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Dynamic control of reflective/diffusive optical surfaces on EGaIn liquid metal

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Abstract

We report a method to dynamically control the surface of gallium-based liquid metal to switch between reflective/diffusive states by removing/depositing oxide films via electrochemical redox reactions. Electrochemical oxidation deposits rough surface oxides that are optically diffusive. Electrochemical reduction returns the metal to a pristine, smooth, and reflective state. This switching is achieved using only ±1.4V. In addition, a meniscus of liquid metal can be pumped to switch between concave/convex shapes. This work expands the range of optical applications of dynamic liquid metal surfaces.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Liquid metals uniquely combine the electrical, thermal and optical properties of metals with the fluidity of a liquid. These properties enable soft electronic circuits [17], energy harvesting [8], actuators [9], microfluidic components [10,11], and thermal conductors [12,13].

In addition to their electrical and thermal properties, liquid metals have useful optical properties. Pools of liquid metal have been used by physicists to create large lenses [14]. At smaller length scales (<1 cm), liquid metals have promise for elastomeric optics [1] because they can easily be patterned in a variety of ways that are not possible with solid metals (such as injection) and create optical components that are completely soft. For example, liquid metals have been used to create dynamic reflective gratings. The surface of a metal-filled elastomer buckled reversibly in response to mechanical strain to switch from a smooth to a corrugated surface [2].

We also have applied liquid metals to optics (lens mold). Liquids naturally form a smooth surface, which is ideal for molding [3]. Prior to molding, the curvature of the metal can be manipulated via pumping inside a (PDMS) cylinder. Using this method, smooth lenses of silicone with a wide range of curvatures (within a radius of about 7 mm) can be fabricated on demand. However, because liquid metals react with oxygen to form a native oxide [15], the surface can wrinkle during changes in surface area and become a rough scattering surface [2].

In this study, we propose a method to dynamically control the reflective and scattering surfaces of liquid metal surfaces by electrochemical redox reactions [16,17]. The advantage of this method is that the formation and removal of the oxide film can be dynamically controlled, and the reflective and scattering surfaces can be freely controlled. This method can be implemented in ambient temperature and pressure, and requires only a low applied voltage. The aqueous electrolyte used is also neutral and safe.

Figure 1 shows photographs and a schematic diagram of the curved surface being changed dynamically by the push-pull method using liquid metal. Here, the metal is oxide-free due to the application of −1 V to the metal. The metal remains pinned to the inner walls of a cylinder. Thus, pushing (pumping) the metal outward generates the convex shape in Fig. 1 (i). Pulling (retracting) the metal changes the curvature and increases the focal length, as shown in Fig. 1 (ii). Further pulling creates a nearly flat state and the surface appears white due to reflections from a white surface, Fig. 1(iii). Figure 1 (iv) shows a slightly concave surface, which inverts the reflection of the photographer. The metal surface can be rendered even more concave by further retraction. Taken in sum, Fig. 1 highlights the ability to change the shape of liquid metal via pumping. We focus the remainder of this paper on using redox reactions to tune the reflectivity of the surface, which could be applied to any of the curvatures shown in Fig. 1.

 figure: Fig. 1.

Fig. 1. An oxide-free EGaIn surface with different curvatures by aspiration. The top row of photographs is top-down and the bottom row shows a side view of the liquid metal, which can be pumped in or out of the reservoir using an embedded microchannel (i): Convex shape with maximum curvature. (ii): Convex shape with a curvature between maximum and zero. (iii): Flat surface with zero curvature. (iv): Concave shape with curvature between 0 and minimum value. (v): Concave shape and curvature at minimum value. The scale bar in (i) is 1.6 mm. See Visualization 1.

