We have two ingredients here.

One of them is the solvent and the other is the solute.

And they're distributed throughout the material.

And as a result of that distribution, will have a random distribution,

in this case, of the solute and the solvent.

As we increase the concentration of solute, at some point,

that terminology flips and the solute ultimately becomes the solvent.

And what was the solvent becomes the solute.

Now we want to think a little bit about, can we come up with a set of rules or

guidelines that will talk about the solubility

limits that we can have in a material?

And Hume-Rothery a number of years ago came along and put together some

empirical rules by looking at what is meant by a solid solution.

So, let's go back and look at a picture of our solid solution.

So that red circle in there is representing an atom that we

have intentionally added to the system.

And we immediately can figure out how we might begin to think

about rules that would govern how much of those red atoms that we can put in there.

First of all, we would expect that in order to make a reasonable substitution,

the radius of the red atom and the radius of the blue atoms need to be the same.

Well, what Hume-Rothery said was, after having looked at a number

of materials that had reasonably extensive solid solubilities,

that is, lots of reds included in with the blues.

That in order for that to be the case, that the volume of this must be, or

the size difference must lie within plus or minus 15%.

So, there's a restriction on the size of the radius of

the solute with respect to that of the solvent.

There are other issues that control how much we can add.

And that basically has to do with

where these elements are located on the periodic chart.

And in the first case, when these materials are close to one another

along the same row,

what we find is that these materials will have the same electronegativities.

Another characteristic is, that the valence will be very similar.

So, now we have three rules, or

observations, that Hume-Rothery has put together.

And they are size, which becomes pretty obvious.

The next two, which have to do with

where they are located on the periodic chart and hence their chemical similarity.

The last one, on the other hand, has to do with the crystal structure.

Now there's certain metallic materials like copper and

nickel, which are completely soluble in one another.

So you can make a whole series of alloys starting out with pure copper,

going all the way over to pure nickel.

So it's a continuous solid solution.

Another example of a material that would behave that way

are the semiconductor materials, silicon and germanium.

Silicon and germanium are both diamond cubic.

If you go back and you look at copper and

nickel, both of those elements have the same crystal structure.

So, in order to have complete extensive solid solubility from one component

over to the other component, you are necessarily then required to have

a material in which the two components have the same crystal structure.

What we're going to do, is using this basic idea of

the iron having two different alatrobes,

we're going to be able to determine what the effect of the coordination number is.

And remember back from an earlier module, when you look at the coordination

number of the body center cubic, it has a coordination number of eight.

Where when we look at the face center cubic structure,

it has a coordination number of 12.

Now focusing once again on the two different coordination numbers.

The radius that we would have iron, when it is in the FCC

structure, we're going to use that as unity.

And when we look at what happens to the effective radius when we go from

a coordination number of 12, face center cubic, to that of body center cubic.

What we find is, that there is a modification to the radius of about 97%.

So that radius is actually smaller in the BCC than it is in the FCC structure.

By looking at a number of other systems in which there

are different coordination numbers, the entire chart then is available.

So we can look at coordination numbers of 12, 8, 6, and 4.

So how do we use this?

Well we use it, for example, when we look at alloys which contain

iron where we have added the intentional addition of chromium.

Now, if we're talking about iron when it's body center cubic,

then it's not a problem, because both iron and chromium are body center cubic.

On the other hand, if we're looking at iron and

we're in a temperature range in which the face center cubic phase is

the stable phase, then what we find is we have to take the chromium.

Which is normally body centered cubic, and we have to modify it, so

its radius is now slightly larger when we put it in the FCC structure.