Interpretations of voltammograms in a typical two-electrode cell: application to complementary electrochromic systems

Interpretations of voltammograms in a typical two-electrode

cell: application to complementary electrochromic systems

Lin-Chi Chen, Kuo-Chuan Ho *

Department of Chemical Engineering, National Taiwan Uni6ersity, Taipei10617, Taiwan Received 21 August 2000; received in revised form 7 November 2000

Abstract

Most electrochemical measurements and characterizations rely on a three-electrode system to obtain useful information. Nevertheless, useful information can still be obtained with a two-electrode system when electrochemical measurements done with a three-electrode system become difficult or impossible. For example, cyclic voltammetry performing on an electrochromic device (ECD) at low scanning rates was proposed as a means to determine the limits of the safe operating voltage. In this study, voltammograms in a typical two-electrode cell is simulated digitally. It was found from the numerical simulation that the voltammograms of the cell could provide ‘fingerprints’ of an ECD. These fingerprints agree well with our earlier experimental observation that the cell characteristics are governed by the electrode with a lower charge capacity. To illustrate the validity of the simulation, complementary electrochromic system comprising the tungsten oxide – polyaniline pair will be discussed. Useful criteria for cell design are also presented. © 2001 Elsevier Science Ltd. All rights reserved.

Keywords:Charge capacity; Electrochromic; Polyaniline; Tungsten trioxide; Two-electrode voltammetry

www.elsevier.nl/locate/electacta

1. Introduction

Cyclic voltammetry has proven to be a powerful diagnostic tool for characterizing many electrochemical systems, including electrochromic materials and devices. In this work, unlike common voltammetry relying on a three-electrode system [1], a two-electrode issue is evolved. Voltammograms in a typical two-elec-trode cell have been interpreted. Although the use of controlled-potential technique in a two-electrode cell has been limited to certain electrochemical systems due to the accompanied counterelectrode’s polarization [2], the technique still plays a very important role in charac-terizing a well-sealed electrochemical cell such as a complementary electrochromic device (ECD), and it also provides much useful information. For example,

Kase et al. [3] used a two-electrode voltammogram to determine the driving voltage of a WO3/Prussian blue

(PB) ECD; similarly, we also took advantage of the same technique to seek the safe operating voltages of WO3/polyaniline (PAni) electrochromic systems in our

previous work [4]. Schmitt and Aegerter [5] employed this method to compare ECDs made with Nb2O5 and

Nb2O5:X (X = Li, Ti, Mo) as colouring electrodes.

Be-sides, two-electrode voltammetry can also be applied to characterize other electrochemical devices, like light-emitting electrochemical cells [6,7] and secondary bat-teries. Furthermore, Wightman [8] and Bruckenstein [9] even recommended the use of the two-electrode system for which the resistance between working and counter electrodes can be easily determined and compensated.

When an electrochemical system is chosen properly, the two-electrode potential-sweep method can provide much needed information by characterizing electro-chemical performances of a cell; however, if it is not * Corresponding author. Fax: + 886-2-23623040.

E-mail address:kcho@ms.cc.ntu.edu.tw (K.-C. Ho).

0013-4686/01/$ - see front matter © 2001 Elsevier Science Ltd. All rights reserved. PII: S0013-4686(01)00369-3

done properly, this technique is not robust and may lead to poor reproducibility making the comparison meaningless. For instance, Jelle et al. [10,11] and Bernard et al. [12] worked on a very similar tungsten oxide – polyaniline electrochromic system comprising the same electrodes and the same ionically conducting

polymer electrolytes; however, the voltammograms ob-tained were quite different. Thus, before applying the two-electrode technique, it would be better for re-searchers to realize what factors influence the voltam-mogram in a typical cell and how to obtain a reproducible voltammogram.

A two-electrode voltammogram is often assumed to be dependent on scanning rates, component electrodes kinetic behaviours, oxidation states, and some undesir-able cell properties such as uncompensated resistances and capacitances; however, a key point to determine what a cell’s voltammogram looks like was verified to be the charge capacity ratio between two component electrodes. From a recent study on the WO3/PAni

ECD, it was found that the voltammogram of the device is strongly affected by the charge capacity ratio between polyaniline and tungsten oxide [4]. Further-more, the electrode with a lower charge capacity limits the performance of the cell. This is illustrated by com-paring the device’s voltammograms (Fig. 1(c)) with the typical voltammograms of an electrodeposited PAni film (Fig. 1(a)) and a sol – gel WO3film (Fig. 1(b)) [4].

