Two-layer Slab Research Articles

Prediction of the transient and steady temperature distributions in a two-layer composite slab in contact with a plasma: exact closed-form solutions

Plasma heating of a one-dimensional two-layer composite slab with a homogeneously imperfect interface is solved by the method of separation of variable. The plasma is composed of a collisionless presheath and sheath on a wall partially reflecting and secondarily emitting ions and electrons. The heating of the composite slab from the plasma accounting for the presheath and sheath is determined from the kinetic analysis. This work proposes an analytical model to calculate the transient and steady temperature, temperature gradient and heating rate profiles inside the composite slab and provides quantitative results applicable to control the temperature evolution and the melting time of it. The predicted surface temperature of the composite slab as a function of time is found to agree well with a finite difference numerical solution. The effects of reflectivities of the ions and electrons on the wall, electron-to-ion source temperature ratio at the presheath edge, ion-to-electron mass ratio, ion charge number, ion ionization energy, plasma flow work-to-heat conduction ratio, solid thermal conductivity and specific heat ratios of the second-to-first layer, dimensionless second layer thickness, Biot numbers for first and second layers, on the temperature profiles are obtained. The thermal contact resistance between the layers to simulate the heat conduction into the second layer is also discussed.

Reflectivity properties of an anisotropic slab waveguide with isolated substrate

The scattering properties for both TE and TM modes of an abruptly ended two-layered slab waveguide with anisotropic core and isolated substrate are examined by an improved iteration technique, which is based on the integral equation method with accelerating parameters. The relative dielectric constants of the core for the three Cartesian directions are considered to be different, but cases with isotropic core are also considered. The electric field distribution on the terminal plane and the reflection coefficients of the dominant TE and TM guided modes, as well as the near-field distribution and the far-field radiation pattern, are computed, while numerical results are presented for several cases of the core anisotropy.

Mass diffusion through two-layer porous media: an application to the drug-eluting stent

A mathematical model for the diffusion–transport of a substance between two porous homogeneous media of different properties and dimensions is presented. A strong analogy with the one-dimensional transient heat conduction process across two-layered slabs is shown and a similar methodology of solution is proposed. Separation of variables leads to a Sturm–Liouville problem with discontinuous coefficients and an exact analytical solution is given in the form of an infinite series expansion. The model points out the role of four nondimensional parameters which control the diffusion mechanism across the two porous layers. The drug-eluting stent constitutes the main application of the present model. Drug concentration profiles at various times are given and analyzed. Also, qualitative considerations and a quantitative description to evaluate feasibility of new drug delivery strategies are provided, and some indicators, such as the emptying time, useful to optimize the drug-eluting stent design are discussed.

Coupled conductive–radiative heat transfer problem for two-layer slab

The coupled conductive radiative transfer problem in two homogeneous layers slab of anisotropic scattering with specularly reflecting boundaries has been considered. A Galerkin-iterative technique is used to solve the coupled conductive radiative heat equations in integral forms for the two layers. Numerical results are obtained for the temperature, the conductive, radiative and the total heat fluxes for the two homogeneous layers with isotropic and anisotropic scattering. The calculations are also carried out for homogeneous plane parallel medium with anisotropic scattering which show good agreement with the published calculations.

Thickness dependence of the atomic and electronic structures ofTiO2rutile (110) slabs and the effects on the electronic and magnetic properties of supported clusters of Pd and Rh

Using first principles calculations with ultrasoft pseudopotentials, we study the thick dependence of the atomic and electronic structures of (110) slabs of $\mathrm{Ti}{\mathrm{O}}_{2}$ rutile having one to five layers. Thin slabs with an even and odd number of layers show significantly different structural characteristics and electronic properties that can affect photocatalysis as well as the catalytic behavior of supported clusters. We discuss the origin of the oscillating band gap for both an even and odd number of layers and the effects on the atomic, electronic, and magnetic properties of octahedral ${\mathrm{M}}_{6}$ and icosahedral ${\mathrm{M}}_{13}$ $(\mathrm{M}=\mathrm{Pd},\mathrm{Rh})$ clusters deposited on stoichiometric slabs with two and three layers. Calculations have also been carried out for a Pd atom deposited on different sites of a two layer slab. These results show that the bridging oxygen atoms are most reactive and preferred for adsorption. The adsorption energy of a Pd atom on the bridging site has only a weak dependence on the thickness of the slab. However, the adsorption energy of a cluster supported on a three-layer slab is significantly higher than the value for a two-layer slab due to significant structural differences and this alters the magnetic and electronic properties of the supported clusters. The magnetic moments of Pd clusters are reduced after interaction with the support. However, for Rh clusters there is an increase in the magnetic moment. In general we find that the cluster-support interactions affect mainly the cluster and support atoms that are in contact at the interface. The variation in the band gap with slab thickness can, however, lead to metallic character of the slab after cluster adsorption and this could have important consequences for catalysis.

