General theory of charge regulation within the Poisson-Boltzmann framework: study of a sticky-charged wall model

TL;DR

This paper presents a new sticky-charge wall model for charge regulation within the Poisson-Boltzmann framework, allowing analysis of electrolyte structural effects on charge regulation through modified boundary conditions.

Contribution

It introduces a simple, intuitive model that modifies boundary conditions in the PB equation to study charge regulation effects.

Findings

Model effectively captures charge regulation behavior.

Structural details of electrolytes influence charge regulation.

Modified boundary conditions provide new insights into electrolyte interactions.

Abstract

This work introduces a sticky-charge wall model as a simple and intuitive representation of charge regulation. Implemented within the mean-field level of description, the model modifies the boundary conditions without affecting the underlying Poisson-Boltzmann (PB) equation of an electrolyte. Employing various modified PB equations, we are able to assess how various structural details of an electrolyte influence charge regulation.

Equations

$$e^{-\beta u_s(x)}=1+l_s\delta (x),$$

$$\beta u_{\rm well}(x)=\left\{\begin{array}[]{l l}-\varepsilon,&\quad\text{for ~{}~{}$0<x<a$}\\ 0,&\quad\text{for ~{}~{}$x>a$}.\end{array}\right.$$

$$\Delta_{a}(x)=\left\{\begin{array}[]{l l}\frac{1}{a},&\quad\text{for ~{}~{}$0<x<a$}\\ 0,&\quad\text{for ~{}~{}$x<0$ or $x>a$},\end{array}\right.$$

$$e^{-\beta u_{\mathrm{well}}(x)}=1+a(e^{\epsilon }-1){\Delta }_a(x).$$

$$\underset{a\to 0}{\lim }{e}^{-\beta u_{\mathrm{well}}(x)}=1+a{e}^{\epsilon }\delta (x).$$

$${\rho }_i(x)=c_i{e}^{-\beta q_i\psi (x)}{e}^{-\beta u_i^s(x)},$$

$$e^{-\beta u_i^s(x)}=1+l_i\delta (x),$$

$$\rho_{i}(x)=c_{i}e^{-\beta q_{i}\psi(x)}\Big{[}1+l_{i}\delta(x)\Big{]}.$$

$$ϵ\frac{d^2\psi (x)}{d{x}^2}=-\sum _{i=1}^K{q}_i{\rho }_i(x)-{\sigma }_c\delta (x),$$

$$ϵ\frac{d^2\psi (x)}{d{x}^2}$$

$$\epsilon\frac{d\psi(x)}{dx}\bigg{|}_{x=0^{+}}=-\sigma_{c}-\sum_{i=1}^{K}q_{i}l_{i}c_{i}e^{-\beta q_{i}\psi(0^{+})}.$$

$$ϵ\frac{d^2\psi (x)}{d{x}^2}=-\sum _{i=1}^K{q}_i{c}_i{e}^{-\beta q_i\psi (x)}.$$

$${\sigma }_{\mathrm{eff}}={\sigma }_c+\sum _{i=1}^K{q}_i{l}_i{c}_i{e}^{-\beta q_i\psi (0^{+})},$$

$$l_{i}^{\rm eff}=l_{i}\bigg{(}\frac{\gamma_{i}-l_{i}^{\rm eff}\rho_{i}(0^{+})}{\gamma_{i}}\bigg{)},$$

$$l_i^{\mathrm{eff}}=\frac{l_i{\gamma }_i}{{\gamma }_i+l_i{c}_i{e}^{-\beta q_i\psi (0^{+})}},$$

$$\epsilon\frac{d\psi(x)}{dx}\bigg{|}_{x=0^{+}}=-\sigma_{c}-\sum_{i=1}^{K}\frac{q_{i}l_{i}c_{i}e^{-\beta q_{i}\psi(0^{+})}}{1+l_{i}c_{i}e^{-\beta q_{i}\psi(0^{+})}/\gamma_{i}}.$$