These findings not only provide a clue to better control the reproducibility of two-electrode voltammetry, but also depict the influences of charge capacity ratio on cell performances, which has been an important issue for designing both ECDs [13 – 16] and lithium-ion sec-ondary battery [17]. For example, ECDs showing dif-ferent voltammograms [18], while composed of the same electrodes, may simply be the effect of having different charge capacity ratios.

To make two-electrode voltammetry a more reliable and robust technique, our aim is to understand how the charge capacity ratio affects the voltammogram in a typical two-electrode cell. As far as we know, very little literature has been devoted to this issue. Although Orlik [18] had simulated cyclic voltammograms in a two-elec-trode system with the rigorous mathematical treatment by considering a non-Faradaic effect, the important problem such as the polarization effect on a countere-lectrode was not considered. To address this issue, one of our main concerns in this work is aimed towards the study of the influence of charge capacity ratio on the counterelectrode’s polarization.

To interpret the effect of charge capacity ratio on a cell’s voltammogram that is to be compared with Fig. 1, two electrodes, each with assumed voltammograms of its own, based on their specific thin-film electrochem-istry are considered [19 – 23]. Moreover, by taking Kirchhoff’s current law [1] and further simplifications into account, slow-scanning voltammograms for a cell comprising the postulated electrodes with different charge capacity ratios are simulated numerically. Fi-nally, the work not only verifies an earlier finding that the lower-capacity electrode limits a cell’s performance but also discusses thoroughly the influence of charge Fig. 1. (a) The typical voltammogram for an electrodeposited

PAni film in 0.1 N HCl. (b) The typical voltammogram for a sol – gel WO3 film in 0.1 N HCl. (c) Two-electrode

voltam-mograms for WO3/PAMPS/PAni ECDs with different charge

capacity ratios: R = (qWO3

/qPAni_{) = 0.47, 0.99, 1.42, and 1.97.}

Fig. 2. The probability density function derived from thermo-dynamics of mixed-valence intercalation reactions proposed by MaCargar and Neff [26]. In the model, a film is viewed as a binary solution of O and R, and m is the Margules constant to account for the effect of mutual interactions between O and R, where m = 1, 1/2, 0, − 1/2, and − 1.

mograms at a PB electrode. Both the Ellis model and the above equations suggest that the slow-scanning voltammetric current of a thin-film electrode is basically proportional to the (dxO/dE) or (dxR/dE)

function, namely the mole fraction density function or the probability density function of O or R. Moreover, the function is directly determined by the Nernst expression (the E – x relationship) of a thin film. As a result, the voltammetric response of a thin film can be determined, if its E – x behaviour is known.

To develop a general E – x expression for a thin-film electrode, thermodynamics of the mixed-valence intercalation reactions proposed by Ellis et al. [20] and McCargar and Neff [26] is taken into account. Accordingly, the E – x expression, by considering the thin film as a binary solid solution of O and R, is given by the following equation

E − E0= RT nF ln

n

where ai(i = O or R) is the activity, f = F/RT = 38.92

V− 1 _{at 25°C; m is the Margules constant, which}

di-rectly reflects the excess free energy for the mixing process of O and R. Simply speaking, the above for-mula is merely the Nernst equation but with a single parameter correction and it results in the following probability density function

m values (m = 1, 1/2, 0, − 1/2, and − 1) at 25°C for n = 1 is given in Fig. 2. It can be seen that the more

negative the m value, the flatter the curve, with the integrated area of each probability density function equal to unity. Hence, it is believed that the voltam-metric shape of a thin film is intrinsically dependent upon its corrected Nernstian expression. Briefly speaking, the curve’s shape is governed by the inter-action parameter m. This leads to voltammograms of different shapes among materials because of different interaction behaviours involved. In fact, this viewpoint was proposed in some classical literature with rigorous treatments by considering the lateral-interaction effects [21 – 23]. Laviron [22] firstly applied a Frumkin-type isotherm with a random dis-tribution assumption and obtained the I – E be-haviours similar to those presented in Fig. 2. By contrast, Matsuda et al. [23] derived a voltammetric model for an organized film and proposed an unusual double-wave voltammogram for some interaction parameters.

capacity ratio on the distribution of a cell’s voltage and its operating voltage.

