Fractal textured surfaces for high temperature corrosion mitigation in molten salts
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Fractal Model for Drag Reduction on Multiscale Nonwetting Rough Surfaces
Rough surfaces in contact with a flow of fluid exhibit alternating no-slip and free shear boundary conditions at the solid–liquid and air–liquid interfaces, respectively, thereby potentially offering drag reduction benefits. The balance between the dynamic pressure in the flow and the restoring capillary pressure in the interasperity spaces determines the stability of the Cassie state of wettability and is a function of the relative extent of no-slip and free shear regions per unit surface area. In the present study, using a fractal representation of rough surface topography, an analytical model is developed to quantify the stability of the Cassie state of wettability as well as drag reduction and the friction factor for laminar flow in a rectangular channel between nonwetting multiscale rough surfaces. A systematic study is conducted to quantify the effects of fractal parameters of the surfaces and the flow Reynolds number on drag reduction and the friction factor. The studies are used to develop friction factor curves extending the classical Moody diagram to hydrophobic and superhydrophobic surfaces. On the basis of the studies, regime maps are derived for estimating the extent of drag reduction offered by hydrophobic and superhydrophobic surfaces, revealing that superhydrophobic surfaces do not always offer the best drag reduction performance. The application of the fractal model to practical topographies of nonwetting surfaces of copper, aluminum, and zinc oxide fabricated via electrodeposition and etching is also discussed.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Fractal analysis on Ag 2 O thin film using a data-driven approach
The synthesis of fractal Ag oxide (Ag 2 O) on the surface of Ag thin film has been achieved at room temperature by using Synchrotron X-ray irradiation. We have performed an automated quantitative analysis of a batch of 1879 fractal Ag 2 O patterns in a scanning electron microscopy (SEM) image within a radius of 2000 mm outward from the center of the X-ray beam. The morphology is similar to that of the diffusion-limited cluster aggregation (DLCA) model. The fractal dimension (D) of Ag 2 O is between 1.7 and 1.5 from the center to the edge. The area distribution density of fractal Ag 2 O follows a quadratic function with radius R. It is found that the branches’ number of fractal Ag 2 O is a key factor affecting the fractal dimension. The more branches the fractal has, the greater the D is. This is the first time that Ag fractal has been investigated by combining automated data analysis methods with batch experimental data. This data-driven approach provides a new research perspective for rationally regulating materials’ fractal morphology and performance.
36 MATERIALS SCIENCE↗
Fractal dimension of optical cirrus in Stripe82
ABSTRACT The geometric characteristics of dust clouds provide important information on the physical processes that structure such clouds. One of such characteristics is the 2D fractal dimension D of a cloud projected on to the sky plane. In previous studies, which were mostly based on infrared (IR) data, the fractal dimension of individual clouds was found to be in a range from 1.1 to 1.7 with a preferred value of 1.2–1.4. In this work, we use data from Stripe82 of the Sloan Digital Sky Survey to measure the fractal dimension of the cirrus clouds. This is done here for the first time for optical data with significantly better resolution as compared to IR data. To determine the fractal dimension, the perimeter-area method is employed. We also consider IR (IRAS and Herschel) counterparts of the corresponding optical fields to compare the results between the optical and IR. We find that the averaged fractal dimension across all clouds in the optical is $\langle D \rangle =1.69^{+0.05}_{-0.05}$ which is significantly larger than the fractal dimension of its IR counterparts $\langle D\rangle =1.38^{+0.07}_{-0.06}$. We examine several reasons for this discrepancy (choice of masking and minimal contour level, image and angular resolution, etc.) and find that for approximately half of our fields the different angular resolution (point spread function) of the optical and IR data can explain the difference between the corresponding fractal dimensions. For the other half of the fields, the fractal dimensions of the IR and visual data remain inconsistent, which can be associated with physical properties of the clouds, but further physical simulations are required to prove it.
