An electrochemical impedance study of the electrochemical doping process of platinum phthalocyanine microcrystals in non-aqueous electrolytes

Journal of Electroanalytical Chemistry 514 (2001) 1 – 15 www.elsevier.com/locate/jelechem

An electrochemical impedance study of the electrochemical doping process of platinum phthalocyanine microcrystals in non-aqueous electrolytes Junhua Jiang, Anthony Kucernak * Department of Chemistry, Imperial College of Science, Technology and Medicine, South Kensington, London SW 7 2AZ, UK Received 12 December 2000; received in revised form 24 July 2001; accepted 4 August 2001

Abstract The electrochemical doping process of platinum phthalocyanine (PtPc) microcrystalline films in acetonitrile electrolyte has been investigated using electrochemical impedance spectroscopy (EIS). The system shows the impedance behaviour expected for a conductive polymer—that is, the appearance of a separate Randles circuit, a Warburg section and purely capacitive behaviour at low frequencies. An equivalent circuit is developed which provides a good fit to experimental impedance data over a wide frequency range of 1 MHz–0.05 Hz. The kinetic parameters of the electrochemical doping process depend strongly upon the doping potential. Analysis of the conductivity of the PtPc film suggests that a percolation effect is responsible for the first-scan discrepancy. At low doping levels, the rate of the first electrochemical step is slow and determined by the conductivity of the microcrystalline film. Once the film becomes conductive, the electrochemical reaction is accelerated abruptly giving rise to a sharp peak. Further increases in doping potential trigger another slow oxidation process. The potential dependence of the diffusion–migration capacitance suggests strong interactions between charge carriers within the microcrystalline film. © 2001 Published by Elsevier Science B.V. Keywords: Electrochemical doping; Electrochemical impedance spectroscopy; Platinum phthalocyanine; Microcrystal

1. Introduction Metallophthalocyanines (MPcs) are robust and highly functionalisable commodity chemicals that are widely used as pigments and oxidation catalysts [1]. Recently, they have become important materials in electronic, optoelectronic and molecular electronic applications owing to their characteristic molecular structure [2–4]. Their physical/chemical properties can be altered significantly via incorporation of different metal atoms at the centre of the phthalocyanine ring or by changing the nature of the substituents on the periphery of the macrocyclic ring, and this has led to their applications in electrochromic devices [5], charge transfer (CT) salts [6–10], switching devices [11] and power

* Corresponding author. Tel.: + 44-20-7594-5831; fax: + 44-207594-5804. E-mail address: [email protected] (A. Kucernak).

sources [12,13]. These applications are mainly based on their rich redox chemistry. MPcs exhibit remarkable redox properties over a wide range of potentials. Usually the electrochemical properties of the monomers are studied in organic solvents [14 –16]. However, their electrochemical characterisation has been limited because of their low solubility even at high temperature in most organic solvents. Recently, the study of insoluble and poorly conducting solids has been carried out by adopting the simple approach of attaching small particles of the solids to electrode surfaces by powder abrasion [17]. Many microcrystals including some MPcs have been investigated via this method and have yielded welldefined voltammograms [18 –20]. Platinum phthalocyanine (PtPc), with higher thermal and electrochemical stability than first-row MPcs, exhibits metal-like conductivity in its partially oxidised state [21 –23]. The electrochemical behaviour of PtPc microcrystals, attached to electrode surfaces by powder abrasion, in a

0022-0728/01/$ - see front matter © 2001 Published by Elsevier Science B.V. PII: S 0 0 2 2 - 0 7 2 8 ( 0 1 ) 0 0 6 2 1 - 0

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non-aqueous medium has been reported in a previous paper [24]. As in the case of conductive polymers, a first-scan discrepancy and an oxidation-state dependent capacitance are observed during electrochemical oxidation. Long-range structural changes and reversible phase transformations coupled to the redox process have been investigated by chronoamperometry at a PtPc microcrystal-rubbed microelectrode [25]. Our in situ UV– visible investigation of the PtPc thin film, reported in a following paper currently under preparation, indicates that it may be a prospective polyelectrochromic material. During the electrochemical oxidation/reduction process, charge-compensating ions have to be either incorporated into or expelled from the MPc microcrystals. In principle, we may have either insertion of anions or expulsion of cations, as the process occurring during oxidation of the PtPc film. Previous work suggests that it is the former process that occurs predominantly [25], and so the incorporated anions must take part in the charge transport process. The kinetic investigation of the electrochemical doping/undoping process is currently of significant theoretical and practical importance. Electrochemical impedance spectroscopy (EIS) is a powerful technique for studying such systems and has been widely used to study the kinetics of electrochemical doping of polymer film electrodes [26– 28]. The advantage of this technique is that it provides a great deal of information in a single experiment: the charge transport rule at the metal film interface; the rate of charge transport in the film; the film conductivity; the double-layer and redox capacitance; and the diffusion coefficients for ionic and electronic charge carriers. The theory of impedance for an electrode with diffusion restricted to a thin layer has been well established [29]. The theoretical impedance functions for an electrode modified by a layer of mixed ionic– electronic conductor have also been analysed recently [30,31]. Similarly to conducting polymers, anion-doped MPc microcrystals have a significant overlapping of delocalised p-electrons along the principal crystal axis. There are two mobile species in both systems—electronic and chargecompensating ionic carriers. It seems, therefore, reasonable to apply the impedance theory developed for conducting polymers to films of powdery anion-doped MPc microcrystals. XPS investigation of the PtPc CT salts indicates that the conduction pathway is not located on the platinum spine but mainly on the ligand chain [23]. PtPc CT salts are characteristic of mixed valence p conductors. In a previous paper, we presented the first example of the study of the undoping process of PtPc microcrystals using EIS and the impedance analysis derived by Deslouis et al. for thin polymer films [31]. This model provides a good description of the characteristics of doped PtPc microcrystalline films. In our previous paper, we examined the dedoping

process of well-conditioned microcrystalline PtPc films [32]. We saw that it was possible to prepare the films in two different chemical states, depending on the pretreatment potential applied to the film, and that these states could be characterised by EIS. Freshly prepared (‘virgin’) films show significantly different electrochemical responses compared to films that have been electrochemically cycled, and which were examined in our previous paper. Our purpose in this paper is to examine the initial doping process that occurs with the freshly prepared virgin films and compare this response to the doping process seen in films that have been electrochemically cycled several times. We analyse these measurements using both approximate solutions to the impedance response over a limited frequency range, and a full non-linear least-squares fit of the impedance spectrum over the entire frequency range measured.

