Electrostatic interaction in the presence of dielectric interfaces and polarization-induced like-charge attraction

Quantitative Theory for Critical Conditions of Like-Charge Attraction between Polarizable Spheres

Despite extensive experimental and theoretical efforts, a concise quantitative theory to predict the occurrence of like-charge attraction (LCA) between polarizable spheres remains elusive. In this work, we first derive a novel three-point image formula, based on a key observation that connects the classical Neumann's image principle with the incomplete beta function. This approach naturally yields simple yet precise critical conditions for LCA, with a relative discrepancy of less than 1% compared to numerical simulations, validated across diverse parameter settings. The obtained critical conditions may provide physical insights into various processes potentially involving LCA, such as self-assembly, crystallization, and phase separation, across different length scales. Additionally, the new image formula is directly applicable to enhance the efficiency of polarizable force field calculations involving polarizable spheres.

Mechanisms of electrostatic interactions between two charged dielectric spheres inside a polarizable medium: an effective-dipole analysis

The mechanisms of electrostatic interactions between two charged dielectric spheres inside a polarizable medium have been investigated, in terms of hypothetical effective dipoles that depict how the positive and negative charge in each particle are separated. Our findings, which revealed that it is possible for polarization-induced opposite-charge repulsion to occur at short interparticle separations if the dielectric constant of the medium is greater than the dielectric constants of both spheres, provide insights into the physics of charge separation in each sphere and of polarization in the medium behind such counterintuitive behaviour.

Effective dipole model for electrostatic interactions between polarizable spherical particles in particle scale simulations

Despite their widespread adoption, particle-scale simulation methods, such as the Discrete Element Method (DEM), for electrically charged particles in several natural processes and industrial transformations do not include realistic polarization effects. At close distances, these can dominate the particle motion and are impossible to predict by the commonly adopted Coulomb point-charge approximation. Sophisticated mathematical tools can account for uneven charge distributions, predicting an attractive force between a charged particle and a neutral particle or possible attraction between two like-charged particles. Such approaches are accurate but too complex for implementation in DEM simulations of many interacting particles. We propose a novel, simpler yet realistically accurate effective dipole model. By attributing a net charge and an induced effective dipole to each sphere, the interaction force between charged polarizable particles is computed in closed form. A comparison of a rigorous solution and the proposed dipole approach for two spherical particles reveals significant improvement over the commonly employed Coulomb law. The effects of particle size ratio and charge ratio on the interaction force are discussed. Then, the dynamic DEM simulation of a shaker filled with a binary mixture of differently sized particles that are all positively charged is shown to predict the counterintuitive formation of fine-on-coarse aggregates.

Image charge effects under metal and dielectric boundary conditions

The image charge (IC) effect is a fundamental problem in electrostatics. However, proper treatment at the continuum level for many-ion systems, such as electrolyte solutions or ionic liquids, remains an open theoretical question. Here, we demonstrate and systematically compare the IC effects under metal and dielectric boundary conditions (BCs), based on a renormalized Gaussian-fluctuation theory. Our calculations for a simple 1:1 symmetric electrolyte in the point-charge approximation show that the double-layer structure, capacitance, and interaction forces between like-charged plates depend strongly on the types of boundaries, even in the weak-coupling regime. Like-charge attraction is predicted for both metal and dielectric BCs. Finally, we comment on the effects of a dielectrically saturated solvent layer on the metal surface. We provide these results to serve as a baseline for comparison with more realistic molecular dynamics simulations and experiments.

Arbitrary-Shape Dielectric Particles Interacting in the Linearized Poisson-Boltzmann Framework: An Analytical Treatment

This work considers the interaction of two dielectric particles of arbitrary shape immersed into a solvent containing a dissociated salt and assuming that the linearized Poisson-Boltzmann equation holds. We establish a new general spherical re-expansion result which relies neither on the conventional condition that particle radii are small with respect to the characteristic separating distance between particles nor on any symmetry assumption. This is instrumental in calculating suitable expansion coefficients for the electrostatic potential inside and outside the objects and in constructing small-parameter asymptotic expansions for the potential, the total electrostatic energy, and forces in ascending order of Debye screening. This generalizes a recent result for the case of dielectric spheres to particles of arbitrary shape and builds for the first time a rigorous (exact at the Debye-Hückel level) analytical theory of electrostatic interactions of such particles at arbitrary distances. Numerical tests confirm that the proposed theory may also become especially useful in developing a new class of grid-free, fast, highly scalable solvers.

