# Development on Solid Polymer Electrolytes for Electrochemical Devices
## 2 Ion Transport Properties and Conduction Mechanism in SPEs
Radial distribution function of lithium also showed weak binding between lithium and oxygen atom of TFSI' in the presence of PC [4]. Therefore, the probability of lithium ions' transportation through the oxygen atom of PEO was higher than that of carbonate and TFSI' oxygen [4]. [PERSON] and co-workers [5] calculated the diffusion of lithium ions in different types of polymers, namely PEO, poly(methylene oxide) (PMO), poly(propylene oxide) (PPO), poly(trimethylene oxide) (PTMO), poly(ethylene oxide-alt-methylene oxide) (P(EO-MO)) and poly(ethylene oxide-alt-trimethylene oxide) (P(EO-TMO), using molecular dynamics and dynamic bond percolation models. They have assigned short molecular dynamics trajectories to identify the distribution of the solvation sites and used it in dynamic bond percolation model to calculate the ionic hopping rate. According to them, the largest number of site density belongs to the polymer with the highest oxygen atom content which is PMO (oxygen to carbon atomic ratio is 1:1). This is followed by P(EO-MO) (2:3), PEO (1:2), P(EO-TMO) (2:5), PPO (1:3) and PTMO (1:3). Even though the ratio of oxygen to carbon atoms for PPO and PTMO are the same, the site density is slightly higher in PPO which may be due to the arrangement of atoms in the PPO [5]. The rate of hopping sites, \(k_{0}\), with distance, \(r\), can be calculated by using the following equation:  
\[k_{0}(r)=\tau^{-1}e^{-E_{dis}(r)/k_{\rm B}T}\,e^{-E_{\lambda}(r)/k_{\rm B}T} \tag{4}\]  
where \(\tau^{-1}\) is frequency, and \(E_{dis}\) and \(E_{\lambda}\) are the dissociation and reorganization energies, respectively. Based on the calculation, the authors found that PMO exhibited the smallest value of \(k_{0}\), whereas that of PPO was the largest [5]. They have used these values to calculate lithium-ion diffusion, using the kinetic Monte Carlo simulation. Among the polyethers studied, the diffusion of lithium ions in P(EO-TMO) was the highest [5]. Diffusion coefficient of ions, \(D\), can also be determined from the following equation:  
\[D=\frac{d^{2}}{4\tau_{2}\delta^{2}} \tag{5}\]  
where \(d\) is the thickness of the electrolyte, \(\tau_{2}=1/\omega\) and \(\delta=dk^{-1}/A\varepsilon e_{0}\) (where \(\omega\) is the angular frequency taken at the minimum value of impedance in Bode plot; \(k^{-1}\) is double layer capacitance; \(\varepsilon\) is dielectric constant of the electrolyte; and \(\varepsilon_{0}\) is the permittivity in free space). Based on Equation (5), \(D\) in PVA-based polymer electrolyte containing potassium iodide salt has been calculated [6]. The \(k^{-1}\) value was obtained by fitting the Nyquist plot of the electrolyte. Since the complex impedance plot shows only a tilted spike, the plot has been fitted based on the series connection of bulk resistance, \(R_{b}\), and constant phase element, CPE. The total impedance (real, \(Z^{\prime}\) and imaginary, \(Z^{\prime\prime}\)) can be described as follows:  
\[Z^{\prime}=R_{b}+cos\ (\pi p/2)/k^{-1}\omega^{p} \tag{6}\]  
\[Z^{\prime\prime}=sin\ (\pi p/2)/k^{-1}\omega^{p} \tag{7}\]  
where \(p\) is a fraction of a right angle. The mobility, \(\mu\), and number density, \(n\), of ions can be obtained by using the following equations:  
\[\mu=qD/k_{b}T \tag{8}\]  
\[n=\sigma/q\mu \tag{9}\]  
It is evident that Equation (1) is the combination of Equations (8) and (9). The calculated transport properties (\(D\), \(n\) and \(\mu\)) have been used to explain the conductivity behavior in PVA-based electrolyte containing different salt concentrations [8]. Relationship between \(\sigma\), \(n\) and \(\mu\) can be expressed as follows:  
\[\sigma=qn\mu \tag{10}\]  
In our earlier studies [8], it was found that the ionic mobility has contributed more in the highest conducting electrolyte, since \(n\) decreased due to the establishment of ion pairs and ionic aggregates. [PERSON] has calculated \(n\) for lithium triflate in gellan-gum-based SPE by using the method from Fourier transform infrared (FTIR) spectroscopy [9; 15].