Calculation Of Distribution Function Research Articles

Application of Kihara potential model to real non-spherical fluids

The calculation of radial distribution function (RDF) of Kihara rigid core fluids is performed using the sequential–analytic (S–A) method and the effect of reduced rigid core diameter on the shape of RDF is demonstrated. A theoretical equation of state for Kihara fluid is presented and used to calculate its reduced critical properties. It has been shown that the critical compressibility factor of a Kihara fluid is solely determined by its reduced rigid core diameter, and hence the critical compressibility factor of any real fluids can be reproduced through adjusting the value of reduced rigid core diameter. Instead of regression of virial coefficient or viscosity data of dilute gases, it is suggested to determine the parameters of Kihara potential model by matching the critical properties of corresponding substances. It has been shown that the Kihara parameters determined by matching critical properties can be used to predict the properties of both dilute and dense fluids while those determined by fitting dilute gas properties cannot.

Combinations of simulation and integral equation theory

Much work has been done to accelerate the prediction of structure and thermodynamics, particularly of the chemical potential of condensed phases by means of molecular simulation, requiring large computational power or simplified simulation procedures. This work demonstrates that the combination of simulation with approximate calculations based on integral equation theory provides the necessary information for quantitatively correcting simulation deficiencies. These ideas are illustrated with two examples: the calculation of artefact-free radial distribution functions from truncated potentials and the accurate prediction of excess chemical potentials from a single trajectory with the full potential.

Calculation of radial distribution function from incomplete experimental diffraction data

A method for calculating atomic radial distribution functions for amorphous materials has been suggested. The method is based on the principle of the maximum informational entropy. Under the condition of a limited amount of the experimental diffraction data, the method allows one to obtain the unique solution and considerably reduces the errors caused by series termination, which are inherent in traditional computational methods based on the Fourier transform. A criterion of using the method of optimum information is formulated, and the reliability of the data obtained are quantitatively estimated. The advantages of the method are demonstrated on the computation of the short-range order in hydrogenated amorphous silicon.

Study of the electron kinetic processes in laser-induced breakdown of electronegative gases over an extended wavelength range

A theoretical investigation of laser-induced breakdown of electronegative gases is presented. The formulations are based on an electron cascade model previously developed by Evans and Gamal (J. Phys. D: Appl. Phys. 13 (1980) 1447–1458). This model solves numerically the time-dependent Boltzmann equation simultaneously with a set of rate equations describing the population density of the formed excited states. It includes the possible kinds of interactions between electrons, molecules and photons. Calculations are carried out under the experimental conditions of Davis et al. (Appl. Optics 30 (1991) 4358–4364) where molecular oxygen over a pressure range of 20–760 Torr is irradiated with the first four harmonics of a Nd : YAG laser source at wavelengths 1064, 532, 355 and 266 nm of pulse duration 8.5, 7.5, 6.5 and 5.5 ns, respectively, and peak irradiance varies between 3.6×10 10 and 3.7×10 11 W/cm 2. Computations revealed that the dependence of threshold irradiance on gas pressure is in quite close agreement with those measured by Davis et al. (1991) for the four laser wavelengths considered in this analysis. It is also shown that at laser wavelengths 532 and 266 nm oxygen breakdown is mainly governed by the combined effect of two and three-body attachment loss processes, while for the wavelengths 335and 1064 nm, the three-body attachment process dominates. In addition, calculation of the electron energy distribution function and its parameters, viz, electron density, ionization rate and electron mean energy predicted the importance of the photoionization of excited states as the main electron generation process over the short wavelength range.

Solution of the plasma-sheath equation with a cool Maxwellian ion source

The plasma-sheath equation with a cool Maxwellian ion source is solved using a hybrid simulation with kinetic ions and Boltzmann electrons. Results are shown for ion source temperatures TS of 0.003 to 1.0 times the electron temperature Te. When TS≲0.1Te, the potential and density profiles in the plasma are well approximated by cold Tonks–Langmuir theory. However, the shape and width of the ion velocity distribution function depend sensitively on the source temperature. The calculated distribution functions agree qualitatively with the experimental measurements of Bachet et al. [Phys. Plasmas 2, 1782 (1995)].

