Construction by Molecular Dynamics Modeling and Simulations of the Porous Structures Formed by Dextran Polymer Chains Attached on the Surface of the Pores of a Base Matrix: Characterization of Porous Structures

Significant increases in the separation of bioactive molecules by using ion-exchange chromatography are realized by utilizing porous adsorbent particles in which the affinity group/ligand is linked to the base matrix of the porous particle via a polymeric extender. To study and understand the behavior of such systems, the M3B model is modified and used in molecular dynamics (MD) simulation studies to construct porous dextran layers on the surface of a base matrix, where the dextran polymer chains and the surface are covered by water. Two different porous polymer layers having 25 and 40 monomers per main polymer chain of dextran, respectively, are constructed, and their three-dimensional (3D) porous structures are characterized with respect to porosity, pore size distribution, and number of conducting pathways along the direction of net transport. It is found that the more desirable practical implications with respect to structural properties exhibited by the porous polymer layer having 40 monomers per main polymer chain, are mainly due to the higher flexibility of the polymer chains of this system, especially in the upper region of the porous structure. The characterization and analysis of the porous structures have suggested a useful definition for the physical meaning and implications of the pore connectivity of a real porous medium that is significantly different than the artificial physical meaning associated with the pore connectivity parameter employed in pore network models and whose physical limitations are discussed; furthermore, the methodology developed for the characterization of the three-dimensional structures of real porous media could be used to analyze the experimental data obtained from high-resolution noninvasive three-dimensional methods like high-resolution optical microscopy. The MD modeling and simulations methodology presented here could be used, considering that the type and size of affinity group/ligand as well as the size of the biomolecule to be adsorbed onto the affinity group/ligand are known, to construct different porous dextran layers by varying the length of the polymeric chain of dextran, the number of attachment points to the base matrix, the degree of side branching, and the number of main polymeric chains immobilized per unit surface area of base matrix. After the characterization of the porous structures of the different porous dextran layers is performed, then only a few promising structures would be selected for studying the immobilization of adsorption sites on the pore surfaces and the subsequent adsorption of the bioactive molecules onto the immobilized affinity groups/ligands.

Molecular dynamics simulation studies of the conformation and lateral mobility of a charged adsorbate biomolecule: Implications for estimating the critical value of the radius of a pore in porous media

The conformations, the values of the lateral transport coefficient of a charged biomolecule (desmopressin) in the adsorbed layer and in the liquid layers above the adsorbed layer, the potential energies of the interaction between the biomolecules located in different liquid layers with the charged solid surface and with the biomolecules in the adsorbed layer, the potential energies of the interaction between water molecules in the hydration layers surrounding the conformations of the biomolecules in different layers, as well as the structure and number of hydration layers between the different conformations of desmopressin, were determined by molecular dynamics simulation studies. The results show that the lateral mobility of the adsorbed desmopressin is approximately equal to zero and the value of the lateral transport coefficient of the biomolecule in the liquid layers located above the adsorbed layer increases as the distance of the liquid layer from the charged solid surface increases. But the values of the lateral transport coefficient of the biomolecule in the liquid layers above the adsorbed layer are lower in magnitude than the value of the transport coefficient of desmopressin along the direction normal to the charged solid surface in the liquid phase located above the vacant charged sites of the solid surface, and these differences in the values of the transport coefficients have important implications with respect to the replenishment of the biomolecules in the inner parts of a channel (pore), the overall rate of adsorption, and the form of the constitutive equations that would have to be used in macroscopic models to describe the mechanisms of mass transfer and adsorption in the pores of adsorbent media. Furthermore, a novel method is presented in this work that utilizes the information about the sizes of the conformations of the biomolecule in the adsorbed layer and in the liquid layers above the adsorbed layer along the direction that is normal to the charged solid surface, as well as the number and size of the hydration layers along the same direction, and could be used to estimate the value of the lower bound of the linear characteristic dimension of a pore (i.e., pore radius) in porous adsorbent media (e.g., porous adsorbent particles; skeletons of porous monoliths) in order to realize effective transport and overall adsorption rate.

Molecular dynamics simulation studies of the transport and adsorption of a charged macromolecule onto a charged adsorbent solid surface immersed in an electrolytic solution

Molecular dynamics simulations were performed in order to study the transport and adsorption of a charged macromolecule (desmopressin) onto a charged solid surface in an electrolytic solution. The strong Coulombic interaction from the charged solid surface represents the major force for accelerating, orienting, entrapping in the electrical double layer, and adsorbing the macromolecule onto the charged solid surface. The macromolecule is flattened as it approaches the charged surface, giving rise to a stronger surface exclusion effect that shields surface sites. When adsorbed, the macromolecule is restrained by a surface interaction more than one hundred times stronger than the thermal energy, of which 99.8% results from the strong dominant Coulombic interaction, and trapped by a hydration layer adjacent to the surface. This leads to zero lateral displacement of the adsorbed macromolecule and indicates that surface diffusion is a physically implausible mechanism in similar systems. Explicit solvent is required for realistic representation of the macromolecular structure and the surface interaction energy. The adsorbed macromolecule also decreased the electrostatic potential gradient perpendicular to the charged solid surface and introduced additional electrostatic potential gradients laterally. The results obtained from the molecular dynamics simulations confirm the importance of electrophoretic migration and support the physical mechanisms used in a macroscopic continuum model that predicts an overshoot in the concentration of a charged macromolecule in the adsorbed phase under certain conditions of pH and ionic strength.

