Lumped Pore Diffusion Research Articles

Experimental investigation and mass transfer modelling of 3D printed monolithic cation exchangers

3D printing has recently found application in chromatography as a means to create ordered stationary phases with improved separation efficiency. Currently, 3D printed stationary phases are limited by the lack of 3D printing materials suitable for chromatographic applications, and require a strict compromise in terms of desired resolution, model size and the associated print time. Modelling of mass transfer in 3D printed monoliths is also fundamental to understand and further optimise separation performance of 3D printed stationary phases. In this work, a novel 3D printing material was formulated and employed to fabricate monolithic cation exchangers (CEXs) with carboxyl functionalities. CEXs were printed with ligand densities of 0.7, 1.4, 2.1 and 2.8 mmol/g and used in batch adsorption experiments with lysozyme as model protein. All CEXs demonstrated high binding strength towards lysozyme, with maximum binding capacities of up to 108 mg/mL. The experimental results were described using mass transfer models based on lumped pore diffusion and lumped solid diffusion mechanisms adapted to reflect the complex geometry of the 3D printed monoliths. An exact 3D model as well as less computationally demanding 1D and 2D approximations were evaluated in terms of their quality to capture the experimental trend of batch adsorption kinetic data. Overall, the model results indicate that mass transfer in the fabricated CEXs is mostly controlled by pore diffusion at high protein concentrations in the mobile phase, with solid diffusion becoming important at low protein concentrations. Also, the kinetic data were approximated equally well by both the full 3D model as well as the 2D approximation, indicating leaner mathematical models of lower dimensionality can be employed to describe mass transfer in complex three dimensional geometries. We believe this work will help spur the development of 3D printable materials for separations and aid in the development of quantitative platforms to evaluate and optimise the performance of 3D printed monoliths.

Competitive sorption of cis-DCE and TCE in silica gel as a model porous mineral solid

The competitive sorption of 1,2- cis-dichloroethene ( cis-DCE) and trichloroethene (TCE) was investigated by means of column experiments using a model porous mineral solid represented by silica gel. The experimental isotherms were obtained by employing a chromatographic method. The competitive sorption isotherms were modelled with the extended Freundlich and extended Langmuir isotherms, using the parameters from single-solute experiments. The breakthrough curves were modelled with the advection–dispersion transport equation coupled with the lumped pore diffusion model. The best results were obtained when the extended Freundlich isotherm was employed. The competitive sorption was revealed with the presence of an overshoot in the breakthrough curve of cis-DCE and a decrease in the degree of sorption of cis-DCE (20%) and TCE (12%). A linear dependency of the overshoot with an increase in the concentration of cis-DCE at a fixed concentration of TCE was observed, between 16% and 20%, and at least at concentrations <6 mg L −1 in the liquid phase. The displaced molecules of cis-DCE by TCE were accumulated through the column causing its overshoot; thus short columns may hinder its observation. Thermodynamic analysis shows an exothermic adsorption process of −34 to −41 kJ mol −1, which is enhanced by sorption in micropores. The Gibbs free energy is positive for cis-DCE in the multi-component case, due to its displacement by TCE.

Influence of microwave irradiation on the intraparticle diffusion of an insulin variant in reversed-phase liquid chromatography under linear conditions

The influence of microwave irradiation on the mass transfer kinetics of an insulin variant in reversed-phase liquid chromatography (RPLC) was investigated. The elution band profiles of insulin were obtained by the pulse-response method, under linear conditions. The RPLC column was placed in a microwave oven and the incremental change in the temperature of the column effluent stream at various microwave energies and mobile phase flow rates were measured. The microwave energy dissipated in the column was set at 15 and 30 W and the mobile phase flow rate was varied from 1.0 to 2.5 mL/min at a mobile phase composition of acetonitrile, water, and trifloroacetic acid (31:69:0.1, v/v/v). The experimental data were analyzed using the conventional method of moment analysis and the lumped pore diffusion model. Regardless of mobile flow rates, the effluent temperatures measured at 15 and 30 W microwave power input were 25 ± 1 and 30 ± 1 °C, respectively. The effect of microwave irradiation on the mass transfer of the variant insulin was determined by comparing the band profiles obtained under the same experimental conditions, at the same column temperature, with and without irradiation. The calculated intraparticle diffusion coefficient, D e, at 30 W (30 ± 1 °C) microwave irradiation was ca. 20% higher than without irradiation at 30 ± 1 °C. These preliminary results suggest that microwave irradiation may have a significant influence on the intraparticle diffusion of insulin in RPLC.

