CONTROLLING VARIABLES OF ELECTRIC DOUBLE-LAYER
CAPACITANCE
(電気二重層の静電容量を決定する変数)
By
Yongdan Hou
A Thesis Submitted to the Materials Engineering,
Graduate School of University of Fukui
In Partial Fulfilment of the Requirements
For the Degree
Of
DOCTOR OF PHILOSOPHY
(MATERIALS ENGINEERING)
@
Department of Applied Physics
September 2014 ABSTRACT
The electric double layers are of crucial significance to the electrochemical researches. To provide a better understanding in the structure/composition of the double-layer, controlling the variables of electric double-layer capacitance were explored. Base on frequency dispersion, this study focused on the role of applied frequency and potential, ions and solvents in controlling the double-layer capacitance in the absence of faradic reactions by impedance measurement. A platinum wire electrode was used as insulator-free electrode, which avoids stray capacitance or irreproducibility of impedance caused by incompleteness of electric shield of electrodes as the conventionally used inlaid electrodes. Findings regarding frequency dependence: the
Nyquist plots showed approximately a straight line, whereas the in-phase component, Z1, extrapolated to zero separation of the electrodes was non-zero; therefore, a resistance was present at the double-layer in parallel form to the capacitance. The parallel resistance was not faradaic resistance due to the absence of any electroactive species and was inversely proportional to the frequency, whereas the capacitance decreases with a linear relation to logarithm of the frequency. The latter was observed to be responsible for the frequency-dependence of the former, which is involved apparently and inevitably in ac-measurements of the capacitance. Results of ions and potential dependence: the capacitance shows almost a value common to concentrations from 1 mM to 2 M (= mol∙dm-3) and four halide ions, and varied ca. 10 % with the dc-potential in 0.1 M KF, KCl and KBr solutions, meanwhile, the resistance is unaffected. Therefore, the capacitance has negligible influence on the localization of ions. Solvent dependence: the capacitance also shows no systematic relation with the dielectric constants, viscosity, boiling temperatures or dipole moments of 13 solvents, but is proportional to the inverse of the lengths of field-oriented molecule. It indicates that the saturated dielectric constants and the lengths of solvent molecules along the dipole are the controlling variables of solvent for the capacitance, and the 13 solvents provide common saturated dielectric constants of 6, which was modelled by introducing image force to Debye’s equation. The analytically evaluated tilts of molecular dipole are mostly parallel to the electrode surface in usual depolarized potential domain, indicating the consistency of the model and the experimental values.
I ACKNOWLEDGEMENT
Foremost, I wish to express my sincere thanks to my supervisor Prof. Koichi Aoki and Prof. Jingyuan Chen, all of the Department of Applied Physics, University of Fukui,
Japan for providing me with the opportunity and continuous support to complete the
PhD study. This work could not have been possible without their guidance. I am grateful for their patience, motivation, enthusiasm and immense knowledge, as well as the financial support I received at the inception of my schooling as a self-financed student.
My sincere thanks also go to Dr. Toyohiko Nishiumi for his guidance and help during my studies.
A special thank also goes to the members in Prof. Aoki’s laboratory for their contributions to my PhD study.
Financial support by Japan Student Services Organization is gratefully acknowledged.
Last but not the least, I dedicate this thesis to my mum Junlan Zhao, and dad
Jingchun Hou for their, both physically and spiritually, and to my husband Lemuel
Gbologah.
II TABLE OF CONTENTS
ABSTRACT ...... I
ACKNOWLEDGEMENT ...... II
TABLE OF CONTENTS ...... III
LIST OF TABLES ...... IV
LIST OF FIGURES...... V
LIST OF ABBREVIATIONS...... X
CHAPTER ONE
INTRODUCTION ...... 1
CHAPTER TWO
REVIEW OF CONVENTIONAL CONCEPTS OF ELECTRIC DOUBLE LAYERS ...... 4
CHAPTER THREE
A PARALLEL RESISTANCE OF ELECTRIC DOUBLE-LAYER CAPACITANCE...... 27
CHAPTER FOUR DOUBLE-LAYER CAPACITANCE – INDEPENDENCE OF DC-POTENTIAL ...... 47
CHAPTER FIVE
DEPENDENCE OF CAPACITANCE ON BASIC PHYSICAL PROPERTIES OF
ELECTROLYTIC SOLVENTS ...... 59
CHAPTER SIX MODIFICATION OF DEBYE’S MODEL FOR DIELECTRIC SATURATION IN ELECTRIC DOUBLE-LAYER ...... 73
CHAPTER SEVEN
SUMMARY AND CONCLUSIONS ...... 84
REFERENCES ...... 86
LIST OF PUBLICATIONS...... 105
CURRICULUM VITAE...... 106
III LIST OF TABLES
Number Page
Table 5 - 1. Values of the EDL capacitance, dmv and dor for 13 solvents...... 66
Table 6 - 1. Compositions of 13 solutions used in impedance measurement...... 81
IV LIST OF FIGURES
Number Page
Figure 2 - 1. Schematic of double layer structure of a) the Helmholtz model, b) the
Gouy-Chapman model, and c) the Stern model ...... 5
Figure 2 - 2. Nyquist plot and equivalent circuit of a) a resistance, b) a capacitance,
c) a resistance in series with a capacitance, d) a resistance parallel to a
capacitance, e) a CPE, f) a CPE in series with a resistance and g) a
CPE parallel to a resistance...... 13
Figure 2 - 3. The capacitance dependence of applied potential for a) Helmholtz
model, b) Gouy-Chapman model and c) Gouy-Chapman-Stern model 19
Figure 2 - 4. Change Classification of 34 publications about the double layer
capacitances vs. potential curves into five shapes; (A) the number of
publications for every available electrode material and (B) ratio of
number of publications belonging to a given shape for simulation and
experimental work...... 23
Figure 2 - 5. The number of experimental capacitance vs. potential curves classified
with values of ∆C/Cav (A) without and (B) with assignment to
electrode materials...... 23
Figure 3 - 1. Illustration of the cell and a pair of the wire electrodes, where L is the
effective length of the wire electrodes immersed in the solution, and d
is the center-to-center distance of the two wires...... 29
Figure 3 - 2. Examination of the delay of the potentiostat by film capacitor (0.1 μF)
in serious with a carbon resistance (a) 1 kΩ, (b) 5 kΩ and (c) 10 kΩ...... 30
Figure 3 - 3. Cyclic voltammograms obtained at the scan rate of 10 mVs-1. In (a)
0.5 M KCl solution and (b) 0.2 mM K3Fe(CN)6 + 0.2 mM K4Fe(CN)6
V at the two parallel Pt wires 0.1 mm in diameter 10 mm in length with
the distances, d = 2 mm. (c) is the magnification of (a), (d) is the plot
of the current at 0.02 V in (a) against scan rate...... 31
Figure 3 - 4. Nyquist plot at the two parallel Pt wires, 0.1 mm in diameter, 10 mm
in length with the distances, d = 2 mm in 0.5 M KCl aqueous solution
obtained for (A) low frequencies at (x mark) 5 Hz and (+ mark) 51 Hz,
and (B) high frequencies at (x mark) 1kHz and (+ mark) 10 kHz. Rs,N
is the value extrapolated for f ...... 32
Figure 3 - 5. Two-dimensional model for potential distribution at the two parallel
cylinders for d/a = 20, and computed potential distribution by the
finite element method in the potential field discretized by triangles...... 34
Figure 3 - 6. Dependence of the dimensionless resistance, RL/, obtained by
(circles) the finite element method, by (solid line) (Eq. 3 - 11) on
ln(d/a – 1), by (dashed line) (Eq. 3 - 12) for the fitting...... 36
Figure 3 - 7. Dependence of Rs,N (= Z1 (f ) ) determined from the Nyquist plots
for concentrations of KCl (a) 0.01, (b) 0.03, and (c) 0.1 M on ln(d/a –
1). The lines are from (Eq. 3 - 12). The dotted line is the recessed line
for (a)...... 37
Figure 3 - 8. Plots of Z1 with log(d/a - 1) for frequencies of (a) 100 Hz, (b) 1000 Hz
and (c) 5000 Hz in 0,5 M LCl. Line (d) is from (Eq. 3 - 12)...... 37
Figure 3 - 9. Equivalent circuits of (A) series combination (Z1 + iZ2), (B)
geometrical assignment of Rs and Cd, and (C) really acting
combination of Rs, Cp and Rp...... 38
Figure 3 - 10. Variations of Rp and Cp in the parallel equivalent circuit with d/a for
frequencies of (a) 100, (b) 1000 and (c) 3000 Hz in 0.5 M KCl
VI solution for L = 10 mm...... 39
Figure 3 - 11. Logarithmic dependence of Rp averaged for d/a on f at [KCl] =
(crosses) 0.01 M, (circles) 0.1 M, and (triangles) 0.5 M KCl. The
slope of the line is -1...... 40
Figure 3 - 12. Variation of Cp averaged for d/a with f at [KCl] = (crosses) 0.01 M,
(circles) 0.1 M, and (triangles) 0.5 M KCl...... 40
Figure 3 - 13. Variation of (Cp)f=1 with concentration of KCl by (circles)
experiments, (a) the Gouy-Chapman's theory, and (a) the ion-cell
model...... 42
Figure 4 - 1. Experimental set up used for evaluating variations of capacitance with
applied electrode potential...... 48
Figure 4 - 2. Equivalent circuit for platinum wire working electrode | electrolyte |
coil counter electrode system ...... 49
Figure 4 - 3. Nyquist plots for 0.01 M (M = mol dm-3) KF and KBr solutions at the
two wire electrodes with 7 mm separation...... 50
Figure 4 - 4. Variations of the parallel capacitances with logarithmic frequency for
(A) KF, (B) KCl, (C) KBr and (D) KI at concentrations of (open
circles) 2 M, (triangle) 0.5 M, (full circles) 0.1 M and (squares) 0.01
M. Values of Cp were obtained at two electrodes without any reference
electrode. M = mol∙dm-3...... 51
Figure 4 - 5. Dependence of Cp at f = 1 Hz on concentrations of (open circles) KF,
(triangles) KCl, (full circles) KBr and (squares) KI...... 52
Figure 4 - 6. Variations of the parallel resistance with frequency on the logarithmic
scale for (circles) 0.5 M KF and (triangles) 0.5 M KCl...... 53
Figure 4 - 7. Dependence of Rp of 1.0 M KCl on the area of the electrode at f =
VII (open circles) 320, (triangles) 541, (squares) 1082 and (full circles)
2475 Hz...... 54
Figure 4 - 8. Dependence of the double-layer capacitance on dc-potential in 0.1 M
solutions of (squares) KF, (circles) KCl and (triangles) KBr, where the
capacitance values are extrapolated to 1 Hz...... 55
Figure 4 - 9. Variations of the double-layer capacitance in KCl solution at (open
circles) 1 M, (filled circles) 0.1 M, (triangles) 0.01 M and (squares)
0.001 M with the dc-potential, where the capacitance values are
extrapolated to 1 Hz...... 56
Figure 4 - 10. Comparison of the potential-dependence of (Cp)f=1 and cyclic
voltammetry in 0.1 M KCl solutionon...... 57
Figure 5 - 1. Nyquist plots obtained in (circles) DMSO, (squares) PC and (triangles)
DCM. High frequencies correspond to large values of Z1...... 62
Figure 5 - 2. Plots of Cp evaluated from Eq. (3) in (circles) DMSO, (squares) PC
-3 and (triangles) DCM solutions including 0.1 mol dm THAClO4
against log(f)...... 63
Figure 5 - 3. Logarithmic plots of Rp evaluated from Eq. (3) in (circles) DMSO,
(squares) PC and (triangles) DCM solutions including 0.1 mol dm-3
THAClO4 against log(f)...... 64
Figure 5 - 4. Variation of Cp with dc-potential and cyclic voltammogram (scan rate
-1 0.02 V s ) in 0.1 M THAClO4 + propylene carbonate at the two
platinum electrodes...... 66
Figure 5 - 5. Variation of the capacitance with dielectric constant of the solvents ...... 67
Figure 5 - 6. Variation of the capacitance with the inverse of molecular diameter
obtained from the molar volume ...... 68
VIII Figure 5 - 7. Variations of the capacitance with (a) dipole moments, (b) boiling
temperatures and (c) viscosity...... 68
Figure 5 - 8. Models of solvent molecules: a) an example of estimating dor of
DMSO molecule from ChemOffice; b) solvent molecules with
orientation at the interface and in bulk...... 70
Figure 5 - 9. Variation of the capacitance with the inverse of molecular diameter
obtained from the length of the oriented molecular dipoles...... 71
Figure 6 - 1. Illustration of molecular dipole in inner layer, where d is thickness of
inner layer; is the angle between direction of molecular dipole and
electric field, E; q and -q are equivalent charges of dipole; (l+2r) is the
length of dipole; r is the radius of two hemispheres...... 74
Figure 6 - 2. Model of a dipole interacting with the electrode by the image force
and external electric field...... 76
Figure 6 - 3. Dependence of averaged values of on E/kBT at = (a) 2 D, (b) 3 D,
(c) 4 D and (d) 5 D, calculated from (Eq. 6 - 12) for a = 0.1 nm, l =
0.4 nm...... 79
Figure 6 - 4. Dependence of the angles of tilted dipoles on the capacitances, where
angles were calculated from the numerical integration of (Eq. 6 - 12)
for a = 0.1 nm at E = (circles) 2×107 and (triangles) 2×108 V m-1. Plots
with squares are from (Eq. 6 - 15) by use of the experimental values of
(r)sat. The numbers corresponds to those in Table 6 - 1...... 80
Figure 6 - 5. Illustration of the state of a molecular dipole in inner layer in the
experimentally polarised potential domain...... 82
Figure 6 - 6. Plot of EDL capacitance against the inverse of thickness of the tilted
dipole...... 82
IX LIST OF ABBREVIATIONS
Abbreviations Meaning
DCM dichloromethane
DMSO dimethyl sulfoxide
EDL electric double-layer
EIS electrochemical impedance spectroscopy
GC Gouy-Chapman
IHP inner Helmholtz plane
OHP outer Helmholtz plane
PC propylene carbonate
Pt platinum pzc potential of zero charge
X Chapter One Introduction
Chapter One
Introduction
1.1 Background to Electric Double-Layer Capacitance Accompanied by the
Inconsistency between the Theoretical Predictions and Experimental Results
When potential is applied to the interface, counter ions will accumulate at the interface to form electric double layers. The electric double-layer is of great significant to the electrochemical research studies. Firstly, it enables any possible electrode reactions: as one volt is applied to the interface, an extremely strong electric field, being of 108 V/m, will be generated across the electric double-layer. Without the existence of the double layer, no ions or molecules can stay at the interface. Secondly, the double-layer also affects dynamics of electrode reactions. The concept of electric double-layer was introduced by Helmholtz [1], and it included a parallel metal plate and a layer of ionic solution phase. Gouy [2] and Chapman [3] developed the concept of the double-layer to distributing ions which are subjected to the balance between electrostatic force and diffusional dispersion force. The Gouy-Chapman theory is valid only for very dilute ionic solutions which are not useful for conventional electrochemical measurements or practical double-layer capacitors. Stern [4] combined the Helmholtz model and the Gouy-Chapman (GC) model into one, to explain data of double-layer capacitances. However, Stern's model could only explain the experimental results qualitatively of NaF solution at mercury electrodes. Problems of the GC theory involved the assumption of a point charge and a loss of charge-charge interaction.
Previous theories were of limited consistency with that of the practical electric
1 Chapter One Introduction double-layer behaviours. Aoki [5] developed a theory of the double-layer from the microscopic point of view by statistical mechanics. The theory sought to solve the problems of point charge assumption and the neglect of charge-charge interaction at the expense of the self-consistence field.
In order to know the structure/composition of the double-layer, the study of electric double layer capacitance is a main way. The variation of capacitance with dc-potential can help to figure out the contribution of ions or/and molecules to the double-layer, and the orientations of them. Practically, the experimental capacitance results reveal many different concepts.
1.2 Objectives
The main objective of the thesis is to attempt to find out the controlling variables of electric double-layer capacitance, which can help people to understand the electric double- layer better, as well as to provide a guide for optimizing the design of capacitors/supercapacitors by pointing out the experimental controlling parameters of the capacitance.
