Geometric frustration in buckled colloidal monolayers

Guest-guided anchoring patterns of cyclodextrin supramolecular microcrystals on droplet surfaces

The growth of cyclodextrin inclusion complexes (ICs) on oil/water interfaces represents a beautiful example of spontaneous pattern formation in nature. How the supramolecules evolve remains a challenge because surface confinement can frustrate microcrystal growth and give rise to unusual phase transitions. Here we investigate the self-assembly of ICs on droplet surfaces using microfluidics, which allows directly visualizing packing, wetting and ordering of the microcrystals anchored on the surface. The oil guests of distinct molecular structures can direct the assembly of the ICs and largely affect anchoring dynamics of the ICs microcrystals, leading to a range of behaviors including orientating, slipping, buckling, jamming, or merging. We discuss the behaviors observed in terms of the flexibility of the building blocks, which offers a new degree of freedom through which to tailor their properties and gives rise to a striking feature of anchoring patterns that have no counterpart in normal colloidal crystals.

Emergent disorder and mechanical memory in periodic metamaterials

Ordered mechanical systems typically have one or only a few stable rest configurations, and hence are not considered useful for encoding memory. Multistable and history-dependent responses usually emerge from quenched disorder, for example in amorphous solids or crumpled sheets. In contrast, due to geometric frustration, periodic magnetic systems can create their own disorder and espouse an extensive manifold of quasi-degenerate configurations. Inspired by the topological structure of frustrated artificial spin ices, we introduce an approach to design ordered, periodic mechanical metamaterials that exhibit an extensive set of spatially disordered states. While our design exploits the correspondence between frustration in magnetism and incompatibility in meta-mechanics, our mechanical systems encompass continuous degrees of freedom, and thus generalize their magnetic counterparts. We show how such systems exhibit non-Abelian and history-dependent responses, as their state can depend on the order in which external manipulations were applied. We demonstrate how this richness of the dynamics enables to recognize, from a static measurement of the final state, the sequence of operations that an extended system underwent. Thus, multistability and potential to perform computation emerge from geometric frustration in ordered mechanical lattices that create their own disorder.

Motility-induced phase separation and frustration in active matter swarmalators

We introduce a two dimensional system of active matter swarmalators composed of elastically interacting run-and-tumble active disks with an internal parameter ϕ_{i}. The disks experience an additional attractive or repulsive force with neighboring disks depending upon their relative difference in ϕ_{i}, making them similar to swarmalators used in robotic systems. In the absence of the internal parameter, the system forms a motility-induced phase separated (MIPS) state, but when the swarmalator interactions are present, a wide variety of other active phases appear depending upon whether the interaction is attractive or repulsive and whether the particles act to synchronize or ant-synchronize their internal parameter values. These phases include a gas-free gel regime, arrested clusters, a labyrinthine state, a regular MIPS state, a frustrated MIPS state for attractive antisynchronization, and a superlattice MIPS state for attractive synchronization.

Self-driven configurational dynamics in frustrated spring-mass systems

Various physical systems relax mechanical frustration through configurational rearrangements. We examine such rearrangements via Hamiltonian dynamics of simple internally stressed harmonic four-mass systems. We demonstrate theoretically and numerically how mechanical frustration controls the underlying potential energy landscape. Then, we examine the harmonic four-mass systems' Hamiltonian dynamics and relate the onset of chaotic motion to self-driven rearrangements. We show such configurational dynamics may occur without strong precursors, rendering such dynamics seemingly spontaneous.

Pseudo-Goldstone Modes and Dynamical Gap Generation from Order by Thermal Disorder

Accidental ground state degeneracies-those not a consequence of global symmetries of the Hamiltonian-are inevitably lifted by fluctuations, often leading to long-range order, a phenomenon known as "order-by-disorder" (ObD). The detection and characterization of ObD in real materials currently lacks clear, qualitative signatures that distinguish ObD from conventional energetic selection. We show that for order by thermal disorder (ObTD) such a signature exists: a characteristic temperature dependence of the fluctuation-induced pseudo-Goldstone gap. We demonstrate this in a minimal two-dimensional model that exhibits ObTD, the ferromagnetic Heisenberg-compass model on a square lattice. Using spin-dynamics simulations and self-consistent mean-field calculations, we determine the pseudo-Goldstone gap, Δ, and show that at low temperatures it scales as the square root of temperature, sqrt[T]. We establish that a power-law temperature dependence of the gap is a general consequence of ObTD, showing that all key features of this physics can be captured in a simple model of a particle moving in an effective potential generated by the fluctuation-induced free energy.

Non-orientable order and non-commutative response in frustrated metamaterials

From atomic crystals to animal flocks, the emergence of order in nature is captured by the concept of spontaneous symmetry breaking^1-4. However, this cornerstone of physics is challenged when broken symmetry phases are frustrated by geometrical constraints. Such frustration dictates the behaviour of systems as diverse as spin ices^5-8, confined colloidal suspensions⁹ and crumpled paper sheets¹⁰. These systems typically exhibit strongly degenerated and heterogeneous ground states and hence escape the Ginzburg-Landau paradigm of phase ordering. Here, combining experiments, simulations and theory we uncover an unanticipated form of topological order in globally frustrated matter: non-orientable order. We demonstrate this concept by designing globally frustrated metamaterials that spontaneously break a discrete [Formula: see text] symmetry. We observe that their equilibria are necessarily heteregeneous and extensively degenerated. We explain our observations by generalizing the theory of elasticity to non-orientable order-parameter bundles. We show that non-orientable equilibria are extensively degenerated due to the arbitrary location of topologically protected nodes and lines where the order parameter must vanish. We further show that non-orientable order applies more broadly to objects that are non-orientable themselves, such as buckled Möbius strips and Klein bottles. Finally, by applying time-dependent local perturbations to metamaterials with non-orientable order, we engineer topologically protected mechanical memories^11-19, achieve non-commutative responses and show that they carry an imprint of the braiding of the loads' trajectories. Beyond mechanics, we envision non-orientability as a robust design principle for metamaterials that can effectively store information across scales, in fields as diverse as colloidal science⁸, photonics²⁰, magnetism⁷ and atomic physics²¹.

