2.2.1.1 Point Defects in Metals/Alloys

Point (or zero-dimensional) defects are associated with imperfections involving an atom or only a few atoms in a localized region. They are often described as “zero­dimensional” defects. There are many types of point defects that one should be aware of. They are schematically shown in Figure 2.25:

Figure 2.25 Various point defects schematically shown on a 2D crystal lattice (1 — monovacancy, 2 — self-interstitial, 3 — interstitial impurity atom, 4 — undersize substitutional atom, and 5 — oversize substitutional atom).

Vacancies

Vacancy is simply a vacant lattice site. Vacancies are present in all crystalline solids, pure or impure, under almost all conditions, predictable by the laws of thermo­dynamics. Vacancies were first imagined to explain the diffusion phenomenon in solids (see Section 2.3). Here we develop a derivation for calculating the equili­brium vacancy concentration. A basic knowledge of thermodynamics is required to understand this approach. We assume that the process of vacancy formation must obey dG = 0 for maintaining a thermodynamic equilibrium condition. One can clearly note here that due to the existence of vacancies, the enthalpy and the entropy of the crystal would be greater than it would have been without the vacancies. Thus, we can write from the definition of the Gibbs free energy,

DGv = DHv — TASv, (2.2)

where DGv is the Gibbs free energy change, DHv is the enthalpy increase, and DSv is the entropy increase due to the presence of vacancies in the crystal.

Therefore, if G(o) is the Gibbs free energy of a perfect lattice and G(n) is the Gibbs free energy of the lattice with n number of vacancies, we can simply write

G(n) — G(o) = AGv. (2.3)

Now we definitely need some discussion on the entropy term. The entropy term is broadly categorized in two ways: vibrational entropy and configurational entropy. The vibrational entropy term is readily intuitional in that the atoms present in the neighborhood of a vacancy are less restrained than the atoms in the perfect portion of the crystal. Thus, each vacancy can provide a very small contribution to the total entropy of the crystal due to the more irregular or ran­dom vibration. Although a detailed theoretical treatment of vacancies would require consideration of the small vibrational entropy contribution, it is gener­ally not considered in this type of derivation as it is of secondary importance. Instead, we should take into account the effect of configurational entropy that arises due to the probabilistic nature of the vacancy creation process. If N is the total number of lattice atoms, there are W different ways of arranging the N atoms and n vacancies on (N + n) lattice sites. Hence, W can be given by the following expression following the combinatorial rules:

where N! = N(N — 1)(N — 2) … 3-2-1, and so forth. On the other hand, the config­urational entropy term is given by Sconf = kln W, where k is the Boltzmann’s con­stant. This constant appears in the science equations quite a bit and has the value on the order of the thermal energy per atom.

Thus, Eq. (2.2) becomes

AGv = ngv — kT ln W’

where gv is the Gibbs free energy associated with forming a vacancy. For thermal equilibrium, the following relation should hold:

@(AGV) @(ln W)

dn gv dn

For large numbers of N and n (in reality their values are in millions), Stirling’s approximation ln(N!) = Nln(N) — N can be used to show that @(ln W)/dn — ln ((N + n)/n).

Thus, Eq. (2.6) becomes

Since N ^ n, we can also write

(2.7)

where Cv is defined as the equilibrium vacancy concentration.

We know from the definition of Gibbs free energy,

gv — Hv — TSv.

So, Eq. (2.7) becomes

Cv — exp (— І)65*© • (2’8)

The above equation presents the compromise between the enthalpy (i. e., energy, Ev, in this case as PV term is negligible, that is, Hv « Ey) and entropy. The vacancy formation energy can also be defined as the energy needed to remove one atom from the lattice and place it on the crystal surface. However, it does not give us any indication of how much time it would take to accomplish it. That is why the ther­mal vacancy formation is guided by the thermodynamic principles, not kinetic ones. It is clear from this that a large vacancy concentration is favored with a decrease in the vacancy formation energy, whereas a large entropy of vacancy for­mation tends to increase the vacancy concentration. Furthermore, it is not only the vacancy formation energy alone that is solely important, but the ratio of Ev to ther­mal energy is also important. At higher temperatures, the thermal energy is high causing a significant probability of strong thermal fluctuations leading to the for­mation of vacancies. Conversely, at low temperatures, the probability of large ther­mal fluctuations is so low that less number of vacancies is created. So, the vacancy concentration strongly depends on the temperature and would increase exponen­tially. In other words, the term exp(—Ev/kT) represents a probability term giving

the chances that a crystal with thermal energy (kT) has to create thermal fluctua­tions sufficient enough to provide the energy needed to produce vacancies. It has been found that the contribution of exp(Sv/k) is very small at all temperatures com­pared to the exp(—Ev/kT) contribution, and hence will not be considered further. Hence, Eq. (2.7) can also be written as