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2. Optical surface control of the EGaIn/electrolyte interface using electrochemical redox reactions

2.1 Experimental setup

Figure 2 shows the schematic diagram of the cell used in this study. Eutectic gallium indium alloy (EGaIn) (mass ratio Ga: 0.75, In: 0.25), which has a melting point of 15.5°C, low viscosity, and low toxicity [18], was used as the liquid metal. Using a syringe, we injected EGaIn into a hole drilled in the cell until it reached a copper ring fit within the hole. The liquid metal pins to this copper ring, which keeps the liquid stationary. As demonstrated in Fig. 1, by pining the metal, the curvature of the EGaIn surface can be controlled by suction and injection with a syringe. A photo-curable resin (Norland optical adhesive) covered top of the copper to prevent it from coming into contact with the electrolyte. This coating isolates the EGaIn surface as the working electrode and optical surface. We used sodium sulfate solution, which is neutral (pH: 7.7), as the electrolyte. We avoided acidic and alkaline solutions because they dissolve the native oxide. Sodium fluoride solution was also tried as a neutral solution, but it produced a black coating on the metal irreversibly when a voltage of about +0.8 V was applied. The black coating impurity is considered as derived from copper, because it wasn't confirmed in the case of a setup without copper as electrodes like section 3 will be described later. While a black surface could be useful in some applications, this product was difficult to remove via electrochemical reduction and thus, we used 0.5 M sodium sulfate instead. A copper wire was used as the counter electrode and immersed in the electrolytic solution. An AC power supply was used so that positive and negative voltages could be applied in a continuously varying manner. We used a square wave with an amplitude of 100 mV, a frequency of 1 kHz, and a duty ratio of 50% using a DC off-set. The offset voltage varied from −0.8 V ∼ +0.8 V. In this paper, offset voltage is defined as a potential difference between average applied voltage across the electrodes and ground.

 figure: Fig. 2.

Fig. 2. Electrochemical cell used for manipulating the optical surface finish on liquid metal.

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Figure 3 shows macroscopic photographs, microscopic images and surface profiles during oxidation or reduction of the EGaIn surface in a flat state. Figure 3(a)∼(d) show the surface states of EGaIn in the oxidized state, and (e)∼(h) show the surface states of EGaIn in the reduced state. These states can be toggled rapidly (response time: from the reduced state to the oxidized state is about 250 ms, from the oxidized state to the reduced state is about 830 ms) by varying the potential between +1 V and −1 V (see Visualization 2). The oxide species on the surface that scatter light (cf. Figure 3(b)) get removed via Faradaic reduction by switching to −1 V. The kinetics of these reactions dictate the response time. The surface naturally wants to oxidize (in the absence of potential), which may explain why removing the oxide takes longer than depositing the oxide. The rapid response time of this method using electricity is shorter than that of methods using physical movement of liquid metal by pneumatics, continuous electrowetting, or gravity since there is no translational movement of the metal mass [17,19,20]. When the oxide film forms on the surface as shown in (a), it scatters light and appears silvery white as shown in (b). Figure 3(c) is a photograph of a part of the surface of (b) observed under a microscope. Figure 3(d) is a profile of (c) scanned by a white interference microscope. It can be seen that the oxidized state produces a rough surface with root mean square (RMS) value of 2.96 µm. On the other hand, when there is no oxide film on the surface, as in (e), the surface reflects light, as in (f). To contrast with (b), it is a metallic reflective surface, although it looks black because it is reflecting black paper. Figure 3(g) is a photograph of a part of the surface of (f) observed under a microscope. Figure 3(h) is the profile of (c) scanned by white interference microscopy. The root mean square (RMS) of the reduced state could not be taken because the slope could not be adjusted, but from the images it is assumed that the roughness is about a few nanometers, as reported previously for a high surface tension liquid [3].

 figure: Fig. 3.

Fig. 3. Cross-sectional schematic of EGaIn surface state (a and e), photographs of diffusing and reflective surfaces on an EGaIn surface (b and f), micrographs of diffusing and reflective surfaces on an EGaIn surface (c and g), and surface profiles of diffusing and reflective surfaces (d and h). In (f), the surface appears black because it is reflecting the color of a black background; in reality, it is mirror-like. Using approximately 1 V, it is possible to switch between the mirror-like and diffuse states.

