Chapter 1

Spectroelectrochemical and

Chronocoulometrical Responses

at Rough Electrodes: An

Introduction

1

2

1 Spectroelectrochemical Responses and Elec-

trode Roughness

The fundamental theoretical and experimental understanding in electrochemistry

is usually about systems with smooth and homogeneous electrode surfaces. The

smooth and homogeneous electrode surfaces may be of single crystal electrodes or

liquid based electrodes. The practical applications along with fundamental studies

of electrochemistry are not possible without encountering the rough, inhomoge-

neous or even porous electrodes. It will be appropriate to say that the irregular

surface and interfaces are ubiquitous in electrochemistry. There are various elec-

trochemical techniques which are used to characterize the surface processes such

as, chronoamperometry, square-wave voltammetry or normal pulse voltammetry

[1, 2], cyclic voltammetry [3] and impedance spectroscopy [3, 4], etc.

The surface morphology or roughness of an electrode affects the spectroelec-

trochemical responses (current, charge, absorbance, impedance, etc.) significantly.

It is almost inevitable to produce an electrode surface which is perfectly smooth

especially, electrodes made of solid metals (Pt, Au, Ru, etc.), doped oxide semi-

conductors (SnO2, InO2, etc.), films, etc. It was reported by Theodore Kuwana [5]

that optical changes may be affected by electrode surface roughness during spec-

troelectrochemistry measurements. Later, Richard L. McCreery and co-workers

[6] pointed out that Absorbance vs time1/2 plot for the minigrid electrodes show

deviation from linearity in the short time (below 50 ms) and said the deviation

from linearity may be due to the electrode surface roughness. Their suggestion

looks convincing because it is prepared by etching rectangular holes in fine metal

foils (usually gold). Also, during the different stages of the sample preparation and

electrochemical measurements, the surface of the sample do changes which causes

unintentional roughness [7] which affects the spectroelectrochemical responses.

The rough electrodes have potential to be used in fundamental and applied

fields [8, 9, 10, 11, 12]. This is possible because rough surfaces show enhanced

3

surface reactions due to the increase in the effective area. Rough surface is also

capable to separate electrons and holes from combination because of generation

of channel-like structure in the vicinity of the electrode. Hence, rough electrode

surface structure is favorable to enhance the performance of light emitting diodes

and the solar cell devices [8].

We do obtain anomalies in our spectroelectrochemical experimental data dur-

ing the study of electrode-electrolyte interfaces. The anomalies are still least

understood due to the mathematical limitations and inadequate theoretical un-

derstanding about the electrode surface roughness. The first real attempt in this

respect, to characterize the rough electrode-electrolyte interface became possible

after the de Gennes scaling conjecture [13]. De Gennes proposed the scaling con-

jecture for diffusion-limited nuclear magnetic spin relaxation processes for porous

media to calculate the fractal dimension in 1982. After this scaling conjecture, Le

Mehaute et al. [14, 15] has successfully introduced the fractal concept into electro-

chemistry. Then, it was Pajkossy and co-workers [16, 17, 18, 19, 20] who popular-

ized the electrochemical methods to characterize the rough electrode-electrolyte

interface.

2 Scope of the Present Thesis

In this thesis, an attempt has been made to study the rough optically transparent

electrodes (OTEs)-electrolyte interface by using spectroelectrochemistry (SEC)

and chronocoulometry techniques. SEC is a coupled technique of spectroscopy

and electrochemistry. So in this technique, the advantages of the two different

techniques are combined which provides the versatile utility to understand the

electron-transfer, spectroscopic study and surface processes. Through the novelty

of SEC and chronocoulometry techniques, we have studied two classes of reaction

taking place at rough electrode-electrolyte interface namely, simple reversible re-

dox reactions and pseudo first-order catalytic coupled reaction (EC’) mechanism.

4

Theoretical models have been proposed to understand the roughness present at

the OTE-electrolyte interface, particularly.

The results obtained have wide range of applicability such as in the develop-

ment of the electrochromic devices [21], light emitting diodes [22, 23, 24], solar

cells [8], photovoltaics [25, 26, 27, 28, 29], carbon nanostructures [30], graphene

[31] and, biomolecules [32]. The catalytic coupled reactions are particularly useful

in fuel cells [33, 34], organo-metallic catalysis [35, 36], electrosynthetic reactions

[35, 37, 38], secondary battery system and studies of metallic corrosion [34], etc.

2.1 Overview of This Chapter

This chapter is all about the introduction of spectroelectrochemistry and chrono-

coulometry techniques. The detailed description of these two techniques with their

significance and advantages to characterize the rough electrode-electrolyte inter-

face is presented. The different kinds of SEC techniques with the experimental

set-up are discussed. Most importantly, the methods used for the characterization

of the rough surface are presented. A comparative analysis of the various meth-

ods and theoretical models to characterize the rough surfaces is presented. The

detailed description of power-spectrum of roughness and its advantages over other

methods such as scaling and fractional-diffusion method is also presented. In the

end, the plan of study and thesis overview are presented.

3 Spectroelectrochemistry (SEC) Techniques to

Study Rough Electrodes

3.1 Introduction

The electrochemistry gives us a better information about the quantity of a redox

species that is, how much of any analyte is present. Conversely, the spectroscopy

(UV-Visible) tells us what kind of that analyte is it in a much more convenient and

5

better way. The marriage of these two techniques is called spectroelectrochem-

istry [32] which provides both the information simultaneously and conveniently.

Therefore, spectroelectrochemistry is very useful to find the reaction intermediates

and products which ease to elucidate the correct mechanism and identification of

organic, inorganic and biological systems [21, 32].

The total current for diffusion-limited phenomena at planar electrode is given

by Cottrell equation as [3]:

Ip(t) =nFCA0D√

πDt, (3.1.1)

where Ip(t) is the Cottrell current, n is number of electron(s) involved in the

redox reaction, F is Faraday constant, C is the concentration of the species, A0 is

geometric area of the electrode, D is diffusion co-efficient and t is the time.

The integration of Cottrellian current for planar electrode gives the charge and

is known as Anson equation given as [3]:

Qp(t) =2nFCA0

√Dt√

π, (3.1.2)

where Qp(t) is the charge at planar electrode. There is linear relationship be-

tween charge and absorbance so once charge is calculated, absorbance can also be

calculated if redox species have chromophore and absorbs radiation in UV-visible

region. The charge and absorbance are related as [3]:

A(t) =ϵQ(t)

nFA0

, (3.1.3)

where A(t) is total absorbance, ϵ is molar absorptivity and Q(t) is the total charge.

Therefore, the absorbance for planar surface is given as [3]:

Ap(t) =2 ϵ C

√Dt√

π, (3.1.4)

6

where Ap(t) is absorbance at planar electrode. The above Anson and absorbance

equations show that both the quantities are proportional to the square root of

time for diffusion controlled reaction at planar electrode surface.

The current-time relationship is known as chronoamperometry and the Cottrell

equation is the basis of this technique. The Cottrell current shows proportionality

with the inverse of square root of time, Eq. 3.1.1. The charge-time relationship

is known as chronocoulometry which shows proportionality with the square root

of time, Eq. 3.1.2. The chronocoulometry technique has experimental advantages

over chronoamperometry [3] because of (i) better “signal to noise” ratio due to

integration of current, (ii) diffusional and non-diffusional components (double-

layer and adsorption) can be easily separated and (iii) charge increases with time

so most meaningful data is obtained at the end of the experiment.

