Ac-electrogravimetry study of electroactive thin films. I. Application to Prussian Blue

Article abstract of DOI:10.1021/jp013924x

Prussian Blue films were investigated by Using ac-electrogravimetry, that is, simultaneous impedance and electrogravimetry spectroscopies at different potentials between the totally reduced form, Everitt's salt, and the mixed-valence compound, Prussian Bl

Full text of DOI:10.1021/jp013924x

3182
J. Phys. Chem. B 2002, 106, 3182-3191
Ac-Electrogravimetry Study of Electroactive Thin Films. I. Application to Prussian Blue
Claude Gabrielli,*,† Jose J. Garc´ıa-Jaren˜o,†,‡ Michel Keddam,† Hubert Perrot,† and
Francisco Vicente‡
UPR 15 du CNRS, Physique des Liquides et Electrochimie, UniVersite´ Pierre et Marie Curie, 4, Place Jussieu,
75252 PARIS CEDEX 05, France, and Department Qu´ımica-F´ısica, UniVersidad de Valencia, Dr. Moliner 50,
46100 Burjassot, Valencia, Spain
ReceiVed: October 23, 2001
Prussian Blue films were investigated by using ac-electrogravimetry, that is, simultaneous impedance and
electrogravimetry spectroscopies at different potentials between the totally reduced form, Everitt’s salt, and
the mixed-valence compound, Prussian Blue, in KCl solutions. Experimental results allow clear information
on the role played by the different ions that are present in the solution to be obtained. It was found that when
the film was reduced potassium ions enter the film whereas these are protons that enter at potentials more
anodic than the formal potential of the redox process in KCl solutions. It was also proved that potassium ions
enter the PB film without water.
Introduction
Investigation of electroactive films coated on the surface of
electrodes has become of special interest in the last years. One
of the most studied materials is Prussian Blue (PB), which is
Fe4[Fe(CN)6]3. Prussian Blue is a well-known compound that
can be easily deposited on the surface of different types of
electrodes.16-21 These films present a well-structured crystal
building that makes this system very interesting to study the
charge transport processes through electroactive films.22-26
Prussian Blue can be obtained in different ways. However,
when PB films are galvanostatically produced following the
procedure described by Itaya et al.,18 it is possible to control in
a very accurate way the quality and the amount of deposited
PB. According to the results of Mortimer and Rosseinsky,20,21
the freshly deposited PB films are in the “insoluble” form,
Fe4[Fe(CN)6]3. After several voltammetric cycles in KCl solu-
tion, they are converted into the “soluble” structure, which is
characterized by the presence of potassium substituting for /4
of the high-spin iron sites in the “insoluble” PB structure
(KFeFe(CN)6).
Prussian Blue films can be reduced to the colorless form,
called Everitt’s salt (KFe4[Fe(CN)6]3 or K2FeFe(CN)6, ES),
or oxidized to the yellow form called Prussian Yellow
(Fe4[Fe(CN)6]3Cl or KFeFe(CN)6Cl, PY). These electrochemical
processes can be easily detected by cyclic voltammetry of PB
films in KCl solution. The reduction process for the “soluble”
PB structure has been described as follows:18
These last years, electroactive thin films were widely
investigated and numerous publications are available. Numerous
potential technological applications of these materials need the
development of thorough studies to understand the relative
contribution of the anions and cations to the charge compensa-
tion process occurring during the switching of the film between
oxidation states. Despite the large quantity of papers, much work
still remains to be carried out to understand the electrochemical
response of these films. In particular, ingress and release of ions
into and from the film can be accompanied by solvent movement
either by solvation, exclusion, or other processes.
Ion and solvent transport in electroactive thin films has been
investigated by using various methods: first, by recording
current and mass changes during potential sweeps by means of
an ammeter and a quartz crystal microbalance,1-4 second, by
measuring the electrochemical impedance of the coated
electrode,5-8 and then by probe beam deflection (mirage
effect).9-14 However, these techniques are not sufficiently
efficient and selective concerning the separation of the transport
of the various species involved in the charge compensation
process. Moreover, usually these techniques give information
on the global response of the electrode/film/solution interface.
As an example, cyclic voltammetry measures the global value
of the current, and it is not possible to discriminate between
anion ingress or cation expulsion for the charge compensation.
In addition, if neutral molecules are involved, their movement
is hardly perceptible. Even when cyclic voltammetry was
coupled with a quartz crystal microbalance (QCM), which adds
the global mass response to the current, their quantitative use
is limited to the case where only one species is involved in the
charge compensation process. The QCM can be also coupled
with other analytical techniques such as radiolabeled techniques;
pertinent information was obtained, but the kinetic aspect was
rather difficult to explore.15
1
kc
KFeIIIFeII(CN)6 + e- + K+ {
\
} K2FeIIFeII(CN)6
(i)
k′
c
where FeIII and FeII refer to the different oxidation states of Fe
atoms in the PB structure.
The impedance spectra and voltammetric response of PB films
in KCl solutions have been the subject of many papers in the
past few years.22,23,27,28 However, many aspects are not yet
clarified about the electrochemical behavior of these films,
especially concerning the entry into and exit from the PB film
of ions and solvent.
