Welcome back, up until this point we've been talking about homogeneous nucleation, and remember that homogeneous nucleation occurs randomly throughout the structure. In this particular lesson we're going to describe a process called heterogeneous nucleation, and what we mean by that is, the nucleation process is going to occur at specific sites. Either sites that we have introduced, or alternatively, at interfaces along the mold wall for example, during a solidification event. So heterogeneous nucleation is an additional type of process where we go through a transformation. And, what we're going to do is to use the basis of homogeneous nucleation, modify that equation just a bit, to come up with a description for the heterogeneous nucleation process. Now we all have some sense of understanding about heterogeneous nucleation process, and one of the examples has to do with seating of clouds. In a particular environment on a day where you may have sufficient amount of moisture in the air to have rain, but in the absence of clouds the rain can't happen. So what is done is to introduce chlorides, or silver chlorides, which are tiny little salt crystals that are put in the airplane and then distributed into the atmosphere. Those crystals have a structure which will facilitate the development of ice, and when that begins to happen the ice is growing at the expense of the liquid that is in the air. And consequently what begins to happen then is the formation of clouds, and then with the presence of the clouds, what we wind up with is rain. So that's a heterogeneous nucleation process. So what we've done then is to go through and seed the clouds. Now what we're going to do is we're going to return to our homogeneous nucleation sphere, and what we're going to be describing here is a situation where I have an interface, and I have formed a droplet of solid at that interface. Now I want you to carefully look at this droplet, and consider the volume of this droplet. In this particular case, what we're looking at is the fact that we have a sphere of some radius r, and what we're going to define is something that we refer to as the wetting angle. And the wetting angle in this particular case is the angle of 180 degrees, and basically what it tells us is that the solid has no interaction with the particular surface that's indicated. Now if we have a surface where the solid will begin to nucleate on that surface, and what we have here is the particular case of the nucleation process, where the interface has stimulated the process of nucleation, and it has reduced the wetting angle. And so now the wetting angle, theta, is something less than 180 degrees. Now if we have another substraight that has more wetting efficiency to the formation of that solid droplet, we have what is up on the visual right now, and that has a wetting angle of 90 degrees. And eventually we get to the point where we begin to see more wetting, illustrated in this set of diagrams. So as we go down what we're seeing is the same radius of curvature associated with each one of those spheres, but when we go from no wetting to more and more wetting, one of the things that we're seeing is that the volume of the sphere is actually decreasing. And now in order for us to begin to understand this particular problem, we're going to define something that we refer to as the spherical cap. And so we can look at each one of those illustrations to the left, and they represent various degrees of wetting. And consequently, what we can do is with this particular geometry, we can calculate the associated surface area and the volume associated with this geometry called the spherical cap. And those particular characteristics, the surface area is given on the visual as well as the volume of that spherical cap. So now what we can do is to relate the geometric construction of the spherical cap, and of course, it's related to the wetting angle. What has not changed, and you need to pay attention to that, is the radius associated with the geometry of the cap. So that cap has the same radius as we go through. So what we've been able to do then is to use this geometry to describe the process of wetting, and the influence that the wetting angle has on the volume and the surface area of the solid that is being generated, thank you.