The specific objectives will be to:
1. Fabricate platinum wire electrodes and reference electrodes;
2. Examine the influence of electrode spinning and surface treatment;
3. Explain frequency/time and voltage dependence of the double-layer;
4. Study the variation of capacitance with ionic concentration and kinds of ions;
5. Investigate the effects of solvents (physical properties);
2 Chapter One Introduction
6. Figure out the relation between capacitive behaviour observed by voltammetry and
by ac-impedance.
1.3 Scope of the Study in this Thesis
The study of this thesis focuses on the electrochemical behaviours of the electric double layers on platinum wire electrode(s).
Chapter 2 gives a comprehensive overview of conventional concepts of the double layers.
Chapters 3, 4, 5 and 6 deals with the analysis of the frequency dependent impedance data, variations of the double-layer capacitance with electrolytic ions and applied potential, the controlling variables of solvent for the capacitance, and a model of dielectric saturation in electric double-layer capacitance, respectively.
Summary and conclusions of the research are dealt with in chapter 7.
3 Chapter Two Review of Conventional Concepts of Electric Double Layers
Chapter Two
Review of Conventional Concepts of Electric Double Layers
2.1 Electric Double Layers
2.1.1 Helmholtz Model
The concept of electric double layers was initially created for metal-metal interface
[6]. It states the presence of the inhomogeneous region of electrolyte near a charged surface caused by charge separation in the electric field, according to Helmholtz in 1879
[1]. It involves a compact monolayer of counter-ions which were adsorbed on the charged metal surface. Figure 2 - 1(a) illustrates the schematic of the Helmholtz model.
Differential capacitance per unit area is often defined by
C (Eq. 2 - 1) E where σ is the charge density and E is the electric potential. In Helmholtz model, the capacitance, CH, was predicted as
ε0εr (Eq. 2 - 2) CH δH where ε0 is the vacuum permittivity, εr is the relative permittivity of solvent; CH is the differential capacitance per area and depends only on dielectric constant and the separation of charge layers, δH, and δH is the separation of the charge layers. The thickness δH of the double-layer can be radius of the solvated ions approximately [7], being of the order of few Angstroms. This approximation is roughly correct for concentrated electrolytic solutions.
4 Chapter Two Review of Conventional Concepts of Electric Double Layers
(a) Helmholtz model (b) Gouy-Chapman model φ(x) H φ(x) Diffuse layer + + + + + + + + + + Electrode Electrode + +
(c) Stern model Inner
+ layer φ(x) Diffuse layer Solvated ion +
Solvent molecule + +
Cation Electrode + Anion +
x1 x2 IHP OHP
Figure 2 - 1. Schematic of double layer structure of a) the Helmholtz model, b) the Gouy-Chapman model, and c) the Stern model
5 Chapter Two Review of Conventional Concepts of Electric Double Layers
2.1.2 Gouy-Chapman Model
In 1910, Gouy [2] and Chapman [3] established the classical theory of electric double layers by solving the Poisson-Boltzmann equation for the bulk electrolyte. It was later coined as “diffuse double-layer”. In Gouy-Chapman’s model, as shown in Figure 2
- 1(b), ions are mobile in the electrolyte solution as point charge, of which the thermal motion are subjected to the balance between electrostatic force and diffusional dispersion force [7]. The distribution of ions in Gouy-Chapman layer is not rigid unlike in the Helmholtz layer, but obeys Boltzmann distribution, depending on the electric potential:
qα (Eq. 2 - 3) nα nα exp( ) kBT
3 where nα is number density of species α (particles/cm ); nα is the value of nα at infinity;
-19 qα is the charge of given ion α, qα = zα∙e, z is the valence of ion α, e = 1.602×10 C;
-23 -1 kB=1.38066×10 JK is the Boltzmann constant, T is Kelvin temperature; Δφ is the potential drop of the double-layer in mean field theory, Δ = (x)- S. Here, it is assumed that the potential and ions are distributed only to the direction, x, perpendicular to the planar electrode. The ionic density of the Gouy-Chapman theory can be expressed by
α (x) qα nα (x) (Eq. 2 - 4) α
The solvent is treated as a continuous dielectric. The potential profile in the electrolyte solution obeys Poisson’s equation:
d2 (x) 1 (x) (Eq. 2 - 5) 2 α d x 0 r α
6 Chapter Two Review of Conventional Concepts of Electric Double Layers where the dielectric constant is assumed to be constant. Inserting (Eq. 2 - 4) into (Eq. 2
- 5) on the simplification of using 1:1 electrolyte yields the Poisson-Boltzmann equation
d2(x) en e(x) e(x) exp exp (Eq. 2 - 6) 2 dx 0 r kBT kBT
This is actually the same as the Debye-Hückel theory for the charge distribution around a central ion if the ions were to be so large that the spherical effects are negligible.
When the first and the second term in the exponential are retained for e/kBT < 1, (Eq. 2
- 6) is reduced to
d2(x) 2en e(x) 2 (Eq. 2 - 7) dx 0 r kBT
1/2 A solution of is exp(e(2n /0rkBT) x). When the potential in the bulk is taken to be
1/2 zero, the accepted solution is restricted to exp(-e(2n /0rkBT) x). Under the condition of (0) = p, the solution is given by
(x) p exp e 2n / k T x p exp x (Eq. 2 - 8) 0 r B
-1 1/2 where the κ = e(2n /0rkBT) is the Debye-Hückel [8] screening length, a quantitative characteristics for the thickness of the Gouy-Chapman double-layer.
The surface charge density, σM, is given by
M d 0 r p 0 r (Eq. 2 - 9) dx x0
Then the potential profile in the solution is given by
M (x) exp(x) (Eq. 2 - 10) 0 r
7 Chapter Two Review of Conventional Concepts of Electric Double Layers
It is obvious that at a certain surface charge density, σM, increases with an increase in κ.
In other words, the thickness of the double-layer increases as the ionic density decreases.
The differential double-layer capacity in the Gout-Chapman theory can be represented as
d M e 0 r (Eq. 2 - 11) CGC cosh d 2kBT
The capacitance, CGC (or CD), resulting from the diffuse charge distribution varies with concentrations and potential. It is worth nothing that, the Debye-Hückel approximation is practically invalid at high ionic concentrations, and hence the Gouy-Chapman theory only approach good agreement with experimental data at low ionic concentrations [9].
Moreover, the variation of CGC with the potential presents a minimum at the potential of zero charge (pzc), but unlimited rises of the capacitance at potentials away from the pzc.
The unlimited rise of capacitance is impractical and could be caused by the assumption of point charge, which allows any possible close-approach of ions toward the electrode surface, and thus, results in the rising of capacity without a limit.
2.1.3 Stern Model
A further development of the double-layer was introduced by Stern [4]. He combined both Helmholtz and Gouy-Chapman’s theories; hence, the electric double-layer is composed of two parts: i) a compact layer (or Helmholtz layer) consisting of a layer of specifically adsorbed ions [7, 10], and ii) a diffuse layer (or
Gouy-Chapman layer) of mobile ions. It is also known as Gouy-Chapman-Stern theory.
Later, Grahame [11] divided the compact layer into two regions, as shown in Figure 2 -
1(c), by denoting the closest approach of small and partially solvated ions to the
8 Chapter Two Review of Conventional Concepts of Electric Double Layers
electrode surface as the inner Helmholtz plane (IHP), x1, and the closest approach of the centres of fully solvated ions to the electrode surface as the outer Helmholtz plane
(OHP), x2. The part outside the OHP is the diffuse double-layer. The capacitance, Cdl, predicted by Stern’s theory, is given by the series combination of the two layers:
1 1 1 4 4 H (Eq. 2 - 12) Cdl CH CD qp cosh 2kBT where CD = CG–C. In concentrated solutions, CH << CD, Cdl, hence, is controlled by CH, the value of which depends slightly on ionic concentrations in the case of no specific adsorption. Up to this model, a gross image has been produced. However, no capacitive components owning to the interactions of dipoles and that between dipoles and the electrode surface (image effects) are considered.
2.1.4 Further Developments
Later researches developed details regarding the functions of solvents in the double-layer. Bockris and co-workers in 1956 [12] discussed the effect of oriented dipoles on frequency dependence of both capacitance and resistance of the double-layer on mercury and copper electrode. In 1961, Mott and co-workers [13] presented the contribution of solvent (water) dipoles to the double-layer capacitance by use of
Graham’s data [11]. In 1963, Bockris’s group [14] proposed a model that included the clear partition of the solvent. They suggested the existence of the monolayer of water within the IHP of the electrode. This layer of molecular dipoles had a fixed orientation because of the charge on the electrode. The capacitance contribution of this layer owning to the variation of electrode potential was negligible. In the presence of specific adsorption, some of the molecules would be replaced by specifically adsorbed ions.
There were also other layers of water molecules outside IHP, but the dipoles in these
9 Chapter Two Review of Conventional Concepts of Electric Double Layers layers would not be as strongly oriented as those in the first layer. Solvated ions could come close to the surface next to the first layer. In 1979, Trasatti [15] reviewed the experimental evidence of the double-layer potential drop resulting from adsorbed water molecules on electrode surface.
The image of the double-layer became clearer with the models building up. The electrochemical properties of the electric double-layer will be introduced in later sections.
2.2 Impedance Measurement
2.2.1 Electrochemical Impedance Spectroscopy
Electrochemical impedance spectroscopy (EIS) is a powerful method of evaluating the performance of an unknown system in the frequency domain. Special equipment is required to apply a sinusoidal voltage or current signal with a small ac-voltage
(amplitude) as a disturbance to measure the responding changes in both magnitude and phase over a range of frequencies. A detailed review of impedance spectroscopic measurement was given by Macdonald [16]. The output of the system with such inputting signal contained two parts: real impedance Z′ and the imaginary impedance
Z′′.
Z Z jZ (Eq. 2 - 13) where j is the imaginary number ( j 1 ). A plot of Z′ (x axis) against Z′′ (y axis) is called Nyquist plot. The admittance is the inverse of impedance, Y = Z-1. Impedance is a vector, of which θ is the phase angle.
Z tan (Eq. 2 - 14) Z
10 Chapter Two Review of Conventional Concepts of Electric Double Layers
There is another characteristic plot of impedance data, called Bode plot using logf (or logω) as x axis, and log│Z│ and phase angle θ as y axis respectively, where f is the ac-frequency, ω is the angular frequency ω = 2πf.
2.2.2 Equivalent Components
Equivalent resistance R
In electrochemistry, R is denoted as the equivalent component of resistance. The impedance of R is: ZR = ZR′, ZR′′ = 0. The value of R is independent of the frequency applied and its phase angle of is also zero. It is presented as a point (R, 0) in the Nyquist plot, in Figure 2 - 2(a), and as a line parallel to x axis of the Bode plot (log│Z│ vs. log f).
Equivalent capacitance C
C is denoted as the equivalent component of capacitance. The impedance of C is: ZC
= -j(1/ωC), ZC′ = 0, ZC′′ = -1/ωC. There is no real impedance of C. C can be presented as a line overlapped with y axis in the Nyquist plot, as shown in Figure 2 - 2(b), and as a line with slope of -1 of the Bode plot (log│Z│ vs. log f).
Series combination of R and C
The expression of the series combination of R and C is
1 1 Z R R j (Eq. 2 - 15) jC C
It can be presented as a line of x = R parallel to y axis in the first quadrant of the Nyquist plot, in Figure 2 - 2(c). The phase angle of such combination is given by
1 tan (Eq. 2 - 16) RC
The value of Z is given by
11 Chapter Two Review of Conventional Concepts of Electric Double Layers
1 Z R2 (Eq. 2 - 17) (C)2
Obviously, when ω→∞, │Z│≈ R, θ ≈ 0, the system behaves like R; when ω→0, │Z│≈
1/ ωC→∞, θ = π/2, the system behaves like C.
In Bode plot (log│Z│ vs. log f), such combination is a line parallel to the x axis at high frequencies, and a line with slope of -1 at low frequencies. Normally, a measured ideal interface reveals such combination of ideal capacitor in series with the solution resistance.
Parallel combination of R and C
The expressions of the parallel combination of R and C are
1 1 jC (Eq. 2 - 18) Z R
R Z (Eq. 2 - 19) 1 (RC)2
R2C Z (Eq. 2 - 20) 1 (RC)2
2 2 R R Z2 RZ Z2 0 Z Z2 (Eq. 2 - 21) 2 2
It can be represented as a semicircle with (R/2, 0) as the centre and R/2 as radius in the first quadrant of the Nyquist plot, as shown in Figure 2 - 2(d). The physical identification can be the faradic resistance, R, parallel to an ideal capacitor C. The phase angle of such combination is given by
Z tan RC (Eq. 2 - 22) Z
12 Chapter Two Review of Conventional Concepts of Electric Double Layers
R C (a) R (b) C (c)
Z′′ Z′′ Z′′
ω ω
R 0 0 0 Z′ Z′ R Z′
R
(d) (e) CPE C
Z′′ Z′′ ω ω
0 R/2 0 Z′ Z′
R CPE R (f) (g)
Z′′ CPE Z′′ θ ω ω
0 R/2 0 R Z′ Z′
Figure 2 - 2. Nyquist plot and equivalent circuit of a) a resistance, b) a capacitance, c) a resistance in series with a capacitance, d) a resistance parallel to a capacitance, e) a CPE, f) a CPE in series with a resistance and g) a CPE parallel to a resistance.
13 Chapter Two Review of Conventional Concepts of Electric Double Layers
The value of Z is given by
R Z (Eq. 2 - 23) 1 (C)2
Obviously, when ω→∞, │Z│≈ 1/ ωC, θ ≈ π/2, the system behaves like C; when ω→0,
│Z│≈ R, θ = 0, the system behaves like R.
In Bode plot (log│Z│ vs. log f), such combination is a line parallel to the x axis at low frequencies, and a line with slope of -1 at high frequencies.
Constant phase element (CPE)
Experimentally, the behaviour of electric double-layer at solid/liquid interface differs from an ideal capacitor. The impedance varies with the applied frequency. The variation of C with the frequency is called “frequency dispersion”. This frequency dispersion was empirically expressed by a constant phase element (CPE). The general expression of the impedance of CPE is given by Lasia [17]:
1 Z ( j)n (Eq. 2 - 24) Y0
n n Z cos (Eq. 2 - 25) Y0 2
n n Z sin (Eq. 2 - 26) Y0 2
where Y0 is a constant, n is related to rotated angle on Nyquist plot of an ideal capacitor,
0 < n < 1. Y0 and n are the parameters present the deviation of C from the ideal capacitor.
The phase angle of such combination is given by
n (Eq. 2 - 27) 2
14 Chapter Two Review of Conventional Concepts of Electric Double Layers
The value of Z is given by
n Z (Eq. 2 - 28) Y0
It can be represented as a line, passing through the origin (0, 0), with the slope of tanθ in the first quadrant of the Nyquist plot, as shown in Figure 2 - 2(e). In Bode plot
(log│Z│ vs. log f), such combination is a line with a slope of –n. when n = 0, CPE is R; when n = 1, CPE is C.
Series combination of CPE and R
The expression of the series combination of CPE and R is
1 n 1 n Z R cos j sin n n (Eq. 2 - 29) Y0 2 Y0 2
It can be presented as a line with a slope of tan(nπ/2) and meet the real axis at x = R in the Nyquist plot, in Figure 2 - 2(f). Practically, this combination was used to present a frequency dependent capacitance in serious with solution resistance.
Parallel combination of CPE and R
The expression of the parallel combination of CPE and R is
2 2 n 2 R cot R 2 R Z Z (Eq. 2 - 30) 2 2 n 2sin 2
It can be represented as circle with [R/2, R∙cot(nπ/2)/2] as the centre, R/(2sin(nπ/2)) as the radius in the Nyquist plot, as shown in Figure 2 - 2(g). Practically, it can represent a faradic resistance parallel to a frequency dependent capacitance.
