Monte Carlo study of the capacitance of the double layer in a model molten salt

Rectification of bipolar nanopores in multivalent electrolytes: effect of charge inversion and strong ionic correlations

Bipolar nanopores have powerful rectification properties due to the asymmetry in the charge pattern on the wall of the nanopore. In particular, bipolar nanopores have positive and negative surface charges along the pore axis. Rectification is strong if the radius of the nanopore is small compared to the screening length of the electrolyte so that both cations and anions have depletion zones in the respective regions. The depths of these depletion zones is sensitive to sign of the external voltage. In this work, we are interested in the effect of the presence of strong ionic correlations (both between ions and between ions and surface charge) due to the presence of multivalent ions and large surface charges. We show that strong ionic correlations cause leakage of the coions, a phenomenon that is absent in mean field theories. In this modeling study, we use both the mean-field Poisson-Nernst-Planck (PNP) theory and a particle simulation method, Local Equilibrium Monte Carlo (LEMC), to show that phenomena such as overcharging and charge inversion cannot be reproduced with PNP, while LEMC is able to produce nonmonotonic dependence of currents and rectification as a function of surface charge strength.

Temperature dependence of differential capacitance in the electric double layer.Symmetric valency 1:1 electrolytes

The differential capacitance of an electric double layer formed by an aqueous solution of KNO₃ on a glassy carbon electrode is measured by impedance analysis at constant frequency. Results are obtained at electrolyte concentrations of 0.1 mol/dm³, 0.5 mol/dm³, and 1.0 mol/dm³, and at a series of temperatures, viz., 288 K, 298 K, 308 K, 318 K, and 328 K. The differential capacitance envelopes reveal a rich, complex pattern of maxima, minima, and local minima, whose magnitude and position change with a change in solution concentration. At the two lower concentrations, the temperature dependence of the capacitance, for example, at zero electrode potential, shows an alternating positive-negative behavior, while at the highest concentration of 1.0 mol/dm³, the slope of the differential capacitance-electrode potential curve is always positive. The experimental results are supplemented by a numerical grand canonical Monte Carlo simulation study of a restricted primitive model double layer but with an off-center cationic charge achieved by displacing the charge center from the ion sphere center toward its surface. The simulations, performed at the electrolyte concentration of 1.0 mol/dm³ and constant cation charge center displacement, and at varying electrode potentials and temperatures, show, in general, a negative temperature dependence of the differential capacitance. However, this temperature dependence can also be positive for a negative electrode charge and for a sufficiently large gradient of the cation charge center displacement with temperature. This feature is seen to be associated with an increase in the entropy of formation of the double layer.

Ferroelectric Phase Behaviors in Porous Electrodes

The phase behavior of ions in porous electrodes is qualitatively different from that in the bulk because of the confinement effect and the interaction between the electrode surface and the electrolyte ions. We found that porous electrodes of which the pore size is close to the size of the electrolyte ions can show ferroelectric phase behaviors in some conditions by Monte Carlo simulations of simple models. The phase behavior of the porous electrodes dramatically changes as a function of the pore size of the porous electrode and that is compared to the phase behavior of typical ferroelectric materials, for which the phase behavior changes as a function of the temperature or the composition. The origin of the phase behavior is discussed in terms of the molecular interaction and the ionic structure inside the porous electrodes. We also found that the density of counterions and that of co-ions inside porous electrodes changes in a nonlinear fashion as a function of the applied voltage, which is in agreement with the experimental results.

Monte Carlo study of molten salt with charge asymmetry near the electrode surface

Results of the Monte Carlo simulation of the electrode | molten salt or ionic liquid interface are reported. The system investigated is approximated by the primitive model of electrolyte being in contact with a charged hard wall. Ions differ in charges, namely anions are divalent and cations are monovalent but they are of the same diameter d = 400 pm. The temperature analysis of heat capacity at a constant volume Cv and the anion radial distribution function, g2-/2-, allowed the choice of temperature of the study, which is T = 2800 K and corresponds to T(*) = 0.34 (definition of reduced temperature T(*) in text). The differential capacitance curve of the interface with the molten salt or ionic liquid at c = 5.79 M has a distorted bell shape. It is shown that with increasing electrolyte concentration from c = 0.4 to 5 M the differential capacitance curves undergo transition from U shape to bell shape.

