field-particle.qmd

---
title: "Field–Particle Interactions"
subtitle: "Diffusiophoresis: gradients drive motion"
---

## Overview

Particles move in response to chemical gradients—a phenomenon called **diffusiophoresis**. This couples the field dynamics to particle transport.

## Velocity Law

$$
\mathbf{v}_i = M_1 \nabla C_1\big|_{\mathbf{x}_i} + M_2 \nabla C_2\big|_{\mathbf{x}_i} + \sqrt{\frac{2}{\text{Pe}}} \boldsymbol{\eta}(t)
$$

where:

- $M_1, M_2$ are **mobility coefficients** (signed: positive = move up gradient, negative = move down)
- $\text{Pe}$ is the Péclet number controlling thermal noise strength
- $\boldsymbol{\eta}(t)$ is white Gaussian noise (Brownian motion)

## Gradient Estimation

Local field gradients at particle positions are computed via weighted interpolation from nearby mesh nodes:

$$
\nabla C\big|_{\mathbf{x}_i} \approx \sum_{j \in \mathcal{N}_i} w_{ij} \frac{C_j - C_i}{|\mathbf{x}_j - \mathbf{x}_i|} \hat{\mathbf{r}}_{ij}
$$

with Gaussian kernel $w_{ij} = \exp(-|\mathbf{x}_j - \mathbf{x}_i|^2 / 2\sigma^2)$.

## Parameters

| Symbol | Description | Typical Value |
|--------|-------------|---------------|
| $M_1$ | Mobility for field 1 | ±4 |
| $M_2$ | Mobility for field 2 | ∓4 |
| Pe | Péclet number | 200 |

::: {.callout-warning}
## Mobility Sweet Spot
$|M| = 4$ is optimal. $|M| = 16$ causes particle escape (0% retention). The ±4 range balances field-particle coupling with particle containment.
:::

## Key Finding

::: {.callout-important}
**Linear mobility is optimal.** Any nonlinearity (saturation, boost) disrupts boundary accumulation. The simple $\mathbf{v} \propto \nabla C$ relationship works best.
:::

## Log-Sensing Variant (Block 8)

A new particle model variant uses **logarithmic gradient sensing** (Weber-Fechner law):

$$
\mathbf{v}_i = M_1 \frac{\nabla C_1}{C_1 + C_0}\bigg|_{\mathbf{x}_i} + M_2 \frac{\nabla C_2}{C_2 + C_0}\bigg|_{\mathbf{x}_i}
$$

This creates concentration-dependent effective mobility ($M_{\text{eff}} \propto M / C$), producing self-limiting aggregation at high concentrations. Early results show LogSensing suppresses hexagonal symmetry breaking at standard mobility ($M=\pm 4$), requiring higher $M$ ($\pm 8$) to restore pattern formation.