Aaron S. Keys

Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra

All hard, convex shapes are conjectured by Ulam to pack more densely than spheres, which have a maximum packing fraction of φ = π/√18 ≈ 0.7405. Simple lattice packings of many shapes easily surpass this packing fraction. For regular tetrahedra, this conjecture was shown to be true only very recently; an ordered arrangement was obtained via geometric construction with φ = 0.7786 (ref. 4), which was subsequently compressed numerically to φ = 0.7820 (ref. 5), while compressing with different initial conditions led to φ = 0.8230 (ref. 6). Here we show that tetrahedra pack even more densely, and in a completely unexpected way. Following a conceptually different approach, using thermodynamic computer simulations that allow the system to evolve naturally towards high-density states, we observe that a fluid of hard tetrahedra undergoes a first-order phase transition to a dodecagonal quasicrystal, which can be compressed to a packing fraction of φ = 0.8324. By compressing a crystalline approximant of the quasicrystal, the highest packing fraction we obtain is φ = 0.8503. If quasicrystal formation is suppressed, the system remains disordered, jams and compresses to φ = 0.7858. Jamming and crystallization are both preceded by an entropy-driven transition from a simple fluid of independent tetrahedra to a complex fluid characterized by tetrahedra arranged in densely packed local motifs of pentagonal dipyramids that form a percolating network at the transition. The quasicrystal that we report represents the first example of a quasicrystal formed from hard or non-spherical particles. Our results demonstrate that particle shape and entropy can produce highly complex, ordered structures.

How do quasicrystals grow

Using molecular simulations, we show that the aperiodic growth of quasicrystals is controlled by the ability of the growing quasicrystal nucleus to incorporate kinetically trapped atoms into the solid phase with minimal rearrangement. In the system under investigation, which forms a dodecagonal quasicrystal, we show that this process occurs through the assimilation of stable icosahedral clusters by the growing quasicrystal. Our results demonstrate how local atomic interactions give rise to the long-range aperiodicity of quasicrystals.

Self Assembly of Soft Matter Quasicrystals and Their Approximants

The surprising recent discoveries of quasicrystals and their approximants in soft-matter systems poses the intriguing possibility that these structures can be realized in a broad range of nanoscale and microscale assemblies. It has been theorized that soft-matter quasicrystals and approximants are largely entropically stabilized, but the thermodynamic mechanism underlying their formation remains elusive. Here, we use computer simulation and free-energy calculations to demonstrate a simple design heuristic for assembling quasicrystals and approximants in soft-matter systems. Our study builds on previous simulation studies of the self-assembly of dodecagonal quasicrystals and approximants in minimal systems of spherical particles with complex, highly specific interaction potentials. We demonstrate an alternative entropy-based approach for assembling dodecagonal quasicrystals and approximants based solely on particle functionalization and shape, thereby recasting the interaction-potential-based assembly strategy in terms of simpler-to-achieve bonded and excluded-volume interactions. Here, spherical building blocks are functionalized with mobile surface entities to encourage the formation of structures with low surface contact area, including non-close-packed and polytetrahedral structures. The building blocks also possess shape polydispersity, where a subset of the building blocks deviate from the ideal spherical shape, discouraging the formation of close-packed crystals. We show that three different model systems with both of these features—mobile surface entities and shape polydispersity—consistently assemble quasicrystals and/or approximants. We argue that this design strategy can be widely exploited to assemble quasicrystals and approximants on the nanoscale and microscale. In addition, our results further elucidate the formation of soft-matter quasicrystals in experiment.

Characterizing Structure Through Shape Matching and Applications to Self-Assembly

Structural quantities such as order parameters and correlation functions are often employed to gain insight into the physical behavior and properties of condensed matter systems. Although standard quantities for characterizing structure exist, often they are insufficient for treating problems in the emerging field of nano- and microscale self-assembly, wherein the structures encountered may be complex and unusual. The computer science field of shape matching offers a robust solution to this problem by defining diverse methods for quantifying the similarity between arbitrarily complex shapes. Most order parameters and correlation functions used in condensed matter apply a specific measure of structural similarity within the context of a broader scheme. By substituting shape matching quantities for traditional quantities, we retain the essence of the broader scheme, but extend its applicability to more complex structures. Here we review some standard shapematching techniques and discuss how they might be used to create highly flexible structural metrics for diverse systems such as self-assembled matter. We provide three proof-of-concept example problems applying shape-matching methods to identifying local and global structures and tracking structural transitions in complex assembled systems. The shape-matching methods reviewed here are applicable to a wide range of condensed matter systems, both simulated and experimental, provided particle positions are known or can be accurately imaged.

Calorimetric glass transition explained by hierarchical dynamic facilitation

The glass transition refers to the nonequilibrium process by which an equilibrium liquid is transformed to a nonequilibrium disordered solid, or vice versa. Associated response functions, such as heat capacities, are markedly different on cooling than on heating, and the response to melting a glass depends markedly on the cooling protocol by which the glass was formed. This paper shows how this irreversible behavior can be interpreted quantitatively in terms of an East-model picture of localized excitations (or soft spots) in which molecules can move with a specific direction, and from which excitations with the same directionality of motion can appear or disappear in adjacent regions. As a result of these facilitated dynamics, excitations become correlated in a hierarchical fashion. These correlations are manifested in the dynamic heterogeneity of the supercooled liquid phase. Although equilibrium thermodynamics is virtually featureless, a nonequilibrium glass phase emerges when the model is driven out of equilibrium with a finite cooling rate. The correlation length of this emergent phase is large and increases with decreasing cooling rate. A spatially and temporally resolved fictive temperature encodes memory of its preparation. Parameters characterizing the model can be determined from reversible transport data, and with these parameters, predictions of the model agree well with irreversible differential scanning calorimetry.

