Perspective: Coulomb fluids—Weak coupling, strong coupling, in between and beyond

Unveiling the Role of Electrostatic Forces on Attraction between Opposing Polyelectrolyte Brushes

Electrostatic interaction and molecular excluded-volume effects are responsible for a plethora of nonintuitive phenomena in soft-matter systems, including local charge inversion and attraction between similar charges. In the current work, we study the surface forces and swelling behavior of opposing polyelectrolyte brushes using a classical density functional theory that accounts for electrostatic and excluded-volume correlations. We observe that the detachment pressure between similarly charged brushes is sensitive to salt concentration in both the osmotic and salted regimes and can be negative in the presence of multivalent counterions. A comparison of the theoretical results with the mean-field predictions unravels the role of correlation effects in determining the surface forces and brush structure. For systems containing multivalent counterions, the detachment pressure attains negative values at an intermediate brush-brush separation, and the attractive region in the pressure vs distance plot is magnified in terms of both the depth and width of attraction with increasing counterion valency. However, the interbrush attraction vanishes when the size-induced correlations are switched off. We also investigated the role of counterion size and polymer chain length on the detachment pressure. It is found that smaller counterions are more effective in neutralizing the polymer charge than bigger counterions, leading to a reduced interbrush repulsion and, in some cases, attraction between like-charged brushes at intermediate distances. Meanwhile, varying the chain length of the grafted polymers only shifts the location of the attraction basin, with little influence on the interaction strength. The theoretical predictions show qualitative agreement with experimental observations and offer valuable insights into the interaction between similarly charged polymer brushes in the presence of multivalent ions.

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.

On the nature of screening in charge-regulated macroion solutions

We present a derivation of the screening length for a solution containing a charge-regulated macroion, e.g. protein, with its counterions. We show that it can be obtained directly from the second derivatives of the total free energy by taking recourse to the "uncertainty relation" of the Legendre transform, which connects the Hessians or the local curvatures of the free energy as a function of density and its Legendre transform, i.e., osmotic pressure, as a function of chemical potentials. Based on the Fowler-Guggenheim-Frumkin model of charge regulation, we then analyze the "screening resonance" and the "overscreening" of the screening properties of the charge-regulated macroion solution.

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.

Variational field theory of macroscopic forces in coulomb fluids

Based on the variational field theory framework, we extend our previous mean-field formalism [Y. A. Budkov and A. L. Kolesnikov, JStatMech 2022, 053205.2022], taking into account the electrostatic correlations of the ions. We employ a general covariant approach and derive a total stress tensor that considers the electrostatic correlations of ions. This is accomplished through an additional term that depends on the autocorrelation function of the local electric field fluctuations. Utilizing the derived total stress tensor and applying the mechanical equilibrium condition, we establish a general expression for the disjoining pressure of the Coulomb fluids, confined in a pore with a slit-like geometry. Using this equation, we derive an asymptotic expression for the disjoining pressure in a slit-like pore with non-electrified conductive walls. Present theory is the basis for future modeling of the mechanical stresses that occur in electrode pores with conductive charged walls, immersed in liquid phase electrolytes beyond the mean-field theory.

Noether's second theorem and covariant field theory of mechanical stresses in inhomogeneous ionic liquids

In this paper, we present a covariant approach that utilizes Noether's second theorem to derive a symmetric stress tensor from the grand thermodynamic potential functional. We focus on the practical case where the density of the grand thermodynamic potential is dependent on the first and second coordinate derivatives of the scalar order parameters. Our approach is applied to several models of inhomogeneous ionic liquids that consider electrostatic correlations of ions or short-range correlations related to packing effects. Specifically, we derive analytical expressions for the symmetric stress tensors of the Cahn-Hilliard-like model, Bazant-Storey-Kornyshev model, and Maggs-Podgornik-Blossey model. All of these expressions are found to be consistent with respective self-consistent field equations.

Mathematical modeling of microscale biology: Ion pairing, spatially varying permittivity, and Born energy in glycosaminoglycan brushes

Biological macromolecules including nucleic acids, proteins, and glycosaminoglycans are typically anionic and can span domains of up to hundreds of nanometers and even micron length scales. The structures exist in crowded environments that are dominated by multivalent electrostatic interactions that can be modeled using mean-field continuum approaches that represent underlying molecular nanoscale biophysics. We develop such models for glycosaminoglycan brushes using steady state modified Poisson-Boltzmann models that incorporate important ion-specific (Hofmeister) effects. The results quantify how electroneutrality is attained through ion electrophoresis, spatially-varying permittivity hydration forces, and ion-specific pairing. Brush-salt interfacial profiles of the electrostatic potential as well as bound and unbound ions are characterized for imposed jump conditions across the interface. The models should be applicable to many intrinsically-disordered biophysical environments and are anticipated to provide insight into the design and development of therapeutics and drug-delivery vehicles to improve human health.

