Working with Common Ceramic Crystal Structures

To this point, we've only worked with structures that possess a single type of atom. But we know that many materials have multiple atoms - forming all sorts of different bond types. Now we'll explore what happens when we put together two sets of ions of opposite charges. Let's explore this scenario considering a random mix of equal concentrations of positive cations and negative anions, and model the situation using an interionic potential. Let's explore - note, these simulations are getting a bit more complex (and still need some fine-tuning for instruction). You may see some interesting thing happen that may or may not be physical - our model might not be completely accurate. Regardless, they are instructive and fun.

Let's see what happens in the scenario described above in Exercise 5.12.1 using NetLogo model 5.12.1. Take about 5 minutes on this exercise.

What do we find? In all cases, we get periodic and regular arrangements (apart from small deviations) of a chessboard array of alternating ions. We get a crystal structure. This emergent crystal structure is similar to what we see when using a Lennard-Jones potential, except that the charges on the ions yield a pattern in which anions are closely coordinated with cations, and vice versa. This is our first simulation of an ionic crystal structure. Over the next section, we'll use the intuition develop in the model above to think about how 3D ionic crystals form (NaCl, MgO, etc.) - these are ceramics - as well as what factors influence the structures we observe. We'll also consider structures comprised of covalently bonded atoms such as silicon or diamond. These materials have many of the mechanical properties that we associated with ceramics - such as brittleness and strength.

We'll be looking now at 3D ionic, which now have more chemical species than elemental metals and can appear more complex. Some examples are shown in Figure 5.12.1, for so-called archetype structures of Rock Salt, Zinc Blende, and Cesium Chloride. You'll notice immediately that the patterns we see in these crystals are not so different from the ones we see in metals. For example, in the Rock Salt crystal there is an array of yellow atoms on a cubic-F lattice, and another array of green atoms that looks suspiciously like they're occupying the interstitial sites of the cubic-F lattice (they indeed are).

Predicting the Structure of Ionic Crystals

As the structure of crystals is so important for their properties, understanding how atoms arrange themselves into these microscopic structures has been a critical endeavor in chemistry, physics, medicine, geology, food science, biology, medicine, and of course, Materials Science and Engineering, since we first learned how to measure crystal structure.

With regards to the structure of ionic solids, powerful and predictive theories have been developed over the past century. One of the most impactful theories was from Linus Pauling in 1929, called, succinctly, "Pauling's Rules". This set of rules, which leveraged knowledge of ion' size, charge, and bonding, as well as other factors. This was a powerful and simple theory which still help scientists predict and understand both simple and relatively complex ionic solids.

This set of rules was indeed so important that in 1951 Pauling received the Nobel Prize in chemistry in part for his work with ionic solids: The citations was: "For research into the nature of the chemical bond," the reporter read from the newswire, "and its application to the elucidation of complex substances".

We're going to distill Pauling's rules down to simpler guidelines for our purposes, but our consideration and Pauling's highlight most of the same factors:

1. The criterion of maintaining compound stoichiometry in the unit cell.
2. The importance of the relative size of the ions (and therefore crystal's interstitial sites).

These two rules are very straight-forward for predicting coordination and structures for a good number of ceramic structures, and we'll limit ourselves to these two considerations.

Let's set out our rules, then, and apply them to predict a crystal structure.

Stoichiometry:

In any compound we're considering, we have a defined chemical stoichiometry. For example, for $\ce{NaCl}$, we must have one sodium for each chlorine. For $\ce{MgO2}$, we have two oxygens for each magnesium, and for $\ce{BaTiO3}$, we must have one titanium for each barium, and three oxygens for each barium.

This will tell us how many of each atom we have in the unit cell. Let's look an the example of cesium chloride in Figure 5.12.1(c). Know that the chemical formula for cesium chloride is $\ce{CsCl}$ - meaning we must have a ratio of cesium-to-chorine in the unit cell of 1:1, or $\frac{N_{\ce{Cs^{+}}}}{N_{\ce{Cl^{-}}}} = \frac{1}{1}$. This makes sense - if it was some other ratio we'd have charge in each unit cell, which would lead to a highly unstable crystal. Test yourself - what are the cation:anion rations in the zinc-blende and rock salt crystal structures in Figure 5.12.1 (note - in the figures, sublattices are in black and yellow, respectively).

Cation/Anion Size Ratio

From our simulation in NetLogo model 5.12.1, you may have noticed that the system tries to maximize the number of oppositely charged-nearest neighbors. However, we'll be limited by the number of nearest neighbor anions that a cation can coordinate with due to the ratio of sizes of the cations/anions. We can see this schematically in Figure 5.12.2. Let's explore this:

We know that ions want to be close together to minimize their potential energy. In our ionic bonding potentials (recall lecture and homework) the equilibrium interionic bond distance is represented in these space-filling models when the two spheres representing the anion and the cation are touching.

• In the Unstable configuration, below, the cation (small, green) is at a value larger than $r_0$ with respect to all of its neighbors. It would clearly shift its position to form bonds. We'd likely view a different structure.
• In the Stability Limit configuration, the cation is just large enough to fit into the interstitial space between the anions. How would we calculate this stability limit? We've done this already in the previous sections: the analysis of the stability limit is the same as calculating the interstitial size!
• In the Stable Configuration the cation is large enough that it starts to displace the surrounding anions. The number of bonds are still maximized, though, at least until the cation becomes so large that another anion to fit into the shell and bond with it. We'll step through this systematically with 3D arrangements.

