A multiple decay-length extension of the Debye-Hückel theory: to achieve high accuracy also for concentrated solutions and explain under-screening in dilute symmetric electrolytes

Exceptionally Strong Double-Layer Barriers Generated by Polyampholyte Salt

Experiments using the surface force apparatus have found anomalously long-range interactions between charged surfaces in concentrated salt solutions. Ion clustering has been suggested as a possible origin of this behavior. In this work, we demonstrate that if such stable clusters indeed form, they are able to induce remarkably strong free energy barriers under conditions where a corresponding solution of simple salt provides negligible forces. Our cluster model is based on connected ions producing a polyampholyte salt containing a symmetric mixture of monovalent cationic and anionic polyampholytes. Ion distributions and surface interactions are evaluated utilizing statistical-mechanical (classical) polymer density functional theory, cDFT. In the Supporting Information, we briefly investigate a range of different polymer architectures (connectivities), but in the main part of the work, a polyampholyte ion is modeled as a linear chain with alternating charges, in which the ends carry an identical charge (hence, a monovalent net charge). These salts are able to generate repulsions, between similarly charged surfaces, of a remarkable strength, exceeding those from simple salts by orders of magnitude. The underlying mechanism for this is the formation of brush-like layers at the surfaces, i.e., the repulsion is strongly related to excluded volume effects, in a manner similar to the interaction between surfaces carrying grafted polymers. We believe our results are relevant not only to possible mechanisms underlying anomalously long-ranged underscreening in concentrated simple salt solutions but also for the potential use of synthesized polyampholyte salt as extremely efficient stabilizers of colloidal dispersions.

Dynamic Modulation of Ions Solvation Sheath by Butyramide as Molecular Additives in Aqueous Batteries

The high activity of water in aqueous battery electrolytes can trigger side reactions, limiting their large-scale application. Additives that form contact pairs (CPs) with cations by coordinating with them can effectively reduce water's activity. However, due to the complex interactions between ions, additives, and solvent molecules and the fact that current strategies for additive screening primarily rely on static physical parameters, the dynamic mechanisms that govern the modulation of ion solvation sheaths are still poorly understood. In this study, we introduce butyramide (BUT) as a molecular additive and employ molecular simulations to demonstrate its regulatory effect on the hydration sheath of Ca2+, which is more pronounced than that for Na+. The dynamic process by which BUT replaces water molecules in the tight hydration sheath of Ca2+ is elucidated by forming a stable [BUT-Ca2+(H2O)7] complex that suppresses water molecule activity. At a 2 M concentration, the free energy barrier for the transition from contact pair (CP) to solvent-shared pair (SP) for Ca2+ is 11.7 kJ/mol higher than that for Na+ at 8.5 kJ/mol, consistent with the cationic Hofmeister series. Furthermore, the stability and dynamic fluctuations among solvent-separated pair (SSP), SP, and CP states are attributed to the balance between electrostatic attractive potential energy and hydration repulsive potential energy, supported by quantum chemical calculations of the ion desolvation process. Using BUT as an additive presents a promising strategy to enhance battery performance by modulating the solvation environment of metal ions, addressing the growing demand for safer and more sustainable energy storage solutions.

Self-consistent electrostatic formalism of bulk electrolytes based on the asymmetric treatment of the short- and long-range ion interactions

We predict the thermodynamic behavior of bulk electrolytes from an ionic hard-core (HC) size-augmented self-consistent formalism incorporating asymmetrically the short- and long-range ion interactions via their virial and cumulant treatment, respectively. The characteristic splitting length separating these two ranges is obtained from a variational equation solved together with the Schwinger-Dyson (SD) equations. Via comparison with simulation results from the literature, we show that the asymmetric treatment of the distinct interaction ranges significantly extends the validity regime of our previously developed purely cumulant-level Debye-Hückel (DH) theory. Namely, for monovalent solutions with typical ion sizes, the present formalism can accurately predict up to molar concentrations the liquid pressure dominated by HC interactions, the internal energies driven by charge correlations, and the local ion distributions governed by the competition between HC and electrostatic interactions. We evaluate as well the screening length of the liquid and investigate the deviations of the macromolecular interaction range from the DH length. In fair agreement with simulations and experiments, our theory is shown to reproduce the overscreening and underscreening effects occurring respectively in submolar mono- and multivalent electrolytes.

