Geometry The Polymer Model

One of the most fruitful applications of the PB equation is in the description of long cylindrical polyelectrolytes, of which DNA is the prototypical example. The same simplifications used in the planar case with a uniform and constant surface charge density and a restricted primitive model of the electrolyte are assumed here. 

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 

DNA and linear polyelectrolyte concentrations usually refer to the concentration of the ionizable sites (for DNA these are the phosphate groups) along the backbone. Denoting this concentration by Q, the volume of our cylindrical disk is just l/Q  

This equation gives the radius of our slice in terms of the ionizable site spacing and polyelectrolyte concentration. Note that there is no correction for the finite size of the cylinder (i.e., — a ) since the total volume of all slices must 

Before solving Eq. [224], we need to specify the concentration of initially bound counterions (added salt concentrations are assumed known). While the definition of the cell model assigns one initially bound counterion to the cell, the magnitude of the charge of the ionizable site is not required to be (although it often is) equal to that of the counterion. With and zo denoting the site and counterion valences, respectively, the number of counterions o that neutralize N. sites is 

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