Chapter 4

Generalization of the Anson

Equation for Fractal and

Nonfractal Rough Electrodes

113

114

Abstract

We have generalized the Anson equation for chronocoulometry at deterministic

and randomly rough electrode-electrolyte interface morphologies. The randomly

rough electrode surface is characterized by the power-spectrum of roughness. An

elegant mathematical formula between the surface roughness power-spectrum and

the charge-transient is obtained. Detailed analysis of this formula is performed

for the chronocoulometric response of fractal and nonfractal electrodes using an

approximate power-law and Gaussian power-spectra, respectively. Realistic finite

fractal roughness consists of the following four morphological characteristics: frac-

tal dimension (DH); the strength of fractality (µ), which is the magnitude of the

power spectrum at the unit roughness wavenumber; and the lower (ℓ) and up-

per (L) cut-off lengths, which are related to cut-off wavenumbers in the power

spectrum. Nonfractal Gaussian roughness consists of two morphological charac-

teristics, the mean square width of roughness (h2) and the transversal correlation

length (a). The chronocoulometric response for such rough electrodes shows three

time regimes in double logarithmic plots. An increase in roughness increases the

magnitude of charge transients, which introduce non-linear behavior in traditional

Anson plots. Finally, this generalization provides insight into the effect of fractal

and nonfractal morphological disorder on the chronocoulometric measurements of

Nernstian charge transfer under the potential step method.

115

116

1 Introduction

Chronocoulometry is a classical electrochemical technique frequently used in elec-

troanalytical chemistry. It is the measurement of the total charge that passes

during the time following a potential step as a function of time. The primary appli-

cations that utilize this technique include the measurement of diffusion-coefficients

and concentration [1, 2], the kinetics of electron transfer reactions [3, 4, 5], the

qualitative and quantitative study of the adsorbed species [6, 7, 8, 9, 10, 11], the

surface charge density [12], the electrode area and the effective time window of an

electrochemical cell [13] and the determination of metals in samples [14, 15, 16].

Chronocoulometry is preferred over chronoamperometry because it offers impor-

tant experimental advantages [17].

For the redox reaction O + ne− R, taking place at an electrode-electrolyte

interface, the total charge produced for the reduction at the smooth electrode for

diffusion- only phenomena is given by the Anson equation as [17]:

QP (t) =2nF C0

O A0

√Dt√

π, (1.0.1)

where QP (t) is the charge at the planar electrode, n is the number of electron(s)

involved in the reaction, F is the Faraday constant, C0O is the concentration of the

oxidized species, A0 is the area of the electrode, D is the diffusion coefficient and

t is the time. Equation (1) can be modified to QP (t) = 2nF CsA0

√Dt/π,

with a more general Nernstian electrode boundary constraint in the presence

of both oxidized (C0O) and reduced (C0

R) species viz, δCO(z = 0, t) = −Cs =

−(C0O − C0

Rθ)/(1 + θ) [18], where θ = e−nf(E−E0′), E0′ is the formal potential and

E is the applied potential.

Fractal geometry is often used to understand the diffusion-limited processes at

the rough electrode-electrolyte interface and the influence of surface disorder. The

simplest theoretical models for the response due to fractal roughness are based on

the scaling argument, which is often used to characterize the anomalous behavior

in the response of the fractal electrode. De Gennes’ scaling law for the diffusion-

controlled current is given as: I ∝ t−β [19, 20, 21] and can be extended by its time

117

integration as the total charge. Thus, the scaling form of the chronocoulometric

response is as follows:

Q(t) ∝ t−β+1 , (1.0.2)

where the exponent, β, depends on interfacial roughness given by β = DH−12

and

varies from 0.5 to 1 and the DH (fractal dimension) varies from 2 (planar surface)

to 3 (nearly porous surface). When DH = 2, the scaling exponent in the charge

becomes 0.5, while at DH = 3, it is 0. If the fractal dimension approaches 3, the

charge-approaches independence in time.

Responses such as current, charge, impedance and absorbance are the quan-

tities that are theoretically analyzed at the rough electrode-electrolyte interface

[18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39,

40, 41, 42, 43]. The general approach was based on the ab initio-method to study

the geometrical disorder (roughness) aspects of the electrode responses, such as

current, charge, impedance and absorbance. Recent interest in random roughness

morphology with a finite fractal and nonfractal nature allowed us to use ab initio

methodology, which has introduced new opportunities to understand the influence

of surface disorder. Some of these opportunities include absorbance transients at

the rough electrode [30], potential sweep methods on an arbitrary topography [31],

Cottrellian current at a rough electrode with a solution resistance effect [32], par-

tial diffusion-limited interfacial transfer/reaction [33, 34], Gerischer admittance

at a rough electrode [35], anomalous Warburg impedance at a rough electrode

[36, 37, 38] and anomalous diffusion reaction rates or diffusion-controlled poten-

tiostatic current transients [18, 39, 40, 41, 42, 43].

