Linearized Poisson-Boltzmann Equation Research Articles

Energetics of the Electron Transfer Pathways in the Homodimeric Photosynthetic Reaction Center.

Photosynthetic reaction centers from a green sulfur bacterium (GsbRC), the PscA/PscA proteins, and photosystem I (PSI), PsaA/PsaB proteins, share structural similarities. Here, we report the redox potential (Em) values of GsbRC by solving the linear Poisson-Boltzmann equation and considering the protonation states of all titratable sites in the entire GsbRC protein and identify the factors that shift the Em values with respect to PSI. The Em values for one-electron reduction of the accessory (A-1) and adjacent (A0) chlorophylls in GsbRC are 100-250 mV higher than those in PSI, whereas the Em values for the Fe4S4 cluster (FX) are at the same level. The PsaA-Trp697/PsaB-Trp677 pair in PSI, which forms the A1-quinone binding site, is replaced with PscA-Arg638 in GsbRC. PsaB-Asp575 in PSI, which is responsible for the Em difference between A1A and A1B quinones in PSI, is absent in GsbRC. These discrepancies also contribute to the upshift in Em(A-1) and Em(A0) in GsbRC with respect to PSI. It seems likely that the upshifted Em for chlorophylls in GsbRC ultimately originates from the characteristics of the electrostatic environment that corresponds to the A1 site of PSI.

Transient Analysis of the Electro-Osmotic Flow of Multilayer Immiscible Maxwell Fluids in an Annular Microchannel

This work investigates the transient multilayer electro-osmotic flow of viscoelastic fluids through an annular microchannel. The dimensionless mathematical model of multilayer flow is integrated by the linearized Poisson-Boltzmann equation, the Cauchy momentum equation, the rheological Maxwell model, initial conditions, and the electrostatic and hydrodynamic boundary conditions at liquid-liquid and solid-liquid interfaces. Although the main force that drives the movement of fluids is due to electrokinetic effects, a pressure gradient can also be added to the flow. The semi-analytical solution for the electric potential distribution and velocity profiles considers analytical techniques as the Laplace transform method, with numerical procedures using the inverse matrix method for linear algebraic equations and the concentrated matrix exponential method for the inversion of the Laplace transform. The results presented for velocity profiles and velocity tracking at the transient regime reveal an interesting oscillatory behavior that depends on elastic fluid properties via relaxation times. The time required for the flow to reach steady-state is highly dependent on the viscosity ratios and the dimensionless relaxation times. In addition, the influence of other dimensionless parameters on the flow as the electrokinetic parameters, zeta potentials at the walls, permittivity ratios, ratio of pressure forces to electro-osmotic forces, number of fluid layers, and annular thickness are investigated. The findings of this study have significant implications for the precise control of parallel fluid transport in microfluidic devices for flow-focusing applications.

Modeling of the Electrostatic Interaction and Catalytic Activity of [NiFe] Hydrogenases on a Planar Electrode.

Hydrogenases are a group of enzymes that have caught the interest of researchers in renewable energies, due to their ability to catalyze the redox reaction of hydrogen. The exploitation of hydrogenases in electrochemical devices requires their immobilization on the surface of suitable electrodes, such as graphite. The orientation of the enzyme on the electrode is important to ensure a good flux of electrons to the catalytic center, through an array of iron-sulfur clusters. Here we present a computational approach to determine the possible orientations of a [NiFe] hydrogenase (PDB 1e3d) on a planar electrode, as a function of pH, salinity, and electrode potential. The calculations are based on the solution of the linearized Poisson-Boltzmann equation, using the PyGBe software. The results reveal that electrostatic interactions do not truly immobilize the enzyme on the surface of the electrode, but there is instead a dynamic equilibrium between different orientations. Nonetheless, after averaging over all thermally accessible orientations, we find significant differences related to the solution's salinity and pH, while the effect of the electrode potential is relatively weak. We also combine models for the protein adsoption-desorption equilibria and for the electron transfer between the proteins and the electrode to arrive at a prediction of the electrode's activity as a function of the enzyme concentration.

Improving the Accuracy of Modelling CO2 Electroreduction on Copper Using Many‐Body Perturbation Theory

AbstractCopper (Cu) remains the most important metal catalyst for the carbon dioxide reduction reaction (CO2RR) into C2 products. Due to limited evidence from in situ experiments, mechanistic studies are often performed in the framework of density functional theory (DFT), using functionals at the generalized gradient approximation (GGA) level, which have fundamental difficulties to correctly describe CO adsorption and surface stability. We employ the adiabatic connection fluctuation dissipation theorem within the random phase approximation (RPA), in combination with the linearized Poisson–Boltzmann equation to describe solvation effects, to investigate the mechanism of CO2RR on the Cu(100) facet. Qualitatively different from the DFT‐GGA results, RPA results propose the formation of *OCCHO as the potential determining step towards C2 products. The results suggest that it is important to use more accurate methods like RPA when modeling reactions involving multiple CO‐related species like CO2RR.

