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.     

Chemico-electromechanical coupling in microporous media

We determine the macroscopic transport properties of isotropic microporous media by volume-averaging the local Nernst-Planck and Navier-Stokes equations in nonisothermal conditions. In such media, the excess of charge that counterbalances the charge deficiency of the surface of the minerals is partitioned between the Gouy-Chapman layer and the Stern layer. The Stern layer of sorbed counterions is attached to the solid phase, while the Gouy-Chapman diffuse layer is assumed to have a thickness comparable to the size of the pores. Rather than using Poisson-Boltzmann distributions to describe the ionic concentrations in the pore space of the medium, we rely on Donnan distributions obtained by equating the chemical potentials of the water molecules and ions between a reservoir of ions and the pore space of the medium. The macroscopic Maxwell equations and the macroscopic linear constitutive transport equations are derived in the vicinity of equilibrium, assuming that the porous material is deformable. In the vicinity of thermodynamic equilibrium, the cross-coupling phenomena of the macroscopic constitutive equations of transport follow Onsager reciprocity. In addition, all the material properties entering the constitutive equations depend only on two textural properties, the permeability and the electrical formation factor.     

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.     

The pressure- and temperature-induced wetting transitions in the binary water + ethylene glycol monoisobutyl ether mixture

A homemade pendent drop/bubble tensiometer is applied to perform the surface/interfacial tension measurements for the binary water + ethylene glycol monoisobutyl ether (iso-C4E1) mixture over the temperature range from 25 to 150 degrees C and over the pressure range up to 100 bar. The symbol C(i)E(j) is the abbreviation of a nonionic polyoxyethylene alcohol C(i)H(2i+1)(OCH2CH2)(j)OH. The wetting behavior of the iso-C4E1-rich phase at the surface of the aqueous phase is systematically examined according to the wetting coefficient determined from the experimental results of surface/interfacial tensions. It is found that the iso-C4E1-rich phase exhibits a sequence of wetting transitions, nonwetting --> partial wetting --> complete wetting, at the water surface in the water + iso-C4E1 system along with increasing temperature. On the other hand, the iso-C4E1-rich phase undergoes a wetting transition from partial wetting to nonwetting at the surface of the aqueous phase by increasing the system pressure at a fixed temperature near the lower critical solution temperature (LCST) of the closed-loop miscibility gap in the water + iso-C4E1 system.     

Flow of Multicomponent Electrolyte Solutions through Narrow Pores of Nanofiltration Membranes

The slow flow of a multicomponent electrolyte solution in a narrow pore of a nanofiltration membrane is considered. The well-known semiempirical method of subdivision of electrical potential into quasi-equilibrium and streaming parts and the definition of streaming concentrations and pressure are discussed. The usefulness of this tool for solving the electrohydrodynamic equations is shown and justified: the use of a small parameter enables a system of electrohydrodynamic partial differential equations to be reduced to a system of ordinary differential equations for streaming functions. Boundary conditions for streaming functions at both the capillary inlet and outlet are derived. The proposed model is developed for the flow of a multicomponent electrolyte solution with an arbitrary number of ions. This is coupled with (i) the introduction of specific interactions between all ions and the pore wall and (ii) the inclusion of the dissociation of water in both conservation and transport equations. Effective distribution coefficients of ions are introduced that are functions of both the specific interaction potentials and the surface potential of the nanofiltration membrane material. The axial dependency of surface potential is expressed by the use of a charge regulation model from which the discontinuity in electric potential and ion pore concentrations at the pore inlet and outlet can be described.A relation between the frequently used capillary and homogeneous models of nanofiltration membranes is developed. An example of application of the homogeneous model for interpretation of experimental data on nanofiltration separation of electrolyte solutions is presented, which shows a reasonable predictive ability for the homogeneous model.     

Wetting behavior of mixtures of water and nonionic polyoxyethylene alcohol

Five binary water + C4Ej mixtures, water + n-C4E0, water + 2-C4E0, water + iso-C4E0, water + n-C4E1, and water + iso-C4E1, were chosen to perform the surface/interfacial tension measurements over the experimental temperature range from 10 to 85 degrees C at the normal pressure by using a homemade pendent drop/bubble tensiometer. The symbol CiEj is the abbreviation of a nonionic polyoxyethylene alcohol CiH(2i+1)(OCH2CH2)jOH. The wetting behavior of the CiEj-rich phase at the interface separating gas and the aqueous phase is systematically examined according to the wetting coefficient resulting from the experimental data of surface/interfacial tensions measurements. For those systems with a lower critical solution temperature, for example, water + n-C6E2, water + n-C4E1, and water + iso-C4E1, a wetting transition from partial wetting to nonwetting is always observed when the system is brought to close to its lower critical solution temperature. On the other hand, to start with a partial wetting CiEj-rich phase, a wetting transition from partial wetting to complete wetting is always observed when the system is driven to approach its upper critical solution temperature. The effect of hydrophobicity of CiEj on the wetting behavior of the CiEj-rich phase at the interface separating gas and the aqueous phase was carefully investigated by using five sets of mixtures: (1) water + n-C4E0, water + n-C5E0, and water + n-C6E0; (2) water + 2-C4E0 and water + 2-C5E0; (3) water + 2-C4E0 and water + n-C4E0; (4) water + n-C4E1, water + n-C5E1, and water + n-C6E1; (5) water + n-C4E0 and water + n-C4E1. The CiEj-rich phase would tend to drive away from complete wetting (or nonwetting) to partial wetting with an increase in the hydrophobicity of CiEj in the binary water + CiEj system. All the wetting behavior observed in the water + CiEj mixtures is consistent with the prediction of the critical point wetting theory of Cahn.     