Charge and absorbance show a linear relationship [30, 44], and the combination

of the two techniques (spectroelectrochemistry) gives the versatile uses in electro-

analytical chemistry [45, 46, 47, 48]. Chronocoulometry describes the amount of

a species produced following a potential step as a function of time. The various

contributing components to the charge such as a double-layer, adsorption and

diffusion can be separated easily in chronocoulometry experiments. This sepa-

ration of components of the charge provides more quantitative information and

the possible effects of the electrode surface configuration on the responses. Addi-

118

RO

l LDt

Diffusion width

DH

Electrolyte

еl L

h

Random fractal electrode

Figure 4.1: Schematic diagram of the diffusion-controlled chronocoulometry atthe rough electrode-electrolyte interface, where the O(sol) + ne− R(sol) redoxreaction occurs. The various morphological and phenomenological characteristicsthat control the electrochemical responses of the electrode-electrolyte interface,( i.e., fractal dimension (DH), lower (ℓ) and upper (L) cut-off length scales offractality, width (h) and diffusion length (

√Dt )) are shown.

tionally, chronoabsorptometry is possible only when the substance in the solution

phase contains a chromophore. The presence of chromophore is not a necessary

condition for the chronocoulometric measurements which provides the amount of

species present in the solution, either adsorbed or diffused. It is important to

discuss this technique with spectroelectrochemistry [30]. Herein, we wish to use

our ab initio approach to analyze (diffusion-controlled) reversible redox reactions

taking place at the rough electrode-electrolyte interface through the charge tran-

sients. As there exists mathematical isomorphism between chronocoulometry and

chronoabsorptometry, we will not provide the detailed calculations and the for-

mulation section which can be found in the spectroelectrochemical description in

chapter 2.

The chapter is organized as follows: the charge transient is obtained for the

surfaces with arbitrary and random roughness profiles; the results for the charge

transients are analyzed for finite self-affine fractal surfaces and nonfractal surfaces;

the results are discussed; and the conclusions are reported.

119

2 Model for Electrodes with Random Roughness

Large numbers of morphological irregularities on surfaces (or interfaces) can be

satisfactorily described as centered Gaussian fields, ζ(r∥), where r∥ = (x, y) is a

two dimensional space vector [49]. The statistical properties of such surfaces can

be written in terms of the mean and correlation function as:

⟨ζ(r∥)

⟩= 0⟨

ζ(r∥)ζ(r′∥)⟩

= h2W (|r∥ − r′∥|) . (2.0.1)

The angular brackets ⟨·⟩ in Eq. 2.0.1 represent an ensemble-average over various

possible surface configurations. h2 = ⟨ζ2⟩ is the mean square departure of the sur-

face from flatness, and W (·) is the normalized correlation function. An additional

simplification was achieved using the assumption of statistical homogeneity and

isotropy; the correlation function depended only on the separation distance of the

two points (|r∥ − r∥′|).

The surface correlation function contained information about the surface mor-

phological characteristics such as the average area, mean square height, gradient,

curvature and correlation length for the random profile. For a given nonfractal

surface correlation function, the randomly rough surface is primarily characterized

by two parameters, the root mean square height (h) and the transverse correla-

tion length (a). If a = 0, no correlation exists between the values assumed by

ζ(r∥) at two different points; however, they may be close to each other. Therefore,

a = 0 implies a ‘white noise’ surface that tends to be discontinuous everywhere.

Similarly, a → 0, implies a surface with very densely packed irregularities. If a is

large, points on the surface will be correlated even if they are far apart. A large

a value implies a gently rolling surface, such as a large distance between the hills

and valleys. Similarly, a → ∞, implies a flat surface at z = h.

Another useful way to understand the statistics for such centered Gaussian

fields (surfaces) uses the surface structure factor, ⟨|ζ(K∥)|2⟩, where ζ(K∥) is the

Fourier transform of the random surface profile ζ(r∥) and K∥ is the wave-vector

in two dimensions. The structure factor is defined as the Fourier transform of the

120

surface two-point correlation function and is useful to represent the electrochemical

response of randomly rough electrodes [42]. The structure factor is dependent on

the spatial wavenumber of roughness, i.e. |K∥|. The graphical representation of

the power-spectrum of roughness in detail can be found in ref. [35]. The ensemble-

averaged Fourier transformed surface profile and its correlation follow the equation

below:

⟨ζ(K∥)⟩ = 0

⟨ζ(K∥)ζ(K ′∥)⟩ = (2π)2δ

(K∥ + K ′

∥

)⟨|ζ(K∥)|2⟩ , (2.0.2)

where δ(K∥) is the Dirac delta function in K∥. In theory, the statistical prop-

erties of the random surface profiles are characterized by a structure factor or

surface roughness power-spectrum, i.e.

⟨∣∣∣ζ(K∥)∣∣∣2⟩. Roughness features with a

slowly varying surface profile are compressed around lower wavenumbers, whereas

a rapidly varying roughness of surface is positioned in the high wavenumber do-

main. Realistic finite fractal electrode roughness uses the power-spectrum with

band limited power-law form, which helps to circumvent the mathematical diffi-

culties of non-differentiability and non-homogeneity of the ideal fractals and to

introduce two finite lateral cut-off length scales. The smaller and larger cut-off

length scales between the surfaces show a self-affine property where the larger cut-

off length is also referred to as the correlation length. A more detailed discussion

on finite fractals is available in ref. [34, 40].

It is noted that the 2k-th moments of the power-spectrum are related to the

mean square k-th gradients of the surface profile correlations [42], which is shown

as follows:

m2k =1

(2π)2

∫d2K∥K

2k∥ ⟨|ζ(K∥)|2⟩ = ⟨(∇k

∥ζ(r∥))2⟩ , (2.0.3)

where ∇k∥ ≡ (∂/∂x, ∂/∂y) is the two dimensional gradient operator. The moments

of the power-spectrum are related to the various morphological features of a rough

surface, including mean square (MS) width (m0), MS gradient (m2) and MS cur-

vature (m4). The surface roughness correlation function, power-spectrum and its

moments are typically obtained from AFM studies.

