The vapor-liquid interface potential of (multi)polar fluids and its influence on ion solvation

Multiscale Modeling of Aqueous Electric Double Layers

From the stability of colloidal suspensions to the charging of electrodes, electric double layers play a pivotal role in aqueous systems. The interactions between interfaces, water molecules, ions and other solutes making up the electrical double layer span length scales from Ångströms to micrometers and are notoriously complex. Therefore, explaining experimental observations in terms of the double layer's molecular structure has been a long-standing challenge in physical chemistry, yet recent advances in simulations techniques and computational power have led to tremendous progress. In particular, the past decades have seen the development of a multiscale theoretical framework based on the combination of quantum density functional theory, force-field based simulations and continuum theory. In this Review, we discuss these theoretical developments and make quantitative comparisons to experimental results from, among other techniques, sum-frequency generation, atomic-force microscopy, and electrokinetics. Starting from the vapor/water interface, we treat a range of qualitatively different types of surfaces, varying from soft to solid, from hydrophilic to hydrophobic, and from charged to uncharged.

Electrokinetic, electrochemical, and electrostatic surface potentials of the pristine water liquid-vapor interface

Although conceptually simple, the air-water interface displays rich behavior and is subject to intense experimental and theoretical investigations. Different definitions of the electrostatic surface potential as well as different calculation methods, each relevant for distinct experimental scenarios, lead to widely varying potential magnitudes and sometimes even different signs. Based on quantum-chemical density-functional-theory molecular dynamics (DFT-MD) simulations, different surface potentials are evaluated and compared to force-field (FF) MD simulations. As well explained in the literature, the laterally averaged electrostatic surface potential, accessible to electron holography, is dominated by the trace of the water molecular quadrupole moment, and using DFT-MD amounts to +4.35 V inside the water phase, very different from results obtained with FF water models which yield negative values of the order of -0.4 to -0.6 V. Thus, when predicting potentials within water molecules, as relevant for photoelectron spectroscopy and non-linear interface-specific spectroscopy, DFT simulations should be used. The electrochemical surface potential, relevant for ion transfer reactions and ion surface adsorption, is much smaller, less than 200 mV in magnitude, and depends specifically on the ion radius. Charge transfer between interfacial water molecules leads to a sizable surface potential as well. However, when probing electrokinetics by explicitly applying a lateral electric field in DFT-MD simulations, the electrokinetic ζ-potential turns out to be negligible, in agreement with predictions using continuous hydrodynamic models. Thus, interfacial polarization charges from intermolecular charge transfer do not lead to significant electrokinetic mobility at the pristine vapor-liquid water interface, even assuming these transfer charges are mobile in an external electric field.

Ionic mobility driven by correlated van der Waals and electrostatic forces

Classical theories of dielectric friction make two critical assumptions: (i) friction due to van der Waals (vdW) forces is described by hydrodynamic drag and is independent of the ionic charge and (ii) vdW and electrostatic forces are statistically independent. Both assumptions turn out to be incorrect when tested against simulations of anions and cations with varying charge magnitude dissolved in water. Both the vdW and electrostatic components of the force variance scale linearly with the ionic charge squared. The two components are strongly anticorrelated producing simple relations for the total force variance in terms of self-variances. The inverse diffusion constant scales linearly with the charge squared. Solvation asymmetry between cations and anions extends to linear transport coefficients.

Competition between Born solvation, dielectric exclusion, and Coulomb attraction in spherical nanopores

