Point Defects in Crystalline Solids

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 http://www.techfak.uni-kiel.de/matwis/amat/def_en/kap_2/illustr/t2_2_1.html

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 http://www.mse.uiuc.edu/info/mse182/t142.html

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 http://www.mse.uiuc.edu/info/mse182/t143.html

11. Impurities in Crystals • Solvent atoms: the predominant atomic species • Solute atoms: the impurity atoms • Solute atoms can be interstitial or substitutional http://www.doitpoms.ac.uk/tlplib/dislocations/dislocations_in_2D.php

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: 0.144 nm and 0.144 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) http://vulcan2.case.edu/groups/ernst/carburization.html

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.