The spinel LiMn₂O₄ sample was synthesized using lithium (Li)- and manganese (Mn)-based reagents in a 1:2 stoichiometric ratio. Lithium acetate, manganese acetate tetrahydrate, and citric acid were employed as precursors in this sol-gel method. All reagents were dispersed in deionized water and stirred constantly. After continuous stirring, the pH of the sample was maintained. The pH value of the resultant sample was maintained between 6 and 7 after vigorous stirring. The resultant mixture was then stirred constantly at 350 rpm at 70°C until it turned into a translucent gel. The obtained gel was then heated to 180°C in a vacuum oven to obtain a powdery sample. This pulverized sample was subsequently calcined in two steps: First, the compound decomposition was calcined at 350°C for 6 h. Second, the final phase formation calcination was carried out at 775°C for 20 h in the muffle furnace. After cooling, the calcined sample was ground again, using a mortar and pestle, and the LiMn₂O₄ sample was abbreviated as LMO.

The synthesized LMO sample was investigated for phase analysis using a RIGAKU X-ray diffractometer (XRD) with λ = 1.540°A. The XRD data were plotted between the angular ranges from 10° to 70°, with a step size of 0.02. The morphological characterization of the LMO sample was obtained using a scanning electron microscope (SEM) operated at 15 kV. In addition, the thermal stability of the LMO sample was analyzed using thermogravimetric analysis (TGA, SII Exostar) in the O₂ atmosphere with a heating rate of 5°C min⁻¹. Then, the crystal bonding of the LMO sample was also examined by Fourier transform infrared spectroscopy (FTIR) in the range of 4,500-400 cm⁻¹, and a pellet was prepared using KBr as a binder to test the pulverized sample. Similarly, to determine the electrical conductivity of the LMO sample, the pellet was prepared by mixing 2.5% PVC (polyvinyl alcohol) and then annealing at 250°C for 3 h. After drying the pellet, a silver paste was coated on both sides of the surface to make the pellet conductive and then dried at 150°C for half an hour. The dried pellet was then placed on a two-point probe assembly to measure the electronic conductivity of the LMO sample using the KEITHLEY 6517A electrometer at high resistance and a variable frequency range of 20 Hz to 1 mHz.

The electronic conductivity (σ _DC ) in Scm⁻¹ was calculated using the standard Eq. 1, and I-V was also plotted on the source measurement unit. σ = 1   R ( L A ) , (1) where L is the thickness of the pellet in cm, A is the cross-sectional area of the pellet in cm², and R is the resistance of the sample in Ω (ohms).

Furthermore, the electronic conductivity of the sample was used in the calculation of activation energy (E_a) in eV using Arrhenius Eq. 2: σ = σ o exp ( − E a K B T ) , (2) where K _B is the Boltzmann constant in J.K⁻¹ and T is the temperature in Kelvin (K).

However, the AC conductivity of the sample was evaluated with a bias voltage of 1.0 V, and the frequency range was also kept between 20 Hz and 1 MHz.

The thermogravimetric analysis (TGA) was performed to analyze the thermal properties of the sample and also to obtain the decomposition temperature at a variable atmosphere. The LiMn₂O₄ sample was characterized by the variation in a mass loss at different temperatures, as shown in Figure 1. The obtained TGA curve represents the two main losses shown in the red solid line. The first loss was shown between the temperature range of 150°C and 235°C (∆m = 5.36%). This calculated mass loss was due to the endothermic reaction caused by decomposition or evaporation of physically absorbed moisture and also by organic or oxalate compounds in the sample released by dehydration of the precursor. The second mass loss was the main loss associated with a 55.49% mass loss in the temperature range of 235°C to 635°C, as shown in Figure 1, which corresponds to the decomposition of all the starting materials such as lithium acetate and manganese acetate. After that, mixed phases of MnO₂ and Mn₂O₃ formed and started to decompose in a stable range (Helan and Berchmans, 2011). In this second region, a pure LMO formed by a synergistic impact of three different compounds MnO₂, Mn₂O₃, and Li₂CO₃ through phase transformation and carbon decomposition in the compound (Fadzil et al., 2017). Therefore, the sample calcination was carried out in two stages: First, the sample was calcined at 350°C for 6 h to decompose the organic compound, and the second calcination was carried out at a temperature of 775°C for 20 h to obtain the proper phase and morphology in the LMO sample.

