Spherical Macroions Research Articles

Electrostatic interactions between charge regulated spherical macroions.

We study the interaction between two charge regulating spherical macroions with dielectric interior and dissociable surface groups immersed in a monovalent electrolyte solution. The charge dissociation is modelled via the Frumkin-Fowler-Guggenheim isotherm, which allows for multiple adsorption equilibrium states. The interactions are derived from the solutions of the mean-field Poisson-Boltzmann type theory with charge regulation boundary conditions. For a range of conditions we find symmetry breaking transitions from symmetric to asymmetric charge distribution exhibiting annealed charge patchiness, which results in like-charge attraction even in a univalent electrolyte-thus fundamentally modifying the nature of electrostatic interactions in charge-stabilized colloidal suspensions.

Theoretical description of the electrical double layer for a mixture of n ionic species with arbitrary size and charge asymmetries. I. Spherical geometry.

In this work, we propose a theoretical finite element description of the ionic profiles of a general mixture of n species of spherical charged particles dissolved in an implicit solvent, with arbitrary size and charge asymmetries, neutralizing a spherical macroion. This approach aims to close the gap between the nano- and the micro-scales in macroion solutions, taking into account the ion correlations and ionic excluded volume effects consistently. When these last two features are neglected, the classical non-linear Poisson-Boltzmann theory for n ionic species-with different ionic closest approach distances to the colloidal surface-is recovered as a limit case. As a proof of concept, we study the electrical double layer of an electroneutral mixture of oppositely charged colloids and small microions, with an asymmetry 1:333 in size and 1:10 in valence, in salt-free and added salt environments. Our theoretical approach displays a good agreement regarding the ionic profiles, the integrated charge, and the mean electrostatic potential obtained from molecular dynamics simulations with explicit-sized microions. Although the non-linear Poisson-Boltzmann colloid-colloid and colloid-microion profiles differ notably from those obtained via molecular dynamics simulations with explicit small-sized ions, the associated mean electrostatic potential agrees well with the corresponding explicit microion simulations.

Monte Carlo simulations and mean-field modeling of electric double layers at weakly and moderately charged spherical macroions.

Monte Carlo simulations are employed to determine the differential capacitance of an electric double layer formed by small size-symmetric anions and cations in the vicinity of weakly to moderately charged macroions. The influence of interfacial curvature is deduced by investigating spherical macroions, ranging from flat to moderately curved. We also calculate the differential capacitance using a previously developed mean-field model where, in addition to electrostatic interactions, the excluded volumes of the ions are taken into account using either the lattice-gas or the Carnahan-Starling equation of state. For both equations of state, we compare the mean-field model for arbitrary curvature with a recently developed second-order curvature expansion. Our Monte Carlo simulations predict an increase in the differential capacitance with growing macroion curvature if the surface charge density is small, whereas for moderately charged macroions the differential capacitance passes through a local minimum. Both mean-field models tend to somewhat overestimate the differential capacitance as compared with Monte Carlo simulations. At the same time, they do reproduce the curvature dependence of the differential capacitance, especially for small surface charge density. Our study suggests that the quality of mean-field modeling does not worsen when weakly or moderately charged macroions exhibit spherical curvature.

Accurate Determination of the Quantity and Spatial Distribution of Counterions around a Spherical Macroion.

The accurate distribution of countercations (Rb+ and Sr2+ ) around a rigid, spherical, 2.9-nm size polyoxometalate cluster, {Mo132 }42- , is determined by anomalous small-angle X-ray scattering. Both Rb+ and Sr2+ ions lead to shorter diffuse lengths for {Mo132 } than prediction. Most Rb+ ions are closely associated with {Mo132 } by staying near the skeleton of {Mo132 } or in the Stern layer, whereas more Sr2+ ions loosely associate with {Mo132 } in the diffuse layer. The stronger affinity of Rb+ ions towards {Mo132 } than that of Sr2+ ions explains the anomalous lower critical coagulation concentration of {Mo132 } with Rb+ compared to Sr2+ . The anomalous behavior of {Mo132 } can be attributed to majority of negative charges being located at the inner surface of its cavity. The longer anion-cation distance weakens the Coulomb interaction, making the enthalpy change owing to the breakage of hydration layers of cations more important in regulating the counterion-{Mo132 } interaction.