121

3 Theory of Chronocoulometry for Electrodes

with Roughness

In the presented ab initio formulation, one must evaluate the concentration field

around an arbitrary electrode profile when calculating the electrode response as a

charge transient at an interface. The mathematical formulation of the diffusion-

controlled interfacial reversible reaction on disordered surfaces has complexity as-

sociated with boundary conditions on the surface, with uncontrolled surface ge-

ometric disorder. The complete diffusion problem for a reaction, O + ne− R,

satisfies the following equation:

∂

∂tδCα(r, t) = Dα∇2 δCα(r, t) , (3.0.1)

where α ≡ O,R, representing the oxidized or reduced species δCα(ζ), is the

difference between the surface and bulk concentration, Dα is the diffusion co-

efficient (for simplicity, we assume in our calculations DO = DR = D) and r

is the three dimensional vector, r ≡ (x, y, z). At initial time, and far from

the interface, a uniform initial and bulk concentration, C0α, is maintained, viz,

δCα(r, t = 0) = δCα(r, z → ∞, t) = 0. The surface boundary condition is deter-

mined by the Nernstian concentration potential characteristics, flux balance and

equal diffusion coefficient conditions. For example, DO = DR = D at the inter-

face, given by a potential dependent surface constraint: δCO(z = ζ, t) = −Cs. At

a large value of the potential difference, Cs approaches a diffusion- limited case,

such as Cs = C0O. The solution of Eq. 3.0.1 and its initial boundary conditions

for flux/current in the case of the smooth electrode is given in Eq. 1.0.1.

The electrochemically observed quantity, the total charge for electrode-electrolyte

interface, is simply given as the time integral of the current at rough interface. The

time integral of current [18, 43] in the Laplace transform domain is the Laplace

transform of the current divided by the Laplace transform variable p [50]. The to-

tal charge (Q) in a Laplace transformed domain for weakly and gently modulating

two-dimensional rough surfaces is obtained as a second order perturbation the-

ory and has the following functional form in Fourier transformed surface profiles

122

ζ (K∥):

Q(p) =

∫S0

d2r∥

[Q0(2 π)

2 δ(K∥) + Q1 ζ (K∥)

+Q2 ζ(K∥) ζ (K∥ − K ′∥); K∥ → r∥

], (3.0.2)

where∫S0

d2r∥ (·) is the integral over the xy plane, S0 is the projection of the surface

ζ on the reference surface z = 0, the notation for the inverse Fourier transform

is{f(K∥); K∥ → r∥

}= 1

(2π)2

∫d2K∥e

jK∥.r∥f(K∥), with j =√−1 and Q0, Q1 and

Q2 are the operators for the smooth surface and the first and second order terms

in the Laplace and Fourier transformed domain, respectively. These operators are

functions of phenomenological diffusion length (1/q =√

D/p), the wave vector of

the surface roughness profile (K∥) and external perturbation potential (E) through

Cs and are defined as follows:

Q0 =nF Cs

q p, Q1 =

nF Cs

q p(q∥ − q), and

Q2 =nF Cs

(2 π)2q p

∫d2 K ′

∥

[q∥ q∥,∥′ −

1

2(q2 + q q∥)

−|K∥ − K ′∥|

2 − K ′∥ · (K ′

∥ − K ′∥)], (3.0.3)

where q = (p/D)1/2 is the diffusion characteristic, q∥ = [q2 + K∥2]1/2.

Eq. 3.0.2 can be used to predict the choronocoulometric response after tak-

ing the inverse Laplace transform and the inverse Fourier transform for a known

surface profile. This expression is key in providing insight to the response of a

nanometer to micrometer scale structured electrode that represents an arbitrary

geometric profile or a rough electrode with a known profile.

The most common form of the surface roughness morphology is random in na-

ture and is necessary to calculate the charge for a random electrode-electrolyte in-

terface. The quantity of interest for a random surface morphology is the ensemble-

averaged (mean) charge transient over all possible surface configurations. The

mean chronocoulometric response senses the surface morphology through the sin-

gle surface roughness characteristics, the surface structure factor or the power-

123

pectrum [42]. The Laplace transformed choronocoulometric response for the (two-

dimensional) statistically homogeneous random rough electrode is obtained by

ensemble-averaging over surface configurations on Eq. 3.0.2 as:

⟨Q(p)⟩ = nF A0Cs

q p

{1 +

q

(2π)2

∫d2K∥[q∥ − q] ⟨|ζ(K∥)|2⟩

}. (3.0.4)

The ensemble-averaged charge transient is obtained by performing the inverse

Laplace transform of Eq. 3.0.4 and is written as the following:

⟨Q(t)⟩ = 2nF A0Cs

√Dt√

π

(1 + R ⟨|ζ(K∥)|2⟩

), (3.0.5)

where the operator R brings in a dynamic effect of surface roughness on the charge

through its effect on the power-spectrum of roughness of the electrode, given by

R = − 1

2(2π)2D t

∫d2K∥

(1− exp(−K2

∥D t)

−√πK2

∥Dt erf(√

K2∥Dt

)). (3.0.6)

Equation 3.0.5 elegantly describes the ensemble-averaged charge transient in terms

of the surface structure factor. This equation possesses information of various

morphological features of the rough electrode surface (or interface) and provides

physical insights into anomalies and possible interpretations for the electrochemi-

cal responses.

