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.     

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).     

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.     

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.     

Perfusive flow and intraparticle distribution of a neutral analyte in capillary electrochromatography

The relevance and magnitude of an electroosmotic perfusion mechanism in electrochromatography is analyzed. To systemize our studies we first considered the transport of an electroneutral and nonadsorbing tracer. Based on the refractive index matching in a microfluidic setup containing fixed spherical porous particles, we conducted a quantitative analysis in real time of the spatio-temporal distribution of fluorescent tracer molecules during their uptake by (and a release from) single particles using confocal laser scanning microscopy. Even under conditions of a significant electrical double layer overlap the intraparticle electroosmotic flow produces due to its unidirectional nature and in striking contrast to the symmetric (spherical) distributions typical for purely diffusive transport strongly asymmetric concentration profiles inside spherical particles as the locally charged pore liquid begins to respond to the externally applied electrical field. The profiles retain an axisymmetric nature, i.e., rotational symmetry with respect to the field direction. Results of our measurements could be successfully interpreted and further analyzed by a compact mathematical model. Intraparticle Peclet numbers of up to 150 have been realized and found to significantly enhance the mass transport on particle scale towards the convection-dominated regime when compared to a conventional (diffusion-limited) kinetics.     

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.     

Flow-through particles for the high-performance liquid chromatographic separation of biomolecules: perfusion chromatography

This paper reports a new technique for reducing resistance to stagnant mobile phase mass transfer without sacrificing high adsorbent capacity or necessitating extremely high pressure operation. The technique involves the flow of liquid through a porous chromatographic particle, and has thus been termed "perfusion chromatography". This is accomplished with 6000-8000 A pores which transect the particle. Data from electron microscopy, column efficiency, frontal analysis and theoretical modelling all suggest that mobile phase will flow through these large pores. In this manner, solutes enter the interior of the particles through a combination of convective and diffusional transport, with convection dominating for Peclet numbers greater than one. The implications of flow through particles on bandspreading, resolution and dynamic loading capacity are examined. It is shown that the rate of solute transport is strongly coupled to mobile phase velocity such that bandspreading, resolution of proteins and dynamic loading capacity are unaffected by increases in mobile phase velocity up to several thousand centimeters per hour. The surface area of this very large-pore diameter material is enhanced by using a network of smaller, 500-1500 A interconnecting pores between the throughpores. Scanning electron micrographs show that the pore network is continuous and that no point in the matrix is more than 5000-10,000 A from a through-pore. As a consequence, diffusional path lengths are minimized and the large porous particles take on the transport characteristics of much smaller particles but with a fraction of the pressure drop. Capacity and resolution studies show that these materials bind and separate an amount of protein equivalent to that of conventional high-performance liquid chromatography as well as low performance agarose-based media at greater than 10-100 times higher mobile phase velocity with no loss in resolution.     

Partitioning and adsorption in gas-liquid-solid chromatography: Models of temporal moments for retention and variance

Summary The resolution of gaseous chemical species in gasliquid-solid chromatography is influenced by absorption (partitioning) in the liquid and adsorption at the liquid-solid interface. We consider fundamental mass transfer models with adsorption and partitioning effects for solid chromatographic supports covered with thin films of stationary liquid. The dynamic models, based on mass-balance partial-differential equations, include the significant phenomena: convection, axial dispersion, gas-liquid mass transfer, intraparticle diffusion, and liquid-solid adsorption. Expressions for retention time and band variance (first and second temporal moments) are presented and evaluated for four distinct models: (1) capillary tube with inner surface covered with a uniform-thickness liquid film, (2) column of nonporous spheres covered with a uniform-thickness liquid film, (3) porous spherical particles with intraparticle pores covered with a uniform-thickness liquid film, (4) porous spherical particles with intraparticle pores completely filled with liquid.     

