  In this work a dynamical system approach is taken to systematically
investigate the one-dimensional classical Poisson-Boltzmann (PB) equation with
various boundary conditions. This framework, particularly, the phase space
portrait, has a unique advantage of a geometric view of the dynamical systems,
which allows one to reveal and examine critical features of the PB models. More
specifically, we are able to reveal the mechanism of Gouy-Chapman layer: the
presence of an {\em equilibrium} for the PB equation, including
equilibrium-at-infinity for Gouy-Chapman's original setup as the limiting case.
Several other critical, somehow counterintuitive, features revealed in this
work are the saturation phenomenon of surface charge density, the uniform
boundedness of electric pressure (given length) and of length (given electric
pressure) in surface charge, and the critical length for a reversal of electric
force direction. All have a common mechanism: the presence of an equilibrium
for the PB equation. We believe that the critical features presented from the
classical PB models persist for modified PB systems up to a certain degree. On
the other hand, any qualitative change in these features as the sophistication
of the model is increasing is an indication of new phenomena with new
mechanisms.
