Nonlinear Poisson-Boltzmann Research Articles

Electrical perturbations of ultrathin bilayers: role of ionic conductive layer.

The effect of electrostatic force on the dynamics, morphological evolution, and drainage time of ultrathin liquid bilayers (<100 nm) are investigated for perfect dielectric-perfect dielectric (PD-PD) and ionic liquid-perfect dielectric (IL-PD) bilayers. The weakly nonlinear "thin film" equation is solved numerically to obtain spatiotemporal evolution of the liquid-liquid interface responses to transverse electric field. In order to predict the electrostatic component of conjoining/disjoining pressure acting on the interface for IL-PD bilayers, an analytical model is developed using the nonlinear Poisson-Boltzmann equation. It is found that IL-PD bilayers with electric permittivity ratio of layers (lower to top), εr, greater than one remain stable under an applied electric field. An extensive numerical study is carried out to generate a map based on εr and the initial mean thickness of the lower layer. This map is used to predict the formation of various structures on PD-PD bilayer interface and provides a baseline for unstable IL-PD bilayers. The use of an ionic liquid (IL) layer is found to reduce the size of the structures, but results in polydispersed and disordered pillars spread over the domain. The numerical predictions follow similar trend of experimental observation of Lau and Russel. (Lau, C. Y.; Russel, W. B. Fundamental Limitations on Ordered Electrohydrodynamic Patterning; Macromolecules 2011, 44, 7746-7751).

Compact Model of a pH Sensor with Depletion-Mode Silicon-Nanowire Field-Effect Transistor

A compact model of a depletion-mode silicon-nanowire (Si-NW) pH sensor is proposed. This drain current model is obtained from the Pao-Sah integral and the continuous charge-based model, which is derived by applying the parabolic potential approximation to the Poisson's equation in the cylindrical coordinate system. The threshold-voltage shift in the drain-current model is obtained by solving the nonlinear Poisson-Boltzmann equation for the electrolyte. The simulation results obtained from the proposed drain-current model for the Si-NW field-effect transistor (SiNWFET) agree well with those of the three-dimensional (3D) device simulation, and those from the Si-NW pH sensor model also agree with the experimental data.

Rotating electroosmotic flow of power-law fluids at high zeta potentials

In this paper, rotating electroosmotic flow (EOF) of power-law fluids at high zeta potentials in a slit microchannel is analyzed. The electric double layer (EDL) potential distribution is considered by using nonlinear Poisson–Boltzmann equation. Based upon the analytical charge density distribution, the finite difference method is used to compute numerically rotating EOF velocity profiles of power-law fluids. Results of the present analysis are compared with a simplified analysis obtained by Debye–Hückel linear approximation when the fluid is Newtonian. The classical steady plug-shape EOF velocity of non-Newtonian fluid is reduced for large enough time if the rotating effect is ignored. Additionally, the influences of the flow behavior index n, the rotating angular velocity Ω, wall Zeta potential ψw and the electrodynamic width K on the velocity profiles are discussed in detail.

An Iterative Discontinuous Galerkin Method for Solving the Nonlinear Poisson Boltzmann Equation

AbstractAn iterative discontinuous Galerkin (DG) method is proposed to solve the nonlinear Poisson Boltzmann (PB) equation. We first identify a function space in which the solution of the nonlinear PB equation is iteratively approximated through a series of linear PB equations, while an appropriate initial guess and a suitable iterative parameter are selected so that the solutions of linear PB equations are monotone within the identified solution space. For the spatial discretization we apply the direct discontinuous Galerkin method to those linear PB equations. More precisely, we use one initial guess when the Debye parameterλ=(1), and a special initial guess forλ≫1 to ensure convergence. The iterative parameter is carefully chosen to guarantee the existence, uniqueness, and convergence of the iteration. In particular, iteration steps can be reduced for a variable iterative parameter. Both one and two-dimensional numerical results are carried out to demonstrate both accuracy and capacity of the iterative DG method for both cases ofλ=(1) andλ≪ 1. The (m+ 1)th order of accuracy forL2andmth order of accuracy forH1forPmelements are numerically obtained.

