In this lesson, we're going to be describing a phenomenon

of diffusion that we refer to as self diffusion.

And an example for us to be able to consider,

how do we measure the diffusion pf aluminum in aluminum, or copper in copper?

We do that by using, for

example, radioactive isotopes when they are available.

So what is illustrated up here is a simulation of

the structure where the light blue circles represent the material itself,

the darker blue circles represent the radioactive elements.

The reason that we wanted to do this is that we can actually detect

the diffusion process by measuring at

different locations in the sample the concentration of the radioactive isotope.

So it starts out as two thin lines of the radioactive isotope.

At the initial time to use equal to zero.

We wait a period of time at an elevated temperature.

And because we have a concentration gradient here,

we're going to see the diffusion of those two different

lines going in the directions of positive and negative x.

Now the direction of the arrows then represent the direction of transport or

the direction of the flux of the atoms.

So the flux to the left and

the flux to right are going to be equal to one another.

Now at some later time, we see that the sphere,

or the circles of the radio that represent the radioactive isotopes,

have begun to become even more uniformly distributed throughout the structure.

In this particular process, we can refer to it as a self diffusion process,

even though we know that these radioactive,

elements have a slightly different mass than the non-radioactive isotope.

We can assume that, for all intents and purposes,

these measurements, these differences are going to be very small between the two.

And as a result, by tracking the elements, we can then come up with a value for

the tracer or the self diffusion of the material.

Why did we want to make these measurements?

Well, it turns out that we can begin to use the self diffusion measurements and

come up with some physical principles that underline the diffusion

process in a pure substance.

What I have done here is to plot the activation energy for

self diffusion Of these various elements and

I've done it as a function of the melting temperature of the material.

So for example, if we look at lead.

Lead has a lower melting temperature than aluminum than does

ultimately platinum, which has the highest melting temperature.

And you see this nice linear relationship between the melting temperature and

the activation energy for self diffusion, and

because all of these values are closely scattered around the line.

It's telling us something about the mechanism of diffusion in these materials.

The other point that I want to make is that all of these elements happened

to be in the form of a face centered cubic material.

So the variation then in the activation energy follows this nice plot.

And recall we have looked at other temperature dependent plots

earlier in the course where we looked at the melting temperature,

and we correlated it to the coefficient of thermal expansion.

And we've also looked at melting temperature and

its effect on the elastic modulus of the different elements.

And in that case, we related the kinds of behaviors to the bond strengths

that can be assigned to the temperature of melting.

So here this is telling us something about the bond strength which is

ultimately controlling the diffusion of the material.

The data that we're looking at comes from published material in the literature.

Now, if we go back and

we calculate the diffusivity of a certain number number of elements.

Again these diffusivities are aluminum through platinum,

where aluminum represents the lowest melting temperature and

platinum represents the highest melting temperature.

In these particular cases I've used data from the literature.

That allows me to calculate the diffusivity in the solid state,

but just at the melting temperature for each one of these elements.

And I have illustrated that on the plot that I have here.

So, what we have is aluminum, cooper, nickel, and platinum.

Again, all face centered materials, and when we look at

the melting temperature diffusivities, they're all fairly close to one another.

They're not too far away in terms of the values that they have.

And this, along with what we have just described previously,

as this linear behavior between melting temperature and the activation energy for

self-diffusion allows us then to come up with something that we refer to

as the homologous temperature theta, which is the dimensionless temperature.

And we can look at it from the point of view of the following type of equation.

Let's take a look at aluminum.

And if we look at aluminum and copper and nickel and platinum,

we can calculate what the diffusivity would be of all those different elements,

and they should be the same value at and equivalent homologous temperature.

In other words, if we define the homologous temperature as

the temperature of interest, which is below the melting temperature,

and the melting temperature of the particular material,

we would then come up with a diffusivity of all these different

elements based upon this homologous temperature.

And that's going to be equal to some constant divided by the homologous

temperature for the material.

So if we know something about the diffusivity of aluminum and

we know its melting temperature, we should then be able to calculate what

the diffusivity of copper would be at an equivalent homologous temperature.

Thank you.

4.7 Self Diffusion

Material Behavior

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