The recent measurement of a very low dielectric constant, ε, of water confined in nanometric slit pores leads us to reconsider the physical basis of ion partitioning into nanopores. For confined ions in chemical equilibrium with a bulk of dielectric constant ε_{b}>ε, three physical mechanisms, at the origin of ion exclusion in nanopores, are expected to be modified due to this dielectric mismatch: dielectric exclusion at the water-pore interface (with membrane dielectric constant, ε_{m}<ε), the solvation energy related to the difference in Debye-Hückel screening parameters in the pore, κ, and in the bulk κ_{b}, and the classical Born solvation self-energy proportional to ε^{-1}-ε_{b}^{-1}. Our goal is to clarify the interplay between these three mechanisms and investigate the role played by the Born contribution in ionic liquid-vapor (LV) phase separation in confined geometries. We first compute analytically the potential of mean force (PMF) of an ion of radius R_{i} located at the center of a nanometric spherical pore of radius R. Computing the variational grand potential for a solution of confined ions, we then deduce the partition coefficients of ions in the pore versus R and the bulk electrolyte concentration ρ_{b}. We show how the ionic LV transition, directly induced by the abrupt change of the dielectric contribution of the PMF with κ, is favored by the Born self-energy and explore the decrease of the concentration in the pore with ε both in the vapor and liquid states. Phase diagrams are established for various parameter values and we show that a signature of this phase transition can be detected by monitoring the total osmotic pressure as a function of R. For charged nanopores, these exclusion effects compete with the electrostatic attraction that imposes the entry of counterions into the pore to enforce electroneutrality. This study will therefore help in deciphering the respective roles of the Born self-energy and dielectric mismatch in experiments and simulations of ionic transport through nanopores.

Condensed Phase Water Molecular Multipole Moments from Deep Neural Network Models Trained on Ab Initio Simulation Data

Ionic solvation phenomena in liquids involve intense interactions in the inner solvation shell. For interactions beyond the first shell, the ion-solvent interaction energies result from the sum of many smaller-magnitude contributions that can still include polarization effects. Deep neural network (DNN) methods have recently found wide application in developing efficient molecular models that maintain near-quantum accuracy. Here we extend the DeePMD-kit code to produce accurate molecular multipole moments in the bulk and near interfaces. The new method is validated by comparing the DNN moments with those generated by ab initio simulations. The moments are used to compute the electrostatic potential at the center of a molecular-sized hydrophobic cavity in water. The results show that the fields produced by the DNN models are in quantitative agreement with the AIMD-derived values. These efficient methods will open the door to more accurate solvation models for large solutes such as proteins.

Renormalized site density functional theory for models of ion hydration

The development of accurate statistical mechanics models of molecular liquid systems is a problem of great practical and fundamental importance. Site-density functional theory (SDFT) is one of the promising directions in this area, but its success hinges upon the ability to efficiently reconcile the co-existence of two distinct intra- and inter-molecular interaction regimes in a molecular liquid. The renormalized formulation of SDFT (RSDFT), which we have recently developed, resolves this problem by introducing an additional potential field variable that decouples two interaction scales and maps the molecular liquid problem onto the effective simple liquid mixture. This work provides a critical assessment of RSDFT for the hydrated ion system-a problem that historically has always been one of the most difficult cases for SDFT applications. Using a two-site model of water, we perform a comprehensive analysis of hydrated alkali metal and halogen ions, including both structural and free energy based characteristics. The results indicate that RSDFT provides a significant improvement over conventional three-dimensional reference interaction site model implementations and may prove useful in coarse grained simulations based on two-site solvent models.

Quadrupole-mediated dielectric response and the charge-asymmetric solvation of ions in water

Treating water as a linearly responding dielectric continuum on molecular length scales allows very simple estimates of the solvation structure and thermodynamics for charged and polar solutes. While this approach can successfully account for basic length and energy scales of ion solvation, computer simulations indicate not only its quantitative inaccuracies but also its inability to capture some basic and important aspects of microscopic polarization response. Here, we consider one such shortcoming, a failure to distinguish the solvation thermodynamics of cations from that of otherwise-identical anions, and we pursue a simple, physically inspired modification of the dielectric continuum model to address it. The adaptation is motivated by analyzing the orientational response of an isolated water molecule whose dipole is rigidly constrained. Its free energy suggests a Hamiltonian for dipole fluctuations that accounts implicitly for the influence of higher-order multipole moments while respecting constraints of molecular geometry. We propose a field theory with the suggested form, whose nonlinear response breaks the charge symmetry of ion solvation. An approximate variational solution of this theory, with a single adjustable parameter, yields solvation free energies that agree closely with simulation results over a considerable range of solute size and charge.

