## Description

The behavior of battery cells is intrinsically multiscale, as the multiphysics phenomena involving diffusion, migration, intercalation, and the accompanying mechanical effects take place at the characteristic length scale of the electrode compound, which are three orders of magnitudes smaller than the battery cell size. Directly resolving all scales and modeling all particles in the electrodes is computationally unfeasible. Instead, the nano-scale effects are incorporated into the microscale problem through constitutive models that are derived from advanced homogenization methods. This contribution deals with a computational homogenization approach. Originating from the fundamental balance laws (of mass, force, charge, electrolysis) at both scales, the multiscale analysis roots itself on an energy-based weak formulation, which allows to extend the Hill–Mandel energy averaging theorem to the problem at hand. In the batteries modeling literature, it is generally assumed that the electromagnetic fields and their interactions are static. This assumption implies vanishing interference effects between the electric and magnetic phenomena. As a consequence, the set of Maxwell’s equations are replaced by their electrostatic counterparts, as for the steady current case. Here the electromagnetics is explicitly taken in to account via the electroquasi-static formulation. Capacitive but inductive effects are included, making the electroquasi-statics different from the static approach, because Maxwell’s correction is preserved within Ampere’s law, i.e., the effect of the magnetizing field is still taken into account. The conditions at which the quasi-static solution to Maxwell’s equations becomes exact have been verified for battery cells. Electrolyte in batteries is a typical system involving n different species. The set of n mass balance equations contains n + 1 unknowns, i.e., n mass concentrations plus the electric potential. An additional relation is sufficient to solve the set of equations and the most common selection in battery modeling is the electroneutrality condition. In several studies, the electroneutrality condition is thus used instead of Gauss’s law for the electric field. Here on the contrary, electroneutrality has been taken into account as an assumption within the balance equations. For some of the latter, typically the force balance, the impact of electroneutrality is major because the effects of the Lorentz interactions bulk forces are minuscule (although they do not vanish) with respect to the mechanical effects due to the constrained swelling. On the contrary, electroneutrality has no influence on Maxwell’s law because the bulk terms cannot be disregarded, at least according to the current numerical and experimental evidences. The methodology above described leads to formulate balance equations and boundary conditions governing the displacement, electric, chemical, and electrochemical potentials, derived macroscopically for the whole battery cell as well as for the RVE. Time dependent scale transitions are formulated, as required by the length/time scales involved in Li-ion batteries processes. Scale separation in time is investigated, leading to a concurrent time modeling between the macro- and microscales. Computational procedures and simulations are finally presented assuming suitable constitutive prescriptions.

## Recommended Citation

Salvadori, A.,
Grazioli, D.,
&
Geers, M.
(2014).
Multiscale and Multiphysics modeling of Li-ion battery cells.
In A. Bajaj, P. Zavattieri, M. Koslowski, & T. Siegmund (Eds.).
*
Proceedings of the Society of Engineering Science 51st Annual Technical Meeting, October 1-3, 2014
*,
West Lafayette: Purdue University Libraries Scholarly Publishing Services, 2014.
https://docs.lib.purdue.edu/ses2014/mms/esm/22

Multiscale and Multiphysics modeling of Li-ion battery cells

The behavior of battery cells is intrinsically multiscale, as the multiphysics phenomena involving diffusion, migration, intercalation, and the accompanying mechanical effects take place at the characteristic length scale of the electrode compound, which are three orders of magnitudes smaller than the battery cell size. Directly resolving all scales and modeling all particles in the electrodes is computationally unfeasible. Instead, the nano-scale effects are incorporated into the microscale problem through constitutive models that are derived from advanced homogenization methods. This contribution deals with a computational homogenization approach. Originating from the fundamental balance laws (of mass, force, charge, electrolysis) at both scales, the multiscale analysis roots itself on an energy-based weak formulation, which allows to extend the Hill–Mandel energy averaging theorem to the problem at hand. In the batteries modeling literature, it is generally assumed that the electromagnetic fields and their interactions are static. This assumption implies vanishing interference effects between the electric and magnetic phenomena. As a consequence, the set of Maxwell’s equations are replaced by their electrostatic counterparts, as for the steady current case. Here the electromagnetics is explicitly taken in to account via the electroquasi-static formulation. Capacitive but inductive effects are included, making the electroquasi-statics different from the static approach, because Maxwell’s correction is preserved within Ampere’s law, i.e., the effect of the magnetizing field is still taken into account. The conditions at which the quasi-static solution to Maxwell’s equations becomes exact have been verified for battery cells. Electrolyte in batteries is a typical system involving n different species. The set of n mass balance equations contains n + 1 unknowns, i.e., n mass concentrations plus the electric potential. An additional relation is sufficient to solve the set of equations and the most common selection in battery modeling is the electroneutrality condition. In several studies, the electroneutrality condition is thus used instead of Gauss’s law for the electric field. Here on the contrary, electroneutrality has been taken into account as an assumption within the balance equations. For some of the latter, typically the force balance, the impact of electroneutrality is major because the effects of the Lorentz interactions bulk forces are minuscule (although they do not vanish) with respect to the mechanical effects due to the constrained swelling. On the contrary, electroneutrality has no influence on Maxwell’s law because the bulk terms cannot be disregarded, at least according to the current numerical and experimental evidences. The methodology above described leads to formulate balance equations and boundary conditions governing the displacement, electric, chemical, and electrochemical potentials, derived macroscopically for the whole battery cell as well as for the RVE. Time dependent scale transitions are formulated, as required by the length/time scales involved in Li-ion batteries processes. Scale separation in time is investigated, leading to a concurrent time modeling between the macro- and microscales. Computational procedures and simulations are finally presented assuming suitable constitutive prescriptions.