A molecular dynamics study on the transport of a charged biomolecule in a polymeric adsorbent medium and its adsorption onto a charged ligand.

The transport of a charged adsorbate biomolecule in a porous polymeric adsorbent medium and its adsorption onto the covalently immobilized ligands have been modeled and investigated using molecular dynamics modeling and simulations as the third part of a novel fundamental methodology developed for studying ion-exchange chromatography based bioseparations. To overcome computational challenges, a novel simulation approach is devised where appropriate atomistic and coarse grain models are employed simultaneously and the transport of the adsorbate is characterized through a number of locations representative of the progress of the transport process. The adsorbate biomolecule for the system studied in this work changes shape, orientation, and lateral position in order to proceed toward the site where adsorption occurs and exhibits decreased mass transport coefficients as it approaches closer to the immobilized ligand. Furthermore, because the ligands are surrounded by counterions carrying the same type of charge as the adsorbate biomolecule, it takes the biomolecule repeated attempts to approach toward a ligand in order to displace the counterions in the proximity of the ligand and to finally become adsorbed. The formed adsorbate-ligand complex interacts with the counterions and polymeric molecules and is found to evolve slowly and continuously from one-site (monovalent) interaction to multisite (multivalent) interactions. Such a transition of the nature of adsorption reduces the overall adsorption capacity of the ligands in the adsorbent medium and results in a type of surface exclusion effect. Also, the adsorption of the biomolecule also presents certain volume exclusion effects by not only directly reducing the pore volume and the availability of the ligands in the adjacent regions, but also causing the polymeric molecules to change to more compact structures that could further shield certain ligands from being accessible to subsequent adsorbate molecules. These findings have significant practical implications to the design and construction of polymeric porous adsorbent media for effective bioseparations and to the synthesis and operation of processes employed in the separation of biomolecules. The modeling and analysis methods presented in this work could also be suitable for the study of biocatalysis where an enzyme is immobilized on the surface of the pores of a porous medium.

Adsorption of a single protein interacting with multiple ligands: inner radial humps in the concentration profiles induced by non-uniform ligand density distributions.

The dynamic behavior of the concentration profiles of a single protein in the pore solution and the adsorbed phase is studied in different adsorbent media when the spatial density distribution of the immobilized ligands is either uniform or non-uniform and at the same time the single protein is forming one-site and two-site adsorbate-ligand complexes with the immobilized monovalent ligands. The competition for the formation of one-site and two-site interaction complexes leads to the formation of inner radial humps in the concentration profiles of the two-site adsorbate-ligand complex in adsorbent particles having either uniform or non-uniform spatial ligand density distributions. The results show that inner radial humps in the concentration profiles of the adsorbed protein (total concentration of adsorbed protein by one-site and two-site interactions) occur only in adsorbent media whose spatial ligand density distributions are non-uniform and have maxima or minima occurring in radial positions located between the center and the outer surface of the particles. The non-uniform spatial ligand density distributions satisfying this property provide the cause for the occurrence of inner radial humps in the concentration profiles of a single adsorbed protein, while the multi-site adsorption interactions affect the magnitude and the rate of propagation of the inner radial humps in the concentration profiles of the single adsorbed protein. It is also demonstrated that adsorbent media having certain non-uniform functional forms of spatial distribution in the density of immobilized ligands could provide more efficient adsorption of a protein than an adsorbent medium whose spatial distribution of the density of immobilized ligands is uniform. Furthermore, the results in this study suggest the type of information that could be obtained from finite bath experiments and could be used to (i) determine whether multi-site adsorbate-ligand complexes are formed during the adsorption of a single adsorbate and (ii) select between alternative adsorbent media the adsorbent particles that could provide the highest overall adsorption rate for a given adsorbate of interest. The results clearly indicate that it is very important to study the dynamic behavior of an adsorption system having a non-uniform spatial ligand density distribution and where the values of the pH and ionic strength are such that the electrophoretic effects are active.

The design by molecular dynamics modeling and simulations of porous polymer adsorbent media immobilized on the throughpore surfaces of polymeric monoliths.

