www.elsevier.nl/locate/jelechem
Journal of Electroanalytical Chemistry 499 (2001) 112– 120
Theory of the electrochemical impedance of anomalous diffusion
Juan Bisquert a,*, Albert Compte b
b
a
Departament de Ciències Experimentals, Uni6ersitat Jaume I, 12080 Castelló, Spain
Brandeis Uni6ersity, Volen Center MS 013, 415 South Street, Waltham, MA 02454, USA
Received 5 May 2000; received in revised form 6 October 2000; accepted 2 November 2000
Abstract
This paper addresses the electrochemical impedance of diffusion in a spatially restricted layer. A physically grounded framework
is provided for the behavior Z(i…)8 (i…) − i/2 (0BiB2), thus generalising the Warburg impedance (i=1). The analysis starts
from the notion of anomalous diffusion, which is characterized by a mean squared displacement of the diffusing particles that has
a power law dependence on time r 28t i. Using a theoretical approach to anomalous diffusion that employs fractional calculus,
several models are presented. In the first model, the continuity equation is generalised to a situation in which the number of
diffusing particles is not conserved. In the second model the constitutive equation is derived from the stochastic scheme of a
continuous time random walk. And in the third, the generalised constitutive equation can be interpreted within a non-local
transport theory as establishing a relationship of the flux to the previous history of the concentration through a power-law
behaving memory kernel. This third model is also related to diffusion in a fractal geometry. The electrochemical impedance is
studied for each of these models, and the representation in terms of transmission lines is established. The main finding is that,
while models with quite different non-trivial diffusion mechanisms behave similarly in a semi-infinite situation, the consideration
of the effect of the boundaries gives rise to neatly different impedance spectra. © 2001 Elsevier Science B.V. All rights reserved.
Keywords: Ac impedance; Diffusion; Ionic conductivity; Anomalous diffusion; Equivalent circuit
1. Introduction
Electrochemical impedance is widely used to investigate the interfacial and bulk properties of materials and
to measure the relevant physico-chemical parameters.
The Warburg impedance
Z(i…)8 (i…) − 1/2
(1)
signals the occurrence of diffusion-controlled processes
in electrochemical systems [1 – 5], solid-state electronic
devices [6,7], and solid-state ionics [8]. It has long been
recognized, however, that in many cases impedance
measurements of diffusive processes give rise to power
laws in frequency which deviate more or less from the
exact 1/2 exponent law [9]. Such behaviour, namely,
Z(i…)8 (i…)
− i/2
(0 B i B2)
(2)
* Corresponding author. Tel.: +34-964-728057; fax: + 34-964728066.
E-mail address: bisquert@uji.es (J. Bisquert).
is discussed in this paper in terms of the theory of
anomalous diffusion [10,11].
Ordinary diffusion is described by the equation
(Fick’s second law)
#c
#2c
=D 2
#t
#x
(3)
which gives rise to the impedance of Eq. (1). When
diffusion takes place in a layer of thickness L, driven by
a diffusion overvoltage at x =0, the observed behaviour is not solely the Warburg impedance. In a
spatially restricted situation there are at least two distinct domains in the impedance spectra, which are
separated by the characteristic frequency
d = D/L 2
(4)
The Warburg impedance (1) occurs in the high frequency regime, …\ \ …d. At frequencies lower than
d, the impedance behaviour depends on whether the
diffusing species are reflected or extracted at the end of
the diffusion region (denoted here by x= L). The reason for this is that the frequency …d corresponds to the
0022-0728/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 0 2 2 - 0 7 2 8 ( 0 0 ) 0 0 4 9 7 - 6
J. Bisquert, A. Compte / Journal of Electroanalytical Chemistry 499 (2001) 112–120
transient time for a diffusing particle injected at x = 0
to cover a distance L. For … \ \…d the particles will
not sense the boundary at x = L, so that the system
will behave as semi-infinite. In fact, using a general
framework it was shown [5,12] that in ordinary diffusion, Warburg behaviour (1) occurs exactly at frequencies in excess of …d, irrespective of the boundary
condition at x= L. Consequently, the specific properties of the system at x = L cannot account for fractional exponents in the impedance as in Eq. (2), and
one is led to search for a description of transport
mechanisms which differs from ordinary diffusion.
