Thermodynamics of ions and water transport in porous media

The thermodynamic framework of Prigogine, de Groot, and Mazur is extended to study the transport of ions and water in thermoporoelastic materials assuming infinitesimal deformations. New expressions are developed for the first and second principles of nonequilibrium thermodynamics of multicomponent systems and a generalized power balance equation is derived. For porous materials, all the components cannot be treated on a symmetric basis. A Lagrangian framework associated with deformation of the solid phase is introduced and, in this framework, Curie's principle is used to set up the form of the linear constitutive equations describing the transport of ions, water, and heat through the pore network. The material properties entering these equations were recently obtained by Revil and Linde [J. Colloid Interface Science 302 (2006) 682-694] using a volume-averaging approach based in the Nernst-Planck and Stokes equations. This provides a way to relate the material properties entering the constitutive equations to two textural parameters characterizing the topology of the pore space of the material (namely the tortuosity of the pore space and the permeability). The generalized power balance equation is used to derive the linear poroelastic constitutive equations (including the osmotic pressure) to describe the reversible contribution of deformation of the medium in response to ions and water transport through the connected porosity.     

Introducing interacting diffuse layers in TLM calculations: a reappraisal of the influence of the pore size on the swelling pressure and the osmotic efficiency of compacted bentonites

The truncation of the Gouy-Chapman diffuse part in compacted clay-rocks and bentonite is introduced into the electrical triple-layer model (TLM) recently developed by P. Leroy and A. Revil [J. Colloid Interface Sci. 270 (2004) 371]. The new model is used to explain the dependence of the osmotic efficiency and the swelling pressure as functions of the mean pore size of the medium, determined from the porosity and the specific surface. The truncation of the diffuse layer introduces a new variable in the system of equations to be solved, the electrical potential at the midplane between adjacent charged surfaces. This new variable is evaluated through a Taylor expansion of the electrical potential. The present model is able to capture the variation of the osmotic efficiency and the swelling pressure with the mean pore size. The partition of counterions between the Stern layer and the diffuse layer as a function of the pore size calculated by the TLM also shows a good consistency with the model. This implies that more than 90% of the counterions are located in the Stern layer.     

Diffusion of ions in unsaturated porous materials

In a salinity gradient, the diffusion of ions through the connected porosity of a porous and charged material is influenced by the charged nature of the interface between the pore water and the solid. This influence is exerted through the generation of a macroscopic electrical field termed the diffusion or membrane potential. This electrical field depends on the excess of counterions located in the pore space counterbalancing the charge density of the surface of the solid. In unsaturated porous materials, we have to consider (1) the effect of the charged nature of the air/water interface, (2) the increase of the counterion density as the counterions are packed in a smaller volume when the saturation of the nonwetting phase (air) increases, and (3) the influence of the water saturation upon the tortuosity of the water phase. The volume average of the Nernst-Planck equation is used to determine the constitutive equations for the coupled diffusion flux and current density of a multicomponent electrolyte in unsaturated conditions. We assume that water is the wetting phase for the solid phase. We neglect the electro-osmotic flow in the coupled constitutive equations and the deformation of the medium (the medium is assumed to be both isotropic and rigid). This model explains well the observed tendency of strong decreases of the apparent diffusion coefficient of ions with the decrease of the saturation of the water phase under steady-state conditions. This decrease is mainly due to the influence of the saturation upon the tortuosity of the water phase.     

Diffusion of ionic species in bentonite

We present a macroscopic model of ionic diffusion in bentonites including the effect of the hydraulic-electrical-chemical couplings expected in such charged porous medium. The anomalous concentrations of the ions in the pore water of the bentonite are modeled with a modified Donnan model in which we account for the partition of the counterions between the Stern and Gouy-Chapman layers. This is accomplished using an electric triple layer (TLM) model combined with an explicit complexation model at the mineral/water interface. The porosity entering into the determination of the formation factor of the medium is an effective porosity obtained by removing the fraction of hydration water covering the surface of the clay minerals. We investigate two different cases of diffusion. In the first case, we consider a salinity gradient between two reservoirs in contact with a cylindrical sample of bentonite. The model predicts an increase of the diffusivity of the salt with the salinity of the solution in contact with the bentonite in agreement with experimental data. In the second case, we analyze a self-diffusion experiment of an ionic tracer. The model predicts an increase of the diffusivity of anions with the effective porosity and with the ionic strength. This is also in good agreement with experimental data.     

