Poisson-Boltzmann (PB) theory is the classic approach to soft matter electrostatics which has been applied to numerous problems of physical chemistry and biophysics. Its essential limitations are the neglect of correlation effects and of fluid structure. Recently, several theoretical insights have allowed the formulation of approaches that go beyond PB theory in a systematic way. In this topical review we provide an update on the developments achieved in self-consistent formulations of correlation-corrected Poisson-Boltzmann theory. We introduce the corresponding system of coupled nonlinear equations for both continuum electrostatics with a uniform dielectric constant and a structured solvent, a dipolar Coulomb fluid, including nonlocal effects. While the approach is only approximate and also limited to corrections in the so-called weak fluctuation regime, it allows to include physically relevant effects, as we show for a range of applications of these equations.
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.
Poisson-Boltzmann theory allows one to study soft matter and biophysical systems involving point-like charges of low valencies. The inclusion of fluctuation corrections beyond the mean-field approach typically requires the application of loop expansions around a mean-field solution for the electrostatic potential \(ϕ({\bf r})\), or sophisticated variational approaches. Recently, Poisson-Boltzmann theory has been recast, via a Legendre transform, as a mean-field theory involving the dielectric displacement field \({\bf D}({\bf r})\). In this paper we consider the path integral formulation of th…
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…
The equilibrium properties of ionic microgels are investigated using a combination of the Poisson-Boltzmann and Flory theories. Swelling behavior, density profiles, and effective charges are all calculated in a self-consistent way. Special attention is given to the effects of salinity on these quantities. It is found that the equilibrium microgel size is strongly influenced by the amount of added salt. Increasing the salt concentration leads to a considerable reduction of the microgel volume, which therefore releases its internal material -- solvent molecules and dissociated ions -- into the s…
In this paper we develop a simple theory to study the effects of ionic size on ionic distributions around a charged spherical particle. We include a correction to the regular Poisson-Boltzmann equation in order to take into account the size of ions in a mean-field regime. The results are compared with Monte Carlo simulations and a Density Functional Theory based on the Fundamental Measure approach and a second-order bulk expansion which accounts for electrostatic correlations. The agreement is very good even for multivalent ions. Our results show that the theory can be applied with very good a…
We present a method aimed at sampling charge density fluctuations in Coulomb systems. The derivation follows from a functional integral representation of the partition function in terms of charge density fluctuations. Starting from the mean-field solution given by the Poisson-Boltzmann equation, an original approach is proposed to numerically sample fluctuations around it, through the propagation of a Langevin like stochastic partial differential equation (SPDE). The diffusion tensor of the SPDE can be chosen so as to avoid the numerical complexity linked to long-range Coulomb interactions, ef…
We develop a double mean-field theory for charged macrogels immersed in electrolyte solutions in the spirit of the cell model approach. We first demonstrate that the equilibrium sampling of a single explicit coarse-grained charged polymer in a cell yields accurate predictions of the swelling equilibrium if the geometry is suitably chosen and all pressure contributions have been incorporated accurately. We then replace the explicit flexible chain by a suitably modeled penetrable charged rod that allows to compute all pressure terms within the Poisson-Boltzmann approximation. This model, albeit …
Analytical estimate of effective charges at saturation in Poisson-Boltzmann cell models
我们提出了一个简单的近似方案来计算高电荷胶体（球形或圆柱形的无限长）的有效电荷。在非线性Poisson-Boltzmann理论中，我们从无限稀释极限中的有效电荷的表达式开始，该表达式对于大的盐浓度是渐近有效的;然后将该结果扩展到有限的胶体浓度，近似盐分配效应，该效应将悬浮液中的盐含量与透析水库的盐含量相关联。这导致的有效电荷作为胶体体积分数和盐浓度的函数的解析表达式。这些结果与根据Alexander { \it et al.}提出的标准公式数值计算的有效电荷{ \em at saturation}（即在大裸电荷的极限）相比是有利的。在细胞模型中。
Mobile charge in an electrolytic solution can in principle be represented as the divergence of ionic polarization. After adding explicit solvent polarization a finite volume of electrolyte can then be treated as a composite non-uniform dielectric body. Writing the electrostatic interactions as an integral over electric field energy density we show that the Poisson-Boltzmann functional in this formulation is convex and can be used to derive the equilibrium equations for electric potential and ion concentration by a variational procedure developed by Ericksen for dielectric continua (Arch. Ratio…
Structure and thermodynamics in restricted primitive model electrolytes are examined using three recently developed versions of a linear form of the modified Poisson-Boltzmann equation. Analytical expressions for the osmotic coefficient and the electrical part of the mean activity coefficient are obtained and the results for the osmotic and the mean activity coefficients are compared with that from the more established mean spherical approximation, symmetric Poisson-Boltzmann, modified Poisson-Boltzmann theories, and available Monte Carlo simulation results. The linear theories predict the the…
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 simulations with 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-co…
