## 2 Materials and Methods
### Modelling Methods  
Adsorption kinetics of antibiotics on NC and Si-PILC were performed using an initial fixed antibiotic concentration (110 mg.L\({}^{-1}\) for CPX and 480 mg.L\({}^{-1}\) TC, respectively) with contact times varying between 0.5 and 24 h. The kinetics studies were useful to assess the contact time required to reach equilibrium. In order to acquire additional information about adsorption mechanisms, experimental data were fitted to the pseudo-first-order, pseudo-second-order, and intraparticle diffusion models [16; 18; 32; 33].  
The [PERSON] pseudo-first order equation can be expressed as (Equation (2)):  
\[q_{t}=q_{e}(1-e^{(-k_{1}t)}) \tag{2}\]  
where \(q_{t}\) is the amount of antibiotic adsorbed at time \(t\) (mg.g\({}^{-1}\)), \(q_{e}\) is the equilibrium adsorption capacity of the adsorbent (mg.g\({}^{-1}\)), and \(k_{1}\) is the rate constant of pseudo first-order kinetics (min\({}^{-1}\)).  
Similarly, if the mechanism is thought to be pseudo-second order kinetic, the equation can be expressed as (Equation (3)):  
\[q_{t}=\frac{k_{2}q_{e}^{2}t}{1+k_{2}q_{e}t} \tag{3}\]  
where \(q_{t}\) is the amount of antibiotic adsorbed at time \(t\) (mg.g\({}^{-1}\)), \(q_{e}\) is the equilibrium adsorption capacity of the adsorbent (mg.g\({}^{-1}\)), and \(k_{2}\) is the rate constant of pseudo second-order kinetics (g.mg.min\({}^{-1}\)).  
The intraparticle diffusion model was derived from [PERSON]'s second law of diffusion and assumes that intraparticle diffusion is the rate limiting step, the film diffusion can be negligible at the beginning of diffusion and the intraparticle diffusivity is constant. The equation of the intraparticle diffusion model can be expressed as it is shown in Equation (4). Thus, the plot of \(q_{t}\) versus \(t^{0.5}\) can give one straight line or shows multilinearity which can be related to different diffusion mechanisms [18; 33].  
\[q_{t}=k_{d}\ t^{0.5} \tag{4}\]  
where \(q_{t}\) is the amount adsorbed (mg.g\({}^{-1}\)) at time \(t\) and \(k_{d}\) is the intraparticle-diffusion parameter (mg.g\({}^{-1}\)min\({}^{-0.5}\)).  
The adsorption equilibrium data were fitted to Langmuir, Freundlich, and Sips isotherm models [34]. The Langmuir model assumes one adsorption energetically equivalent without interactions among the adsorbed neighbor molecules on a homogeneous surface. The mathematical expression of the Langmuir model is shown in Equation (5):  
\[q=\frac{q_{m}kC_{eq}}{1+kC_{eq}} \tag{5}\]  
where \(q_{m}\) is the maximum adsorbed amount within a monolayer of adsorptive (mg.g\({}^{-1}\)), k (mg\({}^{-1}\).L) is the Langmuir dissociation constant, which is related to the adsorption energy, and \(C_{eq}\) has the same meaning as above.  
The Freundlich equation is commonly used for multisite adsorption isotherms, it is an empirical method related to adsorption on heterogeneous surfaces and its mathematical expression is (Equation (6)):  
\[q=k_{f}C_{e}^{1/n} \tag{6}\]  
where \(k_{f}\) (L.g\({}^{-1}\)) and \(n\) (dimensionless) are the Freundlich characteristic constants, indicating the adsorption capacity and adsorption intensity, respectively.  
The third isotherm model is considered as a combination of the [PERSON] and Freundlich equations and is called the Sips model. It is also an empirical equation, related to more heterogeneous adsorption systems without adsorbate-adsorbate interactions; its mathematical expression is shown in Equation (7).  
\[q=q_{m}\frac{\left(bC_{e}\right)^{1/n}}{1+\left(bC_{e}\right)^{1/n}} \tag{7}\]  
where \(q_{m}\) and \(C_{e}\) have the same meanings as above, \(b\) is a parameter related to the affinity of the adsorbate towards the surface, and \(n\) is a parameter that represents the heterogeneity of the system.  
The Scatchard plots were built by transformation of the isotherm data to obtain a plot of \(q/C_{e}\) versus \(q\) (where \(q\) and \(C_{e}\) have the same meanings indicated above). The shape of the Scatchard plot can be useful to acquire complementary information about the adsorption phenomena [25; 35; 36].