Analysis and parametric sensitivity of the behavior of overshoots in the concentration of a charged adsorbate in the adsorbed phase of charged adsorbent particles: practical implications for separations of charged solutes

In this work, an analysis of the parametric sensitivity of the overshoot in the concentration of the adsorbate in the adsorbed phase, which occurs under certain conditions during an ion-exchange adsorption process, is presented and used to suggest practical implications of the concentration overshoot phenomenon on operational policies and configurations of chromatographic columns and finite bath adsorption systems. The results presented in this work demonstrate and explain how the development of an overshoot in the concentration of the adsorbate in the adsorbed phase could be enhanced or suppressed by (i) varying the diffusion coefficient, D3, of the adsorbate relative to the diffusion coefficients, D1 and D2, of the cations and anions, respectively, of the background/buffer electrolyte, (ii) altering the initial surface charge density, delta0, of the charged adsorbent particles, (iii) varying the Debye length, lambda, and (iv) changing the initial concentration, Cd3(0), of the adsorbate in the bulk liquid of the finite bath. The influence of the pH and ionic strength, Iinfinity, of the liquid solution on the development of an overshoot in the concentration of the adsorbate in the adsorbed phase is also presented and discussed through the relationships of these parameters to delta0 and lambda, respectively. Furthermore, a detailed explanation of the effects of each parameter on the interplay between the diffusive and electrophoretic molar fluxes, as well as on the structure and functioning of the electrical double layer, which are responsible for the concentration overshoot phenomenon, is presented.

Expressions for evaluating the possibility of slip at the liquid-solid interface in open tube capillary electrochromatography

In this work, expressions are constructed and solved that describe the velocity field of electroosmotic flow (EOF) in open tube capillary electrochromatography (CEC) systems when the possibility of having unequal tangential velocities at the liquid-solid interface is considered and a slip condition is employed as a boundary condition for the velocity of the EOF at the capillary wall. The coupled equations of hydrodynamics (momentum balance equation) and electrostatics (Poisson equation) are solved numerically in order to obtain the distribution of the velocity field as well as the value of the volumetric flow rate in the open tube. Also, expressions for the velocity field and the volumetric flow rate of the EOF are presented that are valid for certain electrolytic systems and for certain parameter values for which analytical solutions to the momentum balance and Poisson equations could be obtained. The results presented in this work indicate that having slip in the velocity of the EOF at the wall of the capillary could (i) substantially increase the electroosmotic velocity in the plug-flow region of the radial domain of the open capillary tube and (ii) increase the portion of the radial domain of the open capillary tube where the velocity of the EOF has a plug-flow profile, which in turn could increase the average velocity and volumetric flow rate of the EOF in the open capillary tube. Furthermore, the modeling approach and the results presented in this work indicate a method for experimentally evaluating the possibility of having slip in the velocity of the EOF at the capillary wall.

Novel general expressions that describe the behavior of the height equivalent of a theoretical plate in chromatographic systems involving electrically-driven and pressure-driven flows

Novel general expressions are constructed and presented that describe the behavior of the height equivalent of a theoretical plate (plate height), H, as a function of the linear velocity, Vx, along the axis, x, of the column and the kinetic parameters that characterize the mass transfer and adsorption mechanisms in chromatographic columns. Open tube capillaries as well as columns packed with either non-porous or porous particles are studied. The porous particles could have unimodal or bimodal pore-size distributions and intraparticle convective fluid flow and pore diffusion are considered. The expressions for the plate height, H, presented in this work could be applicable to high-performance liquid chromatography (HPLC) and capillary electrochromatography (CEC) systems, and could be used together with experimental plate height, H, versus linear velocity, Vx, data to determine the values of the parameters that characterize intraparticle convective fluid flow and pore diffusion. Furthermore, chromatographic systems under unretained as well as under retained conditions are examined. The experimental values of the plate height, H, versus the linear velocity, Vx, for a CEC system involving charged porous silica C8 particles and an uncharged analyte are compared with the theoretical results for the plate height, H, obtained from the expressions presented in this work. The agreement between theory and experiment is good, and the results indicate that the magnitude of the intraparticle electroosmotic flow (EOF) in the pores of the particles is substantial while the pore diffusion coefficient was of small magnitude. But the overall intraparticle mass transfer resistance in these particles was low because of the significant contribution of the intraparticle EOF. Simulation results are also presented (i) for a hybrid HPLC-CEC system, and (ii) for different CEC systems involving open capillaries as well as packed columns having non-porous or porous particles. The analysis of the results indicates (a) the reasons for the superior performance exhibited by the hybrid HPLC-CEC system over the performance obtained when the system is operated only in the HPLC mode, and (b) the operational configuration and the properties that the structure of the porous particles would have to have in CEC systems involving uncharged or charged analytes under unretained or retained conditions in order to obtain high CEC efficiency (low values of the plate height, H).