Multi-layer transient heat conduction using transition time scales

The solution of multi-layer transient heat conduction problems may be simplified by analyzing the different transition times of the various layers of a composite slab. These transition times, in fact, allow the ‘disturbance’ effect of each layer on the eigenvalues of the composite slab to be analyzed and estimated. It was found that the eigenvalues may be obtained in the first approximation by merging in increasing order the suitable corrected eigenvalues of each layer for which explicit equations are available. The errors in the resulting dimensionless temperatures are of one order of magnitude larger than the deviations between the exact and approximate eigenvalues. In particular, when one of the two outer boundaries is kept at constant temperature and the transition times of the layers are quite different, the eigenvalues may be written down very simply as the eigenvalues of the layer whose exposed surface is not at prescribed temperature. In this paper the temperature solution for the case of a two-layer slab with interlayer thermal contact resistance is presented.

Structure, Energetics, and Dynamics of Water Adsorbed on the Muscovite (001) Surface: A Molecular Dynamics Simulation

Molecular dynamics (MD) computer simulations of liquid water adsorbed on the muscovite (001) surface provide a greatly increased, atomistically detailed understanding of surface-related effects on the spatial variation in the structural and orientational ordering, hydrogen bond (H-bond) organization, and local density of H2O molecules at this important model phyllosilicate surface. MD simulations at constant temperature and volume (statistical NVT ensemble) were performed for a series of model systems consisting of a two-layer muscovite slab (representing 8 crystallographic surface unit cells of the substrate) and 0 to 319 adsorbed H2O molecules, probing the atomistic structure and dynamics of surface aqueous films up to 3 nm in thickness. The results do not demonstrate a completely liquid-like behavior, as otherwise suggested from the interpretation of X-ray reflectivity measurements and earlier Monte Carlo simulations. Instead, a more structurally and orientationally restricted behavior of surface H2O molecules is observed, and this structural ordering extends to larger distances from the surface than previously expected. Even at the largest surface water coverage studied, over 20% of H2O molecules are associated with specific adsorption sites, and another 50% maintain strongly preferred orientations relative to the surface. This partially ordered structure is also different from the well-ordered 2-dimensional ice-like structure predicted by ab initio MD simulations for a system with a complete monolayer water coverage. However, consistent with these ab initio results, our simulations do predict that a full molecular monolayer surface water coverage represents a relatively stable surface structure in terms of the lowest diffusional mobility of H2O molecules along the surface. Calculated energies of water adsorption are in good agreement with available experimental data.

Forced Vibration of a Prestretched Two-Layer Slab on a Rigid Foundation

Within the framework of a piecewise homogeneous body model, with the use of the three-dimensional linearized theory of elastic waves in initially stressed bodies, the forced vibration of a prestretched two-layer slab resting on a rigid foundation is studied. To the upper plane of the slab, a harmonic point force is applied. It is assumed that the layer materials are incompressible, and their elastic properties are characterized by Treloar’s potential. Numerical results are presented for the case where the material stiffness of the lower layer is greater than that of the upper one. The influence of prestretching the layers on the frequency dependences of the normal stresses operating on the interface between the layers and between the slab and the rigid foundation are analyzed.

Fault reactivation in brittle–viscous wrench systems–dynamically scaled analogue models and application to the Rhine–Bresse transfer zone

Analogue experiments on oblique rifting and subsequent transpressional reactivation were performed with two-layer slabs of sand and silicone. In this brittle–viscous system, transtensional and transpressional wrench faulting was induced by movements of a basal rigid plate. The dynamically scaled analogue models are confronted with the structural evolution of the Rhine–Bresse transfer zone (RBTZ) that linked Palaeogene rifting in the Upper Rhine and Bresse Grabens and that appears to have been transpressively reactivated in Neogene to recent times. Fault patterns produced in the sand layer above the basal silicone layer are compared with structural elements in the sedimentary cover, separated from the basement by an evaporitic décollement layer. In order to investigate strain-rate dependence of fault reactivation and graben inversion, transpressional shortening was performed under different displacement rates. Experimental results suggest that the reactivation of pre-existing structures in a brittle cover above a viscous décollement is strongly dependent on the strain rate within the viscous layer. Under low to intermediate displacement rates (2.6 cm h −1), deformation concentrates within the basal viscous layer and former normal faults within the cover are not reactivated. The reactivation at higher displacement rates (5 cm h −1) results in a complete inversion of graben structures within the cover. Ongoing shortening produces lobed thrust fronts, which crosscut pre-existing normal faults. Late Pliocene to recent en-échelon aligned folds and isolated thrust faults in the cover of the RBTZ are attributed to thick-skinned reactivation of basement faults. A comparison of natural and experimentally obtained structures suggests that fault reactivation occurred under low displacement rates (<1 mm/a). This results in a low mechanical coupling between basement and cover in areas with significantly thick décollement layers, providing an explanation for decoupled stresses between basement and cover, such as observed in the northern Jura Mountains.