2. Experimental

2.1. Chemicals and instrumental Tetrabutylammonium perchlorate (TBAClO4, Fluka), tetrabutylammonium tetrafluoroborate (TBABF4, Aldrich) and tetrabutylammonium hexafluorophosphate (TBAPF6, Fluka) were of electrochemical grade and used as received. Acetonitrile (AN, Analar) was distilled under vacuum into a flask filled with 4A, molecular sieves. PtPc was synthesised and purified as described in our previous paper [32]. Voltammetric measurements were performed at room temperature (20 °C) using an Autolab general purpose electrochemical system (GPES). EIS experiments were performed using an Autolab frequency response analyser (FRA2). Solutions were deoxygenated with high purity argon.

2.2. Electrode structure The construction and preparation of a powdery PtPc microcrystalline electrode used in voltammetric and EIS experiments were described in our previous report [32]. Powdery PtPc microcrystals were packed into a cylindrical cavity with a diameter of 0.3 mm and a depth of 0.18 mm by mechanical abrasion. Successful packing was evidenced by a shiny purple colour at the outer surface of the packed layer. Electrochemical and microscopic investigation revealed that the cavity was filled along its entire length and that the PtPc material was in good contact with the platinum current collector. This preparation shows good reproducibility judged by dissolving the packed microcrystals in 1-chloronaphthalene and measuring the absorbance at umax (650 nm) on the basis of the measured extinction coefficient (log10[m/cm2 mol − 1]= 5.18) [33,34].

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2.3. Electrochemical measurements The majority of electrochemical measurements were performed in an oxygen-free three-compartment electrochemical cell. The reference half-cell was Ag 10 − 3 mol dm − 3 AgNO3 + 0.1 mol dm − 3 TBAClO4 +AN, and the counter electrode was a platinum foil. Two different states of the film will be considered in this paper. The virgin film has been freshly prepared and has not undergone any potential cycling at all. The cycled film has undergone two successive scans between 1.25 and −0.5 V at a scan rate of 2 mV s − 1, with the potential scan stopped at 0.0 V on the positive going scan. Measurements of the electrode impedance were performed for a frequency range from 0.05 Hz to 1 MHz (about 100 frequency data points per spectrum). Each frequency scan took approximately 20 min to complete. Care was taken to minimise stray capacitances and inductances, which become troublesome at frequencies greater than 100 kHz. A 10 mV amplitude sine wave was applied to the electrode under potentiostatic control. The electrode was pretreated at the very beginning for 20 s at a reducing potential (0 V). Each experimental measurement consisted of polarising the electrode, waiting for the system to equilibrate for 20 s, and then making the EIS measurement. The potential would then be stepped to the next value. As in our previous paper [32], the effect of equilibration time was tested, and it was found that steady-state responses were obtained within 10 s of polarising the electrode, much

Fig. 1. The first and second cyclic voltammograms for PtPc microcrystal electrodes in an AN medium containing 0.1 mol dm − 3 tetra− butylammonium salt of the following anions: (a) BF− 4 ; (b) ClO4 ; and −1 (c) PF− ). 6 . 1st scan, —; 2nd scan,--- (w =2 mV s

3

more quickly than conductive polymer systems for which a typical equilibration time of 600 s has been found [35–38]. The CNLS program developed by Boukamp was used to fit experimental results in terms of the admittance response [39,40]. Employing a range of initial guess values checked the robustness-of-fit. The EIS measurements were made on cycled films unless otherwise stated.

3. Results and discussion

3.1. Cyclic 6oltammetry Fig. 1 shows the first and second cyclic voltammograms for freshly prepared PtPc powdery microcrystalline electrodes in acetonitrile medium containing 0.1 mol dm − 3 tetrabutylammonium with (a) BF− 4 ; (b) − ClO− 4 ; or (c) PF6 as the anion. The processes involved in the electrochemistry of this film have been described in our previous papers [24,25,32], and we will highlight only the major attributes here. During the first scan, the virgin PtPc microcrystalline film demonstrates a large first-scan discrepancy and large capacitance. This firstscan discrepancy only occurs on a freshly prepared (virgin) electrode. It is not possible to regenerate this response through any combination of electrode polarisation or relaxation. The electrochemical response and variation of response with scan number for these films are highly reproducible across independently prepared electrodes. The responses on successive scans although quite different from those for the first scan, remain relatively consistent amongst themselves. On repetitive cycling of an initially virgin film, the ratio of total cathodic to total anodic charge increases from 0.85 during the first scan to 0.95 in the second scan, indicating that some irreversible charge transport occurred. Successive scans show total charge reversibility (i.e. a ratio of 1.0). The incomplete remove of anions from the PtPc film after potential cycling leads to a significant enhancement in the microcrystal conductivity. Variation of the scan limits of voltammograms for the cycled PtPc system has been used to assign the peaks [24]. The first redox process (1/1%) shows a peak potential difference of ca. 0.15 V. During the first scan, peak 1 is not visible, and occurs only during subsequent scans, although its corresponding reduction peak is seen at all times. When the applied potential exceeds 0.8 V, the film starts to undergo further oxidation and this is associated with incorporation of large quantities of anions. The resultant material is much more conductive than the precursor. Electrochemical oxidation proceeds at the boundary between the oxidised conductive material and the reduced relatively non-conductive material. The response mimics a nucleation/growth controlled

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phase transformation, and results in a sharp peak (2). The corresponding reduction peak (2%) is rather broad, and partially buried under peak (1%). The peak– peak potential difference is very large, at 0.71 V, and the peak current shows a w 0.7 – 0.8 dependence, as would be expected for a nucleation/growth process [24]. Between this peak and the next oxidation process (0.85– 1.0 V), a large capacitance is evident in the scan. When the potential exceeds 1.0 V, the film undergoes another phase transformation. Additional amounts of anions are incorporated. This redox process also produces a large peak–peak (3/3%) potential difference of 0.42 V. − For PF− 6 , and to a lesser extent ClO4 , a second reduction peak is seen at (3¦). That the large peak potential differences between 2–2%, and to a lesser extent 3– 3%, are due to the significant overpotential necessary to initiate a nucleation/growth phase transformation, is supported by the total chemical reversibility of these processes, the unusual scan rate dependence of the peak current, and the manifestation of peaks during chronoamperometric transients [25]. As will be shown later, the chemical reaction that produces peak 1 is also that which produces peak 2, although in the case of the latter peak, there is a significant overpotential required to initiate the reaction. The large potential difference between the aforementioned peak couples means that it is possible to prepare the PtPc microcrystalline films in two distinct states which are nonetheless stable at the same potential. Such films will be in either an oxidised, or unoxidised form, and are stable relative to each other because of the significant overpotential required to nucleate the transformation of the film from one form into the other [32].

