Simple Scaling Laws Research Articles

Rapid solidification forming of glassy and crystalline ribbons by planar flow casting

Emerging products for an ultra-energy-efficient economy demand new amorphous and nanocrystalline thin metallic glass (TMG) alloys. Planar flow casting (PFC) is the rapid solidification process of choice for forming TMGs. In this paper, TMG and crystalline alloys are demonstrated to process similarly by PFC despite their well-known different solid-state atomic structures. The high-fragility of the TMG alloy is argued to be responsible. A simple scaling law for ribbon thickness is reported. Thickness depends on the thermal properties of the alloy and heat transfer characteristic of the PFC machine. Perhaps surprisingly, explicit knowledge of the viscosity of the glass is not needed. Dimensional reasoning, mathematical modeling and experiment for two different alloys on two different PFC machines are the methods employed.

Fractal dimension as a scaling law for nuclear quantum effects: a neutron Compton scattering study on carbon allotropes

In this work, we tackle the problem of the sensitivity of neutron Compton scattering, measured through the widths of nuclear momentum distributions, to the degree of complexity and ordering of the structural motif characterising the surrounding environment felt by a particular nucleus (carbon). In doing so, we replace the usual concept of the bond strength categorised in terms of its thermodynamical or electronic properties with a novel observable inspired by the language of mathematical topology, the Hausdorff-Besicovitch fractal dimension. We derive a relatively straightforward connection between the fractal dimension of a given system under consideration and the nuclear kinetic energy. To achieve this, we modify the concept of the energy equipartition theorem for solid-state systems composed of carbon atoms where the atom-ordering topology does not follow a simple two or three-dimensional order, but rather atoms are placed along curves in space that have an intermediate dimension related to the varying amounts of information they contain. A series of results from past neutron Compton scattering studies, as well as new results on Buckminsterfullerene (C60), correlate with the topological measures of surface roughness and bending, as categorised quantitatively by the fractal dimension of the system. Namely, for the same formal chemical binding motif (sp2 C) and with decreasing system dimensionality from nearly 3 towards 1, the quantum nature of the system becomes more pronounced. The simple scaling law developed in this work allows for relatively simple assessment of the nuclear “quantumness” of a given system with potentially important ramifications in the ab initio modelling of nuclear quantum effects in condensed matter.

Hyperviscosity-based stabilization for radial basis function-finite difference (RBF-FD) discretizations of advection–diffusion equations

We present a novel hyperviscosity formulation for stabilizing RBF-FD discretizations of the advection–diffusion equation. The amount of hyperviscosity is determined quasi-analytically for commonly-used explicit, implicit, and implicit–explicit (IMEX) time integrators by using a simple 1D semi-discrete Von Neumann analysis. The analysis is applied to an analytical model of spurious growth in RBF-FD solutions that uses auxiliary differential operators mimicking the undesirable properties of RBF-FD differentiation matrices. The resulting hyperviscosity formulation is a generalization of existing ones in the literature, but is free of any tuning parameters and can be computed efficiently. To further improve robustness, we introduce a simple new scaling law for polynomial-augmented RBF-FD that relates the degree of polyharmonic spline (PHS) RBFs to the degree of the appended polynomial. When used in a novel ghost node formulation in conjunction with the recently-developed overlapped RBF-FD method, the resulting method is robust and free of stagnation errors. We validate the high-order convergence rates of our method on 2D and 3D test cases over a wide range of Peclet numbers (1–1000). We then use our method to solve a 3D coupled problem motivated by models of platelet aggregation and coagulation, again demonstrating high-order convergence rates.