Streaming potential generated by a long viscous drop in a capillary

The streaming potential generated by motion of a long drop of viscosity mu(d) = lambdamu in a uniform circular capillary filled with fluid of viscosity mu is investigated by means of a model previously used to study electrophoresis of a charged mercury drop in water. The capillary wall is at potential zeta c relative to the bulk fluid within it, and the surface of the drop is at potential zeta(d). Potentials are assumed to be sufficiently small so that the charge cloud is described by the linearized Poisson-Boltzmann equation, and the Debye length characterizing the thickness of the charge cloud is assumed to be thin compared with the gap h(0) between the drop and the capillary wall. Ions in the external fluid are not allowed to discharge at the surface of the drop, and the wall of the capillary has a nonzero surface conductivity sigma c. The drop is assumed to be sufficiently long so that end effects can be neglected. Recirculation of fluid within the drop gives rise to an enhanced streaming current when zeta(d) is nonzero, leading to an anomalously high streaming potential. This vanishes as the drop viscosity becomes large. If V is the velocity of the drop and gamma is the coefficient of interfacial tension between the two fluids, then the capillary number is Ca = mu V/gamma, and the gap varies as h(0)planck'sCa(2/3). When Ca is small, the gap h(0) is small and electrical conduction along the narrow gap is dominated by the surface conductivity sigma(c) of the capillary wall, which is constant. The electrical current convected by flowing fluid is proportional to Ca, as is the change in streaming potential caused by the presence of the drop. If sigma(c) = 0, then the electrical conductance of the gap depends on its width h(0) and on the bulk fluid conductivity sigma and becomes small as h(0) approximately equal to Ca(2/3) --> 0. The streaming potential required to cancel the O(Ca) convection current therefore varies as Ca(1/3). If sigma(c) = 0 and the drop is rigid (lambda --> infinity), then the change in streaming potential over and above that expected due to the change in pressure gradient is proportional to the difference in potentials zeta(c)-zeta(d).     

A microfluidic approach for measuring capillary pressure in PEMFC gas diffusion layers

A dearth of experimental capillary pressure data limits our understanding and optimization of liquid water transport in PEMFC gas diffusion layers (GDLs). A microfluidic device and method is described for measuring the capillary pressure as a function of liquid water saturation for these thin porous materials with complex, heterogeneous wetting properties. A sample sandwich (hydrophilic membrane–GDL–hydrophobic membrane) is key for probing the entire hydrophilic and hydrophobic pore volume of the GDL during sequential liquid intrusion and gas intrusion experiments. The capillary pressure curves for an as-purchased Toray 090 and two differentially-processed Avcarb P75T GDLs were evaluated; each material displayed highly repeatable, but quantitatively different, room temperature capillary pressure curves that matched qualitative differences in their macroscopic wettability. The measurements show that hysteresis between the liquid intrusion and gas intrusion curves is significant. For example, both the Toray and fully wet-proofed Avcarb GDLs appear hydrophobic during most of the liquid intrusion curve and hydrophilic during most of the gas intrusion curve. The implications of this work for water management, and future device designs and experiments are described.     

Experimental study of the immiscible displacement of shear-thinning fluids in pore networks

