0:04

We're beginning a new lesson here and finally after all these time we

are going to be introducing Fick's First Law, which is really the first time we

are going to begin to talk about the process of diffusion in solid state.

We're going to introduce here Fick's First Law.

In the case of Fick's First Law, it's an empirical relationship, and

we start out by looking at a cubic element.

We're going to be describing something we refer to as the flux, and

we're looking at in this cubic element the flux in x, y, and z directions.

And what we mean by that idea of flux, it is a measure of the number of

atoms that we have that cross a particular plane that's in a unit area.

0:53

Now, if we're in a situation and we're going to be in this as an approximation

as we go through the lectures that talk about solutions of problems in diffusion.

We're going to make an assumption that we're going to reduce the problem

to one dimension.

And the way we do, of course,

is we're going to assume that the flux in two of the directions,

namely in the y and z direction they're going to be equal to one another, and

we have a flux consequently in the x direction, which is different.

Now when we have a situation where the two fluxes in the y-z plane are equal,

it means that the composition in that plane is the same,

unlike the composition along the x direction, which is different.

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So when we look at a one dimensional problem then we look at the flux coming

out of the plane and as a result of the uniform composition we can

reduce fixed law rather than considering fixed law in three dimensional space.

We're just going to be looking at a one-dimensional problem here,

where we're only looking at the diffusion in the plane that is

given by J of X, which represents the flux in the X direction.

2:17

Okay, so let's look at an example.

We're going to be looking at a plane which contains two different types of atoms,

a blue atom and a light blue atom.

We're looking at the A atoms and we're looking at the B atoms.

And what's happening here is that, as we go from left to right,

we are seeing a concentration gradient, associated with the A atoms.

In other words, the number of atoms on the plane at one.

A blue is going to be greater than that of plane one, or

plane two containing the blue atoms.

So now what it tells us is we have a concentration gradient between

plane one and plane two.

3:07

Now, what's going to determine how quickly the atoms are going to move

from that plane one throughout the microstructure

will depend on the concentration gradient, how large the difference is between

the atoms that are on plane one and the atoms on plane two.

We're also interested in the concept of a term that we refer to as the diffusivity.

The diffusivity is how fast are the atoms moving at a given temperature.

We're therefore, also going to be interested in the temperature.

The higher the temperature, what we expect is that the process of diffusion,

which will occur much more readily as the temperature goes up.

The other thing that is important is how far the atoms have to move or

the jump distance, and of course the crystal structure itself.

And in this case,

we're just looking at it from a point of view of a simple two dimensional crystal.

4:11

And all of these factors combine to lead to an expression that was developed

again in curricula by Fick.

And what he said was that the flux of atoms in the X direction

is going to be proportional to the diffusivity D and the difference

in composition and the difference between the distance between the planes.

And when we write this in the form of a differential what we have

is that the flux in the x direction is going to be equal to

the negative of the diffusion coefficient times the C-D-X.

The reason that the negative is included here is because we're moving down

a concentration gradient.

We'll see in more advanced courses that the flux of atoms in

a particular direction more generally is related to.

The concentration gradient, so down a concentration gradient.

But we're going to find in more advanced courses that's not a necessary condition.

We just simply need to have a concentration gradient and

it doesn't necessarily mean that it is down the concentration gradient.

5:40

equal to some consonant D0 times the exponential of minus Q,

and here we're talking about an activation energy divided by rt.

What we mean by the activation energy of the defusion process is how much energy

is necessary in order to start the atoms moving from one place to another place.

So how much energy do we need to put in into the structure in order for

that defusion process to occur.

Again we have the gas constant and we have t as the temperature.

The expression that we have here is referred to as an erroneous expression,

and we have what's on the left hand side related to

this exponential dependence of the activation energy and D0.

What becomes important is, what is D0 and what are Q?

And depending upon the particular diffusion process and

the particular system D0 and Q will be specified.

So if we make this particular behavior linear so now we have the log of D

is going to be related to the log of D zero which becomes our intercept.

Now we have the linear relationship where we have negative q over r time one over t.

And I've put it that way that one over t because when we

starting looking at these erroneous plots the way we

usually plot them is based upon 1 over the temperature,

or the reciprocal temperature in units of kelvin to the minus 1.

So if we have our equation of Fick's First Law and

we look at this as a function of temperature, what we can do is

we can measure the diffusivity, and these can be done experimentally,

and we can measure the diffusivity at one temperature,

and the diffusivity of another temperature.

And over this entire relationship the expressions that I have written here hold.

And we're assuming that at these two different temperatures

the process of atomic motion is exactly the same.

So, therefore, we're going to have the same activation energy.

8:07

And, when we write the equation then,

knowing the diffusivity at two different temperatures, we can actually

calculate the activation energy cue, because everything else is known.

We have the diffusivity at one temperature,

the diffusivity of another temperature.

And, hence, we can solve for the value of Q.

So, let's go ahead and use an example calculation here, where,

what we are going to be looking at, is zinc dissolved in copper.

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The concentration of zinc is going to be relatively low.

And we're interested in the diffusivities of the zinc and the copper,

and what we're going to be given is how much at this two different temperatures

will be the diffusivity and by doing that, then we can come up with a value for

the activation energy for Zinc diffusing inside of Copper.

So the required data that we need to have is, what is the diffusivity

at one temperature, and the diffusivity of another temperature?

And if we measure these diffusivities, we can use those values or we

can use tabulated values of diffusivities at two different temperatures.

Once we have the diffusivities at two different temperatures,

we can put those back into our relationship and as a result of that,

we can write the expression that I have indicated up on the screen.

And we have again the constant which is our gas constant

in the appropriate units and the temperature again used in Kelvin.

9:51

And we can come up with a value of the activation energy.

And it's on the order of 190.8 kilo calories or kilo joules per mole.

Now, it is important for you to take this calculations and actually carry them out.

The reason that you want to do this is because you're looking at numbers or

you're using numbers that are extremely small like numbers like ten to the 11 and

ten to the 16 and

you're not maybe necessarily use to seeing that dimension number.

So it's important for you to actually try some of these calculations out and

make sure that you understand and

come up with what are orders of magnitude and typical values that you would have.

Now we're going to be doing these types of calculations throughout the remainder of

this module so you want to make sure that you understand this particular process.

Thank you.