Imposed Electrolyte Concentration Gradient Research Articles

Diffusiophoresis of a charged porous shell in electrolyte gradients

The diffusiophoresis of a charged spherical porous shell or permeable microcapsule with arbitrary inner and outer radii, fluid permeability, and electric double layer thickness in an ionic solution is analyzed. With the assumption that the imposed electrolyte concentration gradient is relatively weak and the transport system is only perturbed slightly from equilibrium, the electrostatic potential, ionic concentration (electrochemical potential energy), and fluid velocity fields are determined as power-series expansions of the small fixed charge density of the porous shell by solving the relevant linearized electrokinetic equations. An explicit expression for the diffusiophoretic velocity of the porous shell correct to the second order of the fixed charge density results from balancing the exerted electrostatic and hydrodynamic forces. Both the electrophoretic (first-order) and chemiphoretic (second-order) mobilities increase with an increase in the normalized thickness of the porous shell, and these increases are significant when the porous shell is thin. However, the diffusiophoretic velocity of the porous shell, which is the sum of the electrophoretic and chemiphoretic velocities, may not be a monotonic function of its normalized thickness. The variation in the normalized fixed charge density of the porous shell from negative to positive values can lead to more than one reversals in the direction of the diffusiophoretic velocity.

Diffusioosmosis of Electrolyte Solutions around a Circular Cylinder at Arbitrary Zeta Potential and Double-Layer Thickness

The steady diffusioosmotic flow of an electrolyte solution around a charged circular cylinder caused by a constant concentration gradient imposed in the direction along the axis of the cylinder is investigated theoretically. The cylinder can have either a constant surface potential or a constant surface charge density of an arbitrary value. The electric double layer surrounding the cylinder can have an arbitrary thickness, and its electrostatic potential distribution is determined by an analytical approximation to the solution of the Poisson-Boltzmann equation. Solving a modified Navier-Stokes equation with the constraint of no net electric current arising from the cocurrent diffusion, electric migration, and diffusioosmotic convection of the electrolyte ions, the macroscopic electric field and the fluid velocity along the axial direction induced by the imposed electrolyte concentration gradient are obtained semianalytically as functions of the radial position in a self-consistent manner. The direction of the diffusioosmotic flow relative to the concentration gradient is determined by the combination of the zeta potential (or surface charge density) of the cylinder, the properties of the electrolyte solution, and other relevant factors. For a prescribed concentration gradient of an electrolyte, the magnitude of the fluid velocity generally increases with increasing distance from the surface of the cylinder, but there are exceptions. The effects of the radial distribution of the induced macroscopic electric field and of the ionic convection in the double layer on the diffusioosmotic flow are quite significant in practical situations.

Diffusioosmosis of electrolyte solutions in fibrous porous media

An analytical study is presented for the diffusioosmotic flow of an electrolyte solution in the fibrous medium constructed by an ordered array of parallel charged circular cylinders at the steady state. The prescribed electrolyte concentration gradient is constant but can be oriented arbitrarily with respect to the axes of the cylinders. The electric double layer surrounding each cylinder may have an arbitrary thickness relative to the radius of the cylinder. A unit cell model which allows for the overlap of the double layers of adjacent cylinders is employed to account for the effect of fibers on each other. The electrostatic potential distribution in the fluid phase of a cell is obtained by solving the linearized Poisson–Boltzmann equation, which applies to the case of low surface potential of the cylinders. The macroscopic electric field induced by the imposed electrolyte concentration gradient through the fluid phase in a cell is determined as a function of the radial position. A closed-form formula for the fluid velocity profile of the electrolyte solution due to the combination of electroosmotic and chemiosmotic contributions as a function of the porosity of the array of cylinders correct to the second order of their surface charge density or zeta potential is derived as the solution of a modified Navier–Stokes equation. The diffusioosmotic velocity can have more than one reversal in direction over a small range of the zeta potential. For a given electrolyte concentration gradient in a cell, the fluid flow rate does not necessarily increase with an increase in the electrokinetic radius of the cylinder, which is the cylinder radius divided by the Debye screening length. The effect of the radial distribution of the induced axial electric field in the double layer on the diffusioosmotic flow is found to be of dominant significance in most practical situations.