$$\epsilon\frac{d\psi(x)}{dx}\bigg{|}_{x=0^{+}}=-\sigma_{c}-\sum_{i=1}^{K}\frac{q_{i}l_{i}c_{i}e^{-\beta q_{i}\psi(0^{+})}}{1+\sum_{i=1}^{K}l_{i}c_{i}e^{-\beta q_{i}\psi(0^{+})}/\gamma}.$$

$${ϵ}_0\frac{d^2\psi (x)}{d{x}^2}=-\sum _{i=1}^K{q}_i{\rho }_i(x)+\frac{dP(x)}{dx},$$

$$\epsilon_{0}\frac{d\psi(x)}{dx}\bigg{|}_{x=0^{+}}=-\sigma_{c}+P(0^{+})-\sum_{i=1}^{K}l_{i}q_{i}\rho_{i}(0^{+}).$$

$$P(x)=\bigg{(}\frac{c_{d}\sinh\big{(}p_{0}\beta\psi^{\prime}(x)\big{)}}{p_{0}\beta\psi^{\prime}(x)}\bigg{)}\bigg{(}p_{0}L\big{(}p_{0}\beta\psi^{\prime}(x)\big{)}\bigg{)},$$

$$ϵ={ϵ}_0-\frac{P(x)}{{\psi }^{\prime }(x)},$$

$${\rho }_d(x)=c_d,$$

$$P(x)=c_{d}p_{0}L\big{(}p_{0}\beta\psi^{\prime}(x)\big{)},$$

$${\rho }_i(x)=c_i{e}^{-\beta q_i\psi (x)}{e}^{\beta {\alpha }_i{\psi }^{\prime 2}(x)/2},$$

$$\epsilon\frac{d^{2}\psi(x)}{dx^{2}}=-\sum_{i=1}^{K}q_{i}\rho_{i}(x)-\frac{d}{dx}\bigg{[}\psi^{\prime}(x)\sum_{i=1}^{K}\alpha_{i}\rho_{i}(x)\bigg{]},$$

$$P(x)=-{\psi }^{\prime }(x)\sum _{i=1}^K{\alpha }_i{\rho }_i(x).$$

$$\epsilon\frac{d\psi(x)}{dx}\bigg{|}_{x=0^{+}}=-\sigma_{c}-\psi^{\prime}(0^{+})\sum_{i=1}^{K}\alpha_{i}\rho_{i}(0^{+})-\sum_{i=1}^{K}l_{i}q_{i}\rho_{i}(0^{+}).$$

$$\psi (0^{+})-\psi (0^{-})$$

Full text

I Introduction

This work introduces a sticky-charged wall model as an intuitive tool for describing charge regulation, a view that a surface charge is not fixed but an outcome of two competing dynamic processes at an interface: ion dissociation and ion binding. In consequence, the effective surface charge is a function of the environment in which it is embedded.

The sticky potential is an idealization of a short-range surface potential that arises from nonelectrostatic interactions, such as the van der Waals interactions in the case of physisorption or the covalent bindings in the case of chemisorption. It is formally obtained by taking the width of the square-well potential to zero and its depth to infinity Baxter68 . By eliminating the range of the potential, the surface stickiness is characterized by a single parameter; any microscopic details are suppressed. Consequently, ions interact with a surface only on direct contact. This leads to a simple proportionality between the number of adsorbed ions and the contact density of an ionic species.

Within the mean-field level of description, the sticky-charged wall model modifies the boundary conditions without affecting the underlying Poisson-Boltzmann (PB) equation. This transforms the study of charge regulation to the obtaining of a solution to a Poisson-Boltzmann equation with an alternative boundary conditions. By considering a number of different modified Poisson- Boltzmann equations Frydel16 , we carry out a systematic study of how different descriptions of an electrolyte influence adsorption.

The sticky-wall model, furthermore, can be modified to incorporate a limited number of binding sites by renormalizing the parameter of stickiness. This leads to the boundary conditions as formulated within the Ninham-Parsegian model of charge regulation based on chemical equilibrium and formulated in terms of equilibrium constants Ninham71 .

