Finite Ion Size Research Articles

Finite ion size effect on the stability ratio of colloidal dispersions

An algorithm is presented for calculating the stability ratio of a dispersion of spherical colloidal particles on the basis of the Derjaguin–Landau–Verwey–Overbeek theory. The finite ion size effect is taken into account by using the approximate form of ionic activity coefficients given by Carnahan and Starling. A simple approximate analytic expression for the stability ratio without involving numerical integration is also derived.

Transport Properties in Nanochannels: Ionic Size-, Permittivity-, and Viscosity-Related Effects

The transport properties in nanochannels are examined using a modified electrokinetic model (MEM) that takes into account the finite ion size by modeling the hydrated ions as charged dielectric spheres so that the electrolyte solution becomes a nonhomogeneous fluid characterized by permittivity and viscosity values that are both functions of the local ionic concentrations. We show that the simultaneous consideration of ionic size, permittivity, and viscosity effects always improve on the predictions of the standard electrokinetic model for point-like ions (SEM), with corrections that are significant in the nanofluidic system’s parameter range. The results reported herein are expected to provide a deeper understanding of the ion transport and fluid flow in nanofluidic devices.

Asymmetric double-layer charging in a cylindrical nanopore under closed confinement.

This article presents a physical-mathematical treatment and numerical simulations of electric double layer charging in a closed, finite, and cylindrical nanopore of circular cross section, embedded in a polymeric host with charged walls and sealed at both ends by metal electrodes under an external voltage bias. Modified Poisson-Nernst-Planck equations were used to account for finite ion sizes, subject to an electroneutrality condition. The time evolution of the formation and relaxation of the double layers was explored. Moreover, equilibrium ion distributions and differential capacitance curves were investigated as functions of the pore surface charge density, electrolyte concentration, ion sizes, and pore size. Asymmetric properties of the differential capacitance curves reveal that the structure of the double layer near each electrode is controlled by the charge concentration along the pore surface and by charge asymmetry in the electrolyte. These results carry implications for accurately simulating cylindrical capacitors and electroactuators.

Finite-Size Charged Species Diffusion and pH Change in Nanochannels.

Molecular transport through nanofluidic structures exhibits properties that are unique at the nanoscale. The high surface-to-volume ratio of nanometer-sized confined spaces renders particle interactions with the surface of central importance. The electrical double layer (EDL) at the solid-liquid interface of charged surfaces generates an enrichment of counterions and the exclusion of co-ions that lead to a change in their diffusivity. In addition, the diffusive transport is altered by steric and hydrodynamic interactions between fluid molecules and the boundaries. An extensive body of literature investigates molecular transport at the nanoscale. However, most studies account for ionic species as point charges, severely limiting the applicability of the results to "large" nanofluidic systems. Moreover, and even more importantly, the change of pH in the nanoconfined region inside nanochannels has been completely overlooked. Corroborated by experimental data, here we present an all-encompassing analysis of molecular diffusion from the micro- to the ultra-nanoscale. While accounting for finite-size ions, we compute self-consistently the pH inside the channels. Surprisingly, we found that the concentration of ions H+ can change by more than 2 orders of magnitude compared to the bulk, hugely affecting molecular transport. Further, we found that counterions exhibit both enrichment and exclusion, depending on the size of nanochannels. Achieving a greater understanding of the effective transport properties of fluids at the nanoscale will fill the gap in knowledge that still limits development of innovative systems for medicine and industrial applications alike.

Steric-effect-induced enhancement of electrokinetic energy conversion efficiency in curved nanochannels with rectangular sections at high zeta potentials

In this work, we investigate the electrokinetic energy conversion in curved rectangular nanochannels at high zeta potentials, intending to increase the efficiency of electrokinetic generation. As we demonstrate, the geometry effect can have an important role in the investigations of the conversion efficiency. In addition, we consider the influences of steric effect (the effect of finite ion size) on the energy conversion efficiency, which is significant and can’t be ignored, especially at high surface potentials. The finite difference method is used to obtain numerical solutions for the corresponding efficiency. Our results show that, within optimized parameters ranges, the conversion efficiency can be increased by up to 50 percent by considering the influences of the steric effect. Additionally, the conversion efficiency η shows a increasing trend with the curvature ratio δ and a reversed trend with the aspect ratio AR of the rectangular cross section. These results are useful for estimating and designing more effective electrokinetic energy conversion devices.

