Reconsidering porous electrode theory for Lithium-Ion batteries: Rigorous upscaling of localized fluctuations as a consequence of locally anisotropic microstructures

Kurzfassung

A class of numerical models known under different names (Newman model, DFN, P2D, porous electrode theory) has been widely used in the modeling of electrochemical systems with porous electrodes, in particular to predict the behaviour of lithium-ion cells. The models treat the composite materials as effective homogeneous media. Detailed phenomena on the pore and particle scale have to be treated within microstructure-resolving models which distinguish the transport and reaction processes on the scale of the individual phases. The DFN-type models are relatively inexpensive computationally and intuitive, and thus help get simulation results and their interpretations faster. The problem of their relation to the microscopic transport-reaction equations was addressed in the literature by applying formal volume averaging rules to the latter; the mathematical correctness of the volume averaging was investigated with the help of the asymptotic homogenization ansatz for partial differential equations (PDEs). Due to the lack of strict time scale separation in lithium-ion cells some phenomena can not however be mathematically rigorously homogenized. Even in the simplest basic DFN model the lithium mass transport in the active material does not satisfy the necessary homogenization criteria and is treated heuristically as transport in some "effective" spherical particle per volume element of the homogenized models. One important deviation between the DFN-based and the microstructure-resolving simulations has been found in the form of spatially localized fluctuations of the overpotential on the active material interface. In this talk we will present a mathematical analysis that demonstrates that these fluctuations are closely related to the homogenization application bottlenecks in the DFN derivation, which cannot be accounted for by the basic DFN assumptions. The analysis strongly relies on the theory of PDEs and on the asymptotic properties of their solutions. The properties of the fluctuation dynamics are derived in a semi-analytical manner, an agreement with the numerical results is demonstrated. As a next step, building on this analysis, a new reduced-order lithium-ion cell model is proposed that can be considered as a DFN modification and that can reproduce the local fluctuations at the same time. In conclusion, we will comment on the possible role of our findings in the future applications and cover the following questions: when and why the original DFN model's predictions agree well with the microscopic simulations, despite the lack of mathematical rigour, and when not; how our theory can assist in developing mathematically rigorous upscaled DFN-like models to include more phenomena (like side reaction, binder influence, mechanics, anisotropic transport), both in lithium-ion context and generally for similar transport-reaction systems.