In the previous lesson, we talked about boundaries and we talked about the importance of boundaries in material systems. Now, what I'd like to do is to describe these boundaries and how they influence the process of diffusion, and we often refer to this process as a short circuit path. So let's go back to our model of a grain boundary again. We have this open region. And consequently, we're going to expect to see easier transport throughout the grain boundary. Than you would in the matrix, and what you find is, of course, the openness of this structure will allow for diffusion to occur much more readily. This slide contains an important amount of information. Let's take it apart and think about it. When we look at the blue circles, the blue circles that are filled in represent the apparent diffusivity. That is, we're looking at the possibility of more than one type of diffusion process occurring. Now, we know that the activation energy for the diffusion of solute throughout the matrix is going to have a higher activation energy than does the diffusion along the short circuit paths of these open structures. And so what we see here are two straight lines. One straight line is related to the behavior associated with that steep slope, which is associated with the behavior of diffusion in the matrix. And the one with a more shallow slope is what's associated with the diffusion of the solute through the grain boundary regions. Now, I've circled the region where those two curves cross and I'm referring to this region as the zone of mixed results, because both of these phenomenon, grain boundary diffusion as well as bulk diffusion, are both contributing to the overall diffusion process. Now, when we look at how the data fit, with respect to these two different mechanisms, what we're seeing is a steeper slope for the bulk or matrix diffusion, and the less steep for the more open, grain boundary diffusion region. To finish off this section on grain boundary and bulk diffusion, there is some discussion that needs to be made regarding the two processes. What I've done is to plot the effective process of grain boundary diffusion and bulk diffusion over a temperature range for a particular set polycrystalline data. What we see here is, if we look at the high temperature range that is the data that lies to the left, what we see is that the process of diffusion is dominated by what's happening at that high temperature, namely bulk diffusion. So bulk diffusion is going to be the process that is important at the high temperatures. Now, when we get to the lower temperature region, what we find is the process that dominates at the low temperature is the process of grain boundary diffusion. And there is a crossover point where we go from one type of behavior to the other type of behavior. And if you recall, when we looked at the plot of the diffusivity as a function of the reciprocal of temperature, we found that rather than exhibiting a straight line, it exhibits the behavior associated with two diffusion processes. And in the region where they come close, that zone of mixed results, we see a slight curvature where we're going from one mechanism to the other mechanism. Now, this crossover point is very important and it depends upon the grain size of the material. For materials which have very large gran sizes, that intersection winds up moving to lower temperatures. So as we increase the grain size, the amount of the diffusivity that is controlled by the bulk increases, and hence it's occurring over a larger temperature range. On the other hand, when we have finer and finer grains, what we're going to see is the process becomes more dominated by the grain boundary diffusion all the way up to much higher temperatures. And this really becomes important when we start dealing with structures like nanomaterials where the scale of the structure is very, very fine. Now again, the activation energy is larger for bulk than it is for grain boundary. And if we use the erroneous type equation that describes for us the diffusivities, you see that we have the diffusivity associated with bulk, and that associated with grain boundaries. Typically, the activation energy for bulk diffusion is about twice that of the activation energy for diffusion along the grain boundaries. So, what we'll then do is to reiterate the point of the grain size and how the grain size ultimately affects the behavior with respect to the diffusivity. So over this whole temperature range, we're going to see two different processes that are recurring and we must be very careful to ensure that we only look at certain regions and don't extrapolate outside of those regions. So for example, if we had only the bulk diffusion data available at high temperature and we took that and extrapolated the data down to low temperature where the dotted line is, what we would see is values that we predict from high temperatures not being what the low temperature data actually tells us. Alternatively, if we were to collect at the low temperature regime and we would extrapolate the data into high temperature regime, we would find that there would be, again, a discrepancy in the predicted and the actual values. So we need to be very careful of this type of behavior that's illustrated by the diffusion process looking at grain boundaries and the bulk in a polycrystalline material. Thank you