Approximations to the Poisson-Boltzmann Equation

The two most common approximations used in applying the PB equation are the assumption of a bulk electrolyte and linearization of the equation. 

Together these essentially replace the Poisson-Boltzmann cell model with the Debye-Hiickel bulk model, allowing many more systems to be treated analytically, although not necessarily accurately, and providing considerable insight into the physical characteristics of electrolyte solutions. 

The most commonly used simplification when applying the PB equation to a polyelectrolyte is to assume that the solvent environment of the macroion extends to infinity. Doing so allows one to ignore any counterions initially bound to the polyelectrolyte (co 0) and to replace concentrations at the outer boundary by bulk concentrations (cf cf). This simplification increases the rate at which the iterative solution to the PB equation converges since ion normalization is no longer a constraint. It is readily shown that the condition under which a finite system may be treated as infinite is  

For a system in which R defines the distance to the closest outer boundary enclosing a single central polyelectrolyte, and if the dielectric coefficient is constant or increases toward some bulk value at R, Eq. [433] can be recast into the more expressive form 

Because Kq describes the length scale over which the potential decays away from the polyelectrolyte, Eq. [434] essentially says that if the potential distal from the polyelectrolyte falls to zero before the outer boundary is reached, this boundary may then be removed to infinity without affecting the solution to the PB equation. In practice, the restriction kqR 5 seems sufficient for invoking an infinite model of a finite system containing mono- and divalent ions.  

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In 1923, E Hiickel and P Debye, winner of the 1936 Nobel Prize in Chemistry, adapted the Poisson-Boltzmann theory to explain the nonidealities of dilute solutions of strong electrolytes. To visualize Na ions surrounded by CD ions and C1 surrounded by Na at the same time, think of a NaCl crystal that is expanded uniformly. Now add fluctuations, Debye and Hhckel focused on one ion as a charged sphere, and used the linear approximation to the Poisson-Boltzmann equation to compute the electrostatic free energy of creating the nonuniform distribution of its surrounding counterions and co-ions. 

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