Heat and mass transfer analysis of s-PTT nanofluid in microchannels under combined electroosmotic and pressure-driven flows with wall slip using the homotopy perturbation method

The heat and mass transfer of the electroosmotic flow in microchannel transporting viscoelastic nanofluid is investigated considering Brownian motion of nanoparticles and slip boundary conditions. The simplified Phan-Thien-Tanner model is employed to describe the rheological behavior of fluid and the nonlinear Navier model with non-zero slip critical shear stress is considered at walls. The governing nonlinear momentum, mass, and heat transfer equations are solved using the Homotopy Perturbation Method. The study reveals that increasing the fluid elasticity, nanoparticle concentration, and size significantly enhances the flow rate, heat and mass transfer. Additionally, elasticity and Reynolds number decrease the friction factor. Reducing the double-layer thickness and increasing the Reynolds number lead to higher flow rates and fluid velocities. Notably, the findings emphasize the critical role of the slip conditions on the Sherwood and Nusselt numbers.

Cluster Formation Induced by Local Dielectric Saturation in Restricted Primitive Model Electrolytes

Experiments using the Surface Force Apparatus (SFA) have found anomalously long-ranged charge-charge underscreening in concentrated salt solutions. Meanwhile, theory and simulations have suggested ion clustering to be a possible origin of this behavior. The popular Restricted Primitive Model of electrolyte solutions, in which the solvent is represented by a uniform relative dielectric constant, εr, is unable to resolve the anomalous underscreening seen in experiments. In this work, we modify the Restricted Primitive Model to account for local dielectric saturation within the ion hydration shell. The dielectric "constant" in our model locally decreases from the bulk value to a lower saturated value at the ionic surface. The parameters for the model are deduced so that typical salt solubilities are obtained. Our simulations for both bulk and slit geometries show that our model displays strong cluster formation and these give rise to long-ranged density correlations between charged surfaces, at distances similar to what has been observed in SFA measurements. An electrolyte model wherein the dielectric constant remains uniform does not display similar clusters, even with εr equal to the low saturated value at ion contact. Hence, the observed behaviors are not simply due to an enhanced Coulomb interaction. In the latter case, cluster growth is counteracted by long-ranged repulsions between like-charged ions within clusters; this is an effect that is considerably reduced when the dielectric response drop is local. Our results imply that long-ranged interactions in these systems are mainly due to cluster-cluster correlations, rather than large electrostatic screening lengths.

Nonlinear Poisson-Boltzmann solutions for charged parallel plates: When opposite charges repel

I present an exact solution of the Poisson-Boltzmann equation for two parallel plates and discuss the solution properties. I discuss in more detail plates with opposite charges: In this case, there are two critical separations, Lc,1 < Lc,2. For separations less than Lc,1, the force between plates is repulsive. It switches to attractive at Lc,1, but with the electric potential having the same sign on both plates. For L > Lc,2, the force remains attractive, and the potential at the plates has the same sign as the charge on each plate. I also describe charge regulation, determined by pKa, and provide formulas for both the critical distance where oppositely charged plates repel and their charging process. The implications of these results for the nanoparticle assembly, as driven by electrostatic interactions, are also discussed.

