Thin layer infinite diffusion

Experimental measurements of DH in a-Si H using SIMS were first performed by Carlson and Magee (1978). A sample is grown that contains a thin layer in which a small amount (=1-3 at. %) of the bonded hydrogen is replaced with deuterium. When annealed at elevated temperatures, the deuterium diffuses into the top and bottom layers and the deuterium profile is measured using SIMS. The diffusion coefficient is obtained by subtracting the control profile from the annealed profile and fitting the concentration values to the expression, valid for diffusion from a semiinfinite source into a semi-infinite half-plane (Crank, 1956),... 

As in Sect. 2, another experiment is required to evaluate all the diffusivities, i.e., to obtain the correct value for Deiam for use with Table 2. Bai and Miller [4] repeated the contacting experiment of Fig. 13 except that only a thin layer of AOT was present initially. As a result, the similarity solution, which assumes a semi-infinite AOT phase, is not valid after a short transient. Instead the governing equations must be solved numerically with the boundary... 

In a typical spectroelectrochemical measurement, an optically transparent electrode (OTE) is used and the UV/vis absorption spectrum (or absorbance) of the substance participating in the reaction is measured. Various types of OTE exist, for example (i) a plate (glass, quartz or plastic) coated either with an optically transparent vapor-deposited metal (Pt or Au) film or with an optically transparent conductive tin oxide film (Fig. 5.26), and (ii) a fine micromesh (40-800 wires/cm) of electrically conductive material (Pt or Au). The electrochemical cell may be either a thin-layer cell with a solution-layer thickness of less than 0.2 mm (Fig. 9.2(a)) or a cell with a solution layer of conventional thickness ( 1 cm, Fig. 9.2(b)). The advantage of the thin-layer cell is that the electrolysis is complete within a short time ( 30 s). On the other hand, the cell with conventional solution thickness has the advantage that mass transport in the solution near the electrode surface can be treated mathematically by the theory of semi-infinite linear diffusion. 

Occasionally (e.g., thin-layer electrochemistry, porous-bed electrodes, metal atoms dissolved in a mercury film), diffusion may be further confined by a second barrier. Figure 2.7 illustrates the case of restricted diffusion when the solution is confined between two parallel barrier plates. Once again, the folding technique quickly enables a prediction of the actual result. In this case, complete relaxation of the profile results in a uniform finite concentration across the slab of solution, in distinct contrast to the semi-infinite case. When the slab thickness t is given, the time for the average molecule to diffuse across the slab is calculable from the Einstein equation such that... 

All of the principles of semi-infinite potential-step experiments discussed so far apply to thin-layer work. Some modification in the quantitative response is necessary to account for the presence of a diffusion barrier. Figure 3.11 illustrates the diffusion phenomena occurring during a chronoamperometric experiment in a thin-layer cell of typical dimensions. It will be useful to compare this figure with the semi-infinite situation depicted in Figure 3.1. Notice that the supply of reactant in the bulk solution phase is effectively infinite in Fig-... 

Problems in designing cells for semi-infinite diffusion techniques are well known to electrochemists. When the same considerations must be applied to a layer of solution less than 100 pm thick, the difficulties are significantly enhanced. Like most real-world components, thin-layer cells do not always conform to theory (Chap. 3), and design trade-offs are necessary to optimize performance in a particular application. 

Figure 7-4 a) Two sided diffusion from an infinitely thin layer (source), b) Distribution of concentrations for different values of the product Dt. 

At the beginning of diffusion, time t = 0 and the total amount of color having mass m is located at position x = 0. Because of the theoretically infinitely thin layer 5x of the color source, the initial concentration there is infinite and the concentrations at all other positions of the bar are zero. The solution of the diffusion equation (7-12) is immediately given as ... 

Finite diffusion — Finite (sometimes also called -> limited) diffusion situation arises when the -> diffusion layer, which otherwise might be expanded infinitely at long-term electrolysis, is restricted to a given distance, e.g., in the case of extensive stirring (- rotating disc electrode). It is the case at a thin film, in a thin layer cell, and a thin cell sandwiched with an anode and a cathode. Finite diffusion causes a decrease of the current to zero at long times in the - Cottrell plot (-> Cottrell equation, and - chronoamperometry) or for voltammetric waves (see also - electrochemical impedance spectroscopy). Finite diffusion generally occurs at -> hydrodynamic electrodes. 

Finite-space diffusion takes place during the charging of insertion electrodes at moderate frequencies, transforming into mainly capacitive behavior within the limit of very low frequencies, in contrast to the semi-infinite diffusion for solution redox-species (except for thin-layer solution electrochemistry) electrochemical impedance spectroscopy becomes a very useful diagnostic tool for the characterization of insertion mechanisms ... 

