Colloquium: Understanding ion motion in disordered solids from impedance spectroscopy scaling

ac impedance measurements of ion-conducting glasses provide considerable insight into the nature of the ionic motions in disordered solids. However, interpreting the ac impedance has been a matter of considerable debate, particularly in regards to how best to represent the relaxation process that is the result of a transition from correlated to uncorrelated ion hopping. Although interpretations based upon the electric modulus have featured prominently in the earlier literature, direct analysis of the complex conductivity in the frequency domain is gaining popularity as it provides direct (via Fourier transform) information regarding the microscopic mean-squared displacement of ions. Here many recent findings are summarized that emphasize the scaling features present in the impedance spectra to demonstrate how the casual researcher might interpret ac impedance results without elaborate curve fitting.

Figure 1

Covalent network of pure $\mathrm{Si}{\mathrm{O}}_2$. (b) Formation of nonbridging oxygen sites on addition of ${\mathrm{Li}}_2\mathrm{O}$.

Figure 2

Schematic representation of how the ac conductivity and dielectric constant typically depend upon frequency for an ionic material. Several of the limiting features are labeled.

Figure 3

Schematic representation of the time dependence for mean-squared displacement of ions corresponding to the ac conductivity in Eq. (6). Dashed line $A$ corresponds to approximation used in Eq. (6). Dashed line $B$ shows a more appropriate approach to a constant at long times as indicated by Eq. (14).

Figure 4

Real and imaginary parts of the electric modulus for $\mathrm{Li}\mathrm{P}{\mathrm{O}}_3$ glass near room temperature. The solid lines are a fit to Eqs. (12, 13). The dashed line shows a fit to the KWW relaxation function with $\beta =0.6$.

Figure 5

Scaling plot of both the ac conductivity and ac permittivity of ${\left({\mathrm{Na}}_2\mathrm{O}\right)}_{10}{\left(\mathrm{Ge}{\mathrm{O}}_2\right)}_{90}$ demonstrating the collapse of individual spectra collected at different temperatures onto a master curve. Inset: The proportionality that is maintained between the vertical and horizontal scales of the ac conductivity in this instance of canonical scaling.

Figure 6

Scaling plot of both the ac conductivity and ac permittivity of a series of potassium thioborate glasses demonstrating the collapse of the master curves of individual spectra collected at different temperatures for given ion concentration onto a master master curve for all ion concentrations. Inset: The variation of $\Delta \epsilon T$ with ion concentration. The dashed line corresponds to a variation of $\Delta \epsilon T\approx x^{0.4}$.

Figure 7

The dc conductivity (multiplied by the temperature) vs the scaling frequency for a series of potassium thioborate glasses showing how canonical scaling is present in each composition, but that the proportionality constant $K(N,q,\xi )$ is systematically changing from one concentration to the next.

Figure 8

Real part of the electric modulus for the series of potassium thioborate glasses shown in previous figures. Here the modulus is multiplied by the high-frequency dielectric constant and the frequencies have been scaled as before in Fig. 6. Inset: The compositional variation of the stretching exponent.

Figure 9

Scaling plot of both the ac conductivity and ac permittivity of CKN demonstrating the collapse of individual spectra collected at different temperatures, both above and below the glass transition, onto a master curve. The two lines indicate the approximate power-law exponents seen near $f_0$.

Figure 10

The dc conductivity (multiplied by the temperature) (vs) the scaling frequency for CKN showing how a canonical scaling is present both above and below the glass transition, but that the proportionality constant $K(N,q,\xi )$ is systematically changing in the transition range by roughly a factor of 4.3. Inset: The corresponding variation of $\Delta \epsilon T$ with temperature near $T_g\approx 60°\mathrm{C}$.

Figure 11

The real part of the electric modulus for the CKN data shown in Fig. 10. Here the modulus is scaled to its maximum value and the frequencies have been scaled to the frequency of the maximum in $M^{″}$. Inset: The temperature variation of the stretching exponent reported by Howell (22).

Figure 12

Construction of the imaginary part of the electric modulus from the scaling forms for the conductivity [Eq. (18)] and permittivity [Eq. (22)]. This demonstrates how the shape of the modulus is sensitive to the quantity $\lambda ={\epsilon }_{\infty }∕\Delta \epsilon$. Corresponding values of the stretching exponent are also provided for comparison.

Figure 13

The conductivity exponent observed in metaphosphate glasses as a function of the constriction (i.e., ratio of the cations' size to the mean spacing between phosphate chains).

Figure 14

The conductivity exponent vs the effective dimensionality of the local conduction space.

Figure 15

The dielectric loss (multiplied by the temperature) for series of ${\left({\mathrm{Na}}_2\mathrm{O}\right)}_x{\left(\mathrm{Ge}{\mathrm{O}}_2\right)}_{1-x}$ glasses including the undoped $\mathrm{Ge}{\mathrm{O}}_2$. Solid symbols are raw data. Open symbols are with the loss from $\mathrm{Ge}{\mathrm{O}}_2$ subtracted. Inset: The vertical displacement of the loss at 100 and $200\mathrm{K}$ proportional to the value of $K(N,q,\xi )$ when the nonionic loss due to $\mathrm{Ge}{\mathrm{O}}_2$ is first removed.

Figure 16

The dielectric loss at five selected frequencies (multiplied by the temperature) for both $\mathrm{Na}\mathrm{P}{\mathrm{O}}_3$ and $\mathrm{K}\mathrm{P}{\mathrm{O}}_3$ glass. The individual isochronal spectra have been shifted along the ordinate axis so as to produce the best collapse to the curves at $3\mathrm{kHz}$. Inset: The required shift factors and corresponding dc activation energy.