To correlate the average charge with the various morphological features of

the surface, we first consider the short time behavior of Eq. 3.0.5. Expansion of

Eq. 3.0.5 for the small t results in an expression that clearly interrelates the charge

to the several morphological features of roughness. The equation of the charge

transient of electrogenerated species in terms of moments of power-spectrum or

124

mean square derivatives of roughness is as follows:

⟨Q(t)⟩ =2 nF A0Cs

√Dt√

π

(1 +

∞∑k=0

(−1)k m2k+2

2(k + 1)(2k + 1)k!(Dt)k

)

=2 nF A0Cs

√Dt√

π

(1 +

1

2⟨(∇∥ζ)

2⟩ − 1

12⟨(∇2

∥ζ)2⟩Dt

+1

60⟨(∇3

∥ζ)2⟩ (Dt)2 − · · ·

). (3.0.7)

Equation 3.0.7 may be recast in terms of explicit physical features of roughness,

viz, the excess surface-area and mean square mean curvature (⟨H2⟩) as:

⟨Q(t)⟩ = 2 nF A0Cs

√Dt√

π

(1 +

△A

A0

− ⟨H2⟩3

Dt+ · · ·)

, (3.0.8)

where the excess relative area or excess roughness factor is given by △A/A0 =

⟨(∇∥ζ)2⟩/2, the mean curvature [42, 43] is given approximately by H ≈ 1

2∇2

∥ζ(r∥)

and the ensemble-averaged mean-curvature for a random surface is

⟨H2⟩ ≈ 1

4⟨(∇2

∥ζ(r∥))2⟩ . (3.0.9)

Equation 3.0.8 suggests that at very early times, the charge transient behaves

as a smooth surface electrode (with excess-area), and it starts sensing the local

shape of the electrode. As time progresses, there is a contribution of the higher

derivatives of the surface profile. Quantities △AA0

and ⟨H2⟩ are strongly dependent

on the smallest lateral length scale of roughness, and the transition from the short

time to the intermediate time will be dependent on the smallest lateral length

scale of the roughness (ℓ). The inner crossover transition time can be estimated

as ti ∼ ℓ2/2D.

Next, we consider the long-time asymptotic expansion of Eq. 3.0.5, which is

obtained with an asymptotic for erf(√K2

∥Dt) ∼ 1− exp(−K2∥D t)/

√πK2

∥Dt [50].

The following equation was obtained:

⟨Q(t)⟩ = 2 nF A0Cs

√Dt√

π

(1− m0

2Dt+ · · ·

). (3.0.10)

125

Thus, in the long time limit, when the diffusion layer becomes very thick, the first

leading term, the Anson response, in the mean charge is independent of roughness

features, but the second term provides morphological correction. This result is

dependent on the mean square width of roughness, i.e. m0 = h2. Crossover from

intermediate time regions to the Anson response will be dependent on h2, and

the outer crossover time (to) is estimated as to ≈ h2/2D. Equations 3.0.8 and

Eq.3.0.10 clearly demonstrate that charge transients and their crossover region

will be strongly dependent on morphological characteristics such as mean square

(MS) width, MS gradient and MS curvature.

4 Chronocoulometric Response of Finite Self-Affine

Fractal Electrode

Fractal geometry is a useful tool for describing many natural and artificial surface

topographies. Most surfaces are described in terms of idealized fractals for the

sake of simplicity. Idealized fractal models take into account the self-similarity in

roughness over an infinite length scale. As mentioned earlier, these idealized fractal

roughness profiles cannot be handled rigorously for practical purposes because

of their peculiar mathematical properties, among which non-differentiability and

non-homogeneity restrict their applicability in the physical systems. Therefore, for

practical purposes and realistic modeling of roughness, the band-limited self-affine

fractals are often taken into consideration [51].

Fractal surfaces, which exhibit statistical self-resemblance over all length scales,

can be described by the complete power-law power-spectrum. However, any real

fractal surface is characterized by the power-law spectrum over a few decades

in wavenumbers with a high and a low wavenumber cut-off. In several cases, the

band-limited power-law spectrum shows a gradual flattening for the low wavenum-

bers and a sharp decrease for the high wavenumbers. Such power spectra can be

approximated by a function with a sharp cut-off at low wavenumbers and a sharp

cut-off at high wavenumbers (i.e. the power at a high wavenumber is very small,

and a low wavenumber feature does not contribute significantly to flux hetero-

126

geneity in a diffusion-controlled mass transfer), which is given by [51, 52]

⟨|ζ(K∥)|2⟩ = µ |K∥|2DH−7, 1/L ≤ |K∥| ≤ 1/ℓ . (4.0.1)

From the statistical framework for finite fractals, we obtained four fractal mor-

phological characteristics of a power-spectrum of roughness, including fractal di-

mension (DH), the strength of fractality (µ) as an extrapolated value of Eq. 4.0.1

at |K∥| = 1(arbitrary units) and the lower and upper cut-off length scales (ℓ) and

(L), which are represented as a range over which surface has self-affine fractal

properties. The fractal dimension (DH) is the global property that describes the

scale invariance property of the roughness and is largely associated with anoma-

lous behavior in the diffusion-controlled response. The time exponent is typically

assumed to be a function of roughness. The strength of fractality (µ) is related to

the topothesy. The geometrical interpretation of the topothesy is the length scale

over which the profile has a mean slope of 1 [53]. The lower cut-off length scale

(ℓ) is the length above which we observe roughness, and the upper cut-off length

scale (L) is the length above which we do not sense the roughness features. An

example of a finite fractal is a Pd electrode with L/ℓ ≈ 12 [54]. It is important

to emphasize that a finite fractal will become a nonfractal when L/ℓ ∼ 1, this

behaviour is seen in the power spectrum density of ref. [26] for a rough glassy

carbon electrode (L/ℓ ≈ 6).

