A close look into the excluded volume effects within a double layer

Exact analytical solution of the Flory-Huggins model and extensions to multicomponent systems

The Flory-Huggins theory describes the phase separation of solutions containing polymers. Although it finds widespread application from polymer physics to materials science to biology, the concentrations that coexist in separate phases at equilibrium have not been determined analytically, and numerical techniques are required that restrict the theory's ease of application. In this work, we derive an implicit analytical solution to the Flory-Huggins theory of one polymer in a solvent by applying a procedure that we call the implicit substitution method. While the solutions are implicit and in the form of composite variables, they can be mapped explicitly to a phase diagram in composition space. We apply the same formalism to multicomponent polymeric systems, where we find analytical solutions for polydisperse mixtures of polymers of one type. Finally, while complete analytical solutions are not possible for arbitrary mixtures, we propose computationally efficient strategies to map out coexistence curves for systems with many components of different polymer types.

Polar liquids at charged interfaces: A dipolar shell theory

The structure of polar liquids and electrolytic solutions, such as water and aqueous electrolytes, at interfaces underlies numerous phenomena in physics, chemistry, biology, and engineering. In this work, we develop a continuum theory that captures the essential features of dielectric screening by polar liquids at charged interfaces, including decaying spatial oscillations in charge and mass, starting from the molecular properties of the solvent. The theory predicts an anisotropic dielectric tensor of interfacial polar liquids previously studied in molecular dynamics simulations. We explore the effect of the interfacial polar liquid properties on the capacitance of the electrode/electrolyte interface and on hydration forces between two plane-parallel polarized surfaces. In the linear response approximation, we obtain simple formulas for the characteristic decay lengths of molecular and ionic profiles at the interface.

Relaxation dynamics of two interacting electrical double-layers in a 1D Coulomb system

We consider an out-of-equilibrium one-dimensional model for two electrical double-layers. With a combination of exact calculations and Brownian dynamics simulations, we compute the relaxation time (τ) for an electroneutral salt-free suspension, made up of two fixed colloids, withNneutralizing mobile counterions. ForNodd, the two double-layers never decouple, irrespective of their separationL; this is the regime of like-charge attraction, whereτexhibits a diffusive scaling inL²for largeL. On the other hand, for evenN,Lno longer is the relevant length scale for setting the relaxation time; this role is played by the Bjerrum length. This leads to distinctly different dynamics: forNeven, thermal effects are detrimental to relaxation, increasingτ, while they accelerate relaxation forNodd. Finally, we also show that the mean-field theory is recovered for largeNand moreover, that it remains an operational treatment down to relatively small values ofN(N> 3).

Surface tension in liquids containing antagonistic ions

We use a modified Poisson-Boltzmann formalism to examine immiscible electrolytes containing dissolved antagonistic ions with arbitrary preferential solubilities. We solve the nonlinear equation and obtain an analytical expression for the potential profile, ion densities, and surface tension. Our model takes into account the dependence of the Debye lengths in the two liquids on the preferential solvation. In the limit of point-like ions, the surface tension scales with the average ion density n0 as the classical n01/2 power law. At larger densities ion crowding at the interface leads to a crossover to smaller exponents. The dependence of the surface tension on the Gibbs transfer energy is non-monotonic and exhibits a maximum or a minimum. We further look at a liquid bilayer confined in a parallel-plate capacitor and subject to an external potential. The ion distribution depends on whether the external potential has the same or opposite sign as the Donnan potential. Lastly, we calculate the surface tension of a liquid-liquid interface that has a sinusoidal height modulation and find that surface modulations increase the energy.

General theory of charge regulation within the Poisson-Boltzmann framework: Study of a sticky-charged wall model

This work introduces a sticky-charge wall model as a simple and intuitive representation of charge regulation. Implemented within the mean-field level of description, the model modifies the boundary conditions without affecting the underlying Poisson-Boltzmann (PB) equation of an electrolyte. Employing various modified PB equations, we are able to assess how various structural details of an electrolyte influence charge regulation.

Lattice model of ionic liquid confined by metal electrodes

We study, using Monte Carlo simulations, the density profiles and differential capacitance of ionic liquids confined by metal electrodes. To compute the electrostatic energy, we use the recently developed approach based on periodic Green's functions. The method also allows us to easily calculate the induced charge on the electrodes permitting an efficient implementation of simulations in a constant electrostatic potential ensemble. To speed up the simulations further, we model the ionic liquid as a lattice Coulomb gas and precalculate the interaction potential between the ions. We show that the lattice model captures the transition between camel-shaped and bell-shaped capacitance curves-the latter characteristic of ionic liquids (strong coupling limit) and the former of electrolytes (weak coupling). We observe the appearance of a second peak in the differential capacitance at ≈0.5 V for 2:1 ionic liquids, as the packing fraction is increased. Finally, we show that ionic size asymmetry decreases substantially the capacitance maximum, when all other parameters are kept fixed.

