## 3 Results and Discussion
### Intraparticle Diffusion Study  
To elucidate the adsorption and mass transfer mechanisms of the dye onto ZGO, the Weber-Morris intraparticle diffusion (IPD) model was applied. The results revealed that the adsorption process involved multiple diffusion stages at varying rates, as reflected in the distinct linear regions of the plot. Each of these regions can be described by the following equation [38; 50]:  
\[\mathrm{qt}=\mathrm{k_{p}t^{1/2}+C} \tag{8}\]  
where \(\mathrm{kp}\) is the intraparticle diffusion (IPD) rate constant, and \(C\) represents the intercept at \(\mathrm{qt}=0\), which is a constant related to the process. Thus, the linearized plot of \(\mathrm{qt}\) (mg g\({}^{-1}\)) as a function of \(\mathrm{t^{1/2}}\) (min\({}^{1/2}\)) provides key insights into the different stages of the adsorption mechanism.  
The contribution of the first diffusion step could be evaluated using the characteristic ratio, known as the initial adsorption ratio (Ri), defined by the following equation:  
\[\mathrm{R_{i}}=\frac{\mathrm{q_{ref}+C}}{\mathrm{q_{ref}}} \tag{9}\]  
\begin{table}
\begin{tabular}{l c c c} \hline  & & **Adsorbents** & \\ \hline
**Parameters** & **ZGO2.5** & **ZGO5.0** & **ZGO10** \\ \hline Pseudo-first-order model & & & \\ \(\mathrm{q_{e(exp)}}\) (mg g\({}^{-1}\)) & 90.92 & 91.28 & 97.95 \\ \(\mathrm{q_{e(cal)}}\) (mg g\({}^{-1}\)) & 92 \(\pm\) 2 & 90 \(\pm\) 2 & 93 \(\pm\) 2 \\ \(\mathrm{k_{1}}\) (min\({}^{-1}\)) & 0.21 \(\pm\) 0.020 & 0.105 \(\pm\) 0.009 & 0.14 \(\pm\) 0.01 \\ R\({}^{2}\) & 0.9764 & 0.9805 & 0.9755 \\ SSE & 221 & 185 & 241 \\ Pseudo-second-order model & & & \\ \(\mathrm{q_{e(cal)}}\) (mg g\({}^{-1}\)) & 101 \(\pm\) 4 & 101 \(\pm\) 2 & 104 \(\pm\) 3 \\ \(\mathrm{k_{2}}\) (g mg\({}^{-1}\) min\({}^{-1}\)) & 0.0028 \(\pm\) 0.0006 & 0.0013 \(\pm\) 0.0002 & 0.0018 \(\pm\) 0.0004 \\ R\({}^{2}\) & 0.9503 & 0.9864 & 0.9833 \\ SSE & 169 & 129 & 164 \\ Elovich model & & & \\ a (mg g\({}^{-1}\) min\({}^{-1}\)) & 153 \(\pm\) 80 & 29 \(\pm\) 6 & 58 \(\pm\) 10 \\ b (g mg\({}^{-1}\)) & 0.068 & 0.052 & 0.055 \\ R\({}^{2}\) & 0.8536 & 0.9457 & 0.9313 \\ SSE & 152 & 57 & 75 \\ \hline \end{tabular}
\end{table}
Table 1: Kinetic parameters of the pseudo-first-order model, the pseudo-second-order model, and the [PERSON] kinetic model.  
where qref is the adsorption capacity at the longest time, and C can be related to the boundary layer thickness. The parameters of the IPD model are summarized in Table 2.  
The diffusion of MB onto the three different composite materials, as observed in Figure 8, was concentrated in the initial stage (i), characterized by a high adsorption rate within the first minutes. This was attributed to the diffusion of the dye towards the external surface of ZGO and its subsequent fixation on the surface active sites. Subsequently, a second linear region (ii) was observed, indicating an intraparticle diffusion process, where the dye penetrated into the pores of the adsorbent following a concentration gradient. The final stage (iii), corresponding to the equilibrium region, was characterized by a very slow diffusion rate, probably due to the filling of the micropores and interactions between the adsorbed molecules. The value of the initial adsorption factor in the three materials fell within the range of 1 \(>\) Ri \(>\) 0.9, indicating weakly initial adsorption [50].