Entropy favors heterogeneous structures of networks near the rigidity threshold.

The dynamical properties and mechanical functions of amorphous materials are governed by their microscopic structures, particularly the elasticity of the interaction networks, which is generally complicated by structural heterogeneity. This ubiquitous heterogeneous nature of amorphous materials is intriguingly attributed to a complex role of entropy. Here, we show in disordered networks that the vibrational entropy increases by creating phase-separated structures when the interaction connectivity is close to the onset of network rigidity. The stress energy, which conversely penalizes the heterogeneity, finally dominates a smaller vicinity of the rigidity threshold at the glass transition and creates a homogeneous intermediate phase. This picture of structures changing between homogeneous and heterogeneous phases by varying connectivity provides an interpretation of the transitions observed in chalcogenide glasses.