3.2. EIS 3.2.1. General characteristics The ideal impedance behaviour of conducting polymer electrodes has been predicted by the models of Armstrong [41], Buck and Mundt [42], Vorotyntsev et al. [30], Gabrielli et al. [43] and others [44– 46] and is composed of a separate Randles circuit, a Warburg section and purely capacitive behaviour at low frequencies. Unfortunately, this ideal behaviour has seldom been observed. Most frequently, the real part of the impedance does not become constant and the imaginary part does not tend to the impedance of an ideal capacitor. In contrast, within this paper, we have observed this ideal behaviour using PtPc and our special electrode structure. As our system exhibits hysteresis in its electrochemical behaviour— that is, it exhibits chemical reversibility but shows large peak potential differences— it is necessary to consider the validity of performing impedance measurements in the potential range at which the two

different forms of the film may exist. In order to perform EIS measurements and be assured that the results contain no artifacts, it is necessary to be certain that the system is under steady-state conditions—i.e. that the system is not evolving with time. For our systems, such a constraint is satisfied over virtually the entire potential range. The exceptions are those potentials at which the phase-transformation voltammetric peaks occur (2 and 3). Utilising chronoamperometry on microelectrodes, we have examined how quickly the PtPc system attains a steady state, and are confident that the system is stationary for virtually the entire data set provided [25]. The only possible exception to that is for those few potentials at the very base of the phasetransformation voltammetric peak, as, at these potentials, phase transformation may occur over a time-scale which has some impact on the EIS measurements. Nonetheless, even for these potentials, the variations of the fitted parameters are seen to be smooth and there is no evidence of spurious points, as might be expected if the system were undergoing some variation in composition with time. We are thus confident that the EIS results contain no artifacts due to the phase transformation processes. Steady-state complex impedance plots over a range of potentials for PtPc microcrystalline films in 0.1 mol dm − 3 TBABF4 + AN solution are depicted in Fig. 2. Because the physical/chemical properties of the microcrystalline film are mainly determined by the doping potential, the spectra are highly potential-dependent. All the spectra show a semi-circle at high frequencies (92 kHz –1 MHz). At 0.3 V, a large semi-circle in the middle frequency range and a quasi-capacitive behaviour at very low frequencies are seen. There is no Warburg impedance transition between them. The two semi-circles at high and intermediate frequencies shrink as the potential is increased up to 0.6 V. The very low frequency dispersion indicates quasi-capacitive behaviour. However, an obvious Warburg impedance is produced at low frequencies as the potential is increased. At potentials more positive than 0.6 V, the oxidation of the microcrystals produces a sharp peak in the CV (Fig. 1). Within the impedance spectrum, the semi-circle in the mid-frequency range almost disappears. The spectra still show a Warburg impedance at low frequencies and a quasi-capacitive line at very low frequencies. In the potential region between the sharp peak and the next oxidation peak, only a capacitive current is seen in the voltammetric scan. In contrast, the impedance spectra show a large change. As the potential is increased, the semi-circle at high frequencies grows and the semi-circle at intermediate frequencies reappears, followed by a Warburg resistance at low frequencies and capacitance-like behaviour at very low frequencies. A further increase in the doping potential triggers the next oxidation process. The semi-circle at

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linear extrapolation fails to pass precisely through the origin, although the intercepts are fairly close. The deviation from an ideal capacitive response may be attributed to the inhomogeneity and roughness of the film. At more positive potentials (E\ 1.0 V), the response deviates from this relationship and the Im Z versus 1/… plot deviates from linearity leading to a large error in the determination of the low-frequency capacitance. In the diffusion-controlled regime (ca. 0.2–10 Hz), the magnitude of the ideal Warburg impedance is given by [47] 1/2 Z 8 L/(D 1/2 ) s CL…

(2)

where Ds is the apparent diffusion coefficient for charge transport in the polymer film, L the film thickness and CL is defined in Eq. (1). Z should be inversely proportional to the square-root of …. Fig. 4 shows a linear relationship between Z and … − 1/2. Even though the impedance phase angles in the Warburg section show a small deviation from the theoretical value of p/4, suitable estimates of Ds may be made from the slope of the linear plots for potentials less than 1.15 V.

3.2.2. Equi6alent circuit In contrast to the above analysis, obtained over narrow frequency ranges of our EIS results, our intention now is to apply the impedance model presented in our last paper [32].

Fig. 2. Steady-state impedance spectra at different potentials for PtPc microcrystal electrodes in an AN medium containing 0.1 mol dm − 3 TBABF4. O, 0.3 V; D, 0.5 V; 9, 0.6 V; , 0.75 V; + , 0.85 V; *, 1.0 V; ", 1.15 V; × , 1.25 V.

very high frequencies becomes a little larger and the semi-circle at intermediate frequencies expands rapidly. The Warburg impedance and quasi-capacitive behaviour disappear, and the very low-frequency dispersion shifts to a higher real axis value. At very low frequencies (B 0.2 Hz), the impedance is dominated by the quasi-capacitance, and the imaginary component of the impedance should have the following relationship with frequency Im Z =1/(j…CL)

(1)

where j2 = − 1 and CL is the low-frequency capacitance. Fig. 3 shows linear plots of Im Z versus 1/… for the low-frequency impedance. At potentials more negative than 1.0 V the lines are linear and converge to the same point on the y-axis, and from the slope a potential-dependent capacitance can be obtained. But the

Fig. 3. Plot of imaginary component of the impedance, Im Z vs. … − 1 for the PtPc microcrystal electrode described in Fig. 2.

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account for the deviation from the ideal p/4 diffusion line at low frequency and it greatly improves the quality-of-fit. Ce was considered as one component of the redox capacitance of polymer films by Bobacka et al. [35]. Ra and Ca are the resistance and the adsorption pseudocapacitance connected with the charging/discharging process in the first layer of the film, respectively. Both of these parameters improve the quality-of-fit of the frequency dispersion transition be-

Fig. 4. Plot of complex impedance magnitude Z vs. … − 1/2 for the PtPc microcrystal electrodes described in Fig. 3. Data taken from Warburg sections.

