Zitao Ni

Ph.D. in Materials Science and Engineering

Learning Plan — P2D Model

P2D Model Learning Plan

1. Learning Objectives

Overall Goal: Master lithium-ion battery electrochemical simulation from scratch, and independently develop P2D models for battery performance analysis.

Core Competencies:

  • Understand fundamental electrochemical principles and battery operating mechanisms
  • Master the mathematical foundations of SPM, SPMe, and P2D models
  • Implement complete battery simulations in Python
  • Analyze simulation results and interpret battery behavior

2. Learning Pathway (8 Weeks Total)

Week 1: Electrochemical Fundamentals & SPM Model Concepts

# Topic Objective Status
1.1 Electrochemical Simulation Overview Understand the digital twin concept ✅ Complete
1.2 Li-ion Battery Operating Principles Understand deintercalation–migration–intercalation ✅ Complete
1.3 SPM Core Assumptions Understand single-particle simplification ✅ Complete
1.4 Parameter Naming Conventions Master self.{quantity}_{phase}_{position} ✅ Complete
1.5 SOC & N/P Ratio Understand the relationship between cell-level and electrode-level SOC ✅ Complete

Week 2: OCV Curves & Thermodynamic Foundations

# Topic Objective Status
2.1 Nernst Equation Understand thermodynamic equilibrium potential ✅ Complete
2.2 Ideal vs. Real OCV Understand the influence of material characteristics on OCV ✅ Complete
2.3 Graphite Anode OCV Master the tanh empirical fitting method ✅ Complete
2.4 NMC Cathode OCV Master polynomial fitting method ✅ Complete
2.5 OCV Visualization Learn to plot curves with matplotlib ✅ Complete

Week 3: Diffusion Equation (Fick’s Second Law)

# Topic Objective Status
3.1 Fick’s First Law Understand the relationship between diffusion flux and concentration gradient ✅ Complete
3.2 Fick’s Second Law Understand concentration evolution over time ✅ Complete
3.3 Spherical Diffusion Understand the diffusion equation in spherical coordinates ✅ Complete
3.4 Numerical Discretization Methods Master the implicit Euler method ✅ Complete
3.5 Tridiagonal Matrix Solver Master the Thomas algorithm ✅ Complete

Week 4: Butler-Volmer Kinetics

# Topic Objective Status
4.1 Electrode Kinetics Overview Understand interfacial reaction rates ⏳ Current
4.2 Butler-Volmer Equation Master the relationship between current density and overpotential ⏳ Current
4.3 Exchange Current Density Understand the influence of reaction rate constants ⏳ Current
4.4 Overpotential Calculation Learn to compute activation overpotential ⏳ Current
4.5 Numerical Solution Methods Master the Newton-Raphson iteration ⏳ Current

Week 5: Full SPM Implementation & Validation

# Topic Objective Status
5.1 Voltage Coupling Equation Understand battery voltage composition ⏳ Not Started
5.2 SPM Four-Equation Coupling Master the complete model architecture ⏳ Not Started
5.3 Writing simulate.py Implement the simulation main routine ⏳ Not Started
5.4 Constant-Current Discharge Simulation Learn to run basic simulations ⏳ Not Started
5.5 Result Validation & Analysis Compare theoretical and simulation results ⏳ Not Started

Week 6: SPMe Model (Incorporating Electrolyte)

# Topic Objective Status
6.1 Electrolyte Transport Equations Understand the origin of concentration polarization ⏳ Not Started
6.2 Porous Electrode Theory Master the Bruggeman correction ⏳ Not Started
6.3 Electrolyte Concentration Distribution Learn to simulate concentration gradients ⏳ Not Started
6.4 SPMe Model Implementation Extend SPM to SPMe ⏳ Not Started
6.5 High-Rate Performance Analysis Compare SPM vs. SPMe differences ⏳ Not Started

Week 7: Full P2D Model Implementation

# Topic Objective Status
7.1 P2D Model Architecture Understand the pseudo-two-dimensional structure ⏳ Not Started
7.2 Spatial Discretization Master electrode mesh generation methods ⏳ Not Started
7.3 Complete Equation System Assemble all governing equations ⏳ Not Started
7.4 Numerical Solver Implement multi-equation coupled solver ⏳ Not Started
7.5 Parameter Sensitivity Analysis Learn to analyze parameter influence ⏳ Not Started

Week 8: Advanced Applications & Optimization

# Topic Objective Status
8.1 Capacity Fade Models Understand SEI film growth mechanisms ⏳ Not Started
8.2 Thermal Coupling Model Learn to simulate battery heating ⏳ Not Started
8.3 Parameter Identification Master experiment-based data fitting ⏳ Not Started
8.4 Model Validation Compare simulation with experimental results ⏳ Not Started
8.5 Project Summary Complete the full simulation project ⏳ Not Started

