1, 2, 3, 4, 5, 6 slip planes that are basically intersecting every 1, 1, 1 directions. If you look down, for instance, if I were to look down this axis, yes, I would basically

see six 1, 1, 0 planes. So this would be a 1, 1, 1 direction. And these would be three 1, 1, 0 planes, yes?

And in between these 1, 1, 0 planes, there are also 1, 1, 2 planes, yes? So principle, a dislocation that's oriented along this axis

can just move into any one of these planes. There are many options to do. And it does that very easily, because these dislocations are not dissociated. So let's put up the slides here.

And things are very slow today. Must be Friday afternoon.

Even the systems are slow. There's a cross slip.

There we go. OK. So this is what I was telling you. You can have cross slip and the dislocations, screw dislocations in BCC alpha iron will easily do this, because they're not dissociated.

In the case of FCC, it can equally be easy. But the material has to have a high stacking fault energy. Not to be, the dislocations shouldn't be dissociated. And if you remember what was a high stacking fault energy,

well, there was about 100 millijoules per square meter. OK. Now up to now, we've discussed dislocations that dissociate, dislocations that exert forces on each other.

But what happens if dislocation crash into each other? They run into each other. And so what happens? Well, in specific cases, we get junctions.

Two dislocations will form a third dislocation. And whether or not this happens depends on the B square criteria. So one of the junctions that is,

and then when two dislocations react together, usually it's only a piece of their whole length that reacts together where they meet. And so you form a third dislocation.

And it may be that this third dislocation does not move as easily as the two other segments for reasons may be related to this structure or for intrinsic reasons. Because, for instance, the Burgers factor of the dislocation doesn't make it a glide dislocation.

And so that means that these junctions will act as pinning points, more or less strong pinning points. And it's a very important aspect of strain hardening.

What happens when dislocations meet each other? So one of the dislocation junctions is a product of dislocation reaction. They're very important for strain hardening, as I said. And some of these dislocation junctions can be sessile.

Dislocation people will say a dislocation is glissile, that means it can move, or it's sessile. If it's sessile, it's a dislocation that just doesn't move. It's very strongly pinned. And it doesn't matter how much stress you apply,

it just cannot move. So because these junction can be sessile, it will take a lot of force to get the rest of the dislocations to move.

And they will act also as barriers to other dislocation. So that all increases the flow stress, strain hardening, in other words. So one of these common junction in BCC iron is the A upon 2, 1, 0, 0 dislocation junction. And it's formed simply by having two coplanar screw

dislocations react. And the reaction is, I'll illustrate it in a moment. So A upon 2, 1, 1 bar 1 plus A upon 2, 1 bar 1, 1. If you sum these two vectors, you should get A upon 2,

2, 0, 0 or A, 1, 0, 0. So this is a different burger. You can see it's a very different burgers vector than this one. So first of all, question is, you can write thousands of these reactions if you want.

Whether or not they will happen, you check this by doing the b square criteria. So check the b square. So b square of the first dislocation, b square of the second dislocation is the same. Sum that, and you find 1.5 times a square.

And that's obviously larger than a square. So yes, the reaction is energetically favorable. So it will happen. And it does happen in practice. So here I have an array of screw dislocations.

Excuse me. Screw dislocations, these are dislocations lines, yes, on a 0, 1, 1 plane here. So one of these screw sets of dislocations has screw dislocation has this burgers factor, A upon 2, 1,

1 minus 1. And the other set is A upon 2, 1 bar 1, 1. And where they meet, yes, they can form A, 1, 0, 0 dislocation. And you see this actually happens in practice. You see here these screw dislocations in BCC iron,

actually a ferritic steel. And in certain areas, these dislocations will form this kind of honeycomb pattern, yes, where these little segments here are these A, 1, 0, 0 junctions.

So pen here, how does this work? So again, you can use this very simply. Let's see, right. I'm going to put this like this.

And the reason why, I just want you to look. If you look this way, yes, at this dodecahedron, it's square, right? It doesn't look hexagonal, right? And that's because we're looking down a 1, 0, 0 direction.

