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.     

Electrophoresis of soft particles: analytic approximations

An approximate analytic expression is derived for the electrophoretic mobility of a weakly charged spherical soft particle (i.e., a hard particle covered with a weakly charged polyelectrolyte layer) on the basis of the general mobility expression for soft particles (Ohshima, H., J. Colloid Interface Sci. 2000, 228, 190-193). The obtained mobility expression, which reproduces various approximate results so far derived and gives some new mobility formulas, covers all types of weakly charged soft particles with arbitrary values of the thickness of polymer layer, the radius of the particle core, the electrophoretic softness, and the Debye length, including spherical polyelectrolytes with no particle core as well as spherical hard particles with no polyelectrolyte layer.     

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.     

Electrostatic interactions between diffuse soft multi-layered (bio)particles: beyond Debye-Hückel approximation and Deryagin formulation

We report a steady-state theory for the evaluation of electrostatic interactions between identical or dissimilar spherical soft multi-layered (bio)particles, e.g. microgels or microorganisms. These generally consist of a rigid core surrounded by concentric ion-permeable layers that may differ in thickness, soft material density, chemical composition and degree of dissociation for the ionogenic groups. The formalism allows the account of diffuse interphases where distributions of ionogenic groups from one layer to the other are position-dependent. The model is valid for any number of ion-permeable layers around the core of the interacting soft particles and covers all limiting situations in terms of nature of interacting particles, i.e. homo- and hetero-interactions between hard, soft or entirely porous colloids. The theory is based on a rigorous numerical solution of the non-linearized Poisson-Boltzmann equation including radial and angular distortions of the electric field distribution within and outside the interacting soft particles in approach. The Gibbs energy of electrostatic interaction is obtained from a general expression derived following the method by Verwey and Overbeek based on appropriate electric double layer charging mechanisms. Original analytical solutions are provided here for cases where interaction takes place between soft multi-layered particles whose size and charge density are in line with Deryagin treatment and Debye-Hückel approximation. These situations include interactions between hard and soft particles, hard plate and soft particle or soft plate and soft particle. The flexibility of the formalism is highlighted by the discussion of few situations which clearly illustrate that electrostatic interaction between multi-layered particles may be partly or predominantly governed by potential distribution within the most internal layers. A major consequence is that both amplitude and sign of Gibbs electrostatic interaction energy may dramatically change depending on the interplay between characteristic Debye length, thickness of ion-permeable layers and their respective protolytic features (e.g. location, magnitude and sign of charge density). This formalism extends a recent model by Ohshima which is strictly limited to interaction between soft mono-shell particles within Deryagin and Debye-Hückel approximations under conditions where ionizable sites are completely dissociated.     

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.     

Electrical phenomena of soft particles. A soft step function model

Simple analytic expressions are derived for the electrophoretic mobility of a soft particle consisting of the hard particle core covered with an ion-penetrable surface layer of polyelectrolyte for the case where the electric potential is low. The effect of the distribution of the polymer segments is taken into account by modeling the surface layer as a soft step function with the inhomogeneous distribution width δ. It is shown that the electrophoretic mobility becomes lower than that for the hard step function model and that the maximum deviation of the soft step function model from the hard step function model, which is a function of λδ (where 1/λ is the softness parameter) and κ/λ (where κ is the Debye-Hückel parameter), is 2.7% at λδ = 0.1, 5.1% at λδ = 0.2, and 11% at λδ = 0.5. In the limit of very high electrolyte concentrations, the obtained mobility expression tends to the result derived from the conventional hard step function model. In addition, an analytic expression for the interaction energy between two similar soft plates is derived on the basis of the present soft step function model. The magnitude of the interaction energy is shown to decrease by a factor 1/(1 + κδ)(2). Approximate analytic expressions for the interaction energies between two similar soft spheres and between two similar soft cylinders are also derived with the help of Derjaguin's approximation.     

Counterion distributions and effective interactions of spherical polyelectrolyte brushes

We consider the electrosteric repulsion of colloidal particles whose surface carries a dense layer of long polyelectrolyte chains (spherical polyelectrolyte brushes). The theory of electrosteric repulsion of star polyelectrolytes developed recently is augmented to include particles with a finite core radius. It is shown that most of the counterions are confined within the brush layer. The strong osmotic pressure thus created within the brush layer dominates the repulsive interaction between two such particles. Because of this the pair interaction potential between spherical polyelectrolyte brushes can be given in terms of an analytic expression. The theoretical predictions are compared with available experimental data and semi-quantitative agreement between the two is found.     

Electrophoretic mobility of cylindrical soft particles

 A general theory for the electrophoresis of a cylindrical soft particle (i.e., a cylindrical hard colloidal particle coated with a layer of ion-penetrable polyelectrolytes) in an electrolyte solution in an applied transverse or tangential electric field is proposed. This theory unites two different electrophoresis theories for cylindrical hard particles and for cylindrical polyelectrolytes. That is, the general mobility expression obtained in this paper tends to the mobility expression for a cylindrical hard particle for the case where the polyelectrolyte layer is absent or the frictional coefficient in the poly-electrolyte layer becomes infinity, whereas it tends to that for a cylin-drical polyelectrolyte in the absence of the particle core. Simple approximate analytic mobility expressions are also presented. Received: 29 August 1996 Accepted: 7 November 1996     

Small angle neutron scattering studies on non-aqueous dispersions of calcium carbonate. Part III. Concentrated dispersions

Small angle neutron scattering has been used to examine calcium carbonate dispersions in toluene over the w/w concentration range 5 to 40 %. The particles were stabilised by a combination of a surface active agent and a linear polymeric molecule. Analysis of the experimental data gave results for the structure factorS(Q) and the pair correlation functiong(r).Theoretical computations of the structure factor were made using a hard sphere model and the mean spherical approximation approach. These analyses indicated that the interaction was essentially soft at low volume fractions but became hard at the higher volume fractions.     