One of the goals of this work is to study the effects of the solvation energy on the behavior of adsorption. One contribution to the solvation energy is the dielectric constant of a solvent, which screens the electric field and arises as the result of an orientational polarization. Within the standard Poisson-Boltzmann model, the orientational polarization is linear in the electric field, which generally is accurate for low values of field or surface charge. For large surface charges, nonlinear contributions start to modify the dielectric constant in the vicinity of a surface, which, in turn, modifies adsorption. The contributions of nonlinear polarization to the structure of a double-layer had been investigated in Refs. David07 and Frydel16 . The present paper extends this study into behavior of adsorption.

In addition to the solvation effects, we study ion-specific effects of polarizable ions on the behavior of adsorption. To this end, we use the polarizable Poisson-Boltzmann equation Frydel11 and consider an electrolyte that is a mixture of polarizable and nonpolarizable ions.

This paper is organized as follows. In Sec. (II), we briefly overview the Baxter sticky potential and review its derivation as the limiting case of a square-well potential. In Sec. (III), we consider the sticky-charged wall model for the standard Poisson-Boltzmann equation. In the same section, we consider modifications of the sticky surface by limiting the number of sticky sites in order to make contact with the Ninham-Parsegian model. In Sec. (IV), we investigate the contributions of the solvation energy, due to nonlinear polarization, to the behavior of adsorption. To this end, we use the dipolar Poisson-Boltzmann and the Langevin Poisson-Boltzmann equations. In Sec. (V), we study the ion-specific effects of polarizable ions and their effects on preferred adsorption. In this case, we use the polarizable Poisson-Boltzmann equation. Finally, in Sec. (VI), we close the work with conclusions.

II Baxter sticky potential

The first implementation of a sticky potential is generally credited to Baxter who studied thermodynamic properties of sticky hard-spheres Baxter68 . The adhesive interactions between spheres can give rise to clusters Bietry18 and, as clusters grow and combine, eventually lead to phase transition. Since then, the sticky hard-sphere model has been explored by numerous groups, and today, it is considered as a good representation of colloidal gels Fuchs00 ; Piazza07 ; Royall18 . In electrostatics, the Baxter potential was first used to capture surface adsorption within the restricted primitive model Carnie81 . In the series of following papers, Blum86 ; Blum89 ; Cornu89 ; Blum90 ; Blum90b and McQuarrie95 ; McQuarrie96a ; McQuarrie96b , various theoretical aspects and variations of the sticky model were explored, most of the work carried out within the liquid-state theories.

In this work, we consider a sticky wall model within the framework of the Poisson-Boltzmann equation. The absorbing surface is considered to be planar and uniformly sticky with the sticky potential denoted as $u_{s}(x)$. Due to its infinitesimal range, the Boltzmann factor of the sticky potential is given by

[TABLE]

where the length $l_{s}$ determines the strength of the surface stickiness. (The sticky wall is assumed to be at $x=0$.) The above result is most conveniently obtained from the limit of the square-well potential,

[TABLE]

By defining the finite range delta function,

[TABLE]

such that $\lim_{a\to 0}\Delta_{a}(x)=\delta(x),$ the square-well potential is written as $\beta u_{\rm well}(x)=a\epsilon\,\Delta_{a}(x)$, and its Boltzmann factor becomes

[TABLE]

Taking the limit $a\to 0$ and $\varepsilon\to\infty$, such that $ae^{\varepsilon}=l_{s}$, the above result recovers Eq. (1),

[TABLE]

Based on Eq. (1) we deduce that the total density $\rho(x)$ is split into the density of mobile particles, $\rho(x>0)$ (assuming the system is confined to a half-space $x>0$), and the density of adsorbed particles, $l_{s}\rho(0^{+})\delta(x)$, which tells us that the surface density of adsorbed particles is $l_{s}\rho(0^{+})$. Any condition that alters the contact density $\rho(0^{+})$ and the near-field region of a double-layer should, therefore, modify the adsorption behavior.

The link between the near-field region and the adsorption behavior is later investigated in this work. As previous studies of modified Poisson-Boltzmann equations do not indicate significant far-field effects under condition in which the mean-field theory is acceptable, there is an indication that near a surface these effects are not negligible Frydel16 . This raises the possibility for the adsorption to be dependent on the microscopic details of an electrolyte.

