Uploaded by JUSTANEMONERD on 08.12.2012

Transcript:

PROFESSOR CIMA: A lot of interesting things to do today.

Almost too much. I couldn't sleep last night, I was so excited

about this. OK.

So, we ended last time talking about one-dimensional defects in crystals.

And we spent a little time looking at this figure.

And we pointed out that if you carefully look at this figure, you'll

find there's an extra half-plane in it. In other words, if I count the bubbles across the top, I'll get a different

number then I find across the bottom. Because someone stuck an extra half-plane

in this crystal. And as a result of that, if you look carefully

around this region where the half-plane ends, the packing is not like it

is out here. Here it's close packed.

Here I think you can see that that little atom is not in the nook between

atoms below. It's displaced from its original position.

It's been strained. And so we said this region around the end

of the dislocation is such a region of strain to accommodate the fact that

this plane ends halfway through the crystal.

And we talked about last time that there is a strain.

Any type of strain in a real material has an energy associated with it.

The strain-per-unit volume is the stress that I have to apply to get

that strain, times the strain-- remember, this is the percent change in dimensions--

is going to require some energy. So the fact that this dislocation is in this

crystal meant that if I strained things to put in there, it cost me

some energy to put it in. And in fact--

I didn't tell you how this was derived-- but it makes sense that it is proportional

to some material property, in this case, the shear modulus

of the material. Remember, the shear modulus, if I take a block

of material-- let's say my block like this--

and I push this way on the top and push this way on the bottom, this is

going to do that. Just like that.

And the relationship between the stress I apply and the change in

dimension is called the shear modulus. Now, there's this other thing in here called

the Burgers vector. And that's basically derived from this little

diagram here. Here's my edge dislocation in this square

lattice. It ends right here.

You can see all the strain that happens here. And in fact, you can tell that the dislocation

is there simply by walking around this loop.

What do I mean by that? Well, if I walk around this loop what I find

is that the number of steps I take on the top is different than the number

of steps I take on the bottom. So there had to have been an extra half-plane

in there. And we define this B by saying, oh, well,

if this were a perfect lattice, and I took exactly the same number of steps,

I wouldn't end up where I started.

I would be off by, in this case, one lattice unit.

And the difference between where I end and where I started in the perfect

lattice is called the Burgers vector of the dislocation.

And you can see the larger that is, the more strain had

to have been there. So it makes sense that the strain energy would

depend on the size of the Burgers vector of the dislocation.

And that's it. And then finally, I got this here.

This L is the length of the dislocation. That means this plane goes in and out of the

board. And it goes from one end of the crystal to

the other end of the crystal.

So that would be the length. If I double the size of the crystal to put

that half-plane in there, it takes double the amount of energy.

OK, so none of that's too surprising. So here are the things that we were supposed

to remember when we ended the lecture.

Energy is proportional to the amount of the strain.

So that's the Burgers vector squared. It's proportional to the length.

Now the significance of that is if it had its way the dislocation would

actually want to be as straight as possible. Why?

Because if it's not straight, it goes from here to here, it's longer.

And since the energy is proportional to the length, that means a

dislocation that wanders its way through the crystal has more energy

than one that goes straight through it. It makes sense.

And then finally, if you actually plug in the numbers here, you discover

something that's really astounding. And that is that the amount of energy associated

with an edge dislocation is 6 eV per unit length.

And the length is the lattice spacing. So in other words, for each lattice spacing

I go in and out of the board here, it costs me 6 eVs of energy.

So remember, to produce a point defect at room temperature, 1 eV or so, a

couple of eVs, well, at room temperature, you get a very small

number of vacancies. Well here, in order to get a dislocation that

go all the way through the crystal, I would have to multiply

the number of lattice spacings through the thickness of the crystal

times 6 eVs. And if you went through and, of course, did,

then, naively, you would say the number of dislocations divided by

the number of lattice planes is going to be something like E to the minus

E_A over kT. Now this E_A, E_D I'll call it, is this location,

is going to be this number, 6 eVs times the thickness of the crystal.

I'll call it L. And of course, this is just an absolutely

gimongous number, right? So it's E to the minus ginormous number, and

it goes way to zero. It doesn't happen.

It's not thermodynamic. So how do they even form?

How do you even get them? And this is how you get them.

They are extrinsic. You actually have to lean on the crystal and

deform it to create-- you have to do work.

You have to take the crystal and actually plastically deform it to get

a dislocation. So here's our perfect crystal up here.

And what I'm going to do is, again, shear this.

So here it is, my perfect crystal, and what I'm going to do is I'm going to

push on the top half over here, and the bottom half over here.

So that's a shear. The one force goes this way, the other force

goes that way. And the top half wants to move that way.

And the first thing it does is elastically deform.

Just like I said. So it has a shear modulus.

The first thing that happens is it elastically deforms.

And then something happens. A dislocation appears right at this edge.

You can see it in this diagram. Boom.

It just popped in. Just one at a time.

Exactly. Why only one?