A new Green's function Monte Carlo algorithm for the solution of the two-dimensional nonlinear Poisson–Boltzmann equation: Application to the modeling of the communication breakdown problem in space vehicles during re-entry

The objective of this paper is the exposition of a recently-developed, novel Green's function Monte Carlo (GFMC) algorithm for the solution of nonlinear partial differential equations and its application to the modeling of the plasma sheath region around a cylindrical conducting object, carrying a potential and moving at low speeds through an otherwise neutral medium. The plasma sheath is modeled in equilibrium through the GFMC solution of the nonlinear Poisson–Boltzmann (NPB) equation. The traditional Monte Carlo based approaches for the solution of nonlinear equations are iterative in nature, involving branching stochastic processes which are used to calculate linear functionals of the solution of nonlinear integral equations. Over the last several years, one of the authors of this paper, K. Chatterjee has been developing a philosophically-different approach, where the linearization of the equation of interest is not required and hence there is no need for iteration and the simulation of branching processes. Instead, an approximate expression for the Green's function is obtained using perturbation theory, which is used to formulate the random walk equations within the problem sub-domains where the random walker makes its walks. However, as a trade-off, the dimensions of these sub-domains have to be restricted by the limitations imposed by perturbation theory. The greatest advantage of this approach is the ease and simplicity of parallelization stemming from the lack of the need for iteration, as a result of which the parallelization procedure is identical to the parallelization procedure for the GFMC solution of a linear problem. The application area of interest is in the modeling of the communication breakdown problem during a space vehicle's re-entry into the atmosphere. However, additional application areas are being explored in the modeling of electromagnetic propagation through the atmosphere/ionosphere in UHF/GPS applications.

Specific ion effects on the electrophoretic mobility of small, highly charged peptides: a modeling study.

In this work, we use coarse-grained modeling to study the free solution electrophoretic mobility of small highly charged peptides (lysine, arginine, and short oligos thereof (up to nonapeptides)) in NaCl and Na2SO4 aqueous solutions at neutral pH and room temperature. The experimental data are taken from the literature. A bead modeling methodology that treats the electrostatics at the level of the nonlinear Poisson Boltzmann equation developed previously in our laboratory is able to account for the mobility of all peptides in NaCl, but not Na2SO4. The peptide mobilities in Na2SO4 can be accounted for by including sulfate binding in the model and this is proposed as one possible explanation for the discrepancy. Oligo arginine peptides bind more sulfate than oligo lysines and sulfate binding increases with the oligo length.

Polarization effects of dielectric nanoparticles in aqueous charge-asymmetric electrolytes.

Small nanoparticles, globular proteins, viral capsids, and other nanoscopic biomolecules usually display dielectric properties that are different from those of the medium in which they are dispersed. These dielectric heterogeneities can significantly influence the surrounding ion distribution, which determines the self-assembly and colloidal stability of these nanoparticles in solution. Here, we study the impact of a dielectric discontinuity in the structural and thermodynamic properties of a spherical nanoparticle made of different dielectric materials when it is immersed in a charge-asymmetric 1:z supporting electrolyte. The mean electrostatic potential, integrated charge, and ionic profiles are analyzed as a function of both the salt concentration and the nanoparticle's valence via Monte Carlo simulations and the nonlinear Poisson-Boltzmann theory. We observe that the electrostatic screening and charge neutralization near the surface of a nanoparticle increase when the nanoparticle's dielectric permittivity increases in all instances. For 1:1 salts, this effect is small and the nonlinear Poisson-Boltzmann theory displays a good agreement with simulation results. Nevertheless, significant deviations are displayed by the mean field scheme regarding simulation results in the presence of multivalent ions. In particular, for trivalent counterions we observe that increasing the dielectric permittivity or the valence of the nanoparticle decreases the critical salt concentration at which occurs a sign inversion of the mean electrostatic potential at the Helmholtz plane, which is closely related to the behavior of the ζ potential and the electrophoretic mobility. Moreover, we observe that the phenomenon of surface charge amplification, or the augmenting of the net charge of a nanoparticle by the adsorption of like-charged ions on its surface, can be promoted by polarization effects in weakly charged spherical nanoparticles with low dielectric permittivity.