So I'm going to put this like this up here. Yeah? And so this guy is a z direction. This guy is a x direction. This guy is a y direction. And so this plane here is 1, 1, 1, 0 direction, OK?

And so I told you that all these edges here are Burgers factor for this slide plane, OK? So what we basically have here, so let me just turn it around. So say we have screw dislocations

like this that have a Burgers factor as this. And you have screw dislocations like this, which have a Burgers factor like this. All right?

So now I'm going back here. This vector here and this vector here, let's make the sum of these two vectors. So the sum of these two vectors is this is the same one here, right?

So this plus this is not the right reaction. I didn't want to do this, but anyway, it's very good that it happened. So let me correct this.

The Burgers factor of these guys are like this, yes? And the Burgers factor of these guys is in this direction. Then I have this Burgers factor and this Burgers. So the sum of these two is this one here, right? And so it's parallel to this z direction, all right?

OK? And this is an A1, 0, 0, 1 type of Burgers factor, OK? You had seen that originally, I mean, it wasn't intended. I had put this point in this direction.

And when I tried to react them, I get this vector. So that wouldn't happen because the b squared would not be smaller. But it's a possibility. As I said, you can do any type of reactions you want. So this will only happen for these two

specific Burgers factors. And they have to be in the screw orientation. So yes, that's basically what I wanted to say. That's one type of a junction, which is common in the BCC.

In FCC, it's a lot more difficult to make a quick drawing for reasons related to the fact that our dislocations are dissociated.

So I've tried to do this here for what is known as a Lohmer-Cottrell junction, usually short LC junction, yeah? And what you're basically looking at

is the reaction of, let me illustrate this with my, yeah, it's basically like this. So you've got this glide, this white plane is a glide plane.

It's this one here, this plane here. And on my tetrahedron, yes, and I use my tetrahedron to have another 1-1-1 plane. And that's this guy here, this guy.

You can see they intersect here. And they have, so here they have a common line. Right, so what we're basically describing is I have a dislocation here, yes, like this,

that interacts with a dislocation on this plane here, in the way that is shown here.

And in the way it's in the drawing, they intersect here. So this dislocation meets this guy. So in this particular case, what happens, of course,

is so this is the stacking fault. And this is the stacking fault for this dislocation. This dislocation here, when they intersect, this is where it meets this guy.

And in certain cases, there are many possibilities, but in certain cases, this location will be formed here at this intersection, which has the Burgers vector is delta alpha, delta alpha.

And that is a delta alpha. Delta alpha is, you can see this, is a Burgers factor,

yes, this is a dislocation, which does not belong to, which does not lie in the ABC plane. And it doesn't lie on the BCD plane either.

So when you have a dislocation with the Burgers factor, yes, that's not part of the glide plane, then the dislocation cannot move because the dislocation has

to move inside its glide plane. So the Burgers factor must be at least parallel to the glide plane, yes? So when the Burgers factor is perpendicular to the glide plane or at an angle, yes, it's

just stuck because it can't go anywhere. It's not on its glide plane. And you have very sessile junction. Now, I need to check something here because if a delta,

right, so this is, yeah. So do you have your notes? Did somebody print this thing from eClass? If you did, you should check this reaction here because it needs, it should be delta B plus A delta, right?

And it's, there's something else, I don't know why, but OK. OK. And that's what we call a Lomer-Cottrell lock, yeah?

And what is interesting at this Lomer-Cottrell lock is that, this entire piece is called Lomer-Cottrell, is that it consists of three pure edge dislocations.

So that means the Burgers factor is perpendicular to the line direction. OK, very strong junction. And yes.

So dislocations can interact. They do so in BCC iron and gamma iron in a different way. In BCC, we get these A100 type Burgers factor. In FCC, austenitic steels, we get Lomer-Cottrell locks.

In BCC, there are also interesting things that, in FCC, that happen. And those are things called jogs, yeah?

So usually we think of dislocations as being in the glide plane, yes? But parts of it can move to a different glide plane. And we've already seen that that

can happen, for instance, when you get cross-flip. When you get cross-flip. So if, let me go back here, if I have a dislocation here, yes, like this, which cross-slips to this plane here.