Charge/discharge of an electrochemical supercapacitor electrode pore; non-uniqueness of mathematical models

A thermodynamic analysis has been done to enhance understanding of the relation between various mathematical models for electrochemical supercapacitor pores. For the same capacitive charge/discharge experiment a variety of one-dimensional mathematical model equations concerning the transport of ions and double layer charge/discharge along the pore are shown to be indistinguishable. Some of those indistinguishable equations could be interpreted as derived from diffusional mechanisms while others appear as derived from migrational mechanisms. Ohmic resistivities and diffusivities obtained in such case are not contradicting results but characterize identical physical processes. The results are valid as long as the assumptions of irreversible thermodynamics of local equilibrium along the pore and of linearization of the flux equations hold.     

Effect of ion solvation on the diffuse layer structure in mixed electrolytes

A “solvionic” model of a multicomponent electrochemical system (mixed electrolyte) is considered. An ion in the solution is considered as a point charge rigidly fixed inside its solvation shell. The corresponding equations for the diffuse layer on an ideally polarizable electrode are derived, and an effective method of their numerical solution is formulated. The calculations are performed in order to follow the changes in the diffuse layer structure with variations in the electrode charge and electrolyte composition. Far from the zerocharge potential of solution, the dependences of distributions of solution components over the diffuse layer on the electrode charge radically differ from those within the classic Gouy-Chapman theory. Analytical equations (asymptotics at large electrode charges) for concentrations of solvated ions in the plane of their maximum approach and for their “surface excesses” (diffuse adsorption) are determined. Results of numerical calculations for a 0.2 M LiCl + 0.05 M BaCl₂ solution are plotted.     

Application of the hypernetted chain approximation to the electrical double layer for 2:1 and 1:2 electrolytes

The equations needed to estimate the potential drop across the diffuse layer according to the hypernetted chain approximation (HNCA) are derived in this paper for 2:1 and 1:2 electrolytes at the restricted primitive level. It is shown that HNCA results can be expressed in the same format as the corresponding Gouy-Chapman equations with inclusion of two modifying functions. One function depends on the fraction of the solution volume occupied by the ions, and the other depends on the reciprocal thickness of the ionic atmosphere surrounding each ion. In addition, an expression for the potential profile in the diffuse layer for 2:1 and 1:2 electrolyte solutions is derived according to Gouy-Chapman theory. The modifying functions in the HNCA are then estimated using the Henderson-Blum approach for solutions containing ions with diameters of 300 and 400 pm for concentrations in the range from 0.1 to 2 M. It is shown that the Henderson-Blum approach is inadequate for systems with multivalent ions except for charge densities very close to the point of zero charge.     

A statistical model for the heterogeneous structure of porous catalyst pellets

The complex structures of the void space of porous media are often characterised by parameters such as pore network connectivity and lattice size. This paper presents a comparison of the estimates of these parameters obtained from two previous methods based on nitrogen sorption and mercury porosimetry, and also from a new, completely independent approach based on pulsed-gradient spin-echo nuclear magnetic resonance (PGSE NMR). It was found that the new PGSE NMR technique obtains estimates of connectivity and lattice size in agreement with nitrogen sorption but different to mercury porosimetry. This difference was attributed to the various physical processes involved actually probing different aspects of the pore space geometry. It was further suggested that the representation of the pore structure derived from either nitrogen sorption or PGSE NMR is really a mapping of the real pore space onto an equivalent abstract, random pore bond network. However, it has been shown that this mapping does capture some of the characteristic properties of the pore space that control transport over mesoscopic ( < 10 microm) length scales. For materials which additionally possessed macroscopic (> 10 microm) structural heterogeneity, it was found that the model could also be adapted to predict the macroscopic transport properties of the porous medium.     

