1. Introduction

Electrochromic (EC) materials are of great technological interest for a variety of applications, for example, in highly interesting energy-efficient smart windows [1, 2, 3]. An EC device can be viewed as a thin- film electrical battery whose charging state is manifested in optical absorption, i.e., the optical absorption increases with increased state of charge and vice versa. As a result of this changeable absorption, the transmittance of solar energy and visible light in windows can be controlled, hence the name “smart window” [2]. These unique properties of controlling light and solar energy transmittance through windows make the EC technology potentially able to increase indoor comfort and save large amounts of energy for buildings [2, 3]. Most energy savings derive from reducing the amount of solar energy passing through the windows, i.e., reducing the building’s cooling needs. In fact, compared to an efficient low-emissivity window, the EC window shows annual peak cooling load reductions of 19%–26% when the EC window is controlled to minimize solar heat gains and lighting energy use savings of 48%–67% when the EC window is controlled for visual comfort [4].

Today, EC devices like small windows, car rear-view mirrors, and eyewear are being produced as prototypes by various companies [5]. To reach the construction market, it is crucial to be able to produce large-area EC devices (from $1{\mathrm{m}}^2$ upward) with satisfactory performance and low cost. A challenge with upscaling is to design the electrical system of the EC device with a rapid and uniform coloration (charging) and bleaching (discharging) [1, 3]. Al though this is a well-known issue, little work has been done to address and solve these problems [6, 7].

The EC device electrical system design can be improved either by decreasing the transparent conductor resistance or by optimizing the charge and discharge procedures. Decreasing the transparent conductor resistance, i.e., increasing its thickness, will result in a significant increase of the overall device cost since the transparent conductor material is very expensive. To optimize the coloration and bleaching procedures, there has to be a well-founded understanding of the current distribution effects in the EC electrical and electrochemical systems. A further advantage of understanding the current distribution effects is that the EC electrical system design also can be optimized for low cost and lifetime, for example decreasing the amount of transparent conductor material to a minimum without sacrificing too much of the performance and lifetime. A current distribution model, validated with experimental data, is known to be a powerful tool for investigating and understanding current distribution effects of novel designs such as the EC system.

This paper introduces a methodology, based on camera vision, that makes it possible to validate EC current distribution models with experimental data. The methodology consists of two experimental procedures and one model simulation procedure. The first experimental procedure enables the experimental equipment to measure current distribution effects as transmittance distribution during coloration and bleaching procedures. The second experimental procedure enables current distribution models to simulate transmittance distribution over time for different coloration and bleaching procedures. The simulation procedure evaluates EC current distribution models by comparing the model-simulated transmittance distribution data with the experimentally measured transmittance distribution data. To show this methodology’s ability to validate EC current distribution, some simple current distribution models are included. Moreover, some EC window-size coloration predictions are also included to show the potential of using this methodology as a design tool for optimizing the EC system design. This paper is the starting point of the work to develop a well- defined mathematical model for EC systems.

2. Experimental

In this section the material and equipment used in the methodology are presented. The actual procedures used in the methodology are presented in Section 3.

2A. EC Device

The EC devices used in this study are made of flexible polyester foils with sputter-deposited transparent electrical conductors [${\mathrm{In}}_2{\mathrm{O}}_3$:$\mathrm{Sn}{\mathrm{O}}_2$ (ITO)], anodic EC material (NiO), and cathodic EC material (${\mathrm{WO}}_x$) that are joined together with an ion-conducting electrolyte, illustrated in Fig. 1.

All EC devices have an ITO sheet resistance measured to $20\mathrm{Ohm}/\mathrm{sq}$. Two different EC device sizes are used in this study, $5\mathrm{cm}\times 5\mathrm{cm}$ and $20\mathrm{cm}\times 5\mathrm{cm}$, as shown in Figs. 1c, 1d. The purpose of using two different EC device sizes is to use the $5\mathrm{cm}\times 5\mathrm{cm}$ devices for measurements where the current distribution effects should be kept to a minimum and the $20\mathrm{cm}\times 5\mathrm{cm}$ devices to study current distribution effects as transmittance distribution. The current distribution effects can be studied on devices with the width of only $5\mathrm{cm}$ since the current collector resistance is negligible, thus making the distance between the current collectors the critical design parameter.

