Vacancy Concentration - Arrhenius Behavior.

Arrhenius behavior is something that you may have encountered in a chemistry class. It tells us about a rate constant - essentially how the rate and direction of a reaction. In chemistry, you may have learned to compute a reaction rate based on absolute temperature $T$ and the activation energy of a reaction, $E_a$.

We're interested in chemical reactions in materials science as well, but we're also interested in the rates of other reactions with activation energies, including defect formation energies (like vacancies) and diffusional jumps (like interstitial- or vacancy mediated diffusion). As all of these phenomenon occur due to some activation energy and increase with temperature, we can them using a generalized Arrhenius behavior:

$$k = A \exp{\frac{-E_a}{RT}} \tag{8.7.1}$$

Where $k$ is the rate of a reaction, $A$ is a temperature independent pre-factor (more on this later), and $R = 8.314$ J/K/mol is the universal gas constant. (Sometimes $R$ is replaced with the Boltzman constant $k_{\mathrm{B}}$ if we are interested in energy per atom or molecule instead of mole.).

At values in which most physical system of interest operate, where $E_a >> RT$ (low temperatures) the rate constant $k$ increases exponentially with $T$. In this regime the Arrhenius equation - which is largely an empirical formula - does well to describe many temperature-dependent activation processes. At values where $E_a << RT$, which are very high temperatures, the Arrhenius equation approaches the value of $A$ - leveling off. This regime is not of interest in most physical systems of interest to materials science - the model breaks down. For example, when modeling vacancy populations, temperatures associated with the "leveling-off" of the Arrhenius function are well above melting temperatures and so are not useful.

When modeling vacancy concentrations, we use a form of the Arrhenius equation that provides us with the vacancy concentration by finding the equilibrium balance between the formation of vacancies and the destruction of vacancies. An approach to this analysis can be found here, but we'll simply use the result:

$$\frac{N_{\mathrm{v}}}{N} = \exp{\frac{-Q_{\mathrm{v}}}{k_{\mathrm{B}}T}} \tag{8.7.2}$$

where $N_{\mathrm{v}}$ is the number of vacancies per unit volume, $N$ is the number of potential vacancy sites per unit volume (or the number of atoms per unit volume), and $Q_{\mathrm{v}}$ is the formation energy for a vacancy.

We can rearrange Eq. 8.7.2 to give us the vacancy concentration $N_{\mathrm{v}}$:

$$N_{\mathrm{v}}= N\exp{\frac{-Q_{\mathrm{v}}}{k_{\mathrm{B}}T}} \tag{8.7.3}$$

Exponential equations are difficult to plot and work with, but we can create a linear plot from this equation by taking the natural log of each side of the equation (remember our logarithmic identities) and then plotting $\ln{N_{\mathrm{v}}}$ vs $\frac{1}{T}$. This linearizes (in the form of $y = b-mx)$ the equation such that the natural log of the vacancy concentration $\ln{N_{\mathrm{v}}}$ will change linearly with independent variable $\frac{1}{T}$:

$$\ln N_{\mathrm{v}} = \ln N - \frac{Q_{\mathrm{v}}}{k_{\mathrm{B}}}\frac{1}{T} \tag{8.7.4}$$

We can then plot $\ln{N_{\mathrm{v}}}$ vs $\frac{1}{T}$ and get a linear plot. We call this an Arrhenius plot.

Great - we now have a good way that we can change temperature, measure vacancy concentration, and get the formation energy of vacancies. Why would we ever want to do that?

Well, we know that vacancies are important. They exist in nuclear reactors and can lead to serious nuclear events. They influence the electronic properties of materials - indeed, we control these defects closely in the transparent conducting oxides used in personal electronic devices and solar panels. (If you click on that link you'll see some Kröger–Vink notation showing how vacancies lead to free electrons in some transparent conducting oxides.) They're central to vacancy-mediate diffusion behavior and will dictate diffusion rates and therefore processing conditions, heavily influencing the way we process semiconductors and metal alloys. Their existence ends up dictating how the mechanical behavior of creep occurs at high temperatures in steam turbines - the technology behind 80% of our power production in the United States. The presence of vacancies affect degrediation rates of materials, for example in biomedical implant such as joint replacements.

Clearly, understanding vacancies, their energies and concentrations are critical to many of the technologies that sustain our world - energy, electronics, and medicine.

You are comparing the total number of vacancies per unit volume in two metals of equal volumes at room temperature: Metal A and Metal B.

Metal A has a larger activation energy (more difficult to break bonds) than Metal B. Which of the following is true. Select one and explain why:

Metal A will have more total vacancies per volume because it has a higher activation energy.
Metal B will have more vacancies per volume because it has a lower activation energy.
They will have the same number of vacancies because because they are the same temperature.
We don't have enough information to determine this.