The conditions governing the specific adsorption of counter-ions in the interaction of two parallel plate-like colloidal particles

Frens and Overbeek have proposed that during the Brownian collision of two colloidal particles in a hydrophobic sol, the surface charge density due to potential-determining (p.d.) ions remains virtually unchanged. It is argued here that the cause of this behaviour is the low concentration of p.d. ions in the diffuse layer. However, equilibrium can be maintained with respect to counter-ions adsorbed into the Stern region from the supporting electrolyte, because the concentration of such electrolyte in the dispersion medium is considerably greater than that of p.d. ions.A general expression is quoted from earlier work for the electric double layer interaction between two parallel plate-like particles in the case where surface charge due to p.d. ions is fixed, but where counter-ions adsorbed into the Stern region can equilibrate with ions of the same species in the diffuse layer. Incorporating discreteness-of-charge and ion-size effects into the adsorption isotherm of the counter-ions, the double layer interaction energy of the two plates is calculated at contact of the two outer Helmholtz planes (o.h.p.'s). It is shown that although this energy exceeds the classical expression obtained by assuming the potential at the o.h.p. to be independent of plate separation, it remains finite.     

Electrokinetic Characteristics of Porous Glasses in Solutions of Sodium and Iron(III) Chlorides

The structural (structural resistance coefficient, bulk porosity, average pore radius, and specific surface area) and electrokinetic (surface conductivity and electrokinetic potential) characteristics of high-silica micro- and macroporous glasses produced from two-phase sodium borosilicate glass have been compared in solutions of an indifferent electrolyte (sodium chloride) and iron(III) chloride containing iron ions, which have a high specificity to silica surfaces. It has been shown that, in the presence of iron ions, the electrokinetic behavior of porous glasses is governed by two factors. First, the superequivalent adsorption of these ions in the Stern layer leads to positive values of the electrokinetic potential, and, second, their mobility in the pore space decreases, thereby resulting in the appearance of equilibrium solution concentration ranges, in which the specific conductivity of a pore solution becomes lower than that of a free solution.     

Colloido-chemical characteristics of nanoporous glasses with different compositions in solutions of simple and organic electrolytes. 2. Equilibrium electrochemical characteristics of membranes

The regularities of variations in the electrokinetic potential and surface charge of nanoporous glass membranes with different compositions have been studied as depending on the type of an electrolyte (sodium, potassium, ammonium, tetramethylammonium, and tetraethylammonium chlorides) and the structure of pore space. It has been shown that, in solutions containing specifically sorbed organic counterions, the range of positive values of electrokinetic potential arises due to the superequivalent absorption of counterions in the Stern layer. It has been found that the influence of the specific adsorption of counterions on the electrokinetic potential of porous glasses increases with the amount of secondary silica in the pore space. The effects of the counterion specificity, pore channel sizes, and composition of a porous glass on the value of the surface charge have been analyzed. The absolute value of the surface charge has been shown to significantly increase in the presence of organic counterions in comparison with inorganic ions throughout the examined range of background electrolyte concentrations.     

Numerical analysis of electroosmotic flow in dense regular and random arrays of impermeable, nonconducting spheres

