than the width of rough-

375

ness at relatively high frequency of available experimental frequency window. There-

376

fore, for simple sinusoidal and complex Weierstrass surface, increase (or decrease) in

377

frequency after this crossover frequency only decreases (or increases) magnitude of lo-

378

cal impedance but does not introduce change in pattern of distribution. Hence there

379

is onset of steady state activity pattern for nanostructured surfaces.



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380

• It is difficult to relate peak, cliff and valley positions of a complex surface to the

381

local impedance behavior. Hence, the probability distribution function for logarithmic

382

local impedance at Weierstrass surface are studied to understand overall change in

383

surface activity with roughness features. Therefore, overall effect of surface on local

384

impedance is seen through their mean and standard deviations. Distribution function

385

for logarithmic local impedance at Weierstrass surfaces moves towards higher mean

386

impedance and variance with increase in roughness.

387

• Presence of nano scale roughness introduce nano-localization of reaction sites. If L

388

(= λ0 ) is assumed to be 1 µm and ` is about 57 nm, then size of impedance nano-

389

localization (of enhanced activity sites to the depletion sites) is comparable to the size

390

of finest feature of roughness (∼ `).

391

Though experimental mapping of local impedance at nanometer (high) resolution is still a

392

challenging problem. Here we show, it is possible to simulate local impedance at nanoscale

393

resolution. This methodology will be further extended to experimentally scanned surface

394

profiles.

395

Acknowledgement

396

R.K. thanks University of Delhi for financial support under ”Scheme to Strengthen R&D

397

Doctoral Research Programme“. R.K. and S.D. (for SRF fellowship) are grateful to DST-

398

SERB (Project No. SB/S1/PC-021/2013)-India for providing financial assistance.

399

Appendix

400

The admittance y(ω) of an interfacial redox reaction, O + ne− * ) R, driven by a sinusoidal

401

interfacial potential (η(t) = E − Ee = η0 ejωt , where the equilibrium potential is Ee and η0

402

is the magnitude of applied sinusoidal potential.) can be obtained by solving appropriate

403

diffusion equations. The concentration varies locally from Cα0 to δCα0 . The concentration 19 ACS Paragon Plus Environment

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404

profile of the form δCα (~r, t) = δCα (~r)ejωt , satisfies the diffusion equation in the sinusoidal

405

regime, i.e, jωδCα (~r) = Dα 52 Cα (~r)

(A.1)

407

where α = O, R representing the oxidized and reduced species, δCα (~r) is the difference √ between surface and bulk concentration, j = −1 and ω is the angular frequency. The

408

Nernstian boundary condition is valid for systems with a large value of exchange current

409

density. The Nernstian boundary constraint is linearized under assumption of small external

410

perturbation potential and can be written as

406

CO CR − 0 = −nf η(t) CO0 CO

(A.2)

411

There is a local transfer kinetics limitation at the interface (ζ) which can be obtained

412

under Nernstian boundary constraint. Under the identity, given relation between concentra-

413

tion of oxidized and reduced species with unequal diffusion coefficients, at the interface of a

414

gently and weakly fluctuating, is given as 75

p p DO δCO (z = ζ(x, y)) ≈ − DR δCR (z = ζ(x, y))

415

Hence, the linearized Nernstian condition at the interface can be expressed as: . 1 ξ δCO = −nf η0 + CO CR . 1 ξ δCR = nf η0 ξ + CO CR

416

(A.3)

where ξ =

(A.4)

p DO /DR .

