Colloidal charge renormalization in suspensions containing multivalent electrolyte

Nature of Overcharging and Charge Inversion in Electrical Double Layers

Understanding overcharging and charge inversion is one of the long-standing challenges in soft matter and biophysics. To study these phenomena, we employ the modified Gaussian renormalized fluctuation theory, which allows for the self-consistent accounting of spatially varying ionic strength as well as the spatial variations in dielectric permittivity and excluded volume effects. The underlying dependence of overcharging on the electrostatic coupling is elucidated by varying the surface charge, counterion valency, and dielectric contrast. Consistent with simulations, three characteristic regimes corresponding to weak, moderate, and strong coupling are identified. Important features like the inversion of zeta potential, crowding, and ionic layering at the surface are successfully captured. For weak coupling, there is no overcharging. In the moderate coupling regime, overcharging increases with the surface charge. Finally, in the strong coupling regime, ionic crowding and saturation in overcharging are observed. Our theory predicts a nonmonotonic dependence of charge inversion on multivalent salt concentration as well as the addition of monovalent salt, in quantitative agreement with experiments.

Electrostatic Correlation Induced Ion Condensation and Charge Inversion in Multivalent Electrolytes

The study of the electrical double layer lies at the heart of colloidal and interfacial science. The standard mean-field Poisson-Boltzmann (PB) theory is incapable of modeling many phenomena originating from ion correlation. An important example is charge inversion or overcharging of electrical double layers in multivalent electrolyte solutions. Existing theories aiming to include correlations cannot capture the non-monotonic dependence of charge inversion on salt concentration because they have not systematically accounted for the inhomogeneous nature of correlations from surface to the bulk and the excluded volume effect of ions and solvent molecules. In this work, we modify the Gaussian renormalized fluctuation theory by including the excluded volume effect to study ion condensation and charge inversion. A boundary layer approach is developed to accurately model the giant difference in ion correlations between the condensed layer near the surface and the diffuse layer outside. The theory is used to study charge inversion in multivalent electrolytes and their mixtures. We predict a surface charge induced formation of a three-dimensional condensed layer, which is necessary but not sufficient for charge inversion. The value of the effective surface potential is found to depend non-monotonically on the bulk salt concentration. Our results also show a non-monotonic reduction in charge inversion in monovalent and multivalent electrolyte mixtures. Our work is the first to qualitatively reproduce experimental and simulation observations and explains the underlying physics.

Forces between solid surfaces in aqueous electrolyte solutions

This review addresses experimental findings obtained with direct force measurements between two similar or dissimilar solid surfaces in aqueous electrolyte solutions. Interpretation of these measurements is mainly put forward in terms of the classical theory of Derjaguin, Landau, Verwey, and Overbeek (DLVO). This theory invokes a superposition of attractive van der Waals forces and repulsive double layer forces. DLVO theory is shown to be extremely reliable, even in the case of multivalent ions. However, such a description is only successful, when appropriate surface charge densities, charge regulation characteristics, and ion pairing or complexation equilibria in solution are considered. Deviations from DLVO theory only manifest themselves at distances of typically below few nm. More long-ranged non-DLVO forces can be observed in some situations, particularly, in concentrated electrolyte solutions, in the presence of strongly adsorbed layers, or for hydrophobic surfaces. The latter forces probably originate from patch-charge surface heterogeneities, which can be induced by ion-ion correlation effects, charge fluctuations, or other types of surface heterogeneities.

Evolution of Interactions in the Protein Solution As Induced by Mono and Multivalent Ions

The evolution of interactions in the bovine serum albumin (BSA) protein solution on addition of mono and multivalent (di, tri and tetra) counterions has been studied using small-angle neutron scattering (SANS), dynamic light scattering (DLS) and ζ-potential measurements. It is found that in the presence of mono and divalent counterions, protein behavior can be well explained by DLVO theory, combining the contributions of screened Coulomb repulsion with the van der Waals attraction. The addition of mono or divalent salts in protein solution reduces the repulsive barrier and hence the overall interaction becomes attractive, but the system remains in one-phase for the entire concentration range of the salts, added in the system. However, contrary to DLVO theory, the protein solution undergoes a reentrant phase transition from one-phase to a two-phase system and then back to the one-phase system in the presence of tri and tetravalent counterions. The results show that tri and tetravalent (unlike mono and divalent) counterions induce short-range attraction between the protein molecules, leading to the transformation from one-phase to two-phase system. The two-phase is characterized by the fractal structure of protein aggregates. The excess condensation of these higher-valent counterions in the double layer around the BSA causes the reversal of charge of the protein molecules resulting into reentrant of the one-phase, at higher salt concentrations. The complete phase behavior with mono and multivalent ions has been explained in terms of the interplay of electrostatic repulsion and ion-induced short-range attraction between the protein molecules.