The Interplay of Diffusional and Electrophoretic Transport Mechanisms of Charged Solutes in the Liquid Film Surrounding Charged Nonporous Adsorbent Particles Employed in Finite Bath Adsorption Systems

A model that describes the diffusive and electrophoretic mass transport of the cation and anion species of a buffer electrolyte and of a charged adsorbate in the liquid film surrounding nonporous adsorbent particles in a finite bath adsorption system, in which adsorption of the charged adsorbate onto the charged surface of the nonporous particles occurs, is constructed and solved. The dynamic behavior of the mechanisms of this model explicitly demonstrates (a) the interplay between the diffusive and electrophoretic molar fluxes of the charged adsorbate and of the species of the buffer electrolyte in the liquid film surrounding the nonporous adsorbent particles, (b) the significant effect that the functioning of the electrical double layer has on the transport of the charged species and on the adsorption of the charged adsorbate, and (c) the substantial effect that the dynamic behavior of the surface charge density has on the functioning of the electrical double layer. It is found that at equilibrium, the value of the concentration of the charged adsorbate in the fluid layer adjacent to the surface of the adsorbent particles is significantly greater than the value of the concentration of the adsorbate in the finite bath, while, of course, the net molar flux of the charged adsorbate in the liquid film is equal to zero at equilibrium. This result is very different than that obtained from the conventional model that is currently used to describe the transport of a charged adsorbate in the liquid film for systems involving the adsorption of a charged adsorbate onto the charged surface of nonporous adsorbent particles; the conventional model (i) does not consider the existence of an electrical double layer, (ii) assumes that the transport of the charged adsorbate occurs only by diffusion in the liquid film, and (iii) causes at equilibrium the value of the charged adsorbate in the liquid layer adjacent to the surface of the particles to become equal to the value of the concentration of the charged adsorbate in the liquid of the finite bath. Furthermore, it was found that a maximum can occur in the dynamic behavior of the concentration of the adsorbate in the adsorbed phase when the value of the free molecular diffusion coefficient of the adsorbate is relatively large, because the increased magnitude of the synergistic interplay between the diffusive and electrophoretic molar fluxes of the adsorbate in the liquid film allows the adsorbate to accumulate (to be entrapped) in the liquid layer adjacent to the surface of the adsorbent particles faster than the concentrations of the electrolyte species, whose net molar fluxes are significantly hindered due to their opposing diffusive and electrophoretic molar fluxes, can adjust to account for the change in the surface charge density of the particles that arises from the adsorption of the charged adsorbate. The results presented in this work also have significant implications in finite bath adsorption systems involving the adsorption of a charged adsorbate onto the surface of the pores of charged porous adsorbent particles, because the diffusion and the electrophoretic migration of the charged solutes (cations, anions, and charged adsorbate) in the pores of the adsorbent particles will depend on the dynamic concentration profiles of the charged solutes in the liquid film surrounding the charged porous adsorbent particles. The results of the present work are also used to illustrate how the functioning of the electrical double layer could contribute to the development of inner radial humps (concentration rings) in the concentration of the adsorbate in the adsorbed phase of charged porous adsorbent particles.

Pore network modelling of affinity chromatography: determination of the dynamic profiles of the pore diffusivity of beta-galactosidase and its effect on column performance as the loading of beta-galactosidase onto anti-beta-galactosidase varies with time

