Journal of Electroanalytical Chemistry 536 (2002) 11 /18
www.elsevier.com/locate/jelechem
Kramers Kronig transformation, dc behaviour and steady state
response of the Warburg impedance for a disk electrode inlaid in an
insulating surface
/
J. Navarro-Laboulais a,, J.J. Garcı́a-Jareño b, F. Vicente b
a
Depto. Ingenierı́a Quı́mica y Nuclear, Escuela Politécnica Superior de Alcoy, Universidad Politécnica de Valencia,
Paseo del Viaducto 1, 03801 Alcoy (Alicante), Spain
b
Depto. Quı́mica Fı́sica, Universitat de València, C/Dr. Moliner 50, 46100 Burjassot (València), Spain
Received 11 March 2002; received in revised form 17 July 2002; accepted 6 September 2002
Abstract
As the frequency approaches zero, the impedance described by the Warburg function tends to infinity. This means that the
resistance of the equivalent circuit representing the electrochemical process has an infinite resistor and then the current cannot flow
through it. This asymptotic behaviour also prevents the application of the Kramers /Kronig transformations, a set of integrals
which should be fulfilled by any linear system. Using a more general expression of the impedance for a disk electrode inlaid in an
insulating surface developed by Fleischmann and Pons (J. Electroanal. Chem. 250 (1988) 277), the Warburg impedance can be
deduced and the Kramers /Kronig transformation is possible. An expression is also deduced for the mass-transfer resistance and the
frequency at which the Warburg function fails for the representation of the impedance for a disk. A fast algorithm for the
calculation of the generalised impedance function is outlined for its implementation in non-linear impedance fitting programs.
# 2002 Elsevier Science B.V. All rights reserved.
Keywords: Microelectrode impedance; Mass-transport resistance; Composite electrodes
1. Introduction
The development of microelectrode technology for
electrochemistry and the disagreement between the
theoretical predictions based on one-dimensional diffusion equations and the experimental observations [1], led
to theoretical efforts to study the inlaid disk microelectrode problem. Several approaches have been used to
treat this mathematical problem with mixed boundary
conditions. Some of them are based on the properties of
the discontinuous Bessel functions, or the more general
method of Neumann’s integral as a complete solution of
the differential equation describing the inlaid microelectrode [2 /8]. Furthermore, numerical methods and
techniques have been developed to solve a great variety
of geometrical and electrochemical problems in two and
Corresponding author. Tel.: /34-96-652-8479; fax: /34-96-6528409
E-mail address: jnavarla@iqn.upv.es (J. Navarro-Laboulais).
three-dimensions. Besides these methods, other semianalytical methods leading to multidimensional integrals have been developed and should be considered to
calculate the electrochemical response of an electrode of
arbitrary shape.
Although the purpose of the works cited above was to
solve the non-stationary transport equations for microelectrodes, no mention about the size of the electrodes is
made in order to solve the problem mathematically. In
fact, the differential equations and boundary conditions
are not affected or modified by the size of the system,
and therefore the solutions must remain valid for
macroelectrodes. In other words, if a complete solution
for the non-stationary transport problem of a diskshaped electrode inlaid in an insulating surface is
obtained, the steady state (microelectrode behaviour),
the macroscopic and all the possible intermediate states
will be included in that solution.
Numerical methods in electrochemistry are irreplaceable methods to study and analyse the experimental data
0022-0728/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 0 2 2 - 0 7 2 8 ( 0 2 ) 0 1 1 7 3 - 7
12
J. Navarro-Laboulais et al. / Journal of Electroanalytical Chemistry 536 (2002) 11 /18
in an engineering or in a technical context. Here, the
geometry usually is not regular, failing the conventional
mathematical methods to solve the diffusion equations
with a complex chemical reaction scheme, leading to a
coupled system of non-linear partial differential equations that usually does not have an analytical solution.