Outsized Amplitude-Modulated Structure of Very-Long-Range Charge Inversions in Model Colloidal Dispersions

Most theoretical and simulation studies on charged suspensions are at infinite dilution and are focused on the electrolyte structure around one or two isolated particles. Some classic experimental studies with latex particle solutions exhibit interesting phenomenology which imply very-long-range correlations. Here, we apply an integral equation theory to a model charged macroion suspension, at finite volume fraction, and find an amplitude-modulated charge inversion structure, with outsized amplitudes and of very-long-range extension. These inversions are different from the standard charge inversions in that they occur at finite macroions' volume fraction, far away from the central macroion, are outsized, and increase, not decrease, with increasing particle charge and distance to the central particle, which is indicative of long-range correlations. We find our results to be in agreement with our Monte Carlo simulations and qualitatively consistent with existing experimental results.

Bjerrum pairs in ionic solutions: A Poisson-Boltzmann approach

Ionic solutions are often regarded as fully dissociated ions dispersed in a polar solvent. While this picture holds for dilute solutions, at higher ionic concentrations, oppositely charged ions can associate into dimers, referred to as Bjerrum pairs. We consider the formation of such pairs within the nonlinear Poisson-Boltzmann framework and investigate their effects on bulk and interfacial properties of electrolytes. Our findings show that pairs can reduce the magnitude of the dielectric decrement of ionic solutions as the ionic concentration increases. We describe the effect of pairs on the Debye screening length and relate our results to recent surface-force experiments. Furthermore, we show that Bjerrum pairs reduce the ionic concentration in bulk electrolyte and at the proximity of charged surfaces, while they enhance the attraction between oppositely charged surfaces.

Ionic size effects on the Poisson-Boltzmann theory

In this paper, we develop a simple theory to study the effects of ionic size on ionic distributions around a charged spherical particle. We include a correction to the regular Poisson-Boltzmann equation in order to take into account the size of ions in a mean-field regime. The results are compared with Monte Carlo simulations and a density functional theory based on the fundamental measure approach and a second-order bulk expansion which accounts for electrostatic correlations. The agreement is very good even for multivalent ions. Our results show that the theory can be applied with very good accuracy in the description of ions with highly effective ionic radii and low concentration, interacting with a colloid or a nanoparticle in an electrolyte solution.

Facilitated polymer capture by charge inverted electroosmotic flow in voltage-driven polymer translocation

The optimal functioning of nanopore-based biosensing tools necessitates rapid polymer capture from the ion reservoir. We identify an ionic correlation-induced transport mechanism that provides this condition without the chemical modification of the polymer or the pore surface. In the typical experimental configuration where a negatively charged silicon-based pore confines a 1 : 1 electrolyte solution, anionic polymer capture is limited by electrostatic polymer-membrane repulsion and the electroosmotic (EO) flow. Added multivalent cations suppress the electrostatic barrier and reverse the pore charge, inverting the direction of the EO flow that drags the polymer to the trans side. This inverted EO flow can be used to speed up polymer capture from the reservoir and to transport weakly or non-uniformly charged polymers that cannot be controlled by electrophoresis.

Short- and long-time diffusion and dynamic scaling in suspensions of charged colloidal particles

We report on a comprehensive theory-simulation-experimental study of collective and self-diffusion in concentrated suspensions of charge-stabilized colloidal spheres. In theory and simulation, the spheres are assumed to interact directly by a hard-core plus screened Coulomb effective pair potential. The intermediate scattering function, f_c(q, t), is calculated by elaborate accelerated Stokesian dynamics (ASD) simulations for Brownian systems where many-particle hydrodynamic interactions (HIs) are fully accounted for, using a novel extrapolation scheme to a macroscopically large system size valid for all correlation times. The study spans the correlation time range from the colloidal short-time to the long-time regime. Additionally, Brownian Dynamics (BD) simulation and mode-coupling theory (MCT) results of f_c(q, t) are generated where HIs are neglected. Using these results, the influence of HIs on collective and self-diffusion and the accuracy of the MCT method are quantified. It is shown that HIs enhance collective and self-diffusion at intermediate and long times. At short times self-diffusion, and for wavenumbers outside the structure factor peak region also collective diffusion, are slowed down by HIs. MCT significantly overestimates the slowing influence of dynamic particle caging. The dynamic scattering functions obtained in the ASD simulations are in overall good agreement with our dynamic light scattering (DLS) results for a concentration series of charged silica spheres in an organic solvent mixture, in the experimental time window and wavenumber range. From the simulation data for the time derivative of the width function associated with f_c(q, t), there is indication of long-time exponential decay of f_c(q, t), for wavenumbers around the location of the static structure factor principal peak. The experimental scattering functions in the probed time range are consistent with a time-wavenumber factorization scaling behavior of f_c(q, t) that was first reported by Segrè and Pusey [Phys. Rev. Lett. 77, 771 (1996)] for suspensions of hard spheres. Our BD simulation and MCT results predict a significant violation of exact factorization scaling which, however, is approximately restored according to the ASD results when HIs are accounted for, consistent with the experimental findings for f_c(q, t). Our study of collective diffusion is amended by simulation and theoretical results for the self-intermediate scattering function, f_s(q, t), and its non-Gaussian parameter α₂(t) and for the particle mean squared displacement W(t) and its time derivative. Since self-diffusion properties are not assessed in standard DLS measurements, a method to deduce W(t) approximately from f_c(q, t) is theoretically validated.