Diffusioosmosis of Electrolyte Solutions in a Fine Capillary Tube

A theoretical study is presented for the steady diffusioosmotic flow of an electrolyte solution in a fine capillary tube generated by a constant concentration gradient imposed in the axial direction. The capillary wall may have either a constant surface potential or a constant surface charge density of an arbitrary quantity. The electric double layer adjacent to the charged wall may have an arbitrary thickness, and its electrostatic potential distribution is determined by an analytical approximation to the solution of the Poisson-Boltzmann equation. Solving a modified Navier-Stokes equation with the constraint of no net electric current arising from the cocurrent diffusion, electric migration, and diffusioosmotic convection of the electrolyte ions, the macroscopic electric field and the fluid velocity along the axial direction induced by the imposed electrolyte concentration gradient are obtained semianalytically as a function of the radial position in a self-consistent way. The direction of the diffusioosmotic flow relative to the concentration gradient is determined by the combination of the zeta potential (or surface charge density) of the wall, the properties of the electrolyte solution, and other relevant factors. For a prescribed concentration gradient of an electrolyte, the magnitude of fluid velocity at a position in general increases with an increase in its distance from the capillary wall, but there are exceptions. The effect of the radial distribution of the induced tangential electric field and the relaxation effect due to ionic convection in the double layer on the diffusioosmotic flow are found to be very significant.

Diffusioosmosis of electrolyte solutions in a fine capillary slit

The steady diffusioosmotic flows of an electrolyte solution along a charged plane wall and in a capillary channel between two identical parallel charged plates generated by an imposed tangential concentration gradient are theoretically investigated. The plane walls may have either a constant surface potential or a constant surface charge density. The electrical double layers adjacent to the charged walls may have an arbitrary thickness and their electrostatic potential distributions are determined by the Poisson–Boltzmann equation. Solving a modified Navier–Stokes equation with the constraint of no net electric current arising from the cocurrent diffusion, electric migration, and diffusioosmotic convection of the electrolyte ions, the macroscopic electric field and the fluid velocity along the tangential direction induced by the imposed electrolyte concentration gradient are obtained semianalytically as a function of the lateral position in a self-consistent way. The direction of the diffusioosmotic flow relative to the concentration gradient is determined by the combination of the zeta potential (or surface charge density) of the wall, the properties of the electrolyte solution, and other relevant factors. For a given concentration gradient of an electrolyte along a plane wall, the magnitude of fluid velocity at a position in general increases with an increase in its electrokinetic distance from the wall, but there are exceptions. The effect of the lateral distribution of the induced tangential electric field and the relaxation effect in the double layer on the diffusioosmotic flow are found to be very significant.

Diffusioosmosis of electrolyte solutions in a capillary slit with surface charge layers

The steady diffusioosmotic flow of an electrolyte solution in a fine capillary slit with each of its inside walls covered by a layer of adsorbed polyelectrolytes is analytically studied. In this solvent-permeable and ion-penetrable surface charge layer, idealized polyelectrolyte segments are assumed to distribute at a uniform density. The electric double layer and the surface charge layer may have arbitrary thicknesses relative to the gap width between the slit walls. The Debye–Huckel approximation is used to obtain the electrostatic potential distribution on a cross section of the slit. The macroscopic electric field induced by the imposed electrolyte concentration gradient through the slit is determined as a function of the lateral position rather than taken as its constant bulk-phase quantity. Explicit formulas for the fluid velocity profile are derived as the solution of a modified Navier–Stokes/Brinkman equation. The effect of the lateral distribution of the induced axial electric field in the slit on the diffusioosmotic flow is found to be of dominant significance in most practical situations and to drive the fluid towards the end of higher electrolyte concentration. The existence of the surface charge layers can lead to a quite different diffusioosmotic flow from that in a capillary with bare walls.