Investigation of the concentration dependence of mass transfer coefficients in reversed-phase liquid chromatography

In order to investigate the concentration dependence of mass transfer coefficients in RPLC, experimental breakthrough curves obtained by staircase frontal analysis (FA) were fitted to the simplified models such as multiplate (MP) model, equilibrium dispersive (ED) model, and transport model, and the sophisticated models such as lumped pore diffusion (POR) model and general rate (GR) model. The MP model was used to obtain the initial guesses of the parameters of the ED and the transport models. Then the best values were obtained by minimizing the differences between theoretical and experimental values with a nonlinear fitting procedure. The values of the parameters of the POR and the GR models can be calculated by using the expressions derived from the plate height equations, which was further validated by using the fitting method. It was found that the mass transfer coefficients would depend on the solute concentration. This can be ascribed to the surface diffusivity, which correlates with the concentration and is lumped into the mass transfer coefficients for both simplified and sophisticated models.

Isotherm parameters and intraparticle mass transfer kinetics on molecularly imprinted polymers in acetonitrile/buffer mobile phases

The equilibrium isotherm and the intraparticle mass transfer kinetics of the enantiomers of the template were investigated on an Fmoc-L-tryptophan (Fmoc-L-Trp) imprinted polymer at different pHs and water concentrations in acetonitrile/aqueous buffer mobile phases. The equilibrium isotherm data were measured using frontal analysis at 25±2∘C. The adsorption energy distribution was found to be trimodal, with narrow modes. Consistent with this distribution, the adsorption data were modeled using a tri-Langmuir isotherm equation and the best estimates of the isotherm parameters were determined. The intraparticle mass transfer parameters were derived by comparing the profiles of experimental overloaded bands and the profiles calculated using the isotherm model and the lumped pore diffusion (POR) model of chromatography. These results showed that different adsorption and mass transfer mechanisms exist in mobile phases made of acetonitrile/aqueous buffer and of acetonitrile/acetic acid solutions.

Evaluation of the Langmuir Formalism for Modeling the Adsorption Isotherms of Proteins and Polyelectrolytes in Simulations of Ion Exchange Chromatography

AbstractThe adsorption behavior of two cationic proteins (cytochrome c and α‐chymotrypsinogen) and one synthetic polycation (poly(diallylmethylammonium chloride), polyDADMAC) was investigated on a porous particle based and a monolithic cation exchanger column. All isotherms could be fitted to the Langmuir equation. Subsequently, the derived Langmuir equation parameters were combined with two kinetic models, the transport dispersive (TD) and the lumped pore diffusion (POR) models, to simulate breakthrough curves of the substances on the ion exchanger columns. In the case of the proteins, simulations based on the experimentally derived Langmuir parameters showed poor correlation with the experimental breakthrough curves. When the Langmuir parameters were instead fitted in the simulations, adequate agreement could be achieved. In the case of the TD model, the overall mass transfer parameter had to also be fitted as a function of the protein concentration to optimize the fit. In the case of the POR model, one set of transport parameters could be used to simulate the breakthrough curves independent of the mobile phase composition (buffer concentration 75 or 120 mM, NaCl concentration 0–5 %) or the protein concentration (0.001–0.3 mg/mL). As in the TD model, it was not possible to use the experimentally derived Langmuir parameters to simulate the breakthrough curves with the POR model. Instead, the Langmuir parameters had to be fitted for each individual case in order to achieve agreement. Neither the TD nor the POR model could satisfactorily simulate the behavior of the synthetic polyelectrolyte. The Langmuir equation can thus be used to provide standard transport models with an adsorption formalism to simulate breakthrough curves in ion exchange chromatography. However, the existence of a close correlation of the actually observed physico‐chemical adsorption behavior of the involved proteins could not be supported.