Let's try to elucidate this by observing this phenomenon in NetLogo model 5.12.1 and trying exercise Exercise 5.12.2.

Anion Coordination and Geometry in 3D

From Exercise 5.12.2 we find that simply changing the cation radius can yield different structures due to the stability of the cation-anion geometry. In three dimensional crystals, we have the following common interstitial sites:

• Linear $\text{CN} = 2$.
• Triangular $\text{CN} = 3$
• Tetrahedral $\text{CN}= 4$
• Octahedral $\text{CN}= 6$
• Cubic $\text{CN}= 8$

So we have two rules so far: the unit cell has to maintain the stoichiometry of the chemical compound, and ions prefer to sit in the sites defined in Figure 5.12.3 as defined by the size ratio in the same figure. Let's try to see how we might predict a crystal structure for beryllium sulfide: $\ce{BS}$.

First, let's figure out from Rule 2 what type of site the B will position itself in in order to maximize its coordination. We need to fine $r_{\text{cation}}/r_{\text{anion}}$ and then look up the coordination/interstitial site in Figure 5.12.3. The ionic radius of $\ce{B^{2+}}$ is $r_{\ce{B^2+}} = 0.059\,\text{nm}$. The ionic radius of $\ce{S^{2-}} $ is $r_{\ce{S^2-}} = 0.170\,\text{nm}$. So $r_{\ce{B^2+}}/r_{\ce{S^2-}} = \frac{0.059}{0.170} = 0.347$. According to our stability ranges (Figure 5.12.3), we should expect that $\ce{B^{2+}}$ occupies a tetrahedral site.

Now, let's assume that the $\ce{S}$ ions form an FCC sublattice. How do we expect the tetrahedral sites to fill in this lattice? See the FCC sublattice in Figure 5.12.4.

Well, we need to know a few things here:

1. The $\ce{B^{2+}}$ ions will fill the tetrahedral sites.
2. We must maintain the $\ce{BS}$ stoichiometry of the crystal in the unit cell.
3. Like charges repel.

There are 8 tetrahedral interstitial sites in the FCC structure and 4 atoms. This means that in order to maintain the $\ce{BE}$ stoichiometry, we can only fill half of the interstitial sites. Which of the 8 sites will we fill? Well, the $\ce{B^{2+}}$ ions will want to be as far away from each other as possible. That configuration (recall the structure for methane for your chemistry classes) is a tetrahedron. The $\ce{B^{2+}}$ will then occupy (e.g.) the $\frac{1}{4}\frac{1}{4}\frac{1}{4}$, $\frac{3}{4}\frac{3}{4}\frac{1}{4}$, $\frac{3}{4}\frac{1}{4}\frac{3}{4}$, and $\frac{1}{4}\frac{3}{4}\frac{3}{4}$ sites in the lattice to look like Figure 5.12.5. This is indeed the structure!

Now give it a try in Exercise 5.12.3.

Covalent Ceramics

The previous section may have jogged your memory regarding something we mentioned in the chapter on bonding when we classified ceramics as typically having ionic or covalent bonding. Indeed, we really have two classes of crystalline ceramics - ionic solids and covalent solids. The precise degree of ionic character where this cuts off is blurred, but typically covalent ceramic materials are used more frequently in electronic applications because they share their electrons with their neighbors instead of transferring them. This gives many of them semiconducting instead of insulating properties.

We'll look at two examples of covalent ceramics: diamond and graphite. Diamond and graphite are both elemental carbon, but they have different crystal structures and effectively highly different properties. When we have two or more distinct crystal structures for the same composition, we call these materials polymorphs. (When the polymorphs are elemental solids we actually have another term: allotropes.) Diamond is hard, strong, and transparent, and has poor electrical conductivity. Graphite is relatively weak (we'll show why in a minute, opaque, and a decent conductor of electricity. Let's take a looks at the two structures in Figure 5.12.6.

The diamond structure looks familiar. Indeed, it has an cubic-F sublattice, just like the FCC metals we've seen before. However, it also has another set of atoms, also sitting on a cubic-F sublattice, but with an origin of $\frac{1}{4}\frac{1}{4}\frac{1}{4}$ - sitting on the "tetrahedral" sites of the first lattice. So, one way we could describe this structure is that we have a cubic-F Bravais lattice with a two-atom motif, with the atoms positioned at $000$ and $\frac{1}{4}\frac{1}{4}\frac{1}{4}$. What a simple way to describe a complex structure!

The graphite structure is more complicated. We won't delve into the hexagonal Bravais lattice, but you can see the hexagonal pattern from the top view in Figure 5.12.6(c.). The graphite crystals is comprised of stack graphene layers (single sheets of $sp^2$-bonded carbon. The bonding in the plane of the sheet is strong, but the bonding between the sheets is only a weak secondary bonding. This is why graphite sheets can be easlily sheared of lifted past each other. It finds common use as a dry lubricating in moving components that can't tolerate oil. This weak bonding also allowed researchers to isolated a single layer of graphene in 2004, leading to the entirely new MSE field of two dimensional materials.