Poisson-Boltzmann Approximation Research Articles

Polyelectrolyte Brushes with Protein-Like Nanocolloids.

Electrostatic interaction of ampholytic nanocolloidal particles (NPs), which mimic globular proteins, with polyelectrolyte brushes is analyzed within mean-field Poisson-Boltzmann approximation. In accordance with experimental findings, the theory predicts that an electrostatic driving force for the particle uptake by the brush may emerge when the net charge of the particle in the buffer and the charge of the brush are of the same sign. The origin of this driving force is change in the ionization state of weak cationic and anionic groups on the NP surface provoked by interaction with the brush. In experimental systems, the ionic interactions are complemented by excluded-volume, hydrophobic, and other types of interactions that all together control NP uptake by or expulsion from the brush. Here, we focus on the NP-brush ionic interactions. It is demonstrated that deviation between the buffer pH and the NP isoelectric point, considered usually as the key control parameter, does not uniquely determine the insertion free energy patterns. The latter depends also on the proportion of cationic and anionic groups in the NPs and their specific ionization constants as well as on salt concentration in the buffer. The analysis of the free energy landscape proves that a local minimum in the free energy inside the brush appears, provided the NP charge reversal occurs upon insertion into the brush. This minimum corresponds either to a thermodynamically stable or to a metastable state, depending on the pH offset from the IEP and salt concentration, and is separated from the bulk of the solution by a free energy barrier. The latter, being fairly independent of salt concentration in height, may strongly impede the NP absorption kinetically even when it is thermodynamically favorable. Hence, change reversal is a necessary but insufficient condition for the uptake of the NPs by similarly charged polyelectrolyte brushes.

Pressure and isothermal compressibility of complex plasmas in the average Wigner–Seitz cell model with regard to nonlinear screening effect

AbstractWe consider a two‐component electroneutral equilibrium complex plasma of finite‐sized macroions (each with a charge , ) and point oppositely charged microions with unit charges. We take into account a macroions nonlinear screening effect and obtain system pressure in the Wigner–Seitz average cell model within the framework of the Poisson–Boltzmann approximation. It is shown that the obtained pressure and isothermal compressibility are positive over the entire range of macroions concentrations.

Choice of DLVO approximation method for quantifying the affinity between latex particles and membranes

The Derjaguin-Landau-Verwey-Overbeek (DLVO) model, as well as the extended model (XDLVO), is popularly employed to quantify the interfacial interactions underlying membrane fouling. However, disagreements between membrane fouling extent and the interaction energies derived from DLVO or XDLVO models have been reported, which suggests gaps in the understanding. This study demonstrates that the DLVO approximation methods for predicting the interfacial foulant-membrane interaction are sensitive to the boundary condition assumptions (e.g., constant charge versus constant potential). In particular, while both the Poisson-Boltzmann (P–B) and linear superposition approximation (LSA) equations can quantify the electrostatics (EL) interaction energy component, the former assumes constant potential, while the latter additionally considers constant charge scenarios. The relative accuracy of these two equations were evaluated here. For dead-end filtration tests, flux decline trends, OCT analysis results and fouling model parameters were obtained. Regarding fouling of pristine PCTE membrane by latex particles of opposite charge signs, both the P–B and LSA equations contribute to predict relative fouling extents correctly. However, for the case of the BPEI-coated membrane, the P–B equation failed whereas the LSA equation gave good agreements. For cross-flow filtration tests in organic solvents, LSA also out-performed the P–B equation in providing more accurate predictions of membrane fouling. The results here highlight the shortcomings in the commonly used P–B equation and are expected to be potentially valuable in the development of better approximations for quantifying interfacial interaction energies.

Virtual cell model for osmotic pressure calculation of charged biomolecules.

