 Ionics is the topic of dedicated lecture courses,
here we will only deal with two of the fundamental properties and equations  the Debye length and the Nernst equation  in a very simplified way. 
  The most general and most simple situation that we
have to consider is a contact between two materials, at least one of which is a solid ionic
conductor or solid electrolyte. Junctions with liquid electrolytes, while somewhat more complicated,
essentially follow the same line of reasoning. 
 Since this involves that some kind of ion can move,
or, in other words, diffuse in the solid electrolyte, the local concentration c of the mobile ion can respond to two types of driving
forces: 
  1. Concentration
gradients, leading to particle currents j_{diff} (and, for particles with charge
q, automatically to an electrical current j_{elect} = q ·
j_{diff}) given by Ficks laws 
 

  With D = diffusion
coefficient of the diffusing particle. 
  2. Electrical fields E, inducing electrical current
according to Ohms law (or whatever current  voltage  characteristics applies to
the particular case), e.g. 
 
j_{field} = s · E = q · c · µ · E 


  With µ = mobility of the particle. 
 Both driving forces may be
present simultaneously; the total current flow or voltage drop then results from
the combined action of the two driving forces. 
  Note that in one equation
the current is proportional to the gradient of the concentration whereas in the
other equation the proportionality is to the concentration directly. This has
immediate and far reaching consequences for all cases where in equilibrium the two components must cancel each other
as we will see in the next subchapter. 
 In general, the two partial currents will not be
zero and some net current flow is observed. Under equilibrium conditions, however,
there is no net current, this requires that the partial currents either are all zero, or that they must have the same
magnitude (and opposite signs), so that they cancel each other. 
  The equilibrium condition thus is 
 

  The importance of this equation cannot be over emphasized. It
imposes some general conditions on the steady state concentration profile of the
diffusing ion and thus the charge density. Knowing the charge density distribution, the potential distribution can be
obtained with the Poisson equation, and this leads to the Debye length and Nernsts law which we will discuss in
the next paragraphs. 
 
