Modified Poisson-Boltzmann Theory Research Articles

Effects of electrostatic coupling and surface polarization on polyelectrolyte brush structure.

In this work, we perform molecular dynamics simulations to study a spherical polyelectrolyte brush. We explore the effects of surface polarization and electrostatic coupling on brush size and distribution of counterions. The method of image charges is considered to take into account surface polarization, considering a metallic, an unpolarizable, and a dielectric nano-core. It is observed that, for all cases, a moderate shrinking-swelling effect appears with an increase in the electrostatic coupling parameter. This effect occurs under high Manning ratios. The curves relating the average size of polyelectrolyte brush as a function of coupling show a minimum. The results show that the grafting density of polyelectrolytes on the nano-core surface plays an important role in the polarization effect. We consider a modified Poisson-Boltzmann theory to describe the counterion profiles around the brush in the case of unpolarizable nano-cores and weak electrostatic coupling.

Coarse-grained molecular dynamics simulation of cation distribution profiles on negatively charged lipid membranes during phase separation.

Revealing the ion distributions on a charged lipid membrane in aqueous solution under the influence of long-range interactions is essential for understanding the origin of the stability of the bilayer structure and the interaction between biomembranes and various electrolytes. However, the ion distributions and their dynamics associated with the phase separation process of the lipid bilayer membrane are still unclear. We perform coarse-grained molecular dynamics simulations to reveal the Na+ and Cl- distributions on charged phospholipid bilayer membranes during phase separation. During the phase separation, cations closely follow the position of negatively charged lipids on a microsecond timescale and are rapidly redistributed parallel to the lipid bilayer. In the homogenous mixture of zwitterionic and negatively charged lipids, cations weakly follow negatively charged lipids, indicating the strong interaction between cations and negatively charged lipids. We also compare cation concentrations as a function of surface charge density obtained by our simulation with those obtained by a modified Poisson-Boltzmann theory. Including the ion finite size makes the statistical results consistent, suggesting the importance of the ion-ion interactions in aqueous solution. Our simulation results advance our understanding of ion distribution during phase separation.

Influence of concentration and temperature dependent dielectric constants on the thermodynamics of electrolytes

The symmetric Poisson–Boltzmann theory, the modified Poisson–Boltzmann theory, and the mean spherical approximation are utilized to calculate the osmotic coefficient, and the individual and mean activity coefficients of model KCl, LiCl, NaCl, and NaClO4 electrolytes all dissociated in water where the dielectric constant depends on concentration. The theories are also employed to compute the same thermodynamic quantities for HCl and NaCl solutions where the dielectric constant is now temperature dependent. In each situation the theoretical predictions are compared with the corresponding exact benchmark Monte Carlo simulation results. The mean spherical approximation and the modified Poisson–Boltzmann theory reproduce the simulation data for osmotic coefficient quantitatively and that for the mean activity almost quantitatively. The individual activities are accurately represented by the modified Poisson–Boltzmann theory but only qualitatively by the symmetric Poisson–Boltzmann and the mean spherical approximation. In general, the symmetric Poisson–Boltzmann theory shows good agreement with the simulations at low concentrations but deviates at higher concentrations.

Charge asymmetric electrolytes around a rigid cylindrical polyelectrolyte: A generalization of the capacitive compactness

The surface mean electrostatic potential and capacitive compactness of the electrical double layer surrounding an infinite rigid cylindrical polyelectrolyte are analysed for equally-sized -1:z+ aqueous electrolytes. Monte Carlo simulations, the non-linear Poisson–Boltzman equation, the modified Poisson–Boltzmann theory, and the hypernetted chain/mean spherical approximation integral equation are used to examine the role of multivalent cations (coions) when the properties of monovalent anions (counterions) are fixed. A non-zero mean electrostatic potential in the neighbourhood of an uncharged polyelectrolyte is predicted by the simulations and the modified Poisson–Boltzmann theory in the presence of multivalent cations. The concept of capacitive compactness is generalised to overcome the divergence found in the classical definition, when a non-zero potential at the point of zero charge is present. With a highly electrified colloidal surface an inversion is observed, as a function of colloidal charge, in the precedence of the mean electrostatic potential near the polyelectrolyte’s surface and of the capacitive compactness between -1:+1 and -1:z+ electrolytes.

Modified Poisson-Boltzmann theory for polyelectrolytes in monovalent salt solutions with finite-size ions.

