Battery technology plays an important role in energy storage. In particular, lithium–ion (Li–ion) batteries are of great interest, because of their high capacity, long cycle life, and high energy and power density. However, for further improvements of Li–ion batteries, a deeper understanding of physical processes occurring within this type of battery, including transport, is needed. To provide a detailed description of these phenomena, a 3D representation is required for the morphology of composite materials used in Li–ion batteries. In this paper, we develop a stochastic simulation model in 3D, which is based on random marked point processes, to reconstruct real and generate virtual morphologies. For this purpose, a statistical technique to fit the model to 3D image data gained by X-ray tomography is developed. Finally, we validate the model by comparing real and simulated data using image characteristics which are especially relevant with respect to transport properties.
Batteries are a very important and already well-engineered technology. However, not all physical phenomena within batteries are well understood so far. Most models for transport processes only take global parameters into account, so that the detailed 3D morphology of the media within which the transport processes of lithium ions are occurring is not considered, see Ref. [25]. One reason for this is the fact that imaging of the 3D morphology in high resolution is a difficult task. The first 3D images of both positive and negative composite electrodes used in Li–ion batteries have been obtained very recently, see e.g. Refs. [18] and [27]. They allow a deeper insight into the interior of this type of batteries. Hence, in Ref. [18], a first descriptive analysis of the 3D structure of Li–ion batteries has been performed with respect to transport-relevant properties, where the lithium-transporting phase is interpreted as a ‘pore phase’ and the graphite as a ‘solid phase’. We will use this terminology in the present paper as well.
Here, we go one step further than simply analyzing the 3D morphology in a purely descriptive way by developing a stochastic simulation model for the pore phase based on the above mentioned 3D image data of Li–ion batteries, see Fig. 1. More precisely, the model, which we propose in the present paper, is based on tools from stochastic geometry, especially on point-process models. Hence, for image segmentation, we use an algorithmic approach described in Ref. [23] to come up with a suitable representation of the pore phase by unions of overlapping spheres. It can be seen as a realization of a random marked point process, where the centers of spheres are the points and the corresponding radii are the marks. See also Refs. [28] and [29], where unions of overlapping ellipsoids have been considered for structural analyzes of Li–ion electrode materials.
In this paper, we have developed, fitted, and validated a stochastic simulation model for the 3D microstructure of composite materials used in Li–ion batteries which is based on tools from stochastic geometry and spatial statistics. Therefore, following Ref. [18], we consider the Li-transporting phase as pores and the graphite as solid. The microstructure of the pore phase is of essential importance for the lithium-transport processes within batteries, because their efficiency and performance depend on suitable transportation pathways with optimal shapes.
For modeling and especially for fitting our model to real data, the 3D data set described and analyzed in Ref. [18] has been considered.
In a first step, we have applied a preprocessing of image data, where we have smoothed the binarized image using several morphological transformations. Note that the details of the image, removed by this smoothing, are not neglected but included into the model as one of the modeling components.
For modeling the pore phase by appropriate tools from stochastic geometry, we transform the smoothed pore phase into an off-grid representation by unions of overlapping spheres, where we proceed in a similar way as suggested in Ref. [23]. To get a suitable representation of this type, we use a stochastic segmentation technique, i.e., we describe the pore phase as a realization of a random marked point process, where the points are the centers of spheres and the marks are the corresponding radii.
The stochastic model for these unions of spheres is built in two steps. First, the centers are modeled by a 3D Matérn cluster process with elliptically shaped clusters. Then, given a realization of this point-process model, the radii are modeled as marks of these points, where the radii are generated by means of a moving-average procedure and a subsequent MCMC-simulation for rearrangements. Finally, the image details removed by morphological smoothing are integrated into the model, where a (Coxian) point process is considered, combined with several morphological operations, in order to integrate missing connections of the pore phase and to stochastically invert the morphological smoothing.