15 Chapter Two Review of Conventional Concepts of Electric Double Layers
Advantages and disadvantages of equivalent circuits
The application of equivalent circuit is the main analytical method used in the study of electrical impedance spectroscopy for providing an intuitive relation between EIS result and models of kinetics of electrode process. The value of EDL capacitance depends on the equivalent circuit employed for the data analysis. However, an equivalent circuit is not always unique to an electrode process; an EIS result can be fitted by different equivalent circuits according to the IUPAC recommendation for equivocal equivalent circuits [18].
By the EIS measurements, the electrochemical behaviours of EDL were explored.
Generally, if the non-uniqueness of the applied equivalent circuit is not considered, the electrochemical behaviours of EDL can be influenced by the physical components at the interface: electrode material, composition of electrolyte (solute and solvent), the interaction among them and the physical condition of each. The major effect of the above is reviewed in the consequent sections.
2.3 Frequency Dispersion of Electric Double-Layer Capacitance
2.3.1 Expressions
Practically, Nyquist plot of an electrolyte/electrode system reveals a line deviated from the behaviour of an ideal capacitor by angle θ and shifted on the real axis by solution resistance Rs (R), as shown in Figure 2 - 2(f). This impedance result could not be described by simple elements (such as R and/or C). Both real impedance Z′ and imaginary impedance Z′′ vary with frequency. The phenomenon of frequency dependence of both real and imaginary impedance data is called “frequency dispersion”.
This frequency dispersion was attributed to capacitance dispersion, in which the
16 Chapter Two Review of Conventional Concepts of Electric Double Layers capacitance decreases generally with an increase in frequency. The frequency dispersion was investigated as Constant Phase Element (CPE) [19-35], introduced in section 2.2.2.
In the literature, different equations were presented. The expressions given by Lasia have been extensively reviewed in section 2.2.2, where Y0 and n are the characterized parameters for CPE. Brug and co-works [19] proposed similar equations to that of
-1 Lasia’s but referred Y0 to Q , and referred n to (1-α). Zoltowski [23] presented two
-1 definitions for CPE. The first was the similar to Lasia’s expression, but stated that Y0 was directly proportional to the active surface area; and the second expression was
n Z Qb j (Eq. 2 - 31)
-1 -2 n where Qb is also a constant, ohm m s .
The CPE has been considered for the following boundary conditions: limited as a capacitor for n = 1, a resistance for n = 0 (0 ≤ n ≤ 1 is considered). Therefore, CPE is quite a flexible fitting component. Unfortunately, the physical meaning of the distribution of time constant is missing.
With the participation of faradic reactions, the Nyquist plot of the electric double-layer sometimes revealed a 45 degree impedance line shooting from the point of solution resistance (from low to high frequencies). Such impedance is a consequence of semi-infinite diffusion often observed at low frequency.
2.3.2 Origin of Frequency Dispersion
The EDL capacitance is known to exhibit capacitance dispersion even without faradic reactions [36-38]. Reasons adduced for the capacitance dispersion included: dielectric loss [36], cracks on electrode and/or insulator [37], adsorption [21, 39-42], surface roughness [22, 28, 43-45], fractal dimensions of electrode surface [20, 46], cell
17 Chapter Two Review of Conventional Concepts of Electric Double Layers geometry [47], lateral charge spreading in the double-layer [48], harmonic components
[49], and heterogeneities on atomic scale [50]. Though the frequency dispersion diminished at single crystal surface [51], it still can be observed [52, 53]. It becomes significant when adsorption occurs at single crystal surface [54, 55].
2.4 Applied Electrode Potential vs. Electric Double-Layer Capacitance
2.4.1 Theoretical Prediction
As predicted by the Helmholtz theory, the capacitance, CH, expressed by (Eq. 2 - 2), depends only on dielectric constant and the separation of charge layers. In the case of the various applied electrode potential, CH should be constant to any change in the potential, as seen as CH in Figure 2 - 3(c) [7].
The variation of capacitance, CGC, with the electrode potential in the
Gouy-Chapman theory was predicted as (Eq. 2 - 11). CGC shows a minimum in the dilute electrolyte at the potential of zero charge (pzc), as illustrated in Figure 2 - 3(b)
[7]. Moreover, CGC can also be enhanced by increasing ionic concentration in the electrolyte [56].
The Stern (or Gouy-Chapman-Stern) model combined both the inner layer and diffuse layer presents a valley-like shape C vs. E curve, as shown in Figure 2 - 3(c) [7].
The depth of the valley increases with decrease in ionic concentrations. It reveals a constant capacitance as predicted by Helmholtz model and valley shape varied capacitance as predicted by Gouy-Chapman.
However, the experimental data sometimes presented unsymmetrical curves of the negative and the positive part, due to the specific adsorption of cations at the negative potentials [57].
The predicted potential dependence has been classified roughly into three types [58,
18 Chapter Two Review of Conventional Concepts of Electric Double Layers
59]; a U-shape as predicted from the Gouy-Chapman (GC) theory [2, 3], a bell-shape as predicted from the Modified Poisson-Boltzmann equation [60-62], and a camel-shape as predicted from the Gouy-Chapman-Stern model.
(a) 500
400 2 -
300 1.0 M Fcm μ /
d 0.1 M
C 200
0.01 100 M
0 150 100 50 0 -50 -10 -15 0 0 (E-Ez) / mV (b) High ionic concentration 2 - Minimum at pzc Due to CD Fcm μ
/ Low ionic d CH concentration C
+ 0 – (E-Ez) / mV
Figure 2 - 3. The capacitance dependence of applied potential for a) Helmholtz model, b) Gouy-Chapman model and c) Gouy-Chapman-Stern model
19 Chapter Two Review of Conventional Concepts of Electric Double Layers
2.4.2 Experimental Data
Electrocapillary measurement was developed to measure the surface charge density on a liquid electrode (e.g. mercury). It was based on the measurement of surface tension according to Lippmann’s method. Thus, the surface tension of the liquid in a capillary balanced with the force of gravity:
2 2rc cos rc Lhg (Eq. 2 - 32) where rc is the radius of the capillary, γ is the surface tension, θ is the contact angle measured by microscope, h is the height of the liquid column in the capillary, g is the gravitational acceleration, ρL is the density of liquid. When potential, E, is applied to the electrode, the surface tension energy is balanced with the electrical work [11].
Ad qdE (Eq. 2 - 33) where A is the surface area and q is the excess charge on the electrode surface. Then, the charge density on the electrode surface can be obtained by
M q (Eq. 2 - 34) A E
Obviously, the surface charge density is the slope of any capillary vs. potential curves.
The evaluated capacitance is given by (Eq. 2 - 1). Graham [11]carried out a series of experiments using NaF aqueous solutions on mercury electrodes and obtained the value of differential capacitance of compact layer by
1 1 1 CH C CD (Eq. 2 - 35) where C is the measured value, CD is calculated from (Eq. 2 - 11).
As for the solid electrodes, the capacitance can be evaluated accurately by EIS measurements, as introduced in section 2.2. Particularly, variation of capacitance with the electrode potential will be reviewed hereunder.
20 Chapter Two Review of Conventional Concepts of Electric Double Layers
An important property of electric double layers is the dependence of the double-layer capacitance, Cd, on dc-potential, E, at an electrode, from which excess charge on the electrode can be estimated. The potential dependence has been classified roughly into three types [58, 59]; a U-shape as predicted from the Gouy-Chapman (GC) theory [2, 3], a bell-shape as predicted from the Modified Poisson-Boltzmann equation
[60-62], and a camel-shape as predicted from the Gouy-Chapman-Stern model. In addition, some authors reported variations small enough for the classification [21,
63-69]. It is interesting to peruse conditions exhibiting these shapes.
Thirty four documents were perused, and classified shapes of Cd vs. E curves into a valley [2, 11, 21, 63, 67, 70-74], one peak [55, 58, 59, 75-79], two peaks [11, 26, 27, 55,
58, 59, 80-82], a plane [5, 21, 56, 63-69, 83, 84] and exceptions [11, 64, 78, 83, 85, 86].
The classification was based on whether a curve varied by over 30 % of the relative variation, C/Cav, where Cav is the average of the variations of Cd with E. The number of the classified publications is shown in Figure 2 - 4(A) for every available electrode.
The classified distributions depended on the electrode materials; the valley being predominant at mercury [11, 63, 70-72, 74], the one peak being predominant at platinum
[76, 78, 87], a plane being at carbon [64, 65, 67, 68, 83]. The plane was observed when non-ionic substances are adsorbed on the electrodes [11, 70], especially for carbon. The percentage of the classified shapes without distinction of electrodes is shown on the right ordinate in Figure 2 - 4(B). Although the valley-shape has been widely acknowledged as a proof of the GC theory, it accounts for only for a quarter of the number of reports. The percentage reports of computer simulation [5, 58, 59, 79] is also in Figure 2 - 4(B). The simulated results including only a few reports on the plane-shape are not in accordance with the experimental results, probably because most researchers on the simulation have paid attention to ionic interactions which depended
21 Chapter Two Review of Conventional Concepts of Electric Double Layers strongly on potential.
Although the shape is a noticeable feature, the classification depends on values of
C/Cav. It is not significant for small values of C /Cav. We classified experimental Cd, with E curves by values of C /Cav, which is shown in Figure 2 - 5. Seventy percent of the curves belong to the group with ∆C/Cav < 0.9, in Figure 2 - 5(A). Figure 2 - 5(B) shows assignment of ∆C/Cav to electrode materials. Curves with ∆C/Cav > 0.61 are nearly assigned to Hg [11, 63, 70-72, 74] and Ag [26, 75, 80]. Platinum [63, 78, 86], gold [21, 27, 73, 81] and carbon [63-65, 67, 68, 83] show small variations of ∆C/Cav. In contrast, the theoretical prediction is opposite to the above. The mean field approximation presented 11 curves [58, 59] for ∆C/Cav > 0.91, and 2 curves [59, 79] for
0.61 < ∆C/Cav < 0.90. The ion-cell model [5] on the assumption of non-overlap of ions has predicted a plane-shape curve with ∆C/Cav < 0.3 at high salt concentrations.
(A) (B)
22 Chapter Two Review of Conventional Concepts of Electric Double Layers
Figure 2 - 4. Change Classification of 34 publications about the double layer capacitances vs. potential curves into five shapes; (A) the number of publications for every available electrode material and (B) ratio of number of publications belonging to a given shape for simulation and experimental work.
(B) 0 ~ 30 % 30 ~ 60 % 60 ~ 90 % 90 % ~
Figure 2 - 5. The number of experimental capacitance vs. potential curves classified
with values of ∆C/Cav (A) without and (B) with assignment to electrode materials.
2.5 Effect of Ions on Electric Double-Layer Capacitance
2.5.1 Ionic Concentration
Concentration dependence of double-layer capacitance has seldom been discussed.
The Gouy-Chapman theory mentions that the double-layer capacitance has the
23 Chapter Two Review of Conventional Concepts of Electric Double Layers square-root dependence of ionic concentration [2, 3]. The capacitance at the nitrobenzene|water interface by Wandlowski et al. increased with the concentrations, showing the dependence of the square-root of the concentrations [88], similar to the
Gouy-Chapman theory. The ion-cell model has suggested the dependence of 1/3 powers of the concentration [5]. The empirical increase in capacitance due to increase in the ionic concentrations were also reported [11, 80].
2.5.2 Type of Ions
The size effect of ions on the capacitance was widely discussed: the capacitances were larger with a decrease in size of cations [64, 67, 89-91] and anions[51, 53, 66, 80,
84, 89, 92-97].
2.6 Contribution of Solvent to Electric Double-Layer Capacitance
2.6.1 Basic Physical Properties of Solvents
Capacitances of an electric double-layer (EDL) vary empirically with properties of solvents. An electrolyte with a high dielectric constant is preferable in designing a super-capacitor because it seems to enhance the capacitance on the basis of a parallel plate capacitance model. However, dielectric constant near the electric double-layer
(EDL) takes a value much smaller than that in the bulk [64, 98]. Morita [89] observed enhancement of the capacitance with an increase in electrolytic conductivity by mixing organic solvents. Jänes and co-workers [98-100] examined the intensity of the capacitance by varying the organic solvents, and ascribed the variations of capacitance to the solution resistance and relaxation time constant of the EDL. Kim [64] found out that the small solvent molecules can penetrate into the porous carbon electrode to enhance the capacitance, associated with a shift of potential of zero-charge.
24 Chapter Two Review of Conventional Concepts of Electric Double Layers
2.6.2 Dielectric Saturation
The dielectric saturation is an experimental result at which the applied electric field exerts less effect on the EDL than those predicted in the bulk. Born [101] presented equations for the free energy change in an ion moving from bulk to the double-layer.
Debye calculated the effective dipole moments oriented by external field under thermal equilibrium [102]. Webb [103] ascribed the dielectric saturation near an ion to orientation of solvent molecules and inductive polarization by applying the Langevin's theory [104]. Onsager [105] represented the electric field acting on a molecule as a cavity of a molecule and the external field. Kirkwood [106] approximated the dielectric saturation as the bulk value and that in the first shell of neighbouring molecules.
Dielectric constants were estimated theoretically as a function of distance from the first adsorbed ion layer [103, 107], temperature [108] and concentration of ions in aqueous solutions [107, 109]. Booth [110] evaluated dielectric constants in high electric field above 107 V/m by modifying Onsager and Kirkwood’s theory. Conway [111] calculated variations of the potential in the EDL for variable dielectric constants [102, 103, 112].
Macdonald [9, 113] compared reported values of the EDL capacitances in the light of structure of the inner layer including the dielectric saturation and lateral interaction of adsorbed molecules. Macdonald and Barlow [114] evaluated the dielectric saturation through dipoles imaged electrostatically on the electrode and the ionic solution layer.
Models of variable dielectric constants have been proposed, including the continuous variation with the electric field [115], the discrete shell model [116], and the ion surrounded by concentric spherical layers with different dielectric constants [117].
Computer simulations of variations of dielectric constants in the EDL have been carried out under several experimental conditions [118-127]. Clear understanding of the saturation is still under debate.
25 Chapter Two Review of Conventional Concepts of Electric Double Layers
Since dielectric constant is field dependent, the potential drop and the values of εr hence strongly depend on the inner-layer thickness. Monomolecular was considered as the inner layer [114, 128]: taken the diameter of water as the thickness, εr is assumed to be 4 [111] calculated from Stern’s theory by using Webb’s data [103]; whereas, εr is averaged to be 14.9 in I- aqueous solutions [114] considering the distance from the electrode surface to the centre of the first layer of adsorbed ions as inner-layer thickness.
When a distortion polarization, ε∞, and an orientation contribution, εor, compose the dielectric saturation, εinner = ε∞ + εor, the distortion dielectric constant of the water adsorbed on an electrode are obtained to be 6 [14] or 4 [129]. Mott [13] evaluated the dielectric constant of water in the inner layer region is about 15 when the field vanished, whereas, it drops to a constant value within 3 to 5 for strong fields, by use of Graham’s data [11].
2.7 Preferable Electrode Geometry in Electrochemical Measurements
An electrode frequently used for measurements of double-layer capacitances is a disk, the circumference of which is coated with an insulator. However, boundaries between the exposed electrode and the insulating wall provide insufficient shielding yields irreproducible floating-capacitance, depending on fabrication of electrodes, periods of use of electrodes, and dc-potential. A technique of suppressing the floating capacitance is to use a wire electrode without insulator. The area of the exposed surface is determined by the immersion length of the electrode.
26 Chapter Three A Parallel Resistance of Electric Double-Layer Capacitance
Chapter Three
A Parallel Resistance of Electric Double-Layer Capacitance
3.1 Aim
The constant phase element as an expression of frequency-dependent EDL impedance data presents the behaviours of EDL by the two parameters: n and Y0. The former stands for the dispersion level in the scale from 0 to 1. The latter is a constant, and inversely proportional to the surface area. The impedance of measured liquid/solid system is given by
1 1 1 Z Rs (Eq. 3 - 1) Z F Z CPE where Rs is the ohmic solution resistance in series with the parallel combination of the faradic impedance ZF of the electrode reaction and the impedance of CPE, ZCPE. The
1 n expression of the impedance of CPE was given by (Eq. 2 - 24) [ ZCPE Y0 ( j) ].