Moonfalls: collisions between the Earth and its past moons

During the last stages of the terrestrial planet formation, planets grow mainly through giant-impacts with large planetary embryos. The Earth's Moon was suggested to form through one of these impacts. However, since the proto-Earth has experienced many giant-impacts, several moons are naturally expected to form through a sequence of multiple (including smaller scale) impacts. Each impact potentially forms a sub-Lunar mass moonlet that interacts gravitationally with the proto-Earth and possibly with previously-formed moonlets. Such interactions result in either moonlet-moonlet mergers, moonlet ejections or infall of moonlets on the Earth. The latter possibility, leading to low-velocity moonlet-Earth collisions is explored here for the first time. We make use of SPH simulations and consider a range of moonlet masses, collision impact-angles and initial proto-Earth rotation rates. We find that grazing/tidal-collisions are the most frequent and produce comparable fractions of accreted-material and debris. The latter typically clump in smaller moonlets that can potentially later interact with other moonlets. Other collision geometries are more rare. Head-on collisions do not produce much debris and are effectively perfect mergers. Intermediate impact angles result in debris mass-fractions in the range of 2-25% where most of the material is unbound. Retrograde collisions produce more debris than prograde collisions, whose fractions depend on the proto-Earth initial rotation rate. Moonfalls can slightly change the rotation-rate of the proto-Earth. Accreted moonfall material is highly localized, potentially explaining the isotopic heterogeneities in highly siderophile elements in terrestrial rocks, and possibly forming primordial super-continent topographic features. Our results can be used for simple scaling laws and applied to n-body studies of the formation of the Earth and Moon.

Scaling Law of High Harmonic Generation in the Framework of Photon Channels.

A photon channel perspective on high harmonic generation (HHG) is proposed by quantizing both the driving laser and high harmonics. It is shown that the HHG yield can be expressed as a sum of the contribution of all the photon channels. From this perspective, the contribution of a specific photon channel follows a simple scaling law and the competition between the channels is well interpreted. Our prediction is shown to be in good agreement with the simulations by solving the time-dependent Schrödinger equation. It also can explain well the experimental results of the HHG in the noncollinear two-color field and bicicular laser field.

A simple scaling law for the equation of state and the radial distribution functions calculated by density-functional theory molecular dynamics

It is shown that the equation of state (EOS) and the radial distribution functions obtained by density-functional theory molecular dynamics (DFT-MD) obey a simple scaling law. At given temperature, the thermodynamic properties and the radial distribution functions given by a DFT-MD simulation remain unchanged if the mole fractions of nuclei of given charge and the average volume per atom remain unchanged. A practical interest of this scaling law is to obtain an EOS table for a fluid from that already obtained for another fluid if it has the right characteristics. Another practical interest of this result is that an asymmetric mixture made up of light and heavy atoms requiring very different time steps can be replaced by a mixture of atoms of equal mass, which facilitates the exploration of the configuration space in a DFT-MD simulation. The scaling law is illustrated by numerical results.

Electromechanical modeling of eye fatigue detecting using flexible piezoelectric sensors

Eye fatigue has attracted significant interest due to its potential harm to human daily activities. An ultrathin flexible piezoelectric sensor was currently designed and fabricated to detect eye fatigue by deforming together with the eyelid epidermis. Herein we develop a theoretical model to illustrate the correlation between the eyelid motion and the signals output by the piezoelectric sensor. The theoretical predictions on the eyelid motion based on the measured electrical output agree well with the in vivo observations in experiment. A simple scaling law is established to evaluate the impacts of different parameters on the function ability of the flexible piezoelectric sensor. The results may provide useful guidelines for designing and optimizing similar devices for alike biological motions.

Advanced approaches for quantitative characterization of thermal transport properties in soft materials using thin, conformable resistive sensors

Noninvasive methods for precise characterization of the thermal properties of soft biological tissues such as the skin can yield vital details about physiological health status including at critical intervals during recovery following skin injury. Here, we introduce quantitative measurement and characterization methods that allow rapid, accurate determination of the thermal conductivity of soft materials using thin, skin-like resistive sensor platforms. Systematic evaluations of skin at eight different locations and of six different synthetic skin-mimicking materials across sensor sizes, measurement times, and surface geometries (planar, highly curvilinear) validate simple scaling laws for data interpretation and parameter extraction. As an example of the possibilities, changes in the thermal properties of skin (volar forearm) can be monitored during recovery from exposure to ultraviolet radiation (sunburn) and to stressors associated with localized heating and cooling. More generally, the results described here facilitate rapid, non-invasive thermal measurements on broad classes of biological and non-biological soft materials.

