2.5.4 Summary to: Ionic Conductors
- Page ID
- 2776
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)
Electrical current can conducted by ions in
|
Challenge: Find / design a material with a "good" ion conductivity at room temperature
|
Basic principle
Diffusion current jdiffdriven by concentration gradients grad(c) of the charged particles (= ions here) equilibrates with the | \[j_\text{diff}=-D\cdot\text{grad(c)}\] |
Field current jfieldcaused 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 Vec: | \[V_\text{ec}=\text{e}\cdot U(x)+\text{k}T\cdot\text{ln}(c(x))\] |
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: | \[c(x)=\text{exp} - \frac{(Vx-V_\text{ec})-}{\text{k}T}\] |
Electrical field gradients and concentration gradients at "contacts" are coupled and non-zero on a length scale given by the Debye length dDebye |
\[d_\text{Debye}=\left(\frac{\varepsilon\cdot\varepsilon_0\cdot\text{k}T}{\text{e}^2\cdot c_0}\right)^{1/2}\]
|
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 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 : | \[d_\text{Debye}(1)+d_\text{Debye}(2)\] |
... is directly given by the Boltzmann distribution written for the energy: (with the ci=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 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.