Depletion-driven antiferromagnetic, paramagnetic, and ferromagnetic behavior in quasi-two-dimensional buckled colloidal solids

We investigate quasi-two-dimensional buckled colloidal monolayers on a triangular lattice with tunable depletion interactions. Without depletion attraction, the experimental system provides a colloidal analog of the well-known geometrically frustrated Ising antiferromagnet [Y. Han et al., Nature 456, 898-903 (2008)]. In this contribution, we show that the added depletion attraction can influence both the magnitude and sign of an Ising spin coupling constant. As a result, the nearest-neighbor Ising "spin" interactions can be made to vary from antiferromagnetic to para- and ferromagnetic. Using a simple theory, we compute an effective Ising nearest-neighbor coupling constant, and we show how competition between entropic effects permits for the modification of the coupling constant. We then experimentally demonstrate depletion-induced modification of the coupling constant, including its sign, and other behaviors. Depletion interactions are induced by rod-like surfactant micelles that change length with temperature and thus offer means for tuning the depletion attraction in situ. Buckled colloidal suspensions exhibit a crossover from an Ising antiferromagnetic to paramagnetic phase as a function of increasing depletion attraction. Additional dynamical experiments reveal structural arrest in various regimes of the coupling-constant, driven by different mechanisms. In total, this work introduces novel colloidal matter with "magnetic" features and complex dynamics rarely observed in traditional spin systems.

Nonadditive Interactions Unlock Small-Particle Mobility in Binary Colloidal Monolayers

We examine the organization and dynamics of binary colloidal monolayers composed of micron-scale silica particles interspersed with smaller-diameter silica particles that serve as minority component impurities. These binary monolayers are prepared at the surface of ionic liquid droplets over a range of size ratios (σ = 0.16-0.66) and are studied with low-dose minimally perturbative scanning electron microscopy (SEM). The high resolution of SEM imaging provides direct tracking of all particle coordinates over time, enabling a complete description of the microscopic state. In these bidisperse size mixtures, particle interactions are nonadditive because interfacial pinning to the droplet surface causes the equators of differently sized particles to lie in separate planes. By varying the size ratio, we control the extent of nonadditivity in order to achieve phase behavior inaccessible to additive 2D systems. Across the range of size ratios, we tune the system from a mobile small-particle phase (σ < 0.24) to an interstitial solid (0.24 < σ < 0.33) and furthermore to a disordered glass (σ > 0.33). These distinct phase regimes are classified through measurements of hexagonal ordering of the large-particle host lattice and the lattice's capacity for small-particle transport. Altogether, we explain these structural and dynamic trends by considering the combined influence of interparticle interactions and the colloidal packing geometry. Our measurements are reproduced in molecular dynamics simulations of 2D nonadditive disks, suggesting an efficient method for describing confined systems with reduced dimensionality representations.

Explicit demonstration of geometric frustration in chiral liquid crystals

Many solid materials and liquid crystals exhibit geometric frustration, meaning that they have an ideal local structure that cannot fill up space. For that reason, the global phase must be a compromise between the ideal local structure and geometric constraints. As an explicit example of geometric frustration, we consider a chiral liquid crystal confined in a long cylinder with free boundaries. When the radius of the tube is sufficiently small, the director field forms a double-twist configuration, which is the ideal local structure. However, when the radius becomes larger (compared with the natural twist of the liquid crystal), the double-twist structure cannot fill space, and hence the director field must transform into some other chiral structure that can fill space. This space-filling structure may be either (1) a cholesteric phase with single twist, or (2) a set of double-twist regions separated by a disclination, which can be regarded as the beginning of a blue phase. We investigate these structures using theory and simulations, and show how the relative free energies depend on the system size, the natural twist, and the disclination energy. As another example, we also study a cholesteric liquid crystal confined between two infinite parallel plates with free boundaries.

Lattice Induced Short-Range Attraction between Like-Charged Colloidal Particles

We perform experiments and computer simulations to study the effective interactions between like-charged colloidal tracers moving in a two-dimensional fluctuating background of colloidal crystal. By a counting method that properly accounts for the configurational degeneracy of tracer pairs, we extract the relative probability of finding a tracer pair in neighboring triangular cells formed by background particles. We find that this probability at the nearest neighbor cell is remarkably greater than those at cells with larger separations, implying an effective attraction between the tracers. This effective attraction weakens sharply as the background lattice constant increases. Furthermore, we clarify that the lattice-mediated effective attraction originates from the minimization of free energy increase from deformation of the crystalline background due to the presence of diffusing tracers.

Topological Order in an Antiferromagnetic Tetratic Phase

We study lattice melting in two-dimensional systems of spinful particles that interact antiferromagnetically. We argue that, for strong spin interactions, single lattice dislocations are forbidden by magnetic frustration. This leads to a melting scenario in which a tetratic phase, containing free dislocation pairs and bound disclinations, separates the solid from the liquid. We demonstrate this phase numerically in a system of hard spheres confined between parallel plates, where spins are represented by the heights of the spheres. In the tetratic phase, the spins are shown to be as antiferromagnetically ordered as allowed by their spatial configuration.

Wave spectroscopy in a driven granular material

Driven granular media constitute model systems in out-of-equilibrium statistical physics. By assimilating the motions of granular particles to those of atoms, by analogy, one can obtain macroscopic equivalent of phase transitions. Here, we study fluid-like and crystal-like two-dimensional states in a driven granular material. In our experimental device, a tunable magnetic field induces and controls remote interactions between the granular particles. We use high-speed video recordings to analyse the velocity fluctuations of individual particles in stationary regime. Using statistical averaging, we find that the particles self-organize into collective excitations characterized by dispersion relations in the frequency-wavenumber space. These findings thus reveal that mechanical waves analogous to condensed matter phonons propagate in driven granular media. When the magnetic coupling is weak, the waves are longitudinal, as expected for a fluid-like phase. When the coupling is stronger, both longitudinal and transverse waves propagate, which is typically seen in solid-like phases. We model the dispersion relations using the spatial distribution of particles and their interaction potential. Finally, we infer the elastic parameters of the granular assembly from equivalent sound velocities, thus realizing the spectroscopy of a granular material.