Exact analytical solution to the Poisson-Boltzmann equation for semiconductor devices

This paper shows that the nonlinear Poisson-Boltzmann equation for semiconductor devices describing potential distribution in a Double Gate-Metal Oxide Semiconductor Field Effect Transistor (DG-MOSFET) is exactly solvable. The DG-MOSFET represents one of the most advanced device structures in semiconductor technology and is a primary focus of modeling efforts in the semiconductor industry. Furthermore, exact analytical solution is obtained in the implicit form for further physical interpretation and a full discussion is given.

Modeling the Electric Potential across Neuronal Membranes: The Effect of Fixed Charges on Spinal Ganglion Neurons and Neuroblastoma Cells

We present a model for the electric potential profile across the membranes of neuronal cells. We considered the resting and action potential states, and analyzed the influence of fixed charges of the membrane on its electric potential, based on experimental values of membrane properties of the spinal ganglion neuron and the neuroblastoma cell. The spinal ganglion neuron represents a healthy neuron, and the neuroblastoma cell, which is tumorous, represents a pathological neuron. We numerically solved the non-linear Poisson-Boltzmann equation for the regions of the membrane model we have adopted, by considering the densities of charges dissolved in an electrolytic solution and fixed on both glycocalyx and cytoplasmic proteins. Our model predicts that there is a difference in the behavior of the electric potential profiles of the two types of cells, in response to changes in charge concentrations in the membrane. Our results also describe an insensitivity of the neuroblastoma cell membrane, as observed in some biological experiments. This electrical property may be responsible for the low pharmacological response of the neuroblastoma to certain chemotherapeutic treatments.

Effect of oppositely charged hydrophobic additives (alkanoates) on the stability of C14TAB foam films

The impact of the hydrophobicity of oppositely charged additives on the stability of C14TAB foam films has been studied using a thin film pressure balance (TFPB). Disjoining pressure isotherms of 10−4M C14TAB solutions containing negatively charged sodium alkanoates CnOONa with increasing chain length from n=4 to n=12 are measured. The results of disjoining pressure measurements are presented as maximum possible pressure Πmax applied to a stable film as a function of the alkanoate concentration. Furthermore, the surface properties of the mixed systems are described to gain a general understanding about the surface characteristics. Since C14TAB and alkanoates are oppositely charged, the question arises whether charge reversal at the film surfaces occurs and how it depends on the chain length of the alkanoate. The presented results indicate that C14TAB/alkanoate systems can be divided into three groups. Depending on the chain length, alkanoates act as organic salts (n=4), co-surfactant (n=6) or anionic surfactant (8≤n≤12). In case of alkanoates acting as anionic surfactant, the simulation of the isotherms by solving the non-linearized Poisson–Boltzmann equation under assumption of constant potential actually indicates that charge reversal occurs close to the nominal IEP. The film stabilities are markedly affected by the hydrophobicity and the amount of alkanoate.

Overlapping of the Diffuse Double Layer in the Vicinity of the Microporous Electrode

