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All sources NASA NTRS DOE OSTI

DOE OSTI2023

Durable and Low-Cost Fractal Structured Multifunctional Coatings for Next-Generation CSP

Fractal coatings on various low-cost and high-temperature metallic substrates such as SS316, SS347, In800H, In740H, In625, Haynes 230, etc. were fabricated employing our patented novel electrodeposition process at various deposition potentials and deposition times, as well as chemical etching. The best combination of the materials and processing conditions were co-optimized for the reliable and repeatable fabrication of the multifunctional coatings. Surface morphologies were examined using SEM and AFM, and fractal characterization was done using a 3D optical surface profiler. Fractal dimension was used to identify successful fabrication and material parameters. The mechanical durability of the coatings was evaluated using several standardized tests, such as sand abrasion (ASTM D968) and adhesion (ASTM D3359), to fully characterize and demonstrate the mechanical durability of the fabricated surfaces. HTF immersion tests on the fractal multifunctional coatings were conducted based on a modification of the ASTM D870, which is established for static water immersion testing. To calculate the corrosion rates and descaled weight loss of the samples subjected to thermal exposure for different time durations (100 h, 300 h, 500 h, and optionally 750 h) to estimate the corrosion rate. While protecting the substrates from corrosion is an important goal, an allied goals is also to examine the post-corrosion temperature stability and properties of the molten salts themselves due to the dissolution of the constituents of the alloys and the corrosion products from the coatings that cause “fouling” of the molten salts. Fouling characteristics of the different molten salt HTFs exposed to coated and uncoated substrates were investigated. Finally, considering a prototype molten salt/sCO2 heat exchanger, cost/performance model is developed to demonstrate that the coatings on low-cost alloys such as stainless steel provide lower levelized cost of heat exchanger compared to high nickel content Ha230.

14 SOLAR ENERGY↗

DOE OSTI2021

Self-Similarity, Fractality and Entropy Principle in Collisions of Hadrons and Nuclei at Tevatron, RHIC and LHC

$z$-Scaling of inclusive spectra as a manifestation of self-similarity and fractality of hadron interactions is illustrated.The scaling for negative particle production in $Au+Au$ collisionsfrom BES-I program at RHIC is demonstrated. The scaling variable $z$ depends on the momentum fractions of the colliding objects carried by the interacting constituents, and momentum fractions of the produced objects in the scattered and recoil directions, which carries the inclusive particle and its counterpart, respectively. Structures of the colliding objects and fragmentation processes in final stateare expressed by fractal dimensions. Medium produced in the collisions is described by a specific heat. The scaling function $\psi(z)$ reveals energy, angular, multiplicity, and flavor independence.It has a power behavior at high $z$ (high $p_T$). Based on entropy principle and $z$-scaling, energy loss as a function of the collision energy, centrality and transverse momentum of inclusive particle is estimated.New conservation law including fractal dimensions is found. Quantization of fractal dimensions is discussed.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2020

Perfect quantum state transfer on diamond fractal graphs

In the quest for designing novel protocols for quantum information and quantum computation, an important goal is to achieve perfect quantum state transfer for systems beyond the well-known one- dimensional cases, such as 1D spin chains. Here, we use methods from fractal analysis and probability to find a new class of quantum spin chains on fractal-like graphs (known as diamond fractals) which support perfect quantum state transfer and which have a wide range of different Hausdorff and spectral dimensions. The resulting systems are spin networks combining Dyson hierarchical model structure with transverse permutation symmetries of varying order.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Soft matter approach for creating novel protein hydrogels using fractal whey protein assemblies as building blocks

