Cylindrical Poisson-Boltzmann Equation Research Articles

Adsorption of cesium and strontium on mesoporous silicas.

Studies are underway on the adsorption reactions of metal ions in confined spaces at the solid-water interface, but it is unclear how the effects of confinement differ for different types of ions. We investigated the effect of the pore size on the adsorption of two cations with different valence, Cs+ and Sr2+, on mesoporous silicas with different pore size distributions. The amount of Sr2+ adsorbed per unit surface area did not differ significantly among the silicas, whereas that of Cs+ was particularly high for silicas with a larger fraction of micropores. The results of X-ray absorption fine structure analysis showed that both ions form outer-sphere complexes with the mesoporous silicas. The results of adsorption experiments were analyzed by fitting using a surface complexation model with the cylindrical Poisson-Boltzmann equation and optimized capacitance of the Stern layer for different pore sizes, and we found that the intrinsic equilibrium constant for the adsorption of Sr2+ is constant regardless of the pore size, whereas that of Cs+ increases as the pore size decreases. The decrease in the relative permittivity of water inside pores with a decrease of the pore size can be interpreted to cause a change in the hydration energy of Cs+ in the second coordination sphere upon adsorption. The reasons for the different confinement effects on the adsorption reactions of Cs+ and Sr2+ were discussed based on the distance of the adsorbed ions from the surface and the chaotropic and kosmotropic nature of Cs+ and Sr2+, respectively.

A solution of the cylindrical Poisson-Boltzmann equation

A solution of the cylindrical Poisson-Boltzmann equation

A simple algorithm for the calculation of the electric double layer potential distribution in a charged cylindrical narrow pore

A simple algorithm is presented for obtaining an approximate analytic solution to the cylindrical Poisson-Boltzmann equation for the electric double layer potential distribution in a charged cylindrical narrow pore filled with an electrolyte solution. Agreement with the exact numerical solution is excellent for low-to-moderate values of the pore surface potential when the pore radius is less than the Debye length. The obtained results are thus considerably better approximations than those previously obtained by Martynov and Avdeev (Colloid J 44: 626–632 (1983)). Approximate analytic expressions are also derived for the relationship between the pore surface charge density and the pore surface potential.

An Approximate Expression for the Electrostatic Gibbs Energy of Cylindrical Micelles

Based on the extended Langmuir's method, a simple expression for the calculation of the electrostatic contribution to the Gibbs energy of cylindrical micelles formation is derived from the nonlinear cylindrical Poisson–Boltzmann equation. Comparison is made with previous literature results. The expression is found to be accurate for cylindrical micelles with larger values of s or κr. This accuracy will be sufficient for most applications in colloid research.

Electrical Properties of Cylindrical Reverse Micelle

In this article, accurate analytic expressions of the surface charge density/surface potential and adsorption excesses for cylindrical reverse micelle in a solution of symmetric electrolyte are derived from the nonlinear cylindrical Poisson–Boltzmann equation. These expressions are in good agreement with numerical values and Van Aken's results for cylindrical reverse micelle with high surface potentials and large κa.

Electrical Properties of a Highly Charged Cylindrical Particle in a Symmetric Electrolyte Solution

Based on the extended Langmuir's method, accurate analytic expressions for the electrical properties of cylindrical colloidal particle in a solution of symmetric electrolyte are derived from the nonlinear cylindrical Poisson-Boltzmann equation. Comparison is made with previous literature results. The present formulae have good agreement with literature values and are quite simple for practical use.

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.

Exact asymptotic expansions for the cylindrical Poisson–Boltzmann equation

The mathematical theory of integrable Painlevé/Toda type systems sheds new light on thebehaviour of solutions to the Poisson–Boltzmann equation for the potential due to a longrod-like macroion. We investigate here the case of symmetric electrolytes together with thatof 1:2 and 2:1 salts. Small and large scale features are analysed, with particularemphasis on the low salinity regime. Analytical expansions are derived for severalquantities relevant for polyelectrolyte theory, such as the Manning radius. In addition,accurate and practical expressions are worked out for the electrostatic potential,which improve upon previous work and cover the full range of radial distances.

