Generalized Poisson-Boltzmann Equation Research Articles

Dielectric Boundary Force for the Minimization of the Mean-Field Electrostatic Free-Energy Functional Including the Ionic Size Effect

The generalized Poisson–Boltzmann equation with the uniform ionic size is investigated. In mean-field theory, the ionic size effect is added by modifying the entropy effect of solvent molecules in the electrostatic free-energy functional of ion concentrations. Taking minimization of the mean-field electrostatic free-energy functional including the ionic size effect with respect to the ion concentrations, the minimum free-energy is obtained (see (4.6) in Li [(2009a) “Continuum electrostatics for ionic solutions with non-uniform ionic sizes,” Nonlinearity 22(4), 811–833]. Maximizing the minimum free-energy with respect to the electrostatic potential determines the generalized Poisson–Boltzmann equation. By the definition of shape derivative, the first variation of this minimum free-energy with respect to variation of the dielectric boundary is derived. Thus, the dielectric boundary force of an ionic solution with multiple ionic species with the uniform ionic size is obtained.

Effect of pH and salt on the protonation state of fatty aniline investigated by sum-frequency vibrational spectroscopy.

Langmuir monolayers of fatty aniline (C16-aniline) were investigated using sum-frequency vibrational spectroscopy at various pH levels and NaCl concentrations. To analyze the sum-frequency generation (SFG) spectra of aniline, a multi-peak Lorentzian model, in accordance with the theory of SFG for a charged interface, was applied. First, SFG spectra of C16-aniline from pH 2 to 6 at a constant ionic strength of 10mM (where the phase of the complex potential of the dc-induced signal was suppressed to a few degrees) were fitted with the above-mentioned method. The mean-field theory that considers the chemical equilibrium of the aniline headgroup was used to analyze the fitting results to find that the pKa of aniline is 4.4 ± 0.3. The protonation fraction of the aniline headgroup was estimated to be less than 5% at pH 6 and NaCl concentrations were up to 1M. The generalized Poisson-Boltzmann equation in the Gouy-Chapmann model effectively explained the observed SFG spectra in the OH region for fatty aniline at pH as low as 2, even for the systems without addition of any salt.

Generalized chemical model for plasmas with application to the ionization potential depression

On the basis of the generalized Poisson–Boltzmann equation derived from the Bogolyubov chain of equations for the equilibrium distribution functions in the pair correlation approximation, a general expression is proposed for the Helmholtz free energy of a system that contains any number of components and whose particles interact via arbitrary potentials. This opens up an extraordinary opportunity to simultaneously treat a whole range of physical effects including partial ionization, quantum effects of diffraction and electron degeneracy, short- and long-range interactions of charged particles with neutrals, finite size effects, etc. It is shown that all medium constituents are tied together in a single screening matrix, whose determinant and trace determine the excess contribution to the free energy. The approach developed is then applied to the problem of the ionization potential depression (IPD) leading to quite simple analytical expressions, which turn out to be useful for various practical purposes. In particular, for a single ionization from the neutral state the IPD is shown to significantly depend on the ionization degree such that it consists of the difference of charged and neutral contributions for a fully ionized plasma and turns non-zero for an almost neutral medium. On the other hand, for a multiple ionization process finite size effects of atoms and ions are demonstrated to be of great importance and accounted for in order to achieve good agreement with experimental data on the IPD under warm dense matter conditions.

The Dielectric Boundary Force in Molecular Solvation of the Generalized Poisson–Boltzmann Equation with Ionic Sizes

In molecular solvation, the size-modified electrostatic free energy is investigated. With the uniform ionic and solvent molecular sizes, the generalized Poisson–Boltzmann (PB) equation is considered. The first variation of the size-modified electrostatic free energy with respect to the location variation of the interface is derived. The concept of shape derivative is used to define such variations. The explicit formula of the dielectric boundary force is derived.

