In this paper, the distribution of relaxation times (DRTs) functions are calculated numerically in Matlab for synthetic impedance data from single parallel $$RC$$RC circuit and two parallel $$RC$$RC circuits connected in series, experimental impedance data from supercapacitors and α-LiFeO₂ anode based Li ion batteries. The quality of the impedance data is checked with the Kramers–Krönig (KK) relations. The DRTs are calculated within the KK compatible regime for all the systems using Tikhonov regularization (TR) method. Here we use a fast and simple L-curve method to estimate the TR parameter (λ) for regularization of the Fredholm integral equations of first kind in impedance. Estimation of the regularization parameters are performed effectively from the offset of the global corner of the L-curve rather than simply using the global corner. The physical significances of DRT peaks are also discussed by calculating the effective resistances and capacitances coupled with peak fitting program. For instance, two peaks in the DRTs justify the electrical double layer capacitance and ion diffusion phenomena for supercapacitors in low to intermediate frequencies respectively. Moreover, the surface film effect, Li/electrolyte and electrode/electrolyte charge transfer related processes are identified for α-LiFeO₂ anode based Li-ion batteries. This estimation of the offset of the global corner extends the L-curve approach coupled with the Tikhonov regularization in the field of electrochemistry and can also be applied in similar process detection methods.

Among many transition-metal oxides, Fe₃O₄ anode based lithium ion batteries (LIBs) have been well-investigated because of their high energy and high capacity. Iron is known for elemental abundance and is relatively environmentally friendly as well contains with low toxicity. However, LIBs based on Fe₃O₄ suffer from particle aggregation during charge–discharge processes that affects the cycling performance. This study conjectures that iron agglomeration and material performance could be affected by dopant choice, and improvements are sought with Fe₃O₄ nanoparticles doped with 0.2% Ti. The electrochemical measurements show a stable specific capacity of 450 mAh g⁻¹ at 0.1 C rate for at least 100 cycles in Ti doped Fe₃O₄. The stability in discharge capacity for Ti doped Fe₃O₄ is achieved, arising from good electronic conductivity and stability in microstructure and crystal structure, which has been further confirmed by density functional theory (DFT) calculation. Detailed distribution function of relaxation times (DFRTs) analyses based on the impedance spectra reveal two different types of Li ion transport phenomena, which are closely related with the electron density difference near the two Fe-sites. Detailed analyses on EIS measurements using DFRTs for Ti doped Fe₃O₄ indicate that improvement in interfacial charge transfer processes between electrode and Li metal along with an intermediate lithiated phase helps to enhance the electrochemical performance.

The oxygen vacancy evolution and ion dynamics in Dy doped nanoceria have been investigated with microstructural, optical and ionic conductivity studies. The influence of Dy³⁺ ions on the microstructures and the optical and ionic conductivity properties of these nanoparticles has been studied using X-ray diffraction, HR-TEM, EDAX, UV-vis, Raman and impedance spectroscopy. From Rietveld refinement of the XRD profiles, it has been found that the oxygen vacancies, lattice parameters and Ce–O bond lengths increase with Dy³⁺ ion concentration. The Rietveld analysis, together with HR-TEM, confirms the cubic fluorite structure with space group Fmm of all the samples. The EDAX spectra show good stoichiometry of the different atoms in the samples. The direct band gap, calculated from UV-vis spectra, shows a red shift with increasing concentration of Dy³⁺ ions. Raman spectroscopy studies of the samples gave insight into their vibrational properties and pointed to the fact that the oxygen vacancy content increased significantly with the doping concentration. The number of oxygen vacancies and their interaction with dopant cations strongly influence the electrical properties of Dy doped ceria.

Oxygen ion conduction and relaxation mechanisms in bismuth substituted La₂Mo₂O₉ have been reported in this paper. The phase transition observed in La₂Mo₂O₉ is suppressed in bismuth substituted samples, and the cubic β form is stabilized at room temperature. A transition of the dc anionic conductivity from low temperature Arrhenius type to high temperature Arrhenius type has been observed in La₂Mo₂O₉. However, for bismuth doped samples the dc anionic conductivity shows a transition from Arrhenius behavior at low temperatures to Vogel-Tamman-Fulcher in the high temperature regime. The frequency dependence of the ac conductivity shows a power law behavior. The ac conductivity remains constant in low frequency region but shows dispersion at the crossover frequency. The crossover frequency shifted towards higher value with the increase of temperature. The dimensionality of the conduction space does not change with bismuth doping. The scaling behavior of the conductivity spectra for different temperatures and compositions indicates that dynamics of oxygen ions is independent of temperatures and compositions.