The use of the finite element method for calculating the thermophoresis velocity of large aerosol particles

The finite element method has been employed to calculate the thermophoresis velocity of solid aerosol particles, the sizes of which are much larger than the mean free path of molecules in a gas. The thermophoretic velocities of axially symmetric particles moving along their rotation axes have been numerically calculated. Cylindrical particles, particles having a shape resulting from rhomb rotation around one of its diagonals, and spheroidal particles have been considered. The data obtained for spheroidal particles have been compared with the available results of analytical calculations.     

Electrokinetic phenomena of soft particles

A review is given on the recent developments of theories of electrokinetics of soft particles (i.e., hard particles covered with an ion-penetrable surface layer of polyelectrolytes) with particular emphasis on the electrophoretic relaxation effects for highly charged soft particles and the effects of inhomogeneous distribution of polymer segments in the surface layer. The electrostatic interaction between soft particles is also discussed. Not only the interaction before contact of the surface layers of the interacting particles but also the interaction after their contact is considered.     

Interaction between an ion-penetrable particle and a solid particle. 1. Electrical double-layer interaction

Expressions are derived for the force and potential energy of the electrical double layer interaction between two parallel plates of different nature, i. e., an ion-penetrable plate and an ion-impenetrable plate. The latter may have either constant surface potential or constant surface charge density. It is shown that when the ion-impenetrable plate has a constant surface potential, the interaction force may, under certain conditions, become attractive even if the surface potentials of the two plates at infinite separation are of the same sign. In contrast, when the ion-impenetrable plate has a constant surface charge density, the interaction force may, under certain conditions, become repulsive even if the two plates at infinite separation are of opposite sign. This means that an ion-penetrable plate shows a dual behavior. That is, under certain conditions, it behaves like a solid plate with constant surface potential or surface charge density, depending on whether it interacts with a solid plate having a constant surface potential or a constant surface charge density.     

Interaction between an ion-penetrable particle and a solid particle. II. Criteria for heterocoagulation

The potential energy of the total interaction between two spherical colloidal particles of different nature is calculated, i. e., of an ion-penetrable particle and an ion-impenetrable solid particle having a constant surface potential or constant surface charge density. The criteria for heterocoagulation are derived. The obtained results suggest a possibility of selective coagulation in the mixed system.     

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.     

Electrostatic interaction between two spherical colloidal particles

Explicit exact analytic expressions are obtained in the form of infinite series for the potential distribution and the potential energy of the electrostatic interaction for the system of two dissimilar spheres in an electrolyte solution on the basis of the linearized Poisson—Boltzmann equation without recourse to Derjaguin's approximation. The leading term of the expression for the interaction energy (the zeroth order approximation) corresponds to the interaction energy that would be obtained if both spheres were ion-penetrable spheres (“soft” spheres). This term is a screened Coulomb interaction due to a simple linear superposition of the unperturbed potentials of the respective spheres, which is proportional to the product of their unperturbed surface potentials. The first-order approximation corresponds to the interaction energy that would be obtained if either sphere were a soft particle (the other being hard). The first-order correction term consists of two sub-terms, each of which is proportional to the square of the unperturbed surface potential of either sphere and does not depend on the unperturbed surface potential of the other sphere, can be interpreted as the interaction between the soft sphere and its image with respect to the hard sphere. This image interaction is attractive if the surface potential of the hard sphere is constant and repulsive if the surface charge density of the sphere is constant. It is shown that Derjaguin's method as well as its extension to the interaction of unequal spheres by Hogg, Healy and Fuerstenau (HHF) is quite a good approximation.     

The use of the finite-element method for calculating the photophoresis velocity of large aerosol particles

The finite-element method has been employed to calculate the photophoresis velocity of solid aerosol particles, the sizes of which are much larger than the mean free path of molecules in a gas. The thermal electromagnetic radiation from the particle surface and the temperature dependences of the density, viscosity, and thermal conductivity of the gaseous medium and particle material have been taken into account. The photophoresis velocity has been numerically calculated for a number of axially symmetric particles moving along their rotation axes. Cylindrical particles, particles having a shape resulting from rhomb rotation around one of its diagonals, and spheroidal particles have been considered.     

