Influence of anisotropic ion shape on structure and capacitance of an electric double layer: A Monte Carlo and density functional study

Theory and Simulations of Ionic Liquids in Nanoconfinement

Room-temperature ionic liquids (RTILs) have exciting properties such as nonvolatility, large electrochemical windows, and remarkable variety, drawing much interest in energy storage, gating, electrocatalysis, tunable lubrication, and other applications. Confined RTILs appear in various situations, for instance, in pores of nanostructured electrodes of supercapacitors and batteries, as such electrodes increase the contact area with RTILs and enhance the total capacitance and stored energy, between crossed cylinders in surface force balance experiments, between a tip and a sample in atomic force microscopy, and between sliding surfaces in tribology experiments, where RTILs act as lubricants. The properties and functioning of RTILs in confinement, especially nanoconfinement, result in fascinating structural and dynamic phenomena, including layering, overscreening and crowding, nanoscale capillary freezing, quantized and electrotunable friction, and superionic state. This review offers a comprehensive analysis of the fundamental physical phenomena controlling the properties of such systems and the current state-of-the-art theoretical and simulation approaches developed for their description. We discuss these approaches sequentially by increasing atomistic complexity, paying particular attention to new physical phenomena emerging in nanoscale confinement. This review covers theoretical models, most of which are based on mapping the problems on pertinent statistical mechanics models with exact analytical solutions, allowing systematic analysis and new physical insights to develop more easily. We also describe a classical density functional theory, which offers a reliable and computationally inexpensive tool to account for some microscopic details and correlations that simplified models often fail to consider. Molecular simulations play a vital role in studying confined ionic liquids, enabling deep microscopic insights otherwise unavailable to researchers. We describe the basics of various simulation approaches and discuss their challenges and applicability to specific problems, focusing on RTIL structure in cylindrical and slit confinement and how it relates to friction and capacitive and dynamic properties of confined ions.

Structural transitions at electrodes, immersed in simple ionic liquid models

We used a recently developed classical Density Functional Theory (DFT) method to study the structures, phase transitions, and electrochemical behaviours of two coarse-grained ionic fluid models, in the presence of a perfectly conducting model electrode. Common to both is that the charge of the cationic component is able to approach the electrode interface more closely than the anion charge. This means that the cations are specifically attracted to the electrode, due to surface polarization effects. Hence, for a positively charged electrode, there is competition at the surface between cations and anions, where the latter are attracted by the positive electrode charge. This generates demixing, for a range of positive voltages, where the two phases are structurally quite different. The surface charge density is also different between the two phases, even at the same potential. The DFT formulation contains an approximate treatment of ion correlations, and surface polarization, where the latter is modelled via screened image interactions. Using a mean-field DFT, where ion correlations are neglected, causes the phase transition to vanish for both models, but there is still a dramatic drop in the differential capacitance as proximal cations are replaced by anions, for increasing surface potentials. While these findings were obtained for relatively crude coarse-grained models, we argue that the findings can also be relevant in "real" systems, where we note that many ionic liquids are composed of a spherically symmetric anion, and a cation that is asymmetric both from a steric and a charge distribution point of view.

A Monte Carlo study of the electrical double layer of a shape-asymmetric electrolyte around a spherical colloid

In this paper, we present a Monte Carlo simulation study on the structure of the electrical double layer around a spherical colloid surrounded by a binary electrolyte composed of spherical and non-spherical ions. Results are provided for the radial distribution functions between the colloid and ions, the orientation correlations between the colloid and non-spherical particles, and the integrated charge. Work is reported mainly for non-spherical particles modeled as spherocylinders, although a particular comparison is made between spherocylindrical particles and dimers. For the conditions investigated here, spherocylinders and dimers produce essentially the same structural information. Additionally, it is shown that spherocylinders mostly orient tangentially to the colloid at its surface; this preferred orientation disappears for larger distances. We also evidence that, near the colloid, the integrated charge attenuates monotonically when the macroparticle is highly charged, whereas for intermediate and low charged states of the colloid, the integrated charge can display charge reversal, overcharging, or both, with magnitudes that are sensitive to the salt concentration and to the localization of charge inside the spherocylinders.