In this study, mesoscopic sized fractal assembly (FA) particles were prepared using whey protein isolate; the coldset gelation properties of FA particles were investigated in-depth. Two types of FA particles (FA -62 and FA -90) with different mean sizes were synthesized through controlled thermal treatment of whey protein solutions at two concentrations (62 g/L and 90 g/L). Particle characteristics e.g. hydrodynamic radius, ζ-potential and surface hydrophobicity were dependent on pH and structure of FA. Transmission electron microscopy (TEM) observation confirmed the fractal morphology and small-angle X-ray scattering (SAXS) analysis suggested an internal fractal structure for the obtained FA particles. Eight cold-set FA protein gels (2 % w/v) were manufactured by controlling two gelling factors at two levels: pH (5.8 and 7.0) and Ca 2+ content (5 mM and 10 mM). Rheological characteristics in the large amplitude oscillatory shear regime revealed that pH 7.0 gels were softer and elastic while pH 5.8 gels were harder and brittle. Rheology Pipkin diagrams demonstrated that the strain softening/stiffening and the shear-thinning/thickening behaviors may be fine-tuned by manipulating the key gelation factors: e.g. FA structure, pH, and Ca 2+ . The entrainment speed-dependent friction coefficient curves showed that at an intermediate velocity regime (6-250 mm s -1 ), FA -90 particles induced hydrogels had superior lubrication effect compared to FA -62 gels. Further, this work demonstrated a food structure design approach regarding tuning texture and lubrication properties of protein gels without changing protein content and protein composition. The optimized protein hydrogels may be used as texturizer for "cleaner label" food formulas and/or as delivery system for carrying micronutrients.

60 APPLIED LIFE SCIENCES↗

DOE OSTI2022

Dynamical fractal and anomalous noise in a clean magnetic crystal

Fractals—objects with noninteger dimensions—occur in manifold settings and length scales in nature. In this work, we identify an emergent dynamical fractal in a disorder-free, stoichiometric, and three-dimensional magnetic crystal in thermodynamic equilibrium. The phenomenon is born from constraints on the dynamics of the magnetic monopole excitations in spin ice, which restrict them to move on the fractal. This observation explains the anomalous exponent found in magnetic noise experiments in the spin ice compound Dy 2 Ti 2 O 7 , and it resolves a long-standing puzzle about its rapidly diverging relaxation time. Here, the capacity of spin ice to exhibit such notable phenomena suggests that there will be further unexpected discoveries in the cooperative dynamics of even simple topological many-body systems.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2022

Fractal coatings of Ni and NiYSZ for high-temperature corrosion mitigation in solar salt

This paper introduces a novel approach to corrosion mitigation using electrodeposited, fractal Ni and NiYSZ coatings. Highly adherent, durable multiscale coatings with fractal dimensions above 1.80 were fabricated. Corrosion characteristics of coated plain and etched surfaces of In800H, SS310, SS316 and SS347, in comparison to Ha230, were studied in molten 60% NaNO 3 + 40% KNO 3 at 600 °C. The corrosion rate of coated Fe-based alloys reduced by up to 68% compared to uncoated alloys and was on par with or lower than the corrosion rate of Ha230. Furthermore, the corrosion-resistant coatings provide for use of low-cost alloys in high temperature corrosive environments.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Random Knotting in Fractal Ring Polymers

Many ring polymer systems of physical and biological interest exhibit both pronounced topological effects and nontrivial self-similarity, but the relationship between these two phenomena has not yet been clearly established. Here, we use theory and simulation to formulate such a connection by studying a fundamental topological property—the random knotting probability—for ring polymers with varying fractal dimension, d f . Using straightforward scaling arguments, we generalize a classic mathematical result, showing that the probability of a trivial knot decays exponentially with chain size, N, for all fractal dimensions: P 0 (N) ∝ exp(–N/N 0 ). However, no such simple considerations can account for the dependence of the knotting length, N 0 , on d f , necessitating a more involved analytical calculation. This analysis reveals a complicated double-exponential dependence, which is well supported by numerical data. By contrast, functional forms typical of simple scaling theories fail to adequately describe the observations. These findings are equally valid for two-dimensional ring polymer systems, where “knotting” is defined as the intersection of any two segments.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2025

