Moment theory for kinetic study of chromatography

Moment analysis method for determination of rate constants of solute permeation across interface of spherical molecular aggregates by means of high-performance liquid chromatography

A moment analysis method was developed for the study of solute permeation at the interface of spherical molecular aggregates. At first, new moment equations were developed for determining the partition equilibrium constant (Kp) and permeation rate constants (kin and kout) of solutes from the first absolute (μ1A) and second central (μ2C) moments of elution peaks measured by using high-performance liquid chromatography (HPLC). Then, the method was applied to the analysis of mass transfer phenomena of three solutes, i.e., hydroquinone, resorcinol, and catechol, at the interface of sodium dodecylsulfate (SDS) micelles. HPLC data were measured by using an ODS column and an aqueous phosphate buffer solution (pH = 7.0) as the mobile phase solvent. Pulse response experiments were conducted while changing SDS concentration (5 - 20 mmol dm-3) in the mobile phase under the conditions that the surface of ODS stationary phase was dynamically coated by SDS monomers. In order to demonstrate the effectiveness of the moment analysis method using HPLC, the values of Kp, kin, and kout were determined for the three solutes as 35 - 69, 2.4 × 10-8 - 1.4 × 10-6 m s-1, and 7.0 × 10-10 - 2.1 × 10-8 m s-1, respectively. Their values increase with an increase in the hydrophobicity of the solutes. The method has some advantages for the study of interfacial solute permeation of molecular aggregates. For example, neither immobilization nor chemical modification of both solute molecules and molecular aggregates is required when elution peaks are measured by using HPLC. Interfacial solute permeation takes place in the mobile phase without any chemical reaction or physical action on molecular aggregates. The values of Kp, kin, and kout were analytically determined from those of μ1A and μ2C by using the moment equations. The results of this study must contribute to the dissemination of an opportunity for studying the interfacial solute permeation of molecular aggregates to many researchers because of extremely high versatility of HPLC.

Diffusion behavior of rhodamine 6G in single octadecylsilyl-functionalized silica particle revealed by fluorescence correlation spectroscopy

We investigated the diffusion behavior of rhodamine 6G (Rh6G) within single octadecylsilyl-functionalized (ODS) silica particle in an acetonitrile (ACN)/water system using fluorescence correlation spectroscopy (FCS). FCS measurements were conducted at the center of the particle to exclusively determine the intraparticle diffusion coefficient (D). The obtained D values were analyzed based on a pore and surface diffusion model, the results of which indicate that surface diffusion primarily governs the intraparticle diffusion of Rh6G. Furthermore, an increase in the concentration of ACN (CACN) resulted in a corresponding increase in the surface diffusion coefficient (Ds), whereas the addition of NaCl did not significantly affect the Ds values. We attributed this dependence of Ds to the dielectric constant change in the interfacial liquid phase formed on the ODS layer. Specially, Ds values of (4.0 ± 0.5) × 10-7, (7.7 ± 1.1) × 10-7, (1.0 ± 0.3) × 10-6, and (1.1 ± 0.2) × 10-6 cm2 s-1 were obtained for CACN = 20, 30, 40, and 50 vol%, respectively. We anticipate that this approach will contribute to advancing research on intraparticle mass transfer mechanisms.

Moment Analysis Method for Measurement of Reaction Equilibrium and Rate Constants by Using High-Performance Liquid Chromatography

The moment analysis method was developed for the determination of association equilibrium constant (KA) and association (ka) and dissociation (kd) rate constants of intermolecular interactions between solute and ligand molecules. They are accurately determined by using moment equations from elution peak profiles because they are measured by using high-performance liquid chromatography (HPLC) under preferable conditions that neither immobilization nor chemical modification (i.e., fluorescence labeling) of solute and ligand molecules is required. To demonstrate the effectiveness of the method, it was applied to the inclusion complex formation system between dibenzo-18-crown-6 (DB18C6) and alkaline earth metal cations, i.e., Mg2+, Ca2+, and Sr2+, as a concrete example. Because the diameter of the three metal cations is smaller than that of the inner cavity of DB18C6, the values of KA, ka, and kd were analytically determined by assuming the stoichiometry of 1:1 between DB18C6 and the metal cation. They reflected the influence of the difference in the size between the inner cavity of DB18C6 and the metal cations on the inclusion complex formation. It seems that the moment analysis method based on HPLC separation is effective for the multifaceted analysis of chemical reactions because some characteristics of the method are different from those of other conventional methods. It must contribute to the dissemination of an opportunity for the study of chemical reactions to many researchers because of the versatility of HPLC.