Theory of polymer-electrolyte-composite electroactuator sensors with flat or volume-filling electrodes

In reverse actuation, a voltage/electrical-current signal can be generated from applying a mechanical force to an electroactuator. Such processes are of interest in touch sensing and soft robotics applications. We develop a classical density functional theory of reverse actuation for polymer-electrolyte-composite electroactuators, which treats mobile cations in the same spirit as forward actuation (curving in response to applied voltage). The proposed framework is applied to electroactuators with micro-structured porous electrodes (with cylindrical or slit pores) and flat electrodes, the dynamic response of which has to be modelled differently. Open- and short-circuit operation modes are investigated separately. A detailed analysis of the proposed theory indicates the preferred architectures for sensing, depending on the operation regimes.

Modeling the camel-to-bell shape transition of the differential capacitance using mean-field theory and Monte Carlo simulations

Mean-field electrostatics is used to calculate the differential capacitance of an electric double layer formed at a planar electrode in a symmetric 1:1 electrolyte. Assuming the electrolyte is also ion-size symmetric, we derive analytic expressions for the differential capacitance valid up to fourth order in the surface charge density or surface potential. Our mean-field model accounts exclusively for electrostatic interactions but includes an arbitrary non-ideality in the mixing entropy of the mobile ions. The ensuing criterion for the camel-to-bell shape transition of the differential capacitance is analyzed using commonly used mixing models (one based on a lattice gas and the other based on the Carnahan-Starling equation of state) and compared with Monte Carlo simulations. We observe a reasonable agreement between all our mean-field models and the simulation data for the camel-to-bell shape transition. The absolute value of the differential capacitance for an uncharged (or weakly charged) electrode is, however, not reproduced by our mean-field approaches, not even upon introducing a Stern layer with a thickness equal of the ion radius. We show that, if a Stern layer is introduced, its thickness dependence on the ion size is non-monotonic or, depending on the salt concentration, even inversely proportional.

Ionic size effects on the Poisson-Boltzmann theory

In this paper, we develop a simple theory to study the effects of ionic size on ionic distributions around a charged spherical particle. We include a correction to the regular Poisson-Boltzmann equation in order to take into account the size of ions in a mean-field regime. The results are compared with Monte Carlo simulations and a density functional theory based on the fundamental measure approach and a second-order bulk expansion which accounts for electrostatic correlations. The agreement is very good even for multivalent ions. Our results show that the theory can be applied with very good accuracy in the description of ions with highly effective ionic radii and low concentration, interacting with a colloid or a nanoparticle in an electrolyte solution.

Mean-field theory of active electrolytes: Dynamic adsorption and overscreening

We investigate active electrolytes within the mean-field level of description. The focus is on how the double-layer structure of passive, thermalized charges is affected by active dynamics of constituting ions. One feature of active dynamics is that particles adhere to hard surfaces, regardless of chemical properties of a surface and specifically in complete absence of any chemisorption or physisorption. To carry out the mean-field analysis of the system that is out of equilibrium, we develop the "mean-field simulation" technique, where the simulated system consists of charged parallel sheets moving on a line and obeying active dynamics, with the interaction strength rescaled by the number of sheets. The mean-field limit becomes exact in the limit of an infinite number of movable sheets.

Incorporation of ion and solvent structure into mean-field modeling of the electric double layer

An electric double layer forms when the small mobile ions of an electrolyte interact with an extended charged object, a macroion. The competition between electrostatic attraction and translational entropy loss of the small ions results in a diffuse layer of partially immobilized ions in the vicinity of the macroion. Modeling structure and energy of the electric double layer has a long history that has lead to the classical Poisson-Boltzmann theory and numerous extensions that account for ion-ion correlations and structural ion and solvent properties. The present review focuses on approaches that instead of going beyond the mean-field character of Poisson-Boltzmann theory introduce structural details of the ions and the solvent into the Poisson-Boltzmann modeling framework. The former include not only excluded volume effects but also the presence of charge distributions on individual ions, spatially extended ions, and internal ionic degrees of freedom. The latter treat the solvent either explicitly as interacting Langevin dipoles or in the form of effective non-electrostatic interactions, in particular Yukawa interactions, that are added to the Coulomb potential. We discuss how various theoretical models predict structural properties of the electric double layer such as the differential capacitance and compare some of these predictions with computer simulations.

