*2.1. DRT of Frequency Domain Data*

Since the DRT analysis of frequency domain data is well-established and the focus of this study is the DRT of time domain data the method is only briefly described. We refer to [15] for a more detailed explanation of the procedure used.

**Figure 1.** (**a**) Impedance spectra of pristine cell at 40 % SOC. (**b**) Corresponding DRT.

In order to calculate the DRT using frequency domain data, the resistive–capacitive part of the given spectra is reconstructed with a series of *m* RC-elements, a parallel circuit of an ohmic resistor and a capacitor.

$$Z\_{\rm RC,sing\,lc} = \frac{R}{1 + j\omega RC} = \frac{R}{1 + j\omega \tau} \tag{1}$$

$$Z\_{RC,series} = \sum\_{k=1}^{m} \frac{h(\tau\_k)}{1 + j\omega \tau\_k} \tag{2}$$

where *h*(*τk*) are the RC-elements unknown polarization contributions. By pre-defining the time constants *τ* = *R* · *C*, the equation of the RC elements and thus the optimization problem become linear. In view of the nature of electrochemical processes, an equidistant distribution on a logarithmic scale is assumed for the time constants. The range of time constants is expanded by two decades and the number of RC-elements per decade is increased by a factor of three compared to the measured frequencies in order to enhance the accuracy and to obtain an adequate resolution of the DRT (see e.g., [15]).

Since the number of RC-elements per decade is higher than the resolution of the impedance data, the optimization problem becomes ill-posed. Therefore, a regularization technique which was introduced by Tikhonov [28] is used to enable an analytical solution. The Tikhonov regularization for determining the DRT has already been shown in some publications [11,13–15]. The regularization parameter was elected according to [15]. The value is kept constant for the evaluation of all spectra, since the results of the DRT analysis are sensitive to the parameter. Thus, the DRTs of the different spectra are comparable and changes in the dispersion due to the aging process which can be tracked.

The parts of the measured spectra with positive imaginary values cannot be modelled by a series of RC-elements and have to be discarded. Additionally, the internal resistance *Ri* is subtracted from impedance values of the measured spectra prior to the fitting procedure. It is assumed that *Ri* is equal to the zero crossing of the imaginary axis [9]. After

preprocessing of the measurement data, the spectra mainly consist of resistive–capacitive contributions, which can be modelled by RC-elements.

The optimization problem is solved with a non-negative least squares fit because only positive resistances are physically meaningful. Figure 1b depicts the DRT of the spectrum shown in Figure 1a. The position of the peaks indicate the time constants of the processes involved and the area under the peaks correlates to the polarization contribution of the process [14–16]. In contrast to the Nyquist diagram, which shows only a single semicircle at moderate frequencies, various processes with different polarization contributions and time constants are visible in the same frequency range.
