Poisson-Boltzmann theory of the charge-induced adsorption of semi-flexible polyelectrolytes

A model is suggested for the structure of an adsorbed layer of a highly charged semi-flexible polyelectrolyte on a weakly charged surface of opposite charge sign. The adsorbed phase is thin, owing to the effective reversal of the charge sign of the surface upon adsorption, and ordered, owing to the high surface density of polyelectrolyte strands caused by the generally strong binding between polyelectrolyte and surface. The Poisson-Boltzmann equation for the electrostatic interaction between the array of adsorbed polyelectrolytes and the charged surface is solved for a cylindrical geometry, both numerically, using a finite element method, and analytically within the weak curvature limit under the assumption of excess monovalent salt. For small separations, repulsive surface polarization and counterion osmotic pressure effects dominate over the electrostatic attraction and the resulting electrostatic interaction curve shows a minimum at nonzero separations on the Angstrom scale. The equilibrium density of the adsorbed phase is obtained by minimizing the total free energy under the condition of equality of chemical potential and osmotic pressure of the polyelectrolyte in solution and in the adsorbed phase. For a wide range of ionic conditions and charge densities of the charged surface, the interstrand separation as predicted by the Poisson-Boltzmann model and the analytical theory closely agree. For low to moderate charge densities of the adsorbing surface, the interstrand spacing decreases as a function of the charge density of the charged surface. Above about 0.1 M excess monovalent salt, it is only weakly dependent on the ionic strength. At high charge densities of the adsorbing surface, the interstrand spacing increases with increasing ionic strength, in line with the experiments by Fang and Yang [J. Phys. Chem. B 101, 441 (1997)].     

Corrected Debye-Hückel analysis of surface complexation; III. Spherical particle charging including ion condensation

Statistical mechanics has been used to derive a model for the charging of a spherical particle in a salt solution to complement our experimental studies and gain a deeper understanding of the processes involved in surface complexation. Our chosen model goes beyond the equilibrium constants and the Gouy-Chapmann theory currently used in surface complexation models. The proton adsorption is taken to occur at a harmonic potential well on the surface characterized by a frequency v and a well depth u(0). Outside the particle surface there is a capacitor layer of width w(c) which is impenetrable to the salt ions. The diffuse screening of the charged particle is described by a corrected Debye-Hückel analysis accounting for ion size in the ion-ion interactions. To account also for nonlinear electrostatic response a layer of condensed counterions has been introduced. The criterion for the onset of ion condensation is that the electrostatic field exceeds a linear response criterion. Ion size effects are accounted for in terms of hole-corrected electrostatic energies and excluded volume. The model has been applied to titrated surface charge data on goethite (alpha-FeOOH) at various background concentrations and good agreement between the experimental data and the model was obtained. Both the size of the screening ions and the central particle size were shown to be of importance for the surface charge.     

SMPBS: Web server for computing biomolecular electrostatics using finite element solvers of size modified Poisson‐Boltzmann equation

SMPBS (Size Modified Poisson‐Boltzmann Solvers) is a web server for computing biomolecular electrostatics using finite element solvers of the size modified Poisson‐Boltzmann equation (SMPBE). SMPBE not only reflects ionic size effects but also includes the classic Poisson‐Boltzmann equation (PBE) as a special case. Thus, its web server is expected to have a broader range of applications than a PBE web server. SMPBS is designed with a dynamic, mobile‐friendly user interface, and features easily accessible help text, asynchronous data submission, and an interactive, hardware‐accelerated molecular visualization viewer based on the 3Dmol.js library. In particular, the viewer allows computed electrostatics to be directly mapped onto an irregular triangular mesh of a molecular surface. Due to this functionality and the fast SMPBE finite element solvers, the web server is very efficient in the calculation and visualization of electrostatics. In addition, SMPBE is reconstructed using a new objective electrostatic free energy, clearly showing that the electrostatics and ionic concentrations predicted by SMPBE are optimal in the sense of minimizing the objective electrostatic free energy. SMPBS is available at the URL: smpbs.math.uwm.edu © 2017 Wiley Periodicals, Inc.     

