📘 Main Equations in Electrochemical Modeling of Li-Ion Batteries
Electrochemical models rely on a set of fundamental equations that describe mass transport, charge conservation, and reaction kinetics in battery materials. These equations form the core of physics-based modeling approaches like DFN, SPMe, and SPM.
🔋 1. Fick’s Second Law (Solid-Phase Diffusion)
$$ \ \frac{\partial c_s}{\partial t} = \frac{1}{r^2} \frac{\partial}{\partial r} \left( D_s r^2 \frac{\partial c_s}{\partial r} \right) \tag{1} \ $$
This equation governs the diffusion of lithium ions inside solid active material particles in the electrode. It accounts for concentration gradients over time and space within each particle.
✅ Used in:
- DFN (full resolution per particle)
- SPMe (single particle per electrode)
- SPM (simplified single particle)
🌊 2. Nernst–Planck Equation (Electrolyte Diffusion and Migration)
$$ \ \frac{\partial c_e}{\partial t} = \nabla \cdot \left( D_e \nabla c_e - \frac{t_+}{F} i_c \right) \tag{2} \ $$
This equation models the lithium-ion transport in the electrolyte, capturing both diffusion (due to concentration gradients) and migration (due to electric fields).
✅ Used in:
- DFN
- SPMe
❌ Not used in SPM
⚡ 3. Ohm’s Law for Solid Conductors (Charge Conservation)
$$ \ \nabla \cdot (\sigma \nabla \phi_s) = -a j \tag{3} \ $$
This describes charge conservation in the solid matrix of the electrode (e.g., graphite or NMC), where current is conducted by electrons. The term on the right represents the source/sink of current due to electrochemical reactions.
✅ Used in:
- DFN
- SPMe
❌ Not used in SPM (assumes uniform potential)
🔁 4. Butler–Volmer Equation (Electrode Reaction Kinetics)
$$ \ j = j_0 \left( \exp \left( \frac{\alpha_a F \eta}{RT} \right) - \exp \left( \frac{\alpha_c F \eta}{RT} \right) \right) \tag{4} \ $$
This nonlinear equation determines the reaction current density based on overpotential ( \eta ), capturing the forward and reverse electrochemical reactions at the electrode/electrolyte interface.
✅ Used in:
- DFN
- SPMe
⚠️ SPM: Often linearized or simplified
⚖️ 5. Nernst Equation (Open Circuit Voltage)
$$ \ E = E_0 + \frac{RT}{zF} \ln \left( \frac{a_{\text{oxidized}}}{a_{\text{reduced}}} \right) \tag{5} \ $$
This equation provides the equilibrium potential of an electrochemical cell, as a function of species activities or concentrations. It defines the open-circuit voltage under no-load conditions.
✅ Used in:
- DFN
- SPMe
❌ Not explicitly used in SPM
🧾 Summary of Equation Usage by Model
| Equation | Description | DFN | SPMe | SPM |
|---|---|---|---|---|
| Eq. (1) | Solid-phase diffusion | ✅ | ✅ | ✅ |
| Eq. (2) | Electrolyte diffusion/migration | ✅ | ✅ | ❌ |
| Eq. (3) | Solid-phase charge conservation | ✅ | ✅ | ❌ |
| Eq. (4) | Reaction kinetics (Butler–Volmer) | ✅ | ✅ | ⚠️ Simplified |
| Eq. (5) | Equilibrium potential (Nernst) | ✅ | ✅ | ❌ |
These core equations allow electrochemical models to simulate battery behavior with high fidelity across a range of operating conditions, supporting accurate predictions of performance, aging, and thermal behavior.
🔗 Learn more or simulate these models at cathode.energy