Spectroscopy of the fractal Hofstadter energy spectrum

Hofstadter’s butterfly, the predicted energy spectrum for non-interacting electrons confined to a two-dimensional lattice in a magnetic field, is one of the most remarkable fractal structures in nature. At rational ratios of magnetic flux quanta per lattice unit cell, this spectrum shows self-similar distributions of energy levels that reflect its recursive construction. For most materials, Hofstadter’s butterfly is predicted under experimental conditions that are unachievable using laboratory-scale magnetic fields. More recently, electrical transport studies have provided evidence for Hofstadter’s butterfly in materials engineered to have artificially large lattice constants, such as those with moiré superlattices. Yet, so far, direct spectroscopy of the fractal energy spectrum predicted by Hofstadter nearly 50 years ago has remained out of reach. Here, in this work, we use high-resolution scanning tunnelling microscopy/spectroscopy (STM/STS) to investigate the flat electronic bands in twisted bilayer graphene (TBG) near the predicted second magic angle, an ideal setting for spectroscopic studies of Hofstadter’s spectrum. Our study shows the fractionalization of flat moiré bands into discrete Hofstadter subbands and discerns experimental signatures of self-similarity of this spectrum. Moreover, our measurements uncover a spectrum that evolves dynamically with electron density, showing phenomena beyond that of Hofstadter’s original model owing to the combined effects of strong correlations, Coulomb interactions and the quantum degeneracy of electrons in TBG.

electronic properties and materials↗

DOE OSTI2021

Lattice thermal transport in two-dimensional alloys and fractal heterostructures

Abstract Engineering thermal transport in two dimensional materials, alloys and heterostructures is critical for the design of next-generation flexible optoelectronic and energy harvesting devices. Direct experimental characterization of lattice thermal conductivity in these ultra-thin systems is challenging and the impact of dopant atoms and hetero-phase interfaces, introduced unintentionally during synthesis or as part of deliberate material design, on thermal transport properties is not understood. Here, we use non-equilibrium molecular dynamics simulations to calculate lattice thermal conductivity of $${\mathrm {(Mo|W)Se_2}}$$ ( Mo | W ) Se 2 monolayer crystals including $${\mathrm {Mo}}_{1-x}{\mathrm {W}}_x{\mathrm {Se_2}}$$ Mo 1 - x W x Se 2 alloys with substitutional point defects, periodic $${\mathrm {MoSe_2}|\mathrm {WSe_2}}$$ MoSe 2 | WSe 2 heterostructures with characteristic length scales and scale-free fractal $${\mathrm {MoSe_2}}|{\mathrm {WSe_2}}$$ MoSe 2 | WSe 2 heterostructures. Each of these features has a distinct effect on phonon propagation in the crystal, which can be used to design fractal and periodic alloy structures with highly tunable thermal conductivities. This control over lattice thermal conductivity will enable applications ranging from thermal barriers to thermoelectrics.

42 ENGINEERING↗

DOE OSTI2021

Spectra of perfect state transfer Hamiltonians on fractal-like graphs

Abstract In this paper we study the spectral features, on fractal-like graphs, of Hamiltonians which exhibit the special property of perfect quantum state transfer (PQST): the transmission of quantum states without dissipation. The essential goal is to develop the theoretical framework for understanding the interplay between PQST, spectral properties, and the geometry of the underlying graph, in order to design novel protocols for applications in quantum information science. We present a new lifting and gluing construction, and use this to prove results concerning an inductive spectral structure, applicable to a wide variety of fractal-like graphs. We illustrate this construction with explicit examples for several classes of diamond graphs.

Physics↗

DOE OSTI2023

Fractal dimensions of jammed packings with power-law particle size distributions in two and three dimensions

Static structure factors are computed for large-scale, mechanically stable, jammed packings of frictionless spheres (three dimensions) and disks (two dimensions) with broad, power-law size dispersity characterized by the exponent -β. Here, the static structure factor exhibits diverging power-law behavior for small wave numbers, allowing us to identify a structural fractal dimension d f . In three dimensions, d f ≈ 2.0 for 2.5≤ β ≤ 3.8, such that each of the structure factors can be collapsed onto a universal curve. In two dimensions, we instead find 1.0≲ d f ≲1.34 for 2.1≤ β ≤ 2.9. Furthermore, we show that the fractal behavior persists when rattler particles are removed, indicating that the long-wavelength structural properties of the packings are controlled by the large particle backbone conferring mechanical rigidity to the system. A numerical scheme for computing structure factors for triclinic unit cells is presented and employed to analyze the jammed packings.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Mesoscale fractal whey protein particles derived from microscale linear-shaped protein assemblies (Part 1): Manufacturing method and particle characteristics