2. Theoretical aspect and method

2.1. Electrochemistry of electrochromic thin films A slow-scanning voltammetric response of an elec-trochromic thin film is often explained by the thin-film electrochemistry [19 – 23]. For a typical reversible redox process, O + ne−_{UR, occurs on a monolayer}

or in a thin film, the rate of the electrode reaction per area, I/nFA, can then be equated to the rate of reduction of O to yield [19] I nFA= − dGO dt = −w dGO dE= −wGT

or the rate of production of R to give

I nFA= dGR dt =w dGR dE=wGT

whereGO+GR=GT and xO+ xR= 1. The symbols Gi

and xi (i = O or R) represent the surface

concentra-tion of ith species in the film (mol/cm2_{) and its}

corre-sponding mole fraction, respectively. w and E are the scanning rate and the electrode potential, respectively. Other notations maintain their meanings in the elec-trochemical field. Ellis et al. [20] had also applied a very similar equation to compare the

voltam-2.2. Probability density functions of different shapes To observe the shape influence of a charge capacity ratio on a cell’s voltammogram directly, we postulate two electrodes with two different probability density functions instead of the above-mentioned corrected Nernstian forms. For simplicity, one probability density function is a triangular shape as shown in Fig. 3(a) for the positive electrode; the other is a semi-ellipse as given in Fig. 3(b) for the negative electrode. Referring to Fig. 2, the triangular function may be viewed as the case of m = 1, while the semi-elliptic function as the case of m = − 1. In practice, the mathematical expres-sions for these two functions are chosen as Eqs. (4) and (5) after a thorough consideration to fulfil the

charac-teristics of a probability density function, i.e. the inte-grated area of unity.

(1) For the positi6e electrode with a triangular probability

density function:

!

!

(2) For the negati6e electrode with a semi-elliptic

proba-bility density function:

!

!

The superscript P denotes the positive electrode and N for the negative one. It shall be stressed here that Eq. (4) is defined as a function of xO, while Eq. (5) is

written for xR. Since before the discharge process of a

cell, a positive electrode must contain only the oxidized species; conversely, a negative one is expected to exist at a fully reduced state.

2.3. Charge capacity ratio

The charge capacity of thin film i, qi_{can be obtained}

by integrating its voltammetric current density, i = I/A from Eq. (1). While capacities of two electrodes are measured at the same scan rate, their charge capacity ratio, R, is defined as follows

R q P qN=

)&

)

)&

)

For convenience of calculation, electron-transfer numbers nP_{and n}N_{are both assumed to be 1. On the}

other hand, GTN is fixed at 10− 6mol/cm2, while GTP is

varied to make five different charge capacity ratios, say

R = 20, 2, 1, 0.5, and 0.05 for further simulation about

the effect of the charge capacity ratio on a cell’s voltammogram.

2.4. Digital simulation for two-electrode 6oltammograms

Voltammograms of a cell (iCELL_{versus V}CELL₎

com-prising the postulated electrodes (P and N) with differ-ent charge capacity ratios (R = 20, 2, 1, 0.5, and 0.05) are simulated by the following considerations and approaches.

Fig. 3. (a) A postulated positive electrode with a triangular-shape probability density function. (b) A postulated negative electrode with a semi-elliptic probability density function.

Fig. 4. Simulated voltammograms of a cell comprising the postulated positive and negative electrodes with different charge capacity ratios (R = 20, 2, 1, 0.5, and 0.05) at a scanning rate of 1 mV/s.

Briefly speaking, the cell is maintained at its fully charged state due to the fully pre-polarized step before voltammetric scanning.

