Zitao Ni

Ph.D. in Materials Science and Engineering

Study Notes · Part 1: Fundamental Concepts & OCV

P2D Model Study Notes · Part 1: Fundamental Concepts & OCV

Date: 2026-05-21
Task: Build an electrochemical battery simulation model from scratch in Python
Progress: SPM Single Particle Model · OCV Curves Complete

1. What Can Electrochemical Simulation Do?

Electrochemical simulation = creating a “digital twin” of a battery — using mathematical equations to simulate everything happening inside a battery on a computer.

What You Want to Know What Simulation Can Tell You
How much charge remains? Simulate how voltage drops during discharge
How much current can it handle? Simulate whether the battery interior “congests” under different currents
Why does the battery heat up? Simulate internal losses → heat generation
How long will the battery last? Simulate aging processes (cycle life)
How to design better batteries? Adjust parameters and observe performance changes

2. What Happens Inside a Li-ion Battery?

Three key steps during discharge:

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graph LR
    A["Anode Particle<br/>Li → Li⁺ + e⁻"] -->|"Li⁺ through electrolyte"| B["Cathode Particle<br/>Li⁺ + e⁻ → Li"]
    A -->|"e⁻ through external circuit"| C["Work!<br/>Light the bulb"]
    C --> B
  1. Deintercalation: Lithium atoms at the anode particle surface lose electrons, becoming Li⁺ that enter the electrolyte
  2. Migration: Li⁺ travels through the electrolyte (separator) toward the cathode
  3. Intercalation: Li⁺ gains electrons at the cathode particle surface and embeds into the cathode

3. What is SPM (Single Particle Model)?

SPM = Single Particle Model, the simplest electrochemical battery model.

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graph TD
    subgraph Real Battery
        A1["Particle Population<br/>Thousands of particles"]
        A2["Particle Population<br/>Thousands of particles"]
    end
    subgraph SPM Model
        B1["One sphere = Entire anode"]
        B2["One sphere = Entire cathode"]
    end
    A1 -.->|"Simplification"| B1
    A2 -.->|"Simplification"| B2

Three Core Assumptions of SPM

Assumption Meaning
(1) All particles behave identically One sphere represents the entire electrode
(2) Neglect electrolyte concentration variations Li⁺ concentration is uniform throughout the electrolyte
(3) Uniform current distribution Each particle experiences the same current

SPM Capability Boundaries

  • Voltage–time curves (discharge/charge curves)
  • Lithium concentration distribution within particles
  • Battery performance under different currents
  • SOC evolution over time
  • Accurate prediction at high currents (neglects electrolyte gradients)
  • Differences between particles at different positions

4. Parameter Naming Convention

Naming formula: self.{quantity}_{phase}_{position}

Quantity Prefixes

Prefix Meaning Common Units
L Length / Thickness m
R Radius or Ideal Gas Constant m / J·mol⁻¹·K⁻¹
c Concentration mol·m⁻³
D Diffusion Coefficient m²·s⁻¹
eps Volume Fraction / Porosity Dimensionless
k Reaction Rate Constant Context-dependent
i Current Density A·m⁻²
F Faraday Constant 96485 C·mol⁻¹
T Temperature K
Nr Number of Discrete Points Dimensionless

Phase Suffixes & Position Suffixes

Suffix Category Meaning
_s Phase Solid Phase
_e Phase Electrolyte Phase
_n Position Negative Electrode
_p Position Positive Electrode
_s Position Separator

Note: _s can denote either “solid phase” or “separator” depending on context. For example, eps_s_n = negative solid-phase volume fraction, eps_e_s = separator electrolyte volume fraction.

Combination Examples

Variable Name Decomposition Meaning
c_s_max_n c + s + max + n Maximum lithium concentration in negative solid phase
D_s_p D + s + p Positive solid-phase diffusion coefficient
eps_e_s eps + e + s Separator electrolyte volume fraction
i_app i + app(applied) Applied current density

5. Cell SOC vs. Electrode SOC

Key Distinction

Cell SOC ≠ a given electrode’s SOC. During discharge, the two electrodes evolve in opposite directions:


 Negative SOC Positive SOC
Cell 100% (Full) 95% 10%
Cell 0% (Empty) 5% ~79%
Direction (Discharge) ↓ Decreasing ↑ Increasing

SOC Relationship Governed by Lithium Conservation

Moles of lithium lost by the anode = Moles of lithium gained by the cathode:

$$
N_{\mathrm{total}} \cdot \Delta SOC_n = P_{\mathrm{total}} \cdot \Delta SOC_p
$$

$$
\frac{\Delta SOC_p}{\Delta SOC_n} = \frac{N_{\mathrm{total}}}{P_{\mathrm{total}}} \quad \text{(SOC Exchange Rate)}
$$

Key insight: This relationship is exactly linear — it is a direct consequence of mass conservation, not an approximation.

