Gouy-Chapman Research Articles

Double-Layer Capacitances Caused by Ion–Solvent Interaction in the Form of Langmuir-Typed Concentration Dependence

Variations of the double layer capacitances (DLCs) at a platinum electrode with concentrations and kinds of salts in aqueous solutions were examined in the context of facilitating orientation of solvent dipoles. With an increase in ionic concentrations, the DLCs increased by ca. a half and then kept constant at concentrations over 1 mol dm−3. This increase was classically explained in terms of the Gouy–Chapman (GC) equation combined with the Stern model. Unfortunately, measured DLCs were neither satisfied with the Stern model nor the GC theory. Our model suggests that salts destroy hydrogen bonds at the electrode–solution interface to orient water dipoles toward the external electric field. A degree of the orientation depends on the interaction energy between the salt ion and a water dipole. The statistical mechanic calculation allowed us to derive an equation for the DLC as a function of salt concentration and the interaction energy. The equation took the Langmuir-type in the relation with the concentration. The interaction energy was obtained for eight kinds of salts. The energy showed a linear relation with the interaction energy of ion–solvent for viscosity, called the B-coefficient.

Internal configuration and electric potential in planar negatively charged lipid head group region in contact with ionic solution

The lipid bilayer composed of negatively charged lipid 1-palmitoyl-3-oleoyl-sn-glycero-3-phosphatidylserine (POPS) in contact with an aqueous solution of monovalent salt ions was studied theoretically by using the mean-field modified Langevin–Poisson–Boltzmann (MLPB) model. The MLPB results were tested by using molecular dynamic (MD) simulations. In the MLPB model the charge distribution of POPS head groups is theoretically described by the negatively charged surface which accounts for negatively charged phosphate groups, while the positively charged amino groups and negatively charged carboxylate groups are assumed to be fixed on the rod-like structures with rotational degree of freedom.The spatial variation of relative permittivity, which is not considered in the well-known Gouy–Chapman (GC) model or in MD simulations, is thoroughly derived within a strict statistical mechanical approach. Therefore, the spatial dependence and magnitude of electric potential within the lipid head group region and its close vicinity are considerably different in the MLPB model from the GC model.The influence of the bulk salt concentration and temperature on the number density profiles of counter-ions and co-ions in the lipid head group region and aqueous solution along with the probability density function for the lipid head group orientation angle was compared and found to be in qualitative agreement in the MLPB and MD models.

Overlapping of the Diffuse Double Layer in the Vicinity of the Microporous Electrode

Introduction A microporous electrode with nm-sized micropores on the surface is an attractive subject in electrochemistry. Some experimental results show the physicochemical properties of microporous electrodes are not explained only in terms of the surface area. We have an interest in the electrical double layer in the narrow space. When the size of the micropore on the electrode is smaller than the diffuse part of the electrical double layer, the diffuse double layer show overlap across the pore. This overlapping of the diffuse double layer cannot be estimated on the one dimension model, such as Gouy–Chapman (GC) theory for planar electrodes. We propose a model of the diffuse double layers in the vicinity of micropores on the basis of the Poisson-Boltzmann equation without any complicated assumptions. This calculation shows essential features of the diffuse double layer on the electrode with micropores. The size dependence, the permittivity dependence, the potential dependence, and the electrolyte concentration dependence of the diffuse double layer and also the differential capacitance of the electrode with micropores are discussed.Model The model is described with a cylindrical coordinate. The electrode was assumed as a conductor with an equipotential surface. The diameter of a pore on the electrode was varied from 0.2 to 5 nm. To avoid singular points in the calculation, the radius of the corners was assumed to be 50 pm in the geometry of this model. The solution phase contains a 1:1 electrolyte. In the solution phase, the Poisson–Boltzmann equation was applied to obtain the profile of the electric potential. In this calculation, the relative permittivity was set to be from 10 to 78. The nonlinear Poisson–Boltzmann equation was numerically solved by the finite element method. The differential capacitance of the double layer was estimated from the dependence of the total charge on the electrode surface on the surface potential.Results and discussion The calculated potential profile showed the overlapping of the diffuse double layer in the micropore. The electric field in the micropore is weaker than that outside of the micropore, that is, the overlapped diffuse double layer provides the mild electric field. Assuming that the thickness of the diffuse double layer is the division of the surface potential by the gradient of the potential at the electrode surface, the degree of the overlapping can express qualitatively. When the size of the micropore on the electrode is smaller than the thickness of the diffuse double layer, the calculated capacitance agrees with the GC capacitance at the planner electrode. On the other hand, when the size of the micropore on the electrode is larger than the thickness of the diffuse double layer, the double layer capacitance converges on the GC capacitance at the microporous electrode. This means that the surface area of micropore becomes effective only at thinner diffuse double layers. The change of the thickness of the diffuse double layer in the micropore causes the strong dependence of the capacitance on the potential and the electrolyte concentration compared with the situation of planar electrode. The characteristic features of microporous electrodes would be originated from the heterogeneous electric field caused by the overlapping of the diffuse double layer.

