Charged cylinder

The electric potential due to the charge distribution inside the cylinder follows from Equations 3.3 and 3.76 as 

The electric potential is an arbitrary constant at the cylinder surface (r = a), and it must be emphasized that there is no mathematical divergence in the potential 

The ion cloud in the solution modifies the result of Equation 3.77. Exact solution of the Poisson-Boltzmann equation (Equation 3.70) is known for the salt-free solutions containing only the counterions (Alfrey et al. 1951). As expected, the electric potential falls off smoothly with the radial distance, and there exists a counterion cloud near the cylinder. In order to get insight into the basic nature of the electrostatics in salty electrolyte solutions around a charged thin cylinder, we linearize Equation 3.70 to get the Debye-Hiickel theory (Equation 3.71). Solving this equation with the boundary conditions that the electric field vanishes far away from the cylinder and that it is given by Equation 3.76 at the surface of the cylinder, the result is 

We shall call E as the charge density parameter or Coulomb strength parameter interchangeably. The electric potential follows from Equations 3.78 and 3.79 as 

If the Debye length is very large as in the case of extremely dilute electrolyte solutions, and if the radius of the cylinder is very small, the arguments of the 

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How can Equation (11.79) be solved Before computers were available only simple ihapes could be considered. For example, proteins were modelled as spheres or ellipses Tanford-Kirkwood theory) DNA as a uniformly charged cylinder and membranes as planes (Gouy-Chapman theory). With computers, numerical approaches can be used to solve the Poisson-Boltzmann equation. A variety of numerical methods can be employed, including finite element and boundary element methods, but we will restrict our discussion to the finite difference method first introduced for proteins by Warwicker and Watson [Warwicker and Watson 1982]. Several groups have implemented this method here we concentrate on the work of Honig s group, whose DelPhi program has been widely used. 

A.G. Cherstvy and R.G. Winkler, Complexation of semiflexible chains with oppositely charged cylinder. J. Chem. Phys. 120, 9394-9400 (2004). 

Proof of boundedness of the force of interaction between two charged particles of an arbitrary shape in H3, held at a given distance from each other in an electrolyte solution, upon an infinite increase of the particle s charge. (It was shown in 2.2 that the repulsion force between parallel symmetrically charged cylinders saturates upon an infinite increase of the particle s charge. This is also true for infinite parallel charged plane interaction [9]. The appropriate result is expected to be true for particles of an arbitrary shape.)... 

Assume that the polyion is a charged cylinder of contour length L and radius R. On evaluating the surface charge density ct of the cylinder by a self-consistent methodology, the electrostatic free energy pgeiec was expressible as [44]... 

Just as a charged sphere in saltwater surrounds itself with a number of mobile ions different from what would occupy the same region in its absence, so does a charged cylinder. As with spheres, there are low-frequency ionic fluctuations that create attractive forces between like cylinders. In the special case of thin cylinders whose material dielectric response is the same as that of the medium and the distance between cylinders is small compared with the Debye screening length, this ionic-fluctuation force has appealing limiting forms. 

For two charged cylinders, only approximate solutions have been given... 

Equation (6.82) agrees with the unscreened Coulomb potential of a charged cylinder. That is, the surface potential in this case is the same as if the counterions were absent. 

In the limit a O, the particle core vanishes and the particle becomes a cylindrical polyelectrolyte (a porous charged cylinder) of radius b. For the low potential case, Eq.(21.79) gives... 

Condensation in the above sense, a cloud of ions remaining within a finite distance even at infinite dilution, is not unique to the infinitely long charged cylinder, although the phenomenon is not usually known by that name. The mobile ions of the double layer next to a charged infinite plane surface behave in the same way. 

In conclusion, we can say that the Poisson-Boltzmann description of the condensed population of mobile ions near a charged cylinder is similar to the Poisson-Boltzmann description of the mobile ions in the Gouy diffuse double layer at a charged planar surface, a description that has been well known for a long time. In both cases the ions are "bound", or "condensed", in the sense that... 

Describe plasma motion in the electric field E(r) created by a long charged cylinder and uniform magnetic field B parallel to the cylinder. Find out the maximum operational pressure of the plasma centrifuge. Is it necessary or not to trap ions in the magnetic field to provide the plasma rotation ... 

FIGURE 7.6 Cell model for the theoretical calculations of the DNA hybridization-induced ion-concentration redistribution as well as the average concentration of cations and anions in the intermolecu-lar spaces. Both the ssDNA and dsDNA have been modeled as negatively charged cylinders with a radius = 0.5 nm and = 1 nm, respectively, which form a hexagonal lattice with a cell radius of R the DNA molecules are arranged normal to surface of the FED with a center-to-center average separation distance of = 2R n r) is the ion concentration as a function of the coordinate r from the DNA axis and tiq is the bulk-ion concentration. 

The characteristic branching parameter (grafting density), n/m = specifies the onset of counterion localization inside the molecular brush. Note that in the osmotic regime, the spacers get fully extended, /t m. It is therefore not surprising, that the counterion localization in a cylindrical molecular brush coincides (in scaling terms) with the Manning condensation threshold [25] for a charged cylinder, qh = 1. 

Consider the interaction between polyions and small ions. For a polyelectrolyte chain, the charges on the polymer chains repel each other so that the polymer chain tends to assume a more extended configuration. Because the diameter of a polymer chain is very small compared with it length, in many applications, a polyelectrolyte chain can be treated as a charged cylinder. Tlie interaction between small ions and a charged cylinder (rf infinite length can be described by the Poisson-Boltzmann equation 16-18)... 

For the interactions of the fixed charges and mobile ions, we treat the polyions as charged cylinders. We use nonlinear theory to calculate the counterion condensation. After the counterion condensation, the interaction between the free counterions and polyions can be calculated by using an equation suggested by Manning (14). The final result is... 

Fig. 4. Effective linear charge density efr of a charged cylinder derived by superimposing the far field (large r) Poisson-Boltzmann potential field onto the potential field predicted by use of the Debye-Hiickel approximation. The solid lines correspond to different values of ax, where a is the cylinder radius and x is the Debye length the electrol3de is monovalent. For comparison, the counterion condensation prediction for is shown as a dotted line. Adapted from Ref. 51.

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