A new approach for analyzing particle motion near an interface using total internal reflection microscopy

A new method was developed for analyzing the normal motion of a single colloidal particle near an interface. The optical technique of total internal reflection microscopy (TIRM) was used to determine the distribution of vertical displacements of a particle from a specific starting position as a function of time. At very small displacement times, the displacements are normally distributed with a variance that is proportional to the diffusion coefficient times the displacement time. The change in the diffusion coefficient with separation distance between the particle and plate was found to match that predicted by Brenner (Chem. Eng. Sci. 16 (1961) 242). As the sampling time becomes very large, the variance reaches a constant value determined strictly by the shape of the local potential energy profile holding the particle. A major advantage of this approach, relative to other measurement methods, is that the particle's spatially variant diffusion coefficient can be determined without any knowledge of the forces acting on the particle.     

The lateral electric force on an off-axis sphere in a circular cylinder

The electrophoretic velocity of a sphere within a liquid-filled circular cylinder in a direction parallel to the cylinder axis has been studied by Yariv and Brenner (Phys. Fluids 2002, 14, 3354–3357; SIAM J. Appl. Math. 2003, 64, 423–441). We use their analyses of the electric field in order to determine the electrical force on the sphere along the cylinder radius (i.e., perpendicular to its axis) when either the radius of the sphere is small compared to that of the cylinder, or when the radius of the sphere is only slightly smaller than that of the cylinder. In both cases the force acts towards the centreline of the cylinder, and hence this force tends to stabilize electrophoresis of the sphere along the cylinder axis.     

Wall effects on electrophoretic motion of spherical polystyrene particles in a rectangular poly(dimethylsiloxane) microchannel

The wall effects on electrophoretic motion of spherical polystyrene particles in a rectangular poly(dimethylsiloxane) microchannel were studied experimentally. It is found that the particle electrophoretic velocity is insensitive to the trajectory between the channel sidewalls, consistent with the theoretical prediction. We also demonstrate that the electrophoretic motion of larger particles along the channel centerline is more viscously retarded by the sidewalls of a narrower channel. This observation is well predicted by incorporating the analytical models for the particle electrophoresis along the centerline of a slit channel and along the axis of a cylindrical pore.     

Electrophoretic mobility of a cylindrical colloidal particle in a salt-free medium

Approximate expressions are derived for the electrophoretic mobility of dilute cylindrical colloidal particles in a salt-free medium containing only counterions. The cylinder is assumed to be infinitely long. It is shown that as in the case of a spherical particle, there is a certain critical value of the particle surface charge separating two cases. When the particle surface charge is lower than the critical value (case 1), the electrophoretic mobility increases with increasing particle surface charge per unit length. When the particle surface charge is higher than the critical value (case 2), the mobility becomes constant (for a cylinder in a transverse field) or the increase in the electrophoretic mobility with the particle surface charge becomes suppressed (for a cylinder in a tangential field). These phenomena are caused by the effect of counterion condensation in the vicinity of the particle surface. The critical value of the particle charge is essentially independent of the particle volume fraction phi for the dilute case, unlike the case of a sphere, in which case the critical charge value is proportional to ln(1/phi).     

Electrophoretic motion of a spherical particle with a symmetric nonuniform surface charge distribution in a nanotube

The electrophoretic motion of a spherical nanoparticle, subject to an axial electric field in a nanotube filled with an electrolyte solution, has been investigated using a continuum theory, which consists of the Nernst-Planck equations for the ionic concentrations, the Poisson equation for the electric potential in the solution, and the Stokes equation for the hydrodynamic field. In particular, the effects of nonuniform surface charge distributions around the nanoparticle on its axial electrophoretic motion are examined with changes in the bulk electrolyte concentration and the surface charge of the tube's wall. A particle with a nonuniform charge distribution is shown to induce a corresponding complex ionic concentration field, which in turn influences the electric field and the fluid motion surrounding the particle and thus its electrophoretic velocity. As a result, contrary to the relatively simple dynamics of a particle with a uniform surface charge, dominated by the irradiating electrostatic force, that with a nonuniform surface charge distribution shows various intriguing behaviors due to the additional interplay of the nonuniform electro-osmotic effects.     

