2.5.4 Summary to: Ionic Conductors
Electrical current can conducted by ions in

Challenge: Find / design a material with a "good" ion conductivity at room temperature

Basic principle
Diffusion current j_{diff}driven by concentration gradients grad(c) of the charged particles (= ions here) equilibrates with the  \[j_\text{diff}=D\cdot\text{grad(c)}\] 
Field current j_{field}caused by the internal field always associated to concentration gradients of charged particles plus the field coming from the outside  \[j_\text{field}=\sigma\cdot {\color{purple}E}=q\cdot c\cdot \mu\cdot {\color{purple}E}\] 
Diffusion coefficient D and mobility µ are linked via theEinstein relation; concentration c(x) and potential U(x) or field E(x) = –dU/dxby the Poisson equation. 
\[\mu=\text{e}D/\text{k}T \\  \dfrac{\text{d}^2U}{\text{d}x^2}= \frac{\text{d}{\color{purple}E}}{\text{d}x}= \frac{\text{e}\cdot c(x)}{\varepsilon\varepsilon_0}\] 
Immediate results of the equations from above are:
In equilibrium we find a preserved quantity, i.e. a quantity independent of x  the electrochemical potential V_{ec}:  \[V_\text{ec}=\text{e}\cdot U(x)+\text{k}T\cdot\text{ln}(c(x))\] 
If you rewrite the equaiton for c(x), it simply asserts that the particles are distributed on the energy scale according to the Boltzmann distrubution:  \[c(x)=\text{exp}  \frac{(VxV_\text{ec})}{\text{k}T}\] 
Electrical field gradients and concentration gradients at "contacts" are coupled and nonzero on a length scale given by the Debye length d_{Debye} 
\[d_\text{Debye}=\left(\frac{\varepsilon\cdot\varepsilon_0\cdot\text{k}T}{\text{e}^2\cdot c_0}\right)^{1/2}\]

The Debye length is an extremely important material parameter in "ionics" (akin to the space charge region width in semiconductors); it depends on temperature T and in particular on the (bulk) concentration c_{0} of the (ionic) carriers.  
The Debye length is not an important material parameter in metals since it is so small that it doesn't matter much. 
The potential difference between two materials (her ionic conductors) in close contact thus...
... extends over a length given (approximately) by :  \[d_\text{Debye}(1)+d_\text{Debye}(2)\] 
... is directly given by the Boltzmann distribution written for the energy: (with the c_{i}=equilibrium conc. far away from the contact. 
\[\frac{c_1}{c_2}=\text{exp} \frac{\text{e}\cdot\Delta U}{\text{k}T}\;\text{Boltzmann}\] 
The famous Nernst equation, fundamental to ionics, is thus just the Boltzmann distribution in disguise!  \[\Delta U=\frac{\text{k}T}{\text{e}}\cdot\text{ln}\frac{c_1}{c_2}\;\text{Nernst's equation}\] 
"Ionic" sensors (most famous the ZrO_{2}  based O_{2} sensor in your car exhaust system) produce a voltage according to the Nernst equation because the concentration of ions on the exposed side depends somehow on the concentration of the species to be measured.