Ionic Layering and Overcharging in Electrical Double Layers in a Poisson-Boltzmann Model

A network model to predict ionic transport in porous materials

Understanding the dynamics of electric-double-layer (EDL) charging in porous media is essential for advancements in next-generation energy storage devices. Due to the high computational demands of direct numerical simulations and a lack of interfacial boundary conditions for reduced-order models, the current understanding of EDL charging is limited to simple geometries. Here, we present a network model to predict EDL charging in arbitrary networks of long pores in the Debye-Hückel limit without restrictions on EDL thickness and pore radii. We demonstrate that electrolyte transport is described by Kirchhoff's laws in terms of the electrochemical potential of charge (the valence-weighted average of the ion electrochemical potentials) instead of the electric potential. By employing the equivalent circuit representation suggested by these modified Kirchhoff's laws, our methodology accurately captures the spatial and temporal dependencies of charge density and electric potential, matching results obtained from computationally intensive direct numerical simulations. Our network model provides results up to six orders of magnitude faster, enabling the efficient simulation of a triangular lattice of five thousand pores in 6 min. We employ the framework to study the impact of pore connectivity and polydispersity on electrode charging dynamics for pore networks and discuss how these factors affect the time scale, energy density, and power density of capacitive charging. The scalability and versatility of our methodology make it a rational tool for designing 3D-printed electrodes and for interpreting geometric effects on electrode impedance spectroscopy measurements.

Toward Modeling the Structure of Electrolytes at Charged Mineral Interfaces Using Classical Density Functional Theory

The organization of water molecules and ions between charged mineral surfaces determines the stability of colloidal suspensions and the strength of phase-separated particulate gels. In this article, we assemble a density functional that measures the free energy due to the interaction of water molecules and ions in electric double layers. The model accounts for the finite size of the particles using fundamental measure theory, hydrogen-bonding between water molecules using Wertheim's statistical association theory, long-range dispersion interactions using Barker and Henderson's high-temperature expansion, electrostatic correlations using a functionalized mean-spherical approximation, and Coulomb forces through the Poisson equation. These contributions are shown to produce highly correlated structures, aptly rendering the layering of counterions and co-ions at highly charged surfaces and permitting the solvation of ions and surfaces to be measured by a combination of short-range associations and long-ranged attractions. The model is tested in a planar geometry near soft, charged surfaces to reproduce the structure of water near graphene and mica. For mica surfaces, explicitly representing the density of the outer oxygen layer of the exposed silica tetrahedra allows water molecules to hydrogen-bond to the solid. When electrostatic interactions are included, water molecules assume a hybrid character, being accounted for implicitly in the dielectric constant but explicitly otherwise. The disjoining pressure between approaching like-charged surfaces is calculated, demonstrating the model's ability to probe pressure oscillations that arise during the expulsion of ions and water layers from the interfacial gap and predict strong interattractive stresses that form at narrow interfacial spacing when the surface charge is overscreened. This interattractive stress arises not due to in-plane correlations under strong electrostatic coupling but due to the out-of-plane structuring of associating ions and water molecules.

Electro-osmotic flow in nanoconfinement: Solid-state and protein nanopores

Electro-osmotic flow (EOF) is a phenomenon where fluid motion occurs in porous materials or micro/nano-channels when an external electric field is applied. In the particular example of single-molecule electrophoresis using single nanopores, the role of EOF on the translocation velocity of the analyte molecule through the nanopore is not fully understood. The complexity arises from a combination of effects from hydrodynamics in restricted environments, electrostatics emanating from charge decorations and geometry of the pores. We address this fundamental issue using the Poisson-Nernst-Planck and Navier-Stokes (PNP-NS) equations for cylindrical solid-state nanopores and three representative protein nanopores (α-hemolysin, MspA, and CsgG). We present the velocity profiles inside the nanopores as a function of charge decoration and geometry of the pore and applied electric field. We report several unexpected results: (a) The apparent charges of the protein nanopores are different from their net charge and the surface charge of the whole protein geometry, and the net charge of inner surface is consistent with the apparent charge. (b) The fluid velocity depends non-monotonically on voltage. The three protein nanopores exhibit unique EOF and velocity-voltage relations, which cannot be simply deduced from their net charge. Furthermore, effective point mutations can significantly change both the direction and the magnitude of EOF. The present computational analysis offers an opportunity to further understand the origins of the speed of transport of charged macromolecules in restricted space and to design desirable nanopores for tuning the speed of macromolecules through nanopores.

