Finite Ion Size Research Articles

Combined Effects of Interfacial Permittivity Variations and Finite Ionic Sizes on Streaming Potentials in Nanochannels

In this work, we investigate the effects of local permittivity variations, induced by a preferential orientation and exclusion of water dipoles close to channel walls, and the effects of finite-sized ions on the induced streaming potential in nanochannels. We make a detailed analysis of the underlying physicochemical interactions by considering combinations of cases where ions are considered to be point sized/finite sized and permittivity variation effects to be present/absent. By accounting for the dielectric friction (which in turn is a function of the local permittivity) in addition to the classical Stokes friction, we show that for high interfacial potentials and narrow confinements, the induced streaming potential field for the cases in which the polarization effects are considered for finite-sized ions is remarkably higher than for the cases in which the polarization effects are neglected. Thus, by coupling the nonlinear effects of finite-sized ions and water dipole polarization along with the dielectric friction, we open a new paradigm of streaming potential predictions for narrow fluidic confinements, bearing far-ranging scientific and technological consequences in nanoscale science and technology.

Influence of nonelectrostatic ion-ion interactions on double-layer capacitance

Recently a Poisson-Helmholtz-Boltzmann (PHB) model [Bohinc et al., Phys. Rev. E 85, 031130 (2012)] was developed by accounting for solvent-mediated nonelectrostatic ion-ion interactions. Nonelectrostatic interactions are described by a Yukawa-like pair potential. In the present work, we modify the PHB model by adding steric effects (finite ion size) into the free energy to derive governing equations. The modified PHB model is capable of capturing both ion specificity and ion crowding. This modified model is then employed to study the capacitance of the double layer. More specifically, we focus on the influence of nonelectrostatic ion-ion interactions on charging a double layer near a flat surface in the presence of steric effects. We numerically compute the differential capacitance as a function of the voltage under various conditions. At small voltages and low salt concentrations (dilute solution), we find out that the predictions from the modified PHB model are the same as those from the classical Poisson-Boltzmann theory, indicating that nonelectrostatic ion-ion interactions and steric effects are negligible. At moderate voltages, nonelectrostatic ion-ion interactions play an important role in determining the differential capacitance. Generally speaking, nonelectrostatic interactions decrease the capacitance because of additional nonelectrostatic repulsion among excess counterions inside the double layer. However, increasing the voltage gradually favors steric effects, which induce a condensed layer with crowding of counterions near the electrode. Accordingly, the predictions from the modified PHB model collapse onto those computed by the modified Poisson-Boltzmann theory considering steric effects alone. Finally, theoretical predictions are compared and favorably agree with experimental data, in particular, in concentrated solutions, leading one to conclude that the modified PHB model adequately predicts the diffuse-charge dynamics of the double layer with ion specificity and steric effects.

Steric-effect-induced alteration of thermal transport phenomenon for mixed electroosmotic and pressure driven flows through narrow confinements

The present paper addresses, for the first time, the thermal transport process for mixed electroosmotic and pressure-drive flows of electrolyte solutions, through nanoscopic confinements with step-change in the wall temperature, by going beyond the prevalent simplifying assumption of non-interacting, point charge behaviour of the ions in the electrolyte solution. An attempt is made here to delineate the alterations in the heat transfer characteristics induced by the inclusion of the finite ionic size effect or the ‘steric effect’, in the electrokinetic framework, which becomes very significant for electro-hydrodynamic flows through narrow-confinements with high surface charge density. Under such situations the finite size of the ions cannot be trivially precluded from the analysis, thereby rendering the point charge assumption erroneous. The distinctive influence of the ‘steric’ effect of ions on the heat transfer characteristics is numerically investigated by delineating the variations in the local liquid temperature, local Nusselt number and the thermal entrance length for the thermally developing regime. The observed significant influences of the bulk volume fraction of ions, as represented by the steric factor, on the thermal transport phenomenon are physically explained by simultaneously highlighting their intrinsic differences with the effects of other existing parameters, like Joule heating and viscous dissipation. The utilitarian scope of the work lies in the fact that it proposes a more comprehensive methodology for analyzing the heat transfer characteristics in state-of-the-art nanoscale electromechanical devices.

Improving Generalized Born Models by Exploiting Connections to Polarizable Continuum Models. II. Corrections for Salt Effects.