Scanning electrochemical microscopy (SECM) [9], hydrodynamic voltammetry [10], or the current distribution in an electrochemical reactor [11], are examples
where numerical methods are fundamental to understand the information provided by experiments.
On the other hand, analytical expressions for the data
interpretation are fundamental to the electrochemist
when factors such as the geometry of the electrode are
known. Simple experiments lead in these cases to
knowledge of physical quantities of interest such as the
diffusion coefficient from a chronoamperometric experiment or the heretogeneous rate constant for an electrochemical reaction from impedance measurements.
Although the validity of these equations is here beyond
all doubt, it should be noticed that these expressions are
the result of a few simplifications that do not make them
universal in the sense that they cannot be used for all
situations for all electrochemical systems. This is the
case of the Warburg impedance studied here.
The present work was initially inspired by previous
work carried out by the authors on the electrochemistry
of composite materials [12 /15] and some explanation of
this problem to understand why the Warburg equation
should be revised, is advisable. From this starting point,
the results obtained in this analysis are general and
could be applicable to other electrochemical systems.
The electrochemical behaviour of conductor/insulator composite materials is the result of a few interacting
factors. On the electrode surface there could be electrochemically active and inactive conducting particles
surrounded by the insulating phase. These particles are
connected through the solid, so the percolation process
governs the bulk electrical conduction properties in the
solid. This implies the existence of a potential distribution on the surface because the ohmic drop of each
conducting particle across the solid could not be the
same [16 /21]. Furthermore, composites are usually
prepared by dispersing in the continuous insulating
phase the conducting particles which have different
shapes and sizes that could be represented by a
probability density function. All these phenomena imply
that the observed electrochemical properties are the
mean value of a set of distributed electrical and
geometrical properties.
In order to solve this problem, different approaches
have been tried. One is to consider the composite
electrode as a set of independent microelectrodes. This
model is known as the multimicroelectrode hypothesis
[22 /27]. Owing to the difficulties in accomplishing the
independence of electrochemically active sites on a
composite electrode, this hypothesis has not been
corroborated yet [28,29]. The first stage to undertake
this problem is the selection of an impedance function
which fulfils the physical restrictions of the problem,
and then, the application of a probability distribution
function to contemplate the effect of the microelectrode
collective.
The Warburg equation seems an appropriate first
option to try an impedance function for the composite
electrodes but presents two disadvantages which make it
unacceptable. First, using a function of this kind
assumes that the diffusion field near the electrode is
one-dimensional and the existence of a microelectrode
distribution with different sizes leads to a Warburg
impedance, for which the parameters are the mean
values of this distribution [30]. And second, the Warburg impedance is not Kramers /Kronig transformable.
From a theoretical point of view this opposes the
conditions of linearity, causality and stability that a
diffusion controlled process must fulfil and from a
practical point of view this means that an equivalent
circuit which uses the Warburg function for a blocking
electrode could be inappropriate [31].
Additionally, one problem associated with the
Kramers /Kronig transformation of experimental data
is the evaluation of the impedance in the frequency
range from 0 to . If the data set is modelled with a
Warburg function, its value diverges when the frequency
approaches to zero which contradicts the experimental
evidence obtained by potentiodynamic techniques. Few
alternatives have been proposed to avoid the problem of
the evaluation of the limits of the impedance function,
but these solutions, which have in general a mathematical nature, do not help the explanation of the Warburg
impedance behaviour related to other electrochemical
techniques.
The aim of this work is to show these physical
inconsistencies associated with the Warburg impedance,
which make it difficult to relate impedance experiments
with potentiodynamic ones. These problems could be
resolved using a more general description of the
impedance for an inlaid electrode proposed by Fleischmann and Pons [5]. The relation between these functions
and the Warburg impedance is demonstrated and the
derivation of an expression for the steady state mass
transport resistance for an electrode is obtained. Results
from other electrochemical techniques than impedance
and from some experimental observations are also
discussed.
