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Plasmonic fiber-optic sensing system for in situ monitoring the capacitance and temperature of supercapacitors

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Abstract

Compared with ex situ measurement, the in situ measurement is more suitable for inspecting complex electrochemical reactions and improving the intelligent energy storage management. However, most of the in situ investigation instruments are bulky and expensive. Here we demonstrate a miniaturized, portable, and low-cost fiber-optic sensing system for in situ monitoring the capacitance and temperature. It can help evaluate the self-discharge rate in supercapacitors (SCs). The fiber-optic sensing system with two probes are implanted inside the SCs to monitor the capacitance and temperature, respectively. The dual fiber-optic probes can work independently and avoid cross-interference through structure design. The fiber-optic localized surface plasmon resonance (LSPR) probe near the electrode surface can detect the capacitance in real-time by monitoring ion aggregation on the opposite electrode. The fiber-optic surface plasmon resonance (SPR) probe encapsulated in the thermosensitive liquid can independently detect the temperature change. The measurement uncertainties of the two sensing probes are 5.6 mF and 0.08 ℃, respectively. The proposed tiny and flexible fiber-optic sensing system provides a promising method for in situ monitoring the critical parameters. It is also a powerful tool for investigating electrochemical reactions in various energy storage devices.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Supercapacitors (SCs) are promising types of energy storage devices. They can provide a high level of current and power quickly, and offer a fast transient response for electronic devices. The SCs also have a longer lifespan and wider operating temperature than the batteries. Therefore, it is becoming an attractive power solution for renewable energy generation, transportation, and electronic devices [1]. To improve the supercapacitor management systems, it is essential to monitor the critical parameters of SCs in real time. Some traditional in situ methods, such as transmission electron microscope (TEM), scanning electron microscope (SEM), X-ray diffraction (XRD), and Raman are widely used to study the mechanism and performance of energy storage device [2]. Scanning transmission electron microscopy (STEM) can image Li electrodeposition from a typical electrolyte. It is very useful to understand the microstructural evolution of Li electrodes and to identify the mechanism that may lead to cell failure [3]. SEM can offer a detailed surface and cross-section image to invest the microstructure of supercapacitor [4]. By using the diffraction peak, XRD instrument can analyze the composition and crystallinity information of electrode materials [5]. Raman spectroscopy can be used to analyze the structure and phase composition of the electrode. And the charge storage mechanism of nanomaterial is successfully studied by Raman technique [6]. Although these methods can get detailed relevant parameters for analyzing the mechanism. The bulky size and costly price of the instruments still limit their application, especially in field-test. Actually, SCs also have some problems, such as higher self-discharge rate. The real-time capacitance monitoring is a rapid and reliable method to determine the state of charge of electrochemical storage devices [7]. Moreover, although the supercapacitors can work over a wide temperature range, the internal temperature also affects the self-discharge rate [8]. So, capacitance and internal temperature are two critical parameters to evaluate the SCs state of health.

To solve these problems above, the miniaturized plasmonic fiber-optic sensing system is an ideal choice. With tiny size and anti-corrosion characters, the fiber sensors can be implanted in the SCs to monitor the changing parameters. Surface plasmon resonance (SPR) in proposed sensors occurs when the incident photon longitudinal momentum matches that of the oscillating charges at the surface. The resonance wavelength is sensitive to the surrounding refractive index (RI) change [9]. In localized surface plasmon resonance (LSPR), the free electrons confine on the surface of the nanoparticles rather than metal film. Moreover, the LSPR sensors are insensitive to temperature change, which is more suitable for detecting the RI change in a wide temperature fluctuation system [10]. Besides, fiber-optic plasmon sensors can effectively overcome the bulky and expensive defects of traditional prism-based plasmon sensors [11]. SPR and LSPR have been extensively developed in biomolecular research [12,13], environmental detection [14,15], food safety [16]. In recent years, various novel fiber-optics based on SPR and LSPR have been developed to improving sensitivity in biochemical analysis [1719]. However, in energy storage analysis, the research based on surface plasmon is far from enough. Compared with prism-based plasmon sensors, fiber-optic sensors with tiny size can be implanted inside the energy storage device to detect the key parameters in real-times [20,21]. Tilted fiber Bragg grating (TFBG) sensors has been reported for in situ monitoring the state of charge in SCs, the correlation between the real-time charge-discharge circles and the optical sensing signal has been found [22]. The multiple fiber Bragg grating (FBG) sensors were studied to temperature and pressure in commercial 18650 lithium battery to monitoring the temperature and pressure in operando mode [23].

