# On the normal modes of weak colloidal gels

## TL;DR

This study investigates how hydrodynamic interactions influence the normal modes and relaxation dynamics of weak colloidal gels, revealing the importance of long-range interactions in their viscoelastic behavior.

## Contribution

It introduces a comprehensive analysis of hydrodynamic effects on gel normal modes, highlighting the role of long-range interactions in gel dynamics and developing a phenomenological model.

## Key findings

- Long-ranged hydrodynamic interactions significantly alter collectivity and energy dissipation.
- Relaxation rates scale as λ ∼ l^{-2} for models with long-range interactions.
- Stress relaxation matches normal mode predictions and the phenomenological model.

## Abstract

The normal modes and relaxation rates of weak colloidal gels are investigated in computations employing different models of the hydrodynamic interactions between colloids. The eigenspectrum is computed for freely draining, Rotne-Prager-Yamakawa and Accelerated Stokesian Dynamics approximations of the hydrodynamic mobility in a normal mode analysis of a harmonic network representing the gel. The spatial structure of the normal modes suggests that measures of collectivity and energy dissipation in the gels are fundamentally altered by long-ranged hydrodynamic interactions, while hydrodynamic lubrication affects only the relaxation rates of short wavelength modes. Models accounting for long-ranged hydrodynamic interactions exhibit a microscopic relaxation rate for each normal mode, $\lambda$ that scales as $\lambda\sim l^{-2}$, where $l$ is the spatial correlation length of the mode. For the freely draining approximation, $\lambda\sim l^\gamma$, where $\gamma$ varies between 3 and 2 with increasing $\phi$. A simple phenomenological model of the internal elastic response to normal mode fluctuations is developed, which shows that long-ranged hydrodynamic interactions play a central role in the viscoelasticity of the gel network. Dynamic simulations show that the stress decay as measured by the time-dependent shear modulus matches the normal mode predictions and the phenomenological model. Analogous to the Zimm model in polymer physics, our results indicate that long-ranged hydrodynamic interactions play a crucial role in determining the microscopic dynamics and macroscopic properties of weak colloidal gels.