Introduction A microporous electrode with nm-sized micropores on the surface is an attractive subject in electrochemistry. Some experimental results show the physicochemical properties of microporous electrodes are not explained only in terms of the surface area. We have an interest in the electrical double layer in the narrow space. When the size of the micropore on the electrode is smaller than the diffuse part of the electrical double layer, the diffuse double layer show overlap across the pore. This overlapping of the diffuse double layer cannot be estimated on the one dimension model, such as Gouy–Chapman (GC) theory for planar electrodes. We propose a model of the diffuse double layers in the vicinity of micropores on the basis of the Poisson-Boltzmann equation without any complicated assumptions. This calculation shows essential features of the diffuse double layer on the electrode with micropores. The size dependence, the permittivity dependence, the potential dependence, and the electrolyte concentration dependence of the diffuse double layer and also the differential capacitance of the electrode with micropores are discussed.Model The model is described with a cylindrical coordinate. The electrode was assumed as a conductor with an equipotential surface. The diameter of a pore on the electrode was varied from 0.2 to 5 nm. To avoid singular points in the calculation, the radius of the corners was assumed to be 50 pm in the geometry of this model. The solution phase contains a 1:1 electrolyte. In the solution phase, the Poisson–Boltzmann equation was applied to obtain the profile of the electric potential. In this calculation, the relative permittivity was set to be from 10 to 78. The nonlinear Poisson–Boltzmann equation was numerically solved by the finite element method. The differential capacitance of the double layer was estimated from the dependence of the total charge on the electrode surface on the surface potential.Results and discussion The calculated potential profile showed the overlapping of the diffuse double layer in the micropore. The electric field in the micropore is weaker than that outside of the micropore, that is, the overlapped diffuse double layer provides the mild electric field. Assuming that the thickness of the diffuse double layer is the division of the surface potential by the gradient of the potential at the electrode surface, the degree of the overlapping can express qualitatively. When the size of the micropore on the electrode is smaller than the thickness of the diffuse double layer, the calculated capacitance agrees with the GC capacitance at the planner electrode. On the other hand, when the size of the micropore on the electrode is larger than the thickness of the diffuse double layer, the double layer capacitance converges on the GC capacitance at the microporous electrode. This means that the surface area of micropore becomes effective only at thinner diffuse double layers. The change of the thickness of the diffuse double layer in the micropore causes the strong dependence of the capacitance on the potential and the electrolyte concentration compared with the situation of planar electrode. The characteristic features of microporous electrodes would be originated from the heterogeneous electric field caused by the overlapping of the diffuse double layer.

Correlation potential of a test ion near a strongly charged plate.

We analytically calculate the correlation potential of a test ion near a strongly charged plate inside a dilute m:-n electrolyte. We do this by calculating the electrostatic Green's function in the presence of a nonlinear background potential, the latter having been obtained using the nonlinear Poisson-Boltzmann theory. We consider the general case where the dielectric constants of the plate and the electrolyte are distinct. The following generic results emerge from our analyses: (1) If the distance to the plate Δz is much larger than a Gouy-Chapman length, the plate surface will behave effectively as an infinitely charged surface, and the dielectric constant of the plate effectively plays no role. (2) If Δz is larger than a Gouy-Chapman length but shorter than a Debye length, the correlation potential can be interpreted in terms of an image charge that is three times larger than the source charge. This behavior is independent of the valences of the ions. (3) The Green's function vanishes inside the plate if the surface charge density is infinitely large; hence the electrostatic potential is constant there. In this respect, a strongly charged plate behaves like a conductor plate. (4) If Δz is smaller than a Gouy-Chapman length, the correlation potential is dominated by the conventional image charge due to the dielectric discontinuity at the interface. (5) If Δz is larger than a Debye length, the leading order behavior of the correlation potential will depend on the valences of the ions in the electrolyte. Furthermore, inside an asymmetric electrolyte, the correlation potential is singly screened, i.e., it undergoes exponential decay with a decay width equal to the Debye length.

Coarse-Grained Modeling of the Titration and Conductance Behavior of Aqueous Fullerene Hexa Malonic Acid (FHMA) Solutions

The coarse-grained continuum primitive model is developed and used to characterize the titration and electrical conductance behavior of aqueous solutions of fullerene hexa malonic acid (FHMA). The spherical FHMA molecule, a highly charged electrolyte with an absolute valence charge as large as 12, is modeled as a dielectric sphere in Newtonian fluid, and electrostatics are treated numerically at the level of the non-linear Poisson-Boltzmann equation. Transport properties (electrophoretic mobilities and conductances) of the various charge states of FHMA are numerically computed using established numerical algorithms. For reasonable choices of the model parameters, good agreement between experiment (published literature) and modeling is achieved. In order to accomplish this, however, a moderate degree of specific binding of principal counterion and FHMA must be included in the modeling. It should be emphasized, however, that alternative explanations are possible. This comparison is made at 25 °C for both Na(+) and Ca(2+) principal counterions. The model is also used to characterize the different charge states and degree of counterion binding to those charge states as a function of pH.