Influence of surface nanostructure-induced innermost ion structuring on capacitance of carbon/ionic liquid double layers

Ionic liquids have drawn great interest as electrolytes for energy storage applications in which they form characteristic electrical double layers at electrode interfaces. For ionic liquids at carbon electrode interfaces, their double layers are subject to nanoscale structuring of the electrode surface, involving altered ion structure and interactions that significantly influence the double layer capacitance. In this regard, we investigate the modulation of ionic liquid double layers by electrode surface roughness and the resulting effects on the ion structure, interaction, and capacitance. We performed fixed voltage molecular dynamics simulations to compute the differential capacitance profiles for the ionic liquids [BMIm+][TFSI-] and [BMIm+][FSI-] at model carbon electrode interfaces with the surface channel width at subnanometer and nanometer scales. We find that both [BMIm+][TFSI-] and [BMIm+][FSI-] exhibit enhanced differential capacitance for the electrode surface with a subnanometer channel width relative to the flat graphene surface, but the most pronounced enhancements for these two ionic liquids unexpectedly appear at different applied potential regimes. For [BMIm+][TFSI-], the nanostructured electrode shows significant enhancement of capacitance at high positive potential. For [BMIm+][FSI-], on the other hand, this enhancement is small at positive polarization but noticeable at low negative potential. We demonstrate that differences in these capacitance trends is due to differences in ion correlation that arise from a steric constraint of nanostructured electrode surface on the voltage-mediated restructuring of ions closest to the electrode interface. For example, the TFSI- and FSI- anions tend to structure with their charged and nonpolar groups in contact with the positive electrode surface when the constraint on these close-contact anions is relaxed. This anion structuring largely retains the cation association near the nanostructured electrode, resulting in only a slight increase in capacitance at positive polarization. Our simulations highlight the sensitive dependence of the innermost ion structure on the electrode surface nanostructure and applied voltage and the resulting influence on ion correlation and capacitance of ionic liquid double layers.

Influence of Hydrolyzed Metal Ions and Surfactants on the Phase Transfer of Al2O3, SiO2, and SnO2

There are various possibilities for changing the surface properties of particles. In this work, the charge reversal on different metal oxides with different electrolytes is investigated and whether this allows a change in wettability due to a subsequent adsorption of surfactants, e.g., sodium dodecyl sulfate (SDS). It is investigated if the materials of the particles differ only by the isoelectric point or if the surface chemistry of the materials has an influence on the charge reversal as well. Furthermore, the adsorption of SDS as an anionic surfactant is examined, which is also characterized by a second charge reversal and related to a sign change of the electrophoretic mobility μe. Finally, it is examined whether the adsorption of the hydrolyzed metal ions and the subsequent adsorption of SDS are effective enough to hydrophobize the particles and allow phase transfer from the aqueous to second nonaqueous liquid phase. In addition, the influence of pH is investigated because the hydrolyzed metal cations are formed only in a certain pH range, which means that the bridge formed between the particle surface and the surfactant works only in a certain pH range, which would allow pH-selective extraction of the particle system into the second nonaqueous liquid phase.

Nature of Overcharging and Charge Inversion in Electrical Double Layers

Understanding overcharging and charge inversion is one of the long-standing challenges in soft matter and biophysics. To study these phenomena, we employ the modified Gaussian renormalized fluctuation theory, which allows for the self-consistent accounting of spatially varying ionic strength as well as the spatial variations in dielectric permittivity and excluded volume effects. The underlying dependence of overcharging on the electrostatic coupling is elucidated by varying the surface charge, counterion valency, and dielectric contrast. Consistent with simulations, three characteristic regimes corresponding to weak, moderate, and strong coupling are identified. Important features like the inversion of zeta potential, crowding, and ionic layering at the surface are successfully captured. For weak coupling, there is no overcharging. In the moderate coupling regime, overcharging increases with the surface charge. Finally, in the strong coupling regime, ionic crowding and saturation in overcharging are observed. Our theory predicts a nonmonotonic dependence of charge inversion on multivalent salt concentration as well as the addition of monovalent salt, in quantitative agreement with experiments.

