Freezing on a sphere.

The best understood crystal ordering transition is that of two-dimensional freezing, which proceeds by the rapid eradication of lattice defects as the temperature is lowered below a critical threshold. But crystals that assemble on closed surfaces are required by topology to have a minimum number of lattice defects, called disclinations, that act as conserved topological charges-consider the 12 pentagons on a football or the 12 pentamers on a viral capsid. Moreover, crystals assembled on curved surfaces can spontaneously develop additional lattice defects to alleviate the stress imposed by the curvature. It is therefore unclear how crystallization can proceed on a sphere, the simplest curved surface on which it is impossible to eliminate such defects. Here we show that freezing on the surface of a sphere proceeds by the formation of a single, encompassing crystalline 'continent', which forces defects into 12 isolated 'seas' with the same icosahedral symmetry as footballs and viruses. We use this broken symmetry-aligning the vertices of an icosahedron with the defect seas and unfolding the faces onto a plane-to construct a new order parameter that reveals the underlying long-range orientational order of the lattice. The effects of geometry on crystallization could be taken into account in the design of nanometre- and micrometre-scale structures in which mobile defects are sequestered into self-ordered arrays. Our results may also be relevant in understanding the properties and occurrence of natural icosahedral structures such as viruses.

Dynamical Approach to the Jamming Problem.

A simple dynamical model, biased random organization (BRO), appears to produce configurations known as random close packing (RCP) as BRO's densest critical point in dimension d=3. We conjecture that BRO likewise produces RCP in any dimension; if so, then RCP does not exist in d=1-2 (where BRO dynamics lead to crystalline order). In d=3-5, BRO produces isostatic configurations and previously estimated RCP volume fractions 0.64, 0.46, and 0.30, respectively. For all investigated dimensions (d=2-5), we find that BRO belongs to the Manna universality class of dynamical phase transitions by measuring critical exponents associated with the steady-state activity and the long-range density fluctuations. Additionally, BRO's distribution of near contacts (gaps) displays behavior consistent with the infinite-dimensional theoretical treatment of RCP when d≥4. The association of BRO's densest critical configurations with random close packing implies that RCP's upper-critical dimension is consistent with the Manna class d_{uc}=4.

Microswimmers under the spotlight: interplay between agents with different levels of activity.

The ability of active matter to assemble into reconfigurable nonequilibrium structures has drawn considerable interest in recent years. We investigate how active fluids respond to spatial light patterns through simulations and experiments on light-activated self-propelled colloidal particles. We examine the processes of inverse templated assembly, which involves creating a region without active particles through a bright pattern, and templated assembly, which promotes the formation of dense particle regions through a dark pattern. We identify scaling relations for the characteristic times for both processes that quantify the interplay between the dimension of the applied pattern and the intrinsic properties of the active fluid. We also explore the assembly mechanism and dynamics of large clusters and show how assembly and inverse assembly can be combined to create any arbitrarily complex template. In addition to providing protocols for templated assembly via light patterning, our results demonstrate how the local packing fraction can be fine-tuned by modulation of the light intensity. The protocol so obtained exceeds the capabilities of conventional assembly strategies, in which packing fraction is dictated by thermodynamics, and opens the door to arbitrarily precise and programmable nonequilibrium assembly strategies in active matter.

Coupled Dynamical Phase Transitions in Driven Disk Packings.

Under the influence of oscillatory shear, a monolayer of frictional granular disks exhibits two dynamical phase transitions: a transition from an initially disordered state to an ordered crystalline state and a dynamic active-absorbing phase transition. Although there is no reason a priori for these to be at the same critical point, they are. The transitions may also be characterized by the disk trajectories, which are nontrivial loops breaking time-reversal invariance.

Model-Free Measurement of Local Entropy Production and Extractable Work in Active Matter.

Time-reversal symmetry breaking and entropy production are universal features of nonequilibrium phenomena. Despite its importance in the physics of active and living systems, the entropy production of systems with many degrees of freedom has remained of little practical significance because the high dimensionality of their state space makes it difficult to measure. Here we introduce a local measure of entropy production and a numerical protocol to estimate it. We establish a connection between the entropy production and extractability of work in a given region of the system and show how this quantity depends crucially on the degrees of freedom being tracked. We validate our approach in theory, simulation, and experiments by considering systems of active Brownian particles undergoing motility-induced phase separation, as well as active Brownian particles and E.coli in a rectifying device in which the time-reversal asymmetry of the particle dynamics couples to spatial asymmetry to reveal its effects on a macroscopic scale.