Optical tomographic mapping of cerebral haemodynamics by means of time-domain detection: methodology and phantom validation

One of the primary applications of diffuse optical imaging is to localize and quantify the changes in the cerebral oxygenation during functional brain activation. Up to now, data from an optical imager are simply presented as a two-dimensional (2D) topographic map using the modified Beer–Lambert law that assumes homogeneous optical properties beneath each optode. Due to the highly heterogeneous nature of the optical properties in the brain, the assumption is evidently invalid, leading to both low spatial resolution and inaccurate quantification in the assessment of haemodynamic changes. To cope with these difficulties, we propose a nonlinear tomographic image reconstruction algorithm for a two-layered slab geometry that uses time-resolved reflected light. The algorithm is based on the previously developed generalized pulse spectrum technique, and implemented within a semi-three-dimensional (3D) framework to conform to the topographic visualization and to reduce computational load. We demonstrate the advantages of the algorithm in quantifying simulated changes in haemoglobin concentrations and investigate its robustness to the uncertainties in the cortical structure and optical properties, as well as the effects of random noises on image quality. The methodology is also validated by experiments using a solid layered phantom.

Finite element method for diffusive light propagations in index-mismatched media

Near-infrared (NIR) light propagations in strongly scattering tissue have been studied in the past few decades and diffusion approximations (DA) have been extensively used under the assumption that the refractive index is constant throughout the medium. When the index is varying, the discontinuity of the fluence rate arises at the index-mismatched interface. We introduce the finite element method (FEM) incorporating the refractive index mismatch at the interface between the diffusive media without any approximations. Intensity, mean time, and mean optical path length were computed by FEM and by Monte Carlo (MC) simulations for a two-layer slab model and a good agreement between the data from FEM and from MC was found. The absorption sensitivity of intensity and mean time measurements was also analyzed by FEM. We have shown that mean time and absorption sensitivity functions vary significantly as the refractive index mismatch develops at the interface between the two layers.

Analytical solution of transient heat conduction in a two-layer anisotropic cylindrical slab excited superficially by a short laser pulse

This paper presents an exact analytical solution of 2D transient temperature, caused by a non-periodical pulse heating of a cylindrical two-layer slab. The corresponding partial differential equations are solved step-by-step, using the separation of variables technique. Analytical/computational procedure for eigenvalue derivation is considered separately. General initial assumptions, such as existence of thermal contact resistance, thermal conduction and diffusion anisotropy, radiative heat losses, non-uniform heating, and finite absorption depth are accounted in the solution. Numerical simulations of temperature distribution throughout the sample are given at the end of the paper.

Adsorption from alkane+perfluoroalkane mixtures at fluorophobic and fluorophilic surfaces. I. Nature of the noncritical adsorption profiles

Neutron reflection has been applied to probe the nature and extent of adsorption from a mixture of (1−x)n-hexane+xperfluoro-n-hexane against silicon substrates modified with alkylsilane (fluorophobic) or fluoroalkylsilane (fluorophilic) coupled layers. For an equimolar mixture (x=0.5, 60.7 vol %) in the one-phase region at T=30 °C—removed both in temperature and composition from the upper critical point at 22.65 °C and x=0.36—the structure was resolved at both fluorophobic and fluorophilic surfaces. Liquid mixtures with three different refractive index contrasts were used to reduce model ambiguity in the ensuing analysis. For both surfaces the composition profiles of the adsorbed liquids could be represented using two-layer slab models which included interlayer Gaussian roughness. For the fluorophobic surface, the thickness of the layer closest to the substrate is ∼20 Å and composed of ∼83 vol % n-hexane, and the second, more dilute layer has a composition profile which decays smoothly into the bulk over a range of ∼100 Å. A similar result is found for the fluorophilic surface, but in this case the layer closest to the substrate is ∼15 Å thick and composed of ∼95 vol % perfluoro-n-hexane. Qualitatively similar behavior is found for adsorption from a mixture with x=0.7 against a fluorophobic substrate and for a mixture with x=0.2 against a fluorophilic substrate.

