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Proton insertion into thin films of amorphous WO3: kinetics study
Materials Science and Engineering, B I 3 (1992) 243-246
243
Proton insertion into thin films of amorphous W03: kinetics study O. Bohnke and B. Vuillemin Laboratoire d'Electrochimie des Solides (UA 0436 CNRS) Universitd de Franche-Comtd 25030 Besanqon (France) (Received November 7, 1991)
Abstract Thin films of electrochromic materials may greatly contribute to understanding the mechanism of cation insertion into mixed conductors. In this paper, we present the results of electrochemical and optical in-situ measurements during proton insertion performed on thin films of tungsten trioxide deposited by vacuum evaporation onto conductive substrates. The measurements clearly show that the electrochemical insertion of protons and electrons into the oxide, i.e. the current density, is limited by a resistance and not, as generally believed, by the charge transfer reaction at the electrode-electrolyte interface. The coloration of the oxide is limited by the ionic diffusion of the inserted cation into the host matrix. From these experimental results, we propose a model to explain the electrocoloration of tungsten oxide. This model involves simultaneous diffusion and chemical reaction. A finite difference method has been used to solve the differential equations and to obtain the diffusion coefficient of the species diffusing into the oxide.
1. Introduction
The electrochromism of amorphous tungsten trioxide (a-WO~) thin films has been extensively studied [1-6] but there are still uncertainties about the processes governing both the current and coloration rate. However, it seems clear that electrocoloration of a-WO3 is the result of a double injection of cations and electrons into the host matrix. During the electrochemical coloration, the electrons are injected into the oxide from the electronic conductor through the electronic/a-WO~ interface. At the same time, the ions M ÷ coming from the electrolyte are injected through the electrolyte/a-WO 3 interface. The overall electrochromic reactio n may be written as follows: a-WO3(transparent) + xM + + x e - ~
M x W O 3 (blue)
(1) In this paper, we present results of electrochemical and in-situ optical measurements during proton insertion into thin films of tungsten trioxide obtained by vacuum thermal evaporation. A model is proposed to explain the mechanism of electrocoloration of this oxide. This model involves simultaneous diffusion and chemical reaction.
2. Experimental details
Tungsten trioxide thin films were deposited onto conductive substrates by vacuum thermal evaporation
of WO 3 powder (99.99% purity, Johnson Matthey Co) contained in a W boat. The evaporation pressure was 10 -5 mbar and the deposition rate was typically 5 A s-~. The thickness of the films was 4600 A. We used three types of substrate: transparent indium--tin oxide (ITO) coated glass substrates purchased from Balzers Co., with sheet resistances of 140f2 []-~ and 10~ [] -~, and W metal foil (99.9% purity) was purchased from Prolabo. The electrochemical and optical measurements were performed simultaneously in a classical three-electrode cell as previously described [8]. Computer-controlled equipment was used for data acquisition. The working electrode of the electrochemical cell was the electrochromic electrode, the counter electrode was a platinum grid and the reference electrode was a saturated sulfate electrode (SSE). The electrolyte was a continuously deaerated 0.01 N H2SO 4 solution (pH = 2.26).
3. Results
The electrochromic mechanism we propose involves simultaneous diffusion and chemical reaction and is based upon experimental observation. It is directly connected to simultaneous electrochemical and optical measurements. When the electrical pulse applied to the electrode is interrupted, the current crossing the electrode goes down to Zero but the optical density of the film continues to increase for 0.5-1 s. Elsevier Sequoia
244
(). Bo/mke and B. l'tdllemin
/'
l'rot
E L K CTI:IOL"KfE 9"
OI']ICAL DENSI] 5
B-"
•
• • X
7-
Fit) itlll/f:
008
t
WO 3
ELECTRONIC CONOUCTOR
<
= ~')0 ~ c m ) H ÷
015 ~)25
=~
H'
,14
0
im
65-
W, 5,',, H~-t
4
~-Z 3
Fig. 2. Model for the eleclrocoloration of a-WO.
