Free Path Theory Research Articles

Freeness of Signed Graphic Arrangements

Freeness occupies an important position in the study of hyperplane arrangements. In this paper, we conclude the freeness of three special classes of signed graphic arrangements based on the addition–deletion theorem and Abe’s free path theory.

Investigating the validity of the Bosanquet equation for predicting the self-diffusivities of fluids inside nanotubes using equilibrium molecular dynamics simulations

The self-diffusion of fluids in nanotubes generally consists of both molecule–molecule and molecule–wall interactions, which can be quantitatively described by the Knudsen mechanism and the molecular mechanism, respectively. Combining these two effects, the Bosanquet equation is generally used to predict the self-diffusivities of molecules in one-dimensional nanopores. In this work, equilibrium molecular dynamics simulations were employed to investigate the validity of the equation in predicting the self-diffusivities of fluids inside carbon, boron nitride, and silicon carbide nanotubes with diameters from ∼1.0 to 4.3 nm. Our results indicate that although the Bosanquet equation can predict the self-diffusivities of H2, Ar, CH4, CO2, C2H6, and C3H8 in carbon nanotubes in the same order of magnitude, the accuracy of these predictions is generally rather poor. At high and moderate loadings, the large deviation mainly results from the limited accuracy of the simplistic free path model, which tends to neglect the intermolecular forces of fluid molecules. However, at low loadings, the failure of the Bosanquet equation can be traced to the failure of the Knudsen model, which largely underestimates the diffusivity in nanotubes due to the smoothness of the tube wall. Furthermore, the Bosanquet equation fails to predict the self-diffusivities of H2O in confinement since the presence of hydrogen bonding violates the mean free path theory. It is suggested that further modification of this extrapolation should take into account the intermolecular forces of fluid molecules as well as the smoothness of the tube wall.

Analytical expression for predicting the reduced settling velocity of small particles in turbulence

The influence of turbulence on the settling velocity of small particles remains an inconclusive research subject. Both enhanced and retarded particle settling compared to quiescent settling have been reported in previous literature, and several theories have been proposed. Among the mechanisms that account for reduced settling velocities in turbulence, the loitering effect is an important one that generally exists in various conditions. This study focuses on an analytical prediction of reduced particle settling velocities due to the loitering effect. By considering the velocity autocorrelation function as a step function analogous to the free path theory, and using a modified integral time scale to capture the essence of the loitering effect, an analytical expression is derived for predicting the reduction in particle settling velocities in turbulent flows. Calculation results of the expression are then examined by comparing with the results of a random walk model, direct numerical simulations that conditionally captured the reduction of particle settling velocities due to the loitering effect, and several representative experiments. Major possible influencing factors on the prediction of the analytical expression and applicable conditions of the expression are then further discussed. The proposed analytical expression is shown to be suitable for predicting the reduced settling velocities of small particles with relatively weak inertia in turbulent flows and could provide a reasonable explanation for reported cases in which small to moderate reductions in particle settling velocities were observed.

A new turbulence analysis method based on the mean speed and mean free path theory of the molecule thermal motion—correction and Improvement

Based on the mean speed and mean free path theory of the molecule thermal motion, a mathematical formulation of the Reynolds stress by reasonable analysis and derivation has been obtained and the result have been published in the Journal of Physics A. In follow-up studies, certain flaws were detected in the mathematical derivation presented in my previous paper and therefore the final mathematical expressions of the Reynolds shear stress achieved are not very accurate. In this paper, these flaws have been fixed and a new form of mathematical expression of Reynolds shear stress has been proposed. In addition, the mathematical expression of the Reynolds normal stress was also derived.

A new turbulence analysis method based on the mean speed and mean free path theory of the molecule thermal motion

Based on the mean speed and mean free path theory of the molecule thermal motion, we obtain a mathematical formulation of the Reynolds stress by reasonable analysis and derivation. Results show that in the case of non rarefied gas flow, the Reynolds stress is inversely proportional to three parameters: the number of molecule in unit volume, the square root of the molecule mass and the square of the effective diameter of the molecule. Meanwhile, the Reynolds stress is proportional to other two parameters: the square of the temperature; the gradient of time-averaged velocity in the flow field. The essence of turbulence is the chaos resulted from the molecule thermal motion in non uniform flow field.

