Two-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.

Analysis of combined pressure-driven electroosmotic flow through square microchannels

In this paper three-dimensional single-phase liquid flow through microchannels with a square-shaped cross-section driven by simultaneous application of pressure gradient and electroosmotic pumping mechanism is studied. The governing system of equations consists of the electric potential field and flow field equations. The solution procedure involves three steps. First, the net charge distribution on the cross-section of the microchannel is computed by solving two-dimensional Poisson–Boltzmann equation using the finite element method. Then, using the computed fluid’s charge distribution, the magnitude of the resulting body force due to interaction of an external electric field with the charged fluid particles is calculated along the microchannel. In the third step, the flow equations are solved by considering three-dimensional Navier–Stokes equations with an electrokinetic body force. The computations reveal that the flow pattern in the microchannel is significantly different from the parabolic velocity profile of the laminar pressure-driven flow. The effect of the liquid bulk ionic concentration and the external electric field strength on flow patterns through the square-shaped microchannels is also investigated.

Frequency-dependent laminar electroosmotic flow in a closed-end rectangular microchannel

This article presents an analysis of the frequency- and time-dependent electroosmotic flow in a closed-end rectangular microchannel. An exact solution to the modified Navier–Stokes equation governing the ac electroosmotic flow field is obtained by using the Green's function formulation in combination with a complex variable approach. An analytical expression for the induced backpressure gradient is derived. With the Debye–Hückel approximation, the electrical double-layer potential distribution in the channel is obtained by analytically solving the linearized two-dimensional Poisson–Boltzmann equation. Since the counterparts of the flow rate and the electrical current are shown to be linearly proportional to the applied electric field and the pressure gradient, Onsager's principle of reciprocity is demonstrated for transient and ac electroosmotic flows. The time evolution of the electroosmotic flow and the effect of a frequency-dependent ac electric field on the oscillating electroosmotic flow in a closed-end rectangular microchannel are examined. Specifically, the induced pressure gradient is analyzed under effects of the channel dimension and the frequency of electric field. In addition, based on the Stokes second problem, the solution of the slip velocity approximation is presented for comparison with the results obtained from the analytical scheme developed in this study.

Double layer interaction between two plates with polyelectrolyte brushes

When polyelectrolyte chains are grafted to colloidal particles, the electric field between particles is affected by the charges of the chains. In some previous theoretical attempts, the charge density of the polyelectrolyte chains per unit length was considered constant, and its effect was accounted for by introducing an additional constant charge density into the unidimensional Poisson–Boltzmann equation, which was evaluated assuming that it is uniformly distributed in the polyelectrolyte volume of the brush. In this paper, a more detailed model is employed for the calculation of the electrical potential between two plates on which polyelectrolyte brushes are present. In this model, the polyelectrolyte chain is viewed as a rigid cylinder, on the surface of which charges are generated through the dissociation of ionizable sites and adsorption of the cations of the electrolyte. To each of the chains an atmosphere is attached which for simplicity is assumed cylindrical. In the brush region, the electrical potential is described by a two-dimensional Poisson–Boltzmann equation, while in the region free of polyelectrolyte chains by a unidimensional Poisson–Boltzmann equation. Such a model is physically suitable when the charges of the chains are sufficiently large for the repulsion they generate to ensure that the chains are fully extended. Such cases are quite frequent, because relatively low charges lead to an almost complete extension of the chains. In this paper, both the plate surface and the surface of the cylinders are considered charged. The effects of electrolyte concentration, pH, brush thickness and chain coverage density on the repulsion between plates are examined.

Thermal effects on electro-osmotic pumping of liquids in microchannels

In this paper we present a mathematical model predicting combined electro-osmotic- and pressure-driven flow behavior in a straight microchannel for the case when the system is under a non-isothermal condition. The distribution of the charge density is governed by a nonlinear, two-dimensional Poisson–Boltzmann equation, and a body force caused by the interaction between the charge density and the applied electrical potential field is included in the full Navier–Stokes equations. Under non-isothermal conditions, arising from heat sources (Joule heating) within the system and/or a temperature difference between the system and the ambient, the equation of energy conservation describing temperature distribution has to be considered as well. The governing equations, interlinked via temperature, are solved numerically using a finite difference method. The numerical results indicate that for a given cross-sectional mean velocity, there exists an optimal dimensionless parameter (κh), which is the inverse Debye length multiplied by the channel size, which gives the highest hydraulic head generated by the electro-osmotic force. It has also been demonstrated that the pumping performance predicted by the isothermal model deviates substantially from that predicted by the non-isothermal model when Joule heating is significant.