Generalization of the Gouy-Chapman-Stern model of an electric double layer for a morphologically complex electrode: Deterministic and stochastic morphologies

We generalize the linearized Gouy-Chapman-Stern theory of an electric double layer for morphologically complex and disordered electrodes. An equation for capacitance is obtained using a linear Gouy-Chapman or Debye-Hückel equation for the potential near the complex-geometry electrode-electrolyte interface. The effect of the surface morphology of an electrode on an electric double layer is obtained using multiple scattering formalism in surface curvature. The result for capacitance is expressed in terms of the ratio of Gouy screening length to the local principal radii of curvatures of the surface. We also include a contribution of a compact layer, which is significant in the overall prediction of capacitance. Our general results are analyzed in detail for two special morphologies of electrodes, i.e., a nanoporous membrane and a forest of nanopillars. Variations of local shapes and global size variations due to residual randomness in morphology are accounted for as curvature fluctuations over a reference shape element. In particular, the theory shows that the presence of geometrical fluctuations in porous systems causes an enhanced dependence of capacitance on mean pore sizes and suppresses the magnitude of capacitance. This theory is further extended to include contributions to capacitance from adsorption of ions and electrode material due to electronic screening. Our predictions are in reasonable agreement with recent experimental measurements on supercapacitive microporous and mesoporous systems.

Ionic Size Effect on the Double Layer Properties: A Modified Poisson-Boltzmann Theory

On the basis of a simple modified Poisson-Boltzmann (SMPB) theory, taking into account the finite ionic size, the analytic expression for the effect of ionic size on the diffuse layer potential drop at negative charge densities has been given for the simple 1:1 electrolyte. It is shown that the potential drop across the diffuse layer depends on the size of the ions in the electrolyte. For a given electrolyte concentration and electrode charge density, the diffuse layer potential drop in a small ion system is smaller than that in a large ion system. It is also displayed that the diffuse layer potential drop is always less than the value of the Gouy-Chapman (GC) theory, and the deviation increases as the electrode charge density increases for a given electrolyte concentration. These theoretical results are consistent with the results of the Monte-Carlo simulation [Fawcett and Smagala, Electrochimica Acta 53, 5136 (2008)], which indicates the importance of including steric effects in modeling diffuse layer properties.

Monte Carlo studies of ion size effects in the diffuse double layer

Results from Monte Carlo (MC) simulations for restricted electrolytes are presented for both 1:1 and 2:1 electrolytes. It is shown that the potential drop across the diffuse layer for these systems may be expressed by a Taylors series in the Gouy-Chapman (GC) estimate of the same quantity. The coefficients of this series are defined in terms of the MSA volume fraction and the reciprocal thickness of the ionic atmosphere. The series model can also be used to estimate the differential capacity of the diffuse layer. The properties of unrestricted electrolytes are considered at or very close to the potential of zero charge.

Screening and Separation of Charges in Microscale Devices: Complete Planar Solution of the Poisson−Boltzmann Equation

The Poisson-Boltzmann (PB) equation is widely used to calculate the interaction between electric potential and the distribution of charged species. In the case of a symmetrical electrolyte in planar geometry, the Gouy-Chapman (GC) solution is generally presented as the analytical solution of the PB equation. However, we demonstrate here that this GC solution assumes the presence of a bulk region with zero electric field, which is not justified in microdevices. In order to extend the range of validity, we obtain here the complete numerical solution of the planar PB equation, supported with analytical approximations. For low applied voltages, it agrees with the GC solution. Here, the electric double layers fully absorb the applied voltage such that a region appears where the electric field is screened. For higher voltages (of order 1 V in microdevices), the solution of the PB equation shows a dramatically different behavior, in that the double layers can no longer absorb the complete applied voltage. Instead, a finite field remains throughout the device that leads to complete separation of the charged species. In this higher voltage regime, the double layer characteristics are no longer described by the usual Debye parameter kappa, and the ion concentration at the electrodes is intrinsically bound (even without assuming steric interactions). In addition, we have performed measurements of the electrode polarization current on a nonaqueous model electrolyte inside a microdevice. The experimental results are fully consistent with our calculations, for the complete concentration and voltage range of interest.