Accurate Determination of the Quantity and Spatial Distribution of Counterions around a Spherical Macroion

AbstractThe accurate distribution of countercations (Rb+ and Sr2+) around a rigid, spherical, 2.9‐nm size polyoxometalate cluster, {Mo132}42−, is determined by anomalous small‐angle X‐ray scattering. Both Rb+ and Sr2+ ions lead to shorter diffuse lengths for {Mo132} than prediction. Most Rb+ ions are closely associated with {Mo132} by staying near the skeleton of {Mo132} or in the Stern layer, whereas more Sr2+ ions loosely associate with {Mo132} in the diffuse layer. The stronger affinity of Rb+ ions towards {Mo132} than that of Sr2+ ions explains the anomalous lower critical coagulation concentration of {Mo132} with Rb+ compared to Sr2+. The anomalous behavior of {Mo132} can be attributed to majority of negative charges being located at the inner surface of its cavity. The longer anion–cation distance weakens the Coulomb interaction, making the enthalpy change owing to the breakage of hydration layers of cations more important in regulating the counterion–{Mo132} interaction.

Attraction between Like-Charged Macroions Mediated by Specific Counterion Configurations.

Attraction between like-charged macroions is fundamental to many processes in biology, chemistry, and physics. It also plays an important role in industrial applications such as ion-extraction processes or catalysis. In this work, we report a novel mechanism by which attraction can be realized between spherical macroions at high ionic strength. It consists of specific configurations of two, three, and more counterions that appear between macroions with high statistical probability. The attraction is manifested in a minimum in the potential of mean force between the macroions at short distances. Its depth increases with increasing charge of the macroion, demonstrating that the attraction is electrostatic in nature. It is shown that the implicit solvent model with a distance-dependent dielectric constant can capture both the geometry and thermodynamics of charge-stabilized macroion dimers on the qualitative level. The results obtained for a model colloid with a smooth surface are extrapolated to more realistic systems. Evidence is found that the reported mechanism can be observed in small chemical compounds with encapsulated ions such as fullerenes.

Monte Carlo study of the effects of macroion charge distribution on the ionization and adsorption of weak polyelectrolytes and concurrent counterion release

HypothesisAdsorption of weak polyelectrolytes onto charged nanoparticles, and concurrent effects such as spatial partitioning of ions may be influenced by details of the polyelectrolyte structure (linear or star–like) and size, by the mobility of the nanoparticle surface charge, or the valence of the nanoparticle counterions. ExperimentsIonization and complexation of weak polyelectrolytes on spherical macroions with monovalent and divalent countrions has been studied with constant–pH Monte Carlo titrations and primitive electrolyte models for linear and star–like polymers capable, also, of forming charged hydrogen bonds. Nanoparticles surface charge has been represented either as a single colloid–centered total charge (CCTC) or as surface–tethered mobile monovalent spherical charges (SMMSC). FindingsDifferences in the average number of adsorbed polyelectrolyte arms and their average charge, and in the relative amount of macroion counterions (m–CI’s) released upon polymer adsorption are found between CCTC and SMMSC nanoparticles. The amount of the counterions released also depends on the polymer structure. As CCTC adsorbs a lower number of star–like species arms, the degree of condensation of polymer counterions (p–CI’s) onto the polyelectrolyte is also substantially higher for the CCTC colloid, with a concurrent decrease of the osmotic coefficient values.

Expansion and shrinkage of the electrical double layer in charge-asymmetric electrolytes: A non-linear Poisson-Boltzmann description

The spatial extension or thickness of the electrical double layer determines many thermodynamic and electrokinetic properties of charged colloids in solution. In the classical Debye-Hückel formalism of point ions, the thickness of the electrical double layer around a spherical macroion or next to an infinite planar electrode can be characterized by the bulk Debye length of the supporting ionic fluid. As a result, that approach neglects, at least, the influence of the colloidal charge on the spatial extension of the ionic cloud. Given that the Debye-Hückel formalism of point ions is valid only in the limit of very weak colloidal charges, in this work we use the non-linear Poisson-Boltzmann equation to study the thickness of the electrical double layer near a positively charged electrode in spherical and planar geometries, in the presence of several binary charge-asymmetric −1:z+ point-ions electrolytes with monovalent counterions and multivalent coions. The properties of counterions are maintained fixed, whereas the properties of coions (such as their valence and concentration) are varied fulfilling the bulk electroneutrality condition. The thickness of the ionic cloud is quantified here via the recently introduced capacitive compactness idea (Phys. Chem. Chem. Phys. 20 (2018) 262). Physically, this length represents essentially the separation distance between two electrodes associated to the corresponding effective electrical double layer capacitor, in both planar and spherical geometries. Our numerical calculations show that the capacitive compactness obtained via the non-linear Poisson-Boltzmann equation reduces to the Debye length of the supporting bulk electrolyte at the point of zero charge. In the presence of a charge symmetric −1:+1 electrolyte, the capacitive compactness always decreases as a function of the colloidal charge. Contrastingly, the electrical double layer may expand or shrink as a function of the surface charge density, in the presence of multivalent coions, which is confirmed here by primitive model Monte Carlo simulations. This last non-monotonic behaviour of the capacitive compactness for multivalent coions, depending on the colloidal charge, is related to the microscopic behaviour of the local electric field, the mean electrostatic potential, and the net ionic charge per unit volume close to the colloidal surface. At very large colloidal charges, the capacitive compactness of all −1:z+ electrolytes collapses onto a single curve illustrating the dominance of counterions in the non-linear Poisson-Boltzmann theory.

