Anton Souslov

Surface phonons, elastic response, and conformal invariance in twisted kagome lattices

Model lattices consisting of balls connected by central-force springs provide much of our understanding of mechanical response and phonon structure of real materials. Their stability depends critically on their coordination number z. d-dimensional lattices with z = 2d are at the threshold of mechanical stability and are isostatic. Lattices with z < 2d exhibit zero-frequency “floppy” modes that provide avenues for lattice collapse. The physics of systems as diverse as architectural structures, network glasses, randomly packed spheres, and biopolymer networks is strongly influenced by a nearby isostatic lattice. We explore elasticity and phonons of a special class of two-dimensional isostatic lattices constructed by distorting the kagome lattice. We show that the phonon structure of these lattices, characterized by vanishing bulk moduli and thus negative Poisson ratios (equivalently, auxetic elasticity), depends sensitively on boundary conditions and on the nature of the kagome distortions. We construct lattices that under free boundary conditions exhibit surface floppy modes only or a combination of both surface and bulk floppy modes; and we show that bulk floppy modes present under free boundary conditions are also present under periodic boundary conditions but that surface modes are not. In the long-wavelength limit, the elastic theory of all these lattices is a conformally invariant field theory with holographic properties (characteristics of the bulk are encoded on the sample boundary), and the surface waves are Rayleigh waves. We discuss our results in relation to recent work on jammed systems. Our results highlight the importance of network architecture in determining floppy-mode structure.

Isostaticity, auxetic response, surface modes, and conformal invariance in twisted kagome lattices

Model lattices consisting of balls connected by central-force springs provide much of our understanding of mechanical response and phonon structure of real materials. Their stability depends critically on their coordination number z. d-dimensional lattices with z = 2d are at the threshold of mechanical stability and are isostatic. Lattices with z < 2d exhibit zero-frequency “floppy” modes that provide avenues for lattice collapse. The physics of systems as diverse as architectural structures, network glasses, randomly packed spheres, and biopolymer networks is strongly influenced by a nearby isostatic lattice. We explore elasticity and phonons of a special class of two-dimensional isostatic lattices constructed by distorting the kagome lattice. We show that the phonon structure of these lattices, characterized by vanishing bulk moduli and thus negative Poisson ratios (equivalently, auxetic elasticity), depends sensitively on boundary conditions and on the nature of the kagome distortions. We construct lattices that under free boundary conditions exhibit surface floppy modes only or a combination of both surface and bulk floppy modes; and we show that bulk floppy modes present under free boundary conditions are also present under periodic boundary conditions but that surface modes are not. In the long-wavelength limit, the elastic theory of all these lattices is a conformally invariant field theory with holographic properties (characteristics of the bulk are encoded on the sample boundary), and the surface waves are Rayleigh waves. We discuss our results in relation to recent work on jammed systems. Our results highlight the importance of network architecture in determining floppy-mode structure.

Phonons and elasticity in critically coordinated lattices.

Much of our understanding of vibrational excitations and elasticity is based upon analysis of frames consisting of sites connected by bonds occupied by central-force springs, the stability of which depends on the average number of neighbors per site z. When z  <  zc  ≈  2d, where d is the spatial dimension, frames are unstable with respect to internal deformations. This pedagogical review focuses on the properties of frames with z at or near zc, which model systems like randomly packed spheres near jamming and network glasses. Using an index theorem, N0  -NS  =  dN  -NB relating the number of sites, N, and number of bonds, NB, to the number, N0, of modes of zero energy and the number, NS, of states of self stress, in which springs can be under positive or negative tension while forces on sites remain zero, it explores the properties of periodic square, kagome, and related lattices for which z  =  zc and the relation between states of self stress and zero modes in periodic lattices to the surface zero modes of finite free lattices (with free boundary conditions). It shows how modifications to the periodic kagome lattice can eliminate all but trivial translational zero modes and create topologically distinct classes, analogous to those of topological insulators, with protected zero modes at free boundaries and at interfaces between different topological classes.

Impact of single-particle compressibility on the fluid-solid phase transition for ionic microgel suspensions.

We study ionic microgel suspensions composed of swollen particles for various single-particle stiffnesses. We measure the osmotic pressure π of these suspensions and show that it is dominated by the contribution of free ions in solution. As this ionic osmotic pressure depends on the volume fraction of the suspension ϕ, we can determine ϕ from π, even at volume fractions so high that the microgel particles are compressed. We find that the width of the fluid-solid phase coexistence, measured using ϕ, is larger than its hard-sphere value for the stiffer microgels that we study and progressively decreases for softer microgels. For sufficiently soft microgels, the suspensions are fluidlike, irrespective of volume fraction. By calculating the dependence on ϕ of the mean volume of a microgel particle, we show that the behavior of the phase-coexistence width correlates with whether or not the microgel particles are compressed at the volume fractions corresponding to fluid-solid phase coexistence.

Order by disorder in the antiferromagnetic Ising model on an elastic triangular lattice.

Geometrically frustrated materials have a ground-state degeneracy that may be lifted by subtle effects, such as higher-order interactions causing small energetic preferences for ordered structures. Alternatively, ordering may result from entropic differences between configurations in an effect termed order by disorder. Motivated by recent experiments in a frustrated colloidal system in which ordering is suspected to result from entropy, we consider in this paper the antiferromagnetic Ising model on a deformable triangular lattice. We calculate the displacements exactly at the microscopic level and, contrary to previous studies, find a partially disordered ground state of randomly zigzagging stripes. Each such configuration is deformed differently and thus has a unique phonon spectrum with distinct entropy, lifting the degeneracy at finite temperature. Nonetheless, due to the free-energy barriers between the ground-state configurations, the system falls into a disordered glassy state.

