Classical Poisson-Boltzmann Equation Research Articles

EDL impact on mixed magneto-convection in a vertical channel using ternary hybrid nanofluid

Electro-osmotic transportation of conducting ionic nanofluids in vertically bounded arrangements has been expansively researched due to its vast assortment of engineering and medical applications, such as soil study, fluid dialysis, chemical processing, capillary electrophoresis, planar chromatography, separation techniques, and other real interests. On that account, in the manuscript, a theoretical model is constituted to simulate the fully developed mixed convective flow of ionic ternary hybrid nanofluid persuaded by electroosmosis and magnetohydrodynamics in a long vertical non-conducting channel under linearly changing temperature on channel walls. The classical Poisson-Boltzmann equation is employed to extract the electric double layer (EDL) impact on the flow formation via Debye-Hückel linearization conjuncture. The leading partial differential equations delineating the flow are formulated based on the general laws of conservation of momentum and energy. The relevant dimensionless setup transforms the flow model to a simplified equivalent model, which is solved analytically. The new results are comprehensively examined in terms of basic flow, magnetic, and thermal characteristics for various implanted parameters via multiple graphs and tables. Graphical outcomes confessed that the magnetic field and Debye-Hückel parameters have striking impacts on the stream features. Thin EDL supports speeding up the fluid motion through the channel domain. A noteworthy result noted from the examination is that the concentration of tri-hybridized nanoparticles in pure water is an issue that effectively delays the commencement of flow instability in the channel domain under the magnetic setting. The present study’s findings may be valuable for designing electromechanical devices, nanofluidic devices, micropumps, solar energy systems, etc.

Poisson–Boltzmann equation with a random field for charged fluids

The classical Poisson–Boltzmann equation (CPBE), which is a mean field theory by averaging the ion fluctuation, has been widely used to study ion distributions in charged fluids. In this study, we derive a modified Poisson–Boltzmann equation with a random field from the field theory and recover the ion fluctuation through a multiplicative noise added in the CPBE. The Poisson–Boltzmann equation with a random field (RFPBE) captures the effect of the ion fluctuation and gives different ion distributions in the charged fluids compared to the CPBE. To solve the RFPBE, we propose a Monte Carlo method based on the path integral representation. Numerical results show that the effect of the ion fluctuation strengthens the ion diffusion into the domain and intends to distribute the ions in the fluid uniformly. The final ion distribution in the fluid is determined by the competition between the ion fluctuation and the electrostatic forces exerted by the boundaries. The RFPBE is general and feasible for high dimensional systems by taking the advantage of the Monte Carlo method. We use the RFPBE to study a two dimensional system as an example, in which the effect of ion fluctuation is clearly captured.

Electrostatic interactions in charged nanoslits within an explicit solvent theory

Within a dipolar Poisson–Boltzmann theory including electrostatic correlations, we consider the effect of explicit solvent structure on solvent and ion partition confined to charged nanopores. We develop a relaxation scheme for the solution of this highly non-linear integro-differential equation for the electrostatic potential. The scheme is an extension of the approach previously introduced for simple planes (Buyukdagli and Blossey 2014 J. Chem. Phys. 140 234903) to nanoslit geometry. We show that the reduced dielectric response of solvent molecules at the membrane walls gives rise to an electric field significantly stronger than the field of the classical Poisson–Boltzmann equation. This peculiarity associated with non-local electrostatic interactions results in turn in an interfacial counterion adsorption layer absent in continuum theories. The observation of this enhanced counterion affinity in the very close vicinity of the interface may have important impacts on nanofluidic transport through charged nanopores. Our results indicate the quantitative inaccuracy of solvent implicit nanofiltration theories in predicting the ionic selectivity of membrane nanopores.

Solving fluctuation-enhanced Poisson–Boltzmann equations

Electrostatic correlations and fluctuations in ionic systems can be described within an extended Poisson–Boltzmann theory using a Gaussian variational form. The resulting equations are challenging to solve because they require the solution of a high-dimensional nonlinear partial differential equation for the pair correlation function. This has limited existing studies to simple approximations or to one-dimensional geometries. In this paper we show that the numerical solution of the equations is greatly simplified by the use of selective inversion of a finite difference operator which occurs in the theory. This selective inversion preserves the sparse structure of the problem and leads to substantial savings in computer effort. In one and two dimensions further simplifications are made by using a mixture of selective inversion and Fourier techniques. With the help of numerical examples, we validate the accuracy of the numerical method and show that electrostatic correlations and fluctuations could lead to a large deviation from the classical Poisson–Boltzmann equation with the increase of the coupling parameter.

