Liquid-liquid interfacial tension of electrolyte solutions

Frequency and field-dependent response of confined electrolytes from Brownian dynamics simulations

Using Brownian dynamics simulations, we investigate the effects of confinement, adsorption on surfaces, and ion-ion interactions on the response of confined electrolyte solutions to oscillating electric fields in the direction perpendicular to the confining walls. Nonequilibrium simulations allows to characterize the transitions between linear and nonlinear regimes when varying the magnitude and frequency of the applied field, but the linear response, characterized by the frequency-dependent conductivity, is more efficiently predicted from the equilibrium current fluctuations. To that end, we (rederive and) use the Green-Kubo relation appropriate for overdamped dynamics, which differs from the standard one for Newtonian or underdamped Langevin dynamics. This expression highlights the contributions of the underlying Brownian fluctuations and of the interactions of the particles between them and with external potentials. Although already known in the literature, this relation has rarely been used to date, beyond the static limit to determine the effective diffusion coefficient or the DC conductivity. The frequency-dependent conductivity always decays from a bulk-like behavior at high frequency to a vanishing conductivity at low frequency due to the confinement of the charge carriers by the walls. We discuss the characteristic features of the crossover between the two regimes, most importantly how the crossover frequency depends on the confining distance and the salt concentration, and the fact that adsorption on the walls may lead to significant changes both at high and low frequencies. Conversely, our results illustrate the possibility to obtain information on diffusion between walls, charge relaxation, and adsorption by analyzing the frequency-dependent conductivity.

Partial osmotic pressures of ions in electrolyte solutions and the Gibbs-Guggenheim uncertainty principle

The concept of the partial osmotic pressure of ions in an electrolyte solution is critically examined. In principle these can be defined by introducing a solvent-permeable wall and measuring the force per unit area which can certainly be attributed to individual ions. Here I demonstrate that although the total wall force balances the bulk osmotic pressure as required by mechanical equilibrium, the individual partial osmotic pressures are extrathermodynamic quantities dependent on the electrical structure at the wall, and as such they resemble attempts to define individual ion activity coefficients. The limiting case where the wall is a barrier to only one species of ion is also considered, and with ions on both sides the classic Gibbs-Donnan membrane equilibrium is recovered, thus providing a unifying treatment. The analysis can be extended to illustrate how the electrical state of the bulk is affected by the nature of the walls and the container handling history, thus supporting the "Gibbs-Guggenheim uncertainty principle," namely the notion that the electrical state is unmeasurable and usually accidentally determined. Since this uncertainty is conferred also onto individual ion activities, it has implications for the current (2002) IUPAC definition of pH.

Surface Tension of Dielectric-Air Interfaces

Analytical and semianalytical expressions for the surface tension of dielectric-air interfaces are presented after considering local and nonlocal dielectric effects near interfaces. It is shown that the nonlocal effects of dielectrics are significant for highly polar dielectric fluids such as water. Far from the interface, nonlocal dielectric effects are shown to cause not only the oscillatory potential of the mean force but also a reversal of sign at intermediate distances.

Surface tension in liquids containing antagonistic ions

We use a modified Poisson-Boltzmann formalism to examine immiscible electrolytes containing dissolved antagonistic ions with arbitrary preferential solubilities. We solve the nonlinear equation and obtain an analytical expression for the potential profile, ion densities, and surface tension. Our model takes into account the dependence of the Debye lengths in the two liquids on the preferential solvation. In the limit of point-like ions, the surface tension scales with the average ion density n0 as the classical n01/2 power law. At larger densities ion crowding at the interface leads to a crossover to smaller exponents. The dependence of the surface tension on the Gibbs transfer energy is non-monotonic and exhibits a maximum or a minimum. We further look at a liquid bilayer confined in a parallel-plate capacitor and subject to an external potential. The ion distribution depends on whether the external potential has the same or opposite sign as the Donnan potential. Lastly, we calculate the surface tension of a liquid-liquid interface that has a sinusoidal height modulation and find that surface modulations increase the energy.