The chronoabsorptometry resembles chronocoulometry Eq. 3.1.2 and Eq. 3.1.4

but the influence of kinetic process occuring in the diffusion layer on absorbance-

time behavior is quite different from that of charge-time behavior [39]. This differ-

ence in the kinetic process is due to the fact that, absorbance is the characteristics

of the integrated concentration of the species while charge represents the inte-

grated flux of the species. Thus, we get entirely different information from these

two important electroanalytical techniques despite the fact that, both are mathe-

matically similar.

chronoabsorptometry has additional advantages over chronocoulometry [40] of

(i) being unaffected by “non-diffusional” effects (adsorption, double layer etc.),

(ii) independent of electrode side reactions as long as they do not affect the con-

centration of the absorbing species, and (iii) observing directly the existence of

the intermediate product of species studied without previous interference from the

measurement of electrochemical parameters. Keeping in mind the advantages of

chronocoulometry and chronoabsorptometry, we have developed the theoretical

models for the rough electrode-electrolyte interface. Through the models we have

studied two classes of reaction that is, simple facile redox reaction and pseudo-

7

first-order catalytic coupled reaction at rough electrode-electrolyte interface.

3.2 A Brief Review of Spectroelectrochemistry

This important technique was first explored by Kuwana and co-workers [41] in

early 1960s by monitoring the course of an electrochemical reaction by transmission

spectroscopy for diffusion controlled electrode reaction. His group studied various

redox species at different kinds of optically transparent electrodes (OTEs) through

this technique [40, 41, 42, 43, 44, 45, 46, 47]. These early works of Kuwana

and co-workers explored this technique to understand the mechanistic and kinetic

behaviors of the redox species more conveniently and accurately. Other prominent

contributers are R.W. Murray and co-workers [48, 49], Chia-yu Li and George S.

Wilson [50], Richard L. McCreery and co-workers [51], Shouzhuo Yao and co

workers [52], etc.

Recently, Wolfgang Kaim and Axel Klein have published a book, titled ”Spec-

troelectrochemistry” [32] which is a landmark in SEC and shows the growing im-

portance of spectroelectrochemical techniques among the chemists. Since carbon

is very cheap and easy to process therefore, many research groups are also trying to

develop carbon based optically transparent electrodes to study the carbon based

materials which is going to be future material. Ladislav Kavan and Lothar Dunsch

[30] studied carbon nanostructures while Constans M. Weber and co-workers [31]

studied exclusively about graphene based optically transparent electrodes. SEC

techniques have succesfully used in applied chemistry also by several authors such

as in photovoltaics [25, 26, 27, 28, 29], organic light-emitting diodes [22, 23, 24],

metal clusters [53, 54, 55, 56, 57], and biomolecules [58, 59, 60, 61] which further

explored the new horizons in spectroelectrochemistry.

Our group is the first one to account the effect of electrode surface roughness

on absorbance-transient response [62]. The results obtained at the rough OTE can

be crucial for the fabrication and development of the many devices such as: solar

cells, electrochromic devices, LED, supercapacitors, graphene based applications,

8

etc.

3.3 Different Kinds of SEC Techniques

Spectroelectrochemical techniques are of several kinds depending on the wave-

length of the electromagnetic radiation used and its interaction with the redox

species. The simplest and widely used SEC technique is transmission or UV-

visible. Because of its low sensitivity, specular and internal reflectance SEC tech-

niques provide better alternative to transmission technique. The reflectance SEC

technique is used to study the surface of the electrode as well. The detailed de-

scription of these techniques are provided in the coming sections.

We have tried to study the influence of electrode surface roughness on the spec-

troelectrochemical responses, which is the first of its kind. Thus, we have opted

the simplest mode of SEC that is, transmission in our theoretical models. For

further extension of this work one can exclusively use the more sophisticated tech-

niques like SERS (Surface-Enhanced Raman Spectroscopy) along with reflectance

techniques to study the possible electrode surface roughness effect. Due to this

importance of these techniques, a brief discussion regarding SERS and reflectance

are also presented in the coming sections.

3.4 Transmission Spectroelectrochemistry

Figure 1.1 depicts the simplest mode of transmission spectroscopy in which the

light-beam is passed directly through the optically transparent electrode (OTE)

and the adjacent solution [63]. OTE is an essential component of all the trans-

mission experiments. Several types of OTEs are useful in this approach [63, 64,

65, 66, 67, 68, 69, 70].

In this method, change in absorbance is determined as a function of time while

potential is scanned or may involve wavelength scan to get the spectra of the

redox species. Initially, a potential is applied where no redox reaction is occurred

9

Light Beam

Rough OTE

Electrolyte

O

e-Light Beam

Rough OTE

Electrolyte

R

e-

Figure 1.1: Schematic diagram of transmission mode spectroelectrochemistry tech-niques [63].

then potential is stepped to that region of the potential where the desired redox

reactions occur. In recent times, to improve the sensitivity of the UV-Visible

SEC, i.e. to detect short-lived intermediates and to get the voltammetric data

simultaneously, cyclic voltametry with a fast spectrometer may be used in place of

chronoamperometry experiment [71, 72]. A special kind of designed cell is required

for using cyclic voltametry perturbation since general designed cell causes large

ohmic drop which can distort the reliability of the data. UV-Visible SEC is mainly

used for the characterization of the electronic structure and the changes occurring

from charge-transfer reaction.

3.5 Reflectance Spectroelectrochemistry

To generate the electroactive species in the solution, the most common method

is transmission experiment and have proved extremely useful. However, they suf-

fer from several drawbacks such as [51, 73, 74, 75]: (i) the electrode other than

optically transparent can not be used, (ii) they can not detect the chromophores

having low absorptivity since optical path length is smaller hence, sensitivity is

less, (iii) can not detect the chromophores of short life-times with general designed

10

Electrolyte Solution

Rough OTE

Electrolyte Solution

Rough Electrode

Electrolyte Solution

(B) Internal Reflectance

Rough OTE

Electrolyte Solution

(A) Specular Reflectance

Rough Electrode

Figure 1.2: The schematic diagram of two types of reflectance mode spectroelec-trochemistry techniques [63].

cell, (iv) ohmic loss is also large, and (v) not suitable for irreversible reactions thus,

long time or time averaging is needed.

The above problems with the methods based on transmission experiment can

be solved by using specular reflectance spectroscopy (Fig. 1.2 (A) [63]). The nature

of electrode surface is of main concern in this method because phenomena such

as adsorption, chemical composition, thickness of the film, surface roughness, etc.

may affect the spectroelectrochemical responses. Hence, this technique is mainly

used for the surface chemistry and related phenomena, evaluation of the optical

constants of metals and other materials, particularly in the form of films whose

properties may differ markedly from those of bulk solids [3]. The path length

is increased in this method which increases the sensitivity hundred to thousand

folds of magnitude [51, 74]. Thus, species having weak chromophores and short

life-times can be easily detected.

Figure 1.2 (B) [63] depicts the set-up for the internal reflection spectroelec-

trochemistry. In this method, the light beam is introduced from the back side

of an electrode at an angle greater than the critical angle so that the beam is

totally reflected. Specular reflectance spectroelectrochemistry and internal reflec-

11

tion spectroelectrochemistry differ only in their reflection patterns of the light

beam [3, 63]. In later technique, due to total internal reflection, the path-length

increases further which increases the sensitivity of the technique.

