Diffusivities under transient conditions

Monitoring Self-Diffusion under Transient Conditions. 116... 

As a non-invasive method, however, PFG NMR also provides excellent conditions for the measurement of molecular diffusivities under transient conditions. As an example. Fig. 18 shows the results of space- and time-resolved self-diffusion measurements over a bed of activated zeolite Na-X during the uptake of n-hexane [163]. Taking into account that, due to the finite size of... 

Under transient conditions the concentration distribution depends not only on the coordinate but also on time. The relevant functions can be found by considering the linear diffusion occurring along the x-axis in a volume element (iU bounded by the two planes S which are a distance dx apart (Fig. 11.1) it is obvious that dV=S dx. The rate of concentration change dcj/dt in this volume is given by the ratio of -S dJj (the... 

The performance of the BioCD under assay conditions has been tested using several gold standard systems. These are assays of anti-rabbit and anti-mouse IgG systems, prostate specific antigen (PSA), and haptoglobin. Incubations have been performed under equilibrium conditions without transport limitation, and also under transient conditions as ambient assays that are diffusion limited. Ambient assays are performed in practice, while equilibrium assays provide more information about the performance of the antibodies and provides a quantitative estimate for equilibrium dissociation constants. 

From Fig. 2.1a, it can be observed that the Nemst diffusion layer, defined by the abscissa at which the concentration reaches the value r0 in the linear concentration profile, is independent of the potential in all the cases in spite of their having been obtained under transient conditions. This is in agreement with Eqs. (2.20) and... 

It is convenient to distinguish between permeation measurements in which the flux is measured under a known (and constant) pressure gradient and those in which the flux of a component i is driven by a concentration difference between the membrane faces under a constant and equal total pressure at both sides (Wicke-Callenbach [3]). Either of these two main methods may be performed imder steady state or under transient conditions. Whether or not component fluxes cmd diffusivities measured with both methods give similar or different values depends on the conditions and on the type of the dominant diffusion mechanism. 

Evaluation of the data requires time constants for both the heat transfer and diffusion under nonisothermal conditions. It turns out that the initial part of the response curve is mainly determined by the mass transport (transport diffusion), whereas the descending part is mainly governed by the heat transfer. The STIR technique avoids intrusion by the evolution of heat, in that it measures the transient temperature, and has a rapid response (time constant about 10 s). Results obtained for the diffusion of CH3OH into Na-X were essentially consistent with those derived from the ZLC and PFG NMR... 

Very often, the transport of the liquid through the rubber is controlled by diffusion with either a constant or concentration-dependent diffusivity. This transport can be performed under stationary conditions when the concentration of the liquid varies with position, and under transient conditions when the liquid concentration varies with position and time. 

Example 8.2-3 Mass balance equation in a slab geometry medium We take a pure diffusion system having a slab geometry, and the mass balance equation describing the concentration distribution of all species under transient conditions is given by ... 

To operate the diffusion cell under transient condition, the concentration of one of the solutes is perturbed in one chamber and its concentration in the other chamber is monitored. The time-variation of that concentration will depend on the interplay of various processes occurring inside the pellet. Those processes responsible to the flow through the pellet are reflected in the response, while those processes occurring inside the pellet but not directly contributing to the through flux will be reflected in the response as a secondary level. This will be clear later when we deal with the analysis of the transient diffusion cell. 

Within the effective chemical isolation thickness of a cap, as defined by Eq. 4 under transient conditions or Eq. 5 under steady-state conditions, the chemical migration processes are limited to advection and diffusion (that is, no significant bioturbation or erosion). The dynamics of the chemical migration behavior within this layer can be estimated by the advection-diffusion equation. Traditionally, the cap is often approximated as semi-infinite and the transient behavior estimated using an analytical solution of the advection-diffusion equation [1]. The approach can be extended to reactive contaminants using the solution of van Genuchten [2]... 

The diffusion-conduction equation, Eq. (21), expresses the flux density of an ionic species from a strong binary electrolyte in terms of the electrolyte flux J12 and the current density I. Under transient conditions, both 7, and J 2 are position-dependent, but I is not. Thus, taking into account Eq. (21) and the relation c, = v/ci2, the continuity equation, Eq. (11), leads to Pick s second law... 

The same calculation as made in Chapter 1 for the mass transport by diffusion through a thin sheet of thickness dx leads to the main equation of heat conduction under transient conditions ... 