Anomalous diffusion is characterized by a mean
squared displacement of the diffusing particles that
does not follow the ordinary linear law r 2 8t but,
more generally, has a power law dependence on time:
r 2 8 t i
(5)
Not surprisingly, many different mechanisms give
rise to this anomalous behaviour, including complex
flows [13,14], structural complexity in the substrate of
diffusion [15,16] and in the diffusing tracers [17]. Thus,
no single theory of anomalous diffusion can account
for all possible phenomenologies [18].
We explore here the implications for electrochemical
systems of a particular theoretical approach to anomalous diffusion [11], namely one that makes use of
fractional calculus (see Appendix A). The direct relationship of this formalism to a stochastic random walk
picture [10] (the continuous time random walk [19])
has established fractional diffusion equations as the
natural framework to describe a number of experimental situations, notably those involving structural complexity, like carrier diffusion in amorphous materials
[15,16,19] or gel electrophoresis [17]. Fractional diffusion equations also arise when one considers powerlaw memory kernels [11,20] or fractal geometries
[21,22]. It is therefore natural to consider these equations in the context of electrochemical impedance.
Impedance measurements exhibiting fractionary exponents have been previously discussed in terms of
certain phenomenological functions [8,9,23] although a
physical justification was not provided. Several studies
examined the fractional power law (5) using electrochemical techniques of the time domain [22,24 –26],
but some of these reports were concerned with ordinary diffusion toward a fractal surface, as overviewed
in Ref. [27].
In this paper several generalised schemes of diffusion are suggested in terms of fractional derivatives,
and their correspondence with a microscopic picture of
transport is discussed. This fractional derivative formalism provides rather simple expressions in the frequency domain, where it gives the power laws of
frequency that are often observed. This is the basic
reason why the various diffusion models developed
113
below provide reasonably simple analytical expressions
of the impedance, and so could be readily used in the
analysis of the data. It must be noted however that
each of these models assumes a unique diffusion mechanism at all scales of time and frequency, while systems in reality could present different mechanisms, as
is indicated in Section 5. Therefore, these models must
be regarded as a first approximation in the treatment
of a complex problem.
The structure of the rest of the paper is as follows.
In Section 2 an overview is presented of the transport
equations for anomalous diffusion previously derived
[11] on the basis of fundamental considerations. In
Section 3 the impedance of spatially-restricted diffusion is obtained for the various models with different
boundary conditions, and the behaviour of the response in the frequency domain is analysed. In Section
4 the impedance models are stated in terms of transmission lines. Section 5 provides a general discussion
of the results.
2. Theories of anomalous diffusion
Ordinary diffusion theory starts off from two equations, the continuity equation and the constitutive
equation. The continuity equation is a fundamental
conservation law relating the time variation of the
number density c to the macroscopic particle flux J:
#c
#J
=−
#t
#x
(6)
The constitutive equation (or Fick’s law) is a phenomenological equation relating the flux J to the gradient of concentration c through the diffusion
coefficient D
J= − D
#c
#x
(7)
When combined with the continuity equation, Eq. (7)
gives the phenomenological diffusion equation or
Fick’s second law (3).
In an anomalous diffusion situation the two Eqs. (6)
and (7) cannot hold at the same time, since that would
imply ordinary diffusion. Either one, or the other, or
both ought to be generalised. Here we take the fractional generalised equations as discussed in Ref. [11],
and we consider three cases, which are labeled anomalous diffusion Ia (ADIa), anomalous diffusion Ib
(ADIb), and anomalous diffusion II (ADII).
Note that here we do not consider finite relaxation
times, so ~ should be set to 0 in the equations of Ref.
[11]. The impedance with non-vanishing relaxation
times was treated in Refs. [28,29].
