Understanding the Electric Double Layer at the Electrode-Electrolyte Interface: Part I - No Ion Specific Adsorption

In this work, we present a mean-field model that takes into account the key components of electrical double layer theory at the interface between an electrode and an electrolyte solution. The model considers short-range specific interactions between different species, including electrode-ion repulsion, the hydration of ions, dielectric saturation of solvent (water), and excluded volume (steric) interactions between species. By solving a modified Poisson-Boltzmann equation, which is derived from the grand thermodynamic potential of an inhomogeneous electrolyte solution, and using the appropriate results of quantum chemistry calculations on the hydration of ions, we can accurately approximate the differential capacitance profiles of aqueous electrolyte solutions at the boundary with a silver electrode. The specific interactions between the ions and the electrodes in the systems under consideration (aqueous solutions of NaClO4 and KPF6) are assumed to be significantly weaker than their Coulomb interactions. A novel aspect of our research is the investigation of the impact of short-range ion-water interactions on the differential capacitance, which provides new insights into the behavior of the electrical double layer. This model has the potential to be useful for electrochemical engineers working on the development of supercapacitors and related electrochemical energy storage devices. It serves as a basis for future modeling of electrolyte systems on real electrodes, especially in scenarios where chemical ion-electrode interactions are significant.

Application of the symmetric Poisson-Boltzmann theory to a model colloidal mixture

A symmetric Poisson-Boltzmann theory is used to analyse the structure of a primitive model colloidal system which contains either 4 or 6 components. The approach symmetrizes the pair distribution function gij(r) between two asymmetric charged species with respect to an interchange of the indices. Good agreement is found with molecular dynamics simulation structural properties when the exclusion volume term is modelled by the Percus-Yevick uncharged hard sphere radial distribution function.

Electric Double Layer and Orientational Ordering of Water Dipoles in Narrow Channels within a Modified Langevin Poisson-Boltzmann Model

The electric double layer (EDL) is an important phenomenon that arises in systems where a charged surface comes into contact with an electrolyte solution. In this work we describe the generalization of classic Poisson-Boltzmann (PB) theory for point-like ions by taking into account orientational ordering of water molecules. The modified Langevin Poisson-Boltzmann (LPB) model of EDL is derived by minimizing the corresponding Helmholtz free energy functional, which includes also orientational entropy contribution of water dipoles. The formation of EDL is important in many artificial and biological systems bound by a cylindrical geometry. We therefore numerically solve the modified LPB equation in cylindrical coordinates, determining the spatial dependencies of electric potential, relative permittivity and average orientations of water dipoles within charged tubes of different radii. Results show that for tubes of a large radius, macroscopic (net) volume charge density of coions and counterions is zero at the geometrical axis. This is attributed to effective electrolyte charge screening in the vicinity of the inner charged surface of the tube. For tubes of small radii, the screening region extends into the whole inner space of the tube, leading to non-zero net volume charge density and non-zero orientational ordering of water dipoles near the axis.

Rodlike Counterions near Charged Cylinders: Counterion Condensation and Intercylinder Interaction

We study a system composed of like-charged cylinders and dumbbell-like counterions, with the focus laid on the role of the internal structure of counterions, using Monte Carlo simulations. The dumbbell ions are found to exhibit novel counterion condensation behavior governed by their length. Effective electrostatic interactions mediated between charged parallel cylinders also turn out significantly different from the case of pointlike ions, as a result of the complex interplay between the spatially separated charge distribution in the dumbbell counterions, their orientation, and the curvature of the charged cylinder. We show that at a weak-to-moderate electrostatic coupling strength, where effective like-charge interactions are usually found to be repulsive, the intercylinder interaction can become attractive and display a distinctive sensitivity to the cylinder curvature and dumbbell size, proving the significant effect of ion structure.

