The interaction of two charged spheres in the Poisson–Boltzmann equation

1981 ◽  
Vol 59 (13) ◽  
pp. 1860-1864 ◽  
Author(s):  
Joseph E. Ledbetter ◽  
Thomas L. Croxton ◽  
Donald A. McQuarrie
Keyword(s):  
Boltzmann Equation ◽  
Charge Density ◽  
Surface Charge ◽  
Electrostatic Potential ◽  
Surface Charge Density ◽  
Ionic Solution ◽  
Poisson Boltzmann ◽  
Poisson Boltzmann Equation ◽  
Good Agreement ◽  
Charged Spheres

The Poisson–Boltzmann equation for two large charged spheres immersed in an ionic solution with either constant surface charge density or constant surface potential is solved numerically. The repulsion between the spheres is calculated from the electrostatic potential in the double layer surrounding the spheres. Good agreement between this numerically calculated force and the force computed using the Derjaguin formula for spheres with constant surface charge density is found at small separations of the spheres.

scholarly journals A Focus on Two Electrokinetics Issues

Micromachines ◽  
2020 ◽  
Vol 11 (12) ◽  
pp. 1028
Author(s):  
Cheng Dai ◽  
Ping Sheng
Keyword(s):  
Boltzmann Equation ◽  
Zeta Potential ◽  
Flow Field ◽  
Drag Coefficient ◽  
Surface Charge ◽  
Physical Picture ◽  
Debye Layer ◽  
Poisson Boltzmann ◽  
Poisson Boltzmann Equation ◽  
Mobile Ions

This review article intends to communicate the new understanding and viewpoints on two fundamental electrokinetics topics that have only become available recently. The first is on the holistic approach to the Poisson–Boltzmann equation that can account for the effects arising from the interaction between the mobile ions in the Debye layer and the surface charge. The second is on the physical picture of the inner electro-hydrodynamic flow field of an electrophoretic particle and its drag coefficient. For the first issue, the traditional Poisson–Boltzmann equation focuses only on the mobile ions in the Debye layer; effects such as charge regulation and the isoelectronic point arising from the interaction between the mobile ions in the Debye layer and the surface charge are left to supplemental measures. However, a holistic treatment is entirely possible in which the whole electrical double layer—the Debye layer and the surface charge—is treated consistently from the beginning. While the derived form of the Poisson–Boltzmann equation remains unchanged, the zeta potential boundary condition becomes a calculated quantity that can reflect the various effects due to the interaction between the surface charges and the mobile ions in the liquid. The second issue, regarding the drag coefficient of a spherical electrophoretic particle, has existed ever since the breakthrough by Smoluchowski a century ago that linked the zeta potential of the particle to its mobility. Due to the highly nonlinear mathematics involved in the electro-hydrodynamics inside the Debye layer, there has been a lack of an exact solution for the electrophoretic flow field. Recent numerical simulation results show that the flow field comprises an inner region and an outer region, separated by a rather sharp interface. As the inner flow field is carried along by the particle, the measured drag is that at the inner/outer interface rather than at the solid/liquid interface. This identification and its associated physical picture of the inner flow field resolves a long-standing puzzle regarding the electrophoretic drag coefficient.

scholarly journals On triple trigonometrical equations

1973 ◽  
Vol 14 (2) ◽  
pp. 174-178 ◽  
Author(s):  
B. M. Singh
Keyword(s):  
Exact Solution ◽  
Charge Density ◽  
Surface Charge ◽  
Electrostatic Potential ◽  
Surface Charge Density ◽  
Equal Length ◽  
Two Dimensional ◽  
Electrostatic Problem ◽  
Unit Depth

An exact solution of triple trigonometrical equations is obtained by using the finiteHilbert transform. The solution of these equations is used to solve a two-dimensional electrostatic problem. The problem of determining the electrostatic potential due to two parallel coplanar strips of equal length, charged to equal and opposite potentials, each parallel to and equidistant from an earthed strip, is considered. Both the charged strips lie along the x-axis and they are equally spaced with respect to the y-axis. Finally the expression for the surface charge density (per unit depth) of the strip is derived

scholarly journals Applicability of the Linearized Poisson-Boltzmann Theory to Contact Angle problems and Application to the Carbon Dioxide-Brine-Solid Systems

