# Evaluation of Present Accelerated Temperature Testing and Modeling of Batteries
## 3 Case Studies
### Case Study 2  
In this case, study, pouch-shaped lithium-ion batteries are selected to verify if the Arrhenius model applies. The major composition of cathode, anode and electrolyte are LiCoO\({}_{2}\), graphite, and LiPF\({}_{6}\)-salt mixed with the organic solvent, respectively. The nominal capacity and operation voltage range are 3.36 Ah and 3.0-4.4 V, respectively.  
The accelerated temperature testing is being conducted on pouch-shaped lithium-ion batteries at four different temperatures, 10 \({}^{\circ}\)C, 25 \({}^{\circ}\)C, 45 \({}^{\circ}\)C and 60 \({}^{\circ}\)C, with eight samples at each condition. The discharge C-rate and charge cut-off current are fixed at 1C and C/5, respectively. After every 50 cycles, the samples are charged and discharged using a standard profile for discharge capacity characterization. Figure 3 shows the discharge capacity loss at different temperatures.  
At 10 \({}^{\circ}\)C, 25 \({}^{\circ}\)C, and 45 \({}^{\circ}\)C, the slopes of the discharge capacity loss curves steadily decrease over cycles at first, then maintain at a relatively constant level. At 60 \({}^{\circ}\)C, the discharge capacity loss curve in the first 250 cycles shows a similar trend to that at 10 \({}^{\circ}\)C, 25 \({}^{\circ}\)C, and 45 \({}^{\circ}\)C. However, the capacity loss at 60 \({}^{\circ}\)C after the 250 th cycle increases drastically, forming an inflection point on the curve. This phenomenon suggests that the dominant failure mechanisms have possibly changed after the 250 th cycle. It is also found that the eight samples cycled at 60 \({}^{\circ}\)C present large variations after the 250 th cycle, although with similar degradation trends. Different transition times of the failure mechanisms in these samples may account for this variation.  
In most of the literatures where the Arrhenius model is incorporated in the cycle aging degradation model as an AF, the capacity loss is often expressed as in Equations (7) and (8) [12; 14; 15; 34]. To determine whether an Arrhenius model works in this case, the relationship between capacity  
Figure 3: Capacity degradation with cycle number.  
loss at different cycle numbers and temperature will be investigated. For Arrhenius behavior as expressed by Equation (2), the logarithm of \(K\) is supposed to follow a linear relationship with \(1/T\), that is, the logarithm of \(Q_{\mathrm{loss}}\) of different temperatures at the same cycle number changes linearly with \(1/T\).  
\[Q_{\mathrm{loss}}=1-NDC=Ae^{-E_{a}/RT}\times f(N) \tag{7}\]  
\[Q_{\mathrm{loss}}=1-NDC=Ae^{-E_{a}/RT}\times f(Ah) \tag{8}\]  
The logarithm of \(Q_{\mathrm{loss}}\) of each sample is then plotted versus the reciprocal of temperature (\(1/T\times 1000\)) in kelvin as in Figure 4. At both the 250 th and 400 th cycle, the average log(\(Q_{\mathrm{loss}}\)) of eight samples does not show a linear relationship with the reciprocal of kelvin temperature, illustrating that the Arrhenius model does not fit over the entire practical operation range of the battery.  
Even if an Arrhenius model is forced to fit all the data with the R-squared of 0.9512 and 0.8945, respectively, as some research groups did [10; 15; 20], the derived activation energy can change with the cycle number. The analysis above shows that the Arrhenius relationship is not universal to model the effect of temperature on lithium-ion batteries' degradation rates, since the coefficients are no longer fixed when the capacity loss at different times is selected.