J. Bisquert, A. Compte / Journal of Electroanalytical Chemistry 499 (2001) 112–120
114
2.1. Anomalous diffusion Ia
diffusion in a fractal medium. Combining it with the
continuity Eq. (6), one obtains the fractional diffusion
equation:
We generalise the continuity equation as
#J
#c
=−
k
#t
#x
k
(kB1)
(8)
This means that the system is in a situation where the
number of diffusing particles is not conserved. No
particular mesoscopic approach has validated this phenomenological generalisation so far. It yields, however,
interesting impedance properties as we show in the
following section. The constitutive equation remains the
same (Eq. (6)). In this generalisation the diffusion equation reads
#2c
#kc
=
D
#t k
#x 2
(9)
Only the constitutive equation differs from ordinary
diffusion, being
( 1 − k (c
(t 1 − k (x
(k B1)
(10)
In Ref. [11], this constitutive equation is derived from
a stochastic scheme (continuous time random walk) in
the macrosopic limit. There, the random walkers wait
after each jump for a period of time drawn from a
broad power-law distribution. The effect is that some
particles remain stuck for a long time and diffusion is
slower (subdiffusion). An expression for the flux can be
obtained [11] and it is shown that the flux relates to the
concentration gradient through Eq. (10). This is the
situation that would correspond to the carrier diffusion
in amorphous conductors modeled in Refs. [15,16]. The
corresponding diffusion equation can be seen [11] to be
the same as Eq. (9). The interpretation of the macroscopic current is further discussed in Ref. [30].
2.3. Anomalous diffusion II
The continuity equation is taken as usual (Eq. (6))
and the constitutive equation is generalised to
#c
#
k−1
#t
#x
k−1
J = −D
(k B1)
(11)
Note that the fractional operator here is a fractional
integral, since the order of the derivative is negative, it
is also defined by Eq. (A1). As shown in Ref. [11] (see
also Refs. [20,31]), this generalised constitutive equation
can be interpreted within a non-local transport theory
as establishing a relationship of the flux J to the
previous history of the concentration c through a
power-law behaving memory kernel. An equivalent set
of equations is obtained in Ref. [22] for the case of
(12)
3. Linear diffusion impedance
One-dimensional transport by diffusion is considered
in a region 0B xB L. Small amplitude oscillating quantities are denoted by a tilde. The diffusion overvoltage
E [3] is located at the boundary x=0, determining the
concentration of diffusing species at the edge of the
diffusion layer. Therefore,
E0 =
2.2. Anomalous diffusion Ib
J = −D
#2 − kc
#2c
=D 2
2−k
#t
#x
dE
dc
c̃,
(x=0)
(13)
0
where subscript 0 denotes a derivative following local
equilibrium conditions.
The conduction current i0 at x= 0 corresponds entirely to the flux of diffusing particles:
i0 =qAJ0 ,
(x= 0)
(14)
where q is the charge that crosses the interface per
diffusing particle and A is the electrode area. The
amplitude of electrical voltage and current at each
frequency is obtained by Laplace transformation,
L[E0 ] =Uand L[i0 ] =I. The linear impedance is given by
Z(s)=
U(s)
I(s)
(15)
where s is the Laplace variable related to the angular
frequency by s= i…, and …=2yf, where f is the frequency. Using Eqs. (13) and (14) one obtains
Z(s)=RW
D C(s)
L j(s)
(16)
where L[c̃] =C, L[J0 ] =j, and the following parameter
has been introduced
RW =
L dE
qAD dc
(17)
0
Notice that the dimensions of RW depend on the dimensions of the parameter D, which are different in the
various frameworks presented above. RW is a resistance
if D is in cm2 s − 1.
The various diffusion equations for the small-amplitude sinusoidal concentration may be written in the
Laplace domain in the unified form
#2C 1
= C
#x 2 u 2
(18)
where u is a function of frequency, different for each
model, as stated below. Eq. (18) has the general
solution
J. Bisquert, A. Compte / Journal of Electroanalytical Chemistry 499 (2001) 112–120
x
x
C = B1 cosh + B2 sinh
u
u
(19)
The impedance Z(s) is found in each case using Eqs.
(16) and (19) and an additional boundary condition at
x = L. Two different boundary conditions are considered below. In the first one the particle flux is zero at
the boundary
#C
=0
#x
(x= L)
(20)
In the second one the concentration at the boundary
remains at the equilibrium level
C=0
(x =L)
(21)) upon encountering a wall at x= L [32]. A more
general boundary condition has been discussed in previous works [5,12].
In the following paragraphs the impedance functions
arising from different diffusion equations and boundary
conditions are derived and the limiting behaviour in
certain frequency ranges is discussed.
3.1. Ordinary diffusion
From Eqs. (6) and (7):
sC= −
#j
#x
(22)
j= − D
#C
#x
(23)
(21)
These standard conditions refer to a random walker
which is either reflected (Eq. (20)) or absorbed (Eq.
