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Singularidades associadas ao mecanismo de estabilização eletroestérico e seus efeitos sobre o comportamento reológico de suspensões concentradas.; Singularities related to the electrosteric stabilization mechanism and their effects over the rheological behaviour of concentrated suspensions.

Guimarães, Kleber Lanigra
Fonte: Biblioteca Digitais de Teses e Dissertações da USP Publicador: Biblioteca Digitais de Teses e Dissertações da USP
Tipo: Tese de Doutorado Formato: application/pdf
Publicado em 13/11/2008 PT
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O presente projeto de trabalho buscou avaliar e estabelecer relações de causa e efeito que justificassem a ocorrência de singularidades no comportamento reológico de formulações complexas contendo elevada carga de sólidos em suspensão incorporada. O mecanismo de estabilização coloidal empregado para preparação das formulações e posterior análise foi o mecanismo de estabilização eletroestérico. O caso particular estudado correspondeu à dispersão de cargas inorgânicas (óxido de alumínio) em meio aquoso auxiliada pela incorporação de conteúdo orgânico com ação dispersante (poliacrilato de amônio) e ligante (emulsão acrílica) respectivamente. A implementação de técnicas analíticas complementares permitiu o estudo detalhado de uma singularidade associada ao comportamento reológico do sistema, exclusivamente evidenciada em função da elevada carga de sólidos, oferecendo, portanto, subsídios para elaboração de um modelo supramolecular fundamentado em possíveis alterações de configuração das cadeias poliméricas para justificativa do fenômeno. A contribuição referente ao pleno entendimento do fenômeno, considerando-se suas causas e efeitos, reside na possibilidade de se estabelecer diretrizes de formulação que propiciem a otimização das quantidades de aditivos incorporados e o respectivo controle do estado de dispersão e estabilidade do sistema. Considerou-se...

Influência do coeficiente de atrito entre os agregados e da viscosidade da matriz no comportamento reológico suspensões concentradas heterogêneas.; Influence of aggregate's friction coefficient and matrix viscosity in the rheological behavior of heterogeneous concentrated suspensions.

Mendes, Thiago Melanda
Fonte: Biblioteca Digitais de Teses e Dissertações da USP Publicador: Biblioteca Digitais de Teses e Dissertações da USP
Tipo: Dissertação de Mestrado Formato: application/pdf
Publicado em 05/05/2008 PT
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O comportamento reológico de suspensões concentradas heterogêneas como concretos e argamassas possui influência direta na energia requerida e na produtividade durante o processamento e aplicação destes materiais, além de ser um fator determinante na obtenção de um produto final qualidade. De um modo geral pode-se dizer que a fluidez destas suspensões está diretamente relacionada à facilidade de movimentação das partículas, logo pode-se inferir que a redução dos fenômenos atrito é um fator determinante para o bom desempenho destas suspensões durante o fluxo. Deste modo, este trabalho buscou por meio de modelos ideais de esferas e silicone líquido avaliar a influência do coeficiente de atrito dos agregados e da viscosidade da matriz no comportamento reológico de suspensões concentradas. O coeficiente de atrito de agregados esféricos com diferentes granulometrias foi determinado experimentalmente através do ensaio de cisalhamento direto, também conhecido como shear box, e o comportamento reológico de suspensões compostas por estes agregados e silicones líquidos de diferentes viscosidades foram avaliadas por meio do ensaio de squeeze flow. Os resultados revelam que a distribuição granulométrica possui um influência no coeficiente de atrito dos agregados e que esta propriedades está relacionada a área e ao tipo de contato entre as partículas. Entretanto...

Morphological characterization of cells in concentrated suspensions using multispectral diffuse optical tomography

Hajihashemi, Mohammad Reza; Li, Xiaoqi; Jiang, Huabei
Fonte: PubMed Publicador: PubMed
Tipo: Artigo de Revista Científica
Publicado em 01/10/2012 EN
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Based on a non-spherical model of particle scattering, we investigate the capabilities and limitations of a T-matrix based inverse algorithm to morphologically characterize cells in concentrated suspensions. Here the cells are modeled as randomly orientated spheroidal particles with homogenous dielectric properties and suspended in turbid media. The inverse algorithm retrieves the geometrical parameters and the concentration of cells simultaneously by inverting the reduced scattering coefficient spectra obtained from multispectral diffuse optical tomography (MS-DOT). Both round and spheroidal cells are tested and the role of multiple and higher order scattering of particles on the performance of the algorithm is evaluated using different concentrations of cells.