III The standard PB equation

We start by considering a sticky-charged wall model within the standard Poisson-Boltzmann equation. Here ions are represented as structureless point-charges, and solvent as a background medium with a dielectric constant. A sticky-charged wall is placed at $x=0$ where it confines an electrolyte to the region $x>0$. The mean-field density of an ionic species $i$ is given by

[TABLE]

where $\psi(x)$ is an electrostatic potential, $u_{i}^{s}(x)$ is the sticky interaction specific to a species $i$, and $q_{i}$ and $c_{i}$ is the charge and the bulk concentration of a species $i$, respectively. Since the Boltzmann factor of the sticky potential is

[TABLE]

where $l_{i}$ is the parameter of stickiness for a species $i$, the density becomes

[TABLE]

Inserting this into the Poisson equation,

[TABLE]

where $\sigma_{c}$ is the bare surface charge before adsorption, and $K$ is the total number of ionic species, we arrive at the Poisson-Boltzmann equation

[TABLE]

where $\psi(0^{+})$ indicates the contact value of a potential from an electrolyte side. The sole difference between the above equation and the standard Poisson-Boltzmann equation is the modified surface charge in square parenthesis.

Alternatively, the effective surface charge can be incorporated into the boundary conditions. The expression of the boundary conditions is obtained by operating on Eq. (10) with $\lim_{a\to 0}\int_{-a}^{a}dx$. As the potential across a wall is continuous, $\psi(0^{+})=\psi(0^{-})$, this yields

[TABLE]

Within the region $x>0$, an electrolyte is governed by the standard Poisson-Boltzmann equation,

[TABLE]

We note that $\psi(x<0)=\psi(0^{+})$, which ensures continuity of an electrostatic potential across a wall.

The resulting boundary conditions in Eq. (11) suggests that the effective surface charge is

[TABLE]

where the second term accounts for a surface charge due to adsorbed ions. If adsorbed ions have the same charge as the bare surface charge, the effective surface charge will increase. For the opposite situation, it will be reduced.

In consequence, the effective surface charge is not fixed, or determined a priori, as we do not know the value of a potential at a contact with a wall, $\psi(0^{+})$. It can only be determined by solving Eq. (12). The mathematical coupling of the differential equation to the boundary condition reflects the physical coupling between the surface and an electrolyte, which are in contact and in dynamic equilibrium. Any variation in an electrolyte will be reflected in the effective surface charge. Such coupling is a feature and consequence of charge regulation.

The boundary conditions in Eq. (11) are also interesting from mathematical point of view. It combines the values of a function $\psi(x)$ and the values of its derivative on the boundary, however, not in a linear manner, which does not qualify it as the usual Robin boundary condition but can be considered as a highly nonlinear generalization of it.

The solution of the Poisson-Boltzmann equation for a fixed surface charge is obtained by the iterative numerical procedure, starting with an initial guess for an electrostatic potential and then correcting that solution using exact internal relations that check for self-consistency. The same method is used for the sticky wall model. The effective surface charge $\sigma_{\rm eff}$ is simply corrected after each iteration using the value of $\psi(0^{+})$ from the previous iteration. For the case of counterion adsorption, each consecutive correction reduces the effective surface charge and, as a result, charge regulation accelerates convergence and facilitates the numerics.

III.1 alternative interpretation of a sticky potential

In the same way as we define the effective surface charge $\sigma_{\rm eff}$ whose value depends on the number of adsorbed ions, we can define the effective stickiness, whose value changes with the number of adsorbed particles. The microscopic origin of such renormalization is that adsorption occurs at specific discrete sites, where each site can bind with at most one ion. Once occupied, the sites are deactivated and the averaged surface density of active sites becomes reduced. This, in turn, leads to the reduction in a wall stickiness.