Dendritic polyelectrolytes revisited through the Poisson-Boltzmann-Flory theory and the Debye-Hückel approximation

The properties of a dendritic polyelectrolyte in equilibrium with a reservoir of monovalent salts are investigated using the cell model and the Poisson-Boltzmann-Flory theory. Within this approach we use the Debye-Hückel approximation to solve the Poisson-Boltzmann equation and minimize the semi-grand potential of the system with respect to the size of the molecule which enables us to inspect its conformations as well as the electric field, the ionic density profile, the overall charge density, the effective charge of the dendrimer and the osmotic pressure based on their response to the salt concentration and the dendrimer charge. The model predicts pronounced trapping of salt ions, a local charge neutrality and a zero electric field in the volume of the molecule as well as oscillations of the density profiles and the electric field in the vicinity of the dendrimer-bulk interface. As a result of ion trapping and screening of Coulomb interactions monovalent salts are found to have a minor effect on the size of the dendrimer. Specifically, the dendrimer exists in slightly swollen states as compared to the neutral molecule which indicates that the conformational properties of the polyelectrolyte depend weakly on monovalent salts. These observations harmonise with the equilibrium behavior of the dendrimer pressure, the internal pressure and the bulk pressure, respectively.

Strong attractions and repulsions mediated by monovalent salts

Controlling interactions between proteins and nanoparticles in electrolyte solutions is crucial for advancing biological sciences and biotechnology. The assembly of charged nanoparticles (NPs) and proteins in aqueous solutions can be directed by modifying the salt concentration. High concentrations of monovalent salt can induce the solubilization or crystallization of NPs and proteins. By using a multiscale coarse-grained molecular dynamics approach, we show that, due to ionic correlations in the electrolyte, NPs pairs at high monovalent salt concentrations interact via remarkably strong long-range attractions or repulsions, which can be split into three regimes depending on the surface charge densities of the NPs. NPs with zero-to-low surface charge densities interact via a long-range attraction that is stronger and has a similar range to the depletion attraction induced by polymers with radius of gyrations comparable to the NP diameter. On the other hand, moderately charged NPs with smooth surfaces as well as DNA-functionalized NPs with no possibility of hybridization between them interact via a strong repulsion of range and strength larger than the repulsion predicted by models that neglect ionic correlations, including the Derjaguin-Landau-Vervey-Overbeek (DLVO) model. Interactions between strongly charged NPs (>2 e/nm²), both types smooth and DNA-functionalized NPs, show an attractive potential well at intermediate-to-high salt concentrations, which demonstrates that electrolytes can induce aggregation of strongly charged NPs. Our work provides an improved understanding of the role of ionic correlations in NP assembly and design rules to utilize the salting-out process to crystallize NPs.

Mean-field beyond mean-field: the single particle view for moderately to strongly coupled charged fluids

In a counter-ion only charged fluid, Coulomb coupling is quantified by a single dimensionless parameter. Yet, the theoretical treatment of moderately to strongly coupled charged fluids is a difficult task, central to the understanding of a wealth of soft matter problems, including biological systems. We show that the corresponding coupling regime can be remarkably well described by a single particle treatment, which, at variance with previous works, takes due account of inter-ionic interactions. To this end, the prototypical problem of a planar charged dielectric interface is worked out. Testing our predictions against Monte Carlo simulation data reveals an excellent agreement.