2 1 0 0.5
0.7
- 0.9
-1,1
-1,3
- 1,5
Applied Voltage (V/~SSE)
Fig. 1. J vs. V,, at O.D.=0.08 (m), 0.15 (+), 0.25 (O), 0.4 (&),
(1.5 ( x ). This long time indicates that the observed phenomenon is not an electronic relaxation but rather an ionic diffusion process. This phenomenon has been observed in both proton and lithium electrolytes and for all methods used to obtain the oxide, i.e. chemical vapour deposition [7], anodic oxidation [8] and vacuum evaporation. During coloration, the current density (J) decreases with time. It depends strongly on both the electrolyte conductivity and the substrate conductivity; the higher these conductivities, the higher the current crossing the cell. This point has been clearly shown in previous papers [3, 7, 8]. Figure 1 shows the variation of the coloring current density at a given injected charge value (or at a given x value) with respect to the applied potential V,. It is clearly shown that the current varies linearly with V,. The electrode used in this experiment is WO3 deposited onto ITO (10ff2/n). We previously mentioned this behavior with electrodes made of WO 3 obtained by chemical vapor deposition (CVD) deposited onto the SnO2:F substrate [7] and with electrodes made of WO 3 obtained by anodic oxidation of W foil [8]. Kamimori et al. also reported such behavior [3]. However, this result is different from the results reported by Faughnan et al. [6] who postulated that the coloring current is limited by a potential barrier at the electrolyte-WO 3 interface and consequently that it increases exponentially with increasing applied potential.
4.
Discussion
From these experimental results, we propose a model to explain both the current behavior and the coloration process at the tungsten oxide electrode. It is
schematically shown in Fig. 2. Part of this model has been previously proposed by other authors [2, 3, 5] but no one has derived a theoretical expression of the variation of the optical density with time. The mechanism we propose is based on two successive processes occurring during the electrochromic phenomenon. First, a rapid diffusion of electrons through the oxide and a charge transfer reaction at the WO3-electrolyte interface leads to a current J crossing the electrode and to the formation of a species (H+,e-). The species (H+,e - ) then diffuses into the oxide with a diffusion coefficient D. Second, a chemical reaction between the diffusing species (H+,e - ) and the W ~'+ ions takes place, leading to the coloration of WO~ and to the formation of an immobilized proton in the oxide. The observed current in the cell is the consequence of the first reaction and the coloration is the result of the two successive reactions. In a first approximation, if we assume that the coloration reaction is instantaneous and then proceeds very rapidly compared with the diffusion process, a local equilibrium exists between the diffusing species ( H + , e ) and the immobilized proton. The concentration of the immobilized species ((72) is then directly proportional to the concentration of the diffusing species ( C1 ). According to this model, and if we assume that diffusion is one-dimensional along the Z-axis, the diffusion equation is [9]: 0C
1
0t
02C1
-D
i~Cm
- -
0Z 2
(2)
i3t
with
C2 =aC~
(3)
The initial and boundary conditions are:
C~=C 2=0
at
D 0Cl_ J (1 + a) OZ nF i3Ci 0Z
0
at
t = 0 (for all Z) at
Z = 1
Z=0
(4) (5)
(6)
O. Bohnke and B. Vuillemin
/
Proton insertion into thin films of amorphous WO/ kinetics study
with l the thickness of the oxide film, n the number of electrons exchanged per mole of substance reduced (n = 1), F the Faraday's constant and J the current density. Condition (5) implies that there is no accumulation of (H+,e -) species in the oxide at the WO3-electrolyte interface and condition (6) implies that no diffusing species can cross the electronic conductor-WO 3 interface. The effect of the instantaneous reaction is to slow down the diffusion process into the oxide as shown by equation (5). To analyze the coloration process, we can visualize the operation by referring to Fig. 2. As coloration proceeds under a constant potential Vo, applied against a reference electrode, protons are inserted into WO3 and x increases in the host matrix. Consequently, the chemical potential of (H+,e -) in the matrix increases, leading to an electromotive force ~, that opposes the current flow, as described by Crandall et al. [10]. This back emf, which is just equal to the change of the chemical potential of (H+,e - ), is mostly responsible for the current drop in the electrode. Moreover, it varies with x. Since insertion does not proceed under an equilibrium condition, there is a gradient of x in the oxide that varies with time. The most important parameter in the kinetics process is the value of x at the electrolyte-WO 3 interface. It will govern the value of the overpotential r/across the interface. The potential V,, may then be expressed as: 1
V, = rJ(t) + rl(t) + Vb(t) +±/,/H +electr°lyte F
(7)
1
Vh(t)=-~-/~<~+~- /
RT pH F
at
Z=0
3O0-
250-,
200?
