Welcome back. In this lesson, we're going to introduce the concept of random clusters in the liquid. As you recall in the last lecture, we talked about looking at a specific material. And as a consequence of looking at a specific material, we have a fixed interfacial energy, and we have a corresponding melting temperature. And if we choose a particular undercooling, we can wind up calculating the dependence of r, and the free energy as a result of changing r. And what we have on the slide is the net curve that is the compromise between the surface area and the volume terms. Now what we want to do is, before we go any further, we need to describe precisely what it is that we mean regarding this curve. What we need to look at is, we'll go back to this curve. When we look at this curve, what we can do is we can choose a particle, r1. And we can drop that particle r1 into our liquid. When we look at what will happen to that particle, we have a couple of events that are possible. In the first case, what can happen is the particle could grow. And what we see, however, is that when the particle grows from r1 to r1 + a small amount, delta r, what we find is there is a corresponding increase in the free energy of the system. On the other hand, if we take a look at particle r1, and this time rather than growing it shrinks, that is r1- a delta r. So what that tells us is, when we look at the change in free energy, that change in free energy is negative. Hence, the dissolution of the particle would be a spontaneous process. Now, if we look at alternatively another particle that we're going to call r2. What happens in that case is, when the particle grows, the free energy drops. When the particle shrinks, the free energy increases. So consequently, what we find at particle r2 is the growth of that particle. Now if we look at the dotted line which represents r*, r* is the point at which it can go either way. Either spontaneously dissolve, or spontaneously grow. So r* is that critical point. In addition to that, if we come and we take the values of r* that we've calculated, and we insert them into the Delta G(r) curve, we'll wind up calculating the barrier term which is Delta G*(r). So, we now have both of those points, r*, and Delta G*(r). So, we understand what this curve is telling us. Now what we're going to do is to take a look at a liquid. And I'm describing this liquid here as a collection of dimes on a table. And those dimes are randomly dense packed. Now, if I compare the liquid structure with what I'm referring to as a solid, the dimes are periodically arrayed. So in effect, what I have is a two-dimensional lattice. Now, if I decide to go through and color code these, what I've done is to take the solid, put red circles on top of each one of those. And you can clearly see the periodic nature of the dimes. On the other hand, what you see in the liquid is random clusters that you didn't really recognize in the previous slide. But you can see here that they are clusters that resemble the solid that I have on the right. So, rather than dropping these little r values in above and below r*, we actually have them generated inside of the liquid. And as the temperature goes up, the size of these clusters increase. And now we're going to see what impact that's going to have to our process of homogeneous nucleation. So, I'm going to plot cluster size versus undercooling. The point indicated on the diagram of 0 indicates that we're at the equilibrium melting temperature. So, if I go ahead and I plot the cluster size, it follows along that dotted curve. And what it's telling me is that I have clusters that exist, and the size of those clusters are increasing as a function of the amount of undercooling. And so what I'm going to do is I'm going to label that series of clusters as r max. And we'll return to that in just a second. So we're now plotting cluster size, and that's the maximum cluster size that we're going to see in our liquid. Now what I'm going to do is I'm going to plot the relationship between undercooling and r*. And what we showed in the previous lesson was that as we increase the amount of undercooling we have, what we're finding is that the value of r* is decreasing. Now, if we pay attention to where those two curves intersect, that intersection is going to tell us about the amount of undercooling that's necessary in order to have homogeneous nucleation. All temperatures above where those two intersect will not be able to nucleate. And the reason for that is simply that we have below the homogeneous nucleation temperature, the cluster sizes that we have are too small. They don't come up to the values of r*. And as a consequence of that, they're going to wind up dissolving. On the other hand, when the particles that are in the liquid have a value of r max, what we find is those cluster sizes are large enough, and they can therefore spontaneously grow when they appear in the liquid. Now let's return to this discussion regarding the particle sizes. When we look at the particle size distribution, what we're plotting along the y-axis is the number of particles of a particular size. And along the x-axis, we're plotting the radius. What we will see is actually a distribution of particles. And there will be some maximum values and some minimum values, and they have a shape that looks much like the distribution function up on the screen. So consequently, we're going to be identifying the particles that are the largest possible particles that we can see, and those are the ones that ultimately will nucleate. If you were to plot other particles that were smaller than our max, it would shift the homogeneous nucleation temperature to a higher value or to a greater undercooling. So now with this in mind, we have the ability to have a good quantitative understanding in a fairly simple system of the homogeneous nucleation process from random clusters in the liquid. Thank you.