If the total number of sites per unit area is $\gamma_{i}$ (under the assumption that sites are ion specific) and none of the sites are occupied, the effective stickiness parameter is $l_{\rm eff}=l_{i}$. On the other hand, if all the sites are occupied, $l_{\rm eff}=0$. Within the same logic, half the sites occupied implies $l_{\rm eff}=l_{a}/2$. This suggests the following expression for the effective parameter of stickiness

[TABLE]

where $l_{i}^{\rm eff}\rho_{i}(0^{+})$ is the surface density of adsorbed ions of the species $i$, and $\gamma_{i}-l_{i}^{\rm eff}\rho_{i}(0^{+})$ is the number of unoccupied sites. Solving the above equation for $l_{i}^{\rm eff}$ we get

[TABLE]

and the new boundary conditions are

[TABLE]

The surface is now characterized by the parameters $\sigma_{c}$, $\{l_{i}\}$, and $\{\gamma_{i}\}$.

Other versions of a sticky wall are possible. For example, if the binding sites bind indiscriminately to all ions, the corresponding boundary conditions are

[TABLE]

In this case, the surface is characterized by the parameters $\sigma_{c}$, $\{l_{i}\}$, and $\gamma$.

We refer to the sticky boundary conditions in Eq. (11), Eq. (16), and Eq. (17) as the boundary condition $A$, $B$, and $C$. In the subsequent work, we consider mainly the boundary condition of the type $A$.

The boundary conditions $B$ and $C$ are the same as those obtained in the Ninham-Parsegian model of charge regulation Ninham71 ; David16 ; David18 , derived based on the formalism of chemical equilibrium, where adsorption is determined by the dissociation constant $K_{i}^{\rm d}=\gamma_{i}/l_{i}$.

The expressions in Eq. (16) and (17) for the boundary conditions ”B” and ”C” have similar structure as the modified Poisson-Boltzmann equation in Ref. David97 , which accounts for the excluded volume effects using a local approximation; that is, it applies expressions of a homogeneous system to a heterogeneous situation Frydel12 . In consequence, a local approximation fails to produce the usual density oscillations seen in exact systems and leads to unphysical local density saturation.

The saturation of Eq. (16) and Eq. (17) implies that the number of adsorbed particles is bound from above by the value of $\gamma_{i}$. The saturation in this case, however, occurs only within the plane of a sticky surface, thus, the local approximation is applied to a single point at $x=0$ and not the entire region. Consequently, there is no saturation within the density profile itself. This is seen if we consider that the density of ions at a surface is represented by a delta function, $l_{i}^{\rm eff}\rho_{i}(0^{+})\delta(x)$, which clearly precludes any possibility of local saturation.

One can raise the question about the role of distribution of biding sites on the behavior of adsorption. For example, such sites could be distributed randomly or on a regular lattice. To explore such possibilities would require a number of careful simulations for different distributions of the binding sites. As the goal of this work is to focus on the boundary conditions ”A”, and the cases ”B” and ”C” are mentioned to make contact with other models of charge regulation, we do not undertake such a detailed study.

IV Effect of the solvation energy on adsorption

In this part of the paper, we look into how the change in the solvation energy due to nonlinear polarization of a solvent modifies adsorption. The expected trend is that the more favorable solvation energy facilitates ion dissolution, thereby, reduces adsorption. Large dielectric constant, which implies increased screening of electrostatic interactions, is associated with better solvation energy. In consequence, if the nonlinear polarization near a surface leads to increased dielectric constant, the adsorption will be reduced. If it leads to reduced dielectric constant, the adsorption will be enhanced.

In the standard PB equation, the polarization density of a solvent is linear in electrostatic field, $P(x)=-\chi_{e}\psi^{\prime}(x)$, where $\chi_{e}$ is the electric susceptibility, and the resulting dielectric constant is $\epsilon=\epsilon_{0}(1+\chi_{e})$. Nonlinear contributions to $P(x)$ could arise in strong electrostatic field as a result of orientational saturation, when molecular dipole is aligned with a field and fails to respond to its further increase. Furthermore, if a solvent is compressible, nonlinearities could arise due to accumulation of the solvent dipoles near a surface.

IV.1 The Dipolar PB equation

We start with the dipolar Poisson-Boltzmann equation (DPB). The DPB model was first conceived as a more realistic representation of a dipolar solvent David07 . It represents solvent particles as a gas of point-dipoles. The nonlinearities in this model are primarily the result of inhomogeneous distribution of a solvent.