When no faradic reaction is present (ZF→∞), (Eq. 3 - 1) is reduced to
1 n Z Rs Y0 ( jω ) (Eq. 3 - 2) Z CPE
The in-phase component Z1 and the out-phase Z2 are written as:
α (Eq. 3 - 3) Z1 Rs Y0ω cos(nπ / 2)
α Z 2 Y0ω sin(nπ / 2) (Eq. 3 - 4)
Taking the ratio of (Eq. 3 - 3) and (Eq. 3 - 4) leads to
Z 2 tan( / 2) (Eq. 3 - 5) Z1 Rs
27 Chapter Three A Parallel Resistance of Electric Double-Layer Capacitance
When Rs is much smaller than Z1, the Nyquist plot for polarized interfaces would fall on a proportional line, as compared to the CPE in Figure 2 - 2(e). The proportionality in wide frequency domains has been found at the aluminium electrodes
[29], at Au(210) electrode in potassium halides solutions [55], at gold electrode in sulphuric acid [81], at platinum electrode in potassium chloride solution [130], at
+ conducting polymer-coated electrode [131, 132], and at Nb2O5 electrode in Li solution
11 [22]. Then the n value obtained from the proportionality will be determined unequivocally, independent of the frequency. The behaviours of the capacitance have also been studied by slope of the plot of logarithm of out-phase vs. log(ω) [29, 32, 50,
81, 133]. The plot of the in-phase components, i.e. log(Z1-Rs) vs. log(ω), would exhibit a similar variation. However, few authors have paid attention to the latter dependence
[134, 135], probably because the in-phase component is less dependent on the frequency than the out-phase.
Considering the inability of the CPE to present the physical meaning of the distributed time constant, it is necessary to re-examine the properties of CPE. In this chapter, the behaviour of electric double layers at polarized interfaces in KCl solutions is re-examined in order to evaluate the properties of the constant phase element (CPE).
Specific attention was paid to frequency dependence of both the capacitance and the resistance. In order to evaluate solution resistance accurately, a rectangular cell involving parallel electrodes were applied.
3.2 Experimental
Commercially obtained potassium chloride (from NACALAI TESQUE, INC.,
Kyoto, Japan) was used as to prepare aqueous solutions with ion-exchanged distilled water. Potassium hexacyanoferrate (III), K3Fe(CN)6, and potassium hexacyanoferrate (II)
28 Chapter Three A Parallel Resistance of Electric Double-Layer Capacitance
trihydrate, K4Fe(CN)6∙3H2O, were also obtained from the same company. All the chemicals were of analytical grade.
L d
Figure 3 - 1. Illustration of the cell and a pair of the wire electrodes, where L is the effective length of the wire electrodes immersed in the solution, and d is the center-to-center distance of the two wires.
Working and counter electrodes were platinum (Pt) wires (99.95 % up) and 0.1 mm in diameter (Designation No.: 02630296 KOA99), provide by TANAKA KIKINZOKU
KOUGYOU KABUSHIKIGAISHA, Japan. The Pt wire was wound around a tungsten wire shielded in a glass capillary tube, and bonded by conducting paste. The exposed portion of Pt wire was then soaked in acetone and rinsed with ion exchanged-distilled water. One of the Pt wire electrode was fixed vertically with a beam, and the other was mounted on an optical x-y positioner so that the distance, d, between the two wires was adjusted, as illustrated in Figure 3 - 1. The positioner was placed on a jack so that both tips of the wires were adjusted to be on the same level. The active length, L, of the electrodes was controlled with a lift for the aqueous solution. Values of d and L were read through an optical microscope. A typical value of L was 10 mm. Aqueous solution rose up from the aqueous surface along the wire-surface by the surface tension. The rise
29 Chapter Three A Parallel Resistance of Electric Double-Layer Capacitance was at most 0.2 mm from an optical microscope, indicating an error on the surface area of the electrode which was less than 2 %.
0.15
(a) F
0.10 (b) / C (c) 0.05
0.00 0 1 2 3 4 5 log(f / Hz)
Figure 3 - 2. Examination of the delay of the potentiostat by film capacitor (0.1 μF) in serious with a carbon resistance (a) 1 kΩ, (b) 5 kΩ and (c) 10 kΩ.
The potentiostat was Compactstat (Ivium, Netherlands), equipped with a lock-in amplifier. Applied alternating voltage was 10 mV in amplitude. Solution was deaerated by nitrogen gas for 15 min before electrochemical measurements.
Delay of the potentiostat was examined by a series combination of a carbon resistance (1, 5 and 10 k) and a film capacitor (0.1 F) for a frequency range from 1
Hz to 10 kHz, as shown in Figure 3 - 2. The capacitance value was obtained by C =
-1/2∙πfZ2. No abnormality was observed as far as |Z2|/Z1 > 0.04. For frequencies larger than |Z2|/Z1 < 0.04, |Z2| values were over estimated by a few percentages. Most experimental conditions of the double-layer measurements were in the domain of |Z2|/Z1
> 0.04.
30 Chapter Three A Parallel Resistance of Electric Double-Layer Capacitance
3.3 Results and Discussion
3.3.1 Nyquist Plot
9 (c) A (b) 0 /
1 I -9 A (a) -0.3 0.0 0.3 / I 0 E / V vs. Pt (d) 20 / nF I -1 10 0 -0.4 -0.2 0.0 0.2 0.4 0.00 0.06 0.12 E / V vs. Pt / Vs-1 Figure 3 - 3. Cyclic voltammograms obtained at the scan rate of 10 mVs-1. In (a)
0.5 M KCl solution and (b) 0.2 mM K3Fe(CN)6 + 0.2 mM K4Fe(CN)6 at the two parallel Pt wires 0.1 mm in diameter 10 mm in length with the distances, d = 2 mm. (c) is the magnification of (a), (d) is the plot of the current at 0.02 V in (a) against scan rate.
Figure 3 - 3 shows cyclic voltammograms of 0.5 M (= mol dm-3) KCl solution with
3- 4- and without adding redox species (0.2 mM Fe(CN)6 + 0.2 mM Fe(CN)6 ) at the two-wire electrodes in nitrogen atmosphere. The voltammograms were symmetric with respect to the origin because the two-electrode system was geometrical symmetry.
Currents at potential |E| < 0.14 V were proportional to the potential scan rates in the absence of redox species, in Figure 3 - 3(c) and (d). The current must be capacitive current. Voltammograms in the solution with redox species increased by 100 times owing to the redox reaction, as shown in Figure 3 - 3(b). Even if any Faradaic current is included in the voltammogram in the KCl solution (Figure 3 - 3(a) and (c)) as impurities of the solution, the concentration of impurities should be less than 2 M.
31 Chapter Three A Parallel Resistance of Electric Double-Layer Capacitance
60 (A)
40 / k 2
- Z - 20
0 0 1 2 3 4 5 Z1 / k
0.4 (B) / k 2 Z
- Rs,N
0 0 0.05 Z1 / k Figure 3 - 4. Nyquist plot at the two parallel Pt wires, 0.1 mm in diameter, 10 mm in length with the distances, d = 2 mm in 0.5 M KCl aqueous solution obtained for (A) low frequencies at (x mark) 5 Hz and (+ mark) 51 Hz, and (B) high frequencies at (x mark)
1kHz and (+ mark) 10 kHz. Rs,N is the value extrapolated for f .
Figure 3 - 4(A) shows the Nyquist plot at the parallel Pt wire electrodes in 0.5 M
KCl solution. The common Nyquist plot was well reproduced which is the benefit from the unshielded electrodes. Nyquist plots exhibited no variation regardless of whether electrodes were immersed in nitrohydrochloric acid before ac-measurements or not. The linear plot in Figure 3 - 4(A) is different from a conventional shape of the combination of a semicircle and a line. However, straight line-like plots have been reported [22, 29,
32 Chapter Three A Parallel Resistance of Electric Double-Layer Capacitance
55, 130-132, 136, 137] for work on the CPE behaviour. If Z1 represents solution resistance, a change in frequency should vary only Z2 values, retaining Z1 values. Then the plot must fall on a vertical line. The observed variation of Z1 with the frequency indicates that Z1 should include contributions of the double-layer capacity. This corresponds to the CPE behaviour.
When (Eq. 3 - 4) is applied to the slope in Figure 3 - 4 on the assumption of neglected solution resistance, n = 0.95 was obtained for frequencies more than 50 Hz.
Since n = 1 and 0 correspond to only an ideal capacitor and only an ideal resistor, respectively, the impedance at the twin platinum electrode is close to a capacitor. In order to determine n more accurately, the study evaluated the solution resistance from the extrapolation of Z1 to zero Z2 for infinite frequency [7], denoted by Rs,N, as shown in
Figure 3 - 4(B).
3.3.2 Solution Resistance
Z1 seems to include information of capacitance, and hence Rs,N may not stand accurately for the solution resistance. A technique of estimating how much Rs,N represents the solution resistance is to examine dependence of Rs,N on geometry of the electrodes. Solution resistance between two parallel wires is not expressed by a simple proportionality to d. Analytical expressions were derived for the resistance between thin parallel wires.
A model of the resistance is a pair of parallel cylindrical long wires a in radius, separated by the distance, d, between the two centres of the cylinders. Voltage V is applied between the two wires immersed in solution with resistivity, . The cross-section of the wires (circles) was placed on the coordinate of Figure 3 - 5. If current flows out isotropically from a circle located at the origin and flow into the
33 Chapter Three A Parallel Resistance of Electric Double-Layer Capacitance infinity two-dimensionally, the current density vector, j, at the radial distance vector, r, is satisfied with 2rj = constant, where the dot denotes the inner product. When the length of the wire is L, the total current I is given by 2ajL. Consequently, we have
2rj L= I. When the vector of the current density, j1, flows out from the circle at x =
-d/2 into the infinity independently from the circle at x = d/2, it is expressed by
y
r a x O -d/2 d/2
Figure 3 - 5. Two-dimensional model for potential distribution at the two parallel cylinders for d/a = 20, and computed potential distribution by the finite element method in the potential field discretized by triangles.
2πr d / 2 j1L I (Eq. 3 - 6)
Similarly the vector of the current density, j2, flowing out from the infinity enters the circle at x = d/2 independently from the circle at x = - d/2, it is given by
2πr d / 2 j2 L I (Eq. 3 - 7)
Taking the inner product of (Eq. 3 - 6) and (Eq. 3 - 7) with r+d/2, r-d/2, respectively, yields
34 Chapter Three A Parallel Resistance of Electric Double-Layer Capacitance
( r d / 2 )I ( r d / 2 )I j , j 1 2 2 2 2π r d / 2 L 2π r d / 2 L (Eq. 3 - 8)
From (Eq. 3 - 8), the total current density, j = j1 + j2, is given by
I ( r d / 2 ) ( r d / 2 ) j 2πL 2 2 r d / 2 r d / 2 (Eq. 3 - 9)
Electric potential in solution, , is correlated with j through Ohm's law, j =
-(1/)(d/dr). Applying it to (Eq. 3 - 9) and integrating the resulting equation along a current line yields the potential difference, V, between the two electrodes. If we select a current line on the x-axis, and carry out the integration, we obtain
d / 2a I d V jdx ln 1 d / 2a πL a (Eq. 3 - 10)
Then the solution resistance is expressed by
d Rs,G V / I ln 1 πL a (Eq. 3 - 11)
Since the integral contour in (Eq. 3 - 10) was taken to be the shortest path, the resistance should be underestimated. It is important to note that the resistance is proportional to ln(d/a-1) rather than d.
The underestimation of (Eq. 3 - 11) was corrected by means of the numerical computation of the two-dimensional Laplace equation for . The numerical solution was obtained by the finite element method. The boundary conditions were zero flux on all the boundaries except on the electrodes. Figure 3 - 5 shows an example of the divided triangular elements. The software was home-made. Potential profiles were computed, and shown as gradient colours in Figure 3 - 5. Fluxes evaluated on the electrode from the profiles were summed to yield the total current, and hence the conductance. The
35 Chapter Three A Parallel Resistance of Electric Double-Layer Capacitance
computed dimensionless resistance, Rs,GL/, was plotted against ln(d/a -1) in Figure 3 -
6. They were slightly larger than (Eq. 3 - 11) (solid line in Figure 3 - 6), as was predicted. They fell on the line 1.25 times as large as the line of (Eq. 3 - 11) for ln(d/a-1) > 1.5. Then the resistance for monovalent electrolyte with concentration c is given by
ln(d / a 1) log(d / a 1) Rs,G 1.25 0.916 πLc( ) Lc( ) (Eq. 3 - 12) where + and - are molar conductivity of the mono-cation and the mono-anion, respectively.
1.5
1 RL/ 0.5
0 0 1 2 3 4 ln(d/a - 1) Figure 3 - 6. Dependence of the dimensionless resistance, RL/, obtained by (circles) the finite element method, by (solid line) (Eq. 3 - 11) on ln(d/a – 1), by (dashed line) (Eq. 3 - 12) for the fitting.
36 Chapter Three A Parallel Resistance of Electric Double-Layer Capacitance
(a)
1 / k s R 0.5 (b)
(c) 0 0 1 2 3 log(d/a - 1)
Figure 3 - 7. Dependence of Rs,N (= Z1 (f ) ) determined from the Nyquist plots for concentrations of KCl (a) 0.01, (b) 0.03, and (c) 0.1 M on ln(d/a – 1). The lines are from (Eq. 3 - 12). The dotted line is the recessed line for (a)
(a) 100 / 1
Z (b) 50 (c) (d)
0 0 1 2 log(d / a - 1)
Figure 3 - 8. Plots of Z1 with log(d/a - 1) for frequencies of (a) 100 Hz, (b) 1000 Hz and (c) 5000 Hz in 0,5 M LCl. Line (d) is from (Eq. 3 - 12).
37 Chapter Three A Parallel Resistance of Electric Double-Layer Capacitance
Values of Rs,N, were plotted against log(d/a - 1) for some concentrations of KCl in
Figure 3 - 7. Although they are approximately linear to log(d/a - 1), all the plots for concentrations from 0.001 to 3 M had positive values of intercepts. Furthermore, they are slightly smaller than the lines obtained from (Eq. 3 - 12) for all the concentrations.
The smaller shift will be explained in terms of the frequency dependence of the double-layer capacity in section 3.3.3.
Figure 3 - 8 shows dependence of Z1 at some frequencies on log(d/a - 1). The plot for a given frequency fell on a line with the common slope, which is similar to the value
-3 2 -1 of the slope in (Eq. 3 - 12) (Figure 3 - 8(d)), 12 for + = 7.3510 S m mol and -
= 7.6310-3 S m2 mol-1. In contrast, the intercept at log(d/a - 1) = 0, i.e. d = 2a, means the resistance that would appear if the two electrodes were to come in contact each other.
This resistance should be located at the interface or included in the double-layer. It increases with a decrease in frequencies.
(A) Measured (B) Model
Cp Cp RS Z1 Z2 Cd Cd
Rp Rp
Figure 3 - 9. Equivalent circuits of (A) series combination (Z1 + iZ2), (B) geometrical assignment of Rs and Cd, and (C) really acting combination of Rs, Cp and Rp.
3.3.3 Parallel Equivalent Circuit
The above result indicates that the observed double-layer capacitance, Cd, includes not only the double-layer capacitance, Cp, but also the resistance, Rp, in a parallel combination of the equivalent circuit, as illustrated in Figure 3 - 9(B). Two parallel
38 Chapter Three A Parallel Resistance of Electric Double-Layer Capacitance
combinations of Cp and Rp, mean two wire electrodes. The measured quantities, Z1 and
Z2 in Figure 3 - 9(B), do not directly correspond to the solution resistance, Rs, and Cd.
The two equivalent circuits can be converted through
2 Z1 iZ 2 Rs (Eq. 3 - 13) 1/ Rp iCp
Equating the real parts on the both hand sides yields
2 2Rp 2Cp Rp Z R , Z 1 s 2 2 2 (Eq. 3 - 14) 1 (Cp Rp ) 1 (Cp Rp )
Rp and Cp are extracted in the following forms;
Z R 2 Z 2 2Z R 1 s 2 , C 2 p 2 Z R p 2 2 (Eq. 3 - 15) 1 s Z1 Rs Z 2
6 1
} 0.8 ) 5 F
/ 0.6 p R /
(a) p 4 0.4 (b) C log( { 0.2 3 (c) 0 100 200 300 dd/a / a
Figure 3 - 10. Variations of Rp and Cp in the parallel equivalent circuit with d/a for frequencies of (a) 100, (b) 1000 and (c) 3000 Hz in 0.5 M KCl solution for L = 10 mm.