Optical Properties of Hybrid Aligned Short Pitch Cholesterics

Diffractive optical properties of selective reflection in short pitch cholesterics are considered in the confined geometry with different boundary conditions at upper and lower substrates. The main difference in the intensities of back selective reflection at the opposite interfaces has been detected. Additionally new stable cholesteric textures were found and explored. The asymmetric structure obtained may be used as non-reciprocal dichroic element. A new method for calculation of light propagation in stratified inhomogeneous anisotropic media, where optical properties may vary along three major axis is proposed and applied for continuous distribution function calculation of helical axes within hybrid aligned cholesteric layer.

Isentropic scaling analysis of ozone in the upper troposphere and lower stratosphere

We examine ozone concentrations recorded by 7630 commercial flights from August 1994 to December 1997 for spatial scaling properties. The large amount of data allows an approximately isentropic analysis of ozone variability in the upper troposphere and lower stratosphere. Since ozone is a good passive tracer at cruise altitudes, the results provide a strong diagnostic for scalar advection theories and models. Calculations of structure functions and increment probability distribution functions show that ozone variability scales anomalously from ∼2 to ∼2000 km, although not continuously in this interval. We find no evidence for the simple scaling predicted for smooth advection/diffusion, even at the large scales. At mesoscales the upper tropospheric ozone field is rougher and more intermittent than in the lower stratosphere. Within the troposphere the equatorial ozone field is rougher than at higher latitudes, and the intermittency decreases with increasing latitude. In the stratosphere the intermittency and roughness are greater at high latitudes and over land than at midlatitudes and over the ocean.

Influence of secondary ionization processes on the breakdown of nitrogen by a wide range of laser wavelengths

Following the modified model used in our previous paper for the study of laser-induced breakdown of molecular nitrogen, in this work we present the influence of secondary ionization processes on the breakdown phenomenon. The analysis is based on the calculation of the electron energy distribution function and its parameters for laser wavelengths covering the range 266-1064 nm, in irradiated molecular nitrogen at atmospheric pressure. The result of computations revealed that no evidence is observed for collision and photoionization processes of ground-state molecules. Under the experimental conditions applied in this analysis, collision processes lead only to the population of excited states. Breakdown of nitrogen at laser wavelengths 532, 355 and 266 nm proceeds totally via photoionization of the excited states. For = 1064 nm, electron growth is obtained due to multi-photoionization of the excited states, in addition to the contribution of collisional ionization of these states.

Quantitative analysis of preferential orientation components of ferroelectric thin films

A quantitative study of the preferred orientation of lanthanum and calcium modified lead titanate thin films, including the measurement of pole figures and calculation of orientation distribution function (OD), is presented. The main preferential orientation components are identified as <001> and <100> perpendicular to the film surface. Variations of the texture strength and the contribution of these components to the final texture are observed with changes in the film thickness or the substrate used. These variations are correlated to the ferroelectric properties of these films. This information can be used to optimise the preparation process to obtain improved thin films for pyroelectric and piezoelectric applications.

Omega meson cloud and the proton's light anti-quark distributions

We use the meson cloud model of the nucleon to calculate distribution functions for ( d ̄ − u ̄ ) and d ̄ / u ̄ in the proton. Including the effect of the omega meson cloud, with a coupling constant g ω 2/4 π≈8, allows a reasonably good description of the data.

A statistical model for a deformable solid body with a dangerous volume and its certain applications. Report 4

A solution of the problem on the integral similarity criteria is presented as applied to objects operating under conditions of frictional-and-mechanical and contact-and-mechanical fatigue. Using them, formulas have been derived for the calculation of distribution functions of limit stresses for a metal-polymer loading system.

Thermodynamics and transport properties in the transient regime

The nonequilibrium behaviour in the transient regime of metals excited by ultrashort optical pulses is investigated by means of a second order expansion of the Boltzmann equation. By definition, the transition range is located between the time necessary for the establishment of the electron temperature and the time where a description by the standard steady state equations is justified. Relaxation functions are derived for the electrical and thermal currents, and the relaxation times related to them are determined. It is shown that for the electrical transport the relaxation time corresponds to Drude's momentum scattering time whereas the corresponding time for the heat flow is identified as the electron temperature relaxation time. Further, expressions for the electrical and thermal conductivity are obtained in the case of a local thermal nonequilibrium between the electron and phonon subsystems in first and second order, respectively. Consequences for the determination of the temperature distributions inside metals are discussed. The solution of the Boltzmann equation is also used for the calculation of the time dependent energy distribution function of the electrons. The results are in good agreement with the experiment.

