Ionic conductivity calculator from impedance (EIS)

Conductivity from resistance. The electrolyte is sandwiched between two ion-blocking electrodes (stainless steel, Au or Pt) and its resistance R is read from the impedance spectrum. The conductivity is (Front Chem 2021, doi:10.3389/fchem.2020.619832)

$$\sigma =\frac{L}{R\cdot A}$$

with $L$ the electrolyte thickness and $A$ the area of the blocking electrode; the same relation, ${\sigma }^{\prime }=(1/R)\cdot (d/A)$ with $d$ the pellet thickness, gives the bulk conductivity of garnet pellets (J Phys Chem C 2020, doi:10.1021/acs.jpcc.0c03558). With $L$ in cm, $R$ in Ω and $A$ in cm², $\sigma$ is in S/cm; ×1000 gives mS/cm. A disc electrode of diameter $d$ has $A=\pi \cdot d^2/4$.

Which resistance. In the Nyquist plot ($-Z^{″}$ against $Z^{\prime }$) of a solid electrolyte, a high-frequency semicircle comes from the bulk (and grain boundaries), and the steep rise at low frequencies from ions piling up at the blocking electrodes (J Phys Chem C 2020; Materials 2021, doi:10.3390/ma14071671). The high-frequency intercept with the real axis is taken as the resistance of the solid electrolyte (ACS Appl Mater Interfaces 2022, doi:10.1021/acsami.2c10853); when one semicircle covers grains and grain boundaries together, its resistance gives the total conductivity (Materials 2021). Each arc is modelled as a resistor in parallel with a constant phase element (CPE), and that resistor gives the conductivity; a CPE exponent $n=1$ is an ideal capacitor and $n<1$ a depressed arc (ACS Appl Mater Interfaces 2025, doi:10.1021/acsami.4c22390). Arcs with capacitances of a few pF are bulk; a grain-boundary arc in Li garnets had 8.8 nF (J Phys Chem C 2020), and electrode interfaces about 0.1 µF (Chem Mater 2016, doi:10.1021/acs.chemmater.6b00579).

How the tool reads R. High-frequency intercept: from the highest frequency down, the first pair of points where $-Z^{″}$ changes from $\le 0$ to $>0$, with $Z^{\prime }$ interpolated linearly at $-Z^{″}=0$ (if $-Z^{″}$ never crosses zero, the two highest-frequency points are extrapolated, and the tool says so). Semicircle fit: a least-squares circle through the points in the frequency range you choose (the algebraic fit minimising $\sum (x^2+y^2+D\cdot x+E\cdot y+F{)}^2$); its two crossings with the real axis are $R_1$ (high frequency) and $R_2$ (low frequency), and $R_2-R_1$ is the arc’s own resistance. A resistor in parallel with a CPE traces an arc of a circle, so a series resistance $R_{\mathrm{b}}$ plus one such arc gives $R_1=R_{\mathrm{b}}$ and $R_2=R_{\mathrm{b}}+R_{\mathrm{arc}}$ exactly; this is checked against synthetic spectra.

Activation energy. Two Arrhenius forms are in use: $\sigma ={\sigma }_0\cdot \exp (-E_{\mathrm{a}}/kT)$ (RSC Adv 2022, doi:10.1039/d1ra09138g) and $\sigma T=A\cdot \exp (-E_{\mathrm{a}}/kT)$ (Nanomaterials 2021, doi:10.3390/nano11020414), $k$ the Boltzmann constant. The tool fits a straight line to $\ln (\sigma T)$ (default) or ${\log }_{10}\sigma$ against $1000/T$, so $E_{\mathrm{a}}=-\text{slope}\times 1000\times k$ for $\ln (\sigma T)$ and $E_{\mathrm{a}}=-\text{slope}\times 1000\times k\times \ln 10$ for $\log \sigma$, with $k=8.617333\times {10}^{-5}\mathrm{eV}/\mathrm{K}$ ($1.380649\times {10}^{-23}\mathrm{J}/\mathrm{K}$ $÷1.602176634\times {10}^{-19}\mathrm{C}$). $E_{\mathrm{a}}$ per mole is $E_{\mathrm{a}}(\mathrm{eV})\times F/1000$ $=E_{\mathrm{a}}(\mathrm{eV})\times 96.485\mathrm{kJ}/\mathrm{mol}$, with $F=96485\mathrm{C}/\mathrm{mol}$ (doi:10.1021/acsaem.5c01865). Because $\ln (\sigma T)=\ln \sigma +\ln T$, the two forms give $E_{\mathrm{a}}$ values that differ by about $k\cdot T$ ($\approx 0.026\mathrm{eV}$ near room temperature), so report which one you used.

