Electrochemistry of capillary systems with narrow pores. II. Electroosmosis

Manegold and Solf have reported systematic deviations of the electroosmotic properties of collodion membranes with narrow pores from predictions based on the Helmholtz–Smoluchowski model. To interpret the electroosmotic data quantitatively it is necessary to replace the assumption of the Helmholtz–Smoluchowski model that the thickness of the electric double layer is small compared with the pore radius by a new assumption. We have assumed that the counter ions are distributed homogeneously in the pore fluid. In Part I of this series of contributions, equations have been given describing the electroosmotic properties of a membrane with narrow pores based on the new assumption. These equations are derived here in detail and are applied to an analysis of the experimental data given by Manegold and Solf.     

Electrochemistry of capillary systems with narrow pores III. Electrical conductivity

The extension of the Teorell–Meyer–Sievers theory of the dialysis potential to a general theory of capillary systems with narrow pores outlined in Part I of this series of publications has been applied to electroosmotic phenomena in Part II. In this Part, the electrical conductivity, including the electrical convection conductivity, will be treated in terms of the new theory. The corresponding equations already referred to in Part I are derived. In addition, results of measurements of the electrical conductivity of collodion membranes with graded porosity and graded electrochemical activity in aqueous KCl solutions of different concentrations are reported. They are used to test the new theory. It will be shown that it is possible to determine the fixed ion concentration A of the membranes by using electrical conductivity data. The theory predicts that the value of A should be identical with the `selectivity constant' of the Meyer–Sievers theory of the dialysis potential. This prediction will be checked in Part IV of this series of contributions.     

Electrochemistry of capillary systems with narrow pores. I. Overview

In capillary systems with narrow pores the Helmholtz electrochemical double layer located at the pore wall extends over the entire cross section of the pores. It loses its character as the “charge on the wall”. It will be shown that not only the electrokinetic phenomena but also the electrical conductivity and the dialysis potential of membranes with narrow pores can be understood from the same point of view, namely: the electrolyte solution in the pores of a membrane with narrow pores is considered to be an approximately homogeneous solution in contact with immobilised charges located at the pore wall. In this case the electrochemical equations contain the fixed ion concentration as a parameter instead of the ζ potential. This makes it possible to describe quantitatively to a good approximation data on the electroosmosis, the electrical conductivity, the streaming potential and the dialysis potential taken from the literature, as well as results of our own measurements, by using a single membrane constant.     

Electrochemistry of capillary systems with narrow pores. VI. Convection conductivity (theoretical considerations)

Generally, the electrical convection current and the electrical convection conductivity (Smoluchowski's surface conductivity) have to be taken into account to describe transport phenomena across membranes with narrow pores although the electrical charge distribution within the pores cannot be described as a Helmholtz electrical double layer. In collodion membranes, which have a comparatively low fixed ion concentration, the contribution of the convection conductivity to the electrical conductivity of the pore fluid may be neglected. This assumption was made tacitly in the analysis of our data obtained with this type of membrane.In this communication equations are derived which take the convection conductivity into account. They are in agreement with the phenomenological transport equations developed by Staverman on the basis of the thermodynamics of irreversible processes.The electrical convection conductivity can be considered to be the contribution of the fixed ion concentration to the electrical conductivity. It is argued that this contribution cannot be neglected in ion exchange membranes with a high fixed ion concentration and a high mechanical permeability. Neglecting the electrical convection conductivity in such systems could lead to considerable differences between experimental conductivity data and the theoretical predictions. An electrical conductivity term for the fixed ions is proposed which can be used as a correction factor in the equations in which the contribution of the electrical convection conductivity has been neglected. Suggestions are made for the measurement of the electrical convection conductivity in systems with narrow pores and high electrical conductivity (e.g. ion exchange resins). The consequences of the electrical convection conductivity in practical applications of ion-exchange resins are discussed (acceleration of the rates of ion exchange; improvement of the separation properties by the application of a direct electrical current flow).     

