# Investigation of the Limits of the Linearized Poisson-Boltzmann Equation.

TL;DR: In this article , a comparison between a numerical solution of the Poisson-Boltzmann equation and the analytical solution of its linearized version through the Debye-Hückel equations considering both size-dissimilar and common ion diameters approaches is presented.

Abstract: This work presents a comparison between a numerical solution of the Poisson-Boltzmann equation and the analytical solution of its linearized version through the Debye-Hückel equations considering both size-dissimilar and common ion diameters approaches. In order to verify the limits in which the linearized Poisson-Boltzmann equation is capable to satisfactorily reproduce the nonlinear version of Poisson-Boltzmann, we calculate mean ionic activity coefficients for different types of electrolytes as various temperatures. The divergence between the linearized and full Poisson-Boltzmann equations is higher for lower molalities, and both solutions tend to converge toward higher molalities. For electrolytes of lower valencies (1:1, 1:2, 2:1, and 1:3) and higher distances of closest approach, the full version of the Debye-Hückel equation is capable of representing the activity coefficients with a low divergence from the nonlinear Poisson-Boltzmann. The size-dissimilar full version of Debye-Hückel represents a clear improvement over the extended version that uses only common ion diameters when compared to the numerical solution of the Poisson-Boltzmann equation. We have derived a salt-specific index (Θ) to gradually classify electrolytes in order of increasing influence of nonlinear ion-ion interactions, which differentiate the Debye-Hückel equations from the nonlinear Poisson-Boltzmann equation.

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TL;DR: In this paper , the authors considered the interaction of two dielectric particles of arbitrary shape immersed into a solvent containing a dissociated salt and assuming that the linearized Poisson-Boltzmann equation holds.

Abstract: This work considers the interaction of two dielectric particles of arbitrary shape immersed into a solvent containing a dissociated salt and assuming that the linearized Poisson–Boltzmann equation holds. We establish a new general spherical re-expansion result which relies neither on the conventional condition that particle radii are small with respect to the characteristic separating distance between particles nor on any symmetry assumption. This is instrumental in calculating suitable expansion coefficients for the electrostatic potential inside and outside the objects and in constructing small-parameter asymptotic expansions for the potential, the total electrostatic energy, and forces in ascending order of Debye screening. This generalizes a recent result for the case of dielectric spheres to particles of arbitrary shape and builds for the first time a rigorous (exact at the Debye–Hückel level) analytical theory of electrostatic interactions of such particles at arbitrary distances. Numerical tests confirm that the proposed theory may also become especially useful in developing a new class of grid-free, fast, highly scalable solvers.

3 citations

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TL;DR: In this article , a generalized Debye-Hückel model was proposed to calculate mean activity coefficients of electrolytes in water-methanol mixtures with arbitrary percentage of methanol from 0 to 100%.

Abstract: We propose a generalized Debye–Hückel model from Poisson–Fermi theory to calculate mean activity coefficients of electrolytes in water–methanol mixtures with arbitrary percentage of methanol from 0 to 100%. The model accounts for both short and long ion–ion, ion–water, ion–methanol, and water–methanol interactions, the size effect of all particles, and the dielectric effect of mixed-solvent solutions. We also present a numerical algorithm with mathematical and physical details for using the model to fit experimental data. The model has only 3 empirical parameters to fit the experimental data of NaF, NaCl, and NaBr, for example, in pure-water solutions. It then uses another 3 parameters to calculate the activities of these salts in mixed-solvent solutions for any percentage of methanol. Values of these parameters show mathematical or physical meaning of ionic activities under variable mixing condition and salt concentration. The algorithm can automatically determine optimal values for the 3 fitting parameters without any manual adjustments.

1 citations

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TL;DR: In this article , the performance of four implicit solvent models for associative and two models for non-associative electrolyte solutions has been compared and analyzed for the mean ionic activity coefficient (MIAC) of electrolyte in the solution.

Abstract: The performance of four implicit solvent models for associative and two models for non-associative electrolyte solutions has been compared and analyzed for the mean ionic activity coefficient (MIAC) of electrolyte in the solution. Two of these models for associative electrolyte solutions are based on a chemical approach (Fisher-Levin-Guillot-Guissani (FLGG) and Ebeling-Grigo (EG)), and one of them is based on the reference cavity approximation (Zhou-Yeh-Stell (ZYS)). The last one is based on the thermodynamic perturbation theory (Binding mean spherical approximation, BiMSA). Models without ion-pairing also consist of hard sphere (HS) contribution in addition to the electrostatic contribution from the Debye-Hückel (HS+DH) and the mean spherical approximation (HS+MSA) theories. To this aim, the models’ predictions are compared with the numerical solution of the Poisson-Boltzmann equation, the Monte Carlo simulations, and the experimental data. We have shown that considering the ion pairing results in a better agreement of the predicted MIAC against both simulations and experimental data of 2:2 electrolytes. We have also adjusted the ionic diameter and the distance between ion pairs to the MIAC experimental data and validated the estimated fractions of free ions with the electrical conductivity measurements. We have shown that the FLGG model captures the physics of the system more accurately compared to other models.

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TL;DR: In this article , the author put forward a research on the character image setting of 3D animation works and solved the problem of Poisson equation, and put forward an approach to solve it.