Multifunctional Silica-Modified Hybrid Microgels Templated from Inverse Pickering Emulsions

Microgels are regarded as soft colloids with environmental responsiveness. However, the majority of reported microgels are inherently hydrophilic, resulting in aqueous dispersions, and only used in water-based applications. Herein, we reported an efficient method for hybridization of poly(N-isopropylacrylamide) microgel by coating hydrophobic silica nanoparticles on their surface. The resultant hybrid microgel had switchable surface wettability and could be dispersed in both aqueous and oil phases. Meanwhile, the coated hydrophobic silica nanoparticles solved the difficulty in redispersing microgels caused by particle aggregation and film formation during the drying process, providing a significant advantage in dried storage. Furthermore, the introduction of hydrophobic silica nanoparticles endowed the hybrid microgel with a variety of applications, including cargo encapsulation, active release induced by emulsion reversion, and trace water absorption.

Frustrated Superradiant Phase Transition

Frustration occurs when a system cannot find a lowest-energy configuration due to conflicting constraints. We show that a frustrated superradiant phase transition occurs when the ground-state superradiance of cavity fields due to local light-matter interactions cannot simultaneously minimize the positive photon hopping energies. We solve the Dicke trimer model on a triangle motif with both negative and positive hopping energies and show that the latter results in a sixfold degenerate ground-state manifold in which the translational symmetry is spontaneously broken. In the frustrated superradiant phase, we find that two sets of diverging time and fluctuation scales coexist, one governed by the mean-field critical exponent and another by a novel critical exponent. The latter is associated with the fluctuation in the difference of local order parameters and gives rise to site-dependent photon number critical exponents, which may serve as an experimental probe for the frustrated superradiant phase. We provide a qualitative explanation for the emergence of unconventional critical scalings and demonstrate that they are generic properties of the frustrated superradiant phase at the hand of a one-dimensional Dicke lattice with an odd number of sites. The mechanism for the frustrated superradiant phase transition discovered here applies to any lattice geometries where the antiferromagnetic ordering of neighboring sites are incompatible and therefore our work paves the way toward the exploration of frustrated phases of coupled light and matter.

Mutual information and breakdown of the Perron-Frobenius scenario in zero-temperature triangular Ising antiferromagnets on cylinders

A nominally two-dimensional spin model wrapped onto a cylinder can profitably be viewed, especially for long cylinders, as a one-dimensional chain. Each site of such a chain is a ring of spins with a complex state space. Traditional correlation functions are inadequate for the study of correlations in such a system and need to be replaced with something like mutual information. Being induced purely by frustration, the disorder of a cylindrical zero-temperature triangular Ising antiferromagnet (TIAFM) and attendant correlations have a chance of evading the consequences of the Perron-Frobenius theorem which describes and constrains correlations in thermally disordered one-dimensional systems. Correlations in such TIAFM systems and the aforementioned evasion are studied here through a fermionic representation. For cylindrical TIAFM models with open boundary conditions, we explain and derive the following characteristics of end-to-end mutual information: period-three oscillation of the decay length, halving of the decay length compared to what Perron-Frobenius predicts on the basis of transfer matrix eigenvalues, and subexponential decay-inverse square in the length-for certain systems.

New Dimension in Magnetism and Superconductivity: 3D and Curvilinear Nanoarchitectures

Traditionally, the primary field, where curvature has been at the heart of research, is the theory of general relativity. In recent studies, however, the impact of curvilinear geometry enters various disciplines, ranging from solid-state physics over soft-matter physics, chemistry, and biology to mathematics, giving rise to a plethora of emerging domains such as curvilinear nematics, curvilinear studies of cell biology, curvilinear semiconductors, superfluidity, optics, 2D van der Waals materials, plasmonics, magnetism, and superconductivity. Here, the state of the art is summarized and prospects for future research in curvilinear solid-state systems exhibiting such fundamental cooperative phenomena as ferromagnetism, antiferromagnetism, and superconductivity are outlined. Highlighting the recent developments and current challenges in theory, fabrication, and characterization of curvilinear micro- and nanostructures, special attention is paid to perspective research directions entailing new physics and to their strong application potential. Overall, the perspective is aimed at crossing the boundaries between the magnetism and superconductivity communities and drawing attention to the conceptual aspects of how extension of structures into the third dimension and curvilinear geometry can modify existing and aid launching novel functionalities. In addition, the perspective should stimulate the development and dissemination of research and development oriented techniques to facilitate rapid transitions from laboratory demonstrations to industry-ready prototypes and eventual products.

Relaxation to steady states of a binary liquid mixture around an optically heated colloid

We study the relaxation dynamics of a binary liquid mixture near a light-absorbing Janus particle after switching on and off illumination using experiments and theoretical models. The dynamics is controlled by the temperature gradient formed around the heated particle. Our results show that the relaxation is asymmetric: The approach to a nonequilibrium steady state is much slower than the return to thermal equilibrium. Approaching a nonequilibrium steady state after a sudden temperature change is a two-step process that overshoots the response of spatial variance of the concentration field. The initial growth of concentration fluctuations after switching on illumination follows a power law in agreement with the hydrodynamic and purely diffusive model. The energy outflow from the system after switching off illumination is well described by a stretched exponential function of time with characteristic time proportional to the ratio of the energy stored in the steady state to the total energy flux in this state.

