Quantifying microstructural dynamics and electrochemical activity of graphite and silicon-graphite lithium ion battery anodes

Despite numerous studies presenting advances in tomographic imaging and analysis of lithium ion batteries, graphite-based anodes have received little attention. Weak X-ray attenuation of graphite and, as a result, poor contrast between graphite and the other carbon-based components in an electrode pore space renders data analysis challenging. Here we demonstrate operando tomography of weakly attenuating electrodes during electrochemical (de)lithiation. We use propagation-based phase contrast tomography to facilitate the differentiation between weakly attenuating materials and apply digital volume correlation to capture the dynamics of the electrodes during operation. After validating that we can quantify the local electrochemical activity and microstructural changes throughout graphite electrodes, we apply our technique to graphite-silicon composite electrodes. We show that microstructural changes that occur during (de)lithiation of a pure graphite electrode are of the same order of magnitude as spatial inhomogeneities within it, while strain in composite electrodes is locally pronounced and introduces significant microstructural changes.


Supplementary Note 1: Volumetric strain and shear strain in the graphite -graphite battery
In infinitesimal strain theory, strain is described by a linearized strain tensor, which can be derived from a given displacement field 6 : . Here, , i=1,2,3 are the components of the displacement field and , j=1,2,3 are the three space dimensions. may be separated into a symmetric part ( ) and an antisymmetric part ( ), such that . While the eigenvalues of the antisymmetric part are imaginary and describe rigid body rotations around the eigenvector directions, the eigenvalues of the symmetric part are real and represent expansions or contractions along the eigendirections. In general, any strain can be expressed as a superposition of a pure volumetric strain component (i.e. the shape of the deformed object remains unchanged) and a pure shear strain component (i.e. the volume of the deformed object remains unchanged). More generally, in three dimensions, volumetric strain can be described by the divergence of the displacement field or equivalently the trace of the symmetric strain tensor, which is the sum of its eigenvalues: In analogy, the shear strains along the different dimensions correspond to the respective differences of the eigenvalues, i.e.: For our computations, linearized strain theory is applicable because | | holds for the computed displacement fields for any time and position. For more information, we refer the reader to 6 .
Three videos showing the volumetric strain in all 29 time steps are available.
To complement the analysis of volumetric strain presented in the main text, we also investigate the distribution of maximal shear in the electrodes, which we define as: Shear strain is expected to occur in the electrodes because (i) the expansion of graphite upon intercalation of lithium is anisotropic and (ii) the cell housing limits further expansion only along the IP directions. Because of (i), shear strain will be correlated with electrochemical activity and volumetric strain.
To achieve time resolution, we compute the shear for the displacement field correlating subsequent scans (i.e., the initial 0 % SOC scan with the 13 % SOC, the 13 % SOC scan with the 30 % SOC scan, and so on).
Supplementary Figure 1c depicts this time series of shear strains for slices in the top and bottom electrodes and a vertical cut through both electrodes and the separator respectively. In agreement with the observations from Figure 3a in the main text, shear strain in the TP direction progresses through the bottom electrode along with the moving lithiation front, while shear strain is more uniform along the TP direction in the top electrode, indicating that delithiation is occurring more homogeneous here. Moreover, the shear strain is more homogenous along the IP directions of the lithiating bottom electrode, compared to the delithiating top electrode, where a large crack-like feature can be observed. This suggests that the analysis of local strains might enable the identification of sub-domains that become mechanically and electrochemically disconnected from the electrode. This would result in capacity fade.

Supplementary Note 2: Binarization of the bottom graphite electrode
The dynamically cropped volumes are binarized into a particle phase and an electrolyte/binder/carbon black background phase using Huang's automatic thresholding method 7 . In each time step, the threshold that partitions the volumetric grey scale image is calculated by minimizing the Shannon entropy function. Supplementary Figure 2a  In general, the thresholding method may significantly influence the resulting data binarization and all microstructure analysis that is based upon it. In order to test the sensitivity of our binarization on the type of thresholding algorithm applied, we compare the particle volume change as a function of flowed charge for two different thresholding methods: Huang's (used throughout the main paper) and Otsu's 8 .
As evident from Supplementary Figure 3, the results differ insignificantly, demonstrating that our analysis is robust.

Supplementary Note 3: Scale space analysis of the graphitegraphite battery
To determine a representative volume on which the microstructural parameters should be calculated, we performed a scale space analysis. Supplementary Figure 4 shows the porosity (Supplementary Figure 4a) and the through-plane tortuosity (Supplementary Figure  4b) as a function of sub-volume size. For this analysis, a large volume was cropped out from the binarized microstructure of the pristine bottom electrode. and were then computed on non-overlapping regularly allocated sub-volumes of sizes with a=1…10 within this volume. Each point in the Supplementary Figure 4a-b represents the results obtained from one sub-volume with a size as indicated on the x-axis. As the subvolume size is increased, the distributions in the microstructure converge towards a porosity and a tortuosity .
As indicated by the blue arrow, subvolumes with an edge length equal or greater than approximately 3 times the long axis of an average particle (i.e. ) are representative of the full electrode microstructure and reveal similar spatial variations in the microstructural parameters. This indicates that there is some small microstructural inhomogeneity at length scales larger than 200 .
For the tortuosity calculations in the main text, a volume of is considered. We note that the spatial variations in the tortuosities (< 10 %) are on the same order of magnitude as the temporal variations in tortuosities that occur during lithaition. The same holds for the porosities. This underscores the necessity to consider dynamic observation windows.