150
P
1O0
'e. 50
0
-50
.-100 2
4 rim, (,)
Fig. 3. -~w.~-~/F vs. time at (a) -0.07, (b) -0.29, (c) -0.37 V/NHE. 0.7
0.6
OD(I)
= fj
£ C2(Z,I)dZ
o
0.•
OD
E = 5.35 x 106 c m 2 / m o l e ,
theory
0.]
~1- exoelilsle"l °l
O.1
O
, O
F 2
J 4
1line ( , )
where rJ is the ohmic drop across the electrode and the electrolyte, and /~H. Y is the chemical potential of protons in the acid electrolyte. Figure 1 clearly shows that J(t) varies linearly with V, (for J ( t ) > 1 mA cm-2). This result suggests that the current is limited by resistance and not by thermal activation over a barrier. This latter case would lead to a current which depends exponentially on the applied potential [11]. The relationship between J and V~ may then be expressed, for J > 1 mA cm -2, as: V,,=rJ(t)+ V~(t)-2.3
245
with (8)
it-
where R is a gas constant and T is the absolute temperature. The resistance r is given by the inverse of the slope of the straight lines of Fig. 1. r was found to be 75Q cm 2 for WO 3 vacuum deposited onto W sheet, 90ff~ c m 2 for deposition onto ITO (10ff~ rl-I and 145Q c m 2 onto ITO (140f2 []- 1). r is then influenced by the con-
Fig. 4. Optical density vs. time at (a) -0.07, (b) -0.29, (c) -0.37 V/NHE: (--), experimental; (I~), theoretical. ductivity of the substrate. We have previously shown [7, 8] that this resistance is also influenced by the electrolyte conductivity. From eqn. (8) we can determine the values of/~iH*,e-~ at Z = 0 as a function of time. The result is shown in Fig. 3 at different Vo. For each value of the applied potential, there is a time at which the composition of the interface is in equilibrium with hydrogen gas, i.e. tt/H+x-/ ----0. The higher V,,, the smaller this time. Consequently, for greater times, H 2 evolution may occur thermodynamically at this interface. Eqn. (2) may not be solved analytically because of the non-linear boundary condition (5). We solved it by using the finite difference method [9]. Since [W 5÷ ] = c2(g,t), the optical density of the film is given by the relationship: 1
O'D(t) = f eC2(Z,t) dz
(9)
0
where e is the absorption coefficient of the colored film, i.e. e = 5.35 x 1 0 6 c m 2 mo1-1. Figure 4 shows the
246
O. Bohnke and B. Vuillernin
/
Proton insertion into thin fihns qf anunphotts WO ~.kinetics study
results of the above calculations together with the experimental curves at different applied potentials for WO~ deposited onto ITO (lOft2 [] ~). The best fit is obtained with a = 0 . 7 , D = 2 x l 0 -~° cm 2 s -1 and r= 90Q cm 2. However the discrepancy observed at small times may be attributed to a slower coloration reaction. Further studies with a slow chemical reaction instead of an instantaneous one are underway. A better fitting might be obtained.
5. Conclusion During coloration of a-WO3 thin films obtained by vacuum thermal evaporation in acid electrolyte, the current was found to be limited by resistance and not, as generally believed, by the charge transfer reaction at the electrode-electrolyte interface. The current density-potential relationship may be expressed by 1 RT V~= r J ( t ) - F la(H+x-'- 2.3 ~ - pH
where ~(H*x-) is the chemical potential at the interface of the injected (proton-electron) pair. The resistance r depends on the substrate conductivity and on the electrolyte conductivity. Assuming that simultaneous diffusion and chemical reaction are involved in the electrocoloration mechanism, the differential equation may be solved by a finite difference method and the diffusion coefficient of the (H +,e ) species determined.
