# Development and Validation of a Capacitor-Current Circuit Model for Evaporation-Induced Electricity
## 4 Results and Discussion
### Model Fitting Results  
During our study, we measured \(V_{\text{OC}}\) and \(I_{\text{SC}}\) to obtain the device performance. And the \(V_{\text{OC}}\) reached a plateau of 300 mV, while the \(I_{\text{SC}}\) stabilized around 10 \(\mu\)A, agreeing with our previous results [20]. However, we also observed an intriguing phenomenon which was ignored in previous study: while the \(V_{\text{OC}}\) remained stable over time, the \(I_{\text{SC}}\) displayed a distinct decay following an initial peak before reaching a steady state. This observation prompted us to delve deeper into the underlying mechanisms governing these responses. The stable \(V_{\text{OC}}\) indicates that the device can retain charge without significant discharge, suggesting that once charged, the potential difference across the capacitive component holds steady due to minimal leakage. However, the decaying current seen in \(I_{\text{SC}}\) measurements implies that there is a process of charge dissipation or leakage occurring within the system when current is allowed to flow. To explain these contrasting observations, we propose a simplified hybrid model involving both a capacitor and a current source, as we discussed in Section 2.2. This model incorporates the initial rapid charging phase facilitated by ion flow and the subsequent discharge through an inherent internal resistance. The capacitor represents the device's ability to store charge, while the current source reflects the continuous ion-driven charging process. The internal resistance \(R_{\text{int}}\) introduces a pathway for current leakage, which explains the current decay over time. The above discussion lays the foundation for the model fitting presented in this subsection, where we analyze the data using this hybrid capacitor-current circuit model to capture both the transient and steady-state behavior of the system. The experimental validation of the proposed capacitor-current circuit model was conducted using wood-based nanogenerators with citric acid-modified microchannels. This experimental system was chosen as a representative case to demonstrate the feasibility and applicability of the model to describe the transient and steady-state behavior of  
Figure 6: Device setup of the wood-based nanogenerator. (**a**) Single device setup for testing current output. (**b**) Multiple devices series connected and (**c**) parallel connected to showcase the scale-up setup. evaporation-induced electricity generation. The validation provides a solid foundation for understanding the model's performance under these specific conditions. We acknowledge that this validation is limited to a single experimental system, and further studies are required to generalize the model's applicability to other material systems. Future research will focus on validating the model using diverse materials and configurations to ensure its robustness and universality. Now, we need to recall Equation (4), \(I(t)=\mathrm{C}\times\frac{\mathrm{d}V(t)}{\mathrm{d}t}+\frac{V(t)}{R}\). Notice that the right side of this equation does not include a current component, but a voltage component. So first, we need to reformulate this equation. If we assume that \(V(t)\) decays exponentially over time as a response to charging and discharging, i.e., \(V(t)=V_{0}e^{-\frac{t}{t}}\), then we can insert this form into Equation (4). So we have  
\[\frac{\mathrm{d}V(t)}{\mathrm{d}t}=-\frac{V_{0}}{\tau}e^{-\frac{t}{t}} \tag{5}\]  
where \(\tau\) is the time constant, which provides insight into how fast the current stabilizes, in which \(\tau=R_{\mathrm{int}}C\). Here, \(V_{0}\) represents the initial voltage across the system (in volts), which serves as the driving potential for the current response. This parameter is a key factor influencing the transient behavior of the system.