Standard Poisson-Boltzmann Research Articles

Which Interfaces Matter Most? Variability in Grain Boundary Defect Chemistry and Conductivity in a Concentrated Solid Electrolyte

Inorganic solid electrolytes with fast ionic conductivity are critical components of fuel/electrolysis cells, all-solid-state batteries, membrane and separation reactors, sensors and other electrochemical devices. For example, solid oxygen ion (O2-) conductors are used in solid oxide fuel/electrolysis cells (SOFCs/SOECs) for high-efficiency conversion between chemical and electrical energy and chemical production from CO2. One major challenge in engineering such materials is the diminished grain boundary (GB) conductivity in polycrystalline oxide ceramics. This phenomenon, which was initially accredited to highly resistive silica-based glassy impurity phases, is attributed to the formation of intrinsic space charge layers (SCLs) directly adjacent to GB cores. The positive potential at the GB core (due to large concentrations of oxygen vacancies) results in a depletion of oxygen vacancies in the space charge layer as well as the segregation of charge-compensating acceptor solutes from grain interiors which ultimately reduces GB conductivity. Several strategies have been explored to improve GB conductivity by understanding and mitigating the SCL effects. While the standard Poisson–Boltzmann approach models the near-grain boundary defect behavior in dilute solid solutions and recent models by Mebane et al [1] and Virkant et al [2] incorporate the effects of defect-defect interactions (in concentrated solid solutions) and chemomechanical stress, respectively, there is yet to be a model developed that can predict individual GB conductivities in concentrated solid solutions based on direct experimental measurements of near-GB defect chemistry.Here, we present an experimental-computational framework that predicts which GBs in a polycrystalline electrolyte likely facilitate ionic conductivity. We developed a thermodynamic phase-field modeling framework and applied it to a model oxygen electrolyte Gd0.25Ce0.75O2-δ, wherein the GB-to-GB variability in defect concentrations (Gd3+ solutes, electrons localized at Ce3+, and oxygen vacancies) was measured microscopically using scanning transmission electron microscopy electron energy-loss spectroscopy (STEM EELS). These data were used to predict the GB-to-GB variability in cross-GB ionic conduction using a unique model, which prioritizes reproducing microscopically observed GB defect concentrations and considers defect-defect interactions in highly concentrated solid solutions, making the framework applicable to other technologically relevant solid electrolytes. Across the GBs studied, we revealed a non-monotonic relationship between depletion and GB conductivity, with the highest conductivity predicted for intermediate depletion amounts. This work lays the foundation for future experimental-computational research and advanced data-driven design of solid electrolytes needed for functional and energy storage/conversion devices.[1] D. S. Mebane, R. A. D. Souza, Energy & Environmental Science, 2015.[2] K. S. N. Vikrant, R. Edwin García, Energy & Environmental Science, 2018.[3] H. Vahidi, W.J. Bowman (Submitted). Acknowledgments HV and WJB acknowledge primary support from the UCI new faculty startup funding. This research was partially supported by the National Science Foundation Materials Research Science and Engineering Center program through the UC Irvine Center for Complex and Active Materials (DMR-2011967). We acknowledge the use of facilities and instrumentation at the UC Irvine Materials Research Institute (IMRI) supported in part by the same NSF MRSEC (DMR-2011967). AM, DM, and AC acknowledge funding from the NSF [WB1] [hv2] and the WVU MAE research department for the use of the Thorny Flat HPC Cluster for the numerical modeling work and computational resources.

Electroosmotic modulated unsteady squeezing flow with temperature-dependent thermal conductivity, electric and magnetic field effects