Comparison of the mass transfer in totally porous and superficially porous stationary phases in liquid chromatography

The characterization of mass-transfer processes in a chromatographic column during a separation process is essential, since the influence of the mass-transfer kinetics on the shape of the chromatographic band profiles and on the efficiency of the separation is crucial. Several sources of mass transfer in a chromatographic bed have been identified and studied: the axial dispersion in the stream of mobile phase, the external mass-transfer resistance, intraparticle diffusion, and the kinetics of adsorption–desorption. We measured and compared the characteristics and performance of a new brand of shell particles and those of a conventional brand of totally porous silica particles. The shell stationary phase was made of 2.7-μm superficially porous particles (a 1.7-μm solid core is covered with a 0.5-μm-thick shell of porous silica). The other material consisted of totally porous particles of conventional 3.5-μm commercial silica. We measured the first and second central moments of the peaks of human insulin over a wide range of mobile phase velocities (from 0.02 to 1.3 mL/min) at 20°C. The plate height equations were constructed and the axial dispersion, external mass transfer, as well as the intraparticle diffusion coefficients were calculated for the two stationary phases.     

Electroosmotically enhanced mass transfer through polyacrylamide gels

We present an internal pumping strategy to enhance solute fluxes in polymer gels. The method is based on electroosmotic flow driven by an electric field applied across a gel that has been doped with charged colloidal inclusions. This work is motivated by the need to enhance the transport in gel-based biosensor devices whose response dynamics are often mass transfer limited. In this case, polyacrylamide gel slabs were doped with immobilized, charged silica colloids, and the flux of a fluorescent tracer was measured as a function of applied field strength, the volume fraction and size of the colloidal silica inclusions, and the bulk electrolyte composition. Significant flux enhancements were achieved with applied electric currents on the order of a few mA. Control experiments indicated that the flux enhancement was not due to any distortion of the gel diffusional properties in response to the presence of the inclusions. At a constant inclusion volume fraction, the electroosmotic solute flux enhancement was strongest for the smallest particle sizes that provide the highest total surface area, consistent with the electroosmotic mechanism whereby fluid flow is generated along the solid/liquid interface.     

Rapid fabrication of core-shell silica particles using a multilayer-by-multilayer approach

Monodisperse silica core-shell particles with a high surface area and large pore size were rapidly prepared via a multilayer-by-multilayer (ML-b-ML) process, with 6-7 layers of silica nanoparticles deposited per coating cycle onto an oppositely charged polyelectrolyte in 0.1 M NH(4)NO(3) at pH 2.7. The resultant porous shells show much greater porosity compared to particles obtained by the traditional layer-by-layer process.     

The role of pore size and structure on the thermal stability of gold nanoparticles within mesoporous silica

Highly dispersed gold particles (<2 nm) were synthesized within the pores of mesoporous silica with pore sizes ranging from 2.2 to 6.5 nm and different pore structures (2D-hexagonal, 3D-hexagonal, and cubic). The catalysts were reduced in flowing H2 at 200 degrees C and then used for CO oxidation at temperatures ranging from 25 to 400 degrees C. The objective of this study was to investigate the role of pore size and structure in controlling the thermal sintering of Au nanoparticles. Our study shows that sintering of Au particles is dependent on pore size, pore wall thickness (strength of pores), and pore connectivity. A combination of high-resolution TEM/STEM and SEM was used to measure the particle size distribution and to determine whether the Au particles were located within the pores or had migrated to the external silica surface.     

Capillary electrochromatography with monolithic silica column: I. Preparation of silica monoliths having surface-bound octadecyl moieties and their chromatographic characterization and applications to the separation of neutral and charged species

