assumptions that we have to make.

And the major assumption that we're going to make is that the diffusion

of b is going to be independent of its surrounding compositions.

So, when we do that and we wind up summing all those

individual thin films, what we will see is a master plot,

which tells us that the composition is now going to

be a function in any particular interval delta alpha.

It's going to be a function of the contribution that that particular thin

film makes, along with the contributions that are made by the other thin films that

surround it and are in close proximity to that particular thin film region.

So now what we're going to do is when we look at the solution of composition

as a function of position in time, now what we have to do is, because we're

adding up all those contributions to those thin films, we now have a summation.

If we take that summation and re-replace the summation with an interval,

then what we find is a function that has been termed the error function.

And what we will now have is an equation that describes for

us the time dependent and position dependent, change in composition

during diffusion, and that particular solution has this form.

So we have the composition that's given here that we begin with.

And what we have is, in the error function, we have x, which is the position

where we are in space, d is the diffusivity and t represents the time.

Now, we need to go back and consider what is meant by the error function.

And the error function is like any of these functions that you deal with

in engineering like an exponential function or a sign function,

we can use the arrow function precisely the same way.

So what we have plotted here is what the value of z is on the x axis and

the corresponding value of the arrow function of that number, and

the properties of this particular arrow function is that it

increases with increasing z.

And what we also find is the range of values which

seem to be reasonable with respect to z and the error function of z.

Now what is in z?

Well, what is in z is x divided by two onto the square root of Dt.

So our experimental information goes in here, that is, what is the temperature,

and hence the temperature will tell us what the diffusivity is.

And how long is this material sitting in the particular furnace and

x is where we want to look at the particular value

of position to determine the composition at that position.

So we'll look at the values of z that we have on the x axis and

look at the corresponding value of the error function of z.

Now, one of the things that I want you to pay close attention to is the values that

we're looking at, that is x divided by the square root of Dt times two.

Those numbers generally will range between zero and one.

If you get numbers that are outside of that range,

you've done something wrong with your calculations.

The reason that I bring this up is because of the magnitudes

of the numbers like the diffusivity.

They can be on the order of 10 to the -11.

And until you get used to working with these numbers, you don't necessarily have

a feel for the values that are reasonable with respect to the error function.

So generally, the problems that we're going to be working with and

trying to use this error function, we're going to be looking at numbers that lie

within the range of say, .2 up to about .6.

So as long as your z comes into that range,

then you know reasonably that your calculations have

been correct with respect to position diffusivity and time.

We now have a solution that we can apply

to a variety of important problems that we're going to be

using in material science and engineering for processing materials.

Thank you.