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.     

Interaction of interpolyelectrolyte complexes formed by a linear polyelectrolyte and dendrimers or spheres

The interaction between polyelectrolyte complexes formed by a linear polyelectrolyte and a dendrimer or a spherical particle with the opposite charge has been investigated via computer simulation. The influence of the compositions of the complexes on the effective force of the interaction between them has been studied. It has been shown that the effective attraction between the complexes appears at short distances in the vicinity of the isoelectric point. This attraction is correlative in nature and stronger for the complexes of the linear polyelectrolyte with spherical particles than for the complexes with dendrimers.     

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.     

Tunable rheology of dense soft deformable colloids

In the last two decades, advances in synthetic, experimental and modeling/simulation methodologies have considerably enhanced our understanding of colloidal suspension rheology and put the field at the forefront of soft matter research. Recent accomplishments include the ability to tailor the flow of colloidal materials via controlled changes of particle microstructure and interactions. Whereas hard sphere suspensions have been the most widely studied colloidal system, there is no richer type of particles than soft colloids in this respect. Yet, despite the remarkable progress in the field, many outstanding challenges remain in our quest to link particle microstructure to macroscopic properties and eventually design appropriate soft composites. Addressing them will provide the route towards novel responsive systems with hierarchical structures and multiple functionalities. Here we discuss the key structural and rheological parameters which determine the tunable rheology of dense soft deformable colloids. We restrict our discussion to non-crystallizing suspensions of spherical particles without electrostatic or enthalpic interactions.     

Hard versus soft particle electrokinetics of silica colloids

To verify the existence of a gel layer at the surface of silica, dependences of the electrophoretic mobility of fresh and aged colloidal silica particles on the KCl concentration are measured. These dependences, corrected for the relaxation/polarization effect, are fitted by analytical expressions based on the model of hard, soft, and brush surfaces. A bad fit is obtained for both silicas when its surface is considered ideal (hard). Much better fits are achieved with the invariable soft layer model for the fresh silica but especially for the aged silica whose surface is less charged probably as a result of an extension and/or loosening of the layer. A perfect fit is found for aged silica when applying a trivial model of the soft polyelectrolyte layer combined with the scaling model of polyelectrolyte brushes.     

On the General Expression for the Electrophoretic Mobility of a Soft Particle

Electrokinetic equations for electrophoresis of a soft particle (that is, a hard particle covered with a layer of polyelectrolytes) have been solved previously under the conditions that the net force acting on the soft particle as a whole (the particle core plus the polyelectrolyte layer) must be zero and that the electrical force acting on the polymer segment is balanced with a frictional force exerted by the liquid flow (J. Colloid Interface Sci. 163, 474 (1994)). In the present work we replaced the latter condition by the alternative and more appropriate condition that pressure is continuous at the boundary between the surface layer and the surrounding electrolyte solution to solve the electrokinetic equations and obtained the general mobility expression for the electrophoretic mobility of a spherical soft particle. It is found that the general mobility expression thus obtained reproduces all of the approximate mobility expressions derived previously and, in addition, that the continuous pressure condition leads to the correct limiting behavior of the electrophoretic mobility in the case where the frictional coefficient tends to zero (this behavior cannot be derived from the force balance condition for the polyelectrolyte layer). Copyright 2000 Academic Press.     

定标粒子理论计算非水溶液的盐效应常数

本文应用定标粒子理论计算了非电解质溶质在盐(NaI、或KI)和环丁砜组成的非水电解质溶液中溶解度的盐效应常数。硬球作用项采用Masterton-Lee的方法。软球作用项采用胡英等的径向分布函数处理方法, 并考虑进了偶极-偶极、偶极-诱导偶极、电荷-偶极和电荷-诱导偶极等相互作用。分子的硬球直径σ和能量参数∈/k由经验方程计算。由理论值和实验结果比较得出: 当σ_2取0.563 nm、离子半径取电子密度标度时, 理论值与实验值符合得较好。     

Relevance of electrokinetic theory for "soft" particles to bacterial cells: implications for bacterial adhesion

Bacterial cells and other biological particles carry charged macromolecules on their surface that form a "soft" ion-permeable layer. In this paper, we test the applicability of an electrokinetic theory for soft particles to characterize the electrophoretic mobility (EPM) and adhesion kinetics of bacterial cells. The theory allows the calculation of two parameters--the electrophoretic softness and the fixed charged density--that define the characteristics of the polyelectrolyte layer at the soft particle surface. The theory also allows the calculation of an outer-surface potential that may better predict the electrostatic interaction of soft particles with solid surfaces. To verify its relevance for bacterial cells, the theory was applied to EPM measurements of two well-characterized Escherichia coli K12 mutants having lipopolysaccharide (LPS) layers of different lengths and molecular compositions. Results showed that the obtained softness and fixed charge density were not directly related to the known characteristics of the LPS of the selected strains. Interaction energy profiles calculated from Derjaguin-Landau-Verwey-Overbeek (DLVO) theory were used to interpret bacterial deposition (adhesion) rates on a pure quartz surface. The outer surface potential failed to predict the low attachment efficiencies of the two bacterial strains. The lack of success in the application of the theory for soft particles to bacterial cells is attributed to chemical and physical heterogeneities of the polyelectrolyte layer at the cell surface.     

Improvement on the Derjaguin's method for the interaction of spherical particles

A correction to the classical Derjaguin's method has been given for the electrical double layer interaction of two spherical particles. Simple analytic expressions are given. Results obtained respectively for identical and dissimilar spherical particles show that the expressions open up the usable area of familiar Derjaguin's formulas.