Multiionic effects on the capacitance of porous electrodes

The rapid and reversible ionic electrosorption in the electrical double layers (EDLs) of moderately charged micropores in contact with a solution is the main concept underlying capacitive energy and desalination devices. For the usual operating conditions, the ion concentration is large enough for the confinement of ions to play an important role in their distribution in the EDL. On the other hand, although most laboratory experiments have been carried out with simple salt solutions, realistic applications require a proper analysis of the effect of the different ionic species existing in natural waters. Here we focus on the role of multiionic solutions on the double layer structure. For this purpose, a model is presented in which the EDL overlap and the existence of a Stern layer are considered. It is also taken into account that the ions can be tightly packed by using the Carnahan-Starling model. This model is applied to analyze the structure of the EDL with multiionic solutions containing divalent ions. The predictions of this model are found to largely differ from those of the better known Bikerman equation, and are more realistic. It is demonstrated that the presence of tiny amounts of divalent ions in the bulk is enough to dominate the EDL behavior, and hence, its capacitance, energy storage, and desalination properties.

Counter-ion distribution around flexible polyelectrolytes having different molecular architecture

We explore the monovalent counter-ion distribution around flexible highly-charged polyelectrolytes with different molecular architectures (linear chains, stars, and unknotted and trefoil rings) using molecular dynamics simulations that include an explicit solvent that interacts with the polyelectrolyte. In particular, we find that the molecular topology influences the fraction of counter-ions transiently associating with the polyelectrolyte on a scale of the order of the chain segments, forming a "condensed" counter-ion interfacial layer. As with the hydrogen bonding of water to proteins and other polymers, the persistence time of these interfacial "bound" counter-ions is relatively short, O(1 ps), and we characterize the fluctuations in the number of the counter-ions populating the interfacial layer. We also find that the counter-ions are distributed in a non-uniform fashion on the polyelectrolyte backbone, forming dynamical clusters whose form and average size is sensitive to molecular architecture. In addition, we find that the residual bound counter-ions, not located in either the interfacial layer or the bulk solution, form a diffuse ionic cloud around the polyelectrolyte due to the uncompensated polyelectrolyte charge along the backbone. Generally charge valence strongly influences the extent of the diffuse counter-ion cloud, but in the case of monovalent counter-ions, we find that the size of the diffuse counter-ion cloud nearly coincides with the polyelectrolyte radius of gyration, independent of molecular topology.

An electric double layer of colloidal particles in salt-free concentrated suspensions including non-uniform size effects and orientational ordering of water dipoles

Non-uniform size effects and orientational ordering of water dipoles influence the relative permittivity and electric potential in suspension.

Celebrating Soft Matter's 10th Anniversary: Influencing the charge of poly(methyl methacrylate) latexes in nonpolar solvents

Sterically-stabilized poly(methyl methacrylate) (PMMA) latexes dispersed in nonpolar solvents are a classic, well-studied system in colloid science. This is because they can easily be synthesized with a narrow size distribution and because they interact essentially as hard spheres. These PMMA latexes can be charged using several methods (by adding surfactants, incorporating ionizable groups, or dispersing in autoionizable solvents), and due to the low relative permittivity of the solvents (εr ≈ 2 for alkanes to εr ≈ 8 for halogenated solvents), the charges have long-range interactions. The number of studies of these PMMA particles as charged species has increased over the past ten years, after few studies immediately following their discovery. A large number of variations in both the physical and chemical properties of the system (size, concentration, surfactant type, or solvent, as a few examples) have been studied by many groups. By considering the literature on these particles as a whole, it is possible to determine the variables that have an effect on the charge of particles. An understanding of the process of charge formation will add to understanding how to control charge in nonaqueous solvents as well as make it possible to develop improved technologically relevant applications for charged polymer nanoparticles.

Charge Renormalization for Ellipsoidal Macroions

We study the problem of counterion condensation for ellipsoidal macroions, a geometry well-suited for modeling liquid crystals, anisotropic vesicles, and polymers. We find that the ions within an ellipsoid's condensation layer are relatively unrestricted in their motions, and consequently work to establish a quasi-equipotential at its surface. This simplifies the application of Alexander et al.'s procedure, enabling us to obtain accurate analytic estimates for the critical valence of a general ellipsoid in the weak screening limit. Interestingly, we find that the critical valence of an eccentric ellipsoid is always larger than that of the sphere of equal volume, implying that counterion condensation provides a force resisting the deformation of spherical macroions. This contrasts with a recent study of flexible spherical macroions, which observed a preference for deformation into flattened shapes when considering only linear effects. Our work suggests that the balance of these competing forces might alter the nature of the transition.

