Simple Power Law Relationship Research Articles

Simple, near-universal relationships between bond lengths, strengths, and anharmonicities

Harmonic bond force constants and bond lengths are shown to generally obey the simple relationships, ke=ζ2Re−3 (hydrides) and ke=10ζ1/2Re−4 (all other bond types), where ζ is the reduced nuclear charge and Re is the equilibrium bond length. Equally simple power-law relationships are found for higher-order bond force constants. Although not spectroscopically accurate, these models are nonetheless of significant heuristic value for identifying strongly multireference states of diatomic molecules (including electronically coupled excited states ill-suited for inclusion in laser-cooling schemes), rationalizing the observed trends in vibrational frequencies for diatomics and/or local mode oscillators within molecules or complexes and estimating and/or validating covalent bonding parameters within molecular mechanics force fields. Particular advantages of our approach over other bond length-strength scaling relationships proposed in the literature include its simplicity and generality and its appropriate asymptotic behavior. Notably, the relationships derived in this work can be used to predict harmonic and higher-order force constant bonds between any pair of atoms in the Periodic Table (including transition metals and lanthanides) without requiring row- or column-dependent parameterization, to accuracies commensurate with conventional force field transferability errors. We therefore anticipate that they will expedite force field development for metal-containing complexes and materials, which are structurally well-characterized but challenging to parameterize ab initio.

Velocity-dependent J-factors for Milky Way dwarf spheroidal analogues in cosmological simulations

We study the impact of the dark matter velocity distribution modelling on signals from velocity-dependent dark matter annihilation in Milky Way dwarf spheroidal galaxies. Using the high resolution APOSTLE simulations, we identify analogues corresponding to Milky Way dwarf spheroidal galaxies, and from these directly determine the dark matter pair-wise relative velocity distribution, and compare to best-fitting Maxwell-Boltzmann distribution models. For three velocity-dependent annihilation models, p-wave, d-wave, and the Sommerfeld model, we quantify the errors introduced when using the Maxwell-Boltzmann parameterization. We extract a simple power-law relation between the maximum circular velocity of the dwarf spheroidal analogue and the peak speed of the Maxwell-Boltzmann distribution. We show that this relation can be used to accurately calculate the dark matter relative velocity distribution, and find that it allows us to estimate the dark matter annihilation signal without the need to directly calculate the relative velocity distribution for each galaxy. The scatter in the J-factors calculated from the analogues dominates the uncertainty obtained when compared to the J-factor as determined from the observational data for each dwarf spheroidal, with the largest scatter from d-wave models and the smallest from Sommerfeld models.

Chemical reaction of Ni/Al interface associated with perturbation growth under shock compression

The exothermic reaction of Ni/Al laminates always starts from the interface, and the role of interfacial instability in the shock-induced chemical reaction has not been clarified. This work reports the Richtmyer–Meshkov (RM) instability growth, atomic diffusion, and chemical reaction of Ni/Al interface under shock compression based on atomistic simulations. For shocking from Al to Ni, the interface experiences finite collapse and exhibits weak localized reaction. The diffusion of solid Ni to molten Al will be inhibited due to the formation of NiAl phase, and continuous inter-diffusion occurs with the melting of Ni. For shocking from Ni to Al, a small amount of NiAl structure is formed due to the atomic residue during defect collapse. RM instability growth is observed at higher shock intensity, which significantly promotes the atomic mixing and results in a power-law increase in the number of diffusing atoms. Meanwhile, the chemical reaction propagates rapidly from the vortex to the head of the spike accompanied by the decomposition of many clusters, with the nonlinear development of RM instability. The number and the size of Ni clusters no more satisfy the simple power-law relationship for which we propose an improved power-law distribution. Interestingly, the growth of nanoscale perturbation approximately satisfies the logarithmic law with time, but the linear growth stage is inhibited due to significant inter-diffusion, especially for the small wavelength. Thus, the mixing width and the reaction degree are positively correlated with the initial wavelength in our simulation scale, which is contrary to the RM growth law of the free surface.

