Curvature as an external field in Ising-like puckered cylinders*

Recently, free-standing puckered sheets with arrays of bistable units were introduced as a mechanical analog of an Ising antiferromagnet with, however, a strong coupling to flexural phonons [1].  At finite temperature, Ising-like phase transitions and anomalous thermal expansion can be observed. Here, we present evidence that geometry can be used to control the phase behavior of this model: curvature produces a radius-dependent “external field” that suppresses the ordered anti-alignment of neighboring mechanical “spins.” Molecular dynamics simulations of puckered sheets in cylindrical geometries are used to probe the effect of curvature on the stability of configurations at zero temperature as well as phase transitions at finite temperature.  Continuum shallow-shell calculations support the mapping of curvature to a uniform magnetic field that couples to the staggered “magnetization”, defined as the average of the staggered buckled units. Furthermore, a simple phenomenological model predicts that the antiferromagnetic critical temperature decreases with increasing curvature, consistent with our simulation data. Our findings highlight the importance of curvature and temperature for designing meta-cylinders and motivates further study of highly compressible Ising models on curved surfaces.

[1] Paul Z. Hanakata, Abigail Plummer, and David R. Nelson. "Anomalous thermal expansion in Ising-like puckered sheets." arXiv preprint arXiv:2105.10015 (2021).