79 ASTRONOMY AND ASTROPHYSICS↗
Can fractal dimensions objectivize gastropod shell morphometrics? A case study from Lake Lugu (SW China)
1. Morphometrics are fundamental for the analysis of size and shape in fossils, particularly because soft parts or DNA are rarely preserved and hard parts such as shells are commonly the only source of information. Geometric morphometrics, that is, landmark analysis, is well established for the description of shape but it exhibits a couple of shortcomings resulting from subjective choices during landmarking (number and position of landmarks) and from difficulties in resolving shape at the level of micro-sculpture. 2. With the aid of high-resolution 3D scanning technology and analyses of fractal dimensions, we test whether such shortcomings of linear and landmark morphometrics can be overcome. As a model group, we selected a clade of modern viviparid gastropods from Lake Lugu, with shells that show a high degree of sculptural variation. Linear and landmark analyses were applied to the same shells in order to establish the fractal dimensions. 3. The genetic diversity of the gastropod clade was assessed. The genetic results suggest that the gastropod clade represents a single species. The results of all morphometric methods applied are in line with the genetic results, which is that no specific morphotype could be delimited. Apart from this overall agreement, landmark and fractal dimension analyses do not correspond to each other but represent data sets with different information. Generally, the fractal dimension values quantify the roughness of the shell surface, the resolution of the 3D scans determining the level. In our approach, we captured the micro-sculpture but not the first-order sculptural elements, which explains that fractal dimension and landmark data are not in phase. 4. We can show that analyzing fractal dimensions of gastropod shells opens a window to more detailed information that can be considered in evolutionary and ecological contexts. We propose that using low-resolution 3D scans may successfully substitute landmark analyses because it overcomes the subjective landmarking. Analyses of 3D scans with higher resolution than used in this study will provide surface roughness information at the mineralogical level. We suggest that fractal dimension analyses of a combination of differently resolved 3D models will significantly improve the quality of shell morphometrics.
54 ENVIRONMENTAL SCIENCES↗
Novel fractal-textured solar absorber surfaces for concentrated solar power
The solar absorptance of a thermal receiver surface significantly affects the photothermal conversion efficiency of concentrated solar power (CSP) plants. The development of low-cost Gen3 CSP systems calls for increasing the solar absorptance of the thermal receivers at operating temperatures above 750 °C. This study presents an innovative approach to the fractal, multiscale texturing of absorber surfaces to significantly enhance solar energy absorption for Gen3 CSP applications. The absorber surface is described in terms of its fractal parameters that are uniquely determined from surface profile measurements. The interaction of solar radiation with the fractal surface is numerically simulated by solving the governing Maxwell’s equations for electromagnetic wave propagation to investigate the effect of texturing on the absorptance of the surfaces. It is shown that absorption of the solar spectrum increases with an increase in the fractal dimension and the multiscale asperity height of the surface texturing. Here, the theoretical model is demonstrated to be in close agreement with experimental measurements of spectral absorptance of electrodeposited copper (Cu), copper mono-oxide (CuO), and copper-manganese oxide (CuMnO) surfaces that are textured to produce a range of fractal parameters by tailoring the deposition parameters. Fractal surface texturing is shown to reduce reflectance by over one order of magnitude, yielding an absorptance of greater than 0.985 for CuMnO. For the first time, the study presents a significant effective means of increasing solar absorptance and a fundamental theoretical description of the underlying physics.
14 SOLAR ENERGY↗
Water transport properties of virtual fractal porous media: Implications for the unsaturated transport properties of cement-based materials
Cement-based materials are more and more recognized as fractal materials. From a practical point of view, fractality means that the pore size distribution can be described using fractal scaling law (i.e. power functions). Here, the simplest scaling law making use of a single fractal dimension was used to generate simple and virtual fractal porous media (based on bundles of parallel cylindrical pores). The capillary curve and permeability (relative and intrinsic) of the virtual porous media were then estimated through theoretical and numerical approaches. The results show that: (1) although bereft of any physical basis, van Genuchten equation bears some fractal information; (2) although highly flexible, van Genuchten equation fails to perfectly fit the capillary curves and can thus generate error in inverse analysis (for intrinsic permeability evaluation) and (3) there seems to be a relation between the intrinsic permeability and van Genuchten pressure parameter.
