Classifying Crystals of Rounded Tetrahedra and Determining Their Order Parameters Using Dimensionality Reduction

Using
simulations we study the phase behavior of a family of hard
spherotetrahedra, a shape that interpolates between tetrahedra and
spheres. We identify 13 close-packed structures, some with packings
that are significantly denser than previously reported. Twelve of
these are crystals with unit cells of N = 2 or N = 4 particles, but in the shape regime of slightly rounded
tetrahedra we find that the densest structure is a quasicrystal approximant
with a unit cell of N = 82 particles. All 13 structures
are also stable below close packing, together with an additional 14th
plastic crystal phase at the sphere side of the phase diagram, and
upon sufficient dilution to packing fractions below 50–60%
all structures melt. Interestingly, however, upon compressing the
fluid phase, self-assembly takes place spontaneously only at the tetrahedron
and the sphere side of the family but not in an intermediate
regime of tetrahedra with rounded edges. We describe the local environment
of each particle by a set of l-fold bond orientational
order parameters q̅l, which we use in an extensive principal component analysis. We find
that the total packing fraction as well as several particular linear
combinations of q̅l rather than individual q̅l’s are optimally distinctive, specifically the differences q̅4 – q̅6 for separating tetragonal from hexagonal structures and q̅4–q̅8 for distinguishing tetragonal structures. We argue that these
characteristic combinations are also useful as reliable order parameters
in nucleation studies, enhanced sampling techniques, or inverse-design
methods involving odd-shaped particles in general.