Finite volume solution of the modified Poisson-Boltzmann equation for two colloidal particles

The double layer forces between spherical colloidal particles, according to the Poisson-Boltzmann (PB) equation, have been accurately calculated in the literature. The classical PB equation takes into account only the electrostatic interactions, which play a significant role in colloid science. However, there are at, and above, biological salt concentrations other non-electrostatic ion specific forces acting that are ignored in such modelling. In this paper, the electrostatic potential profile and the concentration profile of co-ions and counterions near charged surfaces are calculated. These results are obtained by solving the classical PB equation and a modified PB equation in bispherical coordinates, taking into account the van der Waals dispersion interactions between the ions and both surfaces. Once the electrostatic potential is known we calculate the double layer force between two charged spheres. This is the first paper that solves the modified PB equation in bispherical coordinates. It is also the first time that the finite volume method is used to solve the PB equation in bispherical coordinates. This method divides the calculation domain into a certain number of sub-domains, where the physical law of conservation is valid, and can be readily implemented. The finite volume method is implemented for several geometries and when it is applied to solve PB equations presents low computational cost. The proposed method was validated by comparing the numerical results for the classical PB calculations with previous results reported in the literature. New numerical results using the modified PB equation successfully predicted the ion specificity commonly observed experimentally.     

Treating ion distribution with Gaussian‐based smooth dielectric function in DelPhi

The standard treatment of ions in the framework of the Poisson–Boltzmann equation relies on molecular surfaces, which are commonly constructed along with the Stern layer. The molecular surface determines where ions can be present. In the Gaussian‐based smooth dielectric function in DelPhi, smooth boundaries between the solute and solvent take the place of molecular surface. Therefore, this invokes the question of how to model mobile ions in the water phase without a definite solute‐solvent boundary. This article reports a natural extension of the Gaussian‐based smooth dielectric function approach that treats mobile ions via Boltzmann distribution with an added desolvation penalty. Thus, ion concentration near macromolecules is governed by the local electrostatic potential and the desolvation penalty (from being partially desolvated). The approach is tested against the experimental salt dependence of binding free energy on 7 protein–protein complexes and 12 DNA–protein complexes, resulting in Pearson correlations of 0.95 and 0.88, respectively. © 2017 Wiley Periodicals, Inc.     

Hybrid boundary element and finite difference method for solving the nonlinear Poisson-Boltzmann equation

A hybrid approach for solving the nonlinear Poisson-Boltzmann equation (PBE) is presented. Under this approach, the electrostatic potential is separated into (1) a linear component satisfying the linear PBE and solved using a fast boundary element method and (2) a correction term accounting for nonlinear effects and optionally, the presence of an ion-exclusion layer. Because the correction potential contains no singularities (in particular, it is smooth at charge sites) it can be accurately and efficiently solved using a finite difference method. The motivation for and formulation of such a decomposition are presented together with the numerical method for calculating the linear and correction potentials. For comparison, we also develop an integral equation representation of the solution to the nonlinear PBE. When implemented upon regular lattice grids, the hybrid scheme is found to outperform the integral equation method when treating nonlinear PBE problems. Results are presented for a spherical cavity containing a central charge, where the objective is to compare computed 1D nonlinear PBE solutions against ones obtained with alternate numerical solution methods. This is followed by examination of the electrostatic properties of nucleic acid structures.     

A modified Poisson-Boltzmann model including charge regulation for the adsorption of ionizable polyelectrolytes to charged interfaces, applied to lysozyme adsorption on silica