Reflectance of Metal Grid Lines Calculated by Free Path Theory

Reflectance of Metal Grid Lines Calculated by Free Path Theory

Lattice anharmonicity, phonon dispersion, and thermal conductivity of PbTe studied by the phonon quasiparticle approach

We investigated the vibrational property of lead telluride (PbTe) with a focus on lattice anharmonicity at moderate temperatures $(300lTl800$ K) using the phonon quasiparticle approach which combines first-principles molecular dynamics and lattice dynamics. The calculated anharmonic phonon dispersions are strongly temperature dependent and some phonon modes adopt giant frequency shifts, e.g., transverse optical modes in the long-wavlength regime. As a result, we witness the avoided crossing between transverse optical modes and longitudinal acoustic modes at elevated temperature, in good agreement with experimentation and available theoretical studies. These results, together with the large root-mean-square displacements of atoms, reveal a strong anharmonic effect in PbTe. The obtained phonon lifetimes allow studies of transport properties. For considered temperatures, the phonon mean free paths can be shorter than lattice constants at relatively high temperature, especially for optical modes. This finding goes against the widely employed minimal phonon mean free path concept. As such, the calculated lattice thermal conductivity of PbTe, which is indeed relatively small, does not have the prescribed minima at high temperature, showcasing the breakdown of the minimal mean free path theory. Our study provides a basis for delineating vibrational and transport properties of PbTe and other thermoelectric materials within the framework of the phonon gas model.

Heat transfer in cellulose-based aerogels: Analytical modelling and measurements

A simple analytical approach for estimating the total heat transfer inside new cellulose-based aerogels has been investigated. The model accounts for the characteristic solid matrix at the nanometric scale by using a cellular representation of the nanofoam porous structure.The radiation-conduction heat transfer is taken into account. Previous analytical correlation for the fluid phase is used to model the conduction heat transfer in gas. New analytical formulations based on mean free path theory combined with phonon tracking approach are proposed to model the conduction heat transfer in the solid phase at the nanometric scale. The contribution of radiation heat transfer is obtained from Rayleigh scattering approach combined to the Rosseland approximation.These analytical relations validated experimentally are expected to be useful for researchers aiming at developing new insulating organic aerogels since they permit to determine conduction-radiation equivalent conductivity as a function of cell dimensions, phonon and optical properties of cellulose.

Electron Inelastic Mean Free Path Theory and Density Functional Theory Resolving Discrepancies for Low-Energy Electrons in Copper

We develop the many-pole dielectric theory of UV plasmon interactions and electron energy losses, and couple our advances with recent developments of Kohn-Sham density functional theory to address observed discrepancies between high-precision measurements and tabulated data for electron inelastic mean free paths (IMFPs). Recent publications have demonstrated that a five standard error difference exists between longstanding theoretical calculations and measurements of electron IMFPs for elemental solids at energies below 120 eV, a critical region for analysis of electron energy loss spectroscopy (EELS), X-ray absorption spectroscopy (XAS), and related technologies. Our implementation of improved optical loss spectra and a physical treatment of second-order excitation lifetimes resolves this problem in copper for the first time for energies in excess of 80 eV and substantially improves agreement for lower energy electrons.

Derivation and validation of a binary multi-fluid Eulerian model for fluidized beds

The paper presents a multi-fluid Eulerian model derived from binary kinetic theory of granular flows, free path theory and an empirical friction theory. The effects of the inter- and inner-particle collisions, particle translational motions and particle–particle friction are included. As the effects due to fluiddynamic particle velocity differences and particle–particle friction are considered, some unconventional terms are produced compared with the previous models. Model validation using the data from Mathiesen et al. (2000) shows that the coupling terms give a stronger and more realistic particle–particle coupling because the effects due to the fluiddynamic velocity differences are considered. The model gives reasonable predictions of the particle volume fraction, particle velocities and velocity fluctuations. The model analysis reveals that the basic particle velocity fluctuations constitute 2 terms: the velocity fluctuations of the discrete particles, and the velocity fluctuations of the continuous fluid flow. Furthermore, the simulation results show that the velocity fluctuations of the continuous fluid flow are dominant in a binary riser flow.

