*3.2. Detailed Radial Change and Influencing Factors*

For a better understanding of the volume change, a detailed consideration of the alteration of the radius is useful. This also allows for identifying any influencing factors. Figure 6a shows the change in radius Δ*r* over time *t* for the entire cycle of the position that expands the most, as well as current (*I*), voltage (*U*), and temperature (*T*) characteristics of the test.

The change in radius over the CC charge phase is about Δ*r* = 27 μm. It decreases somewhat as the cell enters the CV charging phase. At the start of the discharge, there is a very brief increase in radius until the decrease in cell thickness due to the mechanical processes predominates and the cell contracts again. At the end of the CC discharge phase, respectively at the beginning of the CV discharge phase, a kink in the curve is visible.

**Figure 6.** (**a**) Radial expansion Δ*r* in μm of the LG INR18650 M29, the corresponding current *I* in A and voltage *U* in V, and the temperature *T* in ◦C over the time *t* in h for charge and discharge cycle for position 1 (α = 10◦ from the current collector tab). The area highlighted in yellow marks the end of the CC phase or the beginning of the CV phase at which the temperature inside the cell continues to increase despite the falling current, resulting in a change of the cell thickness; (**b**) radial expansion Δ*r* in μm of position 1–7 (α {10, 20, ..., 70}◦) from the current collector tab; (**c**) comparison of the radial expansion of position 8 and 32 with negative expansion over the charge and discharge cycle.

If only the voltage was decisive for the expansion of the battery cell, the change in the radius would correspond to the voltage curve. Looking at the voltage curve and the expansion of the cell in Figure 6a, this means that other effects besides lithiation have an influence on the expansion. Particularly noticeable is the sharp peak between the CC and CV charging phases, which is marked in yellow here. Although the applied current drops and the voltage remains constant, the expansion continues to rise. Looking at the temperature curve, the peak of the expansion correlates with the temperature increase of the cell. This is due to the ideal gas law, according to which the gas pressure increases with increasing temperature while the volume remains constant. The increase in gas pressure also causes the battery cell to expand further, although no further electrode growth takes place. By decreasing temperature and thus also gas pressure, the expansion also decreases to an almost constant level until the end of the CV discharge phase.

At the beginning of the discharge phase, the expansion initially decreases analogously to the voltage due to the delithiation of the graphite, which leads to a contraction of the electrode. Shortly after the start of the CC discharge phase, the radius increases again despite progressive delithiation. The reason is presumably the increasing temperature, which causes the internal gas pressure to rise again and superimposes the reduction in the radius, thus leading to expansion again until the effect of delithiation predominates and the radius of the battery cell decreases again. Due to the temperature effect, the slope of the radius change is smaller during discharge than during charge.

Figure 6b,c show that the above effects also apply to lower expansions and even to locations where contraction of the battery cell occurs. At points where the cell contracts, an additional contraction occurs as the temperature rises. This is due to the strong expansion at other points (see positions 1–7 in Figure 6b), which causes the spaces between the bulges to contract due to the mechanical rigidity of the housing.

As already indicated in Figure 5, the expansion exhibits a hysteresis. Figure 7a shows the voltage for charging *U*ch and discharging *U*dch as well as the corresponding change in radius Δ*r*ch and Δ*r*dch plotted versus SoC.

As expected, the voltage exhibits hysteresis due to structural changes in graphite and non-uniform distribution of lithium ions [24,25]. The expansion also shows this hysteresis, with the curves of the radius change for charge and discharge intersecting at SoC = 0.08.

**Figure 7.** (**a**) Change in radius Δ*r* in μm for charge and discharge and the associated voltage *U* in V versus State of Charge SoC showing the hysteresis between charging and discharging; (**b**) temperature curve *T* in ◦C during charging and discharging versus the state of charge SoC.

This hysteresis shows that the lithiation and delithiation of the electrodes results in an asymmetric volume change.

The overlap of the curves of the radius progression can be explained with regard to the temperature. The temperature increases with decreasing SoC due to the high applied current at a discharge rate of 1C to a temperature of *T* > 33 ◦C at the corresponding SoC. This causes the internal gas pressure to increase, which results in the radius not decreasing as fast as it increased during charging. This causes the two curves to meet. This is also the reason why the hysteresis becomes smaller for SoCs close to 1.