The equivalent circuit used is shown inset in Fig. 5(a). The circuit predicts the results that reproduce Randles circuit behaviour, the Warburg section and quasi-capacitive behaviour at low frequencies, over a wide range of potentials. The treatment of the Warburg component is given below in Eq. (3). The pure resistance on the left of the diagram, Rs, is considered as the sum of the solution resistance and contact resistance between the metal and the microcrystalline PtPc film. The cell capacitance is not shown in this equivalent circuit, and it would be expected that the semi-circle caused by Rs and this capacitance would be visible only at frequencies very much higher than those used here. The high-frequency response of conductive polymer systems usually consists of a semi-circle due to the bulk resistance, Rb, of the polymer phase, in parallel with a geometric capacitance, Cg. The physics of the geometric capacitance have been analysed by Buck and Mundt [42]. We found that utilising a constant phase element (CPE), Q1, and Rb gives a better fit to the high-frequency dispersion of the experimental data. Indeed, a bulk CPE and bulk resistance have been used in the simulation of high-frequency dispersion within solid ionic conductors [39,40], within which are a high concentration of grain boundaries and defects. Such a system is quite similar to ours, as we have a large number of crystal–crystal contacts which may produce a transmission line of linked capacitors and resistances which require description via a CPE. At frequencies lower than those at which the highfrequency semi-circle appears, the impedance response is associated with anion transfer at the film solution interface. The semi-circle in this region can be described by the CT resistance Rct, in parallel with a CPE, Q2, due to the irregular geometry of the surface of the PtPc molecular columns. After the charge transfer semi-circle, there will be a Warburg impedance, ZW, due to the diffusion of charged species in the film. A capacitance Ce in series with ZW is necessary in the model to

Fig. 5. Comparison of the calculated and experimental data of representative impedance spectra at (a) 0.3; (b) 0.85; and (c) 1.15 V. Experimental (o, + ); fitted points (*, — ). Bode plots for the spectra are inset in diagrams (b) and (c). The equivalent circuit model of a PtPc microcrystal electrode is inset into (a). Rs =solution resistance and contact resistance; Rb =the resistance of the bulk material; Q1 is assumed to model the geometric capacitance and related phenomena of the bulk material; Q2 is used to model the double layer capacitance and related interfacial phenomena; Ra and Ca are the resistance and the ‘adsorption’ pseudocapacitance connected with the charging/discharging process in the first layer of the film, respectively; Rct = charge transfer resistance at the film electrolyte interface; ZW =finite-length diffusion – migration impedance; and Ce = electronic capacitance.

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tween the charge transfer semi-circle and the Warburg impedance or capacitance line. At the low-frequency limit, the phase angle approaches p/2 and the impedance becomes capacitive and of large magnitude. In mixed electronic– ionic conductors, the migration of electronic and ionic charge carriers also contribute to the charge transport. Theoretical consideration of the diffusion –migration phenomena for the charge transport has been presented recently by Vorotyntsev et al. [30] and Buck et al. [42,48]. A simplified frequency dependence of ZW is given as follows (Grzeszczuk and Poks [37]): ZW(…)=

 

1 ~ Co j… +

1/2

coth(j…~)1/2

 

(te − ti)2 ~% j… C %o

1/2

tanh(j…~%)1/2

(3)

where ~ and Co are the relaxation time and capacitance associated with the coupled diffusion– migration of the electronic and ionic charge carries. ~% and C %o are the relaxation time and capacitance associated with the transport of the species in excess. te and ti are transference numbers for electron and ions, respectively. The relaxation time, ~, is defined as [37,48] ~=

L2 4Ds

(4)

A comparison between experimental and fitted EIS data in the form of Nyquist and Bode plots as a function of potential is shown in Fig. 5. The aforementioned equivalent circuit and impedance theory provides a good description of the frequency dispersions of the experimental data at potentials more negative than 1.15 V. At 1.15 V, the best fit values begin to deviate at low frequencies. This deviation becomes rather large as the doping potential increases further. Therefore, the above model based on the concept of the perfect blocking film solution interface seems invalid at this higher potential. A model considering charge leakage, i.e. a slow reaction with solution species at the polymer solution interface has been established by Lang et al. [49]. They analysed the effect of the slow reaction on the low-frequency dispersion at a polymer film electrode. The imaginary component of the low-frequency impedance deviates greatly from a linear relation with the reciprocal of the frequency. This relation is found in Fig. 3 when the film is doped at potentials more positive than 1.15 V. It suggests that there is a slow reaction occurring at the film solution interface at this higher potential, although in this paper we will not simulate this phenomenon. The equivalent circuit is composed of nine distinct elements, leading to a total of ten independent variables (each CPE counts as two variables) plus the variables

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associated with the Warburg element. The latter has four adjustable parameters, leading to a total of 14 freely adjustable parameters. Obviously this is a very large number of parameters, and the question must be asked as to whether all of these parameters are required. Experimentally, we find that we cannot produce high-quality fits if we remove any of the circuit elements previously mentioned— but is there any theoretical justification for the number of fitting parameters used? Let us consider how many parameters would be necessary to fit a general spectrum. Our purpose is not to specify the form of the equation required, merely to determine the number of parameters that would in general be necessary to fit a spectrum of a given complexity. How can one measure complexity in a spectrum? The measure of complexity that seems the most general is to consider the order of polynomial which would be necessary to be able to show the same ‘shape’ as our spectrum— by this, we do not mean that the polynomial will produce a good fit to our spectrum— merely that the polynomial has the same number of extrema or points of inflexion. We can estimate the order of the polynomial needed, by looking at the number of points of inflexion in our spectrum. For a spectrum with n points of inflexion, we require n+3 control variables to specify a suitable curve adequately (for example, if we have one point of inflexion then we would need a cubic polynomial, and hence four adjustable parameters). In Fig. 5, we estimate conservatively that we have three points of inflexion in each spectrum. Furthermore, for each frequency point we must fit both the real and imaginary component (or phase/magnitude) — this is equivalent to having to fit two independent spectra, and thus doubles the number of variables we require.1 Thus as a conservative estimate, we would require a minimum of 12 variables to represent our spectra adequately. Thus although we require slightly more than the theoretical minimum to fit our spectra, the number of variables which we fit is not unreasonable. Admittedly this is an estimate, but it seems like a good approach to estimate the number of variables required to fit the impedance spectra. All fitting parameters for a representative impedance spectrum of a PtPc film electrode doped at 0.50 V are listed in Table 1. These parameters were obtained by fitting simultaneously all parameters of the admittance equation representing the equivalent circuit given in Fig. 5(a). The values of Rs, Rb, Rct, Ra, Ca and Ce can be read directly from this table. During the fitting process, the Warburg region is fitted to the following equation, the admittance form of Eq. (3): 1 This assumes that real/imaginary or phase/amplitude data are totally independent. An argument may be made that this is not entirely true, although such a discussion is beyond the breadth of the simple analysis advanced here.