A. Assume that an (infinitely) thin layer of 14CC>2 is introduced at the surface of the guard cell. If Dcch is 106 times larger in air than in the cell wall, what are the relative times for 14CC>2 to diffuse across the two barriers ... 

Further experiments have been conducted to confirm whether or not the presumed diffusion layer and its thickness, 8, as estimated from (95) corresponds to physical reality. First AC impedance spectroscopy has been used to find the frequency response of the real and imaginary components of the cell impedance and compared with the theoretical prediction for diffusion across a thinned diffusion layer. At very high AC frequencies, where the AC perturbation had insufficient time to probe to the edge of the diffusion layer, effectively the response expected for semi-infinite diffusion was seen ( Warburgian behaviour ). At lower AC frequencies, as expected, the cell impedance was greatly reduced in the presence of ultrasound. Moreover, not only was the quantitative behaviour as predicted theoretically... 

The diffusion coefficients in solids are typically very low (on the order of 10 to 10" mVs), and thus the diffusion process usually affects a thin layer at the surface. A solid can conveniently be treated as a semi-infinite medium during transient mass diffusion regardless of its size and shape when the penetration depth is small relative to the thickness of the solid. When this is not the case, solutions for one dimensional transient mass diffusion through a plane wall, cylinder, and sphere can be obtained from the solution.s of analogous heat conduction problems using the Heisler charts or one term solutions pieseiited in Chapter 4. 

In all experiments, a constant silver layer thickness of three monolayers was initially deposited. In the multicomponent system, Cd diffuses towards S under thin layer conditions" established by a 2D CdxAgy surface alloy. In the one-component systems, Cd diffuses towards S under semi-infinite conditions. A phase transition... 

Alternatively, the thin-skin model, which assumes an infinite diffusion resistance with only a thin layer on the droplet surface being heated and evaporated, can be used. This second assumption is more appropriate for cases in which the droplet reaches its boiling temperature immediately after the start of the evaporation process. All the other approaches (such as the diffusion-limited model), which are recommended in the case of mulhcompo-nent droplets, require some form of spahal discretization inside the droplet, resulting in much higher computational costs. To close the set of equahons, it is necessary to describe the evolution of the droplet temperature Tp as discussed in Section 5.2.3. For example, upon including the effect of evaporation on the heat flux for the rapid-mixing model, the particle-enthalpy balance yields... 

Figure 17.1.2 A Cell for transmission spectroelectrochemistry involving semi-infinite linear diffusion. Light beam passes along vertical axis. [Reprinted from N. Winograd and T. Kuwana, Electroanal. Chem., 7, 1 (1974), by courtesy of Marcel Dekker, Inc.] B Optically transparent thin-layer system front and side views, (a) Point of suction application in changing solutions (b) Teflon tape spacers (c) 1 X 3 in. microscope slides (d) test solution (e) gold minigrid, 1 cm high ...

Pick s laws also describe diffusion in solid phases. In solids transport properties can be considerably different than in liquid phases. Only one component can mobile diffuse in the matrix of the second component. At higher temperatures the diffusion coefficient can be more similar in size than in liquid phases, but the diffusion coefficient at room temperature can be orders of magnitudes smaller, e.g., D < 10 ° cm s k To overcome the time limitation one must make the diffusion length smaller. Ultra-thin layers or nanoparticles provide such small dimensions. Under such conditions the diffusion is not semi-infinite but has a restricted extension. This has to be considered in the boundary conditions. 

From these expressions it can be deduced that, in contrast to cyclic voltammet-ric responses for solution systems with semi-infinite planar diffusion, for redox processes for confined or thin layer systems close to the electrode or adsorbed molecules at the electrode surface, the peak current expression becomes linear with respect to the scan rate (Eqs. II.1.13a and II.1.13b). 

At increased scan rates, a transition is observed from the thin layer regime, where the peak current varies linearly with scan rate, to semi-infinite diffusion control, where a v dependence is observed. Under semi-infinite diffusion conditions, voltammograms are characteristic of those obtained for solution species. The general voltammogram characteristics are... 

Developments in spectroelectrochem-istry based on TCO electrodes under semi-infinite diffusion conditions were reviewed by Kuwana and Winograd in 1974 [244]. Since 1974, significant developments in absorption spectroelectrochem-istry have occurred under semi-infinite diffusion conditions and in optically transparent thin-layer electrochemistry, several new TCO electrodes have heen characterized, and the indirect coulometric titration technique has been developed and apphed to biological systems [245]. 

Transient Infinite Diffusion of a Thin Layer In this example, we consider the transient diffusion of an infinitely thin layer of a diffusing species i placed in... 

FIGURE 4.11 Transient diffusion of a thin layer between two semi-infinite bodies. 

Problem 4.2. Equation 4.40 in the text provides the solution for the transient diffusion of a thin layer of material between two semi-infinite bodies ... 

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