Diffusion spreadability as a probe of the microstructure of complex media across length scales

Three-dimensional construction of hyperuniform, nonhyperuniform, and antihyperuniform disordered heterogeneous materials and their transport properties via spectral density functions

Rigorous theories connecting physical properties of a heterogeneous material to its microstructure offer a promising avenue to guide the computational material design and optimization. The spectral density function χ[over ̃]_{_{V}}(k), which can be obtained experimentally from scattering data, enables accurate determination of various transport and wave propagation characteristics, including the time-dependent diffusion spreadability S(t) and effective dynamic dielectric constant ε_{e} for electromagnetic wave propagation. Moreover, χ[over ̃]_{_{V}}(k) determines rigorous upper bounds on the fluid permeability K. Given the importance of χ[over ̃]_{_{V}}(k), we present here an efficient Fourier-space based computational framework to construct three-dimensional (3D) statistically isotropic two-phase heterogeneous materials corresponding to targeted spectral density functions. In particular, we employ a variety of analytical functional forms for χ[over ̃]_{_{V}}(k) that satisfy all known necessary conditions to construct disordered stealthy hyperuniform, standard hyperuniform, nonhyperuniform, and antihyperuniform two-phase heterogeneous material systems at varying phase volume fractions. We show that by tuning the correlations in the system across length scales via the targeted functions, one can generate a rich spectrum of distinct structures within each of the above classes of materials. Importantly, we present the first realization of antihyperuniform two-phase heterogeneous materials in 3D, which are characterized by autocovariance function χ_{_{V}}(r) with a power-law tail, resulting in microstructures that contain clusters of dramatically different sizes and morphologies. We also determine the diffusion spreadability S(t) and estimate the fluid permeability K associated with all of the constructed materials directly from the corresponding spectral densities. Although it is well established that the long-time asymptotic scaling behavior of S(t) only depends on the functional form of χ[over ̃]_{_{V}}(k), with the stealthy hyperuniform and antihyperuniform media, respectively, achieving the most and least efficient transport, we find that varying the length-scale parameter within each class of χ[over ̃]_{_{V}}(k) functions can also lead to orders of magnitude variation of S(t) at intermediate and long time scales. Moreover, we find that increasing the solid volume fraction ϕ_{1} and correlation length a in the constructed media generally leads to a decrease in the dimensionless fluid permeability K/a^{2}, while the antihyperuniform media possess the largest K/a^{2} among the four classes of materials with the same ϕ_{1} and a. These results indicate the feasibility of employing parameterized spectral densities for designing composites with targeted transport properties.

Hyperuniformity in the Manna Model, Conserved Directed Percolation and Depinning

Hyperuniformity is an emergent property, whereby the structure factor of the density n scales as S(q)∼q^{α}, with α>0. We show that for the conserved directed percolation (CDP) class, to which the Manna model belongs, there is an exact mapping between the density n in CDP, and the interface position u at depinning, n(x)=n_{0}+∇^{2}u(x), where n_{0} is the conserved particle density. As a consequence, the hyperuniformity exponent equals α=4-d-2ζ, with ζ the roughness exponent at depinning, and d the dimension. In d=1, α=1/2, while 0.6>α≥0 for other d. Our results fit well the simulations in the literature, except in d=1, where we perform our own to confirm this result. Such an exact relation between two seemingly different fields is surprising, and paves new paths to think about hyperuniformity and depinning. As corollaries, we get results of unprecedented precision in all dimensions, exact in d=1. This corrects earlier work on hyperuniformity in CDP.

Anomalous relaxation and hyperuniform fluctuations in center-of-mass conserving systems with broken time-reversal symmetry

