Diffuse Double-Layer Interaction between Two Identical Spherical Colloidal Particles

A simple method is given for calculating the potential energy of the diffuse double-layer interaction between two identical spherical colloidal particles in a symmetrical electrolyte solution with the help of Derjaguin's approximation. This method uses accurate analytic expressions for the corresponding interaction energy between two parallel similar plates obtained previously (Colloids Surf. A Physicochem. Eng. Asp. 146, 213 (1999); J. Colloid Interface Sci. 212, 130 (1999)). Agreement with numerical data provided by Honig and Mul (J. Colloid Interface Sci. 36, 258 (1971)) is excellent particularly for small particle separations. Copyright 2000 Academic Press.     

Electrostatic interaction between soft particles

Electrostatic interaction between two soft particles (i.e., polyelectrolyte-coated particles) in an electrolyte solution is discussed. An approximate analytic expression for the interaction energy between two dissimilar soft spheres is derived by applying Derjaguin's approximation to the corresponding interaction energy between two parallel dissimilar soft plates for the case where the density of fixed charges within the polyelectrolyte layer is low. The obtained expression covers various limiting cases that include hard sphere/hard sphere interaction, spherical polyelectrolyte/spherical polyelectrolyte interaction, soft sphere/spherical polyelectrolyte interaction, soft sphere/hard sphere interaction, and spherical polyelectrolyte/hard sphere interaction.     

Extended DLVO interactions between spherical particles and rough surfaces

An "extended DLVO" approach that includes Lifshitz-van der Waals, Lewis acid-base, and electrostatic double layer interactions is used to describe interaction energies between spherical particles and rough surfaces. Favorable, unfavorable, and intermediate deposition conditions are simulated using surface properties representing common aquatic colloids and polymeric membranes. The surface element integration (SEI) technique and Derjaguin's integration method are employed to calculate interaction energy. Numerical simulations using SEI demonstrate that nanometer scale surface roughness features can produce a distribution of interaction energy profiles. Local interaction energies are statistically analyzed to define representative interaction energy profiles-minimum, average, and maximum-for various combinations of simulated particles and surfaces. In all cases, the magnitude of the average interaction energy profile is reduced, but the reduction of energy depends on particle size, asperity size, and density of asperities. In some cases, a surface that is on average unfavorable for deposition (repulsive) may possess locally favorable (attractive) sites solely due to nanoscale surface roughness. A weighted average of the analytical sphere-sphere and sphere-plate expressions of Derjaguin reasonably approximates the average interaction energy profiles predicted by the SEI model, where the weighting factor is based on the fraction of interactions involving asperities.     

On the role of electrostriction terms in ionic electrostatic repulsion Electrostatic component of the disjoining pressure

Two methods, viz. Derjaguin's and Langmuir's, are considered for a theoretical calculation of the ionic electrostatic repulsion of charged particles in an electrolyte solution.

In a paper by Babchin, Langmuir's method was used based on the calculation of the excess hydrostatic pressure in the symmetry plane. It is shown that the effect of the electrostriction terms, as suggested by Babchin, results from a gross mathematical error made when integrating the Poisson-Boltzmann equation. It has further been shown that even when Derjaguin's method, characterized by a more general applicability, is correctly used, the electrostriction pressure in the equations of hydrodynamics and electrostatics are mutually compensated. Thus, the attempt to revise the fundamentals of the Derjaguin-Landau-Verney-Overbeek's theory is based on a misunderstanding.     

粒度不均一交换剂的离子交换反应动力学(Ⅰ)——粒内扩散或液膜扩散控制反应速度的同位素交换反应

本文研究了包含不同半径离子交换剂体系的同位素交换反应动力学。对于粒内扩散和液膜扩散控制反应速度的不均一体系,导出了表述离子交换剂和有限浴溶液中离子之间异相同位素交换反应交换度。     

平板型高电位胶粒双电层的相互作用

利用线性迭加法,提出了平行平板型高电位颗粒之间的弱相互作用的近似表达式.结合文献[3]给出的强相互作用表达式,对高电位平行平板型颗粒的相互作用给出了完整的描述,和精确数值解吻合相当好.强弱相互作用的接合点在κh=4,误差在接合点处最大,～10％.根据Derjaguin法和改进的Derjaguin法,求出了高电位球颗粒在恒电位条件下的相互作用能.     

Approximate expressions of determining the double layer energy and force of interaction between two charged colloidal spheres

The approximate expressions have been obtained to calculate the electrical double layer energy and force between two spherical colloidal particles based on the improved Derjaguin approximation. Results for identical spheres interacting under constant surface potential, constant surface charge are given. Comparison of present results with numerical results calculated by Carnie and Chan is made. The expressions are found to work quite well for the constant surface potential case, and for the constant charge case, we make correction for the expressions. The results given are satisfactory providedkh0.4.     