Nonlinear Phase-Field Modeling of Lithium Dendritic Growth during Electrodeposition

Lithium batteries have been widely used, but the growth of lithium dendrites does lead to some hazards, such as internal short circuit, thermal runaway and even explosion. In paper, a nonlinear phase field model combined with Butler-Volmer expression was developed to study dendrite growth during electrodeposition. Two types of initial electrode-electrolyte interface are applied in numerical simulations. Results show that the cation concentration enrichment occurs near the dendritic tip due to the electromigration, and the curve of the growth velocity versus time follows a simple power law relation, indicating that the stability of the dendritic array originating from initial electrode-electrolyte interface can be broken with the applied electrical potential increasing. Moreover, we found that applied electric potential gradient can strongly influence the dendritic growth velocity and the dendritic growth during electrodeposition is less relative to the surface tension anisotropy.

Yielding Behavior of Bottlebrush and Linear Block Copolymers

Block copolymers can exhibit a pronounced yield stress, but the impact of molecular architecture, chemistry, and self-assembly on macroscopic rheology remains poorly understood. Here, we study the linear-viscoelastic and yield-stress fluid behavior of two architectures—bottlebrush copolymers (with statistical or blocky sequences) and linear diblocks—that self-assemble into body-centered cubic (BCC) spheres and hexagonally close-packed cylinders (HEX). The dynamic properties of these polymers were probed by oscillatory frequency and amplitude sweeps at temperatures well below the order–disorder transition (TODT) to furnish insights into the yielding transition. All BCC-forming polymers have a similar signature of yielding: smaller yield strains (γy,BCC ≈ 0.053 < γy,HEX ≈ 0.18), sharper solid–liquid transitions, and better reversibility than HEX. Statistical bottlebrushes show the most frequency-independent structural modulus (G0) and no signs of defect relaxation. A simple power-law relationship captures the dependence of the normalized structural modulus (G0/RT) on the inter-micelle distance (d) across different architectures and morphologies [G0/(RT) = 1.31 × 104 (nm2.6 mol/m3) d–2.6]. These studies establish quantitative structure–property relationships that are relevant in contemporary applications, for example, extrusion-based 3D printing.

A Knee Point in the Rotation–Activity Scaling of Late-type Stars with a Connection to Dynamo Transitions

Abstract The magnetic activity of late-type stars is correlated with their rotation rates. Up to a certain limit, stars with smaller Rossby numbers, defined as the rotation period divided by the convective turnover time, have higher activity. A more detailed look at this rotation–activity relation reveals that, rather than being a simple power-law relation, the activity scaling has a shallower slope for the low-Rossby stars than for the high-Rossby ones. We find that, for the chromospheric Ca II H&amp;K activity, this scaling relation is well modeled by a broken two-piece power law. Furthermore, the knee point of the relation coincides with the axisymmetry to nonaxisymmetry transition seen in both the spot activity and surface magnetic field configuration of active stars. We interpret this knee point as a dynamo transition between dominating axi- and nonaxisymmetric dynamo regimes with a different dependence on rotation and discuss this hypothesis in the light of current numerical dynamo models.

Palpation Sensitivity of an Embedded Nodule Using the Finite Element Method

Abstract A physician palpates a tissue to detect an embedded tumor nodule by sensing an increase in local tissue stiffness and nodule size. The Hertz contact model, however, is unable to predict the material or physical properties of a tumor nodule embedded in a healthy tissue of finite thickness. In this study, utilizing a hyperelastic material model, we propose a general methodology to analyze the extent to which the stiffness, size, and depth of a nodule embedded in a tissue affect its detectability. Using dimensional analysis, we generate simple power-law relations to predict physical and material properties of tumor nodules embedded in healthy tissue during indentation. Our results indicate that indenter radius and indentation depth are critical parameters in nodule detection and a thin indenter and large indentation depth increase detection sensitivity of an embedded tumor nodule. Our results also show that anisotropic material properties of either a tissue or an embedded nodule render the embedded tumor nodule undetectable using indentation. We define palpation sensitivity maps that can be used to predict material and physical properties of tumor nodules in healthy tissues. The analysis and results presented in this study might increase accuracy and precision in instrumented probe-based laparoscopic or robotic surgeries.

Scaling laws to predict humidity-induced swelling and stiffness in hydrogels.