Nanoscale Resolution of Electric-field Induced Motion in Ionic Diblock Copolymer Thin Films

Understanding the responses of ionic block copolymers to applied electric fields is crucial when targeting applications in areas such as energy storage, microelectronics, and transducers. This work shows that the identity of counterions in ionic diblock copolymers substantially affects their responses to electric fields, demonstrating the importance of ionic species for materials design. In situ neutron reflectometry measurements revealed that thin films containing imidazolium based cationic diblock copolymers, tetrafluoroborate counteranions led to film contraction under applied electric fields, while bromide counteranions drove expansion under similar field strengths. Coarse-grained molecular dynamics simulations were used to develop a fundamental understanding of these responses, uncovering a nonmonotonic trend in thickness change as a function of field strength. This behavior was attributed to elastic responses of microphase separated diblock copolymer chains resulting from variations in interfacial tension of polymer-polymer interfaces due to the redistribution of counteranions in the presence of electric fields.

Charged Surface-Active Impurities at Nanomolar Concentration Induce Jones-Ray Effect

The electrolyte surface tension exhibits a characteristic minimum around a salt concentration of 1 mM for all ion types, known as the Jones-Ray effect. We show that a consistent description of the experimental surface tension of salts, bases, and acids is possible by assuming charged impurities in the water with a surface affinity typical for surfactants. Comparison with experimental data yields an impurity concentration in the nanomolar range, well below the typical experimental detection limit. Our modeling reveals salt-screening enhanced impurity adsorption as the mechanism behind the Jones-Ray effect: for very low salt concentration added salt screens the  electrostatic repulsion between impurities at the surface, which dramatically increases impurity adsorption and thereby reduces the surface tension.

Colloid–oil–water-interface interactions in the presence of multiple salts: charge regulation and dynamics

The salt-induced dislodgement of charged colloidal particles from an oil–water interface is investigated theoretically and experimentally.

Tuning Colloid-Interface Interactions by Salt Partitioning

We show that the interaction of an oil-dispersed colloidal particle with an oil-water interface is highly tunable from attractive to repulsive, either by varying the sign of the colloidal charge via charge regulation or by varying the difference in hydrophilicity between the dissolved cations and anions. In addition, we investigate the yet unexplored interplay between the self-regulated colloidal surface charge distribution with the planar double layer across the oil-water interface and the spherical one around the colloid. Our findings explain recent experiments and have direct relevance for tunable Pickering emulsions.

Inhomogeneous screening near the dielectric interface

Screening is one of the most important concepts in the study of charged systems. Near a dielectric interface, the ion distribution in a salt solution can be highly nonuniform. Here, we develop a theory that self-consistently treats the inhomogeneous screening effects. At higher concentrations when the bulk Debye screening length is comparable to the Bjerrum length, the double layer structure and interfacial properties are significantly affected by the inhomogeneous screening. In particular, the depletion zone is considerably wider than that predicted by the bulk screening approximation or the WKB approximation. The characteristic length of the depletion layer in this regime scales with the Bjerrum length, resulting in a linear increase of the negative adsorption of ions with concentration, in agreement with experiments. For asymmetric salts, inhomogeneous screening leads to enhanced charge separation and surface potential.

Anomalous system-size dependence of electrolytic cells with an electrified oil-water interface

Manipulation of the charge of the dielectric interface between two bulk liquids not only enables the adjustment of the interfacial tension but also controls the storage capacity of ions in the ionic double layers adjacent to each side of the interface. However, adjusting this interfacial charge by static external electric fields is difficult since the external electric fields are readily screened by ionic double layers that form in the vicinity of the external electrodes. This leaves the liquid-liquid interface, which is at a macroscopic distance from the electrodes, unaffected. In this study we show theoretically, in agreement with recent experiments, that control over this surface charge at the liquid-liquid interface is nonetheless possible for macroscopically large but finite closed systems in equilibrium, even when the distance between the electrode and interface is orders of magnitude larger than the Debye screening lengths of the two liquids. We identify a crossover system-size below which the interface and the electrodes are effectively coupled. Our calculations of the interfacial tension for various electrode potentials are in good agreement with recent experimental data.

Dielectric discontinuity in equilibrium block copolymer micelles

The surface tension between the hydrophobic core and the solvent is known to play a major role in the self-assembly of diblock copolymer micelles in solution. Coulombic forces are also very important in the case of aggregates with weakly charged coronas. The aggregation number and morphology are often tuned by the addition of electrolytes to the solution via electrostatic screening and an eventual change in solvent quality. However, when the surface tension is low enough, dielectric discontinuity between the core and the solvent becomes important enough in comparison to other mechanisms, driving the surface tension at the same time it screens electrostatic interactions in the corona. Below, we demonstrate the importance of this effect for micelles with neutral and weakly charged coronas.