Diffusioosmosis of Electrolyte Solutions along a Charged Plane Wall

The steady diffusioosmotic flow of an electrolyte solution along a dielectric plane wall caused by an imposed tangential concentration gradient is analytically examined. The plane wall may have either a constant surface potential or a constant surface charge density of an arbitrary quantity. The electric double layer adjacent to the charged wall may have an arbitrary thickness, and its electrostatic potential distribution is determined by the Poisson-Boltzmann equation. The macroscopic electric field along the tangential direction induced by the imposed electrolyte concentration gradient is obtained as a function of the lateral position. A closed-form formula for the fluid velocity profile is derived as the solution of a modified Navier-Stokes equation. The direction of the diffusioosmotic flow relative to the concentration gradient is determined by the combination of the zeta potential of the wall and the properties of the electrolyte solution. For a given concentration gradient of an electrolyte along a plane wall, the magnitude of fluid velocity at a position in general increases with an increase in its electrokinetic distance from the wall, but there are exceptions. The effect of the lateral distribution of the induced tangential electric field in the double layer on the diffusioosmotic flow is found to be very significant and cannot be ignored.

Diffusiophoresis and electrophoresis of a charged sphere parallel to one or two plane walls

The diffusiophoretic and electrophoretic motions of a dielectric spherical particle in an electrolyte solution located between two infinite parallel plane walls are studied theoretically. The imposed electrolyte concentration gradient or electric field is constant and parallel to the two plates, which may be either impermeable to the ions/charges or prescribed with the far-field concentration/potential distribution. The electrical double layer at the particle surface is assumed to be thin relative to the particle radius and to the particle–wall gap widths, but the polarization effect of the mobile ions in the diffuse layer is incorporated. The presence of the neighboring walls causes two basic effects on the particle velocity: first, the local electrolyte concentration gradient or electric field on the particle surface is enhanced or reduced by the walls, thereby speeding up or slowing down the particle; second, the walls increase the viscous retardation of the moving particle. To solve the conservative equations, the general solution is constructed from the fundamental solutions in both rectangular and spherical coordinates. The boundary conditions are enforced first at the plane walls by the Fourier transforms and then on the particle surface by a collocation technique. Numerical results for the diffusiophoretic and electrophoretic velocities of the particle relative to those of a particle under identical conditions in an unbounded solution are presented for various values of the relevant parameters including the relative separation distances between the particle and the two plates. For the special case of motions of a spherical particle parallel to a single plate and in the central plane of a slit, the collocation results agree well with the approximate analytical solutions obtained by using a method of reflections. The presence of the lateral walls can reduce or enhance the particle velocity, depending on the properties of the particle–solution system, the relative particle–wall separation distances, and the electrochemical boundary condition at the walls. In general, the boundary effects on diffusiophoresis and electrophoresis are quite significant and complicated, and they no longer vary monotonically with the separation distances for some situations.

Diffusioosmosis of electrolyte solutions in fine capillaries

Abstract An analytical study is presented for the diffusioosmotic flow of an electrolyte solution in a narrow capillary tube or slit caused by a constant concentration gradient imposed along the axial direction at the steady state. The electric double layer adjacent to the charged capillary wall may have an arbitrary thickness relative to the capillary radius. The electrostatic potential distribution on a cross section of the capillary is obtained by solving the linearized Poisson–Boltzmann equation, which applies to the case of low surface potential at the capillary wall. The macroscopic electric field induced by the imposed electrolyte concentration gradient through the capillary is determined as a function of the radial position. Closed-form formulas for the fluid velocity profile due to the combination of electroosmotic and chemiosmotic contributions are derived as the solution of a modified Navier–Stokes equation. The diffusioosmotic velocity can have more than one reversal in direction over a small range of the surface potential. For a given concentration gradient of electrolyte in a capillary, the fluid-flow rate is not necessarily to increase with an increase in the electrokinetic radius of the capillary, which is the capillary radius divided by the Debye screening length. The effect of the radial distribution of the induced axial electric field in the double layer on the diffusioosmotic flow is found to be of dominant significance in most practical situations.

