Interaction between electrical double layers of soil colloids and Fe/Al oxides in suspensions

Phyllosilicates with net negative surface charge and Fe/Al oxides with net positive surface charge coexist in variable-charge soils, and the interaction between these oppositely charged particles affects the stability of mixed colloids, aggregation, and even the surface chemical properties of variable-charge soils. The interaction of the diffuse layers of electrical double layers between the negatively charged soil colloidal particles and the positively charged particles of goethite or gamma-Al(2)O(3) was investigated in this article through the comparison of zeta potentials between single-soil colloidal systems and binary systems containing soil colloids and Fe/Al oxides. The results showed that the presence of goethite and gamma-Al(2)O(3) increased the zeta potential of the binary system containing soil colloids and Fe/Al oxides, which clearly suggests the overlapping of the diffuse layers in soil colloids and Fe/Al oxides. The overlapping of the diffuse layers leads to a decrease in the effective negative charge density on soil colloid and thus causes a shift of pH-zeta potential curves toward the more positive-value side. The interaction of the electrical double layers is also related to the charge characteristics on the Fe/Al oxides: the higher the positive charge density on Fe/Al oxides, the stronger the interaction of the electrical double layers between the soil colloid particles and the Fe/Al oxides.     

Screening and separation of charges in microscale devices: complete planar solution of the Poisson-Boltzmann equation

The Poisson-Boltzmann (PB) equation is widely used to calculate the interaction between electric potential and the distribution of charged species. In the case of a symmetrical electrolyte in planar geometry, the Gouy-Chapman (GC) solution is generally presented as the analytical solution of the PB equation. However, we demonstrate here that this GC solution assumes the presence of a bulk region with zero electric field, which is not justified in microdevices. In order to extend the range of validity, we obtain here the complete numerical solution of the planar PB equation, supported with analytical approximations. For low applied voltages, it agrees with the GC solution. Here, the electric double layers fully absorb the applied voltage such that a region appears where the electric field is screened. For higher voltages (of order 1 V in microdevices), the solution of the PB equation shows a dramatically different behavior, in that the double layers can no longer absorb the complete applied voltage. Instead, a finite field remains throughout the device that leads to complete separation of the charged species. In this higher voltage regime, the double layer characteristics are no longer described by the usual Debye parameter kappa, and the ion concentration at the electrodes is intrinsically bound (even without assuming steric interactions). In addition, we have performed measurements of the electrode polarization current on a nonaqueous model electrolyte inside a microdevice. The experimental results are fully consistent with our calculations, for the complete concentration and voltage range of interest.     

Finite-thickness-enhanced attractions for oppositely charged membranes and colloidal platelets

Within linearized Poisson-Boltzmann theory, we study the disjoining pressure of two oppositely charged parallel objects (membranes and colloidal platelets) in a 1:1 electrolyte, with a focus on the effects of their finite thickness. This extension of the standard Gouy-Chapman model from an interacting pair of double layers to a quartet (one on each side of the two interacting objects) is shown to enhance the regime of attractive interactions significantly, in particular, when the separation and the thickness are on the order of the Debye length of the solvent, provided the dielectric mismatch between objects and solvent is not too extreme. The enhancement of attractions occurs for objects with fixed charge as well as for those that exhibit charge regulation but not for those with a constant surface potential. The underlying mechanism for this enhancement for thin objects is the transfer of net ionic charge from the electrolyte in between to the other sides. For biological membranes in water, this effect is small; however, it is due to strong image charge effects.     

Internal and free energy in a pair of like-charged colloids: Monte Carlo simulations

The effective interaction between two colloidal particles in a bath of monovalent co- and counterions is studied by means of lattice Monte Carlo simulations with the primitive model. The internal electrostatic energy as a function of the colloid distance is studied fixing the position of the colloids. The free energy of the whole system is obtained introducing a bias parabolic potential, that allows us to sample efficiently small separations between the colloidal particles. For small charges, both the internal and free energy increase when the colloids approach each other, resulting in an effective repulsion driven by the electrostatic repulsion. When the colloidal charge is large enough, on the other hand, the colloid-ion coupling is strong enough to form double layers. The internal energy in this case decreases upon approaching the colloids because more ions enter the double layer. This attractive contribution to the interaction between the colloids is stronger for larger charges and larger ionic concentrations. However, the total free energy increases due to the loss of ionic entropy, and resulting finally in a repulsive interaction potential driven by the entropic contributions. The loss of ionic entropy can be almost quantitatively reproduced with the ideal contribution, the same level of approximation as the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory. The overall behavior is captured by the DLVO theory qualitatively, and a comparison is made with the functional form predicted by the theory, showing moderate agreement.     