Interaction between Charged Cylinders in Electrolyte Solution; Excluded Volume Effect

Electrostatic interactions govern the physical properties of charged cylindrical structures in electrolyte solutions. Besides the surface charge on the cylinders, another factor influencing the electrostatic interactions are the mobile ions. The finite size of the mobile ions is included by the excluded volume effect within the lattice statistics, while the electrostatic interactions are considered by means of the mean electrostatic field. In this article we consider charged parallel cylinders embedded into an electrolyte solution of mobile monovalent ions. A modified nonlinear Poisson-Boltzmann equation is proposed via variational procedure, and we implement the finite element method to solve it numerically. Excluded volume effect of the system containing two and multiple charged parallel cylinders are taken into account. Numerical results show that the excluded volume effect decreases the concentration of counterion and increases the electrostatic potential near the charged cylinders. The angular distribution of counterion around the particular cylinder is asymmetric. The study of the electrostatic interaction between two parallel equally charged cylinders reveals that an increase in the free energy is seen when the ionic strength is decreased. The free energy decreases as a function of the cylinders separation distance. On the contrary for two oppositely charged cylinders, the free energy increases with increasing cylinder separation distance, while for two cylinders with different charged density it shows nonmonotonic variation with the increasing cylinders separation distance.

Lattice Model of an Ionic Liquid at an Electrified Interface

We study ionic liquids interacting with electrified interfaces. The ionic fluid is modeled as a Coulomb lattice gas. We compare the ionic density profiles calculated using a popular modified Poisson-Boltzmann equation with the explicit Monte Carlo simulations. The modified Poisson-Boltzmann theory fails to capture the structural features of the double layer and is also unable to correctly predict the ionic density at the electrified interface. The lattice Monte Carlo simulations qualitatively capture the coarse-grained structure of the double layer in the continuum. We propose a convolution relation that semiquantitatively relates the ionic density profiles of a continuum ionic liquid and its lattice counterpart near an electrified interface.

Ion-Image Interactions and Phase Transition at Electrolyte-Metal Interfaces

The arrangement of ions near a metallic electrode is crucial to energy storage in electrical double-layer capacitors. Classic Poisson-Boltzmann theory predicts that the charge stored in the double layer is a continuous function of applied voltage. However, recent experiments and simulations strongly suggest the presence of a voltage-induced first-order phase transition in the electrical double layer, leading to a hysteretic response: the capacitance-voltage relation is dependent on whether the voltage is increasing or decreasing. By developing a simple analytical model, we show that ion-image interaction could explain this phase transition. Moreover, our model shows that the presence of phase transition depends on the bulk energy of the ionic liquid. Our results justify mixing ionic liquids with solvents as a way to achieve large capacitance and avoid hysteresis.

Shells of charge: a density functional theory for charged hard spheres

A functional for the electrostatic excess free-energy for charged, hard sphere fluids is proposed. The functional is derived from two complementary, but equivalent, interpretations of the mean spherical approximation (MSA). The first combines fundamental measure theory (FMT) from hard-core interactions with the idea that MSA can be interpreted in terms of the interaction spherical shells of charge. This formulation gives the free-energy density as a function of weighted densities. When all the ions have the same size, the functional adopts an FMT-like form. The second in effect 'functionalizes' the derivation of MSA; that is, it generalizes the MSA as a functional-based version of MSA (fMSA). This formulation defines the free-energy density as a function of a position-dependent MSA screening parameter and the weighted densities of the FMT approach. This FMT/fMSA functional is shown to give accurate density profiles, as compared to Monte Carlo simulations, under a wide range of ion concentrations, size asymmetries, and valences.

Density functional formulation of the random-phase approximation for inhomogeneous fluids: Application to the Gaussian core and Coulomb particles

Using the adiabatic connection, we formulate the free energy in terms of the correlation function of a fictitious system, h_{λ}(r,r^{'}), in which interactions λu(r,r^{'}) are gradually switched on as λ changes from 0 to 1. The function h_{λ}(r,r^{'}) is then obtained from the inhomogeneous Ornstein-Zernike equation and the two equations constitute a general liquid-state framework for treating inhomogeneous fluids. The two equations do not yet constitute a closed set. In the present work we use the closure c_{λ}(r,r^{'})≈-λβu(r,r^{'}), known as the random-phase approximation (RPA). We demonstrate that the RPA is identical with the variational Gaussian approximation derived within the field-theoretical framework, originally derived and used for charged particles. We apply our generalized RPA approximation to the Gaussian core model and Coulomb charges.