Ion specific effects: decoupling ion-ion and ion-water interactions

Ion-specific effects in aqueous solution, known as the Hofmeister effect, are prevalent in diverse systems ranging from pure ionic to complex protein solutions. The objective of this paper is to explicitly demonstrate how complex ion-ion and ion-water interactions manifest themselves in the Hofmeister effect based on a series of recent experimental observations. These effects are not considered in the classical descriptions of ion effects, such as the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, and therefore they fail to describe the origin of the phenomenological Hofmeister effect. However, given that models considering the basic forces of electrostatic and van der Waals interactions can offer rationalization for the core experimental observations, a universal interaction model stands a chance of being developed. In this perspective, we separately derive the contribution from ion-ion electrostatic interactions and ion-water interactions from second harmonic generation (SHG) data at the air-ion solution interface, which yields an estimate of the ion-water interactions in solution. The Hofmeister ion effect observed for biological solutes in solution should be similarly influenced by contributions from ion-ion and ion-water interactions, where the same ion-water interaction parameters derived from SHG data at the air-ion solution interface could be applicable. A key experimental data set available from solution systems to probe ion-water interactions is the modulation of water diffusion dynamics near ions in a bulk ion solution, as well as near biological liposome surfaces. This is obtained from Overhauser dynamic nuclear polarization (ODNP), a nuclear magnetic resonance (NMR) relaxometry technique. The surface water diffusivity is influenced by the contribution from ion-water interactions, both from localized surface charges and adsorbed ions, although the relative contribution of the former is larger on liposome surfaces. In this perspective, ion-water interaction energy values derived from experimental data for various ions are compared with theoretical values in the literature. Ultimately, quantifying ion-induced changes in the surface energy for the purpose of developing valid theoretical models for ion-water interactions will be critical to rationalizing the Hofmeister effect.

On the theoretical description of weakly charged surfaces

It is widely accepted that the Poisson-Boltzmann (PB) theory provides a valid description for charged surfaces in the so-called weak coupling limit. Here, we show that the image charge repulsion creates a depletion boundary layer that cannot be captured by a regular perturbation approach. The correct weak-coupling theory must include the self-energy of the ion due to the image charge interaction. The image force qualitatively alters the double layer structure and properties, and gives rise to many non-PB effects, such as nonmonotonic dependence of the surface energy on concentration and charge inversion. In the presence of dielectric discontinuity, there is no limiting condition for which the PB theory is valid.

Are antagonistic salts surfactants?

It is well known that surfactants decrease both water/air and water/oil interfacial tensions whereas in contrast inorganic salts increase both. We study a new, third class of surface-active ionic solutes, which have been called antagonistic salts, consisting of an organic group with a small inorganic counterion. These show decreased interfacial tension at the oil/water interface due to a redistribution of the organic group in the oil but do not show any surface activity at the air/water interface and are consequently different from surfactants that lower both tensions. We use a simple modeling using Poisson-Boltzmann theory that accounts for the surface activity of the antagonistic salt at the water/oil interface.

Adsorption of ions at the interface oil|aqueous electrolyte and at interfaces with adsorbed alcohol

We investigate the applicability of the Schmutzer's model for three types of interfaces: aqueous electrolyte|alkane, aqueous electrolyte|long chained alcohol phase, and aqueous electrolyte|alkane with adsorbed alcohol. The model predicts a strong decrease of the electrolyte desorption at water|alcohol interface in comparison with water|alkane, in quantitative agreement with the tensiometric data. The effect is related to the penetration of the alcohol -OH group into the surface layer of the aqueous solution. The same model predicts a decrease with the electrolyte concentration of the chemical potential of the alcohol molecules adsorbed at the water|oil interface, which results in an increase of the adsorption activity of oil-soluble alcohols, again in quantitative agreement with the experiment. The analysis of the deviations of Schmutzer's model predictions from the experimental data for large polarizable anions allows making qualitative conclusions for the magnitude of the ion-specific hydrophobic and dispersion forces at water|gas and water|oil interfaces.

