0:04

Welcome back.

In the previous lesson, what we did was to describe how we determine

the compositions in the single phase field and in the two phase field.

But what's also important is how much

of the phases that we have when we are in the two phase field.

And, consequently,

what we need to do is to develop something that we refer to as the lever rule.

So, let's turn out attention now to a section of the isomorphous

diagram where we have liquid on the left and we have solid on the right.

And at that particular temperature, namely T1, the composition of the liquid is given

by dropping the vertical line down to the x axis and the same thing with the solid.

And we're then able to determine what the compositions of the phases are.

Remember that the composition of the liquidus determines

what the composition of the phases, and the same,

we determine the composition of the solid phase from the solidus boundary.

So, here is the composition of our liquid phase.

Here is the composition of our solid phase.

And the line that separates the two is referred to as the tie line.

This is our horizontal line where we are establishing

thermodynamic equilibrium with respect to temperature.

1:35

Now, what we need do in order for us to determine how much of one phase we have,

we recognize because we're dealing with a fixed amount of material.

What we know is once we get into a two phase field,

what we have are the two phases.

A solid phase and a liquid phase.

In this case what we're going to say is that the fraction

of those two phases has got to be equal to 1.

Or in other words whatever percentage of solid we have, whatever percentage liquid

we have, we have to recognize that we have 100% all the time of those two phases.

Now what we can do is, we know that the composition, that is XB0,

the composition of our alloy, is now divided into two parts.

Some of the material goes into the liquid phase,

some of the material goes into the solid phase.

So that's what the second expression on the visual gives you.

Now once we know that the fraction of the liquid and

the solid have to sum to 1, what we're able to do is to substitute for

the fraction of the liquid, 1 minus the fraction of the solid.

When we do that, and substitute it into that second equation,

what we're then able to do is to write an equation in which we have

eliminated the fraction of liquid.

And so now everything is written in terms of fraction of solid.

And we carry this through.

What we're then able to do is to determine what is the fraction of solid.

If you look at the line the way we're describing the fraction of solid

is the distance between XB naught and

XB with respect to the liquid phase.

So that's going to give us the fraction

of the liquid phase divided by the total distance

between the composition of the solid minus the composition of the liquid.

So, when you look at the picture, what we have is, the solid is on the right hand

side, but what determines the fraction of the solid phase is the distance

between XB0 and the composition of the liquid at that given temperature.

Now, conversely, what we can do is rearrange the equation as we had done

above by just simply then calculating what the fraction of liquid is.

And when we look at the fraction of liquid,

it is just simply given by the expression the distance,

as determined from the composition of the solid minus the composition of the alloy,

then divided by the entire composition range inside of that two phase field.

Although, once you have the composition, or the fraction of solid,

all you then need to do is to simply take the fraction of solid and subtract

that from 1, and of course, that's going to give you the fraction of liquid.

4:42

Now, another way we can think about this is to just simply use a lever rule.

For example, where you're on a playground and

you have a point which represents the balance point between you and a friend.

And, depending upon who is the heavier of the two people,

the way you need to put the fulcrum is, the fulcrum needs to be at the point

that is going to be closer to the heavier object.

So in this particular case,

it is like looking at a moment where it's the force over the distance, and

that has to be equal to the force over the distance with respect to the other mass.

So those two come together, and when we think about the Lever Rule,

we can think about it in this way, and once we see where our pivot point is,

we immediately will know which of the two phases are in the majority.

5:36

So, here is our two phase field again, and this time I want to stress another point.

Let's say that we're going to look at a series of alloys.

We're going to go from Alloy One, Alloy Two, Alloy Three,

all the way across the phase field.

So each time we mix up another composition, and

we take it temperature T1, hold it there until we reach equilibrium.

That is, the compositions are fixed and the temperature is fixed.

Now, when you look at the diagram, and you're interested in determining

the composition of the liquid, once again, you use the line, which

is the liquidus line and that will tell you what the composition of the liquid is.

Now, regardless of which of those several compositions of

alloys that you're interested in, you see that inside of that two phase field,

they all have the same composition of liquid.

So, regardless of what the composition of the alloy is,

the composition of the liquid is always fixed.

Once again, when we're interested in determining what

the composition of the solid is, what we see here is exactly the same thing.

Regardless of which alloy that we're looking at, 1 through 5,

the composition of the solid is always given there at the solidus boundary,

and that's going to be given in this particular case,

a value of 0.8 in terms of the fraction of B that we have in the alloy.

7:07

So, let's take another case and

let's look at our alloy at some elevated temperatures.

So now we're in a range where we have a single phase field,

our material is completely uniform in terms of its composition, and

the composition of the alloy is the composition in that single phase field.

Now, if we go through a process of equilibrium cooling,

what we're going to see is, as we move out of that single phase field and

we come down toward the two phase boundary and

cross the liquidus boundary, we then move in to that two phase field.

As we continue to cool our alloy, we go through the freezing range of the alloy.

And what is happening in that freezing range, as we come down in temperature,

what we see is the fraction of the solid phase is going to be increasing and

[COUGH] the fraction of the liquid phase is decreasing.

The other thing that we see is when we enter that two phase field,

the composition of the liquid phase is the same as the composition of the alloy.

Then when we exit that two phase field, the composition of the alloy and

the composition of the solid are effectively the same.

8:46

Now, what I have inserted over here to the right is,

I've put some numbers with respect to compositions on the phase boundaries.

We're going to use these in the next lesson so

we can actually now take some data, apply it to our analysis,

use the Lever Rule and actually determine how much

of the two phases we have as we move through that two phase field.

Thank you.