2B. Experimental Equipment

The equipment used in this study consists of a digital camera (Sony 10XCDV60CR), high-resolution optics (Goyo 80GMHR31614MCN) with a $532\mathrm{nm}$ laser-line bandpass filter, a cold cathode fluorescent backlight panel, a potentiostat (PAR EG&G Model 273A), a general-purpose interface bus (IEEE 488.1) controller for high-speed USB, and a computer with the software LabView 8.6. A schematic and an actual picture of the experimental setup are shown in Fig. 2.

The camera (with an optical bandpass filter mounted) and the potentiostat are connected to the computer, with the backlight panel providing background light for the camera. To minimize stray light disturbance, the camera and backlight panel are placed in a closed cabinet. In addition, an opaque material, designed to only emit light within the dimensions of the active EC surface, is placed on top of the backlight panel during calibration and the following measurement(s). The camera is placed about $80\mathrm{cm}$ from the backlight panel.

A LabView interface is programmed to simultaneously control the coloration/bleaching (charge/ discharge) of the device and to collect data from the EC device samples. A potentiostat is used for collecting electrical data (current, voltage, and charge), while a digital camera is used for collecting optical data (transmittance and location). As accurate electrical charge measurement is important, the potentiostat’s coulometer is used for charge measurements. Using a digital camera for optical measurements has the advantage of freely choosing quantifying points, lines, or contrast levels without modifying the experimental setup. Yet another advantage is the possibility to capture pictures of the EC surface as images, which makes it possible to capture the actual coloration/bleaching behavior and optical deviations (such as defects) over the EC surface. This study utilizes the option of quantifying discrete points for determining the local transmittance state and utilizes images for showing the actual coloration/bleaching behavior.

It is important to notice the use of an optical bandpass filter. This bandpass filter is the key to making adequate and comparable optical light-transmittance measurements, independent of the background light spectrum. Using an optical bandpass filter, the camera only detects wavelengths of $532\mathrm{nm}$, making the transmittance measurement easy to define and compare with other transmittance measurements. The specific bandpass filter wavelength, $532\mathrm{nm}$, was chosen since EC applications often are see-through applications and the human eye is most sensitive at around that wavelength. However, using a bandpass filter coloration/bleaching information for other wavelengths is neglected, but since it is easy to measure the light-transmittance spectrum at different states of charge (using a spectrometer), the neglected information is possible to obtain anyway.

An Ocean Optics spectrometer was used for verifying the transmittance measurements.

2C. Experimental Setup Accuracy and Precision

As the setup should be able to measure the change of transmittance state and its distribution over a large-area device, the setup has to be optically calibrated to process accurate and precise location and transmittance data from the digital images.

The pixel-to-real-world coordinates are calibrated with a calibration grid. The calibration results in a location measurement accuracy within $1\mathrm{mm}$ with very high precision. The pixel size is about $0.4\mathrm{mm}$ with a camera distance of $80\mathrm{cm}$.

The transmittance measurement is calibrated by first measuring the light intensities at 0% and 100% transmittance state for every measurement point with 50 iterations, followed by calculating the mean linear relationship for every measurement point. Repeating with 50 iterations for every measurement point is important for increasing the accuracy since it minimizes influence of both the fluctuations of light intensity and unevenness of light intensity over the backlight panel area. However, since light scattering and reflection make the camera detect light in tensities from other pixels than desired, the calculated linear relationship between the light intensity and transmittance has to be adjusted to give an accurate transmittance. This light scatter correction procedure is preformed by measuring the light intensities and transmittance through a linearly stepped neutral density filter (11 transmittance states in the range of 1%–91%) and calculating the correction coefficients for the linear relationship. By using the light scatter correction values of 0.92 for the linear coefficient and 1.16 for the zero transmittance point, the transmittance measurement accuracy is within $\pm 1\%$ units and the precision within $\pm 1\%$ [i.e., $\pm (T_{532\mathrm{nm}}\times 0.01)\%$ units] in the transmittance range 5%–85%. The limiting factors of this setup’s accuracy and precision are the light intensity fluctuation over time and the unevenness over the backlight panel.