A previous analytical investigation of the generalized Born (GB) implicit solvation model is extended to solvents of nonzero ionic strength. The GB model with salt effects (GB-SE) is shown to resemble the Debye-Hückel-like screening model (DESMO), a polarizable continuum model (PCM) that we have recently developed for salty solutions. DESMO may be regarded either as a generalization of the conductor-like PCM (C-PCM) that extends C-PCM to electrolyte solutions or alternatively as a generalization of Debye-Hückel theory to arbitrary cavity shapes. The connection between GB-SE and DESMO suggests how the former can be modified to account for the exclusion of mobile ions from the cavity interior, an effect that is typically absent in GB-SE models. We propose two simple GB-SE models that are exact for a point charge in a spherical cavity and that introduce the ability to account, albeit approximately, for the finite size of the mobile ions. The accuracy of these new models is demonstrated by applications to both model systems and real proteins. These tests also demonstrate the accuracy of the DESMO approach, as compared to more sophisticated PCMs developed for electrolyte solutions.

Mesoscale modeling of electric double layer capacitors with three-dimensional ordered structures

This paper presents general mathematical formulations for simulating electric double layer capacitors (EDLCs) with three-dimensional ordered structures. For the first time, a general set of boundary conditions was derived in order to account for the Stern layer without simulating it in the computational domain. These boundary conditions were valid for planar, cylindrical, and spherical electrode particles or pores. They made possible the simulations of EDLCs with complex geometries while rigorously accounting for both the Stern and diffuse layers. The model also simultaneously accounted not only for 3D electrode morphology but also for finite ion size and field-dependent electrolyte dielectric permittivity. It was used to faithfully simulate the complex structure of an EDLC electrode consisting of ordered bimodal mesoporous carbon featuring both macropores and mesopores. Areal and gravimetric capacitances were predicted based on non-solvated and solvated ion diameters. These two cases set the upper and lower bounds for the predicted capacitances. The capacitances predicted using non-solvated ion diameter were found to be in good agreement with experimental data reported in the literature. All surfaces contributed to the overall capacitance of EDLCs. The gravimetric capacitance of different bimodal carbons increased linearly with increasing specific surface area corresponding to constant areal capacitance.

Ion size effects on the electrokinetics of salt-free concentrated suspensions in ac fields

We analyze the influence of finite ion size effects in the response of a salt-free concentrated suspension of spherical particles to an oscillating electric field. Salt-free suspensions are just composed of charged colloidal particles and the added counterions released by the particles to the solution that counterbalance their surface charge. In the frequency domain, we study the dynamic electrophoretic mobility of the particles and the dielectric response of the suspension. We find that the Maxwell–Wagner–O’Konski process associated with the counterions condensation layer is enhanced for moderate to high particle charges, yielding an increment of the mobility for such frequencies. We also find that the increment of the mobility grows with ion size and particle charge. All these facts show the importance of including ion size effects in any extension attempting to improve standard electrokinetic models.

Effect of Solvent Polarization in Nano-Confined Electric Double Layer with Finite Ion Sizes

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Equilibrium properties of charged spherical colloidal particles suspended in aqueous electrolytes: Finite ion size and effective ion permittivity effects

The equilibrium properties of a charged spherical colloidal particle immersed in an aqueous electrolyte solution are examined using an extension of the Standard Electrokinetic Model that takes into account the finite ion size by modeling the aqueous electrolyte solution as a suspension of polarizable insulating spheres in water. We find that this model greatly amplifies the steric effects predicted by the usual modified Poisson–Boltzmann equation, which only imposes a restriction on the ability of ions to approach one another. This suggests that a solution of the presented model under nonequilibrium conditions could have important consequences in the interpretation of dielectric and electrokinetic data in colloidal suspensions.

RNA and Its Ionic Cloud: Solution Scattering Experiments and Atomically Detailed Simulations

RNA molecules play critical roles in many cellular processes. Traditionally viewed as genetic messengers, RNA molecules were recently discovered to have diverse functions related to gene regulation and expression. RNA also has great potential as a therapeutic and a tool for further investigation of gene regulation. Metal ions are an integral part of RNA structure and should be considered in any experimental or theoretical study of RNA. Here, we report a multidisciplinary approach that combines anomalous small-angle x-ray scattering and molecular-dynamics (MD) simulations with explicit solvent and ions around RNA. From experiment and simulation results, we find excellent agreement in the number and distribution of excess monovalent and divalent ions around a short RNA duplex. Although similar agreement can be obtained from a continuum description of the solvent and mobile ions (by solving the Poisson-Boltzmann equation and accounting for finite ion size), the use of MD is easily extended to flexible RNA systems with thermal fluctuations. Therefore, we also model a short RNA pseudoknot and find good agreement between the MD results and the experimentally derived solution structures. Surprisingly, both deviate from crystal structure predictions. These favorable comparisons of experiment and simulations encourage work on RNA in all-atom dynamic models.

