## 3 Results
### Weber-Morris (W-M) Model
The film diffusion rate constants, _k\({}_{\text{ipd}}\)(1)_, were 54.322, 46.860, 15.127, and 7.209 \(\upmu\text{mol}\cdot\text{g}^{-1}\cdot\text{min}^{-1/2}\) for Cs\({}^{+}\), Rb\({}^{+}\), Sr\({}^{2+}\), and Co\({}^{2+}\), respectively, while the intra-particle diffusion rate constants, _k\({}_{\text{ipd}}\)(2)_, were lower: 20.715, 21.597, 4.192, and 4.600 \(\upmu\text{mol}\cdot\text{g}^{-1}\cdot\text{min}^{-1/2}\). Overall, these values indicate that Cs\({}^{+}\) and Rb\({}^{+}\) ions diffused similarly into the composite frameworks, while Sr\({}^{2+}\) and Co\({}^{2+}\) ions diffused more slowly through brick pores before being adsorbed by the material. In the following section, we attempt to identify the various determining factors which both controlled the adsorption of cationic elements onto alkali brick pellets and regulated the interfacial diffusion kinetics. For this purpose, we compiled data from the literature on the thermodynamic and physical properties of the ions Cs\({}^{+}\), Rb\({}^{+}\), Co\({}^{2+}\), and Sr\({}^{2+}\) (as well as other metallic cations) in water and correlated them with the experimental results. The W-M kinetics curves obtained for the metallic cations were drawn and are shown in Figure S1 (see Supplementary Materials), and the calculated intra-particle diffusion rate constants, _k\({}_{\text{ipd}}\)(1)_ and _k\({}_{\text{ipd}}\)(2)_, are listed in Table S1 (see Supplementary Materials). Note that  
Figure 4: Plotting of the adsorption capacity (Q\({}_{\text{t}}\), in \(\upmu\text{mol}\cdot\text{g}^{-1}\)) against t\({}^{1/2}\). Application of Weber-Morris kinetics model to adsorption kinetics of cations (Cs\({}^{+}\), Rb\({}^{+}\), Co\({}^{2+}\), and Sr\({}^{2+}\)) on brick composite in aqueous medium. Note that the first, second and third steps observed from W-M model are represented in black squares, red squares and blue triangles, respectively. the values of ionic potential, Jones-Dole viscosity coefficient (\(B\)), and hydrated radius used in the present work are also given in Table S1. Ionic potential is known to be an indicator of the electrostatic properties of an ion when interacting with other ions, either of similar charge (yielding electrostatic repulsion forces) or of opposite charge (yielding electrostatic attraction forces). Ionic potential is defined as the ratio of the electrical charge of an ion (\(Z\)) to its radius (\(R\)). It was previously considered as an important criterion for the choice of efficient adsorbents for toxic elements [90]. In order to calculate the ionic potential, we fixed \(Z=+1\) for monovalent ions and +2 for divalent ions and took the values of hydrated radii (\(R_{H}\)) proposed in [PERSON] works [91]. According to this author, a cation with an ionic radius, \(R_{i}\), is surrounded with water in a spherical shell, \(\Delta\)R\({}_{\rm H2O}\), in which several H\({}_{2}\)0 molecules are immobilized by the electrostatic field of the cationic entity. The total distance from the centre of the cation to the edge of the shell is then given by \(R_{Total}=R_{H}=R_{i}+\Delta R_{H2O}\). Figure 5 displays the evolution of the intra-particle diffusion rate constants, \(k_{jpd}\)_(1)_ and \(k_{jpd}\)_(2)_, as a function of the ratio \(Z/R_{H}\). The \(k_{jpd}\)_(1)_ rate constant exhibited a relatively good correlation with the ratio \(Z/R_{H}\), at least for divalent cations (R\({}^{2}\) = 0.9365), indicating a noticeable contribution of electrostatic forces to external liquid film diffusion. Conversely, the weak correlation of the \(k_{jpd}\)_(2)_ rate constant with the ratio \(Z/R_{H}\) (R\({}^{2}\) = 0.2176) seemed to show a less important electrostatic contribution to intra-particle diffusion in comparison to that observed in the first stage of the Weber-Morris model (Figure 5). Figure 5 displays the change in the hydrated radius (R\({}_{\rm H}\)) in relation to the intra-particle diffusion rate constants \(k_{jpd}\)_(1)_ and \(k_{jpd}\)_(2)_ for the first and second stages of the Weber-Morris model, respectively.