Depletion Interactions Produced by Nonadsorbing Charged and Uncharged Spheroids

The effect of macromolecule shape on the depletion attraction between two hard spherical particles in a solution with nonadsorbing hard spheroidal macromolecules of arbitrary size and aspect ratio was investigated using a modified form of the force-balance model of J. Y. Walz and A. Sharma (1994, J. Colloid Interface Sci. 168, 495). The macromolecules were represented as general spheroids, which could be either charged or uncharged. For the uncharged case, a set of analytical expressions describing the depletion attraction, valid for particles much larger than the characteristic macromolecule size, was developed. Comparisons with the case of spherical macromolecules were made under the condition of either constant macromolecule number density, rho(b), or constant volume fraction, phi. It was found that increasing the spheroidal macromolecule aspect ratio (major axis length/minor axis length) decreases the depletion attraction at constant rho(b), but increases the interaction at constant phi. In the latter case, the interaction produced by prolate macromolecules is greater than that produced by oblate macromolecules of equal axis lengths, while the opposite is true at constant rho(b). A simple scaling analysis is used to explain these trends. Surface charge is found to increase both the range and the magnitude of the depletion attraction; however, the general trends are the same as those found in the uncharged systems. Finally, the effect of the depletion attraction produced by spherical and spheroidal macromolecules on the stability of a dispersion of charged particles was examined. It was found that charged spheroids at concentrations of order 1% volume can produce secondary energy wells of sufficient magnitude to induce flocculation in a dispersion of charged spherical particles. Copyright 2000 Academic Press.     

Three-dimensional discrete charge effects on the electrostatic interaction between two soft particles

An expression for the electrostatic interaction energy between two parallel plate-like soft particles (i.e., hard particles covered with an ion-penetrable surface layer of polyelectrolytes) in an electrolyte solution is derived by using the linearized Poisson-Boltzmann equation. This expression is based on a discrete charge model in which the surface layer consists of a cubic lattice of fixed point charges. We show that the deviation of the results of the discrete charge model from those of the conventional smeared charge model becomes significant as the ratio of the lattice spacing to the Debye length becomes large. As this ratio decreases, on the other hand, the discrete charge model approaches a smeared charge model, leading to the Donnan-potential regulated interaction model.     

An Exact Solution to the Electrostatic Interaction between an Ion-Penetrable Sphere and an Ion-Penetrable Rod

No exact solution for the free energy of electrostatic interaction for a charged sphere and rod geometry in an electrolyte solution has yet been proposed. This geometry is interesting because it can be applied to describe macromolecules interacting with a random fiber-matrix for modeling of hindered transport in diffusional systems. Here we present an analytical approach that yields an exact solution to the problem for ion-penetrable-also called "soft"-sphere and infinitely long rod. This solution is compared to a published finite-element analysis of the same system with nonpenetrable-also called "hard"-sphere and infinitely long rod maintaining a constant surface charge density restriction. For any ionic strength or ratio of rod radius to sphere radius the ion-penetrable method yields an electrostatic free energy of interaction which is lower than that given by the analysis for hard bodies. This free energy is significantly lower for most parameter value combinations and therefore suggests that one should carefully examine the system being modeled to determine if it is approximated better by a hard body or ion-penetrable body approach. Copyright 2000 Academic Press.     

Schiller layers in β-ferric oxyhydroxide sol as an order—disorder phase separating system

The structure and properties of Schiller layers in β-ferric oxyhydroxide (β-FeOOH) sol are studied by microscopic observation. The structure is determined to be smectic. Rod-like particles of β-FeOOH sol gather to form mat-like assemblies, in each of which the rods are packed with their axes nearly perpendicular to the mat face. The mats are superimposed one on the other, to make up the whole structure. Two spatial periodicities in this structure would be the cause of the iridescence. This ordered structure coexisted in equilibrium with disordered suspension of the same sol. The Schiller layer systems usually have very low pH values (≲ 2) and in these cases attractive interaction or the effect of the secondary minimum is eminent. However, at moderately low pH (∼ 4), the interaction between the particles becomes repulsive, and, in spite of this, Schiller layers are formed when the sol is condensed to a certain extent. Formation of Schiller layers is a phase transition in monodisperse sol.     

