Zitao Ni

Ph.D. in Materials Science and Engineering

Study Notes · Part 3: Butler-Volmer Kinetics

Chapter 3: Butler-Volmer Kinetics

Understanding and implementing the Butler-Volmer equation for SPM battery simulation — the electrochemical reaction kinetics that govern lithium intercalation/deintercalation at particle surfaces.

Learning Materials

Resource Description
Main Learning Notes Complete Q&A-style study notes covering all 3 sessions
Session Background Course background & learning plan for starting a new conversation
Learning Plan Snapshot Chapter 3 → Chapter 4 progress snapshot & preview

Three-Session Structure

Session Topic Core Activity Output
3.1 Physical Intuition Q&A on overpotential concept, dynamic equilibrium, j₀ dependence linear_approximation.png
3.2 Mathematical Derivation Hand-derive arcsinh inversion, linear approximation, Newton iteration newton_bv_iteration.png
3.3 Code Implementation Build BV function from scratch, visualize, verify bv_overpotential.png

Key Concepts Covered

  • Overpotential ($\eta$): Extra voltage needed to drive charge-transfer reactions across the interface energy barrier
  • Butler-Volmer Forward: $j = j_0 \cdot [e^{\alpha F\eta/RT} - e^{-(1-\alpha)F\eta/RT}]$
  • Exchange Current Density: $j_0 = k \cdot c_e^{1-\alpha} \cdot c_s^{\alpha} \cdot (c_{\max} - c_s)^{1-\alpha}$
  • arcsinh Inversion ($\alpha$=0.5): $\eta = \frac{2RT}{F} \cdot \mathrm{arcsinh}(j / 2j_0)$
  • Small-$\eta$ Linear Approximation: $\eta \approx \frac{RT}{F} \cdot \frac{j}{j_0}$ (valid for $|\eta| \ll RT/F$)
  • Newton Iteration: Numerical solution for $\alpha \neq 0.5$ with quadratic convergence
Figure Description
linear_approximation.png sinh(x) vs x comparison, j-η linear vs exact, error growth with η
newton_bv_iteration.png Newton iteration trajectory, correction decay, α asymmetry effect
bv_overpotential.png η→j forward curves, j→η inverse curves, j₀ vs SOC “∩” shape

Next Chapter

→ Chapter 4: Full SPM Implementation & Voltage Coupling

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