Cylindrical polyelectrolyte cylinder

Consider the case where cylinder 1 is a soft cylinder and cylinder 2 is a cylindrical polyelectrolyte. By taking the limit Kd2 1, we obtain from Eqs. (15.66) and (15.71)... 

Note that the following exact expression for the electrostatic interaction between porous cylinder (cylindrical polyelectrolyte) 1 and hard cylinder 2 has been derived [5,10] ... 

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... 

Finally, in the limit u 0, the cylinder core vanishes and the cyhnder becomes a cylindrical polyelectrolyte. In this limit, Eq. (22.35) becomes... 

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. 

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 ... 

Polyelectrolyte cylindrical brushes behave in similar ways as the polyelectrolyte stars in many aspects. Due to their anisotropic architecture, their morphologies can be tuned between worms, helices, and spheres. Polyelectrolyte core-shell cylindrical brushes have been used for the fabrication of inorganic NPs or NWs. In particular, superparamagnetic hybrid cylinders with magnetic NPs in the core of the brushes were prepared. They can be aligned on the substrate in magnetic field. This provides another way for the directed assembly of hybrid materials in a controlled manner. 

Fig. 10 Critical surface charge densities obtained by the WKB approach for polyelectrolyte adsorption onto planar, cylindrical, and spherical surfaces. The asymptotic scaling relations for a cylinder (rod) (45) and a sphere (53) are indicated by dotted lines [48]...

Interactions between polyelectrolyte-coated surfaces in the absence and presence of SDS were examined using the interferometric surface force technique of either the Mark II type [12] or the Mark IV type [13]. Muscovite mica was obtained from Mica New York Corp. in New York (green mica) and from M. Watanabe Co. in Tokyo (ruby mica). It was cleaved to 1-3 m thin pieces and silvered on one side. The pieces were then glued (using Epon 1004 from Shell Chemicals) onto two half-cylindrical silica discs with the silvered side down. The surfaces were mounted in a crossed cylinder configuration inside the surface force apparatus with the upper surface connected to a piezoelectric tube and the lower one on a double cantilever spring. The distance between the surfaces in changed by means of motors or by means of the piezo-electric crystal. 

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