The experimental setup sample rate for time, current, potential, charge, and transmittance points is about $0.3\mathrm{s}$.

3. Results

3A. Experimental Results

3A1. Transmittance Distribution Measurements

The setup is able to measure the transmittance distribution as pictures, which is shown in Fig. 3, where a $20\mathrm{cm}\times 5\mathrm{cm}$ EC device has been colored (charged) with a potential of $6.0\mathrm{V}$ versus ${\mathrm{WO}}_x$ for about $6\mathrm{s}$.

Studying pictures like this, current distribution effects are shown as uneven transmittance distribution, i.e., a color gradient over the surface. This color gradient is seen in Fig. 3 as more colored areas near the current collectors than in the middle. By placing the transmittance measurement points according to Fig. 3, current distribution effects can be measured quantitatively as transmittance distribution, shown in Figs. 4, 5.

The quantitative data show the same results as in Fig. 3; i.e., the areas near the current collectors are coloring/bleaching faster due to current distribution effects. The transmittance is only plotted for half of the device as measurements showed that the transmittance distribution behavior during coloration/bleaching was symmetrical.

3A2. Correlation of Transmittance to State of Charge

To simulate transmittance distribution, using EC current distribution models, this methodology uses experimental data for correlating the transmittance state to the state of charge (SOC). This correlation enables the model to simulate the charge distribution as transmittance distribution, thus making it possible to compare measured and model-simulated transmittance distribution data.

To correlate the experimental transmittance to SOC, small-area EC devices ($5\mathrm{cm}\times 5\mathrm{cm}$) with a controlled coloration/bleaching algorithm are used. The controlled coloration/bleaching algorithm consists of a galvanostatic charge and discharge current of $0.05\mathrm{mA}/{\mathrm{cm}}^2$, with the fully colored and bleached state limits of 15% and 55% transmittance, respectively. The small-area device and galvanostatic coloration/bleaching algorithm are chosen to minimize the current distribution effects. If there are significant current distribution effects during this experimental procedure, the transmittance will not be uniform over the EC surface throughout the procedure, thus making it impossible to relate the SOC to the actual corresponding transmittance state. The current distribution effects are recorded by always measuring the transmittance in nine evenly distributed measurement points.

Fitting the experimental transmittance and SOC data from this procedure results in the polynomial Eq. (1),

The comparison shows that the fitted data differ by no more than $\pm 0.0075\%$ units from the measured data. This shows that the correlation procedure has high accuracy.

3B. Model Simulation Results

3B1. EC Models

There are numerous local kinetic models for the coloration and bleaching behavior of EC materials, especially ${\mathrm{WO}}_x$, including effects of charge transfer, diffusion, and local ohmic effects [8, 9]. However, neither of these models includes effects of current distribution. The current distribution models presented in this study are two-dimensional time- dependent models (one for coloring and one for bleaching) describing the potential, current, and charge distribution over the EC device surface. Ohm’s law [Eqs. (2, 3)] is used to describe current flow through the transparent conductor layers (ITO layers):

These models describe secondary current distribution as they include voltage losses in the electrochemical reaction kinetics. The electrochemical reaction kinetics is described by a Butler–Volmer-based equation. Moreover, the electrolyte is described as a resistance. However, the models do not include any diffusion effects or electrical resistance effects between the current collector and the transparent conductor. An addition to the discharge model is an electrode interface resistance. More details are shown in Appendix A.

Although this model relies on simple kinetics, there are numerous of kinetic models for the coloration and bleaching behavior of EC materials, especially ${\mathrm{WO}}_x$, under different conditions [8].

