Diffusiophoresis and Electrophoresis of Colloidal Spheroids

An analytical study is presented for the diffusiophoresis and electrophoresis of a rigid nonconducting spheroid in a uniform applied field that is oriented arbitrarily with respect to its axis of revolution. The range of the interaction between the solute species and the particle surface is assumed to be small relative to the dimension of the particle, but the effect of polarization of the diffuse species in the thin solute-particle interaction layer is incorporated. To solve the conservative equations governing the system, a slip velocity of fluid and normal fluxes of solute species at the outer edge of the thin diffuse layer which balance convection and diffusion of the solute species along the particle surface are used as the boundary conditions for the fluid domain outside the diffuse layer. Explicit expressions for the migration velocity of the spheroidal particle are obtained for the cases of diffusiophoresis in a nonionic solute concentration gradient, diffusiophoresis in a concentration gradient of symmetric electrolyte, and electrophoresis in an external electric field. An interesting feature is found that the diffusiophoretic or electrophoretic velocity of the particle decreases with the reduction of the maximum length of the particle in the direction of migration. Also, the average migration velocity for an ensemble of identical, non-interacting spheroids with random orientation distribution is obtained for each case considered.     

Boundary Effects on Diffusiophoresis of Cylindrical Particles in Nonelectrolyte Gradients

The diffusiophoretic motion of a long circular cylinder in a transversely imposed gradient of a nonionic solute near a large plane wall parallel to its axis is analyzed. The range of the interaction between the solute and the solid surfaces is assumed to be small relative to the particle radius and to the gap width between the particle and the wall, but the polarization effect of the mobile solute in the thin diffuse layers adjacent to the solid surfaces caused by the strong adsorption of the solute is incorporated. A normal flux of the solute and a slip velocity of the fluid at the outer edge of the diffuse layers are used as the boundary conditions for the fluid domain outside the diffuse layers. Through the use of cylindrical bipolar coordinates along with these boundary conditions, a set of transport equations governing this problem is solved in the quasisteady situation and the wall effects on the diffusiophoresis of the cylinder are computed for various cases. For the diffusiophoretic motion of a cylinder normal to a plane, the particle mobility decreases monotonically with the decrease of the distance of the particle axis from the wall. The stronger the polarization effect in the diffuse layer, the weaker the wall effect on the diffusiophoresis. The effect of the normal plane on the diffusiophoresis of a cylinder is much more significant than that for a sphere at the same separation. For the diffusiophoresis of a cylinder parallel to a plane, the boundary effect is a complicated function of the relevant parameters (not necessarily varies monotonically with the extent of separation) mainly due to the existence of a diffusio-osmotic flow caused by the tangential fluid velocity at the plane wall. Copyright 2000 Academic Press.     

Diffusiophoresis of colloidal spheres in nonelectrolyte gradients at small but finite Péclet numbers

The diffusiophoretic motion of a spherical particle in a uniform imposed gradient of a nonionic solute is analyzed for small but finite Péclet numbers. The range of the interaction between the solute molecules and the particle surface is assumed to be small relative to the radius of the particle, but the polarization effect of the mobile solute in the thin diffuse layer surrounding the particle caused by the strong adsorption of the solute is incorporated. A normal flux of the solute and a slip velocity of the fluid at the outer edge of the diffuse layer are used as the boundary conditions for the fluid domain outside the diffuse layer. Through the use of a method of matched asymptotic expansions along with these boundary conditions, a set of transport equations governing this problem is solved in the quasisteady situation and an approximate expression for the diffusiophoretic velocity of the particle good to O(Pe ²) is obtained analytically. The analysis shows that the first correction to the particle velocity is O(Pe ²). The normalized particle velocity is found to decrease monotonically with the Péclet number and to increase monotonically with the dimensionless relaxation coefficient. The stronger the polarization effect in the diffuse layer, the weaker the convective effect on the diffusiophoresis. Received: 25 May 2000 Accepted: 6 September 2000     

Particle Interactions in Diffusiophoresis and Electrophoresis of Colloidal Spheres with Thin but Polarized Double Layers

