One-dimensional Poisson-Boltzmann Equation Research Articles

Study of nonlinear Poisson-Boltzmann equation for a rodlike macromolecule using the pseudo-spectral method

The pseudo-spectral method with the Chebyshev and Legendre polynomials are used in order to compute the electric potential for a rodlike macromolecule located in salt free solution via the Poisson-Boltzmann equation (PBE). Afterwards, verification of the method is demonstrated for a long macromolecule as well as a large plate for which the behaviors can be properly explained by a one-dimensional PBE. It is concluded that the method based upon the Chebyshev polynomials with the specified collocation points is more accurate than the technique based on the Legendre polynomials with the same number of points. As a macromolecule has a rodlike shape, a two-dimensional PBE is considered, which corresponds to a much more realistic case. To solve the PBE, the concentration of the macromolecule in the solution and the electric field are used to compute the height and radius of the unit cell, which are obtained to be as 16.7 and 8.1 nm, respectively. It is worth noting that numerical computation of the PBE for a macromolecule with a finite length has not been reported previously using the pseudo-spectral method. The results given in this work can be appropriately used for stiff fragments of DNA and actin filaments.

The exact solution to the one-dimensional Poisson–Boltzmann equation with asymmetric boundary conditions

The exact solution to the one-dimensional Poisson–Boltzmann equation with asymmetric boundary conditions can be expressed in terms of the Jacobi elliptic functions. The boundary conditions determine the modulus of the Jacobi elliptic functions. The boundary conditions can not be solved analytically, thus a numerical scheme has been applied.

A unified charge-based model for SOI MOSFETs applicable from intrinsic to heavily doped channel

A unified charge-based model for fully depleted silicon-on-insulator (SOI) metal-oxide semiconductor field-effect transistors (MOSFETs) is presented. The proposed model is accurate and applicable from intrinsic to heavily doped channels with various structure parameters. The framework starts from the one-dimensional Poisson—Boltzmann equation, and based on the full depletion approximation, an accurate inversion charge density equation is obtained. With the inversion charge density solution, the unified drain current expression is derived, and a unified terminal charge and intrinsic capacitance model is also derived in the quasi-static case. The validity and accuracy of the presented analytic model is proved by numerical simulations.

Sensitivity of field-effect biosensors to charge, pH, and ion concentration in a membrane model

In field-effect transistors used to detect charged biomolecules (BioFETs), the biomolecules form a charged membrane on the transistor surface. In this paper, the one-dimensional Poisson–Boltzmann equation is used to calculate the charge sensitivity (the sensitivity of the BioFET to changes in biomolecule charge), ion sensitivity (to changes in ion concentration of the solution), or pH sensitivity (to changes in pH of the solution), both analytically and numerically, and the results are compared to models where the charged molecules are represented as an infinitely thin plane. Complexation of ions with the oxide surface is shown to have a negligible effect on parameters typical of devices, but the layer used to tether the charged molecules to the surface could modify the sensitivity considerably.

Calculation of the Response of Field-Effect Transistors to Charged Biological Molecules

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Robust approximations are presented that allow for the simple calculation of the total charge and potential drop <formula formulatype="inline"><tex>$\psi_{0}$</tex> </formula> across the region of electrolyte containing charged biological macromolecules that are attached to the gate area of a field-effect transistor (FET). The attached macromolecules are modeled as an ion-permeable membrane in contact with the insulator surface, exchanging protons with the electrolyte as described by the site-binding model. The approximations are based on a new screening length involving the Donnan potential in the membrane and are validated by comparison to the results obtained by numerical solution of the one-dimensional Poisson–Boltzmann equation in the electrolyte and membrane. For gates covered with amphoteric materials such as SiO<formula formulatype="inline"> <tex>$_{2}$</tex></formula>, the high surface charge density <formula formulatype="inline"> <tex>$\sigma_{0}$</tex></formula> due to proton exchange at values of <emphasis emphasistype="boldital">p</emphasis>H far from the point-of-zero charge is a nonlinear function of <formula formulatype="inline"><tex>$\psi_{0}$</tex> </formula>, but <formula formulatype="inline"><tex>$\psi_{0}$</tex></formula> and <formula formulatype="inline"><tex>$\sigma_{0}$</tex></formula> are still linear functions of the semiconductor surface potential between the source and drain. Nonlinear expressions for the amphoteric site charge at the contacts can thus be applied effectively with the new approximations to calculate the current-voltage characteristics of the FETs using the strong inversion and charge-sheet models. </para>