A three-dimensional pore network model for diffusion in porous adsorbent particles was employed in a dynamic adsorption model that simulates the adsorption of a solute in porous particles packed in a chromatographic column. The solution of the combined model yielded the dynamic profiles of the pore diffusion coefficient of beta-galactosidase along the radius of porous adsorbent particles and along the length of the column as the loading of beta-galactosidase onto anti-beta-galactosidase immobilized on the surface of the pores of the particles occurred, and, the dynamic adsorptive capacity of the chromatographic column as a function of the design and operational parameters of the chromatographic system. It was found that for a given column length the dynamic profiles of the pore diffusion coefficient were influenced by (a) the superficial fluid velocity in the column, (b) the diameter of the adsorbent particles, and (c) the pore connectivity of the porous structure of the adsorbent particles. The effect of the magnitude of the pore connectivity on the dynamic profiles of the pore diffusion coefficient of beta-galactosidase increased as the diameter of the adsorbent particles and the superficial fluid velocity in the column increased. The dynamic adsorptive capacity of the column increased as (i) the particle diameter and the superficial fluid velocity in the column decreased, and (ii) the column length and the pore connectivity increased. In preparative affinity chromatography, it is desirable to obtain high throughputs within acceptable pressure gradients, and this may require the employment of larger diameter adsorbent particles. In such a case, longer column lengths satisfying acceptable pressure gradients with adsorbent particles having higher pore connectivity values could provide high dynamic adsorptive capacities. An alternative chromatographic system could be comprised of a long column packed with large particles which have fractal pores (fractal particles) that have high pore connectivities and which allow high intraparticle diffusional and convective flow mass transfer rates providing high throughputs and high dynamic adsorptive capacities. If large scale monoliths could be made to be reproducible and operationally stable, they could also offer an alternative mode of operation that could provide high throughputs and high dynamic adsorptive capacities.

Modeling and analysis of the dynamic behavior of mechanisms that result in the development of inner radial humps in the concentration of a single adsorbate in the adsorbed phase of porous adsorbent particles observed in confocal scanning laser microscopy experiments: diffusional mass transfer and adsorption in the presence of an electrical double layer

A theoretical model for adsorption of a single charged adsorbate that accounts for the presence of an electrical double layer in the pores of adsorbent particles is constructed and solved. The dynamic behavior of the mechanisms of the model can result in the development of inner radial humps (concentration rings) in the concentration of a single charged analyte (adsorbate) in the adsorbed phase of porous adsorbent particles. The results of the present work demonstrate the implication of the concept regarding the effect of the presence of an electrical double layer in the pores of adsorbent particles and the induced interactions between the electrostatic potential distribution and the mechanisms of mass transport of the species by diffusion, electrophoretic migration, and adsorption. Furthermore, the mechanisms of the model could explain qualitatively the development of the concentration ring (hump) observed in confocal scanning laser microscopy experiments.

Quantitative analysis and synthesis of the electrokinetic mass transport and adsorption mechanisms of a charged adsorbate in capillary electrochromatography systems employing charged adsorbent particles

The dynamic mathematical model of Grimes and Liapis [J. Colloid Interf. Sci. 234 (2001) 223] for capillary electrochromatography (CEC) systems operated under frontal chromatography conditions is extended to accommodate conditions in CEC systems where a positively charged analyte is introduced into a packed capillary column by a pulse injection (analytical mode of operation) in order to determine quantitatively the electroosmotic velocity, electrostatic potential and concentration profiles of the charged species in the double layer and in the electroneutral core region of the fluid in the interstitial channels for bulk flow in the packed chromatographic column as the adsorbate adsorbs onto the negatively charged fixed sites on the surface of the non-porous particles packed in the chromatographic column. Furthermore, certain key parameters are identified for both the frontal and analytical operational modes that characterize the performance of CEC systems. The results obtained from model simulations for CEC systems employing the analytical mode of operation indicate that: (a) for a given mobile liquid phase, the charged particles should have the smallest diameter, d(p), possible that still provides conditions for a plug-flow electroosmotic velocity field in the interstitial channels for bulk flow and a large negative surface charge density, deltao, in order to prevent overloading conditions; (b) sharp, highly resolute adsorption zones can be obtained when the value of the parameter gamma2min, which represents the ratio of the electroosmotic velocity of the mobile liquid phase under unretained conditions to the electrophoretic velocity of the anions (0>gamma2.min>-1), is very close to negative one, but the rate at which the solute band propagates through the column is slow; furthermore, as the solute band propagates across larger axial lengths, the desorption zone becomes more dispersed relative to the adsorption zone especially when the value of the parameter gamma2,max, which represents the ratio of the electroosmotic velocity of the mobile liquid phase under retained conditions to the electrophoretic velocity of the anions (0>gamma2,max>-1), is significantly greater than gamma2,min; (c) when the value of the equilibrium adsorption constant, K(A),3, is low, very sharp, highly resolved adsorption and desorption zones of the solute band can be obtained as well as fast rates of propagation of the solute band through the column; (d) sharp adsorption zones and fast propagation of the solute band can be obtained if the value of the mobility, v3, of the analyte is high and the value of the ratio v1/v3, where v1 represents the mobility of the cation, is low; however, if the magnitude of the mobility, v3, of the analyte is small, dispersed desorption zones are obtained with slower rates of propagation of the solute band through the column; (e) good separation of analyte molecules having similar mobilities and different adsorption affinities can be obtained in short operational times with a very small column length, L, and the resolution can be increased by providing values of gamma2,min and gamma2,max that are very close to negative one; and (f) the change in the magnitude of the axial current density, i(x), across the solute band could serve as a measurement for the rate of propagation of the solute band.

