2.5.4 Summary to: Ionic Conductors
- Page ID
- 2776
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Electrical current can conducted by ions in
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Challenge: Find / design a material with a "good" ion conductivity at room temperature
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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 |
\[d_\text{Debye}=\left(\frac{\varepsilon\cdot\varepsilon_0\cdot\text{k}T}{\text{e}^2\cdot c_0}\right)^{1/2}\]
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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. | |
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.