Moment equations for partial filling capillary electrophoresis

Moment equations were developed for partial filling CE systems, in which solute dissolution phenomena by spherical molecular assemblies or intermolecular interactions take place. Because experimental conditions of partial filling CE are divided into five categories on the basis of the magnitude relationship between the migration velocity of solute molecules and that of molecular assemblies or ligand molecules, the moment equations were systematically developed for each case by using the Einstein equation for diffusion and the random walk model. In order to demonstrate the effectiveness of the moment equations, they were applied to the analysis of partial filling CE behavior, which is correlated with dissolution phenomena of small solute molecules into spherical molecular assemblies as specific examples. Simulation results only in the case that the migration velocity of solute molecules is faster than that of molecular assemblies were represented in this paper. Detailed explanations about the derivation procedure of the moment equations and the simulation results in other cases can be found in the Supporting Information. The moment equations are theoretical bases for applying partial filling CE to the study on solute permeation kinetics at the interface of spherical molecular assemblies and on reaction kinetics of intermolecular interactions.

Simplification of Moment Analysis Procedure for Kinetic Study of Chromatographic Behavior of Core-shell Particles

The moment analysis method for chromatographic behavior in core-shell columns was simplified. Mass-transfer phenomena other than intra-stationary phase diffusion are analyzed while considering that the packing materials are spherical particles. The manner of intra-stationary phase diffusion is analyzed while assuming a hypothetical flat plate. For most core-shell particles commercially available, the geometry of a spherical thin layer can be supposed as a hypothetical flat plate with a relative error of less than ca. 2% because the thickness of the shell layer is sufficiently smaller than the diameter of whole particle. This supposition makes moment analysis easier because the moment equations for flat plates are simpler than those strictly developed for core-shell particles. Some chromatographic data measured using a core-shell column were analyzed by the simple moment analysis method to confirm its usefulness. It was demonstrated that the method is effective for a preliminary study of mass-transfer kinetics in core-shell columns.

Kinetic study on solute permeation at the interface of molecular aggregates by partial filling capillary electrophoresis

Moment analysis method using partial filling CE was developed for the kinetic study on solute permeation at the interface of spherical molecular aggregates. Moment equations for partial filling CE were developed by classifying CE systems into five categories according to the migration velocities of solute and molecular aggregate. The method was applied to the study on the dissolution of electrically neutral solutes into SDS micelles. Elution peaks were measured by partial filling CE while changing the concentration of SDS and the filling ratio of SDS micellar zone to the capillary (ϕ_M ). Partition equilibrium constants (K_p ) and rate constants of interfacial solute permeation of SDS micelles (k_in and k_out ) were determined from the first absolute and second central moments of the elution peaks by using the moment equations. Their values were comparable irrespective of ϕ_M and were almost the same as those previously measured by complete filling CE. The positive correlation of K_p with the hydrophobicity of the solutes was explained in terms of the change in k_in and k_out . It was demonstrated that the moment analysis method using partial filling CE is effective for studying solute permeation kinetics at the interface of spherical molecular aggregates.

Moment Theory of Chromatography for the Analysis of Reaction Kinetics of Intermolecular Interactions

Moment theory was applied to the kinetic study of intermolecular interactions. The association equilibrium constant (K_A) and association (k_a) and dissociation (k_d) rate constants of chemical reactions were analytically determined on the basis of the moment theory from elution peak profiles measured by high-performance liquid chromatography (HPLC). The HPLC data were measured under the conditions that neither immobilization nor fluorescence labeling of solute and ligand molecules is required. These are the advantages of the moment analysis method for determining accurate values of K_A, k_a, and k_d. Moment equations were developed on the basis of the Einstein equation for diffusion, the random walk model, and the general rate model of chromatography. The moment analysis method was applied to the inclusion complex formation system between dibenzo-18-crown-6 or dibenzo-15-crown-5 and alkali metal cations. It was demonstrated that the values of K_A, k_a, and k_d can be determined on the assumption that the stoichiometry between crown ethers and cations is 1:1 or 2:1. The influence of the difference in the size between the inner cavity of crown ethers and cations on the association and dissociation of the inclusion complex was considered. The moment analysis method using HPLC is effective for analyzing intermolecular interactions from various perspectives because it is based on the separation technique and has different characteristics from other methods such as spectroscopy. The results of this study contribute to the dissemination of an opportunity for studying intermolecular interactions from equilibrium and kinetic points of view to many researchers because HPLC is widespread.