Coulomb energy of uniformly charged spheroidal shell systems

We provide exact expressions for the electrostatic energy of uniformly charged prolate and oblate spheroidal shells. We find that uniformly charged prolate spheroids of eccentricity greater than 0.9 have lower Coulomb energy than a sphere of the same area. For the volume-constrained case, we find that a sphere has the highest Coulomb energy among all spheroidal shells. Further, we derive the change in the Coulomb energy of a uniformly charged shell due to small, area-conserving perturbations on the spherical shape. Our perturbation calculations show that buckling-type deformations on a sphere can lower the Coulomb energy. Finally, we consider the possibility of counterion condensation on the spheroidal shell surface. We employ a Manning-Oosawa two-state model approximation to evaluate the renormalized charge and analyze the behavior of the equilibrium free energy as a function of the shell's aspect ratio for both area-constrained and volume-constrained cases. Counterion condensation is seen to favor the formation of spheroidal structures over a sphere of equal area for high values of shell volume fractions.

DNA under Force: Mechanics, Electrostatics, and Hydration

Quantifying the basic intra- and inter-molecular forces of DNA has helped us to better understand and further predict the behavior of DNA. Single molecule technique elucidates the mechanics of DNA under applied external forces, sometimes under extreme forces. On the other hand, ensemble studies of DNA molecular force allow us to extend our understanding of DNA molecules under other forces such as electrostatic and hydration forces. Using a variety of techniques, we can have a comprehensive understanding of DNA molecular forces, which is crucial in unraveling the complex DNA functions in living cells as well as in designing a system that utilizes the unique properties of DNA in nanotechnology.

Temperature effects on energy production by salinity exchange

In recent years, the capacitance of the interface between charged electrodes and ionic solutions (the electric double layer) has been investigated as a source of clean energy. Charge is placed on the electrodes either by means of ion-exchange membranes or of an external power source. In the latter method, net energy is produced by simple solution exchange in open circuit, due to the associated decrease in the capacitance of the electric double layer. In this work, we consider the change in capacitance associated with temperature variations: the former decreases when temperature is raised, and, hence, a cycle is possible in which some charge is put on the electrode at a certain potential and returned at a higher one. We demonstrate experimentally that it is thus viable to obtain energy from electric double layers if these are successively contacted with water at different temperatures. In addition, we show theoretically and experimentally that temperature and salinity variations can be conveniently combined to maximize the electrode potential increase. The resulting available energy is also estimated.

Electrostatics-driven shape transitions in soft shells

Manipulating the shape of nanoscale objects in a controllable fashion is at the heart of designing materials that act as building blocks for self-assembly or serve as targeted drug delivery carriers. Inducing shape deformations by controlling external parameters is also an important way of designing biomimetic membranes. In this paper, we demonstrate that electrostatics can be used as a tool to manipulate the shape of soft, closed membranes by tuning environmental conditions such as the electrolyte concentration in the medium. Using a molecular dynamics-based simulated annealing procedure, we investigate charged elastic shells that do not exchange material with their environment, such as elastic membranes formed in emulsions or synthetic nanocontainers. We find that by decreasing the salt concentration or increasing the total charge on the shell's surface, the spherical symmetry is broken, leading to the formation of ellipsoids, discs, and bowls. Shape changes are accompanied by a significant lowering of the electrostatic energy and a rise in the surface area of the shell. To substantiate our simulation findings, we show analytically that a uniformly charged disc has a lower Coulomb energy than a sphere of the same volume. Further, we test the robustness of our results by including the effects of charge renormalization in the analysis of the shape transitions and find the latter to be feasible for a wide range of shell volume fractions.