115
and
u(s)=
d
s
1/2
L
(24)
where …d is the characteristic frequency given by Eq.
(4).
3.1.1. OD with reflecting boundary
The impedance with boundary condition (20) is
Z(s)= RW(…d/s)1/2 coth[(s/…d)1/2]
(25)
This well-known function is represented in Fig. 1a.
The limiting behaviour at high frequencies …\…d is
the Warburg impedance Eq. (1). At low frequency one
gets the series combination of a resistance RW/3 and the
diffusion capacitance [5]
CW =
L2
RWD
(26)
3.1.2. OD with absorbing boundary
From boundary condition (21) the following result is
found:
Z(s)= RW(…d/s)1/2 tanh[(s/…d)1/2]
(27)
This is also a familiar result which is represented in
Fig. 2a. At high frequencies …\ …d the Warburg behaviour appears, and at low frequencies …B…d the
impedance coincides with a parallel combination of the
resistance RW and a capacitance C= CW/3 [5].
3.2. Anomalous diffusion Ia
Laplace-transformation of Eqs. (7) and (8) gives the
following equations:
Fig. 1. Complex plane plots of the impedance models for ordinary
and anomalous diffusion (k =0.8 and …d = 1 rad s − 1) discussed in
the text, with the reflecting boundary condition (c̃/(x= 0 at x =L.
The marked points correspond to the frequency (Hz) f= …d.
#j
#x
(28)
#C
#x
(29)
s kC= −
j= − D
J. Bisquert, A. Compte / Journal of Electroanalytical Chemistry 499 (2001) 112–120
116
Z(s)=RW(…d/s)k/2
(33)
For kB 1, Eq. (33) gives a straight line inclined at
less than 45° in the complex plot. At low frequency
(…B …d) Eq. (32) is approximately equal to
1
Z(s)= RW + Qs − k
3
(34)
where Q= RW… kd. The first term in Eq. (34) is a series
resistance; the second is a constant-phase element
(CPE) [5]. Thus at low frequencies the function (32)
also tends in the complex plot to an inclined straight
line, as seen in Fig. 1b. Comparing Eqs. (33) and (34)
one notices that the power-law exponent at low frequencies exactly doubles that at high frequency. These
features are also found in a model for impedance of
porous electrodes in which transport is driven by drift
in the electrical field [33].
3.2.2. ADIa with absorbing boundary
Applying now the boundary condition Eq. (21) it
follows that
Z(s)= RW(…d/s)k/2 tanh[(s/…d)k/2]
(35)
The function is illustrated in Fig. 2b. The behaviour
at high frequency is again Eq. (33). At low frequencies
B …d the admittance can be written, approximately,
as
Z(s) − 1 =
Fig. 2. Complex plane plots of the impedance models for ordinary
and anomalous diffusion (k =0.8 and …d = 1 rad s − 1) discussed in
the text, with the absorbing boundary condition c̃ = 0 at x =L. The
marked points correspond to the frequency (Hz) f= …d.
and
u(s) =
d
s
k/2
L
(30)
where
d =(D/L 2)1/k
(36)
where now Q= 3RW… kd. Eq. (36) is the parallel combination of a resistance and a CPE, giving at low frequency a depressed arc in the complex plot.
The impedance functions of anomalous diffusion Ia
have been previously suggested on a heuristic basis
[9,23] and were used for fitting experimental spectra
[8,23]. Since the model has been related to non-uniform
diffusion [9], it is worth noting that the assumed diffusion equation is spatially uniform, and it is rather the
time dependence that is generalised. Nonetheless, it
must be emphasized again that no particular mesoscopic foundation has been provided for this model of
anomalous diffusion.
(31)
The dimensions of …d in Eqs. (31) and (47) below are
in rad s − 1.
3.2.1. ADIa with reflecting boundary
Using boundary condition Eq. (20) the impedance is
obtained:
Z(s) =RW(…d/s)k/2 coth[(s/…d)k/2]
1
1
+
RW Qs − k
(32)
This function is shown in Fig. 1b. At high frequency
the impedance can be approximated by
3.3. Anomalous diffusion Ib
From Eqs. (6) and (10) one finds
sC= −
#j
#x
#C
j= − Ds 1 − k
#x
(37)
(38)
hence u and …d have the same expression as in Eqs. (30)
and (31).