Limits. The circle only describes one arc: leave out the low-frequency electrode spike and inductive points above the axis crossing, and use the arc’s range of frequencies. Which arc is bulk and which is grain boundary has to come from its capacitance or frequency, not from this fit. An Arrhenius line holds only over a range without phase changes: a polymer electrolyte studied from −90 to 100 °C showed three regimes with $E_{\mathrm{a}}$ of 0.35 and 0.45 eV rather than one line (RSC Adv 2022).

The built-in spectrum is illustrative, not measured: it is computed from a series resistance $R_{\mathrm{b}}=120\mathrm{\Omega }$, an arc of 230 Ω in parallel with a CPE ($n=0.85$), a blocking-electrode CPE and 1.5 µH of cable inductance, from 3.16 MHz to 10 Hz. The pellet is 1.05 mm thick between 10 mm electrodes, so $A=\pi \times (1.0\mathrm{cm}{)}^2/4=0.7854{\mathrm{cm}}^2$.

1. Semicircle fit over 10 kHz–1 MHz (13 points, RMS 0.42 Ω off the circle): $R_1=122.48\mathrm{\Omega }$, $R_2=357.04\mathrm{\Omega }$, arc diameter 234.56 Ω, centre 27.9 Ω below the axis (a depressed arc).
2. Total conductivity from $R_2$: $$\begin{array}{rl}\sigma & =\frac{0.105\mathrm{cm}}{357.04\mathrm{\Omega }\times 0.7854{\mathrm{cm}}^2}\\ & =\mathbf{3.744}\times {\mathbf{10}}^{\mathbf{-}\mathbf{4}}\mathbf{S}\mathbf{/}\mathbf{cm}\\ & =\mathbf{0.3744}\mathbf{mS}\mathbf{/}\mathbf{cm}.\end{array}$$ From $R_1$ (the series, bulk resistance): $1.092\times {10}^{-3}$ S/cm.
3. High-frequency intercept: $-Z^{″}$ crosses zero between 1.468 and 1 MHz, at $Z^{\prime }=123.81\mathrm{\Omega }$. The inductance puts it 3 % above the true 120 Ω; the circle fit (122.48 Ω) is closer.
4. Activation energy from five temperatures (25–65 °C, $R=357.0,246.3,174.1,125.8,92.8\mathrm{\Omega }$): the $\ln (\sigma T)$ line gives $E_{\mathrm{a}}=\mathbf{0.320}\mathbf{eV}\mathbf{=}\mathbf{30.88}\mathbf{kJ}\mathbf{/}\mathbf{mol}$ (R² = 1, $A=2.86\times {10}^4\mathrm{S}\mathrm{K}/\mathrm{cm}$); the same data plotted as $\log \sigma$ give 0.293 eV, the $k\cdot T$ difference between the two forms.

For comparison, Nb-doped Li₇La₃Zr₂O₁₂ pellets measured between Au electrodes from 25 to 50 °C had activation energies of 0.28–0.35 eV (Materials 2021, doi:10.3390/ma14071671), and Al/Ga-substituted LLZO 0.31 eV, within the 0.26–0.37 eV reported before (Chem Mater 2016, doi:10.1021/acs.chemmater.6b00579).