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n2 109A2 n1 109A1

0.15 9 0.03 0.209 0.04 0.25 90.03 0.51 9 0.13 0.7590.21 0.8490.13 0.1190.02 1.09 9 0.03 1.31 9 0.10 1.31 9 0.01 2.01 90.06 2.09 90.05 0.84 9 0.03 0.32 90.04 1.18 9 0.34 1.22 9 0.01 3.0790.05 0.119 0.02 1.11 9 0.02 1.299 0.01 1.3390.01 2.05 90.06 1.02 90.03 0.75 9 0.02 0.29 90.05 2.52 90.87 1.46 90.16 3.559 0.12 0.0890.01 0.8890.02 1.359 0.09 1.319 0.01 2.04 90.03 1.90 90.24 0.81 9 0.01 0.26 90.03 2.45 9 0.59 1.63 90.15 3.629 0.23 BF− 4 ClO− 4 PF− 6

B2 103Y2 103Y1

B1

tanh term coth term

ZW Ca/mF Ra/kV Q2 Rct/kV Q1 Rb/kV Rs/kV

Table 1 − − Fitting parameters of EIS for a PtPc microcrystal electrode doped at 0.5 V in acetonitrile electrolyte containing tetrabutylammonium with BF− 4 , ClO4 or PF6 as anion

Ce/mF

2.42 9 0.33 2.5490.02 1.01 90.09

8

YW(…)= Y1(j…)1/2tanh(B1(j…)1/2) + Y2(j…)1/2coth(B2(j…)1/2)

(5)

The relaxation time ~ can be derived from the fitting parameter B1 (~=B 21). The diffusion–migration capacitance Co is determined by the two fitting parameters Y1 and B1 (Co = Y1B1). Similarly, Y2 and B2 correspond to the tanh function term in Eq. (3) by ~%= B 22 and C %o =Y2B2(te − ti)2. It is usually assumed that ~=~% and Co = C %o [30], or more generally, ~"~% and Co =C %o [42,48]. For our experiments, we find that if we assume that Co = C %o then we obtain values for (te − ti) which imply transference numbers not in the range 0–1. Thus, in our experiments we find that ~" ~% and Co "C %o since te (or ti) must meet 05 te, ti 5 1. It is not possible to calculate the value of C %o independently of the transference numbers. Two CPEs, Q1 and Q2, are expressed by A1 and n1, A2 and n2, respectively. It is surprising to find that n1 \ 1. This observation appears counterintuitive. The CPE is an empirical impedance function explicitly mentioned by Cole and Cole. It is convenient to write at the admittance level. YCPE = A(j…)n

(6)

where A and n are frequency-independent parameters and 05 n5 1. Some discussion of its relation to physical processes, and applicability has been given by Macdonald [50]. It describes an ideal capacitor for n=1 and an ideal resistor for n=0. It is generally thought to arise, when n "0 or 1, from the presence of inhomogeneities in the electrode-material system, and it can be described in terms of a distribution of relaxation times, or it may arise from non-uniform diffusion whose electrical analogue is an inhomogeneously distributed RC transmission line. Although a CPE-like response appears in the majority of experimental data, it is always well approximated only over a finite range of frequency and becomes physically unrealisable for sufficiently low or high frequencies. In our experiments, the semi-circles appearing at the high and middle frequencies seem depressed and can be well simulated using a CPE and Rct. The semi-circles at very high frequencies (1 MHz –92 kHz) have seldom been reported and seem inflated rather than depressed. Although the physical meaning of the CPE is unclear at very high frequencies, we treat Eq. (6) as an empirical expression. It provides a good fit for the very high-frequency dispersion of the PtPc film. In the following sections, we will discuss some important parameters of the film characteristics and electrochemical doping processes.

3.2.3. Effecti6e film dielectric constant In the above section, we suggested that a bulk resistance, Rb, and a CPE, Q1, caused the inflated semi-cir-

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Fig. 6. Variation of effective dielectric constants as a function of doping potentials for the PtPc film in an AN medium containing 0.1 mol dm − 3 tetrabutylammonium salts of the following anions: (a) − − BF− 4 ; (b) ClO4 ; and (c) PF6 .

cle at very high frequency. As the value of n for the very high frequency CPE does not change appreciably with potential, we assume that the value of A follows that of the equivalent capacitance, i.e. Cg 8A1. Thus although the physical meaning of the CPE is ambiguous, it can be logically approximated to that of the geometric capacitance of the bulk film. The geometric capacitance Cg is expressed by the classic formula: Cg =mmoA/L

(7)

where m is the dielectric constant of the bulk film, mo denotes the vacuum dielectric constant (8.85× 10 − 10 F cm − 1) and, A and L are the cross-sectional surface area and thickness of the film, respectively. The effective dielectric constant of the PtPc film is estimated from Eq. (7), assuming we may substitute Cg by A1, and plotted as a function of doping potential in Fig. 6. Although the absolute values of the dielectric constant are calculated from the approximate value of Cg, we may nonetheless consider the trend in dielectric constant as a valuable characteristic. The dielectric constant increases with increasing doping potentials and almost reaches a plateau at higher doping potentials. The values of the dielectric constant measured at very high frequency (100 kHz– 1 MHz) for pure CoPc and FePc obtained by using the transient-photo conductivity measurement technique increase from 20 to about 60 as the temperature is increased from 300 to 360 K [51,52]. The dielectric constant of a conjugated polymer — poly(p-phenylene) is reported to increase with doping level and reaches a plateau at higher doping levels [53], analogous to the PtPc system. These facts suggest that the hypothesis in which the inflated semicircle at very high frequency corresponds to Rb and Q1 is reasonable, although the exact values of m calculated here may be subject to a small error produced by approximating Cg by A1.