The effect of the finite size of ions and Debye layer overspill on the screened Coulomb interactions between charged flat plates.

Screened repulsion between uniformly charged plates with an intervening electrolyte is analyzed for strongly overlapped electrical double layers (EDL), accounting for the steric effect of ions and their expulsion from EDL edges into the surrounding solution. As a generalization of a study by Philipse etal. which does not account for these effects, an analytical expression is derived for the repulsion pressure in the limit of infinitely long plates with a zero-field assumption, which agrees closely with the corresponding numerical solution at low inter-plate separations. Our results show an augmented repulsive pressure for finite-sized ions at strong EDL overlaps. For plates with a finite lateral size, we demonstrate a further extended domain of low inter-plate gaps where the repulsion pressure increases with ion size due to a strong interplay between the steric interaction of ions and the EDL overspill phenomenon, considered earlier in a study by Ghosal & Sherwood limited to the linear Debye-Hückel regime (which cannot account for the steric effect of ions). This investigation on a simple model should enhance our understanding of the interaction between charged particles in electrophoresis, nanoscale self-assembly, active particles, and various other electrokinetic systems.

Modeling Polycrystalline Electrode-electrolyte Interfaces: The Differential Capacitance

We present and analyze a model for polycrystalline electrode surfaces based on an improved continuum model that takes finite ion size and solvation into account. The numerical simulation of finite size facet patterns allows to study two limiting cases: While for facet size diameter we get the typical capacitance of a spatially homogeneous but possible amorphous or liquid surface, in the limit , an ensemble of non-interacting single crystal surfaces is approached. Already for moderate size of the facet diameters, the capacitance is remarkably well approximated by the classical approach of adding the single crystal capacities of the contributing facets weighted by their respective surface fraction. As a consequence, the potential of zero charge is not necessarily attained at a local minimum of capacitance, but might be located at a local capacitance maximum instead. Moreover, the results show that surface roughness can be accurately taken into account by multiplication of the ideally flat polycrystalline surface capacitance with a single factor. In particular, we find that the influence of the actual geometry of the facet pattern in negligible and our theory opens the way to a stochastic description of complex real polycrystal surfaces.

Dynamics of ionic flows via Poisson-Nernst-Planck systems with local hard-sphere potentials: Competition between cations.

We study a quasi-one-dimensional steady-state Poisson-Nernst-Planck type model for ionic flows through a membrane channel with three ion species, two positively charged with the same valence and one negatively charged. Bikerman's local hard-sphere potential is included in the model to account for ion sizes. The problem is treated as a boundary value problem of a singularly perturbed differential system. Under the framework of a geometric singular perturbation theory, together with specific structures of this concrete model, the existence and uniqueness of solutions to the boundary value problem for small ion sizes is established. Furthermore, treating the ion sizes as small parameters, we derive an approximation of individual fluxes, from which one can further study the qualitative properties of ionic flows and extract concrete information directly related to biological measurements. Of particular interest is the competition between two cations due to the nonlinear interplay between finite ion sizes, diffusion coefficients and boundary conditions, which is closely related to selectivity phenomena of open ion channels with given protein structures. Furthermore, we are able to characterize the distinct effects of the nonlinear interplays between these physical parameters. Numerical simulations are performed to identify some critical potentials which play critical roles in examining properties of ionic flows in our analysis.

The Poisson-Boltzmann model for implicit solvation of electrolyte solutions: Quantum chemical implementation and assessment via Sechenov coefficients.