Evaluations of Rp and Cp by substituting Rs,N for Rs from Z1 and Z2 at various values of d/a were done. Figure 3 - 10 shows dependence of log(Rp) and Cp for some values of frequency. Values of Rp and Cp were independent of d/a, indicating that they should
39 Chapter Three A Parallel Resistance of Electric Double-Layer Capacitance
6
5 )
/ 4 p R 3
log( 2
1 1 2 3 4 5 log(f / Hz)
Figure 3 - 11. Logarithmic dependence of Rp averaged for d/a on f at [KCl] = (crosses) 0.01 M, (circles) 0.1 M, and (triangles) 0.5 M KCl. The slope of the line is -1.
1.2 F
1 / p
C 0.8 0.6 0.4 0.2 0 1 2 3 4 5 log(f / Hz)
Figure 3 - 12. Variation of Cp averaged for d/a with f at [KCl] = (crosses) 0.01 M, (circles) 0.1 M, and (triangles) 0.5 M KCl.
40 Chapter Three A Parallel Resistance of Electric Double-Layer Capacitance have interfacial properties without including any information of d.
Values of Rp and Cp averaged over d/a are plotted against the frequency in Figure 3
- 11 and Figure 3 - 12, respectively. The logarithm of Rp shows a linear relation to log(f) for f < 104.5 Hz with a slope -1. The plots varied negligibly with concentrations of KCl in the domain from 0.005 M to 3 M. The empirical equation for Rp is given by
6 Rp = 1.810 / f (Eq. 3 - 16) with a unit in . In contrast, the averaged values of Cp decrease linearly with the logarithms of the frequency. This type of the decrease resembles that of electrolyte capacitors [138, 139]. The slope and the intercept were almost independent of the concentration of KCl, c. Letting the concentration variation of the intercept (at 1 Hz) be
(Cp)f=1, we can express empirically the dependence of Cp on log(f) as
-7 Cp = (Cp)f=1 – 1.2010 log(f) (Eq. 3 - 17) where the unit of Cp is Farad.
2 F /
= 1 = 1 f ) p (b) C ( (a)
0 -4 -3 -2 -1 0 1 log(c / M)
Figure 3 - 13. Variation of (Cp)f=1 with concentration of KCl by (circles) experiments, (a) the Gouy-Chapman's theory, and (a) the ion-cell model.
41 Chapter Three A Parallel Resistance of Electric Double-Layer Capacitance
Figure 3 - 13 shows the variation of (Cp)f=1 with c. Double-layer capacities are predicted intuitively to increase with an increase in concentrations of ions because ions are thought to play a significant role in capacitances. The Gouy-Chapman theory
1/2 mentions that Cd is proportional to c (Figure 3 - 13(a)), whereas the ion-cell model predicts the proportionality to c1/3 (Figure 3 - 13(b)) [5]. However, the present experimental data (Figure 3 - 13) shows concentration-independence of (Cp)f=1 within
-2 experimental errors. The average value of (Cp)f=1 is (1.10 0.04) F or (34 2) F cm .
The independence indicates whether the ions (K+ and Cl-) must be accumulated on the electrode by the electric tension of the ions to the electrode [5] or the water molecules play a significant role in the capacitance, like Stern's model. The independence has been observed for polarized mercury electrodes by Grahame [140].
We explain the experimental result, Rs,N < Rs,G, in Figure 3 - 7. Taking the ratio in
(Eq. 3 - 14) is equal to –Z2/(Z1-Rs) = CpRp. The solution resistance geometrically calculated corresponds to a value determined by a direct current, and is approximated as the value at the low frequency, 1 Hz. Then we have:
Since Cp (Cp)f=1 at low frequency (< 10 Hz), the ratio is approximated to
6 Z 2 /Z1 Rs,G Cp Rp 2π(310 )(Cp ) f 1 (Eq. 3 - 18)
In contrast, the solution resistance by the Nyquist plot for large frequency, e.g. fN
=105 Hz, makes the ratio to yield
6 7 Z 2 /Z1 Rs,N Cp Rp 2π(310 )(Cp ) f 1 1.210 f N (Eq. 3 - 19)
Comparison of (Eq. 3 - 18) with (Eq. 3 - 19) yields obviously
Z 2 /Z1 Rs,G Z 2 /Z1 Rs,N (Eq. 3 - 20)
42 Chapter Three A Parallel Resistance of Electric Double-Layer Capacitance
from which we obtain Rs,N < Rs,G. Consequently, solution resistance evaluated from the
Nyquist plot is smaller by the decrease in the capacitance than the geometrically evaluated solution resistance. The inequality is caused by the decrease in Cp with an increase in frequency.
Chloride is adsorbed specifically on mercury electrodes, depending on electrode potential [141]. It would be desirable to discuss the possibility of specific adsorption on the platinum electrode. The potential control in this study was, however, made in the two electrode system without any reference electrode, because attention was paid to accurate determination of solution resistance. Therefore, it is not prudent to discuss the effects of potential-depending adsorption on Cp and Rp from the present experimental results.
3.3.4 Verification of Equivalent Circuit
When alternating voltage, V0exp(it), at angular velocity is applied to an ideal capacity, Cideal, the responding current is expressed by I = d(CidealV)/dt = iCidealV. Then the current exhibits a phase shift of /2. On the basis of frequency-dependence, capacitance, C, may provide not only the phase shift of /2 but also time-variation of the capacity because of
I iC (dC / dt)V (Eq. 3 - 21)
Here, the time is approximately equivalent to 1/, we have d/dt = -1/t2. Replacing
C by Cp, inserting (Eq. 3 - 17) into (Eq. 3 - 21) and carrying out the integration yields
7 7 I iCp (1.20 10 / 2.3)V iCp 3.310 f V (Eq. 3 - 22)
This equation suggests the parallel equivalent circuit composed of Cd and the frequency-dependent admittance, 3.310-7f. The latter can be written as a resistance
43 Chapter Three A Parallel Resistance of Electric Double-Layer Capacitance
6 310 /f, which should be identical to Rp by definition in Figure 3 - 9(A). Indeed, the parallel resistance is inversely proportional to f, as shown in (Eq. 3 - 16). The proportionality constant is not far from that in (Eq. 3 - 16). The difference may be ascribed to the assumption of t = 1/2f. Since dC/dt in (Eq. 3 - 21) is an in-phase component, it is the non-zero values of dC/dt that provokes the parallel resistance as if
Rp might exist in the double-layer.
In summary, the total current of EDL could be expressed as I = dq/dt. At the presence of frequency dependent capacitance, the time derivative of q = CV is given by
dC()V (t) dV (t) C() d I C() V (t) (Eq. 3 - 23) dt dt dt
The first term on the right hand is the out phase, C, defined by Cp. In contrast, the second term is in phase, a resistive component Rp. They were in a parallel combination as illustrated in the equivalent circuit in Figure 3 - 9(B). The frequency-dependent capacitance is responsible for the parallel resistance, which does not exist physically, but can be observed in impedance measurement. So far as the double-layer impedance is composed of an ideal capacitance, the parallel resistance cannot be taken into account in the analysis. It cannot be incorporated into the analysis including the
Nernst-Planck-Poisson equations [142].
3.3.5 Difference between Parallel Circuit and Conventional Series Circuit
The conventional equivalent circuit is a series combination of solution resistance,
Rsd, and double-layer capacitance, Cd. The former includes the parallel resistance Rp.
Relations between the parallel circuit and the conventional circuit can be given by
Rp R R sd s 2 (Eq. 3 - 24) 1 Cp Rp
44 Chapter Three A Parallel Resistance of Electric Double-Layer Capacitance
1 C C d p 2 2 (Eq. 3 - 25) C p Rp
The conventional solution resistance Rsd is larger than the true solution resistance by
2 Rp/[1+(CpRp) ]. Since Rp is inversely proportional to , Rsd is approximated as Rs + k/ for a constant k. On the other hand, the conventional double-layer capacitance
2 2 contains 1/ CpRp . Since Rp is independent of , Cd may vary only slightly with . If
Rp is independent of dc-potential, Cd should have the dc-potential dependence similar to
Cp, according to (Eq. 3 - 25). So far as the double-layer impedance is composed of an ideal capacitance, the parallel resistance cannot be taken into account in the analysis. It cannot be incorporated into the analysis including the Nernst-Planck-Poisson equation
[142], either.
3.4 Conclusion
The behaviour of electric double layers at polarized interfaces in electrolyte solutions > 0.1 mM is summarized as follows:
A) The Nyquist plot is approximately a straight line because Z1 has roughly the inverse
proportion to frequency.
B) Cd is represented by a parallel combination of a capacitance and a resistance.
C) The resistance parallel to the double-layer is inversely proportional to frequency.
D) The origin of Rp is ascribed to the linear decrease in Cp on the scale of logarithmic
frequency.
E) Values of Cp are independent of concentration of salt.
F) The solution resistance by the Nyquist plot at high frequency is smaller than the
45 Chapter Three A Parallel Resistance of Electric Double-Layer Capacitance
geometrically evaluated resistance.
Without frequency-dependence in Cd, Rp might be infinite. Consequently, the electric properties of the double-layer could be represented by a series combination of an ideal resistance and an ideal capacitance. The frequency-dependence of Z1 is the apparent resistance associated with impedance measurements. If double-layer capacitances were to be determined by direct current measurements, it might include no resistance component. Application of sinusoidal voltage to the double-layer with non-zero values of dC/dt provides the parallel resistance unexpectedly. In other words, measurements of double-layer capacitance belong to breakdown measurements.
46 Chapter Four Double-Layer Capacitance – Independence of dc-Potential
Chapter Four
Double-Layer Capacitance – Independence of dc-Potential
4.1 Aim
The experimental results so far reported show that a) the shape of the resulting curves depends on electrode materials, valley being predominant at Hg, one peak being predominant at Pt, and plane being predominant at carbon; b) the experimental results reveal mostly small values of relative variation, ∆C/Cav (~0.9) and the bigger variations are nearly assigned to Hg and Ag electrodes. In chapter 3, the invariance of differential capacitance with concentration of chloride ions (0.05 ~ 3 M) indicates that 1) the chloride ions were specifically adsorbed on the platinum electrode surface; or 2) the parallel capacitance was caused by solvent rather than ions. Obviously, the evaluated capacitance showed no Gouy-Chapman prediction at low ionic concentrations; therefore, the capacitance was caused by solvent molecules rather than ions.
Generally, it is the believed that F- is non-specifically adsorbed on mercury electrodes whereas Cl-, Br-, and I- are specifically adsorbed [11, 71, 143]. This is inconsistent with the observed result in this study. The study therefore seeks to examine whether this well-known knowledge of that at mercury is valid at platinum electrodes in the polarized potential domain, taking into account the frequency dispersion of the capacitance. The double-layer capacitance used here was the net capacitance without parallel resistance of the double-layer.
47 Chapter Four Double-Layer Capacitance – Independence of dc-Potential
4.2 Experimental
All chemicals were of analytical grade. Haloids potassium salts (KF, KCl, KBr, KI) were used as supporting electrolyte and were commercially obtained from Sigama
Aldrich (KF), Nacalai Tesque, Inc., Kyoto, Japan (KCl), Kanto Chemical Co., Inc. (KBr) and Yoneyama Chemical Industies, Ltd, Osaka, Japan (KI), respectively. Aqueous solutions were prepared with home prepared ion-exchanged distilled water (18 MΩ cm).
HF solutions were stored in polyethylene (PE) containers.
RE WE
CE
Electrolyte
Positioner
Figure 4 - 1. Experimental set up used for evaluating variations of capacitance with applied electrode potential.
The working electrodes used in this study were platinum wires of measurements 0.1 mm in diameter, and 10 mm in length were immersed into a solution controlled by a positioner. Ag|AgCl (in saturated KCl) and a platinum coil (0.5 mm in diameter and 120 mm in length) were used as the reference and a counter electrode, respectively. The platinum wire and coil were cleaned in accordance with section 3.2. A salt bridge including the sample solution was set between the reference electrode and the bulk
48 Chapter Four Double-Layer Capacitance – Independence of dc-Potential solution via a glass filter, as shown in Figure 4 - 1. All electrolytes were deaerated with nitrogen gas for 20 min before the electrochemical experiments.
Applied alternating voltage was 10 mV in amplitude through potentiostat (same as in section 3.2). The dc-potential domain on which the ac-voltage was superimposed was selected on the basis of 30% enhancement of the cyclic voltammetric current from the linear increment of the voltammogram. It ranged from 0.0 to 0.4 V vs. Ag|AgCl.
In-between every impedance measurement, cyclic voltammetry was made in order to confirm consistency of the voltammograms before and after the ac-impedance measurements. Delay of the potentiostat was checked as introduced in section 3.2.
4.3 Results and discussion
4.3.1 Nyquist Plot
Since the surface area of the coil electrode used is much bigger (over 100 times) than that of the platinum-wire working electrode, the equivalent circuit and the expressions are shown in Figure 4 - 2 and (Eq. 4 - 1).
Cp RS Cd Figure 4 - 2. Equivalent circuit for platinum wire working electrode | electrolyte | coil counter electrode system Rp
Z R 2 Z 2 Z R 1 s 2 , C 2 p p 2 2 (Eq. 4 - 1) Z1 Rs Z1 Rs Z 2
Solution resistance, Rs, was evaluated from Nyquist plot, when f → ∞, or Z2 → 0, as shown in Figure 4 - 3. The plots are linear rather than semi-circles, because the
49 Chapter Four Double-Layer Capacitance – Independence of dc-Potential
impedance contains no Faradic process. Values of Cp and Rp were determined by use of
(Eq. 4 - 1).
Figure 4 - 3. Nyquist plots for 0.01 M (M = mol dm-3) KF and KBr solutions at the two wire electrodes with 7 mm separation
4.3.2 Variation of Double-Layer Capacitance with both Logarithmic ac- Frequency
and Ionic Concentration
Two platinum wires of equal lengths were used to obtain dependence of Cp and Rp on different kinds of halides and their concentrations without a reference electrode.
Reasons for using the two-electrode system are (i) to keep ionic concentrations in the cell against leakage of ions from the reference electrode, and (ii) to retain the solution resistance in a given arrangement of electrodes geometrically constant in order to detect
Rp. The equivalent circuit is given by Figure 3 - 9(B). Figure 4 - 4(A)-(D) show the frequency dependence of Cp at different concentrations of KF, KCl, KBr and KI, respectively, All the values of Cp were linear to log(f), i.e. CP = (CP)f=1 - k1 log(f), regardless of different kinds of halides and their concentrations. The ratio, Cd/Cp, ranged from 1.01 to 1.03 in the present frequency domain. Since Cd is numerically similar to Cp,
50 Chapter Four Double-Layer Capacitance – Independence of dc-Potential it has frequency dependence similar to Figure 4 - 4. The frequency dispersion is thought to be relevant to reformation of double-layer by the solvent dipoles at electrodes [144,
145], movements of ions along both lateral and vertical directions and surface roughness[48, 55, 146]. Reason for the frequency dispersion is not discussed here.
1.5 1.5 (A) (B) F F
1 1 / / p p C C
0.5 0.5
0 0 1 2 3 4 1 2 3 4 log( f / s-1) log( f / s-1) 1.5 1.5 (C) (D) F F
1 1 / / p p C C
0.5 0.5
0 0 1 2 3 4 1 2 3 4 log( f / s-1) log( f / s-1)
Figure 4 - 4. Variations of the parallel capacitances with logarithmic frequency for (A) KF, (B) KCl, (C) KBr and (D) KI at concentrations of (open circles) 2 M, (triangle) 0.5
M, (full circles) 0.1 M and (squares) 0.01 M. Values of Cp were obtained at two electrodes without any reference electrode. M = mol∙dm-3
Values of the slope in Figure 4 - 4 were almost common, -0.99 0.04 F, to four kinds of halides in the domain of the concentration. The intercepts of the linear variations in Figure 4 - 4 decreased slightly with a decrease in the concentrations.
