Nonlinear Poisson-Boltzmann Equation Research Articles

Evaluation and improvement of Gay-Berne interaction potential to simulate 3D DLVO interaction of clay particles

This paper first presents a set of DLVO-based energy-separation functions for a pair of finite uniformly charged square platelets of infinitesimal thickness in three elementary configurations: face-to-face, edge-to-edge, and edge-to-face. The novel dataset was generated by summing the electrostatic interaction energy computed numerically by solving the non-linear 3D Poisson-Boltzmann equation and the van der Waals interaction energy calculated analytically. The dataset aims to inform qualitatively and quantitatively the energy/force separation functions used in the Discrete Element Method (DEM) and Coarse-Grained Molecular Dynamics (CGMD) modelling of clays. The same dataset was then used to calibrate and evaluate two Gay-Berne (GB)-type potentials: i) a DLVO-adapted Gay-Berne potential, where the Born-van der Waals branches of the underlying Lennard-Jones (LJ) potential are replaced with van der Waals-Columbic branches to represent DLVO interactions; ii) the Mie potential, where the exponents of the two energy terms are ‘unlocked’ instead of being set equal to 12 and 6 as per the original LJ potential. It is shown that the orientation parameter, µ, and the anisotropy parameter, ν, need to be different from µ = 2 and ν = 1 as adopted in CGMD clay modelling to capture the progression of the shape of the pair energy-separation function from face-to-face to edge-to-face and edge-to-edge configuration. It is also shown that the MIE potential (with exponents m = 3 and n = 1.5) better captures the slow decay of the electrostatic repulsive energy component of the DLVO potential energy Coulombic branch of the interaction potential compared to the DLVO-adapted GB potential, which embeds the Lennard-Jones (LJ) exponents m = 12 and n = 6.

Concentrated aqueous solutions of multivalent macroions enhance the electrical double layer capacitance and differential capacitance of a planar supercapacitor

The electrostatic properties of the ionic cloud, or electrical double layer, neutralizing a charged electrode are relevant to improve the storage of electrical energy and the efficiency of supercapacitors and/or batteries using coulombic fluids. In this work, we study the microscopic profiles of charged particles, integrated surface charge density, and the mean electrostatic potential associated to a pair of infinite parallel plates uncharged or, else, with opposite charge, in the presence of a salt-free aqueous solution constituted by big positive macroions and small anions, which interact via an explicit electrostatic coulombic potential and are dissolved in an implicit solvent. The particle profiles and the integrated surface charge density associated to Monte Carlo simulations and integral equations theory display an oscillatory or non-monotonic behaviour producing the phenomena of charge inversion and charge reversal. These phenomena are not predicted by the non-linear Poisson-Boltzmann equation, which highlights the importance of ion correlations and ionic volume excluded effects. The phenomenon of surface charge amplification is observed near the point of zero charge, which is yielded by the particle size-asymmetry. For a fixed concentration of macroions, a monotonic increase of the mean electrostatic potential difference between the charged parallel plates is predicted by simulations and theories as a function of the electrodes' charge. For a fixed surface charge density, the capacitance of the parallel plates capacitor increases when the ionic strength of the macroion's solution increases. A concavity inversion and an augment of the differential capacitance is also observed for an electrode near the point of zero charge as function of the macroion's concentration.

Understanding electrostatic interaction on strong cation-exchanger via co-ion valency effects