2. Physical inconsistencies of the Warburg impedance
Let SE be the region defined by the electrode and SI
the insulating region which surrounds it. The electrode
surface could have an arbitrary shape but, for simplicity,
J. Navarro-Laboulais et al. / Journal of Electroanalytical Chemistry 536 (2002) 11 /18
let us consider a disk-shaped electrode of radius r . The
electrochemical reaction takes place on the electrode
surface characterised by the forward and backward
heterogeneous rate constants, kf and kb, respectively.
Under these conditions the diffusion problem can be
described by Fick’s equation:
@cox (q; t)
@t
@cred (q; t)
@t
Dox 92q cox (q;
t)
z 0; q SE
Dred 92q cred (q;
(1)
where j0 is the exchange current density and the other
symbols have their usual meaning [32,33].
This solution (5) is the transfer function of an
electrochemical system, and then, must accomplish the
physical restrictions of any transfer function for a lineal
system, that is, linearity, causality, stability and being a
finite function. In other words, the Warburg impedance
must be Kramers /Kronig transformable [31,34 /36],
that is, it must fulfil the equations:
Re[Z(v)] Re[Z()]
t)
with initial conditions:
cox (q; t)c+ox
q; t5 0
+
cred (q; t)cred
2
p
g
w Im[Z(w)] v Im[Z(v)]
(2)
2v
Im[Z(v)] p
(3a)
z0; q SE
@n
@cred (q; t)
@n
z0; q SI
(3b)
The subscripts ‘ox’ and ‘red’ refer to oxidised and
reduced species, respectively, D is the diffusion coefficient, t the time, q the coordinate vector, q /q (x , y, z),
n the normal vector to the electrode surface and i(q, t)
the local current on the electrode that should be
integrated over the whole electrode surface to obtain
the measured current, I (t):
I(t)
g i(q; t) dq
(4)
SE
The Warburg impedance can be evaluated from the
previous equations considering a small amplitude sinusoidal perturbation and some extreme simplifications.
For an electrode of infinite radius (r 0/) the Eq. (1)
can be rewritten as one-dimensional equations and the
boundary condition (3b) and Eq. (4) disappears because
it is not applicable for this kind of problem. Another
required simplification used in electrochemistry, even if
the three-dimensional problem is considered, is to
assume Butler /Volmer kinetics in Eq. (3a). Finally,
the assumption of equal initial concentration and equal
diffusion coefficient for the oxidised and reduced
species, allows the achievement of a compact analytical
solution of the previous system of differential equations.
These simplifications led us to the well-known equation
for the impedance of a macroscopic electrode called the
Warburg equation or the Warburg impedance:
ZW (v)
RT
nF pr2 j
g
2
RT
pffiffiffi
(1j)v1=2
2
2
2
1=2
+
2 n F pr D c
0
(5)
dw
Re[Z(w)] Re[Z(v)]
dw
w2 v2
(6a)
(6b)
In order to determine the above integrals, it can be
demonstrated that the impedance function should previously fulfil the following mathematical conditions:
lim Z(v)Z?0 jZƒ0
(7a)
lim Z(v) Z?
(7b)
v00
0
0
@cox (q; t)
@c (q; t)
Dred red
i(q; t)
@n
@n
@cox (q; t)
w 2 v2
0
and boundary conditions:
Dox
13
v0
where Z?0 ; Zƒ0 and Z? are constants. In other words, the
real and imaginary parts of the impedance should be
bounded between two known values. If the Warburg
impedance is considered, this function diverges when the
frequency approaches zero. Thus, the Warburg impedance function would not be Kramers /Kronig transformable.
Some mathematical transformations have been proposed in order to avoid this problem. One method was
to add a shunt resistance in the impedance calculations
in order to elude the asymptotic behaviour of the
function at null frequency [36]. Another proposal was
to use the KK-transforms in the admittance domain
where the function is well-behaved. Although these
alternatives could work from a practical point of view,
they cannot explain or help to explain why the Warburg
impedance behaves as it does and why it is not KKtransformable.