Geometric frustration in the myosin superlattice of vertebrate muscle

Geometric frustration results from an incompatibility between minimum energy arrangements and the geometry of a system, and gives rise to interesting and novel phenomena. Here, we report geometric frustration in a native biological macromolecular system---vertebrate muscle. We analyse the disorder in the myosin filament rotations in the myofibrils of vertebrate striated (skeletal and cardiac) muscle, as seen in thin-section electron micrographs, and show that the distribution of rotations corresponds to an archetypical geometrically frustrated system---the triangular Ising antiferromagnet. Spatial correlations are evident out to at least six lattice spacings. The results demonstrate that geometric frustration can drive the development of structure in complex biological systems, and may have implications for the nature of the actin--myosin interactions involved in muscle contraction. Identification of the distribution of myosin filament rotations with an Ising model allows the extensive results on the latter to be applied to this system. It shows how local interactions (between adjacent myosin filaments) can determine long-range order and, conversely, how observations of long-range order (such as patterns seen in electron micrographs) can be used to estimate the energetics of these local interactions. Furthermore, since diffraction by a disordered system is a function of the second-order statistics, the derived correlations allow more accurate diffraction calculations, which can aid in interpretation of X-ray diffraction data from muscle specimens for structural analysis.

Accurate detection of spherical objects in a complex background

The automated detection of particles in microscopy images has become a routinely used method for quantitative image analysis in biology, physics, and other research fields. While the majority of particle detection algorithms have been developed for bulk materials, the detection of particles in a heterogenous environment due to surfaces or other objects in the studied material is of great interest. However, particle detection is hindered by a complex background due to the diffraction of light resulting in a decreased contrast and image noise. We present a new heuristic method for the reliable detection of spherical particles that suppresses false detections due to a heterogenous background without additional background measurements. Further, we discuss methods to obtain particle coordinates with improved accuracy and compare with other methods, in particular with that of Crocker and Grier.

Interacting jammed granular systems

More than 30 years ago Edwards and co-authors proposed a model to describe the statistics of granular packings by an ensemble of equiprobable jammed states. Experimental tests of this model remained scarce so far. We introduce a simple system to analyze statistical properties of jammed granular ensembles to test Edwards theory. Identical spheres packed in a nearly two-dimensional geometrical confinement were studied in experiments and numerical simulations. When tapped, the system evolves toward a ground state, but due to incompatible domain structures it gets trapped. Analytical calculations reproduce relatively well our simulation results, which allows us to test Edwards theory on a coupled system of two subsystems with different properties. We find that the joint system can only be described by the Edwards theory if considered as a single system due to the constraints in the stresses. The results show counterintuitive effects as in the coupled system the change in the order parameter is opposite to what is expected from the change in the compactivity.

Dynamics of Confined Microgel Liquids: Weakened Spatial Confinement Effect by Microgel Particle Compliance

Spatial confinement has a great impact on the structures and dynamics of interfacial molecular and polymer liquid films. Most prior research has focused on confined liquids of fixed material compliance and often treated them in approximation to the "hard-sphere" interaction model. In this study, we microscopically investigate the structural dynamics of highly deformable poly(N-isopropylacrylamide) (PNIPAM) microgels confined between two solid surfaces in comparison to that of nearly nondeformable microgels of the same chemistry. We observe that the mobility and structural relaxation of highly deformable PNIPAM microgels at an apparent volume fraction, ϕ = 0.49-0.70, show little change with the reduction of gap spacing, in stark contrast to confinement-induced dynamic retardation of "hard-sphere"-like stiff PNIPAM microgels. The critical gap spacing, defined as the onset of confinement effect to deviate from the bulk behavior, is found to be approximately 17-22 particle layers for highly deformable microgels of ϕ = 0.56-0.70, much smaller than that of approximately 40 particle layers or larger for stiff microgels or model "hard-sphere" colloidal liquids of similar ϕ. Additionally, we observe no evident confinement-enhanced structural reorganization of deformable microgels near the confining surfaces when gap spacing approaches the critical gap spacing. Microgel deformation upon strong confinement is attributed to the disrupted confinement-induced ordering of confined microgels. Hence, it is clearly indicated that spatial confinement exhibits a much weaker effect on highly compliant microgel particles than stiff ones, resulting in a significantly less reduction in microgel interfacial dynamics. It therefore gives insights into the molecular design of polymeric thin films of variable compliance to control friction and lubrication.

Reconfiguring confined magnetic colloids with tunable fluid transport behavior

Collective dynamics of confined colloids are crucial in diverse scenarios such as self-assembly and phase behavior in materials science, microrobot swarms for drug delivery and microfluidic control. Yet, fine-tuning the dynamics of colloids in microscale confined spaces is still a formidable task due to the complexity of the dynamics of colloidal suspension and to the lack of methodology to probe colloids in confinement. Here, we show that the collective dynamics of confined magnetic colloids can be finely tuned by external magnetic fields. In particular, the mechanical properties of the confined colloidal suspension can be probed in real time and this strategy can be also used to tune microscale fluid transport. Our experimental and theoretical investigations reveal that the collective configuration characterized by the colloidal entropy is controlled by the colloidal concentration, confining ratio and external field strength and direction. Indeed, our results show that mechanical properties of the colloidal suspension as well as the transport of the solvent in microfluidic devices can be controlled upon tuning the entropy of the colloidal suspension. Our approach opens new avenues for the design and application of drug delivery, microfluidic logic, dynamic fluid control, chemical reaction and beyond.

Force chains in crystalline and frustrated packing visualized by stress-birefringent spheres

Force networks play an important role in the stability of configurations when granular material is packed into a container. These networks can redirect part of the weight of grains inside a container to the side walls. We employ monodisperse stress-birefringent spheres to visualize the contact forces in a quasi-2D and a nearly-2D configuration of these spheres in a thin cuboid cell. The packing structures are particularly simple: a hexagonal lattice in the ground state when the cell width is equal to the sphere diameter, and a frustrated, slightly distorted lattice in thicker cells. The force redistribution is substantially changed by this geometrical modification. In both cases, we observe an 'inverse' Janssen effect with the pressure decreasing from the top to the bottom of the container when the material is loaded with a weight on top of the vessel.