Species having very short life-times and weak chromophores can be detected

due to increase in the path length in the internal reflection spectroelectrochemistry.

This technique can be used to observe the faradaic and non-faradaic phenomena

at the electrode-electrolyte interface [76], quantitative studies of species [77] along

with other basic advantages as in the case of specular reflectance spectroelectro-

chemistry.

3.6 Surface-Enhanced Raman Spectroscopy (SERS)

The surface-enhanced Raman spectroscopy (SERS) provides the same information

that normal Raman spectroscopy does, but with a greatly enhanced signal. This

technique was accidentally discovered during the normal Raman measurement of

monolayer of pyridine adsorbed on an electrochemically roughened silver electrode

[78]. The initial idea was to generate high surface area on the roughened metal

surface. The Raman spectrum obtained was of unexpectedly high quality. SERS

activity strongly depends on the nature of metal and its degree of surface roughness

[79]. Therefore, SERS active substrate fabrication is a very important field in

SERS research. For further extension of the present work, one can exclusively

use the SERS since it has wide applicability in the fields of chemistry, physics,

materials science, surface science, nanoscience, and the life sciences [80, 81, 82].

3.7 Practical Considerations in SEC

Since SEC is a hybrid technique therefore, there are several practical impleca-

tions also during the measurements since there is no standard experimental set-up

in SEC. It is challenging to combine the advantages of the two different tech-

niques. The electrochemically generated species is measured simultaneously, i.e.

12

in-situ. Thus, the construction and design of the spectroelectrochemical cell and

its connectivity with spectrometer must be done accordingly keeping in mind the

sensitivity and accuracy of the measurement.

The major implications and challenges in spectroelectrochemical measurements

are as [83]: (i) the adjustment of response time is difficult for short-lived interme-

diates or which undergo rapid follow-up reactions i.e. irreversible processes. (ii)

Since longer period is required in SEC measurements due to slow reactions of the

electrogenerated products therefore, reversible reaction becomes less reversible.

(iii) The electrolyte components such as solvent, conducting salt, cell window,

etc., also affect the absorption. Thus, deuterated solvents are used or special

materials such as CaF2 is added in low energy IR measurements. (iv) Due to gen-

eration of the several intermediate species close-lying waves are observed which

add difficulty in knowing the corresponding potential of the peaks. (v) For EPR

spectroelectrochemistry high dielectric constant of the medium is advantageous.

3.8 Role of OTEs in Spectroelectrochemistry

OTEs are valuable platform for a broad range of applications stemming from

fundamental spectroelectrochemical investigations of electron transfer mechanisms

[84, 85] to applied technological applications [22, 23, 24, 25, 26, 27, 28, 29]. The

nature of OTEs determine the successful application of UV-visible SEC. Thus,

SEC requires an electrode which has four main properties such as [86]: electrical

conductivity, optical transparency, robustness and inertness.

Several methods have been used over the year to prepare OTEs of different ma-

terials [63, 64, 65, 66, 67, 68, 69, 70]. The carbon based OTEs are the main material

to be used in future since it is cheap and easy to access with four attractive features

[86]: surface modification is easy, wide potential window range in aqueous media,

high stability in strongly alkaline and acidic conditions and, good electrochemical

activity for a range of redox systems. The carbon based OTEs have broad range

of applications such as in photovoltaics [25, 26, 27, 28, 29], organic light-emitting

13

diodes [22, 23, 24], electrochromic devices [21] and biomolecules [58, 59, 60, 61].

3.9 Spectroelectrochemistry of Catalytic-Coupled Reaction

Mechanism

The term “catalytic” is employed to denote that class of reaction in which the

initial electroactive reactant is regenerated on or close to the electrode surface

by a species (catalyst) say z, involving the electrolysis of product generated in

the first step [33, 34]. The rate of this heterogeneous regenerative process is

strongly dependent upon the nature of the electrode material [34] and on the

nature of catalyst, z. The catalytic processes have broad range of applications

such as: in fuel cells [33, 34], biomolecules [21, 32, 87, 88, 89], organo-metallic

catalysis [35, 36], electrosynthetic reactions [35, 37, 38], secondary battery system

and studies of metallic corrosion [34]. Therefore, it is of great interest to examine

the kinetic behavior of this type of reaction at modified or rough OTEs to gain

further insight into the mechanism of catalysis of reaction.

Winograd et al. [40] calculated the total absorbance change at smooth OTE

for pseudo first-order catalytic coupled reaction scheme and is given as:

AC(E, t) = AP (t)

{2√kt e−kt +

√π(1 + 2 kt) erf(

√kt)

4√kt

}, (3.9.1)

where AC(E, t) is the absorbance for catalytic coupled reaction, k is the homoge-

neous rate constant, “erf” is the error function [90] and AP (t) is the absorbance

for the diffusion-limited process at the planar OTEs without any catalytic coupled

reaction, Eq. 3.1.4.

14

4 Single Potential-step Chronocoulometry

4.1 Introduction

Chronocoulometry (CC) is one of the classical electrochemical techniques which

is used frequently in electroanalytical measurements. CC is nothing but the mea-

surement of the charge flow across the electrode-electrolyte interface as a function

of time. The CC technique was popularized in its present form by Anson and

co-workers [91, 92] and widely employed in place of chronoamperometry because

it offers important experimental advantages [3].

The Anson equation [3] Eq. 3.1.2 always show some positive intercept on the

charge axis even at t = 0 in Q(t) vs√t plot. This indicates that, there are

some other phenomena also which contribute to the total charge. Two distinct

phenomena are responsible for this and these are adsorption and double layer

charging. After accounting the contributions of these two phenomena, the Anson

equation (Eq. 3.1.2) modifies as [3]:

Qtotal = Qp(t) +Qdl +Qads

Qtotal =2nFCA0

√Dt√

π+Qdl + nFA0Γ , (4.1.1)

where Qdl is the double layer charging contribution, Qads = nFA0Γ is the contri-

bution due to adsorption and Γ is the surface excess. Therefore, the intercept of

the Q(t) vs√t plot gives the contribution of (Qdl + Qads). To separate Qdl and

Qads accurately, double potential-step chronocoulometry technique is preferably

used [93]. An analogous separation of the components of a current-transient is

not generally feasible. This later advantage of CC is specially valuable for the

study of surface processes. In fact, to study the adsorbed species at the electrode

surface was the main motivation to invent this technique [93, 94, 95, 96, 97, 98].

Other quantities which can be measured by using CC are: electrode sur-

face area and effective time window of an electrochemical cell [99], simultane-

15

ous diffusion co-efficients and concentration [100], kinetics of both heterogeneous

electron transfer reactions and chemical reactions coupled to electron transfer

[101, 102, 103], and possible mechanism of the redox reaction taking place at an

electrode-electrolyte interface.

4.2 Chronocoulometry of Catalytic-Coupled (EC’) Reac-

tion Mechanism

The psuedo first-order catalytic-coupled (EC’) reaction mechanism scheme is one

of the most important class of catalytic coupled reaction which is represented as

[33, 34]:

O + ne− R + zk−→ O + z′ (4.2.1)

where k is homogeneous rate constant, z and z′ are electrochemically inactive at

the electrode potential at which reaction proceed. Examples of this type of mech-

anism include oxidation of ascorbic acid [104] and aminophenol [105], reduction of

azines [106], azo dyes [107, 108] and certain radicals [109]. An EC’ reaction mecha-

nism is used in many fundamental as well as in applied fields [33, 34, 35, 36, 37, 38].

