Infinite transient diffusion profile

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. 

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. 

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. 

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... 

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. 

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). 

The penetration theory in its simplest form represents the case of transient molecular diffusion into a semi-infinite medium It can be applied to real situations if hydrodynamic conditions exist for which such an assumption is approximately valid This would be the case if flow close to the interface is laminar, concentration profiles there are practically nonaal to the interface and time of contact of the phases is reasonably short ... 

Transient Infinite Diffusion of an Arbitrary Concentration Profile The... 

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