3B2. Model Fitting and Verification

In the models, one parameter, $S·i_0$, was used when fitting the model to experimental results. All other parameters where kept constant. More details are shown in Appendix A.

Figures 7, 8 show both the measured and the simulated data for one case of coloring (charging) and bleaching (discharging), respectively.

One could see that the simulated data catch the transmittance distribution behavior fairly well, i.e., that the areas near the current collectors color/bleach faster than the areas near the middle of the device. However, the coloring model shows a somewhat better match than the bleaching model.

Notice that these results only show that the presented models are validated for transmittance distribution for this particular case. Depending on the purpose of the model simulations, the model should be validated for different coloration/bleaching voltages, sizes, or other properties. The model could also be validated with the total current and charge response as these data are also available.

3B3. Model Simulation Predictions

When having an EC current distribution model able to simulate the transmittance distribution, the model can be used as a design tool. A design tool where numerous parameters such as changed geometry, changed transparent conductor resistance, and different coloration/bleaching voltage behaviors can be evaluated. One example is to use the coloring model to compare the coloration behavior of a full size window at different coloration voltages ($1.5\mathrm{V}$, $3.0\mathrm{V}$, and $6.0\mathrm{V}$), shown in Figs. 9, 10, 11.

The model predicts that coloring a window with $1.5\mathrm{V}$ results in a relatively even transmittance distribution throughout the coloration but slow overall coloration (no point reaches 15% transmittance state even after $1800\mathrm{s}$). A $3.0\mathrm{V}$ coloring, on the other hand, gives a relatively fast coloration but a rather uneven transmittance distribution throughout the coloration (the transmittance distribution becomes more even with time). Finally a $6.0\mathrm{V}$ coloring gives a very fast coloration but very uneven transmittance distribution throughout the coloration (the transmittance distribution increases with time).

Comparing Figs. 7, 9, the model prediction also shows that, by increasing the size more than three times ($20\mathrm{cm}\times 5\mathrm{cm}$ to $70\mathrm{cm}\times 5\mathrm{cm}$), the average coloration speed decreases almost seven times at the edge ($6.7\%\text{\hspace{0.17em} units}/\mathrm{s}$ down to $1\%\text{\hspace{0.17em} units}/\mathrm{s}$) and more than 13 times in the middle ($2\%\text{\hspace{0.17em} units}/\mathrm{s}$ down to $0.15\%\text{\hspace{0.17em} units}/\mathrm{s}$).

4. Discussion

It is known that a challenge for the EC technology is designing a window-size EC device with rapid and uniform coloration (charging) and bleaching (discharging). The presented methodology gives the possibility of measuring and simulating transmittance distribution over large-area EC devices, thus making it possible to experimentally validate EC models. These models can then be used to optimize an EC window design for rapid and uniform coloration/bleaching. Even though this study only shows transmittance distribution measurements on a $20\mathrm{cm}\times 5\mathrm{cm}$ EC device, the methodology could easily measure the transmittance distribution up to the size of $70\mathrm{cm}\times 5\mathrm{cm}$ by utilizing the whole backlight panel and measuring half of the device at a time. In fact, even larger sizes are possible by simply using a larger backlight panel.

Even though the accuracy and precision of the transmittance measurements in this methodology are good enough for the purpose, they could probably be improved somewhat with a more stable backlight source. Choosing a more stable and even light source over surface and time, the present light intensity unevenness and fluctuations would be smaller. This fluctuation and unevenness are inevitable since this methodology’s backlight panel has a single fluorescent lamp as light source providing and distributing light for the whole backlight surface. An example giving more even light is using a backlight source with light-emitting diodes (LEDs). Besides being stable, an LED backlight panel has the advantage of being capable of controlling the backlight spectrum, thus making it possible to utilize the camera’s possibility to measure color coordinates, since with a more standardized backlight spectrum the bandpass filter is not needed. Measuring color coordinates and their distribution during charge and discharge could give more information on the current distribution effects. However, a disadvantage with this type of LED backlight panel is that it is very expensive.