Simple Moment Analysis for a Kinetic Study of the Chromatographic Behavior of Spherical Particles and Silica Monoliths

A simple procedure of moment analysis was proposed for a kinetic study of the rate processes in the columns packed with full-porous spherical particles and silica monoliths. Previous chromatographic data measured in reversed-phase HPLC systems using Mightysil and Chromolith columns were analyzed by a simple moment analysis. The surface of the packing materials is chemically modified with octadecyl alkyl ligands. A mixture of methanol and water (80/20, v/v) and alkylbenzene homologous series (C₆H₅C_nH_2n+1, n = 0 - 7) were used as the mobile-phase solvent and sample probes, respectively. More detailed information about the experimental conditions is provided in Supporting Information. The values of the intra-stationary phase diffusivity (D_e) and the surface diffusion coefficient (D_s), derived by the simple moment analysis, were almost the same as those by the conventional moment analysis. The simple moment analysis is effective for quantitative studies of mass transfer in chromatographic systems. The previous chromatographic data were also analyzed by assuming external porosity (ε_e) as typical values, i.e., 0.40 for spherical particles and 0.70 for silica monoliths. The resulting values of D_e and D_s were of the same order of magnitude as those derived by using ε_e experimentally measured. Even if ε_e is assumed to be typical values, the simple moment analysis is effective for preliminary studies of the mass-transfer kinetics in the columns.

Effect of the polydispersity on the dispersion of polymers through silicas having different morphologies (fully porous and core-shell particles and monoliths)

The effect of the polydispersity of polystyrenes on the dispersion through silicas having different morphologies (fully porous, core-shell particles and monoliths) was investigated. The heights equivalent to a theoretical plate (HETP) of those columns were measured for a small molecule (toluene) and a series of polystyrenes of different sizes in non-adsorbing conditions. The different contributions to the total HETP including polydispersity were determined experimentally. The longitudinal diffusion and the mass transfer resistance term were obtained from peak parking experiments. The eddy dispersion was obtained from models and experiments. The effect of polydispersity on the HETP values (H_poly) can thus be calculated from the total HETP by substraction of the other contributions. The results were compared to the Knox model which surestimates the H_poly values for porous and core-shell particles which is usually explained by an overestimation of the polydispersity index (PDI) given by the manufacturer. The PDI of two polymers (P02, M_w= 690 g.mol^-1 and P03, M_w=1380 g.mol^-1) was verified by liquid chromatography by separating each fraction of the polymer on the silica columns by using adsorbing conditions which are obtained with a mixture of heptane and THF. The PDI obtained are comparable to the PDI given by the manufacturer meaning that the assumptions made by Knox are not entirely valid. A direct method is proposed in this paper in order to determine H_poly. In this method the excess of spreading as compared with a polymer with only one size corresponding to the average size is studied assuming the polymer size distribution is gaussian. The H_poly values obtained by the direct method are comparable to the experimental values.

Mechanistic Modeling of Preparative Column Chromatography for Biotherapeutics

Chromatography has long been, and remains, the workhorse of downstream processing in the production of biopharmaceuticals. As bioprocessing has matured, there has been a growing trend toward seeking a detailed fundamental understanding of the relevant unit operations, which for some operations include the use of mechanistic modeling in a way similar to its use in the conventional chemical process industries. Mechanistic models of chromatography have been developed for almost a century, but although the essential features are generally understood, the specialization of such models to biopharmaceutical processing includes several areas that require further elucidation. This review outlines the overall approaches used in such modeling and emphasizes current needs, specifically in the context of typical uses of such models; these include selection and improvement of isotherm models and methods to estimate isotherm and transport parameters independently. Further insights are likely to be aided by molecular-level modeling, as well as by the copious amounts of empirical data available for existing processes.

Moment theory for the analytical determination of rate constants for solute permeation at the interface of spherical molecular aggregates

Moment equations were developed on the basis of the Einstein equation for diffusion and the random walk model to analytically determine the rate constant for the interfacial solute permeation from a bulk solvent into molecular aggregates (k_in ) and the inverse rate constant from the molecular aggregates to the bulk solvent (k_out ). The moment equations were in good agreement with those derived in a different manner. To demonstrate their effectiveness in one concrete example, the moment equations were used to analytically determine the values of k_in and k_out of three electrically neutral solutes, i.e. resorcinol, phenol, and nitrobenzene, from the first absolute (μ_1A ) and second central (μ_2C ) moments of their elution peaks, as measured by electrokinetic chromatography (EKC), in which the sodium dodecyl sulfate (SDS) micelles were used as a pseudostationary phase. The values of k_in and k_out should be determined with no chemical modifications and no physical action with the molecular aggregates because they are dynamic systems formed through weak interactions between the components. The moment analysis of the elution peak profiles measured by EKC is effective to unambiguously determine k_in , k_out , and the partition equilibrium constant (k_in /k_out ) under appropriate experimental conditions.

