Sedimentation velocity and potential in a concentrated suspension of charged liquid drops

The sedimentation behavior of a concentrated suspension of charged liquid drops is analyzed theoretically at arbitrary surface potential and arbitrary double-layer thickness; that is, the effects of double-layer polarization and double-layer overlapping are taken into account. Kuwabara's unit cell model is employed to model the suspension system, and a pseudospectral method based on the Chebyshev polynomial is adopted to solve the governing electrokinetic equations numerically. Several interesting phenomena, which are of significant influence if the internal flow inside a liquid drop is taken into account, are observed. Key factors are examined such as the thickness of the electric double layer, the magnitude of the surface potential, the volume fraction of liquid drops, and the viscosity of the internal fluid. The results presented here add another dimension to the previous studies, which include concentrated suspensions of rigid particles and mercury drops under low zeta potential, with the consideration of the internal flow of liquid drops and double-layer polarization, characterized by its viscosity and the zeta potential respectively. It is found, among other things, that the smaller the viscosity of the internal fluid is, the higher the sedimentation velocity of liquid drops. The higher the zeta potential is, the larger the decrease in sedimentation velocity. In particular, the sedimentation velocity of an inviscid drop (gas bubble) is about three times higher than that of a rigid one. The decrease in sedimentation velocity resulting from the effect of double-layer polarization achieves about 50% if the zeta potential is sufficiently high.     

Diffusiophoretic mobility of charged porous spheres in electrolyte gradients

The diffusiophoretic motion of a polyelectrolyte molecule or charged floc in an unbounded solution of a symmetrically charged electrolyte with a uniform prescribed concentration gradient is analytically studied. The model used for the particle is a porous sphere in which the density of the hydrodynamic frictional segments, and therefore also that of the fixed charges, is constant. The electrokinetic equations which govern the electrostatic potential profile, the ionic concentration distributions (or electrochemical potential energies), and the fluid velocity field inside and outside the porous particle are linearized by assuming that the system is only slightly distorted from equilibrium. Using a regular perturbation method, these linearized equations are solved for a charged porous sphere with the density of the fixed charges as the small perturbation parameter. An analytical expression for the diffusiophoretic mobility of the charged porous sphere in closed form is obtained from a balance between its electrostatic and hydrodynamic forces. This expression, which is correct to the second order of the fixed charge density of the particle, is valid for arbitrary values of kappaa and lambdaa, where kappa is the reciprocal of the Debye screening length, lambda is the reciprocal of the length characterizing the extent of flow penetration inside the particle, and a is the particle radius. Our result to the first order of the fixed charge density agrees with the corresponding solution for the electrophoretic mobility obtained in the literature. In general, the diffusiophoretic mobility of a porous particle becomes greater as the hindrance to the diffusive transport of the solute species inside the particle is more significant.     

Primary electroviscous effect in a suspension of charged porous spheres

Primary electroviscous effect for a dilute suspension of porous spheres with fixed volumetric charge density is investigated theoretically. In the absence of flow, the electrical potential and solution charge density are assumed to satisfy the linearized Poisson-Boltzmann equation. With incorporation of the electrical body force, the Brinkman equation and the Stokes equation are used to govern the fluid flow inside and outside a sphere. The theory is formulated by assuming weak deviation of the charge cloud from its equilibrium state. However, the electrical body force is not restricted to be small compared to the viscous force in the fluid momentum equation. The results show that the double layer distortion is increased with increasing particle permeability, thereby enhancing the relative importance of its stress contribution. Nonetheless, the intrinsic viscosity remains a decreasing function of permeability, similar to the case of uncharged particles.     