Electrochemistry of capillary systems with narrow pores. I. Overview

In capillary systems with narrow pores the Helmholtz electrochemical double layer located at the pore wall extends over the entire cross section of the pores. It loses its character as the “charge on the wall”. It will be shown that not only the electrokinetic phenomena but also the electrical conductivity and the dialysis potential of membranes with narrow pores can be understood from the same point of view, namely: the electrolyte solution in the pores of a membrane with narrow pores is considered to be an approximately homogeneous solution in contact with immobilised charges located at the pore wall. In this case the electrochemical equations contain the fixed ion concentration as a parameter instead of the ζ potential. This makes it possible to describe quantitatively to a good approximation data on the electroosmosis, the electrical conductivity, the streaming potential and the dialysis potential taken from the literature, as well as results of our own measurements, by using a single membrane constant.     

Mutual diffusion coefficient in binary mixtures in different aggregation states

An equation for the mutual diffusion coefficient for the components of a binary mixture occurring in various aggregation states has been derived. Situations of this type arise in narrow pores in which the adsorbate density sharply changes across the pore cross-section. The equation was derived with the assumption that molecules of the mixture components have a spherical shape and are similar in size. The constructed equations are based on the lattice-gas model applicable to any aggregation state. The migration of particles is described in terms of the transition state theory. At low mixture densities corresponding to an ideal gas phase, the equations are based on expressions of the rigorous kinetic theory of gases. The theory takes into account the change in the mechanism of particle migration in different phases: from pair collisions for the gas to the overcoming of the activation barrier by thermofluctuation for dense phases. __________ Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 8, pp. 1717–1725, August, 2005.     

Electrochemistry of capillary systems with narrow pores. VI. Convection conductivity (theoretical considerations)

Generally, the electrical convection current and the electrical convection conductivity (Smoluchowski's surface conductivity) have to be taken into account to describe transport phenomena across membranes with narrow pores although the electrical charge distribution within the pores cannot be described as a Helmholtz electrical double layer. In collodion membranes, which have a comparatively low fixed ion concentration, the contribution of the convection conductivity to the electrical conductivity of the pore fluid may be neglected. This assumption was made tacitly in the analysis of our data obtained with this type of membrane.

In this communication equations are derived which take the convection conductivity into account. They are in agreement with the phenomenological transport equations developed by Staverman on the basis of the thermodynamics of irreversible processes.

The electrical convection conductivity can be considered to be the contribution of the fixed ion concentration to the electrical conductivity. It is argued that this contribution cannot be neglected in ion exchange membranes with a high fixed ion concentration and a high mechanical permeability. Neglecting the electrical convection conductivity in such systems could lead to considerable differences between experimental conductivity data and the theoretical predictions. An electrical conductivity term for the fixed ions is proposed which can be used as a correction factor in the equations in which the contribution of the electrical convection conductivity has been neglected. Suggestions are made for the measurement of the electrical convection conductivity in systems with narrow pores and high electrical conductivity (e.g. ion exchange resins). The consequences of the electrical convection conductivity in practical applications of ion-exchange resins are discussed (acceleration of the rates of ion exchange; improvement of the separation properties by the application of a direct electrical current flow).     

Electrochemistry of capillary systems with narrow pores III. Electrical conductivity

The extension of the Teorell–Meyer–Sievers theory of the dialysis potential to a general theory of capillary systems with narrow pores outlined in Part I of this series of publications has been applied to electroosmotic phenomena in Part II. In this Part, the electrical conductivity, including the electrical convection conductivity, will be treated in terms of the new theory. The corresponding equations already referred to in Part I are derived. In addition, results of measurements of the electrical conductivity of collodion membranes with graded porosity and graded electrochemical activity in aqueous KCl solutions of different concentrations are reported. They are used to test the new theory. It will be shown that it is possible to determine the fixed ion concentration A of the membranes by using electrical conductivity data. The theory predicts that the value of A should be identical with the ‘selectivity constant' of the Meyer–Sievers theory of the dialysis potential. This prediction will be checked in Part IV of this series of contributions.     

Electroosmotic flow in a capillary annulus with high zeta potentials

The electroosmotic flow through an annulus is analyzed under the situation when the two cylindrical walls carry high zeta potentials. The analytical solutions for the electric potential profile and the electroosmotic flow field in the annulus are obtained by solving the Poisson-Boltzmann equation and the Stokes equation under an analytical scheme for the hyperbolic sine function. A mathematical expression for the average electroosmotic velocity is derived in a fashion similar to the Smoluchowski equation. Hence, a correction formula is introduced to modify the Smoluchowski equation, taking into account contributions due to the finite thickness of the electric double layer (EDL) and the geometry ratio-dependent correction. Specifically, under a circumstance when the two annular walls are oppositely charged, the flow direction can be determined from the sign of such correction formula, and there exists a zero-velocity plane inside the annulus. With the assumption of large electrokinetic diameters, the location of the zero-velocity plane can be estimated from the analytical expression for the velocity distribution. In addition, the characteristics of the electroosmotic flow through the annulus are discussed under the influences of the EDL parameters and geometric ratio of the inner radius to the outer radius of the annulus.     