9

The conductivity of the PtPc microcrystalline film was calculated from the bulk resistance, Rb, the geometric area and the thickness of the PtPc layer. The potential dependence of the conductivity of the film during the doping process as a function of the anion and whether the film is in either the virgin or cycled state is shown in Fig. 7(a–c). The following common features are observed. Firstly, the conductivity of the virgin or the cycled film is mainly determined by the doping potential and is independent of the nature of the anion. The maximum conductivity of both the virgin and the cycled films are ca. 4× 10 − 2 S cm − 1, much smaller than the conductivity of MPc CT salts [7–10,21–23]. The anion-doped MPc crystals have high unidimensionality of charge transport [54], with the conductivity along the stacking axis being some orders of magnitude higher than that perpendicular to this axis. Partially oxidised PtPc(ClO4)0.5 has a conductivity of 102 –103 S cm − 1 along the crystal axis [23]. The resistance of powdery microcrystals can be expressed as a series of inter-chain resistances, intra-chain resistances and inter-crystal resistances [55]. The total behaviour of the film is determined by the largest of these resistances. Once the PtPc microcrystals are doped at higher potentials during the first potential scan, the intra-chain and inter-chain resistance are expected to become small, and the inter-crystal resistance should dominate. Therefore, low experimental conductivity of highly doped PtPc microcrystalline film could be attributed to high inter-crystal resistance. Secondly, for the virgin film, the conductivity–potential relationship shows that the film conductivity increases abruptly once a threshold potential is exceeded, and then reaches a maximum. Further doping leads to a slight decline in the conductivity. The fact that the onset of the conductivity increase requires a threshold potential could in principle be due to percolation effects. In contrast, the cycled film shows no threshold potential and the conductivity change shows a threesegment variation—linear increase, plateau and small decline. Thirdly, the conductivities of the cycled film are much larger than the virgin film at potentials more negative than 1.0 V but become quite comparable at more positive potentials. This large difference can be explained well by percolation effects. The ratio of anodic to cathodic charge is only 0.85 during the first scan, indicating that some irreversible chemical reaction occurs. The incomplete removal of anions from PtPc film after potential cycling leads to a significant enhancement in the microcrystal conductivity. The presence of anions in cycled PtPc films has been confirmed by XPS for systems in which perchlorate is the anion

10

J. Jiang, A. Kucernak / Journal of Electroanalytical Chemistry 514 (2001) 1–15

[24]. The difference between the potential dependence of the virgin and cycled film can be used to explain the first-scan discrepancy in Section 3.1. The identification of the carriers responsible for electric conduction in organic conductors has been a challenging subject. Spin polarons and spinless bipolarons are widely accepted to be the primary source of charge carriers in most p-conductors [56]. XPS investigation of the PtPc CT salts indicates that the conduction pathway is not located on the platinum spine but instead occurs mainly on the ligand chain [23]. Moreover, some theoretical work on NiPc, which is p-conductive, suggests that the highest occupied molecular orbit (HOMO) is composed of the 2pz orbital of carbon atoms and has nodes at the nitrogen and the nickel atom [57,58]. Since this HOMO character is also expected in PtPc, the anion-doped PtPc microcrystals are expected to be examples of mixed valence p-conductors. ESR spectra of PtPc CT salts show narrow line widths and well-defined uniaxial symmetry [23]. These are signatures of a cation radical with the narrow line width indicating a high degree of unidimensional character

Fig. 7. Potential dependence of the conductivity of the virgin and cycled PtPc films in an AN medium containing 0.1 mol dm − 3 tetra− butylammonium salts of the following anions: (a) BF− 4 ; (b) ClO4 ; and (c) PF− 6 .

and only minor interaction of the carrier spin with the anion. A polaron is the combination of a charged site coupled to a free radical via a local lattice distortion [59]. In the case of non-coordinating anions such as − − ClO− 4 , BF4 and PF6 [60], the polaron is not pinned so that it contributes equally with the bipolaron to conduction. Therefore, the well-defined ESR spectra in highly conductive PtPc salts indicate that spin polarons contribute to electrical conduction. At higher doping levels, two polarons on the same chain may combine to produce a bipolaron. Therefore, both polarons and bipolarons are expected to be responsible for the high conductivity of anion-doped PtPc film.

3.2.4. Charge transfer resistance The potential dependence of the CT resistance of the cycled PtPc microcrystal electrode during the doping process is shown as a semi-logarithmic plot in Fig. 8. The CT resistance decreases quite dramatically as the potential is increased and passes a minimum at 0.75 V. Further increases in doping potential result in a rise of the CT resistance. The CT resistance is unaffected by the nature of the doping anion at potentials more negative than 0.85 V but is determined by the anions at more positive potentials. The analysis of the variation of CT resistance with potential is complicated because of the heterogeneous nature of our samples and because of the presence of nucleation/growth controlled phase transformations. The CT resistance decreases by a factor of 250 over the potential range 0.3–0.75 V. It should be noted that two electrochemical peaks occur over this region, although evidence presented below suggests that these two peaks are caused by the same process. The first peak is the rather broad oxidation, only seen in the cycled film (peak 1 in Fig. 1), which starts at 0.25 V and continues up until the base of the second peak (peak 2 in Fig. 1). The latter is initiated at about 0.7 V. This second peak is due to a nucleation/growth controlled

Fig. 8. Semi-logarithmic plot of the change of CT resistance with doping potential for a PtPc film in an AN medium containing 0.1 mol dm − 3 tetrabutylammonium salts of the following anions: (a) − − BF− 4 ; (b) ClO4 ; and (c) PF6 .

J. Jiang, A. Kucernak / Journal of Electroanalytical Chemistry 514 (2001) 1–15

phase transformation, which requires a significant overpotential to be initiated. Thus at potentials less than 0.75 V, the CT resistance is predominantly measuring how facile the oxidation of the PtPc is in an environment in which a significant overpotential is not required. As the nucleation/growth controlled phase transformation is quite rapid once a sufficiently high overpotential is applied, we see a drop in the CT resistance at 0.75 V. In effect, it appears as if the reaction becomes highly reversible, although it does not conform to the normal precepts of electrochemical reversibility. Another proposed aspect, which may have a bearing on the magnitude of the CT resistance, is an increase in surface area due to the enhanced conductivity of the individual microcrystals. As mentioned previously, the conductivity of a microcrystal is highly anisotropic being several orders of magnitude greater along the crystal axis. It would be expected that this would tend to limit reaction sites to the ends of the microcrystal. As the conductivity of the crystal increases, the possibility of reaction sites occurring along the sides of the microcrystal increases, i.e. there is an increase in the interfacial area over which electrochemical reactions can occur. Such a process would appear as a reduction in the CT resistance. However, this change in crystallite conductivity is not mimicked by a significant change in bulk conductivity. Instead, we believe that this change is masked in our sample due to the relatively large crystallite contact–contact resistance. Further oxidation (1– 1.25 V) of the film results in the electrochemical reactions becoming progressively less reversible, indicating a large barrier to the withdrawal of electrons from the partially oxidised phthalocyanine ring.