The equilibrium constant between the immobilized species and the diffusing species may also be determined. The best fit between the theoretical and experimental curves of the optical density variations with time are obtained for a = 0 . 7 and D = 2 x 10 ~'~ cm 2 s- ~at 25 °C.
References 1 R. S. Crandall and B. W. Faughnan, Appl. Phys. Lett., 26 (1975) 120. 2 B. Reichman and A. J. Bard, J. Electrochem. Soe., 126 (1979) 583. 3 T. Kamimori, J. Nagai and M. Mizuhashi, Proc. SPIE, 428
(1983151. 4 R. S. Crandall and B. W. Faughnan, AppL Phys. Lett., 28 (1976195. 5 H. Tada, in C. M. Lampert and C. G. Granqvist (eds.), LargeArea Chromogenics: Materials and Devices for Transmittance Control, SPIE Institute Series, Vol. IS4, 199(/, 230. 6 B. W. Faughnan and R. S. Crandall, in J. Pankove (ed.), Topics in Applied Physics, Springer, Berlin, Vol. 39, 1980, 181. 7 0 . Bohnke, C. Bohnke, A. Donnadieu and D. Davazoglou, J. Appl. Electrochem., 18( 19881447. 8 0 . Bohnke, M. Rezrazi, B. Vuillemin, C. Bohnke, P. A. Gillet and C. Rousselot, Solar Energy-Mater., submitted for publication. 9 J. Crank, Mathematics of Diffusion, Oxford Univ. Press, London, 1957. 10 R. S. Crandall, P. J. Wojtowicz and B. W. Faughnan, Solid State Commun., 18 (1976) 1409. 11 J. O. M. Bockris and A. K. N. Reddy, Modern Electrochemistr)'--2, Plenum Press, New York, 1973.
243
Proton insertion into thin films of amorphous W03: kinetics study O. Bohnke and B. Vuillemin Laboratoire d'Electrochimie des Solides (UA 0436 CNRS) Universitd de Franche-Comtd 25030 Besanqon (France) (Received November 7, 1991)
Abstract Thin films of electrochromic materials may greatly contribute to understanding the mechanism of cation insertion into mixed conductors. In this paper, we present the results of electrochemical and optical in-situ measurements during proton insertion performed on thin films of tungsten trioxide deposited by vacuum evaporation onto conductive substrates. The measurements clearly show that the electrochemical insertion of protons and electrons into the oxide, i.e. the current density, is limited by a resistance and not, as generally believed, by the charge transfer reaction at the electrode-electrolyte interface. The coloration of the oxide is limited by the ionic diffusion of the inserted cation into the host matrix. From these experimental results, we propose a model to explain the electrocoloration of tungsten oxide. This model involves simultaneous diffusion and chemical reaction. A finite difference method has been used to solve the differential equations and to obtain the diffusion coefficient of the species diffusing into the oxide.
1. Introduction
The electrochromism of amorphous tungsten trioxide (a-WO~) thin films has been extensively studied [1-6] but there are still uncertainties about the processes governing both the current and coloration rate. However, it seems clear that electrocoloration of a-WO3 is the result of a double injection of cations and electrons into the host matrix. During the electrochemical coloration, the electrons are injected into the oxide from the electronic conductor through the electronic/a-WO~ interface. At the same time, the ions M ÷ coming from the electrolyte are injected through the electrolyte/a-WO 3 interface. The overall electrochromic reactio n may be written as follows: a-WO3(transparent) + xM + + x e - ~
M x W O 3 (blue)
(1) In this paper, we present results of electrochemical and in-situ optical measurements during proton insertion into thin films of tungsten trioxide obtained by vacuum thermal evaporation. A model is proposed to explain the mechanism of electrocoloration of this oxide. This model involves simultaneous diffusion and chemical reaction.