Constraining self-interacting dark matter with scaling laws of observed halo surface densities

The observed surface densities of dark matter halos are known to follow a simple scaling law, ranging from dwarf galaxies to galaxy clusters, with a weak dependence on their virial mass. Here we point out that this can not only be used to provide a method to determine the standard relation between halo mass and concentration, but also to use large samples of objects in order to place constraints on dark matter self-interactions that can be more robust than constraints derived from individual objects. We demonstrate our method by considering a sample of about 50 objects distributed across the whole halo mass range, and by modelling the effect of self-interactions in a way similar to what has been previously done in the literature. Using additional input from simulations then results in a constraint on the self-interaction cross section per unit dark matter mass of about σ/mχ≲ 0.3 cm2/g. We expect that these constraints can be significantly improved in the future, and made more robust, by i) an improved modelling of the effect of self-interactions, both theoretical and by comparison with simulations, ii) taking into account a larger sample of objects and iii) by reducing the currently still relatively large uncertainties that we conservatively assign to the surface densities of individual objects. The latter can be achieved in particular by using kinematic observations to directly constrain the average halo mass inside a given radius, rather than fitting the data to a pre-selected profile and then reconstruct the mass. For a velocity-independent cross-section, our current result is formally already somewhat smaller than the range 0.5−5 cm2/g that has been invoked to explain potential inconsistencies between small-scale observations and expectations in the standard collisionless cold dark matter paradigm.

Pebble-isolation mass: Scaling law and implications for the formation of super-Earths and gas giants

The growth of a planetary core by pebble accretion stops at the so-called pebble isolation mass, when the core generates a pressure bump that traps drifting pebbles outside its orbit. The value of the pebble isolation mass is crucial in determining the final planet mass. If the isolation mass is very low, gas accretion is protracted and the planet remains at a few Earth masses with a mainly solid composition. For higher values of the pebble isolation mass, the planet might be able to accrete gas from the protoplanetary disc and grow into a gas giant. Previous works have determined a scaling of the pebble isolation mass with cube of the disc aspect ratio. Here, we expand on previous measurements and explore the dependency of the pebble isolation mass on all relevant parameters of the protoplanetary disc. We use 3D hydrodynamical simulations to measure the pebble isolation mass and derive a simple scaling law that captures the dependence on the local disc structure and the turbulent viscosity parameter α. We find that small pebbles, coupled to the gas, with Stokes number τf < 0.005 can drift through the partial gap at pebble isolation mass. However, as the planetary mass increases, particles must be decreasingly smaller to penetrate the pressure bump. Turbulent diffusion of particles, however, can lead to an increase of the pebble isolation mass by a factor of two, depending on the strength of the background viscosity and on the pebble size. We finally explore the implications of the new scaling law of the pebble isolation mass on the formation of planetary systems by numerically integrating the growth and migration pathways of planets in evolving protoplanetary discs. Compared to models neglecting the dependence of the pebble isolation mass on the α-viscosity, our models including this effect result in higher core masses for giant planets. These higher core masses are more similar to the core masses of the giant planets in the solar system.

Localized Enzymatic Degradation of Polymers: Physics and Scaling Laws

The breakdown of polymers by enzymes is of key importance in chemical engineering, biotechnology, and nature. The process involves a localized, moving front of degradation within the polymer, but little is known about its physical character. This analysis shows that the front's width and propagation speed depend on the physical properties (size, structure, and reactivity) of both enzyme and polymer. Based on competition between reaction and transport of enzymes, simple scaling laws are obtained. These results will facilitate applications of such systems in tissue regeneration, targeted drug delivery, and biomass processing for energy generation.

Analysis of the higher-energy structure in strong-field ionization with inhomogeneous electric fields

Recently, the emergence of a higher energy structure (HES) due to a spatial inhomogeneity in the laser electric field, as is typically found close to a nano tip, was reported in Phys.~Rev. Letter {\bf 119}, 053204 (2017). For practical applications, such as the characterization of near-fields or the creation of localized sources of monoenergetic electron beams with tunable energies, further insight into the nature of this higher energy structure is needed. Here, we give a closed form analytical approximation to describe the movement of the electron in the inhomogeneous electric field. In particular, we derive a simple scaling law for the location of the HES peak and give a scheme to analytically tune the width of the peak, both of which will prove useful in optimizing the nanostructure size or geometry for creating the HES in experimental settings.