Modified hybrid closure approximation for prediction of flow-induced fiber orientation

Distribution of fiber orientation in flow molding processes with short-fiber reinforcements is of great importance because it affects the mechanical properties of molded parts. Due to trade-off between the computational efficiency and accuracy, a second-order orientation tensor has been widely used to describe the fiber orientation distribution. For calculation of this fiber orientation tensor, a closure approximation has been introduced to reduce a higher fourth-order orientation tensor to a lower second order. In the present investigation, a hybrid closure approximation has been modified. Two parametric forms of the distribution function, which accurately describe the random-in-space, random-in-plane, and uniaxial distributions of the fiber orientation were linearly interpolated. The interpolating factor was obtained as a function of the fiber interaction coefficient by fitting the distribution function calculations. Test simulation in homogeneous flow and nonhomogeneous flow fields, respectively, showed that the proposed closure approximation gives good performance for a wide range of CI values without showing nonphysical behavior.

Simulation of the anisotropic growth of goss grains in Fe3%Si sheets (grade HiB)

In classical simulation of abnormal grain growth, the microstructure is generally assumed to be homogeneous in size and the texture randomly distributed on such a theoretical microstructure which characterizes the primary recrystallized state. Experimentally, this last assumption could not be verified in Fe3%Si sheets (grade HiB). Consequently, a new simulation procedure has been implemented using a real microstructure measured by Orientation Imaging Microscopy (OIM) as an input data set. In this first work, the influence of the texture inhomogeneity has been analyzed from a theoretical point of view using Orientation Distribution Function (ODF) calculations and Monte-Carlo simulations without correlation with experimental measurements. The present paper deals with firstly the evolution of both microstructural and textural inhomogeneities and secondly the comparison, grain by grain, at several time steps of the experimental and simulated results obtained during secondary recrystallization.

Analysis of thickness profiles of pulsed laser deposited metal films

Practically all theoretical approaches to pulsed laser deposition start on a condition which is extremely hard to realize in practice: that the target surface is smooth and plane. When using liquid (molten) targets, the problem of surface deterioration upon repetitive ablation can completely be solved, allowing for fair comparison of experiment and theory. In this paper measured thickness profiles of metal films deposited in vacuum from molten In, Sn, Bi and Sn–Bi alloy targets are compared with calculated distribution functions. The strictly symmetrical thickness profiles of tin and indium films, derived from two-dimensional optical density maps and Rutherford backscattering data are analysed in terms of Lorentzian-like functions, originating from the so-called shifted Maxwellian velocity distribution. The bismuth profiles show a characteristic deviation from this shape.

UNSTEADINESS IN EFFERVESCENT SPRAYS

The ideal spray theory of Edwards and Marx was used to investigate the dependence of effervescent spray unsteadiness on operating conditions, spatial location, and fluid physical properties. Droplet size, velocity, and arrival time at a particular spray location were measured using a Phase/Doppler Particle Analyzer. The droplet arrival times were used in calculations of interparticle arrival time gaps and interparticle time distribution functions. The spray was determined to be steady (interparticle time distribution function obeying inhomogeneous Poisson statistics) or unsteady (interparticle time distribution function not obeying inhomogeneous Poisson statistics) by comparing experimental and theoretical (steady) interparticle time distribution functions with results reported in terms of the number of deviations between the two. Since the spray was assumed to be a Poisson process, the expected deviation is the inverse of the square root of the number of interparticle events. A chi-square analysis was performed on the discrepancy. Results demonstrate that all droplet size classes, which range from diameters of 3.2 to 60.4 μm, exhibit unsteady behavior. Stokes number calculations show that the largest droplets are incapable of following the turbulent flow field motions. Gas-phase turbulence can therefore be eliminated as a cause of unsteadiness for those drops. Chi-square calculations demonstrate that the probability for obtaining such results from random fluctuations is less than 0.001. Hence, it is concluded that effervescent atomization is an inherently unsteady process. Results also indicate that spray unsteadiness is influenced by the air-to-liquid ratio by mass (ALR) and the liquid mass flow rate, depending on the properties of the liquid used in the spray, and that fluid viscosity and surface tension can affect the level of spray unsteadiness only when the spray is operating in the bubbly or intermittent slug regime. For such conditions, the spray is more unsteady when a lower-viscosity or higher-surface-tension fluid is utilized. When using a liquid that has a low viscosity (0.03 kg/m-s) and high surface tension (0.065 kg/s2), a decrease in ALR or liquid mass flow rate causes the spray to be more unsteady. The use of a high-viscosity (0.124 kg/m-s) liquid lessens the effect of operating conditions on spray unsteadiness. Finally, it was found that the spray is more unsteady at its edge, as well as at greater downstream distances.