36 MATERIALS SCIENCE↗
Soot-particle core-shell and fractal structures from small-angle X-ray scattering measurements in a flame
We have characterized soot particles measured in situ in a laminar co-flow ethylene-air diffusion flame using small-angle X-ray scattering (SAXS). The analysis includes temperature measurements made with coherent anti-Stokes Raman spectroscopy (CARS) and complements soot volume-fraction and maturity measurements made with laser-induced incandescence (LII). We compared the results of fits to the SAXS measurements using a unified model and a fractal core-shell model. Power-law parameters yielded by the unified model indicate that aggregates of primary particles are in the mass-fractal regime, whereas the primary particles are in the surface-fractal regime in the middle of the flame. Higher and lower in the flame, the primary-particle power-law parameter approaches 4, suggesting smooth primary particles. Furthermore, these trends are consistent with fits using the fractal core-shell model, which indicate that particles have an established core-shell structure in the middle of the flame and are internally homogeneous at higher and lower heights in the flame. Primary-particle size distributions derived using the fractal core-shell model demonstrate excellent agreement with distributions inferred from transmission electron microscopy (TEM) images in the middle of the flame. Higher in the flame, a second small mode appears in the size distributions, suggesting particle fragmentation during oxidation. Surface oxidation would explain (1) aggregate fragmentation and (2) loss of core-shell structure leading to smoother primary-particle surfaces by removal of carbon overlayers. SAXS measurements are much more sensitive to incipient and young soot particles than LII and demonstrate significant volume fraction from particles low in the flame where the LII signal is negligible.
42 ENGINEERING↗
Characterization of the fractal dimension of helium plumes in laminar to turbulent transition
Abstract Schlieren imaging in conjunction with a high-speed camera was used to observe the behavior of metered helium plumes as they transition from laminar to turbulent flow in an air environment. The plumes were visualized at twelve jet Reynolds numbers ranging from 200 to 2980. The fractal dimension of the flows was obtained by applying a box counting algorithm to the recorded schlieren images. The results were analyzed to determine the correlation between the Reynolds number of the flow and the fractal dimension of the observed turbulence. A trend of increasing fractal dimension with increasing Reynolds number was observed for several different types of schlieren cutoffs including horizontal cutoff, vertical cutoff, circular cutoff, focused shadowgraphy and de-focused shadowgraphy. The vertical cutoff and focused shadowgraphy imaging methods showed the most consistent results for the fractal dimension characterization during the laminar to turbulent transition. For transitional plumes, it was observed that fractal dimension increased with distance from the jet outlet.
42 ENGINEERING↗
A viscoelastic model for seismic attenuation using fractal mechanical networks
SUMMARY Seismic attenuation (quantified by the quality factor Q) has a significant impact on the seismic waveforms, especially in the fluid-saturated rocks. This dissipative process can be phenomenologically represented by viscoelastic models. Previous seismological studies show that the Q value of Earth media exhibits a nearly frequency-independent behaviour (often referred to as constant-Q in literature) in the seismic frequency range. Such attenuation can be described by the mathematical Kjartansson constant-Q model, which lacks of a physical representation in the viscoelastic sense. Inspired by the fractal nature of the pore fluid distribution in patchy-saturated rocks, here we propose two fractal mechanical network (FMN) models, that is, a fractal tree model and a quasi-fractal ladder model, to phenomenologically represent the frequency-independent Q behaviour. As with the classic viscoelastic models, the FMN models are composed of mechanical elements (spring and dashpots) arranged in different hierarchical patterns. A particular parametrization of each model can produce the same complex modulus as in the Kjartansson model, which leads to the constant-Q. Applying the theory to several typical rock samples, we find that the seismic attenuation signature of these rocks can be accurately represented by either one of the FMN models. Besides, we demonstrate that the ladder model in particular exhibits the realistic multiscale fractal structure of the saturated rocks. Therefore, the FMN models as a proxy could provide a new way to estimate the microscopic rock structure property from macroscopic seismic attenuation observation.
