A variational approach is used to develop a robust numerical procedure for solving the nonlinear Poisson-Boltzmann equation. Following Maggs et al., we construct an appropriate constrained free energy functional, such that its Euler-Lagrange equations are equivalent to the Poisson-Boltzmann equation. We then develop, implement, and test an algorithm for its numerical minimization, which is quite simple and unconditionally stable. The analytic solution for planar geometry is used for validation. Furthermore, some results are presented for a charged colloidal sphere surrounded by counterions.
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 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…
We present an extension to the Poisson-Boltzmann model where the dipolar features of solvent molecules are taken explicitly into account. The formulation is derived at mean-field level and can be extended to any order in a systematic expansion. It is applied to a two-plate system with oppositely charged surfaces. The ion distribution and profiles in the dipolar order parameter are calculated and can result in a large correction to the inter-plate pressure.
The non-linear Poisson-Boltzmann equation for a circular, uniformly charged platelet, confined together with co- and counter-ions to a cylindrical cell, is solved semi-analytically by transforming it into an integral equation and solving the latter iteratively. This method proves efficient, robust, and can be readily generalized to other problems based on cell models, treated within non-linear Poisson-like theory. The solution to the PB equation is computed over a wide range of physical conditions, and the resulting osmotic equation of state is shown to be in fair agreement with recent experim…
A simple technique is proposed for numerically determining equilibrium ion distribution functions belonging to free energies of the Poisson-Boltzmann type. The central idea is to perform a conventional Monte-Carlo simulation using the free energy as the "Hamiltonian" entering the Metropolis criterion and the spatially discretized density as degrees of freedom. This approach is complementary to the possibility of numerically solving the differential equations corresponding to the variational problem, but it is much easier to implement and to generalize. Its utility is demonstrated in two exampl…
The mathematical theory of integrable Painleve/Toda type systems sheds new light on the behavior of solutions to the Poisson-Boltzmann equation for the potential due to a long rod-like macroion. We investigate here the case of symmetric electrolytes together with that of 1:2 and 2:1 salts. Short and large scale features are analyzed, with a particular emphasis on the low salinity regime. Analytical expansions are derived for several quantities relevant for polyelectrolytes theory, such as the Manning radius. In addition, accurate and practical expressions are worked out for the electrostatic p…
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…
Swollen stacks of finite-size disc-like Laponite clay platelets are investigated within a Wigner-Seitz cell model. Each cell is a cylinder containing a coaxial platelet at its centre, together with an overall charge-neutral distribution of microscopic co and counterions, within a primitive model description. The non-linear Poisson-Boltzmann (PB) equation for the electrostatic potential profile is solved numerically within a highly efficient Green's function formulation. Previous predictions of linearised Poisson-Boltzmann (LPB) theory are confirmed at a qualitative level, but large quantitativ…
Screening of a macroion by multivalent counterions is considered. It is shown that ions form strongly correlated liquid at the macroion surface. Cohesive energy of this liquid leads to strong additional attraction of counterions to the surface. Away from the surface this attraction is taken into account by a new boundary condition for the Poisson-Boltzmann equation. This equation is solved with the new boundary condition for a charged flat surface and a long cylinder. For a cylinder Onsager-Manning theory looses its universality so that apparent charge of the cylinder is smaller than their the…
The Poisson-Boltzmann equation is often presented via a variational formulation based on the electrostatic potential. However, the functional has the defect of being non-convex. It can not be used as a local minimization principle while coupled to other dynamic degrees of freedom. We formulate a convex dual functional which is numerically equivalent at its minimum and which is more suited to local optimization.
We formulate the non-linear field theory for a fluctuating counter-ion distribution in the presence of a fixed, arbitrary charge distribution. The Poisson-Boltzmann equation is obtained as the saddle-point, and the effects of fluctuations and correlations are included by a loop-wise expansion around this saddle point. We show that the Poisson equation is obeyed at each order in the loop expansion and explicitly give the expansion of the Gibbs potential up to two loops. We then apply our formalism to the case of an impenetrable, charged wall, and obtain the fluctuation corrections to the electr…
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…
The work reviews on a general level various modified Poisson-Boltzmann equations and demonstrates their use on the specific system of charged particles at an interface. This system is special, first, because it exhibits the long-range interactions on account of an air-water interface, and two, because the strength of these interactions depends on the structure of a double-layer. Thus, this system is particularly sensitive to the type of the modified Poisson-Boltzmann equation used.