Cation valency in water-in-salt electrolytes alters the short- and long-range structure of the electrical double layer

Highly concentrated aqueous electrolytes (termed water-in-salt electrolytes, WiSEs) at solid-liquid interfaces are ubiquitous in myriad applications including biological signaling, electrosynthesis, and energy storage. This interface, known as the electrical double layer (EDL), has a different structure in WiSEs than in dilute electrolytes. Here, we investigate how divalent salts [zinc bis(trifluoromethylsulfonyl)imide, Zn(TFSI)2], as well as mixtures of mono- and divalent salts [lithium bis(trifluoromethylsulfonyl)imide (LiTFSI) mixed with Zn(TFSI)2], affect the short- and long-range structure of the EDL under confinement using a multimodal combination of scattering, spectroscopy, and surface forces measurements. Raman spectroscopy of bulk electrolytes suggests that the cation is closely associated with the anion regardless of valency. Wide-angle X-ray scattering reveals that all bulk electrolytes form ion clusters; however, the clusters are suppressed with increasing concentration of the divalent ion. To probe the EDL under confinement, we use a Surface Forces Apparatus and demonstrate that the thickness of the adsorbed layer of ions at the interface grows with increasing divalent ion concentration. Multiple interfacial layers form following this adlayer; their thicknesses appear dependent on anion size, rather than cation. Importantly, all electrolytes exhibit very long electrostatic decay lengths that are insensitive to valency. It is likely that in the WiSE regime, electrostatic screening is mediated by the formation of ion clusters rather than individual well-solvated ions. This work contributes to understanding the structure and charge-neutralization mechanism in this class of electrolytes and the interfacial behavior of mixed-electrolyte systems encountered in electrochemistry and biology.

An efficient method to establish electrostatic screening lengths of restricted primitive model electrolytes

We present a novel, and computationally cheap, way to estimate electrostatic screening lengths from simulations of restricted primitive model (RPM) electrolytes. We demonstrate that the method is accurate by comparisons with simulated long-ranged parts of the charge density, at various Bjerrum lengths, salt concentrations and ion diameters. We find substantial underscreening in low dielectric solvent, but with an "aqueous" solvent, there is instead overscreening, the degree of which increases with ion size. Our method also offers a possible path to (future) more accurate classical density functional treatments of ionic fluids.

Systematic Incorporation of Ionic Hard-Core Size into the Debye-Hückel Theory via the Cumulant Expansion of the Schwinger-Dyson Equations

The Debye-Hückel (DH) formalism of bulk electrolytes equivalent to the Gaussian-level closure of the electrostatic Schwinger-Dyson identities without the interionic hard-core (HC) coupling is extended via the cumulant treatment of these equations augmented by HC interactions. By comparing the monovalent ion activity and pressure predictions of our cumulant-corrected DH (CCDH) theory with hypernetted-chain results and Monte Carlo simulations from the literature, we show that this rectification extends the accuracy of the DH formalism from submolar to molar salt concentrations. In the case of internal energies or the general case of divalent electrolytes mainly governed by charge correlations, the improved accuracy of the CCDH theory is limited to submolar ion concentrations. Comparison with experimental data from the literature shows that, via the adjustment of the hydrated ion radii, CCDH formalism can equally reproduce the nonuniform effect of salt increment on the ionic activity coefficients up to molar concentrations. The inequality satisfied by these HC sizes coincides with the cationic branch of the Hofmeister series.

Measuring the Capacitance of Carbon in Ionic Liquids: From Graphite to Graphene

The physical electrochemistry of the carbon/ionic liquids interface underpins the processes occurring in a vast range of applications spanning electrochemical energy storage, iontronic devices, and lubrication. Elucidating the charge storage mechanisms at the carbon/electrolyte interface will lead to a better understanding of the operational principles of such systems. Herein, we probe the charge stored at the electrochemical double layer formed between model carbon systems, ranging from single-layer graphene to graphite and the ionic liquid 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide (EMIM-TFSI). The effect of the number of graphene layers on the overall capacitance of the interface is investigated. We demonstrate that in pure EMIM-TFSI and at moderate potential biases, the electronic properties of graphene and graphite govern the overall capacitance of the interface, while the electrolyte contribution to the latter is less significant. In mixtures of EMIM-TFSI with solvents of varying relative permittivity, the complex interplay between electrolyte ions and solvent molecules is shown to influence the charge stored at the interface, which under certain conditions overcomes the effects of relative permittivity. This work provides additional experimental insights into the continuously advancing topic of electrochemical double-layer structure at the interface between room temperature ionic liquids and carbon materials.