3.2.5. Redox capacitance One of the important equilibrium properties of a mixed ionic–electronic conductor is its ability to store

11

charge. This property has been characterised by the redox capacitance. The decomposition of the redox capacitance of a mixed conductor into an ionic and electronic component was recently suggested by Jamnik and Maier [61]. Bobacka et al. introduced an equivalent circuit where the redox capacitance of a polymer film is composed of a diffusional pseudocapacitance in series with an electronic capacitance [35]. Their model provides an excellent description of the frequency dispersion of the experimental data for an oxidised poly(3,4-ethylenedioxythiophene) film electrode. In this paper, we have shown the necessity for a capacitance Ce in series with ZW (in Fig. 5) in order to produce a high-quality fit. A capacitance, called the diffusion–migration capacitance, Co, is derived from the first term in Eq. (3) [37,48], and thus appears as one of the lumped parameters in the Warburg element in Fig. 5(a) inset. We treat the redox capacitance Cred as the combination of two components, Co and Ce, connected in series. Cred = (1/Co + 1/Ce) − 1

(8)

Cred is mainly determined by the smaller component if there is a significant difference in these two capacitances. The variation of the diffusion–migration capacitance, Co, as a function of doping potential is shown in Fig. 9. Co increases with increasing potentials and passes a maximum at 0.5 V. Then it decreases until the potential reaches 0.85 V. Further oxidation produces another peak value at 1.0 V. The diffusion–migration capacitance for a single positively charged site (polaron site) counterion (anion) asymmetric cell has been treated theoretically by Buck et al. [48]. Co is dependent on the concentrations of the charged species. A simple dependence is determined as follows: Co = F 2Vcoxcred/(RTcT)

(9)

with

Fig. 9. Potential dependence of the diffusion –migration capacitance for PtPc film in an AN medium containing 0.1 mol dm − 3 tetrabutylammo− − nium salts of the following anions: (a) BF− 4 ; (b) ClO4 ; and (c) PF6 . Inset: variation of diffusion – migration capacitance with the ratio of reduced sites (cred/cT) over the potential range 0.3 –0.75 V. Dashed line shows the expected parabolic relationship.

12

cT = cox +cred

J. Jiang, A. Kucernak / Journal of Electroanalytical Chemistry 514 (2001) 1–15

(10)

where V is the volume of the electroactive material, cox the concentration of oxidised sites in the film, cT the concentration of total redox sites and cred the concentration of reduced sites. Eqs. (9) and (10) can explain the potential dependence of Co. cred decreases with increasing potential and Co reaches a maximum at cred = cT/2. Further oxidation leads to a decrease in Co. This effect is more obviously displayed in Fig. 9 (inset) where a plot of Co versus cred/cT over the potential range 0.3–0.75 V is presented. Values of cred/cT were obtained by integrating the current for the voltammograms over the aforementioned potential range, and we assume that the sites are completely oxidised at the upper potential limit. As can be seen, there is a curved response with a peak close to the point at which half the sites are oxidised, cred/cT =0.5. The expected parabolic response is shown in this diagram as a dashed line. Too few points are available to show an unequivocal parabolic relationship, although the response does, to a first approximation, show this relationship. Further increases in potential trigger another oxidation process. After this process, there are new redox species (of concentration cred% and c %ox) different from cred and cox. Although analysis of this section to produce a plot of Co versus c %red/c %T plot may very well be fruitful at these higher potentials, the spacing of our data points is too low to allow such a plot to either confirm or deny a parabolic relationship. Nonetheless, qualitatively, we obtain a similar Co – E relation to that seen at lower potentials. Therefore, the potential dependence in Fig. 9 indicates that two different oxidation processes occur over the applied potential range, although the voltammogram in Fig. 1 shows three oxidation peaks. Thus, it is suggested that the sharp peak at 0.75 V and the broad peak over 0.3– 0.6 V are caused

by the same oxidation process. The lower potential peak, peak 1, is caused by oxidation of material that is within an environment that facilitates that process. The latter peak, peak 2, occurs only when a sufficiently high overpotential is applied at which point it is highly accelerated and controlled in a nucleation/growth transformation. The capacitance maximum appears when the film contains equal concentrations of oxidised and reduced redox sites. At this point, the maximum capacitance, Co,max, is given by Co,max = F 2VcT/(4RT)

(11)

and in principle cT can be estimated from this equation. However, Eq. (11) is not valid for any real situation as the system violates electroneutrality. In reality, anions must be incorporated into the film for charge compensation during the doping process. The partition of ions in the redox film and in the solution produces the well-known Nernst–Donnan potential difference. The potential dependence of the other component of the redox capacitance, Ce, is shown in Fig. 10. Ce decreases as the doping potential increases, and then passes a minimum. Further increases in the potential lead to a slight increase in Ce. Bobacka et al. have tentatively given the physical meaning of Ce as the electronic contribution to Cred [35]. Because the experimental fit deviates at low frequencies, it is difficult to determine Ce when the doping potential is more positive than 1.0 V. As a result, we believe that values of Cred calculated using Eq. (8) at higher potentials, i.e. during the second doping process (1.0–1.25 V), are inaccurate and thus we have not plotted them. The variation of Cred as a function of potential during the first doping process is shown in Fig. 11. A volcanic Cred –E relation is observed, as predicted by the model of Chidsey and Murray [62].

Fig. 10. Change of electronic capacitance with doping potential for a PtPc film in an AN medium containing 0.1 mol dm − 3 tetrabutylammonium − − salts of the following anions: (a) BF− 4 ; (b) ClO4 ; and (c) PF6 .

