Self propulsion of droplets driven by an active permeating gel

We discuss the flow field and propulsion velocity of active droplets, which are driven by body forces residing on a rigid gel. The latter is modelled as a porous medium which gives rise to permeation forces. In the simplest model, the Brinkman equation, the porous medium is characterised by a single length scale $\ell$ --the square root of the permeability. We compute the flow fields inside and outside of the droplet as well as the energy dissipation as a function of $\ell$. We furthermore show that there are optimal gel fractions, giving rise to maximal linear and rotational velocities. In the limit $\ell\to\infty$, corresponding to a very dilute gel, we recover Stokes flow. The opposite limit, $\ell\to 0$, corresponding to a space filling gel, is singular and not equivalent to Darcy's equation, which cannot account for self-propulsion

Towards finite-dimensional gelation

We consider the gelation of particles which are permanently connected by random crosslinks, drawn from an ensemble of finite-dimensional continuum percolation. To average over the randomness, we apply the replica trick, and interpret the replicated and crosslink-averaged model as an effective molecular fluid. A Mayer-cluster expansion for moments of the local static density fluctuations is set up. The simplest non-trivial contribution to this series leads back to mean-field theory. The central quantity of mean-field theory is the distribution of localization lengths, which we compute for all connectivities. The highly crosslinked gel is characterized by a one-to-one correspondence of connectivity and localization length. Taking into account higher contributions in the Mayer-cluster expansion, systematic corrections to mean-field can be included. The sol-gel transition shifts to a higher number of crosslinks per particle, as more compact structures are favored. The critical behavior of the model remains unchanged as long as finite truncations of the cluster expansion are considered. To complete the picture, we also discuss various geometrical properties of the crosslink network, e.g. connectivity correlations, and relate the studied crosslink ensemble to a wider class of ensembles, including the Deam-Edwards distribution.Comment: 18 pages, 4 figures, version to be published in EPJ