Continuous self-energy of ions at the dielectric interface

By treating both the short-range (solvation) and long-range (image force) electrostatic forces as well as charge polarization induced by these forces in a consistent manner, we obtain a simple theory for the self-energy of an ion that is continuous across the interface. Along with nonelectrostatic contributions, our theory enables a unified description of ions on both sides of the interface. Using intrinsic parameters of the ions, we predict the specific ion effect on the interfacial affinity of halogen anions at the water-air interface, and the strong adsorption of hydrophobic ions at the water-oil interface, in agreement with experiments and atomistic simulations.

Tuning ion correlations at an electrified soft interface

Ion distributions play a central role in various settings-from biology, where they mediate the electrostatic interactions between charged biomolecules in solution, to energy storage devices, where they influence the charging properties of supercapacitors. These distributions are determined by interactions dictated by the chemical properties of the ions and their environment as well as the long-range nature of the electrostatic force. Recent theoretical and computational studies have explored the role of correlations between ions, which have been suggested to underlie a number of counterintuitive results, such as like-charge attraction. However, the interdependency between ion correlations and other interactions that ions experience in solution complicates the connection between physical models of ion correlations and the experimental investigation of ion distributions. We exploit the properties of the liquid/liquid interface to vary the coupling strength of ion-ion correlations from weak to strong while monitoring their influence on ion distributions at the nanometer scale with X-ray reflectivity and the macroscopic scale with interfacial tension measurements. These data are in agreement with the predictions of a parameter-free density functional theory that includes ion-ion correlations and ion-solvent interactions over the entire range of experimentally tunable correlation coupling strengths (from 0.8 to 3.7). This study provides evidence for a sharply defined electrical double layer for large coupling strengths in contrast to the diffuse distributions predicted by mean field theory, thereby confirming a common prediction of many ion correlation models. The reported findings represent a significant advance in elucidating the nature and role of ion correlations in charged soft matter.

Simulation of charged systems in heterogeneous dielectric media via a true energy functional

For charged systems in heterogeneous dielectric media, a key obstacle for molecular dynamics (MD) simulations is the need to solve the Poisson equation in the media. This obstacle can be bypassed using MD methods that treat the local polarization charge density as a dynamic variable, but such approaches require access to a true free energy functional, one that evaluates to the equilibrium electrostatic energy at its minimum. In this Letter, we derive the needed functional. As an application, we develop a Car-Parrinello MD method for the simulation of free charges present near a spherical emulsion droplet separating two immiscible liquids with different dielectric constants. Our results show the presence of nonmonotonic ionic profiles in the dielectric with a lower dielectric constant.

Influence of specific anions on the orientational ordering of thermotropic liquid crystals at aqueous interfaces

We report that specific anions (of sodium salts) added to aqueous phases at molar concentrations can trigger rapid, orientational ordering transitions in water-immiscible, thermotropic liquid crystals (LCs; e.g., nematic phase of 4'-pentyl-4-cyanobiphenyl, 5CB) contacting the aqueous phases. Anions classified as chaotropic, specifically iodide, perchlorate, and thiocyanate, cause 5CB to undergo continuous, concentration-dependent transitions from planar to homeotropic (perpendicular) orientations at LC-aqueous interfaces within 20 s of addition of the anions. In contrast, anions classified as relatively more kosmotropic in nature (fluoride, sulfate, phosphate, acetate, chloride, nitrate, bromide, and chlorate) do not perturb the LC orientation from that observed without added salts (i.e., planar orientation). Surface pressure-area isotherms of Langmuir films of 5CB supported on aqueous salt solutions reveal ion-specific effects ranking in a manner similar to the LC ordering transitions. Specifically, chaotropic salts stabilized monolayers of 5CB to higher surface pressures and areal densities (12.6 mN/m at 27 Å(2)/molecule for NaClO(4)) and thus smaller molecular tilt angles (30° from the surface normal for NaClO(4)) than kosmotropic salts (5.0 mN/m at 38 Å(2)/molecule with a corresponding tilt angle of 53° for NaCl). These results and others reported herein suggest that anion-specific interactions with 5CB monolayers lead to bulk LC ordering transitions. Support for the proposition that these ion-specific interactions involve the nitrile group was obtained by using a second LC with nitrile groups (E7; ion-specific effects similar to 5CB were observed) and a third LC with fluorine-substituted aromatic groups (TL205; weak dipole and no ion-specific effects were measured). Finally, we also establish that anion-induced orientational transitions in micrometer-thick LC films involve a change in the easy axis of the LC. Overall, these results provide new insights into ionic phenomena occurring at LC-aqueous interfaces, and reveal that the long-range ordering of LC oils can amplify ion-specific interactions at these interfaces into macroscopic ordering transitions.

