In this lesson, we're going to be describing a phenomenon of diffusion that we refer to as self diffusion. And an example for us to be able to consider, how do we measure the diffusion pf aluminum in aluminum, or copper in copper? We do that by using, for example, radioactive isotopes when they are available. So what is illustrated up here is a simulation of the structure where the light blue circles represent the material itself, the darker blue circles represent the radioactive elements. The reason that we wanted to do this is that we can actually detect the diffusion process by measuring at different locations in the sample the concentration of the radioactive isotope. So it starts out as two thin lines of the radioactive isotope. At the initial time to use equal to zero. We wait a period of time at an elevated temperature. And because we have a concentration gradient here, we're going to see the diffusion of those two different lines going in the directions of positive and negative x. Now the direction of the arrows then represent the direction of transport or the direction of the flux of the atoms. So the flux to the left and the flux to right are going to be equal to one another. Now at some later time, we see that the sphere, or the circles of the radio that represent the radioactive isotopes, have begun to become even more uniformly distributed throughout the structure. In this particular process, we can refer to it as a self diffusion process, even though we know that these radioactive, elements have a slightly different mass than the non-radioactive isotope. We can assume that, for all intents and purposes, these measurements, these differences are going to be very small between the two. And as a result, by tracking the elements, we can then come up with a value for the tracer or the self diffusion of the material. Why did we want to make these measurements? Well, it turns out that we can begin to use the self diffusion measurements and come up with some physical principles that underline the diffusion process in a pure substance. What I have done here is to plot the activation energy for self diffusion Of these various elements and I've done it as a function of the melting temperature of the material. So for example, if we look at lead. Lead has a lower melting temperature than aluminum than does ultimately platinum, which has the highest melting temperature. And you see this nice linear relationship between the melting temperature and the activation energy for self diffusion, and because all of these values are closely scattered around the line. It's telling us something about the mechanism of diffusion in these materials. The other point that I want to make is that all of these elements happened to be in the form of a face centered cubic material. So the variation then in the activation energy follows this nice plot. And recall we have looked at other temperature dependent plots earlier in the course where we looked at the melting temperature, and we correlated it to the coefficient of thermal expansion. And we've also looked at melting temperature and its effect on the elastic modulus of the different elements. And in that case, we related the kinds of behaviors to the bond strengths that can be assigned to the temperature of melting. So here this is telling us something about the bond strength which is ultimately controlling the diffusion of the material. The data that we're looking at comes from published material in the literature. Now, if we go back and we calculate the diffusivity of a certain number number of elements. Again these diffusivities are aluminum through platinum, where aluminum represents the lowest melting temperature and platinum represents the highest melting temperature. In these particular cases I've used data from the literature. That allows me to calculate the diffusivity in the solid state, but just at the melting temperature for each one of these elements. And I have illustrated that on the plot that I have here. So, what we have is aluminum, cooper, nickel, and platinum. Again, all face centered materials, and when we look at the melting temperature diffusivities, they're all fairly close to one another. They're not too far away in terms of the values that they have. And this, along with what we have just described previously, as this linear behavior between melting temperature and the activation energy for self-diffusion allows us then to come up with something that we refer to as the homologous temperature theta, which is the dimensionless temperature. And we can look at it from the point of view of the following type of equation. Let's take a look at aluminum. And if we look at aluminum and copper and nickel and platinum, we can calculate what the diffusivity would be of all those different elements, and they should be the same value at and equivalent homologous temperature. In other words, if we define the homologous temperature as the temperature of interest, which is below the melting temperature, and the melting temperature of the particular material, we would then come up with a diffusivity of all these different elements based upon this homologous temperature. And that's going to be equal to some constant divided by the homologous temperature for the material. So if we know something about the diffusivity of aluminum and we know its melting temperature, we should then be able to calculate what the diffusivity of copper would be at an equivalent homologous temperature. Thank you.