Absolute ion hydration free energy scale and the surface potential of water via quantum simulation

With a goal of determining an absolute free energy scale for ion hydration, quasi-chemical theory and ab initio quantum mechanical simulations are employed to obtain an accurate value for the bulk hydration free energy of the Na⁺ ion. The free energy is partitioned into three parts: 1) the inner-shell or chemical contribution that includes direct interactions of the ion with nearby waters, 2) the packing free energy that is the work to produce a cavity of size λ in water, and 3) the long-range contribution that involves all interactions outside the inner shell. The interfacial potential contribution to the free energy resides in the long-range term. By averaging cation and anion data for that contribution, cumulant terms of all odd orders in the electrostatic potential are removed. The computed total is then the bulk hydration free energy. Comparison with the experimentally derived real hydration free energy produces an effective surface potential of water in the range -0.4 to -0.5 V. The result is consistent with a variety of experiments concerning acid-base chemistry, ion distributions near hydrophobic interfaces, and electric fields near the surface of water droplets.

Assessing long-range contributions to the charge asymmetry of ion adsorption at the air-water interface

Anions generally associate more favorably with the air–water interface than cations. In addition to solute size and polarizability, the intrinsic structure of the unperturbed interface has been discussed as an important contributor to this bias. Here we assess quantitatively the role that intrinsic charge asymmetry of water's surface plays in ion adsorption, using computer simulations to compare model solutes of various size and charge. In doing so, we also evaluate the degree to which linear response theory for solvent polarization is a reasonable approach for comparing the thermodynamics of bulk and interfacial ion solvation. Consistent with previous works on bulk ion solvation, we find that the average electrostatic potential at the center of a neutral, sub-nanometer solute at the air–water interface depends sensitively on its radius, and that this potential changes quite nonlinearly as the solute's charge is introduced. The nonlinear response closely resembles that of the bulk. As a result, the net nonlinearity of ion adsorption is weaker than in bulk, but still substantial, comparable to the apparent magnitude of macroscopically nonlocal contributions from the undisturbed interface. For the simple-point-charge model of water we study, these results argue distinctly against rationalizing ion adsorption in terms of surface potentials inherent to molecular structure of the liquid's boundary.

Cations and anions have different affinities for the air-water interface. The intrinsic orientation of surface molecules suggests such an asymmetry, but the bias is dominated by solvent response that is spatially local and significantly nonlinear.

Electrostatic solvation and mobility in uniform and non-uniform electric fields: From simple ions to proteins

A number of observations related to interfacial electrostatics of polar liquids question the traditional assumption of dielectric theories that bulk dielectric properties can be continuously extended to the dividing surface separating the solute from the solvent. The deficiency of this approximation can be remedied by introducing local interface susceptibilities and the interface dielectric constant. Asymmetries of ionic hydration thermodynamics and of the mobility between cations and anions can be related to different propensities of the water molecules to orient their dipole toward and outward from solutes of opposite charges. This electrostatic asymmetry is reflected in different interface dielectric constants for cations and anions. The interface of water with neutral solutes is spontaneously polarized due to preferential water orientations in the interface. This phenomenon is responsible for a nonzero cavity potential directly related to a nonzero surface charge. This connection predicts that particles allowing a nonzero cavity potential must show mobility in an external electric field even if the net charge of the particle is zero. The theory predicts that a positive cavity potential and a positive surface charge translate to an effectively negative solute charge reported by mobility measurements. Passing of the cavity potential through a minimum found in simulations might be the origin of the maximum of mobility vs the ionic size observed experimentally. Finally, mobility of proteins in the field gradient (dielectrophoresis) is many orders of magnitude greater than predicted by the traditionally used Clausius-Mossotti equation. Two reasons contribute to this disagreement: (i) a failure of Maxwell's electrostatics to describe the cavity-field susceptibility and (ii) the neglect of the protein permanent dipole by the Clausius-Mossotti equation. An analytical relation between the dielectrophoretic susceptibility and dielectric spectroscopy of solutions provides direct access to this parameter, confirming the failure of the Clausius-Mossotti equation in application to protein dielectrophresis.

