Nonlinear Poisson-Boltzmann Equation Research Articles

Electroosmotic mechanism of Ellis fluid with Joule heating, viscous dissipation and magnetic field effects in a pumping microtube.

The dynamics of electro-osmotically generated flow of biological viscoelastic fluid in a cylindrical geometry are investigated in this paper. This flux is the result of walls contracting and relaxing sinusoidally in a magnetic environment. The rheology of the fluid is accurately captured with the Ellis fluid (blood) model. Both Joule heating and viscous dissipation are accounted for during thermal analysis. The electric potential induced in the EDL is obtained by applying the Debye-Huckel linearization to the non-linear Poisson-Boltzmann equation. Mathematical modelling is incorporated in cylindrical coordinates in wave frame of reference. Assuming a long wavelength characterized by a low Reynolds number, the Ellis fluid model's governing equations are simplified. Subsequently, the differential equations that result are computed numerically utilizing the NDSolve utility that is integrated into Mathematica. Graphical representations are utilized to visually and comprehensively assess the thermal characteristics, flow features, heat transfer coefficient, and skin friction coefficient. Various factors are taken into consideration, including the impact of Ellis fluid parameters, electric double layer, magnetic field, Brinkman number, and Ohmic dissipation. Ellis fluid's axial velocity boosts with a rise of the electroosmotic parameter and power-law index while decreasing with an increase in the Hartmann number and material fluid parameter. The fluid temperature is directly proportional to EDL parameter and parameters of Ohmic and viscous dissipation. The current model may be used in clinical scenarios involving the gastrointestinal system and capillaries, electro-hydrodynamic therapy, delivery of drugs in pharmacological, and biomedical devices.

Self-consistent Strong Screening Applied to Thermonuclear Reactions

Self-consistent strong plasma screening around light nuclei is implemented in the Big Bang nucleosynthesis (BBN) epoch to determine the short-range screening potential, e ϕ(r)/T ≥ 1, relevant for thermonuclear reactions. We numerically solve the nonlinear Poisson–Boltzmann equation incorporating Fermi–Dirac statistics, adopting a generalized screening mass to find the electric potential in the cosmic BBN electron–positron plasma for finite-sized α particles (4He++) as an example. Although the plasma follows Boltzmann statistics at large distances, Fermi–Dirac statistics is necessary when work performed by ions on electrons is comparable to their rest-mass energy. While self-consistent strong screening effects are generally minor owing to the high BBN temperatures, they can enhance the fusion rates of high-Z (Z > 2) elements while leaving fusion rates of lower-Z (Z ≤ 2) elements relatively unaffected. Our results also reveal a pronounced spatial dependence of the self-consistent strong screening potential near the nuclear surface. These findings about the electron–positron plasma’s role refine BBN theory predictions and offer broader applications for studying weakly coupled plasmas in diverse cosmic and laboratory settings.

Combined electroosmotic and pressure driven flow through soft nanochannel grafted with partially ion-penetrable polyelectrolyte layers

ABSTRACT This study explores the coupled effects of pressure-driven and electroosmotic flows in a soft nanochannel filled with non-Newtonian ionised liquid, where the inner walls are coated with a polymer layer. The ion partitioning effect occurs due to the Born energy, arising from the dielectric permittivity difference between the polyelectrolyte medium and the ionised liquid. The model incorporates the nonlinear Poisson-Boltzmann equation, the modified Darcy-Brinkman equation in the polyelectrolyte layer (PEL), the Cauchy momentum equation in the electrolyte domains and the continuity equation for incompressible fluids. Using the Debye-Hückel approximation for analytical results, the study validates numerical findings. The study considers the no-slip boundary condition on the walls and examines the impact of volume charge density of PEL, bulk electrolyte concentration, non-Newtonian fluid rheology and polymer layer characteristics and applied pressure gradient on flow modulation and ionic transport in the nanochannel. We provide rigorous mathematical results showing how electrohydrodynamic factors like charged PEL, flow behaviour index, EDL thickness, ion partitioning parameters, Debye-Hückel parameter and softness parameters influence the strength of the potential and the total flow modulation. We illustrate ion selectivity in the soft nanochannel and the friction factor across its walls, considering ion partitioning effects.

An Unfitted Finite Element Poisson-Boltzmann Solver with Automatic Resolving of Curved Molecular Surface.

So far, the existing Poisson-Boltzmann (PB) solvers that accurately take into account the interface jump conditions need a pregenerated body-fitted mesh (molecular surface mesh). However, qualified biomolecular surface meshing and its implementation into numerical methods remains a challenging and laborious issue, which practically hinders the progress of further developments and applications of a bunch of numerical methods in this field. In addition, even with a molecular surface mesh, it is only a low-order approximation of the original curved surface. In this article, an interface-penalty finite element method (IPFEM), which is a typical unfitted finite element method, is proposed to solve the Poisson-Boltzmann equation (PBE) without requiring the user to generate a molecular surface mesh. The Gaussian molecular surface is used to represent the molecular surface and can be automatically resolved with a high-order approximation within our method. Theoretical convergence rates of the IPFEM for the linear PB equation have been provided and are well validated on a benchmark problem with an analytical solution (we also noticed from numerical examples that the IPFEM has similar convergence rates for the nonlinear PBE). Numerical results on a set of different-sized biomolecules demonstrate that the IPFEM is numerically stable and accurate in the calculation of biomolecular electrostatic solvation energy.

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.

Comparison of three matched interface and boundary (MIB) schemes for solving the nonlinear Poisson-Boltzmann equation

Comparison of three matched interface and boundary (MIB) schemes for solving the nonlinear Poisson-Boltzmann equation

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.