Effect of Spatial Inhomogeneity in Dielectric Permittivity on DNA Double Layer Formation

In solutions of tetramethylammonium (TMA +) DNA (double stranded) without added low-molecular-weight salt, the counterion radial density is calculated using the cylindrical Poisson-Boltzmann equation with a distance-dependent quasimacroscopic dielectric permittivity. Comparisons with small-angle neutron scattering data indicate that any inhomogeneity in dielectric permittivity is confined to one or two solvent layers from the DNA surface. At least for TMA +, which may be too large to penetrate the grooves of DNA to any significant extent, dielectric inhomogeneity modeled in this way has no detectable effect on the radial distribution.

On exact solutions to the cylindrical Poisson-Boltzmann equation with applications to polyelectrolytes

Using exact results from the theory of completely integrable systems of the Painlevé/Toda type, we examine the consequences for the theory of polyelectrolytes in the (nonlinear) Poisson-Boltzmann approximation.

An Algorithm for the Calculation of the Electrical Potential Distribution in a Charged Capillary with General Electrolytes

Knowledge of the electrical potential distribution is an essential basis for analyzing the flow behavior of electrolytes in a charged capillary, such as electroviscous effects. The cylindrical Poisson–Boltzmann equation (PBE) governs the distribution in the capillary. The PBE is a differential equation that is difficult to solve analytically, especially when the capillary is filled with general electrolytes. In this paper we propose a numerical algorithm to obtain the electrical potential distribution in a charged capillary filled with arbitrary electrolytes. First, we introduce a Poisson–Boltzmann integral equation (PBIE) governing the potential distribution and derived from the physical principles for electrostatic fields and thermodynamic systems. Then we solve the PBIE numerically by iteration. In numerical calculation only the discrete potential is used, and the potential differentials of the first and higher orders are not required. This algorithm essentially removes the difficulty caused by very steep variation of the potential near the wall of the capillary and is easily extended to cover the more general case of arbitrary electrolytes. The results of the examples given in the paper show that the algorithm proposed here is correct, effective, accurate (the relative errors are less than 0.01% when normalized surface potential ξ ≤ 8), and easily implemented on a personal computer.

A molecular theory of cartilage viscoelasticity

Recent work on the subject of cartilage mechanics has begun to focus on the relationship between the microscopic structure of cartilage and its macroscopic mechanical properties (Bader et al., Biochem. Biophys. Acta, 1116 (1992) 147–154; Buschmann, PhD Thesis, Massachusetts Institute of Technology, 1992; Kovach, Biophys. Chem., 53 (1995) 181–187; Lai et al., J. Biochem. Eng., 113 (1991) 245–248; Armstrong and Mow, J. Bone Jt. Surg., 64A (1982) 88; Jackson and James, Biorheology, 19 (1982) 317–330). This paper reviews recent theoretical developments and presents a comprehensive explanation of the viscoelastic properties of cartilage in terms of molecular structure. In doing this, a closed form hybrid solution to the non-linear, cylindrical Poisson-Boltzmann equation is developed to describe the charge-dependent component of the equilibrium elasticity arising from polysaccharide charge (Benham, J. Chem. Phys., 79 (4) (1983) 1969–1973; Einevoll and Hemmer, J. Phys. Chem., 89 (1) (1988) 474–484; Fixman, J. Chem. Phys., 70 (11) (1979) 4995–5001; Ramanathan and Woodburg, J. Chem. Phys., 82 (3) (1985) 1482–1491; Wennerstrom et al., J. Chem. Phys., 76 (9) (1982) 4665–4670). This solution agrees with numerical solutions found in the literature (Buschmann, PhD Thesis, Massachusetts Institute of Technology, 1992). The charge-independent, entropic contribution to the equilibrium elasticity is explained in a manner similar to that recently presented for concentrated proteoglycan solution (Kovach, Biophys. Chem., 53 (1995) 181–187). This approach exploits a lattice model of the solution, subject to a Bragg-Williams type approximation to derive the volume dependence of polysaccharide configuration entropy (Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca, NY, 1953; Huggins, Some properties of Solutions of Long-chain Compounds, 1941, pp. 151–157; Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford University Press, Oxford, 1971). Together, these two contributions accurately reproduce the experimentally determined osmotic pressure of cartilage as previously determined by Maroudas (Maroudas and Bannon, Biorheology, 18 (1981) 619–632). The time-dependent, or creep, phenomena which cartilage exhibits when subject to mechanical load is explained in terms of trictional drag on the polysaccharide chain monomers in terms of a Kirkwood-Riseman type model (Kirkwood and Riseman, J. Chem. Phys., 16 (6) (1948) 573–579). This approach is shown to accurately predict the hydraulic permeability of cartilage as previously determined by Maroudas (Madouras, Ann. Rheum. Dis., 34 (suppl. 3) (1975) 77). By use of a quasi-static approximation (neglecting inertial effects) the time-dependent response to a uniform compressive force is determined and also found to be in good agreement with experimental values from the literature.

