## 3 Results and Discussion
### Inhibitor Releasing Characteristics and Mechanism of OIMs@PAM
#### 3.2.3 The Release Kinetics of GE-EP@OIMs  
The release of the corrosion inhibitor OIM encapsulated in the GE-EP carrier is a complex process influenced by multiple factors. These include the molecular weight, hydrophilicity, and solubility of OIMs; the properties of GE-EP, such as cross-linking, pore size, and degradation; the inhibitor loading method (e.g., in situ coating, adsorption, or cross-linking); and environmental factors like pH, temperature, and ionic strength [33]. Due to this complexity, no single theory can fully explain the release mechanisms. Instead, developing release models based on key factors and fitting cumulative release curves using kinetic models (as shown in Figure 9) provides valuable guidance for predicting performance and enhancing corrosion protection. Higuchi model [33]:  
\[\frac{M_{t}}{M_{\infty}}=K_{H}\;t^{1/2} \tag{6}\]  
[PERSON] model [34]:  
\[\frac{M_{t}}{M_{\infty}}=kt^{n} \tag{7}\]  
Figure 9: Plots of the three kinetic models for the release of OIMs from GE-EP@OIMs: (**a**): [PERSON] model, (**b**): Korsmeyer–Peppas model, and (**c**): [PERSON] model. [PERSON] kinetic model [35; 36]:  
\[\frac{M_{\mathrm{t}}}{M_{\infty}}=\frac{1}{\beta}\ ln(\alpha\beta)+\frac{1}{\beta }\ lnt \tag{8}\]  
In each formula, t is the release time, day; Mt and M\({}_{\infty}\) are the cumulative release of OIMs at time t and infinite time, respectively; \(\alpha\) is the initial adsorption rate, \(\beta\) is the desorption constant, n is the constant of the reaction release mechanism, k\({}_{1}\), k\({}_{\mathrm{H}}\), and k are the release rate constants of the response model, respectively. Figure 9 shows the fitting results of GE-EP@OIMs under different kinetic models. Table 3 shows the release kinetics fitting parameters for each model. From the coefficient of determination (R\({}^{2}\)) of the fitting results, it can be observed that when a single mathematical model is used to fit the release data of GE-EP@OIMs, the Elovich model performs poorly, with only the release data at 20 \({}^{\circ}\)C showing a good fit. The Higuchi model demonstrates good fitting results across the release data, with the lowest R\({}_{2}\) values at 20 \({}^{\circ}\)C, 50 \({}^{\circ}\)C, and 80 \({}^{\circ}\)C being greater than 0.95. It is generally accepted that an R\({}_{2}\) value greater than 0.7 indicates a strong correlation [37]. The Korsmeyer-Peppas model provides the best overall fitting. According to the Korsmeyer-Peppas model fitting parameters in Table 3, n \(<\) 0.45n \(<\) 0.45n \(<\) 0.45 at 20 \({}^{\circ}\)C, and 0.45 \(\leq\) n \(<\) 0.89 at 50 \({}^{\circ}\)C and 80 \({}^{\circ}\)C. This indicates that the release of GE-EP@OIMs follows OIMs in low-temperature solutions, which follows Fick diffusion [34; 38], where only the surface or shallow surface material is released. At medium and high temperatures, the release exhibits an \"abnormal transport mechanism\", where the release is governed by both Fick diffusion and the properties of the matrix material [39]. From the coefficient of determination (R\({}^{2}\)) of the fitting results in Table 3, it is evident that the release of GE-EP@OIMs at different temperatures follows different release models. Figure 10 illustrates the release mechanism of GE-EP@OIMs under various temperature conditions. At 20 \({}^{\circ}\)C, the release follows the Elovich model (R\({}_{2}\) = 0.98), with the release primarily involving OIMs attached to the surface or mixed within the shallow surface layer [33]. This process follows Fick diffusion, and the release rate increases with temperature. This trend is supported by the characteristic parameter \(\alpha\) alpha, which increases with temperature [33]. In the 50 \({}^{\circ}\)C environment, the release conforms to the [PERSON] model (R\({}_{2}\) = 0.99), which indicates an \"abnormal transport\" mechanism [34].