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.

Activation energy, spatial confinement, and mean first passage and escape times of a tracer in a wormlike micellar fluid: an effective potential approach.

Wormlike micelles are long semiflexible cylindrical polymer structures formed by amphiphiles. In solution, these linear micelles percolate in multiconected entangled networks, where cross-links can break and recombine dynamically. Technological applications of wormlike micellar fluids include tunable encapsulation/delivery of molecules or colloids in biomedicine, oil industry, and/or cleaning processes. In this work, we propose that the experimental activation energy, the spatial confinement, and the mean first passage and escape times of a spherical tracer immersed in wormlike micellar network, in which caging effects are observed, can be estimated from economic Brownian dynamics simulations of a single particle interacting with an effective one-dimensional cosine-like potential of amplitudeU0and periodicityL. The proposed one-fitting parameter method has been used to characterize the long-time dynamics of wormlike micellar solutions formed by the self-assembly of a mixture of zwitterionic and anionic surfactants at several temperatures and different concentrations of surfactant and brine. The amplitudeU0has displayed a good agreement regarding the corresponding experimental activation energy at different temperatures. The periodicityLhas shown to be an upper bound of the mesh sizeξand of the same order of magnitude regarding the entanglement lengthle, obtained from rheology and microrheology experiments. The escape time of the tracer in the effective potentialτescapeand the timet*, at which a change of curvature in the mean square displacement occurs, are upper and lower limits, respectively, of the experimental relaxation time. Our method is simple and fast, and we foresee that it should be applicable to model the long-time behaviour of tracers in other polymer systems, in which caging effects are present.

Size and/or charge asymmetry effects in coulombic fluids in the presence of external fields: From simple electrolytes to molten salts.

Charged fluids are ubiquitous in biology and in everyday products such as milk, cosmetics, lubricants, paints, or drugs. The stability and aggregation of charged colloidal solutions are of outstanding relevance in nature and in diverse biomedical applications, including the fabrication of photonic materials and scaffolds for biological assemblies, gene therapy, diagnostics, targeted drug delivery, and molecular labeling. When an electrode or a charged colloidal particle is immersed in a Coulombic fluid, an ionic cloud or electrical double layer forms. One of the most simple theoretical descriptions of the electrical double layer can be provided via the non-linear Poisson-Boltzmann theory of point-ions. In planar geometry, it is possible to derive an analytical description of the electrical double layer for z : z electrolytes. In spite of its simplicity, this mean field approach neglects important characteristics of charged fluids such as ion correlations, ionic excluded volume, polarization effects, etc. Thus, in this review article I would like to discuss some consequences of breaking the symmetry in the ionic size and/or valence of charged fluids in the presence of an external field when ion correlations and ionic excluded volume effects are included consistently. The external field can be produced by a charged electrode or an electrified colloidal particle, and the associated electrical double layer is studied via integral equations theory, and/or molecular simulations beyond the classical non-linear Poisson Boltzmann theory of point ions.

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.

Quantifying the thickness of the electrical double layer neutralizing a planar electrode: the capacitive compactness.

The spatial extension of the ionic cloud neutralizing a charged colloid or an electrode is usually characterized by the Debye length associated with the supporting charged fluid in the bulk. This spatial length arises naturally in the linear Poisson-Boltzmann theory of point charges, which is the cornerstone of the widely used Derjaguin-Landau-Verwey-Overbeek formalism describing the colloidal stability of electrified macroparticles. By definition, the Debye length is independent of important physical features of charged solutions such as the colloidal charge, electrostatic ion correlations, ionic excluded volume effects, or specific short-range interactions, just to mention a few. In order to include consistently these features to describe more accurately the thickness of the electrical double layer of an inhomogeneous charged fluid in planar geometry, we propose here the use of the capacitive compactness concept as a generalization of the compactness of the spherical electrical double layer around a small macroion (González-Tovar et al., J. Chem. Phys. 2004, 120, 9782). To exemplify the usefulness of the capacitive compactness to characterize strongly coupled charged fluids in external electric fields, we use integral equations theory and Monte Carlo simulations to analyze the electrical properties of a model molten salt near a planar electrode. In particular, we study the electrode's charge neutralization, and the maximum inversion of the net charge per unit area of the electrode-molten salt system as a function of the ionic concentration, and the electrode's charge. The behaviour of the associated capacitive compactness is interpreted in terms of the charge neutralization capacity of the highly correlated charged fluid, which evidences a shrinking/expansion of the electrical double layer at a microscopic level. The capacitive compactness and its first two derivatives are expressed in terms of experimentally measurable macroscopic properties such as the differential and integral capacity, the electrode's surface charge density, and the mean electrostatic potential at the electrode's surface.

Control of Selective Ion Transfer across Liquid-Liquid Interfaces: A Rectifying Heterojunction Based on Immiscible Electrolytes.

