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Nernst's equation gives the voltage between two materials in close contact, i.e.
the potential difference between the two
materials. From the foregoing discussion, we know already two important facts
about this potential: |
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It will change from one value to the other over a
distance across the junction that is given by the (two) Debye lengths of the system. |
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The corresponding carrier concentrations are
equilibrium concentrations and thus
governed by the Boltzmann
distribution (considering only classical particles at this point). |
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If the potential difference is
DU, we thus, using the Boltzmann
distribution,
obtain for
the concentration of the carriers c1 in
material 1, and c2 in material 2: |
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This is already
Nernst's equation (or law) - in a somewhat unusual way of writing. |
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Usually we (and everybody else) use
the Boltzmann distribution to compute concentrations as a function of some other known parameters - the energy in this case. But this
is not the only way for using a general
equation! |
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Like any equation, it also works in in reverse: If we know the concentrations, we can calculate the energy
difference that must go with them! |
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The important point now is that the
concentrations of electrons in metals, but also of ions in ionic conductors, or
holes in semiconductors, or any mobile carrier a few
Debye lengths away from the junction, are fixed - there is no need
to compute them! |
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What is not
fixed is the potential difference e
· DU a few Debye lengths away from
the junction, and that is what we now can obtain from the above
equation by rewriting it for DU: |
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This is Nernst's equation in its usual, but somewhat
simplified form. We may briefly consider two complications: |
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1. If the particles carry
z elementary charges, the first factor will now obviously write
kT/(z · e). |
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2. If the interaction between particles is not negligible (which would mean, e.g., that Ficks
law in its simple form would not be usable), the concentrations have to be
replaced by the
activities
a of the
particles. |
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If you want to know in detail what
activities are - use the link. But all you have to know at this point is that
activities are the particle concentrations corrected
for interaction effects. To give an example: If a particle
concentration is 1019 cm–3, the
activity might only be 5 · 1018
cm–3. If you use this factually wrong
number, simple equations like the Boltzmann distribution that do not take into
account particle interactions can still be used. |
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If the activity numbers are very
different from the real concentration numbers, you are no longer doing
Materials Science, but chemistry. |
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Using this, we obtain the general version of Nernst's law |
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| DU = – |
kT
z · e |
· ln |
a1
a2 |
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Nernsts law, being the Boltzmann
distribution in disguise, is of course extremely general. It gives the
potential difference and thus the voltage of any contact between two materials that have
sufficiently large concentrations of mobile carriers so that an equilibrium
distribution can develop. It describes, among other things |
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The contact voltage (Volta potential) between two metals (i.e.
thermocouples). |
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The built-in potential in pn-junctions |
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The voltage of any battery or accumulator. |
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The voltage of fuel cells. |
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The voltage produced by certain kinds of
sensors. |
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The last issue
merits some brief explanation.
Let's assume a material with a sufficiently large concentration of mobile
O– ions at interstitial sites (in other word, mobile
interstitial point defects) at the working temperature - take
Y2O3 stabilized ZrO2 as an
example (whatever that may be). |
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Use it to measure the amount of oxygen in a given
gas mixture with the following oxygen sensor
device: |
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The sensor material experiences two
different oxygen concentrations on its two surfaces, one of which is known
(oxygen in air, a constant for all practical purposes), the other one is the
concentration in the exhaust gas of a car which is supposed to be measured by
the voltmeter |
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Two gas-permeable electrodes have been supplied
which allow oxygen on both sides to react with the sensor material. |
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In equilibrium, we will have some
reaction between the oxygen in the gas and the oxygen in the crystal in the
sense that oxygen will either come out, or diffuse into the material. |
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What we might expect is that the concentration of
interstitial oxygen in the crystal will be larger near to the surface with the
large oxygen gas concentration (air) compared to the surface exposed to a lower
oxygen concentration (exhaust). |
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The gradient in the (negatively
charged) oxygen concentration inside the material then will be determined by
the Debye length of the system (in the real
thing, which is ZrO2, it will be just a few
nm). |
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In total, the concentration
[O]S of mobile O-interstitials right at the surface
will be somehow tied to the partial
pressure pO of the oxygen on both sides; lets say we
have a general relation like |
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æ
è |
const. · pO |
ö
ø |
n |
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But any other (reasonable) relation
you can think of will be just as good. |
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Nernst's law then tells
us immediately, how the voltage between the two electrodes depends on the
oxygen concentration or partial pressure in the exhaust: For the assumed
relation we have |
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| DU = – |
kT
e |
· ln |
c1
c2 |
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| DU = – |
kT
e |
· ln |
(p1)n
(p2)n |
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| DU = – |
n · kT
e |
· ln |
p1
p2 |
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This is quite remarkable: We have an
equation for the voltage that develops across some sensor as a function of the difference of the
oxygen concentration on two sides of the sensor without knowing much about the details of the
sensor! All we have to assume is that there is some mobile O–, no other free carriers, and that establishing
equilibrium does not take forever. |
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Only if you want to know the precise value of n do you have to
delve into the detailed reactions at the interfaces. |
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This is essentially the
working principle of not only the oxygen sensor in the exhaust system of any
modern car ("l -
Sonde"), but of most, if not all, solid state sensors. |
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© H. Föll (Advanced Materials B, part 1 - script)