Empirical adsorption models have been extensively used to design and optimize ion exchange chromatography (IEC) processes for proteins. The equations go 40 years back to the qualitative findings about the electrical double layer (EDL) in ion exchangers and form the basis of the stoichiometric displacement (SD) model widely used in preparative chromatography. While the SD model reduces the experimental effort to find salt-eluting conditions for the separation, knowledge transfer is restricted from one system to another. However, this limitation can be overcome by understanding the physicochemical interaction mechanism between the solid adsorbent and the electrolyte. Via a theoretical and experimental approach, we investigated the physicochemical adsorption mechanism in IEC and developed a methodology to determine it quantitatively by measuring the effective EDL thickness. We performed negative adsorption experiments in high-performance liquid chromatography to measure the excluded volume of co-ions, citrate, or oxalate on strong cation exchange resin. Together with the physical specifications of the column and the deployment of a modified nonlinear Poisson-Boltzmann equation, we identified the effects of the electrolyte composition on the size of the EDL. While it depends on the concentration, valency, and size of the counterion, we derived that the expansion of the EDL is indicated by different valencies of the carboxylate co-ions in trace amounts. Our findings provide a self-consistent theory of the transport phenomena in a solid/fluid system with all parameters specified with the physical properties of the chromatographic process. Further, optimizing the resin design or improving the adsorption and desorption conditions for biomolecules may be facilitated. Altogether, our work may improve material designing and process development and, thereby, help to overcome the concurrent technological and economic bottlenecks of the well-deployed purification step of IEC.

New solution of the non-linear Poisson-Boltzmann differential equation for solid particle dispersions in dissymmetrical electrolytes

A new mathematical solution to the non-linear Poisson-Boltzmann differential equation for solid-liquid dispersions in presence of different dissymmetrical electrolytes was given. The analytical expressions of the surface and charge density of solid particles were given. The variations of electrostatic potential ψ (x) and charge density σ (x) of dispersed particles against the distance x were obtained. For colloidal particles in presence of E(m-n) electrolytes with m≠n with m≥3,n≥3 and for E(2–3) and E(3-2) electrolytes, the mean electrostatic potential as a function of the distance was numerically integrated by Mathematica program version 13.The experimental study of silica suspensions in presence with the following electrolytes NaCl, Na2SO4, CaCl2, Na3PO4, AlCl3, Al2(SO4)3, Ca3(PO4)2, Na4P2O7 and Na5P3O10 led to confirm the theoretical predictions obtained from the analytical solution of Poisson-Boltzmann equation. The results obtained allowed to determine the surface potential as a function of pH of the suspension and the electrostatic potential versus the distance x. The variations of the dissociation coefficient of silica surfaces were determined. An important effect of the anion and cation valences of the dissymmetrical electrolytes on the surface charge density and potential was highlighted.

Electrohydrodynamic peristaltic microfluidic flow with thermal and molecular analysis featuring sloped configurations

This study examines the simultaneous impacts of thermal and molecular transfer of non-Newtonian fluid propelled by the collective impact of electroosmosis and peristaltic pumping via inclined channel. Internal frictional heating and heat generation are also taken into consideration. The electrohydrodynamic phenomenon is described by the nonlinear Poisson–Boltzmann equation. The problem is modulated mathematically through a system of coupled non-linear differential equations in wave frame of reference. Using the regular perturbation method around a small wave number δ, we obtain sequential solutions for stream function ψ, electric potential ϕ, velocity u, pressure gradient dpdx, temperature θ and concentration Ω distributions. The influences of newly arising parameters such as the electroosmotic parameter me, relaxation time parameter λ1, retardation time parameter λ2, wave number δ, Reynold’s number Re, heat generation parameter β, Prandtl number Pr, Schmidt number Sc, Froude number Fr and Soret number Sr on physical characteristics are analyzed graphically. There is an inverse correlation between the parameters of relaxation time and retardation time and their effects on axial velocity, concentration, and temperature distributions. In particular, the temperature of the fluid increases as the EDL parameter, relaxation time parameter, viscous dissipation parameter, and Reynolds number increase, but decreases as the retardation time parameter increases. Moreover, the axial pressure gradient increases as Reynolds number and inclination angle increase, while it decreases as Froude number increases. The current model has applicability in the chemical industry, reservoir engineering, micro-fabrication, and biomedical applications, where electroosmotic effects on heat and mass movement are crucial.