We present the theory and implementation of a Poisson-Boltzmann implicit solvation model for electrolyte solutions. This model can be combined with arbitrary electronic structure methods that provide an accurate charge density of the solute. A hierarchy of approximations for this model includes a linear approximation for weak electrostatic potentials, finite size of the mobile electrolyte ions, and a Stern-layer correction. Recasting the Poisson-Boltzmann equations into Euler-Lagrange equations then significantly simplifies the derivation of the free energy of solvation for these approximate models. The parameters of the model are either fit directly to experimental observables-e.g., the finite ion size-or optimized for agreement with experimental results. Experimental data for this optimization are available in the form of Sechenov coefficients that describe the linear dependence of the salting-out effect of solutes with respect to the electrolyte concentration. In the final part, we rationalize the qualitative disagreement of the finite ion size modification to the Poisson-Boltzmann model with experimental observations by taking into account the electrolyte concentration dependence of the Stern layer. A route toward a revised model that captures the experimental observations while including the finite ion size effects is then outlined. This implementation paves the way for the study of electrochemical and electrocatalytic processes of molecules and cluster models with accurate electronic structure methods.

Screening length for finite-size ions in concentrated electrolytes.

The classical Debye-Hückel (DH) theory clearly accounts for the origin of screening in electrolyte solutions and works rather well for dilute electrolyte solutions. While the Debye screening length decreases with the ion concentration and is independent of ion size, recent surface-force measurements imply that for concentrated solutions, the screening length exhibits an opposite trend; it increases with ion concentration and depends on the ionic size. The screening length is usually defined by the response of the electrolyte solution to a test charge but can equivalently be derived from the charge-charge correlation function. By going beyond DH theory, we predict the effects of ion size on the charge-charge correlation function. A simple modification of the Coulomb interaction kernel to account for the excluded volume of neighboring ions yields a nonmonotonic dependence of the screening length (correlation length) on the ionic concentration, as well as damped charge oscillations for high concentrations.

The Effects of Finite Ionic Sizes and Wall Slip on Entropy Generation in Electroosmotic Flows in a Soft Nanochannel

Abstract The combined effects of finite ionic sizes and boundary slip on the entropy generation in mixed pressure driven and electroosmotic flows (EOFs) in a soft nanochannel are investigated in this study. The soft nanochannel is represented by a rigid nanochannel covered by a charged polyelectrolyte layer (PEL) on its surface. The entropy generation analysis of EOFs in such a soft nanochannel is addressed for the first time. Under the assumption of high zeta potentials, the electric potential, velocity, and temperature distributions are obtained numerically by using the finite difference method. Subsequently, the thermal transport characteristic and the corresponding entropy generation analysis are discussed based on the obtained velocity and temperature distributions. Our results show that the soft nanochannel in the present model is not appropriate for cooling purposes. We also demonstrate that the steric factor v and the PEL thickness d can enhance the entropy generation rate. However, the slip boundary coefficient γ, the drag parameter α, and the equivalent electric double-layer (EDL) thickness λFCL can restrain this entropy generation rate. In addition, the contributions of Joule heating and viscous friction in the entropy generation rate are more prominent than the contribution due to heat transfer. The present theoretical research can be used to design the efficient thermofluidic devices.

Theoretical validation of the step potential electrochemical spectroscopy (SPECS) and multiple potential step chronoamperometry (MUSCA) methods for pseudocapacitive electrodes

This study theoretically and rigorously validates the use of the recently proposed step potential electrochemical spectroscopy (SPECS) and multiple potential step chronoamperometry (MUSCA) methods and their fitting analysis for determining the respective contributions of electrical double layer (EDL) and faradaic reactions to charge storage in pseudocapacitive electrodes. The continuum modified Poisson-Nernst-Planck model coupled with the Frumkin-Butler-Volmer theory were used for simulating interfacial, transport, and electrochemical phenomena in pseudocapacitive electrodes. The model accounted for (i) electron transport in the electrode, (ii) reversible redox reactions, (iii) ion electrodiffusion in binary and symmetric electrolytes, (iv) ion intercalation into the pseudocapacitive electrode, and (v) steric repulsion due to finite ion size. First, typical experimental measurements obtained from the SPECS method were reproduced numerically for a planar pseudocapacitive electrode. The EDL and faradaic currents retrieved from the SPECS fitting procedure were found to be in excellent agreement with those defined from first principles and computed numerically. Here, the faradaic current was modeled in the SPECS method as a diffusion process accounting for interfacial charge transfer kinetics and IR drop. The resistance obtained by SPECS matched the internal resistance obtained from electrochemical impedance spectroscopy. Similarly, the EDL capacitance retrieved by SPECS corresponded to the differential capacitance obtained from cyclic voltammetry (CV). Finally, the CV curves were successfully corrected for ohmic polarization effect using the MUSCA method. Then, the capacitive and diffusive currents retrieved from the electrochemical analysis of CV curves corrected by the MUSCA method were in good agreement with the EDL and faradaic currents reconstructed from the MUSCA method.