* To whom correspondence should be addressed. cg@ccr.jussieu.fr.
† Universite´ Pierre et Marie Curie.
According to the reaction described in scheme i, the reduction
of PB to the ES form is accompanied by the entrance of
‡ Universidad de Valencia.
10.1021/jp013924x CCC: $22.00 © 2002 American Chemical Society
Published on Web 03/01/2002

Electroactive Thin Films: Prussian Blue
J. Phys. Chem. B, Vol. 106, No. 12, 2002 3183
solvated an extra mass nms, where n is the number of solvent
molecules is added to their molar mass), and Ɛc, Ɛa, and Ɛs
are the number of moles exchanged for cationic species, anionic
species, and solvent per surface unit, respectively.
The doping mechanism of an electroactive film can be
described simply by
kc
P + e- + M+ {k′
\
} P,M
c
and
P + A- {kka′
\
} P,A + e-
a
Figure 1. Schema of the electrode/electroactive film/solution system.
potassium ions to maintain the film electroneutrality. However,
the insertion or expulsion of ions from the film can also be
accompanied by an exchange of solvent molecules between the
film and the outer solution.
where P is the host electroactive film and P,M and P,A-
are the inserted cation and anion in the electroactive film matrix.
Therefore, the net instantaneous molar flux of species i (c, a,
or s) is Ji ) dêi/dt in mol cm-2 s-1. It can be also expressed in
terms of concentration as Ci ) êi/dfilm where dfilm is the film
thickness. According to the formalism used in another reference
and by applying the classical kinetic law for heterogeneous
reactions, the time derivative of the concentration Ci is equal
for each species to46, 40
We propose here to use ac-electrogravimetry, that is, simul-
taneous coupling of electrochemical impedance and mass/
potential transfer function measurements, to characterize ion and
solvent motions at the PB film/electrolyte interface. Ac-
electrogravimetry allows the mass response to a small potential
perturbation to be analyzed thanks to a fast quartz crystal
microbalance used in dynamic regime.29-31 This technique has
been already used for studying oxide materials such as WO3,32,33
iron behavior in acidic medium,34 and conducting polymers,35-38
and especially polyaniline films.39-41
dê
dCi
film dt
Ji(dfilm) ) - i ) -d
)
dt
ki(Ci - Ci ) - k′(Ci - Ci)Ci (4)
i
min
max
sol
Theory
where dCi/dt > 0 for inserted species, Ci is the maximum
max
concentration of the sites available for insertion, Ci is the
In this first approach, charge transfer at the electrode/film
interface and mass transport in the solution and in the film are
considered as fast as the charge transfer at the film/solution
interface, which is considered as the rate-limiting step. This
hypothesis is justified for thin films (∼0.1µm) and can be
controlled by electrochemical impedance measurement. Indeed,
when the mass transport process is the rate-limiting step, a
Warburg response is observed with a 45° characterizing slope
in the electrochemical impedance. If this condition is not
fulfilled, theory taking into account limitation by diffusion in
the film or migration inside the film was already presented.42-45
In this theoretical part, only one cation (solvated or not), one
anion, and one free solvent, that is, solvent not in the solvation
shell of an ion, are considered for monocharged ions, (zi ) 1).
Ion Movement. Figure 1 shows the scheme of the electrode/
film/solution. An electrolytic medium was considered from
where cation (c), anion (a), and solvent (s) can eventually move
through the electrolyte/film interface to neutralize the charges
created by the redox reaction of the electroactive film. It should
be noted that each ionic species can be accompanied with
solvent molecule(s). By definition, the fluxes of species i (i )
c, a, or s), Ji, are positive for outgoing ions:
min
minimum concentration of the sites occupied by the species in
the host film, Ci is the concentration of species i in the solution,
sol
and ki and ki′ are the kinetic rate constants of transfers, which
are potential dependent, ki ) ki0 exp bi(E - E0) and k′ ) ki0
i
i
exp b′(E - E0i ) where Ei0 is the normal potential.
i
Steady State and Quasi Steady State. At steady state, the
global flux of species is zero, Ji(x) ) 0; therefore, Ci and E are
constants throughout the film. Thus, the derivative of the
concentration of the species i with respect to the potential is
obtained and is equal to39,40
C
- C
imin
dCi(E) bi - b′
imax
i
)
(5)
(b′ - bi)(E - E0i - Ei)
dE
4
i
cosh2
[
]
2
where Ei was already defined in another paper.40
Dynamic Behavior. In the dynamic regime, when a small
sine wave potential change, ∆E, is imposed across the metal/
film/electrolyte interfaces, low-amplitude sine wave concentra-
tions, ∆Ci, and fluxes, ∆Ji, are observed such as
∆Ji(dfilm) ) -jωdfilm∆Ci ) Ki∆Ci(dfilm) + Gi∆E
where i ) c, a, s (6)
Ji > 0 for x > 0
i ) a, c, s
(1)
If cations (c), anions (a), and solvent (s) are involved in the
redox reaction, the associated mass change and the electric
charge passed through the electrode polymer interface, per unit
surface, are equal to
where
Gi ) [biki(Ci - Ci ) - b′k′(Ci - Ci)Ci ]
(7)
i
i
min
max
sol
and
∆q ) -F∆êc + F∆êa
(2)
(3)
Ki ) ki + k′Ci
(8)
∆m ) mc∆êc + ma∆êa + ms∆ês
i
sol
where mc, ma, and ms are the molar masses of the cation (c),
the anion (a), and the solvent (s), respectively (if ions are
where Gi < 0 for inserting species and Gi > 0 for expelling
species.