Adsorption of acicular particles at liquid-fluid interfaces and the influence of the line tension

In this paper, the adsorption energy of an acicular (prolate and cylindrical) particle onto a liquid-fluid interface and the effect of the line tension are investigated. The results show that, without line tension, acicular particles always prefer to lie flat in the plane of the interface. However, line tension plays a significant role in determining the adsorption of an acicular particle. First, the line tension creates an energy barrier for the adsorption of particles onto an interface. The planar configuration has a larger energy barrier due to the longer contact line. Therefore, the particles prefer to enter the interface in a homeotropic configuration and then rearrange to a planar configuration or an oblique configuration with a small tilt angle. Second, for prolate particles, an energy maximum occurs at some tilt angles when the line tension is large. Therefore, once the prolate particle is adsorbed on the interface in a homeotropic configuration or with a larger tilt angle, it must conquer an energy barrier to rearrange to a planar configuration. For cylindrical particles, when the line tension is higher, the planar configuration will not be the most energy-favorable configuration. The cylindrical particles prefer to stay in the interface with a small tilt angle.     

Hydrodynamic force on a microparticle approaching a wall in a nanoparticle dispersion: observation of a separation-dependent effective viscosity

Colloid probe atomic force microscopy was used to measure the hydrodynamic force exerted on a 30-μm-diameter silica particle being moved toward or away from a silica plate in aqueous dispersions of 22-nm-diameter silica nanoparticles (6 or 8 vol %). Upon comparing the measured force to predictions made using the well-known expression of Cox and Brenner (Cox, R. G.; Brenner, H. Chem. Eng. Sci.1967, 22, 1753-1777) assuming a constant viscosity equal to that of the bulk dispersion, the measured drag force was found to become significantly less than that predicted at smaller particle-plate separation distances (e.g., <500 nm). A recent theoretical paper by Bhattacharya and Blawzdziewicz (Bhattacharya, S.; Blawzdziewicz, J. J. Chem. Phys.2008, 128, 214704) predicted that in a solution of dispersed nanoparticles the effective viscosity characterizing the hydrodynamic force on the particle should vary from that of the solvent at contact to that of the bulk dispersion at large separations. By adjusting the viscosity in the Cox and Brenner expression to make the predicted hydrodynamic force match that measured (i.e., the effective viscosity), a curve showing these exact characteristics was obtained. The effective viscosity profile was not a function of particle speed, and changes in the effective viscosity extended to separation distances of as large as 2 μm (nearly 100 times the hard diameter of the nanoparticles). These results suggest that in the range of typical colloidal forces (on the order of 100 nm), the dynamics of particle motion in such systems are determined by the viscosity of the solvent and not that of the bulk dispersion.     

Electrophoresis of a sphere normal to a plane at arbitrary electrical potential and double layer thickness

The electrophoretic movement of a sphere normal to an uncharged, planar surface is analyzed theoretically, taking the effect of double layer polarization into account. Here, both the surface potential of the particle and the thickness of the double layer surrounding it can be arbitrary. We show that if double layer polarization is neglected, the effect of the surface potential of a particle on its electrophoretic velocity is inappreciable. On the contrary, it becomes significant if double layer polarization is present. However, if the distance between the particle and the surface is sufficiently close, since the hydrodynamic effect dominates, the influence of the surface potential and double layer polarization becomes insignificant.     

Electrophoretic mobility of a spherical colloidal particle in a salt-free medium

A general expression as well as approximate expressions are derived for the electrophoretic mobility of dilute spherical colloidal particles in a salt-free medium containing only counter ions. It is shown that there is a certain critical value of the particle surface charge. When the particle surface charge is lower than the critical value, the electrophoretic mobility is proportional to the particle surface charge or the particle zeta potential, following Hückel's formula. When the particle surface charge is higher than the critical value, the electrophoretic mobility becomes independent of the particle surface charge. This is due to the effect of counter ion condensation in the vicinity of the particle surface.     