To derive the DPB model, we consider the polarization density, $P(x)=\rho_{d}(x)p(x)$, where $\rho_{d}(x)$ is the density of a dipolar species and $p(x)$ is the local dipole moment of a single solvent molecule. The polarization density contributes to the charge density as $-P^{\prime}(x)$, and the Poisson equation is

[TABLE]

and, for a sticky-charged wall case, the boundary conditions are

[TABLE]

In the above equations, $\epsilon_{0}$ is the dielectric constant of a vacuum. For the linear polarization, $P(x)=-\chi_{e}\psi^{\prime}(x)$, we recover the standard PB equation with the dielectric constant $\epsilon=\epsilon_{0}(1+\chi_{e})$. Note that the effective surface charge, in addition to adsorbed ions, includes the polarization surface density, $P(0^{+})$.

The DPB equation is obtained by substituting for $\rho(x)$ and $P(x)$ their respective mean-field expressions given by $\rho_{i}(x)=c_{i}e^{-\beta q_{i}\psi(x)}$ and David07 ; Frydel16

[TABLE]

where $c_{d}$ is the bulk concentration of a dipolar species, $p_{0}$ is the permanent dipole moment, and $L(x)=\coth(x)-x^{-1}$ is the Langevin function. The terms in parenthesis correspond to $\rho_{d}(x)$ and $p(x)$, respectively.

Before considering a sticky wall case, we briefly look into a non-sticky wall model to see how the nonlinear polarization of the DPB model modifies the system with reference to the standard PB model. In Fig. (1) we plot the effective dielectric constant, defined as

[TABLE]

and the density of counterions for a non-sticky wall for a symmetric $1:1$ electrolyte. The parameters $p_{0}$ and $c_{d}$ correspond roughly to those of water, such that the linear regime, $P(x)\approx-\frac{\beta c_{d}p_{0}^{2}}{3}\psi^{\prime}(x)$, recovers the value of water, $\epsilon/\epsilon_{0}\approx 80$. The nonlinear contributions, strongest near a wall, enhance the dielectric constant and the electronic screening as solvent accumulates near a wall. In consequence, the counterion density, in the region around a wall is reduced. Based on this result, we expect the counterion adsorption to become reduced as a consequence of more favorable solvation energy in the vicinity of a wall.

To verify this assertion, in Fig. (2) we plot the surface charge density of adsorbed ions, $\sigma_{\rm ads}$, as a function of the wall stickiness, $l_{s}$, for two different values of a surface charge density, $\sigma_{c}$. The parameter of stickiness are the same for each species, $l_{s}=l_{+}=l_{-}$. As anticipated, the adsorption is reduced (in relation to the PB model with the same parameters). Since nonlinear effects become stronger for large values of electric field, the reduction in adsorption is more pronounced as $\sigma_{c}$ increases.

IV.2 The Langevin PB equation

In the DPB model, the dipolar solvent is represented as compressible. This leads to a rather large and unphysical accumulation of a dipolar species near a wall. A more realistic representation of water solvent should assume incompressibility. This can be done by setting the dipolar density as

[TABLE]

leading to

[TABLE]

and we refer to this model as the Langevin PB equation (LPB) Frydel16 .

Because an incompressible solvent cannot accumulate at a surface, we expect, in contrast to the DPB model, a reduced electrostatic screening in consequence to the saturation effect of the Langevin function in Eq. (23). This is demonstrated by Fig. (3) (a) where the dielectric constant is reduced in the region near a wall. This, in turn, leads to increased density shown in Fig. (3) (b). For a sticky wall model this implies increased adsorption.

When we next look into the adsorption behavior of a sticky-wall model, see Fig. (4), we encounter

enhanced adsorption in relation to the standard PB model with the same sticky parameters. The nonlinear polarization effects become larger with increasing $\sigma_{c}$.

We note that the reduced effective dielectric constant for the LPB model shown in Fig. (3), and the accompanying increase of electrostatic interactions, implies increased role of correlations absent in the mean-field description Yan02 . The question of correlations for a system that incorporates explicit solvent is of course complex. The frequently used random-phase approximation in electrostatics Frydel16b , often presented in the field-theoretical formalism as the Gaussian approximation Frydel15 , is for the present situation not very trustworthy, given that it does not stand a test of comparison even for as simple systems as the Gaussian core Frydel17 or the penetrable sphere model Frydel18 . If correlations were present, it is expected that they should enhance adsorption as a result of counterion ordering in the longitudinal plane Yan02 , thus, the predictions of the present mean-field model can be regarded as the lower bound of what one would see in a more accurate approximation.