Capillary electrochromatography-applicability to the analysis of pesticides and compounds of environmental concern and the theory of electroosmosis

Summary The applicability of capillary electrochromatography to the automated analysis of pesticides and phthalate esters that are of environmental concern was assessed. Reversed phase packing materials were compared. Column to column and run to run reproducibility was established. Peak height with an internal standard gave the best reproducibility. Faster analysis than alternative HPLC methods was demonstrated for a mixture of the insecticide pirimicarb and related pyrimidines. The relationship between the concentration of an analyte in a sample and at the detector was determined by the use of radio-labelled¹⁴C-pirimicarb. The volume fraction of the liquid zone was 0.64. The possibility of electroosmosis through the pores is discussed with reference to the Rice-Whitehead model for electroosmotic flow in a capillary. A new parameter, the effective pore size is used in equations for electroosmosis through porous packings.     

Colloid chemical characteristics of nanoporous glasses with different compositions in solutions of simple and organic electrolytes. 3. Calculation of electrochemical characteristics of membranes in terms of a homogeneous model

The electrosurface characteristics of nanoporous glass membranes–ion concentrations in pores with taking into account the specificity of counterions, electrokinetically mobile charge, the convective component of pore solution electrical conductivity, electroosmotic mobility of a liquid in the field of streaming potential and ion mobilities in pore space–were calculated within the homogeneous model. The effects of the type of counterion (sodium, potassium, ammonium, tetramethylammonium, and tetraethylammonium ions), solution concentration, glass composition, and pore size on the equilibrium and transport characteristics of membrane systems have been analyzed. A method for the determining of electrolyte activity coefficients in the membranes has been proposed.     

Effects of the juxtaposition of carbonaceous slit pores on the overall transport behavior of adsorbed fluids

It is a common approximation in the modeling of adsorption in microporous carbons to treat the pores as slit pores, whose walls are considered to consist of an infinite number of graphitic layers. In practice, such an approximation is appropriate as long as the number of graphitic layers in the wall is greater than three. However, it is understood that pore walls in microporous carbons commonly consist of three or fewer layers. As well as affecting the solid--fluid interaction within a pore, such narrow walls permit the interaction of fluid molecules through the wall, with consequences for the adsorption characteristics. We consider the effect that a distributed pore-wall thickness model can have on transport properties. At low density we find that the only significant deviation in the transport properties from the infinite pore-wall thickness model occurs in pores with single-layer walls. For a model of activated carbons with a distribution of pore widths and pore-wall thicknesses, the transport properties are generally insensitive to the effects of finite walls, in terms of both the solid-fluid interaction within a pore and fluid-fluid interaction through the pore walls.     

Making equation of state models predictive: Part 2: An improved PCP-SAFT equation of state

We use the PCP-SAFT equation of state, which is of the Van der Waals type and has a sound physical basis, to predict mixture properties, such as vapor–liquid and liquid–liquid equilibria, as well as excess enthalpies. We use molecular properties, such as dipole moment, quadrupole moment, polarizability and dispersion interaction coefficients, that have been determined quantum mechanically in Part I of this publication and adjust the remaining three pure compound parameters to pure compound data. We finally present a new combination rule for the dispersion energy parameter ? that is based on the quantum mechanically determined data. The predictions based on quantum mechanically determined pure compound properties along with the new combination rule show an improved performance compared to the original PCP-SAFT combination rule.     