Since chronocoulometry is an effective technique to study homogeneous chem-

ical reactions that are coupled to the heterogeneous electron transfer reaction.

The chronocoulometric study for EC’ reaction mechanism was first done by J. H.

Christie in 1967 for the condition of double potential-step [110]. He proposed an

expression for the charge-transient for pseudo first-order catalytic-coupled reaction

mechanism at smooth electrode which can be written as [110]:

QC(t) = QP (t)

(2√kt e−kt +

√π(1 + 2 kt) erf(

√kt)

4√kt

), (4.2.2)

where QC(t) is the total catalytic charge-transient, t is the time, and“erf” is the

error function [90]. Here QP (t) is the Anson equation Eq. 3.1.2. This pioneer

work of Christie [110] has explored the chronocoulometry technique to study many

16

0.5 1 1.5 2 2.5 3

HtêsL1ê2

20

40

60

80

100

QêmC

Anson

Catalytic

Figure 1.3: Comparative diagram of traditional pure Anson and the pseudo first-order catalytic reaction for the planar responses. The Anson and catalytic curvesare generated by using Eq. 3.1.2 and Eq. 1.0.2, respectively. The various param-eters used to generate these two curves are as: n = 1, A0 = 0.01 cm2, C0

O =5× 10−6 mol cm−3, D = 5× 10−6 cm2 s−1 and k = 1 s−1 .

aspects of catalytic-coupled reactions.

4.3 Comparative Study of pure Anson and pure Catalytic

Responses

It is important to know the quantitative and qualitative comparison between pure

Anson and catalytic responses for the planar case. This kind of comparison will

be helpful to understand the roughness contribution as well. The Anson equation

considers diffusion only while catalytic response accounted the kinetic effect as

well.

17

Figure 1.3 clearly demonstrate that, due to the kinetic effect, the charge-

transient has higher value than that of the pure Anson. This higher value is

due to the fact that, the species R generated at the electrode surface (scheme

4.2.1) undergoes electrolysis again to convert into species O. The charge-transient

for such an electrode process followed by chemical reaction is greater than that of

the corresponding maximum rate of mass transport of the species to the electrode

surface in the absence of any regenerative reaction.

4.4 Practical Considerations in Chronocoulometry

Before going to run any chronocoulometrical experiment, there are some practical

problems also which must be considered to get the reliable data [72]. First one

is consideration of the double layer current which must be accounted properly

so that its contribution to the total charge must be minimized. The decay of

the current belonging to the double layer charging should be very fast to obtain

Q(t) vs√t plot as a straight line as predicted by the Anson equation for relatively

long period of time. Second problem is uncompensated solution resistance which

must be low otherwise will cause problems in the data evaluation because of the

distortion of the reliable part of the chronoamperometric or chronocoulometric

response. Third one is selection of optimum time and potential window because

at long times (t > 20s), contribution of convection becomes prominent which

causes deviation from Cottrell or Anson equation.

Fourth is related to the potentiostat which must have an adequate power re-

serve to supply during the momentarily change in current. Fifth problem is related

to multilayer adsorption of the electroactive species. In the monolayer adsorption

of the electroactive species the oxidation or reduction takes place immediately,

since it is present on the electrode surface. However, if the adsorption is multi-

layer (typical examples are the polymer film electrodes) the pseudo-capacitance

becomes large which leads to the higher value of “Qdl” .

18

5 Modeling of Different Rough Surfaces

5.1 Random Surfaces

Fractal geometry was popularized by Mandelbrot [111] which is also used to char-

acterize the rough surfaces including other random geometries. The rough surfaces

may be natural or artificial. The roughness enhances the effective area of the sur-

face thus, catalytic property also increases. Practically, the electrode surface is

not smooth and electrochemical responses such as current, charge and absorbance

are affected by the roughness present on the electrode surface. Therefore, proper

characterization of rough electrode is crucial to analyze the redox reactions taking

place at rough electrode-electrolyte interface.

These days, the experimental observations in geometry of real systems are an-

alyzed in nanometer and atomic levels which provides new insights to understand

the solid surfaces. The geometry of real systems are always visualized in terms of

random processes. The random processes at natural or material surfaces are char-

acterized by using statistical methods [112, 113]. The random (or rough) surfaces

can be categorized into isotropic, anisotropic and corrugated surfaces. The statis-

tically isotropic surfaces are those surfaces in which fluctuations are independent

of direction on the surface while statistically anisotropic surfaces are dependent on

the direction and the extreme case of anisotropy is known as random corrugated

surface.

The statistics of such type of surface can be characterized by a centered Gaus-

sian random field as [114, 115]:

⟨ζ(r∥)⟩ = 0

⟨ζ(r∥)ζ(r∥′)⟩ = h2W (r∥, r∥′) . (5.1.1)

Often, the correlation method is used to characterize the random surfaces. The

random rough surface is mapped into random homogeneous surface. If the corre-

19

lation function between the two points is independent of the absolute position but

depends upon their relative position, is called homogeneous random surface. The

above correlation statistics equation represents the homogeneous random surface.

Similarly for the isotropic random surfaces, the correlation function is given by

⟨ζ(r∥)ζ(r∥′)⟩ = h2W (|δr∥|) . (5.1.2)

Isotropic random surfaces depend on the magnitude of difference of two positions

vectors, where r∥ = (x, y); W (r∥, r∥′) is the correlation function; |δr∥| = |(r∥− r∥

′)|;

h2 = ⟨ζ2(r∥)⟩ and is known as mean square width of the rough surface.

The surface correlation function gives a measures of the rapidity of variation

of a given random surface. For slowly varying surfaces the correlation between

two positions on the surface ζ(r∥) and ζ(r∥′) will be considerable even for large

values of |δr∥|. As the value of |δr∥| increases, the correlation function dies out

and eventually tends to zero for all surfaces except for periodic and non-centered

surfaces. The correlation must vanish no matter how slowly the surface varies

for very largely separated points on the surface. The surface correlation function

contains information about the surface morphological characteristics i.e., average

area, mean square height, slope (gradient) or roughness factor, curvature and

correlation length etc.

The surface structure factor, ⟨|ζ(K∥)|2⟩, is another equivalent form which is

the Fourier transform of the surface correlation function. This gives as much mor-

phological information as the correlation function. The surface structure factor or

power spectrum of roughness is the ensemble average of all the possible configu-

rations. For two dimensional rough surfaces, ζ(r∥), the surface structure factor is

given as:

⟨ζ(K∥)⟩ = 0

⟨ζ(K∥)ζ(K ′∥)⟩ = (2π)2δ

(K∥ + K ′

∥

)⟨|ζ(K∥)|2⟩ , (5.1.3)

20

where K∥ is a wave-vector in two dimension and δ(K∥) is Dirac delta function. In

the above equation, angular brackets “⟨·⟩” denote an ensemble average over var-

ious possible surface configurations. The random surface profile, ζ(r∥), and their

Fourier transformed form, ζ(K∥), are characterized by surface structure factor, i.e.

⟨|ζ(K∥)|2⟩. The roughness power spectrum can be measured from AFM studies

and are useful in describing both fractal and non-fractal random surfaces.