Effect of Solution Composition on the Energy Production by Capacitive Mixing in Membrane-Electrode Assembly

In this work, we consider the extent to which the presence of multivalent ions in solution modifies the equilibrium and dynamics of the energy production in a capacitive cell built with ion-exchange membranes in contact with high surface area electrodes. The cell potential in open circuit (OCV) is controlled by the difference between both membrane potentials, simulated as constant volume charge regions. A theoretical model is elaborated for steady state OCV, first in the case of monovalent solutions, as a reference. This is compared to the results in multi-ionic systems, containing divalent cations in concentrations similar to those in real seawater. It is found that the OCV is reduced by about 25% (as compared to the results in pure NaCl solutions) due to the presence of the divalent ions, even in low concentrations. Interestingly, this can be related to the "uphill" transport of such ions against their concentration gradients. On the contrary, their effect on the dynamics of the cell potential is negligible in the case of highly charged membranes. The comparison between model predictions and experimental results shows a very satisfactory agreement, and gives clues for the practical application of these recently introduced energy production methods.

Counterion condensation on spheres in the salt-free limit

A highly-charged spherical colloid in a salt-free environment exerts such a powerful attraction on its counterions that a certain fraction condenses onto the surface of a particle. The degree of condensation depends on the curvature of the surface. So, for instance, condensation is triggered on a highly-charged sphere only if the radius exceeds a certain critical radius R*. R* is expected to be a simple function of the volume fraction of particles. To test these predictions, we prepare spherical particles which contain a covalently-bound ionic liquid, which is engineered to dissociate efficiently in a low-dielectric medium. By varying the proportion of ionic liquid to monomer we synthesise nonpolar dispersions of highly-charged spheres which contain essentially no free co-ions. The only ions in the system are counterions generated by the dissociation of surface-bound groups. We study the electrophoretic mobility of this salt-free system as a function of the colloid volume fraction, the particle radius, and the bare charge density and find evidence for extensive counterion condensation. At low electric fields, we observe excellent agreement with Poisson-Boltzmann predictions for counterion condensation on spheres. At high electric fields however, where ion advection is dominant, the electrophoretic mobility is enhanced significantly which we attribute to hydrodynamic stripping of the condensed layer of counterions from the surface of the particle.

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.

Role of adsorbed surfactant in the reaction of aryl diazonium salts with single-walled carbon nanotubes

Because covalent chemistry can diminish the optical and electronic properties of single-walled carbon nanotubes (SWCNTs), there is significant interest in developing methods of controllably functionalizing the nanotube sidewall. To date, most attempts at obtaining such control have focused on reaction stoichiometry or strength of oxidative treatment. Here, we examine the role of surfactants in the chemical modification of single-walled carbon nanotubes with aryl diazonium salts. The adsorbed surfactant layer is shown to affect the diazonium derivatization of carbon nanotubes in several ways, including electrostatic attraction or repulsion, steric exclusion, and direct chemical modification of the diazonium reactant. Electrostatic effects are most pronounced in the cases of anionic sodium dodecyl sulfate and cationic cetyltrimethylammonium bromide, where differences in surfactant charge can significantly affect the ability of the diazonium ion to access the SWCNT surface. For bile salt surfactants, with the exception of sodium cholate, we find that the surfactant wraps tightly enough such that exclusion effects are dominant. Here, sodium taurocholate exhibits almost no reactivity under the explored reaction conditions, while for sodium deoxycholate and sodium taurodeoxycholate, we show that the greatest extent of reaction is observed among a small population of nanotube species, with diameters between 0.88 and 0.92 nm. The anomalous reaction of nanotubes in this diameter range seems to imply that the surfactant is less effective at coating these species, resulting in a reduced surface coverage on the nanotube. Contrary to the other bile salts studied, sodium cholate enables high selectivity toward metallic species and small band gap semiconductors, which is attributed to surfactant-diazonium coupling to form highly reactive diazoesters. Further, it is found that the rigidity of anionic surfactants can significantly influence the ability of the surfactant layer to stabilize the diazonium ion near the nanotube surface. Such Coulombic and surfactant packing effects offer promise toward employing surfactants to controllably functionalize carbon nanotubes.

