Whiteness arises from the random scattering of incident light from disordered structures. [1] Opaque white materials have to contain a sufficiently large number of scatterers and therefore usually require thicker, material-rich nanostructures than structural color arising from the coherent interference of light. [2,3] In nature, bright white appearance arises from the dense arrays of pterin pigments in pierid butterflies, [4] guanine crystals in spiders, [5] or leucophore cells in the flexible skin of cuttlefish. [6] A striking example of such whiteness is found in the chitinous networks of white beetles, e.g., Lepidiota stigma and Cyphochilus sp. [7][8][9] Previous research investigating these beetle structures has shown that the chitinous network is one of the most strongly scattering materials in nature, and therefore the question arises whether this structure is evolutionary optimized for strong scattering while minimizing the Most studies of structural color in nature concern periodic arrays, which through the interference of light create color. The "color" white however relies on the multiple scattering of light within a randomly structured medium, which randomizes the direction and phase of incident light. Opaque white materials therefore must be much thicker than periodic structures. It is known that flying insects create "white" in extremely thin layers. This raises the question, whether evolution has optimized the wing scale morphology for white reflection at a minimum material use. This hypothesis is difficult to prove, since this requires the detailed knowledge of the scattering morphology combined with a suitable theoretical model. Here, a cryoptychographic X-ray tomography method is employed to obtain a full 3D structural dataset of the network morphology within a white beetle wing scale. By digitally manipulating this 3D representation, this study demonstrates that this morphology indeed provides the highest white retroreflection at the minimum use of material, and hence weight for the organism. Changing any of the network parameters (within the parameter space accessible by biological materials) either increases the weight, increases the thickness, or reduces reflectivity, providing clear evidence for the evolutionary optimization of this morphology. amount of employed material, thus reducing the weight of the organism. The brilliant white reflection from Cyphochilus beetles is assumed to be important for camouflage among white fungi and in a shady environment.In contrast to periodic photonic materials, for which the optical response is straightforward to calculate, the reflection of light from such disordered network morphologies requires a detailed knowledge of local geometry. [2,3,9,10] For these complex cases, the validity of the diffusion approximation is limited, since single scattering elements are difficult to be identified. [7] To fully understand the correlation between the structure and (optical) properties of complex materials, the detailed real-space structure in combination with a s...

Heterogeneous materials consisting of different phases are ideally suited to achieve a broad spectrum of desirable bulk physical properties by combining the best features of the constituents through the strategic spatial arrangement of the different phases. Disordered hyperuniform heterogeneous materials are new, exotic amorphous matter that behave like crystals in the manner in which they suppress volume-fraction fluctuations at large length scales, and yet are isotropic with no Bragg peaks. In this paper, we formulate for the first time a Fourier-space numerical construction procedure to design at will a wide class of disordered hyperuniform two-phase materials with prescribed spectral densities, which enables one to tune the degree and length scales at which this suppression occurs. We demonstrate that the anomalous suppression of volume-fraction fluctuations in such two-phase materials endow them with novel and often optimal transport and electromagnetic properties. Specifically, we construct a family of phase-inversion-symmetric materials with variable topological connectedness properties that remarkably achieves a well-known explicit formula for the effective electrical (thermal) conductivity. Moreover, we design disordered stealthy hyperuniform dispersion that possesses nearly optimal effective conductivity while being statistically isotropic. Interestingly, all of our designed materials are transparent to electromagnetic radiation for certain wavelengths, which is a common attribute of all hyperuniform materials. Our constructed materials can be readily realized by 3D printing and lithographic technologies. We expect that our designs will be potentially useful for energy-saving materials, batteries and aerospace applications.

