Infinite charged cylinder

An Infinite Charged Cylinder Consider first an infinite, straight-line charge in a bulk dielectric extended along the z axis. The potential at a radial... 

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.)... 

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 problem 2.4 the movement of charged colloidal particles in water contained in an annular gap between two oppositely charged infinitely long cylinders was considered. The potential distribution across the gap d(j)ldr was assumed constant, where (j) is the potential and r is the radial cylindrical coordinate. It is desired here to take the analysis one step further and determine the potential and concentration distribution within the gap, neglecting any electrode effects. 

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-... 

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)... 

We consider an infinitely long cylinder of radius a oriented along the z-axis of the Cartesian reference frame. It has a uniform charge density cr(z, p>) = —cTc > 0. 

There are a few other geometries that permit analytical solution of the PB equation. For example, results for an infinitely long charged cylinder surrounded by counterions are presented in R. M. Fuoss, A, Katchalsky, and S. Lifson, Pwc. Natl. Acad. Sci. USA, 1951, 37, 579-589. 

We now consider the case of an infinitely long cylinder of fixed radius a (which also includes a common electrolyte ion radius) and surface charge density surrovmded by an electrolyte solution. The corresponding onedimensional PB equation and bovmdary conditions, in a cylindrical coordinate... 

Figure 27 The apparent linear charge density as a function of actual linear charge density for a cylinder of radius 10 A in a mixed 1 1-2 1 electrolyte with concentration 0.1-0.02 M within the NLDH (dashed lines Eq. [263] with the exact ( or approximate (a values) and PGC (solid lines Eq. [365] with exact and approximate Sdgc values) approximations. The NEDH low and high charge density approximations (Eqs. [264]) are shown by dotted lines the infinite-charge density limit is shown by the horizontal dotted-dashed line.

Calculate now a field created by a body with cylindrical symmetry, for example, infinite and uniformly charged cylinder (for simphcity—over a surface) with linear charge density t = dQ/dl = const. (Note that there are no such cylinders or infinite planes in nature. This physical problem is equivalent to the condition where the cylinder has finite length L, however, we consider a field near to the charged surface, i.e., at r L. Then it is possible to neglect the edge effects and solve the problem for an infinite cylinder.)... 

If the brass plate were completely incompressible, the failure thickness so determined would be half that of an unconfined infinite sheet. The failure thickness of an unconfined sheet is less than the failure diameter of a cylinder because rarefactions in a cylinder enter from all sides of the charge and influence the detonation. Thus, the failure diameter may be several times the failure thickness and may vary from one expl to another. More complete details are given in Ref 3... 

Consider a cylindrical soft particle, that is, an infinitely long cylindrical hard particle of core radius a covered with an ion-penetrable layer of polyelectrolytes of thickness d in a symmetrical electrolyte solution of valence z and bulk concentration (number density) n. The polymer-coated particle has thus an inner radius a and an outer radius b = a + d. The origin of the cylindrical coordinate system (r, z, cp) is held fixed on the cylinder axis. We consider the case where dissociated groups of valence Z are distributed with a uniform density N in the polyelectrolyte layer so that the density of the fixed charges in the surface layer is given by pgx = ZeN. We assume that the potential i/ (r) satisfies the following cylindrical Poisson-Boltz-mann equations ... 

FIG URE 14.11 Interaction between two infinitely long charged hard cylinders 1 and 2 of radii ai and A2 at a separation R between their axes. H =R — ai — ai) is the closest distance between their surfaces. 

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 annular gap between two infinitely long concentric cylinders of opposite charge is filled with water. Charged colloidal particles are to be moved from the inner cylinder (the source) to the outer cylinder (the collector) as a result of the voltage drop across the gap. The particle volume concentration is sufficiently small that each particle may be assumed to behave independently of the others. 

Radial losses of mass, momentum, and energy through the lateral surface of the cylinder do not occur. The detonation wave propagates along the axis of the cylindrical charge and is confined laterally by the infinite diameter explosive (the minimum diameter which can support hydrodynamic detonation at its maximum steady-state rate). 

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