Steric-effect induced alterations in streaming potential and energy transfer efficiency of non-newtonian fluids in narrow confinements

In this work, we explore the possibilities of utilizing the combined consequences of interfacial electrokinetics and rheology toward augmenting the energy transfer efficiencies in narrow fluidic confinements. In particular, we consider the exploitation of steric effects (i.e., effect of finite size of the ionic species) in non-Newtonian fluids over small scales, to report dramatic augmentations in the streaming potential, for shear-thickening fluids. We first derive an expression for the streaming potential considering strong electrical double layer interactions in the confined flow passage and the consequences of the finite conductance of the Stern layer, going beyond the Debye-Hückel limit. With a detailed accounting for the excluded volume effects of the ionic species and their interaction with pertinent interfacial phenomena of special type of rheological fluids such as the power law fluids in the above-mentioned formalism, we demonstrate that a confluence of the steric interactions with the non-Newtonian transport characteristics may result in giant augmentations in the energy transfer efficiency for shear-thickening fluids under appropriate conditions.

Distinct conformations of DNA-stabilized fluorescent silver nanoclusters revealed by electrophoretic mobility and diffusivity measurements

Silver-DNA nanoclusters (Ag:DNAs) are novel fluorophores under active research and development as alternative biomolecular markers. Comprised of a few-atom Ag cluster that is stabilized in water by binding to a strand of DNA, they are also interesting for fundamental explorations into the properties of metal molecules. Here, we use in situ calibrated electrokinetic microfluidics and fluorescence correlation spectroscopy to determine the size, charge, and conformation of a select set of Ag:DNAs. Among them is a pair of spectrally distinct Ag:DNAs stabilized by the same DNA sequence, for which it is known that the silver cluster differs by two atoms. We find these two Ag:DNAs differ in size by ∼30%, even though their molecular weights differ by less than 3%. Thus a single DNA sequence can adopt very different conformations when binding slightly different Ag clusters. By comparing spectrally identical Ag:DNAs that differ in sequence, we show that the more compact conformation is insensitive to the native DNA secondary structure. These results demonstrate electrokinetic microfluidics as a practical tool for characterizing Ag:DNA.

Towards an understanding of induced-charge electrokinetics at large applied voltages in concentrated solutions

The venerable theory of electrokinetic phenomena rests on the hypothesis of a dilute solution of point-like ions in quasi-equilibrium with a weakly charged surface, whose potential relative to the bulk is of order the thermal voltage (kT/e approximately 25 mV at room temperature). In nonlinear electrokinetic phenomena, such as AC or induced-charge electro-osmosis (ACEO, ICEO) and induced-charge electrophoresis (ICEP), several V approximately 100 kT/e are applied to polarizable surfaces in microscopic geometries, and the resulting electric fields and induced surface charges are large enough to violate the assumptions of the classical theory. In this article, we review the experimental and theoretical literatures, highlight discrepancies between theory and experiment, introduce possible modifications of the theory, and analyze their consequences. We argue that, in response to a large applied voltage, the "compact layer" and "shear plane" effectively advance into the liquid, due to the crowding of counterions. Using simple continuum models, we predict two general trends at large voltages: (i) ionic crowding against a blocking surface expands the diffuse double layer and thus decreases its differential capacitance, and (ii) a charge-induced viscosity increase near the surface reduces the electro-osmotic mobility; each trend is enhanced by dielectric saturation. The first effect is able to predict high-frequency flow reversal in ACEO pumps, while the second may explain the decay of ICEO flow with increasing salt concentration. Through several colloidal examples, such as ICEP of an uncharged metal sphere in an asymmetric electrolyte, we show that nonlinear electrokinetic phenomena are generally ion-specific. Similar theoretical issues arise in nanofluidics (due to confinement) and ionic liquids (due to the lack of solvent), so the paper concludes with a general framework of modified electrokinetic equations for finite-sized ions.

Electrical double layer around a spherical colloid particle: The excluded volume effect

The influence of the excluded volume effect on both the spatial distribution of ionic species and the electrostatic potential distribution in the neighborhood of a suspended spherical particle is examined on the basis of a modified Poisson-Boltzmann equation, which takes into account the finite ion size by means of a Langmuir-type correction. We find that kappaa (kappa and a being the reciprocal Debye length and the particle radius, respectively) ceases to be a valid parameter for the characterization of the electrical double layer, and that it is necessary to use both parameters kappa and a to characterize adequately the system. We also find that the excluded volume effect considerably increases the surface potential (for a given value of the surface charge density) as compared to the case when ideal ion behavior is assumed. This suggests the use of the particle charge rather than the surface potential in order to characterize the system. Because of this, an approximate equation for the surface charge density of spherical colloid particles, valid for a wide range of system parameter values, is also reported.