Parallel capacitance at f = 1 Hz was chosen as a representative of the capacitance for the
51 Chapter Four Double-Layer Capacitance – Independence of dc-Potential
discussion of dependence on concentrations and kinds of ions. Values of (Cp)f=1 are plotted against concentrations in Figure 4 - 5. Use of low concentrations increases both
Z1 and Rs in (Eq. 4 - 1), and hence the difference Z1-Rs includes large errors. Accurate evaluation of Rs at low concentrations requires high frequency measurements, at which
|Z2| was overestimated by a property of the potentiostat to yield underestimation of Rs.
-2 The average value of (Cp)f=1 in Figure 4 - 5 is (1.2 0.1) F or (38 3) F cm .
1.5 F / =1
f 1 ) p C ( 0.5
0 -3 -2 -1 0 1 log( c / M)
Figure 4 - 5. Dependence of Cp at f = 1 Hz on concentrations of (open circles) KF, (triangles) KCl, (full circles) KBr and (squares) KI
Concentration dependence of double-layer capacitance is seldom discussed. The GC theory mentions that the double-layer capacitance has the square-root dependence of ionic concentration [2, 3]. The capacitance at the nitrobenzene|water interface by
Wandlowski et al. increased with the concentrations, showing the dependence of the square-root of the concentrations [88], similar to the GC theory. The ion-cell model suggests the dependence of 1/3 powers of the concentration [5]. The almost constant
- - - - values of (CP)f=1 common to F , Cl , Br and I indicate that the value of double-layer capacitance should be controlled by solvent molecules rather than ions. This prediction
52 Chapter Four Double-Layer Capacitance – Independence of dc-Potential will be proved in latter discussion. The common values have been already reported [84].
5 ) /
p 4 R
3 log(
2
1 1 2 3 4 5 log( f / s-1) Figure 4 - 6. Variations of the parallel resistance with frequency on the logarithmic scale for (circles) 0.5 M KF and (triangles) 0.5 M KCl.
Figure 4 - 6 shows the variation of the parallel resistance with frequency for KF and KCl. The plots for KBr and KI overlapped with those for KF and KCl. The plots were independent of the concentration range of 0.01 to 2 M for the four halides. Since the slope of the lines is -1, the resistance would be inversely proportional to the frequency. The frequency dependence is caused by Cp/t in (Eq. 3 - 23) through the linear variation of Cp with logf. Slopes of the linearity were common to four halides.
Therefore, the frequency dependence of RP would be common to the halides.
4.3.3 Invariant Double-Layer Capacitance to dc-Potential and Type of Ions
The three-electrode system at the Pt wire working electrode was used in order to obtain variations of the capacitance with dc-potential. Since Cp and Rp were confirmed as independent of ionic concentrations, fluctuation of ionic concentration by the reference electrode had a negligible effect on Cp and Rp. The impedance at the Pt wire electrode in KF, KCl, KBr and KI solutions at concentrations from 1 mM to 1 M were
53 Chapter Four Double-Layer Capacitance – Independence of dc-Potential obtained at ac voltage superimposed on 0.2 V vs. Ag|AgCl. The impedance analysis was made using the model in Figure 4 - 2. It was confirmed that all the values of Rp and Cp were very close to the data obtained at the two electrode system (model (C)). In order to vary the electrode area, changes were made to the immersed length of the Pt wire from
1 to 11 mm by 1 mm per-step. Values of Cp of KCl were proportional to the area of the electrode, as predicted [147]. In contrast, those of Rp were inversely proportional the area, as shown in Figure 4 - 7. The inverse proportionality results from the combination of the proportionality of Cp to the area and Rp = (dt/d)/(Cp/). Therefore Rp had the property of the interface rather than solution.
50 / k p R
0 0 1 2 3 A-1 / mm-2
Figure 4 - 7. Dependence of Rp of 1.0 M KCl on the area of the electrode at f = (open circles) 320, (triangles) 541, (squares) 1082 and (full circles) 2475 Hz.
Figure 4 - 8 show the plots of (Cp)f=1 in 0.1 M KF, KCl and KBr solutions against the dc potential, E, exhibiting independence of the potential. The negligible variance of
Cd vs. E curve was also found in halide ionic solutions [69]. The plots belong actually to the planar shape with Cp/Cp,av ≈ 0.1. The planar shape implies negligible contribution of the electrostatic interaction between ions and the dc-electric field to the capacitance
54 Chapter Four Double-Layer Capacitance – Independence of dc-Potential
[64, 84], which does not satisfied the GC theory. Consequently, the capacitance is caused mainly by solvent molecules rather than ions. This inference agrees with the invariance of the capacitance to the ionic concentrations (in Figure 4 - 4). The plots in
Figure 4 - 8 show no difference in the capacitance among KF, KCl and KBr. It has been widely acknowledged that fluoride is a non-specific adsorbed species whereas chloride and bromide are adsorbed specifically on electrodes. This statement may be valid for mercury electrodes [11, 71, 143], but invalid for platinum ones. Since iodide solution showed a narrow polarized domain (0.15 V), it is insignificant to discuss the shape of Cp vs. E curves.
F 1 / =1 f ) p C (
0 0 0.1 0.2 0.3 0.4 E / V vs. Ag|AgCl Figure 4 - 8. Dependence of the double-layer capacitance on dc-potential in 0.1 M solutions of (squares) KF, (circles) KCl and (triangles) KBr, where the capacitance values are extrapolated to 1 Hz.
Figure 4 - 9 shows dependence of (Cp)f=1 for KCl on the dc-potential for different concentrations. No dependence was found even at the lowest concentration (1 mM).
Therefore, the electrostatic interaction of ions with the electric field plays a minor role in forming the double-layer capacitor. The independence was also observed for KF and
KBr at the concentrations of the above domain.
55 Chapter Four Double-Layer Capacitance – Independence of dc-Potential
F 1 / =1 f ) p C (
0 0 0.1 0.2 0.3 0.4 E / V vs. Ag|AgCl
Figure 4 - 9. Variations of the double-layer capacitance in KCl solution at (open circles) 1 M, (filled circles) 0.1 M, (triangles) 0.01 M and (squares) 0.001 M with the dc-potential, where the capacitance values are extrapolated to 1 Hz.
Dependence of Cp on concentrations is noticeable in Figure 4 - 9 as well as Figure
4 - 4 and Figure 4 - 5. Evaluation of the solution resistance was made by extrapolating
Z1 to infinite frequency conveniently in order to avoid deterioration of electrode surfaces by a long term measurements. Values of Rs by this technique were unfortunately smaller than those evaluated from [40] variations of Z1 with the distance between the two wire electrodes by 30-50 % when halide concentrations were less than
0.1 M. The underestimation of Rs provides smaller values of Cp through (Eq. 4 - 1) than those determined by the latter method. Consequently the values of Cp in Figure 4 - 4,
Figure 4 - 5 and Figure 4 - 9 at low concentrations are smaller than those at high concentrations. The underestimation of Rs by the extrapolation may be due to delay of the potentiostat. Accurate determination of the concentration dependence would require meticulous calibration of a potentiostat, especially at low concentration and high frequency.
56 Chapter Four Double-Layer Capacitance – Independence of dc-Potential
2
2 F A
→ / /
← I
=1 1 f 0 ) p C ( -2 0 0 0.5 1 E / V vs. Ag|AgCl
Figure 4 - 10. Comparison of the potential-dependence of (Cp)f=1 and cyclic voltammetry in 0.1 M KCl solutionon.
Figure 4 - 10 shows the variation of double-layer capacitance in the wide potential domain, showing a valley shape. The cliff-like variation occurred at the potential which gives rise to faradic reactions, as compared with the cyclic voltammogram. The increase of capacitance by faradic currents has been found at Ir(100) in 0.1 M HCl [69], and at
Pt(111) electrode in 0.1 M HClO4 [76]. Capacitances with a valley shape reported so far may include faradaic impedance. Participation in the faradaic currents has been applied to electroactive film-coated electrodes in order to enhance capacitance largely [148,
149].
4.4 Conclusion
The double-layer impedance is expressed by a parallel combination of the capacitance and the resistance. The latter is not a real resistance, but appears noticeably at low frequency. It is necessarily associate the measurements of ac impedance with (Eq.
3 - 23). Since it results from the frequency-dependence of the capacitance, it is a property of the double-layer. Values of Rp are inversely proportional to the electrode
57 Chapter Four Double-Layer Capacitance – Independence of dc-Potential area, are common to four halides and concentrations, and are independent of dc-potential in the polarized domain (from 0.0 to 0.4 V vs. Ag|AgCl).
Cp varies linearly with log(f) in solutions of four halides at concentrations more than
1 mM. This linear variation causes Rp through 1/(Cp/). Cp is independent of both dc-potential in the polarized domain (from 0.0 to 0.4 V vs. Ag|AgCl) and ionic concentration (from 1 mM to 2 M (= mol∙dm-3)) of halides so that the shape of capacitance vs. potential curves is a plane. The capacitance is caused by water molecules rather than ions. A valley shape is possibly ascribed to participation in faradaic currents, i.e., a decrease in the out-phase component of faradaic impedance.
Variations of the capacitance with the dc-potential in 0.1 M KF, KCl and KBr solutions are ca. 10 %. Therefore, the capacitance has negligible influence on the localization of ions by the Gouy-Chapman theory.
58 Chapter Five Dependence of Capacitance on Basic Physical Properties of Electrolytic Solvents
Chapter Five
Dependence of Capacitance on Basic Physical Properties of Electrolytic Solvents
5.1 Aim
Capacitances of an electric double-layer (EDL) vary empirically with different kinds and concentration of ions[11, 80], solvents, applied dc-potential and ac-frequency, materials and structure of electrodes[150], separation of electrodes and conductivity of solutions and electrodes. Researchers have studied the contribution of electrolytic conductivity [89] by mixing organic solvents, kinds of the solvents [98-100], and size of the solvent molecules to the EDL capacitance. The capacitances were larger with a decrease in size of cations [64, 67, 89-91] and anions [51, 53, 66, 80, 84, 89, 92-97]. It sometimes varies with dc-potential when non-ionic substances are adsorbed on such carbon electrodes [64, 65, 68]. Big variations have often been found at ac-frequency, called the frequency dependence of capacitance, or frequency dispersion. The capacitance decreases generally with an increase in frequency. The frequency dispersion has been investigated as Constant Phase Element (CPE) [19-35].
Effects of these variables on the capacitances depend strongly on unpredictable conditions especially when adsorption occurs. The unpredicted conditions are caused not only by specific adsorption but also by insufficient shield of the electrode with an insulator, geometrically uncontrolled solution resistance, limited resolution of instruments, and leakage of possibly disturbing ions from a reference electrode. Some of them have been technically solved when the two parallel wire electrodes, without insulating walls, were used, as indicated in chapter 3 and 4. As a result, the capacitances
59 Chapter Five Dependence of Capacitance on Basic Physical Properties of Electrolytic Solvents were independent of the concentrations and kinds of halide ions, and almost independent of dc-potentials. The invariance to the dc-potential has also been observed in aqueous solutions with halide ions on a well-defined platinum electrode in chapter 4, in 30 wt.% H2SO4 aqueous solutions on glassy carbon electrodes, in 0.1 M HCl at
Ir(100) electrode [69], and in 1-methyl-3-hexylimidazolium chloride at glassy carbon electrode [67]. The independence of the EDL capacitance from the dc-potential at solid metal electrodes differs from that at mercury electrodes, because mercury shows surface tension much larger than platinum does [74, 151]. The frequency-dependence, which was explained phenomenologically in terms of CEP, has been attributed to the logarithmic variation of the capacitance with frequency in chapter 3.
In chapter 4 it was concluded that the EDL capacitance is mainly brought about by solvent rather than the ionic concentration and the type of ions in the absence of specific adsorption. Unfortunately, it is not clear at present which properties of solvent vary systematically with the capacitance. In this chapter, emphasis is laid on seeking the key properties of solvents which influence the capacitance. Impedance is obtained at zero dc potential at room temperature in 13 solvents.
5.2 Experimental
The two-electrode system was used for the ac-impedance measurement. Working and paralleled counter electrodes used were platinum wires, 10 mm in immersion depth,
0.1 mm in diameter. The electrodes were initially cleaned in distilled water and in acetone orderly, after which it was air dried. The manufacturing process of the platinum wire electrode is described in section 3.2. The electrode surface was conditioned by cyclic voltammetry in some polarized potential domains before the impedance
60 Chapter Five Dependence of Capacitance on Basic Physical Properties of Electrolytic Solvents measurements. It was confirmed that the pre-potential application by cyclic voltammetry did not alter the impedance data, as was in accord with that in chapter 4.
The solvents used were selected to cover a wide range of dielectric constants, dipole moments, molar volume and viscosity. All the chemicals were of analytical grades.
Ion-exchange distilled water was used in preparing all aqueous solutions. Each organic solvent was dried by molecular sieves for 20 min before supporting electrolyte was dissolved.
Tetra-n-hexylammonium perchlorate (THAClO4) was dried in a volumetric flask under vacuum at 40 oC for 10 h and stored in an atmospheric desiccator. Sodium tetrafluoroborate was treated at 80 oC by the same method as described above.
Commercially obtained potassium chloride was used. NaBF4 was dissolved in glycerol at 80 oC and cooled to room temperature before use. It was dissolved in ethylene glycol and formamide at room temperature. THAClO4 was dissolved in the other organic solvents at room temperature as well. The dissolution processes were done in vacuum environment, except for glycerol. Delay of the potentiostat was checked in section 3.2.
NaBF4 and all 12 organic solvents were provided by WAKO PURE CHEMICAL
INDUSTRIES, Ltd, Osaka, Japan. THAClO4 was provided by ALFA AESESAR.
5.3 Results and Discussion
5.3.1 Nyquist Plot
Figure 5 - 1 shows the Nyquist plots obtained from DMSO, PC and DCM solutions,
-3 as well as 0.1 mol dm THAClO4, at frequencies ranging from 10 Hz to 50 kHz. The
Nyquist plots fell on each line, suggesting only the participation of the double-layer impedance rather than that of faradaic impedance in the Nyquist plots. The other ten
61 Chapter Five Dependence of Capacitance on Basic Physical Properties of Electrolytic Solvents solvents in Table 5 - 1 showed linear Nyquist plots similar to those in Figure 5 - 1. The extrapolation of the line to Z2 = 0 for infinite frequency yields the solution resistance, Rs.
If the EDL impedance were to be expressed by a simple series combination of an ideal resistance and an ideal capacitance, the Nyquist plot should be a vertical line. The experimental values of the slopes ranging from 4.5 to 9.0 in Figure 5 - 1 indicate that the resistance should vary with frequency. Interpretation of the slope will be discussed later.
100 / k 2 50 - Z
0 0 10 20
Z1 / k Figure 5 - 1. Nyquist plots obtained in (circles) DMSO, (squares) PC and (triangles)
DCM. High frequencies correspond to large values of Z1.
Values of Cp and Rp were determined from Z1 and Z2 at each frequency by using (Eq.
3 - 15). The values of Cp were ca. half values of Cd (= -2/Z2). Figure 5 - 2 shows variations of Cp with logarithm of frequency (f = /2) in DMSO, PC and DCM solutions. Values of Cp show linear relations with the logarithmic frequency, regardless of the kinds of solvents, i.e.
C C k ln f (Eq. 5 - 1) p p 1Hz
62 Chapter Five Dependence of Capacitance on Basic Physical Properties of Electrolytic Solvents
where k is a constant and (Cp)1Hz is the capacitance at f = 1 Hz by the extrapolation in
Figure 5 - 2. The linear variation has been demonstrated to hold for aqueous solutions including several kinds of salts and their concentrations, as have been observed in chapters 3 and 4. Values of (Cp)1Hz and k depended on solvents.
0.5 F / p C
0 0 2 4 log( f / Hz)
Figure 5 - 2. Plots of Cp evaluated from Eq. (3) in (circles) DMSO, (squares) PC and -3 (triangles) DCM solutions including 0.1 mol dm THAClO4 against log(f).
Figure 5 - 3 shows logarithmic variation of Rp against the logarithm of frequency for the three solvents. The plots for all the solvents used showed linear relation with the slope -1. Therefore Rp should be inversely proportional to the frequency. Inserting (Eq.
