The shape memory effect and minimal surfaces

Martensitic transformations, viewed as continuous transformations between triply periodic minimal surfaces (TPMS), as originally proposed by Hyde and Andersson [Z. Kristallogr. 174, 225 (1986)], is extended to include paths between the initial and final phases. Bravais lattices correspond to particular TPMS whose lattice points are flat points, where the Gaussian curvature vanishes. Reversible transformations, which correspond to shape memory materials, occur only if lattice points remain at flat points on a TPMS throughout a continuous deformation. For the shape memory material NiTi, density-functional theory (DFT) yields irreversible and reversible paths with and without energy barriers, respectively. Although there are TPMS for face-centered gamma-Fe) and body-centered (alpha-Fe) cubic lattices, gamma to alpha deformation paths are not reversible, in agreement with non-vanishing energy barriers obtained from DFT.