A Novel, Efficient Approach to Simulate the 3D Resolved Spatial and Temporal Development of Space-Charge Layers in Solid Electrolytes
Authors: Sinzig Stephan , Schmidt Christoph P. , Wall Wolfgang A. 
Published: February 2024

The role of space-charge layers (SCLs) on the overall battery performance of solid-state batteries is heavily discussed in the scientific community ranging from dominant to negligible [1]. Reasons for this are that direct measurements of SCLs in experiments are difficult and so far, simulation models that can efficiently resolve the transient development of the shape of SCLs while being thermodynamically consistent are missing. The different length scales of SCLs on the one hand and of the microstructure of a battery cell on the other hand make a spatial discretization of realistic microstructures with resolved SCLs infeasible. To date, this was circumvented by using phenomenological models that do not require a spatial discretization of the SCL, by applying thermodynamically consistent models to simplified one-dimensional geometries, or by performing simulations on the sub-continuum scale, e.g. DFT simulations or kinetic Monte Carlo simulations. As all models, the mentioned models have limitations. In this case, they are for example either thermodynamically not consistent or limited to small, not representative geometries.
In this work [2], we propose a model that enables the prediction of the spatial and temporal development of SCLs within geometrically resolved microstructures while being thermodynamically consistent [3]. We exploit that effects in SCLs are predominantly one-dimensional. Furthermore, it can be shown, that SCLs develop orthogonal to the interface between the solid electrolyte and the electrode in the steady state. This allows to project the equations in the regions of the solid electrolyte where SCLs are expected in one dimension. In the remaining region of the solid electrolyte, the equations remain three-dimensional. A consistent coupling strategy between the three-dimensional and the one-dimensional region is derived and is irrevocably connected to the fulfilment of conservation properties. This modelling approach allows to drastically reduce the size of the numerical discretization required to resolve SCLs in space, as a fine discretization is only required in one direction embedded into a coarse three-dimensional discretization which represents the geometric microstructure. Thus, the required computational effort significantly reduces.
The continuous equations are discretized in space using the finite element method. The obtained nonlinear system of equations is solved with the Newton-Raphson scheme using one monolithic system, i.e. the concurrent solution of the differently discretized regions to ensure robustness and efficiency of the solver. Tailoring an iterative solver by utilizing physical information of the system (similar to [4]), enables to apply the model to realistic microstructures with more than 10 million unknowns.
With the proposed approach it is possible for the first time to quantify the influence of realistic microstructures on the transient and spatial formation process of SCLs. This work is the first to show that the formation process of SCLs at the interface between solid electrolyte and electrode is strongly influenced by the geometry of the microstructure (see Fig. 1): The dominating trend is a propagation of the SCL through the composite electrode, i.e. from the separator side to the current collector side of each electrode. Simultaneously, an inhomogeneity in the plane being orthogonal to the dominant direction is observed. This can be attributed to different conduction paths within this two-dimensional manifold due to the irregularity of the investigated microstructures and their geometric complexity. Furthermore, characteristic points in the microstructure can be identified where more charge is stored in the SCL during the transient development compared to the steady state. Again, this is attributed to the complex microstructures in interplay with the impedance of the bulk electrolyte and of the SCL. For these observations a modelling approach that resolves SCLs in realistic microstructures is mandatory.

References

[1] de Klerk N. J. J., Wagemaker M., ACS Appl. Energy Mater., 1, 5609 (2018) 10.1021/acsaem.8b00457
[2] Sinzig S., Hollweck T., Schmidt C.P., Wall W. A., J. Electrochem. Soc. (2023) 10.1149/1945-7111/acc692
[4] Braun S., Yada C., Latz A., J. Phys. Chem. C, 119, 22281 (2015), 10.1021/acs.jpcc.5b02679
[4] Fang R., Kronbichler M., Wurzer M., Wall W. A., Comput. Methods Appl. Mech. Eng., 350, 803 (2019) 10.1016/j.cma.2019.03.017

Figure 1