We present a soft-potential-enhanced Poisson-Boltzmann (SPB) theory to efficiently capture ion distributions and electrostatic potential around rodlike charged macromolecules. The SPB model is calibrated with a coarse-grained particle-based model for polyelectrolytes (PEs) in monovalent salt solutions as well as compared to a full atomistic molecular dynamics simulation with the explicit solvent. We demonstrate that our modification enables the SPB theory to accurately predict monovalent ion distributions around a rodlike PE in a wide range of ion and charge distribution conditions in the weak-coupling regime. These include excess salt concentrations up to 1M and ion sizes ranging from small ions, such as Na+ or Cl-, to softer and larger ions with a size comparable to the PE diameter. The work provides a simple way to implement an enhancement that effectively captures the influence of ion size and species into the PB theory in the context of PEs in aqueous salt solutions.

Consistent description of ion-specificity in bulk and at interfaces by solvent implicit simulations and mean-field theory.

Solvent-implicit Monte Carlo (MC) simulations and mean-field theory are used to predict activity coefficients and excess interfacial tensions for NaF, NaCl, NaI, KF, KCl, and KI solutions in good agreement with experimental data over the entire experimentally available concentration range. The effective ionic diameters of the solvent-implicit simulation model are obtained by fits to experimental activity coefficient data. The experimental activity coefficients at high salt concentrations are only reproduced if the ion-specific concentration-dependent decrement of the dielectric constant is included. The dielectric-constant dependent contribution of the single-ion solvation free energy to the activity coefficient is significant and is included. To account for the ion-specific excess interfacial tension of salt solutions, in addition to non-ideal solution effects and the salt-concentration-dependent dielectric decrement, an ion-specific ion-interface interaction must be included. This ion-interface interaction, which acts in addition to the dielectric image-charge repulsion, is modeled as a box potential, is considerably more long-ranged than the ion radius, and is repulsive for all ions considered except iodide, in agreement with previous findings and arguments. By comparing different models that include or exclude bulk non-ideal solution behavior, dielectric decrement effects, and ion-interface interaction potentials, we demonstrate how bulk and interfacial ion-specific effects couple and partially compensate each other. Our MC simulations, which correctly include ionic correlations and interfacial dielectric image-charge repulsion, are used to determine effective ion-surface interaction potentials that can be used in a modified Poisson-Boltzmann theory.

Features of Electrical Double Layers Formed Around Strongly Charged Nanoparticles Immersed in an Electrolyte Solution. The Effect of Ion Sizes

Using the modified Poisson–Boltzmann (PB) theory, which includes restrictions on the maximum attainable concentration of ionic species in a solution Cmax determined by their effective sizes, the distributions of electrostatic potential φ(r) and ion concentration near a spherical nanoparticle with radius a immersed in a 1 : 1 electrolyte solution have been studied under the conditions of constant surface charge density σs. For weakly charged particles, the φ(r) profiles are almost independent of Cmax and coincide with the profile obtained in terms of the classical PB model. Surface potential |φs| gradually increases with a rise in |σs|. Far from the particle, when the potential becomes lower than its thermal value $${{\varphi }_{{\text{T}}}} = \frac{{kT}}{e}$$ (k is the Boltzmann constant, T is the temperature, and e is the elementary charge), the potential decreases exponentially irrespective of the counterion sizes: $$\psi \left( r \right) = \frac{{\varphi \left( r \right)}}{{{{\varphi }_{{\text{T}}}}}} = {{\psi }_{{{\text{eff}}}}}\frac{a}{r}\exp \left( { - \kappa \left( {r - a} \right)} \right),$$ where r is the distance from the particle center and κ is the reciprocal Debye radius. According to the classical PB theory, the growth of the surface charge leads to the saturation of $$\left| {{{\psi }_{{{\text{eff}}}}}} \right|~\left( { \to 4} \right),$$ with the value of $$\left| {{{\psi }_{{\text{s}}}}} \right|$$ obtained by solving the nonlinear PB equation being higher than $$\left| {{{\psi }_{{{\text{eff}}}}}} \right|.$$ In the modified PB theory, which takes into account the size of electrolyte ions in the simplest form, this effect of saturation is absent. Now, $$\left| {{{\psi }_{{{\text{eff}}}}}} \right|$$ depends on both the value of the surface charge and the sizes of counterions. Moreover, at a large size of counterions, $$\left| {{{\psi }_{{{\text{eff}}}}}} \right|$$ substantially exceeds the corresponding value obtained by solving the nonlinear modified PB equation. The difference between the electrical double layer properties obtained by solving the classical and modified PB equations directly follows from the fact that the modified theory predicts the appearance of a condensed layer at a particle surface, with the concentration of counterions in this layer being equal to Cmax. Therewith, the thickness of the layer grows with increasing |σs| (at a constant size of the ions) and the size of the ions (at constant σs).