The osmotic pressure of dilute electrolyte solutions containing charged macro-ions as well as counterions can be computed directly from the particle distribution via the well-known cell model. Originally derived within the Poisson-Boltzmann mean-field approximation, the cell model considers a single macro-ion centered into a cell, together with counterions needed to neutralize the total cell charge, while it neglects the phenomena due to macro-ion correlations. While extensively applied in coarse-grained Monte Carlo (MC) simulations of continuum solvent systems, the cell model, in its original formulation, neglects the macro-ion shape anisotropy and details of the surface charge distribution. In this paper, by comparing one-body and two-body coarse-grained MC simulations, we first establish an upper limit for the assumption of neglecting correlations between macro-ions, and second, we validate the approximation of using a non-spherical macro-ion. Next, we extend the cell model to all-atom molecular dynamics simulations and show that protein concentration-dependent osmotic pressures can be obtained by confining counterions in a virtual, spherical subspace defining the protein number density. Finally, we show the possibility of using specific interaction parameters for the protein-ion and ion-ion interactions, enabling studies of protein concentration-dependent ion-specific effects using merely a single protein molecule.

Thermodynamics of complex plasmas with two different sorts of macroions

Three-component electroneutral systems of finite-size classical macroions with two different charge numbers Z 1 ≫ 1 and Z 2 ≫ 1 and point-like oppositely charged microions are analyzed. Free energy of a mixture of two sorts of macroions is estimated within the Wigner–Seitz cells approximation. The non-linear screening effect is taken into account via the Poisson–Boltzmann approximation within the both cells. The equality of microions pressures at the boundary between the co-existing cells with the different sorts of macroions is used as an equilibrium condition. The difference between the total Helmholtz free energy in equilibrium is shown in comparison with the situation when the Wigner–Seitz cells with macroions with the different charges have the same volumes.

Twisting of Charged Nanoribbons to Helicoids Driven by Electrostatics.

Charged amphiphiles in solution usually self-assemble into flat nanoribbons that spontaneously twist into different shapes. The role of electrostatics in this process is still under strong debate. This work studies the electrostatic free energy of twisting a nanoribbon at the level of the nonlinear Poisson-Boltzmann approximation. It is shown that helicoid-shaped ribbons are more stable than flat ribbons, while other shapes under consideration (cylindrical helixes and bent ribbons) are always less stable than the flat ribbon. The unexpected electrostatics-driven twisting of the ribbon into a helicoid is ascribed to the increase in its perimeter with increasing degree of twisting, as charges near the edge of the ribbon are electrostatically more stable than those near its center. This argument successfully explains the effects of salt concentration and the width of the ribbon on the optimal twisting period and allows us to approximately describe the problem of ribbon twisting in terms of two dimensionless variables that combine the helicoid twisting period, the Debye length of the solution, and the width of the ribbon. The magnitude of the electrostatic twisting energy predicted by our calculations is comparable to that of restoring elastic forces for typical ribbons of self-assembled amphiphiles, which indicates that electrostatics plays an important role in determining the equilibrium shape of charged nanoribbons.

The urokinase plasminogen activator binding to its receptor: a quantum biochemistry description within an in/homogeneous dielectric function framework with application to uPA-uPAR peptide inhibitors.

Despite being recognized as a therapeutic target in the processes of cancer cell proliferation and metastasis for over 50 years, the interaction of the urokinase plasminogen activator uPA with its receptor uPAR still needs an improved understanding. High resolution crystallographic data (PDB ) of the uPA-uPAR binding geometry was used to perform quantum biochemistry computations within the density functional theory (DFT) framework. A divide to conquer methodology considering a mixed homogeneous/inhomogeneous dielectric model and explicitly taking water molecules into account was employed to obtain a large set of uPA-uPAR residue-residue interaction energies. In order of importance, not only were Phe25 > Tyr24 > Trp30 > Ile28 shown to be the most relevant uPA residues binding it to uPAR, but the residues Lys98 > His87 > Gln40 > Asn22 > Lys23 > Val20 also had significant interaction energies, which helps to explain published experimental mutational data. Furthermore, the results obtained with the uPA-uPAR in/homogeneous dielectric function show that a high dielectric constant value ε = 40 is adequate to take into account the electrostatic environment at the interface between the proteins, while using a smaller value of ε (<10) leads to an overestimation of the uPA-uPAR binding energy. Hot spots of the uPA-uPAR binding domain were identified and a quantum biochemistry description of the uPAR blockers uPA21-30 and cyclo21,29uPA21-29[(S21C;H29C)] was performed, demonstrating that cyclization improves the stability of mimetic peptides without compromising their binding energies to uPAR.