Modeling and Analysis of the Electrokinetic Mass Transport and Adsorption Mechanisms of a Charged Adsorbate in Capillary Electrochromatography Systems Employing Charged Nonporous Adsorbent Particles

Mass-transfer systems based on electrokinetic phenomena (i.e., capillary electrochromatography (CEC)) have shown practical potential for becoming powerful separation methods for the biotechnology and pharmaceutical industries. A dynamic mathematical model, consisting of the momentum balance and the Poisson equations, as well as the unsteady-state continuity expressions for the cation and anion of the background electrolyte and of a positively charged analyte (adsorbate), is constructed and solved to determine quantitatively the electroosmotic velocity, the electrostatic potential, the concentration profiles of the charged species in the double layer and in the electroneutral core region of the fluid in the interstitial channels for bulk flow in the packed chromatographic column, and the axial current density profiles as the adsorbate adsorbs onto the negatively charged fixed sites on the surface of the nonporous particles packed in the chromatographic column. The frontal analysis mode of operation is simulated in this work. The results obtained from model simulations provide significant physical insight into and understanding of the development and propagation of the dynamic profile of the concentration of the adsorbate (analyte) and indicate that sharp, highly resolved adsorption fronts and large amounts of adsorbate in the adsorbed phase for a given column length can be obtained under the following conditions: (i) The ratio, gamma(2, 0), of the electroosmotic velocity of the mobile liquid phase at the column entrance after the adsorption front has passed the column entrance to the electrophoretic velocity of the anion is very close to -1. The structure of the equations of the model and model simulations indicate that a stable adsorption front cannot develop when gamma(2, 0) is less than -1 unless the value of the mobility of the cation is less than the value of the mobility of the analyte, which may be a rare occurrence in practical CEC systems. (ii) The ratio of the mobility of the cation to the mobility of the analyte is less than two orders of magnitude. This effect becomes more significant as the value of the equilibrium adsorption constant, K(A, 3), of the analyte increases. (iii) The concentration of the analyte relative to the concentration of the cation is increased (feed solutions with less dilute concentrations of the analyte are employed). Therefore, to obtain good performance for CEC systems operated in the frontal analysis mode (well-resolved adsorption fronts and high adsorbate amounts in the adsorbed phase), one can choose an electrolyte whose cation has a mobility that is not more than one or two orders of magnitude greater than the mobility of the analyte and whose anion has a mobility such that the value of gamma(2, 0) is close to -1; one can then bring the value of gamma(2, 0) closer to -1 by decreasing the particle diameter, d(p), and/or making the value of the surface charge density, delta(0), of the particles more negative (in effect, making the value of the zeta potential, zeta(p), at the surface of the particles more negative at time t=0) to change the value of the velocity, <<upsilon(x)>>|(x=0), of the electroosmotic flow (EOF) at the column entrance (<<upsilon(x)>>|(x=0) is determined after the adsorption front has passed the column entrance). This approach could provide conditions in the column that avoid overloading of the adsorbate. One can obtain faster breakthrough times at the sacrifice of resolution and utilization of the adsorptive capacity of the packed bed if one employs a cation whose mobility is very large relative to the mobility of the analyte and/or an anion that provides a value of gamma(2, 0) significantly greater than -1. If it is possible, one can increase the concentration of the analyte in the feed stream to avoid sacrificing resolution and adsorptive capacity of the packed bed and still decrease the time at which breakthrough occurs. Also, the dynamic behavior of the axial current density, i(x), profiles indicates that the magnitude of i(x) and/or the change in the value of i(x) across the adsorption front could serve as a measurement for the rate of propagation of the adsorption front through the column. Furthermore, the effect of the decreased magnitude of the velocity of the EOF in the region of the column where the analyte is present in the adsorbed phase could act to decrease the effect of tailing when CEC systems are operated in the pulse injection mode (analytical electrochromatography) because the higher velocity of the fluid upstream of the migrating adsorption zone may compress the tail of the peak. Copyright 2001 Academic Press.