Monolithic silica columns with surface-bound octadecyl (C18) moieties have been prepared by a sol-gel process in 100 microm ID fused-silica capillaries for reversed-phase capillary electrochromatography of neutral and charged species. The reaction conditions for the preparation of the C18-silica monoliths were optimized for maximum surface coverage with octadecyl moieties in order to maximize retention and selectivity toward neutral and charged solutes with a sufficiently strong electroosmotic flow (> 2 mm/s) to yield rapid analysis time. Furthermore, the effect of the pore-tailoring process on the silica monoliths was performed over a wide range of treatment time with 0.010 M ammonium hydroxide solution in order to determine the optimum time and conditions that yield mesopores of narrow pore size distribution that result in high separation efficiency. Under optimum column fabrication conditions and optimum mobile phase composition and flow velocity, the average separation efficiency reached 160 000 plates/m, a value comparable to that obtained on columns packed with 3 microm C18-silica particles with the advantages of high permeability and virtually no bubble formation. The optimized monolithic C18-silica columns were evaluated for their retention properties toward neutral and charged analytes over a wide range of mobile phase compositions. A series of dimensionless retention parameters were evaluated and correlated to solute polarity and electromigration property. A dimensionless mobility modulus was introduced to describe charged solute migration and interaction behavior with the monolithic C18-silica in a counterflow regime during capillary electrochromatography (CEC )separations. The mobility moduli correlated well with the solute hydrophobic character and its charge-to-mass ratio.     

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.     

Characteristics of superficially-porous silica particles for fast HPLC: some performance comparisons with sub-2-microm particles

Columns of 2.7-microm fused-core (superficially porous) Type B silica particles allow very fast separations of small molecules at pressures available in most high-performance liquid chromatography instruments. These highly-purified particles with 1.7-microm solid silica cores and 0.5-microm-thick shells of 9 nm pores exhibit efficiencies that rival those of totally porous sub-2-microm particles but at one-half to one-third of the column back pressure. This presentation describes other operating features of fused-core particle columns, including sample loading characteristics and packed bed stability. The superior mass transfer (kinetic) properties of the fused-core particles result in much-improved separation efficiency at higher mobile phase velocities, especially for > 600 molecular weight solutes.     

Nonequilibrium electrokinetic effects in beds of ion-permselective particles

Electrokinetic transport of fluorescent tracer molecules in a bed of porous glass beads was investigated by confocal laser scanning microscopy. Refractive index matching between beads and the saturating fluid enabled a quantitative analysis of intraparticle and extraparticle fluid-side concentration profiles. Kinetic data were acquired for the uptake and release of electroneutral and counterionic tracer under devised conditions with respect to constant pressure-driven flow through the device and the effect of superimposed electrical fields. Transport of neutral tracer is controlled by intraparticle mass transfer resistance which can be strongly reduced by electroosmotic flow, while steady-state distributions and bead-averaged concentrations are unaffected by the externally applied fields. Electrolytes of low ionic strength caused the transport through the charged (mesoporous) beads to become highly ion-permselective, and concentration polarization is induced in the bulk solution due to the superimposed fields. The depleted concentration polarization zone comprises extraparticle fluid-side mass transfer resistance. Ionic concentrations in this diffusion boundary layer decrease at increasing field strength, and the flux densities approach an upper limit. Meanwhile, intraparticle transport of counterions by electromigration and electroosmosis continues to increase and finally exceeds the transport from bulk solution into the beads. A nonequilibrium electrical double layer is induced which consists of mobile and immobile space charge regions in the extraparticle bulk solution and inside a bead, respectively. These electrical field-induced space charges form the basis for nonequilibrium electrokinetic phenomena. Caused by the underlying transport discrimination (intraparticle electrokinetic vs extraparticle boundary-layer mass transfer), the dynamic adsorption capacity for counterions can be drastically reduced. Further, the extraparticle mobile space charge region leads to nonlinear electroosmosis. Flow patterns can become highly chaotic, and electrokinetic instability mixing is shown to increase lateral dispersion. Under these conditions, the overall axial dispersion of counterionic tracer can be reduced by more than 2 orders of magnitude, as demonstrated by pulse injections.     

