*3.1. Validation*

To validate the ionic conductivity results coming from our calculations we compared them with the characteristic time values for transport which are introduced by Jiang and Peng [27]. They have defined three parameters, *ts*, *ti*, and *tl* as:

$$t\_{s(a,c)} = \frac{\left(r\_{0(a,c)}/3\right)^2}{D\_{s(a,c)}}\tag{12}$$

$$t\_{i(a,c)} = \frac{F \epsilon c\_l}{(1 - t\_+^0)|j^{Li}|} \tag{13}$$

$$\mathbf{t}\_{l} = \frac{\mathbf{L}\_{a}^{2}}{\mathbf{D}\_{l,a}^{eff}} + \frac{\mathbf{L}\_{s}^{2}}{\mathbf{D}\_{l,s}^{eff}} + \frac{\mathbf{L}\_{c}^{2}}{\mathbf{D}\_{l,c}^{eff}}.\tag{14}$$

*ts* is describing a characteristic time of the Li diffusion process into solid particles in negative and positive electrodes. *ti* stands for the transport time relating to the local depletion rate of Li ions in electrolyte at the electrode/electrolyte interface, and *tl* is the characteristic time for Li ion transport through the electrolyte. Considering these definitions we can relate *<sup>σ</sup>l*(*<sup>a</sup>*,*c*,*<sup>s</sup>*), *<sup>σ</sup>s*(*<sup>a</sup>*,*<sup>c</sup>*), and *<sup>σ</sup>i*(*<sup>a</sup>*,*<sup>c</sup>*) to *tl*(*<sup>a</sup>*,*c*,*<sup>s</sup>*), *ts*(*<sup>a</sup>*,*<sup>c</sup>*) and *ti*(*<sup>a</sup>*,*<sup>c</sup>*) respectively.

Using the cell parameters reported in the Jiang's article for simulation, we gain the following results for transport times and ionic conductivity calculations in idle state prior to discharge as listed in Table 2. There are slight differences between anode transfer time coming from our calculations and the one reported in Jiang's work. The reason might be (1) due to the differences in parameters assumptions as not all the values are mentioned in the article and (2) in contrast to Jiang's model we included Li-plating and SEI formation (anode aging mechanisms).

**Table 2.** Values of the characteristic times and corresponding ionic conductivities when the battery is in the pause state prior to discharge. Lit. values are extracted from Jiang's article [27].


Comparing the ionic conductivities with their corresponding transport times, we realized that *t* and *σ* values of similar regions in positive and negative electrodes are following the same trend. This compatibility of results suggests that it is valid to compare *<sup>σ</sup>s*,*<sup>a</sup>* to *<sup>σ</sup>s*,*<sup>c</sup>* and additionally *<sup>σ</sup>i*,*<sup>a</sup>* to *<sup>σ</sup>i*,*c*. As shown in Table 2, there is only one value reported for the transport time of electrolyte. Therefore it is not possible to check the trend of our discrete values of electrolyte ionic conductivity in different mediums with the transport time.

#### *3.2. Simulation Results*

The discharge capacity behavior of the simulated Li-ion cell with the mentioned parameters in Table 1 over the cycle number is shown in Figure 3. The relative discharge capacity is defined as the relation of current *Qdis* to the first cycle discharge capacity. During the initial cycles, the discharge capacity decreased faster than the following cycles, which was when the SEI layer initially formed. The almost linear decrease continued until cycle number 230, where the *Qdis* reached 78% of the initial capacity. Then the phase of nonlinear decrease in discharge capacity starts so that in total cycle numbers of 400, the cell lost more than 60% of its initial capacity.

**Figure 3.** Relative discharge capacity of the simulated cell over the life time. A linear ageing phase following by a non-linear aging phase are observable.

Looking at the equivalent thickness of the lithium plating layer on the surface of anode particles in Figure 4A, we can see that from the 116th cycle Li-P started at the separator side of the negative electrode and the layer thickness increased by continuing the cycling of the cell. During the whole 400 simulated cycles, no Li-plating occurred at the current collector side of the anode. The total surface layer, Li-P, and SEI, together with equivalent thickness of each layer is shown in Figure 4B. From cycle 116 until 230, Li-plating showed a more moderate increase rate in comparison to cycles after 230 which is when the cell began nonlinear aging behavior. In contrast to Li-P, the SEI layer grew with a high rate during the first 50 cycles and after that increased more moderately. The decrease in the SEI layer's growth rate is due to the limited electrolyte diffusion through the formed layer as well as the lower EC concentration in the electrolyte as it became consumed through the SEI formation reaction. In contrast to lithium plating, which depends on the location along the anode thickness, SEI layer growth was uniform across the anode. The SEI and Li-P at the separator, grew to around *dSEI* ≈ 800 nm and *dLi*−*<sup>P</sup>* ≈ 120 nm. Kindermann et al. [38] simulated the SEI layer with *dSEI* ≈ 600 nm. Separately Petzl et al. [39] in their experimental low-temperature study measured a *dLi*−*<sup>P</sup>* ≈ 5 μm after 120 cycles at −22 ◦C with 1C.

**Figure 4.** (**A**) Thickness of the plated lithium layer on the surface of negative electrode particles at the current collector (X = 0), 0.9 of the relative anode thickness (X = 0.9), and the separator side (X=1). Over the cycle numbers, no lithium plating happened at the current collector side but it increased moving toward the separator side. (**B**) Total surface layer thickness as well as equivalent thicknesses of plated lithium and SEI (solid electrolyte interface) layers separately. Since the cell has the maximum amount lithium plating at the separator side, only the layers at X = 1 are displayed.