Osmosis through a Fibrous Medium Caused by Transverse Electrolyte Concentration Gradients

The steady diffusioosmotic flow of an electrolyte solution in the fibrous porous medium constructed by a homogeneous array of parallel charged circular cylinders is analytically studied. The imposed electrolyte concentration gradient is constant and normal to the axes of the cylinders. The electric double layer surrounding each cylinder may have an arbitrary thickness relative to the radius of the cylinder. A unit cell model which allows for the overlap of the double layers of adjacent cylinders is employed. The electrokinetic equations that govern the ionic concentration distributions, the electrostatic potential profile, and the fluid flow field in the electrolyte solution surrounding the charged cylinder in a unit cell are linearized assuming that the system is only slightly distorted from equilibrium. Through the use of a regular perturbation method, these linearized equations are solved with the surface charge density (or zeta potential) of the cylinder as the small perturbation parameter. Analytical expressions for the diffusioosmotic velocity of the electrolyte solution as a function of the porosity of the ordered array of cylinders correct to the second order of their surface charge density or zeta potential are obtained from a balance between the electrostatic and the hydrodynamic forces exerted on each cylinder. Comparisons of the results of the cell model with different conditions at the outer boundary of the cell are made. In the limit of maximum porosity, these results can be interpreted as the diffusiophoretic velocity of an isolated circular cylinder caused by the transversely imposed electrolyte concentration gradient.

Diffusioosmosis and electroosmosis in a capillary slit with surface charge layers

Abstract This study analytically examines the steady diffusioosmotic and electroosmotic flows of an electrolyte solution in a fine capillary slit with each of its inside walls covered by a layer of adsorbed polyelectrolytes. In this solvent-permeable and ion-penetrable surface charge layer, idealized polyelectrolyte segments are assumed to distribute at a uniform density. The electric double layer and the surface charge layer may have arbitrary thicknesses relative to the gap width between the slit walls. The electrostatic potential distribution on a cross section of the slit is obtained by solving the linearized Poisson–Boltzmann equation, which applies to the case of low potentials or low fixed-charge densities. Explicit formulas for the fluid velocity profile due to the imposed electrolyte concentration gradient or electric field through the slit are derived as the solution of a modified Navier–Stokes/Brinkman equation. The results demonstrate that the structure of the surface charge layer can lead to an augmented or a diminished electrokinetic flow (even a reversal in direction of the flow) relative to that in a capillary with bare walls, depending on the characteristics of the capillary, of the surface charge layer, and of the electrolyte solution. For the diffusioosmotic flow with an induced electric field, competition between electroosmosis and chemiosmosis can result in more than one reversal in direction of the flow over a range of the Donnan potential of the adsorbed polyelectrolyte in the capillary.

Diffusioosmosis and Electroosmosis of Electrolyte Solutions in Fibrous Porous Media

The steady diffusioosmotic and electroosmotic flows of an electrolyte solution in the fibrous porous medium constructed by a homogeneous array of parallel charged circular cylinders are analyzed under conditions of small Peclet and Reynolds numbers. The imposed electrolyte concentration gradient or electric field is constant and can be oriented arbitrarily with respect to the axes of the cylinders. The thickness of the electric double layers surrounding the cylinders is assumed to be small relative to the radius of the cylinders and to the gap width between two neighboring cylinders, but the polarization effect of the diffuse ions in the double layers is incorporated. Through the use of a unit cell model, the appropriate equations of conservation of the electrochemical potential energies of ionic species and the fluid momentum are solved for each cell, in which a cylinder is envisaged to be surrounded by a coaxial shell of the fluid. Analytical expressions for the diffusioosmotic and electroosmotic velocities of the bulk electrolyte solution as functions of the porosity of the ordered array of cylinders are obtained in closed form for various cases. Comparisons of the results of the cell model with different conditions at the outer boundary of the cell are made. In the limit of maximum porosity, these results can be interpreted as the diffusiophoretic and electrophoretic velocities of an isolated circular cylinder caused by the imposed electrolyte concentration gradient or electric field.