This mathematical analysis does not seem to be
consistent with the empirical evidence of the usefulness
of the Warburg impedance which explains many electrochemical systems. In fact, the reason for this apparent physical inconsistency is found in the way Eq. (5) has
been deduced. The simplification of Fick’s second law, a
second order partial differential equation involving
spatial and time variables, to a spatial one-dimensional
partial differential equation is a common mathematical
operation in electrochemistry. This first step leads to the
Warburg equation, useful from an empirical and
practical point of view but inconsistent from a mathematical or physical one, due to the loss of information in
the simplification of the original differential equation.
J. Navarro-Laboulais et al. / Journal of Electroanalytical Chemistry 536 (2002) 11 /18
14
Another disagreement between the Warburg equation
and the empirical observations is the inability of the
former to explain the dc behaviour of any electrochemical system. It is assumed that Randles’ equivalent
circuit could be used as a representation of an electrochemical process when an alternating potential is
applied at the electrode. If the system is linear, and the
Warburg impedance or the double layer capacitance are
deduced assuming this hypothesis, the electrochemical
process will always be represented by this equivalent
circuit for any potentiodynamic perturbation. The
current /potential curves obtained experimentally (i.e.
polarography, steady-state voltammetry, etc.) and the
impedance experiences, cannot be connected through
the Warburg impedance. As was shown before, the real
part of the Warburg impedance tends to infinity as the
frequency tends to zero. This could be interpreted as an
infinite resistance to the charge transfer, and then, the
current should not be able to flow through the electrode.
Because the equations used to derive the current /
potential dependence in steady-state voltammetry are
the same as those used to derive the Warburg impedance, a conflict between the two techniques occurs. The
mathematical justification of this discrepancy is again
that the Warburg equation is not KK-transformable,
and as will be shown in Section 3, the description of a
more complete definition of the diffusion impedance will
need to explain these apparent inconsistencies.
F5 (x)
g [J (bx
1=2
1
)]2
0
sin(u=2)
b(1 b4 )1=4
db
(10)
where J1(x ) is the Bessel function of the first kind and
order 1, and u /arctan b1. Next we will analyse the
behaviour of this function and its connection with the
Warburg impedance will be shown.
A representation of Eq. (8) in a Nyquist diagram leads
to a flattened semicircle as is shown in Fig. 1. Defining a
new variable x /r2v /D and using the properties of the
trigonometric functions, we can rewrite the impedance
function as:
Z(x)Rct s
x1=2
(F4 (x)jF5 (x))
(11)
with
x1=2
F4 (x) pffiffiffi
2
g
[J1 (y)]2
0
1=2
x
F5 (x) pffiffiffi
2
g
0
[J1 (y)]2
[y2 (x2 y4 )1=2 ]1=2
dy
(12)
[(x2 y4 )1=2 y2 ]1=2
dy
y(x2 y4 )1=2
(13)
y(x2 y4 )1=2
where
Rct RT
pnFr2 j0
(14)
and
3. Electrochemical impedance of an inlaid disk electrode
In the early 1990s the problem of the inlaid disk
electrode stated in Eqs.(1) /(4) was treated, solving the
steady and unsteady state conditions for relatively
simple electrochemical mechanisms [2 /8]. All this
work led to similar analytical or semi-analytical solutions involving multidimensional integral equations [8].
One of these, adequate to our purposes, was provided by
Fleishmann and Pons [5] who proposed the following
equation for the impedance of a disk-microelectrode
inlaid in an insulating surface, assuming Butler /Volmer
kinetics and equal bulk concentrations for the two redox
components:
2 RT
4RT
rv
Re(Z)
(8a)
F4
pnFr2 j0 pn2 F 2 D1=2 v1=2 r2 c+
D
2 4RT
rv
Im(Z)
(8b)
F5
2
2
1=2
1=2
2
+
pn F D v r c
D
where r is the radius of the disk, and F4(x ) and F5(x )
are defined as:
F4 (x)
g [J (bx
1
0
1=2
)]2
cos(u=2)
b(1 b4 )1=4
db
(9)
s
4RT
prn2 F 2 Dc+
(15)
Note that the variable x , relates the characteristic time
of the electrochemical system, r2/D , with the experimental time connected through the angular frequency,
2p/v .