Sensitivity of a field-effect transistor in detecting DNA hybridization, calculated from the cylindrical Poisson-Boltzmann equation

The sensitivity of a DNA biosensor based on a field-effect transistor is calculated from the numerical solution of the Poisson-Boltzmann equation in a cylindrical cell. The sensitivity is typically about half that found from the one-dimensional Poisson-Boltzmann equation for an ion-permeable DNA membrane with the same average charge density. For the combination of low DNA densities and low concentration of ions in solution, however, the discrepancy between the two calculations can be a factor of 10 or higher. The difference in the two calculations is due to accumulation of screening charge around the DNA cylinder, an effect which is ignored in the membrane model.

One-dimensional model of the electrostatic ion acceleration in the ultraintense laser–solid interaction

Effective ion acceleration of picosecond-duration well-collimated bunches in the strong relativistic interaction of a short laser pulse with a thin solid target has been experimentally demonstrated. In this work, with reference to the sharp rear solid–vacuum interface, where ion energization takes place, the one-dimensional Poisson–Boltzmann equation is analytically solved on a finite spatial interval whose extension is determined by requiring electron energy conservation, resulting in the consistent spatial distributions of the hot electrons created by the laser and of the corresponding electrostatic potential. Then, the equation of motions for an ensemble of test ions, initially distributed in a thin layer of the rear target surface, with different initial conditions, is solved and the energy spectrum corresponding to a given initial ion distribution is determined.

The electric double layer at a metal electrode in pure water

Pure water is a weak electrolyte that dissociates into hydronium ions and hydroxide ions. In contact with a charged electrode a double layer forms for which neither experimental nor theoretical studies exist, in contrast to electrolytes containing extrinsic ions like acids, bases, and solute salts. Starting from a self-consistent solution of the one-dimensional modified Poisson–Boltzmann equation, which takes into account activity coefficients of point-like ions, we explore the properties of the electric double layer by successive incorporation of various correction terms like finite ion size, polarization, image charge, and field dissociation. We also discuss the effect of the usual approximation of an average potential as required for the one-dimensional Poisson–Boltzmann equation, and conclude that the one-dimensional approximation underestimates the ion density. We calculate the electric potential, the ion distributions, the pH-values, the ion-size corrected activity coefficients, and the dissociation constants close to the electric double layer and compare the results for the various model corrections.

A contact model for Poisson-based spreading resistance correction schemes incorporating Schottky barrier and pressure effects

Accurate determination of the electrically active dopant profile from spreading resistance measurements requires the application of a carrier spilling correction scheme based on the iterative solution of the one-dimensional Poisson–Boltzmann equation. It will be shown that the results are strongly influenced by the boundary conditions applied describing the probe–silicon contact. An exploratory and somewhat speculative new theoretical probe contact model will be presented which takes into account Schottky barrier effects, surface states on the bevel surface, band-gap narrowing and variations of the dielectric constant due to the large probe pressures applied at the metal–silicon interface. It will be shown that this model can predict adequately the detailed behavior of an experimentally measured spreading resistance low dose implant.