Off-center charge model revisited: Electrical double layer with multivalent cations

The off-center charge model of ions is a relatively simple model for introducing asymmetry in Coulomb interaction while retaining the simplicity and convenience of the spherical hard core geometry. A Monte Carlo simulation analysis of the planar electric double layer formed by this ionic model for 1+:1- valence systems [S. Lamperski et al., Langmuir 33, 11554-11560 (2017)] is extended to include solutions of multivalent (2+, 3+) hard spherical cations and single valence (1-) hard spherical anions near a uniformly charged, planar electrode. The solvent is modelled as a uniform dielectric continuum with a dielectric constant equal to that of the pure solvent, viz., the primitive model. Results are reported for the ion density, the cation charge profile, and the electrostatic potential profile at 1 mol/dm³ salt concentration. Additionally, the double layer potential drop, that is, the electrode potential, and the integral and the differential capacitances are computed as functions of the electrode surface charge density. The latter two quantities show an expected asymmetry as long as the cation valence is not too great and the charge of the off-center ion cannot approach too close to the electrode surface. It is unusual that the integral and differential capacitances are negative for high valence cations and a negatively charged electrode when the off-center charge is large and can be very near the surface of the electrode. The corresponding electrode potential versus surface charge density curve becomes non-monotonic and shows a change of slope, and thus the resultant integral and differential capacitances can become negative. This nonphysical result is the result of an incipient singularity when a large positive charge is too near a negatively charged electrode. Overall, the off-center charge model suggests a useful recipe to model electrical asymmetry within the broader context of the primitive model provided that the off-center charge is not too near the surface of the electrode.

Ionic liquid interface at an electrode: simulations of electrochemical properties using an asymmetric restricted primitive model

We use Monte Carlo simulations of a coarse-grained model to investigate structure and electrochemical behaviours at an electrode immersed in room temperature ionic liquids (RTILs). The simple RTIL model, which we denote the asymmetric restricted primitive model (ARPM), is composed of monovalent hard-sphere ions, all of the same size, in which the charge is asymmetrically placed. Not only the hard-sphere size (d), but also the charge displacement (b), is identical for all species, i.e. the monovalent RTIL ions are fully described by only two parameters (d, b). In earlier work, it was demonstrated that the ARPM can capture typical static RTIL properties in bulk solutions with remarkable accuracy. Here, we investigate its behaviour at an electrode surface. The electrode is assumed to be a perfect conductor and image charge methods are utilized to handle polarization effects. We find that the ARPM of the ionic liquid reproduces typical (static) electrochemical properties of RTILs. Our model predicts a declining differential capacitance with increasing temperature, which is expected from simple physical arguments. We also compare our ARPM, with the corresponding RPM description, at an elevated temperature (1000 K). We conclude that, even though ion pairing occurs in the ARPM system, reducing the concentration of 'free' ions, it is still better able to screen charge than a corresponding RPM melt. Finally, we evaluate the option to coarse-grain the model even further, by treating the fraction of the ions that form ion pairs implicitly, only through the contribution to the dielectric constant of the corresponding dipolar (ion pair) fluid. We conclude that this primitive representation of ion pairing is not able to reproduce the structures and differential capacitances of the system with explicit ion pairs. The main problem seems to be due to a limited dielectric screening in a layer near the electrode surface, resulting from a combination of orientational restrictions and a depleted dipole density.