Partitioning space into cells with certain extreme geometrical properties is a central problem in many fields of science and technology. Here we investigate the Quantizer problem, defined as the optimisation of the moment of inertia of Voronoi cells, i.e., similarly-sized ‘sphere-like’ polyhedra that tile space are preferred. We employ Lloyd’s centroidal Voronoi diagram algorithm to solve this problem and find that it converges to disordered states associated with deep local minima. These states are universal in the sense that their structure factors are characterised by a complete independence of a wide class of initial conditions they evolved from. They moreover exhibit an anomalous suppression of long-wavelength density fluctuations and quickly become effectively hyperuniform. Our findings warrant the search for novel amorphous hyperuniform phases and cellular materials with unique physical properties.

Numerous recent investigations have been devoted to the determination of the equilibrium phase behavior and packing characteristics of hard nonspherical particles, including ellipsoids, superballs, and polyhedra, to name but just a few shapes. Systems of hard nonspherical particles exhibit a variety of stable phases with different degrees of translational and orientational order, including isotropic liquid, solid crystal, rotator and a variety of liquid crystal phases. In this paper, we employ a Monte Carlo implementation of the adaptive-shrinking-cell (ASC) numerical scheme and free-energy calcu- * To whom correspondence should be addressed lations to ascertain with high precision the equilibrium phase behavior of systems of congruent Archimedean truncated tetrahedra over the entire range of possible densities up to the maximal nearly space-filling density. In particular, we find that the system undergoes two firstorder phase transitions as the density increases: first a liquid-solid transition and then a solidsolid transition. The isotropic liquid phase coexists with the Conway-Torquato (CT) crystal phase at intermediate densities, verifying the result of a previous qualitative study [J. Chem. Phys., 2011, 135, 151101]. The freezing-and melting-point packing fractions for this transition are respectively φ F = 0.496 ± 0.006 and φ M = 0.591 ± 0.005. At higher densities, we find that the CT phase undergoes another firstorder phase transition to one associated with the densest-known crystal, with coexistence densities in the range φ ∈ [0.780±0.002, 0.802± 0.003]. We find no evidence for stable rotator (or plastic) or nematic phases. We also generate the maximally random jammed (MRJ) packings of truncated tetrahedra, which may be regarded to be the glassy end state of a rapid compression of the liquid. Specifically, we systematically study the structural characteristics of the MRJ packings, including the centroidal pair 1 correlation function, structure factor and orientational pair correlation function. We find that such MRJ packings are hyperuniform with an average packing fraction of 0.770, which is considerably larger than the corresponding value for identical spheres (≈ 0.64). We conclude with some simple observations concerning what types of phase transitions might be expected in general hard-particle systems based on the particle shape and which would be good glass formers.

Molecular architecture plays a key role in the selfassembly of block copolymers, but few studies have systematically examined the influence of chain connectivity on tetrahedrally close-packed (TCP) sphere phases. Here, we report a versatile material platform comprising two blocks with substantial conformational asymmetry, A = poly(trifluoroethyl acrylate) and B = poly(dodecyl acrylate), and use it to compare the phase behavior of AB diblocks, ABA triblocks, and (AB) n radial star copolymers with n = 3 or 4. Each architecture forms TCP sphere phases at minority A block compositions (f A < 0.5), namely, σ and A15, but with differences in the location of order−order phase boundaries that are not anticipated by mean-field self-consistent field theory simulations. These results expand the palette of polymer architectures that readily self-assemble into complex TCP structures and suggest important design considerations when targeting specific phases of interest.