AFM-porosimetry: density and pore volume measurements of particulate materials

We introduced the novel technique of AFM-porosimetry and applied it to measure the total pore volume of porous particles with a spherical geometry. The methodology is based on using an atomic force microscope as a balance to measure masses of individual particles. Several particles within the same batch were measured, and by plotting particle mass versus particle volume, the bulk density of the sample can be extracted from the slope of the linear fit. The pore volume is then calculated from the densities of the bulk and matrix materials, respectively. In contrast to nitrogen sorption and mercury porosimetry, this method is capable of measuring the total pore volume regardless of pore size distribution and pore connectivity. In this study, three porous samples were investigated by AFM-porosimetry: one ordered mesoporous sample and two disordered foam structures. All samples were based on a matrix of amorphous silica templated by a block copolymer, Pluronic F127, swollen to various degrees with poly(propylene glycol). In addition, the density of silica spheres without a template was measured by two independent techniques: AFM and the Archimedes principle.     

Determination of the interparticle void volume in packed beds via intraparticle Donnan exclusion

Interparticle void volumes and porosities of packed capillaries have been determined using intraparticle Donnan exclusion of a small, unretained, co-ionic tracer (nitrate ions). The operational domain of this approach has been characterized for bare silica, reversed-phase, and strong cation-exchange materials (with different particle sizes and intraparticle pore sizes) in dependence of the mobile phase ionic strength. Interparticle porosities agree well with those analyzed by inverse size-exclusion chromatography (ISEC). Limitations to the use of Donnan exclusion (electrostatic exclusion) and ISEC (mechanical exclusion) arise as either type of exclusion becomes noticeable also in the cusp regions between the particles, or as the intraparticle pores are so large that complete electrostatic and size-exclusion are difficult to realize. Our data confirm that intraparticle Donnan exclusion presents a most simple, fast, and reliable approach for the analysis of packing densities.     

Adsorption in columns packed with porous adsorbent particles having partially fractal structures

A mathematical model is constructed and solved that could describe the dynamic behavior of the adsorption of a solute of interest in single and stratified columns packed with partially fractal porous adsorbent particles. The results show that a stratified column bed whose length is the same as that of a single column bed, provides larger breakthrough times and a higher dynamic utilization of the adsorptive capacity of the particles than those obtained from the single column bed, and the superior performance of the stratified bed becomes especially more important when the superficial velocity of the flowing fluid stream in the column is increased to accommodate increases in the system throughput. This occurs because the stratified column bed provides larger average external and intraparticle mass transfer and adsorption rates per unit length of packed column. It is also shown that increases in the total number of recursions of the fractal and the ratio of the radii between larger and smaller microspheres that make up the partially fractal particles, increase the intraparticle mass transfer and adsorption rates and lead to larger breakthrough times and dynamic utilization of the adsorptive capacity of the particles. The results of this work indicate that highly efficient adsorption separations could be realized through the use of a stratified column comprised from a practically reasonable number of sections packed with partially fractal porous adsorbent particles having reasonably large (i) total number of recursions of the fractal and (ii) ratio of the radii between larger and smaller microspheres from which the partially fractal particles are made from. It is important to mention here that the physical concepts and modeling approaches presented in this work could be, after a few modifications of the model, applied in studying the dynamic behavior of chemical catalysis and biocatalysis in reactor beds packed with partially fractal porous catalyst particles.     

Membrane potential in charged porous membranes

For charged porous membranes, the separation efficiency to charged particles and ions is affected by the electrical properties of the membrane surface. Such properties are most commonly quantified in terms of zeta-potential. In this paper, it is shown that the zeta-potential can be calculated numerically from the membrane potential. The membrane potential expression for charged capillary membranes in contact with electrolyte solutions at different concentrations is established by applying the theory of non-equilibrium thermodynamic to the membrane process and considering the space-charge model. This model uses the Nernst–Planck and Navier–Stokes equations for transport through pores, and the non-linear Poisson–Boltzmann equation, which is numerically solved, for the electrostatic condition of the fluid inside pores. The integral expressions of the phenomenological coefficients coupling the differential flow (solute relative to solvent) and the electrical current with the osmotic pressure and the electrical potential gradients are established and calculated numerically. The mobilities of anions and cations are individually specified. The variations of the membrane potential (or the apparent transport number of ions in the membrane pores) are studied as a function of different parameters: zeta-potential, pore radius, mean concentration in the membrane, ratio of external concentrations and type of ions.