particle-particle.qmd

---
title: "Particle–Particle Interactions"
subtitle: "Attraction and repulsion forces"
---

## Overview

Particles interact directly through short-range forces, independent of the chemical fields. This follows the **PDE_A** model with configurable attraction and repulsion.

## Force Law

$$
\mathbf{F}_{ij} = \left[ p_1 \exp\left( -\frac{d_{ij}^{2p_2}}{2\sigma^2} \right) - p_3 \exp\left( -\frac{d_{ij}^{2p_4}}{2\sigma^2} \right) \right] \hat{\mathbf{r}}_{ij}
$$

where:

- $p_1, p_2$ control **attraction** (strength and range exponent)
- $p_3, p_4$ control **repulsion** (strength and range exponent)
- $\sigma$ is the interaction length scale
- $d_{ij}$ is the inter-particle distance

## Default Configuration

For diffusiophoresis simulations, a simple repulsion-only model is typically used:

$$
\mathbf{F}_{ij} = -\alpha \exp\left( -\frac{5 d_{ij}}{r_{\text{rep}}} \right) \hat{\mathbf{r}}_{ij} \quad \text{if } d_{ij} < r_{\text{rep}}
$$

| Parameter | Description | Value |
|-----------|-------------|-------|
| $\alpha$ | Repulsion strength | 50 |
| $r_{\text{rep}}$ | Repulsion range | 0.04 |
| $\sigma$ | Kernel width | 0.005 |

## Multi-Type Particles

When `particle_params` is provided, each particle type can have distinct interaction parameters:

```
[M1, M2, consumption, production, ar_p1, ar_p2, ar_p3, ar_p4]
```

This enables heterogeneous systems where different particle types attract/repel each other differently.

## Force Visualization

![Particle-particle interaction forces. Left: General attraction-repulsion model with configurable parameters. Right: Simple repulsion used in diffusiophoresis simulations.](assets/pp_interaction.png){.lightbox}

## Role in Pattern Formation

Particle–particle repulsion:

- Prevents particle overlap and clustering collapse
- Maintains minimum spacing for stable structures
- Balances against diffusiophoretic aggregation forces