A semi-analytical solution for multilayer diffusion in a composite medium consisting of a large number of layers

Diffusion in a composite slab consisting of a large number of layers provides an ideal prototype problem for developing and analysing two-scale modelling approaches for heterogeneous media. Numerous analytical techniques have been proposed for solving the transient diffusion equation in a one-dimensional composite slab consisting of an arbitrary number of layers. Most of these approaches, however, require the solution of a complex transcendental equation arising from a matrix determinant for the eigenvalues that is difficult to solve numerically for a large number of layers. To overcome this issue, in this paper, we present a semi-analytical method based on the Laplace transform and an orthogonal eigenfunction expansion. The proposed approach uses eigenvalues local to each layer that can be obtained either explicitly, or by solving simple transcendental equations. The semi-analytical solution is applicable to both perfect and imperfect contact at the interfaces between adjacent layers and either Dirichlet, Neumann or Robin boundary conditions at the ends of the slab. The solution approach is verified for several test cases and is shown to work well for a large number of layers. The work is concluded with an application to macroscopic modelling where the solution of a fine-scale multilayered medium consisting of two hundred layers is compared against an “up-scaled” variant of the same problem involving only ten layers.

Diffusion of Be, Co and Mn impurities in compound semiconductors and glassy carbon studied by the modified radiotracer technique

The main method of modifying properties of semiconductors is to introduce small amount of impurities inside the material. This is used to control magnetic and optical properties of materials and to realize p- and n-type semiconductors out of intrinsic material in order to manufacture fundamental components such as diodes. As diffusion can be described as random mixing of material due to thermal movement of atoms, it is essential to know the diffusion behavior of the impurities in order to manufacture working components. In modified radiotracer technique diffusion is studied using radioactive isotopes of elements as tracers. The technique is called modified as atoms are deployed inside the material by ion beam implantation. With ion implantation, a distinct distribution of impurities can be deployed inside the sample surface with good con- trol over the amount of implanted atoms. As electromagnetic radiation and other nuclear decay products emitted by radioactive materials can be easily detected, only very low amount of impurities can be used. This makes it possible to study diffusion in pure materials without essentially modifying the initial properties by doping. In this thesis a modified radiotracer technique is used to study the diffusion of beryllium in GaN, ZnO, SiGe and glassy carbon. GaN, ZnO and SiGe are of great interest to the semiconductor industry and beryllium as a small and possibly rapid dopant hasn t been studied previously using the technique. Glassy carbon has been added to demonstrate the feasibility of the technique. In addition, the diffusion of magnetic impurities, Mn and Co, has been studied in GaAs and ZnO (respectively) with spintronic applications in mind.

Theory of laminar flame propagation with non-normal diffusion.

A study has been made of the problem of steady, one-dimensional, laminar flame propagation in premixed gases, with the Lewis number differing from (and equal to) unity. Analytical solutions, using the method of matched asymptotic expansions, have been obtained for large activation energies. Numerical solutions have been obtained for a wide range of the reduced activation temperature parameter (n {geometrically equal to} E/RTb), and the Lewis number δ. The studies reveal that the flame speed eigenvalue is linear in Lewis number for first order and quadratic in Lewis number for second order reactions. For a quick determination of flame speeds, with reasonable accuracy, a simple rule, expressing the flame speed eigenvalue as a function of the Lewis number and the centroid of the reaction rate function, is proposed. Comparisons have been made with some of the earlier works, for both first and second order reactions.

Stress concentration and load diffusion in rectangular panels with constant stress flanges

The stress concentration that occurs when load is diffused from a constant stress member into thin sheet is an important problem in the design of light weight structures. By using solutions in biharmonic polar-trigonometric series, the stress concentration can be effectively isolated so that highly accurate information necessary for design can be obtained. A method of analysis yielding high accuracy with limited effort is presented for rectangular panels with transverse edges free or supported by inextensional end ribs. Numerical data are given for panels with length twice the width.

Fugacity coefficients. Enthalpy, and second virial coefficients of boron trifluoride

Fugacity coefficients and isothermal changes of enthalpy have been calculated and reported. The calculations cover a temperature range of 0° to 75°C. up to gas densities of 1.0 gram per cc. The generalized Benedict-Webb-Rubin constants evaluated from generalized PVT relations is found to predict the experimental data with an over-all absolute deviation of 3.1%. Second virial coefficients and potential energy parameters for Lennard-Jones (12-6) potential energy function are reported also.

A generalized equation for diffusion in liquids

The association parameter in the diffuswn equaiior, dye fo Wiike one Chong has been interpreted in deferminable properties, thus permitting easily the calculation of the same for unknown systems. The proposed eqyotion a!se holds goods for water as soiute in organic solvenfs. The over-all percentage error remains the sarrse as that of the original equation.