Dust-Acoustic Wave Dispersion in Thermal Dusty Plasmas at Weak and Moderate Couplings

The dispersion of dust-acoustic waves (DAWs) in weakly and moderately coupled thermal dusty plasmas is studied in the framework of the linear density-response formalism with the static local-field correction for interdust interactions. The plasma medium composition and the charge of dust particles are simultaneously determined within a recently developed chemical model (Physical Review E, vol. 101, 063203, 2020) based on minimizing the Helmholtz free energy of the system under investigation. Stemming from the generalized Poisson-Boltzmann equation, the renormalization procedure is consistently applied to derive an interdust screened potential that takes into account the finiteness of dust grains. Within the framework of the Ornstein-Zernike relationship in the hypernetted chain approximation, the static structure factor of the dust component is evaluated to manifest the appearance of local extrema on its wavenumber dependence, thereby indicating the short-range order formation in the arrangement of dust particles with respect to one another. It is shown that the DAW dispersion law is completely governed by the static structure factor and, therefore, exhibits a nonmonotonic dependence on the wavenumber as well. In the long-wavelength limit, the acoustic-like behavior of the DAW dispersion is strictly confirmed and the corresponding phase speed, reduced in units of the dust thermal velocity, is ultimately expressed via the static structure factor at zero wavenumber.

Finite electric boundary-layer solutions of a generalized Poisson–Boltzmann equation

We solve the nonlinear Poisson–Boltzmann (P–B) equation of statistical thermodynamics for the external electrostatic potential of a uniformly charged flat plate immersed in an unbounded strong aqueous electrolyte. Our rather general variational formulation yields new solutions for the external potential derived from both the classical Boltzmann distribution and its heuristic Eigen–Wicke modification for concentrated symmetric electrolytes. Electrostatic potentials of these mean-field solutions satisfy a homogeneous condition at a free boundary plane parallel to the electrically conducting plate. The preferred position of this plane, characterizing the outer limit of the charged electrolyte, is determined by minimizing electrostatic free energy of the electrolyte. For a given uniform density of surface charge exceeding a well-defined and experimentally accessible threshold, we show that the generalized nonlinear P–B equation predicts a unique sharp interface separating a charged boundary layer or double layer from electroneutral bulk electrolyte. Sharp electric boundary layers are shown to be an essentially nonlinear phenomenon. In the super-threshold regime, the diffuse Gouy–Chapman solution is inapplicable and thus the Derjaguin–Landau–Verwey–Overbeek analysis, predicting electrostatic repulsion between two sufficiently separated and identically charged parallel plates must be rejected. Similar limitations restrict the applicability of the Grahame equation relating surface charge density to surface potential.

Electroneutrality breakdown and specific ion effects in nanoconfined aqueous electrolytes observed by NMR

Ion distribution in aqueous electrolytes near the interface plays a critical role in electrochemical, biological and colloidal systems, and is expected to be particularly significant inside nanoconfined regions. Electroneutrality of the total charge inside nanoconfined regions is commonly assumed a priori in solving ion distribution of aqueous electrolytes nanoconfined by uncharged hydrophobic surfaces with no direct experimental validation. Here, we use a quantitative nuclear magnetic resonance approach to investigate the properties of aqueous electrolytes nanoconfined in graphitic-like nanoporous carbon. Substantial electroneutrality breakdown in nanoconfined regions and very asymmetric responses of cations and anions to the charging of nanoconfining surfaces are observed. The electroneutrality breakdown is shown to depend strongly on the propensity of anions towards the water-carbon interface and such ion-specific response follows, generally, the anion ranking of the Hofmeister series. The experimental observations are further supported by numerical evaluation using the generalized Poisson-Boltzmann equation.

Finite size effects in the static structure factor of dusty plasmas

Based on the previously developed pseudopotential model of the dust particles interaction, which takes into account both the finite size and screening effects, the equilibrium distribution functions are investigated in a broad range of plasma parameters. The treatment stems entirely from the renormalization theory of plasma particles interactions which leads to the so-called generalized Poisson-Boltzmann equation. In particular, an analytical expression for the static structure factor of the dust particles is proposed and its non-monotonic behavior in the hyper-netted chain approximation is found in a specified domain of plasma parameters to indicate the formation of short- or even long-range order in the system.