3. Current Learning Progress

Completed Content

  1. Conceptual Understanding

    • Purpose and application scenarios of electrochemical simulation
    • Three core assumptions of the SPM model
    • Parameter naming rules (quantity_phase_position)
    • SOC conversion principles and N/P capacity ratio

  2. OCV Curves

    • Physical significance of the Nernst equation
    • Graphite anode tanh fitting method
    • NMC cathode polynomial fitting
    • OCV visualization comparison plots

  3. Diffusion Equation

    • Fick’s first law and second law
    • Diffusion equation in spherical coordinates
    • Implicit Euler time discretization method
    • Tridiagonal matrix construction and solution
    • Center and surface boundary condition treatment

Currently Learning

Butler-Volmer Kinetics (coming next)

  • Fundamentals of electrode kinetics
  • Butler-Volmer equation
  • Exchange current density
  • Overpotential inversion via Newton-Raphson iteration

4. Next Study Plan

Topic: Butler-Volmer Kinetics

4.1 Learning Objectives

  1. Understand the fundamentals of electrode kinetics (overpotential, exchange current density, transfer coefficient)
  2. Master the mathematical form and physical meaning of the Butler-Volmer equation
  3. Learn to invert overpotential using the Newton-Raphson method
  4. Understand the voltage calculation workflow of the complete SPM model

4.2 Learning Content

  1. Electrode Kinetics Fundamentals

    • Definition of overpotential: $\eta = \phi_s - \phi_e - OCV(\theta)$
    • Physical significance of activation overpotential

  2. Butler-Volmer Equation

    • Basic form: $j = j_0 \cdot \left[ \exp\left(\frac{\alpha F \eta}{RT}\right) - \exp\left(-\frac{(1-\alpha)F \eta}{RT}\right) \right]$
    • Exchange current density: $j_0 = k \cdot c_e^{1-\alpha} \cdot c_s^{\alpha} \cdot (c_{\mathrm{max}} - c_s)^{1-\alpha}$

  3. Numerical Solution

    • Overpotential inversion (nonlinear equation)
    • Newton-Raphson iteration method

  4. Code Implementation

    • Implement the butler_volmer_overpotential() function
    • Integrate into SPMModel.step() method

4.3 Practical Tasks

Task Description Expected Outcome
4.3.1 Derive the Newton-Raphson iteration formula Handwritten derivation
4.3.2 Implement the overpotential calculation function butler_volmer_overpotential()
4.3.3 Integrate into step() Complete single-step simulation
4.3.4 Test discharge curve Validate the full SPM model

4.4 Pre-Study Suggestions

Before the next session, consider the following:

  1. What is overpotential? What are its constituent components?
  2. Why is the relationship between overpotential and current density nonlinear?
  3. What are the convergence conditions for Newton-Raphson iteration?
  4. Why does overpotential increase sharply at high discharge rates?

Books

  • Electrochemical Methods — Bard & Faulkner
  • Battery Management Systems — Plett
  • Lithium-Ion Batteries: Science and Technologies — Yoshio et al.

Papers

  • Single Particle Model with Electrolyte (SPMe) — Doyle et al.
  • P2D Model — Newman’s original papers

Tools

  • Python: numpy, scipy, matplotlib
  • Numerical computing: sparse matrices, ODE solvers

6. Study Notes Index

File Content Status
learning_notes_01_foundation.md SPM Fundamentals & OCV ✅ Complete
learning_notes_02_diffusion.md Diffusion Equation (Fick’s Second Law) ✅ Complete
learning_notes_03_kinetics.md Butler-Volmer Kinetics ⏳ To Create
learning_notes_04_spm_complete.md Full SPM Implementation ⏳ To Create
learning_notes_05_spme.md SPMe Model ⏳ To Create
learning_notes_06_p2d.md P2D Model ⏳ To Create

7. Learning Logic Map

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graph TD
    A["P2D Model Learning"] --> B["Ch1: SPM Basics & OCV"]
    A --> C["Ch2: Diffusion Equation"]
    A --> D["Ch3: BV Kinetics"]
    A --> E["Ch4: Full SPM Implementation"]
    A --> F["Ch5: SPMe"]
    A --> G["Ch6: P2D"]
    
    B --> B1["Electrochemistry Basics"]
    B --> B2["SPM Assumptions"]
    B --> B3["OCV Curves"]
    
    C --> C1["Fick's First Law"]
    C --> C2["Fick's Second Law"]
    C --> C3["Implicit Euler"]
    C --> C4["Tridiagonal Matrix"]
    C --> C5["Boundary Conditions"]
    
    D --> D1["Overpotential"]
    D --> D2["BV Equation"]
    D --> D3["Exchange Current Density"]
    D --> D4["Newton-Raphson"]
    
    style B fill:#9f9,stroke:#333
    style C fill:#9f9,stroke:#333
    style D fill:#ff9,stroke:#333
    style E fill:#fff,stroke:#333
    style F fill:#fff,stroke:#333
    style G fill:#fff,stroke:#333

Learning Plan Last Updated: 2026-05-26
Next Study Session: As scheduled

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