Session V58: Hyperuniformity and Optimal Tessellations: Structure, Formation and Properties

Hyperuniformity and optimal tessellations: structure, formation and properties

After recent breakthroughs in the search for ordered optimal tessellations (for example, including Frank-Kasper phases in copolymer melts), now findings of the optimal properties of amorphous tessellations are emerging, e.g., in biological tissues.

At the same time, there have been intensive studies of amorphous systems with an anomalous suppression of density fluctuations on large length scales, known as hyperuniformity. This geometric concept qualitatively and quantitatively characterizes a hidden-order in amorphous states that allows for unique physical properties--combining those of crystalline and disordered phases. Thus it offers candidates for optimal amorphous tessellations of space.

This session will foster a discourse between these subfields and about the role of hyperuniformity in the search for tessellations that are optimal with respect to geometrical and physical properties. The session will discuss both a theoretical understanding, computational exploration and experimental verification of the temporal evolution of growing or compressed soft cellular entities and the formation of surprising ordered and amorphous phases and their unexpected structures and physical properties. The applications range from functional designer materials to the cell biology of membrane organelles.   

Disordered Hyperuniform Many-Particle Systems via Tessellations

Disordered hyperuniform many-particle systems are exotic amorphous materials characterized by
an anomalous suppression of long-wavelength density fluctuations that endows them with
novel transport, mechanical and optical properties. I will describe two different
tessellation-based procedures that lead to disordered systems that are hyperuniform.
Both methodologies convert initially disordered nonhyperuniform systems into
disordered hyperuniform ones.

  

Quantifying The Mechanical Energy Landscape of Two-Dimensional Cellular Matter

The local mechanical equilibrium states of cellular materials - consisting of space-filling domains with dominant interfacial energy - are characterized by a spectrum of metastable-state energies on a generally complex landscape. Recent theoretical work quantifies these energies in dramatically simpler ways exploiting geometric and topological statistical information from the structure of the system only, independent of the specific energy functional [1]. We elaborate on and test these theories by analyzing cellular matter samples, particularly in foam monolayer systems, where energies can be directly and independently determined. Two approaches to inferring energies are tested using (i) measured lengths of select boundaries between domains; (ii) the probability distribution of domain sizes and topologies. The experimental results support both theories and elucidate advantages and disadvantages of either approach. It is thus practically demonstrated that the mechanical state of cellular matter is quantifiable by its morphology only, suggesting simple and general diagnostic tools in the study of systems as diverse as biological tissues, foams, or superlattice materials.

[1] S. Kim. Y. Wang, and S. Hilgenfeldt, Phys. Rev. Lett. 120, 24801 (2018).   

Hyperuniformity of the Optimal Multiscale Hashin-Shtrikman Two-Phase Tessellations

Disordered hyperuniform two-phase systems are characterized by anomalous suppression of volume-fraction fluctuations at infinite long wavelengths. They provide fertile ground for fundamental research and have attracted considerable practical interest because they often are endowed with exotic physical properties, including possessing optimal or nearly optimal physical properties. The Hashin-Shtrikman (HS) two-phase multiscale dispersions are derived from special tessellations of space that endow these two-phase systems with the optimal effective transport and elastic properties for given phase properties and phase volume fractions. Using a new tiling formulation that ensures perfect hyperuniformity, we rigorously establish the hyperuniformity of the optimal HS structures. We analytically show that these structures are strongly hyperuniform by deriving the small-wavenumber scaling behavior of their spectral density. We verify these theoretical results numerically in two dimensions by constructing extremely dense polydisperse disk packings. Our work provides insights about the relationship between hyperuniformity in two-phase systems and property optimization.   

General Mechanical Energy Landscape in Two-Dimensional Interacting Systems

Domain systems composed of individual entities, cells, or particles commonly exhibit complex mechanical energy landscape with a plethora of metastable states. Recent studies on a class of domain systems governed by dominant interfacial energy show that an equivalent foam energy (the sum of interfacial lengths), is a strong predictor of metastable state energies, regardless of the exact interfacial energy functionals. The landscape of equivalent foam energies is in turn described accurately by theoretical predictions based either on geometric measures or on statistical measures. Here, we show that this theoretical framework can be extended to systems with particle-based interactions. While the energy of particle-based systems depends on constituent particle positions rather than interfacial structure, the equivalent foam energy of a 2D tessellation generated from particle positions not only yields a strong correlation with the actual energy level but also generates an identical energy landscape. Geometric measures and statistical measures can be obtained from visual information only so this framework can serve as a diagnostic tool for identifying mechanical energy states in a variety of domain systems, including foams, emulsions, granular systems, and confluent tissues.   

