On occasion, we're interested in predicting the behavior of materials. And in this lesson, what I would like to do is to introduce the concept of a phenomenon that is related to adding certain elements to a material. And there is a law that describes what happens during the additions of these materials to the structure of the material. I'm going to first give an example by looking at a simple raft of bubbles, and so that's what's up here on the slide right now. This is the result of an extensive investigation by Bragg and his co-workers early in the 1900s, in 1947. What they did was to take a solution that contains soap. And they bubbled air through that solution. And if they controlled the pressure of the air, and the diameter of the capillary that was blowing the bubbles. They could make a nice uniformed distribution of bubble sizes. And when those sizes were packed together, what you found was the high packing factor that you normally see in either FCC or BCC, so we've seen this packing structure before. We're going to make use of these bubble solutions to illustrate a number of very important points, not just in this module, but in a subsequent module. What I want to point out here is that if we look at various locations in this bubble raft, and what I've done is I've illustrated these regions using an equilateral triangle and what you can see is that this raft is perfect. And what we have is no missing sites, and in addition to having no missing sites, we have perfect alignment. So then what we're looking at is in effect a two dimensional single crystal. Now suppose we take a bubble raft, and this time we control the size so that we can produce a uniform raft of spheres of the same dimension. And then, on occasion, what is introduced is a sphere of larger diameter, using a different diameter capillary. And as a result of that, what we've done is to produce a solid solution with all the radii being the same surrounding one of these bubbles that is slightly larger. And what you can see here is what we've described in previous discussions in this module is the fact that we wind up distorting the structure in the vicinity of this slightly larger radius soap bubble. And so we're going to refer to this as an impurity atom, or alternatively, an intentional addition that we're putting into the material. So here's our material that we've introduced and around that we have the distortion. As we begin to make substitutional solid solutions, let's use Vegard's Law to explore what happens to the lattice parameter of the material as we begin to increase the amount of solute that we have dissolved in a solvent. Vegard's Law's basically an empirical relationship. And it was based upon observations of many different systems when the lattice parameter data were collected as a function of the composition of solid solutions. So you look at two different constituents, and what is found is that there is a rule of mixtures that works approximately closely, where we consider the constituent lattice parameters at the same temperature. So what we do here is we look at the lattice parameter of pure a and the lattice parameter of pure B, and the terms 1- x and x represent how much of the species that we have in our solution. Now let's explore some of the assumptions of Vegard's Law and in that law, what he is assuming is that components a and b are in their pure form. They have exactly the same crystal structures and the lattice parameters of the solid solutions then can be calculated based upon a linear combination of the two components. So when we start looking at the application of Vegard's Law, it turns out that it's not exactly perfect, but in certain cases when you have no data available, sometimes it's easy to go ahead and see what the variation of lattice parameter is as a function of composition using this simple model. Let's turn our attention now to a couple of examples. First of all, what we're going to do is to consider what happens to the copper lattice as we begin to add more zinc. Now when we look at the radius of the copper atom, we find that that radius is actually a bit smaller than the radius of the zinc atom. Now in this particular case I want you to pay close attention because copper is phase center cubic, although zinc is not face center cubic. Zinc is hexagonal close pack but the coordination numbers are the same. Remember that the FCC structure has a coordination number of 12 and so does the coordination number of Zinc, the HCP structure. So they have the same coordination number, so they will have the same effective radius when we're talking about the substitution into Vegard's Law. So what we're going to consider then is the addition of zinc to the solvent, and the solvent here is going to be the copper. The solute is the zinc. And now when we look at the actual experimental data, what we see is a nice linear behavior associated with the addition of zinc to the copper. And because the copper has a smaller radius than does the zinc, the more Zinc we begin to add to our alloy, the larger will be come the lattice parameter. So this is basically doing what we expect it to do and when you look at the data it is reasonably a good fit with respect to linear behavior. Now let's take a look at what happens in the case when we look at another example. Looking at another FCC material, namely silver, and in this case we're going to add zinc to silver. So we start out with our solvent and in this case it's the silver. And the radius of the silver is again, different than that of the zinc. The zinc, now being the solute, has a smaller radius. Again, silver is FCC and zinc is ACP but they still have the same coordination number and we'll be able to use their radii. Now when we look at this plot, again we see a nice relationship between the decrease in the lattice parameter of silver as you begin to add the zinc to To the solution. We've seen how we can follow the addition of solid solution elements in metallic materials. Similar type of behavior occurs in ceramics as well. And it's often very useful when investigating a new material to look at using an approximation such Vegard's law. Thank you.