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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 |
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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 |
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Boltz-
mann |
| DU = –
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kT
e |
· ln |
c1
c2 |
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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)