Electric double layer capacitance of restricted primitive model for an ionic fluid in slit-like nanopores: A density functional approach

We apply recently developed version of a density functional theory [Z. Wang, L. Liu, and I. Neretnieks, J. Phys.: Condens. Matter 23, 175002 (2011)] to study adsorption of a restricted primitive model for an ionic fluid in slit-like pores in the absence of interactions induced by electrostatic images. At present this approach is one of the most accurate theories for such model electric double layers. The dependencies of the differential double layer capacitance on the pore width, on the electrostatic potential at the wall, bulk fluid density, and temperature are obtained. We show that the differential capacitance can oscillate as a function of the pore width dependent on the values of the above parameters. The number of oscillations and their magnitude decrease for high values of the electrostatic potential. For very narrow pores, close to the ion diameter, the differential capacitance tends to a minimum. The dependence of differential capacitance on temperature exhibits maximum at different values of bulk fluid density and applied electrostatic potential.

Molecular Dynamics Simulation Study of the Capacitive Performance of a Binary Mixture of Ionic Liquids near an Onion-like Carbon Electrode

An equimolar mixture of 1-methyl-1-propylpyrrolidinium bis(trifluoromethylsulfonyl)imide ([C3mpy][Tf2N]), 1-methyl-1-butylpiperidinium bis(trifluoromethylsulfonyl)imide ([C4mpip][Tf2N]) was investigated by classic molecular dynamics (MD) simulation. Differential scanning calorimetry (DSC) measurements verified that the binary mixture exhibited lower glass transition temperature than either of the pure room-temperature ionic liquids (RTILs). Moreover, the binary mixture gave rise to higher conductivity than the neat RTILs at lower temperature range. In order to study its capacitive performance in supercapacitors, simulations were performed of the mixture, and the neat RTILs used as electrolytes near an onion-like carbon (OLC) electrode at varying temperatures. The differential capacitance exhibited independence of the electrical potential applied for three electrolytes, which is in agreement with previous work on OLC electrodes in a different RTILs. Positive temperature dependence of the differential capacitance was observed, and it was dominated by the electrical double layer (EDL) thickness, which is for the first time substantiated in MD simulation.

An algorithm for three-dimensional Monte-Carlo simulation of charge distribution at biofunctionalized surfaces

In this work, a Monte-Carlo algorithm in the constant-voltage ensemble for the calculation of 3d charge concentrations at charged surfaces functionalized with biomolecules is presented. The motivation for this work is the theoretical understanding of biofunctionalized surfaces in nanowire field-effect biosensors (BioFETs). This work provides the simulation capability for the boundary layer that is crucial in the detection mechanism of these sensors; slight changes in the charge concentration in the boundary layer upon binding of analyte molecules modulate the conductance of nanowire transducers. The simulation of biofunctionalized surfaces poses special requirements on the Monte-Carlo simulations and these are addressed by the algorithm. The constant-voltage ensemble enables us to include the right boundary conditions; the dna strands can be rotated with respect to the surface; and several molecules can be placed in a single simulation box to achieve good statistics in the case of low ionic concentrations relevant in experiments. Simulation results are presented for the leading example of surfaces functionalized with pna and with single- and double-stranded dna in a sodium-chloride electrolyte. These quantitative results make it possible to quantify the screening of the biomolecule charge due to the counter-ions around the biomolecules and the electrical double layer. The resulting concentration profiles show a three-layer structure and non-trivial interactions between the electric double layer and the counter-ions. The numerical results are also important as a reference for the development of simpler screening models.

Anomalously large capacitance of an ionic liquid described by the restricted primitive model

We use Monte Carlo simulations to examine the simplest model of a room-temperature ionic liquid (RTIL), called the "restricted primitive model," at a metal surface. We find that at moderately low temperatures the capacitance of the metal-RTIL interface is so large that the effective thickness of the electrostatic double layer is up to three times smaller than the ion radius. To interpret these results we suggest an approach which is based on the interaction between discrete ions and their image charges in the metal surface and which therefore goes beyond the mean-field approximation. When a voltage is applied across the interface, the strong image attraction causes counterions to condense onto the metal surface to form compact ion-image dipoles. These dipoles repel each other to form a correlated liquid. When the surface density of these dipoles is low, the insertion of an additional dipole does not require much energy. This leads to a large capacitance C that decreases monotonically with voltage V, producing a "bell-shaped" curve C(V). We also consider what happens when the electrode is made from a semimetal rather than a perfect metal. In this case, the finite screening radius of the electrode shifts the reflection plane for image charges to the interior of the electrode, and we arrive at a "camel-shaped" C(V). These predictions seem to be in qualitative agreement with experiment.