The ratio of diffusion coefficient to mobility for slow electrons in dry air

Using Huxley's solution of the diffusion equation for electron-attaching gases, the ratio of diffusion coefficient D to mobility μ for electrons in dry air was measured over the range 3·06 × 10-17 diffusion equation for non-attaching gases.

Mass Transfer Coefficients in Mixer-Settlers

Mass transfer coefficients have been determined for transfer into a highly viscous phase in a stirred tank involving high Schmidt numbers. The results have been used to compute mass transfer coefficients in the extraction of free fatty acids from oils using alcohol and show good agreement with experimental results

Single-File Diffusion of Confined Water Inside SWNTs: An NMR Study

We report a nuclear magnetic resonance (NMR) study of confined water inside similar to 1.4 nm diameter single-walled carbon nanotubes (SWNTs). We show that the confined water does not freeze even up to 223 K. A pulse field gradient (PFG) NMR method is used to determine the mean squared displacement (MSD) of the water molecules inside the nanotubes at temperatures below 273 K, where the bulk water outside the nanotubes freezes and hence does not contribute to the proton NMR signal. We show that the mean squared displacement varies as the square root of time, predicted for single-file diffusion in a one-dimensional channel. We propose a qualitative understanding of our results based on available molecular dynamics simulations.

Single-File Diffusion of Water Inside Narrow Carbon Nanorings

We use atomistic molecular dynamics (MD) simulations to study the diffusion of water molecules confined inside narrow (6,6) carbon nanorings. The water molecules form two oppositely polarized chains. It is shown that the effective interaction between these two chains is repulsive in nature. The computed mean-squared displacement (MSD) clearly shows a scaling with time similar to t(1/2), which is consistent with single-file diffusion (SFD). The time up to which the water molecules undergo SFD is shown to be the lifetime of the water molecules inside these chains. Simulations of "uncharged" water molecules inside the nanoring show the formation of several water chains and yield SFD. These observations conclusively prove that the diffusion is Fickian when there is a single chain of water and SFD is observed only when two or more chains are present.

Asymptotic behaviour and boundedness of linear systems with time varying coefficients

Motivated by developments in spacecraft dynamics, the asymptotic behaviour and boundedness of solution of a special class of time varying systems in which each term appears as the sum of a constant and a time varying part, are analysed in this paper. It is not possible to apply standard textbook results to such systems, which are originally in second order. Some of the existing results are reformulated. Four theorems which explore the relations between the asymptotic behaviour/boundedness of the constant coefficient system, obtained by equating the time varying terms to zero, to the corresponding behaviour of the time varying system, are developed. The results show the behaviour of the two systems to be intimately related, provided the solutions of the constant coefficient system approach zero are bounded for large values of time, and the time varying terms are suitably restrained. Two problems are tackled using these theorems.

The Context and Diffusion of Knowledge in the Finnish Jewellery Industry - The role of the House of Fabergé

Researchers within the fields of economic geography and organizational management have extensively studied learning and the prerequisites and impediments for knowledge transfer. This paper combines two discourses within the two subjects: the-communities-of-practice and the learning region approaches, merging them through the so-called ecology of knowledge-approach, which is used to examine the knowledge transfer from the House of Fabergé to the Finnish jewellery industry. We examine the pre-revolution St Petersburg jewellery cluster and the post-revolution Helsinki, and the transfer of knowledge between these two locations through the components of communities of people, institutions and industry. The paper shows that the industrial dynamics of the Finnish modern-day goldsmith industry was inherently shaped both through the transfer and the non-transfer of knowledge. It also contends that the “knowledge-economy” is not anchored in and exclusive for the high technology sector of the late 20th century.

Computationally efficient optical tomographic reconstructions through waveletizing the normalized quadratic perturbation equation - art. no.61640R

In this paper, we present a wavelet - based approach to solve the non-linear perturbation equation encountered in optical tomography. A particularly suitable data gathering geometry is used to gather a data set consisting of differential changes in intensity owing to the presence of the inhomogeneous regions. With this scheme, the unknown image, the data, as well as the weight matrix are all represented by wavelet expansions, thus yielding the representation of the original non - linear perturbation equation in the wavelet domain. The advantage in use of the non-linear perturbation equation is that there is no need to recompute the derivatives during the entire reconstruction process. Once the derivatives are computed, they are transformed into the wavelet domain. The purpose of going to the wavelet domain, is that, it has an inherent localization and de-noising property. The use of approximation coefficients, without the detail coefficients, is ideally suited for diffuse optical tomographic reconstructions, as the diffusion equation removes most of the high frequency information and the reconstruction appears low-pass filtered. We demonstrate through numerical simulations, that through solving merely the approximation coefficients one can reconstruct an image which has the same information content as the reconstruction from a non-waveletized procedure. In addition we demonstrate a better noise tolerance and much reduced computation time for reconstructions from this approach.