Planar Geometry The Membrane Model

The simplest polyelectrolyte model of biological relevance is that of a single semiinfinite impenetrable charged plane in equilibrium with a bulk electrolyte solution of known composition. The charged plane also serves as an introduction to the properties of the electric double layer. We simplify the representation of a membrane by assuming that (1) the membrane is impenetrable to ions, (2) the surface charge is uniformly distributed and constant, (3) the electrolyte is modeled as hard-sphere ions of specific size, and (4) the solvent is a structureless continuum described by a uniform dielectric coefficient 

An additional condition on the solution that is often used is to fix the gauge or reference value of the electrostatic potential by (R) = 0. That this choice is only one of convenience and not necessity (two boundary conditions suffice in determining the solution to a second-order differential equation) is easily seen by adding a constant to the potential in Eq. [7] and absorbing the leftover factor into the reference concentration c.  

It is convenient to define three characteristic lengths of the system. The first is a property of the solvent and is called the Bjerrum length  

For water at 298 K, Eq = 78.5 gives Lb = 7.14 A. The second length that we introduce is the Gouy-Chapman length  

If we now introduce the reduced or scaled potential (in units of ksTIcq = 25 mV at 298 K) 

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