Energy-Scaled Debye-Hückel Theory for the Electrostatic Solvation Free Energy in Size-Asymmetric Electrolyte Solutions

In this report, an energy-scaled Debye-Hückel theory is developed for fast and accurate evaluation of the electrostatic solvation free energy in size-asymmetric electrolyte solutions. A size-asymmetric electrolyte solution is mapped to a dielectric continuum medium with Debye-Hückel-like response. Based on the scaling relation of the electrostatic energy of a spherical ion in the small and large size limits, a Padé polynomial is used to interpolate the electrostatic energy at finite size. The Padé polynomial is further interpreted as the electrostatic energy of an effective Debye-Hückel mean field model, depicted by a modified Debye parameter and a surface charge density due to the size asymmetry of the solvent ions. This theory can distinguish the electrostatic energies and the electrostatic solvation free energies of solutes with the same size but opposite charges. Application to charged hard and charged soft spheres demonstrates the accuracy of our approach.

A Gaussian field approach to the solvation of spherical ions in electrolyte solutions

In this work, the electrostatic response of an electrolyte solution to a spherical ion is studied with a Gaussian field theory. In order to capture the ionic correlation effect in concentrated solutions, the bulk dielectric response function is described by a two-Yukawa response function. The modified response function of the solution is solved analytically in the spherical geometry, from which the induced charge density and the electrostatic energy are also derived analytically. Comparisons with results for small ions in electrolyte solutions from the hyper-netted chain theory demonstrate the validity of the Gaussian field theory.

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.

Modifying Kohlrausch's Law to Describe Nonaqueous Electrolytes for Lithium-Ion Batteries

To develop next-generation lithium-ion batteries with enhanced stability and safety, it is crucial to understand the physicochemical principles of nonaqueous electrolytes. Kohlrausch's law describes a linear decrease in the molar conductivity (Λ) with respect to the square root of the molarity of strong electrolytes at lower concentrations. This empirical law explains the impeded ionic mobility at higher concentrations due to ionic interactions, i.e., relaxation and asymmetric effects. However, this law does not hold at higher concentrations due to the ionic association that alleviates the ionic interactions and retards the decrease in the Λ. Especially, the anomalously stagnant decrease in the Λ near the solubility limit has not been clearly explained, calling for the consideration of other concentration-dependent factors such as the mean activity coefficient (γ±), viscosity (η), and dielectric constant (ε). Herein, we develop a systematic method to modify Kohlrausch's law. First, we install the ionic association constant, and the theoretical estimation is compared with the experimental results to induce the correction function that is related with the previously neglected concentration-dependent factors. Thus, the induced correction function was close to the rectified linear unit (ReLU) function, which has been widely used in the field of artificial intelligence. The modified Kohlrausch's law with the ReLU-type correction function provides a highly precise and reliable data fitting, and the fitted parameters showed clear concentration dependency and straightforward interpretability. As a result, this method effectively generalized Kohlrausch's law for nonaqueous electrolytes at higher concentrations up to the solubility limit of 3.0-3.5 M. Moreover, the modified Kohlrausch's law inspired us to discover the physical origins of the anomalously stagnant Λ profiles near the solubility limit; and the most relevant physical origin of the anomaly was the concentration dependency of the γ± and η, which grow exponentially above a critical concentration.

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.