ELECTROSTATICS STUDY OF A SINGLE-STRANDED DNA: A PROSPECTIVE FOR SINGLE MOLECULE SEQUENCING

Single molecule DNA sequencing requires new approaches to identify nucleotide bases. Using molecular dynamics simulations, we investigate the intrinsic electrostatics of single-stranded DNA by solving the nonlinear Poisson–Boltzmann equation. The results show variations of the molecular electrostatic potential (MEP) within 3 nm from the center of the sugar backbone, with suitably differentiable variations at 1.4 nm distance. MEP variations among four nucleotide bases are the most significant near ~ 33.7° and ~ 326.3° angular orientation, while the influences of the neighboring bases on MEPs become insignificant after the 3rd-nearest neighbors. This analysis shows potential for direct electronic sequencing of individual DNA molecules. [Formula: see text]Special Issue Comment: This paper about DNA sequencing based on molecular electrostatic potential maps of the DNA in the channel is related to the Special Issue articles about: measuring enzymes,32 and solving single molecules' trajectories with the RDF approach33 and with the QuB software.34

Extensions to Study Electrochemical Interfaces - A Contribution to the Theory of Ions

In the present study an alternative model allows the extension of the Debye-Huckel Theory (DHT) considering time dependence explicitly. From the Electro-Quasistatic approach (EQS) done in earlier studies time dependent potentials are suitable to describe several phenomena especially conducting media as well as the behaviour of charged particles in arbitrary solutions acting as electrolytes. This leads to a new formulation of the meaning of the nonlinear Poisson-Boltzmann Equation (PBE). If a concentration and/or flux gradient of particles is considered the original structure of the mPBE will be modified leading to a new nonlinear partial differential equation (nPDE) of the third order. It is shown how one can derive classes of solutions for the potential function analytically by application of an algebraic method. The benefit of the mathematical tools used here is the fact that closed-form solutions can be calculated without any numerical approximations.

Sensitivities to parameterization in the size-modified Poisson-Boltzmann equation

Experimental results have demonstrated that the numbers of counterions surrounding nucleic acids differ from those predicted by the nonlinear Poisson-Boltzmann equation, NLPBE. Some studies have fit these data against the ion size in the size-modified Poisson-Boltzmann equation, SMPBE, but the present study demonstrates that other parameters, such as the Stern layer thickness and the molecular surface definition, can change the number of bound ions by amounts comparable to varying the ion size. These parameters will therefore have to be fit simultaneously against experimental data. In addition, the data presented here demonstrate that the derivative, SK, of the electrostatic binding free energy, ΔGel, with respect to the logarithm of the salt concentration is sensitive to these parameters, and experimental measurements of SK could be used to parameterize the model. However, although better values for the Stern layer thickness and ion size and better molecular surface definitions could improve the model's predictions of the numbers of ions around biomolecules and SK, ΔGel itself is more sensitive to parameters, such as the interior dielectric constant, which in turn do not significantly affect the distributions of ions around biomolecules. Therefore, improved estimates of the ion size and Stern layer thickness to use in the SMPBE will not necessarily improve the model's predictions of ΔGel.