Surface conservation laws at microscopically diffuse interfaces

In studies of interfaces with dynamic chemical composition, bulk and interfacial quantities are often coupled via surface conservation laws of excess surface quantities. While this approach is easily justified for microscopically sharp interfaces, its applicability in the context of microscopically diffuse interfaces is less theoretically well-established. Furthermore, surface conservation laws (and interfacial models in general) are often derived phenomenologically rather than systematically. In this article, we first provide a mathematically rigorous justification for surface conservation laws at diffuse interfaces based on an asymptotic analysis of transport processes in the boundary layer and derive general formulae for the surface and normal fluxes that appear in surface conservation laws. Next, we use nonequilibrium thermodynamics to formulate surface conservation laws in terms of chemical potentials and provide a method for systematically deriving the structure of the interfacial layer. Finally, we derive surface conservation laws for a few examples from diffusive and electrochemical transport.

Steric effects in the dynamics of electrolytes at large applied voltages. II. Modified Poisson-Nernst-Planck equations

In situations involving large potentials or surface charges, the Poisson-Boltzman (PB) equation has shortcomings because it neglects ion-ion interactions and steric effects. This has been widely recognized by the electrochemistry community, leading to the development of various alternative models resulting in different sets "modified PB equations," which have had at least qualitative success in predicting equilibrium ion distributions. On the other hand, the literature is scarce in terms of descriptions of concentration dynamics in these regimes. Here, adapting strategies developed to modify the PB equation, we propose a simple modification of the widely used Poisson-Nernst-Planck (PNP) equations for ionic transport, which at least qualitatively accounts for steric effects. We analyze numerical solutions of these modified PNP equations on the model problem of the charging of a simple electrolyte cell, and compare the outcome to that of the standard PNP equations. Finally, we repeat the asymptotic analysis of Bazant, Thornton, and Ajdari [Phys. Rev. E 70, 021506 (2004)] for this new system of equations to further document the interest and limits of validity of the simpler equivalent electrical circuit models introduced in Part I [Kilic, Bazant, and Ajdari, Phys. Rev. E 75, 021502 (2007)] for such problems.

Steric effects in the dynamics of electrolytes at large applied voltages. I. Double-layer charging

The classical Poisson-Boltzmann (PB) theory of electrolytes assumes a dilute solution of point charges with mean-field electrostatic forces. Even for very dilute solutions, however, it predicts absurdly large ion concentrations (exceeding close packing) for surface potentials of only a few tenths of a volt, which are often exceeded, e.g., in microfluidic pumps and electrochemical sensors. Since the 1950s, several modifications of the PB equation have been proposed to account for the finite size of ions in equilibrium, but in this two-part series, we consider steric effects on diffuse charge dynamics (in the absence of electro-osmotic flow). In this first part, we review the literature and analyze two simple models for the charging of a thin double layer, which must form a condensed layer of close-packed ions near the surface at high voltage. A surprising prediction is that the differential capacitance typically varies nonmonotonically with the applied voltage, and thus so does the response time of an electrolytic system. In PB theory, the differential capacitance blows up exponentially with voltage, but steric effects actually cause it to decrease while remaining positive above a threshold voltage where ions become crowded near the surface. Other nonlinear effects in PB theory are also strongly suppressed by steric effects: The net salt adsorption by the double layers in response to the applied voltage is greatly reduced, and so is the tangential "surface conduction" in the diffuse layer, to the point that it can often be neglected compared to bulk conduction (small Dukhin number).