Entropic effects in the electric double layer of model colloids with size-asymmetric monovalent ions

The structure of the electric double layer of charged nanoparticles and colloids in monovalent salts is crucial to determine their thermodynamics, solubility, and polyion adsorption. In this work, we explore the double layer structure and the possibility of charge reversal in relation to the size of both counterions and coions. We examine systems with various size-ratios between counterions and coions (ion size asymmetries) as well as different total ion volume fractions. Using Monte Carlo simulations and integral equations of a primitive-model electric double layer, we determine the highest charge neutralization and electrostatic screening near the electrified surface. Specifically, for two binary monovalent electrolytes with the same counterion properties but differing only in the coion's size surrounding a charged nanoparticle, the one with largest coion size is found to have the largest charge neutralization and screening. That is, in size-asymmetric double layers with a given counterion's size the excluded volume of the coions dictates the adsorption of the ionic charge close to the colloidal surface for monovalent salts. Furthermore, we demonstrate that charge reversal can occur at low surface charge densities, given a large enough total ion concentration, for systems of monovalent salts in a wide range of ion size asymmetries. In addition, we find a non-monotonic behavior for the corresponding maximum charge reversal, as a function of the colloidal bare charge. We also find that the reversal effect disappears for binary salts with large-size counterions and small-size coions at high surface charge densities. Lastly, we observe a good agreement between results from both Monte Carlo simulations and the integral equation theory across different colloidal charge densities and 1:1-electrolytes with different ion sizes.     

Interaction between dissimilar double layers with like signs under charge regulation on the basis of the Gouy-Chapman-Stern-Grahame model

The force of interaction between two flat double layers with dissimilar surface potentials but of the same signs, in contact with 1-1 electrolyte solutions (0.001, 0.01, and 0.1 M), was calculated as a function of the surface separation. The calculation was carried out on the basis of the Gouy-Chapman-Stern-Grahame double-layer model, incorporating the site-dissociation and site-binding model used previously for many oxide studies. The results demonstrated that the interaction force under surface charge regulation was always repulsive, increasing progressively with decreasing surface separation. This result is in contrast to the predictions of previous theories that are based on the Gouy-Chapman (GC) double-layer model. Dependence of surface potentials and surface charge densities on the surface separation is also presented.     

Finite volume solution of the modified Poisson-Boltzmann equation for two colloidal particles

The double layer forces between spherical colloidal particles, according to the Poisson-Boltzmann (PB) equation, have been accurately calculated in the literature. The classical PB equation takes into account only the electrostatic interactions, which play a significant role in colloid science. However, there are at, and above, biological salt concentrations other non-electrostatic ion specific forces acting that are ignored in such modelling. In this paper, the electrostatic potential profile and the concentration profile of co-ions and counterions near charged surfaces are calculated. These results are obtained by solving the classical PB equation and a modified PB equation in bispherical coordinates, taking into account the van der Waals dispersion interactions between the ions and both surfaces. Once the electrostatic potential is known we calculate the double layer force between two charged spheres. This is the first paper that solves the modified PB equation in bispherical coordinates. It is also the first time that the finite volume method is used to solve the PB equation in bispherical coordinates. This method divides the calculation domain into a certain number of sub-domains, where the physical law of conservation is valid, and can be readily implemented. The finite volume method is implemented for several geometries and when it is applied to solve PB equations presents low computational cost. The proposed method was validated by comparing the numerical results for the classical PB calculations with previous results reported in the literature. New numerical results using the modified PB equation successfully predicted the ion specificity commonly observed experimentally.     