On this basis, we propose a dual fiber-optic sensing system with two embedded sensors for in situ monitoring the capacitance and internal temperature in SCs, respectively. Compared with TFBG-SPR and FBG sensors, the multimode fiber sensors we proposed is easy to fabricate and cost-effective. Moreover, by utilizing unpolarized light, optical switch, multimode sensing probes and miniature fiber-optic spectrometer, fiber-optic SPR and LSPR sensing system is small, portable and easy to operate. The fiber-optic LSPR sensor is temperature insensitive, which can precisely monitor the real-time capacitance in charging and discharging process. The silver-based SPR sensor was encapsulated in the temperature-sensitive liquid with high thermo-optic coefficient. This structure design makes it sensitive to temperature change and avoid interference cause by the electrolyte change during cycling. We demonstrate a steady and reproducible relationship between the signal of plasmonic fiber-optic sensors and the target parameters in SCs. The two sensors in the fiber-optic sensing system can work independently and avoid cross-interference in SCs. The real time information collected inside the SCs in essential to understanding the operation mechanism and evaluating the health state of energy storage device.

2. Experimental section

2.1 Preparation of fiber-optic SPR temperature sensor

The optical fiber sensors was made of multimode plastic-clad silica-optical fiber (0.37 NA, 400 µm core, 430 µm cladding diameter, YOFC Co. Ltd). Near the end face of fiber-optic probe, 1cm of coating and cladding was peeled off as the sensing region. Then, the fiber-optic probe was soaked in piranha solution (H2SO4: H2O2 = 3:1) at 70 °C for an hour to increase the surface roughness (as shown in Fig. S1), which can help to strengthen the Ag film on the sensing region. Then, 50 nm Ag was coated on the sensing region by magnetron sputtering (Q150R S Plus, Quorum). In order to ensure the uniformity of the coating, a special fixture is made to realize 360° rotation of the optical fiber during the sputtering process (as shown in Fig. S2). To obtain a terminal reflective mirror, the end face of the fiber probe was polished and covered with a silver layer through a Torrance reaction. In brief, 6 mol/L ammonia solution was added gradually in 0.5 mol/L in AgNO3 solution. The mixed solution turns brown first and later becomes clear and transparent. The end face of fiber was suspended in the solution. Then, a few drops of 30% glucose solution was added, and the sample was kept in 70°C for 30 min to form a silver layer on the end face of the fiber as the reflective mirror. The reflective mirror was encapsulated with Epoxy ab glue adhesive for protecting. Then, the sensing region of the fiber was sealed in glass capillary filled with anhydrous ethanol (Fig. 1(a)).

 figure: Fig. 1.

Fig. 1. Dual fiber-optic dual sensing system. (a) Fiber-optic SPR temperature sensors. (b) Fiber-optic LSPR capacitance sensors. (c) In situ detection system by dual fiber-optic sensing system in supercapacitor.

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2.2 Preparation of gold nanoparticles

The gold nanoparticles were synthesized by the sodium citrate reduction method with slight modification [12]. Before the experiment, all the glassware was soaked in aqua regia (HCl:HNO3 = 3:1) for 1 hours, then rinsed with deionized water and dried. In 250 ml round bottom flask, 1 ml 1% chloroauric acid and 100 ml deionized water were added and heated to boiling under magnetic stirring. Then 1.5 ml of 1% sodium citrate aqueous solution was added and kept it boiling for 15 minutes. During this process, the mixture solution gradually turned to red wine. The UV-Vis absorption spectrum of gold nanoparticles is shown in Fig. S3. The maximum absorption peak is at 525.3 nm.