Comments on "Angular Momentum Transport in Quasi-Keplerian Accretion Disks"

Subramanian, Pujari and Becker (2004) claim that the correct expression for the angular momentum transport in an accretion disc, which is proportional to dΩ/dR, can be derived on the basis of the analysis of the epicyclic motion of gas parcels in adjacent eddies in the disc. We study their argument and show that their derivation contains several fundamental errors: 1) the biased choice of the desired formula from an infinite number of formulae; 2) the biased choice of parcel trajectories; and 3) confusion regarding the reference frames. Following 1) we could derive, for example, a (invalid) formula in which the angular momentum transport is proportional to dv Φ /dR, and from 2) we could even prove that the angular momentum transport is either inward or null. We present the correct approach to the problem of angular momentum transport in an accretion disc in terms of mean free path theory.

On Kinetic Theory Viscosity in a Rotating Gas

Clarke and Pringle (2004) derived a proper viscosity formula in a rotating gas by applying mean free path theory. We study their argument in detail and show that their result can be derived with a much simpler calculational procedure and a physically clearer picture.

Angular momentum transport in quasi-Keplerian accretion disks

We reexamine arguments advanced by Hayashi & Matsuda (2001), who claim that several simple, physically motivated derivations based on mean free path theory for calculating the viscous torque in a quasi-Keplerian accretion disk yield results that are inconsistent with the generally accepted model. If correct, the ideas proposed by Hayashi & Matsuda would radically alter our understanding of the nature of the angular momentum transport in the disk, which is a central feature of accretion disk theory. However, in this paper we point out several fallacies in their arguments and show that there indeed exists a simple derivation based on mean free path theory that yields an expression for the viscous torque that is proportional to the radial derivative of the angular velocity in the accretion disk, as expected. The derivation is based on the analysis of the epicyclic motion of gas parcels in adjacent eddies in the disk.

A New Approach to the Correlation of the Electrical Properties with Interband and Intraband Transitions of Thin Cr Films

Thin chromium films of thickness ranging from 25 to 80 nm are prepared by thermal evaporation under a vacuum of 1.33 × 10−3 Pa. The electrical resistivity was inversely proportional to the thickness of the film. The analysis of the electrical resistivity is treated in the frame of the effective mean free path theory of size effect developed by Pichard et al. Such analysis allows the determination of the mean free path l0, carrier concentration nc, relaxation time τ, and Fermi energy EF. The optical constants, n and k, of chromium thin films are determined in the spectral range of 200 to 25000 nm. The obtained results agree with the optical conductivities predicted theoretically by Moruzzi et al. In addition, the values of nc, σs, l0, and τ obtained electrically are found to match with those obtained optically.

Size and temperature dependence of electrical transport properties of vacuum evaporated Bi2Se3 films

The Bi2Se3 films of thicknesses in the range 50 to 546 nm are vacuum deposited on glass substrates at room temperature. After annealing for 1 h at 423 K, the electrical resistivity, Hall coefficient, and the thermoelectric power are measured. It is observed that the resistivity is strongly affected by both the film thickness and temperature, and the activation energy (ΔE) is found to vary from 0.116 to 0.279 eV with the decrease in film thickness. The Hall coefficient (RH) as well as the mobility increase with increasing film thickness. Also, the thermoelectric power varies in a similar manner. It is found that the deposits were all of n-type and the carrier concentrations (n) measured by different methods have the order of 1020 cm−3. The analysis combining the data from thermoelectric power and electrical conductivity measurements, by using the effective mean free path theory of size effect in thin films developed by Pichard et al. leads to the determination of mean free path, carrier concentration, Fermi energy, and the parameter Ug = (d ln 1g/d ln E), where Ug is a parameter determining the energy dependence of the mean free path of carriers in infinitely thick films.

