Vacancy Concentration - Thermodynamic Argument

Vacancy defects are one of the most important point defects and occur naturally in all crystalline materials. That is, if you look at any crystalline solid at finite temperature, you will find vacancies. This suggests that it is energetically favorable for crystals to have missing atoms at lattice sites, which may seem a bit strange. Energetically, why would a crystal have a lower energy if you form a vacancy (Figure 8.6.1), when the vacancies clearly reduces the number of bonds for the surrounding atoms?

To validate this, we need to recall the concept of Gibbs free energy $\Delta G$ minimization from chemistry, which tells us that tells us about the spontaneity of a reaction: if the Gibbs free energy decreases, and when in equilibrium the change in Gibbs free energy will be zero. There are two contributions to the change in Gibbs free energy: the change in enthalpy $\Delta H$ and the temperature $T$ times the change in entropy $\Delta S$.

In a crystal, the enthalpy can be thought of as the crystals internal energy due to the formation of bonds. If bonds are broken, $\Delta H$ rises. In a way, this is how we were exploring the formation of crystal structures in Ch. 2. We were allowing atoms to interact with each other via an interatomic potential and reduce their internal energy (enthalpy) by forming bonds.

Entropy is more complex in many ways. Here, we'll relate entropy to the number of microstates, or configurations, that the atoms can have when they are in a crystalline arrangement. We call this configurational entropy: If the formation of a defect increases the number of ways that the atoms can pack, then the entropy increases.

This implies that there is some balance of enthalpy and entropy when a defect like a vacancy forms. For example, the vacancy the formed in Figure 8.6.1 required 4 bonds to break. That increases the enthalpy and $\Delta H$ is positive. At the same time, a 9-atom arrangement without the vacancy in Figure 8.6.1 only has one possible configuration. It is a low-entropy state compared to the 8-atom arrangement, which has nine possible configurations and much higher configurational entropy. We trade a bit higher enthalpy with the vacancy for a bit higher entropy. Indeed for a 9 atom system configured in a square like shown in Figure 8.6.1, we have the following number of configurations:

$$ \begin{array}{lccccccc} \hline\hline \text{Vacancies:} & 0 & 1 & 2& 3 & 4 & 5 & ...\\ \text{Configurations} & 1 & 9 & 36 & 84 & 126 & 126 & ... \end{array} $$

As we continue from 5 to 9 vacancies, we reduce the configurational entropy again. A plot of change of enthalpy, entropy, and free energy with vacancy concentration $X_\text{v}$ for a crystal with a large number of atoms the is shown in Figure 8.6.2. For each vacancy that is formed, the same number of bonds is broken, so $\Delta H$ is linear. However, $\Delta X$ is parabolic. When we sum the two contributions together, we get $\Delta G$ which dips to a minimum at some $X_\text{v}$, and then increases.

This minimum tells us that there is some minimum concertation of vacancies $X_v^e$ at which free energy is minimum. Vacancies are expected to exist in equilibrium!