I create, and the dislocation, the screw dislocation,

then goes on to go back to a parallel plane, yes? Like this. So it goes from, it's on this plane. It cross-slips on this one. And then it cross-slips back on a parallel plane, yes? Then these pieces of dislocations here,

these pieces we call jogs, yes? This is the Burgers factor, b. This is the line direction, the original line direction. This is, of course, the line. The dislocation line now goes like this. Now, what is important is that if you

look at these jogs, the dislocation, whatever it does, it always keeps its Burgers factor, yes? It's always the same Burgers factor. Here, it's a screw dislocation. Here, it's a screw dislocation. But at the jog, it's an edge dislocation, right?

It's an edge dislocation, yes? And these, and we call this jogs. So these are very tiny things. I mean, there are not more than a few lattice planes

high, these jogs. So you can't really see them in the TM. But you can see the direct consequence of their presence. And for instance, you get these things, yes?

And this is obviously, something happens to this dislocation, which makes it go here, and then go back, and so forth. Well, what is it? It's just a point where the dislocation is stuck, yes? And it continues to move. And it trails behind it two dislocations, itself,

going towards that pinning point and away from it. And we call this a dipole. Dipole. So dipoles, when you see dipoles in your microstructure, and there are quite a few here. You see one here, a few here, yes?

You know that here, there is a jog on this dislocation. A small bit of edge dislocation that pins the dislocation there.

Another effect of the presence of jogs are these little loops. You see here, this is a structure. And it's lots of loops, lots of little. This is a very nice one here, you can see lots of this. All right? And we call this, it has a name, we call it debris, debris.

So how do we form these jogs? Well, you can form jogs as a result of cross slip, and we've already seen, yes?

Or you can form jogs as a result of dislocations into sections. These are two possibilities. Let's first have a look at the cross slip. So say a dislocation forms, this thing here,

this is in the slip plane of my dislocation. This is not a jog, these are not jogs, these are just kinks. When the dislocation, very careful here,

don't get your words mixed up. So if this is a glide plane, your dislocation, say your screw dislocation here, when it moves, it will move like this. It'll get larger, for instance. But if you look very, very careful,

really high magnification, if you could do this, what you would find out is that the dislocations actually jump a little bit. Pieces of them jump from what we call from one Peierls Valley to the other. And these little pieces of dislocation that you form, those are called kinks.

So they have nothing to do with, they're not related to the jogs here, which are connecting pieces of the dislocation from one glide plane to the other glide plane. And the jogs have very low mobility,

and they act as a pinning point, the edge jogs. So what you get is you can form a kink on the glide plane, but if it's a screw dislocation, there's nothing that prevents it from forming

a kink on another slip plane. So for instance here, this is a screw dislocation. You have these three or six 110 planes. And this screw dislocation forms

a kink on this glide plane, and it forms a kink on this glide plane. Now these two kinks move towards each other, and they form a jog. So this edge jog here, which pins the dislocation. So when the screw dislocation

tries to continue moving, it drags this dislocation behind it, this low mobility dislocation behind it. And you get formation of this dipole, these two parallel dislocations. Another possibility is that dislocations cutting each other.

So here I have a screw dislocation, this guy here. And here I have a screw dislocation on another glide plane, on another 110 type plane.

So let's assume this screw dislocation does not move. It does not move. And this screw dislocation moves from right to left. What does it mean, this Burgers factor? It basically means that the crystals is shifted.

When you pass the glide plane of the dislocation, the crystal gets shifted. So when this dislocation cuts through this guy here,

it gets shifted upward by this amount. And so it's the same dislocation, but I have an edge jog in it. And again, so this is low mobility,

acts as a pinning point. My screw dislocation can continue to move to the left, and it trails these two pieces of dislocation behind it. Now the dipole, the screw dislocation

is a screw dislocation. But the dipole, you can see, has edge components. So it means that you can think of it as an extra half plane inserted in the structure.

And we'll come back to that in a moment. But first, I want to say, for instance, how do we form these so-called debris, these little dislocation loops, in deformed ferritic steels

or BCC iron? Well, this happens when you form multiple jogs. Say, for instance, I have a screw dislocation that moves to the left here.