From pasta to biological tissues to contact lenses, gel and gel-like materials inherently soften as they swell with water. In dry, low-relative-humidity environments, these materials stiffen as they de-swell with water. Here, we use semi-dilute polymer theory to develop a simple power-law relationship between hydrogel elastic modulus and swelling. From this relationship, we predict hydrogel stiffness or swelling at arbitrary relative humidities. Our close predictions of properties of hydrogels across three different polymer mesh families at varying crosslinking densities and relative humidities demonstrate the validity and generality of our understanding. This predictive capability enables more rapid material discovery and selection for hydrogel applications in varying humidity environments.

Hierarchical Synthesis, Structure, and Photophysical Properties of Gallium- and Ruthenium-Porphyrins with Axially Bonded Azo Ligands.

The hierarchical synthesis of three porphyrin and four bisporphyrin derivatives is presented. This strategy relies on the incorporation of linkers based on azo moieties appended with pyridyl and/or acetylenic groups that facilitate axial coordination to Ga- and Ru-metalloporphyrins. These porphyrinic systems allow for a quantitative analysis of the effects of diamagnetic anisotropy (DA) by using 1 H NMR spectroscopic and X-ray crystallographic analyses. A simple power-law relationship between the proton chemical shift and the distance from the porphyrin core is experimentally outlined, which confirms previous theoretical predictions and shows that the limit of DA is about 2 nm. Photophysical properties of the azo-linked porphyrins are analyzed by UV/Vis spectroscopy, showing that significant cis-trans isomerization is not observed for azo ligands bound only to Ga-porphyrins. Incorporation of Ru-porphyrins to an azo ligand facilitates photoswitching behavior, but the process faces competition from decarbonylation of the Ru-porphyrin, and appreciable switching is only documented for GaL1Ru.

A scale invariant model for the expansion of agricultural land and government spending on the agricultural sector

The expansion dynamics of the agricultural surface is a complex process, since it requires decisions among different actors under a multitude of socioeconomic and natural conditions, expectations, and risks. This paper shows that despite this complexity, and at odds with the intuition, the density of agricultural lands and the government spending to foster agriculture around a human settlement displays a simple power-law relationship regarding the distance to such a settlement. The theoretical proof of this relationship is based on theoretical and empirical findings made by several authors on the expansion of population centers, river networks, species and ecosystem’s distribution. The empirical proof is made by fitting a simple power-law model having as response variables: density of agricultural land; government spending coverage, and government spending per unit of agricultural surface, while the distance to the centroid settlement is used as independent variable. Results show that the scaling exponents of the power-law relationship estimated from those models, with data from rural settlements in Mexico, are similar statistically to the expected values from theoretical models for the three expansion processes tested. Three applications of the scale-invariant attribute characterizing the expansion process of agricultural lands are presented:i)monitoring agricultural activities at the territories;ii)a proxy to estimate the expected growth of the agricultural surface; andiii)a typology to characterize government assistance at local level.

Assessing the peatland hummock–hollow classification framework using high-resolution elevation models: implications for appropriate complexity ecosystem modeling

Abstract. The hummock–hollow classification framework used to categorize peatland ecosystem microtopography is pervasive throughout peatland experimental designs and current peatland ecosystem modeling approaches. However, identifying what constitutes a representative hummock–hollow pair within a site and characterizing hummock–hollow variability within or between peatlands remains largely unassessed. Using structure from motion (SfM), high-resolution digital elevation models (DEMs) of hummock–hollow microtopography were used to (1) examine how much area needs to be sampled to characterize site-level microtopographic variation; and (2) examine the potential role of microtopographic shape/structure on biogeochemical fluxes using plot-level data from nine northern peatlands. To capture 95 % of site-level microtopographic variability, on average, an aggregate sampling area of 32 m2 composed of 10 randomly located plots was required. Both site- (i.e. transect data) and plot-level (i.e. SfM-derived DEM) results show that microtopographic variability can be described as a fractal at the submeter scale, where contributions to total variance are very small below a 0.5 m length scale. Microtopography at the plot level was often found to be non-bimodal, as assessed using a Gaussian mixture model (GMM). Our findings suggest that the non-bimodal distribution of microtopography at the plot level may result in an undersampling of intermediate topographic positions. Extended to the modeling domain, an underrepresentation of intermediate microtopographic positions is shown to lead to potentially large flux biases over a wide range of water table positions for ecosystem processes which are non-linearly related to water and energy availability at the moss surface. Moreover, our simple modeling results suggest that much of the bias can be eliminated by representing microtopography with several classes rather than the traditional two (i.e. hummock/hollow). A range of tools examined herein can be used to easily parameterize peatland models, from GMMs used as simple transfer functions to spatially explicit fractal landscapes based on simple power-law relations between microtopographic variability and scale.