Litters of self-replicating origami cross-tiles.

Self-replication and exponential growth are ubiquitous in nature but until recently there were few examples of artificial self-replication. Often replication is a templated process where a parent produces a single offspring, doubling the population in each generation. Many species however produce more than one offspring at a time, enabling faster population growth and higher probability of species perpetuation. We have made a system of cross-shaped origami tiles that yields a number of offspring, four to eight or more, depending on the concentration of monomer units to be assembled. The parent dimer template serves as a seed to crystallize a one-dimensional crystal, a ladder. The ladder rungs are then UV-cross-linked and the offspring are then released by heating, to yield a litter of autonomous daughters. In the complement study, we also optimize the growth conditions to speed up the process and yield a 103 increase in the growth rate for the single-offspring replication system. Self-replication and exponential growth of autonomous motifs is useful for fundamental studies of selection and evolution as well as for materials design, fabrication, and directed evolution. Methods that increase the growth rate, the primary evolutionary drive, not only speed up experiments but provide additional mechanisms for evolving materials toward desired functionalities.

Conching chocolate is a prototypical transition from frictionally jammed solid to flowable suspension with maximal solid content.

The mixing of a powder of 10- to 50-µm primary particles into a liquid to form a dispersion with the highest possible solid content is a common industrial operation. Building on recent advances in the rheology of such "granular dispersions," we study a paradigmatic example of such powder incorporation: the conching of chocolate, in which a homogeneous, flowing suspension is prepared from an inhomogeneous mixture of particulates, triglyceride oil, and dispersants. Studying the rheology of a simplified formulation, we find that the input of mechanical energy and staged addition of surfactants combine to effect a considerable shift in the jamming volume fraction of the system, thus increasing the maximum flowable solid content. We discuss the possible microscopic origins of this shift, and suggest that chocolate conching exemplifies a ubiquitous class of powder-liquid mixing.

Random Close Packing as a Dynamical Phase Transition.

Sphere packing is an ancient problem. The densest packing is known to be a face-centered cubic (FCC) crystal, with space-filling fraction ϕ_{FCC}=π/sqrt[18]≈0.74. The densest "random packing," random close packing (RCP), is yet ill defined, although many experiments and simulations agree on a value ϕ_{RCP}≈0.64. We introduce a simple absorbing-state model, biased random organization (BRO), which exhibits a Manna class dynamical phase transition between absorbing and active states that has as its densest critical point ϕ_{c_{max}}≈0.64≈ϕ_{RCP} and, like other Manna class models, is hyperuniform at criticality. The configurations we obtain from BRO appear to be structurally identical to RCP configurations from other protocols. This leads us to conjecture that the highest-density absorbing state for an isotropic biased random organization model produces an ensemble of configurations that characterizes the state conventionally known as RCP.

Multivalent, multiflavored droplets by design.

Nature self-assembles functional materials by programming flexible linear arrangements of molecules and then folding them to make 2D and 3D objects. To understand and emulate this process, we have made emulsion droplets with specific recognition and controlled valence. Uniquely monovalent droplets form dimers: divalent lead to polymer-like chains, trivalent allow for branching, and programmed mixtures of different valences enable a variety of designed architectures and the ability to subsequently close and open structures. Our functional building blocks are a hybrid of micrometer-scale emulsion droplets and nanoscale DNA origami technologies. Functional DNA origami rafts are first added to droplets and then herded into a patch using specifically designated "shepherding" rafts. Additional patches with the same or different specificities can be formed on the same droplet, programming multiflavored, multivalence droplets. The mobile patch can bind to a patch on another droplet containing complementary functional rafts, leading to primary structure formation. Further binding of nonneighbor droplets can produce secondary structures, a third step in hierarchical self-assembly. The use of mobile patches rather than uniform DNA coverage has the advantage of valence control at the expense of slow kinetics. Droplets with controlled flavors and valences enable a host of different material and device architectures.

Hyperuniform Structures Formed by Shearing Colloidal Suspensions.

