Simple Power Law Relationship Research Articles

Empirical relations for electron mobility in silicon

The electron mobility in silicon has been obtained by the Monte Carlo simulation technique for different applied electric fields and under various physical conditions, such as temperature, impurity concentration, etc. The mobility values have then been expressed empirically by simple power law relationships. It is observed that the mobility values calculated by using these simple power law relations yield values that show agreement within 5 with those obtained from the detailed Monte Carlo simulation technique. It is concluded that these empirical relations can be used very effectively in a device modelling program.

Bubble growth in viscous newtonian and non-newtonian liquids

A model for the dynamics of hydrodynamically controlled spherical bubble growth in quiescent viscous newtonian and non-newtonian liquids is presented. Two constitutive equations were used to describe the behavior of the liquid medium: (1) a simple power law relation and (2) a truncated power law. Application of the truncated power law resulted in four differen cases: (a) at all times the liquid acts as a newtonian liquid; (b) at all times the liquid behaves as a simple power law liquid; (c) at all times the liquid close to the growing bubble obeys a power law relation and the liquid far away from the bubble behaves as a newtonian liquid; (d) all the liquid initially behaves as a newtonian liquid and at some time transforms to a two-region liquid. Analytical expressions were derived for bubble growth as a function of dimensionless system parameters. The relevance of this work to the process of polymer devolatilization is discussed.

Monte Carlo Calculations of Electron Fluence Attenuation Coefficients for Photon General Cavity Theory

Application of photon general cavity theory requires knowledge of the electron mass attenuation coefficient, ß. One approach has been to adopt values of ß derived from the attenuation of beta ray spectra, which are based on a one-to-one correspondence between ß and the maximum energy of the beta ray spectrum. Other approaches are also available, so that, for 60Co generated electrons, for example, six different empirical expressions yield values of ß for LiF from a minimum value of 5.16 cm2.g-1 to a maximum value of 14.26 cm2.g-1. The results are reported of coupled photoelectron Monte Carlo calculations over a wide range of photon energies, 0.1 MeV < E < 1.4 MeV, and wall materials; Al, Cu, Cd and Pb. For the relatively low Z materials and over the energy range 0.3 MeV to 1.4 MeV, ß in LiF is given to within an accuracy of ±10% by the relation ß = 10.5 E-1.82 where E is expressed in MeV. ß is strongly dependent on Z so that for the high Z materials a simple power law relationship does not exist for any extended range of gamma ray energies.

Empirical methods to predict Vickers microhardness

Vickers microhardness variation with indentation size was examined for tool steel (O1). Bulk hardness of tool steel was varied by heat treatment. Several microhardness measurements were made at six different indenter loads. Plots of Vickers microhardness vs. indentation size were correlated with a simple power law relation: H ν = c 1 d ν c2 . The Vickers correlation coefficients c 1 and c 2 have definite relationship with the Brinell hardness. Comparisons with previous work on SS304, Ni200 and two zirconium alloys and current work on a titanium alloy are very good. Empirical correlations between the Vickers correlation coefficients c 1 and c 2 with the Brinell hardness are proposed for a wide range of bulk hardnesses from 1.3 GPa to 7.6 GPa. With these correlations the microhardness of tool steel (01) and four other materials are predicted accurately with just the knowledge of the bulk hardness of the material. The correlation coefficients are found to be interrelated.

Evaluation of Transient Dissolution of Slightly Water‐Soluble Compounds From a Light Nonaqueous Phase Liquid Pool

A transient two‐dimensional mathematical model was developed using anadvection‐dispersion equation to describe the dissolution and transport of slightly watersoluble compounds (SWSCs) from a pool of light nonaqueous phase liquid (LNAPL) into an aquifer. The LNAPL pool is located in a capillary fringe, and the bottom of the pool is in direct contact with the aquifer. In the LNAPL pool, SWSCs are assumed to move by liquid diffusion, and within the aquifer by mass flow under the influence of steady horizontal groundwater flow. The dissolution process along the LNAPL/water interface assumed a local phase equilibrium relation. A formal analytical solution to the mathematical model was first derived in the Laplace transform domain and inverted numerically by means of the Bellman algorithm. This solution was used to calculate thedissolution of a SWSC from an LNAPL pool to groundwater and to identify physicochemical factors that could significantly affect this transient mass transfer process. Results show that within the range of parameters we used in the study, the SWSC flux,which is represented by the modified Damköhler number , can be related to the Fourier number Fo, the transverse hydrodynamic Peclet number Pet, and the ratio of nonaqueous diffusion Peclet number PeN to PeT(or DT/DN) through a simple power lawrelationship.