3184 J. Phys. Chem. B, Vol. 106, No. 12, 2002
Gabrielli et al.
Then, according to eq 6,
It has to be noticed that when only one species, a or c, is
involved in the charge exchange, (∆m/∆q)(ω) is frequency-
independent and equal to ma/F or mc/F, respectively.
∆C
∆E
-Gi
jωdfilm + Ki
i(ω) )
(9)
Partial Electrogravimetric Transfer Functions. Because
generally three quantities are unknown, the cation, anion, and
solvent responses to ∆E, and only two quantities, (∆E/∆I)(ω)
and (∆m/∆E)(ω) are measured, it is not possible to separately
calculate the three unknown quantities. However, it is possible
to eliminate the contribution of one species from the mass/
potential transfer function. First, the anion (a) response is
eliminated from the global electrogravimetric transfer function,
(∆m/∆E)(ω), by considering39,40
Thus, the ∆Ci/∆E(ω) allows the electrochemical impedance,
(∆E/∆I)(ω), and the electrogravimetric transfer function, (∆m/
∆E)(ω) to be calculated.
Indeed, the Faradaic current density change, ∆IF, is related
to the charge change, ∆q, by ∆IF ) jω∆q, and by dividing eq
2 and 3 by ∆E and by using eq 6 it comes
∆E
∆I
1
(ω) )
(10)
∆m
cs(ω) )
ma ∆q
F ∆E
∆C
∆C
∆m
∆E
jωCinterface + jωdfilmF -
c(ω) +
a(ω)
(ω) -
(ω)
(18)
[
]
∆E
∆E
∆E
and
By using eq 9 and 11, the partial electrogravimetric transfer
function (∆mcs/∆E)(ω) relative to cations and solvent alone is
equal to
∆C
∆C
a ∆E
∆C
s ∆E
∆m
(ω) ) dfilm
m
c(ω) + m
a(ω) + m
s(ω) (11)
c ∆E
[
]
∆E
∆m
Gc
Gs
cs(ω) ) -dfilm (mc + ma)
+ m
where Cinterface is the polymer/solution interface capacitance, the
electrolyte resistance being neglected here.
By using eq 9, one obtains
[
]
∆E
jωdfilm + Kc
sjωdfilm + Ks
(19)
In the same way, if a cation contribution is eliminated in the
mass change function, the resulting partial electrogravimetric
transfer function (∆mas/∆E)(ω) is equal to
∆E
(ω) )
∆I
1
(12)
Gc
Ga
∆m
as(ω) )
mc∆q
F ∆E
∆m
∆E
jωCinterface + jωdfilm
F
-
(ω) +
(ω)
(20)
[
]
jωdfilm + Kc jωdfilm + Ka
∆E
and it can be shown that
∆m
and
∆m
∆E
Ga
Gs
sjωdfilm + Ks
as(ω) ) -dfilm (ma + mc)
+ m
(ω) )
[
]
∆E
jωdfilm + Ka
Gc
Ga
Gs
(21)
-dfilm
m
+ m
+ m
cjωdfilm + Kc
ajωdfilm + Ka
sjωdfilm + Ks
(
)
From the plot of the partial electrogravimetric transfer functions,
the number of species involved in the charge compensation
process can be evaluated.41 This elimination of one species over
three is close to the method used by Bruckenstein and Hillman
as a diagnostic for species interference.47-49
Example with One Ion. To explain the basic concepts of
this approach for simple systems, four main transfer functions,
(∆E/∆I)(ω), (∆q/∆E)(ω), (∆m/∆E)(ω), and (∆m/∆q)(ω), were
calculated from eqs 12, 15, 13, and 17 with one interfering ion,
which can be inserted (or expulsed) in (or from) the film. Table
1 presents the two typical results for cation and anion exchanges
in the case of one involved charged species.
To counterbalance the positive charges created by a potential
increasing across the electroactive film, that is, for an oxidation
step, the motion of the ionic species is well defined: expulsion
of cations or insertion of anions. For this example, the solvent
effect is neglected and the molar masses of the ions were given
as in KCl solution. The two parameters Gi and Ki were arbitrarily
chosen to give reasonable results for the various transfer func-
tions. The four transfer functions (∆E/∆I)(ω), (∆q/∆E)(ω), (∆m/
∆E)(ω), and (∆m/∆q)(ω) are plotted in Table 1 and param-
etrized over the frequencies, f, where ω ) 2πf. The arrows
indicate the change of the frequency f from low frequencies to
high frequencies. For the electrochemical impedance, (∆E/
∆I)(ω) (Table 1a and e), the high-frequency loop corresponds
to charge transfer in parallel to double-layer charging and the
straight line, in the lower frequency range, to ionic transfer.
