From the atomic molecular theory, we can describe each element as consisting of identical atoms. Also, we know that compounds are composed of set ratios of several different elements. Our study on Quantum Energy Levels in atoms tells us that the sizes of the atoms increase as we go down a column of the periodic table and decrease as we move across a row. Additionally, we know that differing types of attractions exist between atoms and molecules, and which attractive forces exist depends on the types of atoms and on the type of bonding present. These forces can be in the form of London Dispersion, dipole-dipole, ion-ion, and hydrogen bonding.

The physical properties of solids vary greatly from those of liquids and gases in obvious ways. We seek to to explain the different properties of solids, specifically why solids are fundamentally stronger than liquids or solids. This requires gaining an understanding of the basic structures of solisd and discovering ways to characterize them.

If you spill water, the liquid will spread out as thinly as possible. Gasoues water, or steam, quickly dissipates throughout the air if left in an open container. Solids, however stay compact and typically resist changes in structure. These observations make it clear that, of all the different states of matter, solids have the strongest interatomic or intermolecular forces.

Are all solids created equal when it comes to the strengths of these forces? There are a variety of observations we might make, but a simple approach would be to compare solid melting points, since substances with stronger interatomic or intermolecular forces will require more heat to break apart into liquid.

Notice the very large variations in melting point, covering almost 2000C for these substances. Species with ionic bonds, like NaCl, have much higher melting points than species with covalent bonds, like sucrose, due to the electrostatic attractive forces of the ions present. Metals, on the other hand, can have anywhere from low to high melting points, depending on the type of metal. Recall that each type of metal atom has a different number of valence electrons to contribute to the metallic bonding via the sea of electrons surrounding the species. Larger elements have more electrons to share in this sea. This helps explain why Copper (63.55 g/mol, [Ar]4s23d8[Ar]4s23d8)has a far higher melting point than sodium (22.99 g/mol, [Ne]3s1[Ne]3s1). Likewise, molecular solids can differ greatly in terms of melting points depending on how strong the London dispersion and dipole-dipole forces are.

Nature appears to enjoy creating patterns. Exploring the world reveals that many things are simply composed of countless repeating parts. Take a snowflake for example. These wonders of nature are essentially one big fractal, a repeating small scale geometric pattern that produces an interesting shape.

Figure 1: http://www.clipartspace.com/clipart/snowflakes/snowflake-02.gif

We can apply the idea of small units repeating to form a bigger structure to solids. A solid must consist of some reiterating pattern. If these templates repeat throughout the solid, this would explain why some solids melt at specific points. However, if a solid lacks a set mold, it could explain why no one melting point can be obtained.

In order to determine what these patterns are, we start with the simplest solids, metals. Imagine that each metal atom is a sphere. These spheres should want to be as close together as possible to fully exploit intermolecular attractions. One simply way to do this would be to make a cube out of repeating rows and columns as in Figure 2.

Figure 2

While this provides an easily visualized solution, it does not lead to effective packing. Note all of the empty spaces that occur between each of the sets of four spheres. All of this wasted room means that only about half of the available volume is occupied (geometric calculations show the fraction is 52%), which would make poor use of the intermolecular forces. Force laws tell us that distance is extremely important. In fact, the force experienced is proportional to 1/d2. If nearly half of the system is empty space, a good deal of distance exists between atoms, so they will not experience large enough forces to remain strongly bound.

While this simple cubic packing did not provide an optimal solution, it serves to give us insight into how spheres interact with one another, and we can exploit this knowledge to make a better system of packing. Of course, no set of spheres can come together without empty space developing, so 100% efficient is not possible. What would happen if we alternated where our spheres went depending on their row? We could separate our spheres slightly so that the next row goes into the center of each set of four spheres below it. Our following row could then be the same as the first.

Figure 3

This packing results in a far greater 68% packing efficiency

We could also make it so that each new row goes on top of the previous row’s holes. As a result, we would get repeating patterns consisting of two alternating layers. Each layer should exist in a way which best utilizes the space available. This occurs when they are arranged hexagonally with six spheres surrounding one central sphere. Notice that only small triangles exist between each sphere.