Wavepacket dynamics in one-dimensional system with long-range correlated disorder

We numerically investigate dynamical property in the one-dimensional tight-binding model with long-range correlated disorder having power spectrum 1/fα (α: spectrum exponent) generated by Fourier filtering method. For relatively small α<αc(=2) time-dependence of mean square displacement (MSD) of the initially localized wavepacket shows ballistic spread and localizes as time elapses. It is shown that α-dependence of the dynamical localization length determined by the MSD exhibits a simple scaling law in the localization regime for the relatively weak disorder strength W. Furthermore, scaled MSD by the dynamical localization length almost obeys an universal function from the ballistic to the localization regime in the various combinations of the parameters α and W.

A scaling law for predicting the ballistic limit of aluminium alloy targets perforated in ductile hole formation

The ballistic performance of monolithic targets perforated in ductile formation by rigid projectiles can be reasonably approximated by an energy balance based on the work done in expanding a hole from zero initial radius to that of the penetrating projectile. For a specific projectile, these energy balance laws can be expressed in terms of a simple scaling law which requires an empirical fit to ballistic data. However, once this fit has been performed for a specific projectile against a specific target material, predicting the performance of other materials against this projectile can be done from a simple quasi-static compression stress–strain curve, provided the target is perforated in ductile hole formation. Four variations of ductile hole formation analytical and semi-empirical models have been reformulated as scaling laws and evaluated for their ability to characterise the performance of monolithic aluminium plates impacted by 7.62 mm APM2 projectiles at normal incidence. The database used to fit and evaluate the performance of the scaling laws incorporates more than 1600 V50 measurements against 24 different alloys and tempering/strain-hardening combinations with thicknesses ranging from 16.2 to 63.5 mm. The highest performing scaling law formulation was found to characterize the measured V50 to within 5% of the experimental value for over 97% of the database entries. Further improvements in accuracy are likely impossible when representative stress–strain curves are used for performance characterization rather than tests of each individual plate used in the ballistic tests. The scaling law can be applied to predict the performance of targets perforated in ductile hole formation with a high level of confidence, an example of which is provided for the 12.7 mm APM2 projectile against a range of aluminium alloys.

Deformation of elastomeric pyramid pen arrays in cantilever‐free scanning probe lithography

ABSTRACTCantilever‐free scanning probe lithography (SPL) employs soft elastomeric pen arrays to deliver material or energy to a surface to achieve a high‐resolution, high‐throughput, and low‐cost nanopatterning. In particular, microscale elastomeric pyramid pen arrays are often adopted as the cantilever‐free architecture owing to their distinct structural and mechanical properties. To better understand the mechanical behavior of the elastomeric pyramid pen array during the lithographic printing process, we numerically investigate the compression of an elastomeric pyramid array in a nonadhesive and frictionless contact with a rigid substrate. Simple scaling laws of the width of the contact surface with respect to the compression displacement and force are found and compared with previous models and experiments. By changing the interpyramid distance or the thickness of the base of the pyramid array, increasing deviations from the established scaling laws are observed and explained. Furthermore, we demonstrate that the unique morphology of a compressed pyramid primarily determines the unusual shape of the features fabricated by a specific cantilever‐free SPL technique. © 2018 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2018, 56, 731–738

Theoretical Assessment on Piezoelectric Energy Harvesting in Smart Self-powered Asphalt Pavements

The lighting system along the highway in urban district consume vast amount of electric power, thus leading to a great number of economic cost. We propose to integrate within the road pavement a layer of piezoelectric transducer that is used to transfer mechanical energy stored in the road pavement due to traffic loading to electric power. For this energy harvesting system, we develop a composite beam model in plane strain state with two asphalt mixture layers and a piezoelectric layer in between. The output electric current, voltage, and power due to various exciting loading frequency are derived analytically. The optimal thickness and location of the piezoelectric layer in the pavement are presented. A simple scaling law for the output voltage or power is derived by combining all material and geometric parameters into three independent non-dimensional parameters. The solution here may offer a simple tool for the design of smart asphalt pavement from the point of optimal output electric power.