As mentioned previously, the moments of the power-spectrum of roughness

contain important morphological information. In general, m2k are the 2k-th mo-

ments of the power-spectrum of the band-limited fractal, and its expression is

given as [40]:

m2k =µ

4π

(ℓ−2(δ+k)

δ + k− L−2(δ+k)

δ + k

). (4.0.2)

These moments have physical significance, as they are measures of morphological

features of a rough surface, where m0, m2 and m4 are mean square (MS) width,

the MS gradient and the MS mean curvature of roughness, respectively.

The exact representation of charge for the isotropic self-affine (finite) fractal

surface is obtained by substituting Eq. 4.0.1 in Eq. 3.0.5 and representing vari-

127

ous integrals in terms of special functions. The mean charge for realistic fractal

roughness is as follows:

⟨Q(t)⟩ = QP (t)(1−RF1(t) +RF2(t, ℓ)−RF2(t, L)) (4.0.3)

RF1(t) = − m0

2Dt+

(µ

8π(Dt)DH−3/2

)Γ(DH − 5/2, Dt/ℓ2, Dt/L2

),

RF2(t, λ) =µ

8π(DH − 2)λ2DH−3√t∗

(√π erf(

√t∗)− t∗2−DHγ(DH − 3/2, t∗)

),

where λ is the cut-off length (either ℓ or L) and dimensionless time t∗ = Dtλ2 ,

Γ (α, x0, x1) = Γ (α, x0)−Γ (α, x1) = γ (α, x1)− γ (α, x0), Γ(α, xi) and γ(α, xi) are

the incomplete Gamma functions [50].

An alternative method for the derivation of the leading asymptotic form of

Eq. 4.0.3 is from the leading asymptotic form of the current transient equation

(⟨I(t)⟩) of ref. [39, 40]. The approximate form of the current for an isotropic

self-affine fractal surface can be written as [39, 40]:

⟨I(t)⟩ ≈ nF DA0Cs√πDt

(1 +

µ

8π

(ℓ−2δ

δDt− Γ(δ)

(Dt)δ+1

)), t > ti , (4.0.4)

where δ = DH − 5/2. Equation 4.0.4 is applicable for a time greater than the

inner fractal crossover time, ti. By integrating Eq. 4.0.4 between limits ti and t,

we obtained the asymptotic expression for the charge transient. The charge for

t ≫ ti for an isotropic self-affine fractal electrode-electrolyte interface is given by

⟨Q(t)⟩ ≈ 2 nF A0Cs

√Dt√

π

(1− µ ℓ−2δ

8πδDt+

µΓ(δ)

8π(2δ + 1)(Dt)δ+1

). (4.0.5)

The inner fractal crossover time, ti is the time after which the anomalous inter-

mediate time regime is observed in the current or charge. Contribution of ⟨Q(ti)⟩is small for all t ≫ ti; therefore, it is ignored in the above formula. It is clear

from the above analysis for the finite fractals that the exponent β in Eq. 1.0.2 is

dependent on time regimes. In general, realistic roughness will be time dependent.

128

5 Chronocoulometric Response of Nonfractal Elec-

trode

In most of cases, the electrode roughness is modeled as realistic (finite) fractals.

In some cases, such as a rough glassy carbon electrode [26], roughness has no self-

affine property and has a dominant length scale of randomness. Such roughness

can be modeled with a power-spectrum function with a single lateral correlation

length. One of the statistical models for nonfractal surfaces can be described in

terms of a Gaussian correlation function (W (r∥)) with a MS width (h2) and a

correlation length, a. The Gaussian correlation function is given as [42, 55]

W (r∥) = h2e−r2∥/a2

. (5.0.1)

Fourier transform of the function is the surface structure factor that possesses

various morphological features for the nonfractal surfaces and is given by the

following equation:

⟨|ζ(K∥)|2⟩ = πh2a2e−K2∥a

2/4 . (5.0.2)

Within the presented framework, two morphological parameters are obtained,

viz, the width of the interface, h2, and the transversal correlation length, a. The

transverse correlation length (a) is a measure of the average distance between

consecutive peaks and valleys on the random nonfractal surface, and h is the

measure of mean square departure of the surface from flatness. The excess surface

area, < A >, and the average square gradient are related as 1 + 12⟨(▽∥ζ)

2⟩ =

<A>A0

= R∗ = 1+ 2h2

a2, where A0 is the projected area of the electrode and R∗ is the

roughness factor. The ensemble-averaged mean square curvature is ⟨H2⟩ ≈ 8h2

a4.

The ensemble-averaged charge ⟨Q(t)⟩ for the nonfractal electrode is given by

the following equation:

⟨Q(t)⟩ = 2 nF A0 Cs

√Dt√

π

1 +h2 tan−1

(2√Dta

)a√Dt

. (5.0.3)

The second term in the parentheses of the above equation is the contribution due

129

to roughness. If the roughness (h2) is zero, then only the first term survives, giving

a smooth electrode response. Equation 5.0.3 for charge is related to the current

transient in ref. [18, 42] through an integration with respect to the time of the

mean current or from the mean absorbance [30] for a Gaussian power-spectrum of

roughness.