Electrical interaction between two cylinders with an ion-penetrable charged membrane in an oil/water interface

The electrical interaction between two long, parallel cylinders each is covered by an ion-penetrable charged membrane immersed in an oil/water interface is investigated. The effects of contact angle, radius of cylinder, and membrane thickness on the electrical interaction force are examined. The results of numerical simulation reveal that the following conditions lead to a greater electrical interaction force: (i) a larger contact angle, i.e. a larger fraction of a cylinder in the oil phase; (ii) a larger cylinder radius; and (iii) a thinner membrane. For a fixed ionic strength, the electrical interaction force is insensible to the type of electrolytes in the water phase, in general. However, if two cylinders are close enough, then the higher the valence of counterions the greater the electrical interaction force.     

Effect of ionic size on the deposition of charge-regulated particles to a charged surface

The deposition of charge-regulated particles to a rigid, planar charged surface is modeled theoretically, taking the effects of the excluded area arising from deposited particles and finite ionic sizes into account. Here, a particle comprises a rigid core and an ion-penetrable charged membrane layer, which represents a general type of particle. If the membrane layer has a negligible thickness, the particle simulates a regular inorganic particle, and if the membrane layer has a finite thickness, it simulates biocolloids such as cells. The results of numerical simulation reveal that the rate of particle deposition is faster under the following conditions: (1) lower potential of the planar surface, (2) thicker membrane, (3) higher counterion valance, (4) lower fixed charge density, (5) smaller counterions, (6) larger co-ions, (7) larger functional group, and (8) lower pH. Neglecting the sizes of ionic species may lead to an appreciable deviation in both the electrical repulsive force between particle and surface and the rate of deposition. Typical deviation for the former is approximately 20%, and that for the latter is approximately -75%.     

Electrostatic Interaction between Two Dissimilar Spheres with Constant Surface Charge Density

Explicit exact analytic expressions are obtained in the form of infinite series for the potential energy of the electrostatic interaction for the system of two dissimilar hard spheres with constant surface charge density in an electrolyte solution on the basis of the linearized Poisson-Boltzmann equation. The effects of the particle polarization, that is, the internal fields induced within tim interacting spheres, which are found to be of the order of instead of 1/κa (where κ is the Debye-Hückel parameter and a is the sphere radius), are taken into account. As in the case of the interaction at constant surface potential, the zeroth-order approximation to the interaction energy corresponds to the interaction energy that would be obtained if both spheres were ion-penetrable spheres ("soft" spheres) and to that obtained by the linear superposition approximation. The first-order approximation corresponds to the interaction energy that would be obtained if either sphere were a soft sphere, with the other being a hard sphere with constant surface charge density. The first-order correction term can be interpreted as the image interaction between the soft sphere and its image with respect to the hard sphere.     

Particle interactions in electrophoresis due to inertia

The inertial forces acting on two cylinders and two spheres have been calculated. The cylinders or spheres are of the same radius and zeta potential, and arbitrarily oriented in an electric field. It is found that when two particles are aligned perpendicular to the direction of the electric field, the force between them along the line of the centers is attractive. When they are along the direction of the electric field, the force between them along the line of the centers is repulsive. On a pair of arbitrarily oriented particles, the force perpendicular to the line of the centers tends to rotate the particles around the midpoint between the particle centers such that the pair is aligned normal to the applied electric field. The stable equilibrium orientation of a pair of particles and the attractive interaction forces between the particles when they are stably orientated may give rise to aggregation of particles during electrophoretic motion of a suspension.     

Continuum Field Charging of Dielectric Spheroids

Classical continuum theory for field charging is applied in an analysis of the ionic charging of spheroidal dielectrics. Assuming that the particle orientation is fixed during the charging process, the saturation charge and charging rate are determined as functions of the orientation and aspect ratio of spheroids. For spheroids of small dielectric constants the saturation charge becomes the largest when the electric field is directed perpendicular to the major axis of the spheroid. For an ensemble of randomly oriented spheroids the average saturation charge can be approximated as the arithmetic average of the saturation charges for the spheroid with the electric field directed along the three principal axes of the spheroid. In addition, the ensemble average of the dimensionless charging rate of randomly oriented spheroids of moderate axial ratio approximates the dimensionless charging rate of a sphere. Copyright 2000 Academic Press.     