For (∆q/∆E)(ω) (Table 1b and f), only one low-frequency loop
(13)
Of course, only the charged species, anions and cations, are
implied in the electrochemical impedance, (∆E/∆I)(ω), whereas
the electrogravimetric transfer function, (∆m/∆E)(ω), depends
on cations, anions, and solvent all together.
Two other interesting quantities can be calculated to dis-
criminate the influence of the various species: the electric
charge/potential transfer function, (∆q/∆E)(ω), according to eq
10,
∆I
(ω) )
F(ω)
(14)
(15)
∆q
∆E
1
jω ∆E
Gc
Ga
) dfilm
F
-
[
]
jωdfilm + Kc jωdfilm + Ka
and the mass/electric charge-transfer function, (∆m/∆q)(ω),
where
∆m
∆q
∆E ∆m
(ω) ) jω (ω) (ω)
∆IF ∆E
(16)
Gc
Ga
Gs
- m
+ m
+ m
cjωdfilm + Kc
ajωdfilm + Ka
sjωdfilm + Ks
(
)
)
Gc
Ga
jωF
-
[
]
jωdfilm + Kc jωdfilm + Ka
(17)

Electroactive Thin Films: Prussian Blue
J. Phys. Chem. B, Vol. 106, No. 12, 2002 3185
TABLE 1: Schematic Representation of the Four Transfer
Functions, (∆E/∆I)(ω), (∆q/∆E)(ω), (∆m/∆E)(ω), and
(∆m/∆q)(ω)F
only one ion is taken into account, the theoretical partial
electrogravimetric transfer function takes a zero value.
In this paper, Prussian Blue films (PB films) were investigated
by ac-electrogravimetry at various potentials to obtain informa-
tion on the role of the different ions at these potentials. The
information extracted from the use of this technique will be
analyzed and used to explain several aspects related to the role
of protons and potassium ions during the Prussian Blue S
Everitt’s salt reaction.
Experimental Section
Film Preparation. FeCl3 (chemically pure), K3Fe(CN)6, KCl,
and HCl (p.a.) (A. R. Merck) were used for the synthesis of PB
films. Water was deionized and distilled. One of the Au
electrodes on the quartz crystal was immersed into 0.02 M
K3(Fe(CN)6), 0.02 M FeCl3, and 0.01 M HCl aqueous solutions.
Electrodeposits of PB were galvanostatically carried out by
applying a controlled cathodic current of ic ) 11 µA for 210 s.
The film thickness can be estimated from the electro-
deposition charge:50
Q
4FA
dfilm
)
(l0)3NA
(22)
where Q is the electrodeposition charge (2.31 mC), A is the
electrode area (0.27 cm2), F is the Faraday constant, l0 is the
length of the unit cell (10.17 Å) of the PB crystal, and NA is
the Avogadro number. The value 4 appears because there are
four effective iron atoms per unit cell. The estimated film
thickness for these experiments was dfilm ) 0.14 µm.
Experimental Setup. Electrochemical experiments were
carried out by means of a typical three electrode cell polarized
by using a potentiostat (SOTELEM). PB films were deposited
onto one of the gold electrodes of the crystal used as the working
electrode, a large area grid made of platinum was used as the
counter electrode, and a saturated calomel electrode (SCE) was
the reference electrode.
The electrochemical quartz crystal microbalance takes ad-
vantage of the change of the resonance frequency of a 6 MHz
“AT-cut” quartz crystal (CQE, Troyes, France) due to a minute
mass change of one of its electrodes exposed to the solution.
The two gold electrodes, deposited on the opposite faces of the
quartz crystal, allowed the resonator to be electrically connected
to an oscillator circuit. The validity of the Sauerbrey equation51
was checked with the PB films used in this work through
electroacoustic measurements.52 In the whole potential range,
the microbalance frequency response can be interpreted as a
pure mass change if the film thickness is less than 0.15 µm.
Above this limit, film viscoelastic contributions can drastically
affect the mass change estimation. An experimental value for
the mass/frequency coefficient sensitivity, -7.5 × 107 Hz g-1
cm2, was used for the experimental treatment of all of the
gravimetric data; this coefficient was previously estimated
through copper electrodeposition.