5.2 Gaussian or Nonfractal Surfaces

Generally, the electrode roughness is modeled as realistic (finite) fractals. In some

cases, roughness has no self-affine property such as a rough glassy carbon electrode

[116], 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 [117, 118]:

W (r∥) = h2e−r2∥/a2

. (5.2.1)

Fourier-transform of the above Eq. 5.2.1 gives the surface structure factor or power-

spectrum and it contains various morphological features for the nonfractal surfaces

and is given by the following equation:

⟨|ζ(K∥)|2⟩ = πh2a2e−K2∥a

2/4 . (5.2.2)

From this 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

21

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.

5.3 Finite Fractal Surfaces

Fractal geometry has been used to address such a realistic type of morphology

[111]. Idealized fractal models take into account the roughness over infinite length

scales. However, fractal roughness profile cannot be handled rigorously for prac-

tical purposes because of their peculiar mathematical properties, among which

non-differentiability and non-stationarity restrict their applicability in physical

systems. Therefore, for practical purposes band-limited (finite) self-affine fractals

[119, 120] are taken into consideration. The finite length of roughness is equivalent

to upper length scale cutoff, whereas the smallest features of roughness truncate

the spectrum at high wave-number or lower length scale cutoff. Such types of frac-

tals with constraint make a realistic model of roughness. Yordanov and Atanasov

[121] calculated the structure function for approximately self-affine fractal surface

and identified three intervals in which the behavior of structure function is qual-

itatively different and such surfaces are called band-limited fractal surfaces. The

band-limited (finite) fractals can be classified into two types of surfaces depending

upon the fluctuations of surface, viz. corrugated fractals (fluctuations are along

one direction on the surface) and isotropic surfaces (fluctuations are along two

spatial dimensions but statistical properties are independent of direction on the

surface).

The band-limited corrugated fractal surface has the following power spectrum:

⟨∣∣∣ζ(Kx)∣∣∣2⟩ =

µL−(2DH−5), Kx < 1/L

µK2DH−5x , 1/L ≤ Kx ≤ 1/ℓ

0, Kx > 1/ℓ

, (5.3.1)

where, Kx is the wave number in one dimension. Similarly the band-limited

22

isotropic fractal surface has the following power spectrum:

⟨∣∣∣ζ(K∥)∣∣∣2⟩ =

µL−(2DH−7), |K∥| < 1/L

µ |K∥|2DH−7

, 1/L ≤ |K∥| ≤ 1/ℓ

0, |K∥| > 1/ℓ

, (5.3.2)

where, K∥ is the wave-vector in two dimensions. Equations 5.3.1 & 5.3.2 are useful

in obtaining intermediate frequency behavior in electrochemical response. Apart

from the fractal dimension or Hausdorff-Besicovitch dimension (DH), a fractal-like

surface is characterized by three characteristic length (or time) scales: the large

(L) and small (ℓ) lateral cutoffs and the strength of fractal (µ) which is related

to the topothesy of fractals. Topothesy τ is defined as the horizontal distance

between two points on which structure factor, equal to τ 2, can be expressed as

[112, 121, 122, 123]:

⟨[ζ(x, y)− ζ(x′, y′)]2

⟩≈ τ 2(1−H)X2H , (5.3.3)

where, ζ(x, y) is the centered surface profile function, the angular brackets des-

ignate ensemble averaging and X = [(x − x′)2 + (y − y′)2]1/2. Topothesy of an

‘ideal’ fractal surface is related to the normalizing factor (strength) of the power-

law spectra (µ) by the following relation [121]: τ 2(1−H) = 2µπΓ(1−H)/H22HΓ(1+

H). Here H is Hurst’s exponent which is related to the fractal dimension for cor-

rugated surface as DH = 2 −H and for isotropic surface as DH = 3 −H. Using

these power-spectra, we can develop the theoretical model for corrugated as well

isotropic surfaces and can find spectroelectrochemical expressions for realistic (fi-

nite) fractal electrodes. The ratio L/ℓ < 10 for finite fractals are practically

considered non-fractal in most of the diffusion problems [124].

Interesting characteristics of limited length scales of roughness of surfaces is

encouraged the researchers to consider various physical phenomena using the con-

cept of finite fractal. In our thesis, we consider the concept of finite fractals and

23

surfaces to be corrugated (surface fluctuation along one dimension) and isotropic

(surface fluctuations are in two dimensions) in nature. The realistic nature of

surface depends upon the length scales on which surface is said to be fractal. In

our thesis, two length scales are used defined by lower cutoff length, ℓ (length

above which surface shows the roughness) and upper cutoff length, L (length be-

low which surface shows the roughness). Recently, Compton et al. [116] records

a power-spectrum for rough electrode. The ratio of the two length scales i.e.

L/ℓ, obtained for this power-spectrum comes around L/ℓ ≤ 6 which shows the

non-fractal nature of this power-spectrum.

6 Methods Used to Solve Diffusion Problem at

Fractally Rough Electrodes

6.1 Introduction

The diffusion problem on fractally rough interface has been of interest in electro-

chemistry since it provides crucial clues related to the processes taking place at

rough electrode-electrolyte interface. The anomalous diffusion-limited processes

are observed in fundamental applications to applied fields [13, 14, 16, 117, 123,

125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136].

The main approaches to solve the diffusion problem on fractally rough elec-

trodes are Scaling approach, Fractional diffusion approach and, Power spectrum

based approach. We are using power-spectrum of roughness approach to charac-

terize the rough electrode-electrolyte interface.

6.2 Scaling Approach

In 1982 de Gennes proposed the scaling theory for diffusion-limited nuclear mag-

netic spin relaxation processes for porous media to calculate the fractal dimension

[13]. After his scaling conjecture Le Mehaute et al. [14] has successfully introduced

24

the fractal concept into electrochemistry and latter its applications by Nyikos and

Pajkossy [132, 16] which has made it possible to make progress in understanding

the electrochemical effects of electrode roughness. De Gennes scaling results have

been extensively used in various branches of science. But there are no systematic

experimental and theoretical studies to prove correctness of his scaling conjecture

for fractally rough electrode-electrolyte interface.

In order to capture the complexity arising from the irregular interfaces, i.e.

rough, porous, and partially active interfaces, one often uses the concept of frac-

tals [111]. Fractals arguments are often dealt by using scaling rules [111]. The

irregularity of the electrode surface is characterized as self-affine fractal over lim-

ited length scales. The diffusion controlled transfer process across fractal interface

shows power law behavior in time and is given by

I ∝ t−β , (6.2.1)

where β is known as scaling exponent or de Gennes exponent and is related to

fractal dimension, DH , as:

β =DH − 1

2, (6.2.2)

The scaling exponent, β, varies linearly with fractal dimension, DH (DH = 2 to

3) from 0.5 to 1. In the frequency (ω) domain, i.e. constant phase element (CPE)

behaviour is given by

Z(ω) ∝ (iω)−β , (6.2.3)

where Z and i are real and imaginary part of the impedance respectively. Eq. 6.2.2

is given by De Gennes [13] for diffusion controlled spin relaxation processes in

porous media and its generalized form by Maritan [137]. The scaling in terms of

area in which diffusion layer width, ε ∼√Dt,

√D/ω acts as a dynamic yardstick

length, where D is the diffusion coefficient, t is the time and ω is the frequency, is

25

given by

A(ε) =MH

εDH−2, (6.2.4)

A(t) ∝ 1

(Dt)DH/2−1. (6.2.5)

The scaling argument approach has been used for fractal surface analysis by

several groups [17, 18, 19, 20, 138, 139, 140, 141]. Recently, Srinivasan et al.

