Transient diffusion planar

Transient Finite (Symmetric) Spherical Diffusion So far, we have only examined ID (Cartesian) examples of Fick s second law. Solving Fick s second law in alternative coordinate systems (e.g., for radial, spherical, 2D, or 3D problems) is not really any different. As an example, we examine here the case of transient finite spherical diffusion, which is essentially analogous to the transient finite planar diffusion problem that we just finished discussing. 

Pick s second law is a second-order partial differential equation. Solving it in order to predict transient diffusion processes can be fairly straightforward or quite complex, depending on the specific situation. In this chapter, analytical solutions were discussed for a number of cases, including ID transient infinite and semi-infinite diffusion, ID transient finite planar diffusion, and transient spherical finite diffusion as summarized in Table 4.4. In all cases, solution of Pick s second law requires the specification of a number of boundary conditions and initial conditions. 

Microelectrodes with several geometries are reported in the literature, from spherical to disc to line electrodes each geometry has its own critical characteristic dimension and diffusion field in the steady state. The difhisional flux to a spherical microelectrode surface may be regarded as planar at short times, therefore displaying a transient behaviour, but spherical at long times, displaying a steady-state behaviour [28, 34] - If a... 

Johans et al. derived a model for diffusion-controlled electrodeposition at liquid-liquid interface taking into account the development of diffusion fields in both phases [91]. The current transients exhibited rising portions followed by planar diffusion-controlled decay. These features are very similar to those commonly observed in three-dimensional nucleation of metals onto solid electrodes [173-175]. The authors reduced aqueous ammonium tetrachloropalladate by butylferrocene in DCE. The experimental transients were in good agreement with the theoretical ones. The nucleation rate was considered to depend exponentially on the applied potential and a one-electron step was found to be rate determining. The results were taken to confirm the absence of preferential nucleation sites at the liquid-liquid interface. Other nucleation work at the liquid-liquid interface has described the formation of two-dimensional metallic films with rather interesting fractal shapes [176]. 

Earlier it was pointed out that the use of ultramicroelectrodes could also give a several hundred times increase in iL compared with the diffusion-free currents at planar electrodes. The advantage of increasing the ability to measure at higher current densities by using short times in a transient technique with a planar electrode is that the magnitude of the currents is normal and is not forced down to the difficult-to-measure picoampere region that microelectrodes require. 

Figure 4.14 illustrates the transient solution to a problem in which an inner shaft suddenly begins to rotate with angular speed 2. The fluid is initially at rest, and the outer wall is fixed. Clearly, a momentum boundary layer diffuses outward from the rotating shaft toward the outer wall. In this problem there is a steady-state solution as indicated by the profile at t = oo. The curvature in the steady-state velocity profile is a function of gap thickness, or the parameter rj/Ar. As the gap becomes thinner relative to the shaft diameter, the profile becomes more linear. This is because the geometry tends toward a planar situation. 

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. 

The solution for a diffusion couple in which two semi-infinite ternary alloys are bonded initially at a planar interface is worked out in Exercise 6.1 by the same basic method. Because each component has step-function initial conditions, the solution is a sum of error-function solutions (see Section 4.2.2). Such diffusion couples are used widely in experimental studies of ternary diffusion. In Fig. 6.2 the diffusion profiles of Ni and Co are shown for a ternary diffusion couple fabricated by bonding together two Fe-Ni-Co alloys of differing compositions. The Ni, which was initially uniform throughout the couple, develops transient concentration gradients. This example of uphill diffusion results from interactions with the other components in the alloy. Coupling of the concentration profiles during diffusion in this ternary case illustrates the complexities that are present in multicomponent diffusion but absent from the binary 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. 

Figure 6. Comparison of simulated SECM transients with transients corresponding to different electrode geometries (all processes are diffusion controlled). (A) the SECM transient for a conductive substrate (B) two-electrode thin-layer cell (C) microdisk (D) planar electrode (E) SECM with an insulating substrate (F) one-electrode thin-layer cell. Curves A, B, E, and F were computed with L = d/a = 0.1. Adapted with permission from Ref. [62], Copyright 1991, American Chemical Society.

Figure 32 shows a typical microelectrode voltammogram for an electro-chemically reversible system under near steady-state conditions. Of course at very fast scan rates the behaviour returns to that of planar diffusion and a characteristic transient-type cyclic voltammetric response is obtained as the mass transport changes from convergent to linear diffusion. 