Scaling and universality in the counterion-condensation transition at charged cylinders

Counterions at charged rodlike polymers exhibit a condensation transition at a critical temperature (or, equivalently, at a critical linear charge density for polymers), which dramatically influences various static and dynamic properties of charged polymer solutions. We address the critical and universal aspects of this transition for counterions at a single charged cylinder in two and three spatial dimensions using numerical and analytical methods. By introducing a Monte Carlo sampling method in logarithmic radial scale, we are able to numerically simulate the critical limit of infinite system size (corresponding to the infinite-dilution limit) within tractable equilibration times. The critical exponents are determined for the inverse moments of the counterionic density profile (which play the role of the order parameters and represent the mean inverse localization length of counterions) both within mean-field theory and within Monte Carlo simulations. In three dimensions (3D), we demonstrate that correlation effects (neglected within mean-field theory) lead to an excessive accumulation of counterions near the charged cylinder below the critical temperature (i.e., in the condensation phase), while surprisingly, the critical region exhibits universal critical exponents in accordance with mean-field theory. Also in contrast with the typical trend in bulk critical phenomena, where fluctuations become more enhanced in lower dimensions, we demonstrate, using both numerical and analytical approaches, that mean-field theory becomes exact for the two-dimensional (2D) counterion-cylinder system at all temperatures (Manning parameters), when the number of counterions tends to infinity. For a finite number of particles, however, the 2D problem displays a series of peculiar singular points (with diverging heat capacity), which reflect successive delocalization events of individual counterions from the central cylinder. In both 2D and 3D, the heat capacity shows a universal jump at the critical point and the internal energy develops a pronounced peak. The asymptotic behavior of the energy peak location is used to determine the critical temperature, which is also found to be in agreement with the mean-field prediction.

Measuring inter-DNA potentials in solution

Interactions between short strands of DNA can be tuned from repulsive to attractive by varying solution conditions and have been quantified using small angle x-ray scattering techniques. The effective DNA interaction charge was extracted by fitting the scattering profiles with the generalized one-component method and inter-DNA Yukawa pair potentials. A significant charge is measured at low to moderate monovalent counterion concentrations, resulting in strong inter-DNA repulsion. The charge and repulsion diminish rapidly upon the addition of divalent counterions. An intriguing short range attraction is observed at surprisingly low divalent cation concentrations, approximately 16 mM Mg2+. Quantitative measurements of inter-DNA potentials are essential for improving models of fundamental interactions in biological systems.

Excluded volume driven counterion condensation inside nanotubes in a concave electrical double layer model

The physical properties of organic nanotubes attract increasing attention due to their potential benefit in technology, biology and medicine. We study the effect of ion size on the electrical properties of cylindrical nanotubes filled with electrolyte solution within a modified Poisson-Boltzmann (PB) approach. For comparison purposes, small hollow nanospheres filled with electrolyte solution are considered. The finite size of the particles in the inner electrolyte solution is described by the excluded volume effect within a lattice statistics approach. We found that an increased ion size reduces the number of counterions near the charged inner surface of the nanotube, leading to an enlarged electrostatic surface potential. The concentration of counterions close to the inner surface saturates for higher surface charge densities and larger ions. In the case of saturation, the closest counterion packing is achieved, all lattice sites near the surface are occupied and an actual counterion condensation is observed. By contrast, the counterion concentration at the axis of the nanotube steadily increases with increasing surface charge density. This growth is more pronounced for smaller nanotube radii and larger ions. At larger nanotube radii for small ion size counterion condensation may also be observed according to the Tsao criterion, i.e. the counterion concentration at the centre is independent of the number of counterions in the system. With decreasing radius the Tsao condensation effect is shifted towards physiologically unrealistic surface charge densities.

Incorporation of excluded-volume correlations into Poisson-Boltzmann theory

We investigate the effect of excluded-volume interactions on the electrolyte distribution around a charged macroion. First, we introduce a criterion for determining when hard-core effects should be taken into account beyond standard mean-field Poisson-Boltzmann (PB) theory. Next, we demonstrate that several commonly proposed local-density-functional approaches for excluded-volume interactions cannot be used for this purpose. Instead, we employ a nonlocal excess free energy by using a simple constant-weight approach. We compare the ion distribution and osmotic pressure predicted by this theory with Monte Carlo simulations. They agree very well for weakly developed correlations and give the correct layering effect for stronger ones. In all investigated cases our simple weighted-density theory yields more realistic results than the standard PB approach, whereas all local density theories do not improve on the PB density profiles, but on the contrary, deviate even more from the simulation results.