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2.2 Optical characterization while forming and removing oxide film on EGaIn

Next, the formation and removal of the EGaIn oxide film was verified by optical measurements. Figure 4 shows the schematic diagram of the experiment. The electrochemical cell used in Fig. 2 was placed under an integrating sphere. While keeping the surface of EGaIn flat, we applied an offset voltage of −800 mV to remove oxide such that is reflective. The laser beam incident through the upper hole (DPSS laser λ: 635-650 nm) was adjusted so that it would be reflected by the surface of EGaIn and go out through the original entrance without being scattered by the inner wall of the integrating sphere. When the EGaIn is in a reflective state, the light intensity detected in the integrating sphere is small because no light reflects on the inner wall of the sphere. Conversely, when the EGaIn is in a scattering state, the light intensity entering the detector is large because the light reflects inside the integrating sphere. The procedure of the experiment was to record the current value converted from the light intensity, while change the offset voltage between the working electrode and the counter electrode from −800 mV to +800 mV by 50 mV and from +800 mV to −800 mV by 50 mV. The offset voltage was changed after confirming that the current value indicated by the detector was stable for about 10 seconds. The measurement was performed three times.

 figure: Fig. 4.

Fig. 4. Experimental setup for studying redox reactions optically using an integrating sphere to capture light that reflects from the surface.

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Figure 5 shows the results. The horizontal axis represents the voltage between the working electrode and the counter electrode, and the vertical axis represents the current value corresponding to the light intensity captured by the detector. In the process of changing the voltage from −800 mV to +800 mV, the light intensity increases around −200 mV. The photograph of the - 200 mV surface in the outward path in Fig. 5 shows that the surface has changed from a reflective surface to a diffusing surface. On the other hand, in the process of changing the voltage from +800 mV to −800 mV, the change to the reflective surface occurs not at −200 mV but around −550 mV. The EGaIn surface photograph at −550 mV on the return voltage sweep also shows a change from a diffusing surface to a reflecting surface. This indicates that there is hysteresis in the electrochemical redox reaction of EGaIn, which is a common feature of electrochemical reactions.

 figure: Fig. 5.

Fig. 5. Intensity of scattered light reflected from EGaIn surface versus offset voltage to EGaIn (the upper graph). Photographs of the EGaIn surface versus applied voltage (the bottom figures).

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3. Electrochemical measurement of EGaIn in aqueous sodium sulfate solution

3.1 Experimental setup for electrochemical measurement

Since it was found in the previous section that there is hysteresis in the change of surface state in the electrochemical redox reaction of EGaIn, electrochemical measurements were carried out to obtain the current-voltage characteristics of EGaIn in sodium sulfate solution. Figure 6 shows the experimental setup of electrochemical measurements. The EGaIn was injected from a syringe through a rubber tube into a plastic container. Voltage was applied through the syringe needle to the EGaIn in the tube. In this experiment, a reference electrode (Ag/AgCl) was used to accurately control the potential of the working electrode (EGaIn surface). The measurement conditions are as follows: initial voltage is −0.5 V, maximum voltage is 1.3 V, minimum voltage is −1.4 V, the direction of the first voltage change is negative, sweep speed is 0.05 V/s, and sensitivity is 0.001 A/V. We started with a voltage close to the initial condition (open circuit potential, approximately −0.5 V). The voltage range of −1.4 V to 1.3 V was sufficient to observe the redox reactions with good reproducibility.

 figure: Fig. 6.

Fig. 6. Setup for measuring the current potential characteristics of EGaIn during redox reactions. Figure 6. Setup for measuring the current potential characteristics of EGaIn during redox reactions.

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3.2 Results and discussion

The bottom graph in Fig. 7 shows the current versus potential of the electrochemical measurement. The voltage is lowered from −0.5V to −1.4V and then raised to 1.3V and lowered again to −0.5V. In the initial state, the rough oxide-coated surface is a scattering surface, as shown in photo i) in Fig. 7. In this state, the metal flattens since the oxide film lowers the interfacial tension [21]. At around −1.1V, the current value becomes more negative, indicating the occurrence of a reduction reaction. The photograph in Fig. 7 ii) shows that the oxide film has been removed and is now a reflective surface, as expected for an oxide-free surface. Upon switching to a more positive potential, we observe positive current around −0.8V, which suggests the occurrence of an oxidation reaction. Yet, photo iii) in Fig. 7 shows that there was no obvious change from photo ii). Thus, the deposited oxide film is smooth. With increased potential, the current increased and around 0.2V, the EGaIn surface changed to a scattering surface. The upper graph in Fig. 7 is a part of the graph in Fig. 5 arranged with respect to the position of the redox reaction. From the comparison of the upper and lower graphs in Fig. 7, it can be seen that there is almost no optical loss in the state iii), consistent with the formation of a thin, smooth oxide layer in state iii). This layer starts wrinkling (presumably due to stress from oxidation) near 0.2 V [18].

 figure: Fig. 7.