3-D transient electrophoretic motion of a spherical particle in a T-shaped rectangular microchannel

This paper considers the electrophoretic motion of a spherical particle in an aqueous electrolyte solution in a T-shaped rectangular microchannel, where the size of the channel is close to that of the particle. This is a complicated transient process where the electric field, the flow field, and the particle motion are coupled together. A theoretical model was developed to investigate the influences of the applied electric potentials, the zeta potentials of the channel and the particle, and the size of the particle on the particle motion. A direct numerical simulation method using the finite element method is employed. This method employs a generalized Galerkin finite element formulation that incorporates both equations of the fluid flow and equations of the particle motion into a single variational equation where the hydrodynamic interactions are eliminated. The ALE method is used to track the surface of the particle at each time step. The numerical results show that the electric field in the T-shaped microchannel is influenced by the presence of the particle, and that the particle motion is influenced by the applied electric potentials and the zeta potentials of the channel and the particle. The path of the particle motion is dominated by the local electric field and the ratio of the zeta potential of the channel to that of the particle. The particle's velocity is also dependent on its size in a small channel.     

Dynamic Electrophoretic Mobility of a Soft Particle

A theory of the dynamic electrophoretic mobility of a spherical soft particle (that is, a polyelectrolyte-coated spherical particle) in an oscillating electric field is presented. In the absence of the polyelectrolyte layer a spherical soft particle becomes a spherical hard particle, while in the absence of the particle core it tends to a spherical polyelectrolyte. The present theory thus covers two extreme cases, that is, dynamic electrophoresis of hard particles and that of spherical polyelectrolytes. Simple analytic mobility expressions are derived. It is shown how the dynamic electrophoretic mobility of a soft particle depends on the volume charge density distributed in the polyelectrolyte layer, on the frictional coefficient characterizing the frictional forces exerted by the polymer segments on the liquid flow in the polyelectrolyte layer, on the particle size, and on the frequency of the applied oscillating electric field. Copyright 2001 Academic Press.     

Electrophoresis of a colloidal sphere in a spherical cavity with arbitrary zeta potential distributions and arbitrary double-layer thickness

The electrophoretic motion of a dielectric sphere situated at the center of a spherical cavity with an arbitrary thickness of the electric double layers adjacent to the particle and cavity surfaces is analyzed at the quasisteady state when the zeta potentials associated with the solid surfaces are arbitrarily nonuniform. Through the use of the multipole expansions of the zeta potentials and the linearized Poisson-Boltzmann equation, the equilibrium double-layer potential distribution and its perturbation caused by the applied electric field are separately solved. The modified Stokes equations governing the fluid velocity field are dealt with using a generalized reciprocal theorem, and explicit formulas for the electrophoretic and angular velocities of the particle valid for all values of the particle-to-cavity size ratio are obtained. To apply these formulas, one only has to calculate the monopole, dipole, and quadrupole moments of the zeta potential distributions at the particle and cavity surfaces. In some limiting cases, our result reduces to the analytical solutions available in the literature. In general, the boundary effect on the electrophoretic motion of the particle is a qualitatively and quantitatively sensible function of the thickness of the electric double layers relative to the radius of the cavity.     

Hydrodynamic Permeability of the Membrane as a System of Rigid Particles Covered with Porous Layer (Cell Model)

Problem of uniform flow of viscous, incompressible liquid around a rigid particle covered with porous layer and located inside the spherical cell was solved at small Reynolds numbers. In order to describe the motion of liquid within porous layer, the Brinkman equations were used. Based on the Happel and Brenner cell method, the hydrodynamic permeability of the membrane consisting of the system of porous particles with rigid impermeable cores was calculated.     

Particle interactions in electrophoresis: I. Motion of two spheres along their line of centers

The electrophoretic motion of two charged colloidal spheres with very thin electrical double layers in a constant applied electric field along their line of centers is considered. The particles may differ in radius and in zeta potential at the surface. The electrostatic and hydrodynamic governing equations are solved in the quasi-steady situation using bipolar coordinates and the electrophoretic velocities of particles are calculated for various cases. The interaction effect between particles can be very significant when the distance between particle surfaces gets close to zero. The particle with smaller zeta potential is speeded up by the motion of the other, which is retarded at the same time by the motion of the former one, if the two spheres have unequal zeta potentials of the same electrical sign. For two particles of different signs in zeta potential, motions of both are hindered by each other. The influence of the interaction between particles in general is stronger on the smaller one than on the larger one. For the special case of two electrophoretic spheres with identical zeta potentials, there is no particle interaction for all particle sizes and separations.     