To use this fast QCM in ac mode, the modified working
electrode was polarized at a chosen potential and a sinusoidal
small amplitude potential perturbation was superimposed.41 The
microbalance frequency change, ∆fmicrobalance, corresponding to
the mass response, ∆m, of the modified working electrode, was
detected by means of a special homemade frequency/voltage
converter.41 The resulting signal was simultaneously sent with
the current response, ∆I, of the electrode to a four-channel
frequency response analyzer (Solartron 1254), which allowed
the electrogravimetric transfer function, ∆m/∆V(ω), to be
simultaneously obtained with the electrochemical impedance,
is observed, which is related to the charge compensation process,
either of the cation or of the anion. These two first electro-
chemical transfer functions, (∆E/∆I)(ω) and (∆q/∆E)(ω), do
not allow the ionic species to be chemically identified. The
electrogravimetric transfer function, (∆m/∆E)(ω) (Table 1c and
g), is more informative. First, the location of the loop depends
on the charge of the ion, this loop is in the third quadrant if
cations are expelled and in the first quadrant if anions are
inserted. Indeed, according to eq 17, the low-frequency limit
of (∆m/∆E)(ω) is related to the parameter Gi/Ki. At low
frequency, this limit is equal to (∆m/∆E)(ωf0) ) -dfilmmiGi/
Ki, which is a real positive value when ions are inserted as Gi
< 0 and the loop is located in the first quadrant. On the contrary,
when ions are expelled, Gi > 0, the sign of (∆m/∆E)(ωf0)
becomes negative and the semicircle obtained is located in the
third quadrant. These results are in good agreement with the
electrochemical behavior of the film. Indeed, when this latter
is oxidized (∆E > 0), positive sites are created in the film and
if the cation expulsion is the main reaction (∆m < 0) during
the charge compensation, the low-frequency limit of the
electrogravimetric transfer function becomes negative ((∆m/
∆E)(ωf0) < 0) as illustrated in Table 1c. If the charge
compensation is due to anion insertion, an increase of the mass
of the film is induced (∆m > 0) and, therefore, (∆m/∆E)(ωf0)
reaches a positive value (Table 1g). Second, the diameter of
the (∆m/∆E)(ω) depends also on the mass of the involved ionic
species, mi, because (∆m/∆E)(ωf0) ) -dfilmmiGi/Ki. The last
transfer function, (∆m/∆q)(ω), gives directly the atomic mass
of the ion with a negative value for the cation and a positive
value for the anion. Moreoever, the ratio mass/charge is
independent of the frequencies, f, when only one ionic species
is involved in the charge compensation reaction.
Theoretical partial electrogravimetric transfer functions were
not plotted in these conditions as according to eq 19 and 21; if

3186 J. Phys. Chem. B, Vol. 106, No. 12, 2002
Gabrielli et al.
Figure 2. Curve of current-mass vs potential (a) for a PB film in a
KCl solution during a linear potential scan (experimental conditions;
pH ) 2.8, KCl concentration ) 0.5 M, scan rate ) 20 mV s-1, and
reference SCE (potential limits from +0.6 to -0.2 V and from -0.2
to +0.6 V)) and (b) curve mass vs charge for a cathodic sweep (same
conditions as previously).
Figure 3. Electrochemical impedance spectra of a PB film at different
pH in a KCl 0.5 M solution: (a) E ) +0.2 V; (b) E ) +0.3 V. PB
film has been cycled for five times at each pH before the EIS
measurement takes place to stabilize the system. Changes of the pH
values were obtained by additions of HCl.
occur by the entrance of potassium ions within the film.18 Figure
2a shows a voltammogram of a film of PB in a KCl solution
and the change of mass measured through the quartz micro-
balance accompanying the current variation. The hysteresis
observed shows that the electrochemical process is kinetically
limited. However, this approach does not allow determination
of whether it is mass transport or mass transfer at the film/
solution interface that is responsible for kinetic limitations. The
mass decrease, ∆m, between the reduced (Everitt’s salt) and
∆V/∆I(ω), where ∆V is the raw potential response, which takes
into account the electrolyte resistance.
Correction for the Ohmic Drop. To compare experimental
results with theoretical ones, the raw electrochemical impedance,
∆V/∆I(ω), and the raw electrogravimetric transfer function, ∆m/
∆V(ω), were corrected for the ohmic drop in the solution,
characterized by the electrolyte resistance, Rel, by
∆E
∆I
∆V
∆I
(ω) )
(ω) - Rel
(23)
the oxidized (Prussian Blue) form was about 1.2 µg cm-2
.
Meanwhile, the expected mass change, ∆m, evaluated from the
electric charge enclosed within the voltammogram by consider-
ing that all of the charge compensation involved only the
and
participation of potassium countercations is about 1.6 µg cm-2
.
∆V
∆I
(ω)
∆m
∆E
∆m
∆V ∆V
This discrepancy might show that lighter species were involved
in the charge compensation process or that there was a flux of
water molecules in the opposite sense to the cation flux or that
anions were involved. However, it was not possible to discern
between these three possibilities from the global information
provided by the quartz microbalance operating in quasi steady
state. When the ∆m - ∆q curve is plotted (Figure 2b), the
calculation of the slope in the linear part gives an equivalent
molar mass of 31 g mol-1, which does not correspond to any
species present in the solution.
(ω) )
(ω)
(24)
(ω) - Rel
∆I
to be compared with the corresponding quantities calculated
from the model presented above.
Results and Discussion
In a first step, current and mass responses to a potential scan
were simultaneously measured in KCl aqueous solutions. Then,
measurements of the electrochemical impedance and the elec-
trogravimetric transfer function were carried out.
