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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.

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The Capacity Equation

Electrode capacity is determined by geometry and material properties: Q=cmax⁡⋅εAM⋅L⋅A⋅nelec⋅(θmax⁡−θmin⁡)⋅F3600Q = c_{\max} \cdot \varepsilon_{\text{AM}} \cdot L \cdot A \cdot n_{\text{elec}} \cdot (\theta_{\max} - \theta_{\min}) \cdot \frac{F}{3600} where:
  • QQ is the electrode capacity [A·h]
  • cmax⁡c_{\max} is the maximum lithium concentration [mol/m³]
  • εAM\varepsilon_{\text{AM}} is the active material volume fraction
  • LL is the electrode thickness [m]
  • AA is the electrode area [m²]
  • nelecn_{\text{elec}} is the number of electrodes connected in parallel (defaults to 1)
  • θmax⁡−θmin⁡\theta_{\max} - \theta_{\min} is the usable stoichiometry range
  • F=96485F = 96485 C/mol is Faraday’s constant
This equation relates six parameters (seven when nelecn_{\text{elec}} is specified)—if you know all but one, you can solve for it.

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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.

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Cell Geometry

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Geometry Hierarchy

Battery cells are organized hierarchically: 
Cell
├── Electrode Stack
│   ├── Current Collector (negative)
│   ├── Electrode (negative)
│   ├── Separator
│   ├── Electrode (positive)
│   └── Current Collector (positive)
└── Housing / Packaging

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Key Parameters

ParameterSymbolTypical RangeImpact
Electrode areaAA0.01-1 m²Capacity, current density
Electrode thicknessLL50-150 µmCapacity, rate capability
Number of electrodes connected in parallel to make a cellnelecn_{\text{elec}}1-100+Total capacity
Separator thickness—15-25 µmIonic resistance
Current collector—10-20 µmElectrical 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.

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Cyclable Lithium

Cyclable lithium is the total lithium that shuttles between electrodes during cycling. It sets the upper limit on cell capacity. QLi=θn⋅Qn+θp⋅QpQ_{\text{Li}} = \theta_n \cdot Q_n + \theta_p \cdot Q_p where stoichiometries θn,θp\theta_n, \theta_p are evaluated at a reference state (typically 100% SOC).

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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

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N/P Ratio and Electrode Balancing

The negative-to-positive capacity ratio (N/P ratio) affects how electrodes are utilized: N/P ratio=QnQp\text{N/P ratio} = \frac{Q_n}{Q_p}
ConditionBehavior
N/P > 1 (typical)Positive electrode limits capacity; negative has excess
N/P < 1Negative electrode limits; risk of lithium plating
Lithium-limitedNeither electrode reaches stoichiometry limits
Cells are typically designed with N/P > 1 to prevent lithium plating at the negative electrode during charge.

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Mass Calculations

Mass is needed for gravimetric energy density and thermal modeling.

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Component Mass

Each component’s mass is calculated from: m=ρ⋅A⋅L⋅(1−ε)m = \rho \cdot A \cdot L \cdot (1 - \varepsilon) where ρ\rho is density, AA is area, LL is thickness, and ε\varepsilon is porosity.

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Energy Density

Gravimetric and volumetric energy densities are key cell-level metrics: Egrav=Q⋅Vavgmcell,Evol=Q⋅VavgVcellE_{\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,
})

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Microstructure

Microstructure parameters describe the porous electrode architecture:

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Porosity

The void fraction of the electrode: ε=1−εAM−εbinder−εcarbon\varepsilon = 1 - \varepsilon_{\text{AM}} - \varepsilon_{\text{binder}} - \varepsilon_{\text{carbon}} Higher porosity improves electrolyte transport but reduces energy density.

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Tortuosity

Describes how much longer the effective transport path is compared to the straight-line distance: Deff=D⋅ετD_{\text{eff}} = \frac{D \cdot \varepsilon}{\tau} Common correlations relate tortuosity to porosity:
  • Bruggeman: τ=ε−0.5\tau = \varepsilon^{-0.5}
  • Measured: From electrochemical impedance or other techniques

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Active Material Volume Fraction

The fraction of electrode volume occupied by active material: εAM=VAMVelectrode\varepsilon_{\text{AM}} = \frac{V_{\text{AM}}}{V_{\text{electrode}}} This is a key fitting parameter affecting both capacity and transport.

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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

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Common Calculations

CalculationPurpose
ElectrodeCapacitySolve capacity equation for any unknown
CyclableLithiumCalculate total shuttling lithium
MassComponent mass from geometry and density
MicrostructurePorosity, tortuosity, and volume fractions
GeometricAreaConvert between geometric and electrochemical areas