An experimental/theoretical method to measure the capacitive compactness of an aqueous electrolyte surrounding a spherical charged colloid.

The capacitive compactness has been introduced very recently [G. I. Guerrero-García et al., Phys. Chem. Chem. Phys. 20, 262-275 (2018)] as a robust and accurate measure to quantify the thickness, or spatial extension, of the electrical double layer next to either an infinite charged electrode or a spherical macroion. We propose here an experimental/theoretical scheme to determine the capacitive compactness of a spherical electrical double layer that relies on the calculation of the electrokinetic charge and the associated mean electrostatic potential at the macroparticle's surface. This is achieved by numerically solving the non-linear Poisson-Boltzmann equation of point ions around a colloidal sphere and matching the corresponding theoretical mobility, predicted by the O'Brien and White theory [J. Chem. Soc., Faraday Trans. 2 74, 1607-1626 (1978)], with experimental measurements of the electrophoretic mobility under the same conditions. This novel method is used to calculate the capacitive compactness of NaCl and CaCl2 electrolytes surrounding a negatively charged polystyrene particle as a function of the salt concentration.

Screening of a Small Spherical Macroion by Oppositely Charged Flexible Polyelectrolyte

Electrostatic interactions play important roles in the assembly, the structureand the functions of many biological as well as soft matter systems. In the presence of a macro-ion in aqueous solution, oppositely charged molecules mobilize around it to screen out its electrostatic potential. In this paper, we focus on screening of small macroion by a flexible polyelectrolyteusing Monte-Carlo simulation. It is shown that the condensation of the polyelectrolyte around the macroion shows a first order phase transition from a dense to a dilute concentration of monomers. The width of the condensed region increases with stronger screening or with smaller macro-ion charges. For small macro-ion charge, or when the Coulomb interactions are strongly screened, no complexation between the polyelectrolyte and the macro-ion happens.Additionally, long polyelectrolyte protrudes both tails and loops from the macroion. This is quite different from the case of large macroions where only tails appear. Our results can be used to explain various experimental trends.

The non-dominance of counterions in charge-asymmetric electrolytes: non-monotonic precedence of electrostatic screening and local inversion of the electric field by multivalent coions.

The asymptotic convergence of the thermodynamic and structural properties of unequally-sized charge-symmetric ions in strong electric fields was postulated more than thirty years ago by Valleau and Torrie as the dominance of counterions via the non-linear Poisson-Boltzmann theory [Valleau and Torrie, J. Chem. Phys., 1982, 76, 4623]. According to this mean field prescription, the properties of the electrical double layer near a highly charged electrode immersed in a size-asymmetric binary electrolyte converge to those of a size-symmetric electrolyte if the properties of counterions are the same in both instances. On the other hand, some of the present authors have shown that, in fact, counterions do not dominate the electrical properties of a spherical macroion in the presence of unequally-sized ions, symmetric in valence, if ion correlations and ionic excluded volume effects are taken into account consistently. These ingredients are neglected in the classical Poisson-Boltzmann picture. In the present work, we show the occurrence of the non-dominance of counterions in the opposite scenario, that is, when ions are equally-sized but asymmetric in valence. This is performed in the presence of highly charged colloidal surfaces of spherical and planar geometries for different ionic volume fractions. In addition to the phenomenon of non-dominance of counterions, our simulations and theoretical data also exhibit a non-monotonic order or precedence in the mean electrostatic potential, or electrostatic screening, at the Helmholtz plane of a charged colloid. This interesting behaviour is analyzed as a function of the coion's valence, the ionic volume fraction, and the charge and size of the colloidal particle. All these phenomena are explained in terms of the decay of the electric field near the colloidal surface, and by the appearance of a local inversion of both the electric field and the integrated surface charge density of the colloidal particle in the presence of monovalent counterions and multivalent coions.