The equilibrium adsorption of polyelectrolytes with multiple types of ionizable groups is described using a modified Poisson-Boltzmann equation including charge regulation of both the polymer and the interface. A one-dimensional mean-field model is used in which the electrostatic potential is assumed constant in the lateral direction parallel to the surface. The electrostatic potential and ionization degrees of the different ionizable groups are calculated as function of the distance from the surface after which the electric and chemical contributions to the free energy are obtained. The various interactions between small ions, surface and polyelectrolyte are self-consistently considered in the model, such as the increase in charge of polyelectrolyte and surface upon adsorption as well as the displacement of small ions and the decrease of permittivity. These interactions may lead to complex dependencies of the adsorbed amount of polyelectrolyte on pH, ionic strength, and properties of the polymer (volume, permittivity, number, and type of ionizable groups) and of the surface (number of ionizable groups, pK, Stern capacity). For the adsorption of lysozyme on silica, the model qualitatively describes the gradual increase of adsorbed amount with pH up to a maximum value at pHc, which is below the iso-electric point, as well as the sharp decrease of adsorbed amount beyond pHc. With increasing ionic strength the adsorbed amount decreases (for pH > pHc), and pHc shifts to lower values.     

The Poisson-Boltzmann model for tRNA: Assessment of the calculation set-up and ionic concentration cutoff

Using tRNA molecule as an example, we evaluate the applicability of the Poisson-Boltzmann model to highly charged systems such as nucleic acids. Particularly, we describe the effect of explicit crystallographic divalent ions and water molecules, ionic strength of the solvent, and the linear approximation to the Poisson-Boltzmann equation on the electrostatic potential and electrostatic free energy. We calculate and compare typical similarity indices and measures, such as Hodgkin index and root mean square deviation. Finally, we introduce a modification to the nonlinear Poisson-Boltzmann equation, which accounts in a simple way for the finite size of mobile ions, by applying a cutoff in the concentration formula for ionic distribution at regions of high electrostatic potentials. We test the influence of this ionic concentration cutoff on the electrostatic properties of tRNA.     

Poisson-Boltzmann description of the electrical double layer including ion size effects

The electrical double layer is examined using a generalized Poisson-Boltzmann equation that takes into account the finite ion size by modeling the aqueous electrolyte solution as a suspension of polarizable insulating spheres in water. We find that this model greatly amplifies the steric effects predicted by the usual modified Poisson-Boltzmann equation, which imposes only a restriction on the ability of ions to approach one another. This amplification should allow for an interpretation of the experimental results using reasonable effective ionic radii (close to their well-known hydrated values).     

Effects of ion solvation on phase equilibrium and interfacial tension of liquid mixtures

We study the bulk thermodynamics and interfacial properties of electrolyte solution mixtures by accounting for electrostatic interaction, ion solvation, and inhomogeneity in the dielectric medium in the mean-field framework. Difference in the solvation energy between the cations and anions is shown to give rise to local charge separation near the interface, and a finite Galvani potential between two coexisting solutions. The ion solvation affects the phase equilibrium of the solvent mixture, depending on the dielectric constants of the solvents, reflecting the competition between the solvation energy and translation entropy of the ions. Miscibility is decreased if both solvents have low dielectric constants and is enhanced if both solvents have high dielectric constant. At the mean-field level, the ion distribution near the interface is determined by two competing effects: accumulation in the electrostatic double layer and depletion in a diffuse interface. The interfacial tension shows a nonmonotonic dependence on the salt concentration: it increases linearly with the salt concentration at higher concentrations and decreases approximately as the square root of the salt concentration for dilute solutions, reaching a minimum near 1 mM. We also find that, for a fixed cation type, the interfacial tension decreases as the size of anion increases. These results offer qualitative explanations within one unified framework for the long-known concentration and ion size effects on the interfacial tension of electrolyte solutions.     

Application of the thin-shell formulation to the numerical modeling of Stern layer in biomolecular electrostatics

In this article, the thin-shell formulation is applied to efficiently modeling the Stern layer within computational algorithms oriented toward the boundary element solution of the linearized Poisson-Boltzmann equation. The attention is focused on the calculation of the electrostatic potential in proximity to a biomolecule immersed in an electrolyte medium. Following the proposed approach, the Stern layer is made to collapse to a zero-thickness region (two-dimensional surface) with interface conditions linking the electrostatic potential over the molecular and the bulk ion accessible surfaces. Advantages lie in the limitation of divergent integral problems and in the halving of the unknown number, with a significant impact on computational time and memory requirements when modeling large biomolecules.     