J. Bisquert, A. Compte / Journal of Electroanalytical Chemistry 499 (2001) 112–120
3.3.1. ADIb with reflecting boundary
The impedance with the boundary condition Eq. (20)
is
Z(s) =RW… kd− 1(…d/s)1 − k/2 coth[(s/…d)k/2]
(39)
The function is represented in Fig. 1c. At high frequency one finds
Z(s) =RW… kd− 1(…d/s)1 − k/2
(40)
For kB 1, Eq. (40) gives a straight line inclined at
more than 45° in the complex plot. At low frequency
(…B…d) Eq. (39) gives
1
Z(s) = +Qs − (1 − k)
Cs
(41)
where C= 1/(RW… kd) and Q =RW/3. The first term in
Eq. (41) is the diffusion capacitance, and the second is
a constant-phase element (CPE).
3.3.2. ADIb with absorbing boundary
With the absorbing boundary
impedance adopts the form
Z(s) =RW
k−1
d
(…d/s)
1 − k/2
condition
k/2
tanh[(s/…d) ]
the
(42)
The function is shown in Fig. 2c. The behaviour at
high frequency is provided by Eq. (40). At low frequencies …B…d the admittance can be approximated by the
expression
Z(s) − 1 =Cs +
1
Qs
− (1 − k)
(43)
where C= 3RW…d and Q = RW. Eq. (43) is the parallel
combination of a capacitor and a CPE.
3.4. Anomalous diffusion II
(44)
j = − Ds k − 1
#C
#x
(45)
and in this case
u(s) =
d
s
1 − k/2
L
(46)
where
d =(D/L 2)1/(2 − k)
(48)
The complex plot representation of Eq. (48) is shown
in Fig. 1d. Limiting behaviour at high frequency is
Z(s)=RW… 1d − k(…d/s)k/2
(49)
in accordance with the generalised Warburg impedance
in Ref. [22]. At low frequency (…B…d) Eq. (48) can be
written
Z(s)= Hs 1 − k +
1
Cs
(50)
The capacitance in this last equation is C=1/
(RW… 2d − k). The other element in series H= RW/3 is a
resistance for k=1, but it tends towards inductive
behaviour when the anomalous exponent kB 1. This
element is termed a dispersive inductor here.
3.4.2. ADII with absorbing boundary
The boundary condition (21) provides the result
Z(s)= RW… 1d − k(…d/s)k/2 tanh[(s/…d)1 − k/2]
(51)
This function, shown in Fig. 2d, behaves as Eq. (49)
at high frequency. In the low frequency regime … B…d
the approximating expression is
Z(s) − 1 = Cs +
1
Hs 1 − k
(52)
where C= 1/(3RW… 2d − k) and H=RW. Eq. (52) can be
viewed as the parallel combination of a capacitance C
and a dispersive inductor H. As shown in Fig. 2d, the
impedance traces a loop into the fourth quadrant of the
complex plot at low frequencies.
4. Transmission line representation of diffusion
impedances
The flux and continuity Eqs. (6) and (11) are expressed in the Laplace domain as
#j
sC= −
#x
Z(s)=RW… 1d − k(…d/s)k/2 coth[(s/…d)1 − k/2]
117
(47)
3.4.1. ADII with reflecting boundary
Boundary condition (20) gives rise to the impedance
function
The impedance response of diffusion may be represented as a distributed equivalent circuit (a transmission
line) [5,9,34]. This equivalent circuit representation illustrates intuitively the structure of the diffusion models
and allows for better comparison. In this section the
various diffusion models previously introduced will be
expressed as transmission lines.
The electrical current im and potential ƒm in the
general transmission lines shown in Fig. 3 vary with
position according to the following equations [33]:
#ƒm
= − mim
#x
(53)
#im
1
= − ƒm
#x
nm
(54)
Here m is an impedance per unit length (V m − 1),
and nm is an impedance-length (V m) corresponding to
the whole electrode area A. The subscript m indicates
J. Bisquert, A. Compte / Journal of Electroanalytical Chemistry 499 (2001) 112–120
118
the fact that here these elements represent mass transport by diffusion, as will be evident immediately.