Figure 4

Two types of this packing are possible. The first kind involves every layer being placed directly above two layers below it. This is known as Hexagonal or ABAB packing. Our other option is to arrange as before but offset every third layer. This is known as Cubic or ABCA packing.

Both of these arrangements are equally efficient, using up 74% of the available volume. Each atom will have 12 nearest neighbors (6 on their layer and 3 each from the layers above and below) as opposed to the mere 6 provided by the simple packing we witnessed before and 8 given by the body-centered variety.

How then do solids choose which type of packing they will take on? In principle, we would assume that all solids would be either hexagonal or cubic, since these arrangements are the most efficient. Although a great many metals do pack in one of these ways, in reality metals are spread fairly evenly amongst the body-centered, hexagonal and cubic crystallizations. In order to determine why this is, we must take a more in-depth view of these packings.

Previously, it was said that solids come together in collections of repeating patterns. The packings of these patterns can be further broken down. The simplest of these newfound pieces is referred to as our unit cell and can be used to characterize the entire solid.

Let’s start by deconstructing our simple packing structure. These repeating stacked rows and columns can be viewed as a cube of eight spheres where each sphere exists as a corner of the cube.

Figure 5

Each sphere sits at one corner of the cube and is therefore shared by a total of eight cubes. As such, only 1/8 of each sphere is actually in the unit cell (the remaining pieces are in other unit cells connected to it) so that a total of one atom is in each unit cell. What results is known as a Primitive Cubic Unit Cell. Note that the spheres are actually touching as they were in the packing picture. In this figure, the spheres have been scaled down to avoid clutter.

Next, we take a look at the body-centered packing. Recall from before that body centered packing is an offset of simple packing, so we would expect the unit cells to be similar, and they are.

Figure 6

Two slight differences exist. First, as explained in the packing argument, each corner sphere is slightly separated. Second, there is an additional sphere in the center, which is exclusive to each cell, giving us a total of two atoms per unit cell.

Hexagonal packing does not result in a cubic structure, so we move on to Cubic close packing.

Figure 7

Again, we have a cube with 1/8 of an atom each for the corners, but something new has arisen. There is now ½ of an atom on all of the faces. Each of these face atoms is shared by one additional unit cell. This gives way to a total of 4 atoms per unit cell in an arrangement known as face-centered cubic.

In all, there are fourteen different ways in which solids can stack together. Each of these is some form of a parallelepiped, which features six faces of parallelograms. For the purposes of this study, we will worry only about those which have squares as their parallelograms giving us a cube.

We now have a fundamental understanding of the simple structures in basic metals which consist of only one species. This does not account for the multitude of solids that are ionic or covalent in nature. We must wonder if they follow similar patterns. In metals all atoms in the array are of the same type of atom, but in ionic species, we have two separate atoms—a cation and an anion. Several complications arise because of this.

In our earlier study of Quantum Energy Levels, we found that atoms are separated into valence and inner shells of electrons. These shells are spaced far apart meaning that the size of an atom increases greatly as you move into a new level. As a result, anions tend to be larger than cations since cations typically lose an entire energy level. Anions will increase dramatically in size since having additional negative charges will fight against the stabilizing attractive forces of the positive nucleus. For example, Na has a covalent radius of 0.157 nm, but as a cation it is a mere 0.095 nm. Meanwhile, Cl grows from 0.099 nm to 0.181 nm as it becomes an anion.

Figure 8

This means we will be trying to pack spheres that are not close to the same size. Additionally, we must now deal with charges. In handling a neutral complex, we must make sure that our unit cell is also neutral for it to be a proper representation of the solid as a whole. We can use stoichiometry to ensure this occurs.

Let’s begin by examining the most basic type of ionic solid—salts. Of our salts, LiF is the simplest. Since we are dealing with their respective ions, we should take a look at their sizes. As expected, our fluorine anion is much larger than the corresponding lithium cation. Their respective ionic radii are 0.136 nm and 0.068 nm, so the anion is twice as large as the cation.