Differential capacitance of ionic liquid interface with graphene: The effects of correlation and finite size of ions

We use a mean field model for ionic liquids, which takes into account both the ion correlation and the finite ion size effects, in order to calculate the differential capacitance of the ionic liquid interface with single-layer graphene. Besides choosing ion packing fractions that give rise to the camel-shaped and bell-shaped capacitances of the diffuse layer in ionic liquids, we also explore small packing fractions in the regime of “dilute electrolytes”, as well as different packing fractions for different ion species in asymmetric ionic liquids. We find that the main effect of a graphene electrode arises due to a V-shaped minimum in its quantum capacitance at the point of zero charge (PZC), which is a manifestation of the Dirac cone structure of graphene's π electron bands. As a result, the total capacitance of a graphene–ionic liquid interface exhibits a camel-shaped dependence on the total applied potential, even for large ion packing fractions and finite ion correlation lengths. While the minimum at the PZC in the total capacitance is “inherited” from graphene's quantum capacitance, the two peaks that occur for applied potentials ∼±1 V are sensitive to the presence of the ion correlation and a Stern layer, which both tend to reduce the height and flatten the peaks in the camel-shaped total capacitance. Finally, we have found that the largest fraction of the applied potential goes to charging the graphene electrode.

Enhancing the Ion-Size-Based Selectivity of Capacitive Deionization Electrodes.

Capacitive deionization (CDI) is an emerging water treatment technology often applied to brackish water desalination and water softening. Typical CDI cells consist of two microporous carbon electrodes sandwiching a dielectric separator, and desalt feedwater flowing through the cell by storing ions in electric double layers (EDLs) within charged micropores. CDI cells have demonstrated size-based ion selectivity wherein smaller hydrated ions are preferentially electrosorbed over larger hydrated ions. We demonstrate that such size-based selectivity can be substantially enhanced through the addition of chemical charge to micropores via surface functionalization. We develop a micropore EDL theory that includes both finite ion size effects and micropore chemical charge, which predicts such enhancements and elucidates that they result from denser counterion packing in micropores. With our experimental CDI cell, we desalted an electrolyte consisting of equimolar potassium (K+) and lithium (Li+) ions. We show that use of a surface-functionalized (oxidized) cathode significantly increased the electrosorption ratio of smaller K+ to larger Li+ compared to a cell with a pristine cathode, for example, from ∼1 to 1.84 for a charging voltage of 0.4 V. Our model predicts yet-higher electrosorption ratios are attainable, but our experimental cell suffered from significant cathode chemical charge degradation at applied voltages of ∼1 V.

Induced charge electroosmotic flow with finite ion size and solvation effects

The most common mathematical models for electrolyte flows are based on the dilute solution assumption, leading to a coupled system of the Nernst–Planck–Poisson drift-diffusion equations for ion transport and the Stokes resp. Navier–Stokes equations for fluid flow of the solvent. In charged boundary layers the dilute solution assumption is in general not valid and volume exclusion and solvation effects have to be taken into account in a thermodynamically consistent way. Whenever boundary layer effects have a dominant impact on the global behavior of a certain electrochemical system, an accurate numerical simulation depends on the correct incorporation of these effects. In this contribution we present a novel numerical solution approach which aims at preserving on the discrete level consistency with basic thermodynamic principles and structural properties like independence of flow velocities from gradient contributions to external forces. We illustrate capabilities of the method by an example of vortex generation due to induced charge electroosmotic forces at an electrode inside a nanochannel.