5 - 1) into the definition of Rp (1/(C/t) = Rp) and carrying out the differentiation, we obtain
Rp 1/ 2πfk (Eq. 5 - 2)
This is the verification of the inverse proportionality, which was derived by inserting
(Eq. 5 - 1) into the second term of (Eq. 3 - 23) can carry out the differentiation.
63 Chapter Five Dependence of Capacitance on Basic Physical Properties of Electrolytic Solvents
) 5 / p R log(
0 2 4 log( f / Hz)
Figure 5 - 3. Logarithmic plots of Rp evaluated from Eq. (3) in (circles) DMSO, -3 (squares) PC and (triangles) DCM solutions including 0.1 mol dm THAClO4 against log(f).
Taking the ratio of (Eq. 3 - 14) yields -Z2/(Z1 - Rs) = RpCp. When (Eq. 5 - 1) and
(Eq. 5 - 2) are inserted into the above equation, we have
Z / Z R C / k ln f (Eq. 5 - 3) 2 1 s p 1Hz
The ratio looks to vary logarithmically with the frequency. If the more precise
2 approximation of Cp (Cp)1Hz – k ln f + k1(ln f) is used, the ratio is given by
C k ln f p 1Hz Z 2 /Z1 Rs (Eq. 5 - 4) k k1 ln f
Since values of (Cp)1Hz/k was close empirically to those of k/k1, the logarithmic term is cancelled. Consequently, the Nyquist plot is represented by
Z 2 / Z1 Z 2 /Z1 Rs k / k1 (Eq. 5 - 5)
Thus values of Z2/Z1 are constant. This is the demonstration of providing a line of the
Nyquist plots (in Figure 5 - 1) for the double-layer impedance.
64 Chapter Five Dependence of Capacitance on Basic Physical Properties of Electrolytic Solvents
In summary, the linearity of the Nyquist plots indicates that the EDL impedance can be expressed by the equivalent circuit in Figure 4 - 2, which is based on the frequency-dependence of the EDL capacitance. The linearity was found for all the solvents. The EDL resistance is inversely proportional to the frequency. It does not exist physically, but is observed inevitably through (Eq. 3 - 23) when the impedance is measured as a response of time-variation. The capacitance decreases linearly with the logarithmic frequency, obeying (Eq. 5 - 1). A characteristic variable of the EDL was selected at 1 Hz, denoted as (Cp)1Hz. Values at any other frequencies can be used for the comparisons of Cp of different solvents as far as Cp is linear to logf. Frequency 1 Hz may be convenient for comparison of the capacitance with that obtained by cyclic voltammetry in future. Variations of (Cp)1Hz with solvent properties were examined in the next section.
5.3.2 Variations of Capacitance with Key Properties of Solvents
Variation of Cp at 1 Hz with voltages between two platinum electrodes is shown in
Figure 5 - 4 for propylene carbonate solutions, where the CV is also presented. No variation of (Cp)1Hz was found in the polarized potential domain. Consequently a potential of zero-charge cannot be determined. This is also true in the aqueous solutions, as presented in chapter 4. These observations seems inconsistent with a well-known minimum is a capacitance vs. potential curve at a mercury electrode [11]. The minimum is a intrinsic characteristic of the diffuse double-layer capacitance, CD, predicted from the Gouy-Chapman theory. Since mercury has much higher surface tension than platinum, the Helmholtz (or inner) layer capacitance, CH, at mecury is larger than that platinum. As a result, near PZC, 1/CH < 1/CD at mercury electrodes [11], whereas 1/CH >
1/CD at platinum electrodes. Then the observed capacitance, obeying 1/Cp = 1/CH +
1/CD, at the platinum electrode is controlled by CH, which has no minimum.
65 Chapter Five Dependence of Capacitance on Basic Physical Properties of Electrolytic Solvents
-2 20
100 A / F cm F I 0 / 50 1Hz ) p C
( -20 0 -2 -1 0 1 2 E / V
Figure 5 - 4. Variation of Cp with dc-potential and cyclic voltammogram (scan rate 0.02 -1 V s ) in 0.1 M THAClO4 + propylene carbonate at the two platinum electrodes.
Table 5 - 1. Values of the EDL capacitance, dmv and dor for 13 solvents
(C ) / d d No. Solvents p 1Hz or mv ( ) F cm-2 / nm / nm r sat 1 acetonitrile 11 0.400 0.444 5.0 2 aniline 12 0.640 0.534 8.8 3 benzyl alcohol 10 0.536 0.557 5.8 4 dichloromethane (DCM) 15 0.263 0.475 4.5 5 dimethyl sulfoxide (DMSO) 14 0.369 0.491 5.7 6 ethanol 11 0.316 0.460 4.0 7 ethylene glycol 20 0.415 0.453 9.3 8 formamide 24 0.304 0.405 8.1 9 glycerol 17 0.543 0.495 10.6 10 n-methylformamide 16 0.544 0.461 10.2 11 propylene carbonate (PC) 9 0.614 0.521 6.4 12 tetrahydrofuran 11 0.431 0.513 5.4 13 water 35 0.158 0.312 6.3
66 Chapter Five Dependence of Capacitance on Basic Physical Properties of Electrolytic Solvents
Table 5 - 1 shows a list of values of (Cp)1Hz obtained from 13 solvents. Figure 5 - 5 shows the plot of (Cp)1Hz vs. dielectric constants of the solvents, where the numbers in the circles correspond to those in Table 5 - 1.
40 13 -2
8 F cm F
20 7
/ 9 10
p 4 5 C 2 6 1 12 3 11
0 0 100 200 r Figure 5 - 5. Variation of the capacitance with dielectric constant of the solvents
According to the Helmholtz layer of Stern’s Model, dielectric constant is a crucial variable of the capacitor at a unit area through C = or /d [4], where d is the separation of the parallel plate model of the inner layer, which may be close to a diameter of solvent. Although the capacitance of the parallel plate model should be proportional to the dielectric constant on the assumption of a common value of d, the measured capacitances have no relation with the dielectric constants [64, 98].
Dielectric saturation may occur in the double-layer, as has been pointed out in ref.
[64, 98]. If saturated dielectric constants take a common value to the 13 solvents,
(Cp)1Hz should be inversely proportional to d. Values of d can be estimated from the
1/3 molar volume to be designated as dmv. = (M/NA) . Here M is the molecular weight, ρ is the density of solvent, and NA is the Avogadro constant. Figure 5 - 6 shows dependence of (Cp)1Hz on 1/dmv. Although linearity is found, the plot is not inversely
67 Chapter Five Dependence of Capacitance on Basic Physical Properties of Electrolytic Solvents
proportional to dmv (see the negative value of the intercept). Consequently the distance is not a crucial variable for the capacitance.
40
13 -2
8 F cm F
20 7
/ 9 10 p 5 4
C 2 12 6 1 3 11
0 0 1 2 3 -1 -1 dmv / nm Figure 5 - 6. Variation of the capacitance with the inverse of molecular diameter obtained from the molar volume
40 13 (a) 13 (b) 13 (c) -2
8 8 8 F F cm
7( =16) 20 7 7 → 9(=945) / 10 10 9 10
p 4 4 4 5 6 5 5 C 2 2 2 9 1 12 12 6 11 3 11 11 3 3126 1 1
0 0 2 4 60 100 200 3000 2 4 6 8 p / D o Tbp / C / mPa s Figure 5 - 7. Variations of the capacitance with (a) dipole moments, (b) boiling temperatures and (c) viscosity.
The other possible properties of the solvents are (a) dipole moment, , (b) boiling point, Tbp, and (c) viscosity, . The dipole moment may contribute to molecular orientation in the EDL, while the boiling point can represent interaction among oriented
68 Chapter Five Dependence of Capacitance on Basic Physical Properties of Electrolytic Solvents molecules in the EDL. Viscosity may take part in slow relaxation of the frequency-dependence of the capacitance. Figure 5 - 7 shows the variations of the capacitance with these variables. All the points were scattered. Consequently, the EDL capacitance cannot be determined by these variables.
The values of (Cp)1Hz in Table 5 - 1 and Figure 5 - 5, Figure 5 - 6 and Figure 5 - 7 were the determined capacitance values divided by the geometrical electrode area. If the electrode is microscopically rough, the capacitance values must be smaller than those in
Table 5 - 1 by the roughness factor. The plots in Figure 5 - 5, Figure 5 - 6 and Figure
5 - 7 are, however, not independent essentially of the roughness because the y-axes is inversely proportional to the roughness factor.
The highest correlation in Figure 5 - 5, Figure 5 - 6 and Figure 5 - 7 is the linearity of the capacitance with 1/dmv. The use of dmv for d implies that solvent molecules in the
EDL take thermodynamic average of the diameters by thermal fluctuation or rotation.
This is an unrealistic assumption because the molecules should be arranged in higher order by the strong electric field or solvent|electrode interaction than that in the bulk.
Molecules with strong hydrogen bonds such as water (13) and formamide (8) might be compressed in the EDL to take smaller values of d than dmv. Then true values of d in the
EDL might make the plot in Figure 5 - 6 pass through the origin.
Since molecules in the EDL are oriented by the external field, values of d controlling the capacitance may be the molecular length oriented in the direction of the dipole rather than the diameter estimated from the molar volume. A technique of estimating the controlling d is to use a model of molecules oriented in the direction of the dipole. The 3D structures of molecules were simulated and optimized by means of
Chemoffice software which was based on the AM1 method [152]. Lengths of oriented
69 Chapter Five Dependence of Capacitance on Basic Physical Properties of Electrolytic Solvents
(a) iii) Optimized by AM1 i) Draw in ii) Past in Chem3D H(6) ChemDraw O(3) O(3) H(6) C(4) CH H(5) H(5) 3 S(2) C(4) H(9) H(7) S(2) S H(9) H(7) H C O C(1) 3 C(1) H(8) H(8) H(10) H(10)
(b) dmv = 0.312 dmv = 0.405 dmv = 0.491 nm nm nm Bulk
wate formamid DMSO r e nm nm 304 369 E 0. 0.158 nm 0. = = = or or or d d d
Electrode
Figure 5 - 8. Models of solvent molecules: a) an example of estimating dor of DMSO molecule from ChemOffice; b) solvent molecules with orientation at the interface and in bulk.
solvent molecules, dor, were estimated from bond lengths and bond angles in the optimized structure, and corrected by atomic sizes. Herein, DMSO was taken as an example, as shown in Figure 5 - 8(a). The 2D molecular structure was initially plotted in ChemDraw of ChemOffice, and then pasted in Chem3D. Optimization of the molecular structure yielded the bond length and band angles. The dipole vector was obtained from Model 360 database [153] (Chemical Education Digital Library (online)).
70 Chapter Five Dependence of Capacitance on Basic Physical Properties of Electrolytic Solvents
The length of a molecule along the dipole was an orthogonal projection of the 3D image on the vector. Figure 5 - 8(b) shows some illustrations of the oriented molecular structures, together with dielectric moments which are perpendicular to the electrode surface. The values of dor, listed in Table 5 - 1, are mostly smaller than dmv for molecules.
40
13 -2 F cm F
8
/ 20 7 9 1 Hz 10 4 )
p 5 2 12
C 6 3 ( 11 1
0 0 2 4 6 8 10 12
1/(dor / nm) Figure 5 - 9. Variation of the capacitance with the inverse of molecular diameter obtained from the length of the oriented molecular dipoles.
Figure 5 - 9 shows the variation of the capacitances with the longest lengths of the oriented molecules, dor. The proportionality is found, i.e.
(Cp)1Hz = o(r)sat /dor (Eq. 5 - 6) where (r)sat is the saturated dielectric constant. The proportionality in Figure 5 - 8 indicates that (r)sat is common to the 13 solvents. Since values of dor were evaluated from the orthogonal projection of the molecules on the dipole vectors, the proportionality is supported by the orientation of the solvent molecules. The slope corresponds to the saturated dielectric constant, 6 ± 2. The reported saturated dielectric
71 Chapter Five Dependence of Capacitance on Basic Physical Properties of Electrolytic Solvents
constant for water is 5.8 ± 0.4 [129, 154]. Values of (r)sat determined by (Cp)1Hz and dor for each solvent are listed in Table 5 - 1.
5.4 Conclusion
The Nyquist plots in the 13 solvents take each line, of which slope ranges from 5 to
9. The slope is caused by the linear variation of the capacitance with logarithmic frequency. The parallel resistance is necessarily involved in measurements of the impedance because it results from the frequency-dependence of the capacitance.
Both bulk and interfacial properties of metal-solution system were examined to seek for the variables controlling the EDL capacitance in the absence of specific adsorption.
1. The capacitance cannot be represented by simple relation with the dielectric constant, viscosity, boiling temperature or dipole moment of solvents. It increases linearly with an increase in reciprocal diameter of solvent molecules evaluated from the molar volumes, showing a negative intercept.
2. The capacitance is inversely proportional to the length of a solvent molecule along the dipole moment.
The solvents take an approximately common value of the saturated dielectric constant.
72 Chapter Six Modification of Debye’s Model for Dielectric Saturation in Electric Double-Layer
Chapter Six
Modification of Debye’s Model for Dielectric Saturation in Electric Double-Layer
6.1 Aim
The saturated dielectric constants for 13 solvents at platinum wire electrodes were common by applying the parallel plate model and taking the length of the dipole as the separation. A detailed review of theoretical researches of the dielectric saturation was introduced in chapter 2, but, few of the theories can be consistent with or represented by experimental results. The evaluated saturated dielectric constants in EDL depend on both the thickness of EDL [111, 114] and the intensities of the orientation and/or distortion fields [13, 14, 129]. However, clear understanding of the saturation is still under debate. The concern in this chapter is investigating the origin of dielectric saturation which was observed experimentally by impedance measurement. A saturation model is presented and compared with Debye’s equation, and demonstrated theoretically by modifying Debye’s equation as well.
6.2 Method
The saturated dielectric constants mentioned in chapter 5 were evaluated by applying the parallel plate model,
* C 0 sat / l (Eq. 6 - 1) to the experimentally obtained double-layer capacitance. Herein, there is problem of defining the separation of two layers, l*. Since the EDL capacitance is mainly brought about by solvent rather than the ionic concentration and the type of ions in the absence
73 Chapter Six Modification of Debye’s Model for Dielectric Saturation in Electric Double-Layer of specific adsorption, only the size of solvent molecule was considered as separated distance, l*, in this chapter.
6.3 Theoretical
6.3.1 Model
Molecular dipole
r q d l E
-q
Figure 6 - 1. Illustration of molecular dipole in inner layer, where d is thickness of inner layer; is the angle between direction of molecular dipole and electric field, E; q and -q are equivalent charges of dipole; (l+2r) is the length of dipole; r is the radius of two hemispheres.
A model of dipole molecule is assumed to be composed of two hemispheres with a radius r in and a cylindrical height l, and two equivalent charges being at the centre of the two hemispheres. Capacity of parallel plates per area with separation d is generally given by C = 0r/d. The separation in the inner layer corresponds to the thickness of monolayer of solvent molecules on the electrode, denoted as d. For a particular kind of dipole, d varies with tilted angle , influenced by electric field force, and dielectric constant r which is a function of electric field. So effects of the electric field on C can
74 Chapter Six Modification of Debye’s Model for Dielectric Saturation in Electric Double-Layer
be represented in terms of two definitions; (i) variation of r for a constant value of d, and (ii) variation of d for r = 1.
When this concept is applied to the model of dipole, in Figure 6 - 1, geometry of the model is satisfied by
d l cos 2r (Eq. 6 - 2)
If solvent molecules are sized by the maximum length, (l+2r), then, the first definition leads to
C 0 r /l 2r (Eq. 6 - 3)
The second definition provides
C 0 /l cos 2r (Eq. 6 - 4)
By equating (Eq. 6 - 3) and (Eq. 6 - 4),
l 2r r (Eq. 6 - 5) l cos 2r
This relates the macroscopic measure of r with the microscopic measure of . Very strong electric field makes the dipole oriented so that = 0. Then the extremely
o saturated dielectric constant is r 1. On the other hands, very low field yields 90 to provide the dielectric constant
r l / 2r 1 (Eq. 6 - 6)
This is supposed to be realized in bulk solution. However, this rule provides smaller r than that in the bulk, since the movement of dipole is controlled by electric force rather than thermal fluctuation in the double-layer. For instance, l/2r = 5.4 for aniline at r =
6.4. Moreover, it does not provide a suitable geometrical image of molecules. Since the model has been considered to represent properties of dipoles, it is different from real molecular geometry.