From fine sand to boulders: Examining the relationship between beach-face slope and sediment size

It is a long-standing maxim that coarser sediment sizes are associated with steeper beach-face slopes; but because most work has focused on sandy beaches, few data are available for the pebble, cobble, or boulder size ranges. Little is known, therefore, about how beach morphology and grain size relate at the coarser size grades. We compiled data from the literature—2144 measurements of beach-face slope with associated grain sizes— covering the range from very fine sand to boulders. This meta-analysis shows that beach faces do tend to steepen as average grain size increases, at least up to the cobble size range; but the trend is not simple. Although previous studies suggested a simple power-law relationship between grain size and beach-face slope, in fact the data distribution is best fit by a curve that is steep at the finer grain sizes but is much more gentle as grain size coarsens (and may trend downward for boulders). Fine and medium sands have the greatest range of reported slopes and the data support the importance of effective-weight modification and boundary layer dynamics as primary controls on beach-face steepness at these fine size grades. Around the very coarse sand size grade, a plateau marks a shift in beach dynamics: as hydraulic conductivity increases, beaches switch to being infiltration-dominated. The trend toward steeper slopes continues, but at a lower and steady rate, from granules through cobbles. There appears to be a falloff in slope for the coarsest boulder deposits (but we note that this is based on very few locations, so should be considered preliminary). The overall data distribution is described by a power-law function: tanβ=a(D50−0.125)b+c. The broadness of each transition zone also shows that a range of factors, other than grain size, governs where that tipping point occurs. In the cobble and boulder size ranges there are two deposit categories: boulder beaches, and supratidal boulder ridges. Although not commonly thought of as beaches, boulder ridges record long-term storm deposition and reworking on the highest energy coasts. Supratidal ridges and boulder beaches seem to show different slope/grain-size relationships, with the steepest slopes occurring in boulder ridges, and with boulder beaches showing more gentle gradients than other coarse clastic beaches; but our data set reveals how little quantitative information exists about the coarsest end of the beach spectrum, and invites further work to investigate the sedimentology of cobble- and boulder-dominated systems.

Plumes in a rotating two-layer stratified fluid

Laboratory experiments are conducted to examine a descending plume in a rotating two-layer stratified ambient fluid such that the plume at the interface has moderate to large Rossby number. While the source fluid is more dense than the lower layer, the experiments are designed so that the mean density of the plume fluid impinging upon the interface is less than the lower layer density, as represented by a buoyancy parameter, $$\varLambda $$, being less than unity. In such cases, the discharged plume fluid spreads radially at the interface in the form of an intrusive gravity current at early times. At later times, this intrusion evolves to form an anticyclonic lens due to the influence of the Coriolis force. The measured radial position of the intrusion front, R(t), follows different power law relationships at early and late times during the spread of the intrusion: at early times when rotation does not play a significant role the power law exponent lies between 0.5 and 1.1; at late times when the intrusion acts as a rotationally influenced expanding lens the power law exponent ranges between 0.15 and 0.5, with generally smaller values for larger $$\varLambda $$. The plume fluid reaching the interface progressively increases in density due to re-entraining relatively dense fluid as the plume descends within the thickening lens. Consequently, the plume eventually penetrates through the interface and descends to the bottom of the tank. Faster rotation makes the lens thicker and hence increases the volume of the re-entrained lens-fluid, which decreases the time for the onset of penetration. The penetration time normalized by the rotation rate is found to hold a simple power law relationship with $$\varLambda $$.

