  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.