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.

Projective-truncation-approximation study of the one-dimensional ϕ^{4} lattice model

In this paper, we first develop the projective truncation approximation (PTA) in the Green's function equation of motion (EOM) formalism for classical statistical models. To implement PTA for a given Hamiltonian, we choose a set of basis variables and projectively truncate the hierarchical EOM. We apply PTA to the one-dimensional ϕ^{4} lattice model. Phonon dispersion and static correlation functions are studied in detail. Using one- and two-dimensional bases, we obtain results identical to and beyond the quadratic variational approximation, respectively. In particular, we analyze the power-law temperature dependence of the static averages in the low- and high-temperature limits, and we give exact exponents.

Counterion condensation theory for finite polyelectrolyte and salt concentrations

In the present work we analyze the physical fundamentals of Manning's counterion condensation using his charged line model in a simple salt solution. We extend the theory for the cases of finite saline concentration and polymeric concentration tending to zero and the case of both finite concentrations. To find the equilibrium between the phases of free and condensed counterions, besides minimizing the free energy, we deduce an auxiliary equation to determine the two characteristic parameters of the theory, the fraction of condensed counterions and the volume of condensation. We compare the obtained results in the present work for only one infinite charged line with the ones of counterion condensation theory by Schurr and Fujimoto. We find that the linear density of critical charge depends on the concentration of added salt and takes values higher than one, instead of the unitary value predicted by Manning. We obtain the equations by the activity and osmotic coefficients in function of the critical charge density. We compare them with the corresponding equations by Manning for these parameters. We extend the counterion condensation theory to solutions of linear polyelectrolytes for finite saline and polymeric concentrations using a cell model. We modify the electrostatic contribution to the Gibbs energy adding, to the traditional one calculated by Manning, the energy excess due to the macroion present in a cylindrical cell. We apply the theory to obtain the osmotic coefficient and we compare our results with experimental data of DNA osmotic coefficient and with theoretical adjustment using the Poisson-Boltzmann equation.

On the nature of screening in Voorn–Overbeek type theories

By using a recently formulated Legendre transform approach to the thermodynamics of charged systems, we explore the general form of the screening length in the Voorn–Overbeek-type theories, which remains valid also in the cases where the entropy of the charged component(s) is not given by the ideal gas form as in the Debye–Hückel theory. The screening length consistent with the non-electrostatic terms in the free energy ansatz for the Flory–Huggins and Voorn–Overbeek type theories, derived from the local curvature properties of the Legendre transform, has distinctly different behavior than the often invoked standard Debye screening length, though it reduces to it in some special cases.

Understanding the Electric Double-Layer Structure, Capacitance, and Charging Dynamics

Significant progress has been made in recent years in theoretical modeling of the electric double layer (EDL), a key concept in electrochemistry important for energy storage, electrocatalysis, and multitudes of other technological applications. However, major challenges remain in understanding the microscopic details of the electrochemical interface and charging mechanisms under realistic conditions. This review delves into theoretical methods to describe the equilibrium and dynamic responses of the EDL structure and capacitance for electrochemical systems commonly deployed for capacitive energy storage. Special emphasis is given to recent advances that intend to capture the nonclassical EDL behavior such as oscillatory ion distributions, polarization of nonmetallic electrodes, charge transfer, and various forms of phase transitions in the micropores of electrodes interfacing with an organic electrolyte or ionic liquid. This comprehensive analysis highlights theoretical insights into predictable relationships between materials characteristics and electrochemical performance and offers a perspective on opportunities for further development toward rational design and optimization of electrochemical systems.

Like-Charge Attraction at the Nanoscale: Ground-State Correlations and Water Destructuring

Like-charge attraction, driven by ionic correlations, challenges our understanding of electrostatics both in soft and hard matter. For two charged planar surfaces confining counterions and water, we prove that, even at relatively low correlation strength, the relevant physics is the ground-state one, oblivious of fluctuations. Based on this, we derive a simple and accurate interaction pressure that fulfills known exact requirements and can be used as an effective potential. We test this equation against implicit-solvent Monte Carlo simulations and against explicit-solvent simulations of cement and several types of clays. We argue that water destructuring under nanometric confinement drastically reduces dielectric screening, enhancing ionic correlations. Our equation of state at reduced permittivity therefore explains the exotic attractive regime reported for these materials, even in the absence of multivalent counterions.