The diffusiophoretic and electrophoretic motions of two colloidal spheres in the solution of a symmetrically charged electrolyte are analyzed using a method of reflections. The particles are oriented arbitrarily with respect to the electrolyte gradient or the electric field, and they are allowed to differ in radius and in zeta potential. The thickness of the electric double layers surrounding the particles is assumed to be small relative to the radius of each particle and to the gap width between the particles, but the effect of polarization of the mobile ions in the diffuse layer is taken into account. A slip velocity of fluid and normal fluxes of solute ions at the outer edge of the thin double layer are used as the boundary conditions for the fluid phase outside the double layers. The method of reflections is based on an analysis of the electrochemical potential and fluid velocity disturbances produced by a single dielectric sphere placed in an arbitrarily varying electrolyte gradient or electric field. The solution for two-sphere interactions is obtained in expansion form correct to O(r(12)(-7)), where r(12) is the distance between the particle centers. Our analytical results are found to be in good agreement with the available numerical solutions obtained using a boundary collocation method. On the basis of a model of statistical mechanics, the results of two-sphere interactions are used to analytically determine the first-order effect of the volume fraction of particles of each type on the mean diffusiophoretic and eletrophoretic velocities in a bounded suspension. For a suspension of identical spheres, the mean diffusiophoretic velocity can be decreased or increased as the volume fraction of the particles is increased, while the mean electrophoretic velocity is reduced with the increase in the particle concentration. Generally speaking, the particle interaction effects can be quite significant in typical situations. Copyright 2000 Academic Press.     

Diffusiophoresis and electrophoresis of a charged sphere perpendicular to two plane walls

The problem of diffusiophoretic and electrophoretic motions of a dielectric spherical particle in an electrolyte solution situated at an arbitrary position between two infinite parallel plane walls is studied theoretically in the quasisteady limit of negligible Peclet and Reynolds numbers. The applied electrolyte concentration gradient or electric field is uniform and perpendicular to the plane walls. The electric 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 diffuse ions in the double layer is incorporated. To solve the conservative equations, the general solution is constructed from the fundamental solutions in both cylindrical and spherical coordinates. The boundary conditions are enforced first at the plane walls by the Hankel 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 cases. The collocation results agree well with the approximate analytical solutions obtained by using a method of reflections. The presence of the walls can reduce or enhance the particle velocity, depending on the properties of the particle-solution system and the relative particle-wall separation distances. The boundary effects on diffusiophoresis and electrophoresis of a particle normal to two plane walls are found to be quite significant and complicated, and generally stronger than those parallel to the confining walls.     

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.     

Start‐up of electrophoresis of an arbitrarily oriented dielectric cylinder

An analytical study is presented for the transient electrophoretic response of a circular cylindrical particle to the step application of an electric field. The electric double layer adjacent to the particle surface is thin but finite compared with the radius of the particle. The time‐evolving electroosmotic velocity at the outer boundary of the double layer is utilized as a slip condition so that the transient momentum conservation equation for the bulk fluid flow is solved. Explicit formulas for the unsteady electrophoretic velocity of the particle are obtained for both axially and transversely applied electric fields, and can be linearly superimposed for an arbitrarily‐oriented applied field. If the cylindrical particle is neutrally buoyant in the suspending fluid, the transient electrophoretic velocity is independent of the orientation of the particle relative to the applied electric field and will be in the direction of the applied field. If the particle is different in density from the fluid, then the direction of electrophoresis will not coincide with that of the applied field until the steady state is attained. The growth of the electrophoretic mobility with the elapsed time for a cylindrical particle is substantially slower than for a spherical particle.     

Diffusiophoresis: Migration of Colloidal Particles in Gradients of Solute Concentration

When a rigid colloidal particle is placed in a solution which is not uniform in the concentration of some solute that interacts with the particle, the particle will be propelled in the direction of higher or lower concentration of the solute. The resulting locomotion is called diffusiophoresis. Experimental observations and theoretical predictions of the migration velocity of hydrosoIs are reviewed. Present commercial applications include the formation of rubber gloves and the deposition of paint films onto a steel surface. New applications to the analysis of colloidal mixtures and solid-liquid separation are suggested.     