We study the adsorption-desorption transition of polyelectrolyte chains onto planar, cylindrical and spherical surfaces with arbitrarily high surface charge densities by massive Monte Carlo computer simulations. We examine in detail how the well known scaling relations for the threshold transition-demarcating the adsorbed and desorbed domains of a polyelectrolyte near weakly charged surfaces-are altered for highly charged interfaces. In virtue of high surface potentials and large surface charge densities, the Debye-Hueckel approximation is often not feasible and the nonlinear Poisson-Boltzmann…
We propose a field-theoretical approach based on the thermodynamic perturbation theory and within it derive a grand thermodynamic potential of the inhomogeneous ionic fluid as a functional of electrostatic potential for an arbitrary reference fluid system. We obtain a modified Poisson-Boltzmann equation as the Euler-Lagrange equation for the obtained functional. Applying Noether's theorem to this functional, we derive a general mean-field expression for the stress tensor consistent with the respective modified Poisson-Boltzmann equation. We derive a general expression for the macroscopic force…
A new protocol for linearization of the Poisson-Boltzmann equation is proposed and the resultant electrostatic equation coincides formally with the Debye-Huckel equation, the solution of which is well known for many electrostatic problems. The protocol is examined on the example of electrostatically stabilized nano-bubbles and it is shown that stable nano-bubbles could be present in aqueous solutions of anionic surfactants near the critical temperature, if the surface potential is constant. At constant surface charge non nano-bubble could exist.
The Poisson-Boltzmann (PB) theory is one of the most important theoretical models describing charged systems continuously. However, it suffers from neglecting ion correlations, which hinders its applicability to more general charged systems other than extremely dilute ones. Therefore, some modified versions of the PB theory are developed to effectively include ion correlations. Focused on their applications to ionic solutions, the original PB theory and its variances, including the field-theoretic approach, the correlation-enhanced PB model, the Outhwaite-Bhuiyan modified PB theory and the mea…
The classical Poisson-Boltzmann equation (CPBE), which is a mean field theory by averaging the ion fluctuation, has been widely used to study ion distributions in charged fluids. In this study, we derive a modified Poisson-Boltzmann equation with a random field from the field theory and recover the ion fluctuation through a multiplicative noise added in the CPBE. The Poisson-Boltzmann equation with a random field (RFPBE) captures the effect of the ion fluctuation and gives different ion distributions in the charged fluids compared to the CPBE. To solve the RFPBE, we propose a Monte Carlo metho…
We consider a set of identical mobile point-like charges (counter-ions) confined to a domain with curved hard walls carrying a uniform fixed surface charge density, the system as a whole being electroneutral. Three domain geometries are considered: a pair of parallel plates, the cylinder and the sphere. The particle system in thermal equilibrium is assumed to be described by the nonlinear Poisson-Boltzmann theory. While the effectively 1D plates and the 2D cylinder have already been solved, the 3D sphere problem is not integrable. It is shown that the contact density of particles at the charge…
We extend our previous study of surface tension of ionic solutions and apply it to the case of acids (and salts) with strong ion-surface interactions. These ion-surface interactions yield a non-linear boundary condition with an effective surface charge due to adsorption of ions from the bulk onto the interface. The calculation is done using the loop-expansion technique, where the zero-loop (mean field) corresponds of the non-linear Poisson-Boltzmann equation. The surface tension is obtained analytically to one-loop order, where the mean-field contribution is a modification of the Poisson-Boltz…
The Poisson-Boltzmann (PB) theory is widely used to depict ionic systems. As a mean-field theory, the PB theory neglects the correlation effect in the ionic atmosphere and leads to deviations from experimental results as the concentration or charge valance increases. A modified PB theory including ion correlation effect while retaining its simplicity is critical for many important applications in which ion correlation effect can be significant. In this paper, we present a new model to incorporate ion correlations into the original PB equation by utilizing the Green's function with a non-linear…
The effective electrostatic interaction between a pair of colloids, both of them located close to each other at an electrolyte interface, is studied by employing the full, nonlinear Poisson-Boltzmann (PB) theory within classical density functional theory. Using a simplified yet appropriate model, all contributions to the effective interaction are obtained exactly, albeit numerically. The comparison between our results and those obtained within linearized PB theory reveals that the latter overestimates these contributions significantly at short inter-particle separations. Whereas the surface co…
Ionic solutions are often regarded as fully dissociated ions dispersed in a polar solvent. While this picture holds for dilute solutions, at higher ionic concentrations, oppositely charged ions can associate into dimers, referred to as Bjerrum pairs. We consider the formation of such pairs within the nonlinear Poisson-Boltzmann framework, and investigate their effects on bulk and interfacial properties of electrolytes. Our findings show that pairs can reduce the magnitude of the dielectric decrement of ionic solutions as the ionic concentration increases. We describe the effect of pairs on the…