We study the Oslo model, a paradigm for absorbing-phase transition, on a one-dimensional ring of L sites with a fixed global density ρ[over ¯]; we consider the system strictly above critical density ρ_{c}. Notably, microscopic dynamics conserve both mass and center of mass (CoM), but lack time-reversal symmetry. We show that, despite having highly constrained dynamics due to CoM conservation, the system exhibits diffusive relaxation away from criticality and superdiffusive relaxation near criticality. Furthermore, the CoM conservation severely restricts particle movement, causing the mobility-a transport coefficient analogous to the conductivity for charged particles-to vanish exactly. Indeed, the steady-state temporal growth of current fluctuation is qualitatively different from that observed in diffusive systems with a single conservation law. Remarkably, far from criticality where the relative density Δ=ρ[over ¯]-ρ_{c}≫ρ_{c}, the second cumulant, or the variance, 〈Q_{i}^{2}(T,Δ)〉_{c}, of current Q_{i} across the ith bond up to time T in the steady-state saturates as 〈Q_{i}^{2}〉_{c}≃Σ_{Q}^{2}(Δ)-constT^{-1/2}; near criticality, it grows subdiffusively as 〈Q_{i}^{2}〉_{c}∼T^{α}, with 0<α<1/2, and eventually saturates to Σ_{Q}^{2}(Δ). Interestingly, the asymptotic current fluctuation Σ_{Q}^{2}(Δ) is a nonmonotonic function of Δ: It diverges as Σ_{Q}^{2}(Δ)∼Δ^{2} for Δ≫ρ_{c} and Σ_{Q}^{2}(Δ)∼Δ^{-δ}, with δ>0, for Δ→0^{+}. Using a mass-conservation principle, we exactly determine the exponents δ=2(1-1/ν_{⊥})/ν_{⊥} and α=δ/zν_{⊥} via the correlation-length and dynamic exponents, ν_{⊥} and z, respectively. Finally, we show that in the steady state the self-diffusion coefficient D_{s}(ρ[over ¯]) of tagged particles is connected to activity through the relation D_{s}(ρ[over ¯])=a(ρ[over ¯])/ρ[over ¯].

Universal Hyperuniform Organization in Looped Leaf Vein Networks

The leaf vein network is a hierarchical vascular system that transports water and nutrients to the leaf cells. The thick primary veins form a branched network, while the secondary veins can develop closed loops forming a well-defined cellular structure. Through extensive analysis of a variety of distinct leaf species, we discover that the apparently disordered cellular structures of the secondary vein networks exhibit a universal hyperuniform organization and possess a hidden order on large scales. Disorder hyperuniform systems lack conventional long-range order, yet they completely suppress normalized infinite-wavelength density fluctuations like crystals. Specifically, we find that the distributions of the geometric centers associated with the vein network loops possess a vanishing static structure factor in the limit that the wave number k goes to 0, i.e., S(k)∼k^{α}, where α≈0.64±0.021, providing an example of class III hyperuniformity in biology. This hyperuniform organization leads to superior efficiency of diffusive transport, as evidenced by the much faster convergence of the time-dependent spreadability S(t) to its longtime asymptotic limit, compared to that of other uncorrelated or correlated disordered but nonhyperuniform organizations. Our results also have implications for the discovery and design of novel disordered network materials with optimal transport properties.

Hyperuniformity classes of quasiperiodic tilings via diffusion spreadability

Hyperuniform point patterns can be classified by the hyperuniformity scaling exponent α>0, that characterizes the power-law scaling behavior of the structure factor S(k) as a function of wave number k≡|k| in the vicinity of the origin, e.g., S(k)∼|k|^{α} in cases where S(k) varies continuously with k as k→0. In this paper, we show that the spreadability is an effective method for determining α for quasiperiodic systems where S(k) is discontinuous and consists of a dense set of Bragg peaks. It has been shown in [Phys. Rev. E 104, 054102 (2021)10.1103/PhysRevE.104.054102] that, for media with finite α, the long-time behavior of the excess spreadability S(∞)-S(t) can be fit to a power law of the form t^{-(d-α)/2}, where d is the space dimension, to accurately extract α for the continuous case. We first transform quasiperiodic and limit-periodic point patterns into two-phase media by mapping them onto packings of identical nonoverlapping disks, where space interior to the disks represents one phase and the space in exterior to them represents the second phase. We then compute the spectral density χ[over ̃]_{_{V}}(k) of the packings, and finally compute and fit the long-time behavior of their excess spreadabilities. Specifically, we show that the excess spreadability can be used to accurately extract α for the one-dimensional (1D) limit-periodic period-doubling chain (α=1) and the 1D quasicrystalline Fibonacci chain (α=3) to within 0.02% of the analytically known exact results. Moreover, we obtain a value of α=5.97±0.06 for the two-dimensional Penrose tiling and present plausible theoretical arguments strongly suggesting that α is exactly equal to six. We also show that, due to the self-similarity of the structures examined here, one can truncate the small-k region of the scattering information used to compute the spreadability and obtain an accurate value of α, with a small deviation from the untruncated case that decreases as the system size increases. This strongly suggests that one can obtain a good estimate of α for an infinite self-similar quasicrystal from a modestly sized finite sample. The methods described here offer a simple and general procedure to characterize accurately the large-scale translational order present in quasicrystalline and limit-periodic media in any space dimension that are self-similar. Moreover, the scattering information extracted from these two-phase media encoded in χ[over ̃]_{_{V}}(k), can be used to estimate their physical properties, such as their effective dynamic dielectric constants, effective dynamic elastic constants, and fluid permeabilities.