We present a numerical scheme for analyzing steady-state isothermal electroosmotic flow (EOF) in three-dimensional random porous media, involving solution of the coupled Poisson, Nernst-Planck, and Navier-Stokes equations. While traditional finite-difference methods were used to resolve the Poisson-Nernst-Planck problem, the (electro)hydrodynamics has been addressed with high efficiency using the lattice-Boltzmann method. The developed model allows simulation of electrokinetic transport under most general conditions, including arbitrary value and distribution of electrokinetic potential at the solid-liquid interface, electrolyte composition, and pore space morphology. The approach provides quantitative information on a spatial distribution of simulated velocities. This feature was utilized to characterize EOF fields in regular and random, confined and bulk packings of hard (i.e., impermeable, nonconducting) spheres. Important aspects of pore space morphology (sphere size distribution), surface heterogeneity (mismatch in electrokinetic potentials at confining wall and sphere surface), and fluid phase properties (electrical double layer thickness) were investigated with respect to their influence on the EOF dynamics over microscopic and macroscopic spatial domains. Most important is the observation of a generally nonuniform pore-level EOF velocity profile in the sphere packings (even in the thin double layer limit) which is caused by pore space morphology and which is in contrast to the pluglike velocity distribution in a single, straight capillary under the same conditions.     

Ionic Diffusivity, Electrical Conductivity, Membrane and Thermoelectric Potentials in Colloids and Granular Porous Media: A Unified Model

Ionic diffusivity, electrical conductivity, membrane and thermoelectric potentials in isotropic and homogeneous colloidal suspensions, and granular porous media saturated by a binary symmetric 1:1 electrolyte are four interrelated phenomena. The microstructure and the surface properties of the solid grains-water interface influence directly these properties. The ionic diffusivities (and the electrical conductivity, respectively) in colloids and porous media have contributions from diffusion (and electromigration, respectively) through the bulk solution occupying the pores, together with electromigration occurring at the grains-water interface in the electrical double layer. Surface diffusion in porous materials has no contribution from concentration gradients along the grains-water interface. Instead, surface diffusion is envisioned as a purely electromigration process due to the membrane potential. The tortuosities of the transport of anions and cations are equal to the bulk tortuosity of the pore space only at high ionic strength. As the ionic strength decreases, the dominant paths for transport of the ion corresponding to the counterion of the electrical double layer shift from the pore space to the solid grains-water interface. Because anions and cations do not move independently, the membrane potential created by the charge polarization alters the velocity of the anions and influences the mutual diffusivity coefficient of the salt in the porous material. An electric potential of thermal origin is also produced in nonisothermal conditions. The ionic contributions to the electrical conductivity are based on a differential effective medium approach. These ionic contributions to the electrical conductivity are used to derive the ionic diffusivities and the membrane and thermoelectric potentials. The influence of the temperature and the presence, in the pore space, of a second immiscible and nonwetting phase is also considered in this model. Porosity is shown to affect the membrane potential. Several predictions of the model are checked with success by comparing the model to a set of experimental data previously published. Copyright 1999 Academic Press.     

The water-amorphous silica interface: analysis of the Stern layer and surface conduction

To explain why dynamical properties of an aqueous electrolyte near a charged surface seem to be governed by a surface charge less than the actual one, the canonical Stern model supposes an interfacial layer of ions and immobile fluid. However, large ion mobilities within the Stern layer are needed to reconcile the Stern model with surface conduction measurements. Modeling the aqueous electrolyte-amorphous silica interface at typical charge densities, a prototypical double layer system, the flow velocity does not vanish until right at the surface. The Stern model is a good effective model away from the surface, but cannot be taken literally near the surface. Indeed, simulations show no ion mobility where water is immobile, nor is such mobility necessary since the surface conductivity in the simulations is comparable to experimental values.     

Hydrolysis versus ion correlation models in electrokinetic charge inversion: establishing application ranges