2.3 Fabrication of fiber-optic LSPR capacitance sensors

After removing the cladding and coating, the fiber was soaked in piranha solution at 70 °C for 1 hour to hydroxylate the surface of fiber core. For surface amination, the hydroxylated fiber was immersed in 10% (v/v) 3-aminopropyltriethoxysilane (APTES) aqueous solution for 1 hour and rinsed in ethanol with sonication. Then fiber was transfer to the drying oven at 120 °C for 3 hours to remove residual water and ethanol. The fiber is subsequently immersed in AuNPs solution for 3 hours to form the LSPR sensing layer (as shown in Fig. S4). The AuNPs were deposited on the surface of fiber core. As shown in Fig. S5, the AuNPs is uniformly load the fiber core in LSPR sensors. The average diameter of AuNPs is about 25 nm. Then, the end face of fiber probe was coated with the silver layer as a mirror by Tollens’s reaction (Fig. 1(b)).

2.4 Construction of supercapacitor testing platform

The supercapacitor implanted with dual fiber sensing system is shown in Fig. 1(c). The supercapacitor was made with a two-electrode system. The MnO2 electrode was working electrode and synthesized by electrochemical deposition on carbon fabric (area 2×2 cm) (as shown in Fig. S6). The platinum sheet was the counter electrode. The electrolyte was 1 mol/L LiCl solution. Figure 2 is a schematic diagram of supercapacitor and implanted dual-fibers in situ monitoring system. Both of the fiber-optic LSPR sensor and SPR sensor were connected to the Y-type jumper (400 µm core) through the SMA 905 coupler. The fiber-optic LSPR sensor was placed close to the MnO2 electrode to monitor the surface charge during the charging and discharging process. The fiber-optic SPR sensor is implanted in the electrolyte to detect the change of internal temperature in SCs. The halogen light source (HL 2000, Ocean Optics) as the incident light was coupled to the fiber sensing system and reflected by silver mirror on the end face of fiber-probe. The spectrograph (USB2000+, Ocean Optics) is used to monitor the resonance dips in reflected light. By using the optical switch, it can realize simultaneous double channels operation with one light source and one spectrograph. The signal was analyzed and display by a custom-made LabVIEW program in real-time.

 figure: Fig. 2.

Fig. 2. Schematic diagram of a dual fiber-optic sensing system for detecting capacitance and temperature in SCs.

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3. Results and discussion

3.1 Temperature monitoring by SPR sensors

In this research, we use multimode fiber for SPR sensing. The resulting SPR spectrum is a sum of different angles for excitation. The incident angle at the fiber interface was the average propagation angle. As a consequence, the multimode fiber optic SPR sensor has broader resonant peak. In addition, the multimode fiber-optic SPR probe equipped with a white light source without polarization [24,25]. Compared with single mode fiber sensor, the multimode fiber sensor is easy to fabricate and cost-effective.

To verify the sensitivity and stability, the fiber-optic SPR sensor was calibrated with different refractive index unit (RIU) solutions. The RIU of different NaCl solutions measured by the Abbe refractometer were 1.3427, 1.3516, 1.3561, 1.3610, 1.3645, 1.3678, 1.3716. Figure 3(a) and 3(b) shows that with the RIU increases, the resonance wavelength is red-shifted accordingly. Figure 3(c) is the linear fitting line of different RIU solutions with the resonance wavelength shift. The sensitivity of the SPR sensor to the refractive index unit is 3654.85 nm/RIU (R2 = 0.9914). Each test was repeated 3 times to get the error bar. The result shows that fiber-optic SPR sensor has high sensitivity and stability and can accurately monitor the change of the surrounding RIU.

 figure: Fig. 3.

Fig. 3. Silver-based fiber-optic SPR sensor in different RIU solutions. (a) The SPR sensor at different NaCl solutions. (b) The resonance wavelength shift with RIU change. (c) The linear fitting curves.

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For RI sensitivity, the silver-based SPR sensors exhibit higher RI sensitivity and sharper reflectivity spectrum than those of gold-based sensors [26]. The thermal-optic coefficient reflects the change of RI with the temperature. The thermal-optic coefficient of ethanol is apparently higher than those of water [27,28]. The silver-based SPR sensors was encapsulated in ethanol, which RI change accordingly with ambient temperature. So, the temperature monitoring is realized by monitoring the RI of ethanol by silver-based SPR sensors. At the same, the encapsulation can avoid the oxidation problem of silver film. Therefore, it can monitor the temperature change precisely.