Energy variational analysis of ions in water and channels: Field theory for primitive models of complex ionic fluids

Ionic solutions are mixtures of interacting anions and cations. They hardly resemble dilute gases of uncharged noninteracting point particles described in elementary textbooks. Biological and electrochemical solutions have many components that interact strongly as they flow in concentrated environments near electrodes, ion channels, or active sites of enzymes. Interactions in concentrated environments help determine the characteristic properties of electrodes, enzymes, and ion channels. Flows are driven by a combination of electrical and chemical potentials that depend on the charges, concentrations, and sizes of all ions, not just the same type of ion. We use a variational method EnVarA (energy variational analysis) that combines Hamilton's least action and Rayleigh's dissipation principles to create a variational field theory that includes flow, friction, and complex structure with physical boundary conditions. EnVarA optimizes both the action integral functional of classical mechanics and the dissipation functional. These functionals can include entropy and dissipation as well as potential energy. The stationary point of the action is determined with respect to the trajectory of particles. The stationary point of the dissipation is determined with respect to rate functions (such as velocity). Both variations are written in one Eulerian (laboratory) framework. In variational analysis, an "extra layer" of mathematics is used to derive partial differential equations. Energies and dissipations of different components are combined in EnVarA and Euler-Lagrange equations are then derived. These partial differential equations are the unique consequence of the contributions of individual components. The form and parameters of the partial differential equations are determined by algebra without additional physical content or assumptions. The partial differential equations of mixtures automatically combine physical properties of individual (unmixed) components. If a new component is added to the energy or dissipation, the Euler-Lagrange equations change form and interaction terms appear without additional adjustable parameters. EnVarA has previously been used to compute properties of liquid crystals, polymer fluids, and electrorheological fluids containing solid balls and charged oil droplets that fission and fuse. Here we apply EnVarA to the primitive model of electrolytes in which ions are spheres in a frictional dielectric. The resulting Euler-Lagrange equations include electrostatics and diffusion and friction. They are a time dependent generalization of the Poisson-Nernst-Planck equations of semiconductors, electrochemistry, and molecular biophysics. They include the finite diameter of ions. The EnVarA treatment is applied to ions next to a charged wall, where layering is observed. Applied to an ion channel, EnVarA calculates a quick transient pile-up of electric charge, transient and steady flow through the channel, stationary "binding" in the channel, and the eventual accumulation of salts in "unstirred layers" near channels. EnVarA treats electrolytes in a unified way as complex rather than simple fluids. Ad hoc descriptions of interactions and flow have been used in many areas of science to deal with the nonideal properties of electrolytes. It seems likely that the variational treatment can simplify, unify, and perhaps derive and improve those descriptions.

Differential capacitance of the double layer at the electrode/ionic liquids interface

The differential capacitance of the electrical double layer at glassy carbon, platinum and gold electrodes immersed in various ionic liquids was measured using impedance spectroscopy. We discuss the influence of temperature, the composition of the ionic liquids and the electrode material on the differential capacitance/potential curves. For different systems these curves have various overall shapes, but all include several extremes and a common minimum near the open circuit potential. We attribute this minimum to the potential of zero charge (PZC). Significantly, the differential capacitance generally decreases if the applied potential is large and moving away from the PZC. This is attributed to lattice saturation [A. A. Kornyshev, J. Phys. Chem. B, 2007, 111, 5545] effects which result in a thicker double layer. The differential capacitance of the double layer grows and specific adsorption diminishes with increasing temperature. Specific adsorption of both cations and anions influences the shapes of curves close to the PZC. The general shape of differential capacitance/potential does not depend strongly on the identity of the electrode material.

Local semi-empirical formulae for the contact values of the singlet distribution functions of a double layer

The electrochemical double layer is an important practical and theoretical problem. Generally speaking, experiment gives valuable information about quantities, such as potential differences, that involve integrals of density and charge profiles but does not provide direct information about the profiles themselves. Computer simulations have given numerical information about these profiles. However, explicit expressions are useful in understanding these data. For some years an exact expression has been known for the contact value of total density profile of the ions in the double layer but, until recently, an expression for the contact value of the more important charge profile has been lacking. A few years ago, a semi-empirical local result for the charge profile, valid at low electrode charge, was proposed and, very recently, extended to higher electrode charge. This expression contains a parameter; the effect of variations in this parameter is explored in this paper and the result is compared with a large set of simulation data for the contact values of various profiles that we have accumulated in the past few years. The agreement of the semi-empirical expression with our simulation results is excellent. The best values for this parameter are fairly close to the value suggested by theory.