6 Results and Discussion

Fig. 4.1 depicts the rough electrode-electrolyte interface with fractal morphologi-

cal and phenomenological scales that control the electrochemical responses. The

power-spectrum of such rough electrode-electrolyte interfaces consists of four mor-

phological characteristics (see Fig. 4.12), including the fractal dimension (DH), the

lower (ℓ) and upper (L) cut-off length scales of fractality and a proportionality

factor (µ) or the strength of fractality. Only three of these characteristics (DH , µ

and ℓ) contribute strongly to the electrochemical response of the electrode while

L contributes weakly for the large L/ℓ finite fractals. In this section, the charge

transients (chronocoulograms) will be analyzed in detail for the realistic (finite)

self-affine fractal electrodes and for the nonfractal electrodes.

Fig. 4.2 to Fig. 4.4 and from Fig. 4.5 to Fig. 4.6 show the traditional Anson

plots for the fractal and the nonfractal electrode surfaces, respectively. The first

plot from the bottom in each figure (black line) is for the smooth (planar) electrode

surface. The increase in the fractal dimension (DH) increases the roughness, and

the charge transients increase (Fig. 4.2). Similarly, an increase in the strength of

the power-spectrum (µ) or decrease in the lower cut-off length scale (ℓ) increases

the roughness and the charge-transients in Fig. 4.3 and Fig. 4.4, respectively.

For the nonfractal case, an increase in the MS width of the interface (h2) or

a decrease in the lateral correlation length (a) increases the roughness, which

increases the charge-transients in Fig. 4.5 and Fig. 4.6, respectively. These figures

depict how the increase in roughness increases the charge-transients and brings

in non-linearity into the Anson plots for both the fractal and nonfractal electrode

roughness.

Fig. 4.7 to Fig. 4.9 display the log-log plot of charge and time, which illustrates

130

the change in the charge with time while varying one of the dominant morpholog-

ical parameters (DH , µ or ℓ). The figure illustrates the existence of the three time

regimes: short, anomalous intermediate and long. Of these three time regimes, the

anomalous intermediate time regime is important because the diffusion-layer width

is comparable to the surface irregularities. The effect of roughness is observed only

in this time regime. In the short and the long-time regimes, the diffusion-layer

width is smaller than ℓ and larger than h for the surface irregularities, respec-

tively. Hence, the planar electrode response is observed in thee two time regimes.

These kind of time regimes are not observed in the traditional Anson plots and

lacks the detailed morphological information related to the rough surfaces. All

morphological information pertaining to the self-affine fractal nature is observed

in the intermediate region. Thus, its proper characterization is very important

with the knowledge of transition times. The main observation in these figures is

that, charge increases with the increase in the roughness. After a certain value of

roughness, the charge shows weak dependence on time (compared to the Anson

response) in the intermediate region (see Fig. 4.7 to Fig. 4.9). Though simple scal-

ing result in Eq. (1.0.2) capture the important feature of the fractal electrodes

but is unable to include a complete set of realistic (finite) fractal morphological

characteristics of roughness. Because of this result, it cannot capture all regimes

of the experimental data.

Fig. 4.7 demonstrates the variation of the charge-transient with the fractal

dimension. The charge-transients increase as the fractal dimension is increased.

The anomalous intermediate region starts to show weak dependency on time as the

DH increases. Fig. 4.8 illustrates the variation of the charge with the strength of

the power-spectrum of roughness (µ) (also proportional to the mean square width

of the interface). As the moments of roughness power are directly related to the µ,

the enhancement in the strength of the roughness enhances the charge-transients.

If the lower cut-off length scale decreases roughness factor as well as self-affine

fractal nature of the rough electrode surface increase. The charge increases in the

initial time, but in the anomalous region it starts becoming flatter with an increase

in the roughness due to the lowering of the lower cut-off length in the finite fractal

(see Fig. 4.9). The inset figures are plotted between Qrel and Log t. Here, Qrel

131

0 1 2 3

HtêmsL1ÅÅÅÅ2

0

0.5

1

1.5

QêmC

DH

Figure 4.2: Traditional Anson plot for the fractal electrode surface. The plotsare generated with (a) DH = 2.10, 2.12, 2.14, and 2.15 (from bottom), ℓ =50nm, L = 1µm and µ = 10−4(a.u.). The other parameters used in the abovecalculations include the diffusion coefficient (D = 5 × 10−6 cm2/s), the electrodearea (A0 = 0.01cm2), the concentration (Cs = 10−6mol/cm3) and n = 1.

is the relative charge which is the ratio of the total charge with roughness to the

planar charge, i.e. Qrel = ⟨Q(t)⟩/QP (t). The inset figures explicitly reveal various

time regimes more clearly.

It is evident that the charge is strongly dependent not only on the fractal

dimension but also on the two other morphological parameters, i.e. µ and ℓ. The

rigorous theory based on the power-spectrum of roughness characterizes all the

three time regimes (short, anomalous intermediate and the long time regimes).