Membrane potential in charged porous membranes

For charged porous membranes, the separation efficiency to charged particles and ions is affected by the electrical properties of the membrane surface. Such properties are most commonly quantified in terms of zeta-potential. In this paper, it is shown that the zeta-potential can be calculated numerically from the membrane potential. The membrane potential expression for charged capillary membranes in contact with electrolyte solutions at different concentrations is established by applying the theory of non-equilibrium thermodynamic to the membrane process and considering the space-charge model. This model uses the Nernst–Planck and Navier–Stokes equations for transport through pores, and the non-linear Poisson–Boltzmann equation, which is numerically solved, for the electrostatic condition of the fluid inside pores. The integral expressions of the phenomenological coefficients coupling the differential flow (solute relative to solvent) and the electrical current with the osmotic pressure and the electrical potential gradients are established and calculated numerically. The mobilities of anions and cations are individually specified. The variations of the membrane potential (or the apparent transport number of ions in the membrane pores) are studied as a function of different parameters: zeta-potential, pore radius, mean concentration in the membrane, ratio of external concentrations and type of ions.     

Supramolecular crystal growth in concentrated suspensions of charged monodisperse spherical silica particles

Concentrated suspensions of charged monodisperse spherical silica particles (MSSP) stabilized by alkalis or ammonia are able to crystallize at a certain destabilization. Crystal structures with the particles fixed at certain distances from each other show an isotropic normal mechanism of continuous growth with a rough phase boundary. The crystallization is determined by three parameters,víz. the concentration of particles, temperature, the thickness of the ion atmosphere around the particles and the concentration of counterions. The crystallization of MSSP suspensions is considered as a model of the supramolecular crystallization in the field of synthesis of mesoporous structures.     

Sedimentation of concentrated monodisperse colloidal suspensions: role of collective particle interaction forces

The sedimentation velocities and concentration profiles of low-charge, monodisperse hydroxylate latex particle suspensions were investigated experimentally as a function of the particle concentration to study the effects of the collective particle interactions on suspension stability. We used the Kossel diffraction technique to measure the particle concentration profile and sedimentation rate. We conducted the sedimentation experiments using three different particle sizes. Collective hydrodynamic interactions dominate the particle-particle interactions at particle concentrations up to 6.5 vol%. However, at higher particle concentrations, additional collective particle-particle interactions resulting from the self-depletion attraction cause particle aggregation inside the suspension. The collective particle-particle interaction forces play a much more important role when relatively small particles (500 nm in diameter or less) are used. We developed a theoretical model based on the statistical particle dynamics simulation method to examine the role of the collective particle interactions in concentrated suspensions in the colloidal microstructure formation and sedimentation rates. The theoretical results agree with the experimentally-measured values of the settling velocities and concentration profiles.     

Sedimentation and drying dissipative patterns of binary suspensions of colloidal silica spheres having different sizes

The sedimentation and drying dissipative structural patterns were formed during the course of drying binary mixtures among colloidal silica spheres of 183 nm, 305 nm, and 1.205 μm in diameter in aqueous suspension on a watch glass, a glass dish, and a cover glass, respectively. The broad ring-like sedimentation patterns were formed within several hours in suspension state for all the substrates used. Colorful macroscopic broad ring-like drying patterns were formed for the three substrates. In a watch glass, macroscopic drying patterns were composed of the outer and inner layers of small and large spheres, respectively. The two colored layers were ascribed to the Bragg diffractions of light by the dried colloidal crystals of the corresponding spheres. The width ratio of the layers changed in proportion to the mixing ratio of each spheres. In a glass dish, wave-like macroscopic drying patterns were observed in the intermediate areas between the outside edges of the broad ring and the inner wall of the cell. On a cover glass, the sphere mixing ratios were analyzed from the widths of the drying broad rings of the small spheres at the outside edge. High and distinct broad rings of small spheres and the low and vague broad one formed at the outer edges and in the inner area, respectively. Drying dissipative pattern was clarified to be one of the novel analysis techniques of colloidal size in binary colloidal mixtures.     