In the transmission line, the boundary condition at
x = L is represented by a specific impedance at that
point [5,35]. Considering, for concreteness, the reflecting boundary condition of Eq. (20), which appears as
an open circuit at x = L in Fig. 3a, it can be shown
readily from Eqs. (53) and (54) that the impedance,
defined as
Z=
ƒm(0)
im(0)
(55)
has the form
Z = (nmm)1/2 coth[L(m/nm)1/2]
(56)
Fig. 3. General distributed equivalent circuit representing the
impedance of diffusion in a finite layer. In (a) the reflecting boundary
condition (equivalent to an open circuit) is used at x = L, while (b)
has at x = L the absorbing boundary condition (short circuit).
It must be noticed that Eqs. (53) and (54) are isomorphous, respectively, to the constitutive relationship
and the continuity equation of the diffusion theories
presented above. The isomorphism is established by the
following expressions [5]:
im(x)= qAj
ƒm(x)=
dE
dc
(57)
(58)
C
0
This correspondence implies that the impedance of
the transmission line in Eq. (56), equals the impedance
of diffusion in a layer of thickness L with diffusion
overvoltage at x= 0. Notice that the correspondence
operates by conversion of the flux to electrical current,
Eq. (57), and conversion of the concentration to voltage
in the line, Eq. (58). Therefore, the impedance of any
diffusion theory that consists in equations similar to
Eqs. (53) and (54), can be pictured as a transmission
line, while the specific form of the elements m and nm
in the line depends on the specific features of the
diffusion model. In particular, the models of Eqs. (27),
(32), (39) and (48) can all be expressed as Eq. (56); the
forms of the elements m and nm in each case will be
given shortly.
Similar considerations would apply for the absorbing
boundary condition of Eq. (21), which is represented by
a short-circuit at x= L [5], as shown in Fig. 3b. The
impedance of the line is in that case
Z=(nmm)1/2tanh[L(m/nm)1/2]
(59)
The following correspondences are easily proved; the
various distributed equivalent circuits are shown in Fig.
4.
For ordinary diffusion
m = rm,
nm =
1
,
c ms
(60)
where
rm =
RW
L
(61)
cm =
CW
L
=
L
RWD
(62)
This is the familiar repetitive arrangement of resistors
rm and capacitances cm [5,12], shown in Fig. 4a.
For the model ADIa
m = rm,
nm = qms − k
(63)
where rm is given above in Eq. (61) and
Fig. 4. Transmission line representation of the impedance models for
(a) ordinary diffusion, (b) anomalous diffusion Ia, (c) anomalous
diffusion Ib and (d) anomalous diffusion II (notice that the induction
element is a dispersive inductor defined in Eq. (67)). The reflecting
boundary condition is used at x =L, so these are particular cases of
the general line shown in Fig. 3a.
qm =
RWD
L
(64)
In this model, shown in Fig. 4b, the crosswise capacitor of ordinary diffusion is replaced by a CPE.
J. Bisquert, A. Compte / Journal of Electroanalytical Chemistry 499 (2001) 112–120
For ADIb, as shown in Fig. 4c, the distributed
resistor of ordinary diffusion is replaced by a CPE:
m =qms − (1 − k),
nm =
1
c ms
(65)
where cm is that in Eq. (62) and
qm =
RW
L
(66)
Finally, ADII can be represented by the elements
m =hms 1 − k,
nm =
1
cms
(67)
where cm is defined by Eq. (62) and
hm =
RW
L
(68)
Now dispersive inductors occupy the place of the
resistors of ordinary diffusion, as shown in Fig. 4d.
It is interesting to note in Fig. 4 that the assumed
models of anomalous diffusion do not introduce additional elements in the basic equivalent circuit of ordinary diffusion (Fig. 4a); however, the more general
diffusion mechanisms modify the frequency behaviour
of the resistors and capacitors of Fig. 4a. Furthermore,
it will be noted that when the continuity equation is
generalised in ADIa, the capacitors of Fig. 4a are
affected; while if the constitutive equation is generalised, as in ADIb and ADII, then it is the resistive
elements of Fig. 4a that are modified. This illustrates
the fact that, while the two models labeled here ADI (a
and b) lead to the same diffusion Eq. (9), the two
theories have rather different structure, and consequently give rise to quite different impedance behaviour, as shown in Figs. 1 and 2.
5. Discussion
Anomalous diffusion in semi-infinite space gives rise
to the generalised Warburg behaviour (2), which appears as a straight line in the impedance representation.