Diffusiophoresis in a suspension of charge-regulating colloidal spheres

An analytical study of diffusiophoresis in a homogeneous suspension of identical spherical charge-regulating particles with an arbitrary thickness of the electric double layers in a solution of a symmetrically charged electrolyte with a uniform prescribed concentration gradient is presented. The charge regulation due to association/dissociation reactions of ionogenic functional groups on the particle surface is approximated by a linearized regulation model, which specifies a linear relationship between the surface charge density and the surface potential. The effects of particle-particle electrohydrodynamic interactions are taken into account by employing a unit cell model, and the overlap of the double layers of adjacent particles is allowed. The electrokinetic equations that govern the electric potential profile, the ionic concentration distributions, and the fluid flow field in the electrolyte solution surrounding the particle in a unit cell are linearized assuming that the system is only slightly distorted from equilibrium. Using a regular perturbation method, these linearized equations are solved with the equilibrium surface charge density (or zeta potential) of the particle as the small perturbation parameter. Closed-form formulas for the diffusiophoretic velocity of the charge-regulating sphere correct to the second order of its surface charge density or zeta potential are derived. Our results indicate that the charge regulation effect on the diffusiophoretic mobility is quite sensitive to the boundary condition for the electric potential specified at the outer surface of the unit cell. For the limiting cases of a very dilute suspension and a very thin or very thick electric double layer, the particle velocity is independent of the charge regulation parameter.     

Diffusiophoresis of charged particles and diffusioosmosis of electrolyte solutions

A review is presented on the theoretical basics and recent developments about the diffusiophoresis of charged particles and diffusioosmosis of electrolyte solutions driven by imposed electrolyte concentration gradients with particular emphasis on the principal analytical formulas and their physical interpretations. For diffusiophoresis, migrations of particles with thin polarized electric double layers but arbitrary zeta potentials and with arbitrary double layers but relatively low surface potentials are both discussed in detail, covering not only diffusiophoresis of single particles but also their motions near solid boundaries or other particles. For diffusioosmosis, fluid flows along single plane walls, in micro/nano-channels, and in porous media are considered, in which the solid walls may have arbitrary zeta potentials or surface charge densities, and both the effect of the lateral distribution of the diffuse ions and the relaxation effect in the double layers on the tangential electric field induced by the prescribed electrolyte concentration gradient are included.     

Electrokinetic diffusioosmotic flow of Ostwald-de Waele fluids near a charged flat plate in the thin double layer limit

Electrokinetic diffusioosmotic flow of Ostwald-de Waele, or power-law, fluids near a large charged flat plate is theoretically investigated for very thin double layers. Solutions to the flow velocity both up-close and far from the flat plate as well as the effective viscosity are presented for general values of the flow behavior index. Results show that given a wall zeta potential, ζ, diffusivity difference parameter, β, and constant imposed solute concentration gradient, both the near and far field diffusioosmotic flow velocities obtained for the respective dilatant and pseudoplastic liquids considerably deviate from those obtained for Newtonian liquids as found in previous literature. This likely suggests that the electrokinetic diffusioosmosis and its complementary effect of diffusiophoresis depend sensitively not only on the ζ-β parametric pair, but also on the possible non-Newtonian characteristics of the electrolytic liquid phase of the system. The theory presented herein can also be readily modified to model or describe electrodiffusioosmosis in power-law fluids, which is likely found in flow situations where the fluid non-Newtonian response, imposed solute concentration gradient, and an additional externally applied electric current density (or electric field) are of equal importance.     

Transient electrophoresis of dielectric spheres

The dynamic electrophoretic response of a spherical dielectric particle suspended in an electrolyte solution to a step change in the applied electrics field is analytically studied. The electrical double layer surrounding the particle may have either a small but finite thickness or a very large thickness relative to the particle radius. For the case of electrophoresis of a particle with a thin double layer, the local electroosmotic velocity at the outer edge of the double layer evolving with time after the external field is imposed is used as an apparent slip boundary condition at the particle surface so that the unsteady equation of motion for the fluid flow outside the double layer is solved. Closed-form formulas for the transient electrophoretic mobility of the particle are derived as functions of relevant parameters. The results demonstrate that, when the double layer surrounding the particle is relatively thin, the normalized electrophoretic mobility at a given dimensionless time decreases monotonically with a decrease in the parameter kappaa, where kappa(-1) is the Debye screening length and a is the particle radius. When the double layer of the particle is relatively thick, the particle mobility can have magnitudes comparable to those for a particle with a thin double layer in the initial stage, but will become much smaller afterward. In general, the effect of the relaxation time for transient electrophoresis is negligible, regardless of the value of kappaa.     