Effects of structural properties of the Stern layer on the electrophoretic migration of a highly charged spherical macroion.

The electrophoretic migration of a highly charged spherical macroion suspended in an aqueous solution of NaCl is studied using the molecular dynamic method. The objective is to examine the effects of the colloidal surface charge density on the electrophoretic mobility (μ) of the spherical macroion. The bare charge and the size of the macroion are varied separately to induce changes in the colloidal surface charge density. Our results indicate that μ depends on colloidal surface charge density in a nonmonotonic manner, but that this relationship is independent of the way the surface charge density is varied. It is found that an increase in colloidal surface charge density may lead to the formation of new sublayers in the Stern layer. The μ profile is also found to have a local maximum for a bare charge at which a new sublayer is formed in the Stern layer, and a local minimum for a bare charge at which the outer sublayer becomes relatively dense. Finally, the electrophoretic flow caused by the migration of the spherical macroion is studied to find that one decisive factor causing the electrophoretic flow is the ability of the macroion to carry anions in the electrolyte solution.

Effective charge of cylindrical and spherical colloids immersed in an electrolyte: the quasi-planar limit

We consider the nonlinear Poisson–Boltzmann theory for a single cylindrical or spherical macro-ion in symmetric 1:1, together with asymmetric 1:2 and 2:1 electrolytes. We focus on the regime where , the ratio of the macro-ion radius a over the inverse Debye length in the bulk electrolyte, is large. Analyzing the structure of the analytical expansion emerging from a multiple scale analysis, we uncover a hidden structure for the electrostatic potential. This structure, which appears after a heuristic resummation, suggests a new and convenient expansion scheme that we present and work out in detail. We show that novel exact results can thereby be obtained, in particular pertaining to effective charge properties, in complete agreement with the direct numerical solution to the problem.

Charge Renormalization for Ellipsoidal Macroions.

We study the problem of counterion condensation for ellipsoidal macroions, a geometry well-suited for modeling liquid crystals, anisotropic vesicles, and polymers. We find that the ions within an ellipsoid's condensation layer are relatively unrestricted in their motions, and consequently work to establish a quasi-equipotential at its surface. This simplifies the application of Alexander et al.'s procedure, enabling us to obtain accurate analytic estimates for the critical valence of a general ellipsoid in the weak screening limit. Interestingly, we find that the critical valence of an eccentric ellipsoid is always larger than that of the sphere of equal volume, implying that counterion condensation provides a force resisting the deformation of spherical macroions. This contrasts with a recent study of flexible spherical macroions, which observed a preference for deformation into flattened shapes when considering only linear effects. Our work suggests that the balance of these competing forces might alter the nature of the transition.

The dominance of small ions in the electric double layer of size- and charge-asymmetric electrolytes: a mean-field study on the charge reversal and surface charge amplification

The dominance of counterions in the electric double layer of size-asymmetric semi-punctual ions was proposed more than 30 years ago by Valleau and Torrie. According to their theoretical prescription, at large colloidal surface charges, the double layer properties of a fully asymmetric binary electrolyte become similar to those of a completely symmetric electrolyte if the properties of counterions are the same in both instances. In the same theoretical framework, we propose here that, for a fixed concentration of the smallest ionic species and weakly/moderate colloidal surface charges, the valence of small ions rules or mainly determines the structural and thermodynamic properties of the electric double layer regardless of the colloidal polarity. In other words, we show that the characteristics of the small ions dominate the double layer structure of non-highly charged colloids, independently if the small ions are coions or counterions. This is illustrated by a comprehensive analysis of the ionic and integrated charge profiles around a spherical macroion immersed in a fully size- and charge-asymmetric semi-punctual electrolyte. Charge reversal and surface charge amplification are observed in the regime of low/medium colloidal surface charge densities. The origin of these counterintuitive phenomena, and their corresponding localisation properties in the Helmholtz zone, are explained in terms of the electric double layer structure.

Effective Interactions between Multilayered Ionic Microgels.

Using a one-component reduction formalism, we calculate the effective interactions and the counterion density profiles for microgels that feature a multilayered shell structure. We follow a strategy that involves second order perturbation theory and obtain analytical expressions for the effective interactions by modeling the layers of the particles as linear superpostion of homogeneously charged spheres. The general method is applied to the important case of core–shell microgels and compared with the well-known results for a microgel that can be approximated by a macroscopic, and homogeneously charged, spherical macroion.