General electrokinetic model for concentrated suspensions in aqueous electrolyte solutions: electrophoretic mobility and electrical conductivity in static electric fields

Carrique Fern??ndez, F??lix; Ruiz-Reina, Emilio; Roa, Rafael; Arroyo Rold??n, Francisco J.; Delgado, ??ngel V.
Fonte: Elsevier Publicador: Elsevier
Tipo: Pré-impressão
ENG
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In recent years different electrokinetic cell models for concentrated colloidal suspensions in aqueous electrolyte solutions have been developed. They share some of its premises with the standard electrokinetic model for dilute colloidal suspensions, in particular, neglecting both the specific role of the so-called added counterions (i.e., those released by the particles to the solution as they get charged), and the realistic chemistry of the aqueous solution on such electrokinetic phenomena as electrophoresis and electrical conductivity. These assumptions, while having been accepted for dilute conditions (volume fractions of solids well below 1 %, say), are now questioned when dealing with concentrated suspensions. In this work, we present a general electrokinetic cell model for such kind of systems, including the mentioned effects, and we also carry out a comparative study with the standard treatment (the standard solution only contains the ions that one purposely adds, without ionic contributions from particle charging or water chemistry). We also consider an intermediate model that neglects the realistic aqueous chemistry of the solution but accounts for the correct contribution of the added counterions. The results show the limits of applicability of the classical assumptions and allow one to better understand the relative role of the added counterions and ions stemming from the electrolyte in a realistic aqueous solution...

Well-posedness of a multiscale model for concentrated suspensions

Cancès, Eric; Catto, Isabelle; Gati, Yousra; Bris, Claude Le
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 21/03/2005
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In a previous work [math.AP/0305408] three of us have studied a nonlinear parabolic equation arising in the mesoscopic modelling of concentrated suspensions of particles that are subjected to a given time-dependent shear rate. In the present work we extend the model to allow for a more physically relevant situation when the shear rate actually depends on the macroscopic velocity of the fluid, and as a feedback the macroscopic velocity is influenced by the average stress in the fluid. The geometry considered is that of a planar Couette flow. The mathematical system under study couples the one-dimensional heat equation and a nonlinear Fokker-Planck type equation with nonhomogeneous, nonlocal and possibly degenerate, coefficients. We show the existence and the uniqueness of the global-in-time weak solution to such a system.; Comment: 1 figure

A Shear Thickening transition in concentrated suspensions under Impact

Mukhopadhyay, Shomeek; Allen, Benjamin; Brown, Eric
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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Discontinous Shear Thickening (DST) fluids show a remarkable effect where the suspension behaves like a liquid at low shear rates, but when sheared very hard, resistance to flow increases discontinously with shear rate. This effect has been observed in a large variety of concentrated suspensions of hard, frictional and non attractive particles. DST fluids also show strong impact response such as the ability of a person to run on the surface of a pool filled with a suspension of cornstarch and water. Current models of shear thickening based on lubrication hydrodynamics, dilatancy, and inertial displacement fail to explain the ability of a person to run on the surface and impact response in general. In our experiments we discover a transition at a critical impact velocity in DST fluids above which fronts of solid like regions are generated in the fluid and the collision of these fronts with a solid boundary leads to a shear thickening transition which has not been previously predicted or reported. The large stresses generated above the shear thickening transition are enough to hold up the weight of a person and have potential applications for protective materials.

Local transient rheological behavior of concentrated suspensions

Blanc, Frédéric; Peters, François; Lemaire, Elisabeth
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 29/11/2011
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This paper reports experiments on the shear transient response of concentrated non-Brownian suspensions. The shear viscosity of the suspensions is measured using a wide-gap Couette rheometer equipped with a Particle Image Velocimetry (PIV) device that allows measuring the velocity field. The suspensions made of PMMA particles (31$\mu$m in diameter) suspended in a Newtonian index- and density-matched liquid are transparent enough to allow an accurate measurement of the local velocity for particle concentrations as high as 50%. In the wide-gap Couette cell, the shear induced particle migration is evidenced by the measurement of the time evolution of the flow profile. A peculiar radial zone in the gap is identified where the viscosity remains constant. At this special location, the local particle volume fraction is taken to be the mean particle concentration. The local shear transient response of the suspensions when the shear flow is reversed is measured at this point where the particle volume fraction is well defined. The local rheological measurements presented here confirm the macroscopic measurements of Gadala-Maria and Acrivos (1980). After shear reversal, the viscosity undergoes a step-like reduction, decreases slower and passes through a minimum before increasing again to reach a plateau. Upon varying the particle concentration...