An Electrical Double Layer between Spherical Particles. The Effect of Ion Sizes

Using the modified Poisson–Boltzmann theory, which includes restrictions on the maximum attainable concentration of ionic species in the solution Cmax, determined by their effective dimensions, the spatial distributions of the electric potential and ions between spherical nanoparticles immersed in a 1 : 1 electrolyte solution have been investigated. It has been found that the pattern of the aforementioned distributions is governed by surface electric potential ψs of the particles, interparticle distance, and maximum concentration Cmax. Between weakly charged particles, the electric potential profiles are almost independent of the value of Cmax. As absolute value $$\left| {{{\psi }_{{\text{s}}}}} \right|$$ increases, an effect associated with the ionic sizes manifests itself; i.e., for high $$\left| {{{\psi }_{{\text{s}}}}} \right|$$ values, the electrical double layer expands into the solution, and, the lower Cmax (the larger the effective size of counterions), the higher the degree of expansion. This behavior is related to the fact that, near a particle surface, the profile of counterion concentration is developed with the following distinctly pronounced regions: the region of a plateau with width lc, which corresponds to the condensed part of the profile with C(x) = Cmax for 0 lc. When highly charged particles approach to each other up to such distances that their electrical double layers appear to be strongly overlapped, coions are completely displaced from the interparticle electrolyte layer, and only counterions are present in the layer.

Osmotic pressure between arbitrarily charged planar surfaces: A revisited approach.

The properties of ionic solutions between charged surfaces are often studied within the Poisson-Boltzmann framework, by finding the electrostatic potential profile. For example, the osmotic pressure between two charged planar surfaces can be evaluated by solving coupled equations for the electrostatic potential and osmotic pressure. Such a solution relies on symmetry arguments and is restricted to either equally or oppositely charged surfaces. Here, we provide a different and more efficient scheme to derive the osmotic pressure straightforwardly, without the need to find the electrostatic potential profile. We derive analytical expressions for the osmotic pressure in terms of the inter-surface separation, salt concentration, and arbitrary boundary conditions. Such results should be useful in force measurement setups, where the force is measured between two differently prepared surfaces, or between two surfaces held at a fixed potential difference. The proposed method can be systematically used for generalized Poisson-Boltzmann theories in planar geometries, as is demonstrated for the sterically modified Poisson-Boltzmann theory.

Comparison of zeta potentials and structure for statistical mechanical theories of a model cylindrical double layer

The zeta potential and the structure of a model cylindrical double layer obtained through the Poisson-Boltzmann theory, the hypernetted chain/mean spherical approximation integral equation theory, the modified Poisson-Boltzmann theory, and the density functional theory are compared with the corresponding Monte Carlo simulation results. The cylindrical double layer consists of an infinitely long cylinder with a uniform surface charge immersed in a restricted primitive model electrolyte (equi-sized, rigid spherical ions in a continuum dielectric). Calculations have been performed at room temperature and in a water-like solvent for 1:1, 2:2, 2:1/1:2 valency electrolytes for different electrolyte concentrations, axial charge parameters (surface charge densities), and ionic diameters. The results for the zeta potential, the mean electrostatic potential, and the electrode-ion singlet distributions predicted by the formal statistical mechanical theories reproduce the simulation data to a high level of accuracy overall for the range of physical parameters studied. The theoretical predictions, except that due to the classical mean field theory, also reveal a remarkable consistency among themselves. The results also (a) explore the relationship between the occurrence of charge reversal and the negative derivative of the mean electrostatic potential,(b) provide insight into the connection between reversed and extremal values of the zeta potential versus surfaced charge density curves and some aspects of electrokinetics, viz., inverted electrophoretic velocities and the possible existence of differently charged colloids with the same mobility, and (c) suggest a plausible identification of the integrated surface charge density at a point near the cylindrical polyion surface with the electrokinetic charge.