Fig. 7. Current versus potential between the working electrode (EGaIn) and the reference electrode (lower left plot). i) Initial state (scattered surface), ii) no oxide film (reflective surface), iii) thin oxide film (reflective surface), and iv) change from thin oxide film to thick oxide film (scattered surface). The graph in Fig. 4 aligns the potentials of states ii) and iv) of the graph in the bottom left (top left). The photos and figures on the right are photos and schematic diagrams for states i), ii), iii), and iv).

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Next, we estimated the thickness of the oxide film formed by the oxidation reaction in state iii) of Fig. 6. Figure 8 shows the current vs. potential graph and the charge vs. potential graph superimposed when the oxidation reaction occurred. From the current graph in Fig. 7, the voltage at which the oxidation reaction occurred is calculated to be between −0.799 V and −0.733 V, and the amount of charge transferred in that range is 0.07 mC. From the amount of charge, the thickness of the oxide film can be calculated using the following equation,

$$t[{\textrm{cm}} ]= \frac{{\frac{{C [\textrm{C} ]}}{{F [{\textrm{C}/\textrm{mol}} ]}} \times \textrm{coefficient ratio} \times \textrm{molar mass }[{\textrm{g}/\textrm{mol}} ]}}{{\textrm{density} [{\textrm{g}/\textrm{c}{\textrm{m}^3}} ]\times \textrm{surface area }[{\textrm{c}{\textrm{m}^2}} ]}}$$
where C [C] is the charge, F is the Faraday number (96485.3 [C/mol]), the constant ratio is 1/3 if the reaction equation for gallium is Ga→Ga3++3e-, which is expected if the product is Ga2O3 [16], the molar mass is 187.444 [g/mol], the density is 6.44 [g/cm3] (α-type) or 5.88 [g/cm3] (β-type), and the surface area is 0.25 [cm2] (the surface of a hemisphere with a radius of 0.2 cm). From these values, the estimated film thicknesses are 0.29 nm for the α-type and 0.31 nm for the β-type of gallium oxide. In reality, the film may be amorphous, but nevertheless these give a first order estimate that suggest the film is very thin, consistent with the absence of optical loss.

 figure: Fig. 8.

Fig. 8. An enlarged graph of state iii) in the current potential graph in Fig. 7. The vertical axis on the right represents the charge. Based on the slope of the current graph, the area where the film formed by the oxidation reaction is defined as the integral of the current between the vertical dotted lines.

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4. Conclusion

This paper demonstrates that the surface of a liquid metal surface can be dynamically controlled between a scattering/reflecting state by the formation/removal of an oxide film using electrochemical redox reactions. We demonstrated these changes as a function of potential using optical measurements and correlated the results to electrochemical measurements of current. The measurements show a hysteresis in the voltage needed to toggle the two surface states, which is consistent with the electrochemical measurements. The oxide film forms by two stages: one in which the oxide is thin (<1 nm thick), smooth, and optically reflective and another where it becomes wrinkled, thicker, and rough at higher potentials. This work has implications for creating dynamic surfaces that can toggle between mirror-like to diffuse using low voltages. In addition, the use of liquid metal is interesting because it can be processed at room temperature to create entirely soft devices, patterned in new ways (including injection into optofluidic or 3D printed devices), and undergo shape reconfigurability [4,18,21].

Acknowledgments

We thank ECE at North Carolina State University for allowing us to use their Maker Space for electrochemical cell fabrication.