Shear driven vorticity aligned flocs in a suspension of attractive rigid rods

A combination of rheology, optical microscopy and computer simulations was used to investigate the microstructural changes of a semi-dilute suspension of attractive rigid rods in an imposed shear flow. The aim is to understand the relation of the microstructure with the viscoelastic response, and the yielding and flow behaviour in different shear regimes of gels built from rodlike colloids. A semi-dilute suspension of micron sized, rodlike silica particles suspended in 11 M CsCl salt solution was used as a model system for attractive rods' gel. Upon application of steady shear the gel microstructure rearranges in different states and exhibits flow instabilities depending on shear rate, attraction strength, volume fraction and geometrical confinement. At low rod volume fractions, the suspension forms large, vorticity aligned, particle rich flocs that roll in the flow-vorticity plane, an effect that is due to an interplay between hydrodynamic interactions and geometrical confinement as suggested by computer simulations. Experimental data allow the creation of a state diagram, as a function of volume fraction and shear rates, identifying regimes of stable (or unstable) floc formation and of homogeneous gel or broken clusters. The transition is related to dimensionless Mason number, defined as the ratio of shear forces to interparticle attractive force.

Geometrical Frustration and Planar Triangular Antiferromagnetism in Quasi-Three-Dimensional Artificial Spin Architecture

We present a realization of highly frustrated planar triangular antiferromagnetism achieved in a quasi-three-dimensional artificial spin system consisting of monodomain Ising-type nanomagnets lithographically arranged onto a deep-etched silicon substrate. We demonstrate how the three-dimensional spin architecture results in the first direct observation of long-range ordered planar triangular antiferromagnetism, in addition to a highly disordered phase with short-range correlations, once competing interactions are perfectly tuned. Our work demonstrates how escaping two-dimensional restrictions can lead to new types of magnetically frustrated metamaterials.

Buckling and metastability in membranes with dilation arrays

We study periodic arrays of impurities that create localized regions of expansion, embedded in two-dimensional crystalline membranes. These arrays provide a simple elastic model of shape memory. As the size of each dilational impurity increases (or the relative cost of bending to stretching decreases), it becomes energetically favorable for each of the M impurities to buckle up or down into the third dimension, thus allowing for of order 2^{M} metastable surface configurations corresponding to different impurity "spin" configurations. With both discrete simulations and the nonlinear continuum theory of elastic plates, we explore the buckling of both isolated dilations and dilation arrays at zero temperature, guided by analogies with Ising antiferromagnets. We conjecture ground states for systems with triangular and square impurity superlattices, and comment briefly on the possible behaviors at finite temperatures.

Modal Frustration and Periodicity Breaking in Artificial Spin Ice

Here, an artificial spin ice lattice is introduced that exhibits unique Ising and non-Ising behavior under specific field switching protocols because of the inclusion of coupled nanomagnets into the unit cell. In the Ising regime, a magnetic switching mechanism that generates a uni- or bimodal distribution of states dependent on the alignment of the field is demonstrated with respect to the lattice unit cell. In addition, a method for generating a plethora of randomly distributed energy states across the lattice, consisting of Ising and Landau states, is investigated through magnetic force microscopy and micromagnetic modeling. It is demonstrated that the dispersed energy distribution across the lattice is a result of the intrinsic design and can be finely tuned through control of the incident angle of a critical field. The present manuscript explores a complex frustrated environment beyond the 16-vertex Ising model for the development of novel logic-based technologies.

Topology Restricts Quasidegeneracy in Sheared Square Colloidal Ice

Recovery of ground-state degeneracy in two-dimensional square ice is a significant challenge in the field of geometric frustration with far-reaching fundamental implications, such as realization of vertex models and understanding the effect of dimensionality reduction. We combine experiments, theory, and numerical simulations to demonstrate that sheared square colloidal ice partially recovers the ground-state degeneracy for a wide range of field strengths and lattice shear angles. Our method could inspire engineering a novel class of frustrated microstructures and nanostructures based on sheared magnetic lattices in a wide range of soft- and condensed-matter systems.

Atomic scale investigation of the volume phase transition in concentrated PNIPAM microgels

Combining elastic incoherent neutron scattering and differential scanning calorimetry, we investigate the occurrence of the volume phase transition (VPT) in very concentrated poly-(N-isopropyl-acrylamide) (PNIPAM) microgel suspensions, from a polymer weight fraction of 30 wt. % up to dry conditions. Although samples are arrested at the macroscopic scale, atomic degrees of freedom are equilibrated and can be probed in a reproducible way. A clear signature of the VPT is present as a sharp drop in the mean square displacement of PNIPAM hydrogen atoms obtained by neutron scattering. As a function of concentration, the VPT gets smoother as dry conditions are approached, whereas the VPT temperature shows a minimum at about 43 wt. %. This behavior is qualitatively confirmed by calorimetry measurements. Molecular dynamics simulations are employed to complement experimental results and gain further insights into the nature of the VPT, confirming that it involves the formation of an attractive gel state between the microgels. Overall, these results provide evidence that the VPT in PNIPAM-based systems can be detected at different time- and length-scales as well as under overcrowded conditions.

Frustration and Atomic Ordering in a Monolayer Semiconductor Alloy

Frustrated interactions can lead to short-range ordering arising from incompatible interactions of fundamental physical quantities with the underlying lattice. The simplest example is the triangular lattice of spins with antiferromagnetic interactions, where the nearest-neighbor spin-spin interactions cannot simultaneously be energy minimized. Here we show that engineering frustrated interactions is a possible route for controlling structural and electronic phenomena in semiconductor alloys. Using aberration-corrected scanning transmission electron microscopy in conjunction with density functional theory calculations, we demonstrate atomic ordering in a two-dimensional semiconductor alloy as a result of the competition between geometrical constraints and nearest-neighbor interactions. Statistical analyses uncover the presence of short-range ordering in the lattice. In addition, we show how the induced ordering can be used as another degree of freedom to considerably modify the band gap of monolayer semiconductor alloys.

Emergent collective colloidal currents generated via exchange dynamics in a broken dimer state

Controlling the flow of matter down to micrometer-scale confinement is of central importance in material and environmental sciences, with direct applications in nano and microfluidics, drug delivery, and biotechnology. Currents of microparticles are usually generated with external field gradients of different nature (e.g., electric, magnetic, optical, thermal, or chemical ones), which are difficult to control over spatially extended regions and samples. Here, we demonstrate a general strategy to assemble and transport polarizable microparticles in fluid media through combination of confinement and magnetic dipolar interactions. We use a homogeneous magnetic modulation to assemble dispersed particles into rotating dimeric state and frustrated binary lattices, and generate collective currents that arise from a novel, field-synchronized particle exchange process. These dynamic states are similar to cyclotron and skipping orbits in electronic and molecular systems, thus paving the way toward understanding and engineering similar processes at different length scales across condensed matter.