Ion correlation-driven like-charge attraction in multivalent salt solutions

The electrostatic double layer force is key to determining the stability and self-assembly of charged colloids and many other soft matter systems. Fully understanding the attractive force between two like-charged surfaces remains a great challenge. Here, we apply the modified Gaussian renormalized fluctuation theory to study ion correlation-driven like-charge attraction in multivalent salt solutions. The effects of spatially varying ion correlations on the structure of overlapping double layers and their free energy are self-consistently accounted for. In the presence of multivalent salts, increasing surface charge or counterion valency leads to a short-range attraction. We demonstrate that although both overcharging and like-charge attraction are outcomes of ion correlation, there is no causal relationship between them. Our theory also captures the non-monotonic dependence of like-charge attraction on multivalent salt concentration. The reduction of attraction at high salt concentrations could be a contributing factor toward the reentrant stability of charged colloidal suspensions. Our theoretical predictions are consistent with the observations reported in experiments and simulations.

Predicting Outcomes of Nanoparticle Attachment by Connecting Atomistic, Interfacial, Particle, and Aggregate Scales

Predicting nanoparticle aggregation and attachment phenomena requires a rigorous understanding of the interplay among crystal structure, particle morphology, surface chemistry, solution conditions, and interparticle forces, yet no comprehensive picture exists. We used an integrated suite of experimental, theoretical, and simulation methods to resolve the effect of solution pH on the aggregation of boehmite nanoplatelets, a case study with important implications for the environmental management of legacy nuclear waste. Real-time observations showed that the particles attach preferentially along the (010) planes at pH 8.5 and the (101) planes at pH 11. To rationalize these results, we established the connection between key physicochemical phenomena across the relevant length scales. Starting from molecular-scale simulations of surface hydroxyl reactivity, we developed an interfacial-scale model of the corresponding electrostatic potentials, with subsequent particle-scale calculations of the resulting driving forces allowing successful prediction of the attachment modes. Finally, we scaled these phenomena to understand the collective structure at the aggregate-scale. Our results indicate that facet-specific differences in surface chemistry produce heterogeneous surface charge distributions that are coupled to particle anisotropy and shape-dependent hydrodynamic forces, to play a key role in controlling aggregation behavior.

Incorporating ion-specific van der Waals and soft repulsive interactions in the Poisson-Boltzmann theory of electrical double layers

Electrical double layers (EDLs) arise when an electrolyte is in contact with a charged surface, and are encountered in several application areas including batteries, supercapacitors, electrocatalytic reactors, and colloids. Over the last century, the development of Poisson-Boltzmann (PB) models and their modified versions have provided significant physical insight into the structure and dynamics of the EDL. Incorporation of physics such as finite-ion-size effects, dielectric decrement, and ion-ion correlations has made such models increasingly accurate when compared to more computationally expensive approaches such as molecular simulations and classical density functional theory. However, a prominent knowledge gap has been the exclusion of van der Waals (vdW) and soft repulsive interactions in modified PB models. Although short-ranged as compared to electrostatic interactions, we show here that vdW and soft repulsive interactions can play an important role in determining the structure of the EDL via the formation of a Stern layer and in modulating the differential capacitance of an electrode in an electrolyte. To this end, we incorporate ion-ion and wall-ion vdW attraction and soft repulsion via a 12-6 Lennard-Jones (LJ) potential, resulting in a modified PB-LJ approach. The wall-ion LJ interactions were found to have a significant effect on the electrical potential and concentration profiles, especially close to the wall. However, ion-ion LJ interactions do not affect the EDL structure at low bulk ion concentrations (<1 M). We also derive dimensionless numbers to quantify the impact of ion-ion and wall-ion LJ interactions on the EDL. Furthermore, in the pursuit of capturing ion-specific effects, we apply our model by considering various ions such as Na, K+, Mg2+, Cl-, and SO42-. We observe how varying parameters such as the electrolyte concentration and electrode potential affect the structure of the EDL due to the competition between ion-specific LJ and electrostatic interactions. Lastly, we show that the inclusion of vdW and soft repulsion interactions, as well as hydration effects, leads to a better qualitative agreement of the PB models with experimental double-layer differential capacitance data. Overall, the modified PB-LJ approach presented herein will lead to more accurate theoretical descriptions of EDLs in various application areas.