V The polarizable PB equation and ion-specific effects

In this section we consider the polarizable Poisson-Boltzmann equation (PPB) Frydel11 ; Frydel16 , which represents ions as polarizable point charges, and whose mean-field density is given by

[TABLE]

Note that any finite polarizability $\alpha_{i}$ increases the concentration of ions in the presence of an external field. This indicates that ions with larger polarizability are more likely to be found near a charged surface where electrostatic field is larger. By the same token, polarizable ions are more likely to be adsorbed.

The PPB equation is written as

[TABLE]

where the second term on the right hand side comes from the polarization density due to polarizable ions,

[TABLE]

Then the boundary conditions for the sticky wall model are given by

[TABLE]

Because adsorbed ions are polarizable and an external field can induce dipole moments, the boundary conditions need to account for a surface density of dipoles. Surface dipoles are accounted by a discontinuous electrostatic potential across a surface, and the discontinuity that arises is given by

[TABLE]

where it lowers the potential in the region outside an electrolyte, $\psi(x<0)$, but has no effect on the region inside an electrolyte, consequently, it does not effect adsorption. A jump in the electrostatic potential could be relevant for fluid interfaces with trapped colloids, where it could affect the structure of a double-layer around colloids Frydel07 ; Frydel11b .

To investigate ion specific effects, we consider a system in which half of counterions are polarizable, and the other half is non-polarizable. We start with a non-sticky wall to determine the distribution of different counterions within a double-layer around a wall. These distributions are shown in Fig. (5) for a solvent with a low dielectric constant, $\epsilon/\epsilon_{0}=10$, in order to emphasize the ion-specific effects. For solvents with higher dielectric constant, such as water, the ion-specific effects due to ion polarizability are small. This implies that these effects are relevant for systems with low dielectric constant, such as ionic fluids, and less so for aqueous solutions.

Next, we consider a sticky-wall model for the same system parameters. Fig. (6) plots the surface density of adsorbed polarizable and non-polarizable counterions. As expected, based on the results in Fig. (5), polarizable ions are more likely to be adsorbed than non-polarizable ones. This can be traced to the fact that the density of polarizable ions, see Eq. (24), have larger concentration for any non-zero electric field. Physically this means that the induction of a dipole moment lowers the energy.

VI Conclusion

In conclusion, we have considered the sticky-charged wall model as a simple and intuitive tool to incorporate charge regulation. The model is sufficiently flexible that it can be modified to take into account a limited number of binding sites and allows us to arrive by a different route at the Ninham-Parsegian model of charge regulation Ninham71 . The dissociation constant in that model is inversely proportional to the stickiness parameter in the present sticky wall model.

In the later part of this work, we study various electrostatic effects that may arise as a result of a more detailed microscopic description of an electrolyte. We separate these contributions into those due to the solvation energy and those due to the structure of dissolved ions. In the former case, we consider nonlinear contributions of a solvent due to compressibility and orientational saturation in strong fields. As the saturation makes solvation near a wall unfavorable, adsorption is enhanced as a consequence. For the case of ion structure, we consider polarizability and its effect on adsorption. The observed effect is that polarizable ions are more susceptible to adsorption.

The effects described above are significant under extreme conditions, that is, at large surface charges, where polarization saturation is significant, and/or at a low dielectric constant of a solvent, where polarizability effects become more pronounced. Within the standard conditions and the weak-coupling limit, the microscopic details produce only small variations. Consequently, this study is particularly relevant for systems like ionic liquids where the absence of a dielectric solvent makes electrostatic screening weak, in which case variations in microscopic structure give rise to disparate behaviors.

Acknowledgements.

D.F. would like to thank David Andelman for introducing the topic of charge regulation. D.F. would also like to acknowledge discussions with Yan Levin on some aspects of boundary conditions.