Influence of the dielectrophoretic force in mixed electrical double layers

The equilibrium properties of a charged plane immersed in an aqueous electrolyte solution are examined using a generalized Poisson–Boltzmann equation that takes into account the finite ion size by modeling the solution as a suspension of polarizable insulating spheres in water. This formalism is applied to a general solution composed of two or more counterion species with different valences, sizes, and effective permittivity values. It is shown that, due to the dependence of the dielectrophoretic force on the ion size and effective permittivity value, the concentration of the smaller counterion strongly increases while that of the larger one decreases in the immediate vicinity of the charged surface. As a result the surface potential value strongly increases as compared to the usual modified Poisson–Boltzmann theory that only includes steric interactions among ions. This effect is particularly important in the case of mixtures of univalent and divalent counterions, being significant even for relatively low surface charge values.

Generalized Poisson—Boltzmann Equation Taking into Account Ionic Interaction and Steric Effects

Generalized Poisson—Boltzmann equation which takes into account both ionic interaction in bulk solution and steric effects of adsorbed ions has been suggested. We found that, for inorganic cations adsorption on negatively charged surface, the steric effect is not significant for surface charge density < 0.0032 C/dm2, while the ionic interaction is an important effect for electrolyte concentration > 0.15 mol/l in bulk solution. We conclude that for most actual cases the original PB equation can give reliable result in describing inorganic cation adsorption.

Quantum dynamics in continuum for proton transport—Generalized correlation

As a key process of many biological reactions such as biological energy transduction or human sensory systems, proton transport has attracted much research attention in biological, biophysical, and mathematical fields. A quantum dynamics in continuum framework has been proposed to study proton permeation through membrane proteins in our earlier work and the present work focuses on the generalized correlation of protons with their environment. Being complementary to electrostatic potentials, generalized correlations consist of proton-proton, proton-ion, proton-protein, and proton-water interactions. In our approach, protons are treated as quantum particles while other components of generalized correlations are described classically and in different levels of approximations upon simulation feasibility and difficulty. Specifically, the membrane protein is modeled as a group of discrete atoms, while ion densities are approximated by Boltzmann distributions, and water molecules are represented as a dielectric continuum. These proton-environment interactions are formulated as convolutions between number densities of species and their corresponding interaction kernels, in which parameters are obtained from experimental data. In the present formulation, generalized correlations are important components in the total Hamiltonian of protons, and thus is seamlessly embedded in the multiscale/multiphysics total variational model of the system. It takes care of non-electrostatic interactions, including the finite size effect, the geometry confinement induced channel barriers, dehydration and hydrogen bond effects, etc. The variational principle or the Euler-Lagrange equation is utilized to minimize the total energy functional, which includes the total Hamiltonian of protons, and obtain a new version of generalized Laplace-Beltrami equation, generalized Poisson-Boltzmann equation and generalized Kohn-Sham equation. A set of numerical algorithms, such as the matched interface and boundary method, the Dirichlet to Neumann mapping, Gummel iteration, and Krylov space techniques, is employed to improve the accuracy, efficiency, and robustness of model simulations. Finally, comparisons between the present model predictions and experimental data of current-voltage curves, as well as current-concentration curves of the Gramicidin A channel, verify our new model.

Attractive double-layer forces between neutral hydrophobic and neutral hydrophilic surfaces

The interaction between surface patches of proteins with different surface properties has a vital role to play driving conformational changes in proteins in different salt solutions. We demonstrate the existence of ion-specific attractive double-layer forces between neutral hydrophobic and hydrophilic surfaces in the presence of certain salt solutions. This is performed by solving a generalized Poisson-Boltzmann equation for two unequal surfaces. In the calculations, we utilize parametrized ion-surface potentials and dielectric-constant profiles deduced from recent non-primitive-model molecular dynamics simulations that partially account for molecular structure and hydration effects.