Abstract: Abstract In order to solve the problem of Poisson equation, the author puts forward a research on the character image setting of 3D animation works. 3D animation refers to the three-dimensional virtual image produced by using computer software, also known as 3D animation, is a new technology produced with the development of computer software and hardware technology in recent years. It is the art of photography, set design, and stage lighting reasonable arrangement of various arts and techniques. At the same time, the design and production of 3D animation need more artistic foundation and creativity. A good 3D animation, it requires producers to have a better sense of space and artistic sense, there is must be a good use of all kinds of 3D animation production software. The image design of animated characters from the perspective of content, design styles and styles vary greatly from reality to non-reality. However, whether taken from nature or elsewhere, they are inseparable from their character as a vehicle for expressing human spirit and emotion. Since the birth of animation, its prominent entertainment function has become the value of the existence of this art style, and it has the possibility of development. And animated characters are more iconic than any other element of animation. The purpose of animation character design is to give appeal and vitality to each animation character art.

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TL;DR: Vincze et al. as mentioned in this paper proposed a decoupled II+IW theory that splits the excess chemical potential into two terms corresponding to interactions between ions (II) and interactions between ion and water (IW), which can be computed independently with ϵ(c) being the only link between them.

Abstract: Although Hückel proposed the basic idea of using a concentration dependent dielectric constant, ϵ(c), to compute the activity coefficients of ions in electrolytes in 1925 (Hückel, 1925), a large amount of modeling studies appeared in the literature only after 2010 when we published our II+IW theory (Vincze et al., 2010) that splits the excess chemical potential into two terms corresponding to interactions between ions (II) and interactions between ions and water (IW). In this approach, the two terms are decoupled, which means that they can be computed independently with ϵ(c) being the only link between them. Here, we review our theory and other works based on Hückel’s suggestion by discussing several issues that are partly cornerstones of the theory, partly make it possible to put the theory into a larger context. These issues include the role of ϵ(c) and the ionic radii used in the II and IW term, the statistical mechanical methods to estimate the II term, the existence and interpretation of individual activities, phenomena associated with strong ionic correlations in multivalent electrolytes, and explicit-water models used in molecular dynamics simulations.

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3,131 citations

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TL;DR: In this paper, the authors use the calculus of variations to provide a unique definition of the total energy and to obtain expressions for the total electrostatic free energy for various forms of the Poisson-Boltzmann (PB) equation.

Abstract: The Poisson-Boltzmann (PB) equation is enjoying a resurgence in popularity and usefulness in biophysics and biochemistry due to numerical advances which allow the equation to be rapidly solved for arbitrary geometries and nonuniform dielectrics. The great simplification of PB models is to use the mean electrostatic potential to give an estimate of the potential of mean force (PMF) governing the distribution of the mobile ions in the solvent. This approximation enables both the mean potential and mean ion distribution to be obtained directly from solutions to the PB equation without performing complex statistical mechanical integrations. The nonlinear form of the PB equation has greater accuracy and range of validity than the linear form, but the approximation of the PMF by the mean potential creates theoretical difficulties in defining the total electrostatic energy for the former. In this paper we use the calculus of variations to provide a unique definition of the total energy and to obtain expressions for the total electrostatic free energy for various forms of the PB equation. These expressions involve energy density integrals over the volume of the system. Various equivalent expressions for the total energy are given and the physical meaning of the different terms that appear is discussed. Numerical calculations are carried out to demonstrate the feasibility of our approach and to assess the magnitude of the various terms that arise in the theory. Both the more familiar charging integral and the energy density integral methods can be applied to the PB equation with equal accuracy, but the latter is much more efficient computationally. The energy density integral involves the integral of the excess osmotic pressure of the ion atmosphere. The various forms of the PB equation which have been most widely discussed to date because of the availability of analytical solutions are shown to be special cases where the osmotic term is absent.

551 citations

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TL;DR: The contribution of higher-order electrostatic terms (beyond the Debye-Huckel approximation) to the thermodynamic properties of mixed and pure electrolytes is investigated in this article.

Abstract: The contribution of higher-order electrostatic terms (beyond the Debye-Huckel approximation) to the thermodynamic properties of mixed and pure electrolytes is investigated. It is found that these effects are important for cases of unsymmetrical mixing, especially when one ion has a charge of three units or more. The appropriate correction can be made by a purely electrostatic function since the mutual repulsion of ions of the same sign keeps them far enough apart that short-range forces have little effect. This function is evaluated, and several convenient approximations are also given. Application is made to systems mixing ions of the type 1–2 and 1–3. Higher-order limiting laws exist for symmetrical mixtures and for pure, unsymmetrical solutes, but these effects were not found to be significant in relationship to existing activity or osmotic-coefficient data.

407 citations

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TL;DR: This work is a review of the Poisson–Boltzmann (PB) continuum electrostatics theory and its modifications, with a focus on salt effects and counterion binding, and discusses the conventional PB equation, the corresponding functionals of the electrostatic free energy, including a connection to DFT.

Abstract: This work is a review of the Poisson-Boltzmann (PB) continuum electrostatics theory and its modifications, with a focus on salt effects and counterion binding. The PB model is one of the mesoscopic theories that describes the electrostatic potential and equilibrium distribution of mobile ions around molecules in solution. It serves as a tool to characterize electrostatic properties of molecules, counterion association, electrostatic contributions to solvation, and molecular binding free energies. We focus on general formulations which can be applied to large molecules of arbitrary shape in all-atomic representation, including highly charged biomolecules such as nucleic acids. These molecules present a challenge for theoretical description, because the conventional PB model may become insufficient in those cases. We discuss the conventional PB equation, the corresponding functionals of the electrostatic free energy, including a connection to DFT, simple empirical extensions to this model accounting for finite size of ions, the modified PB theory including ionic correlations and fluctuations, the cell model, and supplementary methods allowing to incorporate site-bound ions in the PB calculations.

306 citations