Electroosmotic flow and heat transfer characteristics of a class of biofluids in microchannels at high Zeta potential

Peristalsis is an important dynamic phenomenon in the field of biomedical research, and has great application prospects in microscale fluids. In recent years, this biomimetic (peristaltic) phenomenon has gained widespread attention due to its large-scale applications in various medical and industrial fields, such as radiation therapy, peristaltic blood pumps, and drug delivery systems. In this study, the electroosmotic flow and heat transfer characteristics are investigated under high wall Zeta potential and slip boundary conditions for a certain type of biological fluid that satisfies the Newtonian fluid model. Fluid flows under the joint action of external electric field, magnetic field, and Joule heating. Firstly, without using the Debye-Hückel linear approximation, the numerical solutions are given by using the Chebyshev spectral method for the nonlinear Poisson-Boltzmann equation, the fourth-order differential equation satisfied by the stream function, and the thermal energy equation. The results are compared with those obtained by using the Debye-Hückel linear approximation to demonstrate the effectiveness of the numerical method used in this study. Secondly, the effects of wall Zeta potential, Hartmann number <inline-formula><tex-math id="M11">\begin{document}$H$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M11.png"/></alternatives></inline-formula>, electroosmotic parameter <inline-formula><tex-math id="M12">\begin{document}$m$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M12.png"/></alternatives></inline-formula>, slip parameter <inline-formula><tex-math id="M13">\begin{document}$\beta $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M13.png"/></alternatives></inline-formula> are discussed on the flow characteristics, peristaltic pumping, and trapping phenomena under electromagnetic environments, and the influence of Joule heating parameter <inline-formula><tex-math id="M14">\begin{document}$\gamma $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M14.png"/></alternatives></inline-formula> and Brinkman number <inline-formula><tex-math id="M15">\begin{document}$Br$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M15.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M15.png"/></alternatives></inline-formula> is explored on heat transfer characteristics. The results show that 1) wall Zeta potential plays an important role in controlling the velocity of fluid peristaltic flow; 2) the increase of electroosmotic parameter <inline-formula><tex-math id="M16">\begin{document}$m$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M16.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M16.png"/></alternatives></inline-formula> and slip parameter <inline-formula><tex-math id="M17">\begin{document}$\beta $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M17.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M17.png"/></alternatives></inline-formula> increases the flow velocity in the central region of the channel, while the increase of Hartmann number <inline-formula><tex-math id="M18">\begin{document}$H$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M18.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M18.png"/></alternatives></inline-formula> hinders the flow of fluid; 3) these flow behaviors exhibit opposite trends near the channel walls; 4) the number of streamlines captured by peristaltic transport decreases with Hartmann number <inline-formula><tex-math id="M19">\begin{document}$H$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M19.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M19.png"/></alternatives></inline-formula> and electroosmotic parameter <inline-formula><tex-math id="M20">\begin{document}$m$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M20.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M20.png"/></alternatives></inline-formula> increasing; 5) the increase of Joule heating parameter <inline-formula><tex-math id="M21">\begin{document}$\gamma $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M21.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M21.png"/></alternatives></inline-formula> and Brinkman number <inline-formula><tex-math id="M22">\begin{document}$Br$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M22.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231685_M22.png"/></alternatives></inline-formula> leads temperature to rise.

A kernel-free boundary integral method for the nonlinear Poisson-Boltzmann equation

This work proposes a boundary integral formulation based Cartesian grid method for the nonlinear Poisson-Boltzmann (PB) interface problem in biophysics. The method (a) does not have the limitation associated with the standard finite difference method for complex interfaces, (b) avoids the generation of any unstructured volume or surface grids needed by the finite element method, and (c) solves the nonlinear and variable coefficient PDE in the framework of boundary integral equations. The method solves the nonlinear PB equation with the Newton iterative method. It first reformulates the linearized PB equation in each Newton iteration as a Fredholm system of two boundary integral (BI) equations of the second kind, which is well-conditioned, and then solves the BI system with a Krylov subspace method. The evaluation of boundary and volume integrals involved in the solution of the BI system is done with a kernel-free method, which does not need to know Green's function associated with the variable coefficient PB equation. The kernel-free method evaluates volume and boundary integrals by solving equivalent simple interface problems on Cartesian grids with either a fast Fourier transform based Poisson solver or a geometric multigrid preconditioned conjugate gradient solver, either of which involves computational work essentially linearly proportional to the number of nodes on the Cartesian grid. Numerical examples in both two and three space dimensions are presented to demonstrate the efficiency and accuracy of the proposed numerical method.