Disordered hyperuniformity (DHU) is a recently proposed new state of matter, which has been observed in a variety of classical and quantum many-body systems. DHU systems are characterized by vanishing infinite-wavelength density fluctuations and are endowed with unique novel physical properties. Here we report the first discovery of disordered hyperuniformity in atomic-scale 2D materials, i.e., amorphous silica composed of a single layer of atoms, based on spectral-density analysis of high-resolution transmission electron microscope images. Subsequent simulations suggest that the observed DHU is closely related to the strong topological and geometrical constraints induced by the local chemical order in the system. Moreover, we show via large-scale density functional theory calculations that DHU leads to almost complete closure of the electronic band gap compared to the crystalline counterpart, making the material effectively a metal. This is in contrast to the conventional wisdom that disorder generally diminishes electronic transport and is due to the unique electron wave localization induced by the topological defects in the DHU state.Disorder hyperuniform (DHU) systems are a unique class of disordered systems which suppress large-scale density fluctuations like crystals and yet possess no Bragg peaks [1, 2]. For a point configuration (e.g., a collection of particle centers of a many-body system), hyperuniformity is manifested as the vanishing structure factor in the infinite-wavelength (or zero-wavenumber) limit, i.e., lim k→0 S(k) = 0, where k = 2π/λ is the wavenumber. In this case of a random field, the hyperuniform condition is given by lim k→0ψ (k) = 0, whereψ(k) is the spectral density [2]. It has been suggested that hyperuniformity can be considered as a new state of matter [1], which possesses a hidden order in between of that of a perfect crystal and a totally disordered system (e.g. a Poisson distribution of points).Recently, a wide spectrum of physical and biological systems have been identified to possess the remarkable property of hyperuniformity, which include the density fluctuations in early universe [3], disordered jammed packing of hard particles [4][5][6][7], certain exotic classical ground states of many-particle systems [8][9][10][11][12][13][14][15], jammed colloidal systems [16][17][18][19], driven non-equilibrium systems [20][21][22][23], certain quantum ground states [24,25], avian photoreceptor patterns [26], organization of adapted im- * These authors contributed equally to this work. † correspondence sent to: xwxfat@gmail.com ‡ correspondence sent to: mohanchen@pku.edu.cn § correspondence sent to: yang.jiao.2@asu.edu ¶ correspondence sent to: hzhuang7@asu.edu mune systems [27], amorphous silicon [28, 29], a wide class of disordered cellular materials [30], dynamic random organizating systems [31-35], and even the distribution of primes on the number axis [36]. In addition, it has been shown that hyperuniform materials can be designed to possess superior physical properties including ...

Disordered hyperuniformity (DHU) is a recently discovered novel state of many-body systems that possesses vanishing normalized infinite-wavelength density fluctuations similar to a perfect crystal and an amorphous structure like a liquid or glass. Here, we discover a hyperuniformity-preserving topological transformation in two-dimensional (2D) network structures that involves continuous introduction of Stone–Wales (SW) defects. Specifically, the static structure factor S(k) of the resulting defected networks possesses the scaling S(k)∼kα for small wave number k, where 1≤α(p)≤2 monotonically decreases as the SW defect concentration p increases, reaches α≈1 at p≈0.12, and remains almost flat beyond this p. Our findings have important implications for amorphous 2D materials since the SW defects are well known to capture the salient feature of disorder in these materials. Verified by recently synthesized single-layer amorphous graphene, our network models reveal unique electronic transport mechanisms and mechanical behaviors associated with distinct classes of disorder in 2D materials.

Disordered hyperuniform materials are a new, exotic class of amorphous matter that exhibits crystal-like behavior, in the sense that volume-fraction fluctuations are suppressed at large length scales, and yet they are isotropic and do not display diffraction Bragg peaks. These materials are endowed with novel photonic, phononic, transport and mechanical properties, which are useful for a wide range of applications. Motivated by the need to fabricate large samples of disordered hyperuniform systems at the nanoscale, we study the small-wavenumber behavior of the spectral density of binary mixtures of charged colloids in suspension. The interaction between the colloids is approximated by a repulsive hard-core Yukawa potential. We find that at dimensionless temperatures below 0.05 and dimensionless inverse screening lengths below 1.0, which are experimentally accessible, the disordered systems become effectively hyperuniform. Moreover, as the temperature and inverse screening length decrease, the level of hyperuniformity increases, as quantified by the "hyperuniformity index". Our results suggest an alternative approach to synthesize large samples of effectively disordered hyperuniform materials at the nanoscale under standard laboratory conditions. In contrast with the usual route to synthesize disordered hyperuniform materials by jamming particles, this approach is free from the burden of applying high pressure to compress the systems.