Super-Flory scaling in compressed micro-gel packings

We perform multi-body finite element simulations of packings of hydrogel particles immersed in a large solvent bath in 2D at various particle volume fraction, φ using the Flory-Rehner constitutive law to model the mechanics of the hydrogel network. The system becomes rigid with finite osmotic pressure, Π and zero-frequency storage modulus, G, only above the random close packing point. At large enough φ regions of pure solvent are completely eliminated, and we find a qualitative change in the G vs φ curve with a weaker dependence on φ. However, in this dense limit, G exceeds the modulus of a monolithic piece of hydrogel made of the same Flory-Rehner material and continues to increase with φ. This result is surprising as we observe strong inhomogeneous non-affine relaxation which helps to reduce the modulus, however, it is in agreement with experiments which also show super-Flory behavior. Furthermore, we show that the slip at the facets between particles is proportional to the transverse gradient of the applied deformation which strongly screens the stress near facets oriented along the applied shear. Our results on the deformation kinematics during shear suggest new measurements to attempt in experiments and should open the way to quantitative theories to estimate G at high φ.