Solvation effects in phase transitions in soft matter

Phase transitions in polar binary mixtures can be drastically altered by even a small amount of salt. This is because the preferential solvation strongly depends on the ambient composition. Together with a summary of our research on this problem, we present some detailed results on the role of antagonistic salt composed of hydrophilic and hydrophobic ions. These ions tend to segregate at liquid-liquid interfaces and selectively couple to water-rich and oil-rich composition fluctuations, leading to mesophase formation. In our two-dimensional simulation, the coarsening of the domain structures can be stopped or slowed down, depending on the interaction parameter (or the temperature) and the salt density. We realize stripe patterns at the critical composition and droplet patterns at off-critical compositions. In the latter case, charged droplets emerge with considerable size dispersity in a percolated region. We also give the structure factors among the ions, accounting for the Coulomb interaction and the solvation interaction mediated by the composition fluctuations.

Concentration and size separation of DNA samples at liquid-liquid interfaces

This report introduces a new analytical concept utilizing the mass transfer resistance of a liquid-liquid interface to concentrate and separate DNA samples. DNA molecules can be electrophoretically accumulated at a liquid-liquid interface of an aqueous two-phase system (ATPS) of poly(ethylene glycol) (PEG) and dextran, two polymers that form two immiscible phases in aqueous electrolyte solutions. The detachment of DNA from the interface into the other phase can be triggered by increasing the applied electric field. We experimentally study the size dependence of the detachment process for a broad spectrum of DNA fragments. In a regime where the coiling of the chains does not play a significant role, the process shows a linear dependence on the diffusion coefficient, with shorter DNA chains detaching at lower electric field strengths than larger ones. The concept may enable novel separation protocols for preparative and analytical purposes.

Precipitation in aqueous mixtures with addition of a strongly hydrophilic or hydrophobic solute

We examine phase separation in aqueous mixtures due to preferential solvation with a low-density solute (hydrophilic ions or hydrophobic particles). For hydrophilic ions, preferential solvation can stabilize water domains enriched with ions. This precipitation occurs above a critical solute density n(p) in wide ranges of the temperature and the average composition, where the mixture solvent would be in a one-phase state without solute. The volume fraction of precipitated domains tends to zero as the average solute density n is decreased to np or as the interaction parameter χ is decreased to a critical value χ(p). If we start with one-phase states with n>n(p) or χ>χ(p), precipitation proceeds via homogeneous nucleation or via heterogeneous nucleation, for example, around suspended colloids. In the latter case, colloid particles are wrapped by thick wetting layers. We also predict a first-order prewetting transition for n or χ slightly below np or χ(p) for neutral colloids.

Fluctuation in electrolyte solutions: the self energy

We address the issue of the self energy of the mobile ions in electrolyte solutions within a general Gaussian renormalized fluctuation theory using a field-theoretic approach. We introduce the Born radii of the ions in the form of a charge distribution allowing for different Born radii between the cations and anions. The model thus automatically yields a theory free of divergences and accounts for the solvation of the ions at the level of continuous dielectric media. In an inhomogeneous dielectric medium, the self energy is in general position dependent and differences in the self energy between cations and anions can give rise to local charge separation in a macroscopically neutral system. Treating the Born radius a as a smallness parameter, we show that the self energy can be split into an O(a(-1)) nonuniversal contribution and an O(a0) universal contribution that depends only on the ion concentration, valency, and the spatially varying dielectric constant. For a weakly inhomogeneous dielectric medium, the nonuniversal part of the self energy is shown to have the form of the Born energy with the local dielectric constant. This self energy is incorporated into the Poisson-Boltzmann equation as a simple means of including this local fluctuation effect in a mean-field theory. We illustrate the phenomenon of charge separation by considering cations and anions of difference sizes and valencies in a periodic dielectric medium.

Ions at the air-water interface: an end to a hundred-year-old mystery?