Supplementary Note 4: Chord length analysis of the bottom graphite electrode
To show that different types of microstructural analysis can be applied to our imaged graphite electrode, we consider another approach that is well-known in statistical image analysis: the so-called chord length distribution 9,10 . The concept is sketched in the left part of Supplementary Figure 5a. If one intersects a random line (green) with a given direction (here, either IP or TP direction) with the foreground in the image (red), so-called chords (blue) originate. The distribution of the length of these chords is called chord length distribution. For more detailed information, we refer the reader to 11  Between 48 % SOC and 91 % SOC the chord lengths continue to increase on average, but this effect is counter-balanced by two other effects: (i) the expansion of the particles in the IP directions gives rise to new short chords at the particle boundaries and (ii) the strong expansion of the electrode in TP direction during the second half of the lithiation gives rise to a separation of chords that have been previously connected. For this reason, the mean chord length stagnates in this regime.
Finally, Supplementary Figure 5b shows the chord length distributions along the different dimensions in the pristine electrode. Chords along the TP direction are on average much shorter than their IP counterparts, demonstrating that the chord length distribution is an excellent tool to characterize the anisotropic shape of our graphite particles.

Supplementary Note 5: Volume expansion of the SiC electrode based on its thickness
If the electrode-current collector interface is assumed to remain at the same position at all times and expansions along the IP directions are neglected, the relative volumetric change of the electrode can be described as its relative thickness change along the TP direction (i.e. the average movement of the electrodeseparator interface along the TP direction).
To quantitatively find the position of the electrode -separator interface in each time step, we computed the 2D IP spatial grey value average as a function of the TP direction. Because the separator always appears brighter than the electrode, this results in a step-like function (see blue curve in Supplementary Figure 6).
We perform a least-squares fit to the data using a function of type: , where , , and are fit parameters (see red curve in Supplementary Figure 6). denotes the TP position corresponding to the average of the separator grey value level and the electrode grey value level (i.e. ). We consider as a measure of the electrode-separator interface position and hence the electrode thickness in each time step (see black bar in Supplementary Figure 6).

Supplementary Note 6: Volume change of silicon and graphite as a function of the state of charge
Graphite has been shown to exhibit an "S-shaped" expansion characteristic (Supplementary Figure 7a) while silicon expands linearly as a function of the lithium content (Supplementary Figure 7b) 12 . We note that the curves in Figures 4b and 5d in the main text showing the volume expansion of the graphite and SiC electrodes, respectively, qualitatively resemble the volume expansion of silicon and graphite as a function of the degree of lithiation. The observed SOC gradients within the electrodes and the particles explain why small features are blurred out in the data presented in Figures 4b and 5d.

Supplementary Note 7: Volumetric fractions of silicon, graphite and pore space in the SiC electrode
The volumetric fractions of (lithiated) silicon, (lithiated) graphite and pore space in the SiC electrode can be estimated based on the mass of the electrode, its chemical composition, its geometric surface area and its thickness. Variables are defined in Supplementary Table 1.
For the pristine electrode, we find: .

Supplementary Note 8: Trinarization procedure for the pristine SiC electrode
We aim to trinarize the tomographic data of the SiC electrode to identify the three phases: silicon, graphite, and pore phase.
Supplementary Figure 8a shows a cutout from a slice through the electrode along the IP directions. The histogram of gray level values (see Supplementary Figure 8c black curve) does not show two clear local minima, indicating that given the noise level, the contrast between the three phases is insufficient to select a set of thresholds.
Therefore, to trinarize the tomographic data, we apply a series of image processing steps using the software MATLAB. . Finally, we correct for the rings around the silicon particles that are inevitable because the silicon-pore interface domains are not infinitely sharp and are thus falsely identified as graphite. We do so by using a set of morphological operations in MATLAB (dilation of the silicon domain using a ball structuring element of radius 3 voxels, closing of the silicon domain using a ball structuring element of radius 2 voxels, dilation of the graphite domain using a ball structuring element of radius 1 voxel) to arrive at the final trinarization shown in Supplementary Figure 8e.
The trinarization (see Figure 5e in the main text) tends to underestimate the silicon fraction (8.8 % instead of the actual 11.7 %) because small silicon particles are either below the resolution limit or are falsely associated with the graphite domain. However, the trinarization captures the major features of large particles well.
A trinarization of the data for (partially) lithiated states is not possible due to insufficient contrast: as seen in Figure 5c in the main text, the X-ray attenuation of fully lithiated silicon particles is comparable to the attenuation of the background.

Supplementary Note 9: Estimate of spatial resolution
Tomographic scans of the graphite-graphite sample have been performed with an effective voxel size of , while scans of the sample with the silicon-graphite electrode were recorded with a voxel size of .
We estimate the effective spatial resolution, i.e. the ability to distinguish two objects that are close to each other from one another, by plotting grey value profiles through supposedly sharp edges in the images (particleelectrolyte interface) and analyzing their width (see Supplementary Figure 9a-b). Depending on the definition of resolution used, we find around spatial resolution for voxel size and spatial resolution for the voxel size.