Spatial variation of permittivity of an electrolyte solution in contact with a charged metal surface: a mini review

Contact between a charged metal surface and an electrolyte implies a particular ion distribution near the charged surface, i.e. the electrical double layer. In this mini review, different mean-field models of relative (effective) permittivity are described within a simple lattice model, where the orientational ordering of water dipoles in the saturation regime is taken into account. The Langevin–Poisson–Boltzmann (LPB) model of spatial variation of the relative permittivity for point-like ions is described and compared to a more general Langevin–Bikerman (LB) model of spatial variation of permittivity for finite-sized ions. The Bikerman model and the Poisson–Boltzmann model are derived as limiting cases. It is shown that near the charged surface, the relative permittivity decreases due to depletion of water molecules (volume-excluded effect) and orientational ordering of water dipoles (saturation effect). At the end, the LPB and LB models are generalised by also taking into account the cavity field.

Electric-double-layer potential distribution in multiple-layer immiscible electrolytes: Effect of finite ion sizes

In a recent study [S. Das and S. Hardt, Phys. Rev. E 84, 022502 (2011).], we provided analytical results for the electric-double-layer (EDL) electrostatic potential distribution in a system of immiscible electrolyte layers confined between plates with gap dimensions comparable to the EDL thickness. We demonstrated that the intrinsic jumps in the ion-solvent interactions across the interface of the immiscible electrolytes lead to nontrivial electrostatic potential distributions that may completely defy or substantially augment the effect of the boundary wall potential. In this Brief Report, I extend this calculation to obtain analytical and numerical results for the case with finite ion sizes (or a finite ionic steric effect). It is found that the finite steric effect substantially enhances the contributions of jump in the ion-solvent interactions in the overall electrostatic potential distribution. More importantly, I demonstrate that such jumps in the ion-solvent interactions, owing to the immiscibility of the electrolytes, ensure that even for very weak wall zeta potentials, the steric effect can significantly affect the electrostatic potential distribution.

Diffuse-charge dynamics of ionic liquids in electrochemical systems

We employ a continuum theory of solvent-free ionic liquids accounting for both short-range electrostatic correlations and steric effects (finite ion size) [Bazant et al., Phys. Rev. Lett. 106, 046102 (2011)] to study the response of a model microelectrochemical cell to a step voltage. The model problem consists of a 1-1 symmetric ionic liquid between two parallel blocking electrodes, neglecting any transverse transport phenomena. Matched asymptotic expansions in the limit of thin double layers are applied to analyze the resulting one-dimensional equations and study the overall charge-time relation in the weakly nonlinear regime. One important conclusion is that our simple scaling analysis suggests that the length scale √(λ*(D)l*(c)) accurately characterizes the double-layer structure of ionic liquids with strong electrostatic correlations where l*(c) is the electrostatic correlation length (in contrast, the Debye screening length λ*(D) is the primary double-layer length for electrolytes) and the response time of λ(D)(*3/2)L*/(D*l(c)(1/2)) (not λ*(D)L*/D* that is the primary charging time of electrolytes) is the correct charging time scale of ionic liquids with strong electrostatic correlations where D* is the diffusivity and L* is the separation length of the cell. With these two new scales, data of both electric potential versus distance from the electrode and the total diffuse charge versus time collapse onto each individual master curve in the presence of strong electrostatic correlations. In addition, the dependance of the total diffuse charge on steric effects, short-range correlations, and driving voltages is thoroughly examined. The results from the asymptotic analysis are compared favorably with those from full numerical simulations. Finally, the absorption of excess salt by the double layer creates a depletion region outside the double layer. Such salt depletion may bring a correction to the leading order terms and break down the weakly nonlinear analysis. A criterion which justifies the weakly nonlinear analysis is verified with numerical simulations.

Poisson–Boltzmann Description of the Electrical Double Layer Including Ion Size Effects

The electrical double layer is examined using a generalized Poisson-Boltzmann equation that takes into account the finite ion size by modeling the aqueous electrolyte solution as a suspension of polarizable insulating spheres in water. We find that this model greatly amplifies the steric effects predicted by the usual modified Poisson-Boltzmann equation, which imposes only a restriction on the ability of ions to approach one another. This amplification should allow for an interpretation of the experimental results using reasonable effective ionic radii (close to their well-known hydrated values).