Effective charges and virial pressure of concentrated macroion solutions

Boon, Niels; Guerrero-Garcia, Guillermo Ivan; van Roij, Rene; de la Cruz, Monica Olvera
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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The stability of colloidal suspensions is crucial in a wide variety of processes including the fabrication of photonic materials and scaffolds for biological assemblies. The ionic strength of the electrolyte that suspends charged colloids is widely used to control the physical properties of colloidal suspensions. The extensively used two-body Derjaguin-Landau-Verwey-Overbeek (DLVO) approach allows for a quantitative analysis of the effective electrostatic forces between colloidal particles. DLVO relates the ionic double-layers, which enclose the particles, to their effective electrostatic repulsion. Nevertheless, the double layer is distorted at high macroion volume fractions. Therefore, DLVO cannot describe the many-body effects that arise in concentrated suspensions. We show that this problem can be largely resolved by identifying effective point charges for the macroions using cell theory. This extrapolated point charge (EPC) method assigns effective point charges in a consistent way, taking into account the excluded volume of highly charged macroions at any concentration, and thereby naturally accounting for high volume fractions in both salt-free and added-salt conditions. We provide an analytical expression for the effective pair potential and validate the EPC method by comparing molecular dynamics simulations of macroions and monovalent microions that interact via Coulombic potentials to simulations of macroions interacting via the derived EPC effective potential. The simulations reproduce the macroion-macroion spatial correlation and the virial pressure obtained with the EPC model. Our findings provide a route to relate the physical properties such as pressure in systems of screened-Coulomb particles to experimental measurements.; Comment: 3 figures

Orientational order in concentrated suspensions of spherical microswimmers

Evans, Arthur A.; Ishikawa, Takuji; Yamaguchi, Takami; Lauga, Eric
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 09/02/2013
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We use numerical simulations to probe the dynamics of concentrated suspensions of spherical microswimmers interacting hydrodynamically. Previous work in the dilute limit predicted orientational instabilities of aligned suspensions for both pusher and puller swimmers, which we confirm computationally. Unlike previous work, we show that isotropic suspensions of spherical swimmers are also always unstable. Both types of initial conditions develop long-time polar order, of a nature which depends on the hydrodynamic signature of the swimmer but very weakly on the volume fraction up to very high volume fractions.

The non-monotonic shear-thinning flow of two strongly cohesive concentrated suspensions

Buscall, Richard; Kusuma, Tiara E.; Stickland, Anthony D.; Rubasingha, Sayuri; Scales, Peter J.; Teo, Hui-En; Worrall, Graham L.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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The behaviour in simple shear of two concentrated and strongly cohesive mineral suspensions showing highly non-monotonic flow curves is described. Two rheometric test modes were employed, controlled stress and controlled shear-rate. In controlled stress mode the materials showed runaway flow above a yield stress, which, for one of the suspensions, varied substantially in value and seemingly at random from one run to the next, such that the up flow-curve appeared to be quite irreproducible. The down-curve was not though, as neither was the curve obtained in controlled rate mode, which turned out to be triple-valued in the region where runaway flow was seen in controlled rising stress. For this first suspension, the total stress could be decomposed into three parts to a good approximation: a viscous component proportional to a plastic viscosity, a constant isostatic contribution, and a third shear-rate dependent contribution associated with the particulate network which decreased with increasing shear-rate raised to the -7/10th power. In the case of the second suspension, the stress could be decomposed along similar lines, although the strain-rate softening of the solid-phase stress was found to be logarithmic and the irreducible isostatic stress was small. The flow curves are discussed in the light of recent simulations and they conform to a very simple but general rule for non-monotonic behaviour in cohesive suspensions and emulsions...

Ion size effects on the electric double layer of a spherical particle in a realistic salt-free concentrated suspension

Roa, Rafael; Carrique, Félix; Ruiz-Reina, Emilio
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 05/05/2011
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A new modified Poisson-Boltzmann equation accounting for the finite size of the ions valid for realistic salt-free concentrated suspensions has been derived, extending the formalism developed for pure salt-free suspensions [Roa et al., Phys. Chem. Chem. Phys., 2011, 13, 3960-3968] to real experimental conditions. These realistic suspensions include water dissociation ions and those generated by atmospheric carbon dioxide contamination, in addition to the added counterions released by the particles to the solution. The electric potential at the particle surface will be calculated for different ion sizes and compared with classical Poisson-Boltzmann predictions for point-like ions, as a function of particle charge and volume fraction. The realistic predictions turn out to be essential to achieve a closer picture of real salt-free suspensions, and even more important when ionic size effects are incorporated to the electric double layer description. We think that both corrections have to be taken into account when developing new realistic electrokinetic models, and surely will help in the comparison with experiments for low-salt or realistic salt-free systems.; Comment: 12 pages, 7 figures