[142] has developed a scaling result for potentiostatic current transient which is

claimed to be applicable to fractal media when the reaction rates is under complete

activation and diffusion, as well as mixed control.

I(t) = nFCb(1/π)1/2A(k

(2−Df )

f (D/t)(Df−1)/2)(π)1/2λξexp(λ2ξ)erf c(λξ) , (6.2.6)

where ξ = Df − 1; Df is defined as the Hausdorff dimension; λ = kf (t/D)1/2; n =

number of equivalence; F = Faraday’s constant; Cb = bulk concentration; A =

true area of electrode and kf = forward rate constant, respectively, and under the

condition of diffusion limit, Eq. 6.2.6 reduces to

I(t) = nFCb(1/π)1/2A(k

(2−Df )

f (D/t)(Df−1)/2) . (6.2.7)

From above discussion it is quite clear that the most of the scaling results use the

scaling argument similar to one developed by de Gennes.

6.3 Fractional Diffusion Approach

The fractional diffusion method was explicitly introduced by Nigmatullin et al.

[143, 144] to describe diffusion across the surface with fractal geometry. How-

ever, Pyun et al. has adopted two approaches viz., scaling approach [145] and

fractional diffusion approach [146] for analyzing the diffusion problem across the

surface with fractal geometry. Le Mehaute and Crepy [14, 15] proposed the TEISI

(Transfer d’Energie sur Interface a Similitude Interne) model which treats the

26

thermodynamics of irreversible processes and provided the first insight into the

use of fractional diffusion approach to analyze the anomalous diffusion problems.

In the linear approximation of thermodynamics of irreversible process, the macro-

scopic flow of an extensive quantity across fractal interface J(t) is described by a

generalized transfer equation and it can be expressed as:

d(1/dF )−1

dt(1/dF )−1J(t) = K0∆X(t) , (6.3.1)

where dF is the fractal dimension, K0 is the constant and ∆X(t) represents the

local driving force.

Dassas and Duby[147] have given the one dimensional diffusion equation as:

d3−dF c(x, t)

dt3−dF= D∗∂

2c(x, t)

∂x2(2 ≤ dF < 3) , (6.3.2)

where c(x, t) is the local concentration of diffusing species, x is the distance from

fractal interface, D∗ is the fractional diffusivity defined as K4−2dF AdF−2ea D3−dF

(K is a constant related to fractal dimension, Aea is the time-independent electro-

chemically active area of flat interface and D represents the chemical diffusivity of

diffusing species) and ∂ν/∂tν means the Riemann-Liouville mathematical operator

of the fractional derivative [148, 149, 150]:

∂νy

∂tν=

1

Γ(1− ν)

d

dt

t∫0

y(ξ)

(t− ξ)νdξ , (6.3.3)

where Γ(·) is the incomplete Gamma function.

The mathematical result based on the concept of generalized transfer equation

(Eq. 6.3.1) the flow at fractal interface JF (x, t) is given as:

JF (x, t) =∂

∂t

[DF (t)

∗∂c(x, t)

∂x

], (6.3.4)

where, ∗ is the convolution operator and DF (t) represents time-dependent diffu-

27

sivity which is defined as:

DF (t) = D∗ t2−dF

Γ(3− dF ). (6.3.5)

By using Eq. 6.3.4, diffusion towards and from fractal electrode is mapped to a

one-dimensional diffusion in Euclidean space as follows:

AeaJF (x, t) = AF (t)JE(x, t) , (6.3.6)

where AF (t) is the time-dependent area of fractal interface defined as

k2−dFAdF /2ea (Dt)(2−dF )/2 (k is a dimensionless constant) and JE(x, t) represents the

flow at planar interface given by Fick’s first law.

The analytical solution of the partial differential equation of Eq. 6.3.2 under

potentiostatic condition with the assumption of semi-infinite diffusion coupled

with facile charge-transfer and the current (I(t)) is given as:

I(t) =zFAea

√D∗cb

Γ(3−dF2

)t−(dF−1)/2

=zFA

dF /2ea K2−dF D(3−dF )/2cb

Γ(3−dF2

)t−(dF−1)/2 , (6.3.7)

where z is the valence of diffusing species.

6.4 Power-Spectrum Based Approach

The roughness power spectrum approach has been used in different kinds of re-

actions and interface systems. This includes diffusion controlled potentiostatic

current-transient on realistic fractals for reversible and quasi-reversible conditions

[151, 152], diffusion controlled impedance on realistic fractals [153], generalized

Warburg impedance [154], generalized Gerischer admittance [155], generalized An-

son charge [124], absorbance [62], ohmic loss effect [156, 157], etc.

For data processing and analysis of stochastic processes, we need power-spectrum

28

which transforms the idealized surface (infinite length) to the real surface with lim-

ited length scales of roughness. In practice objects exhibiting random properties

are encountered. It is often assumed that these objects exhibits the self-affine

properties in a certain range of scales. Self-affinity is a generalization of self-

similarity which is the basic property of most of the rough surfaces. A part of the

self-affine object is similar to whole object after anisotropic scaling. Many rough

surfaces usually belong to the random objects that exhibit the finite self-affine

properties and they are treated as statistical finite self-affine fractals.

The power spectral density (PSD) is a positive real function of spatial fre-

quency wavenumber variable associated with a stationary stochastic process. The

spectral density function captures the frequency content of a stochastic process

and helps in identifying the periodicity on the surface. The power-spectrum itself

is the Fourier-transform of spatial correlation function. Correlation function rep-

resents the relationship of long and short-distance correlation within the surface

profile. The topography of any solid object can be considered as regular (ordered)

or irregular (disordered). Regular surfaces comprise the smooth surface with a

regular pattern whereas irregular surfaces comprise weakly and strongly disor-

dered surfaces [111]. Ordered surfaces can be adequately described by Euclidean

geometry but it is not sufficient to describe disordered surfaces.

7 De Gennes Scaling Conjecture for Diffusion-

Limited Flux and its Critical Analysis: Power-

Spectrum of Roughness Approach

Fractal geometry was introduced by Mandelbrot [111] motivated by nature, is used

to describe the objects of non-Euclidian geometry, like fractals. Fractals are invari-

ant under certain transformations of scale, called scaling and are best described

by scaling rules [111]. One of the most popular scaling rule is de Gennes scaling

29

conjecture [13] proposed for diffusion-limited nuclear magnetic spin relaxation pro-

cesses in porous media to calculate the fractal dimension. This scaling conjecture

states that, the scaling exponent, β, varies linearly with fractal dimension, DH as

Eq. 6.2.2.

De Gennes scaling conjecture is extensively used to characterize anomalous

diffusion-limited processes in various fields like fractured surfaces [19], dental ap-

plications [158], gold film deposition on liquid-liquid and liquid-gas interface [159],

lithium batteries [160, 161] and plastic deformation of Pd [162].

