Infinite transient diffusion

An integration constant has been voluntarily added for allowing this relation to be inverted, as explained earlier in the classical approach, in order to retrieve Equation G7.5. This integration constant is the unperturbed concentration far from the place where the diffusional process occurs, meaning that this model applies to infinite (or semi-infinite) transient diffusion. 

Warburg s impedance When infinite transient diffusion is characterized by AC techniques, such as impedance spectroscopy in electrochemistry, the electrical complex impedance measured by imposing a periodic signal with a constant angular frequency is called Warburg impedance. 

For the semi-infinite transient diffusion problem specified by Equation 4.19, the general solution is given by... 

Transient Interdiffusion in Two Semi-Infinite Bodies The transient diffusion problem illustrated in Figure 4.8, which involves the interdiffusion of two semi-infinite bodies in contact with one another, is closely related to the previous semi-infinite transient diffusion problem. In fact, if you consider just one-half of the problem domain (e.g., consider the evolution of the diffusion profiles for species A for X > 0), diffusion proceeds exactly like the previous semi-infinite diffusion problem. The only difference is that in this case the interfacial concentration of species A is assumed to be pinned at half of its bulk (i.e., pure material A) value. 

Two rather similar models have been devised to remedy the problems of simple film theory. Both the penetration theory of Higbie and the surface renewal theory of Danckwerts replace the idea of steady-state diffusion across a film with transient diffusion into a semi-infinite medium. We give here a brief account of... 

Unlike macroelectrodes which operate under transient, semi-infinite linear diffusion conditions at all times, UMEs can operate in three diffusion regimes as shown in the Figure for an inlaid disk UME following a potential step to a diffusion-limited potential (i.e., the Cottrell experiment). At short times, where the diffusion-layer thickness is small compared to the diameter of the inlaid disc (left), the current follows the - Cottrell equation and semi-infinite linear diffusion applies. At long times, where the diffusion-layer thickness is large compared to the diameter of the inlaid disk (right), hemispherical diffusion dominates and the current approaches a steady-state value. 

The characteristic diffusion time for any UME geometry where the transition from semi-infinite linear diffusion (transient) to hemispherical or spherical diffusion (steady state) occurs may be given as... 

In section 2.5.3 it was shown that the differential equation for transient mass diffusion is of the same type as the heat conduction equation, a result of which is that many mass diffusion problems can be traced back to the corresponding heat conduction problem. We wish to discuss this in detail for transient diffusion in a semi-infinite solid and in the simple bodies like plates, cylinders and spheres. 

We will consider transient diffusion of a substance in a semi-infinite body B. At time t = 0, substance A is stored in the body at a concentration cAa. The desired concentration profile cA = cA(x,t) satisfies the following differential equation, under the assumption cD = const... 

In transient diffusion in a semi-infinite solid with stepwise change in the surface concentration, we find from (2.126)... 

In Section 5.4.2(a) we developed the idea that 8 expresses the ratio of the electrode s radius of curvature to the diffusion-layer thickness. When 8 1, the diffusion layer is small compared to tq, and the system is in the early transient regime where semi-infinite linear diffusion applies. When 6 << 1, the diffusion layer is much larger than tq, and the system is in the steady-state regime. 

This condition implies that iRy < 5 mV, where i = ///2 and is given by the limiting form of (5.9.2). If the sampled-current voltammetry is based on semi-infinite linear diffusion (i.e., on early transients), then i is half of the Cottrell current for sampling time r, and the condition becomes... 

As outlined in section 2.2.1.1, here the diffusion layer thickness depends mainly on the time factor, and the diffusion coefficient of Fe. Section 4.3.1.3 provides more quantitative data on the different values of diffusion layer thickness in a semi-infinite transient state. 

Anomalous diffusion describes a transport process analogous to transient diffusion which does not follow the classical model based on Pick s law. The dependence with the frequency of the electrical impedance featuring this transfer, which is normally 1/2 for the Fickian diffusion in infinite space, is different from this value. The same is true for the degree... 

Transient diffusion in (semi-) infinite space Mixing diffusion and continuity in equal proportions (by complementing paths). 

As shown in both the experimental and theoretical transients, the currents first increase as a result of the formation ofmercury drops and their growth controlled by radial diffusion, decaying at longer times due to the onset of semi-infinite planar diffusion. The diffusion coefficients may be obtained either from i versus plots... 

Transient Semi-Infinite Diffusion The simplest transient diffusions problems are generally those that involve semi-infinite or infinite boundary conditions. Consider, for example, the situation illustrated in Figure 4.6, which represents diffusion of a substance from a surface into a semi-infinite medium. 

When solving Pick s second law for any specific problem, the first step is always to specify the boundary and initial conditions. For the semi-infinite diffusion process illustrated in Figure 4.6 as an example, the concentration of species i is initially constant everywhere inside the medium at a uniform value of c°. At time f = 0, the surface is then exposed to a higher concentration of species i (c ), which causes i to begin to diffuse into the medium (since c > c"). It is assumed that the surface concentration of species i is held constant at this new higher value c during the entire transient diffusion process. Based on this discussion, we can mathematically specify the boundary and initial conditions as follows ... 

In analogy to the reaction half-life that was discussed in Chapter 3, we can specify a diffusion half-depth in transient diffusion problems (S1/2) which is the spatial position at which the concentration of the diffusing species reaches half of its surface value. As an example, the diffusion half-depth for the semi-infinite diffusion process in Equation 4.26 can be obtained as... 

FIGURE 4.10 (a) The transient infinite diffusion of a rectangular eoneentration profile of thickness 21 can be obtained by subtracting the solutions for two semi-infinite step functions located atx = —I and x = +1, respectively, (b) IllusiTation of the subtraction of the two semi-infinite step functions as they evolve with time, yielding the correct evolution of the transient diffusion profile for the rectangular source. 

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. 

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. 

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 ... 

Importance of mutual diffusion Phase separation, polymer blend processing Relation to interaction parameter Transient diffusion Pick s law of diffusion Damped wave diffusion and relaxation Semi-infinite medium, finite slab Periodic boundary condition... 

Initial conditions may also be complex. Suppose, for example, that we wish to solve Pick s equation for transient diffusion into a sphere. The simplest case here is to assume that the sphere is initially "clean," i.e., contains no solute. But what if it is not We would then have to specify an initial concentration distribution Cf=o =/(r,0,(p) and this distribution would have to be entered into the solution process as an initial condition. Evidently there are an infinite number of such distributions hence Pick s equation for this case will have an unlimited niunber of solutions. 

Transient diffusion in a semi-infinite domain, into a solid bounded by parallel planes, in a sphere, and in the radial direction of a cylinder. The domain is assumed to be initially uniformly loaded or uniformly clean, and to have a constant surface concentration. We term these Nonsource Problems. Occasional departures from the stated assumptions are noted as they occur. 

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