The XRD analysis was performed to determine the crystallinity of the sample. Figure 2 demonstrates the XRD pattern for the LMO sample where sharp peaks reflect the crystalline sample. The peak positions also help determine the spinel phase and the cubic structure with the pnma space group of LiMn₂O₄ and also help evaluate the lattice parameters using the standard files (JCPDS 00-035-0782). There were certain other parameters of the LMO sample that were evaluated using the XRD pattern, as shown in Table 1. The lattice parameter for the synthesized sample was 4.779 Å, and the volume of the cubic cell was 109.14 Å³. Furthermore, the Williamson–Hall plot was used to evaluate the influence of the broadening the peak (β_T).

The broadening of the peaks was analyzed to determine the combined influence of crystallite size and the micro-strain. Therefore, the W-H plot was used to calculate the strain and crystallite size. The slope of the W-H plot was 2.039 × 10⁻⁴, shown as loading on the broadening peaks. The grain size of the synthesized sample was 72 nm, which was calculated using the Scherer equation. After structural analysis, morphological analysis was performed using a scanning electron microscope (SEM). The SEM image of the synthesized sample is shown in Figure 3 which depicts the smaller spherical-shaped grains, and also the average grain size of the LMO sample was calculated by ImageJ software, with size ranging from 8 to 10 µm.

Fourier transform infrared spectroscopy (FTIR) reveals the functional group of the organic compound and also exhibits the movement of cations with the aid of vibrational modes. In Figure 4, the FTIR spectrum of the LMO sample represents the four bands in which three bands show strong peaks. The three strong bands/peaks in the spectrum were observed at 3,426, 614, and 524 cm⁻¹ in different regions. In the first region, the peak was between 4,000 and 2,500 cm⁻¹ at 3,426 cm⁻¹ and was reflected due to the absorption of single C–H and O–H bonds in the sample, while in the second region, the two peaks were located in the range of 2,500 to 1,000 cm⁻¹ at 1,628 and 1,378 cm⁻¹, reflecting the weak peaks which associated the C–O and O–H bonds most likely to water bonds and Mn–O bonds at 1,089 cm⁻¹ vibrational frequency (Ali et al., 2020). This high frequency is associated with the asymmetric stretching modes of the MnO₆ group (Hasham et al., 2019; Ali et al., 2020). In the third region, the strong peaks were lying in the vibrational range between 1,000 and 400 cm⁻¹ at 614 and 524 cm⁻¹, and both the peaks were associated due to the transition metals and also stretches relate the Mn⁴⁺–O and Mn³⁺–O, respectively (Chiovoloni et al., 2017) The peaks that appeared below 400 cm⁻¹ in the characteristics determine the alkali metals in the octahedral sites in the sample (LMO).

The electrical characterization of the synthesized sample (LMO) was evaluated using ionic conductivity measurements. This characterization also contributed to the evaluation of the activation energy of the LMO sample. In Figure 5, the ionic conductivity curve of the LMO sample was determined using the KEITHLEY 6517A high resistance electrometer. This plot reflects the direct measurement of the activation energy while calculating the slope of the straight line and also represents the kinetics of lithium ions. The calculated activation energy of the synthesized LMO sample was 401.94 meV, and the electronic conductivity was 4.42 × 10⁻⁸ S/cm. These calculated parameters reflect the LMO sample as a good conductor and also specify a conductive material that shows good properties in lithium-ion batteries.