Hydration Free Energies of Molecular Ions from Theory and Simulation

We present a theoretical/computational framework for accurate calculation of hydration free energies of ionized molecular species. The method is based on a molecular theory, 3D-RISM, combined with a recently developed pressure correction (PC+). The 3D-RISM/PC+ model can provide ∼3 kcal/mol hydration free energy accuracy for a large variety of ionic compounds, provided that the Galvani potential of water is taken into account. The results are compared with direct atomistic simulations. Several methodological aspects of hydration free energy calculations for charged species are discussed.

The Role of Broken Symmetry in Solvation of a Spherical Cavity in Classical and Quantum Water Models

Insertion of a hard sphere cavity in liquid water breaks translational symmetry and generates an electrostatic potential difference between the region near the cavity and the bulk. Here, we clarify the physical interpretation of this potential and its calculation. We also show that the electrostatic potential in the center of small, medium, and large cavities depends very sensitively on the form of the assumed molecular interactions for different classical simple point-charge models and quantum mechanical DFT-based interaction potentials, as reflected in their description of donor and acceptor hydrogen bonds near the cavity. These differences can significantly affect the magnitude of the scalar electrostatic potential. We argue that the result of these studies will have direct consequences toward our understanding of the thermodynamics of ion solvation through the cavity charging process.

Charge asymmetry at aqueous hydrophobic interfaces and hydration shells

Guilty as charged: Water is often modeled as a dielectric continuum, but the molecular structure of water is asymmetric. Two ions that have a virtually identical size, shape, and structure, but an opposite charge sign have been investigated to see whether charge makes a fundamental difference to water structuring. The spectroscopic data for the hydration and interface structures are found to be remarkably different for opposite charges.

An Overview of Electrostatic Free Energy Computations for Solutions and Proteins

Free energy simulations for electrostatic and charging processes in complex molecular systems encounter specific difficulties owing to the long-range, 1/r Coulomb interaction. To calculate the solvation free energy of a simple ion, it is essential to take into account the polarization of nearby solvent but also the electrostatic potential drop across the liquid-gas boundary, however distant. The latter does not exist in a simulation model based on periodic boundary conditions because there is no physical boundary to the system. An important consequence is that the reference value of the electrostatic potential is not an ion in a vacuum. Also, in an infinite system, the electrostatic potential felt by a perturbing charge is conditionally convergent and dependent on the choice of computational conventions. Furthermore, with Ewald lattice summation and tinfoil conducting boundary conditions, the charges experience a spurious shift in the potential that depends on the details of the simulation system such as the volume fraction occupied by the solvent. All these issues can be handled with established computational protocols, as reviewed here and illustrated for several small ions and three solvated proteins.

Ions at hydrophobic interfaces

We review the present understanding of the behavior of ions at the air-water and oil-water interfaces. We argue that while the alkali metal cations remain strongly hydrated and are repelled from the hydrophobic surfaces, the anions must be classified into kosmotropes and chaotropes. The kosmotropes remain strongly hydrated in the vicinity of a hydrophobic surface, while the chaotropes lose their hydration shell and can become adsorbed to the interface. The mechanism of adsorption is still a subject of debate. Here, we argue that there are two driving forces for anionic adsorption: the hydrophobic cavitational energy and the interfacial electrostatic surface potential of water. While the cavitational contribution to ionic adsorption is now well accepted, the role of the electrostatic surface potential is much less clear. The difficulty is that even the sign of this potential is a subject of debate, with the ab initio and the classical force field simulations predicting electrostatic surface potentials of opposite sign. In this paper, we will argue that the strong anionic adsorption found in the polarizable force field simulations is the result of the artificial electrostatic surface potential present in the classical water models. We will show that if the adsorption of anions were as large as predicted by the polarizable force field simulations, the excess surface tension of the NaI solution would be strongly negative, contrary to the experimental measurements. While the large polarizability of heavy halides is a fundamental property and must be included in realistic modeling of the electrolyte solutions, we argue that the point charge water models, studied so far, are incompatible with the polarizable ionic force fields when the translational symmetry is broken. The goal for the future should be the development of water models with very low electrostatic surface potential. We believe that such water models will be compatible with the polarizable force fields, which can then be used to study the interaction of ions with hydrophobic surfaces and proteins.