In this article, we investigate experimentally a wide range of situations where charge inversion (i.e., overcompensation of the surface charge of a colloidal particle by the countercharge) can occur. To that end, the electrophoretic mobility of sodium montmorillonite, silica, and polystyrene latex as functions of pH and concentration of different salts is presented, and conditions are established where charge inversion occurs. The reason for this study is to provide experimental evidence for distinguishing between two existing models for the explanation of charge inversion. One of these is the specific adsorption of ions located in the Stern layer in combination with a Gouy-Chapman diffuse part of the double layer. The other ion-correlation theories explain the phenomenon in terms of purely physical arguments based on Coulombic pair interactions between ions and surface charges and on excluded volume effects. In distinguishing between these two interpretations, the influence of the pH plays a central role because of its effect on the hydrolysis of multivalent cations. In our experiments, it is found that although 1-2 and 2-2 electrolytes provoke a decrease in the absolute values of the electrophoretic mobilities when their concentration in solution is increased, they never lead to charge inversion, whatever the surface charge or the pH. However, in the case of salts of trivalent cations, electrokinetic charge reversal is often observed above a certain critical electrolyte concentration. In addition, the extent of overcharging increases when the concentration is raised above the critical value. This trend occurs for any system in which the surface charge is pH-independent, as in polystyrene latex and montmorillonite. Most of the results presented here are compatible with the specific adsorption of hydrolyzed metal ions as the main driving force for charge inversion. At low pH, when the hydrolysis of trivalent cations is likely to be absent, overcharging can be attributed to ion correlation effects.     

Modeling liquid porosimetry in modeled and imaged 3-D fibrous microstructures

In this paper, an analysis to distinguish the geometric and porosimetric pore size distributions of a fibrous material is presented. The work is based on simulating the intrusion of nonwetting fluid in a series of 3-D fibrous microstructures obtained from 3-D image reconstruction or virtual geometries mathematically generated according to the properties of the media. We start our study by computing the pore size distribution of two typical hydroentangled nonwoven materials and present a theoretical model for their geometric pore size distributions based on Poisson line network model of the fibrous media. It is shown that the probability density function of the geometric pore size distribution can be approximated by a two-parametric Gamma distribution. We also study connectivity of the pore space in fibrous media by computing and comparing the accessible and allowed pore volumes in the form access function graphs. It is shown that the so-called ink-bottle effect can significantly influence the fluid intrusion in a porous material. The pore space connectivity of a homogeneous fibrous media is observed to be a function of thickness, solid volume fraction (SVF), and fiber diameter. It is shown that increasing the materials' thickness or SVF, while other properties are kept constant, reduces the pore space connectivity. On the other hand, increasing the fiber diameter enhances the connectivity of the pores if all other parameters are fixed. Moreover, modeling layered fibrous microstructures; it is shown that the access function graphs can be used to detect the location of the bottle neck pores in a layered/composite porous material.     

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.     

The rheology of pseudoplastic fluids in porous media using network modeling

This paper considers the rheology of pseudoplastic (shear thinning) fluids in porous media. The central problem studied is the relationship between the viscometric behavior of the polymer solution and its observed behavior in the porous matrix. In the past, a number of macroscopic approaches have been applied, usually based on capillary bundle models of the porous medium. These simplified models have been used along with constitutive equations describing the fluid behavior (usually of power law type) to establish semiempirical macroscopic equations describing the flow of non-Newtonian fluids in porous media. This procedure has been reasonably successful in correlating experimental results on the flow of polymer solutions through both consolidated and unconsolidated porous materials. However, it does not allow an interpretation of polymer flow in porous media in terms of the flows on a microscopic scale; nor does it allow us to predict changes in macroscopic behavior resulting from variations at a microscopic level in the characteristics of the porous medium such as pore size distribution. In this work, we use a network approach to the modeling of non-Newtonian rheology, in order to understand some of the more detailed features of polymjer flow in porous media. This approach provides a mathematical bridge between the behavior of the non-Newtonian fluid in a single capillary and the macroscopic behavior as deduced from the pressure drop-flow rate relation across the whole network model. It demonstrates the importance of flow redistribution within the elements of the capillary network as the overall pressure gradient varies. As an example of a pseudoplastic fluid in a porous medium, we consider the flow of xanthan biopolymer. This polymer is important as a displacing fluid viscosifier in enhanced oil recovery applications and, for that reason, a considerable amount of experimental data has been published on the flow of xanthan solutions in various porous media.