Evolution of channel flow and Darcy's law beyond the critical Reynolds number

For incompressible channel flow, there is a critical state, characterized by a critical Reynolds number Rec and a critical wavevector mc along the channel direction, beyond which the channel flow becomes unstable in the linear regime. In this work, we investigate the channel flow beyond the critical state and find the existence of a new fluctuating, quasi-stationary flow that comprises the laminar Poiseuille flow superposed with a counter-flow component, accompanied by vortices and anti-vortices. The net flow rate is reduced by  ~ 15% from the linear, laminar regime. Our study is facilitated by the analytical solution of the linearized, incompressible, three-dimensional (3D) Navier-Stokes (NS) equation in the channel geometry, with the Navier boundary condition, alternatively denoted as the hydrodynamic modes (HMs). By using the HMs as the complete mathematical basis for expanding the velocity in the NS equation, the Rec is evaluated to 5-digit accuracy when compared to the well-known Orszag result, without invoking the standard Orr-Sommerfeld equation. Beyond Rec, the analytical solution is indispensable in offering physical insight to those features of the counter-flow component that differs from any of the pressure-driven channel flows. In particular, the counter flow is found to comprise multiple HMs, some with opposite flow direction, that can lead to a net boundary reaction force along the counter-flow direction. The latter is analyzed to be necessary for satisfying Newton's law. Experimental verification of the predictions is discussed.

Hierarchical Materials from High Information Content Macromolecular Building Blocks: Construction, Dynamic Interventions, and Prediction

Hierarchical materials that exhibit order over multiple length scales are ubiquitous in nature. Because hierarchy gives rise to unique properties and functions, many have sought inspiration from nature when designing and fabricating hierarchical matter. More and more, however, nature's own high-information content building blocks, proteins, peptides, and peptidomimetics, are being coopted to build hierarchy because the information that determines structure, function, and interfacial interactions can be readily encoded in these versatile macromolecules. Here, we take stock of recent progress in the rational design and characterization of hierarchical materials produced from high-information content blocks with a focus on stimuli-responsive and "smart" architectures. We also review advances in the use of computational simulations and data-driven predictions to shed light on how the side chain chemistry and conformational flexibility of macromolecular blocks drive the emergence of order and the acquisition of hierarchy and also on how ionic, solvent, and surface effects influence the outcomes of assembly. Continued progress in the above areas will ultimately usher in an era where an understanding of designed interactions, surface effects, and solution conditions can be harnessed to achieve predictive materials synthesis across scale and drive emergent phenomena in the self-assembly and reconfiguration of high-information content building blocks.

Electrostatic Correlation Induced Ion Condensation and Charge Inversion in Multivalent Electrolytes

The study of the electrical double layer lies at the heart of colloidal and interfacial science. The standard mean-field Poisson-Boltzmann (PB) theory is incapable of modeling many phenomena originating from ion correlation. An important example is charge inversion or overcharging of electrical double layers in multivalent electrolyte solutions. Existing theories aiming to include correlations cannot capture the non-monotonic dependence of charge inversion on salt concentration because they have not systematically accounted for the inhomogeneous nature of correlations from surface to the bulk and the excluded volume effect of ions and solvent molecules. In this work, we modify the Gaussian renormalized fluctuation theory by including the excluded volume effect to study ion condensation and charge inversion. A boundary layer approach is developed to accurately model the giant difference in ion correlations between the condensed layer near the surface and the diffuse layer outside. The theory is used to study charge inversion in multivalent electrolytes and their mixtures. We predict a surface charge induced formation of a three-dimensional condensed layer, which is necessary but not sufficient for charge inversion. The value of the effective surface potential is found to depend non-monotonically on the bulk salt concentration. Our results also show a non-monotonic reduction in charge inversion in monovalent and multivalent electrolyte mixtures. Our work is the first to qualitatively reproduce experimental and simulation observations and explains the underlying physics.

Understanding the Electric Double-Layer Structure, Capacitance, and Charging Dynamics

Significant progress has been made in recent years in theoretical modeling of the electric double layer (EDL), a key concept in electrochemistry important for energy storage, electrocatalysis, and multitudes of other technological applications. However, major challenges remain in understanding the microscopic details of the electrochemical interface and charging mechanisms under realistic conditions. This review delves into theoretical methods to describe the equilibrium and dynamic responses of the EDL structure and capacitance for electrochemical systems commonly deployed for capacitive energy storage. Special emphasis is given to recent advances that intend to capture the nonclassical EDL behavior such as oscillatory ion distributions, polarization of nonmetallic electrodes, charge transfer, and various forms of phase transitions in the micropores of electrodes interfacing with an organic electrolyte or ionic liquid. This comprehensive analysis highlights theoretical insights into predictable relationships between materials characteristics and electrochemical performance and offers a perspective on opportunities for further development toward rational design and optimization of electrochemical systems.

Simultaneous thermoosmotic and thermoelectric responses in nanoconfined electrolyte solutions: Effects of nanopore structures and membrane properties

HYPOTHESIS

Nanofluidic systems provide an emerging and efficient platform for thermoelectric conversion and fluid pumping with low-grade heat energy. As a basis of their performance enhancement, the effects of the structures and properties of the nanofluidic systems on the thermoelectric response (TER) and the thermoosmotic response (TOR) are yet to be explored.