Renormalized charge and dielectric effects in colloidal interactions: a numerical solution of the nonlinear Poisson–Boltzmann equation for unknown boundary conditions

The Derjaguin–Landau–Verwey–Overbeek (DLVO) theory, introduced more than 70 years ago, is a hallmark of colloidal particle modeling. For highly charged particles in the dilute regime, it is often supplemented by Alexander’s prescription (Alexander et al. in J Chem Phys 80:5776, 1984) for using a renormalized charge. Here, we solve the problem of the interaction between two charged colloids at finite ionic strength, including dielectric mismatch effects, using an efficient numerical scheme to solve the nonlinear Poisson–Boltzmann (NPB) equation with unknown boundary conditions. Our results perfectly match the analytical predictions for the renormalized charge by Trizac and coworkers (Aubouy et al. in J Phys A 36:5835, 2003). Moreover, they allow us to reinterpret previous molecular dynamics (MD) simulation results by Kreer et al. (Phys Rev E 74:021401, 2006), rendering them now in agreement with the expected behavior. We furthermore find that the influence of polarization becomes important only when the Debye layers overlap significantly.Graphical

Homogenization of the linearized ionic transport equations in random porous media

In this paper we obtain the homogenization results for a system of partial differential equations describing the transport of a N-component electrolyte in a dilute Newtonian solvent through a rigid random disperse porous medium. We present a study of the nonlinear Poisson–Boltzmann equation in a random medium, establish convergence of the stochastic homogenization procedure and prove well-posedness of the two-scale homogenized equations. In addition, after separating scales, we prove that the effective tensor satisfies the so-called Onsager properties, that is the tensor is symmetric and positive definite. This result shows that the Onsager theory applies to random porous media. The strong convergence of the fluxes is also established. In the periodic case homogenization results for the mentioned system have been obtained in Allaire et al (2010 J. Math. Phys. 51 123103).

Computational and theoretical model of electro-osmotic flow pumping in a microchannel with squeezing walls

Numerical simulation and theoretical solution for the electro-osmotic pumping flow of electrolyte solution in a microchannel with squeezing and charged walls are developed in this study. The mathematical model is derived based on using a strong coupling between the nonlinear Poisson–Boltzmann equation and the flow lubrication theory. The governing equations are integrated numerically using the finite difference method. Moreover, an analytical solution to the problem is also obtained using the lubrication theory and is used to solve the Poisson–Boltzmann equation without any approximation technique. The effects of various parameters such as the wall zeta potential, Debye length, and electric field on the fluid pressure distribution, velocity field, and the net flow rate are investigated in detail. The results show that the induced pumping rate depends strongly on the combined effects of the Helmholtz–Smoluchowski, zeta potential, and electrical double layer. Moreover, the produced net flow directionality can be controlled efficiently by manipulating the Helmholtz–Smoluchowski and/or the wall zeta potential. The results obtained from the numerical simulation are then compared with the theoretical analysis and have shown to be in agreement with the proposed mathematical model.

Solving Stochastic Nonlinear Poisson-Boltzmann Equations Using a Collocation Method Based on RBFs

In this paper, we present a numerical scheme based on a collocation method to solve stochastic non-linear Poisson–Boltzmann equations (PBE). This equation is a generalized version of the non-linear Poisson–Boltzmann equations arising from a form of biomolecular modeling to the stochastic case. Applying the collocation method based on radial basis functions (RBFs) allows us to deal with the difficulties arising from the complexity of the domain. To indicate the accuracy of the RBF method, we present numerical results for two-dimensional models, we also study the stability of this method numerically. We examine our results with the RBF-reference value and the Chebyshev Spectral Collocation (CSC) method. Furthermore, we discuss finding the appropriate shape parameter to obtain an accurate numerical solution besides greatest stability. We have exerted the Newton–Raphson approach for solving the system of non-linear equations resulting from discretization by the RBF technique.