Thermodynamics of Electrolyte Solutions Near Charged Surfaces: Constant Surface Charge vs. Constant Surface Potential

Electric double layers are ubiquitous in science and engineering and are of current interest, owing to their applications in the stabilization of colloidal suspensions and as supercapacitors. While the structure and properties of electric double layers in electrolyte solutions near a charged surface are well characterized, there are subtleties in calculating thermodynamic properties from the free energy of a system with charged surfaces. These subtleties arise from the difference in the free energy between systems with constant surface charge and constant surface potential. In this work, we present a systematic, pedagogical framework to properly account for the different specifications on charged bodies in electrolyte solutions. Our approach is fully variational---that is, all free energies, boundary conditions, relevant electrostatic equations, and thermodynamic quantities are systematically derived using variational principles of thermodynamics. We illustrate our approach by considering a simple electrolyte solution between two charged surfaces using the Poisson--Boltzmann theory. Our results highlight the importance of using the proper thermodynamic potential and provide a general framework for calculating thermodynamic properties of electrolyte solutions near charged surfaces. Specifically, we present the calculation of the pressure and the surface tension between two charged surfaces for different boundary conditions, including mixed boundary conditions.

Explicit characterization of counterion dynamics around a flexible polyelectrolyte

The article presents a comprehensive study of counterion dynamics around a generic linear polyelectrolyte chain with the help of coarse-grained computer simulations. The ion-chain coupling is discussed in the form of binding time, mean-square displacement (MSD) relative to the chain, local ion transport coefficient, and spatiotemporal correlations in the effective charge. We have shown that a counterion exhibits subdiffusive behavior 〈δR^{2}〉∼t^{δ}, δ≈0.9 w.r.t. chain's center of mass. The MSD of ions perpendicularly outward from the chain segment exhibits a smaller subdiffusive exponent compared to the one along the chain backbone. The effective diffusivity of ion is the lowest in chain's close proximity, extending up to the length-scale of radius of gyration R_{g}. Beyond R_{g} at larger distances, they attain diffusivity of free ion with a smooth cross-over from the adsorbed regime to the free ion regime. We have shown that the effective diffusivity drastically decreases for the multivalent ions, while the crossover length scale remains the same. Conversely, with increasing salt concentration the coupling-length scale reduces, while the diffusivity remains unaltered. The effective diffusivity of adsorbed-ion reveals an exponential reduction with electrostatic interaction strength. We further corroborate this from the binding time of ions on the chain, which also grows exponentially with the coupling strength of the ion-polymer duo. Moreover, the binding time of ions exhibits a weak dependence with salt concentration for the monovalent salt, while for multivalent salts the binding time decreases dramatically with concentration. Our work also elucidates fluctuations in the effective charge per site, where it exhibits strong negative correlations at short length-scales.

Overcharging and Free Energy Barriers for Equally Charged Surfaces Immersed in Salt Solutions

The stability of dispersions containing charged particles may obviously be regulated by salt. In some systems, the effective charge, as measured by the potential some small distance away from the particles, can have a sign opposite to the bare surface charge. If charge reversal takes place, there is typically a salt concentration regime within which colloidal stability increases with added salt. These experimental findings on dispersions have been corroborated by atomic force microscopy investigations, where an attraction is found at short separations. This attraction is stronger than expected from standard DLVO theory, and there has been considerable debate concerning its origin. In this work, we use simple coarse-grained models of these systems, where the bare surfaces carry a uniform charge density, and ion-specific adsorption is absent. Our hypothesis is that these experimental observations can be explained by such a simplistic pure Coulomb based model. Our approach entails grand canonical Metropolis Monte Carlo (MC) simulations as well as correlation-corrected Poisson-Boltzmann (cPB) calculations. In the former case, all ions have a common size, while the cPB utilizes a point-like model. We devote significant attention on apparent surface charge densities and interactions between large flat model surfaces immersed in either a 2:1 salt or a 3:1 salt. In contrast to most of the previous theoretical efforts in this area, we mainly focus on the weak long-ranged repulsion and its connection to an effective surface charge. We find a charge reversal and a concomitant development of a free energy barrier for both salts. The experimentally observed nonmonotonic dependence of colloidal stability on the salt concentration is reproduced using MC simulations as well as cPB calculations. A strong attraction is observed at short range for all investigated cases. We argue that in our model, all non-DLVO aspects can be traced to ion-ion correlations.

Molecular Forces in Liquid-Liquid Extraction

The phase transfer of ions is driven by gradients of chemical potentials rather than concentrations alone (i.e., by both the molecular forces and entropy). Extraction is a combination of high-energy interactions that correspond to short-range forces in the first solvation shell such as ion pairing or complexation forces, with supramolecular and nanoscale organization. While the latter are similar to the long-range solvent-averaged interactions in the colloidal world, in solvent extraction they are associated with lower characteristic lengths of the nanometric domain. Modeling of such complex systems is especially complicated because the two domains are coupled, whereas the resulting free energy of extraction is around k_BT to guarantee the reversibility of the practical process. Nevertheless, quantification is possible by considering a partitioning of space among the polar cores, interfacial film, and solvent. The resulting free energy of transfer can be rationalized by utilizing a combination of terms which represent strong complexation energies, counterbalanced by various entropic effects and the confinement of polar solutes in nanodomains dispersed in the diluent, together with interfacial extractant terms. We describe here this ienaics approach in the context of solvent extraction systems; it can also be applied to further complex ionic systems, such as membranes and biological interfaces.