Pore network modelling: determination of the dynamic profiles of the pore diffusivity and its effect on column performance as the loading of the solute in the adsorbed phase varies with time

A three-dimensional pore network model for diffusion in porous adsorbent particles was employed in a dynamic adsorption model that simulates the adsorption of a solute in porous particles packed in a chromatographic column. The solution of the combined model yielded the dynamic profiles of the pore diffusion coefficient of beta-galactosidase along the radius of porous ion-exchange particles and along the length of the column as the loading of the adsorbate molecules on the surface of the pores occurred, and, the dynamic adsorptive capacity of the chromatographic column as a function of the design and operational parameters of the chromatographic system. The pore size distribution of the porous adsorbent particles and the chemistry of the adsorption sites were unchanged in the simulations. It was found that for a given column length the dynamic profiles of the pore diffusion coefficient were influenced by: (i) the superficial fluid velocity in the column, (ii) the diameter of the adsorbent particles and (iii) the pore connectivity of the porous structure of the adsorbent particles. The effect of the magnitude of the pore connectivity on the dynamic profiles of the pore diffusion coefficient increased as the diameter of the adsorbent particles and the superficial fluid velocity in the column increased. The dynamic adsorptive capacity of the column increased as: (a) the particle diameter and the superficial fluid velocity in the column decreased, and (b) the column length and the pore connectivity increased. In preparative chromatography, it is desirable to obtain high throughputs within acceptable pressure gradients, and this may require the employment of larger diameter adsorbent particles. In such a case, longer column lengths satisfying acceptable pressure gradients with adsorbent particles having higher pore connectivity values could provide high dynamic adsorptive capacities. An alternative chromatographic system could be comprised of a long column packed with large particles which have fractal pores (fractal particles) that have high pore connectivities and which allow high intraparticle diffusional and convective flow mass transfer rates providing high throughputs and high dynamic adsorptive capacities. If large scale monoliths could be made to be reproducible and operationally stable, they could also offer an alternative mode of operation that could provide high throughputs and high dynamic adsorptive capacities.

Determination of the pore connectivity and pore size distribution and pore spatial distribution of porous chromatographic particles from nitrogen sorption measurements and pore network modelling theory

The pore connectivity, pore size distribution and pore spatial distribution of the porous structure of native and silanized silica particles were determined by matching the experimental nitrogen sorption data with the theoretical results obtained from pore network model simulations. The agreement between theory and experiment is found to be good. The results clearly indicate that the deposition of the silane layer to the pore surfaces of the native silica particles produces a silanized silica particle with a mean pore diameter and pore connectivity smaller than that of the native silica particle. Furthermore, the evaluation of the pore diffusivity of ribonuclease under unretained conditions shows that the lower values of the pore connectivity found in the samples of silanized silica particles, when compared with the values of the pore connectivity obtained for the native silica particles, increase the diffusional mass transfer resistance within the porous structure of the silanized silica particles.

Film Mass Transfer Coefficient Expressions for Electroosmotic Flows

Expressions are developed and presented that could be used to determine the film mass transfer coefficient of a solute in electroosmotic flows. In contrast to pressure-driven flows at low Reynolds numbers where the film mass transfer coefficient is independent of the linear characteristic dimension of the channel for flow, in electroosmotically driven flows at low Reynolds numbers the film mass transfer coefficient is shown to be a function of the ratio R/lambda, where R is the channel radius and lambda is the Debye length. This result implies that for electroosmotically driven flows in a packed bed or porous monolith with channels for flow having similar geometry but different sizes, the film mass transfer coefficient would vary with the size of the interstitial channels for bulk flow while in pressure-driven flows the film mass transfer coefficient would be the same for all interstitial channels. From the expressions presented in this work, one can show that for the same volumetric flow rate the film mass transfer coefficient of electroosmotically driven flows is proportional to that for pressure-driven flows. Copyright 2000 Academic Press.

Determination of the intraparticle electroosmotic volumetric flow-rate, velocity and Peclet number in capillary electrochromatography from pore network theory

The results obtained from the pore network model employed in this work, clearly show that the magnitudes of the intraparticle electroosmotic volumetric flow-rate, Qintrap, and velocity, (v(intrap,x)), in the pores of the charged porous silica particles considered in this study are greater than zero. The intraparticle Peclet number, Pe(intra, of a solute in these charged porous silica particles would be greater than zero, and, in fact, the magnitude of the intraparticle Peclet number, Pe(intrap), of lysozyme is greater than unity for all the values of the pore connectivity, nT, of the intraparticle pores and of the applied electric potential difference per unit length, Ex, along the axis of the capillary column considered in this work. Furthermore, the values of the intraparticle electroosmotic volumetric flow-rate, Qintrap, and velocity, (v(intrap,x)), as well as the magnitude of the pore diffusion coefficient, Dp, of the solute increase as the value of the pore connectivity, nT, of the intraparticle pores increases. The intraparticle electroosmotic flow can contribute significantly, if the appropriate chemistry is employed in the mobile liquid phase and in the charged porous particles, in (i) decreasing the intraparticle mass transfer resistance, (ii) decreasing the dispersive mass transfer effects, and (iii) increasing the intraparticle mass transfer rates so that high column efficiency and resolution can be obtained.