Modern lubrication systems are increasingly deploying smart (functional) materials. These respond to various external stimuli including electrical and magnetic fields, acoustics, light etc. Motivated by such developments, in the present article unsteady electro-magnetohydrodynamics squeezing flow and heat transfer in a smart ionic viscous fluid intercalated between parallel plates with zeta potential effects is examined. The proposed mathematical model of problem is formulated as a system of partial differential equations (continuity, momenta and energy). Viscous dissipation and variable thermal conductivity effects are included. Axial electrical distribution is also addressed. The governing equations are converted into ordinary differential equations via similarity transformations and then solved numerically with MATLAB software. The transport phenomena are scrutinized for both when the plates move apart or when they approach each other. Also, the impact of different parameters such squeezing number, variable thermal conductivity parameter, Prandtl number, Hartmann number, Eckert number, zeta potential parameter, electric field parameter and electroosmosis parameter on the axial velocity and fluid temperature are analysed. For varied intensities of applied plate motion, the electro-viscous effects derived from electric double-capacity flow field distortions are thoroughly studied. It has been shown that the results from the current model differ significantly from those achieved by using a standard Poisson–Boltzmann equation model. Axial velocity acceleration is induced with negative squeeze number (plates approaching, S < 0) in comparison to that of positive squeeze number (plates separating, S > 0). Velocity enhances with increasing electroosmosis parameter and zeta potential parameter. With rising values of zeta potential and electroosmosis parameter, there is a decrease in temperatures for U e > 0 for both approaching i.e. squeezing plates (S < 0) and separating (S > 0) cases. The simulations provide novel insights into smart squeezing lubrication with thermal effects and also a solid benchmark for further computational fluid dynamics investigations.

Simultaneous microfluidic pumping and mixing using an array of asymmetric 3D ring electrode pairs in a cylindrical microchannel by the AC electroosmosis effect

Abstract The present studies of AC electroosmosis (ACEO) micropumps for simultaneous pumping and mixing are mainly involved in the designs of the planar electrode pairs in a rectangular microchannel, such as the diagonal/herringbone shape of electrodes on top and bottom of the substrates. The two mixing samples usually flow into the left and right half of the inlet, respectively. However, the flow rate is limited because of the less electrodes on the walls, and the mixing is only appropriate for the channel of small diameters owing to the vortices appearing near the surface of electrodes. To solve the problem, we proposed an ACEO micropump with an array of asymmetric ring electrode pairs in a cylindrical microfluidic channel, and established a three-dimensional (3D) theoretical model by the standard Poisson–Boltzmann (PB) theory and convection–diffusion equation. In this paper, the two mixing samples flow into the inner and outer circular pipe of the inlet, respectively. Thus, the samples can be exchanged from the outer pipe area to the inner pipe area, so that it can strengthen the mixing by the vortices around the ring electrode surfaces in the microchannel. The microfluidic velocity field, vorticity field and concentration field are investigated in detail to explain the mechanism of pumping and mixing. To further improve the mixing performance, we designed and researched the combination sequences PnMm of ring electrode pairs based on the pumping mode of “forward driving of electrode pair” and the mixing mode of “reversal driving of electrode pair”. The simulation results show that the ring ACEO micropump with the inner and outer pipe inlet can rapidly pump the microflows and efficiently mix the samples simultaneously compared with the planar multifunctional ACEO micropumps. The numerical simulation of the ring ACEO micropump in this paper is of significant importance for the development of multifunctional microfluidic devices in biochemical field, and feasible fabrication techniques should be experimentally investigated in future studies.

A generalised space-charge theory for extended defects in oxygen-ion conducting electrolytes: from dilute to concentrated solid solutions

The standard Poisson–Boltzmann approach to modeling the near-interface defect behaviour in solid electrolytes performs well at low dopant concentrations but its applicability is questionable at higher dopant levels where interactions become important.

The charge conserving Poisson-Boltzmann equations: Existence, uniqueness, and maximum principle

The present article is concerned with the charge conserving Poisson-Boltzmann (CCPB) equation in high-dimensional bounded smooth domains. The CCPB equation is a Poisson-Boltzmann type of equation with nonlocal coefficients. First, under the Robin boundary condition, we get the existence of weak solutions to this equation. The main approach is variational, based on minimization of a logarithm-type energy functional. To deal with the regularity of weak solutions, we establish a maximum modulus estimate for the standard Poisson-Boltzmann (PB) equation to show that weak solutions of the CCPB equation are essentially bounded. Then the classical solutions follow from the elliptic regularity theorem. Second, a maximum principle for the CCPB equation is established. In particular, we show that in the case of global electroneutrality, the solution achieves both its maximum and minimum values at the boundary. However, in the case of global non-electroneutrality, the solution may attain its maximum value at an interior point. In addition, under certain conditions on the boundary, we show that the global non-electroneutrality implies pointwise non-electroneutrality.