Renormalization group analysis of weakly interacting van der Waals Fermi system

Weak-coupling phenomena of the two-dimensional Hubbard model is gaining momentum as a new interesting research field due to its extraordinarily rich behavior as a function of the carrier density and model parameters. Salmhofer (1998Commun.Math.Phys.194249; 2001Phys.Rev.Lett.87187004) developed a new renormalization-group method for interacting Fermi systems and Metzner (2000Phys.Rev. B617364; 2000Phys.Rev.Lett.855162) implemented this renormalization group analysis of the two-dimensional Hubbard model. In this work, we demonstrate the spin-wave dependent electronic structure and susceptibility behavior of model graphene-phosphorene van der Waals heterostructure in the framework of renormalization group approach. We implement singlet vertex response function for the weakly interacting van der Waals Fermi system with nearest-neighbor hopping amplitudes. This analytical approach is further extended for spin-wave dependent susceptibility behavior. We present the resulting compressibility and phase diagram in the vicinity of half-filling, and also results for the density dependence of the critical energy scale.

From Water Solutions to Ionic Liquids with Solid State Nanopores as a Perspective to Study Transport and Translocation Phenomena

Solid state nanopores are single-molecular devices governed by nanoscale physics with a broad potential for technological applications. However, the control of translocation speed in these systems is still limited. Ionic liquids are molten salts which are commonly used as alternate solvents enabling the regulation of the chemical and physical interactions on solid-liquid interfaces. While their combination can be challenging to the understanding of nanoscopic processes, there has been limited attempts on bringing these two together. While summarizing the state of the art and open questions in these fields, several major advances are presented with a perspective on the next steps in the investigations of ionic-liquid filled nanopores, both from a theoretical and experimental standpoint. By analogy to aqueous solutions, it is argued that ionic liquids and nanopores can be combined to provide new nanofluidic functionalities, as well as to help resolve some of the pertinent problems in understanding transport phenomena in confined ionic liquids and providing better control of the speed of translocating analytes.

Multivalent Ion-Mediated Attraction between Like-Charged Colloidal Particles: Nonmonotonic Dependence on the Particle Charge

Ion-mediated effective interactions are important for the structure and stability of charged particles such as colloids and nucleic acids. It has been known that the intrinsic electrostatic repulsion between like-charged particles can be modulated into effective attraction by multivalent ions. In this work, we examined the dependence of multivalent ion-mediated attraction between like-charged colloidal particles on the particle charge in a wide range by extensive Monte Carlo simulations. Our calculations show that for both divalent and trivalent salts, the effective attraction between like-charged colloidal particles becomes stronger with the increase of the particle charge, whereas it gradually becomes weakened when the particle charge exceeds a "critical" value. Correspondingly, as the particle charge is increased, the driving force for such effective attraction transits from an attractive electrostatic force to an attractive depletion force, and the attraction weakening by high particle charges is attributed to the transition of electrostatic force from attraction to repulsion. Our analyses suggest that the attractive depletion force and the repulsive electrostatic force at high particle charges result from the Coulomb depletion which suppresses the counterion condensation in the limited region between two like-charged colloidal particles. Moreover, our extensive calculations indicate that the "critical" particle charge decreases apparently for larger ions and smaller colloidal particles due to stronger Coulomb depletion and decreases slightly at higher salt concentrations due to the slightly enhanced Coulomb depletion in the intervening space between colloidal particles. Encouragingly, we derived an analytical formula for the "critical" particle charge based on the Lindemann melting law.

Electric-field-based Poisson-Boltzmann theory: Treating mobile charge as polarization

Mobile charge in an electrolytic solution can in principle be represented as the divergence of ionic polarization. After adding explicit solvent polarization a finite volume of an electrolyte can then be treated as a composite nonuniform dielectric body. Writing the electrostatic interactions as an integral over electric-field energy density we show that the Poisson-Boltzmann functional in this formulation is convex and can be used to derive the equilibrium equations for electric potential and ion concentration by a variational procedure developed by Ericksen for dielectric continua [J. L. Ericksen, Arch. Rational Mech. Anal. 183, 299 (2007)AVRMAW0003-952710.1007/s00205-006-0042-4]. The Maxwell field equations are enforced by extending the set of variational parameters by a vector potential representing the dielectric displacement which is fully transverse in a dielectric system without embedded external charge. The electric-field energy density in this representation is a function of the vector potential and the sum of ionic and solvent polarization making the mutual screening explicit. Transverse polarization is accounted for by construction, lifting the restriction to longitudinal polarization inherent in the electrostatic potential based formulation of Poisson-Boltzmann mean field theory.