The model prediction results show the potential of using simulations as a design tool to investigate the current distribution effects of large-area EC devices. The prediction results, such as from this study, can be used as foundation for developing large-area control algorithms. A first suggestion of a rapid and uniform control algorithm strategy would be to use a gradually increasing coloration voltage going from $1.5\mathrm{V}$ to maybe $3.0\mathrm{V}$. Hence, by starting at a lower voltage, the coloration is kept uniform, and by gradually increasing the voltage, the total coloration speed is increased. Suggestions like this are of course possible to simulate and experimentally validate for more detailed and defined algorithms, which is another great strength of this methodology.

An interesting observation is that, even though the kinetic models presented are simple, model simulations fit the experimental data fairly well. This indicates that the current distribution effects over the transparent conductors are the limiting factor for rapid and uniform coloration/bleaching of large-area EC devices. This indication is plausible, considering that experimental and simulated data are obtained at a relatively low sheet resistance, $20\mathrm{Ohm}/\mathrm{sq}$, and relatively small EC device size, $20\mathrm{cm}\times 5\mathrm{cm}$. This is also what the model predicts; the coloration speed goes down dramatically with increased EC device size. Even if the results show that the simulation model fits the experimental data fairly well and that the transparent conductor may be the limiting factor, these models are limited and need further development. The models are based on Butler–Volmer kinetics. This assumption may be appropriate for the coloration process but may be not as appropriate for the bleaching process, which may explain the better fit for the coloration model and the need of an extra electrode interface resistance for the bleaching model. The current distribution model puts large demands on the local kinetic model, since neither local current nor potential is constant.

With the capability to experimentally validate EC current distribution models, future work should concentrate on developing the kinetic models and validating the model for a larger matrix of experimental data. Based on already published findings, the kinetic models should be further developed to include, e.g., diffusion and contact resistance.

5. Conclusions

A good understanding of the current distribution effects is crucial when optimizing the EC window technology for performance, low cost, and lifetime. An experimentally validated current distribution model is a well-suited tool for increasing the understanding of current distribution effects for novel designs such as the EC device system. This paper introduces a methodology that is clearly capable of measuring current distribution effects as transmittance distribution over an EC device, correlating the transmittance state to the SOC with high accuracy so that current distribution models can simulate transmittance distribution over an EC device. This meth odology can thus experimentally validate current distribution models. We conclude that this methodology is crucial for validating current distribution models used to optimize the EC window technology.

The kinetic model is relatively simple, but the model simulations still match the experimental data fairly well; this indicates that the electrical properties of the transparent conductor layers constitute the limiting factor for rapid and uniform coloration (charging) and bleaching (discharging) for large-area EC windows. Although the simulations match the experimental data fairly well, the local kinetic model needs to be developed further.

The potential of this methodology lies in the possibility to both simulate and experimentally validate new construction and control algorithms.

Appendix A

Model Equations

The model uses Butler–Volmer-based kinetics [Eqs. (A1, A2, A3, A4, A5, A6)]:

(A1)$$j=S·j_0\{e^{\left(\frac{{\alpha }_{a}F{\varphi }_{\mathrm{\Sigma }}}{\mathrm{RT}}\right)}-e^{\left(\frac{{\alpha }_{a}F{\varphi }_{\mathrm{\Sigma }}}{\mathrm{RT}}\right)}\},$$

where

(A2)$${\varphi }_{\mathrm{\Sigma }}=E_1-E_2-\mathrm{\Delta }{E}_{iR}-\mathrm{\Delta }{E}_{eq}$$

($X_1$ is for ${\mathrm{WO}}_x$ and $X_2$ is for NiO),

(A3)$$\mathrm{\Delta }{E}_{iR}=j\frac{d}{{\kappa }_{el}},$$

(A4)$$\begin{gathered} \mathrm{\Delta }{E}_{eq}=f(\mathrm{SOC}) \\ =-0.5+\mathrm{SOC}·1.9-{10}^{-3}·e^{(\frac{1}{\mathrm{SOC}+0.15})} \end{gathered}$$