And this part of the dislocation goes through a number of jogs and ends up on this glide plane.

Nothing prevents the other screw dislocation segment to go through also successive jog formations. And so they can meet up back. They can meet again at a higher level.

And then they leave behind, basically, a dislocation loop. Yes? Let me explain this maybe with a simple situation. So I have this.

So say I have a dislocation that forms a jog, a single one. So this piece here is at the higher level, right?

On the higher level. So this dislocation, piece of dislocation, can move on. And it drags this thing behind it, yes? When this moves on, it can also drag this.

But this dislocation here can also form on itself a jog. Yes? And also create a jog. And rejoin this piece, yes? And so as a consequence, they will leave behind a dipole.

A lone dipole. And if I go back to some of the micrographs, you can see that that occurs apparently rather frequently.

Come on. You can see here some of these dipoles, like this dipole here, it seems like it's all by itself, right? So it can get pinched off. They can trail a dipole. And then the dipole can get pinched off.

And these loops here are basically relatively similar processes that create them. Right. Of course, you can imagine that if you have made a very large kink, there is little chance that this dislocation will, by chance, by statistical chance,

will rejoin the other one. Because it would have to make very specific kinks. So what happens to a edge jog, yes,

depends very much on the height of these jogs. One of the things that's interesting is that if you look at the dipole that is behind, that is trailing behind these screw

dislocations with a jog, if you look down the dipole here, this dipole can have an interstitial nature. It can look as if it consists of two edge dislocations oriented like this,

or two edge dislocations oriented like this. This means that the structure is as if you had made there is an extra half plane of atoms inside the dipole. This structure means that there is a missing row of atoms

inside the dipole. So it may sound strange, but dislocations create pointy facts this way. They make vacancies. They make interstitials, yes?

It doesn't happen when the jog is very large. And then we talk about super jogs. And super jogs, they just act as pinning points. And the screw dislocations that are hanging

are part of it, associated with it, rather. They behave as if the jog wasn't there, and they have this pinning point. And actually, by looping around this pinning point,

these screw dislocations can actually act as sources of dislocations, as dislocation sources. We'll talk about this in a moment. OK. So let's say something about this interesting aspect

of dislocations. When you take a piece of iron, alpha iron, pure iron, and you strain it, and you measure the density, yes, you find that the more you strain, the lower the density

is. It's not a big effect, but it's there. So this is the decrease in density. So the more you strain the material, the higher the decrease in the density.

It's a measurable effect. And the reduction in density is due to, first of all, the two effects. It's, first of all, the lattice dilatation that usually surrounds the dislocation core, and the vacancies that are formed by the motion of sessile jogs.

So the jogs that I just drew you tend to be vacancy-producing jogs. So let's read what's on the slide here. Plastic deformation gives you a significant increase

in point defects, in particular, vacancies. And because the concentration of vacancy is much larger than the thermodynamically stable concentration, we talk about excess vacancies. Excess vacancies.

So yes, and you can also form vacancies and self-intesticials, but that happens at higher temperatures when the edge dislocations move out of their glide plane. That happens with edge dislocations. It's a temperature effect usually.

This, if you have dislocation here, and this is the extra half plane, this dislocation doesn't have to stick to its glide plane. It can move to a higher glide plane

without the processes of a jog formation that I just described. That happens at high temperatures. At high temperatures, dislocations can climb out of their glide plane

by absorbing or emitting point defects. So for instance, if this extra half plane emits, let's say, an interstitial atom,

so then it will move up one lattice plane, and it will have created an interstitial atom. That kind of motion, dislocation motion, which is not the glide, we call non-conservative motion

of dislocations. We know from measurements that we produce about 10 times more vacancies as a result of deformation than interstitials.

And the values can reach as high as 10 to the minus 4. And it's a really high value. It's close to melting temperature concentration of vacancies. And so as a result, you get an increase in volume for the same number of atoms.

So let's have a look at whether we can calculate this increase or decrease in density rather. So we'll just simply calculate the effect

of the lattice dilatation associated with dislocations. The lattice dilatation around the core of an edge

dislocation corresponds to per lattice plane that's threaded by the dislocation to an addition of two vacancies. It's as the local density concentrations of vacancies

around the core. When you add its equivalent, the lattice expansion, it's equivalent to adding two vacancies.