Availability of highly reactive halogen ions at the surface of aerosols has tremendous implications for the atmospheric chemistry. Yet neither simulations, experiments, nor existing theories are able to provide a fully consistent description of the electrolyte-air interface. In this Letter a new theory is proposed which allows us to explicitly calculate the ionic density profiles, the surface tension, and the electrostatic potential difference across the solution-air interface. Predictions of the theory are compared to experiments and are found to be in excellent agreement. The theory also sheds new light on one of the oldest puzzles of physical chemistry--the Hofmeister effect.

Inflation of the screening length induced by Bjerrum pairs

Within a modified Poisson-Boltzmann theory we study the effect of Bjerrum pairs on the typical length scale [Formula: see text] over which electric fields are screened in electrolyte solutions, taking into account a simple association-dissociation equilibrium between free ions and Bjerrum pairs. At low densities of Bjerrum pairs, this length scale is well approximated by the Debye length [Formula: see text], with ρ(s) the free-ion density. At high densities of Bjerrum pairs, however, we find [Formula: see text], which is significantly larger than 1/κ due to the enhanced effective permittivity of the electrolyte, caused by the polarization of Bjerrum pairs. We argue that this mechanism may explain the recently observed anomalously large colloid-free zones between an oil-dispersed colloidal crystal and a colloidal monolayer at the oil-water interface.

Dynamics of binary mixtures with ions: dynamic structure factor and mesophase formation

Dynamic equations are presented for polar binary mixtures containing ions in the presence of preferential solvation. In one-phase states, we calculate the dynamic structure factor of the composition accounting for ion motion. Microphase separation can take place for sufficiently large solvation asymmetry of the cations and the anions. We show two-dimensional simulation results of the mesophase formation with an antagonistic salt, where the cations are hydrophilic and the anions are hydrophobic. The structure factor S(q) in the resultant mesophase has a sharp peak at an intermediate wavenumber of the order of the Debye-Hückel wavenumber. As the quench depth is increased, the surface tension nearly vanishes in mesophases due to an electric double layer.

Magnetic stabilization and vorticity in submillimeter paramagnetic liquid tubes

It is possible to suppress convection and dispersion of a paramagnetic liquid by means of a magnetic field. A tube of paramagnetic liquid can be stabilized in water along a ferromagnetic track in a vertical magnetic field, but not in a horizontal field. Conversely, an "antitube" of water can be stabilized in a paramagnetic liquid along the same track in a transverse horizontal field, but not in a vertical field. The stability arises from the interaction of the induced moment in the solution with the magnetic field gradient in the vicinity of the track. The magnetic force causes the tube of paramagnetic liquid to behave as if it were encased by an elastic membrane whose cross-section is modified by gravitational forces and Maxwell stress. Convection from the tube to its surroundings is inhibited, but not diffusion. Liquid motion within the paramagnetic tube, however, exhibits vorticity in tubes of diameter 1 mm or less--conditions where classical pipe flow would be perfectly streamline, and mixing extremely slow. The liquid tube is found to slide along the track almost without friction. Paramagnetic liquid tubes and antitubes offer appealing new prospects for mass transport, microfluidics, and electrodeposition.

Polarizable ions at interfaces

A nonperturbative theory is presented which allows us to calculate the solvation free energy of polarizable ions near water-vapor and water-oil interfaces. The theory predicts that larger halogen anions are adsorbed at the interface, while the alkali metal cations are repelled from it. The density profiles calculated theoretically are similar to those obtained using molecular dynamics simulations with polarizable force fields.

Curvature dependence of the electrolytic liquid-liquid interfacial tension

The interfacial tension of a liquid droplet surrounded by another liquid in the presence of microscopic ions is studied as a function of the droplet radius. An analytical expression for the interfacial tension is obtained within a linear Poisson-Boltzmann theory and compared with numerical results from nonlinear Poisson-Boltzmann theory. The excess liquid-liquid interfacial tension with respect to the pure salt-free liquid-liquid interfacial tension is found to decompose into a curvature-independent part due to short-ranged interfacial effects and a curvature-dependent electrostatic contribution. Several curvature-dependent regimes of different scalings of the electrostatic excess interfacial tension are identified. Symmetry relations of the interfacial tension upon swapping droplet and bulk liquid are found to hold in the low-curvature limit, which, e.g., lead to a sign change of the excess Tolman length. For some systems a low-curvature expansion up to the second order turns out to be applicable if and only if the droplet size exceeds the Debye screening length in the droplet, independent of the Debye length in the bulk.