Ion Counting from Explicit-Solvent Simulations and 3D-RISM

The ionic atmosphere around nucleic acids remains only partially understood at atomic-level detail. Ion counting (IC) experiments provide a quantitative measure of the ionic atmosphere around nucleic acids and, as such, are a natural route for testing quantitative theoretical approaches. In this article, we replicate IC experiments involving duplex DNA in NaCl(aq) using molecular dynamics (MD) simulation, the three-dimensional reference interaction site model (3D-RISM), and nonlinear Poisson-Boltzmann (NLPB) calculations and test against recent buffer-equilibration atomic emission spectroscopy measurements. Further, we outline the statistical mechanical basis for interpreting IC experiments and clarify the use of specific concentration scales. Near physiological concentrations, MD simulation and 3D-RISM estimates are close to experimental results, but at higher concentrations (>0.7 M), both methods underestimate the number of condensed cations and overestimate the number of excluded anions. The effect of DNA charge on ion and water atmosphere extends 20–25 Å from its surface, yielding layered density profiles. Overall, ion distributions from 3D-RISMs are relatively close to those from corresponding MD simulations, but with less Na+ binding in grooves and tighter binding to phosphates. NLPB calculations, on the other hand, systematically underestimate the number of condensed cations at almost all concentrations and yield nearly structureless ion distributions that are qualitatively distinct from those generated by both MD simulation and 3D-RISM. These results suggest that MD simulation and 3D-RISM may be further developed to provide quantitative insight into the characterization of the ion atmosphere around nucleic acids and their effect on structure and stability.

A new Green's function Monte Carlo algorithm for the solution of the three-dimensional nonlinear Poisson–Boltzmann equation: Application to the modeling of plasma sheath layers

Abstract. The objective of this paper is the exposition of a recently-developed Green's function Monte Carlo (GFMC) algorithm for the solution of nonlinear partial differential equations and its application to the modeling of the plasma sheath region around a spherical conducting object, carrying a potential and moving at low speeds through a partially ionized medium. The plasma sheath is modeled in equilibrium through the GFMC solution of the nonlinear Poisson–Boltzmann (NPB) equation. The traditional GFMC-based approaches for the solution of nonlinear equations involve the iterative solution of a series of linear problems. Over the last several years, one of the authors of this paper, K. Chatterjee, has been developing a philosophically different approach, where the linearization of the equation of interest is not required and hence there is no need for iteration. Instead, an approximate expression for the Green's function is obtained using perturbation theory, which is used to formulate the random walk equations within the problem sub-domains, where the random-walker makes its walks. On the other hand, as a trade-off, the dimensions of these sub-domains have to be restricted by the limitations imposed by perturbation theory. The greatest advantage of this approach is the ease and simplicity of parallelization stemming from the lack of the need for iteration, as a result of which the parallelization procedure is identical to the parallelization procedure for the GFMC solution of a linear problem. However, the premise of the algorithm is novel and rigorous mathematical justification has to be established in the future. The application areas of interest include the communication blackout problem during a space vehicle's re-entry into the atmosphere and electromagnetic propagation through the atmosphere/ionosphere in UHF/GPS applications.

A combined MPI-CUDA parallel solution of linear and nonlinear Poisson-Boltzmann equation.

The Poisson-Boltzmann equation models the electrostatic potential generated by fixed charges on a polarizable solute immersed in an ionic solution. This approach is often used in computational structural biology to estimate the electrostatic energetic component of the assembly of molecular biological systems. In the last decades, the amount of data concerning proteins and other biological macromolecules has remarkably increased. To fruitfully exploit these data, a huge computational power is needed as well as software tools capable of exploiting it. It is therefore necessary to move towards high performance computing and to develop proper parallel implementations of already existing and of novel algorithms. Nowadays, workstations can provide an amazing computational power: up to 10 TFLOPS on a single machine equipped with multiple CPUs and accelerators such as Intel Xeon Phi or GPU devices. The actual obstacle to the full exploitation of modern heterogeneous resources is efficient parallel coding and porting of software on such architectures. In this paper, we propose the implementation of a full Poisson-Boltzmann solver based on a finite-difference scheme using different and combined parallel schemes and in particular a mixed MPI-CUDA implementation. Results show great speedups when using the two schemes, achieving an 18.9x speedup using three GPUs.

High-Performance Computing and Big Data in Omics-Based Medicine

High-Performance Computing and Big Data in Omics-Based Medicine