Direct measurements of interaction forces between bodies bearing thin double-layers

A new experimental device is presented for measuring directly the attractive and repulsive specific forces between planar and spherically shaped bodies. Our method was applied for experimentally determining the interaction force—distance diagrams for two fused silica plates (one planar, one spherical) in aqueous potassium chloride solutions. The experimental results were interpreted by using the new numerical algorithm which allows one to calculate the electrostatic interaction forces between regularly shaped bodies bearing thin double-layers immersed in an arbitrary electrolyte. Our numerical procedure enabled the solution of the one-dimensional Poisson—Boltzmann equation in which corrections stemming from the excluded volume, the dielectric saturation and the specific interactions of ions with interfaces can be quantitatively taken into account. These one-dimensional solutions (parallel-plate system) were then integrated according to the modified Derjaguin summation method giving the electrostatic force between plane/spherical bodies in the vicinity of their contact area.

A Consideration of the Charge Density of Polyelectrolytes Based on a Layer Model in Solution

Abstract In a model system where layers formed by aggregations of polyelectrolyte molecules are mutually parallel in a solvent, the surface-charge density of the layers is estimated by solving the one-dimensional Poisson–Boltzmann equation. The surface-charge density at a over temperature, T, can be determined depending on the number, ng, of charged groups per unit of the volume in a layer of polyelectrolytes and on the thickness, Δx, of the layer, if the distance between layers of polyelectrolyts is sufficiently large. This surface-charge density is evaluated by assuming that ng is given by the reciprocal of the cubed mean length between neighboring charged groups in a polyelectrolyte molecule and that Δx is of the order of an ion diameter, possibly including the hydration sheath. The surface-charge density given by the layer model agrees well with that semiempirically given by a rod-like polyelectrolyte model. Also, the ratio of the total charge of a layer to the total charge of the charged groups in the layer agrees well with the degree of the ionic dissociation of ionene polymers in a salt-free solution.

Electrostatic potential between concentric surfaces: Spherical, cylindrical, and planar

An approximate closed-form solution to the one-dimensional Poisson-Boltzmann equation is obtained for any combination of mono-, di-, and trivalent counterions. This solution provides the electrostatic potential between two apposed concentric surfaces, i.e., flat plates or a large sphere or cylinder embedded in a cavity of the same geometry. The solutions for cylindrical and spherical surfaces are quite accurate provided that their radius ( a 1) is much larger than the double layer thickness; specifically, that κa 1 > 10 where κ is the Debye-Hückel reciprocal length. The spherical or cylindrical solutions apply to the cell model of particle aggregates, wherein the electrostatic environment of an average particle within an aggregate may be estimated. Simple approximate formulas for the ion distributions in the double layer are also obtained.

Equilibrium electropotentials in electrolyte systems.

The determinationof electric potentials in finite regions of symmetrical electrolyte in one-dimensional equilibrium situations requires the solution of the one-dimensional Poisson-Boltzmann equation in which the dependent variable is linearly related to the electric potential and contains unknown parameters. These require evaluation as part of the solution to a given boundary value problem. The general solution of the equation is presented. This involves elliptic functions and integrals and is sectionally isomorphic with respect to an integration parameter. The application to problems posed in terms of both initial values and two-point boundary values is discussed. The solution is used to determine the potential and concentration distributions between two flat-faced charged particles immersed in an electrolyte liquid and having interacting double layers.

Analytical solution of the poisson-boltzmann equation for membrane potential

Analytical solution of the one-dimensional Poisson-Boltzmann equation for membrane potential is obtained in an equilibrium state of the Nemst-Planck Poisson system. Approximations, e.g. constant field or Debye-Hückel approximation, need not be used. Two types of solution, arctangenthyperbolic and arccotangenthyperbolic, exist for every value of membrane potential. A new approximate solution is obtained where V and V m are electric potential inside the membrane and membrane potential respectively; R, T and F have their usual thermodynamic meanings, κ is Debye constant, τ the membrane thickness. This approximate solution fits the numerical solution by Runge-Kutta method (maximum error being less than 5 × 10 −4) around the potential being 50 mV. The fitness is considerably more accurate than that of Debye-Huckel approximation (error being c. 15 % around the potential 50 mV.).