Geometry & Capacity

Compute electrode capacity, cyclable lithium, active material loading, and electrode mass from cell geometry and material properties.

Geometric and capacity parameters define the physical structure of a battery cell and its energy storage capability. These calculations are foundational for building accurate battery models.

The Capacity Equation

Electrode capacity is determined by geometry and material properties:

$$Q = c_{\max} \cdot \varepsilon_{\text{AM}} \cdot L \cdot A \cdot n_{\text{elec}} \cdot (\theta_{\max} - \theta_{\min}) \cdot \frac{F}{3600}$$

where:

• $Q$ is the electrode capacity [A·h]
• $c_{\max}$ is the maximum lithium concentration [mol/m³]
• $\varepsilon_{\text{AM}}$ is the active material volume fraction
• $L$ is the electrode thickness [m]
• $A$ is the electrode area [m²]
• $n_{\text{elec}}$ is the number of electrodes connected in parallel (defaults to 1)
• $\theta_{\max} - \theta_{\min}$ is the usable stoichiometry range
• $F = 96485$ C/mol is Faraday’s constant

This equation relates six parameters (seven when $n_{\text{elec}}$ is specified)—if you know all but one, you can solve for it.

Example: Solving for Unknown Parameters

import ionworkspipeline as iwp

# When capacity is known, solve for maximum concentration
calc = iwp.calculations.ElectrodeCapacity("positive")
params = {
    "Positive electrode capacity [A.h]": 3.0,
    "Positive electrode active material volume fraction": 0.65,
    "Positive electrode thickness [m]": 80e-6,
    "Electrode area [m2]": 0.1,
    # Stoichiometry limits from SOH calculation
}
result = calc.run(params)
# Solves for: Maximum concentration in positive electrode [mol.m-3]

This is one example of how geometry and capacity calculations can be used. See the API reference for full details on calculations and parameters.

Cell Geometry

Geometry Hierarchy

Battery cells are organized hierarchically:

Cell
├── Electrode Stack
│   ├── Current Collector (negative)
│   ├── Electrode (negative)
│   ├── Separator
│   ├── Electrode (positive)
│   └── Current Collector (positive)
└── Housing / Packaging

Key Parameters

Parameter	Symbol	Typical Range	Impact
Electrode area	$A$	0.01-1 m²	Capacity, current density
Electrode thickness	$L$	50-150 µm	Capacity, rate capability
Number of electrodes connected in parallel to make a cell	$n_{\text{elec}}$	1-100+	Total capacity
Separator thickness	—	15-25 µm	Ionic resistance
Current collector	—	10-20 µm	Electrical resistance

For pouch and prismatic cells, electrode area is the planar area times the number of layers. For cylindrical cells, it’s the unrolled electrode area.

Cyclable Lithium

Cyclable lithium is the total lithium that shuttles between electrodes during cycling. It sets the upper limit on cell capacity.

$$Q_{\text{Li}} = \theta_n \cdot Q_n + \theta_p \cdot Q_p$$

where stoichiometries $\theta_n, \theta_p$ are evaluated at a reference state (typically 100% SOC).

Why It Matters

• Cell capacity: Cannot exceed cyclable lithium, regardless of electrode capacities
• Degradation tracking: Loss of cyclable lithium indicates SEI growth, plating, or particle cracking
• Electrode balancing: Determines which electrode limits cell capacity

N/P Ratio and Electrode Balancing

The negative-to-positive capacity ratio (N/P ratio) affects how electrodes are utilized:

$$\text{N/P ratio} = \frac{Q_n}{Q_p}$$

Condition	Behavior
N/P > 1 (typical)	Positive electrode limits capacity; negative has excess
N/P < 1	Negative electrode limits; risk of lithium plating
Lithium-limited	Neither electrode reaches stoichiometry limits

Cells are typically designed with N/P > 1 to prevent lithium plating at the negative electrode during charge.

Mass Calculations

Mass is needed for gravimetric energy density and thermal modeling.

Component Mass

Each component’s mass is calculated from:

$$m = \rho \cdot A \cdot L \cdot (1 - \varepsilon)$$

where $\rho$ is density, $A$ is area, $L$ is thickness, and $\varepsilon$ is porosity.

Energy Density

Gravimetric and volumetric energy densities are key cell-level metrics:

$$E_{\text{grav}} = \frac{Q \cdot V_{\text{avg}}}{m_{\text{cell}}}, \quad E_{\text{vol}} = \frac{Q \cdot V_{\text{avg}}}{V_{\text{cell}}}$$

calc = iwp.calculations.Mass("positive")
result = calc.run({
    "Positive electrode thickness [m]": 80e-6,
    "Electrode area [m2]": 0.1,
    "Positive electrode active material volume fraction": 0.65,
    "Positive electrode active material density [kg.m-3]": 4650,
})

Microstructure

Microstructure parameters describe the porous electrode architecture:

Porosity

The void fraction of the electrode:

$$\varepsilon = 1 - \varepsilon_{\text{AM}} - \varepsilon_{\text{binder}} - \varepsilon_{\text{carbon}}$$

Higher porosity improves electrolyte transport but reduces energy density.

Tortuosity

Describes how much longer the effective transport path is compared to the straight-line distance:

$$D_{\text{eff}} = \frac{D \cdot \varepsilon}{\tau}$$

Common correlations relate tortuosity to porosity:

• Bruggeman: $\tau = \varepsilon^{-0.5}$
• Measured: From electrochemical impedance or other techniques

Active Material Volume Fraction

The fraction of electrode volume occupied by active material:

$$\varepsilon_{\text{AM}} = \frac{V_{\text{AM}}}{V_{\text{electrode}}}$$

This is a key fitting parameter affecting both capacity and transport.

Practical Workflow

1. Define Geometry

   Set electrode area, thicknesses, and number of layers

2. Specify Microstructure

   Set active material volume fractions and calculate porosity

3. Calculate Capacities

   Use ElectrodeCapacity for each electrode

4. Determine Cyclable Lithium

   Use CyclableLithium to find the limiting capacity

5. Calculate Mass and Energy Density

   Use Mass calculations for each component

Common Calculations

Calculation	Purpose
ElectrodeCapacity	Solve capacity equation for any unknown
CyclableLithium	Calculate total shuttling lithium
Mass	Component mass from geometry and density
Microstructure	Porosity, tortuosity, and volume fractions
GeometricArea	Convert between geometric and electrochemical areas