## 3 Applications of Crosslinked PEI
### Adsorption of Heavy Metal Ions and Adsorption Models
The adsorption isotherm in this study was best described using the Sips model (Equation (3)), where \(\beta\)s is a model exponent and \(a_{\rm s}\) and \(k_{\rm s}\) are model constants in L mg\({}^{-1}\) and L g\({}^{-1}\), respectively. The Sips equation combines the Freundlich and Langmuir isotherms, and the model has been developed after recognizing the issue of a continual rise in the adsorbed quantity with an increase in concentration in the Freundlich equation [64]. The authors have also noted that the PEI-GA crosslinked material also exhibits high selectivity and repeatability toward Au\({}^{3+}\) and Pd\({}^{2+}\), with the adsorption mechanism involving chelation coordination and electrostatic interactions for both ions. \[\beta s\ln c_{e} =-\ln(\frac{k_{s}}{q_{e}})+\ln a_{s} \tag{3}\]  
A water-insoluble epichlorohydrin-crosslinked PEI resin has also been prepared and used for trace metal and metalloid removal from mining and industrial wastewaters, and its suitability for complex various divalent metal cations were explored [21]. The synthesis and characteristics of the polymer are discussed in Section 2 (Scheme 10). The complexation order for metal ions studied was Cr \(>\) Zn \(>\) Fe \(>\) Ni \(>\) Mn \(>\) Pb. Among the metal cations in this study, chromium was found to have a high affinity toward the crosslinked polymer, while SeO\({}_{3}\)\({}^{2-}\) and AsO\({}_{2}\)\({}^{-}\) were found to be poorly adsorbed. The study found adsorption percentages of Cr 97%, Ni 82%, Mn 79%, Pb 63%, and Zn 96%. The desorption of the ions was successfully achieved using HNO\({}_{3}\) and the recovered polymer was reused for another batch of adsorption, where significant efficiency was recorded. The chemical modification of PEI and subsequent crosslinking for various applications were also investigated. For example, a sulfonated crosslinked PEI was prepared by reacting the crosslinked polymer with 3-chloropropanesulfonyl chloride and heating under reflux overnight. The material was found to exhibit a high efficiency to remove Hg, with the percentage reaching up to 87% in synthetic solutions, with high selectivity even in the presence of competing ions such as Mn, Ni, Fe, Pb, Zn, and Cr. The mechanism of removal was proposed as a mercury ion binds to the functional groups present in the polymer, mainly the sulfate groups. The experimental adsorption data were fitted to adsorption isotherms, of which the Freundlich isotherm (Equation (2)) was found to fit best. The adsorption kinetics of mercury in the adsorbent is best fits a pseudo-second-order model (Equation (4)), which means that adsorption occurred via chemisorption [66]. Moreover, the crosslinked material can be regenerated by HNO\({}_{3}\) treatment and hence is reusable for another batch of adsorption. \[\frac{t}{q_{t}}=\frac{1}{k_{2}q_{e}{}^{2}}+\frac{t}{q_{e}} \tag{4}\]Poly(ethylene glycol) diglycidyl ether (DGE-PEG), 1,4-butanediol diglycidyl ether (DGE-1,4-BD), and glutaratic aldehyde (GA) were used as PEI crosslinkers to prepare cryogels. The crosslinkers were added to PEI in different molar ratios (1:1 and 1:2 molar ratios) and the reactions were performed at 23 \({}^{\circ}\)C (room temperature) and another batch at \(-\)20 \({}^{\circ}\)C. The crosslinked polymers were washed and dried in an oven before usage in adsorption studies. The degree of crosslinking was calculated from elemental studies, with GA giving a higher degree of crosslinking compared to the other crosslinkers (DGE-PEG and DGE-1,4-BD). The pH of the solution in GA crosslinking should be kept between 4.5 to 4.7, as a higher pH value increases the crosslinking extent. The materials used were Hg\({}^{2+}\) as [HgCl\({}_{4}\)]\({}^{2-}\) complex and Cu\({}^{2+}\) metal ions for sorption with DGE-PEG and DGE-1,4-BD, showing a high absorption efficiency.