Modeling the velocity field of the electroosmotic flow in charged capillaries and in capillary columns packed with charged particles: interstitial and intraparticle velocities in capillary electrochromatography systems

Mass transfer systems based on electrokinetic phenomena (i.e., capillary electrochromatography (CEC)) have shown practical potential in becoming powerful separation methods for the biotechnology and pharmaceutical industries. A mathematical model has been constructed and solved to describe quantitatively the profiles of the electrostatic potential, pressure, and velocity of the electroosmotic flow (EOF) in charged cylindrical capillaries and in capillary columns packed with charged particles. The results obtained from model simulations (i) provide significant physical insight and understanding with regard to the velocity profile of the EOF in capillary columns packed with charged porous particles which represent systems employed in CEC, (ii) provide the physical explanation for the experimental results which indicate that the velocity of the EOF in capillary columns packed with charged porous particles is a very weak function (it is almost independent) of the diameter of the particles, and (iii) indicate that the intraparticle velocity, nu(p,i), of the EOF can be greater than zero. The intraparticle Peclet number, Pe(int rap), for lysozyme was found to be greater than unity and this intraparticle convective mass transfer mechanism could contribute significantly, if the appropriate chemistry is employed in the mobile liquid phase and in the charged porous particles, in (a) decreasing the intraparticle mass transfer resistance, (b) decreasing the dispersive mass transfer effects, and (c) increasing the intraparticle mass transfer rates so that high column efficiency and resolution can be obtained. Furthermore, the results from model simulations indicate that for a given operationally permissible value of the applied electric potential difference per unit length, Ex, high values for the average velocity of the EOF can be obtained if (1) the zeta potential, zeta(p), at the surface of the particles packed in the column has a large negative magnitude, (2) the value of the viscosity, mu, of the mobile liquid phase is low, (3) the magnitude of the dielectric constant, epsilon, of the mobile liquid phase is reasonably large, and (4) the combination of the values of the concentration, C(infinity), of the electrolyte and of the dielectric constant, epsilon, provide a thin double layer. The theoretical results for the velocity of the EOF obtained from the solution of the model presented in this work were compared with the experimental values of the velocity of the EOF obtained from a fused-silica column packed with charged porous silica C8 particles. Systems with four different particle diameters and three different concentrations of the electrolyte were considered, and the magnitude of the electric field was varied widely. The agreement between theory and experiment was found to be good.

Modeling and simulation of the dynamic behavior of monoliths. Effects of pore structure from pore network model analysis and comparison with columns packed with porous spherical particles

A mathematical model is presented that could be used to describe the dynamic behavior, scale-up, and design of monoliths involving the adsorption of a solute of interest. The value of the pore diffusivity of the solute in the pores of the skeletons of the monolith is determined in an a priori manner by employing the pore network modeling theory of Meyers and Liapis [J. Chromatogr. A, 827 (1998) 197 and 852 (1999) 3]. The results clearly show that the pore diffusion coefficient, Dmp, of the solute depends on both the pore size distribution and the pore connectivity, nT, of the pores in the skeletons. It is shown that, for a given type of monolith, the film mass transfer coefficient, Kf, of the solute in the monolith could be determined from experiments based on Eq. (3) which was derived by Liapis [Math. Modelling Sci. Comput., 1 (1993) 397] from the fundamental physics. The mathematical model presented in this work is numerically solved in order to study the dynamic behavior of the adsorption of bovine serum albumin (BSA) in a monolith having skeletons of radius r(o) = 0.75x10(-6) m and through-pores having diameters of 1.5x10(-6)-1.8x10(-6) m [H. Minakuchi et al., J. Chromatogr. A, 762 (1997) 135]. The breakthrough curves of the BSA obtained from the monolith were steeper than those from columns packed with porous spherical particles whose radii ranged from 2.50x10(-6) m to 15.00x10(-6) m. Furthermore, and most importantly, the dynamic adsorptive capacity of the monolith was always greater than that of the packed beds for all values of the superficial fluid velocity, Vtp. The results of this work indicate that since in monoliths the size of through-pores could be controlled independently from the size of the skeletons, then if one could construct monolith structures having (a) relatively large through-pores with high through-pore connectivity that can provide high flow-rates at low pressure drops and (b) small-sized skeletons with mesopores having an appropriate pore size distribution (mesopores having diameters that are relatively large when compared with the diameter of the diffusing solute) and high pore connectivity, nT, the following positive results, which are necessary for obtaining efficient separations, could be realized: (i) the value of the pore diffusion coefficient, Dmp, of the solute would be large, (ii) the diffusion path length in the skeletons would be short, (iii) the diffusion velocity, vD, would be high, and (iv) the diffusional response time, t(drt), would be small. Monoliths with such pore structures could provide more efficient separations with respect to (a) dynamic adsorptive capacity and (b) required pressure drop for a given flow-rate, than columns packed with porous particles.