Total longitudinal potential energy of interaction between parallel charged rods of finite size

We present a formula for the total potential energy of interaction between two parallel charged prismatic rods of finite size as a function of longitudinal inter-rod displacement (TLPE), obtained by calculating the repulsive electrostatic potential energy (using the nonlinear Poisson-Boltzmann equation) and the attractive van der Waals (VDW) potential energy between the rods in skewed configurations. The full profile of the TLPE curves is like an inverted bell, whose (top) width is twice the rod length, and depth is the same as that of the secondary minimum of the Derjaguin-Landau-Verwey-Overbeek (DLVO) potential for the rods.Using the TLPE function, we estimated root mean square of longitudinal displacements of fluctuating rods of 250 × 50 × 50 nm3 and a scaled surface potential of 5 for lateral inter-rod separations, d = 20 and 25 nm, e.g., in 5-rod systems, which showed non-negligible values of 24–87 nm.

PCGA: Polynomial collocation genetic algorithm for singular Poisson-Boltzmann equation arising in thermal explosions

Heat generation as a result of the exothermic reaction reaches the environment mainly due to the conduction through the walls of the vessel. The balance between the heat generated and the heat conducted away, resulting in the explosion is described by the Frank-Kamenetzkii (FK) parameter ρ. The critical value of FK for which the explosion occurs depends upon the shape of the vessel, which requires the solution of governing singular nonlinear Poisson-Boltzmann equation. Owing to the exponential nonlinearity and singularity the analytical exact solution for the non-integer k values does not exist. This work focuses on implementing the polynomial collocation by exploiting the global optimization features of the genetic algorithm to solve the Poisson-Boltzmann equation for integer and non-integer shape factors (k). The governing equation was converted into coupled nonlinear algebraic equations and an objective function was formulated. The method was examined for six different configurations of the control parameters of GA to find the best set of parameters. The solution for temperature distribution is obtained for cylindrical (k = 1), parallelepiped (k = 0.438, 0.694), and an arbitrary shape (k = 0.5) respectively. The solution obtained from Polynomial Collocation Genetic Algorithm (PCGA) remained in good agreement with the corresponding analytical results for k = 1, with the minimum absolute error of 10−10. The critical values of the FK are obtained as 1.5,1.4,and1.7 for shape factor k=0.438,0.5,and0.694 respectively with the convergence of the order of 10−6to10−5. The obtained solution is fairly stable over appropriate independent runs with the variation in the fitness value ranging from 10−05to10−03. Further simulations were performed to validate the results through statistical error indices. The diminutive errors of the order of 10−6 confirm reliable optimum solution, accuracy, and stability.

Local inversion of the mean electrostatic potential, maximum charge reversal, and capacitive compactness of concentrated 1:1 salts: The crucial role of the ionic excluded volume and ion correlations

• A local inversion of the electrostatic potential in one and three ionic radii from the macroions’s surface is observed. • If the zeta potential is in the above region, an inversion of the macroion’s mobility could be driven by 1:1 electrolytes. • The capacitive compactness around a macroion immersed in very concentrated indiferent 1:1 salts is studied. Recent restricted primitive model Monte Carlo simulations (Takamichi Terao, Mol. Phys. 119 , e1831634, 2020) have suggested the possibility of observing the phenomenon of charge inversion in highly concentrated monovalent salts, in which ionic specific adsorption is absent. By using the non-linear Poisson-Boltzmann equation supplemented by a hard-spheres contribution, integral equations theory, and Monte Carlo simulations, the associated mean electrostatic potential, maximum charge reversal and capacitive compactness are studied here. The main finding of this study is the observation of a local inversion of the mean electrostatic potential in a region bounded by one and three ionic radii measured from the macroions’s surface. If the zeta potential is located in this region, the above result suggests the possibility of observing an inversion of the macroion’s electrophoretic mobility, driven by 1:1 aqueous electrolytes in the absence of ionic specific adsorption . On the other hand, the maximum charge reversal and the capacitive compactness increases and decreases montonically, respectively, as a function of the ionic volume fraction when the salt concentration and/or the ionic size of the aqueous electrolytes increase.

