> ## Documentation Index
> Fetch the complete documentation index at: https://docs.ionworks.com/llms.txt
> Use this file to discover all available pages before exploring further.

# 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. For example, when capacity is known, the `ElectrodeCapacity` calculation can solve for maximum concentration; supply any five of the six parameters and it returns the missing one.

<Note>
  To configure these calculations in a pipeline, see [Pipelines → Calculations → Geometry & Capacity](/pipelines/calculations/geometry-capacity).
</Note>

## 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     |

<Note>
  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.
</Note>

## 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          |

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

## 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:

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

## 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

<Steps>
  <Step title="Define Geometry">
    Set electrode area, thicknesses, and number of layers
  </Step>

  <Step title="Specify Microstructure">
    Set active material volume fractions and calculate porosity
  </Step>

  <Step title="Calculate Capacities">
    Use `ElectrodeCapacity` for each electrode
  </Step>

  <Step title="Determine Cyclable Lithium">
    Use `CyclableLithium` to find the limiting capacity
  </Step>

  <Step title="Calculate Mass and Energy Density">
    Use `CellMass` to roll the component contributions up to a cell-level mass
  </Step>
</Steps>

## Common Calculations

| Calculation                                                                  | Purpose                                                                                                                                                                  |
| ---------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------ |
| `ElectrodeCapacity`                                                          | Solve capacity equation for any unknown                                                                                                                                  |
| `CyclableLithium`                                                            | Calculate total shuttling lithium                                                                                                                                        |
| `CellMass`                                                                   | Cell mass summed from component densities, thicknesses, and porosities                                                                                                   |
| `ElectrodeVolumeFractionFromLoading` / `ElectrodeVolumeFractionFromPorosity` | Active material volume fraction from loading or porosity                                                                                                                 |
| `PorosityFromElectrodeVolumeFraction`                                        | Porosity from active material volume fraction                                                                                                                            |
| `SurfaceAreaToVolumeRatio`                                                   | Electrochemically active surface area per unit electrode volume — the bridge from geometric electrode dimensions to reactive area used by the porous-electrode equations |