Rapid Assembly of Colloidal Crystals under Laser Illumination on a GeSbTe Substrate

Optical techniques have been actively studied for manipulating nano- to microsized objects. However, long-range attraction and rapid transport of particles within thin quasi-two-dimensional systems are difficult because of the weak thermophoretic forces. Here, we introduce an experimental system that can rapidly generate quasi-two-dimensional colloidal crystals in deionized water, sandwiched between two hard plates. When a pulsed laser is irradiated on a chalcogenide phase-change material spattered on one side of the plates, the induced Marangoni-like flow causes a colloidal self-assembly in the order of tens of micrometers within the laser spot, with a transport velocity of a few tens of micrometers per second. This is due to the large thermal gradient induced by chalcogenide characteristics of high laser absorption and low thermal conductivity, and a strong hydrodynamic slip flow at the hydrophobic chalcogenide interface. Moreover, the colloidal crystals exhibit various lattice structures, depending on the laser intensity and chamber distance. For a certain range of the chamber distance, the colloidal crystal phases can be alternated by tuning the laser intensity in real time. Our system forms and deforms quasi-two-dimensional colloidal crystals at an on-demand location on a GeSbTe substrate.

Phase transitions in phospholipid monolayers: Statistical model at the pair approximation

A Langmuir film, consisting of a phospholipid monolayer at the air-water interface, was modeled as a two-dimensional lattice gas corresponding to a ternary mixture of water molecules (w), ordered-chain lipids (o), and disordered-chain lipids (d). The statistical problem is formulated in terms of a spin-1 model, in which the disordered-chain lipid states possess a high degenerescence ω≫1, and was termed Doniach lattice gas (DLG). Motivated by some open questions in the analysis of the DLG model at the mean-field approximation (MFA) [Phys. Rev. E 90, 052705 (2014)PLEEE81539-375510.1103/PhysRevE.90.052705], we have reconsidered it at the pair-approximation level by solving the model on a Cayley tree of coordination z. The attractors of the corresponding discrete-map problem are associated with the thermodynamic solutions on the Bethe lattice (the central region of an asymptotically infinite Cayley tree). To check the thermodynamic stability of the possible phases, the grand-potential density was obtained by the method proposed by Gujrati [Phys. Rev. Lett. 74, 809 (1995)PRLTAO0031-900710.1103/PhysRevLett.74.809]. In general, the previous MFA results are confirmed at the pair-approximation level, but a novel staggered phase, overlooked in the MFA analysis, was found when the condition ε_{wd}>1/2(ε_{ww}+ε_{dd}) is satisfied, where ε_{xy} represents the nearest-neighbor intermolecular interactions between single-site states x and y. Model parameters obtained by fitting to experimental data for the two most commonly studied zwitterionic phospholipids, 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC) and 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC), yield phase diagrams consistent with the phase transitions observed on Langmuir films of the same lipids under isothermal compression, which present a liquid-condensed to a liquid-expanded first-order transition line ending at a critical point.

Rotational switches in the two-dimensional fullerene quasicrystal

One of the essential components of molecular electronic circuits are switching elements that are stable in two different states and can ideally be switched on and off many times. Here, distinct buckminsterfullerenes within a self-assembled monolayer, forming a two-dimensional dodecagonal quasicrystal on a Pt-terminated Pt3Ti(111) surface, are identified to form well separated molecular rotational switching elements. Employing scanning tunneling microscopy, the molecular-orbital appearance of the fullerenes in the quasicrystalline monolayer is resolved. Thus, fullerenes adsorbed on the 36 vertex configuration are identified to exhibit a distinctly increased mobility. In addition, this finding is verified by differential conductance measurements. The rotation of these mobile fullerenes can be triggered frequently by applied voltage pulses, while keeping the neighboring molecules immobile. An extensive analysis reveals that crystallographic and energetic constraints at the molecule/metal interface induce an inequality of the local potentials for the 36 and 32.4.3.4 vertex sites and this accounts for the switching ability of fullerenes on the 36 vertex sites. Consequently, a local area of the 8/3 approximant in the two-dimensional fullerene quasicrystal consists of single rotational switching fullerenes embedded in a matrix of inert molecules. Furthermore, it is deduced that optimization of the intermolecular interactions between neighboring fullerenes hinders the realization of translational periodicity in the fullerene monolayer on the Pt-terminated Pt3Ti(111) surface.

Switchable geometric frustration in an artificial-spin-ice-superconductor heterosystem

Geometric frustration emerges when local interaction energies in an ordered lattice structure cannot be simultaneously minimized, resulting in a large number of degenerate states. The numerous degenerate configurations may lead to practical applications in microelectronics¹, such as data storage, memory and logic². However, it is difficult to achieve very high degeneracy, especially in a two-dimensional system^3,4. Here, we showcase in situ controllable geometric frustration with high degeneracy in a two-dimensional flux-quantum system. We create this in a superconducting thin film placed underneath a reconfigurable artificial-spin-ice structure⁵. The tunable magnetic charges in the artificial-spin-ice strongly interact with the flux quanta in the superconductor, enabling switching between frustrated and crystallized flux quanta states. The different states have measurable effects on the superconducting critical current profile, which can be reconfigured by precise selection of the spin-ice magnetic state through the application of an external magnetic field. We demonstrate the applicability of these effects by realizing a reprogrammable flux quanta diode. The tailoring of the energy landscape of interacting 'particles' using artificial-spin-ices provides a new paradigm for the design of geometric frustration, which could illuminate a path to control new functionalities in other material systems, such as magnetic skyrmions⁶, electrons and holes in two-dimensional materials^7,8, and topological insulators⁹, as well as colloids in soft materials^10-13.

