# Dynamics in a one-dimensional ferrogel model: relaxation, pairing, shock-wave propagation

## TL;DR

This paper models the dynamic behavior of ferrogels using a one-dimensional chain of magnetic dipoles and springs, revealing relaxation, pairing, and shock-wave phenomena through simulations and theoretical analysis.

## Contribution

It introduces a simple one-dimensional ferrogel model and analyzes its dynamics, including shock-wave propagation and particle pairing, providing insights into more complex systems.

## Key findings

- Long-time relaxation corresponds to shock-wave solutions.
- Initial dynamics involve formation of particle pairs.
- The model offers a framework for understanding ferrogel behavior.

## Abstract

Ferrogels are smart soft materials, consisting of a polymeric network and embedded magnetic particles. Novel phenomena, such as the variation of the overall mechanical properties by external magnetic fields, emerge consequently. However, the dynamic behavior of ferrogels remains largely unveiled. In this paper, we consider a one-dimensional chain consisting of magnetic dipoles and elastic springs between them as a simple model for ferrogels. The model is evaluated by corresponding simulations. To probe the dynamics theoretically, we investigate a continuum limit of the energy governing the system and the corresponding equation of motion. We provide general classification scenarios for the dynamics, elucidating the touching/detachment dynamics of the magnetic particles along the chain. In particular, it is verified in certain cases that the long-time relaxation corresponds to solutions of shock-wave propagation, while formations of particle pairs underlie the initial stage of the dynamics. We expect that these results will provide insight into the understanding of the dynamics of more realistic models with randomness in parameters and time-dependent magnetic fields.