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

Abstract

Here we address the problem of what we can expect from investigations of sorption kinetics on cryogel beads in batch. Does macroporosity of beads indeed help eliminate diffusion limitations under static sorption conditions? Are sorption rate constants calculated using phenomenological kinetic models helpful for predicting sorption properties under dynamic conditions? Applying the rate constants distribution (RCD) model to kinetic curves of Cu(II) ions sorption on polyethyleneimine (PEI) cryogel and gel beads and fines, we have shown that diffusion limitations in highly swollen beads are very important and result in at least ten-fold underestimation of the sorption rate constants. To account for intraparticle diffusion, we have developed the RCD-diffusion model, which yields “intrinsic” kinetic parameters for the sorbents, even if diffusion limitations were important in kinetic experiments. We have shown that introduction of a new variable—characteristic diffusion time—to the RCD model significantly improved the reliability of sorption kinetic parameters and allowed prediction of the minimal residence time in column required for efficient uptake of the adsorbate under dynamic conditions. The minimal residence time determined from kinetic curves simulated using the RCD-diffusion model was in good agreement with experimental data on breakthrough curves of Cu(II) ion sorption on monolith PEI cryogel at different flow rates.

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

“…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 ].…”

Evaluation of Poorly Soluble Drugs’ Dissolution Rate by Laser Scattering in Different Water Isotopologues

“…The particles of the dissolved substance retain all the properties of the phase, so its solution is a heterogeneous system [ 39 , 40 , 41 ]. The heterogeneous dissolution process of substances is limited by diffusion, adsorption, and desorption according to Fick’s law and the dissolution of the Nernst-Shchukarev equation using the following formula [ 42 ]: dC/dt = κS (C saturated − C t ) k = D·S/δ·V where dC/dt is the dissolution rate; k is the rate constant depending on the temperature and nature of substances; C saturated is the concentration of the saturated solution; Ct is the concentration of the solution at a given moment; S is the surface of the solid; D is the diffusion coefficient; δ is the thickness of the diffusion layer; when D, S, δ and V are constant, the dissolution rate constant k is a constant value.…”

Rational Design of Polyamine-Based Cryogels for Metal Ion Sorption

“…Using the RCD-function, one can calculate several other distribution functions, including 2D distribution of sorption sites over constants of sorption rate (K s – ρ(K s )); 2D distribution of sorption sites over constants of desorption rate (K d − ρ(K d )); and 2D affinity—distribution of sorption sites over affinity constants (K AF =K s /K d *Q max – ρ(K AF )). To account for diffusion, the RCD model was modified and an additional parameter—“characteristic diffusion time” was introduced [ 42 ].…”

“…Figure 4 d shows kinetic curves of Cu(II) ion sorption on PAA cryogel cross-linked at the DGEBD:PAA mole ratio of 1:10, which were used to calculate the distribution functions for PAA cryogel ( Figure 5 a) applying the RCD-diffusion model [ 42 ]. Kinetic curves for Cu(II) ions sorption on PEI cryogel are shown in Figure S1, Supplementary Materials .…”