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.     

Rheology of dilute and semidilute suspensions of spherical polyelectrolyte brushes

The viscosity, η, of dilute aqueous suspensions of spherical polyelectrolyte brushes is discussed. The spherical polyelectrolyte brushes consist of a solid polystyrene core of 100-nm diameter onto which linear poly(acrylic acid) chains are grafted. The relative viscosity, η/η_S (η_S is the viscosity of the solvent) of these suspensions is studied as a function of shear rate in the presence of different amounts of added salt. A marked dependence on shear rate is found, in particular when going to higher concentrations. Extrapolation to zero shear rate leads to the relative zero-shear viscosity, η₀/η_S, which can be described in terms of an effective volume fraction, φ_eff, for all salt concentrations under consideration. Moreover, the hydrodynamic radii derived from φ_eff coincide with data obtained by dynamic light scattering in the infinitely dilute regime. Data taken at higher concentrations point to a shrinking of the brush layer owing to mutual interaction.     

Diffusion, sedimentation, and rheology of concentrated suspensions of core-shell particles

Short-time dynamic properties of concentrated suspensions of colloidal core-shell particles are studied using a precise force multipole method which accounts for many-particle hydrodynamic interactions. A core-shell particle is composed of a rigid, spherical dry core of radius a surrounded by a uniformly permeable shell of outer radius b and hydrodynamic penetration depth κ(-1). The solvent flow inside the permeable shell is described by the Brinkman-Debye-Bueche equation, and outside the particles by the Stokes equation. The particles are assumed to interact non-hydrodynamically by a hard-sphere no-overlap potential of radius b. Numerical results are presented for the high-frequency shear viscosity, η(∞), sedimentation coefficient, K, and the short-time translational and rotational self-diffusion coefficients, D(t) and D(r). The simulation results cover the full three-parametric fluid-phase space of the composite particle model, with the volume fraction extending up to 0.45, and the whole range of values for κb, and a/b. Many-particle hydrodynamic interaction effects on the transport properties are explored, and the hydrodynamic influence of the core in concentrated systems is discussed. Our simulation results show that for thin or hardly permeable shells, the core-shell systems can be approximated neither by no-shell nor by no-core models. However, one of our findings is that for κ(b - a) ? 5, the core is practically not sensed any more by the weakly penetrating fluid. This result is explained using an asymptotic analysis of the scattering coefficients entering into the multipole method of solving the Stokes equations. We show that in most cases, the influence of the core grows only weakly with increasing concentration.     

Porous agglomerates in the general linear flow field

We calculate the flow within and around a porous spherical agglomerate suspended in the general linear flow field, and also the flow induced by its rotation. We use the Stokes equations exterior to the particle and the Brinkman equations inside it. The effect of particle permeability on the flow is expressed via the Brinkman parameter beta = r(0)/square root of k, where r0 is particle radius and k is its permeability. With translational creeping motion of porous spheres in a quiet fluid investigated by Debye and Bueche [P. Debye, A.M. Bueche, J. Chem. Phys. 16 (6) (1943) 573-579], this study provides information necessary for investigating dynamics of porous particles moving in creeping shear flows under the action of external forces and torques. The agglomerate flow field solutions are used to calculate the effective viscosity of a dilute suspension of porous solid aggregates, which generalizes the well-known Einstein's equation for solid suspensions. The agglomerate effective viscosity diameter is proposed which allows using the Einstein's formula evaluation of the agglomerates suspension viscosity.     

Efficiency of aerosol particle collection with filtering materials containing dispersed inclusions

During the flow around spherical inclusions in a homogeneous porous medium, the gas flow field is calculated using the Brinkman equation. The expression is derived for the effective permeability of the porous medium containing the rarefied ensemble of randomly arranged dispersed spherical particles of arbitrary sizes. It is shown that the Darcy approximation is applicable to the large-size inclusions. The influence of dispersed inclusions of spheroidal shape on both the pressure drop and the efficiency of aerosol particle collection with a filter is studied in the Darcy approximation. It is shown that dispersed inclusions can both increase and decrease the quality criterion of a filter depending on the values of parameters.     

Longitudinal excitations in Mg-Al-O refractory oxide melts studied by inelastic x-ray scattering

The dynamic structure factor S(Q,omega) of the refractory oxide melts MgAl2O4 and MgAl4O7 is studied by inelastic x-ray scattering with aerodynamic levitation and laser heating. This technique allows the authors to measure simultaneously the elastic response and transport properties of melts under extreme temperatures. Over the wave vector Q range of 1-8 nm-1 the data can be fitted with a generalized hydrodynamic model that incorporates a slow component described by a single relaxation time and an effectively instantaneous fast component. Their study provides estimates of high-frequency sound velocities and viscosities of the Mg-Al-O melts. In contrast to liquid metals, the dispersion of the high-frequency sound mode is found to be linear, and the generalized viscosity to be Q independent. Both experiment and simulation show a weak viscosity maximum around the MgAl4O7 composition.     