2.4.3. The cell’s 6oltage

To simplify the simulation task, two assumptions are made. Firstly, we assume that the slow-scanning voltammetric current is small enough so as to neglect the effect of iR drops. Secondly, the facile kinetics assumption holds true, so one does not need to take overpotentials into consideration. As a result, the cell’s voltage under a cathodic sweep (a discharge process) is merely the potential difference between two electrodes. Furthermore, a slow scan rate (w) of 1 mV/s is applied to the simulated voltammetry for rationalizing the as-sumptions. Thus, we have

VCELL_{= E}P_(x O P_{) − E}N_(x R N_{) = 4 −wt=4−0.001t (9)} and, dVCELL dt = dEP dt − dEN dt = −w= −0.001 (10)

In fact, Eq. (9) was utilized to determine the safe operating voltages for a complementary ECD by Rauh and Cogan [14]. Eqs. (9) and (10) revealed that neither the potential of the working electrode nor counter electrode is well controlled, when performing the two-electrode voltammetry to a cell; instead, all we control is the cell’s voltage, VCELL_{, alone.}

2.4.4. Numerical approach and digital simulation Broadly speaking, to simulate a cell’s voltammogram is to mainly solve Eqs. (7) and (9) along with the initial condition. Since the mathematics involved is non-linear in nature, voltammograms are simulated numerically. The numerical approach to be adopted is the explicit finite difference method [24], and aFORTRAN90 routine is built to complete the numerical work. All calcula-tions are executed by a personal computer with an Intel Pentium-S CPU at 200 MHz. To sum up, iCELL_(t),

VCELL_{(t), E}P_{(t), and E}N_{(t) are simulated for various}

charge capacity ratios from t = 0 s (VCELL_{= 4 V) to}

t = 4000 s (VCELL_{= 0 V).}

3. Results and discussion

3.1. Parameters to determine the shape of a 6oltammogram

Simulated voltammograms (iCELL _{versus V}CELL_{) of}

the cell for different charge capacity ratios (R qP_/qN₎

are shown in Fig. 4. Despite the same positive and negative electrodes being used, the difference in charge capacity ratios causes the voltammograms to look very different from each other. The curves for R = 0.5 and 2.4.1. Go6erning equation

The voltammetric response of a two-electrode cell is governed by the principle of Kirchhoff’s current law [1]. The law states that, a current that leaves from the positive electrode must equal the one that injects into the negative electrode. Therefore, we have

iCELL_{= i}P_{= i}N ₍₇₎

Eq. (7) plays a significant role as the governing equation throughout this work and it implies that the current response of a two-electrode cell must be deter-mined by the minimum value of iP _{and i}N_{. Regarding}

iP_{and i}N_{, they can be easily obtained by applying Eq.}

(4) into Eq. (1) and Eq. (5) into Eq. (1a), respectively. 2.4.2. Initial condition

Before voltammetric scanning, the cell is kept at its as-assembled stage, namely at its initial condition. In this research, the two postulated electrodes are both pre-polarized fully to their unity x values (xO= 1 and

xR= 1 for the positive and negative electrode,

respec-tively) before cell assembling. This pre-treatment is a common pre-colouring step before putting a comple-mentary ECD together. Meanwhile, a fixed charge ca-pacity ratio is assigned to the cell. Hence, the initial condition for the cell’s voltammetric simulation is writ-ten as follows

xOP= xRN= 1; EP= 2 V, EN= − 2 V,

0.05 both exhibit a triangular feature, as seen in Fig. 3(a); on the contrary, when the ratio is larger than unity (R = 2 or 20), a semicircle-like curve is approached, as seen in Fig. 3(b). Only when the charge capacity ratio is equal to 1, a mixed-shape curve is obtained. These curves reveal that the shape of a cell’s voltammogram is determined by the voltammogram (I – E behaviour) of an electrode with the lower charge capacity. This simu-lated result is consistent with our previous observation on a WO3/PAni ECD, as shown in Fig. 1(c). It is

therefore inferred that the shape effect, as seen in both Fig. 4 (the simulated voltammograms) and Fig. 1(c) (the real ones), can be attributed to the same reason, the Kirchhoff’s current law, as outlined by Eq. (7).

