Chapter 2: Diffusion Equation (Fick’s Second Law)
Understanding and implementing the diffusion equation in spherical coordinates for SPM battery simulation.
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 2 → Chapter 3 progress snapshot & preview |
Three-Session Structure
| Session | Topic | Core Activity | Output |
|---|---|---|---|
| 2.1 | Physical Intuition | Derive Fick’s laws, build “onion model” intuition | diffusion_concept.png |
| 2.2 | Numerical Methods | Hand-derive finite differences, Laplacian discretization, L’Hôpital rule | diffusion_discretization.png |
| 2.3 | Code Implementation | Build diffusion solver from scratch, verify against reference | diffusion_simulation.png |
Key Concepts Covered
- Fick’s First Law: $J = -D \cdot \partial c / \partial r$
- Fick’s Second Law (spherical): $\partial c / \partial t = D \cdot [\partial^2 c / \partial r^2 + (2/r) \cdot \partial c / \partial r]$
- Finite Difference Discretization: $\nabla^2 c_i = \alpha_i c_{i-1} + \beta_i c_i + \gamma_i c_{i+1}$
- Implicit Euler: $(I - D\Delta t \cdot L) \cdot c^{new} = c^{old}$
- Center Boundary: $\nabla^2 c|_0 = 6(c_1 - c_0)/dr^2$
- Tridiagonal Matrix & Banded Storage
Next Chapter
→ Chapter 3: Butler-Volmer Kinetics