We investigate the effect of excluded volume interactions on the electrolyte distribution around a charged macroion. First, we introduce a criterion for determining when hard-core effects should be taken into account beyond standard mean field Poisson-Boltzmann (PB) theory. Next, we demonstrate that several commonly proposed local density functional approaches for excluded volume interactions cannot be used for this purpose. Instead, we employ a non-local excess free energy by using a simple constant weight approach. We compare the ion distribution and osmotic pressure predicted by this theory…
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…
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.
We introduce the Poisson-Boltzmann cell model for spherical colloidal particles with a heterogeneous surface charge distribution. This model is obtained by generalizing existing cell models for mixtures of homogeneously charged colloidal spheres. Our new model has similar features as Onsager's second-virial theory for liquid crystals, but it predicts no orientational ordering if there is no positional ordering. This implies that all phases of heterogeneously charged colloids that are liquid-like with respect to translational degrees of freedom are also isotropic with respect to particle orient…
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…
We numerically solve the non-linear Poisson-Boltzmann equation for two cylinders confined by two parallel charged plates. The repulsive electrical double layer component of the cylinder pair potential is substantially reduced by confinement between like-charged plates. While the effective cylinder surface charge is increased by the confinement, the effective interaction screening length is reduced, this effect being dominant so that the repulsive confined cylinder-cylinder interaction potential is reduced.
The Poisson-Boltzmann equation for a strongly charged plate inside a generic charge-asymmetric electrolyte is solved using the method of asymptotic matching. Both near field and far field asymptotic behaviors of the potential are systematically analyzed. Using these expansions, the renormalized surface charge density is obtained as an asymptotic series in terms of the bare surface charge density.
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 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…
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 eq…
Using exact results from the theory of completely integrable systems of the Painleve/Toda type, we examine the consequences for the theory of polyelectrolytes in the (nonlinear) Poisson-Boltzmann approximation.
The behavior of electrolyte solutions close to a charged surface is studied theoretically. A modified Poisson-Boltzmann equation which takes into account the volume excluded by the ions in addition to the electrostatic interactions is presented. In a formal lattice gas formalism the modified Poisson-Boltzmann equation can be obtained from a mean-field approximation of the partition function. In an alternative phenomenological approach, the same equation can be derived by including the entropy of the solvent molecules in the free energy. In order to visualize the effect of the steric repulsion,…
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…
The Poisson-Boltzmann mean-field description of ionic solutions has been successfully used in predicting charge distributions and interactions between charged macromolecules. While the electrostatic model of charged fluids, on which the Poisson-Boltzmann description rests, and its statistical mechanical consequences have been scrutinized in great detail, much less is understood about its probable shortcomings when dealing with various aspects of real physical, chemical and biological systems. These shortcomings are not only a consequence of the limitations of the mean-field approximation per s…
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.
We have investigated a single charged microgel in aqueous solution with a combined simulational model and Poisson-Boltzmann theory. In the simulations we use a coarse-grained charged bead-spring model in a dielectric continuum, with explicit counterions and full electrostatic interactions under periodic and non-periodic boundary conditions. The Poisson-Boltzmann model is that of a single charged colloid confined to a spherical cell where the counterions are allowed to enter the uniformly charged sphere. We compare the simulational results to those of the Poisson-Boltzmann solution and find goo…
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…
The adsorption of large ions from solution to a charged surface is investigated theoretically. A generalized Poisson--Boltzmann equation, which takes into account the finite size of the ions is presented. We obtain analytical expressions for the electrostatic potential and ion concentrations at the surface, leading to a modified Grahame equation. At high surface charge densities the ionic concentration saturates to its maximum value. Our results are in agreement with recent experiments.