(shown in Fig. 12),

(A5)$$\mathrm{SOC}=\frac{q(x,t)}{q_{\max }},$$

(A6)$$q_{\max }=16\mathrm{mC}/{\mathrm{cm}}^2.$$

The potential drop and current flow in the transparent conductors are described by Ohm’s law [Eqs. (A7, A8)]:

(A7)$$(-\sigma \frac{dE}{dx}=i\text{and}-\sigma \frac{di}{dx}=j)\mathrm{gives}-{\sigma }_1\frac{d^2{E}_1}{d{x}^2}=j,$$

(A8)$$-{\sigma }_2\frac{d^2{E}_2}{d{x}^2}=-j.$$

The boundary conditions for the models are

$$x=0\mathrm{and}t=t:i_2=0\mathrm{and}{E}_1=0,$$

and for

$$x=L\mathrm{and}t=t:i_1=0\mathrm{and}{E}_2=V_{\text{app}}.$$

The model calculates the local SOC with integrating the charge [Eq. (A9)],

(A9)$$\frac{dq}{dt}=j,$$

where

$$q(x,0)=0(\text{bleached}/\text{discharged state},T=55\%),$$

$$q(x,0)=q_c(\text{colored}/\text{charged state},T=14.5\%)$$

are used for the coloration and bleaching case, respectively.

For the bleach (discharge) model, a charge- dependent contact resistance at the electrode interface is added Eqs. (A10, A11)]:

(A10)$$\mathrm{\Delta }{E}_{iR}=j(\frac{d}{{\kappa }_{\text{el}}}+R_i),$$

where

(A11)$$R_i=0.05(1-\mathrm{SOC}).$$

Model Constants

The models use the following constants:

• ${\sigma }_{\mathrm{ITO}}={\sigma }_1={\sigma }_2=2·{10}^5\mathrm{S}/\mathrm{m}$.
• $d_{\mathrm{ITO}}=250\mathrm{nm}$ ($d_{\mathrm{ITO}}$ and ${\sigma }_{\mathrm{ITO}}$ give a sheet resistance of $20\mathrm{Ohm}/\mathrm{sq}$).
• ${\kappa }_{\text{el}}=1·{10}^{-3}\mathrm{S}/\mathrm{m}$.
• $d_{\text{el}}=5·{10}^{-6}\mathrm{m}$.
• ${\alpha }_a={\alpha }_c=0.5$.
• $T=298\mathrm{K}$.

Model Adjustment

The $S·j_0$ parameter was fitted to the experimental data, where it was $1·{10}^{-15}\mathrm{A}/{\mathrm{m}}^3$ and $1·{10}^{-7}\mathrm{A}/{\mathrm{m}}^3$ for the color (charge) and bleaching (discharge) models, respectively.

Financial support from the Swedish Research Council Formas is gratefully acknowledged. ChromoGenics AB is acknowledged for financial and material support.

 

Fig. 1 (a) Schematic illustration of an EC device (b) in cross section, (c) the $5\mathrm{cm}\times 5\mathrm{cm}$ EC device, and (d) the $20\mathrm{cm}\times 5\mathrm{cm}$ device. The current collectors are placed $0.5\mathrm{cm}$ from the active EC surface. The darker parts around the active EC surface in (c) and (d) are sealant materials.

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Fig. 2 Experimental setup as (top) schematic illustration and (bottom) actual picture.

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Fig. 3 EC surface and its transmittance measurement points for a $20\mathrm{cm}\times 5\mathrm{cm}$ EC device with uneven transmittance distribution. The six outer measurement points to the left and right are placed $15\mathrm{mm}$ from its nearest current collector, and all other measurement points are evenly spaced between these outer points. The ${\mathrm{WO}}_x$ current collector outside on left side (not shown in the figure) is the origin point for all measurement points. The darker parts at the very edges are sealant glue and mask material.