High-temperature properties of SiC/SiC for fusion applications

Ceramic matrix composites (CMCs) such as SiC/SiC exhibit novel mechanical properties relative to their monolithic counterparts. The crack velocity ( da dt ) versus stress intensity ( K) relationship for monolithic ceramics can be described by a simple power law relationship whereas SiC/SiC was found to exhibit a multistage da dt versus K relationship similar to that for stress corrosion of metals. A K-independent stage II was followed by a strongly K-dependent stage III similar to monolithic materials. Evidence also exists that the fracture resistance of these materials is greater if cracks are produced by subcritical growth processes relative to machined notches. Oxygen was found to increase da dt and decrease the K for the stage II to stage III transition while cyclic loads produced little damage at low K values but there was some evidence for increasing damage with increasing number of cycles and K.

Surface Fractals in Block Copolymers

A surface fractal model was presented to describe the interface in block copolymers. It gives a simple power-law relationship between the scattering intensity I(q) and the wave vector q in a relatively wide range as qxi >> 1, I(q) is-proportional-to q(D-6

Seismic constraints on a model of partial melts under ridge axes

In a recent global scale seismic study, the correlation between S wave velocity under ridge axes and spreading rate was pointed out. The correlation is strong for depths to about 70 km, but it diminishes below this depth. We present the correlation plots at four depths, 38, 66, 90, and 110 km, for which correlation is strong at 38 and 66 km but is weak at 90 km and is almost nonexistent at 110 km. We present a model to explain this behavior, which includes a thermal conduction model for the development of lithosphere and a simple melt percolation. Thermal effects on S wave velocity are assumed to be accounted for entirely by the plate cooling (thermal conduction) model. We point out that the thermal model under this assumption predicts asymptotically no correlation between S wave velocity and spreading rate, specifically for spreading rate larger than about 3 cm yr−1. This contradicts the correlation observed in the data at shallow depths. The existence of partial melt is thus required to explain the observed behavior at 38 and 66 km depths. We start from four basic equations that govern the distribution of partial melt and derive the relation between the amount of partial melt and the spreading rate. We adopt a simple power law relation between permeability (k) and porosity (ƒ) by k(ƒ) = k0ƒn, where k0 and n are constants and assume that pores are filled with melt. We then set up an integral relation between S wave velocity and spreading rate. The final formula indicates that the gradient in the correlation plots is the inverse of the power (1/n) in the permeability‐porosity relation, thus enabling us to constrain n as well as k0 from seismic data. The data also have some sensitivity to the depth to solidus. We show that (1) the depth to solidus is probably within the range 60–100 km and (2) if the power n is n = 2–3, then k0 = 10−8 ‐ 10−10 m2. These parameters predict that porosity and fluid velocity are 1–2% and about 0.5 m yr−1, respectively. The depth to solidus is consistent with previous estimates by petrological data but is perhaps the first and direct seismological evidence of partial melt from surface wave data. Analytical forms for the dependence on depth and spreading rate of porosity, fluid velocity within permeable rocks, and ascent times of magma are also obtained.

Creep behavior of 6 micrometer linear low density polyethelene film

Creep tests were performed to provide material characteristics for a 6.4 μm polyethelene film used to construct high altitude balloons. Results suggest simple power law relationships are adequate for stresses below about 4.83 MPa.

Susceptibility phenomena in a fine particle system

Abstract The concentration dependence of the peak in the dc susceptibility of a weakly interacting fine particle system consisting of Fe 3 O 4 particle has been studied. Measurements of the initial susceptibility of the dispersion in the solid state show that the susceptibility values were depressed at low temperatures. The temperature of the peak ( T g ) in the susceptibility versus temperature curve was found to occur at higher temperatures as the concentration was increased. The variation of T g with concentration shows a simple power law relationship. The influence of dipolar interactions between the particles on the value of T g is discussed by considering the effects of particle interactions on the blocking temperature. In this study we develop an effective blocking temperature model, and apply it to our system.

Ion transport anomalies in glasses.

