[Paper Review] On the well-posedness of a multiscale mathematical model for Lithium-ion batteries

Jesús Ildefonso Díaz Díaz, David Gómez‐Castro+1 more|arXiv (Cornell University)|Jan 1, 2018

Advanced Mathematical Modeling in Engineering40 references17 citations

TL;DR

This paper establishes the first rigorous proof of local and global well-posedness for the full nonlinear multiscale PDE system of the classical Newman model for lithium-ion batteries. By employing Green's operators, fixed-point theory, and sub-super solution techniques in carefully constructed function spaces, the authors prove existence and uniqueness of solutions under mild conditions, with global existence ensured under specific assumptions on nonlinearities and the open-circuit potential function.

ABSTRACT

We consider the mathematical treatment of a system of nonlinear partial differential equations based on a model, proposed in 1972 by J. Newman, in which the coupling between the Lithium concentration, the phase potentials and temperature in the electrodes and the electrolyte of a Lithium battery cell is considered. After introducing some functional spaces well-adapted to our framework, we obtain some rigorous results showing the well-posedness of the system, first for some short time and then, by considering some hypothesis on the nonlinearities, globally in time. As far as we know, this is the first result in the literature proving existence in time of the full Newman model, which follows previous results by the third author in 2016 regarding a simplified case.

Motivation & Objective

• To establish the mathematical well-posedness of the full multiscale nonlinear PDE system governing lithium-ion battery dynamics.
• To prove local and global existence of solutions for the Newman model, which had not been rigorously established before.
• To characterize the conditions under which solutions may blow up, particularly focusing on the role of the open-circuit potential function.
• To ensure physical consistency by verifying mass conservation and thermodynamic compatibility in the model.
• To provide a foundation for reliable numerical simulations and parameter identification in battery modeling.

Proposed method

• Formulation of a coupled macro-micro scale system: macroscopic PDEs for electrolyte concentration, electrode potential, and temperature, and microscopic radial diffusion in spherical particles.
• Use of Green's operators and fixed-point arguments in weighted Sobolev and Hölder spaces to handle nonlinear couplings.
• Application of Browder-Minty theory and sub-super solution techniques to establish existence and bounds on solutions.
• Construction of time-dependent sub- and supersolutions to control blow-up behavior and ensure global existence under specific assumptions.
• Introduction of a truncated potential and temperature model to derive global existence results via comparison principles.
• Use of compatibility conditions and energy estimates to verify mass conservation and physical consistency.

Experimental results

Research questions

• RQ1 Under what conditions does the full Newman model for lithium-ion batteries admit a local-in-time solution?
• RQ2 What structural assumptions on the nonlinearities, particularly the open-circuit potential, are required to ensure global-in-time existence of solutions?
• RQ3 How can sub- and supersolutions be constructed to control solution bounds and prevent blow-up?
• RQ4 In what way do the model's solutions preserve mass and physical consistency, such as total lithium content?
• RQ5 How do the assumptions on the potential function and coefficient behavior affect the long-term behavior of the system?

Key findings

• The paper proves the first local-in-time existence result for the full nonlinear multiscale system of the Newman model, using fixed-point theory in appropriate function spaces.
• Global existence of solutions is established under specific assumptions on the nonlinearities, particularly the open-circuit potential function, which rules out non-physical blow-up.
• The solution satisfies physical constraints: the total lithium concentration in the electrolyte remains constant over time, ensuring mass conservation.
• The model preserves thermodynamic consistency, with the total lithium in the solid electrodes varying only due to intercalation, and the system remains bounded under mild conditions.
• The authors construct explicit sub- and supersolutions that control the evolution of lithium concentration in the electrodes, ensuring positivity and boundedness.
• The analysis confirms that blow-up, if any, is due to the structure of the open-circuit potential, and can be excluded under reasonable physical assumptions on the coefficients.