Manifestations of dynamical facilitation in glassy materials

By characterizing the dynamics of idealized lattice models with a tunable kinetic constraint, we explore the different ways in which dynamical facilitation manifests itself within the local dynamics of glassy materials. Dynamical facilitation is characterized both by a mobility transfer function, the propensity for highly mobile regions to arise near regions that were previously mobile, and by a facilitation volume, the effect of an initial dynamical event on subsequent dynamics within a region surrounding it. Sustained bursts of dynamical activity-avalanches-are shown to occur in kinetically constrained models, but, contrary to recent claims, we find that the decreasing spatiotemporal extent of avalanches with increased supercooling previously observed in granular experiments does not imply diminishing facilitation. Viewed within the context of existing simulation and experimental evidence, our findings show that dynamical facilitation plays a significant role in the dynamics of systems investigated over the range of state points accessible to molecular simulations and granular experiments.

A tale of two tilings

What do you get when you cross a crystal with a quasicrystal? The answer is a structure that links the ancient tiles of Archimedes, the iconic Fibonacci sequence of numbers and a book from the seventeenth century. Quasicrystals are unusual in that though aperiodic, they retain long-range crystalline order. Quasicrystalline surfaces hold tremendous interest due to their structural and chemical complexity leading to exotic properties. Mikhael et al. describe interactions between highly charged colloidal particles and a quasicrystalline surface generated through the exposure of the surface to a quasiperiodic patterned potential. They find that when the charges on the colloids are shielded, the surface acts as a template for the colloidal monolayer, which adopts order from the patterned potential to form a two-dimensional quasicrystal. But when the colloidal interactions are not shielded, transition to Archimedean tiling-like pattern occurs. In this, order is still locally commensurate with quasicrystalline structure but some of the periodicity observed when colloidal interactions dominate remains. However, over the long range, defects are now observed in a quasiperiodic pattern known as a Fibonacci chain. This is a useful colloidal model for atomic quasicrystalline systems and provides new insights into links between periodic and aperiodic crystalline order.

Using the s ensemble to probe glasses formed by cooling and aging

From length scale distributions characterizing frozen amorphous domains, we relate the s ensemble method with standard cooling and aging protocols for forming glass. We show that in a class of models where space-time scaling is in harmony with that of experiment, the spatial distributions of excitations obtained with the s ensemble are identical to those obtained through cooling or aging, but the computational effort for applying the s ensemble is generally many orders of magnitude smaller than that of straightforward numerical simulation of cooling or aging. We find that in contrast to the equilibrium ergodic state, a nonequilibrium length scale characterizes the anticorrelation between excitations and encodes the preparation history of glass states.

Materials science: A tale of two tilings.

What do you get when you cross a crystal with a quasicrystal? The answer is a structure that links the ancient tiles of Archimedes, the iconic Fibonacci sequence of numbers and a book from the seventeenth century. Quasicrystals are unusual in that though aperiodic, they retain long-range crystalline order. Quasicrystalline surfaces hold tremendous interest due to their structural and chemical complexity leading to exotic properties. Mikhael et al. describe interactions between highly charged colloidal particles and a quasicrystalline surface generated through the exposure of the surface to a quasiperiodic patterned potential. They find that when the charges on the colloids are shielded, the surface acts as a template for the colloidal monolayer, which adopts order from the patterned potential to form a two-dimensional quasicrystal. But when the colloidal interactions are not shielded, transition to Archimedean tiling-like pattern occurs. In this, order is still locally commensurate with quasicrystalline structure but some of the periodicity observed when colloidal interactions dominate remains. However, over the long range, defects are now observed in a quasiperiodic pattern known as a Fibonacci chain. This is a useful colloidal model for atomic quasicrystalline systems and provides new insights into links between periodic and aperiodic crystalline order.

A tale of two tilings: Materials science

What do you get when you cross a crystal with a quasicrystal? The answer is a structure that links the ancient tiles of Archimedes, the iconic Fibonacci sequence of numbers and a book from the seventeenth century. Quasicrystals are unusual in that though aperiodic, they retain long-range crystalline order. Quasicrystalline surfaces hold tremendous interest due to their structural and chemical complexity leading to exotic properties. Mikhael et al. describe interactions between highly charged colloidal particles and a quasicrystalline surface generated through the exposure of the surface to a quasiperiodic patterned potential. They find that when the charges on the colloids are shielded, the surface acts as a template for the colloidal monolayer, which adopts order from the patterned potential to form a two-dimensional quasicrystal. But when the colloidal interactions are not shielded, transition to Archimedean tiling-like pattern occurs. In this, order is still locally commensurate with quasicrystalline structure but some of the periodicity observed when colloidal interactions dominate remains. However, over the long range, defects are now observed in a quasiperiodic pattern known as a Fibonacci chain. This is a useful colloidal model for atomic quasicrystalline systems and provides new insights into links between periodic and aperiodic crystalline order.