Test of the Gouy-Chapman Theory for a Charged Lipid Membrane against Explicit-Solvent Molecular Dynamics Simulations

A wealth of experimental data has verified the applicability of the Gouy-Chapman (GC) theory to charged lipid membranes. Surprisingly, a validation of GC by molecular dynamics (MD) simulations has been elusive. Here, we report a test of GC against extensive MD simulations of an anionic lipid bilayer solvated by water at different concentrations of NaCl or KCl. We demonstrate that the ion distributions from the simulations agree remarkably well with GC predictions when information on the adsorption of counterions to the bilayer is incorporated.

Ion Size and Image Effect on Electrokinetic Flows

Electrokinetic phenomena play a major role in microfluidic systems, and such a role becomes even more significant in nanofluidic systems due to the increase of the surface-to-volume ratio. Description of the electric double layer (EDL) at a solid-liquid interface is the key to understand and utilize electrokinetic phenomena. However, the traditional Gouy-Chapman (GC) theory for the EDL, which has been successfully used in many microfluidic applications, does not include some important characteristics such as ion size and image effect. These characteristics are indeed important in nanofluidics. This paper explores the impacts of ion size and the image effect on micro- and nanoscale electrokinetic flows. An advanced theory, the modified Poisson-Boltzmann (MPB) theory proposed by Outhwaite and his co-workers,1,26 is adopted to describe the EDL. Electrokinetic flows in micro- and nanochannels are reinvestigated. The results show that ion size has significant effects on electrokinetic flows in nanosystems in terms of both the flow field and the streaming potential, while the image effect only significantly affects the streaming potential.

Some simulation and modified Poisson–Boltzmann theory results for the contact values of an electrolyte near a charged electrode

Abstract Since the development of a contact value theorem by Henderson et al. it has been known that the prediction of the venerable Gouy–Chapman (GC) theory for the contact value of the density profile of an electrolyte near an electrode is incorrect. The more recent semi-empirical work of Fawcett and Henderson and the simulations of Boda and Henderson have shown that the contact value of the charge profile at low electrode charge is also given incorrectly by the GC theory, at least for a binary symmetric restricted primitive model electrolyte. In the present study we examine the contact values of the profiles of the double layer formed by the electrolyte species in a binary symmetric restricted primitive model electrolyte as a function of the electrode charge by means of simulation. It is difficult to extract detailed information from simulation values of the density and charge profiles at large electrode charge because the contact values of these profiles are dominated by the large quadratic term in the electrode charge. However, we find from our simulations for a fairly wide range of concentrations and electrode charges, that the product of the counterion and coion profiles is a sensitive probe with which to study the double layer because it is a direct test of the basis of the GC approximation that the counterion and coion profiles are exponentials of ± zeψ ( x )/ kT and so have a product that is identically one. We find that this product is not one or even a constant as the electrode charge is varied and, thus, the GC contact values do not agree even qualitatively with the simulation results. In contrast, the modified Poisson–Boltzmann theory is quite successful in accounting for the behavior of this product.

Breakdown of the Gouy−Chapman Model for Highly Charged Langmuir Monolayers: Counterion Size Effect

Deviations from the classic Gouy-Chapman (GC) model due to the finite size of hydrated counterions were tested for negatively charged Langmuir monolayers with different surface charge densities. Monolayers with the largest charge density (>0.6 C.m(-2)) show an increase of the surface potential for a series of alkali metal cations from Li(+) to Cs(+) by 200-250 mV. The increase is similar for different monolayers and suggests that this effect is independent of the particular type of headgroup. The magnitude of variation is comparable with model estimations of the electrical double layer (EDL) potential implying that the deviation from the GC model is drastic. Deviations from the GC model rapidly vanish with decreasing monolayer charge density and become hardly observable below 0.3 C.m(-2). For monolayers with a high charge density on subphases containing different sized counterions, preferential participation of the smallest ions in the EDL should be favorable in terms of electrostatic free energy because of packing density limitations. This effect was demonstrated for behenyl sulfate (BS) monolayers (0.64 C.m(-2)) with the X-ray reflectivity technique. For the Cs(+)-Li(+) system, the fraction of Cs(+) in the EDL is 50-60% compared with only 10% of Cs(+) in the subphase. Providing high surface charge density, a small univalent Cs(+) is capable to compete even with a bulky divalent Mg(2+). For equal concentrations of Cs(+) and Mg(2+) in the subphase, the Cs(+)/Mg(2+) ratio in EDL of BS monolayer is 1.3 to 2.0 (in contrast to 0.04, predicted by the GC model). All experimental results of this study are described in terms of packing density limitations for hydrated counterions in the EDL.