The pore scale mechanisms and network scale transient pattern of the immiscible displacement of a shear-thinning nonwetting oil phase (NWP) by a Newtonian wetting aqueous phase (WP) are investigated. Visualization imbibition experiments are performed on transparent glass-etched pore networks at a constant unfavorable viscosity ratio and varying values of the capillary number (Ca), and equilibrium contact angle (theta(e)). Dispersions of ozokerite in paraffin oil are used as the shear-thinning NWP, and aqueous solutions of PEG colored with methylene blue are used as the Newtonian WP. At high Ca values, the tip splitting and lateral spreading of WP viscous fingers are suppressed; at intermediate Ca values, the primary viscous fingers expand laterally with the growth of smaller capillary fingers; at low Ca values, network spanning clusters of capillary fingers separated by hydraulically conductive noninvaded zones of NWP arise. The spatial distribution of the mobility of shear-thinning NWP over the pore network is very broad. Pore network regions of low NWP mobility are invaded through a precursor advancement/swelling mechanism even at relatively high Ca and theta(e) values; this mechanism leads to irregular interfacial configurations and retention of a substantial amount of NWP along pore walls; it becomes the dominant mechanism in displacements performed at low Ca and theta(e) values. The residual NWP saturation increases and the end WP relative permeability decreases as Ca increases and both become more sensitive to this parameter as the shear-thinning behavior strengthens. The shear-thinning NWP is primarily entrapped in individual pores of the network rather than in clusters of pores bypassed by the WP. At relatively high flow rates, the amplitude of the variations of pressure drop, caused by fluid redistribution in the pore network, increase with shear-thinning strengthening, whereas at low flow rates, the motion of stable and unstable menisci in pores is reflected in strong pressure drop fluctuations.     

Geometry-driven wetting transition

Wetting states are quantitatively described by the number of inflection points on the liquid-vapor interface and by the macroscopic contact angle. The number of inflection points required for complete, partial, and pseudopartial wetting is determined for geometries with positive, zero, and negative capillary pressures. The wetting state of a material system is not always independent of the magnitude of the capillary pressure; for example, the wetting state of a fluid inside a capillary tube may depend on the capillary radius. In particular, a fluid that pseudopartially wets the inside of a tube exhibits a transition to partial wetting (or complete wetting) as the capillary radius is decreased.     

Observing capillarity in hydrophobic silica nanotubes

The development of template-synthesized silica nanotubes has created a unique opportunity for studying confined fluids by providing nanometer-scale containers in which the inner diameter (i.d.) and surface chemistry can be systematically and independently varied. An interesting question to be answered is the following: do solvents wet nanometer-scale tubes in the same way they wet ordinary capillaries? To answer this question, we have conducted studies to explore the wettability of the hydrophobic interiors of individual nanotubes. In these studies, single nanotubes with i.d.'s of either 30 or 170 nm were investigated over a range of water/methanol mixtures. These studies provide a direct route for comparing wetting phenomena in nanotubes with conventional macroscopic theories of capillarity. Our observations reveal four important aspects of capillary wetting in the 30-170 nm regime, a size range where the application of the Young-Laplace theory has not been experimentally investigated for hydrophobic pores. They are (i) a sharp transition between wetting and nonwetting conditions induced by addition of a cosolvent, (ii) invariance of this transition between nanotubes of 30 and 170 nm pore diameter, (iii) failure of the Young-Laplace equation to accurately predict the cosolvent's (methanol) mol fraction where the transition occurs, and (iv) reversibility of the observed wetting. The first two aspects conform to conventional capillarity (Young-Laplace), but the latter two do not. These measurements were complemented with ensemble experiments. The difference between theory and experiment is likely due to reliance on macroscopic values of contact angles or to liquid-phase instability within the hydrophobic pore.     

Prediction of imbibition in unconsolidated granular materials

A new way of modeling imbibition is proposed in this paper. It combines two elements. One is a physically consistent, dynamic criterion for the imbibition of an individual pore originally suggested by Melrose (SPEJ (November 1965) 259-271). The other is the use of a simple but physically representative model of porous media: a dense random packing of spheres that is geometrically predetermined. This approach allows truly a priori predictions of imbibition curves (saturation vs capillary pressure) for different values of contact angle, different initial conditions (e.g., different drainage endpoints), and different macroscopic sample geometries (the ratio of external to internal pores). It also provides a mechanistic basis for understanding the influence of pore-scale phenomena such as "snap-off" of nonwetting phase in the pore throats due to the coalescence of pendular rings. The simulations show that the capillary pressure curve for this unconsolidated packing is very sensitive to the wettability parameters (such as contact angle), whereas the influence of different initial conditions and snap-off is almost negligible. Predicted capillary pressure curves are compared to experimental data presented in the literature, and are consistent with them.     

Liquid nanodroplets spreading on chemically patterned surfaces

Controlling the spatial distribution of liquid droplets on surfaces via surface energy patterning can be used to deliver material to specified regions via selective liquid/solid wetting. Although studies of the equilibrium shape of liquid droplets on heterogeneous substrates exist, much less is known about the corresponding wetting kinetics. Here we present large-scale atomistic simulations of liquid nanodroplets spreading on chemically patterned surfaces. Results are presented for lines of polymer liquid (droplets) on substrates consisting of alternating strips of wetting (equilibrium contact angle theta0 = 0 degrees) and nonwetting (theta0 approximately 90 degrees) material. Droplet spreading is compared for different wavelength lambda of the pattern and strength of surface interaction on the wetting strips. For small lambda, droplets partially spread on both the wetting and nonwetting regions of the substrate to attain a finite contact angle less than 90 degrees. In this case, the extent of spreading depends on the interaction strength in the wetting regions. A transition is observed such that, for large lambda, the droplet spreads only on the wetting region of the substrate by pulling material from nonwetting regions. In most cases, a precursor film spreads on the wetting portion of the substrate at a rate strongly dependent on the width of the wetting region.     

