Fig. 1: Schematics of different mechanisms to achieve negative compressibility.

a Negative compressibility transition in a closed system. The material radius initially decreases as the applied pressure increases. The radius then suddenly increases when the pressure passes a critical threshold, and the material undergoes a negative compressibility transition. b Negative compressibility induced by mass exchange in an open system. When the piston is compressed, the hydrostatic pressure increases and the fluid (blue) more fully penetrates the matrix of the poroelastic material (yellow). As a result of differing responses across the matrix, the effective volume of the material increases, giving rise to a form of negative compressibility. c Negative compressibility induced by energy exchange in an open system. As the material is stressed, it simultaneously absorbs energy from the surrounding environment (grey). This energy is converted into mechanical work in the material, causing it to expand against the applied pressure and exhibit negative compressibility. In all panels, \(R\) and \(P\) indicate radius and applied pressure, respectively, and \(\Delta R\) and \(\Delta P\) indicate the corresponding increments.