Electrical noise in electrolytes: a theoretical perspective

Seemingly unrelated experiments such as electrolyte transport through nanotubes, nano-scale electrochemistry, NMR relaxometry and surface force balance measurements, all probe electrical fluctuations: of the electric current, the charge and polarization, the field gradient (for quadrupolar nuclei) and the coupled mass/charge densities. The fluctuations of such various observables arise from the same underlying microscopic dynamics of the ions and solvent molecules. In principle, the relevant length and time scales of these dynamics are encoded in the dynamic structure factors. However, modelling the latter for frequencies and wavevectors spanning many orders of magnitude remains a great challenge to interpret the experiments in terms of physical processes such as solvation dynamics, diffusion, electrostatic and hydrodynamic interactions between ions, interactions with solid surfaces, etc. Here, we highlight the central role of the charge-charge dynamic structure factor in the fluctuations of electrical observables in electrolytes and offer a unifying perspective over a variety of complementary experiments. We further analyze this quantity in the special case of an aqueous NaCl electrolyte, using simulations with explicit ions and an explicit or implicit solvent. We discuss the ability of the standard Poisson-Nernst-Planck theory to capture the simulation results, and how the predictions can be improved. We finally discuss the contributions of ions and water to the total charge fluctuations. This work illustrates an ongoing effort towards a comprehensive understanding of electrical fluctuations in bulk and confined electrolytes, in order to enable experimentalists to decipher the microscopic properties encoded in the measured electrical noise.

Driving forces of the complex formation between highly charged disordered proteins

Highly disordered complexes between oppositely charged intrinsically disordered proteins present a new paradigm of biomolecular interactions. Here, we investigate the driving forces of such interactions for the example of the highly positively charged linker histone H1 and its highly negatively charged chaperone, prothymosin α (ProTα). Temperature-dependent single-molecule Förster resonance energy transfer (FRET) experiments and isothermal titration calorimetry reveal ProTα-H1 binding to be enthalpically unfavorable, and salt-dependent affinity measurements suggest counterion release entropy to be an important thermodynamic driving force. Using single-molecule FRET, we also identify ternary complexes between ProTα and H1 in addition to the heterodimer at equilibrium and show how they contribute to the thermodynamics observed in ensemble experiments. Finally, we explain the observed thermodynamics quantitatively with a mean-field polyelectrolyte theory that treats counterion release explicitly. ProTα-H1 complex formation resembles the interactions between synthetic polyelectrolytes, and the underlying principles are likely to be of broad relevance for interactions between charged biomolecules in general.

A Gaussian field approach to the planar electric double layer structures in electrolyte solutions

In this work, the planar, electric, double-layer structures of non-polarizable electrodes in electrolyte solutions are studied with Gaussian field theory. A response function with two Yukawa functions is used to capture the electrostatic response of the electrolyte solution, from which the modified response function in the planar symmetry is derived analytically. The modified response function is further used to evaluate the induced charge density and the electrostatic potential near an electrode. The Gaussian field theory, combined with a two-Yukawa response function, can reproduce the oscillatory decay behavior of the electric potentials in concentrated electrolyte solutions. When the exact sum rules for the bulk electrolyte solutions and the electric double layers are used as constraints to determine the parameters of the response function, the Gaussian field theory could at least partly capture the nonlinear response effect of the surface charge density. Comparison with results for a planar electrode with fixed surface charge densities from molecular simulations demonstrates the validity of Gaussian field theory.

Anomalous Underscreening in the Restricted Primitive Model

Underscreening is a collective term for charge correlations in electrolytes decaying slower than the Debye length. Anomalous underscreening refers to phenomenology that cannot be attributed alone to steric interactions. Experiments with concentrated electrolytes and ionic fluids report anomalous underscreening, which so far has not been observed in simulation. We present Molecular Dynamics simulation results exhibiting anomalous underscreening that can be connected to cluster formation. A theory that accounts for ion pairing confirms the trend. Our results challenge the classic understanding of dense electrolytes impacting the design of technologies for energy storage and conversion.