Improving the Accuracy of Modelling CO2 Electroreduction on Copper Using Many-Body Perturbation Theory.

Copper (Cu) remains the most important metal catalyst for the carbon dioxide reduction reaction (CO2 RR) into C2 products. Due to limited evidence from in situ experiments, mechanistic studies are often performed in the framework of density functional theory (DFT), using functionals at the generalized gradient approximation (GGA) level, which have fundamental difficulties to correctly describe CO adsorption and surface stability. We employ the adiabatic connection fluctuation dissipation theorem within the random phase approximation (RPA), in combination with the linearized Poisson-Boltzmann equation to describe solvation effects, to investigate the mechanism of CO2 RR on the Cu(100) facet. Qualitatively different from the DFT-GGA results, RPA results propose the formation of *OCCHO as the potential determining step towards C2 products. The results suggest that it is important to use more accurate methods like RPA when modeling reactions involving multiple CO-related species like CO2 RR.

Predicting the Orientation of Adsorbed Proteins Steered with Electric Fields Using a Simple Electrostatic Model.

Under the most common experimental conditions, the adsorption of proteins to solid surfaces is a spontaneous process that leads to a rather compact layer of randomly oriented molecules. However, controlling such orientation is critically important for the development of catalytic surfaces. In this regard, the use of electric fields is one of the most promising alternatives. Our work is motivated by experimental observations that show important differences in catalytic activity of a trypsin-covered surface, which depended on the applied potential during the adsorption. Even though adsorption results from the combination of several processes, we were able to determine that (under the selected conditions) mean-field electrostatics play a dominant role, determining the orientation and yielding a difference in catalytic activity. We simulated the electrostatic potential numerically, using an implicit-solvent model based on the linearized Poisson-Boltzmann equation. This was implemented in an extension of the code PyGBe that included an external electric field, and rendered the electrostatic component of the solvation free energy. Our model (extensions available at the Github repository) allowed estimating the overall affinity of the protein with the surface, and their most likely orientation as a function of the potential applied. Our results show that the active sites of trypsin are, on average, more exposed when the electric field is negative, which agrees with the experimental results of catalytic activity, and confirm the premise that electrostatic interactions can be used to control the orientation of adsorbed proteins.

Perspectives on the Competition between the Electrochemical Water and N2 Oxidation on a TiO2(110) Electrode.

The electrochemical nitrogen oxidation reaction (NOR) has recently drawn attention due to promising experimental and theoretical results. It provides an alternative, environmentally friendly route to directly synthesize nitrate from N2(g). There is to date a limited number of investigations focused on the electrochemical NOR. Herein, we present a detailed computational study on the kinetics of both the NOR and the competing oxygen evolution reaction (OER) on the TiO2(110) electrode under ambient conditions. The use of grand canonical density functional theory in combination with the linearized Poisson-Boltzmann equation allows a continuous tuning of the explicitly applied electrical potential. We find that the OER may either promote or suppress the NOR on TiO2(110) depending on reaction conditions. The detailed atomistic insights provided on the mechanisms of these competing processes make possible further developments toward a direct electrochemical NOR process.

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.

On the derivations of the Debye–Hückel equations

This work presents the derivations of basic thermodynamic properties and activity coefficients equations from the linearised Poisson–Boltzmann equation. We consider two main approaches, the first one is based in classical thermodynamics, which has been used in the original work of Debye and Hückel, leading to the model which has been an important cornerstone of electrolyte thermodynamics since its original publication in 1923. The second approach relies on more modern derivations based on statistical mechanics, the so-called charging approaches. Both derivation routes have differences and shortcomings. We demonstrate the necessary steps to reach all original models derived from the Debye–Hückel model and further explore their capabilities and limitations concerning individual ion, and mean ionic activity coefficients for different size dissimilarities scenarios between ions. One immediate conclusion is that there is an unnecessary consideration in the Debye and Hückel derivation which is cancelled by another one they have made, leading to a correct expression for the activity coefficient. Also, the long-lasting consideration that both the Debye and Güntelberg charging processes lead to the same thermodynamic properties is demonstrated to be inaccurate, as it is rigorously true only when a common distance of closest approach is used.

Computational Investigation of the Ordered Water System Around Microtubules: Implications for Protein Interactions.

