0:04

Welcome back.

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

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

Â But what's also important is how much

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

Â And, consequently,

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

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

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

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

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

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

Â Remember that the composition of the liquidus determines

Â what the composition of the phases, and the same,

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

Â So, here is the composition of our liquid phase.

Â Here is the composition of our solid phase.

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

Â This is our horizontal line where we are establishing

Â thermodynamic equilibrium with respect to temperature.

Â 1:35

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

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

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

Â what we have are the two phases.

Â A solid phase and a liquid phase.

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

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

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

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

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

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

Â Some of the material goes into the liquid phase,

Â some of the material goes into the solid phase.

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

Â Now once we know that the fraction of the liquid and

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

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

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

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

Â eliminated the fraction of liquid.

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

Â And we carry this through.

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

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

Â is the distance between XB naught and

Â XB with respect to the liquid phase.

Â So that's going to give us the fraction

Â of the liquid phase divided by the total distance

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

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

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

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

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

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

Â And when we look at the fraction of liquid,

Â it is just simply given by the expression the distance,

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

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

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

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

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

Â 4:42

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

Â For example, where you're on a playground and

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

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

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

Â that is going to be closer to the heavier object.

Â So in this particular case,

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

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

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

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

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

Â 5:36

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

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

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

Â all the way across the phase field.

Â So each time we mix up another composition, and

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

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

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

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

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

Â Now, regardless of which of those several compositions of

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

Â they all have the same composition of liquid.

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

Â the composition of the liquid is always fixed.

Â Once again, when we're interested in determining what

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

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

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

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

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

Â 7:07

So, let's take another case and

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

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

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

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

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

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

Â we come down toward the two phase boundary and

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

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

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

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

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

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

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

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

Â the composition of the solid are effectively the same.

Â 8:46

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

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

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

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

Â use the Lever Rule and actually determine how much

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

Â Thank you.

Â