Figure 8 shows a schematic representation of a paper sheet for modeling moisture transport. As is conventional for paper sheets, we assume that the direction into the sheet is represented by the z co-ordinate whereas the two inplane co-ordinates are represented by x and y. It is useful to identify two concentration fields, one for the moisture concentration within the void space, c(x,t) and the moisture content within the fiber matrix, q(x.t) where x = (x,y,z). Diffusive transport under transient conditions through this composite medium is described by the following equations. 

The variation of the diffusion layer thicknesses at planar, cylindrical, and spherical electrodes of any size was quantified from explicit equations for the cases of normal pulse voltammetry, staircase voltammetry, and linear sweep voltammetry by Molina and coworkers (Molina et al., 2010a). Important limiting behaviours for the linear sweep voltammetry current-potential curves were reported in all the geometries considered. These results are of special physical relevance in the case of disk and band electrodes which possess non-uniform current densities since general analytical solutions were derived for the above-mentioned geometries for the first time. Explicit analytical expressions for diffusion layer thickness of disk and band electrodes of any size under transient conditions... 

Anode behaviour is evaluated by d.c. methods under steady state and by impedance spectroscopy under transient conditions. The reaction pathways for hydrogen have been elucidated, and mathematical modelling is providing micro- and nanoscale understanding of electrode processes. At higher current loadings, the diffusion processes have been evaluated showing that the electrochemically active anode thickness is around 10 pm. In practice, however. 

By electrodeposition of CuInSe2 thin films on glassy carbon disk substrates in acidic (pH 2) baths of cupric ions and sodium citrate, under potentiostatic conditions [176], it was established that the formation of tetragonal chalcopyrite CIS is entirely prevalent in the deposition potential interval -0.7 to -0.9 V vs. SCE. Through analysis of potentiostatic current transients, it was concluded that electrocrystallization of the compound proceeds according to a 3D progressive nucleation-growth model with diffusion control. 

Figure 1 shows a deep level transient spectroscopy (DLTS) (Lang, 1974) spectrum from a Au-diffused, n-type Si sample before and after hydrogenation of 300°C for 2h (Pearton and Tavendale, 1982a). The well-known Au acceptor level (Ec - 0.54 eV) was passivated to depths > 10 pm under these conditions and was only partially regenerated by a subsequent... 

In this paper we will first describe a fast-response infrared reactor system which is capable of operating at high temperatures and pressures. We will discuss the reactor cell, the feed system which allows concentration step changes or cycling, and the modifications necessary for converting a commercial infrared spectrophotometer to a high-speed instrument. This modified infrared spectroscopic reactor system was then used to study the dynamics of CO adsorption and desorption over a Pt-alumina catalyst at 723 K (450°C). The measured step responses were analyzed using a transient model which accounts for the kinetics of CO adsorption and desorption, extra- and intrapellet diffusion resistances, surface accumulation of CO, and the dynamics of the infrared cell. Finally, we will briefly discuss some of the transient response (i.e., step and cycled) characteristics of the catalyst under reaction conditions (i.e.,... 

Transport by combined migration—diffusion in a finite planar geometry can achieve a true steady state when only two ions are present, as we saw in Sect. 4.2. The same holds true when there are three or more ions present. Under simplifying conditions [see eqn. (89) below], it is possible to predict the steady-state behaviour with arbitrary concentrations of many ions. However, the corresponding transient problem is much more difficult and we shall not attempt to derive the general transient relationship, as we were able to do in deriving eqn. (82) in the two-ion case. 

In these electrode processes, the use of macroelectrodes is recommended when the homogeneous kinetics is slow in order to achieve a commitment between the diffusive and chemical rates. When the chemical kinetics is very fast with respect to the mass transport and macroelectrodes are employed, the electrochemical response is insensitive to the homogeneous kinetics of the chemical reactions—except for first-order catalytic reactions and irreversible chemical reactions follow up the electron transfer—because the reaction layer becomes negligible compared with the diffusion layer. Under the above conditions, the equilibria behave as fully labile and it can be supposed that they are maintained at any point in the solution at any time and at any applied potential pulse. This means an independent of time (stationary) response cannot be obtained at planar electrodes except in the case of a first-order catalytic mechanism. Under these conditions, the use of microelectrodes is recommended to determine large rate constants. However, there is a range of microelectrode radii with which a kinetic-dependent stationary response is obtained beyond the upper limit, a transient response is recorded, whereas beyond the lower limit, the steady-state response is insensitive to the chemical kinetics because the kinetic contribution is masked by the diffusion mass transport. In the case of spherical microelectrodes, the lower limit corresponds to the situation where the reaction layer thickness does not exceed 80 % of the diffusion layer thickness. 

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