So you'll ask me, well, how do you know this? Well, first of all, there are measurements. There is a technique called positron annihilation spectroscopy, which allows you to study point defects in metals.

And from these studies, we can learn a lot about how many defects we create in materials like steel or iron and find out these things.

So let's see what we do here. Let's see what happens in gamma iron. We take a cube of a very large cube of gamma iron,

one meter cube in size. Lattice parameter is 0.36. And we have dislocations. The density is rho d. And each dislocation creates a lattice dilatation of equal to two atoms volumes per plane that it threads.

So you can calculate the number of iron atoms in a cubic meter, simply with this formula here. So that's a lot of atoms. Then you can calculate number of vacancies.

And basically, you just count the number of 110 planes in a meter here. You count the number of the length of dislocation that you have. So that gives you, and you multiply by 2.

That gives you the amount of vacancies that you've created. So you have basically this number of atoms and this number of vacancies. And then you can calculate the density with and without these vacancies and calculate the decrease in density.

And you get this number. This number multiplied with the dislocation density. So if you work this out, you'll find values that are comparable to what is measured here. And you can also use this method,

and some people have used it, to measure dislocation densities. These density measurements are small, but they're measurable. You just need a good density, but conventional density measurement.

And you measure it as a function of strain. You assume, as I've said here, that you're around two atoms or two vacancies added per threaded plane. And you'll get your dislocation density. Pretty unusual way to determine dislocation density,

but it works. Yes, that's what I wanted to say. Then you probably wonder, well, don't you make an error? Don't you make an error because they're like thermodynamically stable vacancies? You're right, but at room temperature,

their number is very, very, very smaller than the volume change you have generated by deformation.

So now we come to the aspect of the generation of dislocations. And the generation of dislocation is usually what we say is that dislocations are generated

by Frank Reed sources. And Frank Reed source, you can have them in FCC or BCC metals and alloys. Here we'll just illustrate it with the BCC iron.

And this process is called double cross slip. It's double cross slip, which is something I just explained to you. It's one of the mechanisms by which

we can make Frank Reed source in BCC metals and alloys. So how does it work?

So the top picture is, again, something we've illustrated here. So what we have is a, let's do it like this.

So we go back here to our, so here I have a screw, this is again, a 1-0 plane.

This is, OK. Now I want you to imagine that there are 1-1-0 planes parallel, right? This is not the only 1-1-0 planes, as many of them are parallel to each other. And here, this location has made a jog, yes?

And it's decided to glide on this one, this glider. So here, my Burgers factor is like here. My Burgers factor here is like here.

And here I also have Burgers factor like the same Burgers factor. And these are the jogs, right? OK? And I've told you that the jogs are not, they're kind of sessile. They just don't move, yes? Because the dislocation is moving on this plane, yeah?

This little piece of dislocation has a Burgers factor and a line that's not on the glide. It's not a gliding situation, right? So it's basically stuck there, yeah?

And so this dislocation here can expand. When I increase the force, it will expand and become very, very large like this. This is what is happening here. So under the effect of an externally applied stress,

I will have shear stresses on the glide plane, and the dislocation will become half circle, yes? And then I should have brought this with me,

because you can actually illustrate this very nicely, is that once the radius of this dislocation segment is half the distance between these two edge jogs,

the stress needed to increase the diameter of this dislocation loop decreases, right?

It actually decreases. I don't know if you've ever tried to squeeze a balloon between two poles, yes? At the beginning, it's difficult. But once you have reached, the balloon

has reached a certain size on the other side of the poles, it flops through all by itself. I don't know if you've ever, ever done this. I was going to illustrate this. So this is this instability part. That instability point is precisely this.

The loop will spontaneously expand after you've reached this critical value of the radius. So the radius being half the distance between those two pinning points.

So the loop expands, continues to expand, and it curls back like this way. And what happens here, yes? So let's do this as an exercise. So our Burgers factor was this. This is the two pinning points.