Opto-Electrostatic Determination of Nucleic Acid Double-Helix Dimensions and the Structure of the Molecule–Solvent Interface

A DNA molecule is highly electrically charged in solution. The electrical potential at the molecular surface is known to vary strongly with the local geometry of the double helix and plays a pivotal role in DNA–protein interactions. Further out from the molecular surface, the electrical field propagating into the surrounding electrolyte bears fingerprints of the three-dimensional arrangement of the charged atoms in the molecule. However, precise extraction of the structural information encoded in the electrostatic “far field” has remained experimentally challenging. Here, we report an optical microscopy-based approach that detects the field distribution surrounding a charged molecule in solution, revealing geometric features such as the radius and the average rise per basepair of the double helix with up to sub-Angstrom precision, comparable with traditional molecular structure determination techniques like X-ray crystallography and nuclear magnetic resonance. Moreover, measurement of the helical radius furnishes an unprecedented view of both hydration and the arrangement of cations at the molecule–solvent interface. We demonstrate that a probe in the electrostatic far field delivers structural and chemical information on macromolecules, opening up a new dimension in the study of charged molecules and interfaces in solution.

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.

Electric-field-induced oscillations in ionic fluids: a unified formulation of modified Poisson-Nernst-Planck models and its relevance to correlation function analysis

We theoretically investigate an electric-field-driven system of charged spheres as a primitive model of concentrated electrolytes under an applied electric field. First, we provide a unified formulation for the stochastic charge and density dynamics of the electric-field-driven primitive model using the stochastic density functional theory (DFT). The stochastic DFT integrates the four frameworks (the equilibrium and dynamic DFTs, the liquid state theory and the field-theoretic approach), which allows us to justify in a unified manner various modifications previously made for the Poisson-Nernst-Planck model. Next, we consider stationary density-density and charge-charge correlation functions of the primitive model with a static electric field. We predict an electric-field-induced synchronization between emergences of density and charge oscillations. We are mainly concerned with the emergence of stripe states formed by segregation bands transverse to the external field, thereby demonstrating the following: (i) the electric-field-induced crossover occurs prior to the conventional Kirkwood crossover without an applied electric field, and (ii) the ion concentration dependence of the decay lengths at the onset of oscillations bears a similarity to the underscreening behavior found by recent simulation and theoretical studies on equilibrium electrolytes. Also, the 2D inverse Fourier transform of the correlation function illustrates the existence of stripe states beyond the electric-field-induced Kirkwood crossover.

Understanding specific ion effects and the Hofmeister series

Specific ion effects (SIE), encompassing the Hofmeister Series, have been known for more than 130 years since Hofmeister and Lewith's foundational work. SIEs are ubiquitous and are observed across the medical, biological, chemical and industrial sciences. Nevertheless, no general predictive theory has yet been able to explain ion specificity across these fields; it remains impossible to predict when, how, and to what magnitude, a SIE will be observed. In part, this is due to the complexity of real systems in which ions, counterions, solvents and cosolutes all play varying roles, which give rise to anomalies and reversals in anticipated SIEs. Herein we review the historical explanations for SIE in water and the key ion properties that have been attributed to them. Systems where the Hofmeister series is perturbed or reversed are explored, as is the behaviour of ions at the liquid-vapour interface. We discuss SIEs in mixed electrolytes, nonaqueous solvents, and in highly concentrated electrolyte solutions - exciting frontiers in this field with particular relevance to biological and electrochemical applications. We conclude the perspective by summarising the challenges and opportunities facing this SIE research that highlight potential pathways towards a general predictive theory of SIE.

Investigation of the Limits of the Linearized Poisson-Boltzmann Equation

This work presents a comparison between a numerical solution of the Poisson-Boltzmann equation and the analytical solution of its linearized version through the Debye-Hückel equations considering both size-dissimilar and common ion diameters approaches. In order to verify the limits in which the linearized Poisson-Boltzmann equation is capable to satisfactorily reproduce the nonlinear version of Poisson-Boltzmann, we calculate mean ionic activity coefficients for different types of electrolytes as various temperatures. The divergence between the linearized and full Poisson-Boltzmann equations is higher for lower molalities, and both solutions tend to converge toward higher molalities. For electrolytes of lower valencies (1:1, 1:2, 2:1, and 1:3) and higher distances of closest approach, the full version of the Debye-Hückel equation is capable of representing the activity coefficients with a low divergence from the nonlinear Poisson-Boltzmann. The size-dissimilar full version of Debye-Hückel represents a clear improvement over the extended version that uses only common ion diameters when compared to the numerical solution of the Poisson-Boltzmann equation. We have derived a salt-specific index (Θ) to gradually classify electrolytes in order of increasing influence of nonlinear ion-ion interactions, which differentiate the Debye-Hückel equations from the nonlinear Poisson-Boltzmann equation.