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 …
The Poisson-Boltzmann (PB) theory is extensively used to gain insight on charged colloids and biological systems as well as to elucidate fundamental properties of intermolecular forces. Many works were devoted in the past to study PB related features and to confirm them experimentally. In this work we explore the properties of inter-plate forces in terms of different boundary conditions. We treat the cases of constant surface charge, constant surface potential and mixed boundaries. The interplay between electrostatic interactions, attractive counter-ions release, and repulsive van 't Hoff cont…
The cell-model and its treatment on the Poisson-Boltzmann level are two important concepts in the theoretical description of charged macromolecules. In this brief contribution to the Proceedings of the NATO-ASI on ``Electrostatic Effects in Soft Matter and Biophysics'', which took place in Les Houches from Oct. 1-13, 2000, we provide an introduction to both ideas and summarize a few important results which can be obtained from them. We outline the sequence of approximations which ultimately lead to the cell-model. Then we present two exact results, namely, an expression for the osmotic pressur…
In aqueous solutions, dissolved ions interact strongly with the surrounding water, thereby modifying the solution properties in an ion-specific manner. These ion-hydration interactions can be accounted for theoretically on a mean-field level by including phenomenological terms in the free energy that correspond to the most dominant ion-specific interactions. Minimizing this free energy leads to modified Poisson-Boltzmann equations with appropriate boundary conditions. Here, we review how this strategy has been used to predict some of the ways ion-specific effects can modify the forces acting w…
The structure of cylindrical double layers is studied using a modified Poisson Boltzmann theory and the density functional approach. In the model double layer, the electrode is a cylindrical polyion that is infinitely long, impenetrable, and uniformly charged. The polyion is immersed in a sea of equi-sized rigid ions embedded in a dielectric continuum. An in-depth comparison of the theoretically predicted zeta potentials, the mean electrostatic potentials, and the electrode-ion singlet density distributions is made with the corresponding Monte Carlo simulation data. The theories are seen to be…
The behavior of a non-conductive quasi-planar lipid membrane in an electrolyte and in a static (DC) electric field is investigated theoretically in the nonlinear (Poisson-Boltzmann) regime. Electrostatic effects due to charges in the membrane lipids and in the double layers lead to corrections to the membrane elastic moduli which are analyzed here. We show that, especially in the low salt limit, i) the electrostatic contribution to the membrane's surface tension due to the Debye layers crosses over from a quadratic behavior in the externally applied voltage to a linear voltage regime. ii) the …
Surface sensitive synchrotron-X-ray scattering studies reveal the distributions of monovalent ions next to highly charged interfaces. A lipid phosphate (dihexadecyl hydrogen-phosphate) was spread as a monolayer at the air-water interface, containing CsI at various concentrations. Using anomalous reflectivity off and at the $L_3$ Cs$^+$ resonance, we provide, for the first time, spatial counterion distributions (Cs$^+$) next to the negatively charged interface over a wide range of ionic concentrations. We argue that at low salt concentrations and for pure water the enhanced concentration of hyd…
We propose a simple approximation scheme to compute the effective charge of highly charged colloids (spherical or cylindrical with infinite length). Within non-linear Poisson-Boltzmann theory, we start from an expression of the effective charge in the infinite dilution limit which is asymptotically valid for large salt concentrations; this result is then extended to finite colloidal concentration, approximating the salt partitioning effect which relates the salt content in the suspension to that of a dializing reservoir. This leads to an analytical expression of the effective charge as a funct…
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…
Surface sensitive synchrotron x-ray scattering studies were performed to obtain the distribution of monovalent ions next to a highly charged interface. A lipid phosphate (dihexadecyl hydrogen-phosphate) was spread as a monolayer at the air-water interface to control surface charge density. Using anomalous reflectivity off and at the $L_3 $ Cs$^{+}$ resonance, we provide spatial counterion (Cs$^{+}$) distributions next to the negatively charged interfaces. Five decades in bulk concentrations are investigated, demonstrating that the interfacial distribution is strongly dependent on bulk concentr…
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…
We describe a method to simulate the dynamics of charged colloidal particles suspended in a liquid containing dissociated ions and salt ions. Regimes of prime current interest are those of large volume fraction of colloids, highly charged particles and low salt concentrations. A description which is tractable under these conditions is obtained by treating the small dissociated and salt ions as continuous fields, while keeping the colloidal macroions as discrete particles. For each spatial configuration of the macroions, the electrostatic potential arising from all charges in the system is dete…
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…
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 simulate the self-propulsion of devices in a fluid in the regime of low Reynolds numbers. Each device consists of three bodies (spheres or capsules) connected with two damped harmonic springs. Sinusoidal driving forces compress the springs which are resolved within a rigid body physics engine. The latter is consistently coupled to a 3D lattice Boltzmann framework for the fluid dynamics. In simulations of three-sphere devices, we find that the propulsion velocity agrees well with theoretical predictions. In simulations where some or all spheres are replaced by capsules, we find that the asym…