Murchison Widefield Array and XMM-Newton observations of the Galactic supernova remnant G5.9+3.1

Aims. In this paper we discuss the radio continuum and X-ray properties of the so-far poorly studied Galactic supernova remnant (SNR) G5.9 + 3.1. Methods. We present the radio spectral energy distribution (SED) of the Galactic SNR G5.9 + 3.1 obtained with the Murchison Widefield Array (MWA). Combining these new observations with the surveys at other radio continuum frequencies, we discuss the integrated radio continuum spectrum of this particular remnant. We have also analyzed an archival XMM-Newton observation, which represents the first detection of X-ray emission from this remnant. Results. The SNR SED is very well explained by a simple power-law relation. The synchrotron radio spectral index of G5.9 + 3.1 is estimated to be 0.42 ± 0.03 and the integrated flux density at 1 GHz to be around 2.7 Jy. Furthermore, we propose that the identified point radio source, located centrally inside the SNR shell, is most probably a compact remnant of the supernova explosion. The shell-like X-ray morphology of G5.9 + 3.1 as revealed by XMM-Newton broadly matches the spatial distribution of the radio emission, where the radio-bright eastern and western rims are also readily detected in the X-ray while the radio-weak northern and southern rims are weak or absent in the X-ray. Extracted MOS1+MOS2+PN spectra from the whole SNR as well as the north, east, and west rims of the SNR are fit successfully with an optically thin thermal plasma model in collisional ionization equilibrium with a column density NH ~ 0.80 × 1022 cm−2 and fitted temperatures spanning the range kT ~ 0.14–0.23 keV for all of the regions. The derived electron number densities ne for the whole SNR and the rims are also roughly comparable (ranging from ~0.20f−1∕2 to ~0.40f−1∕2 cm−3, where f is the volume filling factor). We also estimate the swept-up mass of the X-ray emitting plasma associated with G5.9+3.1 to be ~46f−1∕2 M⊙.

Foil locomotion through non-sinusoidal pitching motion

We investigate the propulsive characteristics of a neutrally buoyant foil that self propels as it pitches about its quarter chord with a prescribed time periodic waveform. For a fixed pitching frequency and amplitude, we find the foil motion to be strongly dependent on the prescribed waveform. The maximum mean self-propelling speed corresponds to a square shaped waveform for which the foil is held at the two extremes for a significant part of the pitching cycle. In contrast, the cost of transport is minimal, and therefore the energetic efficiency of pitching induced self propulsion a maximum, for a triangle shaped profile in which case the foil’s angular speed is largely held constant over the pitching period. Pitching waveform induced variations in the mean self-propelling speed (u¯p) are surprisingly well predicted with a simple power-law relationship Strms∼Re−0.37, where Reynolds number Re is based on u¯p and the Strouhal number Strms is the ratio of the root mean square foil trailing edge speed and u¯p. This power-law relationship is shown to arise naturally from a balance between the cycle-averaged inertial thrust generated from prescribed pitching and the enhanced drag owing to the boundary layer thinning. The imposed pitching motion induces a Reynolds number independent out of phase heave motion features of which are remarkably well predicted with a simple inviscid model for the foil’s transverse motion. For all the prescribed pitching waveforms, a maximum in the energetic efficiency is always attained over the range 0.3≤St≤0.6. The close correspondence between this Strouhal number range for maximum energetic efficiency and the optimal Strouhal number range for swimming and flying animals is indicative of the similarity between self-propelling flapping foil configurations and undulatory biolocomotion. Our results point to the significance of rigid body pitching profile as an important input parameter that can be varied independently to achieve a desired outcome such as maximization of the mean self-propelling speed or energetic efficiency.

Plasticity of the truth table of low-leakage genetic logic gates

The design and implementation of genetic logic gates is a fundamental component of biological computation. In this article we show that the function of a common class of synthetic genetic AND and NAND gates is not completely dictated by the circuit connectivity, even if promoter leakage is very small. Rather, the logic function is strongly determined by a simple power law relationship between the promoter leakage rate and the binding affinity of the protein complex carrying the information from the input to the output of the gate. Depending on the value of the power law exponent, a circuit designed to be an AND gate can actually operate as a TRUE, OR, AND, or FALSE gate, even if the leakage rate is practically negligible. Surprisingly, all these functionalities are compatible with the physiological range of parameter values showing that the design of genetic logic gates which preserve functionality across cell types and conditions requires careful consideration of both circuit connectivity and parameter values.