Modeling Gel Swelling Equilibrium in the Mean Field: From Explicit to Poisson-Boltzmann Models

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 us to compute all pressure terms within the Poisson-Boltzmann approximation. This model, albeit computationally cheap, yields excellent predictions of swelling equilibria under varying chain length, polymer charge fraction, and external reservoir salt concentrations when compared to coarse-grained molecular dynamics simulations of charged macrogels. We present an extension of the model to the experimentally relevant cases of pH-sensitive gels.

Effect of the non‐linear screening on a modification of the Debye–Hückel plus hole approximation in complex plasma

The authors analyse two‐component electroneutral systems of classical macroions of finite size and point‐like oppositely charged microions. This article deals with the modification of the Debye–Hückel plus hole approximation when a non‐linear screening effect is taken into account in a complex plasma. Parameters of non‐linear screening of the macroions by surrounding microions are calculated within the framework of the Poisson–Boltzmann approximation. Two effects are found as a result of such calculations: (a) subdivision of all microions into two subclasses, free microions and bound microions and (b) a significant reduction of an effective charge Z* of the macroion in comparison with its true value Z due to the non‐linear screening by a thin high‐density envelope of the bound microions. We show that the value of a non‐ideal portion of an internal energy differs considerably in the case when the non‐linear screening effect is taken into account in the vicinity of the macroion.

Effect of non-linear screening on thermodynamic properties of complex plasma

This work is devoted to the analysis of the applicability range of a basic assumption in the phase diagram of complex plasma, i.e., linearized (Debye) screening of macroions by microions, which leads to the Yukawa form of effective interactions between macroions. Parameters of non-linear screening for macroions were calculated within the direct Poisson–Boltzmann approximation. Two effects were revealed as a result of such calculations: (i) decomposition of all microions into two subclasses, free and bound ones, and (ii) significant reduction of an effective charge Z* of the initial bare macroion Z under non-linear screening by a small high-density envelope of the bound microions. The effective charge Z* grows in the direct proportion to Z first and then, the change of Z* is negligibly thin. This renormalization of the initial Z and macroion concentration at the border of the cell leads to corresponding renormalization of initial Γ and κ into Γ* and κ* (Γ* < Γ and κ* < κ). The corresponding calculated shifts of excess internal energy are discussed and illustrated.

Non-linear charge screening and interaction energy of macroions in complex plasma

This paper analyzes the applicability range in the phase diagram of complex plasma, i.e., linearized (Debye) screening of macroions by microions, which leads to the Yukawa form of effective interactions between macroions. The non-linear screening parameters for macroions were calculated using a direct Poisson-Boltzmann approximation. Two effects were revealed in the calculations: (i) decomposition of all microions into two subclasses, free and bound microions, and (ii) the significant reduction of the effective charge (Z*) of the initial bare macroion charge (Z) under non-linear screening by a small high-density envelope of the bound microions. We showed that the excess interaction energy is different when the non-linear screening effect is taken into account.

Coupled Fluid-Kinetic Numerical Method to Model Spacecraft Charging in LEO

Although spacecraft charging in low Earth orbit is generally regarded as insignificant because the cold dense ionospheric plasma prevents the development of strong potential differences, scientific satellites (e.g., Centre National d’Etudes Spatiales TARANIS satellite) carry very sensitive instruments, the measurements by which may be impacted by the charging of the spacecraft. In order to assess the effect of the spacecraft charging on the measurements, simulations will be performed using the Spacecraft Plasma Interaction System (SPIS) software. Different numerical schemes are implemented in the SPIS, but none is perfectly fit to perform very low-noise simulations for satellites in ionospheric plasma within an admissible computation time. In this paper, a coupling between the Poisson–Boltzmann approximation and the particle-in-cell schema is proposed. This results in having the advantages of the analytic methods in terms of statistical noise and computational resources without suffering from some of their limitations.

