## 3 Discussion  
Fits to either Arrhenius or Vogel-Tammann-Fulcher (VTF) models of ion transport were established for phase-based segments of conductivity data to further analyze the transport mode [31]. Regression lines can be seen on plots of temperature dependent DC conductivity in Section 2 (Figures 5-7). In the Arrhenius model, when temperature and conductivity are related using Equation (1), cation transport occurs via a hopping mechanism. This transport mode occurs in crystalline and glassy phases [31]. \[\sigma(T)=\sigma_{0}\,e^{-\varepsilon_{a}/\pi t} \tag{1}\]  
In Equation (1), \(\sigma\) is the DC conductivity resulting from long range ion diffusion, \(\sigma_{0}\) is a pre-exponential factor, \(E_{a}\) is the activation energy, \(R\) is the universal gas constant, and \(T\) is the temperature. This model fits the DC conductivity for most materials in the semicrystalline phase. The DC conductivity for C18 IPS0.05 and C18 IPS0.3 could not be measured at sufficiently low temperatures to fully analyze the temperature dependence of the conductivity for the crystalline phase. Table 6 shows the values of constants \(\sigma_{0}\) and \(E_{a}\) that were obtained for the Arrhenius fits. The pre-exponential factor, \(\sigma_{0}\), is proposed to be affected by both dielectric constant and temperature, and in this instance, also increases with the DC conductivity of the material [32]. The activation energies for ion transport within the studied materials are within the same order of magnitude, and the lower and higher activation energies correspond to higher and lower conductivities, respectively. These values of activation energy are also comparable to that recently reported for a class of sulfonated and crystalline telechelic polyethylenes titrated with various alkali metal cations; the activation energy for conduction in the crystalline C18 MS0.05 is on par with that reported for a sodiated telechelic alkyl sulfonate salt, and approximately 60% less than that reported for a lithiated telechelic alkyl sulfonate salt [33]. Figure 7: DC conductivity of 0.3:1 C18 RS:C18 LiTFSI ratio across temperatures compared to C18 LiTFSI. The isotropic to smectic transitions as measured using DSC are indicated for each material by vertical lines, and regressions calculated based on Arrhenius or VTF models are displayed as fit lines. Data for C18 LiTFSI were originally reported in [21]. Interestingly, at temperatures below the discontinuity (approximately 60 \({}^{\circ}\)C) C18 MS0.05 shows discontinuous conductivity behavior in the crystalline phase, but the data fit the Arrhenius model over a limited temperature range. The conductivity of this mixture is also significantly higher than C18 LiTFSI in the semicrystalline phase. Discontinuities in the conductivity trends at low temperatures for this mixture may be the result of a crystal-crystal transition not visible using DSC. Unfortunately, it was not possible to collect scattering data at temperatures below room temperature. It is also noted that timescales for the conductivity measurements and scattering measurements were not equivalent. Further work is needed to correlate the semicrystalline material structures with the conduction behavior. The smectic portions of conductivity for most materials fit the VTF model. In the VTF model, which follows Equation (2), ion transport dynamics are related to coordination site relaxation dynamics. Increasing free volume at higher temperatures is associated with greater relaxation and greater conductivity [31]. We hypothesized that bulkier additives such as C18 ES or C18 IPS may increase local free volume within the ionic domains and thereby increase conductivity; however, this is not consistently observed in our experimental results.