Welcome back, in this lesson we're going to talk about two processes now. One of them is nucleation, once the nucleation process has been finished, what we now are interested in is the growth process. How does that sphere that is beyond the critical radius, how does it grow? So what we'll do is we're going to examine the behavior below the equilibrium melting temperature. And so what I'm going to do is I'm going to plot here the rate of formation of nuclei inside of my liquid, and there will most likely be a maximum rate of formation at some undercooling, and that's at the tip of the rate curve that's illustrated in the diagram. Now the other thing that's important is that we have the growth of that phase. Once that phase has developed, it's now in the liquid and it will continue to grow. And some of the things, of course, that become important is that when you are at the higher temperatures, the structure is coarse. And even though we have a high rate of diffusion, the distances over which the solute has to be transported is very large. So the growth rate diminishes as we get closer and closer to the temperature. The next issue has to do with the fine structure that we have at lower temperatures. But at the lower temperatures, we're diffusing again over smaller distances, however, the diffusion process is reducing because we're going down to a low temperature. So we have high rates of diffusion at high temperatures, large distances. Low rates of diffusion and smaller distances associated with the smaller particles. Now, because we have nucleated and we have grown, we're actually going to look at these two events together. And what we're going to do is to produce a net curve, which is both the contribution of the process of the nucleation and that of growth. And what we see is a region where both of those are contributing significantly, and we will have some maximum in the rate of nucleation at some temperature that is below the equilibrium melting temperature. And that then describes where the material is most likely to nucleate and grow the both processes. And when we look at the reciprocal of that rate, it's going to tell us something about the time associated with the beginning of the process of nucleation and growth. So we'll just simply invert the curve, and what we'll get is the behavior of the curve that we have up here on the right diagram. So we have taken the maximum, and what we have done is to take the reciprocal. And now what we're looking at is the start of the crystallization. And when we plot the start of crystallization as a function of log time, and we use the access log time because we're dealing with diffusion and, therefore, we need to be considering things in terms of exponential behavior. So we have this characteristic curve that we refer to as a C-curve. Now, there are actually a second way that we can be thinking about the behavior. On the left-hand side I have illustrated again that maximum nucleation rate. And again, this time, it's being made up of two terms. Just like we described previously, but we'll look at it slightly differently than we did in the previous view graphs. In this particular case, we're looking at the mobility term, and what we see, of course, is the mobility term winds up increasing with temperature, so the growth part is going to be affected by that mobility term. Now the other thing that becomes important, that we didn't describe the last time, has to do with the barrier to nucleation. That is, as we get smaller and smaller or larger amounts of undercooling, what we find is the nucleation rate becomes affected by that barrier term. And so there will be a maximum point in the rate of nucleation. And when we put those together, we get this characteristic C. So, when we look at the effect of cooling rate, for example, coming down and trying to have an effect on the process of crystallization, we can look at two different cooling rates, temperature T1 dot, which represents the faster cooling rate, and the lower cooling rate T dot 2. And what we can do is to break each one of these up into little stairs, where we quench down the sum temperature, we hold it for a period of time, quench again, hold it, quench and hold. And what you can see is the hold times at the higher temperatures where a lot of diffusion is occurring is relatively small in comparison to when we take the cooling rate temperature T dot 2. What we see is longer hold times at the higher temperatures. So more things can be occurring at that slower cooling rate. So what we're going to be doing here then, is to go from the isothermal transformation curves and try to construct the cooling curves, the continuous cooling curves. And that's what we have illustrated here. So we can generate these cooling curves by doing experiments to determine where the start and where the end of the process of crystallization is occurring. So we're going to be talking about here not an isothermal transformation curve but we're going to be talking about a continuous transformation curve. So the black line here represents the start of crystallization. And the blue line represents the end of the crystallization process. So what we'll do is we'll start cooling at different rates. The first one is a high rate of cooling. So we're in the region where we should only have uncrystallized material, or an amorphous structure. Now when we cool at the rate that´s indicated by the arrow, and that's the critical cooling rate, any cooling rates that are faster than that cooling rate will be no crystallization. On the other hand, when we cool at a slower rate than the critical cooling rate, we will get crystallization. We see it starts at a temperature where we intersect the black line. And it ends when we intersect the bottom of the blue line. So what we can do is, we can actually do a number of experiments where we can determine the amount of crystallization we have at different cooling rates. And so what we're looking at is these decreases in our cooling rate. And now what we'll do is examine at that circle, that we're looking at the start of the crystallization process at a particular rate of cooling and a particular period of time. So that's the start, and here is the finish. We'll look at another cooling rate, and another cooling rate, as illustrated here. So all of those points wind up determining the start and the end of the crystallization process that we've been describing. So this is what we do when we look at continuous cooling curves. And the important thing here is that, because the continuous cooling curves are plotted and determined as a result of cooling rates, we are then able to actually draw cooling rate curves and predict, based on those cooling rates, whether or not crystallization will occur. Thank you.