Influence of Microwave Irradiation on the Mass-Transfer Kinetics of Propylbenzene in Reversed-Phase Liquid Chromatography

The effect of microwave irradiation on the kinetics of mass transfer in reversed-phase liquid chromatography (RPLC) was studied by measuring its influence on the band profile of propylbenzene in a C18-silica column eluted with an aqueous solution of methanol and placed inside a microwave oven. The elution peaks were measured by the pulse-response method, under linear conditions. The amount of microwave energy induced into the column was varied based on the microwave input power. The experimental data were analyzed using the conventional method of moment analysis and the lumped pore diffusion model. With input powers of 15 and 30 W, the effluent temperatures were 25 ± 1 and 30 ± 1 °C, respectively. The effect of microwave irradiation on the mass transfer of the studied solute was determined by comparing the band profiles obtained under the same experimental conditions, at the same temperature, with and without irradiation. The values of the intraparticle diffusion coefficient, De, measured with microwave i...

Intraparticle mass transfer kinetics on molecularly imprinted polymers of structural analogues of a template

The intraparticle mass transfer kinetics of the structural analogues of a template on a Fmoc- L-Tryptophan (Fmoc- L-Trp) imprinted polymer (MIP) and on the corresponding non-imprinted polymer (NIP) were quantitatively studied using the lumped pore diffusion model (POR) of chromatography. The best equilibrium isotherm models of these compounds were used to calculate the high-concentration band profiles of different substrates on the MIP and the NIP with the POR model. These profiles were compared to experimental band profiles. The numerical values of the intraparticle pore and surface diffusion coefficients were adjusted to determine those that minimized the differences between calculated and experimental profiles. The results of this exercise show that surface diffusion is the dominant intraparticle mass transfer process for the substrates on the polymers and that the energetic heterogeneity of the surface should be considered in accounting for the surface diffusion of the L-enantiomers on the MIP. The surface diffusion coefficient increases with decreasing overall affinity of each substrate for the polymers.

Thermodynamic functions and intraparticle mass transfer kinetics of structural analogues of a template on molecularly imprinted polymers in liquid chromatography

The parameters of the thermodynamics and mass transfer kinetics of the structural analogues ( L-enantiomers) of the template were measured on an Fmoc- L-tryptophan (Fmoc- L-Trp) imprinted polymer, at different temperatures. The equilibrium isotherm data and the overloaded band profiles of these compounds were measured at temperatures of 298, 313, 323, and 333 K. The isotherm data were modeled. The thermodynamic functions of the different adsorption sites were derived from the isotherm parameters, using van’t Hoff plots. The mass transfer parameters were derived by comparing the experimental peak profiles and profiles calculated using the lumped pore diffusion (POR) model for chromatography. These data show that (1) the strength between the substrate molecules and the MIP increases with increasing number of functional groups on the substrates; (2) enthalpy is the driving force for the affinity of the substrates for the MIP; (3) surface diffusion is the dominant mass transfer mechanism of the substrates through the porous MIP. For those substrate molecules that have the same stereochemistry as the template, the energetic surface heterogeneity needs to be incorporated into the surface diffusion coefficients. Heterogeneous surface diffusivities decrease with increasing affinity of the substrates for the MIP.

Mass transfer kinetics on the heterogeneous binding sites of molecularly imprinted polymers

The mass transfer kinetics of the L- and D-Fmoc-Tryptophan (Fmoc-Trp) enantiomers on Fmoc- L-Trp imprinted polymer (MIP) and on its reference polymer (NIP), were measured using their elution peak profiles and the breakthrough curves recorded in frontal analysis for the determination of their equilibrium isotherms, at temperatures of 40, 50, 60, and 70 ∘ C . At all temperatures, the isotherm data of the Fmoc-Trp enantiomers on the MIP were best accounted for by the Tri-Langmuir isotherm model, while the isotherm data of Fmoc-Trp on the NIP were best accounted for by the Bi-Langmuir isotherm model. The profiles of the elution bands of various amounts of each enantiomer were acquired in the concentration range from 0.1 to 40 m M. These experimental profiles were compared with those calculated using the best values estimated for the isotherm parameters and the lumped pore diffusion model (POR), which made possible to calculate the intraparticle diffusion coefficients for each system. The results show that surface diffusion contributes predominantly to the overall mass transfer kinetics on both the MIP and the NIP, compared to external mass transfer and pore diffusion. The surface diffusion coefficients ( D s ) of Fmoc- L-Trp on the NIP does not depend on the amount bound ( q) while the values of D s for the two Fmoc-Trp enantiomers on the MIP increase with increasing q at all temperatures. These positive dependencies of D s on q for Fmoc-Trp on the MIP were fairly well modeled by indirectly incorporating surface heterogeneity into the surface diffusion coefficient. The results obtained show that the mass transfer kinetics of the enantiomers on the imprinted polymers depend strongly on the surface heterogeneity.