Voltammetric and Mass/Potential Curves. During the
reduction of PB films to the colorless Everitt’s salt form in KCl
solutions, the charge compensation is commonly supposed to
Voltammetric studies of PB films in aqueous KCl solutions
showed a Nernstian dependence on the KCl concentration of
the redox potential, evaluated from the mean of the anodic and
cathodic peak potentials.18,19 However, in KCl solution, although
the redox potential slightly varied with the pH, the shape of

Electroactive Thin Films: Prussian Blue
J. Phys. Chem. B, Vol. 106, No. 12, 2002 3187
Figure 4. Experimental and theoretical transfer functions at 0.250 V vs SCE in 0.5 M KCl: (a) experimental and theoretical electrochemical
impedance, (∆E/∆I)(ω), with dfilm ) 0.1 µm, Cinterface ) 8 µF cm-2, Kc ) 4.4 × 10-5 cm s-1, Gc ) 8.8 × 10-7 mol s-1 cm-2 V-1; (b) transfer
function between the electric charge and the potential, (∆q/∆E)(ω), experimental and theoretical data; (c) electrogravimetric transfer function,
(∆m/∆E)(ω), experimental and theoretical data with mc ) 39 g mol-1; (d) partial electrogravimetric transfer function, (∆mas/∆E)(ω), with c ) K+;
(e) transfer function between the mass and the electric charge, (∆m/∆q)(ω)F, experimental and theoretical data; (f) real part of the transfer function
between the mass and the electric charge, Re(∆m/∆q)F, as a function of the frequency, experimental and theoretical data.
the voltammograms was shown to be strongly dependent on
the pH.53
applied potential during the redox processes, that is, with the
oxidation state of the PB film. To quantify the role of each
cation (protons and potassium ions) for each potential value
and to check whether the solvent or anions can also interfere in
the redox process, electrogravimetric transfer function measure-
ments were performed.
Electrogravimetric Transfer Function. The mass response,
∆m, of PB films to a small amplitude potential perturbation,
∆E, was studied at several imposed potentials, with respect to
the frequency, to obtain extra information about the role of
protons and potassium ions during the electrochemical redox
reaction of PB. Before carrying out the experiments, one had
to wait for the steady state (I ) 0) to be reached.
Electrochemical Impedance Measurements: Influence of
the pH. The impedance spectra of the PB film are strongly
dependent on the pH of the solution.24 Figure 3 shows the
electrochemical impedance spectra for a PB film at various pH
at a potential of +0.2 V (Figure 3a), which is the formal
potential for the PB S ES process, and at a potential of 0.3V
(Figure 3b). The pH dependence of the electrochemical imped-
ance was much more important at E ) 0.3 V than at E ) 0.2
V. Because there is no part with a slope equal to 45° in the
impedance response (Figure 3a), the rate-limiting step is not
mass transport in the film but rather ionic transfer between the
solution and the film.
Figures 4 and 5 show the four experimental transfer functions
for a PB film at two potentials, +0.250 V/SCE and 0.375
V/SCE. Electrochemical impedances, (∆E/∆I)(ω), corrected for
ohmic drop according to eq 23, are shown in Figures 4a and
These results indicate, in a qualitative way, that, in addition
to the participation of potassium, protons take part in the charge
compensation process. This participation changes with the

3188 J. Phys. Chem. B, Vol. 106, No. 12, 2002
Gabrielli et al.
Figure 5. Experimental and theoretical transfer functions at 0.375 V vs SCE in 0.5 M KCl: (a) experimental and theoretical electrochemical
impedance, (∆E/∆I)(ω), with dfilm ) 0.1 µm, Cinterface ) 8 µF cm-2, Kc ) 1.2 × 10-5 cm s-1, Gc ) 6.3 × 10-8 mol s-1 cm-2 V-1; (b) transfer
function between the electric charge and the potential, (∆q/∆E)(ω), experimental and theoretical data; (c) electrogravimetric transfer function,
(∆m/∆E)(ω), experimental and theoretical data with mc ) 23 g mol-1; (d) partial electrogravimetric transfer function, (∆mas/∆E)(ω) with mc ) 23
g mol-1; (e) transfer function between the mass and the electric charge, (∆m/∆q)(ω)F, experimental and theoretical data; (f) real part of the transfer
function between the mass and the electric charge, Re(∆m/∆q)F, as a function of the frequency, experimental and theoretical data.