Several authors have frequently used [16, 136, 160] Eq. 6.2.2 to calculate the

fractal dimension since it is easy and less cumbersome also. As reported earlier

[151, 152, 153], the de Gennes scaling conjecture suffers from major limitations

such as: it is valid only for the intermediate regime and there is no proper de-

scription for characterization of this region. It often mixes the data of the scaling

region with the transition region and hence, there is inaccurate prediction of frac-

tal dimension. Since, de Gennes scaling conjecture grossly characterizes fractals

with only one morphological parameter hence, it does not capture the subtle as-

pect of the experimental data. This can not explain the deviation and regime of

time scales from linear behavior in a log-log plot which is often seen in the transi-

tion region of the data towards the large t value [136]. Thus, more sophisticated

theoretical model is needed which can overcome these anomalies.

In this section, de Gennes scaling conjecture for diffusion-limited flux at frac-

tally rough electrode-electrolyte interface is being critically reexamined numeri-

cally. We proposed a new approach to characterize the rough electrode-electrolyte

interface based on rigorous theoretical calculations. This approach may be useful

for other similar systems too. The fractal concept is used to describe the complex-

ity arising from the irregular interfaces (i.e. rough, porous, and partially active

interfaces) [111, 163]. The irregularity of the electrode surface is characterized as

self-affine fractal over limited length scales.

30

Log (wave number)

Log µ

Slope = 2DH - 7Lo

g (p

owe

r sp

ect

rum

)

-Log L -Log ℓ

Figure 1.4: Double logarithmic plot of power spectrum and wave number forisotropic self-affine fractal surfaces. The slope is used to calculate the fractaldimension while intercept gives the strength of fractality value. The two cut-offsgive the lower and upper cut off lengths value.

7.1 Basics of Roughness Power-Spectrum Theory for Dif-

fusion to Rough Surfaces

The power-spectrum of roughness approach is used to describe the diffusion-

limited reaction rates to the random but statistically isotropic surfaces that exhibit

the scale invariance over a limited range of length scales. This power-spectrum

contains the relevant information of the surface morphology in terms of limited

scales of wave-numbers (K) power law function as [112, 121]:

⟨|ζ(K)|2⟩ = µ|K|2DH−7; for1/L ≤ |K| ≤ 1/ℓ . (7.1.1)

The schematic diagram of power spectrum of realistic fractal roughness is rep-

resented diagrammatically in figure 1.4. From this framework, we get four fractal

31

a)

b)

Log tic

Log I /mA

-3

-2

-1

0

1

2

Log t /s

-3 -2 -1 0 1 2

dnLog I / dLog tn

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

anomalous region

transitin region

transition region

Cottrell

Cottrell

n = 1

n = 2

n = 3

toc toa tia tic

a)

b)

Figure 1.5: Schematic diagram for various transition regions and anomalous re-gion. The plot (a) is between LogI and Logt while plot (b) is between derivativesof LogI and Logt. The parameters are used to generate these curves are as:DH = 2.45, µ = 20 × 10−5 (a.u.), ℓ = 10−4 m, L = 50 × 10−4 m. The roughnessfactor is R∗ = 32.4. The various cut-off times are as: tic = inner Cottrellian time,toc = outer Cottrellian time, tia = inner anomalous time, toa = outer anomaloustime.

32

morphological characteristics of power-spectrum roughness namely fractal dimen-

sion (DH), strength of fractality (µ), lower and upper cut-off length scales (ℓ)

and (L), respectively. The log-log plot of power-spectrum and wavenumber gives

a straight line which has slope as 2DH − 7 and intercept as Log µ (Figure 1.4).

Thus from slope and intercept we can calculate the DH and µ, respectively. The

two cut-off values give the two cut-off length scales. The fractal dimension (DH)

is the global property that describes scale invariance property of the roughness

and largely associated with anomalous behavior in flux and its time exponent is

usually assumed to be function of this parameter. The strength of fractality (µ) is

the proportionality constant related to the topothesy of the fractal [112, 121, 123].

The geometrical interpretation of the topothesy as the length scale over which the

profile has a mean slope of 45◦. The unit of topothesy is cm2DH−3 and µ → 0

means no roughness. The lower cut-off length scale (ℓ) is the length above which

we observe roughness and upper cut-off length scale (L) is the length above which

we do not observe the roughness. The scaling region has very weak dependency

on upper cut-off length scale (L) for large L/ℓ ratio.

The total (averaged) flux/current at the stationary Gaussian random surface

is given by [117, 135, 151]

J(t) =

√DA0CS√

πt

[1 +

1

4πDt

∫ ∞

0

dKK(1− eK

2Dt)⟨|ζ (K)|2⟩

], (7.1.2)

where A0 is the area of the surface around which the rough surface fluctuates, CS

is the concentration of the species, D is the diffusion-coefficient and t is the time.

The reaction flux, J(t), is related to the electrode current, I(t) as:

I(t) = −nFJ(t), where n is the number of electron transfer in the redox reaction

and F is the Faraday constant. The second term inside the parenthesis of Eq. 7.1.2

is the contribution due to the roughness present at the electrode surface to the

total flux or current. If there is no roughness present at the electrode surface

then, contribution due to roughness is zero and hence, Eq. 7.1.2 becomes the pure

33

Cottrellian current [3], i.e.

IP (t) =nFA0CS

√D√

πt. (7.1.3)

Substituting Eq. 7.1.1 for the band-limited power law spectrum in Eq. 7.1.2

and we obtained the following equation as [151]:

J(t) =

√DA0CS√

πt

[1 +

µ

8π

(ℓ−2δ − L−2δ

δDt+

Γ(δ), Dt/ℓ2, Dt/L2

(Dt)1+δ

)], (7.1.4)

where δ = DH − 5/2,Γ(α, x0, x1) = Γ(α, x0) − Γ(α, x1) = γ(α, x1) − γ(α, x0),

and Γ(α, xi) and γ(α, xi) are the incomplete Gamma functions [90]. Eq. 7.1.4 is

simple and elegant expression of the total flux or current at the fractally rough

elctrode-electrolyte interface and is valid in all the three time regimes that is,

short, intermediate and long-time regimes. The expression for the time regime is

obtained by expanding the two incomplete gamma functions [90]. We have seen

the dependency of the scaling region on DH , µ and on ℓ graphically, by using

Eq. 7.1.4 [151]. These calculations break the earlier beliefs based on idealized

fractal model that the exponent of the anomalous diffusion region purely depends

on the fractal dimension of roughness only.

Most of the experimental data recorded are for the intermediate and long time

regime, that is, t > ℓ2/D. The expression for this time regime is obtained by

expanding two incomplete gamma functions in Eq. 7.1.4 viz., one for small Dt/L2

and another for large Dt/ℓ2. Eq. 7.1.4 extends the conventional representation of

the Cottrell current transient (1/√t) on the planar electrode in electrochemistry

[3] to the fractally rough electrode. The total mean flux is the summation of

smooth surface flux and an anomalous excess flux due to fractal roughness, or

it can also be looked upon as the product of the (1/√t) current and dynamic

roughness factor (term inside the parenthesis of Eq. 7.1.4).

Eq. 7.1.4 can be used to analyze numerically the various aspects like, Cottrel-

34

lian, transition and anomalous regions and ranges, roughness factor etc. (Fig. 1.5).

The procedure used are as follows: firstly, obtain the theoretical curve by using

Eq. 7.1.4 at fixed fractal morphological characteristics and take derivative of this

LogI vs Logt curve which gives the slope (β), and maxima of this curve gives the

βmax value Fig. 1.5 (b) (n = 1 curve). Then we obtain βmax while varying one

of the parameters and keeping others fixed and plotted βmax against that fractal

morphological characteristics. Then take second (n = 2) and 3rd (n = 3) deriva-

tives of LogI vs Logt curve so that one can find various times and its ranges like

Cottrellian, transition and the anomalous.