Correspondingly, a straight line appears in logarithmic
plots of time transient measurements, Eq. (5). Such
behaviour is not the implication of a particular theoretical framework, and it was demonstrated that various
models starting from quite different assumptions reduce
to Eq. (2) in the frequency range where the ac signal is
not sensing the boundary, i.e. … \ \…d. On the other
hand, an interesting result emerging from our analysis
is that the specific diffusion mechanism heavily influences the intermediate and low frequency behaviour,
and so the effect of the finiteness of size of the diffusion
layer provides a distinction of models in the impedance
representation, as shown in Figs. 1 and 2.
119
The impedance functions of ordinary diffusion can
be stated in terms of two parameters only, namely RW
and …d. Other choices of parameters indicated above,
such as D and L, or the diffusion capacitance CW, are
possible, and these may be helpful for the physical
interpretation. The impedance functions for the anomalous diffusion models have three parameters, as they
contain the fractionary exponent k, in addition to RW
and …d. It is possible to determine k readily from the
slope of the high frequency line of the impedance
spectra. It must be noticed, though, that the high
frequency response is often influenced by other aspects
of the system’s response, for example the double layer
capacitance and charge-transfer resistance at the injecting boundary of the electrode [12], or the junction
capacitance in a semiconductor p–n junction [7]. Obviously, the transmission lines of Fig. 3 would be part of
a wider circuit, e.g. a Randles circuit, as indicated in
Ref. [12].
It has been remarked already that the boundary
condition at x= L cannot influence the high frequency
behaviour of the spectra. Furthermore, it may be
shown that if ordinary diffusion is coupled to a homogeneous reaction (by an additional term − kc in the
conservation equation), a slope of 1 would also be
obtained; detailed arguments in this respect will be
given elsewhere. Therefore, the main criterion to identify the anomalous diffusion behaviour is the slope of
the high frequency line.
Another aspect that should be taken into account in
the interpretation of experiments is the fact that the
diffusion mechanism need not be unique at all the
scales of frequency –time. It has been observed that the
normal behaviour in many disordered systems, consists
in an anomalous diffusion mechanism at high frequencies and short times, which changes to ordinary diffusion as the frequency decreases or the time scale
increases [36,37]. This would occur, for example, in a
material possessing a fractal structure in a small spatial
scale, and a regular euclidean structure when viewed at
a larger scale. Then, the resulting diffusion would be
either anomalous or ordinary depending on whether the
distance travelled by the random walker during the
observation time was shorter or larger than the cutoff
scale of the fractal structure [38]. These situations could
be treated introducing a characteristic crossover time or
frequency that separates the diffusion regimes. Such an
analysis was presented recently for impedance of
porous electrodes [39], and the models suggested here
could be developed along similar lines.
Acknowledgements
This work was supported by la Comisión Interministerial de Ciencia y Tecnologı́a under project PB98-1045.
J. Bisquert, A. Compte / Journal of Electroanalytical Chemistry 499 (2001) 112–120
120
AC acknowledges financial support from the Alfred P.
Sloan Foundation.
Appendix A
Fractional calculus uses fractional differentiation and
integration [40]. A fractional derivative is an integrodifferential operator that generalises the concept of
integer derivative to arbitrary order differentiation:
#hc
1
#n
=
#t h Y(1−h) #t n
&
t
0
dt%
c(x,t%)
(t −t%)h − n + 1
(A1)
where n is a natural number such that n − 1 BhB n.
Alternatively, one can use its Laplace transform
L[( hc/(t h]= s hL[c],
(h B 1)
(A2)
where s is the Laplace variable related to the angular
frequency by s =i…, i= −1. This representation
makes clear that integer derivatives are particular instances of this operator and it also makes their manipulations much easier. It applies for h ]1 as well, except
for some additional initial value terms (see Ref. [40]).
References
[1] E. Warburg, Drud. Ann. der Phys. 6 (1901) 125.
[2] J.E.B. Randles, Disc. Faraday Soc. 1 (1947) 11.
[3] K.J. Vetter, Electrochemical Kinetics, Academic Press, New
York, 1967.
[4] M. Sluythers-Rehbach, J.H. Sluythers, in: E. Yeager, J.O.’M.
Bockris, B.E. Conway, S. Sarangapani (Eds.), Comprehensive
Teatrise of Electrochemistry, vol. 9, Plenum Press, New York,
1984, p. 177.
