### 2.4.4 Summary to: Ionic Conductors

| Electrical current can conducted by ions in Liquid electrolytes (like H2SO4 in your "lead - acid" car battery); including gels Solid electrolytes (= ion-conducting crystals). Mandatory for fuel cells and sensors Ion beams. Used in (expensive) machinery for "nanoprocessing". |  |
Challenge: Find / design a material with a "good" ion conductivity at room temperature
| Basic principle |  |
|  | Diffusion current jdiff driven by concentration gradients grad(c) of the charged particles (= ions here) equilibrates with the |  |
jdiff = – D · grad(c)
jfield = s · E = q · c · µ · E
|  | Field current jfield caused by the internal field always associated to concentration gradients of charged particles plus the field coming from the outside |
|  | 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. |  |
µ = eD/kT
– | d2U dx2 | = | dE dx | = | e · c(x) e e0
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| 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 Vec: |  |
Vec | = const. = | e · U(x) + | kT | · ln c(x)
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|  | 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) = exp – | (V x) – Vec kT
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|  | Electrical field gradients and concentration gradients at "contacts" are coupled and non-zero on a length scale given by the Debye length dDebye |  |
dDebye = | æ ç è | e · e0 · kT e2 · c0 | ö ÷ ø | 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 c0 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 : |  |
dDebye(1) + dDebye(2)
|  | ... is directly given by the Boltzmann distribution written for the energy: (with the ci =equilibrium conc. far away from the contact. |  |
c1 c2 | = exp – | e · D U kT |  | Boltz- mann
--- | --- | --- | --- | ---
D U = – | kT e | · ln | c1 c2 |  | Nernst's equation
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|  | The famous Nernst equation, fundamental to ionics, is thus just the Boltzmann distribution in disguise! |
| "Ionic" sensors (most famous the ZrO2 - based O2 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. |  |
Questionaire
Multiple Choice questions to all of 2.4

© H. Föll (Electronic Materials - Script)