5a. Then, (∆q/∆E)(ω) were calculated by using (∆E/∆I)(ω)
experimental data and plotted in Figures 4b and 5b. A single
loop was obtained for the two potentials. This means that only
one type of species, anions or cations, was involved in the redox
reaction, but the charge of the ion was not determined as it was
previously indicated in Table 1b and f. Moreover, the role of
the solvent was not clarified at this stage. The experimental
electrogravimetric transfer function, (∆m/∆E)(ω), experimen-
tally corrected for ohmic drop (eq 24), is presented in Figures
4c and 5c. Only one semicircle is obtained, and its location in
the third quadrant indicates that the ionic species involved here
was a cation (Table 1c). This single loop confirms the lack of
a contribution of anions for this material. As the relaxation times
are usually different for anions, cations, and free solvent (see
companion paper), if only one loop is observed, it means that
only one species is involved in the redox process. If two ionic
species have the same charge, only one loop is found whatever
their molar mass. But, the chemical identification of the species
is not available at this step. Only, the partial electrogravimetric
transfer function allows an identification of the ions. From eq
20, (∆mas/∆E)(ω) was experimentally calculated and presented
in Figures 4d and 5d. Figure 4d corresponds to the elimination
of the potassium ion contribution (E ) 0.250 V, mc ) 39 g
mol-1) and Figure 5d to the elimination of the hydronium
contribution (E ) 0.375 V, mc ) 19 g mol-1). In both cases,
these partial electrogravimetric functions, (∆mas/∆E)(ω), are
very small showing that it is mainly K+ that is involved in the
charge compensation process at 0.250 V and H3O+ at 0.375 V.
This result implies that, according to the theory, the contribution
of anions and water molecules are always negligible in this
potential range. In addition, the contribution of the potassium
ions was negligible at E ) 0.375 V, while for a potential of E
) 0.250 V, it is the contribution of hydrated protons that was
less important.

Electroactive Thin Films: Prussian Blue
J. Phys. Chem. B, Vol. 106, No. 12, 2002 3189
Figure 6. Dependence of the low-frequency limit of the (∆m/∆q)(ω)
function on the potential. Experimental conditions were pH ) 2.2, [KCl]
) 0.5 M.
At last, (∆m/∆q)(ω) was experimentally calculated according
to eq 16. This transfer function is an attractive parameter to
identify the nature of the ion that was involved in the charge
compensation process, especially when only one cation acts as
counterion. Note that (∆m/∆q)(ω) is defined to be negative
when cations act as countercations (Table 1d) and positive for
anion participation (Table 1b). Therefore, when only one species
is involved in the compensation process, (∆m/∆q)(ω)F is
frequency-independent and directly gives its molar mass as it
is plotted in Figures 4e and 5e. At E ) 0.250 V/SCE, (∆m/
∆q)(ω)F is equal to -39 g mol-1; otherwise at E ) 0.375
V/SCE, (∆m/∆q)(ω)F is around -23 g mol-1. At a potential
of E ) 0.375 V/SCE, the absolute values obtained for this
parameter are between 20 and 25 g mol-1 in a wide range of
frequencies (Figure 4f). This value, not so far from the solvated
proton mass, 19 g mol-1, indicates that at this potential it is
mainly the hydronium ion that participates in the charge
compensation process as a countercation. On the other hand, at
a potential of E ) 0.250 V/SCE, the (∆m/∆q)(ω)F quantity
had values of about 39 g mol-1, also in a wide range of
frequencies (Figure 5f), which are close to the 38.9 g mol-1
corresponding to the potassium atomic mass. It is important to
note that the (∆m/∆q)(ω)F is not dependent on the frequency.
If this function varies with frequency that means that faster and
slower processes take place by the incorporation-expulsion of
different species. For the higher frequencies, the ∆m/∆E limit
is zero. This means that no species is transferred faster than the
experimental time scale. In the PB film case, at both potentials,
it is proved that mainly two species participate in the charge
compensation process. Therefore, at a potential equal to 0.2 V,
it is the nonhydrated potassium ions that played the role of
countercation. This result agrees with those by Zadronecki et
al.,54 which have shown that potasium and ammonium cations
enter dehydrated within the PB film during the reduction
process. For the two potentials, the four main transfer functions,
(∆E/∆I)(ω), (∆q/∆E)(ω), (∆m/∆E)(ω), and (∆m/∆q)(ω), were
calculated from eq 12, 15, 13, and 17 and compared with
experimental results. In all cases, the theoretical plots were in
good agreement in terms of shape and frequency distributions.
Moreover, the basic parameters, mc, Kc, and Gc, chosen for the
calculations were the same for the four transfer functions, which
validates the proposed model.
Figure 7. Variation of the derivative of the insertion law (a) for K+
and H3O+ ions with the potential (dCc(E)/dE) (experimental conditions;
pH ) 2.2, [KCl] ) 0.5 M). Theoretical values were taken in cgs units,
and to simplify the calculations, â, γ, and η are defined as follows: â
) ((bi - b′)/4)(Ci - Ci ); γ ) (b′ - bi)/2; η ) -((b′ - bi)/2)(E0 -
max
min
i
i
i
Ei); for global, â ) -0.026, γ ) 10.5, and η ) -2.2; ifor H3O+, â )
-0.0045, γ ) 5, and η ) -2.2; for K+, â ) -0.024, γ ) 12.5, and
η ) -2.6. Panel b shows the variation of the insertion law for K+ and
H3O+ ions with the potential (∆Cc(E)) with the same experimental
conditions as for panel a.
potassium molar mass. Otherwise, at potentials more anodic than
0.35 V, the obtained molar mass approximated the mass of
hydrated proton. This result explains the smaller values of
apparent molar mass already found in the literature.54 Between
these two potentials, intermediate values of molar mass were
obtained. This shows that the two cations are simultaneously
involved in the charge compensation process.