Electrokinetic study on concentrated suspensions using colloid vibration potential measurements

-potentials of a silica suspension and three types of polystyrene latex suspensions with different surface charge groups were measured, as a function of the particle concentration () in the suspension over a wide range, using the colloid vibration potential (CVP) technique. The concentration dependence of the-potential in silica suspension is explained well by Levine et al.s [1] cell model theory, verifying the applicability of the cell model to the CVP in silica suspension. However, the-potential of latex suspensions ordinarily decreases as the particle concentration increases, even after being corrected by the term of (1-). This tendency is especially noticeable in the systems that have particles with high surface charge densities. Furthermore, the conductivity measurements of these suspensions reveal that the conductivity of these systems, especially in their highly charged state, increases as the particle concentration is increased; opposite in tendency to silica suspensions. These new findings can be explained as follows: on the highly charged surface of a latex particle, a polyelectrolyte-like (hairy) layer is present, which overlaps at some point. This permits interparticle surface conduction and results in the abnormal behavior of CVP in these systems.     

Crystallization in charged two-component suspensions

We report on the crystallization of colloidal crystals comprising of charged particles with different size ratio dispersed in thoroughly deionized water. Single components were characterized carefully and their nucleation behavior was investigated before the preparation of mixtures. Mixtures investigated at constant particle number densities showed body centred cubic structure, conductivity, and shear moduli comply with the assumption of a randomly substituted crystal. Most importantly, for the first time we obtain the dependence of the nucleation rate densities in dependence on the composition and (for one fixed composition) the particle number density. The process of nucleation in random substitutional crystals is found to be similar to the one-component case.     

Sedimentation of a charged colloidal sphere in a charged cavity

An analytical study is presented for the quasisteady sedimentation of a charged spherical particle located at the center of a charged spherical cavity. The overlap of the electric double layers is allowed, and the polarization (relaxation) effect in the double layers is considered. The electrokinetic equations that govern the ionic concentration distributions, electric potential profile, and fluid flow field in the electrolyte solution are linearized assuming that the system is only slightly distorted from equilibrium. Using a perturbation method, these linearized equations are solved for a symmetric electrolyte with the surface charge densities of the particle and cavity as the small perturbation parameters. An analytical expression for the settling velocity of the charged sphere is obtained from a balance among the gravitational, electrostatic, and hydrodynamic forces acting on it. Our results indicate that the presence of the particle charge reduces the magnitude of the sedimentation velocity of the particle in an uncharged cavity and the presence of the fixed charge at the cavity surface increases the magnitude of the sedimentation velocity of an uncharged particle in a charged cavity. For the case of a charged sphere settling in a charged cavity with equivalent surface charge densities, the net effect of the fixed charges will increase the sedimentation velocity of the particle. For the case of a charged sphere settling in a charged cavity with their surface charge densities in opposite signs, the net effect of the fixed charges in general reduces/increases the sedimentation velocity of the particle if the surface charge density of the particle has a greater/smaller magnitude than that of the cavity. The effect of the surface charge at the cavity wall on the sedimentation of a colloidal particle is found to increase with a decrease in the particle-to-cavity size ratio and can be significant in appropriate situations.     

Rheological peculiarities of concentrated suspensions

The rheological properties of concentrated suspensions of metal oxides dispersed in transformer oil, which are used as electrorheological fluids, are systematically studied. Colloidal particles have intermediate sizes between nano- and microsized scales. Low-amplitude dynamic measurements show that the storage moduli of the examined suspensions are independent of frequency and these materials should be considered as solidlike elastic media. The storage modulus is proportional to the five-powdered particle volume concentration. At the same time, a transition through an apparent yield stress with a reduction in the viscosity by approximately six orders of magnitude is distinctly seen upon shear deformation. The character of the rheological behavior depends on the regime of suspension deformation. At very low shear rates, a steady flow is possible; however, upon an increase in the rate, an unsteady regime is realized with development of self-oscillations. When constant shear stresses are preset, in some range of stresses, thickening of the medium takes place, which can also be accompanied by self-oscillations. In order to gain insight into the nature of this effect, measurements are performed for samples with different volume/surface ratios, which show that, in some deformation regimes, suspension is separated into layers and slipping occurs along a low-viscosity layer with a thickness of several dozen microns. Direct observations show a distinct structural inhomogeneity of the flow. The separation and motion of layers with different compositions explain the transition to the flow with the lowest apparent Newtonian viscosity. Thus, the deformation of concentrated suspensions is associated with self-oscillations of stresses and slipping along a low-viscosity interlayer.     

