## 3 Results
### Weber-Morris (W-M) Model  
In order to evidence and differentiate diffusion regions in adsorption kinetics, we decided to test Weber-Morris equation to qualitatively reveal the effects of the heterogeneous and porous nature of alkali-activated brick on the diffusivity of Cs\({}^{+}\), Rb\({}^{+}\), Co\({}^{2+}\), and Sr\({}^{2+}\) ions inside adsorbent frameworks. It has previously been stated that the Weber-Morris (W-M) model is useful for identifying reaction pathways and predicting rate-controlling steps in porous matrices [85]. The model proposed by [PERSON] and [PERSON] [62] was applied in this work to gain deeper knowledge regarding the mechanism of nuclide adsorption onto the porous brick composite in aqueous solutions (see Section 2.9). For each element, the plot of \(Q_{t}\) vs. \(t^{1/2}\) was found to be divided into three stages by two break points (Figure 4). As suggested in previous works regarding adsorption kinetics in relation to the adsorbent pore structures and surface properties of adsorbents [86, 87, 88, 89], it was assumed here that the multi-linearity observed for cation adsorption was due to the effects of the adsorbent porosity on cations. The observed multi-linear plot would then result from the broad pore size distribution of the brick composite, including, essentially, macropores and mesopores and, to a lesser extent, meso/micropores of \"hierarchical\" zeolites (Figure 3b). The higher value of \(k_{\text{jpd}}\)(1) in the first stage of the Weber-Morris model compared to \(k_{\text{jpd}}\)(2) and \(k_{\text{jpd}}\)(3) revealed the importance of external liquid film diffusion (Table 1). The first linear plot represented the fastest adsorption stage. In connection with the porous structure properties of the brick composite (as shown in Figure 3b), the liquid film diffusion occurred not only on the external brick surface, but also inside macropores and even wider mesopores. This indicates that, in the first W-M stage, the ions Cs\({}^{+}\), Rb\({}^{+}\), Co\({}^{2+}\), and Sr\({}^{2+}\) were easily transported into the broader pores of the brick without significant hindrance. As for the second stage of the Weber-Morris model, the lower slopes of the second linear W-M plots (k\({}_{\text{jpd}}\)(2) \(<\) k\({}_{\text{jpd}}\)(1); see Table 1) showed that cationic species diffused more slowly into smaller brick pores. The higher correlation coefficient (R\({}^{2}\)) obtained in the second W-M stage (see Figure 4) suggests, a priori, that intra-particle diffusion was the rate-limiting step of the adsorption process. But, because none of the \begin{table}
\begin{tabular}{c c c c c c c c c} \hline  
**Cs\({}^{*}\)** & 54.322 & 181.98 & 0.7326 & 20.715 & 244.82 & 0.9002 & 1.4778 & 328.86 & 0.4192 \\ \hline
**Rb\({}^{*}\)** & 46.860 & 205.18 & 0.9756 & 21.597 & 258.82 & 0.9712 & 0.9431 & 348.17 & 0.3990 \\ \hline
**Sr\({}^{2*}\)** & 15.127 & 278.79 & 0.8936 & 4.1924 & 304.92 & 0.8660 & 0.5410 & 330.03 & 0.5957 \\ \hline
**Co\({}^{2*}\)** & 7.2088 & 110.35 & 0.9950 & 4.5996 & 116.11 & 0.9704 & 1.5526 & 139.37 & 0.9633 \\ \hline \end{tabular}
\end{table} 
Table 1: Intra-particle diffusion rate constants (k\({}_{\text{jpd}}\)) and film thickness (C) determined for the different stages observed in Weber-Morris curves. second linear W-M plots drawn passed through the origin in Figure 4, it can be concluded that neither the liquid film diffusion nor the intra-particle diffusion was the only rate-controlling step. Finally, in the third stage of the Weber-Morris model, the adsorption process slowed down significantly ((k\({}_{\text{ipd}}\)(3) << k\({}_{\text{ipd}}\)(2)< k\({}_{\text{ipd}}\)(1); see Table 1) due to two features: (i) diffusion through smaller meso/micropores of \"hierarchical\" zeolites generated during brick activation and (ii) establishment of adsorption equilibrium.