2. Experimental details
Tungsten trioxide thin films were deposited onto conductive substrates by vacuum thermal evaporation
of WO 3 powder (99.99% purity, Johnson Matthey Co) contained in a W boat. The evaporation pressure was 10 -5 mbar and the deposition rate was typically 5 A s-~. The thickness of the films was 4600 A. We used three types of substrate: transparent indium--tin oxide (ITO) coated glass substrates purchased from Balzers Co., with sheet resistances of 140f2 []-~ and 10~ [] -~, and W metal foil (99.9% purity) was purchased from Prolabo. The electrochemical and optical measurements were performed simultaneously in a classical three-electrode cell as previously described [8]. Computer-controlled equipment was used for data acquisition. The working electrode of the electrochemical cell was the electrochromic electrode, the counter electrode was a platinum grid and the reference electrode was a saturated sulfate electrode (SSE). The electrolyte was a continuously deaerated 0.01 N H2SO 4 solution (pH = 2.26).
3. Results
The electrochromic mechanism we propose involves simultaneous diffusion and chemical reaction and is based upon experimental observation. It is directly connected to simultaneous electrochemical and optical measurements. When the electrical pulse applied to the electrode is interrupted, the current crossing the electrode goes down to Zero but the optical density of the film continues to increase for 0.5-1 s. Elsevier Sequoia
244
(). Bo/mke and B. l'tdllemin
/'
l'rot
E L K CTI:IOL"KfE 9"
OI']ICAL DENSI] 5
B-"
•
• • X
7-
Fit) itlll/f:
008
t
WO 3
ELECTRONIC CONOUCTOR
<
= ~')0 ~ c m ) H ÷
015 ~)25
=~
H'
,14
0
im
65-
W, 5,',, H~-t
4
~-Z 3
Fig. 2. Model for the eleclrocoloration of a-WO.
2 1 0 0.5
0.7
- 0.9
-1,1
-1,3
- 1,5
Applied Voltage (V/~SSE)
Fig. 1. J vs. V,, at O.D.=0.08 (m), 0.15 (+), 0.25 (O), 0.4 (&),
(1.5 ( x ). This long time indicates that the observed phenomenon is not an electronic relaxation but rather an ionic diffusion process. This phenomenon has been observed in both proton and lithium electrolytes and for all methods used to obtain the oxide, i.e. chemical vapour deposition [7], anodic oxidation [8] and vacuum evaporation. During coloration, the current density (J) decreases with time. It depends strongly on both the electrolyte conductivity and the substrate conductivity; the higher these conductivities, the higher the current crossing the cell. This point has been clearly shown in previous papers [3, 7, 8]. Figure 1 shows the variation of the coloring current density at a given injected charge value (or at a given x value) with respect to the applied potential V,. It is clearly shown that the current varies linearly with V,. The electrode used in this experiment is WO3 deposited onto ITO (10ff2/n). We previously mentioned this behavior with electrodes made of WO 3 obtained by chemical vapor deposition (CVD) deposited onto the SnO2:F substrate [7] and with electrodes made of WO 3 obtained by anodic oxidation of W foil [8]. Kamimori et al. also reported such behavior [3]. However, this result is different from the results reported by Faughnan et al. [6] who postulated that the coloring current is limited by a potential barrier at the electrolyte-WO 3 interface and consequently that it increases exponentially with increasing applied potential.
4.