In order to measure the temperature change, the fiber-optic SPR temperature sensor was implanted in SCs and transferred into a controllable thermostat. The measurement temperature is 25 to 65 °C, and every 5 °C is an interval. Figure 4(a) is the resonant spectrum of the sensor in different temperature. As the temperature increase, resonance wavelength is bule shift (Fig. 4(b)). The linear fitting curve between the temperature and resonance wavelength shift was plotted in Fig. 4(c). The data shows that the sensors proposed has good linearity (R2 = 99.97%), and the sensitivity is 1.06 nm/°C. The measurement uncertainty is based on the standard deviation of baseline in SPR sensor at fixed temperature (as shown in Fig. S9a). The linear fitting curve of different temperatures with resonant wavelength shifts is $\triangle \mathrm{\lambda } ={-} 1.06\textrm{T}$, where △λ is the resonance wavelength shift, T is the ambient temperature. The standard deviation σ is calculated according to the formula [28]:

$${\sigma _t} = \frac{{\sqrt {\mathop \sum \nolimits_{\textrm{i} = 1}^\textrm{n} {{({{\textrm{x}_\textrm{i}} - \bar{x}} )}^2}} }}{{\textrm{n} - 1}} = 0.08^{\circ}\mathrm{C}$$
where xi represents the measured temperature obtained by the proposed sensor with capillary package. And $\bar{x}$ represents average value, n is the number of data point.

 figure: Fig. 4.

Fig. 4. Temperature calibration by fiber-optic SPR sensor. (a)Resonant spectrum at different temperatures. (b) Resonance wavelength shift with temperature change. (c) The linear fitting curve of different temperatures with resonant wavelength shifts.

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3.2 Capacitance monitoring in supercapacitor

The same method was used to calibrate sensitivity of the fiber-optic LSPR sensor (as shown in Fig. S7). Then, the fiber-optic LSPR sensor is implanted near the electrode to monitoring the capacitance. The LSPR insensitive to the bulk RI and ambient temperature changes [29]. In LSPR sensors, the electric field is confined and greatly enhanced on the particle surface, which is only have high sensitivity near the sensing surface (typically tens of nanometers) and rapidly falling off with distance [30]. So, when the LSPR fiber sensors is implanted near the electrode surface, it can monitor the electrical double layer (EDL) precisely. Figure 5(a) is the schematic diagram of the fiber-optic LSPR sensing system in the supercapacitor. In this paper, the electrolyte is LiCl. As shown in Fig. 5(b), when the voltage is 0V, the electrode surface is neutral and ions will not accumulate on the electrode surface. When a positive voltage is applied, the electrode will be positively charged. The negative ion (Cl-) will accumulate on the electrode surface. The charge is proportional to the concentration of Cl- on the electrode surface. The fiber-optic LSPR sensor can detect the capacitance caused by ion accumulation on the electrode surface. The fiber sensor attached near the electrode show a significant optical signal response. It means the refractive index (RI) change near the electrode is much higher than that in the electrolyte during the cycling process. Generally, the thickness of the ion accumulation layer is between 0.1-10 nm [31,32], which matches the detection range of the LSPR evanescent wave. Besides, LSPR is not sensitive to temperature change, so it can avoid temperature interference. Hence, the real-time capacitance on the electrode can be monitored by fiber-optic LSPR sensors.

 figure: Fig. 5.

Fig. 5. Schematic diagram of capacitance real-time detection by the fiber-optic LSPR sensors. (a) detection device. (b) ion aggregation state under different voltages.

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As shown in Fig. 6(a), CV tests were conducted at different scan rates (10 mV/s, 15 mV/s, 20 mV/s). The upper half of the CV curve is the charging process, and the lower part is the discharging process respectively. The capacitance was calculated by the formula: $C = \frac{Q}{{\Delta V}} = \frac{{\smallint Idt}}{{\Delta V}}$, C is the capacitance, I stand for the instantaneous current, t is the corresponding time, ΔV is the operating potential. ΔV was fixed at 0.8V, as the scan rate (in mV/s) increased, the capacitance decreased as well. As shown in Fig. 6(b), in charging process, the charge gradually accumulated on the electrode surface, the resonance peak of LSPR sensors shift to longer wavelength accordingly. In discharging process, the resonance peak gradually shifts back as well. Besides, as the scanning rate increases, the amount of charge on the electrode surface decrease, and the resonance peak decreases accordingly. The optical signal is consistent with the CV curve measured by the electrochemical workstation. So, the fiber-optic LSPR sensor can monitor the capacitance in real-time. Compared with CV electrochemical analysis, which requires integration to calculate the capacitance, the fiber-optic LSPR sensor can monitor the capacitance directly throughout the charging and discharging process. So, in capacitance measurement, the electrochemical technique is an “off-line” method. However, the fiber-optic LSPR sensors offer an “on-line” method, which can directly monitor the capacitance in real-time.

 figure: Fig. 6.

