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TINA CHEN: In this
video, we will

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be exploring the effect of bond
energies on vacancy diffusion

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using a Mathematica simulation.

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Even the most carefully
produced materials have defects.

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These defects often affect
the physical properties

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of the material.

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One type of defect
is a vacancy in which

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an atom or molecule is missing
from a point in the lattice.

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Vacancies can diffuse
or move around.

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In this video, we
will investigate

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diffusion of a single
vacancy in a binary AB alloy.

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To simplify the
simulation process,

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this AB alloy will have
a two-dimensional, square

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lattice.

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In order to simulate the
movement of a vacancy,

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we must consider, on the
atomic level, the interaction

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between A and B.
Specifically, we

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will look at the bond energies.

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Here, we have our AB
alloy square lattice

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with randomly placed atoms A and
B. If we have a vacancy here,

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then the vacancy can potentially
jump to one of four places.

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We look at the energy
required to jump

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to each of the four places.

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If the vacancy wants to
jump to this lattice point,

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then it has to break
each of the bonds

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connecting the atom
to other atoms.

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The same is true for the
vacancy to jump to any

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of the other lattice points.

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It has to break the
bonds connecting

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the atoms to other atoms.

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The frequency with
which the vacancy

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jumps to any of the four
neighboring lattice points

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is proportional to E
to the bond energy.

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The specific equation
relating these two

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is the Arrhensius
equation, which

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shows that the frequency
is also proportional to V,

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the Debye frequency, which is
a property of the material,

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and the temperature.

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For our simulation,
we will only look

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at how the bond energy
affects vacancy diffusion.

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Since we only have two types
of atoms in this alloy,

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there will only be
three types of bonds--

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AA, BB, and AB with
corresponding bond energies,

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VAA, VBB, and VAB.

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For our simulation, we will
start with a 2D square lattice

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with randomly placed
A and B atoms.

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We will see the ratio
of the atoms is 1 to 1.

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We will also have a
randomly placed vacancy.

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We will then simulate a
single jump of the vacancy

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by comparing the
frequencies with which

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a vacancy will jump to the four
neighboring lattice points.

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We will consider only the
case that the vacancy jumps.

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That is, it does not stay
at it's lattice point,

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and it can only move to one of
the four neighboring lattice

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points.

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Then we can calculate the
probability with which

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any of the four jumps occurs.

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This probability will
be the jump frequency

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for a specific lattice point
over the sum of all the jump

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frequencies.

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Note that we could have
considered the possibility

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that the vacancy did not jump,
but then the value of the Debye

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frequency would be needed.

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And when we use the
probabilities instead

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of the actual jump frequency,
the Debye frequency

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cancels out, and we can
consider a general binary alloy.

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Now that we have the
probabilities P1, P2, P3

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and P4 of jumping to the
respective lattice points,

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we can simulate what the
vacancy at this lattice

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point surrounded by
these atoms will do.

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First, we pick a random
number from 0 to 1.

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The value of the random
number will determine

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where the vacancy will go.

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From 0 to P1, the vacancy
will jump to the first lattice

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point.

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From P1 to P1 plus
P2, the vacancy

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will jump to the second
lattice point and so on.

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This is the areas
for which the vacancy

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will jump to the third
and fourth lattice points.

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The number picked by our
random number generator

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will fall in one of
these four areas.

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And the area that
number falls in

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is the lattice point the
vacancy will move to.

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The vacancy then jumps
to the lattice point

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indicated by the random number.

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In this way, we are
using random samples

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to simulate the randomness of
the vacancy moving, given then

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calculated probabilities to
one of the four lattice points.

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To complete the
simulation, we then

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repeat the computation
of the probabilities

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of jumping to the neighboring
atoms, the random number

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sampling that chooses
where the vacancy moves,

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and actually moving the vacancy.

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First, we will look at the case
when atoms of the same type

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have lower bond energies than
atoms of different types.

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The animation allows us to
visualize how the vacancy will

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move within the lattice.

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We can wait and see
as T gets larger,

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something seems to occur.

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And that seems to
be the agglomeration

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of the same type of atom.

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For instance, we can see the
accumulation of red atoms

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in this area.

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We can let the
simulation run or we

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can see the beginning and end
points after 10,000 steps.

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We see here that originally
the lattice was pretty random.

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But after 10,000
steps of the vacancy,

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we can see accumulation of red
atoms here and blue atoms here.

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Next, we will look at the case
when atoms of the same type

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have higher bond energies
and atoms of different types

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have lower bond energies.

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Again, we animate
and see what happens.

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Now, atoms of the same
type seem to be going away

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from each other.

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And instead, order seems to
be appearing in this system.

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We see here red, blue, red,
blue, red, blue, red, blue,

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red, blue.

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Again, we can look at the
initial and final products.

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And we can see the initial
lattice was quite random

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and the end lattice
is very ordered.

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It's almost completely ordered.

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Finally, we will
look at the case

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when atoms of the same type
and atoms of different type

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have similar bond energies.

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We can animate, and
we can wait a while,

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but nothing particularly
special will happen.

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Again, our end
products can show us

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what will happen
after 10,000 steps.

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And we see that there is not
much significant going on.

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The beginning and end lattices
are both rather random.

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So how does a
lattice decide which

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form to approach as
T gets very large?

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The answer is, the
strength of the bonds

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between A and A, between B
and B, and between A and B,

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as we said before.

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That is, do atoms
of the same type

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like or dislike each other?

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Based on the answer, atoms
are the same type will

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either conglomerate, as in the
first case, into regions of A

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and regions of B or atoms will
rearrange into an ordered form,

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as in the second
case, to be closer

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to the type of atom it likes.

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The affinity of an atom
type for other atom types

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is given by the bond energy.

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And the type of lattice
that will be formed

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is given by the effective
interchange energy,

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which is given by VAA
plus VBB minus 2VAB.

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If epsilon is
greater than 0, then

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atoms of one type
like the other type

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more than they like each
other, so the vacancy

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will allow the crystal to
obtain an ordered lattice.

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If epsilon is less than 0,
then atoms of the same type

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like each other more than they
like atoms of the other type,

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and they will conglomerate
by the vacancy diffusion

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mechanism.

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If epsilon equals
0, then disordered,

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solid solution occurs.

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This is because neither the
VAA, VBB, nor the VAB phases

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are favored.

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Instead, they maintain
a random order.

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To create the simulation,
me use the type of modeling

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called Monte Carlo simulation.

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The Monte Carlo method
is simply a class

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of computational algorithms that
use repeated random sampling

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for simulation.

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In this case, we used
random sampling in addition

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to the probabilities
to determine how likely

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a destination for the vacancy
each lattice point is.

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Now take the model of the
single vacancy diffusion

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and imagine many,
many more vacancies.

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We can see now why
defects can cause

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changes in the physical
characteristics

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and even structure, in
this case, of a material.

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I'd like to thank professors
Craig Carter, [INAUDIBLE],,

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and Dwayne Johnson for
their help and advice

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with the coding in this video.