Scaling of poroelastic coarsening and elastic arrest in crosslinked gels

TL;DR

The paper addresses how solvent-rich domains in crosslinked gels coarsen and eventually arrest due to the gel’s elastic response. By unifying Lifshitz–Slyozov–Wagner coarsening with Darcy-like poroelastic transport and a long-range elastic arrest mechanism, it derives scaling laws for domain growth and arrest lengths across fast and slow poroelastic regimes. Two gel models—mesh-scaling and melt-like—yield distinct modulus dependencies, with coarsening exponents $t^{1/3}$ (fast) and $t^{1/4}$ (slow) and arrest lengths scaling with the modulus as $G^{-1/3}$ or $G^{-1/2}$ depending on regime and gel type, broadly consistent with experimental observations on melt-like gels. The framework provides experimentally testable predictions for viscoelastic-to-elastic transitions, solvent drainage times, and the role of viscosity ratios in magnifying arrest lengths, offering a unified view of coarsening and arrest in poroelastic gels.

Abstract

Recent experiments on crosslinked gels quenched from solvent-rich to solvent-poor conditions, show solvent-rich domains surrounded by gel-rich regions that coarsen followed by kinetic arrest at micron scales that persists for hours before full drainage to macroscopic equilibrium occurs on day timescales. Motivated by this, we present a general model for both coarsening and eventual arrest of pressure-driven coarsening in gels with solvent flow driven by interfacial tension of the solvent-rich domains and the gel. In gels in their viscoelastic time regime, this capillary force is dissipated by solvent transport through crosslinked polymers where the polymers develop transient elastic stresses due to solvent flow. At longer times, the long-ranged, elastic response of the gel provides a deterministic force which balances the capillary force thus arresting pressure-driven coarsening at a stiffness-dependent length. For a melt-like, polymer-rich gel, the coarsening length $\sim t^{1/4}$ where $t$ is the time, with an amplitude $ \sim G^{-1/2}$, where $G$ is the shear modulus. The arrest length in this regime also scales $\sim G^{-1/2}$. For low polymer fractions where the dominant length scale is the mesh size, the coarsening length can scale as $t^{1/3}$ varying with $G^{-1/3}$ (as does the arrest length). Our predictions for the arrest-length scaling with $\sim G^{-1/2}$ for the melt-like gel are consistent with the measurements.