Non‐linear screening and phase states of a complex plasma

The applicability limit of the well‐known phase diagram of complex plasmas in the κ–Γ plane (κ is the structural parameter, Γ is the coupling parameter) is under discussion. The present work is devoted to the analysis of the range of applicability of a basic assumption in the initial phase diagram, that is, linearized (Debye) screening of macro‐ions by micro‐ions, which leads to the Yukawa form of effective interactions between macro‐ions. Parameters of non‐linear screening for macro‐ions were calculated within the direct Poisson–Boltzmann approximation in an averaged Wigner–Seitz cell. Two effects were revealed as a result of such calculations: (a) decomposition of all micro‐ions into two subclasses, free and bound ones, and (b) significant reduction of the effective charge Z* of initial bare macro‐ion Z under non‐linear screening by the small high‐density envelope of bound micro‐ions. This effect leads to a re‐normalization of initial Γ and κ into Γ* and κ* (Γ* &lt; Γ, κ* &lt; κ). It is assumed that the phase states of a complex plasma under non‐linear screening are still the same as in the initial phase diagram, but in κ*–Γ* plane instead of the κ–Γ one. The corresponding calculated shifts of the phase states are discussed and illustrated.

Self-consistent field theory of polyelectrolyte brushes with finite chain extensibility.

Polyelectrolyte brushes are formed by charged macromolecules tethered by the end segment to a solid-liquid interface. At low ionic strength of the solution, the intermolecular electrostatic interactions lead to strong stretching of the macromolecules that may, as a result, approach the limit of their extensibility (the contour length). Here, we present an analytical theory of polyelectrolyte brushes developed within the Poisson-Boltzmann approximation which explicitly accounts for finite extensibility of the brush-forming chains. In contrast to earlier theories based on the approximation of Gaussian elasticity of the brush-forming chains, the current approach enables avoiding artificial result of stretching of the chains beyond the contour length at high degrees of ionization or/and large grafting densities.

A semi-analytical calculation of the electrostatic pair interaction between nonuniformly charged colloidal spheres at an air-water interface.

We study the electrostatic pair interaction between two nonuniformly like-charged colloidal spheres trapped in an air-water interface. Under the linear Poisson-Boltzmann approximation, a general form of the electrostatic potential for the system is shown in terms of multipole expansions. After combining the translation-rotation transform of the coordinates with the numerical multipoint collection, we give a semi-analytical result of the electrostatic pair interaction between the colloids. The pair interaction changes quantitatively or even qualitatively with different distributions of the surface charges on the particles. Because of the anisotropic distribution of the surface charge and the asymmetric dielectric medium, the dipole moment of the ion cloud associating with the particle orients diagonally to the air-water interface with an angle α. When the angle is large, the colloids interact repulsively, while they attract each other when the angle is small. The attractive colloids may be "Janus-like" charged and be arranged with some specific configurations. Whatever the repulsions or the attractions, they all decay asymptotically ∝1/d(3) (d is the center-center distance of the particles) which is consistent with our general acknowledge. The calculation results also provide an insight of the effect of the ion concentration, particle size, and the total charge of the particle on the pair interaction between the particles.

Convexity and Stiffness in Energy Functions for Electrostatic Simulations.

We study the properties of convex functionals which have been proposed for the simulation of charged molecular systems within the Poisson-Boltzmann approximation. We consider the extent to which the functionals reproduce the true fluctuations of electrolytes and thus the one-loop correction to mean field theory-including the Debye-Hückel correction to the free energy of ionic solutions. We also compare the functionals for use in numerical optimization of a mean field model of a charged polymer and show that different functionals have very different stiffnesses leading to substantial differences in accuracy and speed.