2021 ◽  
Author(s):  
Mumuni Amadu ◽  
Adango Midanoye
Keyword(s):  
Carbon Dioxide ◽  
Contact Angle ◽  
Charge Density ◽  
Surface Charge ◽  
Surface Charge Density ◽  
Low Ph ◽  
Point Of Zero Charge ◽  
Zero Charge ◽  
Poisson Boltzmann ◽  
Ph Conditions

Abstract In colloidal science and bioelectrostatics, the linear Poisson Boltzmann Equation (LPBE) has been used extensively for the calculation of potential and surface charge density. Its fundamental assumption rests on the premises of low surface potential. In the geological sequestration of carbon dioxide in saline aquifers, very low pH conditions coupled with adsorption induced reduction of surface charge density result in low pH conditions that fit into the LPB theory. In this work, the Gouy-Chapman model of the electrical double layer has been employed in addition to the LPBE theory to develop a contact angle model that is a second-degree polynomial in pH. Our model contains the point of zero charge pH of solid surface. To render the model applicable to heterogeneous surfaces, we have further developed a model for the effective value of the point of zero charge pH. The point of zero charge pH model when integrated into our model enabled us to determine the point of zero charge pH of sandstone, quartz and mica using literature based experimental data. In this regard, a literature based thermodynamic model was used to calculate carbon dioxide solubility and pH of aqueous solution. Values of point of zero charge pH determined in this paper agree with reported ones. The novelty of our work stems from the fact that we have used the LPB theory in the context of interfacial science completely different from the classical approach, where the focus is on interparticle electrostatics involving colloidal stabilization.

scholarly journals Ability of the Poisson–Boltzmann equation to capture molecular dynamics predicted ion distribution around polyelectrolytes

2017 ◽  
Vol 19 (36) ◽  
pp. 24583-24593 ◽  
Author(s):  
Piotr Batys ◽  
Sohvi Luukkonen ◽  
Maria Sammalkorpi
Keyword(s):  
Molecular Dynamics ◽  
Boltzmann Equation ◽  
Charge Density ◽  
Layer Thickness ◽  
Mean Field ◽  
Ion Distribution ◽  
Line Charge ◽  
Poisson Boltzmann ◽  
Poisson Boltzmann Equation

Ion condensation around polyelectrolytes is examined computationally at all-atom and mean field detail levels to extract the practical limits of a PB model; the condensed ion layer thickness is found to depend solely on polyelectrolyte line charge density.

Analytical resolution of the Poisson–Boltzmann equation for a cylindrical polyion immersed in an electrolyte solution: using the optimal linearization method

2019 ◽  
Vol 97 (6) ◽  
pp. 656-661
Author(s):  
Leila Djebbara ◽  
Mohammed Habchi ◽  
Abdalhak Boussaid
Keyword(s):  
Boltzmann Equation ◽  
Electrolyte Solution ◽  
Electrostatic Potential ◽  
Analytical Calculation ◽  
Linearization Method ◽  
General Potential ◽  
Poisson Boltzmann ◽  
Poisson Boltzmann Equation ◽  
Analytical Resolution ◽  
Potential Conditions

By using the optimal linearization method (OLM), the potential of the electrical double layer created by a highly charged cylindrical polyion immersed in an electrolyte reservoir, which is represented by the so-called Poisson–Boltzmann equation (PBE), has been solved analytically under general potential conditions. For this system, three regions must be considered. The first one is in the near-neighborhood of the polyion and it is deprived of coions because of the repulsion phenomenon between the polyion and the coions, as proposed by Fuoss et al. (Proc. Natl. Acad. Sci. 37, 579 (1951). doi: 10.1073/pnas.37.9.579 ). For the second region, where the potential is slightly lower, we propose an OLM for solving the PBE. In the last region, where the potential is sufficiently low, the approximation of Debye–Hückel is adopted. This method allowed us to overcome some shortcomings in the analytical calculation of the electrostatic potential created by a polyion in an electrolyte solution.

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