The existence of an exclusion zone in which particles of a colloidal suspension in water are repelled from hydrophilic surfaces has been experimentally demonstrated in numerous studies, especially in the case of Nafion surfaces. Various explanations have been proposed for the origin of this phenomenon, which is not completely understood yet. In particular, the existence of a fourth phase of water has been proposed by G. Pollack and if this theory is proven correct, its implications on our understanding of the properties of water, especially in biological systems, would be profound and could give rise to new medical therapies. Here, a simple approach based on the linearized Poisson-Boltzmann equation is developed in order to study the repulsive forces mediated by ordered water and involving the following interacting biomolecules: 1) microtubule and a tubulin dimer, 2) two tubulin dimers and 3) a tubulin sheet and a tubulin dimer. The choice of microtubules in this study is motivated because they could be a good candidate for the generation of an exclusion zone in the cell and these models could be a starting point for detailed experimental investigations of this phenomenon.

A study for the longitudinal permeability of fibrous porous media with consideration of electroviscous effects

We investigate theoretically the longitudinal permeability for the electroviscous flow through one-dimensional (1D) fibrous porous media with orderly and disorderly fiber arrays. The structures of fibrous porous media are idealized as circular unit cells. In the cell, the dimensionless net charge density is given by solving the linearized Poisson-Boltzmann equation, and is approximated as 1 by detailed discussions. Based on the approximate value of the net charge density, circular unit cell and Voronoi Tessellation method, we propose mathematical models for the longitudinal permeability with considering electroviscous effects. The effects of the fiber volume fraction, fiber radius and electroviscous resistance number on the longitudinal permeability are analyzed in detail. Moreover, there are good agreements between the present predictions and these results from the literatures.

Applicability of the linearized Poisson\u2013Boltzmann theory to contact angle problems and application to the carbon dioxide\u2013brine\u2013solid systems

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.

Quasi‐periodic electro‐osmotic flow of fractional viscoelastic fluids

In this work, we study the unsteady electro‐osmotic flow of fractional viscoelastic fluids in micro‐parallel channels, where the velocity slip effect on the walls is also considered. The analytical solution is obtained through solving the Cauchy momentum equation, together with the linearized Poisson–Boltzmann equation and the fractional Maxwell constitutive equation. The influences of the normalized electrokinetic width K, Deborah number De, slip length L and fractional derivative α on the velocity and volume flow rate are discussed. The results demonstrate that the dimensionless parameters L, De and K all have a promoting effect on the electro‐osmotic flow and resonance behavior of fractional Maxwell fluids. In addition, compared with the classical Maxwell fluid, the microstructure of the fractional Maxwell fluid weakens the resonance behavior.

Analytical solution of a linear nonlocal Poisson-Boltzmann equation with multiple charges in a spherical solute region surrounded by a water spherical shell.

In this paper, an analytical solution of a linear nonlocal Poisson-Boltzmann equation (NPBE) test model with multiple charges in a spherical solute region surrounded by a water spherical shell is derived as a single series of Legendre polynomials and modified spherical Bessel functions. The classic Kirkwood ball model is then shown to be a special case of the NPBE test model so that its analytical solution is regained from a double series of associated Legendre polynomials (derived by Kirkwood in 1934) to a new single series of Legendre polynomials, sharply reducing its computational cost. As an application of these series solutions, a comparison study is done to demonstrate the differences between the Kirkwood and NPBE test models.

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.

Electrophoretic Mobility and Electric Conductivity of Salt-Free Suspensions of Charged Soft Particles

A unit cell model is employed to analyze the electrophoresis and electric conduction in a concentrated suspension of spherical charged soft particles (each is a hard core coated with a porous polyelectrolyte layer) in a salt-free medium. The linearized Poisson–Boltzmann equation applicable to a unit cell is solved for the equilibrium electrostatic potential distribution in the liquid solution containing the counterions only surrounding a soft particle. The counterionic continuity equation and modified Stokes/Brinkman equations are solved for the ionic electrochemical potential energy and fluid velocity distributions, respectively. Closed-form formulas for the electrophoretic mobility of the soft particles and effective electric conductivity of the suspension are derived, and the effect of particle interactions on these transport characteristics is interesting and significant. Same as the case in a suspension containing added electrolytes under the Debye–Hückel approximation, the scaled electrophoretic mobility in a salt-free suspension is an increasing function of the fixed charge density of the soft particles and decreases with increases in the core-to-particle radius ratio, ratio of the particle radius to the permeation length in the porous layer, and particle volume fraction, keeping the other parameters unchanged. The normalized effective electric conductivity of the salt-free suspension also increases with an increase in the fixed charge density and with a decrease in the core-to-particle radius ratio, but is not a monotonic function of the particle volume fraction.