Odd viscosity in chiral active fluids

We study the hydrodynamics of fluids composed of self-spinning objects such as chiral grains or colloidal particles subject to torques. These chiral active fluids break both parity and time-reversal symmetries in their non-equilibrium steady states. As a result, the constitutive relations of chiral active media display a dissipationless linear-response coefficient called odd (or equivalently, Hall) viscosity. This odd viscosity does not lead to energy dissipation, but gives rise to a flow perpendicular to applied pressure. We show how odd viscosity arises from non-linear equations of hydrodynamics with rotational degrees of freedom, once linearized around a non-equilibrium steady state characterized by large spinning speeds. Next, we explore odd viscosity in compressible fluids and suggest how our findings can be tested in the context of shock propagation experiments. Finally, we show how odd viscosity in weakly compressible chiral active fluids can lead to density and pressure excess within vortex cores.Active chiral fluids are a special case of active matter in which energy is introduced into rotational motion via local application of torque. Here Banerjee et al. develop a hydrodynamic theory of such active fluids and connect it with odd viscosity which was previously considered an abstract concept.

Buckled colloidal monolayers connect geometric frustration in soft and hard matter

Buckled monolayers of diameter-tunable microgel spheres constitute a soft-matter model system for studying geometric frustration in hard-condensed-matter antiferromagnetic materials. In the plane, the spheres self-assemble to form a triangular lattice. By considering the free volume available to two spheres slightly out of the plane, one finds an effective antiferromagnetic interaction; each pair of neighboring spheres prefers to be one up and one down. However, the topology of the triangular lattice prevents all pairs from simultaneously satisfying this rule. The micrometer length scale of the spheres enables direct visualization of the ‘spin’ dynamics at the single-particle level. These dynamics exhibit glassiness, which originates from the in-plane lattice distortions that partially relieve frustration and produce ground states with zigzagging stripes.

Localizing softness and stress along loops in 3D topological metamaterials

Significance Topological mechanical metamaterials are a type of structure, which can be used to design and control softness. They have been proposed as components for many applications, including for cushioning mechanical shocks and isolation of mechanical deformations as well as controlling actuation and failure. We provide a realization of a specific type of topological structure—a gapped topological lattice in three dimensions. We then use this structure to localize soft deformations either on the surface of the material or in the bulk along line defects called dislocations. Topological states can be used to control the mechanical properties of a material along an edge or around a localized defect. The rigidity of elastic networks is characterized by a topological invariant called the polarization; materials with a well-defined uniform polarization display a dramatic range of edge softness depending on the orientation of the polarization relative to the terminating surface. However, in all 3D mechanical metamaterials proposed to date, the topological modes are mixed with bulk soft modes, which organize themselves in Weyl loops. Here, we report the design of a 3D topological metamaterial without Weyl lines and with a uniform polarization that leads to an asymmetry between the number of soft modes on opposing surfaces. We then use this construction to localize topological soft modes in interior regions of the material by including defect lines—dislocation loops—that are unique to three dimensions. We derive a general formula that relates the difference in the number of soft modes and states of self-stress localized along the dislocation loop to the handedness of the vector triad formed by the lattice polarization, Burgers vector, and dislocation-line direction. Our findings suggest a strategy for preprogramming failure and softness localized along lines in 3D, while avoiding extended soft Weyl modes.

Beads on a string: structure of bound aggregates of globular particles and long polymer chains.

Macroscopic properties of suspensions, such as those composed of globular particles (e.g., colloidal or macromolecular), can be tuned by controlling the equilibrium aggregation of the particles. We examine how aggregation - and, hence, macroscopic properties - can be controlled in a system composed of both globular particles and long, flexible polymer chains that reversibly bind to one another. We base this on a minimal statistical mechanical model of a single aggregate in which the polymer chain is treated either as ideal or self-avoiding, and, in addition, the globular particles are taken to interact with one another via excluded volume repulsion. Furthermore, each of the globular particles is taken to have one single site to which at most one polymer segment may bind. Within the context of this model, we examine the statistics of the equilibrium size of an aggregate and, thence, the structure of dilute and semidilute suspensions of these aggregates. We apply the model to biologically relevant aggregates, specifically those composed of macromolecular proteoglycan globules and long hyaluronan polymer chains. These aggregates are especially relevant to the materials properties of cartilage and the structure-function properties of perineuronal nets in brain tissue, as well as the pericellular coats of mammalian cells.

Organization of Strongly Interacting Directed Polymer Liquids in the Presence of Stringent Constraints

The impact of impenetrable obstacles on the energetics and equilibrium structure of strongly repulsive directed polymers is investigated. As a result of the strong interactions, regions of severe polymer depletion and excess are found in the vicinity of the obstacle, and the associated free-energy cost is found to scale quadratically with the average polymer density. The polymer-polymer interactions are accounted for via a sequence of transformations: from the 3D line liquid to a 2D fluid of Bose particles to a 2D composite fermion fluid and, finally, to a 2D one-component plasma. The results presented here are applicable to a range of systems consisting of noncrossing directed lines.