Simulations of Coulomb systems with slab geometry using an efficient 3D Ewald summation method

We present a new approach to efficiently simulate electrolytes confined between infinite charged walls using a 3d Ewald summation method. The optimal performance is achieved by separating the electrostatic potential produced by the charged walls from the electrostatic potential of electrolyte. The electric field produced by the 3d periodic images of the walls is constant inside the simulation cell, with the field produced by the transverse images of the charged plates canceling out. The non-neutral confined electrolyte in an external potential can be simulated using 3d Ewald summation with a suitable renormalization of the electrostatic energy, to remove a divergence, and a correction that accounts for the conditional convergence of the resulting lattice sum. The new algorithm is at least an order of magnitude more rapid than the usual simulation methods for the slab geometry and can be further sped up by adopting a particle-particle particle-mesh approach.

Controlling polymer translocation and ion transport via charge correlations

We develop a correlation-corrected transport theory in order to predict ionic and polymer transport properties of membrane nanopores under physical conditions where mean-field electrostatics breaks down. The experimentally observed low KCl conductivity of open α-hemolysin pores is quantitatively explained by the presence of surface polarization effects. Upon the penetration of a DNA molecule into the pore, these polarization forces combined with the electroneutrality of DNA sets a lower boundary for the ionic current, explaining the weak salt dependence of blocked pore conductivities at dilute ion concentrations. The addition of multivalent counterions to the solution results in the reversal of the polymer charge and the direction of the electroosmotic flow. With trivalent spermidine or quadrivalent spermine molecules, the charge inversion is strong enough to stop the translocation of the polymer and to reverse its motion. This mechanism can be used efficiently in translocation experiments in order to improve the accuracy of DNA sequencing by minimizing the translocation velocity of the polymer.

Disjoining pressure of an electrolyte film confined between semipermeable membranes

We consider an electrolyte solution confined by infinitesimally thin semipermeable membranes in contact with a salt-free solvent. Membranes are uncharged, but since small counter-ions leak-out into infinite salt-free reservoirs, we observe a distance-dependent membrane potential, which generates a repulsive electrostatic disjoining pressure. We obtain the distribution of the potential and of ions, and derive explicit formulas for the disjoining pressure, which are validated by computer simulations. We predict a strong short-range power-law repulsion, and a weaker long-range exponential decay. Our results also demonstrate that an interaction between membranes does strongly depend on the screening lengths, valency of an electrolyte solution, and an inter-membrane film thickness. Finally, our analysis can be directly extended to the study of more complex situations and some biological problems.

Multivalent ion screening of charged glass surface studied by streaming potential measurements

We used streaming potential technique to measure ζ potentials for glass as a function of Co(NH3)6Cl3 concentration, KCl concentration, and pH. Charge inversion was observed only at high surface charge densities and was inhibited by increased KCl concentration. Measured ζ potentials were compared with predictions from a recent theory by dos Santos et al. [J. Chem. Phys. 132, 104105 (2010)] that models multivalent ions adsorbed to the charged surface as a strong coupled liquid (SCL). The location of shear plane was determined independent of the SCL theory, allowing a rigorous experimental test of the theory with no fitting parameters. We found that SCL predictions agree quantitatively with our experimental data.

A simplified representation of anisotropic charge distributions within proteins

Effective coarse-grained representations of protein-protein interaction potentials are vital in the modeling of large scale systems. We develop a method to fit an arbitrary number of effective charges to approximate the electrostatic potential of a protein at a given pH in an ionic solution. We find that the effective charges can reproduce an input potential calculated from a high resolution Poisson-Boltzmann calculation. Since the effective charges used in this model are not constrained to the locations of the original charged groups, the extra degrees of freedom allows us to reproduce the field anisotropy with fewer charges. The fitting procedure uses a number of approximations in the charge magnitudes, initial conditions, and multipoles to speed convergence. The most significant gains are found by fitting the multipole moments of the effective charge potential to the moments of the original field. We show that the Yukawa potential is not only sufficient as a pairwise summation in reproducing the potential, but comes naturally from the linearized expansion of the Poisson-Boltzmann equation. We compute interaction energies and find excellent agreement to the original potential. From the effective charge model we compute the electrostatic contribution to the second virial coefficient.

Vortex distribution in a confining potential

We study a model of interacting vortices in a type II superconductor. In the weak coupling limit, we constructed a mean-field theory which allows us to accurately calculate the vortex density distribution inside a confining potential. In the strong coupling limit, the correlations between the particles become important and the mean-field theory fails. Contrary to recent suggestions, this does not imply failure of the Boltzmann-Gibbs statistical mechanics, as we clearly demonstrate by comparing the results of molecular dynamics and Monte Carlo simulations.