A Systematic Way to Extend the Debye-Hückel Theory beyond Dilute Electrolyte Solutions

An extended Debye-Hückel theory with fourth order gradient term is developed for electrolyte solutions; namely, the electric potential φ(r) of the bulk electrolyte solution can be described by ∇²φ(r) = κ²φ(r) + L_Q²∇⁴φ(r), where the parameters κ and L_Q are chosen to reproduce the first two roots of the dielectric response function of the bulk solution. Three boundary conditions for solving the electric potential problem are proposed based upon the continuity conditions of involving functions at the dielectric boundary, with which a boundary element method for the electric potential of a solute with a general geometrical shape and charge distribution is derived. Solutions for the electric potential of a spherical ion and a diatomic molecule are found and used to calculate their electrostatic solvation energies. The validity of the theory is successfully demonstrated when applied to binary as well as multicomponent primitive models of electrolyte solutions.

Theory of Inhomogeneous Rod-like Coulomb Fluids

A field theoretic representation of the classical partition function is derived for a system composed of a mixture of anisotropic and isotropic mobile charges that interact via long range Coulomb and short range nematic interactions. The field theory is then solved on a saddle-point approximation level, leading to a coupled system of Poisson–Boltzmann and Maier–Saupe equations. Explicit solutions are finally obtained for a rod-like counterion-only system in proximity to a charged planar wall. The nematic order parameter profile, the counterion density profile and the electrostatic potential profile are interpreted within the framework of a nematic–isotropic wetting phase with a Donnan potential difference.

Anisotropic electrostatic screening of charged colloids in nematic solvents

The physical behavior of anisotropic charged colloids is determined by their material dielectric anisotropy, affecting colloidal self-assembly, biological function, and even out-of-equilibrium behavior. However, little is known about anisotropic electrostatic screening, which underlies all electrostatic effective interactions in such soft or biological materials. In this work, we demonstrate anisotropic electrostatic screening for charged colloidal particles in a nematic electrolyte. We show that material anisotropy behaves markedly different from particle anisotropy. The electrostatic potential and pair interactions decay with an anisotropic Debye screening length, contrasting the constant screening length for isotropic electrolytes. Charged dumpling-shaped near-spherical colloidal particles in a nematic medium are used as an experimental model system to explore the effects of anisotropic screening, demonstrating competing anisotropic elastic and electrostatic effective pair interactions for colloidal surface charges tunable from neutral to high, yielding particle-separated metastable states. Generally, our work contributes to the understanding of electrostatic screening in nematic anisotropic media.

Electroosmosis as a probe for electrostatic correlations

We study the role of ionic correlations on the electroosmotic flow in planar double-slit channels, without salt. We propose an analytical theory, based on recent advances in the understanding of correlated systems. We compare the theory with mean-field results and validate it by means of dissipative particle dynamics simulations. Interestingly, for some surface separations, correlated systems exhibit a larger flow than predicted by mean-field. We conclude that the electroosmotic properties of a charged system can be used, in general, to infer and weight the importance of electrostatic correlations therein.

Structural and Double Layer Forces between Silica Surfaces in Suspensions of Negatively Charged Nanoparticles

Direct force measurements between negatively charged silica microparticles are carried out in suspensions of like-charged nanoparticles with atomic force microscopy (AFM). In agreement with previous studies, oscillatory force profiles are observed at larger separation distances. At smaller distances, however, soft and strongly repulsive forces are present. These forces are caused by double layer repulsion between the like-charged surfaces and can be quantitatively interpreted with the Poisson-Boltzmann (PB) model. However, the PB model must be adapted to a strongly asymmetric electrolyte to capture the nonexponential nature of these forces. Thereby, the nanoparticles are modeled as highly charged co-ions, while the counter ions are monovalent. This model permits extraction of the effective charge of the nanoparticles, which is well comparable to the one obtained from electrophoresis. The PB model also explains the presence of a particle-free layer close to the interface.