Anson regimes are more efficient and accurate than those based on the scaling

approach. The double logarithmic plots of the charge and time provide more

insights into the physical picture of all the regimes and explain the transition times

as well. These characteristics cannot be observed in non-logarithmic traditional

Anson plots, i.e. Q(t) vs√t (see Fig. 4.2 to Fig. 4.6) [30]. As expected, charge-

132

0 1 2 3

HtêmsL1ÅÅÅÅ2

0

0.5

1

1.5

2

2.5

QêmC

m

Figure 4.3: Traditional Anson plot for the stregth of fractality. The plots are gen-erated with µ = µ0, 3×µ0, 5×µ0 and 7×µ0(a.u.) (from bottom) where, µ0 =10−6 , DH = 2.3, ℓ = 50nm and L = 1µm . The other parameters used in theabove calculations include the diffusion coefficient (D = 5×10−6 cm2/s), the elec-trode area (A0 = 0.01cm2), the concentration (Cs = 10−6mol/cm3) and n = 1.

133

0 1 2 3

HtêmsL1ÅÅÅÅ2

0

1

2

3

4

5

QêmC

�

Figure 4.4: Traditional Anson plot for the lower cut-off length scale. The plots aregenerated with ℓ = 30, 60, 90 and 150nm (from top), DH = 2.3, µ = 10−5(a.u.)and L = 1µm . The other parameters used in the above calculations include thediffusion coefficient (D = 5× 10−6 cm2/s), the electrode area (A0 = 0.01cm2), theconcentration (Cs = 10−6 mol/cm3) and n = 1.

134

0.1 0.3 0.5

HtêsL1ÅÅÅÅ2

1

3

5

7

QêmC

h

Figure 4.5: Traditional Anson plot for nonfractal surfaces. The effect of root meansquare width on charge-transient.The plots are generated with h = 5, 7, 10, 15 µm(from bottom) and a = 6 µm. The other parameters used in the above calcu-lations include the diffusion coefficient (D = 5 × 10−6 cm2/s), the electrode area(A0 = 0.01cm2), the concentration (Cs = 10−6mol/cm3) and n = 1.

135

0.05 0.1 0.15 0.2 0.25

HtêsL1ÅÅÅÅ2

0.5

1

1.5

2

2.5

3

QêmC

a

Figure 4.6: Traditional Anson plot for nonfractal surfaces with the effect oftransversal correlation length on charge-transient.The plots are generated witha = 2, 3, 4, 6 µm (from top) and h = 5 µm. The other parameters used in theabove calculations include the diffusion coefficient (D = 5×10−6 cm2/s), the elec-trode area (A0 = 0.01cm2), the concentration (Cs = 10−6mol/cm3) and n = 1.

136

-3 -2 -1 0 1

LogHtêsL

-7.5

-6.5

-5.5goL

HQêC

L

DH

-6 -2 2

LogHtêsL

20

60

80

120

XQHtL\êQ

PHtL

D

Figure 4.7: The effect of the fractal dimension, DH on the charge-transients.The above plots are generated with DH = 2.00, 2.15, 2.20, 2.25 and 2.3 (frombottom), ℓ = 50nm, L = 1µm and µ = 10−5(a.u.). The other parameters used inthe above calculations are include the diffusion coefficient (D = 5 × 10−6cm2/s),the concentration (Cs = 10−6mol/cm3), the electrode area (A0 = 0.01cm2) andn = 1.

transient is linear for the smooth electrode surface (first plot from the bottom).

At a higher value of roughness, chronocoulograms start to deviate from the linear

response due to the roughness contribution. The deviation is larger with the

enhancement in the morphological irregularities especially in the short and the

intermediate time regimes.

Fig. 4.10 to Fig. 4.11 is the double log plot of the charge vs time for the

nonfractal surfaces while Qrel(t) vs Log t are shown in the inset. Three time

regions, i.e. the short time√Dt < ℓ region, the intermediate crossover region

and the long-time√t region are observed. Fig. 4.10 indicates that if the width of

the interface increases, the charge and hence the outer crossover time (to ≈ h2/D)

137

-3 -2 -1 0 1

LogHtêsL

-7.5

-6.5

-5.5

-4.5

goL

HQêC

L

m

-6 -2 2

LogHtêsL

50

150

250

XQHtL\êQ

PHtL

Figure 4.8: The effect of the strength of fractality, µ on the charge-transients.The above plots are generated with µ = µ0, 5 × µ0, 10 × µ0 and 20 × µ0(a.u.)(from bottom) where, µ0 = 10−6 , DH = 2.3, ℓ = 50nm and L = 1µm . Theother parameters used in the above calculations are include the diffusion coefficient(D = 5× 10−6cm2/s), the concentration (Cs = 10−6mol/cm3), the electrode area(A0 = 0.01cm2) and n = 1.

138

-3 -2 -1 0 1

LogHtêsL

-7.5

-6.5

-5.5

-4.5

goL

HQêC

L

�-6 -2 2

LogHtêsL

50

150

250

350XQ

HtL\êQ

PHtL

Figure 4.9: The effect of the lower cut-off length scale, ℓ on the charge-transients.The above plots are generated with ℓ = 30, 60, 90 and 150nm (from top),DH = 2.3, µ = 10−5(a.u.) and L = 1µm . The other parameters used in theabove calculations are include the diffusion coefficient (D = 5 × 10−6cm2/s), theconcentration (Cs = 10−6mol/cm3), the electrode area (A0 = 0.01cm2) and n = 1.

139

-2 -1 0 1

LogHtêsL

-7.5

-6.5

-5.5

goL

HQêC

L

h

-3 -1 1

LogHtêsL

2

6

10

14

XQHtL\êQ

PHtL

Figure 4.10: Charge-transients for the nonfractal electrode surface showing theeffects of the the root mean square (rms) width of the interface, h, at a fixed cor-relation length. The plots were generated with h = 5, 7, 10, 15 µm (from bottom)and a = 6 µm. The other parameters used in the above calculations are as follows:the diffusion coefficient (D = 5× 10−6 cm2/s), the electrode area (A0 = 0.01cm2),the concentration (Cs = 10−6 mol/cm3) and n = 1.

