Penetration depth gradient

Observation of a Penetration Depth Gradient in ATR FT-IR Spectroscopic Imaging Applications... 

Wessel, E. et al. (2006) Qbservation of a penetration depth gradient in attenuated total reflection Fourier transform infrared spectroscopic imaging applications. Appl. Spectrosc., 60 (12), 1488-1492. 

First, we will consider thin objects - more specifically, those that can be approximated as having no spatial, internal temperature gradients. This class of problem is called thermally thin. Its domain can be estimated from Equations (7.11) to (7.12), in which we say the physical thickness, d, must be less than the thermal penetration depth. This is illustrated in Figure 7.7. For the temperature gradient to be small over region d, we require... 

The diffusion theory states that interpenetration and entanglement of polymer chains are additionally responsible for bioadhesion. The intimate contact of the two substrates is essential for diffusion to occur, that is, the driving force for the interdiffusion is the concentration gradient across the interface. The penetration of polymer chains into the mucus network, and vice versa, is dependent on concentration gradients and diffusion coefficients. It is believed that for an effective adhesion bond the interpenetration of the polymer chain should be in the range of 0.2-0.5 pm. It is possible to estimate the penetration depth (/) by Eq. (5),... 

The fact that ATR-IR spectroscopy uses an evanescent field and therefore probes only the volume very close to the IRE has important consequences for its application in heterogeneous catalysis, in investigations of films of powder catalysts. The catalyst particle size and packing affect the size of the detectable signals from the catalyst and bulk phase. Furthermore, if the catalyst layer is much thicker than the penetration depth of the evanescent field, diffusion of reactants and products may influence the observed signals. In fast reactions, gradients may exist within the catalyst layer, and ATR probes only the slice closest to the IRE. 

Yttria stabilized zirconia formed by this reaction fills the air electrode pores. The dynamics of this CVD stage has been modeled by Carolan and Michaels [120], who observed that films produced in this manner penetrate the substrate no more than 2-3 pore diameters from the chloride face of the substrate. It has also been shown that the penetration depth is independent of water concentration. The second step of this method is the EVD step. Once pore closure is achieved, the reactants are not longer in contact. Electrochemical semipermeability related to the existence of small electronic conductivity and large gradient of oxygen partial pressure leads to oxygen transport from the steam side to the chloride side through the deposited electrolyte. The electrochemical reactions involved are ... 

The shape of the profile alters with time, until it becomes completely flat and the inner concentration of fluorine corresponds to the one measured on the periosteal surface. This equation is used to describe diffusion processes if there is a concentration gradient only along one axis, i.e. if diffusion is one-dimensional. If diffusion and environmental conditions are constant (e.g. a constant supply of fluorine due to invariant soil humidity), the profile shape and its penetration depth carry the information on exposure time t. The profile will be more developed if the sample has been exposed to this environmental system for a longer time (Fig. 6). This fact leads to the idea of a mathematical evaluation of the diffusion length Dt (the parameter that describes how far the diffusion front has penetrated into the material), which allows the calculation of the burial time t, i.e. the age of the archaeological sample, if the diffusion constant is known [80],... 

The external electric field is in the direction of the pore axis. The particle is driven to move by the imposed electric field, the electroosmotic flow, and the Brownian force due to thermal fluctuation of the solvent molecules. Unlike the usual electroosmotic flow in an open slit, the fluid velocity profile is no longer uniform because a pressure gradient is built up due to the presence of the closed end. The probability of the particle position is obtained by solving the Fokker-Planck equation. The penetration depth is found to be dependent upon the Peclet number, which is a measure of significance of the convective electroosmotic flow relative to the Brownian diffusion, and the Damkohler number, which is a ratio of the characteristic diffusion-to-deposition times. 

In the AC impedance method, a modulation of potential causes a modulation of concentration at the interface. As a consequence there is a modulation of the concentration gradient and hence of the current density. The modulation of concentration decays with distance away from the interface. Thus there is a characteristic penetration depth into the solution phase of the concentration wave, SAC, where SAc = (2Dlw)m. The time scale for chemical processes, either in the solution or at the interface, probed by the perturbation, is simply / >. If the concentration boundary... 

In the EHD impedance method, modulation of the flow velocity causes a modulation of the velocity gradient at the interface which, in turn, causes a modulation in the concentration boundary layer thickness. As demonstrated previously in Section 10.3.3 and Fig. 10.3 the experiment shows a relaxation time determined solely by the time for diffusion across the concentration boundary layer. Although there is a characteristic penetration depth, 8hm, of the velocity oscillation above the surface, and at sufficiently high modulation frequencies this is smaller than the concentration boundary layer thickness, any information associated with the variation of hm with w is generally lost, unless the solution is very viscous. The reason is simply that, at sufficiently high modulation frequencies, the amplitude of the transfer function between flow modulation and current density is small. So, in contrast to the AC impedance experiment, the depth into the solution probed by the EHD experiment is not a function... 

If in the limit of CA 4 0, [CA / (CA) = 0 but R(CA) 0, the concentration CA will become zero for a certain penetration depth Xp (Figure A.2b). For values of X larger than Xp Equation A.2. yields a solution without any physicochemical significance. Since no reactant is converted on the right-hand side of the plane X=Xp, no reactant will penetrate beyond this plane. Therefore, there can be no concentration gradient ... 

A quantity of interest in mass diffusion processes is the depth of diffusion at a given time. This is usually characterized by the penetration depth defined as the location where the tangent to the concentration profile at the surface (x = Oj intercepts the Q, line, as shown in Figure 14-27. Obtaining the concentration gradient at. r = 0 by differentiating Eq. 14-36, the penetration deptli is determined to be... 

Generally, for a qualitative analysis of heterogeneous samples the results will hardly be influenced by this phenomenon. For quantitative analysis, however, the effect must be taken into account. By using another reflection element, for example, germanium with a higher refractive index (4.0 instead of 2.4 for ZnSe), the gradient would be much smaller (about one-third), because of the lower penetration depth under these experimental conditions. 

Analytical solution of the fluid temperature distribution. The wall gradient of this distribution gives the heat transfer coefficient. The exact analytical solution of convection problems is rather involved and is beyond the scope of this text. The concept of boundary layer (penetration depth) provides a convenient tool for approximate analytical solutions and will be considered in Sections 5.1 and 5.2. 

Another circumstance concerns the depth of penetration as it relates to the size of the piece that is heated. If the piece is several times larger than the depth of poietration, then the temperature gradient will resemble conventional gradients, with a cooler interior and a warmer exterior. However, if the piece is small in comparison with the penetration depth, for example, only one or two times greater, then there may be a focused accumulation of the electromagnetic field in the center of the piece due to the multiple passes of the waves and internal reflections. In this case, the center may be the hottest place, and in fact, if it is overheated, the center may bum, whereas the surface remains cool. 

For the determination of residual stress-depth gradients, material has to be removed gradually, e.g., by electrolytical polishing. This arises from the low penetration (information) depth of the X-rays into the material, which lies in the range of few microns only. Based upon the sin v /- method, several new methods have been developed, which allow nondestructive determination of residual stress-depth gradients in depths... 

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