Investigation of the steric mass action formalism in the simulation of breakthrough curves on a monolithic and a packed bed column

The simulation of the behaviour of two proteins (cytochrom c and α-chymotrypsinogen) on two types of stationary phases (monolithic and porous particle-based) was attempted for non-linear (simulation of breakthrough curves) and linear (simulation of elution peaks) cation exchange chromatography. It was found that the combination of a stoichiometric model (steric mass action, SMA) with the lumped pore diffusion (POR) model allows a simulation of high predictive value. Using one set of SMA and transport parameters for a given column morphology and/or protein, breakthrough curves and peaks could be simulated that agreed well with the experimental data while no dependency on either the protein load or the mobile phase composition (salt content) was observed. One of the SMA parameters, namely the characteristic charge needed some slight adjustment (within the error of the experimental determination of this parameter) in order to optimise the fit.

Measurement of intraparticle diffusion in reversed phase liquid chromatography

The intraparticle diffusion coefficient was measured using a method based on the fitting of a set of experimental chromatographic profiles to the lumped pore diffusion model. For this purpose, both the analytical solution of the model in the Laplace domain and a numerical method were used. There was an excellent agreement between the results given by the two methods. These results are compared to those obtained by moment analysis of the same set of chromatographic profiles and by the determination of the intraparticle diffusion coefficient from the second central moment of these bands. Nearly identical results were obtained with these two independent methods. The values of the intraparticle diffusion coefficient, D e , for rubrene in pure methanol was found to be 7.89×10 −7 cm 2/ s by the modeling method and 7.23×10 −7 cm 2/ s by the moment analysis method. These values increase with increasing water concentration, to 1.10×10 −6 and 1.24×10 −6 cm 2/ s , respectively, in a 96 :4 v/ v methanol/water solution and to 1.63×10 −6 and 2.38×10 −6 cm 2/ s , respectively, in a 90 :10 v/ v solution. These results confirm the validity and the consistency of the lumped pore model and the moment analysis theory. They show that both approaches describe correctly the mass transfer kinetics in the particles of packing material during the chromatographic process. Systematic determinations of the intraparticle diffusion coefficient can now be undertaken and the influence of various experimental parameters on this important property of packing materials can be investigated.

Prediction of the band profiles of the mixtures of the 1-indanol enantiomers from data acquired with the single racemic mixture

The adsorption isotherm data of R- and S-1-indanol and of their racemic mixture on cellulose tribenzoate were measured by frontal analysis. These data were then fitted to the Langmuir, the Bilangmuir, the Toth, and the Langmuir–Freundlich isotherm models. The single component data fitted well to both the Bilangmuir and the Toth models. Combined with the lumped pore diffusion model (POR) of chromatography, these isotherms were used to calculate overloaded elution profiles of the pure enantiomers. The calculated and the experimental profiles agree excellently in all cases if the former are derived from the Bilangmuir model. The competitive experimental data also gave excellent agreement with the Bilangmuir model. The simultaneous fit of all the data, for the single components and the racemic mixture, gave again superior agreement with the bilangmuir model. The overloaded elution profiles of samples of the racemic mixture calculated with the Bilangmuir isotherm model combined with the POR model of chromatography gave results in very good agreement with the experimental band profiles of large samples of the racemic mixture. This confirms that in numerous cases the whole set of competitive isotherms of two enantiomers can be derived from the experimental data obtained only with the racemic mixture.

