A singularly perturbed phase field model used to model lithium-ion batteries including chemical and elastic effects is considered. The underlying energy is given by Iε[u,c]:=∫Ω1εf(c)+ε‖∇c‖2+1εC(e(u)-ce0):(e(u)-ce0)dx,
The lithium-ion battery is a fundamental tool in modern technology and the intertwined challenge of harnessing renewable energy, with applications extending from mobile phones to hybrid cars. In recognition of this importance, the 2019 Nobel Prize in Chemistry was awarded to Goodenough, Whittingham, and Yoshino for their pioneering works in the development of lithium-ion batteries [42]. Motivated by the eminence of lithium-ion batteries, we study a mathematical model that underlies their capacity. A prominent performance limitation of lithium-ion batteries is their short life-cycle resulting from the electrochemical processes governing the battery which induce phase transitions. Elaborating on this, during the process of charging, lithium-ions intercalate into the host structure of the cathode. This intercalation is not homogeneous and undergoes phase separation, that is, lithium-ions form areas of high concentration and low concentration with sharp phase transitions between these regions. These phase transitions induce a strain on the host material which, ultimately, leads to its degradation. Damage of the cathode’s host material leads to a decrease in battery performance and limited life-cycle (see [8, 21], and references therein).