Moment analysis for mass transfer kinetics at the interface of spherical molecular aggregates

New moment equations were developed on the basis of the principle of relativity for explaining some characteristics of elution peaks measured by electrokinetic chromatography (EKC) using spherical molecular aggregates. Basic equations representing mass balance and mass transfer kinetics in EKC system in a Galilean coordinate system S were transformed to those in another coordinate system S', which imaginarily moved with respect to S. Moment equations for EKC peaks in S' in the time domain were derived from the analytical solution of the modified basic equations in the Laplace domain. Moment equations for EKC peaks in S were derived from those in S' by the inverse Galilean transformation. The moment equations were used to the re-analysis of EKC data previously measured. The values of permeation rate constants of thymol at the interface of sodium dodecylsulfate micelles were fairly in agreement with those determined in a previous study. The moment equations were also used to the numerical simulation of elution peaks in EKC systems. The influence of some experimental parameters on elution peak profiles was quantitatively analyzed. The moment equations are useful for determining the rate constants of interfacial solute permeation from elution peak profiles measured by EKC.

Moment analysis for reaction kinetics of intermolecular interactions

Moment equations were developed on the basis of the principle of relativity for analyzing elution peak profiles measured by ACE to analytically determine the association (k_a ) and dissociation (k_d ) rate constants of intermolecular interactions. Basic equations representing the mass balance, mass transfer rate, and reaction kinetics in ACE system in a Galilean coordinate system S were transformed to those in another coordinate system S', which imaginarily moved with respect to S. Moment equations for ACE peaks in S' in the time domain were derived from the analytical solution of the modified basic equations in the Laplace domain. Moment equations for ACE peaks in S were derived from those in S' by the inverse Galilean transformation. The moment equations were used for analyzing some ACE data previously published to determine k_a and k_d values. It was demonstrated that the moment equations were effective for extracting the information about affinity kinetics of intermolecular interactions from the elution peak profiles measured by ACE. The moment equations were also used to discuss the influence of mass transfer and reaction kinetics on ACE peak profiles. Some results of the numerical calculations are also indicated in Supporting Information.

Moment Analysis of Mass Transfer Kinetics in Micellar Electrokinetic Chromatography Systems

Moment equations were developed for quantitatively studying the separation characteristics of micellar electrokinetic chromatography (MEKC). They explain how the first absolute and second central moments of elution peaks are correlated with some fundamental parameters of the partition equilibrium and mass transfer kinetics in MEKC systems. In order to discuss the influence of the mass transfer kinetics on peak broadening, the moment equations were used to analyze the separation behavior in MEKC systems. Separation conditions were chosen on the basis of practical MEKC experiments previously conducted. It was quantitatively clarified that both the solute permeation at the interfacial boundary of surfactant micelles and axial diffusion of solute molecules in a capillary had a predominant contribution to the spreading of the elution peaks in MEKC systems. This is a preliminary study for the analytical determination of rate constants concerning solute permeation at the interface of surfactant micelles from elution peak profiles measured by MEKC. In addition, it was also indicated that the experimental conditions of MEKC systems could be controlled so that the interfacial solute permeation would have a predominant role for the band broadening. For example, the contribution of the interfacial permeation was about 33 times larger than that of the axial diffusion of solute molecules under the MEKC conditions in a previous study. This means that the rate constants could appropriately be determined for the interfacial solute permeation.

Moment Analysis Theory for Size Exclusion Capillary Electrochromatography with Chemical Reaction of Intermolecular Interaction

New moment equations were developed for size exclusion capillary electrochromatography (SECEC), in which intermolecular chemical reactions simultaneously took place. They explain how the first absolute and second central moments of elution peaks are correlated with some fundamental equilibrium and kinetic parameters of mass transfer and chemical reaction in SECEC column. In order to demonstrate the effectiveness of the moment equations, they were used to predict chromatographic behavior under hypothetical SECEC conditions. It was quantitatively studied how the association and dissociation rate constants of intermolecular interaction affected the position and spreading of elution peaks. It was indicated that both the intermolecular reaction kinetics and axial dispersion of solute molecules in a capillary column had a predominant contribution to the band broadening.