Inertial Hydrodynamic Effects in Electrophoresis of Particles in an Alternating Electric Field

The influence of the effects associated with the inertia of particles and the surrounding fluid on the electrophoresis in an alternating electric field has been theoretically investigated. From solving the hydrodynamic equations the electrophoretic velocity of a spherical particle was found to depend on the frequency of the external electric field and on the particle-to-fluid-density ratio. It is shown that, due to inertial effects, the liquid flow around particles with a thin electrical double layer (EDL) is no longer potential. A mechanism of the formation of steady-state flow in the vicinity of oscillating particles with a thin EDL is proposed. Using numerical methods, a picture of the fluid streamlines in such a flow is obtained. The spatial distribution of the fluid velocity in the vicinity of a particle is also found. It was established that with an increasing frequency of the electric field the steady-state flow velocity passes through a maximum. The flow direction depends on the ratio between the densities of a particle and the surrounding fluid. The reversal of direction takes place when this ratio is about 0.7. The case of a thick EDL has also been considered, and a comparative analysis of the flow distributions around the particles with a thin and those with a thick EDL has been carried out.     

Diffusiophoresis of a moderately charged spherical colloidal particle

An analytic expression is obtained for the diffusiophoretic mobility of a charged spherical colloidal particle in a symmetrical electrolyte solution. The obtained expression, which is expressed in terms of exponential integrals, is correct to the third order of the particle zeta potential so that it is applicable for colloidal particles with low and moderate zeta potentials at arbitrary values of the electrical double-layer thickness. This is an improvement of the mobility formula derived by Keh and Wei, which is correct to the second order of the particle zeta potential. This correction, which is related to the electrophoresis component of diffusiophoresis, becomes more significant as the difference between the ionic drag coefficients of electrolyte cations and anions becomes larger and vanishes in the limit of thin or thick double layer. A simpler approximate mobility expression is further obtained that does not involve exponential integrals.     

Electrophoresis of a colloidal sphere in a spherical cavity with arbitrary zeta potential distributions

An analytical study is presented for the quasi-steady electrophoretic motion of a dielectric sphere situated at the center of a spherical cavity when the surface potentials are arbitrarily nonuniform. The applied electric field is constant, and the electric double layers adjacent to the solid surfaces are assumed to be much thinner than the particle radius and the gap width between the surfaces. The presence of the cavity wall causes three basic effects on the particle velocity: (1) the local electric field on the particle surface is enhanced or reduced by the wall; (2) the wall increases the viscous retardation of the moving particle; and (3) a circulating electroosmotic flow of the suspending fluid exists because of the interaction between the electric field and the charged wall. The Laplace and Stokes equations are solved analytically for the electric potential and velocity fields, respectively, in the fluid phase, and explicit formulas for the electrophoretic and angular velocities of the particle are obtained. To apply these formulas, one has to calculate only the monopole, dipole, and quadrupole moments of the zeta-potential distributions at the particle and cavity surfaces. It is found that the contribution from the electroosmotic flow developing from the interaction of the imposed electric field with the thin double layer adjacent to the cavity wall and the contribution from the wall-corrected electrophoretic driving force to the particle velocities can be superimposed as a result of the linearity of the problem.     

Diffusiophoresis of a nonuniformly charged sphere in an electrolyte solution

The diffusiophoresis of a rigid, nonuniformly charged spherical particle in an electrolyte solution is analyzed theoretically focusing on the influences of the thickness of double layer, the surface charge distribution, the effect of electrophoresis, and the effect of double-layer polarization. We show that the nonuniform charge distribution on the particle surface yields complicated effect of double-layer polarization, leading to interesting diffusiophoretic behaviors. For example, if the sign of the middle part of the particle is different from that of its left- and right-hand parts, then depending upon the charge density and the fraction of the middle part, the particle can move either to the high-concentration side or to the low-concentration side. Both the diffusiophoretic velocity and its direction can be manipulated by the distribution of the surface charge density. In particular, if the electrophoresis effect is significant, then those properties are governed by the averaged surface charge density of the particle. A dipolelike particle, where its left- (right-) hand half is negatively (positively) charged, always migrates toward the low-concentration (left-hand) side, that is, it has a negative diffusiophoretic velocity. In addition, that diffusiophoretic velocity has a negative local minimum as the thickness of double layer varies.     