Dielectrophoretic force on a sphere near a planar boundary

The small gap distance separating a spherical colloidal particle in electrophoretic motion from a planar nonconducting surface is a required parameter for calculating its electrophoretic mobility. In the presence of an externally applied electric field, this gap distance is determined by balancing the van der Waals, electrical double layer interaction, and gravitational forces with a dielectrophoretic (DEP) force. Here, the DEP force was determined analytically by integration of the Maxwell stress over the surface of the particle. The account of this force showed that its previous omission from the analysis always resulted in underpredicted gap distances. Furthermore, the DEP force dominated under conditions of low particle density or high electric field strength and led to much higher gap distances on the order of a few microns. In one particular case, a combination of low particle density and small particle size produced two possible equilibrium gap distances for the particle. However, the particle was unstable in the second equilibrium position when subjected to small perturbations. In general, larger particles had smaller gap sizes. The effects of four other parameters on gap distance were studied, and gap distances were found to increase with lower particle density, higher electric field strength, higher particle and wall zeta potentials, and lower Hamaker constants. Retardation effects on van der Waals attraction were considered.     

Measuring the electrophoretic mobility of concentrated suspensions in nonaqueous media

We present a new method of measuring the electrophoretic mobility of a particle in a concentrated suspension. The method is used to measure the electrophoretic mobility of PMMA particles (diameter 10 microm) suspended in a mixture of liquid hydrocarbons. The particle volume fraction of the suspension is varied from 0 up to 0.30 and the resulting variation of the electrophoretic mobility is discussed. The suspending liquid is such that its refractive index is very close to that of the particles. Thus the suspension is almost transparent and it is possible to follow through a microscope the motion of one particle. The suspension is subjected to a low-frequency electric field (0.5 Hz). The cell containing the suspension is mounted on a piezoelectric crystal. The displacement that compensates for the particle motion (when the particle image is steady) is determined.     

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.     

Size-dependent electrophoretic motion of polystyrene particles at polyethylene glycol–dextran interfaces

Currently, there is very limited information on the electrophoretic behavior of particles at a liquid–liquid interface formed by two conducting liquid solutions. Here, electrophoretic velocities of polystyrene particles at a polyethylene glycol (PEG)–dextran (DEX) interface were investigated in this paper. Experimental results show that the particle at the interface moves in the opposite direction to the applied electric field, with a velocity much lower than that in the PEG-rich phase and a litter larger than that in the DEX-rich phase. Similarly to the movement in Newtonian fluids, the velocity increases linearly with the increase in the applied electric field. Different to particle electrophoresis in Newtonian fluids, the velocities of the particles at the PEG–DEX interface increase linearly with the decrease in particle's diameters, implying a possible size-based particle differentiation at an interface.     

Effect of surface tension and surface elasticity of a fluid-fluid interface on the motion of a particle immersed near the interface

The motion of a particle immersed in a fluid near a fluid-fluid interface is studied on the basis of the linearized Navier-Stokes equations. The motion is influenced by surface tension, dilatational surface elasticity modulus, and surface shear modulus, as well as by gravity. The backflow at the location of the particle after a sudden impulse has some universal features that are the same as for a rigid wall with stick boundary conditions. At short times the flow depends only on the mass densities of the two fluids. The nature of the short-time flow is calculated from potential flow theory. At a somewhat later time the particle shows a pronounced rebound. The maximum value of the rebound and the time at which the maximum occurs depend on the elastic properties of the interface.     

Dynamic electrophoretic mobility of a concentrated dispersion of particles with a charge-regulated surface at arbitrary potential

The dynamic electrophoretic mobility of a concentrated dispersion of biocolloids such as cells and microorganisms is modeled theoretically. Here, a biological particle is simulated by a particle, the surface of which contains dissociable functional groups. The results derived provide basic theory for the quantification of the surface properties of a biocolloid through an electroacoustic device, which has the merit of making direct measurement on a concentrated dispersion without dilution. Two key parameters are defined to characterize the phenomenon under consideration: the first, A, is associated with the pH of the dispersion, and the second, B, is associated with the equilibrium constant of the dissociation reaction of the functional group. We show that if A is large and/or B is small, the surface potential is high, and the effect of double-layer polarization becomes significant. In this case the dynamic electrophoretic mobility may have a local maximum and a phase lead as the frequency of the applied electric field varies. Due to the hydrodynamic interaction between neighboring particles, the dynamic electrophoretic mobility decreases with the concentration of dispersion.