Fluid Flows Created by Swimming Bacteria Drive Self-Organization in Confined Suspensions

Lushi, Enkeleida; Wioland, Hugo; Goldstein, Raymond E
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 14/07/2014
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Concentrated suspensions of swimming microorganisms and other forms of active matter are known to display complex, self-organized spatio-temporal patterns on scales large compared to those of the individual motile units. Despite intensive experimental and theoretical study, it has remained unclear the extent to which the hydrodynamic flows generated by swimming cells, rather than purely steric interactions between them, drive the self-organization. Here we utilize the recent discovery of a spiral-vortex state in confined suspensions of \textit{B. subtilis} to study this issue in detail. Those experiments showed that if the radius of confinement in a thin cylindrical chamber is below a critical value the suspension will spontaneously form a steady single-vortex state encircled by a counter-rotating cell boundary layer, with spiral cell orientation within the vortex. Left unclear, however, was the flagellar orientation, and hence the cell swimming direction, within the spiral vortex. Here, using a fast simulation method that captures oriented cell-cell and cell-fluid interactions in a minimal model of discrete-particle systems, we predict the striking, counterintuitive result that in the presence of collectively-generated fluid motion the cells within the spiral vortex actually swim upstream against those flows. This is then confirmed by new experiments reported here...

Granular front formation in free-surface flow of concentrated suspensions

Leonardi, A.; Cabrera, M.; Wittel, F. K.; Kaitna, R.; Mendoza, M.; Wu, W.; Herrmann, H. J.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 19/02/2015
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Granular fronts are a common yet unexplained phenomenon emerging during the gravity driven free-surface flow of concentrated suspensions. They are usually believed to be the result of fluid convection in combination with particle size segregation. However, suspensions composed of uniformly sized particles also develop a granular front. Within a large rotating drum, a stationary recirculating avalanche is generated. The flowing material is a mixture of a visco-plastic fluid obtained from a kaolin-water dispersion, with spherical ceramic particles denser than the fluid. The goal is to mimic the composition of many common granular-fluid materials, like fresh concrete or debris flow. In these materials, granular and fluid phases have the natural tendency to segregate due to particle settling. However, through the shearing caused by the rotation of the drum, a reorganization of the phases is induced, leading to the formation of a granular front. By tuning the material properties and the drum velocity, it is possible to control this phenomenon. The setting is reproduced in a numerical environment, where the fluid is solved by a Lattice-Boltzmann Method, and the particles are explicitly represented using the Discrete Element Method. The simulations confirm the findings of the experiments...

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

Abade, Gustavo C.; Cichocki, Bogdan; Ekiel-Jezewska, Maria L.; Naegele, Gerhard; Wajnryb, Eligiusz
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 19/03/2011
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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.; Comment: 4 figures

Jamming, two-fluid behaviour and 'self-filtration' in concentrated particulate suspensions

Haw, M. D.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 29/08/2003
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We study the flow of model experimental hard sphere colloidal suspensions at high volume fraction $\Phi$ driven through a constriction by a pressure gradient. Above a particle-size dependent limit $\Phi_0$, direct microscopic observations demonstrate jamming and unjamming--conversion of fluid to solid and vice versa--during flow. We show that such a jamming flow produces a reduction in colloid concentration $\Phi_{x}$ downstream of the constriction. We propose that this `self-filtration' effect is the consequence of a combination of jamming of the particulate part of the system and continuing flow of the liquid part, i.e. the solvent, through the pores of the jammed solid. Thus we link the concept of jamming in colloidal and granular media with a 'two-fluid'-like picture of the flow of concentrated suspensions. Results are also discussed in the light of Osborne Reynolds' original experiments on dilation in granular materials.; Comment: 4 pages, 3 figures

DC electrokinetics for spherical particles in salt-free concentrated suspensions including ion size effects

Roa, Rafael; Carrique, Félix; Ruiz-Reina, Emilio
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 31/10/2011
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We study the electrophoretic mobility of spherical particles and the electrical conductivity in salt-free concentrated suspensions including finite ion size effects. An ideal salt-free suspension is composed of just charged colloidal particles and the added counterions that counterbalance their surface charge. In a very recent paper [Roa et al., Phys. Chem. Chem. Phys., 2011, 13, 3960- 3968] we presented a model for the equilibrium electric double layer for this kind of suspensions considering the size of the counterions, and now we extend this work to analyze the response of the suspension under a static external electric field. The numerical results show the high importance of such corrections for moderate to high particle charges, especially when a region of closest approach of the counterions to the particle surface is considered. The present work sets the basis for further theoretical models with finite ion size corrections, concerning particularly the ac electrokinetics and rheology of such systems.; Comment: 14 pages, 11 figures