Ionic density distributions near the charged colloids: Spherical electric double layers

We have studied the structure of the spherical electric double layers on charged colloids by a density functional perturbation theory, which is based both on the modified fundamental-measure theory for the hard spheres and on the one-particle direct correlation functional (DCF) for the electronic residual contribution. The contribution of one-particle DCF has been approximated as the functional integration of the second-order correlation function of the ionic fluids in a bulk phase. The calculated result is in very good agreement with the computer simulations for the ionic density distributions and the zeta potentials over a wide range of macroion sizes and electrolyte concentrations, and compares with the results of Yu et al. [J. Chem. Phys. 120, 7223 (2004)] and modified Poisson-Boltzmann approximation [L. B. Bhuiyan and C. W. Outhwaite, Condens. Matter Phys. 8, 287 (2005)]. The present theory is able to provide interesting insights about the charge inversion phenomena occurring at the interface.

Ion Correlation-Induced Phase Separation in Polyelectrolyte Blends.

Inhomogeneous polyelectrolyte materials have been of both longstanding and recent interest; polymer blends exhibit technologically advantageous properties for adhesives and fuel cell membranes and serve as an ideal model system to study more complicated behaviors in polyelectrolyte materials. However, the physics governing the phase behavior of polyelectrolyte blends remains poorly understood. Traditional self-consistent field theory (SCFT) can include Coulombic interactions that arise in polyelectrolytes but can only reproduce Poisson-Boltzmann behavior or perturbations thereof due to the mean-field nature of the SCFT calculation. Recently, tools have been developed to couple SCFT with liquid state (LS) integral equation theory, which can calculate ion correlations in a quantitative fashion. This permits the articulation of ion effects in very low dielectric εr constant regimes that are relevant to polymer blends in nonaqueous conditions. We show that the inclusion of local ion correlations can give rise to marked enhancement of phase separation, contrary to theories invoking the Poisson-Boltzmann approximation, even to the extent of driving phase separation when two polymers are fully miscible (χN = 0). We provide both a demonstration of this effect as well as a conceptual explanation.

Electrostatic interactions between Janus particles

In this paper we study the electrostatic properties of "Janus" spheres with unequal charge densities on both hemispheres. We introduce a method to compare primitive-model Monte Carlo simulations of the ionic double layer with predictions of (mean-field) nonlinear Poisson-Boltzmann theory. We also derive practical Derjaguin Landau Verwey Overbeek (DLVO)-like expressions that describe the Janus-particle pair interactions by mean-field theory. Using a large set of parameters, we are able to probe the range of validity of the Poisson-Boltzmann approximation, and thus of DLVO-like theories, for such particles. For homogeneously charged spheres this range corresponds well to the range that was predicted by field-theoretical studies of homogeneously charged flat surfaces. Moreover, we find similar ranges for colloids with a Janus-type charge distribution. The techniques and parameters we introduce show promise for future studies of an even wider class of charged-patterned particles.

OS1-2-6 界面動電現象による流れを解析するための格子ボルツマン法(OS1 マルチスケール現象のシミュレーション技術(2))

We present a coupled lattice Boltzmann (LB) method to solve a set of model equations for electrokinetic flows in micro-/nano-channels. The model consists of the Poisson equation for the electrical potential, the Nernst-Planck equation for the ion concentration, and the Navier-Stokes equation for the flows of the electrolyte solution. One of the features that distinguishes the present method from the existing LB methods for electrokinetic flows lies in the scheme for the Nernst-Planck equation, in which the electrochemical migration and the convection of the electrolyte solution contributing to the ion flux are incorporated into the collision operator of the LB algorithm. As a validation of the present method, a one-dimensional microchannel is considered, and the simulation results in the steady state are compared with the semi-analytical solution of the model with the Poisson-Boltzmann approximation.