417

The admittance density expression for diffusion controlled adsorption process for arbi-

418

trary surface profile is related with interfacial local current density (i(ω)). 69,70 The interre-

419

lation of local admittance density (y(ω)) and interfacial current density can be expressed



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420

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as: y(ω) =

jω i(ω) η0

(A.5)

421

The current density at the interface in Fourier and Laplace transform domain is useful

422

quantity in formulating admittance density. The expression for current density for unequal

423

diffusion coefficient at weakly and gently fluctuating corrugated (1-D) rough surface in fre-

424

quency (ω) domain is obtained by similar procedure described in Ref. 69 and can be written

425

as:

i(Kx , ω) =

426

nF Dα δCα ω0α ˆ x) + O ˆ x − K 0 )ζ(K ˆ 0) ˆ 1α ζ(K ˆ 2α ζ(K 2πδ(Kx ) + O x x jω

(A.6)

ˆ 1 and O ˆ 2 are defined in equation 7. where O

427

Now, knowing local current-admittance relation (equation A.5) and using equation A.6

428

gives admittance expression (unequal diffusion coefficient) for diffusion controlled process at

429

arbitrary surface profile, which can be represented in operator notation as: √ n2 F 2 jω

y(Kx , ω) = RT

√ 1 0 DO CO

+

√ 1 0 DR CR

ˆ x − K 0 )ζ(K ˆ x) + O ˆ 0) ˆ 2α ζ(K ~x ) + O ˆ 1α ζ(K 2πδ(K x x (A.7)

430

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431

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(48) Eftekhari, A. Fractal Study of LiMn2 O4 Film Electrode Surface for Lithium Batteries Application. Electrochim. Acta 2002, 47, 4347-4350. (49) Eftekhari, A. On the Fractal Study of LiMn2 O4 Electrode Surface. Electrochim. Acta 2003, 48, 2831-2839.

545

(50) Dogaru, T.; Carin, L. Time-Domain Sensing of Targets Buried Under a Gaussian,

546

Exponential, or Fractal Rough Interface. IEEE Transactions on Geoscience and Remote

547

Sensing 2001 39, 1807-1819.

548

¯ (51) Weierstrass K. Uber continuirliche Functionen eines reelles Arguments, die f¯ ur

549

keinen Werth des letzteren einen Bestimmten Differentialquotienten besitzen; Königl.

550

Akademie der Wissenschaften, Berlin, 1872; Reprinted in: Weierstrass K. Mathematis-

551

che Werke II ; Johnson, New York, 1967.

552

553

(52) Mandelbrot, B. B. Fractals: Form, Chance and Dimension; Freeman, San Francisco, 1977.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

554

555

556

557

558

559

(53) Mandelbrot, B. B. Gaussian Self-Affinity and Fractals: Globality, The Earth, 1/f Noise, and R/S ; Springer-Verlag, New York, 2002. (54) Hardy, G. H. Weierstrasss non-differentiable function. Trans. Amer. Math. Soc. 1916 17, 301-325. (55) Kant, R. Statistics of Approximately Self-Affine Fractals: Random Corrugated Surface and Time Series. Phys. Rev. E 1996, 53, 5749-5763.

560

(56) Feder, J. Fractals; Plenum, New York, 1988.

561

(57) Kant, R.; Rangarajan, S. K. J. Diffusion to Rough Interfaces: Finite Charge Transfer

562

563

564

565

566

567

568

569

570

Rates. J. Electroanal. Chem. 1995, 396 285-301. (58) Kant, R. Diffusion-Limited Reaction Rates on Self-Affine Fractals. J. Phys. Chem. B 1997, 101 3781-3787. (59) Abramowitz, M.; Stegan, A. Handbook of Mathematical Functions, Dover Publications Inc., New York, 1972. (60) De Levie, R. The Influence of Surface Roughness of Solid Electrodes on Electrochemical Measurements. Electrochim. Acta 1965, 10, 113-130. (61) Jacquelin, J. Theoretical Impedance of Rough Electrodes with Smooth Shapes of Roughness. Electrochim. Acta 1994, 39, 2673-2684.

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(62) Pototskaya, V. V.; Evtushenko, N. E.; Gichan, O. I. Electrode Reactions Involving

572

Arsenic and Its Inorganic Compounds. Russ. J. Electrochem. 2001, 37, 997-1011.