Transverse eigenproblem of steady-state heat conduction for multi-dimensional two-layered slabs with automatic computation of eigenvalues

The paper analyses the transverse eigenvalue problem of nonconventional Sturm–Liouville type associated to the steady-state heat conduction in 3-D two-component slabs with imperfect thermal contact. In particular, it describes how the physical insight deriving from the transverse direction of six suitable ‘homogeneous parallelepipeds’ inherent to the considered two-layered parallelepiped is capable of providing useful and reasonably accurate information about the best bracketing bounds (lower and upper) for the roots (eigenvalues) of the transverse eigencondition. This information, in fact, enables one to establish starting points (initial guesses) for a root-finding iteration (e.g., Müller’s method) so that convergence of the iteration may absolutely be guaranteed. Representative test examples are computed to illustrate the accuracy, reliability, and efficiency of the proposed fully automated solution algorithm.

Discrete ordinates solution of coupled conductive radiative heat transfer in a two-layer slab with Fresnel interfaces subject to diffuse and obliquely collimated irradiation

The coupled conductive radiative heat transfer in a two-layer slab with Fresnel interfaces subject to diffuse and obliquely collimated irradiation is solved. The collimated and diffuse components problems are treated separately. The solution for diffuse radiation is obtained by using a composite discrete ordinates method and includes the development of adaptive directional quadratures to overcome the difficulties usually encountered at the interfaces. The complete radiation numerical model is validated against the predictions obtained by using the Monte Carlo method.

The effects of internal refractive index variation in near-infrared optical tomography: a finite element modelling approach

Near-infrared (NIR) tomography is a technique used to measure light propagation through tissue and generate images of internal optical property distributions from boundary measurements. Most popular applications have concentrated on female breast imaging, neonatal and adult head imaging, as well as muscle and small animal studies. In most instances a highly scattering medium with a homogeneous refractive index is assumed throughout the imaging domain. Using these assumptions, it is possible to simplify the model to the diffusion approximation. However, biological tissue contains regions of varying optical absorption and scatter, as well as varying refractive index. In this work, we introduce an internal boundary constraint in the finite element method approach to modelling light propagation through tissue that accounts for regions of different refractive indices. We have compared the results to data from a Monte Carlo simulation and show that for a simple two-layered slab model of varying refractive index, the phase of the measured reflectance data is significantly altered by the variation in internal refractive index, whereas the amplitude data are affected only slightly.

A Seminumerical Approach for Heat Diffusion in Heterogeneous Media: One Extension of the Analytical Quadrupole Method

The analytical thermal quadrupole method is suitable for the modeling of multidimensional transient heat diffusion in homogeneous media, especially when applied to multilayered media. Here, we propose a new approach in order to extend the quadrupole frame to heterogeneous media. A seminumerical general solution is proposed for transient heat transfer in finite or semi-infinite media in both axial and radial coordinate systems, when the variation of thermal properties is one-dimensional. The presentation of the method is explained with a 2-D two-layer slab case. Some application examples are then presented from this basic case. The analytical expressions allow deep insight about the physical phenomenon.

Heat transfer mechanisms during short-pulse laser heating of two-layer composite thin films

Using a simple perturbation technique, an analytical investigation is presented for the heat transfer mechanisms during ultrafast laser heating of two-layer composite thin slabs from a microscopic point of view. The composite slab consists of two thin metal films which are in perfect thermal contact. The microscopic parabolic two-step model is adopted to describe the behavior of the composite slab. In the microscopic two-step model, the heating process is modeled by the deposition of radiation energy on electrons, the transport of energy by electrons, and the heating of the material lattice through electron–phonon interactions. The proposed perturbation technique is used when the normalized temperature difference between the solid lattice and the electron gas is relatively a small perturbed quantity.

Thermal Behavior of Perfect and Imperfect Contact Composite Slabs Under the Effect of the Hyperbolic Heat Conduction Model

This work studies the heat transfer mechanisms during rapid heating of two-layer composite thin slabs from a macroscopic point of view using the hyperbolic heat conduction model. The composite slabs consist of two thin metal layers which may be in perfect or imperfect thermal contact. The effects of parameters such as the two films' thickness ratio, thermal conductivity ratio, heat capacity ratio, thermal relaxation time, and interfacial heat transfer coefficient on the thermal behavior of the composite slabs are investigated.

Nonlinear Temperature Effects on Multilayered Concrete Pavements

This study considers a plate consisting of one or more layers, resting on a general elastic foundation, and subjected to a nonlinear through-the-thickness temperature distribution, together with arbitrary wheel loads. The resultant bending stress distribution in the plate is presented as the sum of the bending stresses due to the applied loading and to an equivalent linear temperature gradient, plus the pure thermal stresses due to the nonlinear part of temperature distribution. The resulting algorithm has been implemented into finite-element code ILSL2, capable of accommodating one- or two-layered slabs with a temperature distribution through the individual layers described by a linear function, a quadratic function, or explicitly provided by the user at a number of individual depths. Several examples are used to illustrate application of the method.