Ion-exchange porous adsorbent media having intermediate and low surface densities of dextran polymer grafted on the surface of the throughpores of polymeric monoliths are constructed and characterized by a molecular dynamics modeling and simulation approach that has also been shown to be effective in the construction and characterization of porous ion-exchange adsorbent media whose number of immobilized dextran polymer chains per unit surface area is high. The activation step that prepares the surface of the pores of the dextran polymer layer for the immobilization of the charged ligands insignificantly affected the pore structure of the dextran polymer layer, while this was found to not be the case for previously studied systems that involved high dextran polymer surface densities. Compared to the high dextran polymer density system studied previously, the intermediate dextran polymer density system can generate significantly larger pores but still possesses relatively high interconnection and mutual steric support between dextran chains to exhibit similar structural characteristics and responses to charged ligand immobilization, including dextran layer thickness, stability, monomer distribution, ligand-induced compact chain structures, dextran layer shrinkage, distributions of ligands and counterions, and local nonelectroneutrality. The low dextran polymer density system having relatively isolated dextran chains and insufficient mutual steric support can result in even larger pores than those obtained in the intermediate dextran polymer density system, but a significantly thinner porous dextran polymer layer and different dextran monomer distributions are obtained in the low dextran polymer density system. More importantly, the gradient of the local nonelectroneutrality after the immobilization of the charged ligands is significantly smaller in magnitude in the low dextran polymer density system than that obtained in the system having intermediate dextran polymer density, and, despite a lack of porous layer depth to accommodate adsorbate biomolecules in large amounts, it could still be useful for the effective transport and adsorption of very large biomolecules. Compared with the polymeric monoliths without a porous dextran polymer layer grafted on the surface of their throughpores, the intermediate and low dextran polymer density systems explored and studied in this work provide pore structures with desirable characteristics for the effective transport of adsorbate biomolecules and substantially larger effective surface areas and throughput capacities for the adsorption of the adsorbate biomolecules.

Porous polymer adsorbent media constructed by molecular dynamics modeling and simulations: the immobilization of charged ligands and their effect on pore structure and local nonelectroneutrality.

A molecular dynamics modeling and simulation approach is presented and employed to construct porous dextran polymer ion-exchange adsorbent media. Both the activation step of the surface of the pores of the dextran polymer layer grafted on an agarose surface and the immobilization of charged ligands on the activated surface of the porous dextran polymer layer are considered. For the systems studied in this work, the activation step modifies slightly the pore structure of the base, nonactivated porous dextran polymer, while the immobilization of the ligands on the activated pore surface of the dextran layer changes significantly the pore structure of the activated dextran layer. The density distributions of the counterions and immobilized charged ligands along the direction of net transport in the adsorbent media constructed in this study are found to be nonuniform. The variables that affect the shape and magnitude of the density distributions of the counterions and immobilized charged ligands as well as the total number of charged ligands that can be immobilized on the activated porous dextran layer are identified and presented in this work. Furthermore, the data clearly show that there is local nonelectroneutrality in the porous dextran polymer ion-exchange adsorbent media, and this result has very important practical implications for the operation and performance of separation systems involving ion-exchange adsorbent media (e.g., ion-exchange chromatography systems). Also, the results of this work suggest approaches for (1) controlling the immobilization process of charged ligands and (2) constructing and studying the behavior of chromatographic polymeric monoliths and packed bed columns having a gradient of density of functionalities along the axis of the chromatographic polymeric monolith or packed bed column.

Rational surface design for molecular dynamics simulations of porous polymer adsorbent media.

The construction and use of nonflat agarose surfaces in a simulation box, together with the employment of criteria for the immobilization of a set of dextran polymer chains on the nonflat agarose surfaces whose mathematical physics is compatible with that of the criteria used for the immobilization of the same set of dextran polymer chains on flat agarose surfaces, are shown to generate, through the use of molecular dynamics simulations whose simulation box has linear dimensions along the lateral directions that are the same when flat and nonflat agarose surfaces are used, dextran porous polymer structures whose pore sizes at the outermost surface and in the vicinity of the outermost surface of the porous medium can be controlled by an indirect manner through the variation of the parameters that characterize the nonflat surface. The use of a nonflat surface for the generation of desired large pores requires only a small or modest increase in the number of solvent molecules in the simulation box, while the use of a flat surface for the construction of the same desired large pores requires significant increases in the size of the linear dimensions of the flat surface. This increases so substantially the number of solvent molecules that the computational loads become intractable. The results in this work show that through the use of nonflat surfaces porous dextran polymer layers having pores of desired sizes can be effectively constructed, and this approach could be used for the design and construction of polymer-based porous adsorbent media that could effectively facilitate the transport and adsorption of an adsorbate biomolecule of interest that must be separated from a mixture of components. A useful definition about the properties that a porous polymer structure must have in order to become, for an adsorbate biomolecule of interest of known molecular size, a useful adsorbent medium, is presented and is used to (1) evaluate the porous polymer structures generated through the employment of different nonflat surface models and (2) determine and select the nonflat surface model from a set of nonflat surface models that is effective in producing promising porous structures. Then a procedure is presented by which a set of porous polymer media is generated through the use of the selected nonflat surface model, and the desired porous structure from this set is determined and could be considered to be used for the transport and immobilization of the selected affinity groups/ligands and the subsequent transport and adsorption of the desired to be separated adsorbate.

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.

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 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.

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 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.

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, <>|(x=0), of the electroosmotic flow (EOF) at the column entrance (<>|(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.

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.

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 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.