Whey protein isolates (WPI) are widely used in processed foods for their versatile functional properties. Modifying the structural properties of proteins by assembling them into mesoscale or microscale particles may improve their functionality and broaden their applications. This study aims to manufacture and characterize mesoscale whey protein particles (WPP) derived from WPI. Two types of WPP, WPP1 (0.05 mL/min) and WPP2 (0.25 mL/min), were prepared through a multistep approach involving liquid antisolvent (LAS) precipitation, heat treatment, and microfluidization. Liquid antisolvent precipitation was performed by injecting a 20% (wt/vol) WPI dispersion (pH 7) into an ethanol-glycerol mixture (75:25, vol/vol) under laminar flow, followed by heat treatment at 80°C for 20 min as a particle hardening step. This process produced stable fiber- and ribbon-shaped whey protein assemblies (WPA), which served as precursors to WPP. Subsequent microfluidization (150 MPa, 6 passages) reduced the size of WPA, yielding mesoscale WPP with irregular morphologies and a more uniform size distribution, as revealed by microscopy and dynamic light scattering. ζ-Potential and fluorescence labeling indicated higher surface charge and surface hydrophobicity of WPP compared with untreated WPI. The WPP showed internal mass fractal and surface fractal structures at larger length scales, analyzed using small-angle X-ray scattering. Fourier transform infrared spectroscopy demonstrated an increased fraction of intermolecular β-sheets in WPP, suggesting that hydrogen bonding contributed to their formation. Gel electrophoresis confirmed that disulfide bonds served as the primary cross-links stabilizing the WPP structure. Furthermore, turbidity measurements showed that WPP exhibited superior colloidal phase stability compared with untreated WPI and maintained high colloidal stability under both acidic and neutral pH conditions.

Antisolvent precipitation↗

DOE OSTI2025

Linking Lattice Strain and Fractal Dimensions to Non‐monotonic Volume Changes in Irradiated Nuclear Graphite

Graphite's resilience to high temperatures and neutron damage makes it vital for nuclear reactors, yet irradiation alters its microstructure, degrading key properties. We used small- and wide-angle X-ray scattering to study neutron-irradiated fine-grain nuclear graphite (Grade G347A) across varied temperatures and fluences. Results show significant shifts in internal strain and porosity, correlating with radiation-induced volume changes. Notably, porosity volume distribution (fractal dimensions) follows non-monotonic volume changes, suggesting a link to the Weibull distribution of fracture stress.

X-ray scattering↗

DOE OSTI2025

Fractal Scaling of Explosively Driven Product Gases

ABSTRACT Characterization of the interface between explosive product gases and ambient air in an explosion is a complicated task due to the turbulent mixing and inherently three‐dimensional expansion of the interface. This study aims to quantify the evolution of the interface as a temporally varying Hausdorff dimension. Two test series were conducted with Composition C‐4 charges with masses of 105 and 880 g. Imaging data were collected from the time of detonation until shock wave detachment using ultra‐high‐speed cameras. Gas cloud profiles were extracted using automated image processing algorithms, and the Hausdorff dimension of these two‐dimensional slices of the gas cloud was then estimated using boxcounting algorithms. When scaled with standard gas dynamic nondimensional scalings, the Hausdorff dimension of all explosive events appears to collapse towards a single curve. The fireball was initially nonfractal and began to develop fractal properties as the shock wave separated from the detonation products. Artificial perturbation of the charged surface had no detectable impact on the evolution of the Hausdorff dimension in the early development of the fireball outside of error, despite visible phenomenological differences in the early development of mixing on the fireball surface.