And now, so these are the two pinning points. And this location has now expanded to this shape. So Burgers factor is the same everywhere, right? So this is the Burgers factor of these two pieces.

So now let's say we have chosen this as our line direction. So that means that I follow the line. So on this side, this is the line direction.

On this side, the line direction is in this direction. So let's look on this side. I have line direction is in this direction. The Burgers factor points to the right here.

So if I look in this direction, down the slip plane, this here is an edge dislocation with the extra half plane below the glide plane.

This one here, so this is the line direction, the Burgers factor. So my extra half plane is on top, comes from the top.

So like this. What do we remember? Two dislocations, edge dislocations, on the same glide plane. On the same glide plane.

On the same glide plane. So they will, different signs, they will attract each other. They will attract each other.

So you remember this curve, right? Like this and this one here. This is actually the one for this condition. Here we have a situation where y is 0.

y is 0. So I think this was x over y. So x over y is infinite. So it's basically an attractive interaction. And of course, when they attract each other, these two will just annihilate.

They will recreate. And when they do this, so your dislocation ends up, it basically is pinched off. So this is also very interesting about dislocation. They can disappear suddenly.

And this is what happens. These two edge parts will just disappear. They leave behind a dislocation loop. And they leave behind this segment here, which is actually nothing else than the original segment.

And the process can repeat itself. Now, can it repeat itself forever? It's a good question. Well, let's see. This is just in preparation for something in the future.

But if you have a dislocation loop, and then this thing creates another dislocation loop and another dislocation loop. If I look on this side, so if I look on this side,

here I have edge dislocations. And this was my vector along the dislocation line. So all the dislocations here look like this.

So they all repel each other. So if we reach, for instance, a grain boundary here, these dislocations will stop moving.

And they will repel each other, making it harder and harder for us, because of this back stress, making it harder and harder for us to make new dislocations. And that basically stops the production

of new dislocations. So this is a very important mechanism in iron and steel, ferritic steel, BCC iron, related to the internal generation

of dislocations in BCC iron. Right. So what is this stress relation? Well, we have seen this before, right?

What does it take? What shear stress does it take to get a certain radius? We know this formula. We've used it to calculate, for instance, what was the stress in this fatigued sample. I used this with the TM picture. So we use the same formula, which

relates the shear stress on the dislocation and the glide plane on the dislocation and the radius. And say if we have 82 gigapascal shear modulus

and a burgers factor of 0.248, which would be about the situation for alpha iron, this formula basically tells us about 10 divided by R gigapascal.

So that means that the shear stress required to activate Frank Reed's source will decrease with increasing length here.

When they're far apart, it's easier to generate dislocation than when the points are very close. So the diagram here is you have an increase with a decrease in R. So this is

1 because it's linear with 1 over R. So when your edge jogs are very close to each other, it's harder to initiate to get Frank Reed's sources

to activate Frank Reed's sources than when they're far apart. It's kind of interesting, right? Isn't it interesting? Because if you make your grain size, for instance, very, very small, really small, like a few hundred

or 50 nanometers, what happens then? Obviously, these two points have to be within the crystal, right?

So it basically also means that when you make very small crystals, it becomes very hard to generate dislocations, right? And if you could generate these whatever pinning points would be so close to each other that you'd really need huge forces to get

them to make dislocations, right? It's one of the reasons, among many, why nanostructured materials, their plasticity cannot depend only

on dislocations because there is a fundamental problem about even making them, OK? The other thing is, of course, because the grain size is so small, very quickly, even if you manage to make,

even if you manage to make, activate Frank Reed's sources, they quickly run into barriers, fundamental barriers, grain boundaries, to their glide, right? And again, you cannot generate many dislocations,

as many dislocations. And so it has an impact on your plasticity. So this is a real simple formula, but it has a key meaning. And

a key meaning and it's also actually quite useful because the kind of results you get from it, numbers you get from it, are very realistic. So let's close this bit here by talking, say a few things about point defects in

iron, in particular vacancies and interstitials. But before I do this, I think we'll take a break, yes, until twenty past five and then we'll finish, we'll restart it in about less than ten minutes

so I get a chance to drink something.