Aggregation kinetics and gel formation in modestly concentrated suspensions of oppositely charged model ceramic colloids: a numerical study

Aggregation kinetics and gel formation in aqueous suspensions that undergo heteroaggregation are studied by means of Brownian dynamics simulations. The simulated system, described in a previous paper [M. A. Piechowiak, A. Videcoq, F. Rossignol, C. Pagnoux, C. Carrion, M. Cerbelaud, R. Ferrando, Langmuir, 2010, 26(15), 12540-12547.], is constituted of two kinds of synthesized, almost equally sized colloids: silica particles that are negatively charged and alumina-coated silica particles that are positively charged. The interactions between colloids are modeled by the DLVO potential. Several compositions are analyzed, from silica-rich to alumina-rich cases. The particle volume fraction φ is varied in the range 6-12%. The study of the aggregation kinetics allows us to clarify the effect of those variations on the clustering process. Gelation is analyzed by detection of spanning clusters in each x-, y-, z-direction of the cubic simulation box. Percolating networks start to be observed from φ = 7%, a low value of the volume fraction close to the solid volume fraction experimentally measured in sediments of those suspensions.     

Colloidal stabilization of nanoparticles in concentrated suspensions

The stabilization of nanoparticles in concentrated aqueous suspensions is required in many manufacturing technologies and industrial products. Nanoparticles are commonly stabilized through the adsorption of a dispersant layer around the particle surface. The formation of a dispersant layer (adlayer) of appropriate thickness is crucial for the stabilization of suspensions containing high nanoparticle concentrations. Thick adlayers result in an excessive excluded volume around the particles, whereas thin adlayers lead to particle agglomeration. Both effects reduce the maximum concentration of nanoparticles in the suspension. However, conventional dispersants do not allow for a systematic control of the adlayer thickness on the particle surface. In this study, we synthesized dispersants with a molecular architecture that enables better control over the particle adlayer thickness. By tailoring the chemistry and length of these novel dispersants, we were able to prepare fluid suspensions (viscosity < 1 Pa.s at 100 s-1) with more than 40 vol % of 65-nm alumina particles in water, as opposed to the 30 vol % achieved with a state-of-the-art dispersing agent. This remarkably high concentration facilitates the fabrication of a wide range of products and intermediates in materials technology, cosmetics, pharmacy, and in all other areas where concentrated nanoparticle suspensions are required. On the basis of the proposed molecular architecture, one can also envisage other similar molecules that could be successfully applied for the functionalization of surfaces for biosensing, chromatography, medical imaging, drug delivery, and aqueous lubrication, among others.     

Sedimentation analysis of latex spheres and gold sols

Summary Analytical centrifugation was used for the analysis of latex spheres. It appeared that the latex spheres sedimented according to Stokes' law. Therefore the diameters calculated from the sedimentation coefficients deviated on average less than 5 % from the diameters observed with the electron microscope.Colloidal gold was also investigated and it was shown that the gold sol consisted of particles with discrete sizes. The masses of the particles expressed in the mass of the smallest particles formed the arithmetic series 1, 2, 3, 4...n. The fact that apparently colloidal gold particles cannot exist below a certain definite size was confirmed by gel filtration.It was concluded that analytical centrifugation can be safely used for the determination of the size distribution of relatively large particles, membrane vesicles included.

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Zusammenfassung Die Brauchbarkeit der analytischen Zentrifugation zur Bestimmung der Größenverteilung von relativ großen Partikeln wurde untersucht. Latexkügelchen sedimentierten entsprechend dem Stokes'schen Gesetz. Dementsprechend stimmten die aus den Sedimentationskoeffizienten berechneten Durchmesser binnen 5 % mit den elektronenphonetisch ermittelten Durchmessern überein.Kolloidales Gold ergab eine diskrete Größenverteilung der im Kolloid enthaltenen Goldpartikel. Die Massen der Partikel waren ganzzahlige Vielfache (1, 2, 3, 4...n) der Masse der kleinsten Partikel. Durch Gelfiltration wurde bestätigt, daß die Partikel eines Goldkolloids nicht unter einer bestimmten Mindestgröße vorkommen.Aus den beschriebenen Versuchen wurde abgeleitet, daß die analytische Zentrifugation eine sichere Methode zur Bestimmung der Größenverteilung relativ großer Partikel - Membranbläschen inbegriffen - darstellt.