Selective Counterion Binding to Linear Polyelectrolytes in Aqueous Solutions

The selective counterion condensation induced by a charged cylindrical polyelectrolyte in aqueous solution containing different mono- and di-valent ions was investigated by means of the numerical solutions of the non linearized cylindrical Poisson-Boltzmann equation. The critical charge fraction fc, above which the condensation predicted by the Manning theory occurs, has been evaluated in some selected cases of relevance in biopolyelectrolyte systems and a selective binding of counterions has been shown to occur. To take into account the effects of different types of mono- and di-valent salts, an extension of the Manning theory recently proposed by Paoletti et al. and based on Gibbs free energy minimization, was considered. Discrepancies between the charge distribution of bound ions calculated by means of numerical solutions of Poisson-Boltzmann equation and those predicted by the extended Manning theory, were found.

On the limiting law solution of the cylindrical Poisson-Boltzmann equation for polyelectrolytes

Upper and lower bounds are given for the behaviour of the Poisson-Boltzmann potential of a highly charged, cylindrical polyelectrolyte in excess salt solution as the salt correntration or cylinder radius tends to zero. The exact singular behaviour of the potential very close to the cylinder is given in terms of a delta function.

The cylindrical Poisson–Boltzmann equation. I. Transformations and general solutions

In this paper the cylindrical Poisson–Boltzmann equation in reduced coordinates is transformed into an algebraically nonlinear second order ordinary differential equation, which is a particular case of Painlevé’s third equation. The only singularities of solutions to this equation are movable poles of second order. Series expansions are developed which express the general solution locally about points of analyticity and about a pole. Back transformation yields local solutions to the cylindrical Poisson–Boltzmann equation. The solutions throughout an interval generally require analytic continuation of these local expressions. In addition to developing exact solutions, asymptotic properties are analyzed for the case of an isolated cylindrical polyelectrolyte. Further, an alternative technique of solution is sketched in which the cylindrical Poisson–Boltzmann equation is transformed into the sine-Gordon equation, which may then be solved by standard methods to give a one parameter family of solutions. A bibliography of references to the third Painlevé equation is presented. The application of these results to the analysis of the theoretical behavior of cylindrical polyelectrolytes in ionic solutions will be presented in a subsequent contribution.

Statistical mechanics of electrolytes and polyelectrolytes. III. The cylindrical Poisson–Boltzmann equation

The Poisson–Boltzmann equation for the potential due to an infinitely long, cylindrical polyion in a dilute ionic solution is studied and certain properties of its solution are proved. It is shown that when the charge density on the polyion exceeds a critical value, the surrounding ionic solution separates into two essentially independent phases. The inner phase which is close to the polyion consists predominantly of counterions. An explicit solution to the distribution function in this phase is obtained to the leading order and its validity is proved. This function depends only on the polyion charge density, and not on the concentration of the ionic solution. The distribution function in the outer phase depends upon the ionic strength but, to the leading order, not on the polyion charge density as long as it exceeds the critical value. It is shown that as the charge density on the polyion is increased, a proportionate number of counterions appears in the inner phase so as to neutralize the effect of this increase on the outer phase. It is proved that the potential, to its leading order, in the outer phase for any polyion charge density higher than the critical value, is the same as that due a polyion carrying a charge density equal to the critical value. The nonlinear dependence of the surface value of the potential on the polyion charge density is proved.