Structure of electric double layers in capacitive systems and to what extent (classical) density functional theory describes it

Ongoing scientific interest is aimed at the properties and structure of electric double layers (EDLs), which are crucial for capacitive energy storage, water treatment, and energy harvesting technologies like supercapacitors, desalination devices, blue engines, and thermocapacitive heat-to-current converters. A promising tool to describe their physics on a microscopic level is (classical) density functional theory (DFT), which can be applied in order to analyze pair correlations and charge ordering in the primitive model of charged hard spheres. This simple model captures the main properties of ionic liquids and solutions and it predicts many of the phenomena that occur in EDLs. The latter often lead to anomalous response in the differential capacitance of EDLs. This work constructively reviews the powerful theoretical framework of DFT and its recent developments regarding the description of EDLs. It explains to what extent current approaches in DFT describe structural ordering and in-plane transitions in EDLs, which occur when the corresponding electrodes are charged. Further, the review briefly summarizes the history of modeling EDLs, presents applications, and points out limitations and strengths in present theoretical approaches. It concludes that DFT as a sophisticated microscopic theory for ionic systems is expecting a challenging but promising future in both fundamental research and applications in supercapacitive technologies.

Monte Carlo Study of a Planar Electric Double Layer Formed by Ions with Off-Center Charge

Grand canonical Monte Carlo simulation results are reported for an electric double layer (EDL) modeled by a planar charged hard wall, hard sphere cations with an off-center charge, and spherical anions with a charge at the center of the sphere. The ion charge numbers are Z₊ = +1 and Z_- = -1, and the diameter, d, of a hard sphere is the same for anions and cations. The ions are immersed in a solvent mimicked by a continuum dielectric medium at standard temperature. The results are obtained for three values of charge displacement, s₊₀ = d/16, d/4, 7d/16 from the center of the sphere and the following electrolyte concentrations: 0.5, 1.0, 2.0, and 3.0 M. The profiles of electrode-ion singlet distributions, cation reduced charge density, angular function, and mean electrostatic potential are reported for an electrode surface charge density σ = -0.30 C m^-2, whereas the electrode potential and the differential capacitance of EDL are shown as functions of the electrode charge density varying from -1.00 to +1.00 C m^-2. At negative electrode charges and with increasing values of the charge separation, the differential capacitance curve rises. As the electrolyte concentration increases, the shape of the differential capacitance curve changes from that of a minimum surrounded by two maxima into that of a distorted single maximum.

Atomistic simulations of cation hydration in sodium and calcium montmorillonite nanopores

During the last four decades, numerous studies have been directed to the swelling smectite-rich clays in the context of high-level radioactive waste applications and waste-liners for contaminated sites. The swelling properties of clay mineral particles arise due to hydration of the interlayer cations and the diffuse double layers formed near the negatively charged montmorillonite (MMT) surfaces. To accurately study the cation hydration in the interlayer nanopores of MMT, solvent-solute and solvent-clay surface interactions (i.e., the solvation effects and the shape effects) on the atomic level should be taken into account, in contrast to many recent electric double layer based methodologies using continuum models. Therefore, in this research we employed fully atomistic simulations using classical molecular dynamics (MD) simulations, the software package GROMACS along with the CLAYFF forcefield and the SPC/E water model. We present the ion distributions and the deformation of the hydrated coordination structures, i.e., the hydration shells of Na⁺ and Ca²⁺ in the interlayer, respectively, for MMT in the first-layer, the second-layer, the third-layer, the fourth-layer, and the fifth-layer (1W, 2W, 3W, 4W, and 5W) hydrate states. Our MD simulations show that Na⁺ in Na-MMT nanopores have an affinity to the ditrigonal cavities of the clay layers and form transient inner-sphere complexes at about 3.8 Å from clay midplane at water contents less than the 5W hydration state. However, these phenomena are not observed in Ca-MMT regardless of swelling states. For Na-MMT, each Na⁺ is coordinated to four water molecules and one oxygen atom of the clay basal-plane in the first hydration shell at the 1W hydration state, and with five to six water molecules in the first hydration shell within a radius of 3.1 Å at all higher water contents. In Ca-MMT, however each Ca²⁺ is coordinated to approximately seven water molecules in the first hydration shell at the 1W hydration state and about eight water molecules in the first hydration shell within a radius of 3.3 Å at all higher hydration states. Moreover, the MD results show that the complete hydration shells are nearly spherical with an orthogonal coordination sphere. They could only be formed when the basal spacing d₀₀₁ ≥ 18.7 Å, i.e., approximately, the interlayer separation h ≥ 10 Å. Comparison between DFT and MD simulations shows that DFT failed to reproduce the outer-sphere complexes in the Stern-layer (within ∼5.0 Å from the clay basal-plane), observed in the MD simulations.