The current rectification displayed by solid-state p-n semiconductor diodes relies on the abundance of electrons and holes near the interface between the p-n junction. In analogy to this electronic device, we propose here the construction of a purely ionic liquid-state electric rectifying heterojunction displaying an excess of monovalent cations and anions near the interface between two immiscible solvents with different dielectric properties. This system does not need any physical membrane or material barrier to show preferential ion transfer but relies on the ionic solvation energy between the two immiscible solvents. We construct a simple device, based on an oil/water interface, displaying an asymmetric behavior of the electric current as a function of the polarity of an applied electric field. This device also exhibits a region of negative differential conductivity, analogous to that observed in brain and heart cells via voltage clamp techniques. Computer simulations and mean field theory calculations for a model of this system show that the application of an external electric field is able to control the bulk concentrations of the ionic species in the immiscible liquids in a manner that is asymmetric with respect to the polarity or direction of the applied electric field. These properties make possible to enhance or suppress selective ion transport at liquid-liquid interfaces with the application of an external electric field or electrostatic potential, mimicking the function of biological ion channels, thus creating opportunities for varied applications.

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.

Effective charges and virial pressure of concentrated macroion solutions.

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.

Charged hydrophobic colloids at an oil-aqueous phase interface.

Hydrophobic poly(methyl methacrylate) (PMMA) colloidal particles, when dispersed in oil with a relatively high dielectric constant, can become highly charged. In the presence of an interface with a conducting aqueous phase, image-charge effects lead to strong binding of colloidal particles to the interface, even though the particles are wetted very little by the aqueous phase. We study both the behavior of individual colloidal particles as they approach the interface and the interactions between particles that are already interfacially bound. We demonstrate that using particles which are minimally wetted by the aqueous phase allows us to isolate and study those interactions which are due solely to charging of the particle surface in oil. Finally, we show that these interactions can be understood by a simple image-charge model in which the particle charge q is the sole fitting parameter.

Excluded volume and ion-ion correlation effects on the ionic atmosphere around B-DNA: theory, simulations, and experiments.

The ionic atmosphere around a nucleic acid regulates its stability in aqueous salt solutions. One major source of complexity in biological activities involving nucleic acids arises from the strong influence of the surrounding ions and water molecules on their structural and thermodynamic properties. Here, we implement a classical density functional theory for cylindrical polyelectrolytes embedded in aqueous electrolytes containing explicit (neutral hard sphere) water molecules at experimental solvent concentrations. Our approach allows us to include ion correlations as well as solvent and ion excluded volume effects for studying the structural and thermodynamic properties of highly charged cylindrical polyelectrolytes. Several models of size and charge asymmetric mixtures of aqueous electrolytes at physiological concentrations are studied. Our results are in good agreement with Monte Carlo simulations. Our numerical calculations display significant differences in the ion density profiles for the different aqueous electrolyte models studied. However, similar results regarding the excess number of ions adsorbed to the B-DNA molecule are predicted by our theoretical approach for different aqueous electrolyte models. These findings suggest that ion counting experimental data should not be used alone to validate the performance of aqueous DNA-electrolyte models.

Polarization effects of dielectric nanoparticles in aqueous charge-asymmetric electrolytes.

Small nanoparticles, globular proteins, viral capsids, and other nanoscopic biomolecules usually display dielectric properties that are different from those of the medium in which they are dispersed. These dielectric heterogeneities can significantly influence the surrounding ion distribution, which determines the self-assembly and colloidal stability of these nanoparticles in solution. Here, we study the impact of a dielectric discontinuity in the structural and thermodynamic properties of a spherical nanoparticle made of different dielectric materials when it is immersed in a charge-asymmetric 1:z supporting electrolyte. The mean electrostatic potential, integrated charge, and ionic profiles are analyzed as a function of both the salt concentration and the nanoparticle's valence via Monte Carlo simulations and the nonlinear Poisson-Boltzmann theory. We observe that the electrostatic screening and charge neutralization near the surface of a nanoparticle increase when the nanoparticle's dielectric permittivity increases in all instances. For 1:1 salts, this effect is small and the nonlinear Poisson-Boltzmann theory displays a good agreement with simulation results. Nevertheless, significant deviations are displayed by the mean field scheme regarding simulation results in the presence of multivalent ions. In particular, for trivalent counterions we observe that increasing the dielectric permittivity or the valence of the nanoparticle decreases the critical salt concentration at which occurs a sign inversion of the mean electrostatic potential at the Helmholtz plane, which is closely related to the behavior of the ζ potential and the electrophoretic mobility. Moreover, we observe that the phenomenon of surface charge amplification, or the augmenting of the net charge of a nanoparticle by the adsorption of like-charged ions on its surface, can be promoted by polarization effects in weakly charged spherical nanoparticles with low dielectric permittivity.

Large counterions boost the solubility and renormalized charge of suspended nanoparticles.