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With 3 figures and 2 tables     

The viscoelastic properties of concentrated suspensions

The viscoelastic properties of concentrated, well dispersed, sterically stabilised polymethyl methacrylate particles were studied using a specially developed low frequency, controlled shear stress rheometer. The experimental results show a well defined viscoelastic transition over a narrow frequency range. Evidence is produced to show that the particles act as hard, non-interacting spheres. It is argued that the transition represents a change from a randomly packed assemblage with particle motion dominated by Browniandiffusion processes to a lower packing density shear flow regime.     

Diffusion of spheres in crowded suspensions of rods

Translational tracer diffusion of spherical macromolecules in crowded suspensions of rodlike colloids is investigated. Experiments are done using several kinds of spherical tracers in fd-virus suspensions. A wide range of size ratios L/2a of the length L of the rods and the diameter 2a of the tracer sphere is covered by combining several experimental methods: fluorescence correlation spectroscopy for small tracer spheres, dynamic light scattering for intermediate sized spheres, and video microscopy for large spheres. Fluorescence correlation spectroscopy is shown to measure long-time diffusion only for relatively small tracer spheres. Scaling of diffusion coefficients with a/xi, predicted for static networks, is not found for our dynamical network of rods (with xi the mesh size of the network). Self-diffusion of tracer spheres in the dynamical network of freely suspended rods is thus fundamentally different as compared to cross-linked networks. A theory is developed for the rod-concentration dependence of the translational diffusion coefficient at low rod concentrations for freely suspended rods. The proposed theory is based on a variational solution of the appropriate Smoluchowski equation without hydrodynamic interactions. The theory can, in principle, be further developed to describe diffusion through dynamical networks at higher rod concentrations with the inclusion of hydrodynamic interactions. Quantitative agreement with the experiments is found for large tracer spheres, and qualitative agreement for smaller spheres. This is probably due to the increasing importance of hydrodynamic interactions as compared to direct interactions as the size of the tracer sphere decreases.     

Sedimentation of spheres at small Reynolds number

The effect of fluid inertia on the settling of spheres in a viscous incompressible fluid is studied in the limit of small Reynolds number. The kinetic energy of flow depends on the positions of the spheres, and gives rise to forces on the spheres. In the dilute limit it suffices to study the corresponding pair interaction. The interaction is calculated from the Stokes flow for two spheres settling between plane walls in the point particle limit. The dissipative interaction between a pair of spheres is calculated from the Proudman-Pearson [I. Proudman and J. R. A. Pearson, J. Fluid Mech. 2, 237 (1957)] solution of the Navier-Stokes equations for flow about a sphere in unbounded geometry. The combination of kinetic and dissipative interaction gives rise to a repulsive force of range of the order of the sphere diameter divided by the Reynolds number.     

Rheology of concentrated carbon nanotube suspensions

The rheological properties of non-Brownian carbon nanotube suspensions are measured over a range of nanotube volume fractions spanning the transition from semidilute to concentrated. The polymer-stabilized nanotubes are "sticky" and form a quiescent elastic network with a well-defined shear modulus and yield stress that both depend strongly on nanotube volume fraction with different but related critical exponents. We compare controlled-strain-rate and controlled-stress measurements of yielding in shear flow, and we study the effect of slow periodic stress reversal on yielding and the arrest of flow. Our measurements support a universal scaling of both the linear viscoelastic and steady-shear viscometric response. The former allows us to extract the elastic shear modulus of semidilute nanotube networks for values that are near or below the resolution limit of the rheometers used, while the latter provides a similar extrapolation of the yield stress. A simple scaling argument is used to model the dependence of yield stress and elastic modulus on concentration.