Design of chromatographic separations on reversed phase

The accurate determination of adsorption isotherms is of key importance in the design and optimization of preparative chromatographic bulk separations and purifications. Issues to be considered involve other things: limited amount of test material, particularly in the case of the development of drugs; multicomponent mixture with widely different concentration levels of the main product and of the impurities. In this work, a rational procedure for the characterization of adsorption equilibria in reversed phase chromatography is proposed. This is applied to two different model separations, both involving an ascomycin derivative of industrial interest. The procedure is based on the peak fitting method and the use of a lumped pore diffusion model to simulate column dynamics. The accuracy of the obtained results is assessed thoroughly.

Adsorption equilibria of butyl- and amylbenzene on monolithic silica-based columns

The adsorption isotherms of butyl- and amylbenzene on silica monolithic columns were measured by frontal analysis. The external, internal and total porosities of these columns were determined by inverse size-exclusion chromatography. The adsorption isotherms are concave upward in the entire concentration range investigated. They were fitted to the anti-Langmuir model, an unusual model in liquid–solid and liquid–liquid phase equilibria. Band profiles under overloaded conditions were recorded. They were in good agreement with the profiles calculated using the lumped pore diffusion model of chromatography and these adsorption isotherms.

Modeling of preparative reversed-phase HPLC of insulin.

The adsorption isotherms of three recombinant proteins, human insulin, porcine insulin, and Lispro, were measured by frontal analysis on a YMC-ODS C18 column with an aqueous solution at 31% acetonitrile (0.1% TFA) as the mobile phase. The retention behavior of insulin, its related molecular structure, its conformation, and its aggregation in this phase system are discussed. The experimental isotherm data were fitted to the Langmuir, the Langmuir-Freundlich, and the Toth models. The results allow for a quantitative comparison of the saturation capacities, the equilibrium constants, and the exponents that represent the heterogeneity of the stationary phase obtained for the different insulin variants studied. The Toth model provided the best fit of the experimental data. The overloaded band profiles were calculated using the lumped pore diffusion and the equilibrium-dispersive model of chromatography. An excellent agreement between calculated and experimental profiles was demonstrated.

Modeling of the Separation of the Enantiomers of 1-Phenyl-1-propanol on Cellulose Tribenzoate

The competitive adsorption isotherms of rac-1-phenyl-1-propanol on cellulose tribenzoate were measured by competitive frontal analysis. The experimental data were fitted to four different isotherm models: Langmuir, Bilangmuir, Langmuir-Freundlich, and Tóth. The fittings of the experimental data to all four models were satisfactory. It was excellent in the case of the Langmuir-Freundlich and the Tóth models. Overloaded elution profiles calculated with the Tóth isotherm were in good agreement with the experimental profiles in all the different experimental conditions investigated. This work extends to the case of binary mixtures the equivalence between the general rate and the lumped pore diffusion models already demonstrated for pure compounds when the ratio between the Stanton and the Biot numbers exceeds 5. The adsorption energy distribution for the Tóth isotherm was also calculated.

Comparative modeling of breakthrough curves of bovine serum albumin in anion-exchange chromatography

The experimental results of a previous study of the mass transfer kinetics of bovine serum albumin (BSA) in ion-exchange chromatography, under nonlinear conditions, were reevaluated using the general rate model of chromatography. Solutions of this model were obtained numerically. The influences of axial dispersion, the resistance to mass transfer from the bulk mobile phase to the surface of the packing particles, and the intraparticle mass transfer resistances on the profiles of the breakthrough curves of BSA were investigated. The results obtained are compared to those of a previous investigation of the same data, using the simple transport–dispersive model and the lumped pore diffusion model. The results obtained show that the use of an oversimplified model for the analysis of chromatographic data can lead to erroneous interpretations of the experimental data and to misunderstandings of the fundamentals of the processes involved. Finally, a theoretical comparison between the properties and the range of application of the three models is provided.

Modified Rouchon and Rouchon-like algorithms for solving different models of multicomponent preparative chromatography

Modified Rouchon and Rouchon-like algorithms were used to solve multicomponent equilibrium-dispersive chromatographic models as well as a more general lumped pore diffusion model. The modified algorithms enable remarkable reduction of computation time and avoid computation errors that result from the original Rouchon approach for some cases of multicomponent chromatography. A comparison is given between the general chromatographic model, the lumped pore diffusion model and the equilibrium-dispersive model.