Adaptive Mesh Enrichment for the Poisson–Boltzmann Equation

An adaptive mesh enrichment procedure for a finite-element solution of the two-dimensional Poisson–Boltzmann equation is described. The mesh adaptation is performed by subdividing the cells using information obtained in the previous step of the solution and next rearranging the mesh to be a Delaunay triangulation. The procedure allows the gradual improvement of the quality of the solution and adjustment of the geometry of the problem. The performance of the proposed approach is illustrated by applying it to the problem of two identical colloidal particles in a symmetric electrolyte.

Electroosmotic Flow in Microchannels

The electroosmotic flow induced by an applied electrostatic potential field through microchannels between two parallel plates and a 90° bend is analyzed in this work. A nonlinear, two-dimensional Poisson–Boltzmann equation governing the electrical double-layer field and the Laplace equation governing the electrostatic field distribution in microchannels are numerically solved using a finite-difference method. A body force caused by the interaction between the electrical double-layer field and the applied electrostatic field is included in the full Navier–Stokes equations. The effects of the electrical double-layer field and the applied electrostatic field on the fluid velocity distribution, pressure drop, and skin friction are discussed. A small pressure drop along the parallel plates is detected, although it is always neglected in the literature. Pressure is not a constant across the channel height. The axial velocity profile is no longer flat across the channel height when the Reynolds number is large. A separation bubble is detected near the 90° junction when the Reynolds number is large.

Analysis of electrokinetic effects on the liquid flow in rectangular microchannels

The effects of the electrical double layer near the solid/liquid interface on liquid flow through a rectangular microchannel are analyzed in this work. Based on the Debye–Hückel approximation, a linear solution of a two-dimensional Poisson–Boltzmann equation governing the electrical potential distribution in the cross-section of rectangular channels is presented to describe the electrical double-layer field near the solid/liquid interface. An additional body force originating from the presence of the electrical double layer is considered to modify the conventional Navier–Stokes equation. By using the Green's function formula, an exact solution to this modified Navier–Stokes equation in rectangular microchannels is obtained. The effects of the electrical double-layer field, channel size and the electrical static charge on the fluid velocity distribution, streaming potential, volume flow rate, friction coefficient and apparent viscosity have been discussed. The results show that, for diluted liquid, the liquid flow in rectangular microchannels is influenced significantly by the electrical double layer and hence deviates from the flow characteristics described by classical fluid mechanics.

Electrokinetic Effects on Pressure-Driven Liquid Flows in Rectangular Microchannels

The effects of the electrical double layer near the solid-liquid interface and the induced electrokinetic field on the pressure-driven liquid flow through a rectangular microchannel are analyzed in this work. A nonlinear, two-dimensional Poisson–Boltzmann equation governing the electrical double layer field in the cross section of rectangular channels is numerically solved with the use of a finite-difference scheme. A body force caused by the electrical double field and the flow-induced electrokinetic field is considered in the equation of motion. An exact solution to this equation of motion in rectangular microchannels is obtained by employing the Green function formulation. The effects of the ionic concentration of the liquid, the zeta potential of the solid surface, and the size and the shape of microchannels on the fluid velocity distribution, streaming potential, volumetric flow rate, friction coefficient, and apparent viscosity are discussed. The results clearly show that for a liquid solution of low ionic concentration and a solid surface of high zeta potential the liquid flow in rectangular microchannels is significantly influenced by the presence of the electrical double layer field and hence deviates from the flow characteristics described by classical fluid mechanics.