Shear thickening in concentrated suspensions: phenomenology, mechanisms, and relations to jamming

Brown, Eric; Jaeger, Heinrich M.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 01/07/2013
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Shear thickening is a type of non-Newtonian behavior in which the stress required to shear a fluid increases faster than linearly with shear rate. Many concentrated suspensions of particles exhibit an especially dramatic version, known as Discontinuous Shear Thickening (DST), in which the stress suddenly jumps with increasing shear rate and produces solid-like behavior. The best known example of such counter-intuitive response to applied stresses occurs in mixtures of cornstarch in water. Over the last several years, this shear-induced solid-like behavior together with a variety of other unusual fluid phenomena has generated considerable interest in the physics of densely packed suspensions. In this review, we discuss the common physical properties of systems exhibiting shear thickening, and different mechanisms and models proposed to describe it. We then suggest how these mechanisms may be related and generalized, and propose a general phase diagram for shear thickening systems. We also discuss how recent work has related the physics of shear thickening to that of granular materials and jammed systems. Since DST is described by models that require only simple generic interactions between particles, we outline the broader context of other concentrated many-particle systems such as foams and emulsions...

Quantitative imaging of concentrated suspensions under flow

Isa, Lucio; Besseling, Rut; Schofield, Andrew B; Poon, Wilson C K
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 30/07/2009
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We review recent advances in imaging the flow of concentrated suspensions, focussing on the use of confocal microscopy to obtain time-resolved information on the single-particle level in these systems. After motivating the need for quantitative (confocal) imaging in suspension rheology, we briefly describe the particles, sample environments, microscopy tools and analysis algorithms needed to perform this kind of experiments. The second part of the review focusses on microscopic aspects of the flow of concentrated model hard-sphere-like suspensions, and the relation to non-linear rheological phenomena such as yielding, shear localization, wall slip and shear-induced ordering. Both Brownian and non-Brownian systems will be described. We show how quantitative imaging can improve our understanding of the connection between microscopic dynamics and bulk flow.; Comment: Review on imaging hard-sphere suspensions, incl summary of methodology. Submitted for special volume 'High Solid Dispersions' ed. M. Cloitre, Vol. xx of 'Advances and Polymer Science' (Springer, Berlin, 2009); 22 pages, 16 figs

Electric double layer for spherical particles in salt-free concentrated suspensions including ion size effects

Roa, R.; Carrique, F.; Ruiz-Reina, E.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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The equilibrium electric double layer (EDL) that surrounds the colloidal particles is determinant for the response of a suspension under a variety of static or alternating external fields. An ideal salt-free suspension is composed by the charged colloidal particles and the ionic countercharge released by the charging mechanism. The existing macroscopic theoretical models can be improved by incorporating different ionic effects usually neglected in previous mean-field approaches, which are based on the Poisson-Boltzmann equation (PB). The influence of the finite size of the ions seems to be quite promising because it has been shown to predict phenomena like charge reversal, which has been out of the scope of classical PB approximations. In this work we numerically obtain the surface electric potential and the counterions concentration profiles around a charged particle in a concentrated salt-free suspension corrected by the finite size of the counterions. The results show the large importance of such corrections for moderate to high particle charges at every particle volume fraction, specially, when a region of closest approach of the counterions to the particle surface is considered. We conclude that finite ion size considerations are obeyed for the development of new theoretical models to study nonequilibrium properties in concentrated colloidal suspensions...

On the viscosity of concentrated suspensions of charged colloids

Berli,C. L. A.; Deiber,J. A.; Quemada,D.
Fonte: Latin American applied research Publicador: Latin American applied research
Tipo: Artigo de Revista Científica Formato: text/html
Publicado em 01/01/2005 EN
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This work is concerned with the theoretical estimation of the low-shear viscosity of concentrated suspensions of charged-stabilized latex particles. Calculations are based on the assumption that particles interacting through purely repulsive potentials behave as equivalent hard-spheres (HS), and suspension viscosity may be analyzed in the framework of HS systems. In order to predict numerically the HS radius, the pair potential due to double-layer interaction, as a function of particle concentration, was investigated by using Poisson-Boltzmann theory and the cell model. Calculations explain appropriately experimental data for a wide range of particle sizes, volume fractions and salt concentrations. The problem concerning the effective surface charge of latex particles is also discussed.