[Paper Review] Some comments on the Butler-Volmer equation for modeling Lithium-ion batteries

Ángel Ramos, Please, C. P.|arXiv (Cornell University)|Mar 19, 2015

Advanced Battery Technologies Research10 references4 citations

TL;DR

This paper critiques common misapplications of the Butler-Volmer equation in macro-homogeneous models of lithium-ion batteries, showing that improper functional forms can lead to unphysical behavior near electrode saturation or electrolyte depletion. It proposes physically consistent formulations for the exchange current density and open-circuit voltage that ensure charge transfer remains valid across all concentration extremes, preserving model integrity during full charge/discharge cycles.

ABSTRACT

In this article the Butler-Volmer equation used in describing Lithium-ion (Li-ion) batteries is discussed. First, a complete mathematical model based on a macro-homogeneous approach developed by Neuman is presented. Two common mistakes found in the literature regarding a sign in a boundary conditions and the use of the transfer coefficient are mentioned. The paper focuses on the form of the Butler-Volmer equation in the model. It is shown how practical problems can be avoided by taking care in the form used, particularly to avoid difficulties when the solid particle in the electrodes approaches a fully charged or discharged state or the electrolyte gets depleted. This shows that the open circuit voltage and the exchange current density must depend on the lithium concentration in both the solid and the electrolyte in a particular way at the extremes of the concentration ranges.

Motivation & Objective

• To identify and correct two widespread errors in the literature involving boundary conditions and the transfer coefficient in Neumann-type macro-homogeneous battery models.
• To ensure the Butler-Volmer equation remains physically valid when solid-phase lithium concentration approaches its maximum or minimum values.
• To guarantee the model allows for reversible charging and discharging even when the electrolyte is depleted or electrodes are fully charged/discharged.
• To establish functional forms for exchange current density and open-circuit voltage that prevent unphysical concentration states in numerical simulations.

Proposed method

• Derives a macro-homogeneous mathematical model based on Neumann's framework, incorporating conservation of mass and charge in solid and electrolyte phases.
• Analyzes the Butler-Volmer equation in the context of lithium intercalation, emphasizing the role of exchange current density and overpotential.
• Introduces constraints on the functional dependence of exchange current density and open-circuit voltage on solid and electrolyte concentrations.
• Proposes specific functional forms for exchange current density that vanish at concentration extremes, ensuring physical consistency.
• Applies asymptotic analysis to show behavior near saturation limits, demonstrating that flux must reverse direction to avoid unphysical states.
• Validates the proposed form using a simplified model with piecewise-defined rate constants and derives explicit expressions for exchange current and open-circuit voltage.

Experimental results

Research questions

• RQ1 How do common misapplications of the Butler-Volmer equation lead to unphysical behavior in lithium-ion battery models?
• RQ2 What functional form of the exchange current density ensures the model remains valid when solid-phase lithium concentration approaches its maximum or minimum?
• RQ3 How should the open-circuit voltage be defined to prevent unphysical concentration states during full charge or discharge?
• RQ4 Why is the standard assumption of constant exchange current density invalid in extreme concentration regimes?
• RQ5 What conditions must the Butler-Volmer equation satisfy to allow reversible charging from a fully depleted state?

Key findings

• A common error in the literature involves an incorrect boundary condition for the electrolyte potential, which can lead to unphysical solutions even if numerical solvers appear to converge.
• The transfer coefficient is often misapplied in the charge conservation equation, leading to inconsistent electrochemical potential gradients.
• The exchange current density must vanish as the solid-phase lithium concentration approaches its maximum or minimum value, ensuring no unphysical flux occurs.
• The open-circuit voltage must diverge to negative infinity as the solid phase approaches full lithiation, preventing further intercalation beyond physical limits.
• A physically consistent form of the Butler-Volmer equation requires the exchange current density to scale with (c_s,max - c_s)^α and c_s^β, where α and β depend on charge transfer symmetry.
• The proposed functional form ensures that even at extreme concentrations, the system can still charge or discharge by allowing flux to reverse direction appropriately.