An analysis of the fluctuation potential in the modified Poisson-Boltzmann theory for restricted primitive model electrolytes

An approximate analytical solution to the fluctuation potential problem in the modified Poisson-Boltzmann theory of electrolyte solutions in the restricted primitive model is presented. The solution is valid for all inter-ionic distances, including contact values. The fluctuation potential solution is implemented in the theory to describe the structure of the electrolyte in terms of the radial distribution functions, and to calculate some aspects of thermodynamics, viz., configurational reduced energies, and osmotic coefficients. The calculations have been made for symmetric valence 1:1 systems at the physical parameters of ionic diameter $4.25 \times 10^{-10}$ m, relative permittivity 78.5, absolute temperature 298 K, and molar concentrations 0.1038, 0.425, 1.00, and 1.968. Radial distribution functions are compared with the corresponding results from the symmetric Poisson-Boltzmann, and the conventional and modified Poisson-Boltzmann theories. Comparisons have also been done for the contact values of the radial distributions, reduced configurational energies, and osmotic coefficients as functions of electrolyte concentration. Some Monte Carlo simulation data from the literature are also included in the assessment of the thermodynamic predictions. Results show a very good agreement with the Monte Carlo results and some improvement for osmotic coefficients and radial distribution functions contact values relative to these theories. The reduced energy curve shows excellent agreement with Monte Carlo data for molarities up to 1 mol/dm$^{3}$.

Charge reversal and surface charge amplification in asymmetric valence restricted primitive model planar electric double layers in the modified Poisson-Boltzmann theory

The modified Poisson-Boltzmann theory of the restricted primitive model double layer is revisited and recast in a fresh, slightly broader perspective. Derivation of relevant equations follow the techniques utilized in the earlier MPB4 and MPB5 formulations and clarifies the relationship between these. The MPB4, MPB5, and a new formulation of the theory are employed in an analysis of the structure and charge reversal phenomenon in asymmetric 2:1/1:2 valence electrolytes. Furthermore, polarization induced surface charge amplification is studied in 3:1/1:3 systems. The results are compared to the corresponding Monte Carlo simulations. The theories are seen to predict the "exact" simulation data to varying degrees of accuracy ranging from qualitative to almost quantitative. The results from a new version of the theory are found to be of comparable accuracy as the MPB5 results in many situations. However, in some cases involving low electrolyte concentrations, theoretical artifacts in the form of un-physical "shoulders" in the singlet ionic distribution functions are observed.

Lattice Model of an Ionic Liquid at an Electrified Interface.

We study ionic liquids interacting with electrified interfaces. The ionic fluid is modeled as a Coulomb lattice gas. We compare the ionic density profiles calculated using a popular modified Poisson-Boltzmann equation with the explicit Monte Carlo simulations. The modified Poisson-Boltzmann theory fails to capture the structural features of the double layer and is also unable to correctly predict the ionic density at the electrified interface. The lattice Monte Carlo simulations qualitatively capture the coarse-grained structure of the double layer in the continuum. We propose a convolution relation that semiquantitatively relates the ionic density profiles of a continuum ionic liquid and its lattice counterpart near an electrified interface.

Ion-Conserving Modified Poisson–Boltzmann Theory Considering a Steric Effect in an Electrolyte

The modified Poisson–Nernst–Planck (MPNP) and modified Poisson–Boltzmann (MPB) equations are well known as fundamental equations that consider a steric effect, which prevents unphysical ion concentrations. However, it is unclear whether they are equivalent or not. To clarify this problem, we propose an improved free energy formulation that considers a steric limit with an ion-conserving condition and successfully derive the ion-conserving modified Poisson–Boltzmann (IC-MPB) equations that are equivalent to the MPNP equations. Furthermore, we numerically examine the equivalence by comparing between the IC-MPB solutions obtained by the Newton method and the steady MPNP solutions obtained by the finite-element finite-volume method. A surprising aspect of our finding is that the MPB solutions are much different from the MPNP (IC-MPB) solutions in a confined space. We consider that our findings will significantly contribute to understanding the surface science between solids and liquids.

Reversed electrophoretic mobility of a spherical colloid in the Modified Poisson-Boltzmann approach

Abstract The electrophoretic mobility of a spherical colloid particle immersed in a binary electrolyte is studied by using the Primitive Model Electrophoresis formalism, in combination with the Modified Poisson-Boltzmann equilibrium theory. This approach differs from the classical descriptions of electrophoresis, based on the Poisson-Boltzmann equation for point ions, in that the Primitive Model Electrophoresis theory takes into account consistently the ionic correlations and excluded volume contributions due to the finite size of the ions. Using the equilibrium radial distribution functions of the Modified Poisson-Boltzmann, the primitive model electrophoresis theory predicts a non-universal behavior of the reduced mobility, as a function of the reduced ζ -potential, in contrast to the universal behavior exhibited by the well-known treatment of Wiersema, O’Brien and White. The primitive model electrophoresis mobilities calculated with the Modified Poisson-Boltzmann theory compare favorably with the previously reported mobilities obtained through the hypernetted-chain/mean spherical approximation equilibrium theory. The agreement between these two theories is very good, both quantitatively and qualitatively for univalent electrolytes. For 2:2 electrolytes they qualitatively agree. In particular, the Modified Poisson-Boltzmann mobilities confirm the occurrence of reversed electrophoretic mobilities. Overall the results presented here show the adequacy of the Modified Poisson-Boltzmann equilibrium theory to calculate the electrophoretic mobility of colloidal particles.