Finite Element Approximation to a Finite-Size Modified Poisson-Boltzmann Equation

The inclusion of steric effects is important when determining the electrostatic potential near a solute surface. We consider a modified form of the Poisson-Boltzmann equation, often called the Poisson-Bikerman equation, in order to model these effects. The modifications lead to bounded ionic concentration profiles and are consistent with the Poisson-Boltzmann equation in the limit of zero-size ions. Moreover, the modified equation fits well into existing finite element frameworks for the Poisson-Boltzmann equation. In this paper, we advocate a wider use of the modified equation and establish well-posedness of the weak problem along with convergence of an associated finite element formulation. We also examine several practical considerations such as conditioning of the linearized form of the nonlinear modified Poisson-Boltzmann equation, implications in numerical evaluation of the modified form, and utility of the modified equation in the context of the classical Poisson-Boltzmann equation.

Ion distributions, exclusion coefficients, and separation factors of electrolytes in a charged cylindrical nanopore: A partially perturbative density functional theory study

The structural and thermodynamic properties for charge symmetric and asymmetric electrolytes as well as mixed electrolyte system inside a charged cylindrical nanopore are investigated using a partially perturbative density functional theory. The electrolytes are treated in the restricted primitive model and the internal surface of the cylindrical nanopore is considered to have a uniform charge density. The proposed theory is directly applicable to the arbitrary mixed electrolyte solution containing ions with the equal diameter and different valences. Large amount of simulation data for ion density distributions, separation factors, and exclusion coefficients are used to determine the range of validity of the partially perturbative density functional theory for monovalent and multivalent counterion systems. The proposed theory is found to be in good agreement with the simulations for both mono- and multivalent counterion systems. In contrast, the classical Poisson-Boltzmann equation only provides reasonable descriptions of monovalent counterion system at low bulk density, and is qualitatively and quantitatively wrong in the prediction for the multivalent counterion systems due to its neglect of the strong interionic correlations in these systems. The proposed density functional theory has also been applied to an electrolyte absorbed into a pore that is a model of the filter of a physiological calcium channel.

The Effects of Chemical Substitution and Polymerization on the pKa Values of Sulfonic Acids

The effects of ring substitution on the pK(a) value of benzenesulfonic acid (BSA) were investigated using a combined quantum mechanical and classical approach. Ring substitution with strong electron-withdrawing elements such as F, Cl, and Br is found to enhance the acidity of the BSA. More importantly, ring substitution with -NO(2) groups which form an extended conjugated pi-system with the benzene ring exhibits the strongest enhancement of the acidity. The effects of polymerization on the styrenesulfonic acid (SSA) were also investigated by solving the classical Poisson-Boltzmann equation. It is found that polymerization significantly decreases the acidity of SSA due to the alteration of the electrostatic environment surrounding the acid group upon polymerization. The average pK(a) value converges to 2.9 from the corresponding monomer value of -0.53 at a degree of polymerization of 8-12. These results shed significant light on how to design sulfonic-acid-based solid acid catalysts to achieve desired catalytic properties.

Squeeze-Flow Electroosmotic Pumping Between Charged Parallel Plates

In the present work, the squeezing flow between two charged parallel plates is theoretically investigated, with a provision of accounting for the electric double layer overlap effects. The electroviscous effects arising from the distortion of the electric double layer flow field are investigated in detail, for different strengths of the imposed plate motion. It is revealed that there can be a significant deviation between the predictions from the present model and those obtained by employing a classical Poisson-Boltzmann equation based model. This discrepancy can be attributed to some of the over-simplified assumptions associated with the standard models that might only remain valid for large separation distances between the two plates. Many of these simplified assumptions are found to hold inappropriate in case the squeezing flow occurs in such a narrow gap that the instantaneous liquid layer thickness becomes of the same order or less than the order of the characteristic electric double layer thickness. In such cases, there is likely to be a deficit of counterions within the bulk liquid due to an excess accumulation of those in the electrical double layer. On the other hand, there may occur a surplus of coions in the bulk liquid region due to a rejection of those in the electrical double layer. As a consequence of this presence of excess net charges in the bulk liquid region, strong electro-hydrodynamic interactions are likely to occur between the squeezing motion and the electroosmotic transport, which cannot be accurately captured by the classical theory.