Differential geometry based solvation model. III. Quantum formulation

Solvation is of fundamental importance to biomolecular systems. Implicit solvent models, particularly those based on the Poisson-Boltzmann equation for electrostatic analysis, are established approaches for solvation analysis. However, ad hoc solvent-solute interfaces are commonly used in the implicit solvent theory. Recently, we have introduced differential geometry based solvation models which allow the solvent-solute interface to be determined by the variation of a total free energy functional. Atomic fixed partial charges (point charges) are used in our earlier models, which depends on existing molecular mechanical force field software packages for partial charge assignments. As most force field models are parameterized for a certain class of molecules or materials, the use of partial charges limits the accuracy and applicability of our earlier models. Moreover, fixed partial charges do not account for the charge rearrangement during the solvation process. The present work proposes a differential geometry based multiscale solvation model which makes use of the electron density computed directly from the quantum mechanical principle. To this end, we construct a new multiscale total energy functional which consists of not only polar and nonpolar solvation contributions, but also the electronic kinetic and potential energies. By using the Euler-Lagrange variation, we derive a system of three coupled governing equations, i.e., the generalized Poisson-Boltzmann equation for the electrostatic potential, the generalized Laplace-Beltrami equation for the solvent-solute boundary, and the Kohn-Sham equations for the electronic structure. We develop an iterative procedure to solve three coupled equations and to minimize the solvation free energy. The present multiscale model is numerically validated for its stability, consistency and accuracy, and is applied to a few sets of molecules, including a case which is difficult for existing solvation models. Comparison is made to many other classic and quantum models. By using experimental data, we show that the present quantum formulation of our differential geometry based multiscale solvation model improves the prediction of our earlier models, and outperforms some explicit solvation model.

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).

Microphase Separation and Phase Diagram of Concentrated Diblock Copolyelectrolyte Solutions Studied by Self-Consistent Field Theory Calculations in Two-Dimensional Space

The self-consistent field theory (SCFT) is applied to the microphase separation of concentrated solutions of weakly charged polyelectrolytes. The generalized Poisson–Boltzmann equation describing the electrostatic interactions at the mean-field level is numerically solved by a full multigrid algorithm, which enables one to solve the SCFT equations of polyelectrolyte systems in real space as efficient as neutral polymer systems. To demonstrate the power of the real-space numerical scheme, we consider a diblock copolyelectrolyte consisting of a charged block and a neutral block in two-dimensional space. The phase diagram in the Flory–Huggins interaction parameter—the composition space is constructed by numerical calculations. The density distribution of polymer segments, the counterions, and the net charge of the ordered structures, namely the lamellar phase and the hexagonal phase, are intensively examined. The effects of the interaction parameter and the degree of ionization are examined carefully. The numerical scheme can be easily extended to 3D calculations, various chain architectures, various charge distribution models, and other statistics chain models such as the worm-like chain model without losing any computational efficiency.

Quantum dynamics in continuum for proton transport II: Variational solvent–solute interface

Proton transport plays an important role in biological energy transduction and sensory systems. Therefore, it has attracted much attention in biological science and biomedical engineering in the past few decades. The present work proposes a multiscale/multiphysics model for the understanding of the molecular mechanism of proton transport in transmembrane proteins involving continuum, atomic, and quantum descriptions, assisted with the evolution, formation, and visualization of membrane channel surfaces. We describe proton dynamics quantum mechanically via a new density functional theory based on the Boltzmann statistics, while implicitly model numerous solvent molecules as a dielectric continuum to reduce the number of degrees of freedom. The density of all other ions in the solvent is assumed to obey the Boltzmann distribution in a dynamic manner. The impact of protein molecular structure and its charge polarization on the proton transport is considered explicitly at the atomic scale. A variational solute-solvent interface is designed to separate the explicit molecule and implicit solvent regions. We formulate a total free-energy functional to put proton kinetic and potential energies, the free energy of all other ions, and the polar and nonpolar energies of the whole system on an equal footing. The variational principle is employed to derive coupled governing equations for the proton transport system. Generalized Laplace-Beltrami equation, generalized Poisson-Boltzmann equation, and generalized Kohn-Sham equation are obtained from the present variational framework. The variational solvent-solute interface is generated and visualized to facilitate the multiscale discrete/continuum/quantum descriptions. Theoretical formulations for the proton density and conductance are constructed based on fundamental laws of physics. A number of mathematical algorithms, including the Dirichlet-to-Neumann mapping, matched interface and boundary method, Gummel iteration, and Krylov space techniques are utilized to implement the proposed model in a computationally efficient manner. The gramicidin A channel is used to validate the performance of the proposed proton transport model and demonstrate the efficiency of the proposed mathematical algorithms. The proton channel conductances are studied over a number of applied voltages and reference concentrations. A comparison with experimental data verifies the present model predictions and confirms the proposed model.