Discussion
From these experimental results, we propose a model to explain both the current behavior and the coloration process at the tungsten oxide electrode. It is
schematically shown in Fig. 2. Part of this model has been previously proposed by other authors [2, 3, 5] but no one has derived a theoretical expression of the variation of the optical density with time. The mechanism we propose is based on two successive processes occurring during the electrochromic phenomenon. First, a rapid diffusion of electrons through the oxide and a charge transfer reaction at the WO3-electrolyte interface leads to a current J crossing the electrode and to the formation of a species (H+,e-). The species (H+,e - ) then diffuses into the oxide with a diffusion coefficient D. Second, a chemical reaction between the diffusing species (H+,e - ) and the W ~'+ ions takes place, leading to the coloration of WO~ and to the formation of an immobilized proton in the oxide. The observed current in the cell is the consequence of the first reaction and the coloration is the result of the two successive reactions. In a first approximation, if we assume that the coloration reaction is instantaneous and then proceeds very rapidly compared with the diffusion process, a local equilibrium exists between the diffusing species ( H + , e ) and the immobilized proton. The concentration of the immobilized species ((72) is then directly proportional to the concentration of the diffusing species ( C1 ). According to this model, and if we assume that diffusion is one-dimensional along the Z-axis, the diffusion equation is [9]: 0C
1
0t
02C1
-D
i~Cm
- -
0Z 2
(2)
i3t
with
C2 =aC~
(3)
The initial and boundary conditions are:
C~=C 2=0
at
D 0Cl_ J (1 + a) OZ nF i3Ci 0Z
0
at
t = 0 (for all Z) at
Z = 1
Z=0
(4) (5)
(6)
O. Bohnke and B. Vuillemin
/
Proton insertion into thin films of amorphous WO/ kinetics study
with l the thickness of the oxide film, n the number of electrons exchanged per mole of substance reduced (n = 1), F the Faraday's constant and J the current density. Condition (5) implies that there is no accumulation of (H+,e -) species in the oxide at the WO3-electrolyte interface and condition (6) implies that no diffusing species can cross the electronic conductor-WO 3 interface. The effect of the instantaneous reaction is to slow down the diffusion process into the oxide as shown by equation (5). To analyze the coloration process, we can visualize the operation by referring to Fig. 2. As coloration proceeds under a constant potential Vo, applied against a reference electrode, protons are inserted into WO3 and x increases in the host matrix. Consequently, the chemical potential of (H+,e -) in the matrix increases, leading to an electromotive force ~, that opposes the current flow, as described by Crandall et al. [10]. This back emf, which is just equal to the change of the chemical potential of (H+,e - ), is mostly responsible for the current drop in the electrode. Moreover, it varies with x. Since insertion does not proceed under an equilibrium condition, there is a gradient of x in the oxide that varies with time. The most important parameter in the kinetics process is the value of x at the electrolyte-WO 3 interface. It will govern the value of the overpotential r/across the interface. The potential V,, may then be expressed as: 1
V, = rJ(t) + rl(t) + Vb(t) +±/,/H +electr°lyte F
(7)
1
Vh(t)=-~-/~<~+~- /
RT pH F
at
Z=0
3O0-
250-,
200?
150
P
1O0
'e. 50
0
-50
.-100 2
4 rim, (,)
Fig. 3. -~w.~-~/F vs. time at (a) -0.07, (b) -0.29, (c) -0.37 V/NHE. 0.7
0.6
OD(I)
= fj
£ C2(Z,I)dZ
o
0.•
OD
E = 5.35 x 106 c m 2 / m o l e ,
theory
0.]