An Empirical Approach to Modeling Ion Size Effects in Diffuse Double Layer Theory

A novel empirical model for the diffuse double layer is found by generalizing the simple analytical equations of Gouy–Chapman theory. Two adjustable parameters are introduced into the Boltzmann equation for the exponential dependence of the ion–wall correlation functions on the diffuse layer potential. Optimal parameter values and model validation for 1:1 and 2:1 electrolytes are provided by Monte Carlo simulations. Simple relationships are obtained between these empirical parameters and those commonly associated with the mean-spherical approximation. The new empiricism accurately models diffuse layer potential profiles and ion–wall correlation functions for a restricted 1:1 electrolyte in a primitive solvent.

The electric double-layer differential capacitance at and near zero surface charge for a restricted primitive model electrolyte

The behavior of the differential capacitance of a planar electric double layer containing a restricted primitive model electrolyte in the neighborhood of zero surface charge is investigated by theory and simulation. Previous work has demonstrated that at zero surface charge the differential capacitance has a minimum for aqueous electrolytes at room temperature but can have a maximum for molten salts and ionic liquids. The transition envelope separating the two situations is found for a modified Poisson-Boltzmann theory and a Poisson-Boltzmann equation corrected for the exclusion volume term. Good agreement is found between simulation and the modified Poisson-Boltzmann theory in the neighborhood of the envelope at the reduced temperature of 0.8, while the exclusion volume corrected Poisson-Boltzmann theory shows correct qualitative trends.

Insights from theory and simulation on the electrical double layer

Despite the fact that our conceptual understanding of the electrical double layer has advanced during the past few decades, the interpretation of experimental and applied work is still largely based on the venerable Poisson-Boltzmann theory of Gouy, Chapman and Stern. This is understandable since this theory is simple and analytic. However, it is not very accurate because the atomic/molecular nature of the ions/solvent and their correlations are ignored. Simulation and some theoretical studies by ourselves and others that have advanced our understanding are discussed. These studies show that the GCS theory predicts a narrow double layer with monotonic profiles. This is not correct. The double layer is wider, and there can be substantial layering that would be even more pronounced if explicit solvent molecules are considered. For many years, experimental studies of the double layer have been directed to the use of electrochemistry as an analytical tool. This is acceptable for analytic chemistry studies. However, the understanding of electrochemical reactions that typically occur at the electrode surface, where simulation and theory indicate that the GCS theory can have substantial errors, requires modern approaches. New, fundamental experimental studies that would lead to deeper insights using more novel systems would be desirable. Further, biophysics is an interesting field. Recent studies of the selectivity of ion channels and of the adsorption of ions in a binding sites of a protein have shown that the linearized GCS theory has substantial errors.

Steric selectivity in Na channels arising from protein polarization and mobile side chains

Monte Carlo simulations of equilibrium selectivity of Na channels with a DEKA locus are performed over a range of radius R and protein dielectric coefficient epsilon(p). Selectivity arises from the balance of electrostatic forces and steric repulsion by excluded volume of ions and side chains of the channel protein in the highly concentrated and charged (approximately 30 M) selectivity filter resembling an ionic liquid. Ions and structural side chains are described as mobile charged hard spheres that assume positions of minimal free energy. Water is a dielectric continuum. Size selectivity (ratio of Na+ occupancy to K+ occupancy) and charge selectivity (Na+ to Ca2+) are computed in concentrations as low as 10(-5) M Ca2+. In general, small R reduces ion occupancy and favors Na+ over K+ because of steric repulsion. Small epsilon(p) increases occupancy and favors Na+ over Ca2+ because protein polarization amplifies the pore's net charge. Size selectivity depends on R and is independent of epsilon(p); charge selectivity depends on both R and epsilon(p). Thus, small R and epsilon(p) make an efficient Na channel that excludes K+ and Ca2+ while maximizing Na+ occupancy. Selectivity properties depend on interactions that cannot be described by qualitative or verbal models or by quantitative models with a fixed free energy landscape.