Size and temperature dependence of thermoelectric power and electrical resistivity of vacuum-deposited antimony thin films

Thin antimony films of thicknesses in the range 30 to 200 nm have been vacuum deposited on glass substrates at room temperature. After annealing for about an hour at 500 K, the thermoelectric power and electrical resistivity were measured in vacuum as a function of temperature. The thermoelectric power and electrical conductivity data were combined and simultaneously analysed using the effective mean free path theory of size effect in thin films developed by Tellier and Pichard et al. In addition, their temperature dependence was also analysed. It was found that the thermoelectric power is positive and increases with increasing temperature and is inversely proportional to the thickness of the film. The electrical resistivity was found to be temperature dependent with the temperature coefficient of resistivity being positive, and inversely proportional to the thickness of the film. Analysis combining the data from the thermoelectric power and electrical conductivity measurements has led to the determination of mean free path, carrier concentration, effective mass, Fermi energy and the parameter \(U_g = (d ln l_g /d ln E)_{E = E_F } \) The data were analysed for least squares fitting by local functions, such as the spline functions, which eliminates possible errors in conventional least squares fitting of data using non-local functions valid throughout the range.

Thermoelectric power of Se 20Te 80 alloy thin films: Temperature and size effects

Thin films of the alloy Se 20Te 80 of different thicknesses between 600 and 1250 Å were deposited on clean glass substrates at room temperature in a vacuum of 5 × 10 -5 Torr by the vacuum deposition technique in individual evaporations. The thermoelectric power of these films was determined by measuring the developed thermo-e.m.f. at different temperature differences between the ends using the integral method. It was found that the thermoelectric power decreased with reciprocal temperature linearly. From the slope of the plot, the height of the Fermi level, which is pinned, was evaluated to be about 100 meV. It was also found that the room temperature thermoelectric power of the various films is an inverse function of thickness of the films. A similar inverse thickness dependence was also observed at 350 and 400 K. However, the thickness dependence is very small and is thought to be due to the specularity parameter's being close to unity. The reciprocal thickness dependence of the thermoelectric power of the films is explained on the basis of the size effect theory of Tellier (C.R. Tellier, Thin Solid Films, 51 (1978) 311) and Pichard et al. (C.R. Pichard, C.R Tellier and A.J. Tosserm J. Phys. F, 10 (1980) 2009) termed the “effective mean free path theory” which takes into account the effects of both the surface and the grain boundary scattering in thin films.

Some identities and relations for transport coefficients of dense fluids

The transport coefficients appearing in the dense fluid kinetic theory by the author are cast in the forms of time correlation functions suitable for numerical simulation methods. The Einstein relation for the diffusion coefficient and friction constant is found to follow exactly from the collision integral for the diffusion coefficient in the dense fluid kinetic theory. By assuming that the momentum and the force (or configuration) relax at different rates in the dilute and dense regimes of density, it is shown that the self-diffusion coefficient and the viscosity have two different relations, i.e., the one predicted by the mean free path theory and the Stokes–Einstein-type relation. The intrinsic viscosity of a polymer solution is shown to scale like mν/2a, where ν is an index and ma is the molecular weight, if the momentum relaxation time of the polymer scales like (∼(l−2a))ν/2 with ∼(l−2a) denoting the mean square end-to-end distance. If ν=1, the scaling follows Flory’s prediction for instrinsic viscosity.

The use of mean free path theory to develop the binary diffusion coefficient of dilute gases

The author presents a new way of deriving the binary diffusion coefficient for dilute gases that is clearer and based on more solid conceptual frameworks than the usual method.

Thermoelectric power of vacuum-evaporated tin films

The thermoelectric power (TEP) of tin films was measured in the thickness range 400–2200 Å between 30 and 100°C. It was observed that the TEP for films deposited onto glass substrates can be expressed as S=aT+n and the mean free path theory can generally be applied to analyse the data. The effect of grain boundary scattering was also critically studied.