Accurate Simulations of Electric Double Layer Capacitance of Ultramicroelectrodes

This paper aims to develop rigorous and accurate numerical tools for simulating electric double layers formed near ultramicroelectrodes. It also aims to assess the validity of existing models and to identify the dominant physical phenomena that must be accounted for. The electric double layer capacitance was numerically predicted for spherical ultramicroelectrodes of various radii in aqueous electrolyte. The model accounted for the Stern and diffuse layers, the finite size of ions, and the dependency of the electrolyte dielectric permittivity on the local electric field. This study reveals that models reported in the literature suffer from severe limitations. First, it demonstrates that the electrolyte field-dependent dielectric permittivity significantly affects the predicted Stern layer and total specific capacitances and must be accounted for. The finite ion size and the Stern layer also need to be considered in simulating electric double layers under high concentrations and surface potential. This study also establishes that the Helmholtz model predicts the Stern layer capacitance for all electrode radii if the electrolyte permittivity is assumed to be constant. However, it underestimates the Stern layer capacitance for sphere radii less than 40 nm when accounting for field-dependent permittivity. The electrode curvature was found to have negligible effect on the predicted specific capacitance for sphere radii larger than 40 nm. Then, the latter is equal to that of planar electrodes. Results of this study can be used to design electrodes for electrochemical sensors and electrical energy storage.

Simulating Electric Double Layer Capacitance of Mesoporous Electrodes with Cylindrical Pores

This paper aims to numerically assess the effects of electrolyte properties and electrode morphology on the capacitance of electric double layer capacitors (EDLCs) made of mesoporous electrodes consisting of ordered cylindrical pores in non-aqueous electrolytes. Simulations solved a three-dimensional modified Poisson-Boltzmann model. They accounted for the finite size of ions and field-dependent electrolyte permittivity while the pores were perpendicular to the current collector. The effects of pore radius, porosity, effective ion diameter, and electrolyte field-dependent permittivity on the diffuse layer gravimetric capacitance were investigated systematically in order to determine key parameters affecting EDLCs’ performance. The simulations showed that reducing the ion effective diameter and the pore radius resulted in the strongest increase in diffuse layer gravimetric capacitance up to a critical radius below which the capacitance reaches a plateau. Increasing the electrode porosity also increased the diffuse layer gravimetric capacitance. Accounting for more realistic field-dependent permittivity was found to significantly reduce the predicted diffuse layer gravimetric capacitance. Finally, accounting for the contribution of the Stern layer to the total capacitance was essential in predicting experimental data for a wide range of porous activated carbon electrodes and non-aqueous electrolytes.

Simulation of an ac electro-osmotic pump with step microelectrodes

Pumps with step microelectrodes subjected to an ac voltage are known to have faster pumping rates than those with planar asymmetric microelectrodes. The driving force for pumping in these systems is ac electro-osmosis. This paper aims to understand the flow behaviors of pumps with step microelectrodes by using a realistic model applicable to high external voltages. This model takes the steric effect due to the finite sizes of ions into account and copes with the exponential sensitivity of the counterion concentration to voltage. The effects on the pumping flow rate of varying the pump parameters were investigated. The geometrical parameters were optimized, and the effects of varying the ac frequency and amplitude were examined. The electrical potential of the fluid and the electrical charge at the electrode surface were solved simultaneously, and the Stokes equation was used to describe the fluid flow.

Ion size effects upon ionic exclusion from dielectric interfaces and slit nanopores

