Semi-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. 

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. 

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 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. 

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... 

Some transient problems tend to a trivial (and useless) steady-state solution without flux and concentration profiles. For instance, concentration profiles due to limiting diffusion towards a plane in an infinite stagnant medium always keep diminishing. Spherical and disc geometries sustain steady-state under semi-infinite diffusion, and this can be practically exploited for small-scale active surfaces. 

As a first example, the transient case with Henry isotherm can be considered. Expressions developed in Section 2.3 apply with D replacing Dm,ct m replacing cM (including the substitution of c v M by < M and cfsM by c f ) and Ku (defined as r/cM(r0,t) in both cases, i.e. with or without the presence of L) by AT i / (1 + Kc ). Other cases with analytical solutions arise from the steady-state situation. The supply flux under semi-infinite steady-state diffusion is [57] ... 

In cyclic voltammetry, simple relationships similar to equations (1.15) may also be derived from the current-potential curves thanks to convolutive manipulations of the raw data using the function 1 /s/nt, which is characteristic of transient linear and semi-infinite diffusion.24,25 Indeed, as... 

FIGURE 1.11. Convolution of the cyclic voltammetric current with the function I j Jnt, characteristic of transient linear and semi-infinite diffusion. Application to the correction of ohmic drop, a —, Nernstian voltammogram distorted by ohmic drop , ideal Nernstian voltammogram. b Convoluted current vs. the applied potential, E. c Correction of the potential scale, d Logarithmic analysis. 

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... 

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. 

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... 

Equation (3) has been solved for many cases, e.g., -> semi-infinite diffusion (subentry of - diffusion), and also in combination with the kinetic equation (potential-dependent rate) of - electrode reaction for most of the transient experimental techniques [ii-iv]. 

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. 

C In transient mass dilliision analysis, can we treat the diffusion of a solid into another solid of finite thickness (such as the diffusion of carbon into an ordinary steel component) as a diffusion process in a semi-infinite medium Explain. 

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. 

Transient mass diffusion in a semi-infinite solid... 

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)... 

Transient heat conduction or mass transfer in solids with constant physical properties (diffusion coefficient, thermal diffusivity, thermal conductivity, etc.) is usually represented by a parabolic partial differential equation. For steady state heat or mass transfer in solids, potential distribution in electrochemical cells is usually represented by elliptic partial differential equations. In this chapter, we describe how one can arrive at the analytical solutions for linear parabolic partial differential equations and elliptic partial differential equations in semi-infinite domains using the Laplace transform technique, a similarity solution technique and Maple. In addition, we describe how numerical similarity solutions can be obtained for nonlinear partial differential equations in semi-infinite domains. 

Analytical solutions for cases of temperature-dependent thermal conductivity are available [22, 23]. In cases where the solid s thermophysical properties vary significantly with temperature, or when phase changes (solid-liquid or solid-vapor) occur, approximate analytical, integral, or numerical solutions are oftentimes used to estimate the material thermal response. In the context of the present discussion, the most common and useful approximation is to utilize transient onedimensional semi-infinite solutions in which the beam impingement time is set equal to the dwell time of the moving solid beneath the beam. The consequences of this approximation have been addressed for the case of a top hat beam, p 1 = K = 0 material without phase change [29] and the ratios of maximum temperatures predicted by the steady-state 2D analysis. Transient ID analyses have also been determined. Specifically, at Pe > 1, the diffusion in the x direction is negligible compared to advection, and the ID analysis yields predictions of Umax to within 10 percent of those associated with the 2D analysis. 

Equation 6 together with Eqs. 10 and 11 describe a process of onedimensional diffusion, initiated by a change in the surrounding atmosphere so that the corresponding equilibrium concentration varies from Co to Coo-Equation 10 requires that immediately after the pressure step, the concentration at the boundary (namely for y = 0) assumes the new equilibrium value. This means that the existence of additional transport resistances at the surface of the system is excluded. The second term in Eq. 11 indicates that the process has to proceed as in a semi-infinite medium. This means in particular that the transient adsorption or desorption profiles originating from different crystal faces must not yet have met each other. 

At the interface, the slopes for the Fe " and Fe concentration profiles have opposite signs. For an oxidation half-reaction, the slope for Fe is positive, while that for Fe " is negative, with the direction of the r -axis pointing from the electrode towards the electrolyte (I>0). Figure 4.15 illustrates the shape of the Fe + and Fe + concentration profiles, at a given instant, when the thickness of the diffusion layer is low compared to the dimensions of the overall system. For instance, this applies to situations involving semi-infinite mass transport in a transient state (see section 4.3.1.3). 

Example of a transient state semi-infinite diffusion... 

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