4 web view chapter 5 diffusion in solids 5.1 diffusion in nuclear processes 1 5.2. macroscopic view

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Chapter 5 Diffusion in Solids

15.1 Diffusion in Nuclear Processes

25.2. Macroscopic View of Diffusion

2Species Conservation

3Fick’s Laws

45.3 Useful Mathematical Solutions

4Constant-Source Method

7Instantaneous Source Method

7Diffusion in Finite Solids

9Fission Gas Release from Nuclear Fuel

105.4. Atomic Mechanisms of Diffusion in Solids

15The Einstein Equation

17The Vacancy Mechanism in Metals

195.5. Types of Diffusion Coefficients

20Relations between the Types of Diffusion Coefficients

215.6. Diffusion in Ionic Crystals

22The NaCl-type Structure with Schottky Defects

245.7. Diffusion in the Fluorite Structure of UO2

24Oxygen Diffusion

27Uranium Diffusion

28Uranium Self-Diffusion in UO2

32Interdiffusion in Mixed Ionic Solids with the Fluorite Structure

355.8. Thermal Diffusion

38Appendix 5A Dimensionless Variables and the Similarity Transformation Solution to Eq s (5.4) – (5.7)

40Appendix 5B Laplace Transform Solution to Eqs (5.16) – (5.18)

42Problems

46References

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).

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:

Qdx

dx

x

J

J

J

)

cdx

(

t

+

÷

ø

ö

ç

è

æ

+

-

=

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

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.

Fick’s Laws

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

x

c

D

J

-

=

(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

c

D

x

t

c

+

÷

ø

ö

ç

è

æ

=

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

c

D

t

c

2

2

+

=

(5.3a)

If the concentration is nonuniform in the directions transverse to x, additional second derivative terms are required on the right hand side. However, mathematical solutions of multidirectional diffusion equations are considerably more complicated than those involving only one spatial dimension. Analogous equations for cylindrical and spherical geometry, involving one direction only, are:

Cylindrical geometry:

Q

r

c

r

r

1

D

t

c

+

÷

ø

ö

ç

è

æ

=

(5.3b)

Spherical geometry:

Q

r

c

r

r

1

D

t

c

2

2

+

÷

ø

ö

ç

è

æ

=

(5.3c)

5.3 Useful Mathematical Solutions

There are a number of analytic solutions to the time-dependent, one-spatial-dimension diffusion equation. Compendiums of such solutions are contained inn books by Carslaw and Jaeger (1) and Crank (2). Diffusion problems that are not amenable to closed-form solutions can be solved by numerical techniques¸ for which numerous computer codes are available.

Constant-Source Method

Two common methods of measuring diffusivities in solids that are amenable to analytic solutions are presented in this section. The first, depicted in Fig. 5.2, represents two closely related common techniques. The sketch in the upper left shows the so-called surface source method, in which a layer of material containing the diffusing solute is deposited on a thick block of pure solid. The solute in the surface layer can dissolve and diffuse into the solid substrate. In the diagram on the u

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