In this characterization, the electrochemical impedance spectroscopy (EIS) readings of the LMO material were plotted for validation and comparison of simulation results. This curve was plotted as a reference to simulate the internal parameters and to develop and control the other internal parameters for the enhancement in simulated results. This EIS curve was plotted within the frequency range of 20 Hz to 1 MHz with a 5 mV AC pulse to determine the AC impedance of the synthesized sample. In Figure 6, the semi-circle was intercepted on the z-axis to determine the AC resistance of the LMO material. The AC resistance was 59.65 ohm, and the AC conductivity of the sample was also observed, that is, 1.8*10⁻⁶ S/cm using the diffusion coefficient.

The schematic 1-D electrochemical model was designed for an all-solid-state lithium-ion battery (Liu and Hui, 2007; Liu et al., 2020b). This model was formulated in 1-D with three variable thickness domains, as demonstrated in Figure 7. In this model, the double-layer capacitance was taken on the active electrode and the electronic conductor on the positive electrode. Furthermore, the additional double layer on the electronic conductor was considered to describe the electrochemical impedance spectra (EIS) between the 1 mHz and 1 kHz frequency range on the LMO electrode (Darling and Newman, 1998).

The simulation model was validated using an experimental setup (Hasnain, 1998; Liu et al., 2020a). In this model, the comparative study of EIS spectra between experimental and simulation data was plotted. The optimization was performed using the global least-squares method (Zhang et al., 2020). This method was used to minimize the error between simulated and measured LMO impedance. There is a certain variable parameter that needs to be controlled with a cell voltage of 3.10 V. The control parameters are listed in Table 2.

After choosing control parameters with narrowly limited intervals, the optimization accuracy was improved and the optimization time shortened (Sangros Gimenez et al., 2020). Furthermore, the SNOPT method with a numerical gradient was used to study the optimization solver and also added the optimization node to study the steady-state AC impedance (Raijmakers et al., 2020).

The battery pack model was computed for 3-D thermal analysis during a 4C discharge. This pack was simulated by first coupling two cylindrical batteries in parallel, and six parallel-connected pairs were in series to create a full battery pack. This type of battery pack has been widely used for portable devices.

In this pack, three instances of the lumped battery interfaces were used to create the appropriate heat sources, which were further coupled to the heat transfer interface in a 3-D geometry. All the geometrical and internal and external parameters of the battery pack are illustrated in Table 3.

The electrochemical model of solid-state lithium-ion batteries was formulated with the help of governing equations and certain boundary conditions. To evaluate the steady-state AC impedance study, some variables need to be shifted from time-dependent to frequency-dependent as given in Eq. 3. n = n 0 + Re { n . e 2 Π f . it } , (3) where n denotes the variable and 0 subscript denotes the initial value at which perturbation occurs, I denote the imaginary unit, f denotes the frequency in Hz, and t is for time (Liu et al., 2020).

The electrochemical reactions of the positive and Πnegative electrodes were represented using the Butler–Volmer kinetic Eq. 4. i = i 0 ( exp ( α a Fη RT ) − exp ( − α c Fη RT ) ) (4)

Describing all the variables with their standard norms and this equation can be described in linear form due to small perturbations (Liu and Hui, 2007).

Initially, a sinusoidal perturbation was set at the boundary of the positive electrode current collector with an amplitude of 10mV and grounded at the negative electrode current collector with 0V, while the total impedance (Z- Ω.m²) of the cell was measured at the boundary of the positive electrode current collector as shown in Eq. 5. Z = ϕ s / ( n . I s ) , (5) where ϕ s denotes the potential in volts and I s denotes the current density in Am⁻² (Li, 2015).

In addition, this model was calculated in a frequency range from 10 mHz to 1 kHz in order to minimize the EIS errors. Therefore, a model was designed to include all necessary parameters and satisfy all boundary conditions to obtain the EIS spectra (Liu et al., 2020 and Sharma et al., 2021).

The simulated results were obtained in the form of Nyquist spectra. In these spectra, real and imaginary components of EIS were obtained, which were used to investigate the mass transport, capacitive, and kinetic properties of the battery. The Nyquist plot was plotted between the real and imaginary components with a pulse magnitude of 20 mV AC to obtain the experimental and simulated impedance spectra in the LMO positive-electrode, as shown in Figure 7.

