Extended Rate Constant Distribution Model for Sorption in Heterogeneous Systems. 1: Application to Kinetics of Metal Ion Sorption on Polyethyleneimine Cryogels

Abstract

Here, we suggest a theoretical approach to investigations of sorption kinetics based on determination of intrinsic characteristics of heterogeneous sorbents (affinity, quantity, and distribution of the sorption sites in the space of constants of sorption and desorption rates, distribution of the adsorbate on sorption sites at any arbitrary time, and the theoretical sorption isotherm) via calculation of the rate constant distribution (RCD) functions using experimental data obtained by the batch method. The effect of random errors in the experimental data on the stability of the calculated parameters was evaluated using simulation modeling that enables one to reduce time-and labor-consuming experimental procedures without loss in the reliability of sorbent characteristics. The applicability of the suggested approach to real experimental data was demonstrated on the sorption of transition metal ions on supermacroporous polyethylenimine cryogels. We have also shown how experimental conditions in the kinetics batch test affect determination of the sorption and desorption rates and calculation of the theoretical isotherm and how they can be optimized to yield reliable parameters to predict performance of supermacroporous monolith under dynamic conditions.

Composite Zn(II) Ferrocyanide/Polyethylenimine Cryogels for Point-of-Use Selective Removal of Cs-137 Radionuclides

“…Taking into account our earlier negative experience with the fabrication of composite PEI cryogel for cesium ion sorption via the sequential loading of PEI monolith with Cu(II) and [Fe(CN) 6 ] 4− ions [ 20 ], and the assumption that weaker binding of precursor metal ions to PEI will be beneficial for the fabrication of the composite cryogel with good selectivity to cesium ions, we have tested the same approach (Method I, Figure 1 ) with Zn(II), Ni(II), and Co(II) ions. The affinity of PEI cryogel to these ions is notably lower than to Cu(II) ions [ 13 , 21 ]; besides, in the case of a Co(II)-loaded monolith, it is important to treat cryogel with [Fe(CN) 6 ] 4− ions as soon as possible to avoid the oxidation of Co(II) to Co(III), which binds to PEI irreversibly [ 13 ]. The level of the composite loading with inorganic sorbent using the Method I will be determined by the sorption capacity of the original PEI cryogel, which changes in the order Cu(II) > Zn(II) > Ni(II) > Co(II) ( Figure S1, Supplementary Material ), meaning that the composite cryogel with Zn(II) ferrocyanide would be the preferable option in terms of maximal loading with the sorption active component.…”

“…The presence of sorption sites with different sorption energies in the composite containing potassium–nickel ferrocyanide was hypothesized in [ 11 ], since the bi-site Langmuir equation provided a better fit to the experimental Cs + sorption isotherm. Recently, we developed the extended rate-constant distribution (RCD) model for sorption in heterogeneous systems [ 13 , 14 , 21 ], which provides complete information about sorption properties for material with a continuous distribution of the sorption sites via the calculation of the RCD function from experimental data—sorption breakthrough (dynamics) or sorption kinetic (batch) curves experimentally obtained at different solid:liquid ratios or flow rates, and/or initial concentrations of the adsorbate. All experimental data are processed simultaneously to find the RCD function, which equally well describes a full set of experimental data.…”

“…As shown earlier [ 13 ], due to the different flow conditions and the entrapment of the solution inside the highly swellable supermacroporous matrix, the sorption kinetics on porous beads in fixed bed is notably worse than the kinetics on monoliths of the same materials [ 13 , 14 ]. This explains why sorption on freeze-dried discs was efficient only under high flow rates, forcing the solution to go through the porous structure [ 11 ].…”

Rational Design of Polyamine-Based Cryogels for Metal Ion Sorption

“…Recently, we have developed the extended rate constant distribution model (RCD-model) for sorption in heterogeneous systems and applied it to investigations of metal ion sorption on PEI cryogels [ 41 ]. The RCD-function, which provides complete information about sorption properties for the material with continuous distribution of the sorption sites, which can be calculated from sorption kinetic curves recorded at different solid:liquid ratios or initial concentrations of the adsorbate.…”

Extended Rate Constants Distribution (RCD) Model for Sorption in Heterogeneous Systems: 2. Importance of Diffusion Limitations for Sorption Kinetics on Cryogels in Batch

“…Here, using the extended rate constants distribution (RCD) model [27] to determine Cu(II) ion sorption and desorption rates on cryobeads and gel beads of polyethyleneimine-based sorbent, we try to answer the question of whether we can use knowledge of sorption kinetic properties of cryobeads in batch to predict minimal efficient residence time of the adsorbate in monolith column of the same material. In addition, if yes, how do conditions of the kinetic experiments in batch affect the reliability of such predictions?…”