Electrostatic interaction forces in aqueous salt solutions of variable concentration andvalency

We use atomic force microscopy (AFM) to determine electrostatic interactions between Sitips and Si wafers in aqueous electrolytes of variable composition. We demonstrate thatdynamic force spectroscopy (DFS) in the frequency modulation (FM) mode with stiffcantilevers and sharp tips allows for a continuous detection of the tip–sample interactionswithout mechanical jump-to-contact instability and with substantially higher lateral resolutionthan standard colloidal probe measurements. For four different species of salt (NaCl, KCl,MgCl2,CaCl2) we find repulsive electrostatic forces at the lowest salt concentrations (1 mM) that becomeprogressively screened until they are dominated by attractive van der Waals forces atthe highest concentration (100 mM). For the divalent cations the crossover fromrepulsive to attractive forces occurs at lower concentrations than for monovalentcations. Surface charges extracted from fits to standard Poisson–Boltzmann doublelayer theory indicate a rather weak dependence of the surface charge on the ionconcentration. The high lateral resolution of our approach is illustrated by a 2D force fieldmeasurement over a patterned surface of a supported lipid bilayer on a mica substrate.

Polyelectrolyte-mediated bridging interactions: columnar macromolecular phases

We present a mean-field theory for charged polymer chains in an external electrostatic fieldin the weak and strong coupling limits. We apply the theory to describe the statisticalmechanics of flexible polyelectrolyte chains in a hexagonal columnar lattice of stiffcylindrical macroions, such as DNA, in a bathing solution of a uni-univalent salt(e.g. NaCl). The salt effects are first described in the Debye–Hückel framework. This yieldsthe macroion electrostatic field in the screened Coulomb form, which we taketo represent the mean field into which the chains are immersed. We introducethe Green’s function for the polyelectrolyte chains and derive the correspondingEdwards equation which we solve numerically in the Wigner–Seitz cylindrical cellusing the ground state dominance ansatz. The solutions indicate the presence ofpolyelectrolyte bridging, which results in a like-charge attraction between stiffmacroions. Then we reformulate the Edwards theory for the strong coupling caseand use the standard Poisson–Boltzmann picture to describe the salt solution.We begin with the free energy which we minimize to obtain the Euler–Lagrangeequations. The solutions yield self-consistently determined monomer density andelectrostatic fields. We furthermore calculate the free energy density as well as the totalosmotic pressure in the system. We again show that bridging implicates like-chargeattractions of entropic origin between stiff cylindrical macroions. By analyzing theosmotic pressure we demonstrate that, in certain parts of the parameter space, aphase transition occurs between two phases of the same hexagonal symmetry.

Communications: Monovalent ion condensation at the electrified liquid/liquid interface

X-ray reflectivity studies demonstrate the condensation of a monovalent ion at the electrified interface between electrolyte solutions of water and 1,2-dichloroethane. Predictions of the ion distributions by standard Poisson-Boltzmann (Gouy-Chapman) theory are inconsistent with these data at higher applied interfacial electric potentials. Calculations from a Poisson-Boltzmann equation that incorporates a nonmonotonic ion-specific potential of mean force are in good agreement with the data.

Overcharging and charge reversal in the electrical double layer around the point of zero charge