Setting Boundaries for Statistical Mechanics

Statistical mechanics has grown without bounds in space. Statistical mechanics of noninteracting point particles in an unbounded perfect gas is widely used to describe liquids like concentrated salt solutions of life and electrochemical technology, including batteries. Liquids are filled with interacting molecules. A perfect gas is a poor model of a liquid. Statistical mechanics without spatial bounds is impossible as well as imperfect, if molecules interact as charged particles, as nearly all atoms do. The behavior of charged particles is not defined until boundary structures and values are defined because charges are governed by Maxwell's partial differential equations. Partial differential equations require boundary structures and conditions. Boundary conditions cannot be defined uniquely 'at infinity' because the limiting process that defines 'infinity' includes such a wide variety of structures and behaviors, from elongated ellipses to circles, from light waves that never decay, to dipolar fields that decay steeply, to Coulomb fields that hardly decay at all. Boundaries and boundary conditions needed to describe matter are not prominent in classical statistical mechanics. Statistical mechanics of bounded systems is described in the EnVarA system of variational mechanics developed by Chun Liu, more than anyone else. EnVarA treatment does not yet include Maxwell equations.

Manipulating Interactions between Dielectric Particles with Electric Fields: A General Electrostatic Many-Body Framework

We derive a rigorous analytical formalism and propose a numerical method for the quantitative evaluation of the electrostatic interactions between dielectric particles in an external electric field. This formalism also allows for inhomogeneous charge distributions, and, in particular, for the presence of pointlike charges on the particle surface. The theory is based on a boundary integral equation framework and yields analytical expressions for the interaction energy and net forces that can be computed in linear scaling cost, with respect to the number of interacting particles. We include numerical results that validate the proposed method and show the limitations of the fixed dipole approximation at small separation between interacting particles. The proposed method is also applied to study the stability and melting of ionic colloidal crystals in an external electric field.

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.

Dynamic simulations of many-body electrostatic self-assembly

Two experimental studies relating to electrostatic self-assembly have been the subject of dynamic computer simulations, where the consequences of changing the charge and the dielectric constant of the materials concerned have been explored. One series of calculations relates to experiments on the assembly of polymer particles that have been subjected to tribocharging and the simulations successfully reproduce many of the observed patterns of behaviour. A second study explores events observed following collisions between single particles and small clusters composed of charged particles derived from a metal oxide composite. As before, observations recorded during the course of the experiments are reproduced by the calculations. One study in particular reveals how particle polarizability can influence the assembly process.This article is part of the theme issue 'Modern theoretical chemistry'.

Like Charges Attract?

Using multiscale first-principles calculations, we show that two interacting negatively charged B12I9(-) monoanions not only attract, in defiance of the Coulomb's law, but also the energy barrier at 400 K is small enough that these two moieties combine to form a stable B24I18(2-) moiety. Ab initio molecular dynamics simulations further confirm its stability up to 1500 K. Studies of other B12X9(-) (X = Br, Cl, F, H, Au, CN) show that while all of these B24X18(2-) moieties are stable against dissociation, the energy barrier, with the exception of B24Au18(2-), is large so as to hinder their experimental observation. Our results explain the recent experimental observation of the "spontaneous" formation of B24I18(2-) in an ion trap. A simple model based upon electrostatics shows that this unusual behavior is due to competition between the attractive dipole-dipole interaction caused by the aspherical shape of the particle and the repulsive interaction between the like charges.

Image Charge Effects in the Wetting Behavior of Alkanes on Water with Accounting for Water Solubility

Different types of surface forces, acting in the films of pentane, hexane, and heptane on water are discussed. It is shown that an important contribution to the surface forces originates from the solubility of water in alkanes. The equations for the distribution of electric potential inside the film are derived within the Debye-Hückel approximation, taking into account the polarization of the film boundaries by discrete charges at water-alkane interface and by the dipoles of water molecules dissolved in the film. On the basis of above equations we estimate the image charge contribution to the surface forces, excess free energy, isotherms of water adsorption in alkane film, and the total isotherms of disjoining pressure in alkane film. The results indicate the essential influence of water/alkane interface charging on the disjoining pressure in alkane films, and the wettability of water surface by different alkanes is discussed.

Progress in the theory of electrostatic interactions between charged particles

In this perspective we examine recent theoretical developments in methods for calculating the electrostatic properties of charged particles of dielectric materials.

