First, we should ask, what is the meaning of the term diffusion? In its most general sense, it is the movement of foreign, or impurity atoms (generally referred to as solute species) with respect to the atoms of the host crystal*. The flow of solute atoms is called a flux, although strictly speaking, it is a current. In either terminology, it represents the number of atoms that pass a plane of unit area per unit time. The flux of solute atoms is driven by some nonuniformity or gradient, generically referred to as a force. The most common driving force is a nonuniformity of the concentration of the solute atoms, or a concentration gradient. Other forces can result in movement of solute atoms relative to the host crystal. These include a temperature gradient and an electric field gradient.
We concentrate almost exclusively on diffusion generated by a concentration gradient, referred to as ordinary, or molecular diffusion. Diffusion can occur in two or three dimensions. The most common is 3D diffusion, or migration of solute atoms in the bulk of a solid. 2D diffusion occurs on the surfaces of solids or along internal surfaces that separate the grains of polycrystalline solids. This is termed grain boundary diffusion. Molecular diffusion controls the rate of many important chemical and physical processes that take place in a nuclear fuel rod. A few are summarized in Table 4.1.
In the fast-neutron and gamma field inside a reactor core, many (but not all) diffusion processes are accelerated. This mobility enhancement results from the point defects (Frenkel pairs) created in copious quantities by collisions of the energetic particles with the host atoms. In addition to enhancing mobility of atoms in the solid, the point defects also diffuse. This motion is responsible for agglomeration of vacancies into voids and self interstitials into disks called loops. The presence of these large defects in the solid profoundly affects the mechanical and dimensional properties of the structural metals in which they form. Self-interstitial diffusion exploits the preferred orientation (texture) of Zircaloy to produce the phenomena of radiation growth (in the absence of stress) and irradiation creep (with stress present).
Macroscopic View of Diffusion
Just as thermodynamics can be described from a macroscopic, classical viewpoint or in a microscopic, statistical setting, so can the process of diffusion. The macroscopic laws of diffusion are combinations of a species conservation equation with a mathematical specification of the flux of the solute relative to the host substance.
Conservation of a species whose volumetric concentration is c atoms (or moles) per unit volume is shown in Fig. 4.1. The diagram shows a volume element that is unit area and dx thick. The flux of diffusing species, J, is the number of atoms (or moles) crossing the unit plane per unit time. There may also be a source or sink of the species inside the volume element. The statement of species conservation is: time rate of change of atoms (or moles) in the volume element = net influx of species + creation of the species in the volume element.
In mathematical terms, this word statement is:
t = time
x = distance
Q = source term of the diffusing species, atoms (or moles) per unit volume
This conservation statement applies no matter what force is driving the flux J. Among the common forces are gradients in the x direction of the solute concentration, the temperature, and the electric field. These lead to fluxes describing ordinary molecular diffusion, thermal diffusion, and ionic transport, respectively.