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(63) Pototskaya, V. V.; Evtushenko, N. E.; Gichan, O. I.; Budnikov, G.K.; Maistrenko, V.N.;

574

Vyaselev, M.R. Osnovy sovremennogo elektrokhimicheskogo analiza (Fundamentals of

575

Modern Electrochemical Analysis), Moscow: Mir, 2003. Russ. J. Electrochem. 2004,

576

40, 475-475. 27 ACS Paragon Plus Environment

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577

578

579

580

581

582


(64) Fedkiw; P. S.; Nolen, T. R. The Diffusional (Warburg) Impedance at a Sinusoidal Shape Electrode. J. Electrochem. Soc. 1990, 137, 158-162. (65) Kant, R.; Rangarajan, S. K. Effect of Surface Roughness on Interfacial ReactionDiffusion Admittance. J. Electroanal. Chem. 2003, 552, 141-151. (66) Kumar, R.; Kant, R. Theory of Generalized Gerischer Admittance of Realistic Fractal Electrode. J. Phys. Chem. C 2009, 113, 19558-19567.

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(67) Kumar, R.; Kant, R. Admittance of Diffusion Limited Adsorption Coupled to Re-

584

versible Charge Transfer on Rough and Finite Fractal Electrodes. Electrochim. Acta

585

2013, 95, 275-287.

586

587

588

589

590

591

592

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(68) Srivastav, S., Kant, R. Anomalous Warburg Impedance: Influence of Uncompensated Solution Resistance. J. Phys. Chem. C 2011, 115, 12232-12242. (69) Kant, R; Rangarajan, S. K. Effect of Surface Roughness on Diffusion-Limited Charge Transfer. J. Electroanal. Chem. 1994, 368, 1-21. (70) Kant, R. Electrochemistry at Complex Interfacial geometries, Ph.D. Thesis, Indian Institute of Science, Bangalore, 1993. (71) Champaney, D. C. Fourier Transforms and their Physical Applications; Academic Press, New York, 1973.

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(72) Baker, G. A.; Graves-Morris, P. Pade Approximants Part II: Extensions and Applica-

595

tions; Gian-Carlo Roto (Ed.), Encyclopedia of Mathematics and its Applications, Vol.

596

14, Addison-Wesley, London, 1981.

597

(73) Yadav, V. K.; Kant, R.(Research Guide) Role of Self-Affine Fractal Roughness in

598

the Reversible Charge Transfer Admittance, M.Phil. Dissertation, University of Delhi,

599

Delhi, 2005.



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(74) Louch, D. S.; Pritzker, M. D. Transport to Rough Electrode Surfaces; Part 1. Pertur-

601

bation Solution for Two-Dimensional Steady State Diffusion-Limited Transport to a

602

Surface with Arbitrary Small Amplitude Features, J. Electroanal. Chem. 1991, 319,

603

33-53.

604

(75) Parveen; Kant, R. Theory for Anomalous Response in Cyclic Staircase Voltammetry:

605

Electrode Roughness and Unequal Diffusivities. J. Phys. Chem. C 2014, 118, 26599-

606

26612.


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Table 1: Effect of size of finest feature of roughness (` = λ0 b−N2 ) on distribution function of local impedance at Weierstrass surface. ` (nm) 318 179 101 57 32 18 10

Logarithmic mean h Log zi/Ω µm2 10.484 10.612 10.751 10.896 11.045 11.195 11.346

Table 2: Effect of width of interface (h = at Weierstrass surface. h (µm) 0.4 0.5 0.6 0.7 0.8

√

Logarithmic variance Log (z/hzi) 0.2000 0.2099 0.2153 0.2177 0.2188 0.2186 0.2185

m0 ) on distribution function of local impedance

Logarithmic mean h Log zi/Ω µm2 10.614 10.904 11.179 11.429 11.851

Logarithmic variance Log (z/hzi) 0.2066 0.2221 0.2298 0.2336 0.2367



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 1: Plot of single sinusoidally√varying surface profile (ζ(x)) and its gradient (ζx (x)) at h= 0.1 µm, λ0 = 0.1 µm and b = π.