Fundamental measure theory for the electric double layer: implications for blue-energy harvesting and water desalination

Capacitive mixing (CAPMIX) and capacitive deionization (CDI) are promising candidates for harvesting clean, renewable energy and for the energy efficient production of potable water, respectively. Both CAPMIX and CDI involve water-immersed porous carbon (supercapacitors) electrodes at voltages of the order of hundreds of millivolts, such that counter-ionic packing is important for the electric double layer (EDL) which forms near the surfaces of these porous materials. Thus, we propose a density functional theory (DFT) to model the EDL, where the White-Bear mark II fundamental measure theory functional is combined with a mean-field Coulombic and a mean spherical approximation-type correction to describe the interplay between dense packing and electrostatics, in good agreement with molecular dynamics simulations. We discuss the concentration-dependent potential rise due to changes in the chemical potential in capacitors in the context of an over-ideal theoretical description and its impact on energy harvesting and water desalination. Compared to less elaborate mean-field models our DFT calculations reveal a higher work output for blue-energy cycles and a higher energy demand for desalination cycles.

Strong coupling electrostatics for randomly charged surfaces: antifragility and effective interactions

We study the effective interaction mediated by strongly coupled Coulomb fluids between dielectric surfaces carrying quenched, random monopolar charges with equal mean and variance, both when the Coulomb fluid consists only of mobile multivalent counterions and when it consists of an asymmetric ionic mixture containing multivalent and monovalent (salt) ions in equilibrium with an aqueous bulk reservoir. We analyze the consequences that follow from the interplay between surface charge disorder, dielectric and salt image effects, and the strong electrostatic coupling that results from multivalent counterions on the distribution of these ions and the effective interaction pressure they mediate between the surfaces. In a dielectrically homogeneous system, we show that the multivalent counterions are attracted towards the surfaces with a singular, disorder-induced potential that diverges logarithmically on approach to the surfaces, creating a singular but integrable counterion density profile that exhibits an algebraic divergence at the surfaces with an exponent that depends on the surface charge (disorder) variance. This effect drives the system towards a state of lower thermal 'disorder', one that can be described by a renormalized temperature, exhibiting thus a remarkable antifragility. In the presence of an interfacial dielectric discontinuity, the singular behavior of counterion density at the surfaces is removed but multivalent counterions are still accumulated much more strongly close to randomly charged surfaces as compared with uniformly charged ones. The interaction pressure acting on the surfaces displays in general a highly non-monotonic behavior as a function of the inter-surface separation with a prominent regime of attraction at small to intermediate separations. This attraction is caused directly by the combined effects from charge disorder and strong coupling electrostatics of multivalent counterions, which dominate the surface-surface repulsion due to the (equal) mean charges on the two surfaces and the osmotic pressure of monovalent ions residing between them. These effects can be quite significant even with a small degree of surface charge disorder relative to the mean surface charge. The strong coupling, disorder-induced attraction is typically much stronger than the van der Waals interaction between the surfaces, especially within a range of several nanometers for the inter-surface separation, where such effects are predicted to be most pronounced.

Ion-specific adsorption and electroosmosis in charged amorphous porous silica

Aqueous electrolyte solutions (NaCl, KCl, CsCl, and SrCl₂) confined in a negatively charged amorphous silica slit pore.

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

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

Large counterions boost the solubility and renormalized charge of suspended nanoparticles

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

Lattice-gas Poisson-Boltzmann approach for sterically asymmetric electrolytes

We derive a modification of the Poisson-Boltzmann (PB) equation brought about by the finite and different sizes of mobile ions. The equation we obtain contains explicitly the size asymmetry of the electrolyte, enabling one to apply the PB approach to the cases where this effect may be of importance. We show that the modification we propose predicts the asymmetry of the double-layer capacitance. Our approach thus provides a transparent analytical framework for a previously proposed heuristic modification of the PB equation, which also predicts the same effect. In addition, a detailed exposition of the lattice-based calculation of the entropy, accounting for steric, spatial correlations, enables us to clearly pinpoint some of the drawbacks of such an approach, which are not entirely obvious in previous attempts to include the steric effects in the PB equation.