Approximate analytic expression for the stability ratio of colloidal dispersions

Approximate analytic expressions are derived for the stability ratios of dispersions of spherical colloidal particles with and without viscous interactions between particles on the basis of the DLVO theory of the potential energy of the electrostatic and van der Waals interactions between two approaching particles. The obtained approximate stability expressions agree with exact numerical results with negligible errors and are applicable irrespective of the magnitude of the potential maximum unlike the previous approximate stability expressions, which are applicable only when the potential maximum is much greater than the thermal energy.     

The Effect of Protein Concentration on Electrophoretic Mobility

A theory of sedimentation in a concentrated suspension of spherical soft particles (i.e., polyelectrolyte-coated particles) is developed to obtain general expressions for sedimentation velocity of soft particles and sedimentation potential in the suspension. An Onsager relation between sedimentation potential and electrophoretic mobility of spherical soft particles in concentrated suspensions is derived for the case of low potentials and nonoverlapping electrical double layers of adjacent particles. Copyright 2000 Academic Press.     

Effective medium theory expressions for the effective diffusion in chromatographic beds filled with porous, non-porous and porous-shell particles and cylinders. Part I: Theory

Using the permeability analogue of the diffusion and partitioning processes occurring in a chromatographic column, the different Effective Medium Theory (EMT) models that exist in literature for the electrical and thermal conductivity have been transformed into expressions that accurately predict the B-term band broadening in chromatographic columns. The expressions are written in such a form that they hold for both fully porous and porous-shell particles, and both spherical and cylindrical particles are considered. Mutually comparing the established EMT-expressions, it has been found that the most basic variant, i.e., the Maxwell-based expression, is already accurate to within 5% for the typical conditions encountered in liquid phase chromatography, independently of the exact microscopic morphology of the packing. For most typical values of the intra-particle diffusion rate and the species retention factors, it is even accurate to within 1%. If even higher accuracies are needed, more elaborate EMT-expressions are available. The modelling accuracy of all explicit EMT-expressions is much better than the residence time weighted (RTW) B-term expressions that have been used up to now in the field of chromatography, where the error is typically on the order of 10% and more. The EMT-models have also been used to establish expressions for the obstruction and tortuosity factor in packings of non-porous particles. The EMT has also been applied to the meso-porous zone only, yielding an expression for the intra-particle diffusion coefficient that can be used without having to specify any obstruction factor. It has also been shown that the EMT also provides a very simple but exact expression to represent the way in which the solid core obstructs the effective intra-particle diffusion in the case of porous-shell particles. This obstruction factor is given by γ(part)=2/(2+ρ3) for spherical particles and γ(part)=1/(1+ρ3) for cylinders. Back-transforming the obtained expressions, a set of simple explicit expressions has been obtained that allow to directly obtain the intra-particle diffusion coefficient (D(part)) from peak parking or B-term constant measurements. Using these expressions, it could be demonstrated that the traditionally employed RTW-model yields D(part)-values that display an erroneous retention factor dependency, even in cases where the RTW-model appears to be able to closely fit the peak parking measurements.     

Two interacting particles in a spherical pore

In this work we analytically evaluate, for the first time, the exact canonical partition function for two interacting spherical particles into a spherical pore. The interaction with the spherical substrate and between particles is described by an attractive square-well and a square-shoulder potential. In addition, we obtain exact expressions for both the one particle and an averaged two particle density distribution. We develop a thermodynamic approach to few-body systems by introducing a method based on thermodynamic measures [I. Urrutia, J. Chem. Phys. 134, 104503 (2010)] for nonhard interaction potentials. This analysis enables us to obtain expressions for the pressure, the surface tension, and the equivalent magnitudes for the total and Gaussian curvatures. As a by-product, we solve systems composed of two particles outside a fixed spherical obstacle. We study the low density limit for a many-body system confined to a spherical cavity and a many-body system surrounding a spherical obstacle. From this analysis we derive the exact first order dependence of the surface tension and Tolman length. Our findings show that the Tolman length goes to zero in the case of a purely hard wall spherical substrate, but contains a zero order term in density for square-well and square-shoulder wall-fluid potentials. This suggests that any nonhard wall-fluid potential should produce a non-null zero order term in the Tolman length.     