As was stated in Section 2, a well-behaved impedance
function must be Kramers /Kronig transformable, so,
the limits of (8a) and (8b) should be evaluated in order
to prove that Eq. (11) is bounded. From Eqs. (12) and
(13) we obtain (see Appendix A for details):
Fig. 1. Nyquist diagram for a Butler /Volmer kinetic process in a
inlaid disk electrode. This graph corresponds to Eq. (11) in the text.
For high frequencies, Warburg behaviour is observed and plotted by a
line with a 458 slope. For low frequencies, diffusive transport resistance
is observed. The apex of the graph is evaluated numerically as vmax /
2.5119(D /r2).
J. Navarro-Laboulais et al. / Journal of Electroanalytical Chemistry 536 (2002) 11 /18
1
lim F4 (x) pffiffiffi
2 2
1
lim F5 (x) pffiffiffi
x0
2 2
x0
(16)
(17)
4 1=2
x
3p
(18)
1
lim F5 (x) pffiffiffi x
x00
4 2
(19)
lim F4 (x)
x00
The first two limits are related to the high frequency and
the macroelectrode response, while the other two are
related to the dc response and the steady state microelectrode behaviour. Let us analyse them separately.
No simplification or mention of the size of the
electrode has been made in the mathematical procedure
used to obtain Eq. (8). So, the equation remains valid
for micro- and macro-electrodes, where the distinction
between both types depends on the characteristic time
(r2/D ) and the experimental time. As the radius of the
disk electrode increases, the variable x approaches
infinity. Then, using Eqs. (16) and (17), the impedance
function (Eq. (11)) approaches:
s
lim Z(x)Rct pffiffiffi x1=2 (1j)
2 2
x0
(20)
Using the definition of s and x , we can prove that:
2
RT
Z(v)Rct pffiffiffi
(1j)v1=2
2 n2 F 2 pr2 D1=2 c+
From this equation we deduce an important consequence because it is related to the dc behaviour of a
microelectrode. The limit of Eq. (22) is a real value,
which is the sum of two contributions, the charge
transfer resistance and the diffusive transport resistance:
4s
lim Z(x)Rct Rct RDt
x00
3p
RT
3RT
2
2
pnFj0 r
p rn2 F 2 Dc+
Eq. (23) shows that for low frequencies the impedance of
an electrochemical system can be considered as two
resistances connected in series, one related to the
electron transfer kinetics, and the other to the mass
transport towards the electrode. Eq. (23) is also the
lower bound of the impedance (11), which then allows
the Kramers /Kronig transformation.
The value of the mass transport resistance, RDt, is
several orders of magnitude higher than the charge
transport resistance. This value is experimentally available only for low frequencies using small electrodes. For
example, considering a bulk concentration equal to
106 mol cm 3, and a diffusion coefficient of 10 5
cm s 2, at 298.15 K the mass transport resistance is
8094 V for a disk electrode with a radius of 1 cm and
8094 kV for a microelectrode of 10 mm.
Summarising, the impedance function (11) which may
be called the ‘generalised Warburg impedance’ is
bounded and reproduces the micro- and macroelectrode
behaviour. This function is a more general impedance
function that the Warburg impedance and proves the
importance of the geometrical aspects in the electrochemical response of an electrode. Finally, the deduction of Eq. (5) from Eq. (11) explains the apparent
anomaly of the Warburg impedance against KK-transformation. The complete function (11) is KK-transformable but its asymptotic behaviour for macroelectrodes is
not.