The bimolecular reaction rate for particles constrained on a planar surface has been studied using continuum diffusion theory " and lattice models. In this section it will be shown how two features which are not taken account of in those studies are incorporated in the encounter theory of this chapter. These are the influence of the potential K(R) and the inclusion of the dependence on mean free path. In most instances it is expected that surface corrugation and strong coupling of the reactants to the surface will give the diffusive limit for the steady-state rate. Nevertheless, as stressed above, the initial rate is the kinetic theory, or low-friction limit, and transient exp)eriments may probe this rate. It is noted that an adaptation of low-density gas-phase chemical kinetic theory for reactions on surfaces has been made. The theory of this section shows how this rate is related to the rate of diffusion theory. 

During this stage of the growth of the deposit, the nuclei develop diffusion zones around themselves, and as these zones overlap, the hemispherical mass transfer gives way to linear mass transfer to an effectively planar surface. The current then falls, and the transient approaches that corresponding to the total electrode surface. 

The potentiostatic current transient (PCT) technique has been known as the most popular method to understand lithium transport through an intercalation electrode, based on the assumption that lithium diffusion in the electrode is the rate-determining process of lithium intercalation/deintercalation [45]. By solving Eick s second equation for planar geometry with I.C. in Equation (5.28), impermeable B.C. in Equation (5.29), and potentiostatic B.C. 

The substrate generation/tip collection (SG/TC) mode with an ampero-metric tip was historically the first SECM-type measurement performed (32). The aim of such experiments was to probe the diffusion layer generated by the large substrate electrode with a much smaller amperometric sensor. A simple approximate theory (32a,b) using the well-known c(z, t) function for a potentiostatic transient at a planar electrode (33) was developed to predict the evolution of the concentration profile following the substrate potential perturbation. A more complicated theory was based on the concept of the impulse response function (32c). While these theories have been successful in calculating concentration profiles, the prediction of the time-de-pendent tip current response is not straightforward because it is a complex function of the concentration distribution. Moreover, these theories do not account for distortions caused by interference of the tip and substrate diffusion layers and feedback effects. 

The mechanism of nucleation and growth was determined by analysis of deposition current transients as a function of potential. Figure 2 shows a series of current transients for copper deposition on TiN from 50 mM Cu(II) solution for potential steps from the open-circuit potential to deposition potentials in the range from -0.9 V to —1.5 V plotted on a semi-log plot. The nucleation and growth process is characterized by a current peak where the deposition current first increases due to the nucleation of copper clusters and three-dimensional diffusion-controlled growth, and then decreases as the diffusion zones overlap resulting in one-dimensional diffusion-controlled growth to a planar surface [3-... 

Figure 5. Theoretical estimate of the transport of propidium iodide (Pi) across an artificial planar bilayer membrane. The molecule was treated as a circular disk with charge zs = +2. Only transient aqueous pores are used in this version of the model Future versions should include metastable pores and estimates of the contributions of diffusion and convection.

We made a TAP-like reactor as an attachment for TM+ (Fig.29) to analyze the NSR reaction. A quartz tube with a 4.5mm internal diameter and a length of 3Hmm is inserted into the TAP-like reactor with TM+. The time resolution of a transient reaction analysis with TAP is lower than that with TM-t-, because the time profile of reactants and products with TAP is obtained as the result of multistcp reaction and diffusion, which arc caused by the tube reactor shape instead of the planar catalyst. The most important advantages of TAP arc to evaluate the amount of decrease of reactant gas and product quantitatively. 

Bartlett et al. presented three basic models to explain the time-dependent transient resistivity response R (t) of PPy sensors to alcohol vapors [22-24]. Two of the models assume a diffusion controlled penetration of the analyte molecules into the polymer film. More specifically, they consider bounded planar as well as bounded spherical diffusion. For bounded diffusion in a plane sheet with thickness / and a diffusion coefficient D they find [23]... 

Assuming interfacial equilibrium between the condensed phase and the adsorbed noncondensed monomer species on the surface and within the Helmholtz layer, de Levie [83] described the dissolution transients of an ordered camphor film at a stationary mercury electrode in very dilute solution (10 pM) by semiinfinite planar diffusion according to... 

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