Well it could be two. But it would be one, and it would move a little--

let's just save the next one for a moment. We push one in.

And then we continue pushing, and look what it did.

It moved to the right. And we continue pushing, and it moved to the

right. What is it doing?

STUDENT: Breaking and making new bonds? PROFESSOR: In the end, look at what I've done.

The crystal went right off the end. And look at it.

I've totally changed the crystal. It went from elastically deforming to, all

a sudden, I've got an extra lattice plane over here, and it came from

over here. So it's as if I broke the crystal in half,

picked it up, and moved it one lattice point at the end.

Now, of course, think about doing that. If I break the crystal in half, I have to

break all those bonds, every single bond at the same time.

And then, of course, I make them up the other end.

But that's not how it happens. What happens is this dislocation gets pushed

in and slips along. And in fact, the best analogy is when you

were a kid, and your mom asks you to pick up the rug and move it.

What did you do? Did you pick it up all at once and move it?

No. You pushed on one end, you get a little wrinkle

in it, and then push like this, and the wrinkle runs all the way

across. That's exactly what's happening in this phenomenon

called slip. The dislocation gets pushed in once.

And now as I push, what happens is it moves one lattice plane at a time,

right across. And in the end, I've made a permanent deformation

to the crystal. It's no longer like elasticity.

Remember elasticity? It comes right back.

So that's the stress-strain curve. Now for the macroscopic material.

Remember, this is at the micro level, but what you observe on the macro

scale is something that looks like this. So we start with our block of material and

it behaves elastically. STUDENT: So what are your --?

PROFESSOR: Stress. So this is the force per unit area that I

put it on this crystal. Force, and I got an area.

So if I divide that force by an area that gives me stress.

The strain is delta L over L. So in the early elastic case, when I move it

like this, it moved small distance. If I divide that distance by this, I get the

string strain. So this is elastic.

And then at a certain point, right up there, I punch in this location.

This is called the yield stress. And then what happens?

Well, it starts moving. It moves across.

But now think about that bump in the rug. As you push that bump in the rug, it doesn't

take much force to keep it moving.

And so in fact what happens is you just stay at the yield stress.

Each time you push, instead of building more stress,

the dislocation moves. And you push a little harder, and instead

of the stress getting larger, it just moves.

And so you do this until you stop. Take the stress off.

And now what happens? If I let off the stress, it comes right back

down elastically to this. And look at what I've done.

I've made a permanent deformation. We call it a plastic deformation.

Same thing. Plastic means that it's no longer elastic.

Once I'm done doing my work on it, it's now permanently changed shape.

Almost too much. I couldn't sleep last night, I was so excited

about this. OK.

So, we ended last time talking about one-dimensional defects in crystals.

And we spent a little time looking at this figure.

And we pointed out that if you carefully look at this figure, you'll

find there's an extra half-plane in it. In other words, if I count the bubbles across the top, I'll get a different

number then I find across the bottom. Because someone stuck an extra half-plane

in this crystal. And as a result of that, if you look carefully

around this region where the half-plane ends, the packing is not like it

is out here. Here it's close packed.

Here I think you can see that that little atom is not in the nook between

atoms below. It's displaced from its original position.

It's been strained. And so we said this region around the end

of the dislocation is such a region of strain to accommodate the fact that

this plane ends halfway through the crystal.

And we talked about last time that there is a strain.

Any type of strain in a real material has an energy associated with it.

The strain-per-unit volume is the stress that I have to apply to get

that strain, times the strain-- remember, this is the percent change in dimensions--

is going to require some energy. So the fact that this dislocation is in this

crystal meant that if I strained things to put in there, it cost me

some energy to put it in. And in fact--

I didn't tell you how this was derived-- but it makes sense that it is proportional

to some material property, in this case, the shear modulus

of the material. Remember, the shear modulus, if I take a block

of material-- let's say my block like this--

and I push this way on the top and push this way on the bottom, this is

going to do that. Just like that.

And the relationship between the stress I apply and the change in

dimension is called the shear modulus. Now, there's this other thing in here called

the Burgers vector. And that's basically derived from this little

diagram here. Here's my edge dislocation in this square

lattice. It ends right here.

You can see all the strain that happens here. And in fact, you can tell that the dislocation

is there simply by walking around this loop.

What do I mean by that? Well, if I walk around this loop what I find

is that the number of steps I take on the top is different than the number

of steps I take on the bottom. So there had to have been an extra half-plane

in there. And we define this B by saying, oh, well,

if this were a perfect lattice, and I took exactly the same number of steps,

I wouldn't end up where I started.

I would be off by, in this case, one lattice unit.

And the difference between where I end and where I started in the perfect

lattice is called the Burgers vector of the dislocation.

And you can see the larger that is, the more strain had

to have been there. So it makes sense that the strain energy would

depend on the size of the Burgers vector of the dislocation.

And that's it. And then finally, I got this here.

This L is the length of the dislocation. That means this plane goes in and out of the

board. And it goes from one end of the crystal to

the other end of the crystal.

