## 3 Results and Discussion
### Identification of Equilibrium Isotherm Models  
The removal of MB from the aqueous solution may depend on various mechanisms such as reduction or oxidation, or degradation in the presence of OH- radicals. In the presence of a reducing agent, MB is reduced to leuco methylene blue (LMB), which is a colorless compound; upon oxidation by agitation, LMB is transformed back to its colored state, MB [43]. However, in this study, after removal of the particles, the colorless MB solution was not re-colored upon constant agitation, which confirms no participation of reduction in MB removal. The removal of MB was mainly governed by adsorption.  
To get insight into the adsorption mechanisms of nZVISLP for MB removal and to verify the performance of the adsorbent, various isotherms, as indicated in Table S1 (Supplementary Materials), were validated. The isotherm parameters were evaluated using DEO with an objective functionto minimize statistical metrics including \(\chi^{2}\) and RMSE. The performance of the optimal isotherm parameters is presented in Figure 6. These results indicate that linearizing the non-linear equation underestimated the features of the actual isotherm. It can be observed that the values predicted by DEO using optimized parameters were pretty close to the experimental values. To further confirm the performance and verify the statistical significance, the previously discussed prominent statistical metrics were evaluated and compared for both linear and non-linear model forms, as shown in Table 2. Again, these results further indicated that the parameters estimated using the non-linear model provided a higher \(R^{2}\). For all the evaluated isotherm models, \(R^{2}\) was higher than 0.94, and the \(R^{2}\) values for the [PERSON] and Sips isotherm models were close to 0.98, thus validating their better description of nZVISLP adsorption characteristics for MB removal. Another essential statistical metric, \(\chi^{2}\), indicated that the values were \(\geq\)0.98, thus confirming the suitability of these isotherms. All other significant statistical measurements that were evaluated showed a similar trend. As a result, the optimal isotherm parameters estimated by implementing DEO better captured the adsorption process of MB onto the surface of nZVISLP.  
Figure 6: Scatter plots depicting the performance of differential evolution optimization (DEO) in terms of linear transformation for various isotherm models.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c} \hline  & **Models \(\rightarrow\)** & \multicolumn{2}{c}{**[PERSON]**} & \multicolumn{2}{c}{**[PERSON]**} & \multicolumn{2}{c}{**[PERSON]**} & \multicolumn{2}{c}{**[PERSON]**} & \multicolumn{2}{c}{**[PERSON]**} & \multicolumn{2}{c}{**[PERSON]**} \\ \hline
**Metrics 1** & **Epressions 1** & **L** & **NL** & **L** & **NL** & **L** & **NL** & **L** & **NL** & **L** & **NL** & **L** & **NL** \\ \hline  & & & & & & & & & & & & & & & \\  & & & & & & & & & & & & & & & \\ \end{tabular}
\end{table}
Table 2: Comparison of prominent statistical metrics using linear (L) and non-linear (NL) approaches for evaluating the optimal isotherm model parameters.