A model for ion transport in glass is presented which is based on the experimental evidence that cations in glass create and maintain their own characteristic environments. By postulating the existence of a site memory effect, we are able to visualize the formation of conducting pathways. We develop a quantitative theory which explains the general occurrence of the mixed cation (alkali) effect in glassy material and, in addition, shows that the anomalous dependence of conductivity on the modifier content in single alkali glasses follows a simple power-law relation.

Effect of Contact Residual Stress on Proof Testing

Theoretical predictions are made of strength distributions after proof testing accounting for the possibility of a contact residual stress. Assuming a simple power-law relationship between subcritical crack velocity and the stress intensity factor, the amount of crack growth, hence strength degradation, that occurs during a proof stress cycle is determined. The results clearly show that the contact residual stress can strongly influence the after-proof strength distribution so that after-proof strengths can become weaker than the initial. The results are consistent with experimental observations with soda–lime glass; thus, the importance of accounting for the contact residual stress is emphasized.

Temperature profile of a boundary-layer arc

The interaction between an arc immersed in a high-speed gas flow and an adjacent plane surface has been studied experimentally. The high-speed flow is generated in a conventional pressure-driven shock tube and the arc is supplied by a charged capacitor bank.The thickness of the arc layer has been monitored using a streak camera, and it is found that the distance from the surface to the point of maximum brightness in the arc is, in fact, within about 30% of theoretical predictions. Microdensitometric analysis of the streak photographs together with the use of a simple power-law relationship between plasma luminosity and plasma temperature generate temperature profiles through the arc layer; again these are very similar to their theoretical counterparts.

A Parameterization of the Effect of Surface Roughness on Microwave Emission

A simple model is developed to represent the net effect of surface roughness on the microwave emission from soils. The reflectivity of a rough soil surface is defined in a theoretical model that includes both coherent and incoherent reflectivities in terms of the statistical properties of the rough surface, i.e., the surface height standard deviation σ and its horizontal correlation length l. It is shown that the rough surface reflectivity obtained from this theoretical model can be presented in a form that is simply the reflectivity R <L xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</L> of a smooth surface attenuated by a "rough thickness" G. It is found that the rough thickness G can be parameterized as a function of the statistical slope ratio m (=σ/l) of a rough surface by a simple power-law relationship. Since the slope of a rough surface can be determined experimentally, the rough thickness G can be quantitatively estimated from the parametric representation. Model calculations show that this simple model can provide reasonably accurate results of predicted brightness temperatures that agree well with field measurements within experimental uncertainty.

Dependence of the shape of the resistive transition on composite inhomogeneity in multifilamentary wires

The shape of the resistive critical current transition is an important factor in the specification of a multifilamentary superconducting composite. We have made a detailed analysis of the relation of the shape of the resistive transition to the scale and distribution of inhomogeneity within a composite. In particular we are able to show that the 'n-value' of a transition often has a simple inverse power law relation to the standard deviation of the spatial critical current distribution. The expression for the voltage at a particular current level depends on both the geometry of the composite and the detailed form of the critical current distribution. Two limits are discernable; the 'single filament' limit, applicable when a given filament conserves its transport current and the 'coupled filament' limit, applicable when current transfer between filaments dominates. The problem of deconvolution of an experimental transition to give the critical current distribution is discussed with specific reference to Nb 3 Sn multifilaments. A method of analysis is outlined which combines data from both the resistive transition in a multifilament, and the resistive transition in single filaments extracted from the multifilament.

Numerical study of turbulent blood flow through a caged-ball prosthetic heart valve using a boundary-fitted co-ordinate system.

A numerical model is developed for steady turbulent flow through a fully open Starr-Edwards caged-ball prosthetic heart valve in the aortic position. An orthogonal boundary-fitted co-ordinate system is generated for the axisymmetric flow domain in the vicinity of the valve. The boundary lines follow the left ventricular wall, an idealised sinus, the aortic wall, and the ball occluder. The governing partial differentiation equations, written in a stream function-vorticity formulation, are recast into their curvilinear equivalents before being discretised into finite-difference equations. The equations are then solved iteratively. Regions of separated flow and elevated fluid stress are identified at several flow rates. Analysis of the numerical solutions reveals a simple power-law relationship between the computed turbulent shear stress and the steady flow rate at important flow field locations. The maximum turbulent shear stress occurs consistently near the sewing-ring tip. However, the peak turbulent shear stress in the sinus separation zone is observed to increase significantly with higher flow rates, exceeding values in many other regions. The numerical solutions compare satisfactorily with experimental measurements.

