275 research outputs found
Complexity in surfaces of densest packings for families of polyhedra
Packings of hard polyhedra have been studied for centuries due to their
mathematical aesthetic and more recently for their applications in fields such
as nanoscience, granular and colloidal matter, and biology. In all these
fields, particle shape is important for structure and properties, especially
upon crowding. Here, we explore packing as a function of shape. By combining
simulations and analytic calculations, we study three 2-parameter families of
hard polyhedra and report an extensive and systematic analysis of the densest
packings of more than 55,000 convex shapes. The three families have the
symmetries of triangle groups (icosahedral, octahedral, tetrahedral) and
interpolate between various symmetric solids (Platonic, Archimedean, Catalan).
We find that optimal (maximum) packing density surfaces that reveal unexpected
richness and complexity, containing as many as 130 different structures within
a single family. Our results demonstrate the utility of thinking of shape not
as a static property of an object in the context of packings, but rather as but
one point in a higher dimensional shape space whose neighbors in that space may
have identical or markedly different packings. Finally, we present and
interpret our packing results in a consistent and generally applicable way by
proposing a method to distinguish regions of packings and classify types of
transitions between them.Comment: 16 pages, 8 figure
Shape and symmetry determine two-dimensional melting transitions of hard regular polygons
The melting transition of two-dimensional (2D) systems is a fundamental
problem in condensed matter and statistical physics that has advanced
significantly through the application of computational resources and
algorithms. 2D systems present the opportunity for novel phases and phase
transition scenarios not observed in 3D systems, but these phases depend
sensitively on the system and thus predicting how any given 2D system will
behave remains a challenge. Here we report a comprehensive simulation study of
the phase behavior near the melting transition of all hard regular polygons
with vertices using massively parallel Monte Carlo simulations
of up to one million particles. By investigating this family of shapes, we show
that the melting transition depends upon both particle shape and symmetry
considerations, which together can predict which of three different melting
scenarios will occur for a given . We show that systems of polygons with as
few as seven edges behave like hard disks; they melt continuously from a solid
to a hexatic fluid and then undergo a first-order transition from the hexatic
phase to the fluid phase. We show that this behavior, which holds for all
, arises from weak entropic forces among the particles. Strong
directional entropic forces align polygons with fewer than seven edges and
impose local order in the fluid. These forces can enhance or suppress the
discontinuous character of the transition depending on whether the local order
in the fluid is compatible with the local order in the solid. As a result,
systems of triangles, squares, and hexagons exhibit a KTHNY-type continuous
transition between fluid and hexatic, tetratic, and hexatic phases,
respectively, and a continuous transition from the appropriate "x"-atic to the
solid. [abstract truncated due to arxiv length limitations]
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