[5] J. Bisquert, G. Garcia-Belmonte, F. Fabregat-Santiago, P.R.
Bueno, J. Electroanal. Chem. 475 (1999) 152.
[6] J. Lindmayer, C.Y. Wrigley, Fundamentals of Semiconductor
Devices, Van Nostrand, New York, 1965.
[7] J. Bisquert, G. Garcia-Belmonte, Electron. Lett. 33 (1997) 900.
[8] J. van Erle, A.J. McEvoy, J. Phys. Chem. Solids 55 (1994) 339.
.
[9] J.R. Macdonald (Ed.), Impedance Spectroscopy, Wiley, New
York, 1987.
[10] A. Compte, Phys. Rev. E 53 (1996) 4191.
[11] A. Compte, R. Metzler, J. Phys. A: Math. Gen. 30 (1997) 7277.
[12] J. Bisquert, G. Garcia-Belmonte, P.R. Bueno, E. Longo, L.O.S.
Bulhões, J. Electroanal. Chem. 452 (1998) 229.
[13] G. Matheron, G. de Marsily, Water Resour. Res. 16 (1980) 901.
[14] T. Solomon, E. Weeks, H. Swinney, Phys. Rev. Lett. 71 (1993)
3975.
[15] E.W. Montroll, H. Scher, J. Stat. Phys. 9 (1973) 101.
[16] H. Scher, E.W. Montroll, Phys. Rev. B 7 (1973) 4491.
[17] E. Yarmola, P.P. Calabrese, A. Chrambach, G.H. Weiss, J.
Phys. Chem. B 101 (1997) 2381.
[18] J.P. Bouchaud, A. Georges, Phys. Rep. 195 (1990) 127.
[19] E.W. Montroll, G. Weiss, J. Math. Phys. 6 (1965) 167.
[20] R.R. Nigmatullin, Phys. Stat. Sol. (b) 124 (1984) 389.
[21] R.R. Nigmatullin, Phys. Stat. Sol. (b) 133 (1986) 425.
[22] Y. Dassas, P. Duby, J. Electrochem. Soc. 142 (1995) 4175.
[23] R. Cabanel, G. Barral, J.-P. Diard, B. Le Gorrec, C. Montella,
J. Appl. Electrochem. 23 (1993) 93.
[24] T. Pajkossy, L. Nyikos, Electrochim. Acta 34 (1989) 171.
[25] T. Pajkossy, L. Nyikos, Electrochim. Acta 34 (1989) 181.
[26] M. Strömme, G.A. Niklasson, C.G. Granqvist, Phys. Rev. B 52
(1995) 14192.
[27] R. Kant, J. Phys. Chem. B 101 (1997) 3781.
[28] J.R. Ramos-Barrado, P. Galán Montenegro, C. Criado Cambón,
J. Chem. Phys. 105 (1996) 2813.
[29] C. Criado, P. Galán Montenegro, P. Velásques, J.R. RamosBarrado, J. Electroanal. Chem. 488 (2000) 59.
[30] D.H. Zanette, Physica A 252 (1998) 159.
[31] R.R. Nigmatullin, Phys. Stat. Sol. (b) 123 (1984) 739.
[32] S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1.
[33] J. Bisquert, G. Garcia-Belmonte, F. Fabregat-Santiago, N.S.
Ferriols, P. Bogdanoff, E.C. Pereira, J. Phys. Chem. B 104
(2000) 2287.
[34] G.C. Barker, J. Electroanal. Chem. 41 (1973) 201.
[35] J. Bisquert, Phys. Chem. Chem. Phys. 2 (2000) 4185.
[36] D.L. Sidebottom, P.F. Green, R.K. Brow, Phys. Rev. B 51
(1995) 2770.
[37] M. Ben-Chorin, F. Möller, F. Koch, W. Schirmacher, M. Eberhard, Phys. Rev. B 51 (1995) 2199.
[38] Y. Gefen, A. Aharony, S. Alexander, Phys. Rev. Lett. 50 (1983)
77.
[39] J. Bisquert, G. Garcia-Belmonte, F. Fabregat-Santiago, A.
Compte, Electrochem. Commun. 1 (1999) 429.
[40] K.B. Oldham, J. Spanier, The Fractional Calculus, Academic
Press, New York, 1974.