METHODS

The simultaneous TER and TOR of electrolyte solutions in nanofluidic membrane pores on which an axial temperature gradient is exerted are investigated numerically and semi-analytically. A semi-analytical model is developed with the consideration of finite membrane thermal conductivity and the reservoir/entrance effect.

FINDINGS

The increase in the access resistance due to the nanopore-reservoir interfaces accounts for the decrease of short circuit current at the low concentration regime. The decrease in the thermal conductivity ratio can enhance the TER and TOR. The maximum power density occurring at the nanopore radius twice the Debye length ranges from several to dozens of mW K^-2 m^-2 and is an order of magnitude higher than typical thermo-supercapacitors. The surface charge polarity can heavily affect the sign and magnitude of the short-circuit current, the Seebeck coefficient and the open-circuit thermoosmotic coefficient, but has less effect on the short-circuit thermoosmotic coefficient. Furthermore, the membrane thickness makes different impacts on TER and TOR for zero and finite membrane thermal conductivity.

Structural features and electrostatic energy storage of electric double layers in confined polyelectrolyte solutions under low-salt conditions

Self-consistent field theory is used to systematically study the characteristics and electrostatic energy storage of electric double layers in confined polyelectrolyte solutions for salt-free and low salt concentration systems.

Charging dynamics of electrical double layers inside a cylindrical pore: predicting the effects of arbitrary pore size

Porous electrodes are found in energy storage devices such as supercapacitors and pseudocapacitors. However, the effect of electrode-pore-size distribution on their energy storage properties remains unclear. Here, we develop a model for the charging of electrical double layers inside a cylindrical pore for arbitrary pore size. We assume small applied potentials and perform a regular perturbation analysis to predict the evolution of electrical potential and ion concentrations in both the radial and axial directions. We validate our perturbation model with direct numerical simulations of the Poisson-Nernst-Planck equations, and obtain quantitative agreement between the two approaches for small and moderate potentials. Our analysis yields two main characteristic features of arbitrary pore size: (i) a monotonic decrease of the charging timescale with an increase in relative pore size (pore size relative to Debye length); (ii) large potential changes for overlapping double layers in a thin transition region, which we approximate mathematically by a jump discontinuity. We quantify the contributions of electromigration and charge diffusion fluxes, which provide mechanistic insights into the dependence of charging timescale and capacitance on pore size. We develop a modified transmission circuit model that captures the effect of arbitrary pore size and demonstrate that a time-dependent transition-region resistor needs to be included in the circuit. We also derive phenomenological expressions for average effective capacitance and charging timescale as a function of pore-size distribution. We show that the capacitance and charging timescale increase with smaller average pore sizes and with smaller polydispersity, resulting in a gain of energy density at a constant power density. Overall, our results advance the mechanistic understanding of electrical-double-layer charging.

Electrolyte structure near electrodes with molecular-size roughness

Understanding electrodes' surface morphology influence on ions' distribution is essential for designing supercapacitors with enhanced energy density characteristics. We develop a model for the structure of electrolytes near the rough surface of electrodes. The model describes an effective electrostatic field's increase and associated intensification of ions' spatial separation at the electrode-electrolyte interface. These adsorption-induced local electric and structure properties result in notably increased values and a sharpened form of the differential capacitance dependence on the applied potential. Such capacitance behavior is observed in many published simulations, and its description is beyond the capabilities of the established flat-electrodes theories. The proposed approach could extend the quantitatively verified models providing a new instrument of the electrode surface-parameter optimization for specific electrolytes.

Anion-Assisted Delivery of Multivalent Cations to Inert Electrodes

To understand and control key electrochemical processes-metal plating, corrosion, intercalation, etc.-requires molecular-scale details of the active species at electrochemical interfaces and their mechanisms for desolvation from the electrolyte. Using free energy sampling techniques we reveal the interfacial speciation of divalent cations in ether-based electrolytes and mechanisms for their delivery to an inert graphene electrode interface. Surprisingly, we find that anion solvophobicity drives a high population of anion-containing species to the interface that facilitate the delivery of divalent cations, even to negatively charged electrodes. Our simulations indicate that cation desolvation is greatly facilitated by cation-anion coupling. We propose anion solvophobicity as a molecular-level descriptor for rational design of electrolytes with increased efficiency for electrochemical processes limited by multivalent cation desolvation.