The ionic adsorption around a weakly charged spherical colloid, immersed in size-asymmetric 1:1 and 2:2 salts, is studied. We use the primitive model (PM) of an electrolyte to perform Monte Carlo simulations as well as theoretical calculations by means of the hypernetted chain/mean spherical approximation (HNC/MSA) and the unequal-radius modified Gouy-Chapman (URMGC) integral equations. Structural quantities such as the radial distribution functions, the integrated charge, and the mean electrostatic potential are reported. Our Monte Carlo "experiments" evidence that near the point of zero charge, the smallest ionic species is preferentially adsorbed onto the macroparticle, independently of the sign of the charge carried by this tiniest electrolytic component, giving rise to the appearance of the phenomena of charge reversal (CR) and overcharging (OC). Accordingly, colloidal CR, due to an excessive attachment of counterions, is observed when the macroion is slightly charged and the coions are larger than the counterions. In the opposite situation, i.e., if the counterions are larger than the coions, the central macroion acquires additional like-charge (coions) and hence becomes "overcharged," a feature theoretically predicted in the past [F. Jiménez-Angeles and M. Lozada-Cassou, J. Phys. Chem. B 108, 7286 (2004)]. In other words, here we present the first simulation data on OC in the PM electrical double layer, showing that close to the point of zero charge, this novel effect surges as a consequence of the ionic size asymmetry. We also find that the HNC/MSA theory captures well the CR and OC phenomena exhibited by the computer experiments, especially as the macroion's charge increases. On the contrary, even if URMGC also displays CR and OC, its predictions do not compare favorably with the Monte Carlo data, evidencing that the inclusion of hard-core correlations in Monte Carlo and HNC/MSA enhances and extends those effects. We explain our findings in terms of the energy-entropy balance. In the field of electrophoresis, it has been generally agreed that the charge of a colloid in motion is partially decreased by counterion adsorption. Depending on the location of the macroion's slipping surface, the OC results of this paper could imply an increase in the expected electrophoretic mobility. These observations aware about the interpretation of electrokinetic measurements using the standard Poisson-Boltzmann approximation beyond its validity region.

Correlation effects, image charge effects and finite size in the macro-ion-electrolyte system: A field-theoretic approach

We consider a model of a macro-ion surrounded by small ions of an electrolyte solution. The finite size of ionic charge distributions of ions, and image charge effects are considered. From such a model it is possible to construct a statistical field theory with a single fluctuating field and derive physical interpretations for both the mean field and two-point correlation function. For point-like charges, at the level of a Gaussian (or saddle point) approximation, we recover the standard Poisson-Boltzmann equation. However, to include ionic correlation effects, as well as image charge effects of individual ions, we must go beyond this. From the field theory considered, it is possible to construct self-consistent approximations. We consider the simplest of these, namely the Hartree approximation. The Hartree equations take the form of two coupled equations. One is a modified Poisson-Boltzmann equation; the other describes both image charge effects on the individual ions, as well as correlations. Such equations are difficult to solve numerically, so we develop an (a WKB-like) approximation for obtaining approximate solutions. This, we apply to a uniformly charged rod in univalent electrolyte solution, for point like ions, as well as for extended spherically symmetric distributions of ionic charge on electrolyte ions. The solutions show how correlation effects and image charge effects modify the Poisson-Boltzmann result. Finite-size charge distributions of the ions reduce both the effects of correlations and image charge effects. For point charges, we test the WKB approximation by calculating a leading-order correction from the exact Hartree result, showing that the WKB-like approximation works reasonably well in describing the full solution to the Hartree equations. From these solutions, we also calculate an effective charge compensation parameter in an analytical formula for the interaction of two charged cylinders.

What role do surfaces play in GB models? A new‐generation of surface‐generalized born model based on a novel gaussian surface for biomolecules

We have developed a version of our surface generalized Born (SGB) model that employs a Gaussian surface, as opposed to the van der Waals surface used previously. The Gaussian surface is smooth and its properties are analytically differentiable with respect to the positions of atoms. A significant advantage of a solvent model based on this analytically differentiable surface is the availability of analytical gradients of the surface and solvation forces. An efficient and robust algorithm is designed to construct and triangulate the Gaussian surface for large biomolecules with arbitrary shapes, and to compute the various terms required for energy gradients. The Gaussian surface is shown to better mimic the boundary between the solute and solvent by properly addressing solvent accessibility, as is demonstrated by comparisons with standard Poisson-Boltzmann calculations for proteins of different sizes. These results also demonstrate that surface definition is a dominant contribution to differences between GB and PB calculations, especially if the system is large. Application of the new surface to prediction of long loop regions is presented, and significant improvement in the energetics is seen compared with results obtained using the van der Waals surface, even in the absence of optimized empirical correction terms that were used in the latter calculations.