The effect of the discrete nature of ions on the excess adsorption of Na + and Cl − at the silica–aqueous interface

Wet pastes of silica, prepared by centrifugation of dispersions (with a salt concentration of 100 mM NaCl) that subsequently were decanted the supernatant, were analysed using X-ray photoelectron spectroscopy. The uptake of Na and Cl ions by the wet pastes were quantified and compared with predictions based on the Gouy–Chapman (GC) concept, the triple layer model (TLM), and a modified TLM, which takes into account the discrete ion effect. The most striking feature of the results is that the Cl uptake increases by a factor of ∼2 with increasing surface counter charge, which has been explained on the basis of the discrete nature of ions.

Behavior of mixtures of symmetric and asymmetric electrolytes near discretely charged planar surfaces: A Monte Carlo study

Canonical Monte Carlo (CMC) simulations are employed in this work in order to study the structure of the electrical double layer (EDL) near discretely charged planar surfaces in the presence of symmetric and asymmetric indifferent electrolytes within the framework of a primitive model. The effects of discreteness and strength of surface charge, charge asymmetry, and size asymmetry are specific focuses of this work. The CMC simulation protocol is initially tested against the classical theory, the modified Gouy-Chapman (GC) theory, in order to assess the reliability of the simulation results. The CMC simulation results and the predictions of the classical theory show good agreement for 1:1 electrolytes and low surface charge, at which conditions the GC theory is valid. Simulations with symmetric and asymmetric electrolytes and mixtures of the two demonstrate that size plays an important role in determining the species present in the EDL and how the surface charge is screened. A size-exclusion effect could be consistently detected. Although it is energetically favorable that higher-valence ions screen the surface charge, their larger size prevents them from getting close to the surface. Smaller ions with lower valences perform the screening of the charge, resulting in higher local concentrations of small ions close to the surface. The simulations also showed that the strength of the surface charge enhances the size-exclusion effect. This effect will definitely affect the magnitude of the forces between interacting charged surfaces.

Interaction between dissimilar double layers with like signs under charge regulation on the basis of the Gouy–Chapman–Stern–Grahame model

The force of interaction between two flat double layers with dissimilar surface potentials but of the same signs, in contact with 1-1 electrolyte solutions (0.001, 0.01, and 0.1 M), was calculated as a function of the surface separation. The calculation was carried out on the basis of the Gouy-Chapman-Stern-Grahame double-layer model, incorporating the site-dissociation and site-binding model used previously for many oxide studies. The results demonstrated that the interaction force under surface charge regulation was always repulsive, increasing progressively with decreasing surface separation. This result is in contrast to the predictions of previous theories that are based on the Gouy-Chapman (GC) double-layer model. Dependence of surface potentials and surface charge densities on the surface separation is also presented.

Monte Carlo, density functional theory, and Poisson–Boltzmann theory study of the structure of an electrolyte near an electrode

Monte Carlo (MC) and density functional theory (DFT) results are reported for an electrolyte, consisting of charged hard spheres of diameter 3 Å with the solvent modeled as a dielectric continuum, near a charged flat uniformly charged electrode. These results are more interesting than the earlier MC results of Torrie and Valleau [J. Chem. Phys. 73, 5807 (1980); J. Phys. Chem. 86, 3251 (1982)] for 4.25 Å spheres because the popular Gouy–Chapman (GC) theory is less successful for this system. The DFT results are in good agreement with the MC results. Both the MC and DFT results show particularly interesting features when the counterions are divalent. For such divalent counterions, the diffuse layer potential passes through a maximum magnitude, then declines, and ultimately has a sign that is opposite to that of the electrode charge. The consequences of this behavior are discussed. In contrast, the well-known GC theory consistently overestimates the magnitude of the diffuse layer potential, does not have any unusual behavior, and is in poor agreement with the simulation results.

Simulation and theory of the electrochemical double layer for strong ionic interactions

Abstract In order to obtain agreement with experiment, the Gouy-Chapman (GC) theory of the electrochemical interface must be parametrized empirically. Surprisingly, this difficulty does not seem to have detracted from the popularity of the GC theory. However, the use of empirical parameters precludes a definitive experimental test of the GC theory. Very recently, computer simulations have been employed to examine the electrochemical interface. If the ionic coupling is not to strong (large ions, low valence, large dielectric constant, high temperature), the GC theory, although lacking, is qualitatively satisfactory. However, for strongly coupled ions the GC theory fails completely. Regrettably, so do the ‘better’ theories. Recently, a theory that is partially satisfactory has been developed. Our simulations, this theory, and what is needed for a more satisfactory theory is discussed.