42 ENGINEERING↗

DOE OSTI2021

Dynamics of a fractal set of first-order magnetic phase transitions in frustrated Lu 2 CoMnO 6

The axial next-nearest-neighbor Ising model predicts a fractal (infinite) set of phases with incommensurate wave vectors that are separated by first-order phase boundaries. This complexity results from a simple frustration condition between nearest- and next-nearest-neighbor interactions along a chain of Ising spins. Using x-ray photon correlation spectroscopy (XPCS), we investigate the surprising antiferromagnetic dynamics that emerge from such a complex phase diagram over a wide range of temperatures. In this work, we present XPCS measurements of the frustrated magnetic chain compound Lu 2 CoMnO 6 and Monte Carlo simulations. Incommensurate magnetic Bragg peaks slide towards commensurate “up-up-down-down” spin order with decreasing temperature and increasing time. Both simulation and experiment support a counterintuitive “upside-down” temperature dependence of the magnetic dynamics: at higher temperatures in the region of first-order phase boundaries, slower dynamics are observed where the speckle maintains its coherence. At the lowest temperatures, where part of the sample adopts commensurate order, the dynamics speed up and result in fast decoherence.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Fractal Characterization of Multimodal, Multiscale Images of Shale Rock Fracture Networks

An array of multimodal and multiscale images of fractured shale rock samples and analogs was collected with the aim of improving the numerical representation of fracture networks. 2D and 3D reconstructions of fractures and matrices span from 10 –6 to 100 m. The origin of the fracture networks ranged from natural to thermal maturation to hydraulically induced maturation. Images were segmented to improve fracture identification. Then, the fractal dimension and length distribution of the fracture networks were calculated for each image dataset. The resulting network connectivity and scaling associations are discussed at length on the basis of scale, sample and order of magnitude. Fracture network origin plays an important role in the resulting fracture systems and their scaling. Rock analogs are also evaluated using these descriptive tools and are found to be faithful depictions of maturation-induced fracture networks.

58 GEOSCIENCES↗

DOE OSTI2024

Angular fractals in thermal QFT

We show that thermal effective field theory controls the long-distance expansion of the partition function of a d-dimensional QFT, with an insertion of any finite-order spatial isometry. Consequently, the thermal partition function on a sphere displays a fractal-like structure as a function of angular twist, reminiscent of the behavior of a modular form near the real line. As an example application, we find that for CFTs, the effective free energy of even-spin minus odd-spin operators at high temperature is smaller than the usual free energy by a factor of 1/2 d . Near certain rational angles, the partition function receives subleading contributions from “Kaluza-Klein vortex defects” in the thermal EFT, which we classify. We illustrate our results with examples in free and holographic theories, and also discuss nonperturbative corrections from worldline instantons.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2020

Light scattering study of highly absorptive, non-fractal, hematite aggregates

In this paper, we present measurements of the scattered light intensity by aerosolized hematite aggregate particles. The measurements were made at a wavelength of 532 nm in the scattering angle range from 0.32° to 157°. Hematite has high values of the real and imaginary parts of the refractive index m = n + iκ = 3 + i0.5 at the studied wavelength. Scanning electron micrographs (SEM) indicated that the particles were aggregates whereas the optical microscope pictures showed that the aerosol had a bimodal distribution with effective mean diameters of roughly 1 and 10 µm. This is consistent with the light scattering results which displayed two Guinier regimes. The aggregates were composed of smaller grains with an approximate size of 200 nm. Ultra Small-Angle X-ray Scattering (USAXS) indicate that the aggregates were uniform and non-fractal. Mie calculations for a sphere equivalent to the aggregate size were compared to the experimentally observed results. The observed results showed an enhanced backscattering, whereas the Mie calculations did not due to the large imaginary part of the refractive index. Hematite aggregates were simulated by assuming they were composed of spherical monomers inside a spherical volume. Then the light scattering was calculated using the T-matrix method for these simulated aggregates. The calculated results show an enhanced backscattering. We present a dimensional analysis to estimate the extent of multiple scattering within the aggregate and find a correlation between the average number of scattering events within the aggregate and the enhancement in the backscattering.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