scholarly journals Finite electric boundary-layer solutions of a generalized Poisson–Boltzmann equation

2015 ◽  
Vol 471 (2178) ◽  
pp. 20150024
Author(s):  
Bon M. N. Clarke ◽  
Peter J. Stiles
Keyword(s):  
Boundary Layer ◽  
Surface Charge ◽  
Mean Field ◽  
Boltzmann Distribution ◽  
Uniform Density ◽  
Nonlinear Phenomenon ◽  
Parallel Plates ◽  
Generalized Poisson ◽  
Poisson Boltzmann ◽  
Poisson Boltzmann Equation

We solve the nonlinear Poisson–Boltzmann (P–B) equation of statistical thermodynamics for the external electrostatic potential of a uniformly charged flat plate immersed in an unbounded strong aqueous electrolyte. Our rather general variational formulation yields new solutions for the external potential derived from both the classical Boltzmann distribution and its heuristic Eigen–Wicke modification for concentrated symmetric electrolytes. Electrostatic potentials of these mean-field solutions satisfy a homogeneous condition at a free boundary plane parallel to the electrically conducting plate. The preferred position of this plane, characterizing the outer limit of the charged electrolyte, is determined by minimizing electrostatic free energy of the electrolyte. For a given uniform density of surface charge exceeding a well-defined and experimentally accessible threshold, we show that the generalized nonlinear P–B equation predicts a unique sharp interface separating a charged boundary layer or double layer from electroneutral bulk electrolyte. Sharp electric boundary layers are shown to be an essentially nonlinear phenomenon. In the super-threshold regime, the diffuse Gouy–Chapman solution is inapplicable and thus the Derjaguin–Landau–Verwey–Overbeek analysis, predicting electrostatic repulsion between two sufficiently separated and identically charged parallel plates must be rejected. Similar limitations restrict the applicability of the Grahame equation relating surface charge density to surface potential.

scholarly journals Review and Modification of Entropy Modeling for Steric Effects in the Poisson-Boltzmann Equation

Entropy ◽  
2020 ◽  
Vol 22 (6) ◽  
pp. 632
Author(s):  
Tzyy-Leng Horng
Keyword(s):  
Free Energy ◽  
Lattice Gas ◽  
Steric Effect ◽  
Mean Field ◽  
Boltzmann Distribution ◽  
Experimental Conditions ◽  
Ion Concentrations ◽  
Poisson Boltzmann ◽  
Poisson Boltzmann Equation ◽  
Ion Size

The classical Poisson-Boltzmann model can only work when ion concentrations are very dilute, which often does not match the experimental conditions. Researchers have been working on the modification of the model to include the steric effect of ions, which is non-negligible when the ion concentrations are not dilute. Generally the steric effect was modeled to correct the Helmholtz free energy either through its internal energy or entropy, and an overview is given here. The Bikerman model, based on adding solvent entropy to the free energy through the concept of volume exclusion, is a rather popular steric-effect model nowadays. However, ion sizes are treated as identical in the Bikerman model, making an extension of the Bikerman model to include specific ion sizes desirable. Directly replacing the ions of non-specific size by specific ones in the model seems natural and has been accepted by many researchers in this field. However, this straightforward modification does not have a free energy formula to support it. Here modifications of the Bikerman model to include specific ion sizes have been developed iteratively, and such a model is achieved with a guarantee that: (1) it can approach Boltzmann distribution at diluteness; (2) it can reach saturation limit as the reciprocal of specific ion size under extreme electrostatic conditions; (3) its entropy can be derived by mean-field lattice gas model.

scholarly journals Computing Protein pKas Using the TABI Poisson–Boltzmann Solver

2020 ◽  
pp. 2042006
Author(s):  
Jiahui Chen ◽  
Jingzhen Hu ◽  
Yongjia Xu ◽  
Robert Krasny ◽  
Weihua Geng
Keyword(s):  
Boltzmann Distribution ◽  
Dielectric Medium ◽  
Boundary Integral ◽  
Acid Dissociation ◽  
Low Dielectric ◽  
Poisson Boltzmann ◽  
Computing Procedure ◽  
Poisson Boltzmann Equation ◽  
High Dielectric ◽  
Electrostatic Calculations

A common approach to computing protein pKas uses a continuum dielectric model in which the protein is a low dielectric medium with embedded atomic point charges, the solvent is a high dielectric medium with a Boltzmann distribution of ionic charges, and the pKa is related to the electrostatic free energy which is obtained by solving the Poisson–Boltzmann equation. Starting from the model pKa for a titrating residue, the method obtains the intrinsic pKa and then computes the protonation probability for a given pH including site–site interactions. This approach assumes that acid dissociation does not affect protein conformation aside from adding or deleting charges at titratable sites. In this work, we demonstrate our treecode-accelerated boundary integral (TABI) solver for the relevant electrostatic calculations. The pKa computing procedure is enclosed in a convenient Python wrapper which is publicly available at the corresponding author’s website. Predicted results are compared with experimental pKas for several proteins. Among ongoing efforts to improve protein pKa calculations, the advantage of TABI is that it reduces the numerical errors in the electrostatic calculations so that attention can be focused on modeling assumptions.

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 Nonlinear screening of charged macromolecules

2011 ◽  
Vol 369 (1935) ◽  
pp. 322-334 ◽  
Author(s):  
Gabriel Téllez
Keyword(s):  
Boltzmann Equation ◽  
Mean Field ◽  
Field Approach ◽  
Interesting Insight ◽  
Hückel Theory ◽  
Charge Renormalization ◽  
Poisson Boltzmann ◽  
Charged Macromolecules ◽  
Poisson Boltzmann Equation ◽  
Physical Phenomena

We present several aspects of the screening of charged macromolecules in an electrolyte. After a review of the basic mean field approach, based on the linear Debye–Hückel theory, we consider the case of highly charged macromolecules, where the linear approximation breaks down and the system is described by the full nonlinear Poisson–Boltzmann equation. Some analytical results for this nonlinear equation give some interesting insight on physical phenomena like the charge renormalization and the Manning counterion condensation.

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