Electrostatic Interaction and Partitioning of Ion-Penetrable Spheres

Electrostatic interaction between two ion-penetrable spheres near a horizontal plate or in a slit pore is investigated theoretically. The orientation of the line connecting the two particle centers can be arbitrary relative to the plate(s). The electrostatic interaction energy and force on each particle are obtained analytically by the method of images. Emphasis is placed on the effect of the presence of the second particle, compared to the case of a single particle or the case without any plate(s). It is found that the horizontal electrical force on each particle is always repulsive. This repulsive force is enhanced by the plate(s) of constant surface charge density, while it is reduced by the plate(s) of constant surface potential. The electrostatic interaction together with the steric effect is used to determine the partition coefficient for the case of a slit pore, correct to O(C(infinity)), where C(infinity) is the volume fraction of particles in the bulk solution. The positive correction coefficient is larger for conducting plates than for insulating plates. Copyright 2001 Academic Press.     

Electrostatic interactions in the adhesion of an ion-penetrable and ion-impenetrable bacterial strain to glass

Deposition to glass of Streptococcus salivarius HB-C12 and Staphylococcus epidermidis 3399 in a parallel plate flow chamber has been studied as a function of ionic strength. Electrophoretic mobility measurements revealed that S. epidermidis 3399 possesses a thick ion-penetrable layer, probably associated with its encapsulation, while S. salivarius HB-C12 has an ion-impenetrable surface. Streaming potential measurements indicated that also the glass surface was covered with a relatively thin, ion-penetrable layer. Theoretical initial deposition rates of both strains to glass were obtained by numerically solving the convective-diffusion equation, while accounting for the ion-penetrability of the interacting surfaces. Experimentally, the initial deposition rate of the ion-penetrable strain S. epidermidis 3399 was found to be higher and less dependent on ionic strength than of the ion-impenetrable S. salivarius HB-C12, in accordance with theoretical expectations. Agreement between theoretical and experimental deposition rates could be obtained when glass was considered ion-penetrable when interacting with the ion-penetrable organism S. epidermidis 3399, while glass behaved as an ion-impenetrable surface when interacting with the ion-impenetrable S. salivarius HB-C12. Probably, interaction with an ion-impenetrable strain drives the diffuse double layer charges into the limited volume of the thin ion-penetrable layer on the glass, readily filling it up and making it appear ion-impenetrable. During interaction of glass with another ion-penetrable surface, as of S. epidermidis 3399, diffuse double layer charges move into both ion-penetrable surfaces, resulting in a much lower mobile charge density in the ion-penetrable layer on the glass which consequently continues to behave as ion-penetrable.     

Silver nanodisks: size selection via centrifugation and optical properties

Silver nanodisks, having two different sizes, and spherical particles are synthesized by soft chemistry. By using centrifugation, nanodisks are mainly selected. The experimental absorption spectra of these nanodisks with different sizes are compared to those simulated using the discrete dipole approximation method. For small nanodisk sizes, the nanodisk shape is neglected and the simulated spectra closest to the experiments are obtained by assuming a spheroidal particle. Conversely, for larger nanodisks, the precise geometries represented by snip and aspect ratio are needed for good agreement between experiments and simulations.     

Computer simulations of nematic drops: coupling between drop shape and nematic order

We perform Monte Carlo computer simulations of nematic drops in equilibrium with their vapor using a Gay-Berne interaction between the rod-like molecules. To generate the drops, we initially perform NPT simulations close to the nematic-vapor coexistence region, allow the system to equilibrate and subsequently induce a sudden volume expansion, followed with NVT simulations. The resultant drops coexist with their vapor and are generally not spherical but elongated, have the rod-like particles tangentially aligned at the surface and an overall nematic orientation along the main axis of the drop. We find that the drop eccentricity increases with increasing molecular elongation, κ. For small κ the nematic texture in the drop is bipolar with two surface defects, or boojums, maximizing their distance along this same axis. For sufficiently high κ, the shape of the drop becomes singular in the vicinity of the defects, and there is a crossover to an almost homogeneous texture; this reflects a transition from a spheroidal to a spindle-like drop.