Unsteady Electroosmotic Flow of Jeffrey Fluid in a Circular Microchannel under the Combined Action of Vertical Magnetic Field, External Electric Field, and Pressure at High Zeta Potential

The unsteady electroosmotic flow (EOF) for one kind of linear viscoelastic fluid, which is Jeffrey type fluid, is investigated under the common impact of vertical magnetic field, external electric field, and pressure at high Zeta potential in a circular microchannel. The numerical solutions of the potential and velocity distributions are obtained by solving the nonlinear Poisson-Boltzmann equation, the constitutive equation of the Jeffrey fluid, and the Cauchy momentum equation applying the Chebyshev spectral method and the finite difference method. By contrast, the Chebyshev spectral method has higher accuracy and less computation. The flow characteristics of Jeffrey fluid at high Zeta potential are analyzed with the numerical solution obtained by the Chebyshev spectral method. The results show that the velocity of Jeffrey fluid increases with the increase of the wall Zeta potential and electric width. The oscillation amplitude of velocity distribution increases with the increase of relaxation time but decreases with the increase of retardation time. When the Hartmann number is smaller, the increase of relaxation time leads to the increase of velocity; when the Hartmann number is larger, the increase of relaxation time leads to the decrease of velocity. No matter what the Hartmann number is, the velocity always decreases with the increase of the retardation time. The velocity tends to be stable gradually with the increase of time.

Investigation of the Limits of the Linearized Poisson–Boltzmann Equation

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.

Reduced basis method for the nonlinear Poisson–Boltzmann equation regularized by the range-separated canonical tensor format

Abstract The Poisson–Boltzmann equation (PBE) is a fundamental implicit solvent continuum model for calculating the electrostatic potential of large ionic solvated biomolecules. However, its numerical solution encounters severe challenges arising from its strong singularity and nonlinearity. In (P. Benner, V. Khoromskaia, B. Khoromskij, C. Kweyu, and M. Stein, “Regularization of Poisson-Boltzmann type equations with singular source terms using the range-separated tensor format,” SIAM J. Sci. Comput., vol. 43, no. 1, pp. A415–A445, 2021; C. Kweyu, V. Khoromskaia, B. Khoromskij, M. Stein, and P. Benner, “Solution decomposition for the nonlinear Poisson-Boltzmann equation using the range-separated tensor format,” arXiv:2109.14073, 2021), the effect of strong singularities was eliminated by applying the range-separated (RS) canonical tensor format (P. Benner, V. Khoromskaia, and B. N. Khoromskij, “Range-separated tensor format for many-particle modeling,” SIAM J. Sci. Comput., vol. 40, no. 2, pp. A1034–A1062, 2018; B. N. Khoromskij, “Range-separated tensor representation of the discretized multidimensional Dirac delta and elliptic operator inverse,” J. Comput. Phys., vol. 401, p. 108998, 2020) to construct a solution decomposition scheme for the PBE. The RS tensor format allows deriving a smooth approximation to the Dirac delta distribution in order to obtain a regularized PBE (RPBE) model. However, solving the RPBE is still computationally demanding due to its high dimension N $\mathcal{N}$ , where N $\mathcal{N}$ is always in the millions. In this study, we propose to apply the reduced basis method (RBM) and the (discrete) empirical interpolation method ((D)EIM) to the RPBE in order to construct a reduced order model (ROM) of low dimension N ≪ N $N\ll \mathcal{N}$ , whose solution accurately approximates the nonlinear RPBE. The long-range potential can be obtained by lifting the ROM solution back to the N $\mathcal{N}$ -space while the short-range potential is directly precomputed analytically, thanks to the RS tensor format. The sum of both provides the total electrostatic potential. The main computational benefit is the avoidance of computing the numerical approximation of the singular electrostatic potential. We demonstrate in the numerical experiments, the accuracy and efficacy of the reduced basis (RB) approximation to the nonlinear RPBE (NRPBE) solution and the corresponding computational savings over the classical nonlinear PBE (NPBE) as well as over the RBM being applied to the classical NPBE.