Physical interactions of fish protamine and antisepsis peptide drugs with bacterial membranes revealed by combination of specular x-ray reflectivity and grazing-incidence x-ray fluorescence

As a defined model of outer membranes of gram negative bacteria, we investigated the interaction of monolayers of lipopolysacchrides from Salmonella enterica rough strains R90 (LPS Ra) with natural and synthetic peptides. The fine structures perpendicular to the membrane plane and the ion distribution near the interface were determined by specular x-ray reflectivity (XRR) and grazing-incidence x-ray fluorescence (GIXF) in the presence and absence of divalent cations. The unique combination of XRR and GIXF allows for the quantitative identification of different modes of interactions in a high spatial resolution, which cannot be assessed by other experimental methods. Natural fish protamine disrupts the stratified membrane structures in the absence of Ca(2+) ions, while staying away from the membrane surface in the presence of Ca(2+) ions. In contrast, synthetic antisepsis peptide Pep 19-2.5 weakly adsorbs to the membrane and stays near the uncharged sugar units even in the absence of Ca(2+). In the presence of Ca(2+), Pep 19-2.5 can reach the negatively charged inner core without destroying the barrier capability against ions.

Electric-field-induced crack patterns: experiments and simulation

We report a study of crack patterns formed in laponite gel drying in an electric field. The sample dries in a circular petri dish and the field is radial, acting inward or outward. A system of radial cracks forms in the setup with the center terminal positive, while predominantly cross-radial cracks form when the center is at a negative potential. The laponite accumulates near the negative terminal making the layer thicker at this end. A spring model on a square lattice is used to simulate the desiccation crack formation, with an additional radial force acting due to the electric field. With the radial force acting outward, radial cracks form and for the reversed field cross-radial cracks form. This conforms to the observation that laponite platelets become effectively positive due to overcharging and are attracted towards the negative terminal.

Weak and strong coupling theories for polarizable colloids and nanoparticles

A theory is presented which allows us to accurately calculate the density profile of monovalent and multivalent counterions in suspensions of polarizable colloids or nanoparticles. In the case of monovalent ions, we derive a weak-coupling theory that explicitly accounts for the ion-image interaction, leading to a modified Poisson-Boltzmann equation. For suspensions with multivalent counterions, a strong-coupling theory is used to calculate the density profile near the colloidal surface and a Poisson-Boltzmann equation with a renormalized boundary condition to account for the counterion distribution in the far field. All the results are compared with the Monte Carlo simulations, showing an excellent agreement between the theory and the simulations.

Electrostatics at the nanoscale

Electrostatic forces are amongst the most versatile interactions to mediate the assembly of nanostructured materials. Depending on experimental conditions, these forces can be long- or short-ranged, can be either attractive or repulsive, and their directionality can be controlled by the shapes of the charged nano-objects. This Review is intended to serve as a primer for experimentalists curious about the fundamentals of nanoscale electrostatics and for theorists wishing to learn about recent experimental advances in the field. Accordingly, the first portion introduces the theoretical models of electrostatic double layers and derives electrostatic interaction potentials applicable to particles of different sizes and/or shapes and under different experimental conditions. This discussion is followed by the review of the key experimental systems in which electrostatic interactions are operative. Examples include electroactive and "switchable" nanoparticles, mixtures of charged nanoparticles, nanoparticle chains, sheets, coatings, crystals, and crystals-within-crystals. Applications of these and other structures in chemical sensing and amplification are also illustrated.

Insights from Monte Carlo simulations on charge inversion of planar electric double layers in mixtures of asymmetric electrolytes

Monte Carlo simulations of a planar negatively charged dielectric interface in contact with a mixture of 1:1 and 3:1 electrolytes are carried out using the unrestricted primitive model under more realistic hydrated ion sizes. Two typical surface charge densities are chosen to represent the systems from the weak to strong coupling regimes. Our goal is to determine the dependence of the degree of charge inversion on increasing concentration of both mono- and trivalent salts and to provide a systematic study on this peculiar effect between short-range and electrostatic correlations. The numerical results show that addition of monovalent salt diminishes the condensation of trivalent counterions due to either the favorable solvation energy or the available space constraints. As the concentration of trivalent salt increases, on the other hand, the inclusion of the ionic size and size asymmetry results in a damped oscillatory charge inversion at low enough surface charge and another counterintuitive surface charge amplification. It is proposed that both of the anomalous events in the weak coupling regime are thought to be entropic in origin which is completely different from the electrostatic driven charge inversion in the strong coupling regime. In addition, the electrostatic images arising from the dielectric mismatch lead to a decaying depletion effect on the structure of double layer with growing salt concentration in the case of low charged interface but have no effect at high surface charge values. The microscopic information obtained here points to the need for a more quantitative theoretical treatment in describing the charge inversion phenomenon of real colloidal systems.