(21)
This equation matches the Warburg impedance exactly,
confirming that the asymptotic behaviour of Eq. (11) is
a more general form for the impedance of a macroelectrode. In this situation, the impedance approaches Rct as
the frequency tends to zero, which is the higher bound of
Eq. (11).
Conversely, in the limit when x approaches zero, the
expression matches the microelectrode behaviour. Using
Eqs. (18) and (19), the impedance (11) is reduced to:
4
x1=2
lim Z(x)Rct s
j pffiffiffi
(22)
x00
3p
4 2
15
4. Frequency limit of the Warburg impedance
In Section 3, the connection between the general
impedance function (Eq. (11)) and the microelectrode
and Warburg impedances has been shown. Thus, it is
advisable to have a criterion to distinguish between both
types of extreme behaviour. This rule could be deduced
from the value of the apex in Fig. 1 which depends on
the adimensional frequency, x , defined in Eq. (11), or
also from the maximum of the imaginary part of the
function (11).
Using this last procedure, the problem consists in
calculating the frequency at the maximum, vmax, solving
the equation:
dF5 (x)
2x0
F5 (x0 )0
(24)
dx
xx0
where
x0 (23)
This consequence agrees with the empirical observations
and the steady state behaviour of the microelectrodes.
r2
vmax
D
(25)
Eq. (24) has been deduced by applying the maximum
condition to the imaginary part of Eq. (11), and its
solution can be obtained by series expansion. Eq. (10)
can be rewritten as:
16
J. Navarro-Laboulais et al. / Journal of Electroanalytical Chemistry 536 (2002) 11 /18
0
F5 (x0 )
g
1=2
J21 (bx0 ) sin
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
u
tan(u) sin(u) du
2
(26)
p=2
with u /arctan b1. Making use of the properties of
the derivatives of the Bessel function, the solution of Eq.
(24) is the solution of:
1=2
F5 (x0 )2x0
0
g
1=2
1=2
J1 (bx0 ) J0 (bx0 ) sin
pffiffiffiffiffiffiffiffiffiffiffiffi
1
u
sin(u) du
2
p=2
(27)
0
The previous equation can be solved by expanding the
Bessel functions in series and evaluating the resulting
integrals. For the lower order terms of these expansions,
the integrals can be easily evaluated but this is not the
case for higher order terms. Separately, the programming of Eq. (27) allows its numerical evaluation
accurately enough. The details of both methods are
given in Appendix B leading to the following result:
vmax 2:5119
D
r2
(28)
For an electroactive substance with a coefficient of
diffusion of 105 cm2 s 1, the apex of the impedance in
a Nyquist diagram is located at 4 mHz for an electrode of
1.0 cm of radius, and 4 mHz for an electrode of 10 mm.
Eq. (28) allows the prediction of the effect of the
diffusion at the edges of the electrode because this
geometrical phenomenon is responsible to some extent
for the experimental deviation of macroelectrodes from
Warburg behaviour. Furthermore, the frequencies calculated from Eq. (28) are, from an experimental point of
view, very low to be attained with sufficient reliability to
avoid factors such as the unsteady state or natural mass
convection towards the electrode.
equation derived from the three-dimensional one to
describe the Warburg impedance is the consequence that
makes the KK-transformation impossible.
The derivation of the Warburg function from the
generalised one allows also the obtention of other
magnitudes useful from a practical point of view. The
use of this function is advisable when small electrodes,
not necessarily microelecrodes are used or when the
impedance is measured at very low frequencies. A fast
algorithm to be used in non-linear fitting programs has
been outlined in appendices. The use of the Fleischmann
and Pons impedance functions allows the calculation of
magnitudes of interest such as the charge transfer
resistance, Rct, the constant related to the mass transport, s , and the relaxation time that relates the size of
the electrode and the diffusion coefficient, r2/D . An
expression for a mass-transfer resistance using the
limiting properties of the generalised impedance function has also been obtained. This resistance is several
orders of magnitude greater than the charge transfer
resistance and was not always experimentally accessible.