Charged dielectric spheres interacting in electrolytic solution: A linearized Poisson-Boltzmann equation model.

We present an analytical theory of electrostatic interactions of two spherical dielectric particles of arbitrary radii and dielectric constants, immersed into a polarizable ionic solvent (assuming that the linearized Poisson-Boltzmann framework holds) and bearing arbitrary charge distributions expanded in multipolar terms. The presented development entails a novel two-center re-expansion analytical theory that expands upon and improves the existing ones, bypassing the conventional expansions in modified Bessel functions. On this basis, we develop a specific matrix formalism that facilitates the construction of asymptotic expansions in ascending order of Debye screening terms of potential coefficients, which are then employed to find exact closed-form expressions for the total electrostatic energy. In particular, this work allows us to explicitly and precisely quantify the k-screened terms of the potential coefficients and mutual interaction energy. Specific cases of monopolar and dipolar distributions are described in particular detail. Comprehensive numerical examples and tests of series convergence and the relative balance of leading and higher-order terms of the mutual interaction energy are presented depending on the inter-particle distance and particles' radii. The results of this work find application in soft matter modeling and, in particular, in computational biophysics and colloid science, where the availability of increasingly larger experimental structures at the atomic-level resolution makes numerical treatment challenging and calls for more efficient expressions and an increased range of validity.

Electrophoresis and electric conduction in a salt-free suspension of charged particles.

The electrophoresis and electric conduction of a suspension of charged spherical particles in a salt-free solution are analyzed by using a unit cell model. The linearized Poisson-Boltzmann equation (valid for the cases of relatively low surface charge density or high volume fraction of the particles) and Laplace equation are solved for the equilibrium electric potential profile and its perturbation caused by the imposed electric field, respectively, in the fluid containing the counterions only around the particle, and the ionic continuity equation and modified Stokes equations are solved for the electrochemical potential energy and fluid flow fields, respectively. Explicit analytical formulas for the electrophoretic mobility of the particles and effective electric conductivity of the suspension are obtained, and the particle interaction effects on these transport properties are significant and interesting. The scaled zeta potential, electrophoretic mobility, and effective electric conductivity increase monotonically with an increase in the scaled surface charge density of the particles and in general decrease with an increase in the particle volume fraction, keeping each other parameter unchanged. Under the Debye-Hückel approximation, the dependence of the electrophoretic mobility normalized with the surface charge density on the ratio of the particle radius to the Debye screening length and particle volume fraction in a salt-free suspension is same as that in a salt-containing suspension, but the variation of the effective electric conductivity with the particle volume fraction in a salt-free suspension is found to be quite different from that in a suspension containing added electrolyte.

ARGOS: An adaptive refinement goal-oriented solver for the linearized Poisson-Boltzmann equation.

An adaptive finite element solver for the numerical calculation of the electrostatic coupling between molecules in a solvent environment is developed and tested. At the heart of the solver is a goal-oriented a posteriori error estimate for the electrostatic coupling, derived and implemented in the present work, that gives rise to an orders of magnitude improved precision and a shorter computational time as compared to standard finite difference solvers. The accuracy of the new solver ARGOS is evaluated by numerical experiments on a series of problems with analytically known solutions. In addition, the solver is used to calculate electrostatic couplings between two chromophores, linked to polyproline helices of different lengths and between the spike protein of SARS-CoV-2 and the ACE2 receptor. All the calculations are repeated by using the well-known finite difference solvers MEAD and APBS, revealing the advantages of the present finite element solver.

The Casimir Interaction between Spheres Immersed in Electrolytes

We investigate the Casimir interaction between two dielectric spheres immersed in an electrolyte solution. Since ionized solutions typically correspond to a plasma frequency much smaller than kBT/ħ at room temperature, only the contribution of the zeroth Matsubara frequency is affected by ionic screening. We follow the electrostatic fluctuational approach and derive the zero-frequency contribution from the linear Poisson-Boltzmann (Debye-Hückel) equation for the geometry of two spherical surfaces of arbitrary radii. We show that a contribution from monopole fluctuations, which is reminiscent of the Kirkwood-Shumaker interaction, arises from the exclusion of ionic charge in the volume occupied by the spheres. Alongside the contribution from dipole fluctuations, such monopolar term provides the leading-order Casimir energy for very small spheres. Finally, we also investigate the large sphere limit and the conditions for validity of the proximity force (Derjaguin) approximation. Altogether, our results represent the first step towards a full scattering approach to the screening of the Casimir interaction between spheres that takes into account the nonlocal response of the electrolyte solution.