TINA CHEN: In this video, we will
be exploring the effect of bond energies on vacancy diffusion
using a Mathematica simulation.
Even the most carefully produced materials have defects.
These defects often affect the physical properties
of the material.
One type of defect is a vacancy in which
an atom or molecule is missing from a point in the lattice.
Vacancies can diffuse or move around.
In this video, we will investigate
diffusion of a single vacancy in a binary AB alloy.
To simplify the simulation process,
this AB alloy will have a two-dimensional, square
lattice.
In order to simulate the movement of a vacancy,
we must consider, on the atomic level, the interaction
between A and B. Specifically, we
will look at the bond energies.
Here, we have our AB alloy square lattice
with randomly placed atoms A and B. If we have a vacancy here,
then the vacancy can potentially jump to one of four places.
We look at the energy required to jump
to each of the four places.
If the vacancy wants to jump to this lattice point,
then it has to break each of the bonds
connecting the atom to other atoms.
The same is true for the vacancy to jump to any
of the other lattice points.
It has to break the bonds connecting
the atoms to other atoms.
The frequency with which the vacancy
jumps to any of the four neighboring lattice points
is proportional to E to the bond energy.
The specific equation relating these two
is the Arrhensius equation, which
shows that the frequency is also proportional to V,
the Debye frequency, which is a property of the material,
and the temperature.
For our simulation, we will only look
at how the bond energy affects vacancy diffusion.
Since we only have two types of atoms in this alloy,
there will only be three types of bonds--
AA, BB, and AB with corresponding bond energies,
VAA, VBB, and VAB.
For our simulation, we will start with a 2D square lattice
with randomly placed A and B atoms.
We will see the ratio of the atoms is 1 to 1.
We will also have a randomly placed vacancy.
We will then simulate a single jump of the vacancy
by comparing the frequencies with which
a vacancy will jump to the four neighboring lattice points.
We will consider only the case that the vacancy jumps.
That is, it does not stay at it's lattice point,
and it can only move to one of the four neighboring lattice
points.
Then we can calculate the probability with which
any of the four jumps occurs.
This probability will be the jump frequency
for a specific lattice point over the sum of all the jump
frequencies.
Note that we could have considered the possibility
that the vacancy did not jump, but then the value of the Debye
frequency would be needed.
And when we use the probabilities instead
of the actual jump frequency, the Debye frequency
cancels out, and we can consider a general binary alloy.
Now that we have the probabilities P1, P2, P3
and P4 of jumping to the respective lattice points,
we can simulate what the vacancy at this lattice
point surrounded by these atoms will do.
First, we pick a random number from 0 to 1.
The value of the random number will determine
where the vacancy will go.
From 0 to P1, the vacancy will jump to the first lattice
point.
From P1 to P1 plus P2, the vacancy
will jump to the second lattice point and so on.
This is the areas for which the vacancy
will jump to the third and fourth lattice points.
The number picked by our random number generator
will fall in one of these four areas.
And the area that number falls in
is the lattice point the vacancy will move to.
The vacancy then jumps to the lattice point
indicated by the random number.
In this way, we are using random samples
to simulate the randomness of the vacancy moving, given then
calculated probabilities to one of the four lattice points.
To complete the simulation, we then
repeat the computation of the probabilities
of jumping to the neighboring atoms, the random number
sampling that chooses where the vacancy moves,
and actually moving the vacancy.
First, we will look at the case when atoms of the same type
have lower bond energies than atoms of different types.
The animation allows us to visualize how the vacancy will
move within the lattice.
We can wait and see as T gets larger,
something seems to occur.
And that seems to be the agglomeration
of the same type of atom.
For instance, we can see the accumulation of red atoms
in this area.
We can let the simulation run or we
can see the beginning and end points after 10,000 steps.
We see here that originally the lattice was pretty random.
But after 10,000 steps of the vacancy,
we can see accumulation of red atoms here and blue atoms here.
Next, we will look at the case when atoms of the same type
have higher bond energies and atoms of different types
have lower bond energies.
Again, we animate and see what happens.
Now, atoms of the same type seem to be going away
from each other.
And instead, order seems to be appearing in this system.
We see here red, blue, red, blue, red, blue, red, blue,
red, blue.
Again, we can look at the initial and final products.
And we can see the initial lattice was quite random
and the end lattice is very ordered.
It's almost completely ordered.
Finally, we will look at the case
when atoms of the same type and atoms of different type
have similar bond energies.
We can animate, and we can wait a while,
but nothing particularly special will happen.
Again, our end products can show us
what will happen after 10,000 steps.
And we see that there is not much significant going on.
The beginning and end lattices are both rather random.
So how does a lattice decide which
form to approach as T gets very large?
The answer is, the strength of the bonds
between A and A, between B and B, and between A and B,
as we said before.
That is, do atoms of the same type
like or dislike each other?
Based on the answer, atoms are the same type will
either conglomerate, as in the first case, into regions of A
and regions of B or atoms will rearrange into an ordered form,
as in the second case, to be closer
to the type of atom it likes.
The affinity of an atom type for other atom types
is given by the bond energy.
And the type of lattice that will be formed
is given by the effective interchange energy,
which is given by VAA plus VBB minus 2VAB.
If epsilon is greater than 0, then
atoms of one type like the other type
more than they like each other, so the vacancy
will allow the crystal to obtain an ordered lattice.
If epsilon is less than 0, then atoms of the same type
like each other more than they like atoms of the other type,
and they will conglomerate by the vacancy diffusion
mechanism.
If epsilon equals 0, then disordered,
solid solution occurs.
This is because neither the VAA, VBB, nor the VAB phases
are favored.
Instead, they maintain a random order.
To create the simulation, me use the type of modeling
called Monte Carlo simulation.
The Monte Carlo method is simply a class
of computational algorithms that use repeated random sampling
for simulation.
In this case, we used random sampling in addition
to the probabilities to determine how likely
a destination for the vacancy each lattice point is.
Now take the model of the single vacancy diffusion
and imagine many, many more vacancies.
We can see now why defects can cause
changes in the physical characteristics
and even structure, in this case, of a material.
I'd like to thank professors Craig Carter, [INAUDIBLE],,
and Dwayne Johnson for their help and advice
with the coding in this video.