
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. 