The possible origin of high frequency quasi-periodic oscillations in low mass X-ray binaries

AbstractWe summarize our model that high frequency quasi-periodic oscillations (QPOs) both in the neutron star low mass X-ray binaries (NS-LMXBs) and black hole LMXBs may originate from magnetohydrodynamic (MHD) waves. Based on the MHD model in NS-LMXBs, the explanation of the parallel tracks is presented. The slowly varying effective surface magnetic field of a NS leads to the shift of parallel tracks of QPOs in NS-LMXBs. In the study of kilohertz (kHz) QPOs in NS-LMXBs, we obtain a simple power-law relation between the kHz QPO frequencies and the combined parameter of accretion rate and the effective surface magnetic field. Based on the MHD model in BH-LMXBs, we suggest that two stable modes of the Alfv́en waves in the accretion disks with a toroidal magnetic field may lead to the double high frequency QPOs. This model, in which the effect of the general relativity in BH-LMXBs is considered, naturally accounts for the 3:2 relation for the upper and lower frequencies of the QPOs and the relation between the BH mass and QPO frequency.

A possible explanation of the parallel tracks in kilohertz quasi-periodic oscillations from low-mass-X-ray binaries

We recalculate the modes of the magnetohydrodynamics (MHD) waves in the MHD model (Shi, Zhang & Li 2014) of the kilohertz quasi-periodic oscillations (kHz QPOs) in neutron star low mass X-ray binaries (NS-LMXBs), in which the compressed magnetosphere is considered. A method on pointby- point scanning for every parameter of a normal LMXBs is proposed to determine the wave number in a NS-LMXB. The dependence of the twin kHz QPO frequencies on accretion rates ( DOt(M) ) is obtained with the wave number and magnetic field (B?) determined by our method. Based on the MHD model, a new explanation of the parallel tracks, i.e. the slowly varying effective magnetic field leads to the shift of parallel tracks in a source, is presented. In this study, we obtain a simple power-law relation between the kHz QPO frequencies and Dot(M) /B^2 ? in those sources. Finally, we study the dependence of kHz quasi-periodic oscillation frequencies on the spin, mass and radius of a neutron star. We find that the effective magnetic field, the spin, mass and radius of a neutron star lead to the parallel tracks in different sources.

A Study of Tensile Flow and Work-Hardening Behavior of Alloy 617

The simple power relationship σ = Κe satisfactorily expresses the tensile flow behavior of many metals and alloys in their uniform plastic strain regime. However, many FCC materials with low stacking fault energy have opposed such power law relationship. Alloy 617, an age-hardenable Ni-based superalloy is also observed not to obey the simple power law relationship neither in its solution-treated nor in its aged conditions. Various flow relationships were used to obtain the best fit for the tensile data, and different relationships were identified for the different aged conditions. The work-hardening rate (θ) demonstrates three distinct regions for all aged conditions, and there is an obvious change in the trend of θ versus σ. In the initial portion, θ decreases rapidly followed by a gradual increase in the second stage and again a decrease in its third stage is perceived in the Alloy 617. These three-stage characteristics are attributed to a commonly known precipitate, γ′: Ni3(Ti, Al) which evolves during aging treatment and well recognized under transmission electron microscopy (TEM) observation. TEM results also reveal a slight degree of coarsening in γ′ over aging. The tensile flow and the work-hardening behavior are well correlated with other microstructural evolution during the aging treatments.

Rheology and friction loss of raw and digested sewage sludge with high TSS concentrations: a case study.

High total suspended solids (TSS) digestion of municipal sewage sludge reduces the required space and volume for digestion plants. However, an important consequence of high TSS is the major influence on sludge rheology. The present case study investigates the rheology of sludge from a 130 m3 high solids digestion pilot plant at Vienna's main wastewater treatment plant (4 M PE120). Raw sludge ranged from 6 to 8% TSS and digested sludge from 3.2 to 4.6%. TSS show an exponential impact on rheological parameters. Increasing raw sludge TSS from 6 to 8% at least doubles the shear stress and increases friction loss by a factor of three. However, under real operating conditions simulated at the pilot plant, there are additional impact factors. The mixing ratio between waste activated and primary sludge influences raw sludge rheology, while solids retention time and loss on ignition affects digested sludge rheology. Nevertheless, friction loss calculations based on a simple power law relationship between shear rate and shear stress proved to be applicable and sufficiently accurate for both raw and digested sludge with high TSS. Altogether, this case study underlines the relevance of comprehensive rheological considerations, measurements and calculations when designing high TSS sludge digestion.