According to Eq. (7), when the lower-capacity elec-trode runs out and contributes no currents, i.e. xij= 0

(i = O or R; j = P or N), the larger-capacity side must terminate its redox reaction to maintain the current’s continuity. As a result, the lower-capacity electrode limits the cell’s voltammetric response and dominates the voltammogram’s shape. For this reason, to com-pare voltammograms for cells made of the same ca-thodic and anodic materials, one needs to consider the simple effect of charge capacity ratio. For example, three very different I – V behaviours, a PAni-like [10], a WO3-like [11], and a mixed-featured [12]

voltam-mogram, were reported, all for very similar WO3/PAni

ECDs. This can be explained by the charge capacities between WO3and PAni, namely, q

WO3\qPAni_{, q}WO3B

qPAni_{, and q}WO3_{= q}PAni_.

3.2. The limiting electrode in a two-electrode cell Fig. 4 suggests that the lower-capacity electrode dominates a cell’s voltammogram due to the constraint of current’s continuity. This principle also implies that the lower-capacity electrode must play a limiting role in the charge/discharge process of a cell. Thus, to further prove the existence of the limiting electrode in a cell, a relationship between the maximum cell-discharge ca-pacity, qCELL_{, and the charge capacity ratio is shown in}

Fig. 5, which is obtained by integrating Fig. 4 against time. It is found that the maximum discharge capacity of the cell increases linearly with the capacity ratio when RB1; by contrast, when R]1, it arrives at a ceiling value, 9.65 × 104_mC/cm2_{. As a consequence, the}

qCELL_{– R relationship can be formulated by the}

follow-ing equation

qCELL_{= Min(9.65 × 10}4_{× R,9.65 × 10}4_{) (11)}

Since we have set GTN= 10− 6mol/cm2 and defined

R qP/qN, clearly, Eq. (11) states that the maximum

discharge capacity of the cell is equal to the charge capacity of the lower-capacity electrode. That is,

qCELL_{= Min(q}P_,qN_{) (12)}

Thus, the lower-capacity electrode is the limiting electrode, which not only dominates a cell’s voltam-mogram but also limits a cell’s Faradaic characteristics. This is against the common wisdom that the electrode with a larger capacity shall determine a cell’s perfor-mance. Accordingly, Eq. (12) provides a thumb rule for maximizing the utilization of electrodes, that is, to maximize the utilization of cathode and anode, one needs to equalize charge capacity on both electrodes. Concerning related experimental data to support the rule; evidences can be found in our previous works [4,16] for complementary systems of WO3/PAni and

WO3/PB couples.

3.3. Distribution of a cell’s 6oltage between two

electrodes

Since the polarization effect at the counter electrode cannot be ignored while performing the two-electrode voltammetry, the distribution of a cell’s voltage be-tween two electrodes is still another important issue in this work. For this reason, the potentials of two com-ponent electrodes in Eq. (9), EN_{(t) and E}P_{(t), are also}

calculated during the cell’s voltammetric simulation. A plot of EN_{(t) versus E}P_{(t) for different capacity}

ratios is shown in Fig. 6. The plot reveals that the distribution of a cell’s voltage is strongly influenced by the charge capacity ratio. The applied potential is not equally distributed on two-electrodes, even for the case of R = 1. (Unless a cell is composed of two electrodes with the same I – E behaviour.)

Fig. 5. The maximum discharge capacity of the cell, qCELL

(mC/cm2_{), as a function of charge capacity ratio, which is}

Fig. 6. Digital simulation for the distribution of a cell’s voltage on two electrodes for R = 20, 2, 1,0.5, and 0.05 corresponding to Fig. 4.

on a cell’s voltage window. It shows that a relatively high voltage with a narrow voltage window tends to be obtained as qP_{− q}N _{gets larger. For example, the}

non-zero voltammetric current is ranged from 4 to 0 V and peaks at 2 V for R = 1; in comparison, the voltam-mogram for R = 0.5 peaks at ca. 2.4 V and has a voltage ranging from 4 to 1 V.