Figure 2: Plot of band-limited Weierstrass surface profile (ζ(x)) and its gradient (ζ√ x (x)) at h= 1.0 µm, λ0 = 1 µm, N1 =0, N2 = 3, ` = 18 nm, L = 1 µm, DH = 1.3 and b = π.


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0.2

DH =1.2

ΖHxL

0.1 0.0 -0.1 -0.2 -0.3 0

2

4

6

8

10

6

8

10

6

8

10

xΛ0 0.2

DH =1.5

ΖHxL

0.1 0.0 -0.1 -0.2 -0.3 0

2

4

xΛ0

DH =1.8

0.2 0.1

ΖHxL

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.0 -0.1 -0.2 -0.3 0

2

4

xΛ0 Figure 3: Effect of fractal dimension (DH ) is studied. Plot is generated at h= 0.45 µm, λ0 √ = 1 µm, N1 = -2, N2 = 7, ` = 182 nm, L = 3.14 µm and b = π. DH is varied as 1.2, 1.5 and 1.8 from top to bottom.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 4: Effect of surface features on impedance magnitude is studied. This figure illustrate the effect of aspect ratio (h/λ0 ) on impedance magnitude at sinusoidally varying surface −1 profile. Here, parameters used are: λ0 = 0.1 µm, ω = 101 s√ , D= 5 ×10−8 cm2 /sec, CO = 5 −6 3 −6 3 × 10 mol/cm , CR = 5 × 10 mol/cm , n = 1 and b = π at T = 298 K. Surface profile is scaled for our convenience.


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Figure 5: Effect of surface features on impedance magnitude is studied. This figure illustrate the effect of frequency (ω) on impedance magnitude at sinusoidally varying surface profile. Here, parameters used are: λ0 = 0.1 µm, h = 0.05 µm, √ D= 5 ×10−8 cm2 /sec, CO = 5 × 10 −6 mol/cm3 , CR = 5 × 10 −6 mol/cm3 , n = 1 and b = π at T = 298 K. Unit of ω is s−1 . Surface profile is scaled for our convenience.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 6: Effect of surface features on phase is studied. This figure illustrate the effect of aspect ratio (h/λ0 ) on impedance magnitude at sinusoidally varying surface profile. Here, parameters used are: λ0 = 0.1 µm, ω = 101 s−1 , D=√5 ×10−8 cm2 /sec, CO = 5 × 10 −6 mol/cm3 , CR = 5 × 10 −6 mol/cm3 , n = 1 and b = π at T = 298 K. Surface profile is scaled for our convenience.


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Figure 7: Effect of surface features on phase is studied. This figure illustrate the effect of frequency (ω) on impedance magnitude at sinusoidally varying surface profile. λ0 = 0.1 µm, h = 0.05 µm,√D= 5 ×10−8 cm2 /sec, CO = 5 × 10 −6 mol/cm3 , CR = 5 × 10 −6 mol/cm3 , n = 1 and b = π at T = 298 K. Unit of ω is s−1 . Surface profile is scaled for our convenience.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 8: Effect of surface features on impedance magnitude is studied. This figure illustrate the effect of aspect ratio (h/λ0 ) on impedance magnitude at Weierstrass surface profile. Other parameters are: N1 = 0, N2 = 3, ` = 180 nm, L = 1 µm, DH = 1.3, λ0 = 1 µm, ω = 101 s−1 , √ D= 5 ×10−8 cm2 /sec, CO = 5 × 10 −6 mol/cm3 , CR = 5 × 10 −6 mol/cm3 , n = 1 and b = π at T = 298 K. Surface profile is scaled for our convenience.