Figure 1.5 clearly elucidates various cut-off times and regions such as pure

Cottrellian regions (before tic and after toc), inner transition region (between tic

and tia) and outer transition region (between toa and toc) and in between these two

transition regions there is an anomalous region (between tia and toa). The value

of anomalous range, i.e. ρt = toa − tia = 2.22 in this case, while inner (tic − tia)

and outer (toa − toc) transition ranges are 0.81 and 1.25, respectively. We observe

toa and hence ρt, only when R∗ ≥ 14, in this case R∗ is 32.4. The maximum and

minimum of the second derivative curve (n = 2) gives inner transition time (ti)

and outer transition time (to), respectively.

7.2 Comparison of Roughness Power-Spectrum and Fractional-

Diffusion Theory

The problem of diffusion towards self-similar interfaces is mapped to a one dimen-

sional diffusion problem in Euclidean space by using the following relationship

[146, 147]:

Jf (x, t) =∂

∂t[(Df (t)

∗∂C(x, t)

∂x)] , (7.2.1)

35

Log (t/s)

-6 -4 -2 0 2

Log[Df(t)/D]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

µ µ µ µ = 2*10-6a.u.

µ =µ =µ =µ = 2*10-5 a.u.

µ = µ = µ = µ = 2*10-4 a.u.

µ µ µ µ

Log (t/s)

-6 -4 -2 0 2

Log[Df(t)/D]

0.0

0.5

1.0

1.5

2.0

2.5

DH = 2.20

DH = 2.25

DH = 2.30

DH

Log (t/s)

-8 -6 -4 -2 0

Log[Df(t)/D]

0.0

0.5

1.0

1.5

2.0

2.5

= 0.05 µ µ µ µ m

= 0.03 µ µ µ µ m

= 0.01 µµµµ m

l

l

l

l

(a)

(b)

(c)

Figure 1.6: Dependency of log of ratio of dynamic diffusion coefficient to the planardiffusion coefficient on (a) lower cut-off length ℓ, (b) fractal dimension DH and (c)strength of fractality µ.

36

where Jf is the flux and its unit is (mol/cm2s), Df (t) is a time-dependent diffusion

coefficient which has unit (cm2/s) and ∗ is the convolution operator

Df (t)∗∂C(x, t)

∂x=

t∫0

Df (t− ξ)∂C(x, ξ)

∂xdξ . (7.2.2)

The time dependent diffusion-coefficient, Df (t) is defined as:

Df (t) = D∗ t2−DH

Γ(3−DH), (7.2.3)

where Γ(3−DH) is the gamma function andD∗ is the fractional diffusion-coefficient

expressed in (cm2/s3−DH ) which is given as:

D∗ = K4−2DHR2DH−40 D3−DH , (7.2.4)

where K is a constant related to k and DH . The expression for the simplest case

of spatially one dimension diffusion is given by

∂3−DHC(x, t)

∂t3−DH= D∗∂

2C

∂x2. (7.2.5)

The diffusion towards a fractal electrode is thus mapped to one-dimensional

Euclidean diffusion equation where the derivative of the concentration vs time

is substituted by a fractional derivative and the diffusion-coefficient (D) is sub-

stituted by a fractional diffusion-coefficient D∗. When DH equals 2, Eq. 7.2.5

becomes the usual Fick’s second law for diffusion towards the planar surface.

Figure 1.6 is the plot of Log[Df (t)

D] vs Log(t) at different fractal dimension (DH)

figure 1.6 (a), at different strength of fractality (µ) figure 1.6 (b) and at different

lower cut-off length scale (ℓ) figure 1.6 (c). These figures clearly elucidates that dy-

namic diffusion-coefficient, Df (t), depends not only on fractal dimension but also

on strength of fractality and on lower cut-off length scale obtained from rough-

ness power-spectrum. The fractional diffusion theory accounts only intermediate

37

region and it does not consider short and long-time regimes. Also, intermedi-

ate regime is characterized by only one parameter, i.e. fractal dimension. Thus,

roughness power-spectrum model is the better approach to characterize the rough

electrode-electrolyte systems.

8 Plan of Study and Thesis Overview

The work presented in this thesis contributes towards the understanding of charge

and absorbance-transients at rough electrode-electrolyte interface. The detailed

description of spectroelectrochemical responses contain both, the quantitative and

qualitative information. The thesis contains the following chapters:

Chapter 1: This chapter contains the introduction of spectroelectrochem-

istry and chronocoulometry techniques. The detailed description of these two

techniques with their significance and advantages to study the rough electrode-

electrolyte interface is presented. The possible SEC techniques which can be

used to study the effect of surface roughness are described namely, UV-visible,

reflectance and SERS. The most important part of our theoretical model that is,

power-spectrum of roughness with comparison with other methods is also pre-

sented.

Chapter 2: In this chapter, a quantitative description of the role of OTE

roughness on potential step transmission chronoabsorptometry measurements are

presented to understand the various anomalies in their response. In view of this, a

theoretical model for the potentiostatic absorbance-transient for a simple reversible

redox reaction at the rough electrode-electrolyte interface is developed. A general

operator structure between concentration, surface absorbance density, absorbance-

transients and arbitrary roughness profile of electrode is emphasized. An elegant

mathematical formula between the average spectroelectrochemical absorbance-

transient and surface structure factor or power-spectrum of roughness is obtained.

Through this, the questions like, how OTE roughness affects the absorbance-

38

transient for redox reaction and why power-spectrum of roughness approach is

advantageous to characterize the rough OTE surface are tried to understand.

Chapter 3: The theoretical findings of chapter 2 is extended to understand the

pseudo first-order catalytic reaction mechanism at rough OTE. In this chapter,

a theoretical model for absorbance-transient is developed for the pseudo first-

order catalytic coupled reaction (EC’) mechanism at rough OTE. A second-order

perturbation solution in surface profile for an arbitrary rough surface is obtained.

The expression for the absorbance-transient for the band limited self-affine fractal

roughness of an OTE is obtained. Through this, the kinetic study of the pseudo

first-order reaction mechanism is also presented in this chapter.

Chapter 4: In this chapter, the Anson equation is generalized for fractal and

nonfractal rough electrode-electrolyte interface. The fractal and nonfractal rough

electrode surfaces are characterized using an approximate power-law and Gaussian

power spectra, respectively. The finite fractal roughness consists of the following

four morphological characteristics: fractal dimension (DH); the strength of frac-

tality (µ), which is the magnitude of the power spectrum at the unit roughness

wavenumber; and the lower (ℓ) and upper (L) cut-off lengths, which are related

to cut-off wavenumbers in the power-spectrum. Nonfractal Gaussian roughness

consists of two morphological characteristics, the mean square width of roughness

(h2) and the transversal correlation length (a).

Chapter 5: In this chapter, the rough electrode-electrolyte interface is char-

acterized through chronocoulometry technique for pseudo first-order catalytic cou-

pled reaction (EC’) mechanism. The possible kinetic effects due to rough surface

on EC’ reaction mechanism is analyzed. In this respect, a theoretical model for

chronocoulometric measurements is developed for EC’ reaction at rough electrode-

electrolyte interface. An expression for the charge-transient for band limited self-

affine fractal roughness is obtained. The possible kinetic along with the morpho-

logical studies are presented.

39

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