To arrive at a mathematical expression for the rela-tionship betweenDVCELL_{and R, Eq. (9) is re-written in}

terms of qj _{( j = P or N) based on the principle of}

current continuity VCELL_{= E}P

cathodic sweep. Because ViCELL (where qRxn= 0) has

been given as 4 V for the fully pre-polarized electrodes, DVCELL _{can be determined after calculating V}

cut CELL_{. In}

fact, VcutCELLis the voltage at which qRxn= qCELL. As a

result, by applying Eqs. (6) and (12) to Eq. (13) we have

VcutCELL= EP(0) − EN(1 − R) = − EN(1 − R)

for qP_5qN_{; R51 (14)}

VcutCELL= EP(1 − R− 1) − EN(0) = EP(1 − R− 1)

for qP_]qN_{; R]1 (15)}

Based on the definition of DVCELL_{, the relationship}

betweenDVCELL_{and R is given correspondingly.}

DVCELL_{= [E}P_{(1) − E}N_{(1)] − [E}P_{(0) − E}N_{(1 − R)]}

= 4 + EN_{(1 − R) (R51) (16)}

DVCELL_{= [E}P_{(1) − E}N_{(1)] − [E}P_{(1 − R}− 1_{) − E}N_(0)]

= 4 − EP_{(1 − R}− 1_{) (R]1) (17)}

Finally, applying Eqs. (4) and (5) to Eqs. (14) – (17), shows why a higher voltage with a narrower voltage window tends to be obtained with R-values deviated far from unity. These equations also provide a criterion for determining operating voltages for a cell. According to the criterion, one can assemble a higher-voltage battery by enlarging the difference in charge capacity between two electrodes. Nonetheless, one needs to remember that the criterion is valid only when applied to cells composed of the fully pre-polarized electrodes with pseudo-equilibrium behaviours.

4. Conclusions

So far, we have successfully interpreted the effects of a capacity ratio on a cell’s voltammogram and found that the simulated results are in good agreement with our previous works. To sum up, these effects are con-cluded as follows:

1. The lower-capacity electrode dominated the shape of a cell’s voltammogram.

In the case of R = 1, EN_{and E}P_{are confined in their}

non-zero probability density function domains ( − 2 V5EN_{50 V, 0 V5E}P_{52 V) during the whole}

scanning period; that is, neither the positive electrode nor the negative one runs out (reaches at xji= 0) before

the final stage of the cathodic sweep. In the case when

R-values are deviated from unity, we find that either

the positive or negative electrode is exhausted before the end of scanning. As the electrode is exhausted, the cell’s Faradaic process will be terminated, and the counter electrode’s potential will be kept at a constant value.

These phenomena imply that the lower-capacity elec-trode in a cell is more prone to polarization than the larger-capacity one. Accordingly, when performing the two-electrode voltammetry for characterizing a working electrode, it would be better to maximize the charge capacity of the counter electrode, so that the countere-lectrode can serve as a quasi-reference ecountere-lectrode more properly.

3.4. The operating6oltages of a cell

In principle, a cell shall be operated in a voltage window corresponding to the non-zero current region in its i – V curve; hence, we define a voltage window (DVCELL_{) of a cell as the cell’s voltage difference}

be-tween the initial stage voltage (ViCELL) and the cutoff

stage voltage (VcutCELL). The cutoff voltage is the voltage

at which the voltammetric response decays to zero. By this definition, Fig. 4 not only tells how a charge capacity ratio affects the shape of a cell’s voltam-mogram, but also depicts the effect of a capacity ratio

2. The lower-capacity electrode also limits the maxi-mum cell-discharge capacity.

3. The larger-capacity electrode is more difficult to be polarized than the lower-capacity one.

4. The larger the capacitis difference (qP_{− q}N_{) is, the}

higher the voltage, and the narrower the voltage window to be obtained

In theory, the above results are attributed to Kirch-hoff’s current law, which ensures current continuity in a cell and renders the lower-capacity electrode to limit a cell’s performance. Despite the naı¨ve and simple (dx/ dE) functions to be chosen, it is inferred that the conclusions of this work have the generality and can be applied to any two-electrode cell including an ECD. Of course, one can introduce the more realistic electrode thermodynamics [25 – 27] or kinetics [28,29] instead of proceeding with a rigorous cell’s voltammetric simula-tion. In conclusion, this work provides a criterion to assign a proper charge capacity ratio, realizing a way to better control the reproducibility for a two-electrode voltammetry, and giving an insight to further studies in the related areas.

Acknowledgements

This work was sponsored by the National Research Council of the Republic of China under contract num-bers NSC 87-2214-E002-034 and NSC 89-2214-E002-017.

