Poisson-Boltzmann equation cylindrical

The solution of the linearized Poisson-Boltzmann equation around cylinders also requires numerical methods, although when cylindrical symmetry and the Debye-Hiickel approximation are assumed the equation can be solved. The solution, however, requires advanced mathematical techniques and we will not discuss it here. It is nevertheless useful to note the form of the solution. The potential for symmetrical electrolytes has been given by Dube (1943) and is written in terms of the charge density a as... 

Study of counterion condensation as a limiting property of the solutions of the Poisson-Boltzmann equation for arbitrary, charged cylindrical manifolds in H3 (see 2.3). 

G. V. Ramanathan, Statistical mechanism of electrolytes and polyelectrolytes. III. The cylindrical Poisson-Boltzmann equation, J. Chem. Phys., 78 (1983), p. 3223. 

Because in region I, the boundary of the atmosphere around each polymer chain was approximated with a cylinder, the electrical potential is related to the electrolyte concentration via a Poisson-Boltzmann equation in cylindrical... 

The physical problem is shown in Fig. 2, where the geometry is described by using a cylindrical coordinate system with its origin on the wall. The linear Poisson-Boltzmann equation is... 

W.R. Bowen and A.O. Sharif, Adaptive finite element solution of the non-linear Poisson-Boltzmann equation—a charged spherical particle at various distances from a charged cylindrical pore in a charged planar surface, J. Colloid Interface Sci. 187 (1997)... 

A similar approximation method can be applied for the case of infinitely long cylindrical particles of radius a in a general electrolyte composed of N ionic species with valence z, and bulk concentration n, (/ = 1, 2,. . . , N). The cylindrical Poisson-Boltzmann equation is... 

So far we have treated uniformly charged planar, spherical, or cylindrical particles. For general cases other than the above examples, it is not easy to solve analytically the Poisson-Boltzmann equation (1.5). In the following, we give an example in which one can derive approximate solutions. 

The Poisson-Boltzmann equation for the potential distribution around a cylindrical particle without recourse to the above two assumptions for the limiting case of completely salt-free suspensions containing only particles and their counterions was solved analytically by Fuoss et al. [1] and Afrey et al. [2]. As for a spherical particle, although the exact analytic solution was not derived, Imai and Oosawa [3,4] smdied the analytic properties of the Poisson-Boltzmann equation for dilute particle suspensions. The Poisson-Boltzmann equation for a salt-free suspension has recently been numerically solved [5-8]. 

Numerical solution of the Poisson and Poisson-Boltzmann equations is more complicated since these are three dimensional partial differential equations, which in the latter case can be non-linear. Solutions in planar, cylindrical and spherical geometry, are... 

We summarize recent work showing that condensation can be derived as a natural consequence of the Poisson-Boltzmann equation applied to an infinitely long cylindrical polyelectrolyte in the following sense Nearly all of the condensed population of counter-ions is trapped within a finite distance of the polyelectrolyte even when the system is infinitely diluted. Such behavior is familiar in the case of charged plane surfaces where the trapped ions form the Gouy double layer. The difference between the plane and the cylinder is that with the former all of the charge of the double layer is trapped, while with the latter only the condensed population is trapped. 

The system that we consider is an infinitely long cylindrical poly-ion enclosed in an outer concentric cylindrical container filled with solvent and counter-ions of valence z but with no added salt this is one case in which analytic, as opposed to numerical, solutions of the Poisson-Boltzmann equation are available. Numerical solutions for the case where added salt is present show much the same picture, however, so this limiting case with counter-ions only is still of general interest. The Poisson-Boltzmann equation for this system was solved long ago (9,10). 

The radial potential distribution inside the capillary, (r), is then obtained by solving the Poisson-Boltzmann equation for cylindrical symmetry (30). The resulting potential depends on a single adjustable constant which is fixed by the boundary condition on the potential which relates the p gential gradient at r=l/2Dp to the surface charge density, J c. Then we define... 

A rational description of ionic atmosphere binding is provided by the Poisson-Boltzmann equation and the cylindrical cell model. Figure 1 is an example of such computations and shows the variation of the local concen-... 

The solution of the Poisson-Boltzmann equations for the local charge density p(r) of a double layer at the internal wall of a cylindrical capillary is given by [16],... 