Viscosity of nanofluids: particle shape and fractal aggregates

Colloidal dispersions of nanoparticles in thermal base fluids are known to alter their spherical shapes thereby affecting their surface properties. This aspect has been investigated with respect to the effective viscosity of nanofluids presuming the particles to acquire the shape of prolate spheroid. Also, the contributions of the interfacial layer formed around these particles and their possible agglomeration has been taken in to account. The analysis has been carried out by modifying the Krieger and Dougherty model. The relative viscosity of these nanofluids has been computed as a function of volume fraction, particle size and the eccentricity of the particle. The model also incorporates the concept of fractal dimensions. The results thus obtained compare significantly well with the available experimental data and reaffirms an improvement over earlier models.     

Primary electroviscous effect in a moderately concentrated suspension of charged spherical colloidal particles

On the basis of the standard theory of the primary electroviscous effect in a moderately concentrated suspension of charged spherical particles in an electrolyte solution presented by Ruiz-Reina et al. (Ruiz-Reina, E.; Carrique, F.; Rubio-Hernández, F. J.; Gómez-Merino, A. I.; García-Sánchez, P. J. Phys. Chem. B 2003, 107, 9528), which is applicable for the case where overlapping of the electrical double layers of adjacent particles can be neglected, the general expression for the effective viscosity or the primary electroviscous coefficient p of the suspension is derived. This expression is applicable for a suspension of spherical particles of radius a carrying arbitrary zeta potentials zeta at the particle volume fraction phi < or = 0.3 for the case of nonoverlapping double layers, that is, at kappaalpha > 10 (where kappa is the Debye-Hückel parameter). A simple approximate analytic expression for p applicable for particles with large kappaalpha and arbitrary zeta is presented. The obtained viscosity expression is a good approximation for moderately concentrated suspensions of the particle volume fraction phi < or = 0.3, where the relative error is negligible for kappaalpha > or =100 and even at kappaalpha = 50 the maximum error is approximately 20%. It is shown that a maximum of p, which appears when plotted as a function of the particle zeta potential, is due to the relaxation effect as in the case of the electrophoresis problem.     

Dynamic viscosity of dilute solutions of poly(L-glutamic acid)

The dynamic viscosity η′ of a dilute solution of poly(L-glutamic acid) (DP = 1370) in a mixed solvent made up of aqueous 0.2M NaCl and dioxane (2:1 by volume) is measured over the pH range 4.2–10 and in the frequency range 2–500 kHz. The frequency dependence of η′ in the helix region (low pH) is interpreted in terms of a model molecule consisting of n rigid helical segments connected by universal joints. The steady-flow viscosity, relaxation time, and high-frequency limiting viscosity at pH 4.75 (helical content 80%) are well explained by this model with n = 5. This value of n is consistent with that estimated from the nucleation parameter σ = 1.4 × 10^?3 obtained from the relation between reduced steady-flow viscosity and helical content. The high-frequency values of η′ in the coil region (high pH) are fitted by Peterlin's theory. The internal viscosity seems to arise in part from the polyelectrolytic character of the molecule. An additional relaxation at low frequencies in the coil region is ascribed to rotation of molecules elongated by the electrostatic interaction. The lower value of reduced steady-flow viscosity in the coil region in the mixed solvent compared with that in water is interpreted in terms of the lower degree of effective ionization and the selective solvation of water by the polypeptide. No anomaly is observed in the helix–coil transition region, indicating that the relaxation time for helix–coil equilibrium is less than 10^?6sec.     

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.     

A reference interaction site model approach to depletion forces induced by hard rodlike particles

We study the effective interactions among large hard spherical colloidal particles induced by small hard rodlike particles and compare them with those induced by small hard spherical particles to highlight the specific effects due to the anisotropic shape of the former. This is done by determining the effective pair potentials within the framework of the reference interaction site model approach. The rodlike particles are modeled as N nonoverlapping spherical units arranged in a straight line, so that their total length is N times their transversal diameter. These results are compared against those obtained in the Asakura-Oosawa limit.     

A Cell Model of a Membrane with Allowance for Variable Viscosity of Liquid in Porous Shells of Spherical Grains

The Happel–Brenner cell method has been employed to calculate the hydrodynamic permeability of a porous medium (membrane) composed of a set of partially porous spherical particles with solid impermeable cores. This representation is used to describe the globular structure of membranes containing soluble grains. The apparent viscosity of a liquid is suggested to increase as a power function of the depth of the porous shell from the viscosity of the pure liquid at the porous medium–liquid shell interface to some larger value at the boundary with the impermeable core. All known boundary conditions used for the cell surface, i.e., those proposed by Happel, Kuwabara, Kvashnin, and Cunningham, have been considered. Important limiting cases have been analyzed.     

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.     

