## 3 Results and Discussion
### Pd(II) Sorption in Mono Metal System
#### 3.2.3. Sorption Isotherms  
The sorption isotherms were conducted at pH 1 to compare the sorption capacities of LD/PEI beads and CC/PEI beads, as well as the corresponding blank beads (i.e., LD and CC beads, respectively). As mentioned above, due to the amount of hydrochloric acid (1.1 M) in the mother solution, the predominant palladium species in those diluted working solutions is PdCl\({}_{4}\)\({}^{2-}\) (96%). Therefore, the complexation of free Pd(II) onto carboxylate groups in _L. digitata_ or sulfate groups in _C. crispus_ should be limited. The protein in those algae (8-15% in _L. digitata_, and around 10% in _C. crispus_[50]) should play the main role in Pd(II) sorption; indeed, proteins bear a certain amount of amino-acid moieties that can bind the anion complex through electrostatic attraction. Figure 5 shows that the sorption capacities of the two blank beads are quite low: 0.11 mmol g\({}^{-1}\) for LD beads and 0.04 mmol g\({}^{-1}\) for CC beads, while those of LD/PEI and CC/PEI beads are significantly improved due to the loading of PEI-GA. Moreover, CC/PEI beads show a much higher sorption capacity when compared to LD/PEI beads. Although the same amount of PEI was used for modifying both beads during the synthesis procedure, as mentioned above, the repulsive force of the outer Ca(II)-alginate barrier decreases the amount of PEI penetrating into the core of LD/PEI beads and thus results in a lower metal binding. To have a better understanding of the sorption mechanism and the relationship between the amount of adsorbate adsorbed on the sorbents and the concentration of adsorbate in the liquid at the equilibrium, several models (i.e., Langmuir, Freundlich, and Sips models) were applied to simulate the experimental data. While the Langmuir isotherm describes monolayer adsorption, the Freundlich equation assumes non-ideal and reversible adsorption (not restricted to the formation of a monolayer). As a combined form of these two models, the Sips isotherm predicts the heterogeneous sorption systems and circumvents the limitation of the rising adsorbate concentration associated with the Freundlich isotherm model; at low metal concentrations, it reduces to the Freundlich isotherm, while at high concentrations, it represents the monolayer sorption characteristic of the Langmuir isotherm [51]. The determination coefficients (R\({}^{2}\), shown in Table 2) show that the Sips equation with a greater number of parameters can describe all the experimental isotherms well, but it overestimates the maximum sorption capacity for CC and CC/PEI beads. Conversely, although the Langmuir model presents a slightly lower R\({}^{2}\) for CC-based beads, it predicts more precisely maximum sorption capacities.  
Figure 5: Pd(II) sorption isotherms using raw algal biomass, LD/PEI, and CC/PEI beads (SD: 0.5 g L\({}^{-1}\); pH: 1, adjusted by H\({}_{2}\)SO\({}_{4}\) or NaOH; Contact time: 72 h; T: 20 \({}^{\circ}\)C). Note: CC beads were degraded after the sorption process.  
In Supplementary Materials, Table S2 compares the Pd(II) sorption capacity of a variety of sorbents. Both LD/PEI and CC/PEI beads show a much higher sorption capacity than unmodified biosorbents such as _R. languinosum_ biomass [52], raw alginate beads, and algal beads [53], and comparable sorption levels compared to chemically modified alginate beads [54]. While modified XAD-7 resins [55; 56; 57] shown in the table present lower sorption capacities than CC/PEI beads, _p_-Sulfonatothiacilix [6]arene-impregnated IRA-411 or IRA-400 resins [57] show a much higher sorption capacity. However, the beads are environmentally friendly and low-cost; these properties also make them attractive for practical application.