Different routes into the glass state for soft thermo-sensitive colloids

We report an experimental and theoretical investigation of glass formation in soft thermo-sensitive colloids following two different routes: a gradual increase of the particle number density at constant temperature and an increase of the radius in a fixed volume at constant particle number density. Confocal microscopy experiments and the non-equilibrium self-consistent generalized Langevin equation (NE-SCGLE) theory consistently show that the two routes lead to a dynamically comparable state at sufficiently long aging times. However, experiments reveal the presence of moderate but persistent structural differences. Successive cycles of radius decrease and increase lead instead to a reproducible glass state, indicating a suitable route to obtain rejuvenation without using shear fields.

Lipid-protein interaction induced domains: Kinetics and conformational changes in multicomponent vesicles

The spatio-temporal organization of proteins and the associated morphological changes in membranes are of importance in cell signaling. Several mechanisms that promote the aggregation of proteins at low cell surface concentrations have been investigated in the past. We show, using Monte Carlo simulations, that the affinity of proteins for specific lipids can hasten their aggregation kinetics. The lipid membrane is modeled as a dynamically triangulated surface with the proteins defined as in-plane fields at the vertices. We show that, even at low protein concentrations, strong lipid-protein interactions can result in large protein clusters indicating a route to lipid mediated signal amplification. At high protein concentrations, the domains form buds similar to that seen in lipid-lipid interaction induced phase separation. Protein interaction induced domain budding is suppressed when proteins act as anisotropic inclusions and exhibit nematic orientational order. The kinetics of protein clustering and resulting conformational changes are shown to be significantly different for the isotropic and anisotropic curvature inducing proteins.

Attraction Controls the Entropy of Fluctuations in Isosceles Triangular Networks

We study two-dimensional triangular-network models, which have degenerate ground states composed of straight or randomly-zigzagging stripes and thus sub-extensive residual entropy. We show that attraction is responsible for the inversion of the stable phase by changing the entropy of fluctuations around the ground-state configurations. By using a real-space shell-expansion method, we compute the exact expression of the entropy for harmonic interactions, while for repulsive harmonic interactions we obtain the entropy arising from a limited subset of the system by numerical integration. We compare these results with a three-dimensional triangular-network model, which shows the same attraction-mediated selection mechanism of the stable phase, and conclude that this effect is general with respect to the dimensionality of the system.

Frustrated packing in a granular system under geometrical confinement

Optimal packings of uniform spheres are solved problems in two and three dimensions. The main difference between them is that the two-dimensional ground state can be easily achieved by simple dynamical processes while in three dimensions, this is impossible due to the difference in the local and global optimal packings. In this paper we show experimentally and numerically that in 2 + ε dimensions, realized by a container which is in one dimension slightly wider than the spheres, the particles organize themselves in a triangular lattice, while touching either the front or rear side of the container. If these positions are denoted by up and down the packing problem can be mapped to a 1/2 spin system. At first it looks frustrated with spin-glass like configurations, but the system has a well defined ground state built up from isosceles triangles. When the system is agitated, it evolves very slowly towards the potential energy minimum through metastable states. We show that the dynamics is local and is driven by the optimization of the volumes of 7-particle configurations and by the vertical interaction between touching spheres.

Curvature instability of chiral colloidal membranes on crystallization

Buckling and wrinkling instabilities are failure modes of elastic sheets that are avoided in the traditional material design. Recently, a new paradigm has appeared where these instabilities are instead being utilized for high-performance applications. Multiple approaches such as heterogeneous gelation, capillary stresses, and confinement have been used to shape thin macroscopic elastic sheets. However, it remains a challenge to shape two-dimensional self-assembled monolayers at colloidal or molecular length scales. Here, we show the existence of a curvature instability that arises during the crystallization of finite-sized monolayer membranes of chiral colloidal rods. While the bulk of the membrane crystallizes, its edge remains fluid like and exhibits chiral ordering. The resulting internal stresses cause the flat membrane to buckle macroscopically and wrinkle locally. Our results demonstrate an alternate pathway based on intrinsic stresses instead of the usual external ones to assemble non-Euclidean sheets at the colloidal length scale.

Buckling and wrinkling are instabilities which involve thin elastic sheets and are well-investigated phenomena at the macroscale. Here Saikia et al. investigate curvature instabilities at the colloidal lengthscale in quasi-2D monolayers of rod-like viruses across the fluid-crystal phase transition.

Crystallization and vitrification of electrons in a glass-forming charge liquid

Charge ordering (CO) is a phenomenon in which electrons in solids crystallize into a periodic pattern of charge-rich and charge-poor sites owing to strong electron correlations. This usually results in long-range order. In geometrically frustrated systems, however, a glassy electronic state without long-range CO has been observed. We found that a charge-ordered organic material with an isosceles triangular lattice shows charge dynamics associated with crystallization and vitrification of electrons, which can be understood in the context of an energy landscape arising from the degeneracy of various CO patterns. The dynamics suggest that the same nucleation and growth processes that characterize conventional glass-forming liquids guide the crystallization of electrons. These similarities may provide insight into our understanding of the liquid-glass transition.

Cuts Guided Deterministic Buckling in Arrays of Soft Parallel Plates for Multifunctionality

Harnessing buckling instability in soft materials offers an effective strategy to achieve multifunctionality. Despite great efforts in controlling the wrinkling behaviors of film-based systems and buckling of periodic structures, the benefits of classical plate buckling in soft materials remain largely unexplored. The challenge lies in the intrinsic indeterminate characteristics of buckling, leading to geometric frustration and random orientations. Here, we report the controllable global order in constrained buckling of arrays of parallel plates made of hydrogels and elastomers on rigid substrates. By introducing patterned cuts on the plates, the randomly phase-shifted buckling in the array of parallel plates transits to a prescribed and ordered buckling with controllable phases. The design principle for cut-directed deterministic buckling in plates is validated by both mechanics model and finite element simulation. By controlling the contacts and interactions between the buckled parallel plates, we demonstrate on-demand reconfigurable electrical and optical pathways, and the potential application in design of mechanical logic gates. By varying the local stimulus within the plates, we demonstrate that microscopic pathways can be written, visualized, erased, and rewritten macroscopically into a completely new one for potential applications such as soft reconfigurable circuits and logic devices.