N/P Capacity Ratio

$$
N_{\mathrm{total}} = c_{s,\mathrm{max}}^n \cdot \varepsilon_s^n \cdot L_n \cdot A
$$

$$
P_{\mathrm{total}} = c_{s,\mathrm{max}}^p \cdot \varepsilon_s^p \cdot L_p \cdot A
$$

$$
\mathrm{N/P} = \frac{N_{\mathrm{total}}}{P_{\mathrm{total}}}
$$

N/P Value Implication
> 1 Anode capacity > Cathode capacity → Cathode is the limiting electrode during charge → Prevents lithium plating
< 1 Cathode capacity > Anode capacity → Anode is the limiting electrode during discharge
Does not determine Initial SOC pairing, but constrains how far discharge/charge can proceed

Cell-Level SOC → Electrode SOC Automatic Conversion

In parameters.py, simply set battery_soc_initial = 0.3 (cell starts at 30%), and the program computes automatically.

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graph TD
    A["battery_soc_initial = 0.3"] --> B["Endpoint Definition<br/>SOC_n_full=0.95, SOC_n_empty=0.05<br/>SOC_p_full=0.10"]
    B --> C["N/P Conservation<br/>SOC_p_empty = 0.10 + 0.90×(N/P)"]
    C --> D["Linear Interpolation<br/>SOC_n = 0.95 − 0.7×0.90 = 0.32<br/>SOC_p = 0.10 + 0.7×(N/P exchange rate)"]
    D --> E["Concentration Conversion<br/>c_init = SOC × c_max"]

6. OCV Curves (Open-Circuit Voltage)

OCV is the “fingerprint” of an electrode material — each material has a unique OCV–SOC relationship.

6.1 The Nernst Equation (Thermodynamic Foundation)

$$
OCV(\theta) = U_0 + \frac{RT}{F} \cdot \ln!\left(\frac{\theta}{1-\theta}\right)
$$

Symbol Name Value Meaning
$U_0$ Reference Potential ~0.12 V (graphite) / ~3.85 V (NMC) Characteristic potential of the material
$R$ Ideal Gas Constant 8.314 J·mol⁻¹·K⁻¹
$T$ Temperature 298.15 K
$F$ Faraday Constant 96485 C·mol⁻¹
$RT/F$ Thermal Voltage 0.02569 V The “thermal ruler” of one electron
$\theta$ Surface SOC $c_{s,\mathrm{surf}} / c_{s,\mathrm{max}}$

Physical Origin: The Nernst equation derives from the Gibbs free energy $\Delta G = \Delta G_0 + RT \cdot \ln(\mathrm{activity\ ratio})$; it generates a complete curve with only 2 parameters. However, it assumes the material is an “ideal solution” — atoms do not interact with each other.

6.2 Ideal vs. Real

Nernst Comparison


 Ideal Solution (Nernst) Real Graphite Real NMC
Curve Shape Smooth S-shape Staircase plateaus Smooth but asymmetric
Cause No inter-atomic interactions LiC₆→LiC₁₂→LiC₁₈ phase transitions Multi-transition-metal synergy
Fitting Method 2 parameters Nernst + tanh steps Nernst + polynomial

6.3 Nernst Equation Breakdown

Nernst Breakdown

The core of Nernst is simply $\ln(\theta/(1-\theta))$, multiplied by the thermal voltage $RT/F \approx 0.026\mathrm{V}$:

  • $\theta = 0.5$: $\ln(1) = 0$, $OCV = U_0$ (equilibrium)
  • $\theta \to 0$: $\ln \to -\infty$, OCV drops steeply
  • $\theta \to 1$: $\ln \to +\infty$, OCV rises steeply

Key Insight: The “logarithmic scale” is compressed into the “millivolt scale” by the thermal voltage — the shape of the curve is entirely a mathematical expression of the entropy of mixing.

6.4 Graphite Anode OCV (ocv_negative)

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def ocv_negative(theta):
    U = (0.6379                                          # baseline
         + 0.5416 * np.exp(-305.5309 * theta)             # exponential spike (θ<3%)
         - 0.0440 * np.tanh((theta - 0.1958) / 0.1088)    # tanh step 1 (~20%)
         - 0.1978 * np.tanh((theta - 1.0571) / 0.0854)    # tanh step 2 (~100% cliff)
         - 0.6875 * np.tanh((theta + 0.0117) / 0.0529)    # tanh step 3 (main step!)
         - 0.0175 * np.tanh((theta - 0.5692) / 0.0875))   # tanh step 4 (minor correction)
    return U

Essentially, mathematical “building blocks” (tanh smooth steps + exp spike) are used to approximate the staircase-like OCV produced by graphite phase transitions:

$$
OCV_{\mathrm{graphite}}(\theta) = \mathrm{constant} + \sum_k a_k \cdot \tanh!\left(\frac{\theta - \theta_k}{w_k}\right) + \mathrm{exp\ term}
$$

Each $\tanh$ step corresponds to one phase-transition stage of graphite during lithiation/delithiation.