Using an effective surface charge to explain surface potentials of Langmuir monolayers from dialkyldimethylammonium halides with the Gouy–Chapman theory

The use of an effective surface charge density has allowed the Gouy–Chapman (GC) theory to explain surface potential isotherms of Langmuir monolayers of dioctadecyldimethylammonium bromide (DODAB). The effective surface charge density of DODAB monolayer increases with the electronegativity of the counterions in the subphase. The pressure–area isotherms indicate a very condensed monolayer for DODAB spread on an I −-containing subphase, which exhibits the lowest surface charge density, whereas the monolayer on a F −-containing subphase is extremely expanded owing to the high surface charge density or electrostatic repulsion between headgroups.

A thermodynamic test of the diffuse layer theory and its implications on electrode kinetics

AbstractThe thermodynamic compatibility of the diffuse layer theory with the chemical potential of ionic species in the solution bulk is introduced. From the electrocapillary equation, a compatibility equation, which relates the potential drop across the diffuse layer to the chemical potential of the ionic species in the solution bulk, is deduced for a binary electrolyte. When the Gouy–Chapman (GC) theory is introduced in the compatibility equation, the latter only becomes an identity if the ionic activity is taken as the ionic concentration, as expected. A compatible theory of the diffuse layer which has GC as the limit case at dilute solutions and fulfills the compatibility equation for whatever physical description of the ionic activity coefficients is stated. The significance of the compatible theory is that the electrode reaction rates calculated by means of the electrode kinetic equation with the Frumkin correction using this theory differ from those calculated using GC. In order to test both theories, we have studied the electroreduction of peroxodisulfate at an amalgamated copper rotating disk electrode. The experimental results fit our compatible theory better than GC does.

Ionic Transport in Compacted Bentonite: Preliminary Equilibrium Results

ABSTRACTAn attempt is made to decompose some quantities, most commonly integrated over the chemical composition, the structure of the pore network and the time scale, into their differential counterparts. Into this category typically fall, say, the apparent diffusivities of ionic species. To serve this objective, a two-dimensional unsteady molecular-level electrokinetic transport model for ionic species in bentonite clays will be developed. The model incorporates additional features to the conventional Gouy-Chapman (GC) theory: ionic hydration, dielectric saturation, and the volume exclusion of ions. The governing equations for the flow of electrolyte solution through the pores are solved by an iterative numerical scheme to relate the characteristics of the flow to the characteristics of the pores and to the composition of the external solution in contact with the clay. The pore geometry of the clay is modelled as an array of non-interconnected tortuous channels with no parallel or serial-type non-uniformities along the pathway. This roughly corresponds to the picture of clay particles of infinite extent aligned in parallel and spaced apart by a constant distance. The model aims to simulate and interpret equilibrium and transport experiments for bentonite clays containing different types of background electrolytes at various compactions. Specifically, emphasis is placed on quantifying the extent of co-ion exclusion and understanding the postulated surface diffusion mechanism on the basis of the well-established electric double-layer (EDL) theory. This contribution presents and discusses some preliminary results, based on the modified Boltzmann statistics, for the equilibrating part of the model.

Hydration of polar interfaces. A generalised mean-field model

A generalized non-local electrostatic model has been proposed for the description of the hydration of arbitrary polar surfaces, including interfaces with a finite thickness. This model is suitable for the description of complex biological surfaces, such as the surfaces of lipid bilayer membranes. It is designed so as to resemble as closely as possible the Gouy–chapman diffuse double-layer theory, for the sake of the simplicity of model use. The molecular meaning of the model parameters is discussed and the chief determinants of the surface hydration are identified. The concept of simple solvent polarization is shown to be less suitable for the description of the interfacial hydration than the local excess-charge density approach, which corresponds to a generalized, submolecular polarization. A general scheme for the calculation of the hydration between two thick, structured interfaces has been developed. The effects of surface structure on the hydration-dependent interfacial repulsion have been investigated. The magnitude and the range of the hydration pressure are shown to increase dramatically as a consequence of the water penetration and binding into the interfacial region. Interfacial swelling and dynamics, consequently, may affect the properties of, and the interactions between, structured surfaces.