Bulk and interfacial wetting behavior of binary mixtures induced by associating between unlike-pair molecules

The statistical associating fluid theory of Wertheim is applied to describe binary mixtures with associating between unlike-pair molecules. The phase behavior of this binary mixture would fall into five different types (I, II, III, V, and VI) of the classification scheme of van Konynenburg and Scott by varying the associating strength and the energy parameters. Both interfacial wetting behavior and wetting transitions are carefully examined in all the vapor-liquid-liquid (gamma-beta-alpha) three-phase-coexisting regions of the binary mixtures. The global wetting behavior and wetting transitions are delineated by scanning the parameter space. In certain regions, the middle beta phase exhibits interfacial phase transitions sequentially, nonwetting --> partial-wetting --> nonwetting, at the interface separating lower alpha and upper gamma phases along with increasing temperature.     

Frequency-Dependent Streaming Potentials

An experimental apparatus and data acquisition system was constructed to measure the streaming potential coupling coefficients as a function of frequency. The purpose of the experiments was to measure, for the first time, the real and imaginary portion of streaming potentials. In addition, the measured frequency range was extended beyond any previous measurements. Frequency-dependent streaming potential experiments were conducted on one glass capillary and two porous glass filters. The sample pore diameters ranged from 1 mm to 34 μm. Two frequency-dependent models (Packard and Pride) were compared to the data. Both Pride's and Packard's models have a good fit to the experimental data in the low- and intermediate-frequency regime. In the high-frequency regime, the data fit the theory after being corrected for capacitance effects of the experimental setup. Pride's generalized model appears to have the ability to more accurately estimate pore sizes in the porous medium samples. Packard's model has one unknown model parameter while Pride's model has four unknown model parameters, two of which can be independently determined experimentally. Pride's additional parameters may allow for a determination of permeability. Copyright 2001 Academic Press.     

Capillary adhesion in the limit of saturation: thermodynamics, self-consistent field modeling and experiment

We introduce a simple thermodynamic argument for capillary adhesion forces, for various geometries, in the limit of saturation of the bulk phase. For one specific geometry (i.e., the sphere-plate geometry such as that found in the colloidal probe AFM technique), we provide evidence of the validity of our model by comparison with experiment and self-consistent field calculations. With this latter numerical technique, we also discuss deviations from the macroscopic argument both when the system is moved away from saturation and when the capillary bridge becomes so small that macroscopic thermodynamics is no longer accurate.     

Constitutive modeling of contact angle hysteresis

We introduce a phase field model of wetting of surfaces by sessile drops. The theory uses a two-dimensional non-conserved phase field variable to parametrize the Gibbs free energy of the three-dimensional system. Contact line tension and contact angle hysteresis arise from the gradient term in the free energy and the kinetic coefficient respectively. A significant advantage of this approach is in the constitutive specification of hysteresis. The advancing and receding angles of a surface, the liquid-vapor interfacial energy and three-phase line tension are the only required constitutive inputs to the model. We first simulate hysteresis on a smooth chemically homogeneous surface using this theory. Next we show that it is possible to study heterogeneous surfaces whose component surfaces are themselves hysteretic. We use this theory to examine the wetting of a surface containing a circular heterogeneous island. The contact angle for this case is found to be determined solely by the material properties at the contact line in accord with recent experimental data.     

Study of Salt Effects on the Micelle-Monomer Exchange Process of Octyl-, Decyl-, and Dodecyltrimethylammonium Bromide in Aqueous Solutions by Means of Ultrasonic Relaxation Spectroscopy

Spreading of a tiny macroscopic droplet of a nonvolatile, completely wetting liquid over a flat solid is considered. A liquid in creeping is subjected to capillary forces and long-range molecular forces. The droplet may be surrounded with a precursor wetting film. This paper deals with the problem of determining of the microscopic parameter that influences the interface shape near the apparent line of wetting; this is regarded as the inverse problem in the hydrodynamics of wetting. If the system includes a precursor film, the microscopic parameter coincides with the maximum thickness of the film. A series of inverse equations for the microparameter is obtained, which relate it to, first, the current geometric parameters of the macroscopic drop part and, second, the spreading time. A method for determining how the microparameter depends on the wetting line speed is proposed. The theory expands the opportunity to perform macroscopic measurements and reveals additional parameters. The inverse relations may be used to experimentally study the growth of the maximum thickness of a precursor film during drop spreading. Copyright 2000 Academic Press.