Cv = exp (- І) • (2’9)

The results of experimental and theoretical studies have shown that the vacancy formation energies are typically on the order of 1 eV. There are several methods of measuring vacancy concentration — one being the electrical resistiv­ity measurement. The electrical resistivity of a metal generally increases because of the presence of vacancies, and the change in resistivity is proportional to the vacancy concentration. The vacancy formation energy is generally obtained from the slope of the semilog plot of Cv versus 1/T. Vacancies play a major role in the diffusion processes and thus affect various phase transformation, deformation, and physical processes. Sometimes, the nonequilibrium concentration of vacan­cies can be sustained at room temperature by heating a metal followed by quenching (i. e., fast cooling). Quenching ensures that the high concentration of vacancies characteristic of higher temperature is retained at room temperature without being depleted (as the migration of the vacancies become slower at lower temperatures).

The type of vacancies that we discussed previously should be called “monova­cancy” since only one (i. e., mono) lattice atom is missing (Figure 2.26a). At very high temperatures, many vacant lattice sites find the neighboring sites vacant, too.

If two vacancies come side by side, a “divacancy” is formed (Figure 2.26b). The divacancy formation energy (E®) can be expressed as (2Ev — B), where B is the binding energy of a divacancy (this energy is basically the energy required to sepa­rate a divacancy into two isolated monovacancies). It is generally hard to measure or calculate the binding energy. One estimate for copper has shown values of the

Figure 2.26 Configuration of (a) a monovacancy and (b) a divacancy.

binding energy of a divacancy in the range of 0.3-0.4 eV. The equilibrium divacancy concentration (C®) is given by the following equation:

(С21) — b exp ; (2.10)

where b is the coordination number. Equation (2.10) can also be expressed in the following form:

Self-Interstitial Atom (SIA)

A self-interstitial atom is a type of point defect where a lattice atom occupies an interstitial site instead of its regular position (recall the interstitial space as dis­cussed in Section 2.1). For example, if a copper atom is in one of the interstitial positions, a self-interstitial type of point defect would be created. It can also be thought of as removing one atom from the crystal surface and moving it to an inter­stitial site. Any interstitial site is smaller than its own atom size, and the presence of such an interstitial atom could badly strain the lattice surrounding it. As a result, the formation energy of an SIA (£;) in copper even under equilibrium conditions is quite high (~4eV) compared to the similar quantity for a monovacancy (~1 eV). The equilibrium concentration of SIA (C;) is given by

(2.12)

where ni is the number of interstitial atoms and N; is the number of interstitial sites. Thus, thermal energy (kT) is not sufficient to create self-interstitials. The self­interstitials can be generated more readily under energetic particle radiation, such as fast neutron irradiation. Furthermore, the actual configuration of SIAs could be much different from the simple model that we just discussed (see Chapter 3).

■ Example 2.4

Calculate the concentrations of thermal monovacancies, divacancies, and self-interstitials in copper (FCC crystal structure, coordination number 12) at 20 °C, 500 °C, and 1073 °C. Comment on the results. Assume Ev — 20 kcal mol-1, B — 7 kcal mol-1, and £; — 90 kcal mol-1.

Solution

Since the activation energies are given as per mole, it is convenient to use the gas constant R (—1.987 cal mol-1 K-1) rather than Boltzmann’s constant. We use the above formulations to evaluate the concentration of monovacancies (C), divacancies (C2)), and self-interstitials (C;). It will be easier to use a standard spreadsheet software in which you can write in the equations, and then calculate for all three temperatures. The results are summarized in the table below.

T (in °C/K)	Cv	cv2)	Ci
20/293 500/773 1073/1346	1.20 x 10-15 2.21 x 10~6 5.65 x 10~4	1.45 x 10~24 2.80 x 10~9 2.63 x 10~5	7.30 x 10~68 3.57 x 10~26 2.43 x 10-15

Note that the melting temperature of copper is 1083 °C. So, if the temper­ature rises to that level or above, the concentration of point defects will lose their physical meaning as the liquid state of copper would not contain any crystal point defects.