Density-functional theory of spherical electric double layers and zeta potentials of colloidal particles in restricted-primitive-model electrolyte solutions

A density-functional theory is proposed to describe the density profiles of small ions around an isolated colloidal particle in the framework of the restricted primitive model where the small ions have uniform size and the solvent is represented by a dielectric continuum. The excess Helmholtz energy functional is derived from a modified fundamental measure theory for the hard-sphere repulsion and a quadratic functional Taylor expansion for the electrostatic interactions. The theoretical predictions are in good agreement with the results from Monte Carlo simulations and from previous investigations using integral-equation theory for the ionic density profiles and the zeta potentials of spherical particles at a variety of solution conditions. Like the integral-equation approaches, the density-functional theory is able to capture the oscillatory density profiles of small ions and the charge inversion (overcharging) phenomena for particles with elevated charge density. In particular, our density-functional theory predicts the formation of a second counterion layer near the surface of highly charged spherical particle. Conversely, the nonlinear Poisson-Boltzmann theory and its variations are unable to represent the oscillatory behavior of small ion distributions and charge inversion. Finally, our density-functional theory predicts charge inversion even in a 1:1 electrolyte solution as long as the salt concentration is sufficiently high.     

A new outer boundary formulation and energy corrections for the nonlinear Poisson-Boltzmann equation

The nonlinear Poisson-Boltzmann equation (PBE) has been successfully used for the prediction of numerous electrostatic properties of highly charged biopolyelectrolytes immersed in aqueous salt solutions. While numerous numerical solvers for the 3D PBE have been developed, the formulation of the outer boundary treatments used in these methods has only been loosely addressed, especially in the nonlinear case. The de facto standard in current nonlinear PBE implementations is to either set the potential at the outer boundaries to zero or estimate it using the (linear) Debye-Hückel (DH) approximation. However, an assessment of how these outer boundary treatments affect the overall solution accuracy does not appear to have been previously made. As will be demonstrated here, both approximations can, under certain conditions, produce completely erroneous estimates of the potential and energy salt dependencies. A related concern for calculations carried out on grids of finite extent (e.g., all current finite difference and finite element implementations) is the contribution to the energy and salt dependence from the exterior region outside the computational grid. This too is shown to be significant, especially at low salt concentration where essentially all of the contributions to the excess osmotic pressure and ion stress energies originate from this exterior region. In this paper the authors introduce a new outer boundary treatment that is valid for both the linear and nonlinear PBE. The authors also formulate energy corrections to account for contributions from outside the computational domain. Finally, the authors also consider the effects of general ion exclusion layers upon biomolecular electrostatics. It is shown that while these layers tend to increase the surface electrostatic potential, under physiological salt conditions and high net charges their effect on the excess osmotic pressure term, which is a measure of the salt dependence of the total electrostatic free energy, is weak. To facilitate presentation and allow very fine resolutions and/or large computational domains to be considered, attention is restricted to the 1D spherically symmetric nonlinear PBE. Though geometrically limited, the modeling principles nevertheless extend to general PBE solvers as discussed in the Appendix. The 1D model can also be used to benchmark and validate the salt effect prediction capabilities of existing PBE solvers.     

A Novel Charge Distribution Method for the Numerical Solution of the Poisson-Boltzmann Equation

A charge distribution method to solve the linearized Poisson-Boltzmann equation numerically through use of the finite difference method is proposed, The molecules are mapped by 1 and 0.25 Å grid systems. Each atom is modeled as a point charge and a weighted sum of point charge of every atom that is within its van der Waals radius with a grid point is assigned to the grid point. Depending on a charge distribution factor determined, the charge/grid (q/g) ratio calculated for every grid point inside a molecule can be fixed to a certain value. A grid size of the I Å grid is often fixed for mapping a small or large molecular system. Solvation energies for a group of small molecules calculated by the method arc comparable with those calculated by other methods and the grid energy calculated by the method is also reduced.     