Vibrational properties of disordered stealthy hyperuniform 1D atomic chains

Disorder hyperuniformity is a recently discovered exotic state of many-body systems that possess a hidden order in between that of a perfect crystal and a completely disordered system. Recently, this novel disordered state has been observed in a number of quantum materials including amorphous 2D graphene and silica, which are endowed with unexpected electronic transport properties. Here, we numerically investigate 1D atomic chain models, including perfect crystalline, disordered stealthy hyperuniform (SHU) as well as randomly perturbed atom packing configurations to obtain a quantitative understanding of how the unique SHU disorder affects the vibrational properties of these low-dimensional materials. We find that the disordered SHU chains possess lower cohesive energies compared to the randomly perturbed chains, implying their potential reliability in experiments. Our inverse partition ratio (IPR) calculations indicate that the SHU chains can support fully delocalized states just like perfect crystalline chains over a wide range of frequencies, i.e.ω∈(0,100)cm-1, suggesting superior phonon transport behaviors within these frequencies, which was traditionally considered impossible in disordered systems. Interestingly, we observe the emergence of a group of highly localized states associated withω∼200cm-1, which is characterized by a significant peak in the IPR and a peak in phonon density of states at the corresponding frequency, and is potentially useful for decoupling electron and phonon degrees of freedom. These unique properties of disordered SHU chains have implications in the design and engineering of novel quantum materials for thermal and phononic applications.

Designer pair statistics of disordered many-particle systems with novel properties

The knowledge of exact analytical functional forms for the pair correlation function g2(r) and its corresponding structure factor S(k) of disordered many-particle systems is limited. For fundamental and practical reasons, it is highly desirable to add to the existing database of analytical functional forms for such pair statistics. Here, we design a plethora of such pair functions in direct and Fourier spaces across the first three Euclidean space dimensions that are realizable by diverse many-particle systems with varying degrees of correlated disorder across length scales, spanning a wide spectrum of hyperuniform, typical nonhyperuniform, and antihyperuniform ones. This is accomplished by utilizing an efficient inverse algorithm that determines equilibrium states with up to pair interactions at positive temperatures that precisely match targeted forms for both g2(r) and S(k). Among other results, we realize an example with the strongest hyperuniform property among known positive-temperature equilibrium states, critical-point systems (implying unusual 1D systems with phase transitions) that are not in the Ising universality class, systems that attain self-similar pair statistics under Fourier transformation, and an experimentally feasible polymer model. We show that our pair functions enable one to achieve many-particle systems with a wide range of translational order and self-diffusion coefficients D, which are inversely related to one another. One can design other realizable pair statistics via linear combinations of our functions or by applying our inverse procedure to other desirable functional forms. Our approach facilitates the inverse design of materials with desirable physical and chemical properties by tuning their pair statistics.

Hyperuniformity of maximally random jammed packings of hyperspheres across spatial dimensions

The maximally random jammed (MRJ) state is the most random (i.e., disordered) configuration of strictly jammed (mechanically rigid) nonoverlapping objects. MRJ packings are hyperuniform, meaning their long-wavelength density fluctuations are anomalously suppressed compared to typical disordered systems, i.e., their structure factors S(k) tend to zero as the wave number |k| tends to zero. Here we show that generating high-quality strictly jammed states for Euclidean space dimensions d=3,4, and 5 is of paramount importance in ensuring hyperuniformity and extracting precise values of the hyperuniformity exponent α>0 for MRJ states, defined by the power-law behavior of S(k)∼|k|^{α} in the limit |k|→0. Moreover, we show that for fixed d it is more difficult to ensure jamming as the particle number N increases, which results in packings that are nonhyperuniform. Free-volume theory arguments suggest that the ideal MRJ state does not contain rattlers, which act as defects in numerically generated packings. As d increases, we find that the fraction of rattlers decreases substantially. Our analysis of the largest truly jammed packings suggests that the ideal MRJ packings for all dimensions d≥3 are hyperuniform with α=d-2, implying the packings become more hyperuniform as d increases. The differences in α between MRJ packings and the recently proposed Manna-class random close packed (RCP) states, which were reported to have α=0.25 in d=3 and be nonhyperuniform (α=0) for d=4 and d=5, demonstrate the vivid distinctions between the large-scale structure of RCP and MRJ states in these dimensions. Our paper clarifies the importance of the link between true jamming and hyperuniformity and motivates the development of an algorithm to produce rattler-free three-dimensional MRJ packings.