Adsorption of pH-responsive polyelectrolyte chains onto spherical macroions.

The adsorption of pH-responsive polyelectrolyte chains onto oppositely charged spherical macroions is investigated through Metropolis Monte Carlo simulations and the Weighted Histogram Analysis Method. In this case, the polymer charge density is susceptible to the solution conditions, such as salt concentration and pH, as well as the presence of other charged species. Thus, the pH and ionic strength variations leads to abrupt variations of the conformational and electric properties of the chain, as a result of first-order-like transition between the adsorbed and desorbed states. The diagram of states as a function of ionic strenght and p H is provided. Despite the inhomogeneities in the polyelectrolyte charge distribution induced by the macroion presence and its dependence on ionic strength, the scaling relation between the macromolecular charge densities and the critical Debye length is obtained in agreement with experimental investigations.

Liquid Crystals in Spheroid Macroions and Spherocylindrical Biomacroions Suspended into Water using Sogami-Ise potential

Structure factor S(Q) and its Fourier transform pair correlation function g(r) have been calculated for the stabilized spheroid bovine serum albumin BSA protein molecules and spherocylindrical biomacroions of tobacco mosaic virus TMV and fd viruses suspended into water using SogamiIse SI potential by reducing the spherocylindricity into spherical macroions of volume equal to the volume of spherocylindrical biomacroions. The nature of the first peaks in S(Q)-Q and g(r)-r curves corroborates presence of liquid crystals in the aqueous suspension of BSA macroions and fd biomacroions. The results obtained compare well with that obtained using screened coulomb repulsive Derjaguin, Landau, Verwey and Overbeek DLVO potential and the experimental results. Liquid crystals of the body centered cubic BCC type with coordination number nearly eight have been found amongst the present system of study of aqueous suspension of spheroid BSA macroions and spherocylindrical fd biomacroions.

Ionic asymmetry and solvent excluded volume effects on spherical electric double layers: A density functional approach

In this article, we present a classical density functional theory for electrical double layers of spherical macroions that extends the capabilities of conventional approaches by accounting for electrostatic ion correlations, size asymmetry, and excluded volume effects. The approach is based on a recent approximation introduced by Hansen-Goos and Roth for the hard sphere excess free energy of inhomogeneous fluids [J. Chem. Phys. 124, 154506 (2006); Hansen-Goos and Roth, J. Phys.: Condens. Matter 18, 8413 (2006)]. It accounts for the proper and efficient description of the effects of ionic asymmetry and solvent excluded volume, especially at high ion concentrations and size asymmetry ratios including those observed in experimental studies. Additionally, we utilize a leading functional Taylor expansion approximation of the ion density profiles. In addition, we use the mean spherical approximation for multi-component charged hard sphere fluids to account for the electrostatic ion correlation effects. These approximations are implemented in our theoretical formulation into a suitable decomposition of the excess free energy which plays a key role in capturing the complex interplay between charge correlations and excluded volume effects. We perform Monte Carlo simulations in various scenarios to validate the proposed approach, obtaining a good compromise between accuracy and computational cost. We use the proposed computational approach to study the effects of ion size, ion size asymmetry, and solvent excluded volume on the ion profiles, integrated charge, mean electrostatic potential, and ionic coordination number around spherical macroions in various electrolyte mixtures. Our results show that both solvent hard sphere diameter and density play a dominant role in the distribution of ions around spherical macroions, mainly for experimental water molarity and size values where the counterion distribution is characterized by a tight binding to the macroion, similar to that predicted by the Stern model.

Effect of Discrete Macroion Charge Distributions on Electric Double Layer of a Spherical Macroion

The effect of replacing the conventional uniform macroion surface charge density with discrete macroion charge distributions on the structure of electric double layer (EDL) of a spherical macroion has been investigated by Monte Carlo (MC) simulations. Two discrete models have been investigated in addition to the central macroion charge: point charges localized on the macroion surface and finite-sized charges protruding into the solution. Both models have been studied with fixed and mobile macroion charges. The radial functions of local densities and electrostatic potential in EDL, are calculated and compared to the results obtained for the central macroion charge distribution. It is concluded that the model of charge distribution significantly affects the EDL structure close to the macroion, while the effect is much weaker at larger distances. With point charges localized on the macroion surface, counterions become stronger accumulated to the macroion, as a result the absolute values of surface potential ϕ0 and zeta ξ potential are decreased. With protruding charges, the excluded volume effect dominates over the increased correlation ability; hence the counterions are less accumulated near the macroions and the absolute values of ϕ0 and ξ potentials are increased.