The primitive model in classical density functional theory: beyond the standard mean-field approximation

The primitive model describes ions by point charges with an additional hard-core interaction. In classical density-functional theory (DFT) the mean-field electrostatic contribution can be obtained from the first order of a functional perturbation of the pair potential for an uncharged reference system of hard spheres. This mean-field electrostatic term particularly contributes at particle separations that are forbidden due to hard-core overlap. In this work we modify the mean-field contribution such that the pair potential is constant for distances smaller than the contact distance of the ions. We motivate our modification by the underlying splitting of the potential, which is similar to the splitting of the Weeks-Chandler-Andersen potential and leads to higher-order terms in the respective expansion of the functional around the reference system. The resulting formalism involves weighted densities similar to the ones found in fundamental measure theory. To test our modifications, we analyze and compare density profiles, direct and total correlation functions, and the thermodynamic consistency of the functional via a widely established sum rule and the virial pressure formula for our modified functional, for established functionals, and for data from computer simulations. We found that our modifications clearly show improvements compared to the standard mean-field functional, especially when predicting layering effects and direct correlation functions in high concentration scenarios; for the latter we also find improved consistency when calculated via different thermodynamic routes. In conclusion, we demonstrate how modifications toward higher order corrections beyond mean-field functionals can be made and how they perform, by this providing a basis for systematic future improvements in classical DFT for the description of electrostatic interactions.

Ionic screening in bulk and under confinement

Recent experiments have shown that the repulsive force between atomically flat, like-charged surfaces confining room-temperature ionic liquids or concentrated electrolytes exhibits an anomalously large decay length. In our previous publication [J. Zeman, S. Kondrat, and C. Holm, Chem. Commun. 56, 15635 (2020)], we showed by means of extremely large-scale molecular dynamics simulations that this so-called underscreening effect might not be a feature of bulk electrolytes. Herein, we corroborate these findings by providing additional results with more detailed analyses and expand our investigations to ionic liquids under confinement. Unlike in bulk systems, where screening lengths are computed from the decay of interionic potentials of mean force, we extract such data in confined systems from cumulative charge distributions. At high concentrations, our simulations show increasing screening lengths with increasing electrolyte concentration, consistent with classical liquid state theories. However, our analyses demonstrate that-also for confined systems-there is no anomalously large screening length. As expected, the screening lengths determined for ionic liquids under confinement are in good quantitative agreement with the screening lengths of the same ionic systems in bulk. In addition, we show that some theoretical models used in the literature to relate the measured screening lengths to other observables are inapplicable to highly concentrated electrolytes.

Inter-droplet force between magnetically polarizable Pickering oil-in-water nanoemulsions stabilized with γ-Al2O3 nanoparticles: Role of electrostatic and electric dipolar interactions

HYPOTHESIS

The presence of nanoparticles at oil-water interface influences the interaction forces between Pickering emulsions. When charged nanoparticles are at the oil-water interface of an electrostatically stabilized emulsion, in addition to the screened Coulombic interaction, electric dipolar force also influences the total inter-droplet force profiles. An in-depth understanding of the effects of such electric dipolar forces is essential for designing colloidally stable Pickering nanoemulsions for various applications.

EXPERIMENTS

Inter-droplet forces between γ-Al₂O₃ nanoparticle stabilized oil-in-water nanoemulsion, containing superparamagnetic nanoparticles (magnetically polarizable) in the oil phase, are measured using the magnetic-chaining technique at different pH and salt concentrations. The role of mono-, di- and tri-valent salts on the inter-droplet force profiles are assessed.

