## 3 Applications of Crosslinked PEI  
### Adsorption of Heavy Metal Ions and Adsorption Models  
The crosslinking of PEI/or biomass-modified PEI using an appropriate crosslinker is necessary to reduce the water solubility of PEI and hence to make it useful for an adsorption experiment with heavy metals [9,63]. [PERSON] et al. prepared a magnetic gel material (PEI/Fe\({}_{3}\)O\({}_{4}\)) consisting of a biomass (corncob) modified by glutaraldehyde-crosslinked PEI for the adsorption of Cu\({}^{2+}\) and Pb\({}^{2+}\) from aqueous solutions [9]. The materials prepared were characterized by different techniques, such as FTIR, SEM, and so on. The maximum adsorption of Cu\({}^{2+}\) and Pb\({}^{2+}\) was measured under different conditions: contact time, initial concentration, amount of the adsorbent, pH, and temperature. A fast adsorption in the first 1 h was recorded, while the following 1 h was slow and the adsorption reached a saturation point after 2 h. The decrease in the adsorption rate in the second 1 h is attributed to the decrease in the concentration gradient between the ions found at the surface of the adsorbent and the ions in solution and or due to ion-ion repulsion. The adsorption experimental data collected were fitted using the [PERSON] (Equation (1)) and [PERSON] (Equation (2)) isothermal adsorption models [64] and the adsorption turned out to be better fitted by the Langmuir equation. \[\frac{c_{e}}{q_{e}}=\frac{1}{K_{L}q_{0}}+\frac{c_{e}}{q_{0}} \tag{1}\]  
\(q_{e}\) is the adsorbed amount at equilibrium in mg g\({}^{-1}\), \(c_{e}\) is the adsorbate's equilibrium concentration in mg L\({}^{-1}\), \(q_{0}\) is the maximum amount of monolayer adsorption in mg g\({}^{-1}\), and \(K_{L}\) is the Langmuir adsorption constant. A plot between \(c_{e}/q_{e}\) versus \(c_{e}\) generates a straight line with a slope of \(1/q_{0}\) and a y-intercept equal to \(1/K_{L}q_{0}\). Similarly, \(b\) and \(n\) in Equation (2) are the Freundlich adsorption equilibrium constants. \[\ln q_{e}=\ln b+1/n\ln c_{e} \tag{2}\]  
A plot of \(\ln q_{e}\) versus \(\ln c_{e}\) produces a straight line with a slope equal to \(1/n\) and a y-intercept of \(\ln b\). The Langmuir isothermal adsorption model best fits the adsorption data for both ions, as revealed from a slightly higher value of the R\({}^{2}\) (a statical value) as compared to the Freundlich isothermal adsorption model. One of the bases of the Langmuir adsorption model is the presence of a chemical adsorption on the surface of the adsorbent [64]. A Langmuir adsorption model of the ions in the PEI-CC@Fe\({}_{3}\)O\({}_{4}\) implies that it is mainly chemical adsorption [9]. When it comes to the adsorption kinetics, a pseudo-second order model was found to best fit the experimental data, giving an R\({}^{2}\) value of >0.99 for both Cu\({}^{2+}\) and Pd\({}^{2+}\). A pseudo-first order gave a slightly lower R\({}^{2}\) value in relative terms to the previous model. The adsorption thermodynamics have also been studied for both ions at 308 K, and \(\Delta\)G values were measured. Moreover, the mechanisms of interaction of both ions with the adsorbing material with the analytes were proposed. The interaction involves the nitrogen of the amine and the OH of the carbohydrate (corn) and forms four bonds around the ions [9]. Glutaraldehyde-crosslinked PEI (PEI-GA) was used to adsorb Au\({}^{3+}\) and Pd\({}^{2+}\) in a wide pH range of 1 to 9 and at 25 \({}^{\circ}\)C, giving an adsorption amount of 2575 and 497 mg g\({}^{-1}\), respectively [65]. Au\({}^{3+}\) was adsorbed completely within 10 min for 8.3 mg L\({}^{-1}\) and 20 min was needed for complete adsorption of Pd\({}^{2+}\) from 9.7 mg L\({}^{-1}\) concentrations in a water sample. In total, 2 h and 9 h adsorption equilibrium times were required for 523.9 mg L\({}^{-1}\) Au\({}^{3+}\) and for 565.6 mg L\({}^{-1}\) Pd\({}^{2+}\), respectively.