[MUSIC] Dear students, previously we have talked about the dislocations their concepts and their geometry. And today's lecture we are going to talk about dislocations in real crystals. So, from the previous lecture we know that the dislocation motion or the plastic deformation of a crystal is highly in-homogeneous, Okay, which means the dislocation motion or the deformation will only occur across certain crystallographic planes and certain crystallographic orientations, which are called slip planes and slip directions okay. So in a deformed metal you will have, all these kind of parallel sleep traces were a series of these traces is costly bands and the direction of these traces are corresponding to their sleep lies. Okay. So we also talked about the critical result of shear stress that can drive your dislocation motion which is the Peierls Nabaro stress is associated with your Burgers vector magnitude B, your interplane spacing, D and your shear modulus g as well as your poisons, ratio new. Okay. So the conclusion from this equation is that, the sleep playing asleep directions are the most densely packed the planes, and directions in your crystal. And you have already learned crystal structures. In conventional crystals of FCC structure, HCP structure and BC structure using these criterion, the slip planes and slip directions are kind of determined. For example in FCC structure, your most densely packed planes are those 111 type of planes, and the most densely packed directions in the 111 plane, are 110 type, the directions. And the most closely packed a plane and directions in HCP metals is 0001 plane and these directions and this will also depend on the c/a or a ratio in HCP structures. And in BCC structures the situation is more complex which is associated with the operating temperature of your crystal. So, now you know that you have certain slip planes and slip directions, and the combination of them are called slip system. So with the most closely packed slip planes and slip directions, and now you can calculate the total number of slip systems in certain crystal structures. For example, in FCC crystals you have four 111 typed slip planes. And on each of these four 111 type of planes, you have three slip directions, 110 typed slip directions. So you In total, you'll have 4* 3 =12 slip systems in FCC structures. And similarly you can estimate the total number of slip systems in HCP structures which are three, so is very limited and in BCD structures is 48. So, quite a bit. So, in general, the more slip systems you have, the easier the plastic deformation. So this explains the reason why, for HDPE metals such as magnesium and titanium. The usual way to fabricate these metals is by casting. Rather than by deformation because the deformation of these HCP crystals are limited because of limited number of slip systems. Okay, the next topic is how to calculate a critical resolved shear stress with an arbitrary loading direction. So, suppose you have a single crystal sample like this, and you load your sample with a force F. The initial cross section of your sample is A here, okay? And for some reason we just assume we have a slip plane of this shaded plane. And have this slip direction in this direction, and the slip plane normal forms a angle five relative to your loading force F, and your slip direction forms that angle lambda with respect to your loading force F. Okay. So, a notice is that, your three factors. So these three factors are not necessarily in the same plane. In other words, the summation you are of your angle, Phi and Lambda, and not necessarily 90 degrees. Okay. And then we are going to calculate the resolve their sheer stress. Resolved, honestly play in the sleep direction. Okay? So because we know the cross sectional area that in vertical to your loading forces a. So where's the slip plane normal fine, we can calculate the area of this slip plane as A divided by cosine phi, right? So essentially the force resolved onto the slip plane will be F over A times this cosine phi. And then to calculate the force resolved in the slip direction, you just multiply this, by another cos lambda value. Okay, essentially, your resolve the shear stress in the slip direction is proportional to your force divided by cross sectional area, A times this cos phi, cos lambda value. These two cosines products are called Schimid factor which is an important factor correlating your sleep system with your loading force, okay? So essentially in a very special case that these three vectors are on the same plane, then you will have Phi plus lambda as 90 degrees, then your resolve the shear stress takes its maximum value of one and half the force divided by the cross sectional area. So in general for plastic deformation to take place or in other words for dislocations to move, honestly playing in the slip direction, you have to fulfill this condition of resolve the shear stress calculated by this one. Larger than the critical resolve the shear stress, as determined by your opponent borrow stress okay. So, if you still recall the critical resolve the shear stress or the Peierls Nabaro stress is a material property which is associated with the shear modulus poisons ratio and the crystal structure of your crystal. So, to visualize real dislocations, you must use some advanced Microscopes. And in particular it is a transmission electron microscope called TEM, where you have your specimen or your sample single crystal here for example, you have a high energy electron beam illuminating from the upper to the downside of your, of your sample and then you have some complex arrangement of lenses. So that in the in the final screen you will be able to visualize the dislocation lies. And for example, if you have a three dimensional dislocation structure like this. So you have dislocation lines this way you have dislocation lines this way, the TEM or Transmission Electron Microscope is a projection of your three dimensional thing on a two dimensional imaging plane. So this is your imaging plane, and eventually you'll be able to visualize these dislocation lines. However, because it's a two dimensional image, it doesn't necessarily represent, how these dislocation lines are arranged in three dimensional space. So another important to quantify the dislocation structure in real crystals, is this dislocation density? By definition, it is the total dislocation length per unit one of a crystal. So, if you have a crystal like this, which are full of dislocations, the dislocation density by definition is the summation of all these dislocation lenses per unit one of your crystal, it has a unit of centimeter minus square or meter minus square. Because it's very difficult to measure the exact length of dislocations, the alternative way or more practical way to determine the dislocation density is to count the total number of dislocation lines per unit area under your TEM images, right? It's much easier to count the numbers, rather than count the length. So for example, here shows a dislocation structure in a graphene-Al composite, fabricated in my laboratory, so that you see these dislocation lines. And the way to count your dislocation density is by counting the number of these dislocation lines in this certain area of your image. So, finally, I will show you a video on a real dislocation structure, which is an in situ TEM measurement. So in this measurement, you are deforming this by micro pillar compression or nanopillar compression by deforming this small lead in M nanopillar, which is,very tiny. So recall that here is a scale bar of only 50 nanometers. So this pillar is, really small and you can see a lot of dislocations inside the pillar before deformation. And during the formation, you kind of compress the pillar. And you can see clearly that these dislocations move across the pillar, forming very complex, very dark structures. And eventually, as you can imagine when dislocation move across the entire crystal. You will end up with surface steps on the crystal surface. Okay, so, for today's lecture we have talked about the dislocations in real crystals. Thank you very much [MUSIC]