FINDINGS

Force measurement studies reveal a lowering of inter-droplet spacing, within the linear chains, for higher salt concentrations due to an increased screening. Strong interfacial attachment of the charged nanoparticles results in the formation of an asymmetric charge cloud leading to an electric dipolar interaction. Incorporating the contributions of electric dipolar and screened Coulombic interactions, the theoretically estimated total repulsive force magnitudes are in good agreement with the experimental data. The obtained results offer better insights into the nature of colloidal force between charged particle stabilized nanoemulsions.

Fast Calculation of Electrostatic Solvation Free Energy in Simple Ionic Fluids Using an Energy-Scaled Debye-Hückel Theory

Continuum theories are useful to compute the solvation free energy in ionic fluids. Herein, the electrostatic solvation free energy (ESFE) in simple ionic fluids is studied with an energy-scaled Debye-Hückel (ESDH) theory. Given the ESFEs of simple spherical ions as input, the ESDH theory is applicable to molecules with various complex geometries and charge distributions. Specifically, the ESDH theory is applied to molecules in a molten salt system, where the predicted ESFEs are in good agreement with molecular dynamics simulation results. Our study sheds light on accurately predicting the ESFE in ionic fluids with phenomenological continuum theories.

On the modified Poisson-Boltzmann closure for primitive model electrolytes at high concentration

The modified Poisson-Boltzmann closure is applied to the Kirkwood hierarchy of integral equations to investigate high concentration primitive model electrolytes. Two approximations are considered in the two sphere fluctuation potential problem. The derived damped oscillatory mean electrostatic potentials suggest that this closure should be of use in providing a basis for understanding the large experimental decay lengths found at high electrolyte concentrations.

Bayesian unsupervised learning reveals hidden structure in concentrated electrolytes

Electrolytes play an important role in a plethora of applications ranging from energy storage to biomaterials. Notwithstanding this, the structure of concentrated electrolytes remains enigmatic. Many theoretical approaches attempt to model the concentrated electrolyte by introducing the idea of ion pairs, with ions either being tightly "paired" with a counter-ion or "free" to screen charge. In this study, we reframe the problem into the language of computational statistics and test the null hypothesis that all ions share the same local environment. Applying the framework to molecular dynamics simulations, we find that this null hypothesis is not supported by data. Our statistical technique suggests the presence of two distinct local ionic environments at intermediate concentrations, whose differences surprisingly originate in like charge correlations rather than unlike charge attraction. Through considering the effect of these "aggregated" and "non-aggregated" states on bulk properties including effective ion concentration and dielectric constant, we identify a scaling relation between the effective screening length and theoretical Debye length, which applies across different dielectric constants and ion concentrations.

Primitive model electrolytes in the near and far field: Decay lengths from DFT and simulations

Inspired by recent experimental observations of anomalously large decay lengths in concentrated electrolytes, we revisit the Restricted Primitive Model (RPM) for an aqueous electrolyte. We investigate the asymptotic decay lengths of the one-body ionic density profiles for the RPM in contact with a planar electrode using classical Density Functional Theory (DFT) and compare these with the decay lengths of the corresponding two-body correlation functions in bulk systems, obtained in previous Integral Equation Theory (IET) studies. Extensive Molecular Dynamics (MD) simulations are employed to complement the DFT and IET predictions. Our DFT calculations incorporate electrostatic interactions between the ions using three different (existing) approaches: one is based on the simplest mean-field treatment of Coulomb interactions (MFC), while the other two employ the Mean Spherical Approximation (MSA). The MSAc invokes only the MSA bulk direct correlation function, whereas the MSAu also incorporates the MSA bulk internal energy. Although MSAu yields profiles that are in excellent agreement with MD simulations in the near field, in the far field, we observe that the decay lengths are consistent between IET, MSAc, and MD simulations, whereas those from MFC and MSAu deviate significantly. Using DFT, we calculated the solvation force, which relates directly to surface force experiments. We find that its decay length is neither qualitatively nor quantitatively close to the large decay lengths measured in experiments and conclude that the latter cannot be accounted for by the primitive model. The anomalously large decay lengths found in surface force measurements require an explanation that lies beyond primitive models.

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.