Diffusion in a semi-infinite solid

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... 

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

This equation is valid as long as the time (t) is short enough, i.e., assuming a constant diffusivity of the species in a semi-infinite solid [17]. 

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. 

Consider the absorption of oxygen from air in the aeration of a lake or the solid surface diffitsion in the hardening of mild steel in a carburizing atmosphere. Both these processes involve diffusion in a semi-infinite medium. Assume that a semi-infinite medium has a uniform initial concentration of Cao and is subjected to a constant surface concentration of Cas- Derive the equation for the concentration profiles for a preheated piece of mild steel with an initial concentration of 0.02 wt% carbon. This mild steel is subjected to a carburizing atmosphere for 2 h, and the surface concentration of carbon is 0.7%. If the diffusivity of carbon through the steel is 1 X 10" m /s at the process temperature and pressure, estimate the carbon composition at 0.05 cm below the surface. 

Write the solution to Pick s second law for diffusion into a semi infinite solid when the concentration of diffusing species at the surface is held constant. Define all parameters in this equation. 

We now turn to a discussion of diffusion in a semi-infinite slab, which is basic to perhaps 10 percent of the problems in diffusion. We consider a volume of solution that starts at an interface and extends a long way. Such a solution can be a gas, liquid, or solid. We want to find how the concentration varies in this solution as a result of a concentration change at its interface. In mathematical terms, we want to find the concentration and fiux as a function of position and time. 

FIGURE 26.18 Theoretical temperature rise in the contact area of a pad sliding over a semi-infinite solid for different depths from the surface. Width 2b 2 mm, speed 3 m/s, pressure 2 Mp, p—l, heat conductivity 0.15 W/m/K, heat diffusivity 10 " m /s. 

The diffusion coefficients for Rb, Cs and Sr in obsidian can be calculated from the aqueous rate data in Table 1 as well as from the XPS depth profiles. A simple single-component diffusion model (9j characterizes onedimensional transport into a semi-infinite solid where the diffusion coefficient (cm2-s 1) is defined by ... 

The solution to this partial differential equation depends upon geometry, which imposes certain boundary conditions. Look np the solution to this equation for a semi-infinite solid in which the surface concentration is held constant, and the diffusion coefficient is assumed to be constant. The solution should contain the error function. Report the following the bonndary conditions, the resulting equation, and a table of the error function. 

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. 

These simple examples show that the case of diffusion into a semi-infinite medium can yield rapid answers in a relatively straightforward fashion. Furthermore, the geometry is not trivial. It can often be used to approximate finite geometries, particularly if the diffusion process is a slow one, as it is in liquid or solid media. Penetration will then be confined to short distances from the surface, at least initially, and the medium can consequently be regarded as a semi-infinite one for the short period imder consideration (note the similarity to Illustration 4.1). 

Solutions to Fick s second law can be extracted using boimdary conditions. One practically important solution is for a semi-infinite solid in which surface concentration is held constant. These conditions are met for chemical tempering when the glass is immersed in a molten salt solution (CTiapter 6) as well as for glass surface hydration (Chapter 5 diffusion depth being much less than the thickness, the two opposite surfaces can be considered separately). We shall assume that before diffusion, solute atoms are uniformly distributed with... 

Figure 14. Simple model demonstrating how adsorption and surface diffusion can co-Urnit overall reaction kinetics, as explained in the text, (a) A semi-infinite surface establishes a uniform surface coverage Cao of adsorbate A via equilibrium of surface diffusion and adsorption/desorption of A from/to the surrounding gas. (b) Concentration profile of adsorbed species following a step (drop) in surface coverage at the origin, (c) Surface flux of species at the origin (A 4i(t)) as a function of time. Points marked with a solid circle ( ) correspond to the concentration profiles in b. (d) Surface flux of species at the origin (A 4i(ft>)) resulting from a steady periodic sinusoidal oscillation at frequency 0) of the concentration at the origin.

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. 

Pick s laws also describe diffusion in solid phases. In solids transport properties can be considerably different than in liquid phases. Only one component can mobile diffuse in the matrix of the second component. At higher temperatures the diffusion coefficient can be more similar in size than in liquid phases, but the diffusion coefficient at room temperature can be orders of magnitudes smaller, e.g., D < 10 ° cm s k To overcome the time limitation one must make the diffusion length smaller. Ultra-thin layers or nanoparticles provide such small dimensions. Under such conditions the diffusion is not semi-infinite but has a restricted extension. This has to be considered in the boundary conditions. 

Solutions of the Fick law for diffusion in the a -direction may be divided into those for infinite, semi-infinite, and finite solids. The infinite solid extends to infinity in -h and —x directions, the semi-infinite solid extends from a bounding plane at x = 0 to a = +cx), and the finite solid is bounded by planes a,t x = 0, x = I and sometimes x = l + h). All these solutions are exemplified by physical systems, which will be illustrated in the text. 

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