Microchannel flow control through a combined electromagnetohydrodynamic transport

A mathematical model is developed to study the combined influences of electromagnetohydrodynamic forces in controlling the fluid flow through parallel plate rectangular microchannels. The electric double layer (EDL) effects are modelled by employing the classical Poisson–Boltzmann equation. The governing fluid flow equations are subsequently solved, in an effort to obtain closed form expressions depicting the variations in the overall flow rate as a function of various influencing system parameters. It is revealed that, with the aid of a relatively low-magnitude magnetic field, a substantial augmentation in the volumetric flow rates can be achieved. However, with magnetic fields of higher strengths, strongly opposing volumetric forces might offset any further possibilities of flow rate augmentation. Certain critical non-dimensional parameters are also identified, which can play significant roles in the overall flow augmentation mechanism.

Ion concentrations and velocity profiles in nanochannel electroosmotic flows

Ion distributions and velocity profiles for electroosmotic flow in nanochannels of different widths are studied in this paper using molecular dynamics and continuum theory. For the various channel widths studied in this paper, the ion distribution near the channel wall is strongly influenced by the finite size of the ions and the discreteness of the solvent molecules. The classical Poisson–Boltzmann equation fails to predict the ion distribution near the channel wall as it does not account for the molecular aspects of the ion–wall and ion–solvent interactions. A modified Poisson–Boltzmann equation based on electrochemical potential correction is introduced to account for ion–wall and ion–solvent interactions. The electrochemical potential correction term is extracted from the ion distribution in a smaller channel using molecular dynamics. Using the electrochemical potential correction term extracted from molecular dynamics (MD) simulation of electroosmotic flow in a 2.22 nm channel, the modified Poisson–Boltzmann equation predicts the ion distribution in larger channel widths (e.g., 3.49 and 10.00 nm) with good accuracy. Detailed studies on the velocity profile in electro-osmotic flow indicate that the continuum flow theory can be used to predict bulk fluid flow in channels as small as 2.22 nm provided that the viscosity variation near the channel wall is taken into account. We propose a technique to embed the velocity near the channel wall obtained from MD simulation of electroosmotic flow in a narrow channel (e.g., 2.22 nm wide channel) into simulation of electroosmotic flow in larger channels. Simulation results indicate that such an approach can predict the velocity profile in larger channels (e.g., 3.49 and 10.00 nm) very well. Finally, simulation of electroosmotic flow in a 0.95 nm channel indicates that viscosity cannot be described by a local, linear constitutive relationship that the continuum flow theory is built upon and thus the continuum flow theory is not applicable for electroosmotic flow in such small channels.

Osmotic Properties of Aqueous Ionene Solutions

Osmotic coefficients of aqueous solutions of 3,3-, 4,5-, 6,6-, and 6,9-ionenes with bromide and chloride counterions were measured at 298 K and in the concentration range from 0.001 to 0.1 mol/dm 3 . The measured osmotic coefficients are very low, indicating large deviations from ideality caused by strong interactions in these systems. The solutions with chloride counterions have a higher osmotic coefficient than the corresponding bromide solutions, which is in agreement with experimental data for the counterion activity coefficient of these solutions. The experimental results for osmotic coefficients were analyzed using a cylindrical cell model where a fixed charge is assumed to be uniformly distributed along the z-axis of a rigid and impenetrable polyion, in conjunction with the classical Poisson-Boltzmann equation. More refined models in which discrete charges are located periodically along the z-axis of the polyion were also studied. In the latter case the canonical Monte Carlo method was used to calculate the osmotic coefficient. In agreement with previous studies of weak and moderately charged polyelectrolytes, we found the theoretically predicted osmotic coefficients to be too high in comparison with the experimental data for aqueous ionene solutions.

Salt exclusion from charged and uncharged micropores

The exclusion of electrolytes from charged and uncharged capillaries is studied using (a) the Poisson-Boltzmann equation and (b) Grand Canonical Monte Carlo computer simulations. The thermodynamics and structure are examined as a function of the charge fixed on the inner capillary surfaces. The results for 2:2 electrolytes indicate that the classical Poisson-Boltzmann equation breaks down under these conditions, and in fact its predictions are in serious disagreement with the simulation results for the same system over the whole range of surface charge densities. An additional interesting result of this study concerns the ability of microporous materials to exclude electrolytes. The salt exclusion first increases by increasing the charge density, passes through a maximum, and then decreases with further increase of the surface charge. This behavior (completely missed by the Poisson-Boltzmann approximation) can be explained by the influence of the interionic correlations on the distribution of electrolyte inside the capillary.