A previously developed field-theoretic model (Coalson et al 1995 J. Chem. Phys. 102 4584)that treats core collisions and Coulomb interactions on the same footing is investigated inorder to understand ion size effects on the partition of neutral and charged particles atplanar interfaces and the ionic selectivity of slit nanopores. We introduce a variationalscheme that can go beyond the mean-field (MF) regime and couple in a consistent waypore-modified core interactions, steric effects, electrostatic solvation and image-chargeforces, and surface charge induced electrostatic potential. Density profiles of neutralparticles in contact with a neutral hard wall, obtained from Monte Carlo (MC) simulationsare compared with the solutions of mean-field and variational equations. A recentlyproposed random-phase approximation (RPA) method is tested as well. We show that inthe dilute limit, the MF and the variational theories agree well with simulation results, incontrast to the RPA method. The partition of charged Yukawa particles at a neutraldielectric interface (e.g. an air–water or protein–water interface) is investigated. It isshown that as a result of the competition between core collisions that push theions toward the surface, and repulsive solvation and image forces that excludethem from the interface, a concentration peak of finite size ions sets in close tothe dielectric interface. This effect is amplified with increasing ion size and bulkconcentration. An integral expression for the surface tension that accounts forexcluded volume effects is computed and the decrease of the surface tension withincreasing ion size is illustrated. We also characterize the role played by the ionsize in the ionic selectivity of neutral slit nanopores. We show that the complexinterplay between electrostatic forces, excluded volume effects induced by corecollisions and steric effects leads to an unexpected reversal in the ionic selectivity ofthe pore with varying pore size: while large pores exhibit a higher conductivityfor large ions, narrow pores exclude large ions more efficiently than small ones.

Efficiently accounting for ion correlations in electrokinetic nanofluidic devices using density functional theory

The electrokinetic behavior of nanofluidic devices is dominated by the electrical double layers at the device walls. Therefore, accurate, predictive models of double layers are essential for device design and optimization. In this paper, we demonstrate that density functional theory (DFT) of electrolytes is an accurate and computationally efficient method for computing finite ion size effects and the resulting ion–ion correlations that are neglected in classical double layer theories such as Poisson–Boltzmann. Because DFT is derived from liquid-theory thermodynamic principles, it is ideal for nanofluidic systems with small spatial dimensions, high surface charge densities, high ion concentrations, and/or large ions. Ion–ion correlations are expected to be important in these regimes, leading to nonlinear phenomena such as charge inversion, wherein more counterions adsorb at the wall than is necessary to neutralize its surface charge, leading to a second layer of co-ions. We show that DFT, unlike other theories that do not include ion–ion correlations, can predict charge inversion and other nonlinear phenomena that lead to qualitatively different current densities and ion velocities for both pressure-driven and electro-osmotic flows. We therefore propose that DFT can be a valuable modeling and design tool for nanofluidic devices as they become smaller and more highly charged.

A New Poisson-Nernst-Planck Equation (PNP-FS-IF) for Charge Inversion Near Walls

The plasmas of biology are interacting mixtures of ions - often charged spheres - that do not behave like the ideal solutions of textbooks. Interactions are always present because of strong electrical forces. Flows are usually present. Life without flow is death. We analyze ionic solutions as complex fluids with an approach that has successfully analyzed complex systems like liquid crystals that are dominated by interactions between composite components. The finite size of ions is particularly important in biology in crowded environments like channels, active sites of enzymes, or charged surfaces. We here deal with surfaces and try to capture the essential features of charge inversion (layering) near a charged wall. Charge inversion (layering) near walls is a characteristic phenomenon resulting from the electrostatic interactions in systems with charged walls. The mathematical model is derived by the energy variational approach (EnVarA) _ J.Chem.Phys. (2010) 133:104104 _ that combines the action of conservative (Hamiltonian) systems and the dissipation of Onsager and Rayleigh. Both are written in the same laboratory coordinates after variational derivatives of variables are taken. The generalized energy and dissipation include entropic and electrostatic components, and repulsion between spheres. An interfacial electroneutrality constraint between bulk and charged wall captures some essential features of charge inversion. Taking variational derivatives yields a field theory of partial differential equations and boundary conditions that are appropriate for life's solutions _ that interact and flow _ as well as thermodynamic equilibrium. The new equations, PNP-FS-IF, include (1) a nonlocal contribution of finite size (FS) and (2) an interfacial constraint (IF) of electroneutrality. PNP-FS-IF produces charge inversion near walls. We compare the charge inversion seen with PNP-FS-IF and Monte-Carlo simulations.

Double Layer in Ionic Liquids: Overscreening versus Crowding

We develop a simple Landau-Ginzburg-type continuum theory of solvent-free ionic liquids and use it to predict the structure of the electrical double layer. The model captures overscreening from short-range correlations, dominant at small voltages, and steric constraints of finite ion sizes, which prevail at large voltages. Increasing the voltage gradually suppresses overscreening in favor of the crowding of counterions in a condensed inner layer near the electrode. This prediction, the ion profiles, and the capacitance-voltage dependence are consistent with recent computer simulations and experiments on room-temperature ionic liquids, using a correlation length of order the ion size.