140

-2 -1 0 1

LogHtêsL

-7.5

-6.5

-5.5

goL

HQêC

L

a

-3 -1 1

LogHtêsL

2

6

10

14XQ

HtL\êQ

PHtL

Figure 4.11: Charge-transients for the nonfractal electrode surface showing theeffects of thetransversal correlation length, a. The plots were generated witha = 2, 3, 4, 6 µm (from top) and h = 5 µm. The other parameters used in theabove calculations are as follows: the diffusion coefficient (D = 5 × 10−6 cm2/s),the electrode area (A0 = 0.01cm2), the concentration (Cs = 10−6mol/cm3) andn = 1.

141

3 4 5 6

LogHKêcm-1L

-19

-17

-15

-13

goL

HX»z`HKØ

»»L»2\êmc

4L

1ÅÅÅÅÅÅL

1ÅÅÅÅÅ�

Figure 4.12: Comparison of the power spectra for the fractal and nonfractal sur-faces. The solid lines show fractal surfaces, while the broken curves show nonfractalsurfaces with same MS width (h2) and correlation length (L and a). The first line(from bottom) shows the plot of the Anson equation. The solid curves displaythe fractal surfaces with DH = 2.1 and 2.2 (from bottom), while ℓ = 30 nm,L = 2 µm and µ = 10−4(a.u.). The broken curves indicate nonfractal surfacesthat are generated with h = 1.452 µm and 3.836 µm, respectively (from bottom),while a = 2 µm is same for both curves. The other parameters used are as follows:the diffusion coefficient (D = 5× 10−6 cm2/s), the electrode area (A0 = 0.01cm2),the concentration (Cs = 10−6mol/cm3) and n = 1. The ratio L/ℓ remained fixed(= 66.7) for both the fractal curves.

142

-2 -1 0 1

LogHtêsL

-7

-6

-5

goL

HQêC

L

-5 -3 -1 1LogHtêsL

0

1

2

goL

HXQHtL\êQ

PHtLL

Figure 4.13: Comparison of the chronocoulometric response for the fractal andnonfractal surfaces. These curves of chronocoulometric responses are generatedfor the same parameters as used to generate curves in Fig. 4.12. The inset figureshows double logarithmic plots for the ratio of “generalized Anson” over Ansonresponse.

143

increase. In the short time regime, the charge transient increases by the roughness

factor while it follows√t Anson behavior. In the intermediate transition regime,

there was a slow decrease in the slope of the double log plots of the charge-transient

and it eventually become weakly dependent on time at the higher value of the

roughness factor. Finally, it merges with the planar Anson response in the long-

time regime. Fig. 4.11 shows that, if we decrease the transverse correlation length,

the roughness factor and the charge increase. Thus, in the short time regime, the

charge-transient is increased by the roughness factor while in the intermediate

crossover regime of the charge, it shows a weaker dependency on the time. These

effects are very similar to those seen in the fractal electrode surface but has an

enhanced and prolonged response at the intermediate regime. In the long time

regime, all the responses merge with the smooth (planar) surface charge.

Fig. 4.12 represents the comparison of the power-law and Gaussian power spec-

tra of roughness and Fig. 4.13 the chronocoulometric responses for the fractal and

nonfractal surfaces. The same correlation lengths (i.e. L = a) and the width of the

interface (h2) for these surfaces are used in the calculations. The inner crossover

time (ti) occurred significantly earlier for the fractal electrode than that of the

nonfractal surface (see inset in Fig. 4.13). There is a longer short time Anson re-

gion for the nonfractal surface compared to the fractal surface. In the case of the

fractal surface, the intermediate anomalous power-law region starts earlier com-

pared to the nonfractal surface which has crossover primarily between these two√t

regimes. Early onset of the inner crossover time in the fractal may have caused

its disappearance in some of the experimental data for the diffusion-controlled

phenomena (here, we have not accounted any double-layer contribution). Addi-

tionally, the ratio of the two cut-off lengths, L/ℓ(= 66.67), must be large enough

to differentiate the response of the fractal and nonfractal surfaces.

7 Conclusions

The research presented here enables us to understand the effect of the surface mor-

phology of electrodes on the diffusion-controlled reactions and their influences on

the chronocoulometric response. Analysis of the fractal and nonfractal electrode

144

surfaces enabled us to draw the following conclusions: (1) the charge-transient

is increased with an increase in the roughness of the surface, (2) the roughness

brings non-linearity into the Anson plots, (3) the intermediate region is strongly

influenced by the fractal nature of the electrodes and shows anomalous power-

law behavior in the charge-transients and become almost time independent for

the large roughness surfaces, (4) the scaling law for the charge-transient at the

fractal electrode captures some features of the intermediate time regime but is

not sufficient for the characterization of all the time regimes, and (5) our ab ini-

tio methodology based on the power-spectrum of roughness characterizes all the

three time regimes in the charge-transients successfully. This theoretical model

provides a physical picture that relates the electrochemical responses to the phe-

nomenological length (like diffusion length) and morphological length scales that

characterizes the nature of the rough surfaces. Finally, the theory is an indis-

pensable step in the quantitative description of the role of roughness on a single

potential step chronocoulometric response.

145

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