Temperature dependence of the intrinsic remanent magnetization and anisotropy energy in the spin glass Cu-Mn.

We have measured the temperature and dc magnetic field dependence of the parallel and perpendicular ac susceptibilities of the spin glass Cu--5 at. % Mn at a frequency of 30 Hz. The temperature range is above and below ${T}_{g}^{0}$, the spin-glass transition temperature in zero field. From an analysis of these ac data, in conjunction with that for the dc magnetization, we are able to determine the T and H dependence of a quantity we identify as the intrinsic magnetic remanence, ${M}_{\mathrm{IR}}$. The advantage of studying ${M}_{\mathrm{IR}}$ is that it is determined at each measurement frequency by time-independent measurements. We find ${M}_{\mathrm{IR}}$ obeys the simple power-law relation ${M}_{\mathrm{IR}\mathrm{\ensuremath{\propto}}(1\mathrm{\ensuremath{-}}\mathrm{T}/{T}_{g}^{0}{)}^{1/2}}$ for all T${T}_{g}^{0}$. We also have determined the T and H dependence of the total anisotropy constant and find that it obeys a linear relation which extrapolates to zero near ${T}_{g}^{0}$, in contrast to a much higher temperature found from ESR measurements at 9 GHz. We suggest that a systematic study of ${M}_{\mathrm{IR}}$ over an extended frequency range should provide a valuable set of data for gaining insight into the complex time dependence of spin-glass variables.

Thermal conductivity of benzene and cyclohexane in the temperature range 36?90�C at pressures up to 0.33 GPa

The paper describes new, accurate measurements of the thermal conductivity of benzene and cyclohexane in the temperature range 36–88°C and at pressures up to 0.33 GPa. The experimental data have an estimated accuracy of ±0.3%. The density dependence of the thermal conductivity for both liquids is well represented by a simple power law relationship which is almost independent of temperature. The connection of this correlation to one previously established for normal alkanes is examined.

Relationships between cyclic creep and hysteresis loop energies in metals

Push-pull cyclic tests were carried out on four metals in the plastic region at a number of different stress amplitudes and tensile mean stresses. The resulting cyclic creep strains could be related to the differences in energies dissipated between the tensile and compressive parts of the cycle by simple power law relationships. Two types of behaviour were observed for each material.

Estimating the yields of the Amchitka tests by waveform intercorrelation

Summary. A new method of estimating the yields of events at a nuclear test site is introduced and tested against the Amchitka experience. The intercorrelation method is a relative waveform comparison technique, which involves convolution of the first few seconds of each short-period P-wave recorded at a given station for two events with estimates of the effective source function (including pP) for the other event. This procedure accounts for common path and receiver effects as well as differences in time functions and near-source surface interactions. If the source parameters for a master event are determined independently, the assumed source parameters for a second event can be adjusted to minimize the residual differences in the inter-correlated signal pairs for a large number of stations simultaneously. Yield estimates can then be made using empirical relations between the source parameters and yield. The source time function representation used is ψ(t) =ψ{1 − exp(−Kt)[1 +Kt+ (Kt)2/2 −B(Kt)3]}. Due to the narrowband character of short-period P-waves and the possibility of yield dependence of overshoot of the source spectrum (controlled by B), it is generally not possible to resolve uniquely the static source strength, ψ∞. However; for fixed values of B, an alternate parameter, ψ′∞, can be determined, which gives the correct source strength in the period range of the signals. For the Amchitka events ψ′∞ is found to have a simple power law relation with yield. With B= 1, intercorrelation of MILROW with CANNIKIN gives a ψ′∞ of 4.0 ± 0.6 × 1011 cm3 for CANNIKIN, compared with 4.5±0.5 × 1011 cm3 determined from near-field data. Intercorrelations of MILROW and CANNIKIN with LONGSHOT give consistent ψ′∞ values of 2.0 ± 0.3 × 1010cm3 for LONGSHOT. In all cases the pP lag times and amplitudes obtained in the procedure are consistent with previous determinations. The effects of underparameterization of the effective source functions are investigated and found not to bias the ψ′∞ determinations strongly. The use of complete waveform information is shown to have several advantages over standard amplitude comparisons for estimating yield.