Fig. 6. CV test (a) at different scan rates. (b) The corresponding response of fiber-optic LSPR sensors.

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Ordinarily, the temperature change in SCs will influence other parameters measurement. To verify that the LSPR sensor is not sensitive to temperature change, both of the fiber-optic LSPR sensor and SPR sensor were implanted in SCs at the same time, and slowly heated the SCs from 25 °C to 65 °C. The resonance peak of SPR sensors has a blue shift of approximately 8 nm, while the signal in LSPR sensor is almost stable (as shown in Fig. S8). Therefore, the fiber-optic LSPR sensors can effectively avoid cross-interference caused by the temperature change.

In fiber-optic LSPR sensor implanted SCs, similar GCD tests were conducted at different constant currents (1.0 mA, 1.5 mA, 2.0 mA, Fig. 7(a)). The capacitance is calculated as follows: $C = \frac{Q}{{\Delta V}}$, $Q = I \times t$. With the current increased, the charge on the electrode increased as well. As shown in Fig. 7(b), in charging/discharging process, with surface charge change, the resonant wavelength increases or decreases proportionally. When the voltage is up to set threshold (0.8 V), the supercapacitor is fully charged, the resonance wavelength shift reaches its peak. Moreover, the LSPR signal response time is the same with charge-discharge time. The linear calibration curve between the calculated capacitance (C) and the peak of resonant wavelength shift (Δλ) in fiber-optic LSPR sensors is shown in Fig. 7(c). The formula of fitting curve is $\triangle \lambda = 0.015 \times C - 0.965$ (R2 = 99.9%), and the sensitivity is 0.015 nm/mF. The measurement uncertainty of capacitance is based on the standard deviation of LSPR baseline (as shown in Fig. S9b). The standard deviation is found using the baseline of LSPR sensors. The standard deviation ${\mathrm{\sigma }_C}$ is calculated as the formula.

$${\mathrm{\sigma }_C} = \frac{{\sqrt {\mathop \sum \nolimits_{\textrm{i} = 1}^\textrm{n} {C_i} - {{\bar{C}}^2}} }}{{\textrm{n} - 1}} = 5.67\; mF$$
where ${C_i}$ is the measured capacitance, $\bar{C}$ is the average value, n is the number of date points.

 figure: Fig. 7.

Fig. 7. GCD test (a) at different constant currents. (b)The corresponding response of fiber-optic LSPR sensors. (c) The linear calibration curve between Δλ and C

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3.3 Detection of capacitance and temperature in SCs

To verify that the fiber-optic LSPR and SPR sensors can detect capacitance and temperature independently without cross-interference. The SCs with dual-fiber sensing system were placed in a thermostat. Capacitance is a function of surface area and volume of the electrodes, and ionic conductivity of electrolyte, which is strongly depends on temperature variations. The GCD tests with 0.5 mA were conducted at different temperature (25 °C, 45 °C, 65 °C, Fig. 8(a)). The data shows that as the temperature increases, the capacitance increases as well. The capacitance depends on the number of ions aggregated at the interface between the electrode and electrolyte. Higher temperature contributes the ions to the innermost pores of electrodes and has a higher capacitance, which can be monitored by LPSR sensors [33]. Figure 8(b) is the resonance wavelength shift of LPSR sensor during charge and discharge test at different temperature. The optical signal is proportional to the surface charge on the electrode. In higher temperature, peak of wavelength shift increases as well, which is consist with the result calculated by GCD test. At 25 °C, 45 °C, and 65 °C, the maximum capacitance is about 106.4 mF, 113.4 mF, 123.4 mF, respectively.

 figure: Fig. 8.