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Fig. 4 Transmittance as a function of time when coloring (charging) a $20\mathrm{cm}\times 5\mathrm{cm}$ EC device with $6.0\mathrm{V}$ versus ${\mathrm{WO}}_x$. The measurement points are placed $15\mathrm{mm}$ (squares), $37\mathrm{mm}$ (crosses), $60\mathrm{mm}$ (circles), $82\mathrm{mm}$ (asterisks), and $105\mathrm{mm}$ (triangles) from the ${\mathrm{WO}}_x$ current collector.

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Fig. 5 Transmittance as a function of time when bleaching (discharging) a $20\mathrm{cm}\times 5\mathrm{cm}$ EC device with $-6.0\mathrm{V}$ versus ${\mathrm{WO}}_x$. The measurement points are placed $15\mathrm{mm}$ (squares), $37\mathrm{mm}$ (crosses), $60\mathrm{mm}$ (circles), $82\mathrm{mm}$ (asterisks), and $105\mathrm{mm}$ (triangles) from the ${\mathrm{WO}}_x$ current collector.

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Fig. 6 Comparison between some typical experimentally measured data and the calculated data from the transmittance versus SOC polynomial.

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Fig. 7 Transmittance as a function of time when coloring (charging) a $20\mathrm{cm}\times 5\mathrm{cm}$ EC device with $6.0\mathrm{V}$ versus ${\mathrm{WO}}_x$. The experimental (markers) and model simulation (solid lines) measurement points are placed $15\mathrm{mm}$ (squares), $37\mathrm{mm}$ (crosses), $60\mathrm{mm}$ (circles), $82\mathrm{mm}$ (asterisks), and $105\mathrm{mm}$ (triangles) from the ${\mathrm{WO}}_x$ current collector.

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Fig. 8 Transmittance as a function of time when bleaching (discharging) a $20\mathrm{cm}\times 5\mathrm{cm}$ EC device with $-6.0\mathrm{V}$ versus ${\mathrm{WO}}_x$. The experimental (markers) and model simulation (solid lines) measurement points are placed $15\mathrm{mm}$ (squares), $37\mathrm{mm}$ (crosses), $60\mathrm{mm}$ (circles), $82\mathrm{mm}$ (asterisks), and $105\mathrm{mm}$ (triangles) from the ${\mathrm{WO}}_x$ current collector.

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Fig. 9 Transmittance as a function of time when coloring (charging) a $70\mathrm{cm}\times 5\mathrm{cm}$ EC device with $1.5\mathrm{V}$ versus ${\mathrm{WO}}_x$. The model simulation measurement points are placed $15\mathrm{mm}$ (squares), $98\mathrm{mm}$ (crosses), $180\mathrm{mm}$ (circles), $263\mathrm{mm}$ (asterisks), and $345\mathrm{mm}$ (triangles) from the ${\mathrm{WO}}_x$ current collector. The coloration is stopped when the first point reaches 15% transmittance state.

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Fig. 10 Transmittance as a function of time when coloring (charging) a $70\mathrm{cm}\times 5\mathrm{cm}$ EC device with $3.0\mathrm{V}$ versus ${\mathrm{WO}}_x$ The model simulation measurement points are placed $15\mathrm{mm}$ (squares), $98\mathrm{mm}$ (crosses), $180\mathrm{mm}$ (circles), $263\mathrm{mm}$ (asterisks), and $345\mathrm{mm}$ (triangles) from the ${\mathrm{WO}}_x$ current collector. The coloration is stopped when the first point reaches 15% transmittance state.

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Fig. 11 Transmittance as function of time when coloring (charging) a $70\mathrm{cm}\times 5\mathrm{cm}$ EC device with $6.0\mathrm{V}$ versus ${\mathrm{WO}}_x$. The model simulation measurement points are placed $15\mathrm{mm}$ (squares), $98\mathrm{mm}$ (crosses), $180\mathrm{mm}$ (circles), $263\mathrm{mm}$ (asterisks), and $345\mathrm{mm}$ (triangles) from the ${\mathrm{WO}}_x$ current collector. The coloration is stopped when the first point reaches 15% transmittance state.

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Fig. 12 Relation between $E_{eq}$ and f (SOC) used in the models.

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