An estimate of the order of magnitude of the angular
frequency at which this resistance is attainable or at
which the Warburg impedance begins to fail has also
been derived. This magnitude depends on the mass
transport towards the electrode and the size of the
electrode also.
Acknowledgements
J.J. Garcia-Jareño acknowledges financial support
from the Ministerio de Ciencia y Tecnologı́a (Programa
Ramon y Cajal). Financial support from the Ministerio
de Ciencia y Tecnologı́a (Project ‘MUVE’ MAT20000100-P4-03) is gratefully acknowledged by the authors.
Appendix A: Limits of functions F4(x ) and F5(x)
5. Conclusions
Any linear system can be characterised by a transfer
function resulting from the system of differential
equations, that includes the initial and boundary conditions, describing the physical problem. If the transfer
function is lineal, causal, stable and bounded, this
function is Kramers /Kronig transformable. In this
work the difficulties in carrying out these transformations on the Warburg impedance have been demonstrated. Using a more general model for the impedance
of a disk-shaped electrode inlaid in an insulating surface,
it was possible to demonstrate the Warburg impedance
as a limiting case of this generalised electrochemical
impedance with the integral transformations. Then, the
loss of information in the one-dimensional differential
In order to calculate the Kramers /Kronig transform
of the impedance function (11), the limits of functions
F4(x ) and F5(x ) must be evaluated. Using the definition
of F4(x ) (Eq. (11)) the limit of the argument of the
integral for x 0/ is:
2
2
2 1=2 1=2 ]
x1=2
2 [y (x y )
(A1)
[J1 (y)]2
lim [J1 (y)]
1=2
x0
y(x2 y2 )
yx
Then, the limit of F4(x ) is:
x1=2
lim F4 (x)$ pffiffiffi
x0
2
g
0
1
pffiffiffi
2 2
x1=2
1
dy pffiffiffi
[J1 (y)]
2
yx
2
g
[J1 (y)]2
dy
y
0
(A2)
J. Navarro-Laboulais et al. / Journal of Electroanalytical Chemistry 536 (2002) 11 /18
In a similar way, the limit of F5(x ) can be calculated:
(A3)
The limits of both functions for x 0/0 are calculated
using the same properties of the limits of the arguments
in the integrals. Using Eq. (11) again, we have:
[y2 (x2 y2 )1=2 ]1=2
[J1 (y)]2
lim [J1 (y)]2
(A4)
x00
y(x2 y2 )1=2
y2
Thus, the limit of F4(x ) is:
lim F4 (x)x
1=2
x00
g
[J1 (y)]2
y2
dy
4
3p
1=2
x
(A5)
Using the same procedure for F5(x ), this function tends
to zero for x 0/0, but a more interesting asymptotic
behaviour of this function can be obtained if the Bessel
function is expanded as a series:
x2 x4 5x6
4 16 768
(A6)
As x approaches zero, the faster the higher terms of the
above expression approach zero and become negligible.