References

[1] A.J. Bard, L.R. Faulkner, Electrochemical Methods: Fundamentals and Applications, Wiley, New York, 1980. [2] P.T. Kissinger, W.R. Heineman (Eds.), Laboratory Tech-niques in Electroanalytical Chemistry, chap. 6, 2nd ed., Marcel Dekker, New York, 1996.

[3] T. Kase, T. Miyamoto, T. Yoshimoto, Y. Ohsawa, H. Inaba, K. Nakase, in: C.M. Lampert, C.G. Granqvist (Eds.), Large-Area Chromogenics: Materials and Devices for Transmittance Control, SPIE Proceedings Series, vol. IS4, 1988, pp. 504.

[4] K.-C. Ho, L.-C. Chen, C.-C. Lee, in: C.M. Lampert (Ed.), Switchable Materials and Flat Panel Displays, SPIE Proceedings Series, vol. 3788, pp. 120.

[5] M. Schmitt, A. Aegerter, in: C.M. Lampert (Ed.), Switch-able Materials and Flat Panel Displays, SPIE Proceedings Series, vol. 3788, 1999, pp. 75.

[6] Y. Greenwald, F. Hide, J. Gao, F. Wudl, A.J. Heeger, J. Electrochem. Soc. 144 (1997) L70.

[7] Y. Greenwald, F. Hide, A.J. Heeger, J. Electrochem. Soc. 144 (1997) L241.

[8] R.M. Wightman, Anal. Chem. 53 (1981) 1125A. [9] S. Bruckenstein, Anal. Chem. 59 (1987) 2098.

[10] B.P. Jelle, G. Hagen, S.M. Hesjevik, R. Ødega˚rd, Solid State Ionics, in: M. Balkanski, T. Takahashi, H.L. Tuller (Eds.), Proceedings of Symposium A2, International Con-ference on Advanced Materials, ICAM-91, Strasbourg, France, 27 – 31 May 1991, Elsevier/North-Holland, Am-sterdam, 1992, pp. 661.

[11] B.P. Jelle, G. Hagen, R. Ødega˚rd, Electrochim. Acta 37 (1992) 1377.

[12] M.-C. Bernard, A.H.-L. Goff, W. Zeng, Electrochim. Acta 44 (1998) 781.

[13] H. Tada, Y. Bito, K. Fujino, H. Kawahara, Solar Energy Mater. 16 (1987) 509.

[14] R.D. Rauh, S.F. Cogan, J. Electrochem. Soc. 140 (1993) 378.

[15] K.-C. Ho, Electrochim. Acta 44 (1999) 3227.

[16] L.-C. Chen, K.-C. Ho, Electrochim. Acta 46 (2001) 2151 – 2158.

[17] P. Arora, R.E. White, M. Doyle, J. Electrochem. Soc. 145 (1998) 3647.

[18] M. Orlik, J. Electroanal. Chem. 434 (1997) 139. [19] A.J. Bard, Integrated Chemical Systems, Wiley, New

York, 1994.

[20] D. Ellis, M. Eckhoff, V.D. Neff, J. Phys. Chem. 85 (1981) 1225.

[21] A.P. Brown, F.C. Anson, Anal. Chem. 49 (1977) 1589. [22] E. Laviron, J. Electroanal. Chem. 100 (1979) 263. [23] H. Matsuda, K. Aoki, K. Tokuda, J. Electroanal. Chem.

217 (1987) 15.

[24] G.D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Method, chaps. 1 – 2, 3rd ed., Oxford University Press, Oxford, 1984.

[25] R.S. Crandall, P.J. Wojtowicz, B.W. Faughnan, Solid State Commun. 18 (1976) 1409.

[26] J.W. McCargar, V.D. Neff, J. Phys. Chem. 92 (1988) 3598.

[27] B.P. Jelle, G. Hagen, S.M. Hesjevik, R. Ødega˚rd, Elec-trochim. Acta 38 (1993) 1643.

[28] B. Reichman, A.J. Bard, D. Laser, J. Electrochem. Soc. 127 (1980) 647.

[29] F. Miomandre, M.N. Bussac, E. Vieil, L. Zuppiroli, Electrochim. Acta 44 (1999) 2019.