Dense ionic fluids confined in planar capacitors: in- and out-of-plane structure from classical density functional theory

The ongoing scientific interest in the properties and structure of electric double layers (EDLs) stems from their pivotal role in (super)capacitive energy storage, energy harvesting, and water treatment technologies. Classical density functional theory (DFT) is a promising framework for the study of the in- and out-of-plane structural properties of double layers. Supported by molecular dynamics simulations, we demonstrate the adequate performance of DFT for analyzing charge layering in the EDL perpendicular to the electrodes. We discuss charge storage and capacitance of the EDL and the impact of screening due to dielectric solvents. We further calculate, for the first time, the in-plane structure of the EDL within the framework of DFT. While our out-of-plane results already hint at structural in-plane transitions inside the EDL, which have been observed recently in simulations and experiments, our DFT approach performs poorly in predicting in-plane structure in comparison to simulations. However, our findings isolate fundamental issues in the theoretical description of the EDL within the primitive model and point towards limitations in the performance of DFT in describing the out-of-plane structure of the EDL at high concentrations and potentials.

Structure of an electric double layer containing a 2:2 valency dimer electrolyte

The structure of a planar electric double layer formed by a 2:2 valency dimer electrolyte in the vicinity of a uniformly charged planar hard electrode is investigated using density functional theory and Monte Carlo simulations. The dimer electrolyte consists of a mixture of charged divalent dimers and charged divalent monomers in a dielectric continuum. A dimer is constructed by two tangentially tethered rigid spheres, one of which is divalent and positively charged and the other neutral, whereas the monomer is a divalent and negatively charged rigid sphere. The density functional theory reproduces well the simulation results for (i) the singlet distributions of the various ion species with respect to the electrode, and (ii) the mean electrostatic potential. Comparison with earlier results for a 2:1/1:2 dimer electrolyte shows that the double layer structure is similar when the counterion has the same valency.

Structure of spherical electric double layers with fully asymmetric electrolytes: a systematic study by Monte Carlo simulations and density functional theory

A systematic investigation of the spherical electric double layers with the electrolytes having size as well as charge asymmetry is carried out using density functional theory and Monte Carlo simulations. The system is considered within the primitive model, where the macroion is a structureless hard spherical colloid, the small ions as charged hard spheres of different size, and the solvent is represented as a dielectric continuum. The present theory approximates the hard sphere part of the one particle correlation function using a weighted density approach whereas a perturbation expansion around the uniform fluid is applied to evaluate the ionic contribution. The theory is in quantitative agreement with Monte Carlo simulation for the density and the mean electrostatic potential profiles over a wide range of electrolyte concentrations, surface charge densities, valence of small ions, and macroion sizes. The theory provides distinctive evidence of charge and size correlations within the electrode-electrolyte interface in spherical geometry.

Structure of fluid mixtures near a solute: a density functional approach

The structure of fluid mixtures near a spherical solute is studied using a density functional approach and computer simulation. The input direct correlation function is obtained from integral equation theory with an accurate closure relation. The density and concentration profiles of binary as well as ternary hard-sphere mixtures near a large hard-spherical solute compare quite well with the computer simulation results over a wide range of parametric conditions.