Diffusion layer time-dependent thickness

The last condition redefines a semi-infinite diffusion. Eq. (5.13) for these conditions describes the development of a concentration profile with time as shown in Figure 5.5. A time-dependent thickness of a diffusion layer describes the growth of the concentration profile into the electrolyte given by the equation ... 

The attack of most glasses in water and acid is diffusion controlled and the thickness of the porous layer formed on the glass surface consequently depends on the square root of the time. There is ample evidence that the diffusion of alkali ions and basic oxides is thermally activated, suggesting that diffusion occurs either through small pores or through a compact body. The reacted zone is porous and can be further modified by attack and dissolution, if alkali is still present, or by further polymerisation. Consolidation of the structure generally requires thermal treatment. 

We see that the expression for the current consists of two terms. The first term depends on time and coincides completely with Eq. (11.14) for transient diffusion to a flat electrode. The second term is time invariant. The first term is predominant initially, at short times t, where diffusion follows the same laws as for a flat electrode. During this period the diffusion-layer thickness is still small compared to radius a. At longer times t the first term decreases and the relative importance of the current given by the second term increases. At very long times t, the current tends not to zero as in the case of linear diffusion without stirring (when is large) but to a constant value. For the characteristic time required to attain this steady state (i.e., the time when the second term becomes equal to the first), we can write... 

In a reversible process that occurs under diffusion control, the time-dependent drop of the faradaic current is due to the gradual increase in diffusion-layer thickness. According to Eq. (11.14), we have, for reactants. 

Figure 4 illustrates the dependence of on Aq for the case when r = 1 at several different values of [Fig. 4(a)] and when = 0.5 and at several different values of r [Fig. 4(b)]. From Fig. 4(a), one can see that takes a maximum around y = 0, i.e., Aq The volume ratio affects strongly the value of as shown in Fig. 4(b), which is ascribed to the dependence of the equilibrium concentration on r through Eq. (25). This simple example illustrates the necessity of taking into account the variation of the phase-boundary potential, and hence the adsorption of i, when one tries to measure the adsorption properties of a certain ionic species in the oil-water two-phase systems by changing the concentration of i in one of the phases. A similar situation exists also in voltammetric measurements of the transfer of surface-active ions across the polarized O/W interface. In this case, the time-varying thickness of the diffusion layers plays the role of the fixed volume in the above partition example. The adsorption of surface-active ions is hence expected to reach a maximum around the half-wave potential of the ion transfer. 

All these experiments were carried out at such low scan rates that the outside diffusion layer of the cosubstrate (on the order of 105 A) is much larger than the film thickness. An experimental test for knowing whether this condition is fulfilled is that the plateau of S-shaped catalytic current then observed is much larger than the reversible cosubstrate peak observed in the absence of substrate i icat. Under these conditions, the concentration profiles within the film (bottom of Figure 5.30) do not depend on time. 

The speed of response of the photodiode depends on the diffusion of carriers, the capacitance of the depletion layer, and the thickness of the depletion layer. The forward bias itself increases the width of the depletion layer thus reducing the capacitance. Nevertheless, some design compromises are always required between quantum efficiency and speed of response. The quantum efficiency of a photodiode is determined largely by the absorption coefficient of the absorbing semiconductor layer. Ideally all absorption should occur in the depletion region. This can be achieved by increasing the thickness of the depletion layer, but then the response time increases accordingly. 

The time variation of 8 before the onset of natural convection depends on how the diffusion process is provoked. If a constant current density is switched on at t = 0, then the time variation of the effective diffusion-layer thickness can be obtained from Eqs. (7.179) and (7.202)... 

Very much more is known about the theory of concentration gradients at electrodes than has been mentioned in this brief account. Experimental methods for observing them have also been devised, based on the dependence of refractive index on concentration (the Schlieren method) by means of interferometry (O Brien, 1986). Nevertheless, the basic concept of an effective diffusion-layer thickness, treated here as varying in thickness with fi until the onset of natural convection and as constant with time after convection sets in (though decreasing in value with the degree of disturbance, Table 7.10), is a useful aid to the simple and approximate analysis of many transport-controlled electrodic situations. A few of the uses of the concept of 8 will now be outlined. 

Such a consideration, then, turns attention to the time dependence of iL. Would there be a time domain in which iL is very large The basis of an answer to this question has been given in Section 7.9.10, where discussion shows two time regions in which iL is to be considered. To understand the first of these, one must recall Eq. (7.206), the relation between the limiting current density, iL, and the diffusion layer thickness, 8. It is... 

FIGURE 17.2 Dissolution pro Jes of ne (open circles) and coarse ( lied circles) hydrocortisone (Lu et al., 1993). Simulated curves were drawn using spherical geometry without (a) and with (b) a time-dependent diffusion-layer thickness and cylindrical geometry without (c) and with (d) a time-dependent diffusion-layer thickness. Error bars represent 95% 6I=(= 3). (Reprinted from Lu, A. T. K., Frisella, M. E., and Johnson, K. C. (1993pharm. Res., 10 1308-1314. Copyright 1993. With permission from Kluwer Academic Plenum Publishers.)... 

On the other hand, as the Nernst diffusion layer model is applied to an unstirred solution, it is expected that the passage of current will cause formation of the depletion layer (Fig. 7.1), whose thickness 5o will increase with time. In time, this layer will extend from the electrode surface to the bulk of the solution over tens of pm. In order to estimate the time-dependence of So, we can use the approximate Einstein... 

The mass transport coefficient is, in general, a complex time and potential-dependent function through the linear diffusion layer thickness, <% ,. Only under certain conditions does this dependence disappear (as, for example, for nemstian... 

This different behavior can be explained by considering that for a CE mechanism (the reasoning is similar for an EC one), C species is required by the chemical reaction whose equilibrium is distorted in the reaction layer (whose thickness in the simplified dkss treatment is <5r = jDj(k + 2)) and by the electrochemical reaction, which is limited by the diffusion layer (of thickness 8 = yfnDt). For a catalytic mechanism, C species is also required for both the chemical and the electrochemical reactions, but this last stage gives the same species B, which is demanded by the chemical reaction such that only in the reaction layer do the concentrations of species B and C take values significantly different from those of the bulk of the solution. In summary, the catalytic mechanism can reach a true steady-state current-potential response under planar diffusion because its perturbed zone is restricted to the reaction layer <5r, which is independent of time, whereas the distortion of CE (or EC) mechanism is extended until the diffusion layer 8, which depends on time, and a stationary current-potential response will not be reached under these conditions. 

A voltammetric experiment in a microelectrode array is highly dependent on the thickness of the individual diffusion layers, <5, compared with the size of the microelectrodes themselves, and with the interelectrode distance and the time experiment or the scan rate. In order to visualize the different behavior of the mass transport to a microelectrode array, simulated concentration profiles to spherical microelectrodes or particles calculated for different values of the parameter = fD Ja/r s can be seen in Fig. 5.17 [57] when the separation between centers of... 

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. 

Simulations of three-component time-dependent diffusion were made based on two slightly different models using Matlab software [33]. Basically, fluid layer thicknesses are predicted, which determine diffusion distances. In this way, the H+ and OH- concentrations are revealed which can be related back to the pH. By the known fluorescence intensity-pH relationship, the quantum yield is thus given. 

If the layer thickness-time dependence is well described by these equations, then the growth process is undoubtly diffusion controlled. A plot of the layer thickness against the square root of the annealing time is shown in Fig. 1.20. As seen in Fig. 1.20, the experimental points yield three straight lines. Thus, the NiBi3 layer growth is indeed diffusion controlled. 

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