Startup of electrophoresis in a suspension of colloidal spheres

The transient electrophoretic response of a homogeneous suspension of spherical particles to the step application of an electric field is analyzed. The electric double layer encompassing each particle is assumed to be thin but finite, and the effect of dynamic electroosmosis within it is incorporated. The momentum equation for the fluid outside the double layers is solved through the use of a unit cell model. Closed‐form formulas for the time‐evolving electrophoretic and settling velocities of the particles in the Laplace transform are obtained in terms of the electrokinetic radius, relative mass density, and volume fraction of the particles. The time scale for the development of electrophoresis and sedimentation is significantly smaller for a suspension with a higher particle volume fraction or a smaller particle‐to‐fluid density ratio, and the electrophoretic mobility at any instant increases with an increase in the electrokinetic particle radius. The transient electrophoretic mobility is a decreasing function of the particle volume fraction if the particle‐to‐fluid density ratio is relatively small, but it may increase with an increase in the particle volume fraction if this density ratio is relatively large. The particle interaction effect in a suspension on the transient electrophoresis is much weaker than that on the transient sedimentation of the particles.     

A fresh view on phoresis and self-phoresis

Phoresis, a classic example of particle transport driven by thermodynamic gradients, is enjoying a resurgent research interest motivated both by technological developments and by its relevance to the motility of chemically active particles. Here we succinctly review, using the case of chemophoresis (also called diffusiophoresis), the general framework of phoresis and self-phoresis formulated as a Stokes-flow problem for a liquid solution (solvent and solute) maintained out of thermodynamic equilibrium by solute gradients. Within the constraints of local equilibrium, we discuss the simplest extension of the theory in order to account for correlations in the fluid. We show that this leads to a shift from the paradigm based on the ideal case, in that self-phoresis can no longer be represented as phoresis in a self-generated composition gradient. Our review concludes with a concise overview of a few directions which we think hold the potential to reveal a rich behavior in future investigations.     

Diffusiophoresis in a suspension of spherical particles with arbitrary double-layer thickness

The diffusiophoresis in a homogeneous suspension of identical dielectric spheres with an arbitrary thickness of the electric double layers in a solution of a symmetrically charged electrolyte with a constant imposed concentration gradient is analytically studied. The effects of particle interactions (or particle volume fraction) are taken into account by employing a unit cell model, and the overlap of the double layers of adjacent particles is allowed. 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 sphere in a unit cell are linearized assuming that the system is only slightly distorted from equilibrium. Using a perturbation method, these linearized equations are solved with the surface charge density (or zeta potential) of the particle as the small perturbation parameter. Analytical expressions for the diffusiophoretic velocity of the dielectric sphere in closed form correct to the second order of its surface charge density or zeta potential are obtained from a balance between its electrostatic and hydrodynamic forces. Comparisons of the results of the cell model with different conditions at the outer boundary of the cell are made.     

Thermophoretic Motion of a Sphere Parallel to an Insulated Plane

An analytical study is presented for the thermophoresis of a sphere in a constant applied temperature gradient parallel to an adiabatic plane. The Knudsen number is assumed to be small so that the fluid flow can be described by a continuum model with a thermal creep and a hydrodynamic slip at the particle surface. A method of reflections is used to obtain the asymptotic formulas for the temperature and velocity fields in the quasisteady situation. The thermal insulated plane may be a solid wall (no-slip) and/or a free surface (perfect-slip). The boundary effect on the thermophoretic motion is found to be weaker than that on the axisymmetric thermophoresis of a sphere normal to a plane with constant temperature. In comparison with the motion driven by gravitational force, the interaction between the particle and the boundary is less significant under thermophoresis. Even so, the interaction between the plane and the particle can be very strong when the gap thickness approaches zero. For the thermophoretic motion of a particle parallel to a solid plane, the effect of the plane surface is to reduce the translational velocity of the particle. In the case of particle migration parallel to a free surface due to thermophoresis, the translating velocity of a particle can be either greater or smaller than that which would exist in the absence of the plane surface, depending on the relative thermal conductivity and the surface properties of the particle and its relative distance from the plane. Not only the translational velocity but also the rotational velocity of the thermophoretic sphere near the plane boundary is formulated analytically. The rotating direction of the particle is strongly dominated by its surface properties and the internal-to-external thermal conductivity. Besides the particle motion, the thickness of the thermophoretic boundary layer is evaluated by considering the thermophoretic mobility. Generally speaking, a free surface exerts less influence on the particle movement than a solid wall. Copyright 2000 Academic Press.