On the theory of electric double layer with explicit account of a polarizable co-solvent

We present a continuation of our theoretical research into the influence of co-solvent polarizability on a differential capacitance of the electric double layer. We formulate a modified Poisson-Boltzmann theory, using the formalism of density functional approach on the level of local density approximation taking into account the electrostatic interactions of ions and co-solvent molecules as well as their excluded volume. We derive the modified Poisson-Boltzmann equation, considering the three-component symmetric lattice gas model as a reference system and minimizing the grand thermodynamic potential with respect to the electrostatic potential. We apply present modified Poisson-Boltzmann equation to the electric double layer theory, showing that accounting for the excluded volume of co-solvent molecules and ions slightly changes the main result of our previous simplified theory. Namely, in the case of small co-solvent polarizability with its increase under the enough small surface potentials of electrode, the differential capacitance undergoes the significant growth. Oppositely, when the surface potential exceeds some threshold value (which is slightly smaller than the saturation potential), the increase in the co-solvent polarizability results in a differential capacitance decrease. However, when the co-solvent polarizability exceeds some threshold value, its increase generates a considerable enhancement of the differential capacitance in a wide range of surface potentials. We demonstrate that two qualitatively different behaviors of the differential capacitance are related to the depletion and adsorption of co-solvent molecules at the charged electrode. We show that an additive of the strongly polarizable co-solvent to an electrolyte solution can shift significantly the saturation potential in two qualitatively different manners. Namely, a small additive of strongly polarizable co-solvent results in a shift of saturation potential to higher surface potentials. On the contrary, a sufficiently large additive of co-solvent shifts the saturation potential to lower surface potentials. We obtain that an increase in the co-solvent polarizability makes the electrostatic potential profile longer-ranged. However, increase in the co-solvent concentration in the bulk leads to non-monotonic behavior of the electrostatic potential profile. An increase in the co-solvent concentration in the bulk at its sufficiently small values makes the electrostatic potential profile longer-ranged. Oppositely, when the co-solvent concentration in the bulk exceeds some threshold value, its further increase leads to decrease in electrostatic potential at all distances from the electrode.

The influence of excluded volume and excess ion polarisability on the capacitance of the electric double layer

ABSTRACTWe apply a modified Poisson–Boltzmann theory which permits ions of different sizes and excess polarisabilities to the study of these properties’ effects on the differential capacitance of the electric double layer. For a planar electrode, we find an analytical expression for the differential capacitance, which is examined in the limits of low- and high-applied potential. In the low potential limit, a reduction of the solution relative permittivity caused by the ion polarisability causes the differential capacitance to decrease above a certain concentration, relative to the Gouy–Chapman–Stern theory. A similar effect is observed for the excluded volume, but only if the ions are of different sizes. In the high potential limit, the differential capacitance decreases inversely with the square root of the applied voltage. In a mixed electrolyte, asymmetries in both ion size and excess polarisability alter the surface adsorption of species: at high potentials, smaller ions displace larger ions and less polarisable ions displace more polarisable ions. The extent of the displacement agrees favourably with experimental data. A further consequence of this displacement is the appearance of a second peak in the differential capacitance, which is enhanced by excess ion polarisability.