Network modeling of the convective flow and diffusion of molecules adsorbing in monoliths and in porous particles packed in a chromatographic column

A cubic lattice network of interconnected pores was constructed to represent the porous structure existing in a monolith (continuous bed) or in a column packed with porous chromatographic particles. Expressions were also constructed and utilized to simulate, through the use of the pore network model, the intraparticle interstitial velocity and pore diffusivity of adsorbate molecules in porous chromatographic particles or in monoliths under retained and unretained conditions. The combined effects of steric hindrance at the entrance to the pores and frictional resistance within the pores, as well as the effects of pore size, pore connectivity, nT, of the porous network, molecular size of adsorbate and ligand (active site), and the fractional saturation of adsorption sites (ligands), have been considered. The results for the adsorption systems studied in this work, indicate that the obstruction effects on the intraparticle interstitial velocity, due to (a) the thickness of the immobilized layer of active sites and (b) the thickness of the adsorbed layer, are small and appear to be insignificant when they are compared with the very significant effect that the value of the pore connectivity, nT, has on the magnitude of the intraparticle interstitial velocity. The effective pore diffusion coefficient of the adsorbate molecules was found to decline with increasing molecular size of ligand, with increasing fractional saturation of the active sites or with diminishing pore size, and with decreasing pore connectivity, nT. The results also show that the magnitude of the interstitial fluid velocity is many times larger than the diffusion velocity of the adsorbate molecules within the porous adsorbent particles. Furthermore, the results clearly show that the intraparticle interstitial velocity and the pore diffusivity of the adsorbate increase significantly as the value of the pore connectivity, nT, of the porous medium increases. The results of this work indicate that the pore network model and the expressions presented in this work, could allow one, for a given porous adsorbent, adsorbate, ligand (active site), and interstitial column fluid velocity, to determine in an a priori manner the values of the intraparticle interstitial velocity and pore diffusivity within the monolith or within the porous adsorbent particles as the fractional saturation of the active sites changes. The values of these transport parameters could then be employed in the macroscopic models that could predict the dynamic behavior, scale-up, and design of chromatographic systems. The theoretical results could also have important implications in the selection of a ligand as well as in the selection and construction of an affinity porous matrix.

Modeling of the primary and secondary drying stages of the freeze drying of pharmaceutical products in vials: numerical results obtained from the solution of a dynamic and spatially multi-dimensional lyophilization model for different operational policies

A rigorous unsteady state and spatially multidimensional model is presented and solved to describe the dynamic behavior of the primary and secondary drying stages of the lyophilization of a pharmaceutical product in vials for different operational policies. The results in this work strongly motivate the aggressive control of freeze drying and it is found that heat input control that runs the process close to the melting and scorch temperature constraints yields (i) faster drying times, and (ii) more uniform distributions of temperature and concentration of bound water at the end of the secondary drying stage.

Frontal chromatography of proteins. Effect of axial dispersion on column performance

A mathematical model describing the dynamic adsorption of proteins in columns packed with spherical porous adsorbent particles is used to study the effect of axial dispersion on the performance of chromatographic systems. The values of the axial dispersion coefficient, DL, are estimated from a correlation based on a model describing axial dispersion in packed beds that provides satisfactory results when compared with experiment. Simulations of frontal chromatography in systems including axial dispersion and in systems without axial dispersion are made and compared to determine the effect of axial dispersion on the efficiency of the adsorption process; also, the system parameters that influence axial dispersion are examined. It is found that the reduction in the efficiency of the adsorption process due to axial dispersion is small (< 1%) for columns of length 10 cm or greater. However, for short columns, this efficiency reduction can be as large as 10%. Increasing the adsorbent particle diameter, dp, increases the magnitude of the reduction in efficiency due to axial dispersion; the effect of increasing the adsorbent particle diameter, dp, is much more pronounced in a short column than in a long column.

Model discrimination and estimation of the intraparticle mass transfer parameters for the adsorption of bovine serum albumin onto porous adsorbent particles by the use of experimental frontal analysis data

Experimental data from a chromatographic system involving the adsorption of bovine serum albumin (BSA) onto porous anion-exchange adsorbent particles packed in a column are presented. The parameters that characterize the mass transfer mechanisms of intraparticle diffusion and convection are estimated by fitting the predictions of dynamic mathematical models describing adsorption in column systems having spherical perfusive and purely diffusive adsorbent particles to the experimental breakthrough data obtained from the column adsorption system. Both linear and nonlinear expressions for the equilibrium isotherm are considered. The values of the transport parameters are estimated in the time domain for the nonlinear adsorption models and in the Laplace transform domain for the linear adsorption models. The capabilities of the different models to describe satisfactorily the dynamic behavior of the adsorption system are compared. The dynamic nonlinear adsorption model for purely diffusive particles is found to describe most appropriately the dynamic behavior of the experimental chromatographic system studied in this work.