Electrostatic double layer force between a sphere and a planar substrate in the presence of previously deposited spherical particles

A finite element model of the electrostatic double layer interaction between an approaching colloidal particle and a small region of a charged planar surface containing four previously deposited particles is presented. The electrostatic interaction force experienced by the approaching particle is obtained by solving the Poisson-Boltzmann equation with appropriate boundary conditions representing this complex geometry. The interaction forces obtained from the detailed three-dimensional finite element simulations suggest that for the many-body scenario addressed here, the electrostatic double layer repulsion experienced by the approaching particle is less than the corresponding sphere-plate interaction due to the presence of the previously deposited particles. The reduction in force is quite significant when the screening length of the electric double layer becomes comparable to the particle radius (kappaa approximately 1). The results also suggest that the commonly used technique of pairwise addition of binary interactions can grossly overestimate the net electrostatic double layer interaction forces in such situations. The simulation methodology presented here can form a basis for investigating the influence of several previously deposited particles on the electrostatic repulsion experienced by a particle during deposition onto a substrate.     

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.     

Analysis of particle-wall interactions during particle free fall

In this study, the vertical motion of a particle in a quiescent fluid falling toward a horizontal plane wall is analyzed, based on simplified models. Using the distance between the particle and wall as a parameter, the effects of various forces acting on the particle and the particle motion are examined. Without the colloidal and Brownian forces being included, the velocity of small particles is found to be approximately equal to the inverse of the drag force correction function used in this study as the particle approaches the near-wall region. Colloidal force is added to the particle equation of motion as the particle moves a distance comparable to its size. It is found that the particle might become suspended above or deposited onto the wall, depending on the Hamaker constant, the surface potentials of the particle and wall, and the thickness of the electrical double layer (EDL). For strong EDL repulsive force and weaker van der Waals (VDW) attractive force, the particle will become suspended above the wall at a distance at which the particle velocity is zero. This location is referred to as the equilibrium distance. The equilibrium distance is found to increase with increased in EDL thickness when a repulsive force barrier appears in the colloidal force interaction. For the weak EDL repulsive force and strong VDW attractive force case, the particle can become deposited onto the wall without the Brownian motion effect. The Brownian jump length was found to be very small. Many Brownian jumps would be required in a direction toward the wall for a suspended particle to become deposited.     

Irreversible electrostatic deposition of Prussian blue from colloidal solutions

Prussian blue (PB) can be deposited from colloidal solutions (5.4?×?10^?3?mol_PB?L^?1, 0.01?mol?L^?1 KNO₃) on glassy carbon, either by potential cycling or potentiostatically, provided that the deposition potential is more positive than ?0.2?V vs. Hg/Hg₂Cl₂. Depending on the deposition potential, the PB particles form either a single layer of Everitt??s salt, of PB, or multilayers of Berlin green. Also depending on the electrode potential, the deposition was accompanied by currents which were either only of capacitive nature, or represent the sum of capacitive and faradaic currents. The currents were always limited by the diffusion of the colloidal particles to the electrode surface, i.e., they obeyed the Cottrell equation. The PB layers were characterized by in situ atomic force microscopy.     

Electrostatic interaction between ion-penetrable membranes in a salt-free medium

A set of coupled equations is given which determines the distributions of the electric potential and counterions in a system of two interacting identical ion-penetrable membranes of thickness d at separation h immersed in a salt-free medium containing only counterions. The solution to these coupled equations also gives the electrostatic repulsive force between the membranes. It is shown that the interaction force remains finite at h-->0, unlike the case of the interaction between two planar charged surfaces (d-->0), and that the interaction force becomes independent of the membrane fixed charge and membrane thickness d at very large h. Finally, an approximate single transcendental equation giving the solution to the coupled equations is derived.     

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

The equilibrium electric double layer (EDL) that surrounds colloidal particles is essential for the response of a suspension under a variety of static or alternating external fields. An ideal salt-free suspension is composed of charged colloidal particles and ionic countercharges released by the charging mechanism. 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 counterion concentration profiles around a charged particle in a concentrated salt-free suspension corrected by the finite size of the counterions. The results show the high importance of such corrections for moderate to high particle charges at every particle volume fraction, especially 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 non-equilibrium properties in concentrated colloidal suspensions, particularly salt-free ones with small and highly charged particles.     