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 9: Effect of surface features on impedance magnitude is studied. This figure illustrate the effect of frequency (ω) on impedance magnitude at Weierstrass surface profile. Other parameters are: N1 = 0, N2 = 3, ` = 180 nm, L = 1 µm, DH = 1.3, λ0 = 1 µm, h = 0.5 µm, √ D= 5 ×10−8 cm2 /sec, CO = 5 × 10 −6 mol/cm3 , CR = 5 × 10 −6 mol/cm3 , n = 1 and b = π at T = 298 K. Unit of ω is s−1 . Surface profile is scaled for our convenience.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 10: Effect of surface features on phase is studied. This figure illustrate the effect of aspect ratio (h/λ0 ) on impedance magnitude at Weierstrass surface profile. Other parameters are: N1 = 0, N2 = 3, ` = 180 nm, L = 1 µm, DH = 1.3, λ0 = 1 µm, ω = 101 s−1 , √ D= 5 −8 2 −6 3 −6 3 ×10 cm /sec, CO = 5 × 10 mol/cm , CR = 5 × 10 mol/cm , n = 1 and b = π at T = 298 K. Surface profile is scaled for our convenience.


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Figure 11: Effect of surface features on phase is studied. This figure illustrate the effect of frequency (ω) on impedance magnitude at Weierstrass surface profile. Other parameters are: N1 = 0, N2 = 3, ` = 180 nm, L = 1 µm, DH = 1.3, λ0 = 1 µm, h = 0.5 µm, √ D= 5 ×10−8 cm2 /sec, CO = 5 × 10 −6 mol/cm3 , CR = 5 × 10 −6 mol/cm3 , n = 1 and b = π at T = 298 K. Unit of ω is s−1 . Surface profile is scaled for our convenience.



2.0

1.5

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1.0

0.5

0.0 10.5

11.0

11.5

12.0

Log zW Μm2 Figure 12: Effect of lower length scale cutoff, `, on statistical distribution of local impedance at Weierstrass surface is plotted. Black line represents statistical distribution of local impedance in absence of roughness while colored lines represent statistical distribution of local impedance at Weierstrass surface. ` is decreased or N2 is increased while moving from left to right. ` and N2 details are given in table 1. Other parameters are L = 1 µm (N1 = 0), h = 0.1 µm, DH = 1.3, ω = 101 s−1 , λ0 = 1.0 √ µm, D= 5 ×10−8 cm2 /sec, CO = 5 × 10 −6 3 −6 3 mol/cm , CR = 5 × 10 mol/cm , n =1, b = π and T = 298 K.


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1.5

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1.0

0.5

0.0 10.5

11.0

11.5

12.0

12.5

Log zW Μm2 √ Figure 13: Effect of width of interface, m0 , on statistical distribution of local impedance at Weierstrass surface is plotted. Black line represents statistical distribution of local impedance in absence of roughness while colored lines represent statistical distribution of local impedance at Weierstrass surface. Width of interface increases from left to right and details are given in table 1. Other parameters are L = 1 µm (N1 = 0), ` = 101 nm (N2 = 4), DH = 1.3, ω = 101 s−1 , λ0 = 1.0 √ µm, D= 5 ×10−8 cm2 /sec, CO = 5 × 10 −6 mol/cm3 , −6 3 CR = 5 × 10 mol/cm , n =1, b = π and T = 298 K.



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1.0

0.5

0.0 10.2

10.6

11.0 2

11.4

Log zW Μm

Figure 14: Effect of fractal dimension, DH , on statistical distribution of local impedance at Weierstrass surface is plotted. Black line represents statistical distribution of local impedance in absence of roughness while colored lines represent statistical distribution of local impedance at Weierstrass surface. Fractal dimension increases from left to right as 1.1, 1.3, 1.5, 1.7 and 1.9. Other parameters are L = 1 µm (N1 = 0), ` = 179 nm (N2 = 3), h = 0.1 µm, ω = 101 s−1 , λ0 = 1.0√µm, D= 5 ×10−8 cm2 /sec, CO = 5 × 10 −6 mol/cm3 , CR = 5 × 10 −6 mol/cm3 , n =1, b = π and T = 298 K.


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Anomalous Localization of Electrochemical Activity in Reversible

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