# Development and Validation of a Capacitor-Current Circuit Model for Evaporation-Induced Electricity
## 4 Results and Discussion
### Model Fitting Results
Furthermore, if we insert \(V(t)\) into Equation (4), we have  
\[I(t)=C\left(-\frac{V_{0}}{\tau}e^{-\frac{t}{t}}\right)+V_{0}\frac{e^{-\frac{t} {t}}}{R} \tag{6}\]  
Simplifying the above equation, we have  
\[I(t)=\left(\frac{1}{R}-\frac{C}{\tau}\right)V_{0}e^{-\frac{t}{t}}=\left(\frac {1}{R_{\mathrm{int}}}-\frac{1}{R}\right)V_{0}e^{-\frac{t}{t}} \tag{7}\]  
Hence, we can set \(I_{0}\), representing the combined effect of capacitance and resistance components, i.e., \(\left(\frac{C}{\tau}-\frac{1}{R}\right)\), scaled by \(V_{0}\). And we can also set the steady-state current \(I_{\mathrm{steady}}\) as \(\frac{V}{R}\), corresponding to the long-term current value as \(t\) approaches infinity, where \(\mathrm{d}V/\mathrm{d}t\) becomes negligible. Therefore, now we have the reformulated model's equation as  
\[I(t)=I_{0}e^{-\frac{t}{t}}+I_{\mathrm{steady}} \tag{8}\]  
where the initial current peak, \(I_{0}\), captures the initial rise and peak of the current, which is indicative of the initial charge buildup in the device. The decay part of the curve can be fitted to determine the time constant \(\tau\), which provides insight into the capacitance and internal resistance of the device. And the model will also identify the plateau of the current curve. In Figure 7 we show the evaporation-induced electric current vs. time, with data fitting using Equation (8). Figure 7 presents the time-dependent variation of the output current \(I(t)\), which is a key performance metric of the system. This analysis highlights the transient behavior and eventual steady-state current, providing insights into how the system responds to the input voltage \(V(t)\). Understanding \(I(t)\) is crucial for validating the model and optimizing system design for practical applications. The fitting results are listed in Table 1. In short, we obtained \(I_{\mathrm{steady}}\approx 9.832\)\(\mu\)A, \(I_{0}\approx 16.168\)\(\mu\)A, and time constant \(\tau\approx 621.395\) s. The exponential decay fitting was performed using OriginPro 2021. While the fitting quality was strong, with an \(R^{2}\) value of 0.934, the reduced chi-square value is 0.234 suggesting that the model may slightly overfit the data. While these values may appear typical for experimental systems, their significance lies in their role in validating the proposed capacitor-current circuit model. The agreement between the experimental data and the model demonstrates the latter's ability to capture both the transient and steady-state behavior of the system. This validation provides confidence in the model's applicability to other systems and highlights its utility in guiding the design and optimization of evaporation-induced electricity generation devices. The steady-state current and time constant, in particular, are critical parameters for understanding the energy output and response time of the system. By accurately predicting these values, the model provides a quantitative framework for analyzing and optimizing material and device properties. The initial current peak \(I_{0}\) represents the rapid charge accumulation due to ion flow, while the time constant \(\tau\) characterizes the rate at which the current decays due to the internal resistance and capacitance of the system. The steady-state current \(I_{\text{steady}}\) reflects the balance between charging and leakage, showing the sustained current after the initial decay phase. These results validate our simplified capacitor-current model by effectively capturing the transient steady-state behavior of the evaporation-induced electricity. Based on our previous study, the device has an internal resistance \(R_{\text{int}}\approx 0.1\)\(\text{M}\Omega\)[20], so we have the capacitance \(C=\frac{\tau}{R_{\text{int}}}\approx 3.214\)\(\mu\)F. While the data presented in Figure 7 is from a single dataset, identical measurements were conducted on four independently prepared samples. The observed trends were consistent across all samples, indicating the reliability and repeatability of the results.