The question of how densely regular tetrahedra can pack in three dimensions has attracted many researchers in recent years. In the first thermodynamic study of dense phases of hard tetrahedron system, we reported the spontaneous formation of a dodecagonal quasicrystal that could be further compressed to a surprisingly high packing fraction of 83.36% . The first-order approximant of the quasicrystal, with an 82-particle unit cell, was compressed to a then-record packing fraction of 85.03% . Very shortly after, a family of dense packings of dimers of tetrahedra were propsed with a slightly higher packing fraction (85.47%) , which were further optimized to a packing fraction of 85.63%, the world's current record . Since the dimer crystal, which has a much simpler, four-particle unit cell, packs more densely than the quasicrystal or its approximant, it is thermodynamically favored in the limit of infinite pressure. However, which structure is stable at finite pressures is an open question. Using thermodynamic integration and free energy calculations, we explore the relative thermodynamic stability of these very different ordered phases as a function of packing density .
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 Haji-Akbari A, Engel M, Petschek R, Glotzer S C, preprint.