Self-Consistent Theory for Predicting Reaction Mechanisms and Potentials in Acid and Base: H(ads) and OH(ads) Deposition on the Pt(111) Electrode

It has recently been shown that the reversible potentials, U rev, for electron transfer reactions such as reduction of H+(aq) to form under potential deposited H(ads) and oxidation of an OH bond of H2O(ads) in acid electrolyte can be predicted accurately by modeling reactants and products with a comprehensive theory for the electrochemical interface.1,2 Predictions are based on calculated Gibbs energies for the reactant and product during a reaction being equal at the reversible potential. The reversible potential is identified as the potential at the crossing point for reaction and product Gibbs energies plotted as functions of electrode potential. Using two-dimensional density functional band theory, the electrode potential is changed by adding charge, q, to the translational cell: U(q) = [-4.454 - Ef(q)]V/eV, where 4.454 eV is the calculated thermodynamic work function of the standard hydrogen electrode and E f is the Fermi energy. Solutions at pH = 0 are modeled by a hydrated hydronium ion, H9O4 +, immersed in dielectric continuum containing 1 M cations and anions whose distribution is calculated with a modified Poisson-Boltzmann theory. Because the theory is fully self-consistent, specifying hydronium as the cation in the electrolyte for pH = 0 automatically gives the electrode potential by the above equation for whatever adsorbate and surface charge are present. For pH = 14 base, a hydrated hydroxyl anion, OH-(H2O)3, is used, and again the electrode surface potential is found self-consistently by the above equation. A significant measure of the accuracy of our theory is the predicted autoionization energy of water. Using the hydrated hydronium ion and hydroxyl ion models, we calculate the autoionization energy for water in bulk solution to be 0.845 eV at 298 K, corresponding to Kw = 0.52 x 10-14, in good agreement with experiment. The autoionization energy also is the difference between the reaction energy for water oxidation to OH(aq) + H+ + e- and the reaction energy for OH-(aq) oxidation to OH(aq) + e-. When these two reactions place the OH on the electrode surface as OH(ads), the difference in reaction energies is still 0.845 eV, but the difference in reversible potentials between acid and base is no longer the solution value 0.845 V. This is because the reversible potentials are influenced by the adsorption energies of OH, which depend on surface conditions. The figure shows predictions of reversible potentials on the standard hydrogen electrode (SHE) scale for the reduced system consisting of 2/3 monolayer (ML) H2O(ads) being oxidized to 1/6 ML OH(ads) + 1/2 ML H2O(ads) in acid and OH-(aq) being oxidized to OH(ads) in base, for which the difference is 0.74 V(SHE). Converting to the reversible hydrogen electrode (RHE) scale, the potentials become 0.64 V(RHE) in acid and 0.75 V(RHE) in base. The shift to higher potential for base shows clearly in the measured onset potentials for OH(ads) formation in acid and base.3 This presentation will introduce the theory and outline several applications to forming OH(ads) and H(ads) on Pt(111) electrodes in acid and base. The theory opens the door for finding mechanisms for electrocatalytic reactions in strong base.

DFT calculations on H, OH and O adsorbate formations on Pt(322) electrode

Reversible potentials of H, OH and O adsorbate formations on a Pt(322) electrode in an acid aqueous solution were calculated using density functional theory combined with a modified Poisson–Boltzmann theory and compared with those on Pt(111) and Pt(332) electrodes. The calculations explain well two distinctive trends observed in experimental cyclic voltammograms: (110)-steps on Pt(332) < (111)-terraces on Pt(111) < (100)-steps on Pt(322) for the H-formation potential and Pt(111) < Pt(322) ≈ Pt(332) for the OH-formation potential. At (110)-steps on Pt(332), H adsorbates repulsively interact with water molecules adsorbed at steps, and the interactions lower the H-formation potential at the steps. The situation is similar at (100)-steps on Pt(322), but because of the strong Pt–H bond strength at the (100)-steps, the H-formation potential becomes high on the steps. At (100)-steps and (110)-steps on Pt(322) and Pt(332), respectively, OH formation potentials become higher than that at (111)-terraces on Pt(111) because a large number of H-bond defects, which destabilize OHs, are created at the step-terrace interfaces on the stepped Pt surfaces although Pt–OH bond strengths are stronger at the steps than at the terraces. The observed formation potentials are, therefore, results of strong influences of both the surface geometry and solvation environment.

A modified Poisson-Boltzmann theory: Effects of co-solvent polarizability

In this paper within a field-theoretical approach taking into account explicitly a co-solvent with a nonzero dipole and a polarizability tensor, we derive a modified Poisson-Boltzmann equation. Applying the modified Poisson-Boltzmann equation, we formulate a generalized Gouy-Chapman theory for the case when an electrolyte solution is mixed with a polar co-solvent having a large polarizability. We show that an increase of the co-solvent concentration as well as the co-solvent polarizability lead to a significant increase of differential capacitance at sufficiently high surface potentials of the electrode, whereas the profile of the electrostatic potential becomes considerably more long-ranged. On the contrary, an increase in the permanent dipole of the co-solvent only weakly affects the differential capacitance.