6.5 NMC Cathode OCV (ocv_positive)

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def ocv_positive(theta):
    return 4.15 - 0.9 * theta + 0.2 * theta**2

$$
OCV_{\mathrm{NMC}}(\theta) = 4.15 - 0.9\theta + 0.2\theta^2
$$

Control Point OCV
$\theta = 0$ 4.15 V (fully delithiated, highest potential)
$\theta = 0.5$ 3.75 V (half-full)
$\theta = 1$ 3.45 V (fully lithiated, lowest potential)

6.6 The Full Picture: Four-Term Superposition

OCV Deep Dive

Focus on the upper-right subplot — this is the OCV trajectory when discharging from 30% SOC to 0%:

$$
V_{\mathrm{cell}} = OCV_p(SOC_p) - OCV_n(SOC_n)
$$

From ~3.65 V (start) down to ~3.45 V (end).

Note: This is only the OCV contribution; the overpotential $\eta$ has not been included yet (adding $\eta$ will depress the voltage further).

7. The Four OCV Functions at a Glance

Function Type Purpose
ocv_negative() Empirical Fit Used at runtime, real graphite curve
ocv_positive() Quadratic Polynomial Used at runtime, real NMC curve
ocv_negative_nernst() Pure Physics Teaching use, comparing “ideal vs. real”
ocv_positive_nernst() Pure Physics Teaching use, comparing “ideal vs. real”

8. The SPM Four-Equation Panorama

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graph TD
    P["parameters.py<br/>Battery physical parameters"] --> M["model.py"]

    subgraph model.py
        direction TB
        O["(1) OCV Curves<br/>U(θ): Electrode material fingerprint"]
        D["(2) Diffusion Equation<br/>∂c/∂t = D/r²·∂/∂r(r²·∂c/∂r)"]
        B["(3) Butler-Volmer<br/>j ⇄ η: Interfacial reaction kinetics"]
        V["(4) Voltage Coupling<br/>V_cell = U_p+η_p − U_n−η_n"]
        O --> B
        D --> V
        B --> V
    end

    M --> R["simulate.py<br/>Run & plot analysis"]

9. Project File Structure

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c:\Project\P2D\
├── requirements.txt                # numpy, scipy, matplotlib
├── visualize_ocv.py                # OCV visualization script
├── learning_notes_01_foundation.md  # Study notes
├── ocv_comparison.png              # Figure 1: Nernst vs. empirical fit
├── nernst_breakdown.png            # Figure 2: Nernst equation breakdown
├── ocv_deep_dive.png               # Figure 3: Four-term superposition + discharge trajectory
├── electro-ref/                    # Reference books
└── spm/
    ├── __init__.py
    ├── parameters.py               # Battery physical parameters + automatic SOC conversion
    ├── model.py                    # SPM core engine (4 governing equations)
    └── simulate.py                 # To be created

10. Key Formula Quick Reference

Formula Meaning
$V_{\mathrm{cell}} = OCV_p - OCV_n$ Cell open-circuit voltage = Cathode potential − Anode potential
$\frac{\Delta SOC_p}{\Delta SOC_n} = \frac{N_{\mathrm{total}}}{P_{\mathrm{total}}}$ SOC exchange rate (lithium conservation)
$Q_{\mathrm{Ah}} = \frac{F \cdot \min(cap_n, cap_p)}{3600}$ Cell capacity [Ah]
$i_{1C} = \frac{Q_{\mathrm{Ah}}}{A}$ 1C current density
$j = \frac{i}{a_s \cdot L \cdot F}$ Current density → reaction flux
$a_s = \frac{3 \cdot \varepsilon_s}{R}$ Active specific surface area (spherical particles)
$OCV(\theta) = U_0 + \frac{RT}{F}\ln!\left(\frac{\theta}{1-\theta}\right)$ Nernst equation
$OCV_{\mathrm{NMC}}(\theta) = 4.15 - 0.9\theta + 0.2\theta^2$ NMC cathode OCV

11. To-Do / Next Steps

  • Understand SPM model concepts
  • Master parameter naming rules and physical meanings
  • Understand N/P ratio and SOC conversion
  • Understand OCV curves (Nernst + empirical fitting)
  • Study the diffusion equation (Fick’s Second Law in spherical coordinates)
  • Study Butler-Volmer kinetics
  • Study voltage coupling
  • Write simulate.py, run a complete simulation
  • Plot discharge curves, analyze results

End of Notes · Next Part: Diffusion Equation & Butler-Volmer Kinetics

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