Rotational and translational self-diffusion in concentrated suspensions of permeable particles

In our recent work on concentrated suspensions of uniformly porous colloidal spheres with excluded volume interactions, a variety of short-time dynamic properties were calculated, except for the rotational self-diffusion coefficient. This missing quantity is included in the present paper. Using a precise hydrodynamic force multipole simulation method, the rotational self-diffusion coefficient is evaluated for concentrated suspensions of permeable particles. Results are presented for particle volume fractions up to 45% and for a wide range of permeability values. From the simulation results and earlier results for the first-order virial coefficient, we find that the rotational self-diffusion coefficient of permeable spheres can be scaled to the corresponding coefficient of impermeable particles of the same size. We also show that a similar scaling applies to the translational self-diffusion coefficient considered earlier. From the scaling relations, accurate analytic approximations for the rotational and translational self-diffusion coefficients in concentrated systems are obtained, useful to the experimental analysis of permeable-particle diffusion. The simulation results for rotational diffusion of permeable particles are used to show that a generalized Stokes-Einstein-Debye relation between rotational self-diffusion coefficient and high-frequency viscosity is not satisfied.     

Short-time transport properties in dense suspensions: from neutral to charge-stabilized colloidal spheres

We present a detailed study of short-time dynamic properties in concentrated suspensions of charge-stabilized and of neutral colloidal spheres. The particles in many of these systems are subject to significant many-body hydrodynamic interactions. A recently developed accelerated Stokesian dynamics (ASD) simulation method is used to calculate hydrodynamic functions, wave-number-dependent collective diffusion coefficients, self-diffusion and sedimentation coefficients, and high-frequency limiting viscosities. The dynamic properties are discussed in dependence on the particle concentration and salt content. Our ASD simulation results are compared with existing theoretical predictions, notably those of the renormalized density fluctuation expansion method of Beenakker and Mazur [Physica A 126, 349 (1984)], and earlier simulation data on hard spheres. The range of applicability and the accuracy of various theoretical expressions for short-time properties are explored through comparison with the simulation data. We analyze, in particular, the validity of generalized Stokes-Einstein relations relating short-time diffusion properties to the high-frequency limiting viscosity, and we point to the distinctly different behavior of de-ionized charge-stabilized systems in comparison to hard spheres.     

Short-time rheology and diffusion in suspensions of Yukawa-type colloidal particles

A comprehensive study is presented on the short-time dynamics in suspensions of charged colloidal spheres. The explored parameter space covers the major part of the fluid-state regime, with colloid concentrations extending up to the freezing transition. The particles are assumed to interact directly by a hard-core plus screened Coulomb potential, and indirectly by solvent-mediated hydrodynamic interactions. By comparison with accurate accelerated Stokesian Dynamics (ASD) simulations of the hydrodynamic function H(q), and the high-frequency viscosity η(∞), we investigate the accuracy of two fast and easy-to-implement analytical schemes. The first scheme, referred to as the pairwise additive (PA) scheme, uses exact two-body hydrodynamic mobility tensors. It is in good agreement with the ASD simulations of H(q) and η(∞), for smaller volume fractions up to about 10% and 20%, respectively. The second scheme is a hybrid method combining the virtues of the δγ scheme by Beenakker and Mazur with those of the PA scheme. It leads to predictions in good agreement with the simulation data, for all considered concentrations, combining thus precision with computational efficiency. The hybrid method is used to test the accuracy of a generalized Stokes-Einstein (GSE) relation proposed by Kholodenko and Douglas, showing its severe violation in low salinity systems. For hard spheres, however, this GSE relation applies decently well.     

Hydrodynamic permeability of biporous membrane

This paper concerns the hydrodynamic permeability of biporous medium built up by porous cylindrical particles located in another porous medium by using cell model technique. It is continuation of the previous work of authors where biporous membrane was built up by porous spherical particles embedded in accompanying porous medium. Four known boundary conditions, namely, Happel’s, Kuwabara’s, Kvashnin’s and Cunningham/Mehta-Morse’s, are considered on the outer surface of the cell. The variation of hydrodynamic permeability of biporous medium (membrane) with viscosity ratio, Brinkman constants, and solid fraction are presented and discussed graphically. Comparison of the resulting hydrodynamic permeability is undertaken. Some previous results for dimensionless hydrodynamic permeability have been verified.     

Evaluation of light scattering models to characterize concentrated polymer particles embedded in a solid polymer matrix

This work deals with the application of the static light scattering (SLS) model of Vrij (VM) for the characterization of a spherical polydisperse concentrated polymer particle system. This model is the exact solution for the SLS of such mixture of particles in the Percus–Yevick approximation. The analyzed polymer particle samples are obtained by solution polymerization of isobornyl methacrylate in polyisobutylene. At the end of the polymerization, as a result of phase separation, a particle system of micrometer sized particles with a moderate distribution of sizes and a volume fraction between 5 and 10% is formed. The SLS data were also analyzed using the local monodisperse approximation (LMA), a well‐known approximation to the model of Vrij. As expected, the estimations with the VM gave better results than those performed with the LMA model for the parameters related to the shape of the particle size distribution as compared with independent determinations of these quantities obtained from scanning electron microscopy micrographs. However, the main motivation to use the more rigorous model seems to be the fact that the volume fraction of particles can be extracted from the data even when relative SLS measurements are used. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 958–963, 2010