[MUSIC] Dear students, so in today's lecture, we move on to talk about the 2-dimensional defects, which are essentially surfaces and interfaces. So essentially, all materials or all crystals, they will have free surfaces. And as a matter of fact, free surfaces of a crystal is actually a solid vapor interface or a solid vacuum interface if the crystal is put into vacuum, right? Another type of interface is so called grain boundaries. And it can be simplified to alpha/alpha interface where you will have an interface between two crystals of different crystallographic orientations. And then you may have a so called interphase interface or hydrogenases interface, which is the boundary between two crystals of different structures or chemical compositions. And we will see these different types of 2-dimensional defects one by one. The first one is the free surface, and we all heard of this so called surface energy concept, right? And you can notice that in some surfaces, the liquid form very spherical bubbles like here or like here. However, in some other cases, you will have a water spread out over the surface and they are called hydrophobic or hydrophilic, and this is actually the result of different surface energies. And essentially, when you have a low surface energy, if you have a low surface energy, your water or your liquid tend to wet, okay, tend to wet your surface. Or in other words, the liquid wants to spread across the surface. And when you have a high surface energy, the liquid tend to form bubbles onto the surface. And then these very different behaviors are caused by the surface energies and we need to know what is the physical origin of surface energy. And this is actually the physical origin of surface energy. And suppose you have a microscopic surface like this. And in the small scale or in the atomic scale, this relatively flat or apparently flat microscopic surface are never flat or never sharp. They are actually formed by having these steps, which are corresponding to the most densely packed planes. The reason is that in the most densely packed planes, you have the maximum number of chemical bonds in the plane, right? And then you will have a smaller number or the minimum number of chemical bonds pointing out of the plane. And when you have a surface like this, the surface energy comes from additional energy of this unfulfilled dangling chemical bonds. And that is the reason why in the atomic scale, your free surface is always bounded by these most densely packed planes. And you will have a surface energy minimum of these crystallographic orientations. And here are the descriptions of what I have just discussed. So the surface energy comes from the unfulfilled dangling chemical bonds on the surface and the surface are bonded by the most densely packed planes in a crystal to minimize the surface energy. Okay, for example, in the face-centered cubic, FCC structure, you will have the surface bounded by these 111 type planes. And that is also the reason why when you see natural crystals or natural gems, they are always faceted, which means the surfaces are bonded and shown as certain crystallographic orientations and planes, right? The surfaces are really sharp, which corresponding to certain crystallographic planes. And another type of important interface or 2 dimensional defects is the grain boundary. The grain boundary is the boundary between two crystals of the same composition and crystal structure, but having different orientations. This happens a lot in casting or in thin film deposition, where you have the so called nucleation of these single crystal islands. And then when the islands grow during further solidification of thin film deposition, they contact each other and forming these kind of a 2 dimensional interfaces, right, between these single crystals. Although the single crystals are of the same type, so they have the same crystal structure, the same lattice constants, they are having different orientations because they are nucleated differently. And this is exactly the case where you have one crystal here, another crystal here. However, although they have same crystal structure in this region, in this region, you will have a misfit in their orientations, and this is called your grain boundary. Okay, and there are different types of grain boundaries, as you can imagine. Two simplest type of grain boundaries is one is this so called the tilt boundary and this is your twist boundary where you just bend the two crystals or just twist the two crystals. And this is an example for a simple so called low angle tilt boundary where you can even calculate the average distance between misfit dislocations to accommodate to the lattice misfit in their orientations, okay? So this is going to be left for your solution in your homework. And this is grain boundary energy versus misorientation angle at the grain boundary. And for so called low angle grain boundaries where the misorientation angle is smaller than 10 to 15 degrees, the misorientation of the two crystals are so small that your green boundary energy, which is associated with the dangling bounds of the other grain boundary, they just increase with increasing misorientation angle theta. However, when you have sufficiently large misorientation, the grain boundary structure becomes so distorted and then so disordered. So then, the grain boundary energy is no longer very sensitive to your misorientation angle. Another type of important boundary is so called twin boundary, where this is your twin boundary. And on either side of your twin boundary, you have a mirror image of the crystal lattice and they are called twin boundary. We will have more discussions on this in the mechanical property chapter. But for now, we will just know it's a so called coherent boundary, where there is a one to one correspondence across the twin boundary. And usually, the coherent twin boundary have the energy minimum. So it's very stable in terms of energy And the final type of 2 dimensional defect is the more general case, which is the heterogeneous interfaces. So the interface form between two different materials. And there are also special cases where the special case is that the two crystals, although they are having different chemical composition, they are having the same crystal structure, and they are having the same lattice constants. So that there is no strain or no dislocations at an interaction so that the interface is coherent. So one black circle is connected with one white circle, right? So that's called coherent heterogeneous interface or strain-free heterogeneous interface. And this is usually the very special case. So in general, for heterogeneous interfaces, you need to consider the total interfacial energy of your boundary of your interface is the summation of your strain energy, which is rendered by the lattice mismatch, as well as your chemical energy, which is associated with the chemical bonding across the interface. And as you can imagine, for the special case of strain-free coherent interface just shown the previous page, we only have this chemical energy contribution because we have a strain-free and there is no strain energy and lattice misfit. But this is just the special case in a more general case. You'll have two crystals with different lattice constants and crystal structures. And a parameter describing the difference in lattice constants is so called this lattice mismatch, which is the relative difference in their lattice constants. For example, if you have your alpha crystal lattice constant is smaller than your beta crystal lattice constant, then the mismatch factor delta is calculated as their difference divided by the alpha lattice constant. And because you have lattice misfit and then you have the strain energy and if your strain energy is too large, then one way to release or relax your strain energy at an interface is to form the so called misfit dislocations. And with this knowledge of your lattice constant in your alpha and beta crystal, you can even calculate the average distance between two neighboring misfit dislocations and interface. I will just show you how to calculate it. And between two neighboring misfit dislocations, you will have (N+1) alpha items and N is an integer, and this will be equal to N times the lattice constant of beta crystal, right? Then this difference in this integer 1 is from the mismatch between the two lattice constants, right? And then you can come into this conclusion that the average distance between the two misfit dislocations on the interface can be calculated as the lattice constant of the crystal having larger lattice constant divided by their lattice mismatch. So to summarize, so in general, when you have a heterogeneous interface with two different materials having different crystal structures, the total interfacial energy, again, is the summation of your strain energy and your chemical energy. And for incoherent interfaces, usually the chemical energy will dominate, right, because the chemical bonding is having more energies. And usually, interfacial energy of incoherent interfaces is larger than the semi-coherent interfaces. And the coherent interfaces is usually having the minimum interface energy. And today, we have talked about the 2 dimensional defects of all kinds of interfaces. Thank you very much. [MUSIC]