In this plot, the “ref” is represented as the reference point probe which was placed in the center of the separator to retrieve the potential (Liu et al., 2020). The simulation results show that the cell impedance and a referenced positive electrode LMO have a large semi-circle, as shown in Figure 8, with dark blue and green solid lines, respectively, while the experimental spectra of referenced LMO have a smaller semi-circle represented as a light blue solid line. The semi-circle represents the value of the charge transfer resistance (R_ct) of the whole-cell impedance, a positive electrode (LMO), a negative electrode (carbon), and the experimental impedance of LMO, which are 2.2 × 10⁻³ Ω-m², 1.55 × 10⁻³ Ω-m², 0.8 × 10⁻³ Ω-m², and 1.22 × 10⁻³ Ω-m², respectively. This plot shows the difference in charge transfer resistance of both the experimental and simulated curves, and this difference was most evident in the experiments due to the short scale process. Therefore, to overcome this difference, the four control parameters were plotted using a parametric sweep to compare experimental and simulation plots. The parametric sweep was used to study the control parameters and also to determine their influence on the cell impedance. Figure 9 shows the Nyquist spectra for testing the control parameters in a frequency range of 10 mHz to 1 kHz. This result shows that there was a change in the charge transfer resistance (R_ct) from both experimental and simulated spectra of positive electrode LMO. The R_ct value of the positive electrode LMO in the experiment and in the simulation was 1.29 × 10⁻³ Ωm² and 1.6 × 10⁻³ Ωm², respectively. This value of charge transfer resistance shows a long-timescale diffusion process in the particles. Its effect is therefore more evident in the lower frequency regions.

Therefore, after the parametric sweep function, the LMO positive electrode has greater electronic conductivity. It is also clear from the simulation that the electrochemical reaction was also confined to their electrodes. This mainly depends on the other control parameters due to which the overall cell impedance has been affected. Furthermore, in order to improve electronic conductivity and reduce cell impedance, the other parameters should be appropriately controlled.

The LMO material was further characterized to evaluate the SOC and SOD characteristics of the synthesized material using a computational approach. A 2-D electrochemical model was simulated to understand the rate capability behavior of the battery. During this phenomenon, the battery was initially discharged from its charged state and then fully charged from its discharged state at various C-rates. However, the discharge characteristics of LMO material are shown in Figure 9 at several C-rates including 0.01, 0.1, 0.5, 1, 2, 5, 10 C, and open circuit.

Figure 10 shows the LMO discharge curve, which shows the discharge capacity of the battery and also shows that the battery capacity reached 89% in the no-load test. With this method, as the load current progressively increases relative to the C-rates, the voltage drop across the battery also increases. Therefore, in this graph, the load current was increased at a rate of 10 C, and the battery capacity was increased up to 88% of its full capacity.

The SOC of the lithium-ion battery was also simulated, and it was found that the battery’s charged capacity was lower than the discharged capacity. The reason for this is that both the medium to high current loading of the electrode active material and the cell voltage of the battery are within the stability window of the electrolyte. The SOC of the LMO material at various C-rates is shown in Figure 11. Although the charge capacity was lower than the discharge capacity, polarization was noticed in the discharge curve, which was on the same plateau at 3.2 V. It is evident that the LMO charge curve was stable during the oxidation and reduction processes.

The temperature profile was modeled using a heat transfer interface, where the heat sources stemmed from the battery models by using electrochemical heating Multiphysics nodes. Convection in the air-filled domain enclosing the batteries has been neglected, assuming quiescent conditions. The outer boundaries of the pack were cooled using convective cooling conditions. The battery pack was discharged from 100 to 20% state of charge at a rate of 4 C for 12 min.

Figure 12 shows the temperature distribution of the battery pack as the color code distribution represents the blue color as higher and the red color as a lower temperature. The inner part of the battery pack was about 2°C higher than the outermost part of the battery pack. The outermost cell of the battery pack had a slightly lower discharge voltage and also the ohmic drop and exchange current density were slightly lower, while the diffusion time constant for the lower temperature was slightly higher.