Irreversible electrostatic deposition of Prussian blue from colloidal solutions

Prussian blue (PB) can be deposited from colloidal solutions (5.4 × 10⁻³ mol_PB L⁻¹, 0.01 mol L⁻¹ KNO₃) on glassy carbon, either by potential cycling or potentiostatically, provided that the deposition potential is more positive than −0.2 V vs. Hg/Hg₂Cl₂. Depending on the deposition potential, the PB particles form either a single layer of Everitt’s salt, of PB, or multilayers of Berlin green. Also depending on the electrode potential, the deposition was accompanied by currents which were either only of capacitive nature, or represent the sum of capacitive and faradaic currents. The currents were always limited by the diffusion of the colloidal particles to the electrode surface, i.e., they obeyed the Cottrell equation. The PB layers were characterized by in situ atomic force microscopy.

     

Interaction between colloids with grafted diblock polyampholytes

The interaction between composite colloidal particles composed of a spherical core and grafted AB-diblock polyampholytes (diblock copolymers with oppositely charged blocks) are investigated by using a coarse-grained model solved with Monte Carlo simulations. The B block is end-grafted onto the core of the colloid and its linear charge density is varied, whereas the linear charge density of the A block is fixed. The brush structure of a single colloid, the mean force between two colloids, and the structure of solutions of such colloids have been determined for different linear charge densities of the B blocks and block lengths. Many features of the present system are controlled by the charge of the B blocks. In the limit of uncharged B blocks, (i) the grafted chains are stretched and form an extended polyelectrolyte brush, (ii) a strong repulsive force is operating between two colloids, (iii) and the solution is thermodynamic stable and displays strong spatial correlation among the colloids. In the limit where the charges of the two types of blocks exactly compensate each other, (i) the chains are collapsed and form a polyelectrolyte complex surrounding the cores, (ii) an attractive force appears between two colloids, and (iii) strong colloid clustering appears in the solution. These features become more pronounced as the length of the polymer blocks is increased, and a phase instability occurs at sufficiently long chains. A comparison with properties for other related colloidal particles is also provided.     

Electrostatic double-layer interaction between spherical particles inside a rough capillary

Predictions of electrostatic double-layer interaction forces between two similarly charged spherical colloidal particles inside an infinitely long "rough" capillary are presented. A simple model of a rough cylindrical surface is proposed, which assumes the capillary wall to be a periodic function of axial position. The periodic roughness of the wall is characterized by the wavelength and amplitude of the undulations. The electrostatic double-layer interaction force between two spherical particles located axially inside this rough capillary is determined by solving the nonlinear Poisson-Boltzmann equation employing finite element analysis. The effect of surface roughness of the cylindrical enclosure on the interaction force between two particles is extensively studied on the basis of this model. The simulations are carried out for dimensionless amplitudes (amplitude/particle radii) ranging from 0.05 to 0.15 and scaled wavelengths (wavelength/particle radii) ranging from 0.4 to 4.0. The interaction force between the particles is significantly modified by the proximity of the rough capillary wall. Generally, the interaction force for rough capillaries oscillates around the corresponding interaction force in a smooth capillary depending on the magnitudes of the scaled amplitude and wavelength of the roughness. The influence of roughness on the electrostatic interactions becomes more pronounced when the surface potential of the cylinder wall is different from the sphere surface potentials. When the cylinder and the particle surfaces have large potential differences, the axial force experienced by a particle is dominated by the capillary roughness. There are dramatic oscillations of the force, which alternately becomes repulsive and attractive as the particle moves from the crest to the trough of the rough capillary wall. These results suggest that manipulation of colloidal particles in narrow microchannels may be subject to significant force variations owing to the roughness inherent in microfabricated channels etched on metal films.     

The foam analogy: from phases to elasticity

By mapping the interactions of colloidal particles onto the problem of minimizing areas, the physics of foams can be used to understand the phase diagrams of both charged and fuzzy colloids. We extend this analogy to study the elastic properties of such colloidal crystals and consider the face-centered cubic, body-centered cubic and A15 lattices. We discuss two types of soft interparticle potentials corresponding to charged and fuzzy colloids, respectively, and we analyze the dependence of the elastic constants on density as well as on the parameters of the potential. We show that the bulk moduli of the three lattices are generally quite similar, and that the shear moduli of the two non-close-packed lattices are considerably smaller than in the face-centered cubic lattice. We find that in charged colloids, the elastic constants are the largest at a finite screening length, and we discuss a shear instability of the A15 lattice.