Image-charge forces in thin interlayers due to surface charges in electrolyte

The surface forces arising in wetting films of nonpolar liquids or in thin air interlayers between an electrolyte and a nonpolar medium in the case of discrete charging of the dielectric-electrolyte interface are considered. The contributions of polarization effects to the distribution of the electrostatic potential in the three contacting media were calculated. Within the Debye-Hückel approximation, the analytical solutions were derived for the disjoining pressure in thin films, for the case of either dilute or relatively concentrated electrolyte solutions in the aforementioned systems. Analysis of the analytical and numerical results demonstrated that for dilute solutions the contribution of image forces to the disjoining pressure may significantly exceed the van der Waals forces for films from a few to tens of nanometers thick.

Computing the Coulomb interaction in inhomogeneous dielectric media via a local electrostatics lattice algorithm

The local approach to computing electrostatic interactions proposed by Maggs and adapted by Rottler and Pasichnyk for molecular-dynamics simulations is extended to situations where the dielectric background medium is inhomogeneous. We furthermore correct a problem of the original algorithm related to the correct treatment of the global dipole moment, provide an error estimate for the accuracy of the algorithm, and suggest a different form of the treatment of the self-energy problem. Our implementation is highly scalable on many cores, and we have validated and compared its performance against theoretical predictions and simulation data obtained by other algorithmic approaches.

Low Al-composition p-GaN/Mg-doped Al0.25Ga0.75N/n+-GaN polarization-induced backward tunneling junction grown by metal-organic chemical vapor deposition on sapphire substrate

Low Al-composition p-GaN/Mg-doped Al_0.25Ga_0.75N/n⁺-GaN polarization-induced backward tunneling junction (PIBTJ) was grown by metal-organic chemical vapor deposition on sapphire substrate. A self-consistent solution of Poisson-Schrödinger equations combined with polarization-induced theory was used to model PIBTJ structure, energy band diagrams and free carrier concentrations distribution. The PIBTJ displays reliable and reproducible backward tunneling with a current density of 3 A/cm² at the reverse bias of −1 V. The absence of negative differential resistance behavior of PIBTJ at forward bias can mainly be attributed to the hole compensation centers, including C, H and O impurities, accumulated at the p-GaN/Mg-doped AlGaN heterointerface.

Self-energy-modified Poisson-Nernst-Planck equations: WKB approximation and finite-difference approaches

We propose a modified Poisson-Nernst-Planck (PNP) model to investigate charge transport in electrolytes of inhomogeneous dielectric environment. The model includes the ionic polarization due to the dielectric inhomogeneity and the ion-ion correlation. This is achieved by the self energy of test ions through solving a generalized Debye-Hückel (DH) equation. We develop numerical methods for the system composed of the PNP and DH equations. Particularly, toward the numerical challenge of solving the high-dimensional DH equation, we developed an analytical WKB approximation and a numerical approach based on the selective inversion of sparse matrices. The model and numerical methods are validated by simulating the charge diffusion in electrolytes between two electrodes, for which effects of dielectrics and correlation are investigated by comparing the results with the prediction by the classical PNP theory. We find that, at the length scale of the interface separation comparable to the Bjerrum length, the results of the modified equations are significantly different from the classical PNP predictions mostly due to the dielectric effect. It is also shown that when the ion self energy is in weak or mediate strength, the WKB approximation presents a high accuracy, compared to precise finite-difference results.

Wetting in electrolyte solutions

Wetting of a charged substrate by an electrolyte solution is investigated by means of classical density functional theory applied to a lattice model. Within the present model the pure, i.e., salt-free solvent, for which all interactions are of the nearest-neighbor type only, exhibits a second-order wetting transition for all strengths of the substrate-particle and the particle-particle interactions for which the wetting transition temperature is nonzero. The influences of the substrate charge density and of the ionic strength on the wetting transition temperature and on the order of the wetting transition are studied. If the substrate is neutral, the addition of salt to the solvent changes neither the order nor the transition temperature of the wetting transition of the system. If the surface charge is nonzero, upon adding salt this continuous wetting transition changes to first-order within the wide range of substrate surface charge densities and ionic strengths studied here. As the substrate surface charge density is increased, at fixed ionic strength, the wetting transition temperature decreases and the prewetting line associated with the first-order wetting transition becomes longer. This decrease of the wetting transition temperature upon increasing the surface charge density becomes more pronounced by decreasing the ionic strength.