Geochemistry & Geophysics↗
Scaling Analysis of Two–Phase Flow in Fractal Permeability Fields
Fluid mixing in permeable media is essential in many practical applications. The mixing process is a consequence of velocity fluctuations owing to geological heterogeneities and mobility contrast of fluids. Heterogeneities in natural rocks are often spatially correlated, and their properties, such as permeability, may be described using fractal distributions. This work models the fractal characteristics of such permeability fields in which the covariance function is expressed as a power-law function. A generalized scaling relation is derived relating various fractal permeability fields using the magnitude of their fluctuations. Here, this relation reveals the self-similar behavior of two-phase flow in such permeable media. To that end, a recently developed, high-resolution numerical simulator is employed to validate the analytically derived scaling relations. Two flow problems are considered in which flow is governed by 1) a linear, and 2) a nonlinear transport equation. Due to the probabilistic representation of the fractal permeability fields, a sensitivity study is conducted for each flow scenario to determine the number of realizations required for statistical convergence. Scaling analysis is performed using ensemble averages of simulated saturation profiles and their mixing lengths. Results support the validity of the developed scaling relation across the range of investigated flow conditions at intermediate times. The dynamics of linear flow in the asymptotic regime is affected by the correlation structure of heterogeneity. In nonlinear flow, scaling behavior appears to be dominated by the degree of nonlinearity.
58 GEOSCIENCES↗
Construction of Fractal Order and Phase Transition with Rydberg Atoms
We propose the construction of a many-body phase of matter with fractal structure using arrays of Rydberg atoms. The degenerate low energy excited states of this phase form a self-similar fractal structure. This phase is analogous to the so-called “type-II fracton topological states.” The main challenge in realizing fractonlike models in standard condensed matter platforms is the creation of multispin interactions, since realistic systems are typically dominated by two-body interactions. In this work, we demonstrate that the van der Waals interaction and experimental tunability of Rydberg-based platforms enable the simulation of exotic phases of matter with fractal structures, and the study of a quantum phase transition involving a fractal ordered phase.
74 ATOMIC AND MOLECULAR PHYSICS↗
Skyrmion‐Excited Spin‐Wave Fractal Networks
Magnetic skyrmions exhibit unique, technologically relevant pseudo-particle behaviors which arise from their topological protection, including well-defined, 3D dynamic modes that occur at microwave frequencies. During dynamic excitation, spin waves are ejected into the interstitial regions between skyrmions, creating the magnetic equivalent of a turbulent sea. However, since the spin waves in these systems have a well-defined length scale, and the skyrmions are on an ordered lattice, ordered structures from spin-wave interference can precipitate from the chaos. This work uses small-angle neutron scattering (SANS) to capture the dynamics in hybrid skyrmions and investigate the spin-wave structure. Performing simultaneous ferromagnetic resonance and SANS, the diffraction pattern shows a large increase in low-angle scattering intensity, which is present only in the resonance condition. This scattering pattern is best fit using a mass fractal model, which suggests the spin waves form a long-range fractal network. The fractal structure is constructed of fundamental units with a size that encodes the spin-wave emissions and are constrained by the skyrmion lattice. Furthermore, these results offer critical insights into the nanoscale dynamics of skyrmions, identify a new dynamic spin-wave fractal structure, and demonstrate SANS as a unique tool to probe high-speed dynamics.
36 MATERIALS SCIENCE↗
High-Order Fractal Quantum Oscillations in Graphene/BN Superlattices in the Extreme Doping Limit
Recent studies of van der Waals (vdW) heterostructures and superlattices have shown intriguing quantum phenomena, but these have been largely explored only in the moderate carrier density regime. Here, we report the probe of high-temperature fractal Brown-Zak (BZ) quantum oscillations through magnetotransport in the extreme doping regimes by applying a newly developed electron beam doping technique. Further, this technique gives access to both ultrahigh electron and hole densities beyond the dielectric breakdown limit in graphene/BN superlattices, enabling the observation of nonmonotonic carrier-density dependence of fractal BZ states and up to fourth-order fractal BZ features despite strong electron-hole asymmetry. Theoretical tight-binding simulations qualitatively reproduce all observed fractal BZ features and attribute the nonmonotonic dependence to the weakening of superlattice effects at high carrier densities.
36 MATERIALS SCIENCE↗
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↗
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↗
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↗
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↗