The impact of solvation and dissociation on the transport parameters of liquid electrolytes: continuum modeling and numerical study

Electro-thermodynamics provides a consistent framework to derive continuum models for electrochemical systems. For the application to a specific experimental system, the general model must be equipped with two additional ingredients: a free energy model to calculate the chemical potentials and a kinetic model for the kinetic coefficients. Suitable free energy models for liquid electrolytes incorporating ion–solvent interaction, finite ion sizes and solvation already exist and have been validated against experimental measurements. In this work, we focus on the modeling of the mobility coefficients based on Maxwell–Stefan setting and incorporate them into the general electro-thermodynamic framework. Moreover, we discuss the impact of model parameter on conductivity, transference numbers and salt diffusion coefficient. In particular, the focus is set on the solvation of ions and incomplete dissociation of a non-dilute electrolyte.

Numerical Solution of the Electrokinetic Equations for Multi-ionic Electrolytes Including Different Ionic Size Related Effects.

One of the main assumptions of the standard electrokinetic model is that ions behave as point-like entities. In a previous work (López-García, et al., 2015) we removed this assumption and analyzed the influence of finite ionic size on the dielectric and electrokinetic properties of colloidal suspensions using both the Bikerman and the Carnahan–Starling equations for the steric interactions. It was shown that these interactions improved upon the standard model predictions so that the surface potential, electrophoretic mobility, and the conductivity and permittivity increment values were increased. In the present study, we extend our preceding works to systems made of three or more ionic species with different ionic sizes. Under these conditions, the Bikerman and Carnahan–Starling expressions cease to be valid since they were deduced for single-size spheres. Fortunately, the Carnahan–Starling expression has been extended to mixtures of spheres of unequal size, namely the “Boublik–Mansoori–Carnahan–Starling–Leland” (BMCSL) equation of state, making it possible to analyze the most general case. It is shown that the BMCSL expression leads to results that differ qualitatively and quantitatively from the standard electrokinetic model.

Mathematical studies of Poisson–Nernst–Planck model for membrane channels: Finite ion size effects without electroneutrality boundary conditions

We study a quasi-one-dimensional steady-state Poisson–Nernst–Planck model with a local hard-sphere potential for ionic flows of two oppositely charged ion species through a membrane channel. Of particular interest are qualitative properties of ionic flows in terms of individual fluxes J k without assuming electroneutrality boundary conditions . For the first order approximation J k 1 (in diameter of charged particle) of J k , our result shows that the quantities ∂ V J k 1 and ∂ V λ 2 J k 1 that play critical roles in characterizing ion size effects on ionic flows can be both negative depending sensitively on boundary concentrations and relative ion valences while they are always positive under electroneutrality conditions. This new observation indicates that, for fixed ion species, opposite finite size effects on ionic flows can occur determined further by boundary concentrations. Numerical simulation is performed to detect some critical potentials closely related to ion size effects. Numerical simulation results are consistent with our analytical predictions.

Finite ion size effect on the force and energy of the double-layer interaction between two parallel similar plates at arbitrary separations in an electrolyte solution

Expressions are derived for the force and energy of the double-layer interaction between two parallel similar plates with arbitrary surface charge densities or surface potentials at arbitrary separations in an electrolyte solution, by taking into account the finite ion size effect on the basis of the full expression of the activity coefficients of electrolyte ions derived by Carnahan and Starling. The results for two models of the double-layer interaction, that is, the constant surface potential and constant surface charge density models, are given. It is shown that in the constant surface potential model, at large plate separations, the magnitudes of the double-layer interaction force and energy increase as the total ion volume fraction increases but this tendency reverses at small plate separations, while in the constant surface charge density model, they always increase as the total ion volume fraction increases.

Asymmetric Finite Size of Ions and Orientational Ordering of Water in Electric Double Layer Theory Within Lattice Model.

In the present review, a brief historical survey of the mean-field theoretical description of Electric Double Layer (EDL) is presented. Special attention is devoted to asymmetric finite size of ions and orientational ordering of water dipoles. A model of Wicke and Eigen, who were first to explicitly derive the ion distribution functions for finite size of ions, is discussed. Arguments are given in favour of changing the recently adopted name of the mean-field EDL model for finite size of ions from Bikerman model to Bikerman-Wicke-Eigen model. Theoretically predicted asymmetric and symmetric camel-like shape of the voltage dependence of the differential capacitance is also discussed.