The Irregular Behavior of a Counter-Ion Density Described by the Application of the Two-Dimensional Poisson-Boltzmann Equation

The density of a counter-ion atmosphere, as described by the application of the two-dimensional Poisson-Boltzmann equation, around an ionized cylindrical model-molecule, can change irregularly. This cylindrical model-molecule is immersed in an aqueous solvent of infinite volume. Its radius ( a ), the packing density ( n g ) of charged groups in it, and temperature are parameters for determing the counter-ion density. It is found that as a or n g increase, the counter-ion density being zero at small values of a or n g increases irregularly, even at constant temperature. If temperature rises, it is found that the counter-ion atmosphere around the ionized cylindrical model-molecule may be subject to a transition from a state of accumulated counter ions to one in which the accumulation of counter ions is lost, even though a and n g are constant.

EFFECTS OF BORN REPULSION ON PARTICLE DETACHMENT†

The effect of short range Born repulsion on the total interaction energy of unlike spheres has been investigated, with superimposed London-van der Waals and electrostatic contributions. The latter was calculated using the recently derived model based on the two-dimensional Poisson-Boltzmann equation. Specifically, the energy profile as a function of separation is discussed as it relates to the particle detachment process.

Double layer interactions of unlike spheres. III. Nonlinear and two-dimensional effects

The problems of electrostatic interactions of systems consisting of two unlike spheres or of the plate/sphere configuration are discussed. Comparisons between the results based on the formulation of R. Hogg, T. W. Healy, and D. W. Fuerstenau [ Trans. Faraday. Soc. 62, 1638 (1966)] and the recently developed theory using the two-dimensional Poisson-Boltzmann equation are offered. The reasons for distinct differences in electrostatic energies, found at sufficiently close separations between surfaces of different potentials, are offered.

Heterocoagulation. VI. Interactions of a monodispersed hydrous aluminum oxide sol with polystyrene latex

The stability of binary dispersions consisting of an aluminum hydrous oxide sol and an anionic polystyrene (PS) latex or of the same anionic PS latex in a mixture with a cationic polystyrene latex was studied at constant ionic strength as a function of pH, and of the ratio of the particle number concentration. The aluminum hydrous oxide sol contained spherical particles of narrow size distribution (modal diameter 0.57 μm), whereas the anionic and cationic latex particles had modal diameters of 0.38 and 0.42 μm, respectively. The rate of coagulation was followed by the low-angle laser scattering. The experimental stability constants were compared with the theoretical values calculated using the Hogg, Healy, and Fuerstenau model as well as the recently derived expression, based on the two-dimensional Poisson-Boltzmann equation. In the range of slow coagulation the latter model gave a better agreement with the experimental data.

Heterocoagulation. Part 5.—Adsorption of a carboxylated polymer latex on monodispersed hydrated metal oxides

The adsorption of a monodispersed polymer latex [poly(methyl methacrylate co-methacrylic acid)] with a radius of 40 nm on much larger spherical hydrated aluminium oxide and haematite particles of narrow size distribution was studied as a function of pH, temperature, ionic strength and particle number concentration. The thermodynamics and the kinetics of this heterosystem have been analysed taking the double layer properties of both types of particles into consideration. By measuring the adsorption isotherms of the latex on the (hydrated) metal oxide particles, it was possible directly to compare the experimentally determined interaction energies with the calculated values. The expression based on the classical theory of the electrical double layer greatly overestimates the electrostatic energy. However, the recently derived expression using the two-dimensional Poisson—Boltzmann equation is in excellent agreement with the experimental findings.

Properties of Double-Stranded DNA as a Polyelectrolyte

The stability of the structure of double-stranded DNA in the salt-free solution is discussed on the basis of the polyelectrolyte theory. Assuming that DNA is an infinitely long rod, and the formation of double strands is divided into combining process and folding process, the free energy changes required in these processes are calculated by the use of the exact solutions of two-dimensional Poisson-Boltzmann equation for the one rod and the two rod systems.By strong depression of electrostatic interaction due to counter-ion condensation phenomena, the free energy change is remarkably decreased so that the double-stranded structure of DNA can be stabilized by energy of hydrogen bonds between base pairs. The increase of the activity coefficient of a counterion upon heat denaturation of DNA is also explained.