Calculating the binding free energies of charged species based on explicit-solvent simulations employing lattice-sum methods: an accurate correction scheme for electrostatic finite-size effects

The calculation of a protein-ligand binding free energy based on molecular dynamics (MD) simulations generally relies on a thermodynamic cycle in which the ligand is alchemically inserted into the system, both in the solvated protein and free in solution. The corresponding ligand-insertion free energies are typically calculated in nanoscale computational boxes simulated under periodic boundary conditions and considering electrostatic interactions defined by a periodic lattice-sum. This is distinct from the ideal bulk situation of a system of macroscopic size simulated under non-periodic boundary conditions with Coulombic electrostatic interactions. This discrepancy results in finite-size effects, which affect primarily the charging component of the insertion free energy, are dependent on the box size, and can be large when the ligand bears a net charge, especially if the protein is charged as well. This article investigates finite-size effects on calculated charging free energies using as a test case the binding of the ligand 2-amino-5-methylthiazole (net charge +1 e) to a mutant form of yeast cytochrome c peroxidase in water. Considering different charge isoforms of the protein (net charges -5, 0, +3, or +9 e), either in the absence or the presence of neutralizing counter-ions, and sizes of the cubic computational box (edges ranging from 7.42 to 11.02 nm), the potentially large magnitude of finite-size effects on the raw charging free energies (up to 17.1 kJ mol(-1)) is demonstrated. Two correction schemes are then proposed to eliminate these effects, a numerical and an analytical one. Both schemes are based on a continuum-electrostatics analysis and require performing Poisson-Boltzmann (PB) calculations on the protein-ligand system. While the numerical scheme requires PB calculations under both non-periodic and periodic boundary conditions, the latter at the box size considered in the MD simulations, the analytical scheme only requires three non-periodic PB calculations for a given system, its dependence on the box size being analytical. The latter scheme also provides insight into the physical origin of the finite-size effects. These two schemes also encompass a correction for discrete solvent effects that persists even in the limit of infinite box sizes. Application of either scheme essentially eliminates the size dependence of the corrected charging free energies (maximal deviation of 1.5 kJ mol(-1)). Because it is simple to apply, the analytical correction scheme offers a general solution to the problem of finite-size effects in free-energy calculations involving charged solutes, as encountered in calculations concerning, e.g., protein-ligand binding, biomolecular association, residue mutation, pKa and redox potential estimation, substrate transformation, solvation, and solvent-solvent partitioning.

Charge and electric field fluctuations in aqueous NaCl electrolytes

Crystalloluminescence, the long-lived emission of visible light during the crystallization of certain salts, was first observed over 200 years ago; however, the origin of this luminescence is still not well understood. The observations suggest that the process of crystallization may not be purely classical but also involves an essential electronic structure component. Strong electric field fluctuations may play an important role in this process by providing the necessary driving force for the observed electronic structure changes. The main objective of this work is to provide a basic understanding of the fluctuations in charge, electric potentials, and electric fields for concentrated aqueous NaCl electrolytes. Our charge analysis reveals that the water molecules in the first solvation shell of the ions serve as a sink for electron density originating on Cl(-). We find that the electric fields inside aqueous electrolytes are extremely large (up to several V/Å) and thus may alter the ground and excited electronic states in the condensed phase. Furthermore, our results show that the potential and field distributions are largely independent of concentration. We also find the field component distributions to be Gaussian for the ions and non-Gaussian for the O and H sites (computed in the lab frame of reference), however, these non-Gaussian distributions are readily modeled via an orientationally averaged nonzero mean Gaussian plus a zero mean Gaussian. These calculations and analyses provide the first steps toward understanding the magnitude and fluctuations of charge, electric potentials, and fields in aqueous electrolytes and what role these fields may play in driving charge redistribution/transfer during crystalloluminescence.