So that would be the length. If I double the size of the crystal to put

that half-plane in there, it takes double the amount of energy.

OK, so none of that's too surprising. So here are the things that we were supposed

to remember when we ended the lecture.

Energy is proportional to the amount of the strain.

So that's the Burgers vector squared. It's proportional to the length.

Now the significance of that is if it had its way the dislocation would

actually want to be as straight as possible. Why?

Because if it's not straight, it goes from here to here, it's longer.

And since the energy is proportional to the length, that means a

dislocation that wanders its way through the crystal has more energy

than one that goes straight through it. It makes sense.

And then finally, if you actually plug in the numbers here, you discover

something that's really astounding. And that is that the amount of energy associated

with an edge dislocation is 6 eV per unit length.

And the length is the lattice spacing. So in other words, for each lattice spacing

I go in and out of the board here, it costs me 6 eVs of energy.

So remember, to produce a point defect at room temperature, 1 eV or so, a

couple of eVs, well, at room temperature, you get a very small

number of vacancies. Well here, in order to get a dislocation that

go all the way through the crystal, I would have to multiply

the number of lattice spacings through the thickness of the crystal

times 6 eVs. And if you went through and, of course, did,

then, naively, you would say the number of dislocations divided by

the number of lattice planes is going to be something like E to the minus

E_A over kT. Now this E_A, E_D I'll call it, is this location,

is going to be this number, 6 eVs times the thickness of the crystal.

I'll call it L. And of course, this is just an absolutely

gimongous number, right? So it's E to the minus ginormous number, and

it goes way to zero. It doesn't happen.

It's not thermodynamic. So how do they even form?

How do you even get them? And this is how you get them.

They are extrinsic. You actually have to lean on the crystal and

deform it to create-- you have to do work.

You have to take the crystal and actually plastically deform it to get

a dislocation. So here's our perfect crystal up here.

And what I'm going to do is, again, shear this.

So here it is, my perfect crystal, and what I'm going to do is I'm going to

push on the top half over here, and the bottom half over here.

So that's a shear. The one force goes this way, the other force

goes that way. And the top half wants to move that way.

And the first thing it does is elastically deform.

Just like I said. So it has a shear modulus.

The first thing that happens is it elastically deforms.

And then something happens. A dislocation appears right at this edge.

You can see it in this diagram. Boom.

It just popped in. Just one at a time.

Exactly. Why only one?

Well it could be two. But it would be one, and it would move a little--

let's just save the next one for a moment. We push one in.

And then we continue pushing, and look what it did.

It moved to the right. And we continue pushing, and it moved to the

right. What is it doing?

STUDENT: Breaking and making new bonds? PROFESSOR: In the end, look at what I've done.

The crystal went right off the end. And look at it.

I've totally changed the crystal. It went from elastically deforming to, all

a sudden, I've got an extra lattice plane over here, and it came from

over here. So it's as if I broke the crystal in half,

picked it up, and moved it one lattice point at the end.

Now, of course, think about doing that. If I break the crystal in half, I have to

break all those bonds, every single bond at the same time.

And then, of course, I make them up the other end.

But that's not how it happens. What happens is this dislocation gets pushed

in and slips along. And in fact, the best analogy is when you

were a kid, and your mom asks you to pick up the rug and move it.

What did you do? Did you pick it up all at once and move it?

No. You pushed on one end, you get a little wrinkle

in it, and then push like this, and the wrinkle runs all the way

across. That's exactly what's happening in this phenomenon

called slip. The dislocation gets pushed in once.

And now as I push, what happens is it moves one lattice plane at a time,

right across. And in the end, I've made a permanent deformation

to the crystal. It's no longer like elasticity.

Remember elasticity? It comes right back.

So that's the stress-strain curve. Now for the macroscopic material.

Remember, this is at the micro level, but what you observe on the macro

scale is something that looks like this. So we start with our block of material and

it behaves elastically. STUDENT: So what are your --?

PROFESSOR: Stress. So this is the force per unit area that I

put it on this crystal. Force, and I got an area.

So if I divide that force by an area that gives me stress.

The strain is delta L over L. So in the early elastic case, when I move it

like this, it moved small distance. If I divide that distance by this, I get the

string strain. So this is elastic.

And then at a certain point, right up there, I punch in this location.

This is called the yield stress. And then what happens?

Well, it starts moving. It moves across.

But now think about that bump in the rug. As you push that bump in the rug, it doesn't

take much force to keep it moving.

And so in fact what happens is you just stay at the yield stress.

Each time you push, instead of building more stress,

the dislocation moves. And you push a little harder, and instead

of the stress getting larger, it just moves.

And so you do this until you stop. Take the stress off.

And now what happens? If I let off the stress, it comes right back

down elastically to this. And look at what I've done.

I've made a permanent deformation. We call it a plastic deformation.

Same thing. Plastic means that it's no longer elastic.

Once I'm done doing my work on it, it's now permanently changed shape.