Attraction Controls the Inversion of Order by Disorder in Buckled Colloidal Monolayers

We show how including attraction in interparticle interactions reverses the effect of fluctuations in ordering of a prototypical artificial frustrated system. Buckled colloidal monolayers exhibit the same ground state as the Ising antiferromagnet on a deformable triangular lattice, but it is unclear if ordering in the two systems is driven by the same geometric mechanism. By a real-space expansion we find that, for buckled colloids, bent stripes constitute the stable phase, whereas in the Ising antiferromagnet straight stripes are favored. For generic pair potentials we show that attraction governs this selection mechanism, in a manner that is linked to local packing considerations. This supports the geometric origin of entropy in jammed sphere packings and provides a tool for designing self-assembled colloidal structures.

Programmable Kiri-Kirigami Metamaterials

Programmable kirigami metamaterials with controllable local tilting orientations on demand through prescribed notches are constructed through a new approach of kiri-kirgami, and their actuation of pore opening via both mechanical stretching and temperature, along with their potential application as skins for energy-saving buildings, is discussed.

Harnessing Geometric Frustration to Form Band Gaps in Acoustic Channel Lattices

We demonstrate both numerically and experimentally that geometric frustration in two-dimensional periodic acoustic networks consisting of arrays of narrow air channels can be harnessed to form band gaps (ranges of frequency in which the waves cannot propagate in any direction through the system). While resonant standing wave modes and interferences are ubiquitous in all the analyzed network geometries, we show that they give rise to band gaps only in the geometrically frustrated ones (i.e., those comprising of triangles and pentagons). Our results not only reveal a new mechanism based on geometric frustration to suppress the propagation of pressure waves in specific frequency ranges but also open avenues for the design of a new generation of smart systems that control and manipulate sound and vibrations.

Frustration of crystallisation by a liquid-crystal phase

Frustration of crystallisation by locally favoured structures is critically important in linking the phenomena of supercooling, glass formation, and liquid-liquid transitions. Here we show that the putative liquid-liquid transition in n-butanol is in fact caused by geometric frustration associated with an isotropic to rippled lamellar liquid-crystal transition. Liquid-crystal phases are generally regarded as being “in between” the liquid and the crystalline state. In contrast, the liquid-crystal phase in supercooled n-butanol is found to inhibit transformation to the crystal. The observed frustrated phase is a template for similar ordering in other liquids and likely to play an important role in supercooling and liquid-liquid transitions in many other molecular liquids.

Colloidal molecules assembled from binary spheres under an AC electric field

Colloidal particles are envisioned as analogues of atoms and molecules, however they often lack the complexities present in their counterparts. In this work, we report the assembly of colloidal molecules from a binary mixture of polystyrene spheres (1, 1.6, 2, and 4 μm) under an alternating current electric field. The rich family of assembled oligomers typically consists of a large sphere that is closely surrounded by a number of smaller petal particles, driven by the dipolar attraction between large and small particles. In deionized water, the number of satellite particles, i.e., the coordination number increases with the increasing size ratio of the constituent particles. For a given size ratio, the coordination number decreases with the increasing frequency of the applied field. These trends have also been correctly captured by computing the electric energy of different oligomers based on induced dipolar and double-layer interactions. By suspending the particles in polyvinylpyrrolidone aqueous solution, we can further tune the bond length of the oligomers independent of their coordination numbers. The addition of polyvinylpyrrolidone also allows us to lock the assembled colloidal molecules so that they remain intact after the electric field is turned off. Our method provides a robust way to produce a family of colloidal molecules with well-defined geometry and high yield.

Collective dynamics of time-delay-coupled phase oscillators in a frustrated geometry

We study the effect of time delay on the dynamics of a system of repulsively coupled nonlinear oscillators that are configured as a geometrically frustrated network. In the absence of time delay, frustrated systems are known to possess a high degree of multistability among a large number of coexisting collective states except for the fully synchronized state that is normally obtained for attractively coupled systems. Time delay in the coupling is found to remove this constraint and to lead to such a synchronized ground state over a range of parameter values. A quantitative study of the variation of frustration in a system with the amount of time delay has been made and a universal scaling behavior is found. The variation in frustration as a function of the product of time delay and the collective frequency of the system is seen to lie on a characteristic curve that is common for all natural frequencies of the identical oscillators and coupling strengths. Thus time delay can be used as a tuning parameter to control the amount of frustration in a system and thereby influence its collective behavior. Our results can be of potential use in a host of practical applications in physical and biological systems in which frustrated configurations and time delay are known to coexist.

Relating microstructure and particle-level stress in colloidal crystals under increased confinement

The mechanical properties of crystalline materials can be substantially modified under confinement. Such modified macroscopic properties are usually governed by the altered microstructures and internal stress fields. Here, we use a parallel plate geometry to apply a quasi-static squeeze flow crushing a colloidal polycrystal while simultaneously imaging it with confocal microscopy. The confocal images are used to quantify the local structure order and, in conjunction with Stress Assessment from Local Structural Anisotropy (SALSA), determine the stress at the single-particle scale. We find that during compression, the crystalline regions break into small domains with different geometric packing. These domains are characterized by a pressure and deviatoric stress that are highly localized with correlation lengths that are half those found in bulk. Furthermore, the mean deviatoric stress almost doubles, suggesting a higher brittleness in the highly-confined samples.

Defect Dynamics in Artificial Colloidal Ice: Real-Time Observation, Manipulation, and Logic Gate

We study the defect dynamics in a colloidal spin ice system realized by filling a square lattice of topographic double well islands with repulsively interacting magnetic colloids. We focus on the contraction of defects in the ground state, and contraction or expansion in a metastable biased state. Combining real-time experiments with simulations, we prove that these defects behave like emergent topological monopoles obeying a Coulomb law with an additional line tension. We further show how to realize a completely resettable "nor" gate, which provides guidelines for fabrication of nanoscale logic devices based on the motion of topological magnetic monopoles.