Ordering of adsorbed rigid rods mediated by the Boussinesq interaction on a soft substrate

Orientational ordering driven by mechanical distortion of soft substrates plays a major role in material transformation processes such as elastocapillarity and surface anchoring. We present a theoretical model of the orientational response of anisotropic rods deposited onto a surface of a soft, elastic substrate of finite thickness. We show that anisotropic rods exhibit a continuous isotropic-nematic phase transition, driven by orientational interactions between surface deposited rods. This interaction is mediated by the deformation of the underlying elastic substrate and is quantified by the Boussinesq solution adapted to the case of slender, surface deposited rods. From the microscopic rod-rod interactions, we derive the appropriate Maier-Saupe mean-field description, which includes the Boussinesq elastic free energy contribution due to the substrate elasticity, derive the conditions for the existence of a continuous orientational ordering transition, and discuss the implication of results in the soft (bio)system context.

Critical behavior of charge-regulated macro-ions

Based on a collective description of electrolytes composed of charge-regulated macro-ions and simple salt ions, we analyze their equilibrium charge state in the bulk and their behavior in the vicinity of an external electrified surface. The mean-field formulation of mobile macro-ions in an electrolyte bathing solution is extended to include interactions between association and dissociation sites. We demonstrate that above a critical concentration of salt, similar to the critical micelle concentration, a non-trivial distribution of charge states sets in. Such a charge state can eventually lead to a liquid-liquid phase separation based on charge regulation.

Multivalent counterions accumulate in star-like polyelectrolytes and collapse the polymer in spite of increasing its ionization

We used computer simulations to explore the dissociative and conformational behaviour of branched weak polyelectrolytes with multivalent counterions. We compared simulated titration curves and chain sizes in the presence of added salt of various valencies, keeping the total charge of salt constant. We showed that multivalent counterions enhance ionization of the weak polyelectrolytes, in spite of collapsing of the chains. We provided evidence that such an effect is absent in systems with only monovalent counterions at the same ionic strength, and thus cannot be attributed to electrostatic screening. We attributed it to strong ion-ion correlations that we quantified by comparing potentials of mean force with the mean electrostatic potentials. Finally, we used the partition coefficient to quantify the ability of star-like polyelectrolytes to capture multivalent ions, that is important for water-treatment applications. Our work provides fundamental understanding of the mechanism of polyelectrolyte collapse and ionization response upon addition of multivalent ions.

Charge oscillations in ionic liquids: A microscopic cluster model

In spite of their enormous applications as alternative energy storage devices and lubricants, room-temperature ionic liquids (ILs) still pose many challenges from a pure scientific viewpoint. We develop an IL microscopic theory in terms of ionic clusters, which describes the IL behavior close to charged interfaces. The full structure factor of finite-size clusters is considered and allows us to retain fine and essential details of the system as a whole. Beside the reduction in the screening, it is shown that ionic clusters cause the charge density to oscillate near charged boundaries, with alternating ion-size thick layers, in agreement with experiments. We distinguish between short-range oscillations that persist for a few ionic layers close to the boundary, as opposed to long-range damped oscillations that hold throughout the bulk. The former can be captured by finite-size ion pairs, while the latter is associated with larger clusters with a pronounced quadrupole (or higher) moment. The long-wavelength limit of our theory recovers the well-known Bazant-Storey-Kornyshev (BSK) equation in the linear regime, and elucidates the microscopic origin of the BSK phenomenological parameters.

Statistical field theory of ion–molecular solutions

Schematic representation of the multipolar molecule surrounded by salt ions in a dielectric solvent medium.

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.

Transverse Density Fluctuations around the Ground State Distribution of Counterions near One Charged Plate: Stochastic Density Functional View

We consider the Dean–Kawasaki (DK) equation of overdamped Brownian particles that forms the basis of the stochastic density functional theory. Recently, the linearized DK equation has successfully reproduced the full Onsager theory of symmetric electrolyte conductivity. In this paper, the linear DK equation is applied to investigate density fluctuations around the ground state distribution of strongly coupled counterions near a charged plate, focusing especially on the transverse dynamics along the plate surface. Consequently, we find a crossover scale above which the transverse density dynamics appears frozen and below which diffusive behavior of counterions can be observed on the charged plate. The linear DK equation provides a characteristic length of the dynamical crossover that is similar to the Wigner–Seitz radius used in equilibrium theory for the 2D one-component plasma, which is our main result. Incidentally, general representations of longitudinal dynamics vertical to the plate further suggest the existence of advective and electrical reverse-flows; these effects remain to be quantitatively investigated.

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.

Poisson-Boltzmann equation with a random field for charged fluids

The classical Poisson-Boltzmann equation (CPBE), which is a mean field theory by averaging the ion fluctuation, has been widely used to study ion distributions in charged fluids. In this study, we derive a modified Poisson-Boltzmann equation with a random field from the field theory and recover the ion fluctuation through a multiplicative noise added in the CPBE. The Poisson-Boltzmann equation with a random field (RFPBE) captures the effect of the ion fluctuation and gives different ion distributions in the charged fluids compared to the CPBE. To solve the RFPBE, we propose a Monte Carlo method based on the path integral representation. Numerical results show that the effect of the ion fluctuation strengthens the ion diffusion into the domain and intends to distribute the ions in the fluid uniformly. The final ion distribution in the fluid is determined by the competition between the ion fluctuation and the electrostatic forces exerted by the boundaries. The RFPBE is general and feasible for high dimensional systems by taking the advantage of the Monte Carlo method. We use the RFPBE to study a two dimensional system as an example, in which the effect of ion fluctuation is clearly captured.