From these results, it was concluded that the only contribu-
tions to the (∆m/∆E)(ω) function that have to be considered
for this system were those of the potassium and proton ions,
and therefore, the partial contribution of both cations can be
evaluated from eqs 10 and 11.
Figure 7a shows the plot of the quantity -Gc/Kc, which is
equal, from eq 9, to (∆Cc/∆E)(0), that is, the derivative of the
insertion law (eq 5), with respect to the potential. The contribu-
tion of both ions at different potentials, also plotted in Figure
7a, is obtained by taking into account the relative proportion of
these ions in the compensation process known from the apparent
molar mass shown in Figure 6. It is observed that the role of
hydrated protons is especially important at potentials between
E ) 0.3 V and E ) 0.4 V, where the film is in the Prussian
Blue form. At potentials more cathodic than 0.3 V, the
hydronium contribution became negligible if compared with the
potassium contribution. This confirms that the charge compen-
sation process during reduction mainly took place by incorpora-
Figure 6 shows the dependence of the low-frequencies limit
of (∆m/∆q)(ω)F, which gives an apparent molar mass, mc, of
the cations expelled from Prussian Blue, with respect to the
potential. It is observed that in a wide range of potentials,
between 0 and 0.25 V, this function reaches values near the

3190 J. Phys. Chem. B, Vol. 106, No. 12, 2002
Gabrielli et al.
Figure 8. Variation of the derivative of the insertion law for K+ and
H3O+ ions with the pH. Experimental conditions were E ) +0.375 V
vs SCE and [KCl] ) 0.5 M.
Figure 9. Experimental and theoretical electrogravimetric transfert
function at 0.250 V vs SCE with Rc parameter (Rc ) 0.85 and same
values for the other parameters as in Figure 4).
tion of potassium ions. Figure 7b shows the isotherm laws for
both ions obtained by integration of Figure 7a. This allows the
quantities Cc - Cc to be estimated. It should be noted that
both potassium and proton insertion takes place when PB films
are reduced from the PB form to the ES form. However, the
involved quantity of H3O+ ions is six times lower than the
quantity of K+ ions when PB is switched between oxidized
states.
with Rc ) 0.85. When this assumption is considered, the model
predicts a mass/potential transfer function very close to the
corresponding experimental data. The kinetic parameters used
to calculate this quantity are not very different from the
quantities obtained without taking into account a CPE in the
model.
Concerning the impedance, a CPE is generally due to
nonuniformity at the electrode surface; for mass measurements,
similar causes can be evoked. First of all, even if the quartz
microbalance has a radial dependence of the mass sensitivity,
this should not affect the frequency dependence of the mass/
potential transfer function. On the contrary, if the thickness of
the polymer layer is not uniform, such a behavior can be
observed. Some distribution of the thickness of the polymer film
has already been shown to affect the impedance of the modified
electrode by a polymer.55,56
max
min
These results help to explain the pH dependence of the
electrochemical impedance, although the redox potential of the
PB S ES process mainly depends on the KCl concentration. In
fact, it is observed that the contribution of the K+ is very much
larger than the contribution of the hydrated proton in a wide
range of potentials, which agrees with the reaction of the scheme
i. However, it has been proposed that the hydrated protons
participate in the conduction mechanism through the film.24,53
This electronic conduction through the film has been described
as an electron-hopping process where the hydrated proton ions
must be fixed on the appropriate site to allow the electron
transfer between two neighboring iron sites in the PB crystal
structure. Then, at potentials between 0.3 and 0.4 V, a
protonation (H3O+ insertion) of the film takes place. For lower
potentials, the oxidation or reduction processes occur by the
participation of potassium ions.
Figure 8 shows the influence of pH on the derivative of the
insertion/expulsion isotherm. It confirms that K+ insertion/
expulsion is practically pH-independent whereas the H3O+
insertion/expulsion rate decreases when pH increases, that is,
when H+ concentration decreases in the bathing solution.
Finally, a remark can be made concerning the adequateness
of the model with the experimental results. As shown in Figures
3 and 4, the calculated quantities from the model have
semicircular shapes, whereas the experimental results show
depressed semicircular loops. This is a very common feature
of the electrochemical impedances measured on solid electrodes.
This discrepancy is phenomenologically explained in that field
by using a constant phase element (CPE) in the equivalent circuit
instead of a first-order time constant:
Conclusion
The ac-electrogravimetric measurements on PB films at
different potentials give valuable information on the chemical
nature and the amount of ions that can enter or leave the PB
film at a given potential, whereas the information provided by
the mass changes measurement by means of a quartz crystal
microbalance in quasi steady state during a voltammetric scan
is only global.
It is possible to discern between the role of hydrated protons
and potassium ions by means of ac-electrogravimetry. At
potentials where the film is in the mixed-valence form (PB
form), the hydrated proton plays a very important role while
potassium ions do not act as countercation. At potentials more
cathodic, the contribution of protons is almost negligible
compared with the contribution of potassium ions.
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