Making use of the trigonometric properties, the integral
(10), an equivalent equation to Eq. (13), can be written
as:
1
F5 (x) pffiffiffi
2
g
4 1=2
[J1 (bx1=2 )]2
0
[(1 b )
2 1=2
b ]
b(1 b4 )1=2
db (A7)
Thus, substituting the first term of Eq. (A6) and
neglecting the others terms, we have:
x
F5 (x): pffiffiffi
4 2
g
b[(1 b4 )1=2 b2 ]1=2
0
g
b2
b4 2
x0 x0
4
16
sin
u
(tan(u) sin(u))1=2 du
2
p=2
1
4
u
sin(u) 1=2
sin
du
2 tan(u)
0
x0
g
p=2
1
16
0
x20
g
sin
u
sin(u) 1=2
2
tan3 (u)
du
(A10)
p=2
0
[J1 (x)]2 :
can be expressed as:
0
1
lim F5 (x) pffiffiffi
x0
2 2
17
(1 b4 )1=2
db
(A8)
The integral in the right hand side of this equation can
be solved and is equal to unity. So the asymptotic
behaviour of F5(x ) for x 0/0 is:
1
lim F5 (x) pffiffiffi x
x00
4 2
(A9)
Appendix B: Solution of Eq. (24)
The solution of Eq. (24), or its equivalent Eq. (27),
implies the evaluation of the integrals which include the
unknow, x0. If the Bessel functions are expanded in
series, we obtain a series of rational integrals that can be
solved and then, a polynomial in x0 is obtained. Using
the first two terms of the expression (A6) and neglecting
the rest, the integral involved in the definition of F5(x )
where the relation u /arctan b1 has been used. The
integrals in Eq. (A11) can be calculated leading to the
following expression in x0:
pffiffiffi
1
1
1 2
x0
F5 (x0 ): 1 pffiffiffi ln pffiffiffi
4
2 2
21
1 pffiffiffi
( 2 2)x20
(A11)
64
For the evaluation of the integral in the second term in
the left hand side of Eq. (27), the expansion of the
following function was used:
J1 (x) J0 (x)
1
3 3
5
x
x x5 2
16
192
(A12)
Carrying out the same procedure as before and rewriting
Eq. (27), we obtain an algebraic equation that allows us
to calculate the apex of the impedance function (11):
pffiffiffi pffiffiffi
3
1
1 2
22
1 pffiffiffi ln pffiffiffi
x0
4
2 2
21
64
pffiffiffi
3( 2 2) 2
x0 0
(A13)
32
This equation has two solutions, one of them has no
physical meaning because it is negative, and the other
one is:
x0 2:35325 . . .
If the number of terms used in the series expansion in
Eq. (A10) or in the integral in Eq. (27) is increased, the
accuracy of this result will be increased. Unfortunately,
the resulting integrals are too complicated and the
convergence of the resulting series should be tested.
So, an alternative and faster solution is the calculation
of x0 by numerical methods. Then, an efficient algorithm for the calculation of F4(x), F5(x ) or any integral
involved in the problem is necessary.
The Gaussian quadrature is a numerical method that
allows the calculation of integrals with high precision
using only a few values of the abcissae. The choice of
one or other quadrature scheme depends on the weighting function and the limits of the integral. In our case,
we have adopted the Gauss /Legendre quadrature
J. Navarro-Laboulais et al. / Journal of Electroanalytical Chemistry 536 (2002) 11 /18
18
scheme where the integration interval extends from 0 to
1. For this reason, the function F4(x ) must be written in
a more convenient form:
F4 (x)
g
[J1 (bx1=2 )]2
0
1
g [J (bx
1
0
1
g
0
1=2
)]2
cos(u=2)
b(1 b4 )1=4
cos(u=2)
b(1 b4 )1=4
db
db
1=2 2
x
cos(tan1 (1 g)2 =2)
J1
dg
[1 (1 g)4 ]1=4
1g
(A14)
The gaussian quadrature methods are very efficient
numerical methods for continuous and monotonic
functions, needing a small number of abcissae evaluations, ranging from 10 to 20. In our case, the integrands
of the functions F4(x) and F5(x ) are oscillating functions and the selection of a small number of intervals of
integration will lead to a lack of accuracy. So, in order
to avoid this inconvenience a number of points large
enough has been chosen in the interval. In particular our
integration interval, [0, 1] has been divided into 10 000
steps.
The evaluation of these integrals is time consuming
and the procedure given before is not appropriate for its
implementation in non-linear least-squares fitting programs for impedance. One solution of this problem is to
tabulate a number of values of F4(x ) and F5(x) large
enough for different values of x . Then the calculation of
both functions at any value of x could be performed by
rational interpolation. Once the algorithm is built, its
implementation in a program to obtain the solution of
Eq. (24) is straightforward. The algorithm written in
standard C can be requested via e-mail from the
authors.
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