Electrostatic Screening Length in "Soft" Electrolyte Solutions

Using the Surface Forces Apparatus on solutions of polymeric ions, the effect of specific ion-ion correlations is evaluated on the characteristic decay length, λ, of screened electrostatic interactions between charged surfaces. Electrolyte solutions composed of point charges surrounded by repulsive polymeric shells were used to elucidate the role of ions size and size asymmetry between co- and counterions on the screening of electrostatic forces. In electrolytes composed of large polymeric cations and small point-charge anions, because of the steric and excluded volume effects, the screening length follows the simple scaling relation λ ∼ d, where d is the characteristic size of the large cation. It is also reported that both co- and counterion sizes affect the thickness of the electrical double layer and influence the screened electrostatic interactions. In solutions of polymeric cation/anion pairs, the screening length is shown to depend on an asymmetry factor. These results provide insights into correlation effects in electrolytes, which were so far unreachable experimentally and help elucidate such effects in electronics, energy storage devices, and biomedical systems.

Orientational transition and complexation of DNA with anionic membranes: Weak and intermediate electrostatic coupling

We characterize the role of charge correlations in the adsorption of a short, rodlike anionic polyelectrolyte onto a similarly charged membrane. Our theory reveals two different mechanisms driving the like-charge polyelectrolyte-membrane complexation: In weakly charged membranes, repulsive polyelectrolyte-membrane interactions lead to the interfacial depletion and a parallel orientation of the polyelectrolyte with respect to the membrane; while in the intermediate membrane charge regime, the interfacial counterion excess gives rise to an attractive "salt-induced" image force. This furthermore results in an orientational transition from a parallel to a perpendicular configuration and a subsequent short-ranged like-charge adsorption of the polyelectrolyte to the substrate. A further increase of the membrane charge engenders a charge inversion, originating from surface-induced ionic correlations, that act as a separate mechanism capable of triggering the like-charge polyelectrolyte-membrane complexation over an extended distance interval from the membrane surface. The emerging picture of this complexation phenomenon identifies the interfacial "salt-induced" image forces as a powerful control mechanism in polyelectrolyte-membrane complexation.

Role of metallic core for the stability of virus-like particles in strongly coupled electrostatics

Electrostatic interactions play important roles in the formation and stability of viruses and virus-like particles (VLPs) through processes that often involve added, or naturally occurring, multivalent ions. Here, we investigate the electrostatic or osmotic pressure acting on the proteinaceous shell of a generic model of VLPs, comprising a charged outer shell and a metallic nanoparticle core, coated by a charged layer and bathed in an aqueous electrolyte solution. Motivated by the recent studies accentuating the role of multivalent ions for the stability of VLPs, we focus on the effects of multivalent cations and anions in an otherwise monovalent ionic solution. We perform extensive Monte-Carlo simulations based on appropriate Coulombic interactions that consistently take into account the effects of salt screening, the dielectric polarization of the metallic core, and the strong-coupling electrostatics due to multivalent ions. We specifically study the intricate roles these factors play in the electrostatic stability of the model VLPs. It is shown that while the insertion of a metallic nanoparticle by itself can produce negative, inward-directed, pressure on the outer shell, addition of only a small amount of multivalent counterions can robustly engender negative pressures, enhancing the VLP stability across a wide range of values for the system parameters.

Measuring Inner Layer Capacitance with the Colloidal Probe Technique

The colloidal probe technique was used to measure the inner layer capacitance of an electrical double layer. In particular, the forces were measured between silica surfaces and sulfate latex surfaces in solutions of monovalent salts of different alkali metals. The force profiles were interpreted with Poisson-Boltzmann theory with charge regulation, whereby the diffuse layer potential and the regulation properties of the interface were obtained. While the diffuse layer potential was measured in this fashion in the past, we are able to extract the regulation properties of the inner layer, in particular, its capacitance. We find systematic trends with the type of alkali metal ion and the salt concentration. The observed trends could be caused by difference in ion hydration, variation of the binding capacitance, and changes of the effective dielectric constant within the Stern layer. Our results are in agreement with recent experiments involving the water-silica interface based on a completely independent method using X-ray photoelectron spectroscopy in a liquid microjet. This agreement confirms the validity of our approach, which further provides a means to probe other types of interfaces than silica.

