## 1 Introduction
The first objective of this research was to characterize the brick-based composite by conducting X-ray diffraction (XRD) and micro-porosity measurements, as well as environmental scanning electron microscopy (ESEM). In the lab, the aqueous adsorptive characteristics of the brick composite towards some cations (Cs\({}^{+}\), Rb\({}^{+}\), Co\({}^{2+}\), and Sr\({}^{2+}\)) were examined, with special emphasis on the implication of surface properties on ionic motion within composite frameworks as well as on adsorption advancement. For this purpose, it was necessary to estimate preliminary equilibrium adsorption data and adsorption kinetics parameters: (i) for providing insight into the underlying mechanism; (ii) for predicting the reaction time to accomplish adequately specific radioactive wastewater treatment; and (iii) for optimizing the design of (future) industrial batch/column adsorption systems. These objectives could be accomplished by imperatively identifying and modelling heterogeneous reactions involved in the the brick composite/water interface. However, by applying pseudo-first- and -second-order models, the adsorption kinetics process was over-simplified because intra-particle diffusion and external mass transfer were not taken into consideration in the adsorption kinetics. The knowledge and evidencing of intra-particle diffusion and mass-transfer phenomena at the water-solid interface were essential for us to better understand the heterogeneous kinetics process being studied. Thereby, the second objective of this research was to apply to our system the kinetics-based model proposed by [PERSON] and [PERSON] [57] to reveal reaction pathways and rate-controlling steps. In this step, the effects of pore distribution and the importance of intrinsic properties of hydrated cations on diffusion kinetics were also studied. The Weber-Morris model is, nevertheless, considered more as a descriptive model than a predictive/theoretical model for quantifying diffusion/mass transfer phenomena. The third objective was, therefore, to use diffusion-based models, enabling the estimation of diffusion or mass transfer parameters, which are essential for designing adsorption processes under static (batch) and dynamic (fixed-bed column) conditions. In these modelling approaches, adsorption kinetics were investigated by assuming the occurrence of the following sequential steps: (i) film diffusion/mass transfer; (ii) intra-particle diffusion/mass transfer; and (iii) adsorption on the pore surface of the adsorbent (surface reaction). The corresponding mathematical equations were applied to our heterogeneous system for the purpose of modelling/interpreting the adsorption kinetics process and evidencing/quantifying rate-limiting steps. Should obtained (batch) information about diffusion kinetics and thermodynamic (interfacial) equilibria be relevant and valuable for predicting the removal performance of the brick composite towards Cs\({}^{+}\), Rb\({}^{+}\), Co\({}^{2+}\), and Sr\({}^{2+}\) cations in fixed-bed columns? To verify this, our fourth objective was to use a mass-transfer model to describe the dynamics of the cation/Na\({}^{+}\)-exchange process in packed-bed columns. Breakthrough curves were experimentally determined and modelled using the homogeneous surface diffusion model (HSDM). The FAST 2.1 program [58] was used for simulating breakthrough curves and solving HSDM calculations. The involved mass-transfer mechanism was investigated in detail by modelling the adsorptive and diffusional behaviour of the elements (Cs, Rb, Sr, and Co) through the porous frameworks of the brick composite. In this part, the dimensionless analysis of the involved diffusional regimes and the implication of the connectivity of pores on ion diffusivity were also examined.