Fig. 2: Cycle of intrusion and extrusion in a spring-fluid metastable model exhibiting negative compressibility.

The black dashed line marks the boundary of a hydrophobic cavity, the blue (white) background represents the water (vapor), and the dark blue line indicates the gas-liquid interface. a, b Positive compressibility in a vapor-filled cavity, where the cavity radius is determined by the balance between the applied hydrostatic pressure and the internal restorative force (represented by the springs). The radius decreases continuously as the pressure is progressively increased. c Negative compressibility transition as fluid intrusion occurs, where the cavity expands as the pressure passes a critical threshold. d A second negative compressibility transition as fluid extrusion occurs upon pressure reduction. e End of the cycle following a second leg of positive compressibility, in which the radius increases upon pressure reduction. Intrusion and extrusion cycles allow the system to switch between states,  leading to repeatable negative compressibility transitions. The mathematical symbols indicate radii and pressures, mirroring the notation in Fig. 1. (Image developed based on ref. ²).