Zitao Ni

Ph.D. in Materials Science and Engineering

Learning Plan Snapshot: Ch2 Complete → Ch3 Start

Learning Plan Snapshot: Chapter 2 Complete → Chapter 3 Start

Snapshot Date: 2026-05-25
Completed: Chapter 2 (Diffusion Equation)
Next: Chapter 3 (Butler-Volmer Kinetics)
Overall Progress: 3/8 Weeks Complete

I. Overall Learning Roadmap

Chapter Topic Status
Ch 1 Electrochemical Fundamentals & OCV ✅ Complete
Ch 2 Diffusion Equation (Fick’s Second Law) ✅ Complete
Ch 3 Butler-Volmer Kinetics ← Current
Ch 4 Full SPM Implementation & Validation ⏳
Ch 5 SPMe Model ⏳
Ch 6 P2D Model ⏳
Ch 7 Advanced Applications ⏳

II. Chapter 2 Completion Report

2.1 Knowledge Points Reviewed

# Knowledge Point Mastery
2.1 Fick’s First Law (J = −D·∂c/∂r) ✅ Understand flux driven by concentration gradient
2.2 Fick’s Second Law (mass conservation → PDE) ✅ Hand-derived both Cartesian and spherical forms
2.3 Spherical “onion model” (volume ∝ r²) ✅ Understand geometric focusing effect
2.4 Boundary conditions (center symmetry + surface Neumann) ✅ Master physical meaning of both BCs
2.5 Spatial discretization (uniform grid, dr) ✅ Can convert continuous function to array
2.6 Taylor expansion → finite differences (1st + 2nd order) ✅ Hand-derived central difference formulas
2.7 Spherical Laplacian discretization (α_i, β_i, γ_i) ✅ Hand-derived coefficient combination process
2.8 Explicit vs. Implicit Euler (CFL condition) ✅ Verified necessity of implicit with actual parameters
2.9 L’Hôpital center boundary (origin of 6) ✅ Hand-derived the complete 3×2=6 chain
2.10 Tridiagonal matrix + scipy banded storage ✅ Understand ab[0/1/2] storage mapping
2.11 Building diffusion solver from scratch ✅ Matrix difference vs model.py = 0

2.2 Core Formulas

  1. Fick’s First Law: J = -D · ∂c/∂r
  2. Fick’s Second Law (spherical): ∂c/∂t = D · [∂²c/∂r² + (2/r)·∂c/∂r]
  3. Laplacian Discretization: ∇²c_i = α_i·c_{i-1} + β_i·c_i + γ_i·c_{i+1}
  4. Center Laplacian: ∇²c|₀ = 6(c₁−c₀)/dr²
  5. Implicit Euler Matrix: (I − DΔt·L)·c_new = c_old

2.3 Code Implementation

File Function/Module Status
diffusion_solver_my.py build_diffusion_matrix() ✅ Self-written, matrix diff = 0
diffusion_solver_my.py solve_one_step() ✅ Self-written
diffusion_solver_my.py simulate_diffusion() ✅ Self-written
plot_diffusion_results.py Simulation result visualization ✅
verify_comparison.py Numerical comparison verification ✅ Diff = 0

2.4 Visualization Outputs

Image Content
diffusion_concept.png Physical concept diagram (concentration gradient + onion model + discharge distribution)
diffusion_discretization.png Discretization diagram (node grid + tridiagonal matrix + boundary conditions)
diffusion_simulation.png Simulation results (profile evolution + D sensitivity + center vs. surface)
diffusion_simulation_ref.png Reference comparison plot (verification)

III. Chapter 3 Key Focus Overview

3.1 Learning Objectives

  1. Understand fundamental concepts of electrochemical interfacial reaction kinetics
  2. Master forward and reverse forms of the Butler-Volmer equation
  3. Understand the physical meaning of exchange current density j₀ and its dependencies
  4. Master overpotential η calculation (arcsinh inverse solution at α=0.5)
  5. Understand how the BV equation maps “current → overpotential” and ultimately affects terminal voltage

3.2 Core Content

  1. Electrode Kinetics Overview: Chemical reaction rate of Li⁺ intercalation/deintercalation at particle surface
  2. Butler-Volmer Equation: j = j₀ · [exp(αFη/RT) − exp(−(1−α)Fη/RT)]
  3. Exchange Current Density j₀: j₀ = k · c_e^(1-α) · c_s^α · (c_max−c_s)^(1-α)
  4. Inverse Overpotential (α=0.5): η = 2RT/F · arcsinh(j / 2j₀)
  5. Connection to Diffusion Equation: Diffusion provides c_s_surf → BV computes η → voltage coupling outputs V_cell

3.3 Key Formulas

$$
j = j_0 \cdot \left[ \exp!\left(\frac{\alpha F \eta}{RT}\right) - \exp!\left(-\frac{(1-\alpha)F \eta}{RT}\right) \right]
$$

$$
\eta = \frac{2RT}{F} \cdot \mathrm{arcsinh}!\left(\frac{j}{2j_0}\right) \quad (\alpha = 0.5)
$$

$$
j_0 = k \cdot c_e^{1-\alpha} \cdot c_s^{\alpha} \cdot (c_{\mathrm{max}} - c_s)^{1-\alpha}
$$

3.4 Expected Outcomes

Outcome Description
Understand BV equation physical meaning Understand that exp terms represent oxidation and reduction driving forces respectively
Hand-derive arcsinh inverse Understand why α=0.5 allows analytical inverse solution
Independently implement bv_overpotential() Write BV inverse function from scratch, compare against model.py
Generate BV visualization figures η-j curves at different SOC, j₀ dependency relationships
Understand complete voltage chain Current → j → η → V_cell = OCV ± η

IV. New Conversation Entry Point

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Please read learning/learning_notes_03_kinetics/session_03_background.md to begin this session

Plan Snapshot Date: 2026-05-25
Next Chapter: Course #03 - Butler-Volmer Kinetics

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