Strong and Weak Interactions of Colloidal Particles with High Surface Potentials

A simple method for calculating the interaction force and energy per unit area between two dissimilar plates with high potentials at constant surface potential presented. Using Derjaguin's method and the improved Derjaguin' method, the expressions of the interaction free energy between two dissimilar spheres with high surface potentials are derived. These formulae may be divided into two groups: those for the strong interaction and those for the weak interaction. The juncture of strong and weak interactions is at kappah=4 for dissimilar plates and at kappah=2.8 for spheres The relative error is largest at this point, about 10%. Copyright 2001 Academic Press.     

Diffusiophoresis in a concentrated suspension of colloidal spheres in nonelectrolyte gradients

The diffusiophoretic motion of a homogeneous suspension of identical spherical particles is considered under conditions of small Reynolds and Peclet numbers. The effects of interaction of the individual particles are taken into explicit account by employing a unit cell model which is known to provide good predictions for the sedimentation of monodisperse suspensions of spherical particles. The appropriate equations of conservation of mass and momentum are solved for each cell, in which a spherical particle is envisaged to be surrounded by a concentric shell of suspending fluid, and the diffusiophoretic velocity of the particle is calculated for various cases. Analytical expressions of this mean particle velocity are obtained in closed form as functions of the volume fraction of the particles. Comparisons between the ensemble-averaged diffusiophoretic velocity of a test particle in a dilute suspension and our cell-model results are made. Received: 30 June 1999 Accepted: 8 December 1999     

球胶态颗粒在中等电位条件下双电层相互作用能和力

根据有效表面电位的定义,使用改进的Derjaguin法,推导出在中等电位条件下等同和不等同球形胶态颗粒相互作用能和力的近似表达式.与精确的数值解相比,对等同球颗粒,表面电位≤100mV;对不等同球颗粒,表面电位≤75mV;最大相对误差均小于±10%,表明该近似式是目前较好的.     

The Electrical Double-Layer Interaction between a Spherical Particle and a Cylinder

Based on the well-known Debye-Hückel approximation and the Derjaguin's integration method, this paper presents an integral solution for the electrical double-layer (EDL) interaction between a spherical particle and a cylinder. The effects of the relative dimensions of the cylinder to the sphere on the EDL interaction are studied using this numerical solution. The detailed numerical results indicate that, in general, the curvature effect on the EDL interaction cannot be neglected at small separation distances. The widely used sphere-flat plate approximation will considerably overestimate the actual EDL interaction between a spherical particle and a cylinder. The ratio of the radius of the particle to the EDL thickness, tau=kappaa(p), also plays an important role in determining the EDL interaction at small dimensionless separation distances (相似文献   

Electrophoretic Mobility of Soft Particles in Concentrated Suspensions

A general theory is developed for the electrophoretic mobility of spherical soft particles (i.e., spherical hard colloidal particles of radius a coated with a layer of polyelectrolytes of thickness d) in concentrated suspensions in an electrolyte solution as a function of the particle volume fraction φ on the basis of Kuwabara's cell model. In the limit d-->0, the mobility expression obtained tends to that for spherical hard particles in concentrated suspensions, whereas in the limit a-->0, it becomes that for spherical polyelectrolytes (charged porous spheres with no particle core). Simple approximate analytic mobility expressions are derived for the case where relaxation effect is negligible. It is found that in practical cases, the φ dependence of the mobility is negligible for da, the mobility strongly decreases with increasing φ. Copyright 2000 Academic Press.     

Electrostatic potentials,fields and field gradients from a general crystalline charge density

Explicit expressions for the electrostatic potential, the electric field and the electric field gradient at the nuclear positions of a crystalline lattice are presented. They are derived for a charge density given as an expansion in terms of spherical harmonics around the nuclear sites and as a Fourier series in the interstitial. These expressions can be decomposed into contributions from the spherical region centered around the lattice site of interest, from spherical regions surrounding all the other lattice sites and a contribution from the interstitital region.     

The concept of negative hamaker coefficients. 1. history and present status

Based on Hamaker's classical paper (1937) on the London - Van der Waals attraction between spherical particles, the concept of negative Hamaker coefficients is introduced and the conditions for attaining negative values are given. This concept remains also valid against the background of Lifshitz' macroscopic theory of interaction between solid bodies. Numerical examples calculated on the basis of this theory are presented. Experimental verification of the concept as well as simplified expressions to establish the conditions for attaining negative values are also given. Further refinements of the theory leading to the concept of Hamaker coefficients rather than to that of Hamaker constants are dealt with and the relevant role of interfacial separation is discussed.     

Electrostatic interaction between an ion-penetrable sphere and an ion-impenetrable sphere

The exact expression for the electrostatic interaction between an ionpenetrable sphere and an ion-impenetrable rigid sphere is derived on the basis of the linearized Poisson-Boltzmann equation without recourse to Derjaguin's approximation.