Weak formulations of the nonlinear Poisson-Boltzmann equation in biomolecular electrostatics

We consider the nonlinear Poisson-Boltzmann equation in the context of electrostatic models for a biological macromolecule, embedded in a bounded domain containing a solution of an arbitrary number of ionic species which is not necessarily charge neutral. The resulting semilinear elliptic equation combines several difficulties: exponential growth and lack of sign preservation in the nonlinearity accounting for ion mobility, measure data arising from point charges inside the molecule, and discontinuous permittivities across the molecule boundary. Exploiting the modeling assumption that the point sources and the nonlinearity are active on disjoint parts of the domain, one can use a linear decomposition of the potential into regular and singular components. A variational argument can be used for the regular part, but the unbounded nonlinearity makes the corresponding functional not differentiable in Sobolev spaces. By proving boundedness of minimizers, these are related to standard H1 weak formulations for the regular component and in the framework of Boccardo and Gallouët for the full potential. Finally, a result of uniqueness of this type of weak solutions for more general semilinear problems with measure data validates the strategy, since the different decompositions and test spaces considered must then lead to the same solution.

Nonmonotonic Pair Potentials in the Interaction of Like-Charged Objects in Solution.

We consider the long-standing like-charge attraction problem, wherein under certain conditions, similarly charged spheres suspended in aqueous electrolyte have been observed to display a minimum in their interaction potential, contrary to the intuitively expected monotonically varying repulsion. Recently, we described an interfacial mechanism invoking the molecular nature of the solvent that explains this anomalous experimental observation. In our model for the interaction of negatively charged particles in water, the minimum in the pair potential results from the superposition of competing contributions to the total free energy. One of these contributions is the canonical repulsive electrostatic term, whereas the other is a solvation-induced attractive contribution. We find that whereas both contributions grow approximately exponentially with decreasing interparticle separation, the occurrence of a stable, long-ranged minimum in the pair potential arises from differences in the precise interparticle separation dependence of the two terms. Specifically, the interfacial solvation term exhibits a more gradual decay with distance than the electrostatic repulsion, permitting the attractive contribution to dominate the interaction at large distances. Importantly, these disparities become evident in quantities calculated from exact numerical solutions to the governing nonlinear Poisson-Boltzmann (PB) equation for the spatial electrical potential distribution in the system. In marked contrast, we find that the linearized PB equation, applicable in the regime of low surface electrical potentials, does not support nonmonotonic trends in the total interaction free energy within the present model. Our results point to the importance of exact descriptions of electrostatic interactions in real systems that most often do not subscribe to particular mathematical limits where analytical approximations may provide a sufficiently accurate description of the problem.

Ionization equilibrium in a highly imperfect smoke plasma

The ionization equilibrium in a heterogeneous strongly nonideal smoky plasmas containing condensed particles and an easily ionized addition of cesium atoms in the gas phase is considered. To determine the charges of particles, the nonlinear Poisson-Boltzmann equation was used, and for the ionization of atoms of the gas phase, the Saha equation taking into account the effect of the displacement of the ionization equilibrium. The dependences of the concentration of electrons and particle charges, as well as the interface between the regions of positive and negative charges of particles, on the concentration of cesium atoms and the concentration of aluminum oxide particles are obtained.