Coulomb Floppy Networks: from soft to hard matter.*

Floppy Networks (FNs) play a prominent role in soft condensed matter physics, from polymeric gels and rubber to biomolecules, glasses, and granular materials. We demonstrate how the very same concept emerges in the context of a family of open-framework ionic solids, e.g. ScF₃, which can be conceptualized as Coulomb FNs. They exhibit unusual properties, including quantum structural phase transition near ambient pressure and negative thermal expansion (NTE). We connect these phenomena to FN-like crystalline architecture, stabilized by the net electrostatic repulsion, playing a role similar to the osmotic pressure in a polymeric gel. Our theory provides an accurate quantitative description of NTE and structural transition. Entropic stabilization of criticality explains the observed phase behavior. In addition, a significant entropic contribution to elasticity accounts for the marked discrepancy between numerical and experimentally observed compressibilities.