In periodically sheared suspensions there is a dynamical phase transition, characterized by a critical strain amplitude γ_{c}, between an absorbing state where particle trajectories are reversible and an active state where trajectories are chaotic and diffusive. Repulsive nonhydrodynamic interactions between "colliding" particles' surfaces have been proposed as a source of this broken time reversal symmetry. A simple toy model called random organization qualitatively reproduces the dynamical features of this transition. Random organization and other absorbing state models exhibit hyperuniformity, a strong suppression of density fluctuations on long length scales quantified by a structure factor S(q→0)∼q^{α} with α>0, at criticality. Here we show experimentally that the particles in periodically sheared suspensions organize into structures with anisotropic short-range order but isotropic, long-range hyperuniform order when oscillatory shear amplitudes approach γ_{c}.

Correlation Lengths in the Language of Computable Information.

Computable information density (CID), the ratio of the length of a losslessly compressed data file to that of the uncompressed file, is a measure of order and correlation in both equilibrium and nonequilibrium systems. Here we show that correlation lengths can be obtained by decimation, thinning a configuration by sampling data at increasing intervals and recalculating the CID. When the sampling interval is larger than the system's correlation length, the data becomes incompressible. The correlation length and its critical exponents are thus accessible with no a priori knowledge of an order parameter or even the nature of the ordering. The correlation length measured in this way agrees well with that computed from the decay of two-point correlation functions g_{2}(r) when they exist. But the CID reveals the correlation length and its scaling even when g_{2}(r) has no structure, as we demonstrate by "cloaking" the data with a Rudin-Shapiro sequence.

Mix and match-a versatile equilibrium approach for hybrid colloidal synthesis.

Colloidal synthesis is a powerful bottom-up approach for programmed self-assembly which holds promise for both research and industry. While diverse, each synthetic process is typically restricted to a specific chemistry. Many applications however require composite materials, whereas a chemical equilibrium can typically only match one material but not the other. Here, a scalable general approach is presented, alleviating the dependency on a specific chemical reaction, by resorting to a mechanical equilibrium; an isopycnic density-gradient-step is tailored to form clusters with prescribed composition. Valence control is demonstrated, making dimers, trimers, and tetramers with purity as high as 96%. The measured kinetics shows a scaleable throughput. The density gradient step plays a dual role of both filtering out undesired products and concentrating the target structures. The "Mix-and-Match" approach is general, and applies to a broad range of colloidal matter: diverse material compositions (plastics, glasses, and emulsions); a range of colloidal interactions (van der Waals, Coulomb, and DNA hybridization); and a spectrum of sizes (nanoscale to multiple micrometers). Finally, the strength of the method is displayed by producing a monodisperse suspension from a highly polydisperse emulsion. The ability to combine colloids into architectures of hybrid materials has applications in pharmaceuticals, cosmetics, and photonics.

Enhanced hyperuniformity from random reorganization.

Diffusion relaxes density fluctuations toward a uniform random state whose variance in regions of volume [Formula: see text] scales as [Formula: see text] Systems whose fluctuations decay faster, [Formula: see text] with [Formula: see text], are called hyperuniform. The larger [Formula: see text], the more uniform, with systems like crystals achieving the maximum value: [Formula: see text] Although finite temperature equilibrium dynamics will not yield hyperuniform states, driven, nonequilibrium dynamics may. Such is the case, for example, in a simple model where overlapping particles are each given a small random displacement. Above a critical particle density [Formula: see text], the system evolves forever, never finding a configuration where no particles overlap. Below [Formula: see text], however, it eventually finds such a state, and stops evolving. This "absorbing state" is hyperuniform up to a length scale [Formula: see text], which diverges at [Formula: see text] An important question is whether hyperuniformity survives noise and thermal fluctuations. We find that hyperuniformity of the absorbing state is not only robust against noise, diffusion, or activity, but that such perturbations reduce fluctuations toward their limiting behavior, [Formula: see text], a uniformity similar to random close packing and early universe fluctuations, but with arbitrary controllable density.

Long-wavelength fluctuations and the glass transition in two dimensions and three dimensions.

Phase transitions significantly differ between 2D and 3D systems, but the influence of dimensionality on the glass transition is unresolved. We use microscopy to study colloidal systems as they approach their glass transitions at high concentrations and find differences between two dimensions and three dimensions. We find that, in two dimensions, particles can undergo large displacements without changing their position relative to their neighbors, in contrast with three dimensions. This is related to Mermin-Wagner long-wavelength fluctuations that influence phase transitions in two dimensions. However, when measuring particle motion only relative to their neighbors, two dimensions and three dimensions have similar behavior as the glass transition is approached, showing that the long-wavelength fluctuations do not cause a fundamental distinction between 2D and 3D glass transitions.