Fig. 8. GCD tests (a) at different temperatures (25 °C, 45 °C, 65 °C). (b) The corresponding response of fiber-optic LSPR sensors. (c) The calculated stored charge and response of fiber-optic LSPR sensors at different temperature (d) Simultaneously detection of stored charge and temperature by fiber-optic LSPR and sensors.

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The relationship between the calculated capacitance in GCD test and the highest wavelength shit in fiber-optic LSPR sensors are shown in Fig. 8(c). The test result of LSPR sensors is consisted with the result tested by electrochemical workstation. It verifies that the fiber-optic LSPR sensors can monitor capacitance precisely at different temperatures precisely. Figure 8(d) is the graph of capacitance and temperature measurement by dual fiber-optic LSPR and SPR sensing system. The encapsulated fiber-optic SPR sensor can steadily monitor the internal temperature in SCs without being disturbed electrolyte changes in charging and discharging process. Similarly, the fiber-optic LSPR sensor implanted on the surface of electrode, which only have high sensitivity near the sensing surface and insensitive to the ambient temperature. So, it can monitor the ion accumulation on the electrode precisely without disturbed by the temperature change.

4. Conclusion

In summary, we demonstrate a miniaturized plasmonic fiber-optic sensing system for in situ monitoring the capacitance and temperature. The dual fiber-optic sensing system can work independently without cross-interference. The sensing signal was calibrated by the calculated data in electrochemical testing. The result shows that sensing signal has a linear relationship with the target parameters. Compared with traditional electrochemical methods, the fiber-optic plasmonic sensing system can realize “online” monitoring the capacitance in real-time. It also demonstrates that the capacitance increases accordingly as the working temperature increases. The optical signal can directly reflect the parameter change without calculation process. This paper provides a convenient technique for researchers and engineers to monitor the critical parameters of SCs in real-time. Finally, with tiny size and flexible shape, the fiber-optic sensor can be applied in different kinds of energy storage devices. It is a promising tool for in situ and in operando monitoring in energy storage systems in the future.

Funding

National Natural Science Foundation of China (61905069, 11774081); Postdoctoral Research Foundation of China (2020M670936, 2020T130177); Natural Science Foundation of Heilongjiang Province (RCCXYJ201902, TD2021F001).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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Supplementary Material (1)

NameDescription
Supplement 1       Supplementary test data and pictures

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (8)

Fig. 1.
Fig. 1. Dual fiber-optic dual sensing system. (a) Fiber-optic SPR temperature sensors. (b) Fiber-optic LSPR capacitance sensors. (c) In situ detection system by dual fiber-optic sensing system in supercapacitor.
Fig. 2.
Fig. 2. Schematic diagram of a dual fiber-optic sensing system for detecting capacitance and temperature in SCs.
Fig. 3.
Fig. 3. Silver-based fiber-optic SPR sensor in different RIU solutions. (a) The SPR sensor at different NaCl solutions. (b) The resonance wavelength shift with RIU change. (c) The linear fitting curves.
Fig. 4.
Fig. 4. Temperature calibration by fiber-optic SPR sensor. (a)Resonant spectrum at different temperatures. (b) Resonance wavelength shift with temperature change. (c) The linear fitting curve of different temperatures with resonant wavelength shifts.
Fig. 5.
Fig. 5. Schematic diagram of capacitance real-time detection by the fiber-optic LSPR sensors. (a) detection device. (b) ion aggregation state under different voltages.
Fig. 6.
Fig. 6. CV test (a) at different scan rates. (b) The corresponding response of fiber-optic LSPR sensors.
Fig. 7.
Fig. 7. GCD test (a) at different constant currents. (b)The corresponding response of fiber-optic LSPR sensors. (c) The linear calibration curve between Δλ and C
Fig. 8.
Fig. 8. GCD tests (a) at different temperatures (25 °C, 45 °C, 65 °C). (b) The corresponding response of fiber-optic LSPR sensors. (c) The calculated stored charge and response of fiber-optic LSPR sensors at different temperature (d) Simultaneously detection of stored charge and temperature by fiber-optic LSPR and sensors.

Equations (2)

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σ t = i = 1 n ( x i x ¯ ) 2 n 1 = 0.08 C
σ C = i = 1 n C i C ¯ 2 n 1 = 5.67 m F
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