1. Point Defects in Crystalline Solids

2. Point Defects
The properties of crystalline materials are heavily influenced by the presence of “defects”
In this chapter we will discuss:
Point defects
Impurities
Diffusion

3. Introduction
In crystalline materials, it is IMPOSSIBLE to have a “perfect crystal”
On a macroscopic scale, we can see cracks, for example, which are generally cause for concern for structural integrity

4. Introduction
On a microscopic scale, there are also imperfections, called defects, that can only be seen with electron microscopes.
Some defects, such as vacancies, are unavoidable.

5. Categories of Defects “Zero-dimensional” point defects:
Vacancies, interstitials, and impurity atoms.
1-dimensional defects – dislocations
2-D defects – grain boundaries
3-D defects – precipitates, inclusions

6. Point Defects - crystals

7. Production of Point Defects
Since moving atoms away from their equilibrium position requires an increase in the internal energy of the system (enthalpy, H), why do point defects exist?
Because we must consider Gibbs free energy (G) which is a balance between the enthalpy and the randomness (entropy, S) of the system

8. Production of Point Defects
We can predict mathematically the number of defects, e.g. vacancies, via the Arrhenius expression:
Where
NV = number of vacancies at temp T(K)
NT = total number of lattice sites
Qfv = activation energy for vacancy formation, units of J/mole
R = 8.31 J/mol-K
CV = concentration of vacancies

9. Point defects – ionic solids
Schottky defect: electrically neutral cation-anion vacancy cluster
Ex: NaCl

10. Point defects – ionic solids
Frenkel defect: electrically neutral vacancy-interstitial pair
Frenkel defects can also consist of anion vacancy/anion interstitial pairs, but these are less common than cation vacancy, cation interstitial pairs.
Why?
Ex: MgO

11. Impurities in Crystals
Solvent atoms: the predominant atomic species
Solute atoms: the impurity atoms
Solute atoms can be interstitial or substitutional

12. Impurities in Crystals
In order for solute atoms to be substitutional, they must obey the Hume-Rothery rules

13. Hume-Rothery Rules
The Hume-Rothery rules for substitutional solid solution:
There must be less than ~15% difference in atomic radii.
Atoms must have same crystal structure.
Atoms must have similar electronegativities.
Atoms must have the same valence.
If one or more of the Hume-Rothery rules are violated, only partial solubility is possible.

14. Example of solid solution
Au-Ag is an example of a solid solution:
Atomic radii: nm and nm
Electronegativities: (1.90/2.54)
Valences for Au and Ag: +1
Both metals have FCC structure and almost identical lattice parameters (0.408 nm)
The Hume-Rothery rules thus predict mutual solubility for Ag and Au

15. Example of solid solution
Au-Ag solid solution:
50% gold, 50% silver – 12-karat gold
100% gold – 24-karat gold

16. Impurities in Ionic Crystals
Solute atoms in IONIC crystals can be either cations or anions.
Cation impurities, being generally smaller, can be in both substitutional (i.e. replacing a cation in its normal cation site) or interstitial positions.
Anions however are usually too large to fit into interstitial positions, so they are mainly substitutional

17. Impurities in Ionic Crystals
Charge neutrality needs to be maintained for both vacancies and impurities in ionic crystals.
For example, one missing Ti4+ cation in calcium titanate requires there to be two anion vacancies (since oxygen only has a charge of -2).
If one Na+ substitutes for one Ca2+ there is a need for ½ of an oxygen vacancy. Thus each 2 Na+ impurity atoms result in one oxygen vacancy.

18. Intro to Diffusion
Point defects are able to move around within a crystal. One method by which atoms or molecules move is known as diffusion.
Diffusion: a mass transport process involving the movement of one atomic species into another.

19. Diffusion, carburization
The carburization of steel is an example of a diffusion process.
Carburization: the process of increasing the carbon content of steels in the near-surface region in order to increase the wear-resistance of the steel. It is done for applications where the part experiences friction – e.g., gears, crankshafts.

20. Diffusion, carburization
During carburization, steels are heated to high temps in a CO/CO2 atmosphere, which deposits carbon on the surface of the steel. With time and temperature, the carbon can diffuse and form a carbon-rich layer near the surface.

21. Diffusion, carburization
The microstructure of carburized steel – note diffusion layer of carbon on left side (near surface)

22. Fick’s First Law Consider two adjacent atomic planes of atoms A and B.
Diffusion can be modeled as the jumping of atoms from one plane to another.
The net number of A atoms moving from plane 1 to plane 2 per unit area and unit time is called the diffusion flux, J, with units of atoms/cm2-s.

23. Fick’s First Law Where: J = flux, atoms/cm2-s
C = concentration (e.g., wt%)
x = distance
D = diffusion coefficient
Fick’s first law assumes the concentration gradient is constant

24. Diffusion Coefficient
The diffusion coefficient, D, obeys the Arrhenius relationship:
In plot of ln D vs. 1/T, the slope is
-Qv/R

25. Mechanism of Diffusion
One example of how diffusion works is when an atom swaps places with a vacancy, getting closer to where it wants to be, and the vacancy moves back the other direction.

26. Mechanisms of Diffusion
There are two types of diffusion:
Interstitial
Substitutional
Interstitial diffusion – e.g. the carburizing of steel
Diffusion in substitutional solid solution (e.g. Au/Ag) – requires vacant lattice site
In general, Qi < Qv – recall Q is activation energy

27. Self-Diffusion Coefficients
Self-diffusion coefficients are measured via radioactive tracers
E.g., Ni:
D0 = 1.30 x 10-4 m2/s
Q = 279 kJ/mol

28. Impurity Diffusion Coefficients
Impurity diffusion coefficients are easier to measure, since there is a difference in chemistry between materials.
Examples:
C in BCC Fe: 2.00 x 10-6 m2/s; Q = 84 kJ/mol – Ferrite
C in FCC Fe: 2.00 x 10-5 m2/s; Q = 142 kJ/mol – Austenite

29. Impurity Diffusion Coefficients
Why is the activation energy bigger for C in FCC Fe than for C in BCC Fe?
BCC Packing Factor: 0.68
FCC Packing Factor: 0.74
Harder for C to squeeze through

30. Fick’s Second Law
We recall that Fick’s first law assumed that the concentration gradient was independent of time.
How can we predict the rate at which the concentration of an atomic species varies with time and position? We need to be able to predict how dC/dt changes as a function of distance.

31. Fick’s Second Law Fick’s second law states that:
There are various solutions for C(x,t) that depend on the initial and boundary conditions for the differential equation

32. Summary
We have learned about diffusion and Fick’s first and second laws. We now know how to predict the rate of movement of impurity species in a solid material. This is an important skill with wide-reaching applications.