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

References

1. J. L. Wilbur, R. J. Jackman, G. M. Whitesides, E. L. Cheung, L. K. Lee, and M. G. Prentiss, “Elastomeric optics,” Chem. Mater. 8(7), 1380–1385 (1996). [CrossRef]  

2. M. G. Mohammed and M. D. Dickey, “Strain-controlled diffraction of light from stretchable liquid metal micro-components,” Sens. Actuators, A 193, 246–250 (2013). [CrossRef]  

3. K. Nakakubo, H. Nomada, H. Yoshioka, K. Morita, and Y. Oki, “Gallium and polydimethylsiloxane molding for self-organized spherical lens surface fabrication,” Appl. Opt. 56(36), 9900–9906 (2017). [CrossRef]  

4. M. D. Dickey, “Stretchable and soft electronics using liquid metals,” Adv. Mater. 29(27), 1606425 (2017). [CrossRef]  

5. R. Guo, X. Wang, W. Yu, J. Tang, and J. Liu, “A highly conductive and stretchable wearable liquid metal electronic skin for long-term conformable health monitoring,” Sci. China: Technol. Sci. 61(7), 1031–1037 (2018). [CrossRef]  

6. M. Kim, Hommood Alrowais, and O. Brand, “All-soft physical and chemical microsystems based on liquid metal for wearable electronics applications,” 2017 IEEE 30th International Conference on Micro Electro Mechanical Systems (MEMS), 1162–1165 (2017).

7. C. B. Cooper, K. Arutselvan, Y. Liu, D. Armstrong, Y. Lin, M. R. Khan, J. Genzer, and M. D. Dickey, “Stretchable capacitive sensors of torsion, strain, and touch using double helix liquid metal fibers,” Adv. Mater. 27(20), 1605630 (2017). [CrossRef]  

8. T. Krupenkin and J. A. Taylor, “Reverse electrowetting as a new approach to high-power energy harvesting,” Nat. Commun. 2(1), 448 (2011). [CrossRef]  

9. S. Y. Tang, V. Sivan, P. Petersen, W. Zhang, P. D. Morrison, K. Kalantar-zadeh, A. Mitchell, and K. Khoshmanesh, “Liquid metal actuator for inducing chaotic advection,” Adv. Funct. Mater. 24(37), 5851–5858 (2014). [CrossRef]  

10. D. P. Parekh, C. Ladd, L. Panich, K. Moussa, and M. D. Dickey, “3D printing of liquid metals as fugitive inks for fabrication of 3D microfluidic channels,” Lab Chip 16(10), 1812–1820 (2016). [CrossRef]  

11. K. Khoshmanesh, S. Tang, J. Y. Zhu, S. Schaefer, A. Mitchell, K. Kalantar-zadeh, and M. D. Dickey, “Liquid metal enabled microfluidics,” Lab Chip 17(6), 974–993 (2017). [CrossRef]  

12. D. Yu, Y. Liao, Y. Song, S. Wang, H. Wan, Y. Zeng, T. Yin, W. Yang, and Z. He, “A super-stretchable liquid metal foamed elastomer for tunable control of electromagnetic waves and thermal transport,” Adv. Sci. 7(12), 2000177 (2020). [CrossRef]  

13. M. D. Bartlett, N. Kazem, M. J. Powell-Palma, X. Huang, W. Sun, J. A. Malen, and C. Majidi, “High thermal conductivity in soft elastomers with elongated liquid metal inclusions,” Proc. Natl. Acad. Sci. U. S. A. 114(9), 2143–2148 (2017). [CrossRef]  

14. I. D. Joshipura, H. R. Ayers, C. Majidi, and M. D. Dickey, “Methods to pattern liquid metals,” J. Mater. Chem. C 3(16), 3834–3841 (2015). [CrossRef]  

15. T. Liu, P. Sen, and C. C. Kim, “Characterization of nontoxic liquid-metal alloy galinstan for applications in microdevices,” J. Microelectromechanical Syst. 21(2), 443–450 (2012). [CrossRef]  

16. M. R. Khan, C. Trlica, and M. D. Dickey, “Recapillarity: electrochemically controlled capillary withdrawal of a liquid metal alloy from microchannels,” Adv. Func. Mater. 25(5), 671–678 (2015). [CrossRef]  

17. C. B. Eaker and M. D. Dickey, “Liquid metal actuation by electrical control of interfacial tension,” Appl. Phys. Rev. 3(3), 031103 (2016). [CrossRef]  

18. M. D. Dickey, R. C. Chiechi, R. J. Larsen, E. A. Weiss, D. A. Weitz, and G. M. Whitesides, “Eutectic gallium-indium (EGaIn): a liquid metal alloy for the formation of stable structures in microchannels at room temperature,” Adv. Funct. Mater. 18(7), 1097–1104 (2008). [CrossRef]  