Free energy partitioning analysis of the driving forces that determine ion density profiles near the water liquid-vapor interface

Free energy partitioning analysis is employed to explore the driving forces for ions interacting with the water liquid-vapor interface using recently optimized point charge models for the ions and SPC/E water. The Na(+) and I(-) ions are examined as an example kosmotrope/chaotrope pair. The absolute hydration free energy is partitioned into cavity formation, attractive van der Waals, local electrostatic, and far-field electrostatic contributions. We first compute the bulk hydration free energy of the ions, followed by the free energy to insert the ions at the center of a water slab. Shifts of the ion free energies occur in the slab geometry consistent with the SPC/E surface potential of the water liquid-vapor interface. Then the free energy profiles are examined for ion passage from the slab center to the dividing surface. The profiles show that, for the large chaotropic I(-) ion, the relatively flat total free energy profile results from the near cancellation of several large contributions. The far-field electrostatic part of the free energy, largely due to the water liquid-vapor interface potential, has an important effect on ion distributions near the surface in the classical model. We conclude, however, that the individual forms of the local and far-field electrostatic contributions are expected to be model dependent when comparing classical and quantum results. The substantial attractive cavity free energy contribution for the larger I(-) ion suggests that there is a hydrophobic component important for chaotropic ion interactions with the interface.     

Adsorption of humic acid on goethite: isotherms, charge adjustments, and potential profiles

The adsorption of natural organic matter (NOM) on mineral (hydr)oxide plays an important role in the evaluation of the speciation of toxic metal ions in the environment. Because both NOM and mineral oxide have variable charges that adjust upon adsorption, a good understanding of proton binding is required before the binding of metal ions can be understood. In this study, the adsorption of purified Aldrich humic acid (PAHA) on goethite was examined as a function of the environmental conditions (pH, salt concentration, and free concentration of PAHA) together with the proton adsorption to PAHA, goethite, and their mixtures. The induced charges on both components were separated on the basis of the difference between the charge/pH curves of the mixture and those of the single components. The electrostatic potential profile across the adsorbed layer was obtained as a numerical solution of the Poisson-Boltzmann equation using the charge density of the adsorbed PAHA and the goethite surface. From the quantitative evaluation of the induced charge on both components, it is revealed that the degree of the charge adjustment is related to the electrostatic affinity between the PAHA segments and the goethite surface, the electrostatic repulsion between the PAHA segments, and the electrostatic shielding by salt ions. Considering the charge distribution of the adsorbed PAHA at the goethite surface, it is concluded that the change of the charge adjustment is sensitive to that of the conformation of the adsorbed PAHA. From the detailed inspection of the assumptions made and the comparison with the reported theoretical calculations, the obtained potential profiles are considered to broadly reflect the true potential profiles. Because a charge adjustment is not frequently considered in detail in relation to the NOM adsorption on metal (hydr)oxides, the obtained results can form the basis for the further development of modeling of the adsorption of NOM on (hydr)oxide surfaces.     

Ion-specific hydration effects: Extending the Poisson-Boltzmann theory

In aqueous solutions, dissolved ions interact strongly with the surrounding water and surfaces, thereby modifying 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 within and between charged interfaces immersed in salt solutions.     

On the relationship between the chemical bond energy and the electron energy levels of strongly ionic compounds

The analysis of experimental data suggests that the energy levels of ions in a crystal shift rigidly by an amount equal to the Madelung energy (electrostatic bond energy) maintaining the same separation as in the free ion. This leads to the conclusion that the electrostatic bonding energies of ions in a crystal is taken up partly by the orbital electrons and partly by the nuclear charge.     

The water-amorphous silica interface: analysis of the Stern layer and surface conduction

To explain why dynamical properties of an aqueous electrolyte near a charged surface seem to be governed by a surface charge less than the actual one, the canonical Stern model supposes an interfacial layer of ions and immobile fluid. However, large ion mobilities within the Stern layer are needed to reconcile the Stern model with surface conduction measurements. Modeling the aqueous electrolyte-amorphous silica interface at typical charge densities, a prototypical double layer system, the flow velocity does not vanish until right at the surface. The Stern model is a good effective model away from the surface, but cannot be taken literally near the surface. Indeed, simulations show no ion mobility where water is immobile, nor is such mobility necessary since the surface conductivity in the simulations is comparable to experimental values.