75 Chapter Six Modification of Debye’s Model for Dielectric Saturation in Electric Double-Layer
Modified Debye’s model by image force
The dielectric saturation is observed when a dipole moment is not satisfactorily reflected in the capacitance owing to displacement of electrons, that of ions and orientation of dipole moments [155]. It was expressed by Claisius-Mossotti's law,
Debye's equation, Onsager's equation and Kirkwood's equation [155]. These models are valid when dipoles and/or ions are in bulk solution.
Since electrochemical measurements of capacitance are made under interaction of dipoles and an electrode, the dipoles oriented by the external electric field are complicated by the interaction with the electrode. The simplest interaction is the electrostatic force of a dipole and the conductor, known as an image force. A model is illustrated in Figure 6 - 2, where two charges, q and -q, , at both ends of the dipole as introduced in Figure 6 - 1, are attracted electrostatically to the electrode by the image force, as shown in a text book on electrodynamics [156].
q
l E a -q a a q a l
Electrode -q
Figure 6 - 2. Model of a dipole interacting with the electrode by the image force and external electric field.
76 Chapter Six Modification of Debye’s Model for Dielectric Saturation in Electric Double-Layer
6.3.2 Derivation
In this study, the dipole moment and the length of the dipole are denoted as and l
(= /q), respectively. When the dipole tilts from the direction of the external electric field, E, by the angle , the energy of the dipole by the electric field is given by
U fld E cos (Eq. 6 - 7)
The electrostatic energy of the charge -q, which is attracted by the image force charge q in the electrode, is expressed by
2 U img q /16π 0 a (Eq. 6 - 8) where a is the closest approach of the charge to the electrode. Similarly, the positive charge is also attracted to the electrode with the energy:
2 U img q /16π 0 a l cos (Eq. 6 - 9)
The numerical values of the energies were compared using of the following typical values: = 2 D = 6.6×10-30 Cm, l = 0.4 nm, a = 0.2 nm, = 45o and E = 5×107 V m-1
-21 -21 ( = 0.01 V/a). Then, Ufld = -0.23×10 J and Uimg- + Uimg+ = -4.3×10 J.
Luque and co-workers [157] extended Lang’s theory [158] and calculated the effective position of the image plane on platinum metal to be a half of top lattice plane spacing of the crystal, whose numerical value was c.a. 0.05 nm. If a = 0.05 nm is taken
-22 -20 in the above discussion, then, Ufld = -9.3×10 J and Uimg- + Uimg+ = -1.41×10 J.
Consequently, the image force works more strongly than the electric field force.
Three energies are balanced with thermal energy, kBT, where kB is the Boltzmann constant. Additionally, the interactive energies between dipoles are smaller than Uimg by more than 5 orders of magnitude.
77 Chapter Six Modification of Debye’s Model for Dielectric Saturation in Electric Double-Layer
The probability of taking is expressed by the Boltzmann distribution
p exp U fld U img U img / k BT (Eq. 6 - 10)
Since ranges from 0 to , the partition function of the tilted dipole is given by
2 π 1 Ex q 1 Z psin d exp 1 dx (Eq. 6 - 11) 0 1 k T 16 ak T B π 0 B 1 (l / a) x where x = cos(). The absolute sign in (Eq. 6 - 11) means that the positive charge can be exchanged by the negative charge for > /2.
The average value of cos is expressed by
2 1 Ex 1 cos x exp 1 dx / Z (Eq. 6 - 12) 1 k T 2 1 (l / a) x B 16π 0 al kBT
2 2 It is a function of three variables, E/kBT, /(160al kBT) and a/l. Only the first term in the integrand of (Eq. 6 - 12) is nothing but Debye’s equation.
The integral was evaluated numerically for given values of a, l, at various values of E. Figure 6 - 3 shows the variation with E/kBT for some values of . With an increase in E: the dipole is oriented to the direction perpendicular to the electrode. In contrast, small values of E/kBT make the dipole tilt to the direction parallel to the electrode. Then the dipole is controlled by the image force. Comparing the variation with the Debye's expression, given by
E k T cos coth B Debye (Eq. 6 - 13) kBT E
(Eq. 6 - 13) is a limiting case of (Eq. 6 - 11) and (Eq. 6 - 12) for ainfinity. The variation by Debye is shown as the dashed curve in Figure 6 - 3. With an increase in
78 Chapter Six Modification of Debye’s Model for Dielectric Saturation in Electric Double-Layer
2/al2, the curves by (Eq. 6 - 12) deviate largely from the Debye's curve. The Debye's approximate equation for small values of E/kBT, given by
E cos Debye (Eq. 6 - 14) 3kBT deviates further from (Eq. 6 - 12) than (Eq. 6 - 13).
90 (d) (c) >
(b)
< 60 (a)
30 Debye
Debye approx 0 0 5 10 15
E / kBT Figure 6 - 3. Dependence of averaged values of on E/kBT at = (a) 2 D, (b) 3 D, (c) 4 D and (d) 5 D, calculated from (Eq. 6 - 12) for a = 0.1 nm, l = 0.4 nm.
6.4 Application
The theta angles, calculated from (Eq. 6 - 12) for some possible values of E, a, and l, were plotted in Figure 6 - 4 against Cp. Most dipoles are tilted by angles close to
90 degree, indicating an important role of image force in the dipole orientation in the double-layer.
The numerical theta angle can also be obtained from saturated dielectric constant, which was evaluated by the experimentally determined capacitance. The saturated dielectric constant, sat, was demonstrated to be of an approximately common value, 6 ±
79 Chapter Six Modification of Debye’s Model for Dielectric Saturation in Electric Double-Layer
2, by using the maximum length of molecule along the dipole, (l+2r), as l* of each molecule. The experimental dielectric constant is a little larger than that by (Eq. 6 - 6).
Here, (Eq. 6 - 1) includes a mixed concept of (i) and (ii) discussed in section 6.3, in that
* both sat and l vary with the intensity of electric field and individual molecule. If the concept (ii) is followed, (Eq. 6 - 1) should be rewritten as C = 0/(l/6). Applying (Eq. 6
- 4), cos = 1/6 - 2r/l can be obtained. The value of is ca. 80o at 2r/l = 0.
Consequently, all the dipoles in the inner layer lie along the electrode surface. This is close to that obtained by (Eq. 6 - 12).
1 3 9 12 5 8 13 90 11 6 2 4 10 7 (degree)
60
0 10 20 30 40 -2 Cp / F cm Figure 6 - 4. Dependence of the angles of tilted dipoles on the capacitances, where angles were calculated from the numerical integration of (Eq. 6 - 12) for a = 0.1 nm at E = (circles) 2×107 and (triangles) 2×108 V m-1. Plots with squares are from (Eq. 6 - 15) by use of the experimental values of (r)sat. The numbers corresponds to those in Table 6 - 1.
We assume that the tilts of orientated dipoles are responsible for the saturation of the dielectric constants. Then the following can be obtained:
80 Chapter Six Modification of Debye’s Model for Dielectric Saturation in Electric Double-Layer
* * C 0 sat / l 0 / l cos (Eq. 6 - 15)
cos 1/ r sat (Eq. 6 - 16)
was calculated from (Eq. 6 - 16) using experimental values of (r)sat, and plotted in
Figure 6 - 4 (squares). The theta-values take (80 4) degree in average.
Table 6 - 1. Compositions of 13 solutions used in impedance measurement
No. Solvents Salts No. Solvents Salts
1 acetonitrile A 9 glycerol B 2 aniline A 10 n-methylformamide A propylene 3 benzyl alcohol A 11 A carbonate 4 dichloromethane A 12 tetrahydrofuran A dimethyl 5 A 13 water C sulfoxide 6 ethanol A A: tetra-n-hexylammonium perchlorate; 7 ethylene glycol B B: sodium tetrafluoroborate; 8 formamide B C: potassium chloride.
Obviously, almost all of the solvent dipoles in the inner layer lie parallel along the electrode. Accordingly, if the molecule is taken as a dipole being oriented by both electric field and image force; whiles being balanced by thermal motion, the state of the molecular dipole contacting with the electrode surface becomes clear. It can be simulated into a dipole oriented almost parallel to the electrode surface and rotating like a cylinder, as illustrated in Figure 6 - 5.
81 Chapter Six Modification of Debye’s Model for Dielectric Saturation in Electric Double-Layer E
θ
Electrode
Figure 6 - 5. Illustration of the state of a molecular dipole in inner layer in the experimentally polarised potential domain.
1000 1 )] - / nm) / cos( ′ 500 d or ( d 1 / [ 0 0 10 20 30 40 C / F cm-2 Figure 6 - 6. Plot of EDL capacitance against the inverse of thickness of the tilted dipole.
The occupied volume, Vor, of such a molecule takes
2 d Vor dor cos (Eq. 6 - 17) 2
82 Chapter Six Modification of Debye’s Model for Dielectric Saturation in Electric Double-Layer
where d′ and dor are the thickness of the tilted dipole and the effective length of the dipole, evaluated from the molecular geometry, respectively. The d′ equals to the value of dor∙cosθ in real molecular geometry.
With this image in the head, we plot the EDL capacitance against (d′)-1, instead of
-1 using (dor) in Figure 5 - 9, in Figure 6 - 6. The proportionality in has been significantly improved. Consequently, a source of the dielectric saturation is the tilt of oriented dipoles of solvent molecules owing to the image force against the external force.
6.5 Conclusion
The dielectric saturation of solvent molecules in the electric double-layer can be demonstrated by introducing image force to Debye’s model. The dipoles are turned mostly parallel to the electrode surface by the image force between the dipole and the electrode, although they are apt to be oriented perpendicularly to the electrode, at the amplitude of 10 mV. The saturation of dielectric constants is ascribed to the tilts of dipoles caused by the image force against the external field.
83 Chapter Seven Summary and Conclusions
Chapter Seven
Summary and Conclusions
7.0 Summary and conclusions
Systematic experiments were carried out to figure out the controlling variables of electric double-layer capacitance to help a better understanding of electric double layers, as well as to provide a guide for optimizing the design of capacitors/supercapacitors in pointing out the experimentally controlling parameters of the capacitance.
The fundamental research enumerated in chapter 3, 4 and 5 investigated the electric double-layer impedance from the basic frequency dispersion point of view. The results theoretically and numerically ascribed the frequency-dependant impedance to the logarithmic frequency-dependence of the EDL capacitance accompanied by an in-phase element in a parallel form. This element was provoked by the frequency-dependent capacitance and of an inverse proportionality to the ac-frequency.
This apparently and inevitably in-phase portion as an ohmic resistance had not been noticed because there is no resistance portion in the EDL. The EDL capacitance, therefore, is composed of a resistance parallel to a capacitance, of which the electric performance should not be inexplicit by the accompaniment of the former. The total current can be expressed by (Eq. 3 - 26):
dC()V (t) dV (t) C() d I C() V (t) dt dt dt
Secondly, the parallel capacitance, hereunder called capacitance, exhibited invariance to ionic concentrations, type of ions and applied electrode potential in the
84 Chapter Seven Summary and Conclusions polarized potential domain. It indicates that the capacitance was mainly caused by solvents rather than ions in the electrolyte, at the absence of specific adsorptions.
Thirdly, the evaluated capacitances showed no systematic relation with the dielectric constants, viscosity, boiling temperatures or dipole moments of the solvents, but were proportional to the inverse of the lengths of field-oriented molecule. The proportionality indicates common saturated dielectric constants, 6, of 13 solvents. The variables controlling the capacitance were the saturated dielectric constants and the lengths of solvent molecules along the dipole.
Finally, a tilted molecular model was proposed to explain dielectric saturation of solvent molecules in the electric double-layer, at which a molecule is oriented by image force against electric field force in chapter 6. The tilts were demonstrated theoretically by modifying the Debye's equation, where the tilted angles were mostly 90 degree at 0 dc-potential, 10 mV ac-potential. Meanwhile all of 13 solvents exhibited tilted angles of c.a. 80 degree in average, experimentally. The capacitance revealed a better proportionality to the inverse of the length of such dipole in the direction perpendicular to the electrode surface than that obtained similarly in chapter 5.
The aim of this thesis has been achieved completely.
85 References
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104 LIST OF PUBLICATIONS
1. Yongdan Hou, Koichi Jeremiah Aoki, Jingyuan Chen, Toyohiko Nishiumi
Title: Solvent Variables Controlling Electric Double Layer Capacitance at
Metal-Solution Interface, Journal of Physical Chemistry C, vol. 118(19),
10153-10158 (2014).
DOI: 10.1021/jp5018289
Link: http://pubs.acs.org/doi/abs/10.1021/jp5018289
2. Yongdan Hou, Koichi Jeremiah Aoki, Jingyuan Chen, Toyohiko Nishiumi
Title: Invariance of Double Layer Capacitance to Polarized Potential in Halide
Solutions, Universal Journal of Chemistry, vol. 1(4): 162-169 (2013)
DOI: 10.13189/ujc.2013.010404
Link: www.hrpub.org/journals/article_info.php?aid=741
3. Koichi Aoki, Yongdan Hou, Jingyuan Chen, Toyohiko Nishiumi
Title: Resistance Associated with Measurements of Capacitance in Electric Double
Layers, Journal of Electroanalytical Chemistry, 689:124-129 (2013)
DOI: 10.1016/j.jelechem.2012.10.004
Link: www.sciencedirect.com/science/article/pii/S1572665712004171
105 CURRICULUM VITAE
YONGDAN HOU-GBOLOGAH
E-mail: [email protected], [email protected] Nationality: Chinese Date of birth: 29th September, 1984 Gender: Female Marital status: Married
Education
University of Fukui, Japan (2011-2014)
Ph.D. in Materials Engineering (2014)
Central South University, China (2008-2011)
MSc. in Materials Science and Engineering (2011)
Xiangtan University, China (2003-2007)
BSc. Material Forming and Control Engineering (2007)
Honors and Awards
Graduate:
One of the Top-Ten Excellent Poster Presentations – Fourteenth International
Symposium on Electroanalytical Chemistry, Changchun, China, August, 2013.
Honors Scholarship for Privately Financed International Students, The Japan
Student Services Organization (JASSO), Japan, April, 2013.
106 Honors Scholarship for Privately Financed International Students, The Japan
Student Services Organization (JASSO), Japan, August, 2011.
National Second-Class Scholarship, Central South University, China, October,
2008 ~ 2011,
Professional Experiences
Research Assistant, University of Fukui, Fukui, Japan (11/2011-2014)
Poster Presentation, “An In-Phase Component of Electric Double Layer
Capacitance”, 14th International Symposium on Electroanalytical Chemistry,
Changchun, China (08/2013)
Oral presentation, “Capacitance of Electric Double Layer”, Biennial Academic
Meeting on Electrochemistry, Fukui, Japan (07/2012)
Publications
Ph.D.:
Yongdan Hou, Koichi Jeremiah Aoki, Jingyuan Chen and Toyohiko Nishiumi,
“Solvent Variables Controlling Electric Double Layer Capacitance at
Metal-Solution Interface”, Journal of Physical Chemistry C, vol. 118(19),
10153-10158 (2014).
Yongdan Hou, Koichi Jeremiah Aoki, Jingyuan Chen and Toyohiko Nishiumi,
“Invariance of Double Layer Capacitance to Polarized Potential in Halide
Solutions”, Universal Journal of Chemistry, vol. 1(4): 162-169 (2013)
107 Koichi Aoki, Yongdan Hou, Jingyuan Chen and Toyohiko Nishiumi,
“Resistance Associated with Measurements of Capacitance in Electric Double
Layers”, Journal of Electroanalytical Chemistry, 689:124-129 (2013)
Master:
Yongdan Hou, Yao Jiang, Ting Lei and Yuehui He, “Effects of Sintering
Temperature on Microstructure and Performance of Ti Based Ti-Mn Anodic
Material”, Journal of Central South University Technology, 18:966-971 (2011)
108