## 3 Results and Discussion
### Ionic Conductivity and Dielectric Relaxation Investigations
#### 3.2.2 Broadband Dielectric Permittivity Measurements
Furthering the investigation on dissipation processes and polarization, one of the most commonly used phenomenological models, the Cole-Cole model (CCM), was applied to the complex permittivity data [78]:  
\[\xi=\varepsilon_{\infty}+\frac{\varepsilon_{s}-\varepsilon_{\infty}}{1+(i \omega\tau)^{1-\alpha}} \tag{8}\]  
where \(\varepsilon_{s}\) and \(\varepsilon_{\infty}\) are the low-frequency and the high-frequency permittivity limit values, respectively, and \(c=1-\alpha\) is the CCM exponent (\(0\leq c\leq 1\)) describing the broadness of distribution of the relaxation time. For \(c=1\), the CCM converges to the Debye equation, which assumes a single value of the relaxation time, thus, the CCM can be viewed as a superposition of multiple Debye models [78]. Since the CCM employs the inverse critical frequency (defined as the peak position on the imaginary part of the complex conductivity) to determine relaxation times, the investigation proceeded by constructing Cole-Cole plots of dielectric loss against the dielectric constant for all in-lab prepared samples over the full temperature scanning range from 200 \({}^{\circ}\)C to \(-\)100 \({}^{\circ}\)C. For the 345 MPa and the 540 MPa samples (Figure 12c,d), the semicircle arcs were well-defined and showed symmetric broadening, marking the spread between the low- and high-frequency permittivity limits as in agreement with the frequency series plots. The maxima in the extrapolated arcs were observed at the mid-points, with the presence of linear segments over the low frequency zones implying heterogeneous relaxation processes. In both cases, the arcs diminished mildly until saturation with decreasing temperature, while higher temperatures enhanced dissipation by secondary relaxation processes. This image was heavily distorted in the case of the 245 MPa sample (Figure 12b), where the semicircle arc did not vary proportionally with temperature. Instead, the primary relaxation process appeared unstable, with its arc maximizing over the low temperature regime, becoming evidently replaced by thermally induced processes that migrate from the lower frequencies to dominate over the higher temperatures. This implies that the 245 MPa sample suffered from inhomogeneity issues (i.e., zirconia phases forming distinct zones that interfere with the NASICON lattice structure), or even mechanical instability issues arising from insufficient applied pressure and the ensuing loose packing order. In Figure 13, the evolution of the loss tangent tan\(\delta\) over frequency and temperature is visualized more comprehensibly in high-resolution 2D and 3D phase-space mappings. The primary zone of relaxation coincides with the frequency-dependent dispersion region identified as the narrow diagonal ridge in conductivity phase-space mappings (Figure 8). Likewise, the relaxation zone was transposed toward higher temperatures as frequency increased; dual dependency on both temperature and frequency revealed both thermal and polarization sources of energy dissipation in the NASICON system. The relative width of the relaxation zone appeared diminished for the 245 MPa sample, confining its dissipation over a narrower range of frequencies and temperatures. Relaxation zones for the 345 MPa and 540 MPa samples instead extended over significantly wider zones. Another observation that was not readily made apparent in the conductivity phase-space mappings concerns the presence of a secondary relaxation zone beginning at high temperatures (above 150 \({}^{\circ}\)C) and propagating diagonally over increasing frequencies. The initiatory temperature of this zone coincides with the transition temperature of monoclinic-to-rhombohedral lattice structure for NASICON, suggesting dual relaxation processes at elevated temperatures.