A SCALING LENGTH FOR CHARGED-PARTICLES

The Poisson-Boltzmann problem is considered from the point of view of a scaling length for the electric field: l//e equals (d phi /dx)/(d**2 phi /dx**2). The introduction of l//e leads to a unified description of the non-linear (highly charged surface; dilute solution) and of the linear cases. In the linear case, the scaling length is simply the Debye length lambda . In the non-linear case, it becomes independent of salt concentration. For non-planar surfaces, the use of the planar solution as a starting point is valid if the radius of curvature is larger than the scaling length. In the nonlinear regime this condition is equivalent to xi greater than 1/(2 z), where xi is the linear charge parameter, and z the counterion valency. Thus the condition is satisfied by highly charged polyelectrolytes. This explains the success of our prior approach to this problem, via an empirical perturbative treatment. A novel approximate solution of the Poisson-Boltzmann equation in the non-linear regime is presented in the appendix. The solution leads directly to the scaling length and is convenient for computations.