(3ak) Hard Tetrahedra: Entropy, Geometrical Anisotropy and Structural Complexity

It is the dream of every material scientist to design materials with desirable tunable properties that spontaneously assemble themselves from smaller bits. Self-assembly of such nano- and colloidal-scale bits is driven by the laws of thermodynamics rather than laborious top-down manufacturing. Recent progress in fabrication techniques has made possible a wide variety of anisotropic nano- and colloidal particles that mimic atoms and molecules in the self assembly of complicated structures. These novel building blocks can be anisotropic in both interaction and shape and can form structures not observed in atomic and molecular systems. A deep understanding of how anisotropy translates into structural complexity is pivotal in devising engineering guidelines for designing self-assembling bits. I am particularly interested in how shape anisotropy alone can give rise to structural complexity and my research plan is to use computational statistical mechanics to understand this relationship. 

Hard particles are the most suitable models for studying the exclusive effect of building block geometry on thermodynamic phase behavior. In the realm of atoms and molecules the hard potential mimics excluded volume interactions, and has, for decades, been successfully used for modeling dense states of matters such as liquids and solids. Since all permissible configurations of a hard particle system have the same potential energy, the thermodynamic behavior of such systems is purely determined by entropy. Although entropy is generally associated with 'lack of order', it is a well-known fact that hard particles can assemble into ordered structures that are stabilized by entropy. Such 'entropy-driven disorder-order transitions' have been observed in a wide variety of systems and seem to be ubiquitous in nature [1]. Computational studies of hard particle systems are also interesting due to the intimate relationship that exists between the problem of thermodynamic stability of hard particle systems and the packing problem in geometry. At the limit of infinite pressure, the densest possible arrangement of hard particles is thermodynamically stable. However finding the densest packing(s) of a particular geometric object is often analytically unapproachable, and thus numerical approaches are becoming popular ways of studying packing problems.

Hard tetrahedra, the main focus of my PhD dissertation, are the simple yet perplexing examples of geometrically anisotropic building blocks that self-assemble into complicated structures.  The regular tetrahedron is the simplest of the five Platonic solids.  In my research, Monte Carlo simulations revealed that these geometrically simple building blocks assemble into a dodecagonal quasicrystal, at sufficiently high densities due to entropy alone [2]. These quasi-periodic arrangements are surprisingly dense. An approximant of the quasicrystal that we constructed manually is an even denser packing and constituted the world record at the time of its discovery.  Quasicrystals are an exotic class of solids with long-range order that lack periodicity. In 2011, the Nobel Prize in Chemistry was awarded to Danieh Schechtman for the discovery of the first quasicrystal [3]. We found that the quasicrystal forms robustly from hard tetrahedra, and subsequently found the phase again in a system of hard triangular bipyramids (TBPs), which are dimers of hard tetrahedra [4]. We also confirm that in both systems, the quasicrystal approximant is more stable at intermediate densities than the now known densest packing of these polyhedra [4, 5].  This research suggests the rich and subtle role that entropy plays in stabilizing complex structures and the potential that geometric anisotropy has in inducing complex forms of order.

1. Daan Frenkel. Entropy-driven phase transitions. Physica A, 263:26–38, 1999.

2. Amir Haji-Akbari, Michael Engel, Aaron S. Keys, Xiaoyo Zheng, Rolfe Petschek, Peter Palffy-Muhoray, and Sharon C. Glotzer. Disordered, quasicrystalline and crystalline phase of densely-packed tetrahedra. Nature, 462:773-777, 2009.

3. D. Shechtman, I. Blech, D. Gratias, and J. W.Cahn. Metallic phase with long- range orientational order and no translational symmetry. Phys. Rev. Lett., 53:1951–1953, 1984.

4. Amir Haji-Akbari, Michael Engel, and Sharon C Glotzer. Degenerate quasicrystal of hard triangular bipyramids. Phys. Rev. Lett., 107:215702, 2011.

5. Amir Haji-Akbari, Michael Engel, and Sharon C Glotzer. Phase diagram of hard tetrahedra. J. Chem. Phys., 135:194101, 2011.