Computational design of anisotropic stealthy hyperuniform composites with engineered directional scattering properties

Disordered hyperuniform materials are an emerging class of exotic amorphous states of matter that endow them with singular physical properties, including large isotropic photonic band gaps, superior resistance to fracture, and nearly optimal electrical and thermal transport properties, to name but a few. Here we generalize the Fourier-space-based numerical construction procedure for designing and generating digital realizations of isotropic disordered hyperuniform two-phase heterogeneous materials (i.e., composites) developed by Chen and Torquato [Acta Mater. 142, 152 (2018)1359-645410.1016/j.actamat.2017.09.053] to anisotropic microstructures with targeted spectral densities. Our generalized construction procedure explicitly incorporates the vector-dependent spectral density function χ[over ̃]_{_{V}}(k) of arbitrary form that is realizable. We demonstrate the utility of the procedure by generating a wide spectrum of anisotropic stealthy hyperuniform microstructures with χ[over ̃]_{_{V}}(k)=0 for k∈Ω, i.e., complete suppression of scattering in an "exclusion" region Ω around the origin in Fourier space. We show how different exclusion-region shapes with various discrete symmetries, including circular-disk, elliptical-disk, square, rectangular, butterfly-shaped, and lemniscate-shaped regions of varying size, affect the resulting statistically anisotropic microstructures as a function of the phase volume fraction. The latter two cases of Ω lead to directionally hyperuniform composites, which are stealthy hyperuniform only along certain directions and are nonhyperuniform along others. We find that while the circular-disk exclusion regions give rise to isotropic hyperuniform composite microstructures, the directional hyperuniform behaviors imposed by the shape asymmetry (or anisotropy) of certain exclusion regions give rise to distinct anisotropic structures and degree of uniformity in the distribution of the phases on intermediate and large length scales along different directions. Moreover, while the anisotropic exclusion regions impose strong constraints on the global symmetry of the resulting media, they can still possess structures at a local level that are nearly isotropic. Both the isotropic and anisotropic hyperuniform microstructures associated with the elliptical-disk, square, and rectangular Ω possess phase-inversion symmetry over certain range of volume fractions and a percolation threshold ϕ_{c}≈0.5. On the other hand, the directionally hyperuniform microstructures associated with the butterfly-shaped and lemniscate-shaped Ω do not possess phase-inversion symmetry and percolate along certain directions at much lower volume fractions. We also apply our general procedure to construct stealthy nonhyperuniform systems. Our construction algorithm enables one to control the statistical anisotropy of composite microstructures via the shape, size, and symmetries of Ω, which is crucial to engineering directional optical, transport, and mechanical properties of two-phase composite media.

Observation of magnetic structural universality and jamming transition with NMR

Nuclear magnetic resonance (NMR) has been instrumental in deciphering the structure of proteins. Here we show that transverse NMR relaxation, through its time-dependent relaxation rate, is distinctly sensitive to the structure of complex materials or biological tissues at the mesoscopic scale, from micrometers to tens of micrometers. Based on the ideas of universality, we show analytically and numerically that the time-dependent transverse relaxation rate approaches its long-time limit in a power-law fashion, with the dynamical exponent reflecting the universality class of mesoscopic magnetic structure. The spectral line shape acquires the corresponding non-analytic power law singularity at zero frequency. We experimentally detect the change in the dynamical exponent as a result of the transition into maximally random jammed state characterized by hyperuniform correlations. The relation between relaxational dynamics and magnetic structure opens the way for noninvasive characterization of porous media, complex materials and biological tissues.

Asymptotics and Summation of the Effective Properties of Suspensions, Simple Liquids and Composites

We review the problem of summation for a very short truncation of a power series by means of special resummation techniques inspired by the field-theoretical renormalization group. Effective viscosity (EV) of active and passive suspensions is studied by means of a special algebraic renormalization approach applied to the first and second-order expansions in volume fractions of particles. EV of the 2D and 3D passive suspensions is analysed by means of various self-similar approximants such as iterated roots, exponential approximants, super-exponential approximants and root approximants. General formulae for all concentrations are derived. A brief introduction to the rheology of micro-swimmers is given. Microscopic expressions for the intrinsic viscosity of the active system of puller-like microswimmers are obtained. Special attention is given to the problem of the calculation of the critical indices and amplitudes of the EV and to the sedimentation rate in the vicinity of known critical points. Critical indices are calculated from the short truncation by means of minimal difference and minimal derivative conditions on the fixed points imposed directly on the critical properties. Accurate expressions are presented for the non-local diffusion coefficient of a simple liquid in the vicinity of a critical point. Extensions and corrections to the celebrated Kawasaki formula are discussed. We also discuss the effective conductivity for the classical analog of graphene and calculate the effective critical index for superconductivity dependent on the concentration of vacancies. Finally, we discuss the effective conductivity of a random 3D composite and calculate the superconductivity critical index of a random 3D composite.