Scaling of poroelastic coarsening and elastic arrest in crosslinked gels
Recent experiments on crosslinked gels quenched from solvent-rich to solvent-poor conditions show solvent-rich domains embedded in a gel-rich matrix. These domains coarsen and then undergo kinetic arrest at micron scales for hours, before macroscopic drainage to equilibrium over even longer times. Motivated by these observations, we develop a minimal model that couples capillarity-driven Darcy permeation to the viscoelastic-to-elastic crossover of the polymer network. In the viscoelastic regime, the Young--Laplace traction at curved solvent--gel interfaces generates a pressure gradient in the solvent pores of the gel that drives solvent flow and coarsening. In the elastic regime, the same interfacial traction is balanced by elastic stress. This force balance eliminates pressure gradients in the solvent-filled pores of the gel, removing the Darcy driving force and arresting coarsening. Using the kinetic criterion $t(λ_{\rm arrest}) \sim τ_{\rm el}$, we predict stiffness-dependent coarsening and arrest laws. For melt-like, polymer-rich gels, $λ(t)\sim G^{-1/2} t^{1/4}$ and $λ_{\rm arrest}\sim G^{-1/2}$. For low polymer fractions where the mesh size controls transport, $λ(t)\sim G^{-1/3} t^{1/3}$ and $λ_{\rm arrest}\sim G^{-1/3}$. The predicted $G^{-1/2}$ arrest scaling for melt-like gels agrees with experiment.