A coupled effect of dehydration and electrostatic interactions on selective ion transport through charged nanochannels

Selective ion transport is an essential feature of biological ion channels. Due to the subnanometer size and negatively charged surface of ion channels, the ion selectivity is affected by both dehydration effects and electrostatic interactions. Their coupled effect on selective ion transport, however, has been elusive. Here, using molecular dynamics simulations, we study ion (Li⁺ and Mg²⁺) transport through subnanometer carbon nanotubes (CNTs) with varying charge densities. Our results indicate that the dehydration effect governs the ionic transport at low surface charge densities, hence the nanochannel shows a selectivity for Li⁺ ions. In contrast, the nanochannel switches to a selectivity for Mg²⁺ ions as the electrostatic interaction between the cations and the negatively charged wall dominates the transport at high surface charge densities.

Charged nanorods at heterogeneously charged surfaces

We study the spatial and orientational distribution of charged nanorods (rodlike counterions) as well as the effective interaction mediated by them between two plane-parallel surfaces that carry fixed (quenched) heterogeneous charge distributions. The nanorods are assumed to have an internal charge distribution, specified by a multivalent monopolar moment and a finite quadrupolar moment, and the quenched surface charge is assumed to be randomly distributed with equal mean and variance on the two surfaces. While equally charged surfaces are known to repel within the traditional mean-field theories, the presence of multivalent counterions has been shown to cause attractive interactions between uniformly charged surfaces due to the prevalence of strong electrostatic couplings that grow rapidly with the counterion valency. We show that the combined effects due to electrostatic correlations (caused by the coupling between the mean surface field and the multivalent, monopolar, charge valency of counterions) as well as the disorder-induced interactions (caused by the coupling between the surface disorder field and the quadrupolar moment of counterions) lead to much stronger attractive interactions between two randomly charged surfaces. The interaction profile turns out to be a nonmonotonic function of the intersurface separation, displaying an attractive minimum at relatively small separations, where the ensuing attraction can exceed the maximum strong-coupling attraction (produced by multivalent monopolar counterions between uniformly charged surfaces) by more than an order of magnitude.

Complex electric double layers in charged topological colloids

Charged surfaces in contact with liquids containing ions are accompanied in equilibrium by an electric double layer consisting of a layer of electric charge on the surface that is screened by a diffuse ion cloud in the bulk fluid. This screening cloud determines not only the interactions between charged colloidal particles or polyelectrolytes and their self-assembly into ordered structures, but it is also pivotal in understanding energy storage devices, such as electrochemical cells and supercapacitors. However, little is known to what spatial complexity the electric double layers can be designed. Here, we show that electric double layers of non-trivial topology and geometry -including tori, multi-tori and knots- can be realised in charged topological colloidal particles, using numerical modelling within a mean-field Poisson-Boltzmann theory. We show that the complexity of double layers -including geometry and topology- can be tuned by changing the Debye screening length of the medium, or by changing the shape and topology of the (colloidal) particle. More generally, this work is an attempt to introduce concepts of topology in the field of charged colloids, which could lead to novel exciting material design paradigms.

Rodlike Counterions near Charged Cylinders: Counterion Condensation and Intercylinder Interaction

We study a system composed of like-charged cylinders and dumbbell-like counterions, with the focus laid on the role of the internal structure of counterions, using Monte Carlo simulations. The dumbbell ions are found to exhibit novel counterion condensation behavior governed by their length. Effective electrostatic interactions mediated between charged parallel cylinders also turn out significantly different from the case of pointlike ions, as a result of the complex interplay between the spatially separated charge distribution in the dumbbell counterions, their orientation, and the curvature of the charged cylinder. We show that at a weak-to-moderate electrostatic coupling strength, where effective like-charge interactions are usually found to be repulsive, the intercylinder interaction can become attractive and display a distinctive sensitivity to the cylinder curvature and dumbbell size, proving the significant effect of ion structure.

Ionic structure around polarizable metal nanoparticles in aqueous electrolytes

Metal nanoparticles are receiving increased scientific attention owing to their unique physical and chemical properties that make them suitable for a wide range of applications in diverse fields, such as electrochemistry, biochemistry, and nanomedicine. Their high metallic polarizability is a crucial determinant that defines their electrostatic character in various electrolyte solutions. Here, we introduce a continuum-based model of a metal nanoparticle with explicit polarizability in the presence of different kinds of electrolytes. We employ several, variously sophisticated, theoretical approaches, corroborated by Monte Carlo simulations in order to elucidate the basic electrostatics principles of the model. We investigate how different kinds of asymmetries between the ions result in non-trivial phenomena, such as charge separation and a build-up of a so-called zero surface-charge double layer.