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Chapter 5 Diffusion in Solids

5.1 Diffusion in Nuclear Processes.............................................................................. 1 5.2. Macroscopic View of Diffusion ............................................................................. 2

Species Conservation .............................................................................................. 2 Fick’s Laws .............................................................................................................. 3

5.3 Useful Mathematical Solutions .............................................................................. 4 Constant-Source Method ......................................................................................... 4 Instantaneous Source Method ................................................................................. 7 Diffusion in Finite Solids........................................................................................... 7 Fission Gas Release from Nuclear Fuel................................................................... 9

5.4. Atomic Mechanisms of Diffusion in Solids ........................................................ 10 The Einstein Equation ............................................................................................ 15 The Vacancy Mechanism in Metals........................................................................ 17

5.5. Types of Diffusion Coefficients........................................................................... 19 Relations between the Types of Diffusion Coefficients .......................................... 20

5.6. Diffusion in Ionic Crystals ................................................................................... 21 The NaCl-type Structure with Schottky Defects ..................................................... 22

5.7. Diffusion in the Fluorite Structure of UO2 .......................................................... 24 Oxygen Diffusion.................................................................................................... 24 Uranium Diffusion................................................................................................... 27 Uranium Self-Diffusion in UO2................................................................................ 28 Interdiffusion in Mixed Ionic Solids with the Fluorite Structure ............................... 32

5.8. Thermal Diffusion................................................................................................. 35 Appendix 5A Dimensionless Variables and the Similarity Transformation Solution to Eq s (5.4) – (5.7).................................................................................... 38 Appendix 5B Laplace Transform Solution to Eqs (5.16) – (5.18) ........................ 40

Problems ...................................................................................................................... 42 References ................................................................................................................... 46

Light Water Reactor Materials, Draft 2006 © Donald Olander and Arthur Motta 1 8/30/2009 1

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5.1 Diffusion in Nuclear Processes

First, we should ask, what is the meaning of the term diffusion in the solid state? 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 5.1. Table 5.1 Solid-State Diffusion Processes in Nuclear Materials

Process Diffusing Species Host Solid Corrosion of cladding: -By water (normal operation) - By steam (severe accident)

O2- O

ZrO2 Zr

Hydriding of cladding H Zr Fission gas release from fuel or bubble formation in fuel

Xe, Kr UO2

Sintering and creep of fuel U4+ UO2

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).

Light Water Reactor Materials, Draft 2006 © Donald Olander and Arthur Motta 1 * Diffusion occurs in gases and liquids, but our interest is exclusively in solid-state diffusion

8/30/2009 1

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5.2. 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.

Species Conservation

Conservation of a species whose volumetric concentration is c atoms (or moles) per unit volume is shown in Fig. 5.1. The diagram shows a volume element that

Fig. 5.1 Species conservation in a differential volume

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:

Qdxdx x JJJ)cdx(

t +⎟

⎠ ⎞

⎜ ⎝ ⎛

∂ ∂

+−= ∂ ∂

or

Q x J

t c

+ ∂ ∂

−= ∂ ∂

(5.1)

where t = time x = distance Q = source term of the diffusing species, atoms (or moles) per unit volume

Light Water Reactor Materials, Draft 2006 © Donald Olander and Arthur Motta 2

This conservation statement applies no matter what force is driving the flux J. The most common forces is a chemical potential gradient in the x direction. However temperature, and electric field gradients can also cause a particle flux. The three forces above lead, respectively, to fluxes describing ordinary molecular diffusion, thermal diffusion, and ionic transport.

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Fick’s Laws

When the concentration gradient drives J, the flux is given by Fick’s First Law:

x cDJ

∂ ∂

−= (5.2)

This equation follows the universal observation that matter diffuses from regions of high concentration to regions of low concentration, hence the minus sign.. The flux J and the concentration gradient are in principle measurable quantities, so Eq (5.2) effectively defines the diffusion coefficient D. The definition is not the only one possible; for example, J could have been considered to be proportional to the square of the concentration gradient. The reason that Eq (5.2) is the appropriate definition is that the quantity D is a function of temperature and concentration only, but not of the concentration gradient. Any other flux – concentration gradient relation would not have this essential property. The units of D are length squared per unit time, usually cm2/s provided that J, c, and x are in consistent units. Substituting Eq (5.2) into Eq (5.1) gives Fick’s Second Law:

Q x cD

xt c

+⎟ ⎠ ⎞

⎜ ⎝ ⎛

∂ ∂

∂ ∂

= ∂ ∂

This equation is also called the diffusion equation, by analogy to its heat transport counterpart, the heat conduction equation. In the common case of an isothermal system and D independent of solute concentration(and hence of x), the diffusion equation simplifies to:

Q x

cD t c

2

2 +

∂

∂ =

∂ ∂