Calculation of the adsorption-desorption hysteresis curves in narrow-pore systems

The gas and liquid spinodal branches for an adsorbate located in narrow slit-shaped, cylindrical, and spherocylindrical pores were calculated. The adsorbate is modeled by Lennard-Jones spherical particles. The calculation was based on the lattice gas model taking into account the intermolecular interactions of nearest neighbors in the quasichemical approximation. The density-temperature diagrams for the gas and liquid spinodal branches in the pores are similar to the equilibrium vapor-liquid phase diagrams: they have a common critical point; the dense-phase branches are shifted to lower pore fillings, while the rarefied-phase branches are shifted toward higher pore fillings. The width of adsorption-desorption hysteresis loop in the adsorption isotherms for Lennard-Jones particles was analyzed as a function of the pore size and the interaction potential of the adsorbate with the pore walls. The effect of pore wall roughness and the accuracy of isotherm calculation on the width of the adsorption-desorption hysteresis loop in cylindrical pores is discussed Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 5, pp. 813–823, May, 2007.     

Some considerations concerning the technical use of narrow pores

Carrier transport and diffusion through narrow pores (single-file diffusion) are basic transport mechanisms that have been proposed to explain the specific passive permeation properties of biological membranes. Unlike carrier diffusion, the concept of single-file diffusion has been scarcely considered in connection with technical purposes. Taking into account, however, that extensive studies have been made in recent years on artificial lipid membranes with single-file pores and that cylindrical pores with a very narrow diameter distribution can be produced in thin plates by etching nuclear tracks, it seems useful to discuss more carefully the specific properties and conceivable applications of the single-file pore. As properties of obvious technical interest one finds high permselectivity and high transport rate. In addition, the voltage-dependent block of narrow pores provides an intriguing possibility of ionic flux control. A therapeutic system functioning as an artificial suprarenal gland is briefly outlined as an example.     

Modelling of the pore flow in capillary electrochromatography

Pore flow in capillary electrochromatography (CEC) on porous silica particles has been investigated. To that end the migration behaviour of narrow polystyrene (PS) standards dissolved in di-methylformamide (DMF) with lithium chloride in 1 and 10 mmol/l concentration has been measured. These data have been compared to theoretical predictions. The latter were based on a model comprising cylindrical pores of varying diameter as measured experimentally by porosimetry, while the flow in each set of pores was calculated with the expression given by Rice and Whitehead. A reasonable to good agreement between experimental and predicted data was observed, provided it was assumed that pores of differing diameter occur in series. It was found that the flow in pores with a nominal size of 100 A can be considerable compared to the interstitial flow, especially at 10 mmol/l ionic strength. It is concluded that pore flow within porous particles in CEC, of great importance for improved efficiency in both interactive and exclusion type CEC, can be predicted fairly reliably by means of the Rice and Whitehead expression.     

Estimation of diffusion parameters in functionalized silicas with modulated porosity. Part II: pore network modeling

In this work, the pore structure of those five (5) silicas SiO2-X (see Part I) which have suffered gradual functionalization with functional groups X of increasing length (X = psi, [triple bond]Si-H, [triple bond]Si-CH2OH, [triple bond]Si-(CH2)3OH, [triple bond]Si-(CH2)11CH3), is modeled as a three-dimensional cubic network of cylindrical pores. Those hybrids organic-inorganic SiO2-X samples are characterized by different extent of pore blocking effects. The pores of samples are represented in a 9 x 9 x 9 lattice by the nodes as well the bonds that are interconnected in a so-called dual site-bond model, DSBM, network. The pore network is developed using a Monte Carlo statistical method where the cylindrical pores (nodes and bonds) are randomly assigned into the lattice, until matching of the theoretical results to the experimental data of N2 adsorption-desorption measurements. Thus, a visual picture of the porous solid is possible. This realistic network is used next in order to study the steady-state gas transport (Knudsen gas-phase and viscous diffusion) properties for the examined materials and how flow processes depend on the morphology of the pore structure. The pore diffusivity Dp and total permeability P of each porous medium is determined based on theoretical calculations and the structural statistical parameters, such as porosity epsilonp, tortuosity factor tau and connectivity c of pores is discussed with the corresponding experimental data described in Part I of this work. The results indicate clearly that the diffusion model made it possible to predict pore effective diffusivity in these porous media in very good agreement with the corresponding experimental results for all the examined solids (Part I). The pore diffusivity increases significantly as the value of the pore connectivity increases but the transport properties of the network are influenced strongly at lowest connectivity. Also the predicted tortuosity factor is related inversely to the extent of interconnection of pores in these solids, which indicates that the influence of pore branching to the tortuosity factor of the pore network decreases, as connectivity increases.