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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".
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| Challenge: Find / design a material with a "good" ion conductivity at room temperature
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Basic principle | |
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Diffusion current
jdiff driven by concentration gradients grad(c) of the charged particles (= ions here) equilibrates with the
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| jfield = s · E
= q · c · µ · E |
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Field current
jfield caused by the internal field always associated to concentration gradients of charged particles
plus the field coming from the outside | |
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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. |
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| – | d2U
dx2 | =
| dE
dx
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e · c(x)
ee0
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Immediate results of the equations from above are: |
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In equilibrium we find a preserved quantity, i.e. a quantity independent of x
- the electrochemical potential Vec: |
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| 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: |
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| c(x) = exp – |
(Vx) – 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 |
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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. |
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The Debye length is not an important material parameter in metals since it is so small that
it doesn't matter much. | |
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The potential difference between two materials (her ionic conductors) in close
contact thus... | | |
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... extends over a length given (approximately) by : |
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... is directly given by the Boltzmann distribution written for the energy:
(with the
ci =equilibrium conc. far away from the contact. |
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c1
c2 |
= exp – | e · DU
kT | |
Boltz-
mann |
| DU = – |
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! | |
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"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. |
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© H. Föll (Electronic Materials - Script)