Ion Binding and Adsorption Ion Condensation Model. In the ion condensation model the solution of the Poisson-Boltzmann equation for a charged infinitely long cylinder (cylindrical polymer ion) in an electrolyte solution leads to the following important result the counterion con-... 

The EDL potential i/r(r) can be determined by the solving the well-known Poisson-Boltzmann equation for cylindrical capillary ... 

For more complex geometries, it is sometimes possible to obtain approximate analytic expressions [24]. However, in general, the desired potential-to-charge relationship must be obtained via numerical solution of the Poisson-Boltzmann equation. In particular, for a long cylindrical cavity, the case most relevant to this study, the desired relation can be obtained by solving the system [25],... 

We will recall in this paragraph, in our rationalised notation, a non-exhaustive list of the most important results arising from the principal theoretical models which have been proposed and which may be adapted to the case of DNA. A critical comparison of these various models has already been published by Manning [16]. All the models suppose, as a starting point, that DNA is rigid rod with cylindrical symmetry and proceed, in general, by a resolution of the Poisson-Boltzmann equation for the system polyion plus electrolyte (first treated by Alfrey et al. [17] and Fuoss et al. [18]). 

Benham, C. J. 1983. The cylindrical Poisson-Boltzmann equation. I. Transformations and general solutions. The Journal of Chemical Physics 79 1969-1973. 

Rice, R. E. 1985. Exact solution to the linearized Poisson-Boltzmann equation in cylindrical coordinates. The Journal of Chemical Physics 82, no. 9 4337 340. doi 10.1063/1.448826. 

Tellez, G., and E. Trizac. 2006. Exact asymptotic expansions for the cylindrical Poisson-Boltzmann equation. Journal of Statistical Mechanics Theory and Experiment 2006 P06018. 

Tuinier, R. 2003. Approximate solutions to the Poisson-Boltzmann equation in spherical and cylindrical geometry. Journal of Colloid and Interface Science 258, no. 1 45-49. doi 10.1016/S0021-9797(02)00142-X. 

Van, H., and J. A. M. Smit. 1995. Approximative analytical solutions of the Poisson-Boltzmann equation for charged rods in the presence of salt An analysis of the cylindrical cell model. Journal of Colloid and Interface Science 170, no. 1 134—145. doi 10.1006/ jcis.1995.1081. 

The reduced electrostatic potential y>(r) in the cylindrical zone (1) satisfies the Poisson-Boltzmann equation... 

The solution of the nonlinear Poisson-Boltzmann equation (eqn [42]) with the bormdaiy conditions (eqns [44] and [47]) leads to the following cormterion density profile in the cylindrical zone as a function of the distance r from the polyion... 

In terms of cylindrical coordinates, the Poisson-Boltzmann equation (57) turns into [78] ... 

Odijk T. On the limiting law solution of the cylindrical Poisson-Boltzmann equation for polyelectrolytes. Chem Phys Lett 1983 100 145-150. 

If we consider long enough pores to ignore end effects in the interior of the pore, there is a cylindrical symmetry about the axis of the pore. For this two-dimensional problem, the above Poisson-Boltzmann equation is solved in the cylindrical coordinate system of Figure 3.13, where r is the radial distance from the pore axis. Rewriting the Laplacian in the cylindrical coordinate system,... 

Analytical results for an ideal EO flow within the Debye-Hiickel approximation can also be derived for a cylindrical channel of circular cross section with radius a. Similar to equation (6.65) for flat plate, the Poisson-Boltzmann equation, (r), for cylindrical geometry in the Debye-Hiickel approximation is... 

Equation (68) gives the electrostatic potential at the surface of a solute ion in a pure dielectric solvent. To find the electrostatic potential, Tg, for electrostatic charge, e, in a dielectric solution requires that, Tg, be a solution of the Poisson— Boltzmann equation [21, 25]. Owing to the spherical and planar geometry of a charge in an interfacial system, Onsager and Samaras chose cylindrical co-ordinates (, X) for the Poisson—Boltzmann equation. 

Sims, S. M., Higuchi, W. L, Peck, K. Ionic partition coefficients and electroosmotic flow in cylindrical pores Comparison of the predictions of the Poisson-Boltzmann equation with experiment. /. Colloid Interface Sci. 1993, 155, 210-220. 

---