## 3 Results
### Diffusion-Based Models
#### 3.3.2 Evaluation of Pore and Surface Diffusion Coefficients
As seen in this table, the values of the surface diffusion coefficients ranged globally from 3.89 \(\times\) 10\({}^{-12}\) m\({}^{2}\).s\({}^{-1}\) to 4.22 \(\times\) 10\({}^{-11}\) m\({}^{2}\).s\({}^{-1}\), indicating that they were in the same order of magnitude as those of the effective diffusion coefficients (4.79 \(\times\) 10\({}^{-12}\)\(<\)\(D_{eff}\) < 1.18 \(\times\) 10\({}^{-11}\) m\({}^{2}\).s\({}^{-1}\); see Table 3); however, they were lower than the pore diffusion coefficients (8.77 \(\times\) 10\({}^{-11}\)\(<\)\(D_{p}\) < 1.40 \(\times\) 10\({}^{-10}\) m\({}^{2}\).s\({}^{-1}\); see Table 3). These findings seemed to reveal a predominance of surface diffusion on the (intra-particle diffusion) adsorption kinetics. The mathematical method described above for the determination of \(D_{p}\), \(D_{\mathit{eff\!f}}\), and \(D_{s}\) relative to the second Weber-Morris stage, was also used for the experimental points of the third W-M stage. By plotting \(-ln\left[1-X(t)^{2}\right]\) against time, we obtained relatively good straight lines with correlation coefficients varying from 0.9137 to 0.9909 for the different adsorption reactions studied (see Figure 7). In this case, from the kinetic data points used for determining the Weber-Morris rate constant, \(k_{\mathit{ipld}}(3)\), it was noticed that, except for caesium, the \(D_{\mathit{eff}}\) value was found to be closer to the \(D_{s}\) one for each element studied (see Table 3). Moreover, for long adsorption times, one could approximate Equation (14) by retaining only the first term in the series, which led to the following equation [93]:  
\[-\ln[(1-X(t))\cdot(\frac{\pi^{2}}{6})]=\frac{\pi^{2}D_{\mathit{eff}}}{R_{b}^{2 }}t+C\mathit{st} \tag{22}\]  
As explained above for Equation (16), it was also necessary to take into account a constant Cst' in Equation (22). This Cst' constant could be assessed from the formula Cst' = -ln[(1-X(t')\cdot(\pi^{2}/6)]\cdot\pi^{2}D_{\mathit{eff}}(t_{\circ})/R_{b}^{2}\), where the reaction time t\({}_{\circ}\)' must be taken at the beginning of the third W-M stage. As for plot calculation, Equation (22) ought to cover only the kinetic data points which were used for determining the Weber-Morris rate constant \(k_{\mathit{ipld}}(3)\). Hence, considering the experimental points of the third W-M stage, the term \(-Ln(1-X(t)\cdot(\pi^{2}/6)\) was plotted against the time t (Figure 8), and the value of the slope permitted us to attain the \(D_{\mathit{eff}}\) coefficient. All the \(D_{\mathit{eff}}\) values evaluated from Equation (22) were found to be comparable enough with those calculated from Equation (16) (see Table 3). Particularly, for each element, the \(D_{\mathit{eff}}\) value was closer to the \(D_{s}\) one, suggesting that the pore diffusion contribution was negligible at the end of the kinetics process. On the other hand, in order to evidence the importance of ion dehydration for the diffusivity of cationic species inside composite frameworks, the surface diffusion coefficient (\(D_{s}\)) was plotted against the molar Gibbs energy of hydration of the cation M\({}^{+}\) or M\({}^{2+}\) (\(\Delta G_{H}\)). See Figure 9 (data used for plotting Figure 9 are given in Table S3). As seen in this figure, Sr\({}^{2+}\) and Co\({}^{2+}\) ions (and, more generally, heavy metal cations) diffused more slowly through the brick pores, not only because of hindered or restricted  
Figure 8: Plotting of \(-Ln[(1-X(t))\cdot(\pi^{2}/6)]\) against time by covering only the kinetic data points (in third W-M step) used for determining the Weber–Morris rate constant, \(k_{\mathit{ipld}}(3)\). effects caused by their larger radius sizes (as shown in Figure 5), but also due to their elevated free energy of dehydration (as also shown in Figure 5). In the next paragraph, we use batch adsorption data to achieve the modelling of breakthrough profiles for lab-scale fixed-bed adsorption columns by considering a simplified mathematical approach involving interfacial diffusion phenomena.