J. Jiang, A. Kucernak / Journal of Electroanalytical Chemistry 514 (2001) 1–15

13

Fig. 11. Potential dependence of the redox capacitance during the first doping process for a PtPc film in an AN medium containing 0.1 mol dm − 3 − − tetrabutylammonium salts of the following anions: (a) BF− 4 ; (b) ClO4 ; and (c) PF6 . Table 2 Comparison of cyclic volammetric redox capacitance (Ccv), redox capacitance (Cred), diffusion–migration capacitance (CD), electronic capacitance (Ce) and low-frequency limit capacitance (CL) obtained at 0.85 V Anion

Ccv/mF

Cred/mF

CL/mF

CD/mF

Ce/mF

BF− 4 PF− 6 ClO− 4

1.3 1.0 1.1

0.65 0.59 0.66

4.2 4.0 4.5

1.6 2.0 1.6

1.1 1.4 1.0

A large capacitance effect has been widely observed in the doping process of conductive polymers and the corresponding value has been estimated by cyclic voltammetry. The voltammetric capacitance Ccv has been suggested as arising from the redox capacitance and can be obtained from the voltammetric response by simple calculations based on the expression Ccv =I/w

(12)

where w is the scan rate and I the double layer current. This large capacitance effect is revealed clearly in Fig. 1 between 0.8 and 1.0 V. Moreover, as might be expected, the double layer current is proportional to the scan rate. For comparison purposes, Ccv obtained for the PtPc microcrystal electrode at 0.85 V is listed in Table 2. The scan rate of 2 mV s − 1 used in Fig. 1 corresponds to an ac frequency of about 0.05 Hz for a 10 mV amplitude sine wave ac perturbation. Around this frequency, the impedance plots already deviate from an ideal p/2 capacitive line. Inzelt and co-workers [63] have suggested the inhomogeneity of the surface film as the reason for the deviation from an ideal capacitance at low frequencies for polymer film electrodes. Ccv is found to be different from the capacitance obtained by impedance measurements. In conductive polymers, this discrepancy is related to conformational changes occurring during the doping process [26,64]. Phenomenologically, Ccv is closest to Ce in Table 2.

Fig. 12. Variation of apparent diffusion coefficients for a PtPc film in an AN medium containing 0.1 mol dm − 3 tetrabutylammonium salts of the − − following anions: (a) BF− 4 ; (b) ClO4 ; and (c) PF6 .

J. Jiang, A. Kucernak / Journal of Electroanalytical Chemistry 514 (2001) 1–15

14

where CL can be determined from Eq. (1), and RL is the low-frequency film resistance and can be obtained by extrapolating the low frequency data (0.2– 0.05 Hz) to the real axis. The diffusion coefficients obtained from Eqs. (2), (4) and (13) for the doping of the PtPc microcrystals at 0.85 V, are shown in Table 3 for comparison. The diffusion coefficient calculated by fitting the entire impedance spectrum is very close to those from the Warburg section and from the low-frequency film resistance. The small difference in Ds obtained by the three methods can probably be accounted for in terms of film non-idealities and experimental error.

from an ideal p/4 diffusion line at low frequencies. The diffusion–migration impedance theory developed by Vorotyntsev and co-workers is applicable to the PtPc system. Fourteen parameters are used to fit the EIS spectra, slightly more than the number derived from an analysis of the complexity of the spectra (12). The kinetic parameters of the electrochemical doping process depend strongly upon doping potential. The conductivity analysis of the PtPc film suggests that there are significant changes in the CT resistance, Rct, and the conductivity, |, between the virgin and cycled films. This effect appears to be due to percolation effects and is likely to be responsible for the first-scan discrepancy in cyclic voltammograms. It seems that the first oxidation process undergoes two successive steps, resulting in a broad and a sharp peak in the cyclic voltammograms. The first peak, visible only for cycled films appears to be due to oxidation of material that is present in an environment that facilitates the oxidation process. The second peak occurs for material in an environment that requires a significant overpotential to drive the phase transformation process. The rate of the first step is slow and determined by the conductivity of the microcrystalline film at low doping levels. Once the film becomes highly conductive at high doping levels, the rate of the second step is greatly enhanced. The accelerated electrochemical reaction produces a sharp peak. Further increases in potential trigger a second oxidation process, during which the reaction rate is relatively slow. This indicates that intercalation of anions into the film becomes more difficult at higher doping potentials.

4. Conclusions

Acknowledgements

The impedance spectrum of a microcrystalline PtPc electrode displays a separate Randles circuit, a Warburg section and a purely capacitive line at low frequencies. An electronic equivalent circuit properly describes the impedance response where the redox capacitance is composed of a diffusion– migration capacitance in series with a second bulk capacitance (Ce). Ce is necessary in the model to account for the deviation

We thank Dr N. Long and Dr C. Mongay-Batalla for providing PtPc samples. We wish to thank Johnson Matthey for the loan of precious metals. This project has been funded by the United Kingdom EPSRC and MOD/DERA under grant GR/L 57920.

Table 3 Comparison of apparent diffusion coefficients obtained at 0.85 V by different methods

[1] C.C. Leznoff, A.B.P. Lever, Phthalocyanines —Properties and Applications, vol. 1, VCH, New York, 1993. [2] R.A. Collins, K. Mohammed, J. Phys. D2 (1988) 154. [3] J. Friedrich, D. Haarer, Angew. Chem., Int. Ed. Engl. 23 (1984) 113. [4] A.O. Rong, L. Kummerl, D. Haarer, Adv. Mater. 5 (1995) 2950. [5] M. Kimura, T. Horai, K. Hanabusa, H. Shirai, Chem. Lett. 7 (1997) 653. [6] M.Y. Ogawa, J. Martinsen, S.M. Palmer, J.L. Stanton, J. Tanaka, R.L. Greene, B.M. Hoffman, J.A. Ibers, J. Am. Chem. Soc. 109 (1987) 1115. [7] M. Almeida, M.G. Kanatzidis, L.M. Tonge, T.J. Marks, H.O. Maracy, W.J. McCarthy, C.R. Kannewurf, Solid State Commun. 63 (1987) 457.

3.2.6. Diffusion coefficient The strong potential dependence of the apparent diffusion coefficients for charge transfer during the doping process is shown in Fig. 12. The appearance of a minimum at 0.5 V in the curves and an increase of Ds with doping potential are observed for all anions during the first oxidation process. In addition, Ds depends on the nature of the anion. The reliability of the measured apparent diffusion coefficients is initially considered. The apparent diffusion coefficient may be calculated from the Warburg section in the complex plane over a frequency range of 0.2–14 Hz according to the well-known relationship (Eq. (2)). Alternatively, Ds may also be determined from the fit for the impedance data using Eq. (4). Furthermore, Ds is also given by [65] Ds = L 2/(3CLRL)

(13)

Anion

105Ds a/cm2 s−1

105Ds b/cm2 s−1

105Ds c/cm2 s−1

BF− 4 PF− 6 ClO− 4

1.3 2.4 2.0

2.1 1.6 2.0

1.6 2.1 1.0

a

Determined from Eq. (2). Determined from Eq. (13). c Determined from Eq. (4). b

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