## 2 Materials and Methods
### Theoretical Approach of Adsorption Kinetics  
Under static (batch) experimental conditions, the Weber-Morris model allowed us to gain evidence and follow up with time film mass transfer and pore/surface diffusion phenomena over the course of the kinetics process. Moreover, the combined use of the Weber-Morris model and diffusion-based models (as described in Section 3.3) allowed us to quantify the different diffusional regimes taking place on the brick surface during the adsorption process. Pore surface diffusion modelling of the cations' breakthrough curves was performed using the dynamic packed-bed model HSDM (homogeneous surface diffusion model). Using this model, two subsequent steps were performed: (i) the mass transport from the solution bulk to the brick's external surface (which was represented here by a mass transport coefficient, k\({}_{\text{f}}\)); and (ii) the intra-particle diffusion (which was attributable to both pore volume diffusion, D\({}_{\text{p}}\), and surface diffusion, D\({}_{\text{s}}\)). On the basis of one-dimensional solute transport in the fixed bed, a series of fundamental differential convective-diffusive equations for a packed-bed setup was established to quantify interfacial phenomena, and hence, to explain the adsorber dynamics. In the presently discussed work, HSDM simulations were undertaken by means of the FAST 2.1 software developed by Schimmelpfennig and [PERSON] (2018). On the other hand, a dimensionless analysis of the HDSM model and its application for the simulation of breakthrough curves of modified brick was undertaken by considering the dimensionless parameters: Biot, Stanton, and solute distribution. Adsorption kinetics models: Experimental data were described using the following adsorption kinetics equations (Weber-Morris adsorption kinetics model ([PERSON], 1962)):  
\[Q_{t}=k_{ipd}t^{1/2}+C \tag{1}\]  
where \(Q_{t}\) (in \(\upmu\text{mol}\cdot\text{g}^{-1}\)) is the adsorption capacity at any time \(t\); \(k_{ipd}\) (in \(\upmu\text{mol}\cdot\text{g}^{-1}\cdot\text{mn}^{-1}\)) corresponds to the intra-particle diffusion rate constant; and \(C\) (in \(\upmu\text{mol}\cdot\text{g}^{-1}\)) is a coefficient indicative of the film thickness (film-surface diffusion model ([PERSON], 1962)). \[r_{o}=\left(\frac{Q_{t}}{dt}\right)_{t=0}=\frac{3k_{f}C_{o}}{R_{b}p_{b}} \tag{2}\]where \(r_{o}\) represents the initial adsorption rate; \(k_{f}\) is the film mass transfer coefficient; \(C_{o}\) is the initial concentration of adsorbate; \(R_{b}\) is the brick-grain radius; and \(\rho_{b}\) is the brick density. Knowing the \(r_{o}\) value, the \(k_{f}\) coefficient can then be determined from the following equation:  
\[k_{f}=\frac{R_{b}\rho_{b}r_{o}}{3C_{o}} \tag{3}\]  
According to [PERSON] and [PERSON] [63], at the early stage of the adsorption process, kinetic data can be fitted with an equation equivalent to that used for pseudo-second-order kinetics:  
\[Q_{t}=\frac{r_{o}t}{1+at} \tag{4}\]  
or  
\[\frac{t}{Q_{t}}=\frac{1}{r_{o}}+\frac{at}{r_{o}} \tag{5}\]  
where \"\(r_{o}\)\" and \"\(a\)\" represent two fitting parameters. Pore diffusion coefficient: The pore diffusion coefficient (\(D_{p}\)) was estimated from the ionic diffusivity of the radioactive cation in water (\(D\)) using the following equation [64]:  
\[D_{p}=\frac{D\varepsilon_{bc}}{\tau} \tag{6}\]  
where \(\tau\) is the tortuosity factor, which is defined as:  
\[\tau=\frac{(2-\varepsilon_{bc})}{\varepsilon_{bc}} \tag{7}\]  
Note that in all kinetics experiments, we used nitrate salts. The ionic diffusivity of each cation could then be calculated from the following equation [65]:  
\[\frac{1}{D}=\frac{|z_{1}|D_{1}^{o}+|z_{2}|D_{2}^{o}}{D_{1}^{o}D_{2}^{o}(|z_{1} |+|z_{2}|)} \tag{8}\]  
where \(D_{1}\)\({}^{\circ}\) and \(D_{2}\)\({}^{\circ}\) represent the infinite dilution diffusion coefficients for radioactive cations and anions (NO\({}_{3}\)\({}^{-}\)), respectively.