A multilayered approach to approximating solute polarization

A hybrid multilayered "ONIOM"-type approach to solvation is presented in which the basic free energy of hydration is taken from the Poisson Boltzmann method and the contribution to the solute polarization is taken from a quantum mechanical implementation of the Born method. The method has been tested on the 52 neutral molecules used in the AM1-SM2 parameterization, and the polarized continuum method is taken as the standard by which the results are assessed. Regression analysis shows that the method gives a small improvement over the standard Poisson Boltzmann method or a dramatic improvement over the Born method. The system presented here represents one of the more straightforward applications of the multilayered approach to solvation, but other more sophisticated approaches are discussed.

Long-range electrostatic interactions between like-charged colloids: Steric and confinement effects

Within the framework of a modified Poisson-Boltzmann theory accounting for steric effects of microions, we prove analytically that the effective pair interactions between like-charge colloids immersed in a confined electrolyte are repulsive. Our approach encompasses and extends previously known results to the case of complete confinement, and further incorporates the finite size of the microions which is absent in the standard Poisson-Boltzmann theory.

Prediction of titration properties of structures of a protein derived from molecular dynamics trajectories.

This paper explores the dependence of the molecular dynamics (MD) trajectory of a protein molecule on the titration state assigned to the molecule. Four 100-ps MD trajectories of bovine pancreatic trypsin inhibitor (BPTI) were generated, starting from two different structures, each of which was held in two different charge states. The two starting structures were the X-ray crystal structure and one of the solution structures determined by NMR, and the charge states differed only in the ionization state of N terminus. Although it is evident that the MD simulations were too short to sample fully the equilibrium distribution of structures in each case, standard Poisson-Boltzmann titration state analysis of the resulting configurations shows general agreement between the overall titration behavior of the protein and the charge state assumed during MD simulation: at pH 7, the total net charge of the protein resulting from the titration analysis is consistently lower for the protein with the N terminus assumed to be neutral than for the protein with the N terminus assumed to be charged. For most of the ionizable residues, the differences in the calculated pKaS among the four trajectories are statistically negligible and remain in good agreement with the data obtained by crystal structure titration and by experiment. The exceptions include the N terminus, which responds directly to the change of its imposed charge; the C terminus, which in the NMR structure interacts strongly with the former; and a few other residues (Arg 1, Glu 7, Tyr 35, and Arg 42) whose pKaS reflect the initial structure and the limited trajectory lengths. This study illustrates the importance of the careful assignment of protonation states at the start of MD simulations and points to the need for simulation methods that allow for the variation of the protonation state in the calculation of equilibrium properties.

Towards an exact theory of polyelectrolytes

Abstract Within the framework of the gas approximation an exact method is developed for calculating the electric potential distribution around a polyion. The polyion is modelled as an infinitely long impermeable cylinder and the mobile ions are assumed to be impermeable spheres. Equations are derived for two potentials ψ min and ψ max so that the genuine potential ψ lies in between. The equations are numerically solved for different parameter values. The results are compared with data obtained on the basis of the standard Poisson-Boltzmann equation.

Electrical double layer forces. A Monte Carlo study

Using a novel method the force between two charged surfaces with an intervening electrolyte solution has been determined from Monte Carlo simulations. We find large deviations from the standard Poisson–Boltzmann treatment of the so called double layer force for divalent counterions at high surface charge densities and at short separations. The deviations have two causes: (i) Due to the inclusion of the effect of ion–ion correlations the counterions concentrate more towards the charged wall reducing the overlap between the double layers; and (ii) correlated fluctuations in the ion clouds of the two surfaces lead to an attractive interaction of a van der Waals type. For some realistic values of the parameters the attraction overcomes the repulsive part and there is a net attractive force between similarly charged surfaces. This finding leads to a modification of our conceptual understanding of the interaction between charged particles and it shows that the DLVO theory is qualitatively deficient under some, realistic, conditions.