~1- exoelilsle"l °l
O.1
O
, O
F 2
J 4
1line ( , )
where rJ is the ohmic drop across the electrode and the electrolyte, and /~H. Y is the chemical potential of protons in the acid electrolyte. Figure 1 clearly shows that J(t) varies linearly with V, (for J ( t ) > 1 mA cm-2). This result suggests that the current is limited by resistance and not by thermal activation over a barrier. This latter case would lead to a current which depends exponentially on the applied potential [11]. The relationship between J and V~ may then be expressed, for J > 1 mA cm -2, as: V,,=rJ(t)+ V~(t)-2.3
245
with (8)
it-
where R is a gas constant and T is the absolute temperature. The resistance r is given by the inverse of the slope of the straight lines of Fig. 1. r was found to be 75Q cm 2 for WO 3 vacuum deposited onto W sheet, 90ff~ c m 2 for deposition onto ITO (10ff~ rl-I and 145Q c m 2 onto ITO (140f2 []- 1). r is then influenced by the con-
Fig. 4. Optical density vs. time at (a) -0.07, (b) -0.29, (c) -0.37 V/NHE: (--), experimental; (I~), theoretical. ductivity of the substrate. We have previously shown [7, 8] that this resistance is also influenced by the electrolyte conductivity. From eqn. (8) we can determine the values of/~iH*,e-~ at Z = 0 as a function of time. The result is shown in Fig. 3 at different Vo. For each value of the applied potential, there is a time at which the composition of the interface is in equilibrium with hydrogen gas, i.e. tt/H+x-/ ----0. The higher V,,, the smaller this time. Consequently, for greater times, H 2 evolution may occur thermodynamically at this interface. Eqn. (2) may not be solved analytically because of the non-linear boundary condition (5). We solved it by using the finite difference method [9]. Since [W 5÷ ] = c2(g,t), the optical density of the film is given by the relationship: 1
O'D(t) = f eC2(Z,t) dz
(9)
0
where e is the absorption coefficient of the colored film, i.e. e = 5.35 x 1 0 6 c m 2 mo1-1. Figure 4 shows the
246
O. Bohnke and B. Vuillernin
/
Proton insertion into thin fihns qf anunphotts WO ~.kinetics study
results of the above calculations together with the experimental curves at different applied potentials for WO~ deposited onto ITO (lOft2 [] ~). The best fit is obtained with a = 0 . 7 , D = 2 x l 0 -~° cm 2 s -1 and r= 90Q cm 2. However the discrepancy observed at small times may be attributed to a slower coloration reaction. Further studies with a slow chemical reaction instead of an instantaneous one are underway. A better fitting might be obtained.
5. Conclusion During coloration of a-WO3 thin films obtained by vacuum thermal evaporation in acid electrolyte, the current was found to be limited by resistance and not, as generally believed, by the charge transfer reaction at the electrode-electrolyte interface. The current density-potential relationship may be expressed by 1 RT V~= r J ( t ) - F la(H+x-'- 2.3 ~ - pH
where ~(H*x-) is the chemical potential at the interface of the injected (proton-electron) pair. The resistance r depends on the substrate conductivity and on the electrolyte conductivity. Assuming that simultaneous diffusion and chemical reaction are involved in the electrocoloration mechanism, the differential equation may be solved by a finite difference method and the diffusion coefficient of the (H +,e ) species determined.
The equilibrium constant between the immobilized species and the diffusing species may also be determined. The best fit between the theoretical and experimental curves of the optical density variations with time are obtained for a = 0 . 7 and D = 2 x 10 ~'~ cm 2 s- ~at 25 °C.
References 1 R. S. Crandall and B. W. Faughnan, Appl. Phys. Lett., 26 (1975) 120. 2 B. Reichman and A. J. Bard, J. Electrochem. Soe., 126 (1979) 583. 3 T. Kamimori, J. Nagai and M. Mizuhashi, Proc. SPIE, 428
(1983151. 4 R. S. Crandall and B. W. Faughnan, AppL Phys. Lett., 28 (1976195. 5 H. Tada, in C. M. Lampert and C. G. Granqvist (eds.), LargeArea Chromogenics: Materials and Devices for Transmittance Control, SPIE Institute Series, Vol. IS4, 199(/, 230. 6 B. W. Faughnan and R. S. Crandall, in J. Pankove (ed.), Topics in Applied Physics, Springer, Berlin, Vol. 39, 1980, 181. 7 0 . Bohnke, C. Bohnke, A. Donnadieu and D. Davazoglou, J. Appl. Electrochem., 18( 19881447. 8 0 . Bohnke, M. Rezrazi, B. Vuillemin, C. Bohnke, P. A. Gillet and C. Rousselot, Solar Energy-Mater., submitted for publication. 9 J. Crank, Mathematics of Diffusion, Oxford Univ. Press, London, 1957. 10 R. S. Crandall, P. J. Wojtowicz and B. W. Faughnan, Solid State Commun., 18 (1976) 1409. 11 J. O. M. Bockris and A. K. N. Reddy, Modern Electrochemistr)'--2, Plenum Press, New York, 1973.