19. W. Zhang, Q. Song, W. Zhu, Z. Shen, P. Chong, D. P. Tsai, C. Qiu, and A. Q. Liu, “Metafluidic metamaterial: a review,” Adv. Phys. 3(1), 1417055 (2018). [CrossRef]  

20. F. Yang, Y. Fan, R. Yang, J. Xu, Q. Fu, F. Zhang, Z. Wei, and H. Li, “Controllable coherent perfect absorber made of liquid metal-based metasurface,” Opt. Express 27(18), 25974 (2019). [CrossRef]  

21. M. R. Khan, C. B. Eaker, E. F. Bowden, and M. D. Dickey, “Giant and switchable surface activity of liquid metal via surface oxidation,” Proc. Natl. Acad. Sci. U. S. A. 111(39), 14047–14051 (2014). [CrossRef]  

Supplementary Material (2)

NameDescription
Visualization 1       An oxide free EGaIn surface with changing curvature by aspiration and injection. EGaIn is applied negative voltage (AC -1 V) through the copper electrode in 0.5 M sodium sulfate solution as electrolyte. This video shows at real speed.
Visualization 2       Dynamic control of reflective/ diffusive surface of EGaIn by redox reaction. EGaIn is applied negative or positive voltage (AC -1 V / +1 V) through the copper electrode in 0.5 M sodium sulfate solution as electrolyte. This video shows at real speed.

Data availability

No data were generated or analyzed in the presented research.

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Figures (8)

Fig. 1.
Fig. 1. An oxide-free EGaIn surface with different curvatures by aspiration. The top row of photographs is top-down and the bottom row shows a side view of the liquid metal, which can be pumped in or out of the reservoir using an embedded microchannel (i): Convex shape with maximum curvature. (ii): Convex shape with a curvature between maximum and zero. (iii): Flat surface with zero curvature. (iv): Concave shape with curvature between 0 and minimum value. (v): Concave shape and curvature at minimum value. The scale bar in (i) is 1.6 mm. See Visualization 1.
Fig. 2.
Fig. 2. Electrochemical cell used for manipulating the optical surface finish on liquid metal.
Fig. 3.
Fig. 3. Cross-sectional schematic of EGaIn surface state (a and e), photographs of diffusing and reflective surfaces on an EGaIn surface (b and f), micrographs of diffusing and reflective surfaces on an EGaIn surface (c and g), and surface profiles of diffusing and reflective surfaces (d and h). In (f), the surface appears black because it is reflecting the color of a black background; in reality, it is mirror-like. Using approximately 1 V, it is possible to switch between the mirror-like and diffuse states.
Fig. 4.
Fig. 4. Experimental setup for studying redox reactions optically using an integrating sphere to capture light that reflects from the surface.
Fig. 5.
Fig. 5. Intensity of scattered light reflected from EGaIn surface versus offset voltage to EGaIn (the upper graph). Photographs of the EGaIn surface versus applied voltage (the bottom figures).
Fig. 6.
Fig. 6. Setup for measuring the current potential characteristics of EGaIn during redox reactions. Figure 6. Setup for measuring the current potential characteristics of EGaIn during redox reactions.
Fig. 7.
Fig. 7. Current versus potential between the working electrode (EGaIn) and the reference electrode (lower left plot). i) Initial state (scattered surface), ii) no oxide film (reflective surface), iii) thin oxide film (reflective surface), and iv) change from thin oxide film to thick oxide film (scattered surface). The graph in Fig. 4 aligns the potentials of states ii) and iv) of the graph in the bottom left (top left). The photos and figures on the right are photos and schematic diagrams for states i), ii), iii), and iv).
Fig. 8.
Fig. 8. An enlarged graph of state iii) in the current potential graph in Fig. 7. The vertical axis on the right represents the charge. Based on the slope of the current graph, the area where the film formed by the oxidation reaction is defined as the integral of the current between the vertical dotted lines.

Equations (1)

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t [ cm ] = C [ C ] F [ C / mol ] × coefficient ratio × molar mass  [ g / mol ] density [ g / c m 3 ] × surface area  [ c m 2 ]
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