Self-consistent chemical model of partially ionized plasmas

A simple renormalization theory of plasma particle interactions is proposed. It primarily stems from generic properties of equilibrium distribution functions and allows one to obtain the so-called generalized Poisson-Boltzmann equation for an effective interaction potential of two chosen particles in the presence of a third one. The same equation is then strictly derived from the Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy for equilibrium distribution functions in the pair correlation approximation. This enables one to construct a self-consistent chemical model of partially ionized plasmas, correctly accounting for the close interrelation of charged and neutral components thereof. Minimization of the system free energy provides ionization equilibrium and, thus, permits one to study the plasma composition in a wide range of its parameters. Unlike standard chemical models, the proposed one allows one to study the system correlation functions and thereby to obtain an equation of state which agrees well with exact results of quantum-mechanical activity expansions. It is shown that the plasma and neutral components are strongly interrelated, which results in the short-range order formation in the corresponding subsystem. The mathematical form of the results obtained enables one to both firmly establish this fact and to determine a characteristic length of the structure formation. Since the cornerstone of the proposed self-consistent chemical model of partially ionized plasmas is an effective pairwise interaction potential, it immediately provides quite an efficient calculation scheme not only for thermodynamical functions but for transport coefficients as well.

Differential geometry based solvation model I: Eulerian formulation

This paper presents a differential geometry based model for the analysis and computation of the equilibrium property of solvation. Differential geometry theory of surfaces is utilized to define and construct smooth interfaces with good stability and differentiability for use in characterizing the solvent–solute boundaries and in generating continuous dielectric functions across the computational domain. A total free energy functional is constructed to couple polar and nonpolar contributions to the solvation process. Geometric measure theory is employed to rigorously convert a Lagrangian formulation of the surface energy into an Eulerian formulation so as to bring all energy terms into an equal footing. By optimizing the total free energy functional, we derive coupled generalized Poisson–Boltzmann equation (GPBE) and generalized geometric flow equation (GGFE) for the electrostatic potential and the construction of realistic solvent–solute boundaries, respectively. By solving the coupled GPBE and GGFE, we obtain the electrostatic potential, the solvent–solute boundary profile, and the smooth dielectric function, and thereby improve the accuracy and stability of implicit solvation calculations. We also design efficient second-order numerical schemes for the solution of the GPBE and GGFE. Matrix resulted from the discretization of the GPBE is accelerated with appropriate preconditioners. An alternative direct implicit (ADI) scheme is designed to improve the stability of solving the GGFE. Two iterative approaches are designed to solve the coupled system of nonlinear partial differential equations. Extensive numerical experiments are designed to validate the present theoretical model, test computational methods, and optimize numerical algorithms. Example solvation analysis of both small compounds and proteins are carried out to further demonstrate the accuracy, stability, efficiency and robustness of the present new model and numerical approaches. Comparison is given to both experimental and theoretical results in the literature.

Forces between air-bubbles in electrolyte solution

Ion specificity plays a key role in solution chemistry and many biological processes. However, the classical DLVO theory has not been able to explain the experimentally observed ion specific forces acting between air-bubbles in electrolyte solutions. We resolve this problem by using a generalized Poisson–Boltzmann equation. We demonstrate that inclusion of both short-range potentials obtained from simulation (acting between ions and the air–water interface) and the spatial variation of the local dielectric constant near the air–water interface may be essential to obtain correct results.

Specific Ion Adsorption and Surface Forces in Colloid Science

Mean-field theories that include nonelectrostatic interactions acting on ions near interfaces have been found to accommodate many experimentally observed ion specific effects. However, it is clear that this approach does not fully account for the liquid molecular structure and hydration effects. This is now improved by using parametrized ionic potentials deduced from recent nonprimitive model molecular dynamics (MD) simulations in a generalized Poisson-Boltzmann equation. We investigate how ion distributions and double layer forces depend on the choice of background salt. There is a strong ion specific double layer force set up due to unequal ion specific short-range potentials acting between ions and surfaces.