By stoichiometry, we know that in order to maintain charge neutrality, there must be an equal number of lithiums and fluorines in each unit cell. We now also know that the majority of the space will be taken up by the fluorines. How will this affect our arrangement? Due to the 1:1 ratio of atoms and 2:1 size difference, we can focus on the fluorines first.

It is known that salts arrange themselves in Face-centered cubic unit cells, so let’s make the basis of our cell the fluorine. What happens when we try to pack them together?

Figure 9

Due to the size of the fluorine ions, large holes exist within the structure. These holes turn out to be perfect for fitting our much smaller lithium ions. These cations are referred to as edge atoms since they reside on the sides or edges of the cube, and ¼ of an atom is in each unit cell. Every hole they reside in can be referred to as octahedral since it leads to the lithium having six nearest neighbor fluorines.

We now have the basis for our salt unit cell. Each face will look like that, but what happens in the center? Since every face centered ion contributes half an atom, we can say they contribute one radius of fluorine each. Given that the shape is cubic, we know

that all sides must be equal, so every measurement of depth must be a total of two fluorine radii (one from each corner atom) and the diameter of a lithium ion from the edge. Because the inside already has two fluorine radii, the exact center must be comprised of a lithium. This would give us the following unit cell:

Figure 10

Figure 11

But does our stoichiometry work? The fluorines make up the 8 edges at 1/8 of an atom each plus half an atom for each of the six faces yielding 4 total. Our lithiums reside at 12 edges at ¼ a piece plus one atom at the center also giving 4 total and verifying the accuracy of the unit cell.

Is this structure feasible for all salts? While it satisfies a considerable amount, it does not account for cases where the alkali metal is comparable in size to the anion. CsCl, for example, features ions of 0.169 nm and 0.181 nm, respectively. This means the cesium cations are far too large to fit in the holes created by the chlorines. Instead, the chlorines will arrange via the primitive cubic method allowing the cesium to reside in the center of the cell as such

Figure 12

The new unit cell still has the correct ratio of atoms, but there is one major difference now. Each cesium has 8 nearest neighbors giving it a coordination number of 8. This is an increase from the coordination number of 6 experienced by our lithiums before. As the size of the atoms involved increases, the unit cell has a tendency to become simpler and coordination number rises.

We have now seen several examples of ionic species in 1:1 ratios of cations to anions, but we have yet to determine how having more of one kind would affect our unit cell. An easy example to start with would be that of BaCl2BaCl2. Since barium chloride breaks up into Ba2+Ba2+ and 2 Cl-Cl-, we have to come up with an arrangement that would allow double the amount of chlorine atoms. We know that the species occupying corner spots contributes less, so perhaps we should now base a face centered structure around our smaller cation.

Figure 13

This gives us a total of 4 barium ions (1/8 along 8 edges and ½ on 6 faces) meaning we need to somehow fit 8 chlorine atoms in. The easiest method of getting a large number of atoms in is to have them be completely within the unit cell like we see in body centered. Simply putting a chlorine in the middle would not work. Instead, we must find the holes formed within our barium structure.

Previously, we saw octahedral holes formed in our LiF, now we can utilize the tetrahedral holes that develop inside of Ba. If we break down our unit cell into smaller cubes made up of three face cells and one corner cell, we can see these tetrahedral holes allowing for four nearest neighbors. There are 8 holes of this type overall, which is perfect for fitting our eight necessary chlorine ions.

Figure 14

The preceding unit cells represent a fair portion of the unit cells which occur naturally, but others exist. Even though the unit cells seem complicated, they are all just variations of the few main types we have discussed resulting from size and charge differences.

1) Information on chemical and physical properties of molecules was from Chemfinder.

2) D. Oxtoby, H. Gillis, and N. Nachtrieb. Principles of Modern Chemistry. 5th edition. Thompson Brooks-Cole. 2002. pp. 748-777.

3) J. McMurry and R. Fay Chemistry. 4th edition. Piarson Prentice-Hall. 2004. pp. 381-428

4) T. Brown, H. Lemay, and B. Bursten. Chemistry: The Central Science. 10th edition. Prentice Hall. 2006