Colloidal particles are ubiquitous in biology and in everyday products such as milk, cosmetics, lubricants, paints, or drugs. The stability and aggregation of colloidal suspensions are of paramount importance in nature and in diverse nanotechnological applications, including the fabrication of photonic materials and scaffolds for biological assemblies, gene therapy, diagnostics, targeted drug delivery, and molecular labeling. Electrolyte solutions have been extensively used to stabilize and direct the assembly of colloidal particles. In electrolytes, the effective electrostatic interactions among the suspended colloids can be changed over various length scales by tuning the ionic concentration. However, a major limitation is gelation or flocculation at high salt concentrations. This is explained by classical theories, which show that the electrostatic repulsion among charged colloids is significantly reduced at high electrolyte concentrations. As a result, these screened colloidal particles are expected to aggregate due to short-range attractive interactions or dispersion forces as the salt concentration increases. We discuss here a robust, tunable mechanism for colloidal stability by which large counterions prevent highly charged nanoparticles from aggregating in salt solutions with concentrations up to 1 M. Large counterions are shown to generate a thicker ionic cloud in the proximity of each charged colloid, which strengthens short-range repulsions among colloidal particles and also increases the corresponding renormalized colloidal charge perceived at larger separation distances. These effects thus provide a reliable stabilization mechanism in a broad range of biological and synthetic colloidal suspensions.

Inversion of the Electric Field at the Electrified Liquid-Liquid Interface.

The thermodynamics of the general system of two immiscible electrolytes in the presence of an electric field depends strongly on the distribution of ions near the liquid interface. Here, we calculate the corresponding electrostatic potential difference, excess surface tension, and differential capacity via Monte Carlo simulations, which include ion correlations and polarization effects, and via a modified nonlinear Poisson-Boltzmann theory. Macroscopically, we find good agreement between our results and experimental data without needing any fitting parameter. At higher salt concentrations, charge overcompensation in the lower-permittivity region is observed, which results in a local inversion of the electric field accompanied by charge inversion near the interface. We find that these interesting phenomena are mainly driven by the excluded-volume effects associated with large organic ions in the oil phase, although polarization effects and between-layer ion correlations have a significant impact in the adsorption of ions close to the liquid interface. In addition, our Monte Carlo simulations predict that the differential capacity is maximal at the point of zero charge, in contrast with the classical Poisson-Boltzmann theory results.

Potential of mean force between identical charged nanoparticles immersed in a size-asymmetric monovalent electrolyte.

In a previous theoretical and simulation study [G. I. Guerrero-García, E. González-Tovar, and M. Olvera de la Cruz, Soft Matter 6, 2056 (2010)], it has been shown that an asymmetric charge neutralization and electrostatic screening depending on the charge polarity of a single nanoparticle occurs in the presence of a size-asymmetric monovalent electrolyte. This effect should also impact the effective potential between two macroions suspended in such a solution. Thus, in this work we study the mean force and the potential of mean force between two identical charged nanoparticles immersed in a size-asymmetric monovalent electrolyte, showing that these results go beyond the standard description provided by the well-known Derjaguin-Landau-Verwey-Overbeek theory. To include consistently the ion-size effects, molecular dynamics (MD) simulations and liquid theory calculations are performed at the McMillan-Mayer level of description in which the solvent is taken into account implicitly as a background continuum with the suitable dielectric constant. Long-range electrostatic interactions are handled properly in the simulations via the well established Ewald sums method and the pre-averaged Ewald sums approach, originally proposed for homogeneous ionic fluids. An asymmetric behavior with respect to the colloidal charge polarity is found for the effective interactions between two identical nanoparticles. In particular, short-range attractions are observed between two equally charged nanoparticles, even though our model does not include specific interactions; these attractions are greatly enhanced for anionic nanoparticles immersed in standard electrolytes where cations are smaller than anions. Practical implications of some of the presented results are also briefly discussed. A good accord between the standard Ewald method and the pre-averaged Ewald approach is attained, despite the fact that the ionic system studied here is certainly inhomogeneous. In general, good agreement between the liquid theory approach and MD simulations is also found.

Coulomb interactions in charged fluids.

The use of Ewald summation schemes for calculating long-range Coulomb interactions, originally applied to ionic crystalline solids, is a very common practice in molecular simulations of charged fluids at present. Such a choice imposes an artificial periodicity which is generally absent in the liquid state. In this paper we propose a simple analytical O(N(2)) method which is based on Gauss's law for computing exactly the Coulomb interaction between charged particles in a simulation box, when it is averaged over all possible orientations of a surrounding infinite lattice. This method mitigates the periodicity typical of crystalline systems and it is suitable for numerical studies of ionic liquids, charged molecular fluids, and colloidal systems with Monte Carlo and molecular dynamics simulations.

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.

A Graphics Processing Unit Implementation of Coulomb Interaction in Molecular Dynamics.

We report a GPU implementation in HOOMD Blue of long-range electrostatic interactions based on the orientation-averaged Ewald sum scheme, introduced by Yakub and Ronchi (J. Chem. Phys. 2003, 119, 11556). The performance of the method is compared to an optimized CPU version of the traditional Ewald sum available in LAMMPS, in the molecular dynamics of electrolytes. Our GPU implementation is significantly faster than the CPU implementation of the Ewald method for small to a sizable number of particles (∼10(5)). Thermodynamic and structural properties of monovalent and divalent hydrated salts in the bulk are calculated for a wide range of ionic concentrations. An excellent agreement between the two methods was found at the level of electrostatic energy, heat capacity, radial distribution functions, and integrated charge of the electrolytes.