Size of graphene sheets determines the structural and mechanical properties of 3D graphene foams

Graphene is recognized as an emerging 2D nanomaterial for many applications. Assembly of graphene sheets into 3D structures is an attractive way to enable their macroscopic applications and to preserve the exceptional mechanical and physical properties of their constituents. In this study, we develop a coarse-grained (CG) model for 3D graphene foams (GFs) based on the CG model for a 2D graphene sheet by Ruiz et al (2015 Carbon 82 103–15). We find that the size of graphene sheets plays an important role in both the structural and mechanical properties of 3D GFs. When their size is smaller than 10 nm, the graphene sheets can easily stack together under the influence of van der Waals interactions (vdW). These stacks behave like building blocks and are tightly packed together within 3D GFs, leading to high density, small pore radii, and a large Young’s modulus. However, if the sheet sizes exceed 10 nm, they are staggered together with a significant amount of deformation (bending). Therefore, the density of 3D GFs has been dramatically reduced due to the loosely packed graphene sheets, accompanied by large pore radii and a small Young’s modulus. Under uniaxial compression, rubber-like stress–strain curves are observed for all 3D GFs. This material characteristic is dominated by the vdW interactions between different graphene layers and slightly affected by the out-of-plane deformation of the graphene sheets. We find a simple scaling law between the density ρ and Young’s modulus E for a model of 3D GFs. The simulation results reveal structure–property relations of 3D GFs, which can be applied to guide the design of 3D graphene assemblies with exceptional properties.

A scaling law for short term load forecasting on varying levels of aggregation

This paper proposes a simple empirical scaling law that describes load forecasting accuracy at varying levels of aggregation. We show that for many forecasting methods, aggregating more customers improves the relative forecasting performance up to specific point. Beyond this point, no more improvement in relative performance can be obtained. A benchmarking procedure for applying the scaling law to different forecasting models is presented. The aggregation model is evaluated with year long consumption profiles of over 180 thousand Pacific Gas & Electric customers. A theoretical model based on a bias variance decomposition of the forecast error is used to model the Aggregation Error Curves (AECs) that are empirically explored.

On simple scaling laws for pumping fluids with electrically-charged particles

As 3D printing has matured, the use of more complex materials has increased. In particular, particle-functionalized inks are now relatively wide-spread. Because of the desire to have faster throughput for industrial-scale printing, electrically-driven flow of such materials is being pursued. In many cases, such fluids consist of an electrically-neutral base solvent with embedded charged particles. As one increases the volume fraction of particles, two effects effects arise: (1) an increase in effective overall viscosity fluid and (2) an increase in the induced electrical force that can be applied. In the present analysis, the governing equations for the required pressure gradient in a pipe to move the fluid with a constant flow rate are derived. A key nondimensional scaling ratio governing the relative contribution of electrical and viscous fluid forces to the system behavior is also identified. Numerical examples are provided to illustrate the results.

Simple scaling laws for the evaporation of droplets pinned on pillars: Transfer-rate- and diffusion-limited regimes.

The evaporation of droplets can give rise to a wide range of interesting phenomena in which the dynamics of the evaporation are crucial. In this work, we find simple scaling laws for the evaporation dynamics of axisymmetric droplets pinned on millimeter-sized pillars. Different laws are found depending on whether evaporation is limited by the diffusion of vapor molecules or by the transfer rate across the liquid-vapor interface. For the diffusion-limited regime, we find that a mass-loss rate equal to 3/7 of that of a free-standing evaporating droplet brings a good balance between simplicity and physical correctness. We also find a scaling law for the evaporation of multicomponent solutions. The scaling laws found are validated against experiments of the evaporation of droplets of (1) water, (2) blood plasma, and (3) a mixture of water and polyethylene glycol, pinned on acrylic pillars of different diameters. These results shed light on the macroscopic dynamics of evaporation on pillars as a first step towards the understanding of other complex phenomena that may be taking place during the evaporation process, such as particle transport and chemical reactions.