Exponential growth and selection in self-replicating materials from DNA origami rafts.

Self-replication and evolution under selective pressure are inherent phenomena in life, and but few artificial systems exhibit these phenomena. We have designed a system of DNA origami rafts that exponentially replicates a seed pattern, doubling the copies in each diurnal-like cycle of temperature and ultraviolet illumination, producing more than 7 million copies in 24 cycles. We demonstrate environmental selection in growing populations by incorporating pH-sensitive binding in two subpopulations. In one species, pH-sensitive triplex DNA bonds enable parent-daughter templating, while in the second species, triplex binding inhibits the formation of duplex DNA templating. At pH 5.3, the replication rate of species I is  ∼1.3-1.4 times faster than that of species II. At pH 7.8, the replication rates are reversed. When mixed together in the same vial, the progeny of species I replicate preferentially at pH 7.8; similarly at pH 5.3, the progeny of species II take over the system. This addressable selectivity should be adaptable to the selection and evolution of multi-component self-replicating materials in the nanoscopic-to-microscopic size range.

Magneto-capillary dynamics of amphiphilic Janus particles at curved liquid interfaces.

A homogeneous magnetic field can exert no net force on a colloidal particle. However, by coupling the particle's orientation to its position on a curved interface, even static homogeneous fields can be used to drive rapid particle motions. Here, we demonstrate this effect using magnetic Janus particles with amphiphilic surface chemistry adsorbed at the spherical interface of a water drop in decane. Application of a static homogeneous field drives particle motion to the drop equator where the particle's magnetic moment can align parallel to the field. As explained quantitatively by a simple model, the effective magnetic force on the particle scales linearly with the curvature of the interface. For particles adsorbed on small droplets such as those found in emulsions, these magneto-capillary forces can far exceed those due to magnetic field gradients in both magnitude and range. This mechanism may be useful in creating highly responsive emulsions and foams stabilized by magnetic particles.

Precisely cyclic sand: self-organization of periodically sheared frictional grains.

The disordered static structure and chaotic dynamics of frictional granular matter has occupied scientists for centuries, yet there are few organizational principles or guiding rules for this highly hysteretic, dissipative material. We show that cyclic shear of a granular material leads to dynamic self-organization into several phases with different spatial and temporal order. Using numerical simulations, we present a phase diagram in strain-friction space that shows chaotic dispersion, crystal formation, vortex patterns, and most unusually a disordered phase in which each particle precisely retraces its unique path. However, the system is not reversible. Rather, the trajectory of each particle, and the entire frictional, many-degrees-of-freedom system, organizes itself into a limit cycle absorbing state. Of particular note is that fact that the cyclic states are spatially disordered, whereas the ordered states are chaotic.

Wrinkles and splay conspire to give positive disclinations negative curvature.

Recently, there has been renewed interest in the coupling between geometry and topological defects in crystalline and striped systems. Standard lore dictates that positive disclinations are associated with positive Gaussian curvature, whereas negative disclinations give rise to negative curvature. Here, we present a diblock copolymer system exhibiting a striped columnar phase that preferentially forms wrinkles perpendicular to the underlying stripes. In free-standing films this wrinkling behavior induces negative Gaussian curvature to form in the vicinity of positive disclinations.

Shape-sensitive crystallization in colloidal superball fluids.

Guiding the self-assembly of materials by controlling the shape of the individual particle constituents is a powerful approach to material design. We show that colloidal silica superballs crystallize into canted phases in the presence of depletants. Some of these phases are consistent with the so-called "Λ1" lattice that was recently predicted as the densest packing of superdisks. As the size of the depletant is reduced, however, we observe a transition to a square phase. The differences in these entropically stabilized phases result from an interplay between the size of the depletants and the fine structure of the superball shape. We find qualitative agreement of our experimental results both with a phase diagram computed on the basis of the volume accessible to the depletants and with simulations. By using a mixture of depletants, one of which is thermosensitive, we induce solid-to-solid phase transitions between square and canted structures. The use of depletant size to leverage fine features of the shape of particles in driving their self-assembly demonstrates a general and powerful mechanism for engineering novel materials.