A test-particle method for the calculation of thethree-particle distribution function of the hard-sphere fluid:density functional theory and simulation

The three-particle distribution function g(3) of the homogeneous hard-sphere fluid is studied by means of the test-particle method of Percus. We consider an inhomogeneous situation in which the position of two particles is fixed and the inhomogeneous density profile that arises is related to g(3). The profile is calculated via Monte Carlo (MC) simulation and density functional theory (DFT). The results of the simulation are compared with standard MC results for the homogeneous fluid in order to check the numerics. The results of DFT are compared with MC data, showing a significant improvement over the well-known superposition approximation of Kirkwood.

Fast modelling of low-pressure radio-frequency collisional capacitively coupled discharge and investigation of the formation of a non-Maxwellian electron distribution function

The principles of fast modelling (FM) of a low-pressure radio-frequency capacitively coupled discharge are presented. They are based on averaging over fast electron and ion motions and on eliminating a small spatial scale, the Debye radius. As a result, the solution of a self-consistent system of the electron kinetic equation, Poisson and ion continuity equations takes approximately 10 min on a 486 PC. The calculation of discharge parameters has been performed for a wide range of current and pressure. The comparison with full-scale Monte Carlo calculations and experimental data has been performed. The performed comparisons demonstrate that the developed method of fast modelling has a good accuracy for calculating the global parameters such as central plasma density, applied voltage, sheath thickness, ionization rate, etc, and the profiles of plasma density and the electric fields. The accuracy of the electron distribution function (EDF) calculation is high when the EDF form is not enriched by slow electrons, and seems satisfactory in the case of a strongly peaked EDF. The results are in qualitatively good agreement with the experiment. The quantitative agreement is mainly within a factor of two. This discrepancy can be attributed to the fact that the EDF form is very sensitive to the details of plasma description, e.g. small variation of cross-sections results in considerable changes in the EDF. The mechanism of non-Maxwellian EDF formation due to non-locality effects has been analysed. The evolution of the low-pressure radio-frequency collisional capacitively coupled discharge with current and pressure variation has been investigated.

Improved Monte Carlo method for ion transport in ion-molecule asymmetric systems at high electric fields

An improved Monte Carlo method is developed for the simulation of the ion transport in classical drift tube in the case of ion-molecule asymmetric systems such as O−/O2 or N+/N2. The aim of this new method is to overcome the problem of incident ions which vanish at relative high electric field due to asymmetric charge transfer or electron detachment. These ion removal processes are compensated by a fictitious ion creation which improves the accuracy of the ion distribution function and swarm coefficient calculations. The classical ion-molecule collision processes occurring in weakly ionized gases at room temperature (elastic collisions including energy exchange and thermal motion of background gases and also inelastic collisions) are taken into account. This new method is then validated and the transport and reaction coefficients have been given for a large range of E/N (a part of them for the first time in the literature) in O−/O2 and N+/N2 systems.

Electron scattering loss in Earth's inner magnetosphere: 1. Dominant physical processes

Pitch angle diffusion rates due to Coulomb collisions and resonant interactions with plasmaspheric hiss, lightning‐induced whistlers and anthropogenic VLF transmissions are computed for inner magnetospheric electrons. The bounce‐averaged, quasi‐linear pitch angle diffusion coefficients are input into a pure pitch angle diffusion equation to obtain L and energy dependent equilibrium distribution functions and precipitation lifetimes. The relative effects of each scattering mechanism are considered as a function of electron energy and L shell. Model calculations accurately describe the enhanced loss rates in the slot region, as well as reduced scattering in the heavily populated inner radiation belt. Predicted electron distribution function calculations in the slot region display a characteristic “top hat” distribution which is supported by observations. Inner zone electron lifetimes based on observed decay rates of the Starfish electron population are in approximate agreement with model predictions.