Nano-Real Space Analysis Takes Dynamics of Colloids Three Decades Closer to the Glass Transition

Glasses are among the most widely used of everyday materials, yet the process by which a liquid’s viscosity increases by 14 decades to become a glass remains unclear, as often contradictory theories provide equally good descriptions of the available data. Knowledge of emergent lengthscales and higher-order structure could help resolve this, but this requires time-resolved measurements of dense particle coordinates—previously only obtained over a limited time interval, and a some means of determining local free energy in model glassformers such as hard spheres [1]. Here we present an experimental study of a model colloidal system over a dynamic window significantly larger than previous measurements, revealing structural ordering more strongly linked to dynamics than previously found. Furthermore we find that immobile regions and domains of local structure grow concurrently with density, and that these regions have low configurational entropy [2].

[1] Robinson JE, Turci F, Roth R and Royal CP “Morphometric approach to many-body correlations in hard spheres” ArXiV 1809.02401 (2018).
[2] Hallett JE, Turci F and Royall CP, “Local structure in deeply supercooled liquids exhibits growing lengthscales and dynamical correlations”, Nature Comms. 9 3272 (2018).   

Experiments on periodically sheared colloidal suspensions with diffusion

Periodically sheared dilute, non-Brownian suspensions explore new configurations through collisions in an otherwise reversible flow. Below a critical strain, the particles remain active until they find a configuration with no collisions and reach an absorbing state. However, in a colloidal system at finite temperature, Brownian motion ensures that no state is ever truly absorbed and simulations by Hexner et. al. have shown that a small amount of diffusion enhances the order of the structures near the critical strain. We built a compact rotational shear cell to drive Brownian colloidal suspensions to explore the effect of diffusion on structures (S(q)) and dynamics (mean squared displacement) simultaneously using a confocal microscope. From dynamical measurements, we see that the effective diffusion constant is equal to the self-diffusion of the particles below the transition and increases linearly with strain amplitude above. For a 20% volume fraction suspension, we see S(q) at small q in the strain direction drop by a factor of 3 from thermal equilibrium in the strain direction with hyperuniform scaling S(q) ~ q^0.5 while in the vorticity direction, there is no hyperuniform scaling and S(q) at small q only decreases by 50%.   

Dynamic hyperuniformity of driven suspensions

We show that externally forced particles in a suspension self-organize hyperuniformly in certain directions relative to the force. For this phenomenon to occur, the force-induced coupling between the concentration of particles and their velocity field should satisfy three conditions : (i) it should be long-ranged, decaying with distance r as 1/r^β, with β smaller than the dimensionality d; (ii) it should have a finite divergence, i.e., the flow of particles should not conserve volume; (iii) it should not be inversion-symmetric. These conditions are fulfilled, for example, for particles which tend to align with the force, or in the presence of a steady concentration gradient. The directions of hyperuniformity depend on the specific coupling. The structure factor for wavevectors q along these directions tends to zero for small q as q^α, α = d – β. We check these analytical results in a simple two-dimensional simulation, showing that they are robust against particle collisions and diffusion.   

Complex Spherical Micelle Packings in Aqueous Dispersions of Diblock Polymers

Aqueous self-assembly of diblock polymers enables access to a plethora of concentration- and temperature- dependent morphologies including lamellae, hexagonally-packed cylindrical micelles (H), and high-symmetry packings of spherical micelles. Herein, we describe the aqueous lyotropic phase behaviour of poly(ethylene–block–ethylene oxide) using small angle X-ray scattering. The temperature versus amphiphile concentration phase diagram exhibits multiple micellar packings, including well-known body- and face-centered cubic structures as well as the lower symmetry Frank-Kasper (FK) A15 phase. On heating, the A15 phase undergoes a transition into the H-phase. Rapidly quenching this sample from high temperature to 25 °C results in the nucleation and growth of a FK σ phase, which contains 30 quasispherical micelles situated at five different symmetry equivalent positions in a tetragonal lattice. We find that the σ phase is metastable with respect to A15, and that the metastability strongly depends on the rate of cooling from the H phase and the quench depth. We rationalize our findings based on a subtle energy balance which maximizes interchain cohesion while minimizing both variations in coronal chain stretching and the interfacial penalty for the hydrophobic/hydrophilic contacts.   

Hyperuniformity of generalized random organization models

Studies of random organization models of monodisperse spherical particles have shown that a hyperuniform state is achievable when the system goes through an absorbing phase transition. Here we investigate to what extent hyperuniformity is preserved when the model is generalized to particles with a size distribution and/or nonspherical shapes. We examine binary disks, disks with a size distribution and hard rectangles of various aspect ratios. We show that the systems are hyperuniform as two-phase media at their respective critical points. This analysis reveals that the redistribution of ``mass" of the particles rather than the particle centroids is central to this dynamical process. Our results suggest that general particle systems subject to random organization can be a robust way to fabricate a wide class of hyperuniform states of matter by tuning the structures via different particle-size and -shape distributions. This in turn enables the creation of multifunctional hyperuniform materials with desirable optical, transport and mechanical properties.