WEBVTT

00:00:04.880 --> 00:00:06.390
STUDENT: Today
we're going to look

00:00:06.390 --> 00:00:08.560
at Hooke's Law in Cubic Solids.

00:00:08.560 --> 00:00:11.049
Hooke's law describes
behavior of springs.

00:00:11.049 --> 00:00:12.590
What we're going to
do is we're going

00:00:12.590 --> 00:00:16.450
to model the solid as
a collection of springs

00:00:16.450 --> 00:00:19.780
connecting a whole bunch of
atoms in a cubic lattice.

00:00:19.780 --> 00:00:23.160
Now, disclaimer-- I'm probably
going to do this all wrong.

00:00:23.160 --> 00:00:25.830
But what's science if we don't
make a few mistakes, right?

00:00:25.830 --> 00:00:26.580
Let's get started.

00:00:29.610 --> 00:00:32.040
The potential that
describes fairly well

00:00:32.040 --> 00:00:35.340
the behavior of the
interaction between two atoms

00:00:35.340 --> 00:00:36.690
is the Lennard-Jones potential.

00:00:36.690 --> 00:00:39.210
It describes the van
der Waals interaction,

00:00:39.210 --> 00:00:41.770
and it looks
something like this.

00:00:41.770 --> 00:00:47.550
It's a potential with
a term which goes 1

00:00:47.550 --> 00:00:51.450
over r to the 12th, and one term
which goes 1 over r to the 6th.

00:00:51.450 --> 00:00:54.270
And as you can see, as
the atoms get very close,

00:00:54.270 --> 00:00:57.570
they repel greatly, and then
they have a sweet spot here,

00:00:57.570 --> 00:00:59.810
which is the distance
they prefer to be at.

00:00:59.810 --> 00:01:02.820
As they get farther away, they
start to repel more again.

00:01:02.820 --> 00:01:06.020
The atom like to sit right there
in that little potential well.

00:01:06.020 --> 00:01:08.340
It's written up in Mathematica.

00:01:08.340 --> 00:01:12.560
You can see the equation
here, what each term means.

00:01:12.560 --> 00:01:15.660
And when I come here as
I take in the derivative.

00:01:15.660 --> 00:01:19.350
What that does is give me the
force on the particle when

00:01:19.350 --> 00:01:20.620
it's in this potential.

00:01:20.620 --> 00:01:24.240
So as we see, the force
is high and positive

00:01:24.240 --> 00:01:27.720
when it's closer
to the other atom,

00:01:27.720 --> 00:01:30.780
then the equilibrium
distance and the force

00:01:30.780 --> 00:01:32.430
is negative when it's farther--

00:01:32.430 --> 00:01:34.800
so it gets pulled
towards that sweet spot

00:01:34.800 --> 00:01:36.550
that I mentioned earlier.

00:01:36.550 --> 00:01:39.760
And here are the
two plus together.

00:01:39.760 --> 00:01:42.450
So with Hooke's Law, we're
modeling the atomic bonds

00:01:42.450 --> 00:01:44.191
as springs.

00:01:44.191 --> 00:01:45.690
And the way that's
going to work is,

00:01:45.690 --> 00:01:47.660
we have Hooke's Law which
is that the force is

00:01:47.660 --> 00:01:52.470
proportional to the
spring constant times

00:01:52.470 --> 00:01:53.489
the displacement.

00:01:53.489 --> 00:01:55.530
So the further you pull
it, the greater the force

00:01:55.530 --> 00:01:57.254
in its linear relationship.

00:01:57.254 --> 00:01:59.670
And the problem is, though,
our Lennard-Jones potential is

00:01:59.670 --> 00:02:00.180
not--

00:02:00.180 --> 00:02:01.710
it's not a normal
potential array.

00:02:01.710 --> 00:02:03.499
It's this weird wobbly shape.

00:02:03.499 --> 00:02:05.040
But for Hooke's Law
to work, you have

00:02:05.040 --> 00:02:07.164
to have something that
behaves like a spring, which

00:02:07.164 --> 00:02:08.767
has a parabolic
potential well, which

00:02:08.767 --> 00:02:10.850
is how you get that linear
force relationship when

00:02:10.850 --> 00:02:12.700
you take the derivative.

00:02:12.700 --> 00:02:16.460
So what I've done is
I've used Mathematica

00:02:16.460 --> 00:02:19.920
and its wonderful math tools
to take the Taylor expansion

00:02:19.920 --> 00:02:22.270
about that equilibrium point.

00:02:22.270 --> 00:02:24.770
And you can see here, this
is the second order Taylor

00:02:24.770 --> 00:02:27.310
expansion, so it's a parabola.

00:02:27.310 --> 00:02:30.480
And if you look at this
little manipulate here,

00:02:30.480 --> 00:02:33.400
I have set it up so you
can adjust the potential

00:02:33.400 --> 00:02:36.994
while depth can choose
the equilibrium distance,

00:02:36.994 --> 00:02:38.910
And you can look at the
different order Taylor

00:02:38.910 --> 00:02:42.570
expansion, so that's the
first order-- it's a line.

00:02:42.570 --> 00:02:44.225
The second order is
a parabola-- that's

00:02:44.225 --> 00:02:45.540
what we're going to be using.

00:02:45.540 --> 00:02:49.756
The other ones are just better
and better approximations

00:02:49.756 --> 00:02:53.410
of the action potential.

00:02:53.410 --> 00:02:56.540
Mathematica supposedly
can do negative powers,

00:02:56.540 --> 00:02:58.560
but I'm not seeing
that here, so could

00:02:58.560 --> 00:03:02.697
be the actual perfect model
would be obviously one over r

00:03:02.697 --> 00:03:04.530
to the 12th, there's r
to the negative 12th.

00:03:04.530 --> 00:03:10.556
So I differentiated that second
order potential like that.

00:03:10.556 --> 00:03:12.180
And you can see here
the force response

00:03:12.180 --> 00:03:18.920
for our generalized for an
approximated spring bond.

00:03:18.920 --> 00:03:23.210
And so you can look at
our force response here.

00:03:23.210 --> 00:03:27.239
Notice like before,
the force is positive

00:03:27.239 --> 00:03:29.405
when the atoms are close
together, and negative when

00:03:29.405 --> 00:03:31.279
they're farther apart--
so it gets pulled out

00:03:31.279 --> 00:03:32.737
of that sweet spot.

00:03:32.737 --> 00:03:34.570
And I've plotted it
here with the potential,

00:03:34.570 --> 00:03:38.250
so you can see how exactly it
interacts with that potential.

00:03:38.250 --> 00:03:40.704
Here is the
approximate potential

00:03:40.704 --> 00:03:42.370
with the actual
Lennard-Jones potential.

00:03:42.370 --> 00:03:45.200
So you can see that it's not
a very good approximation,

00:03:45.200 --> 00:03:49.580
but very small displacements
on the order of about 0.1 times

00:03:49.580 --> 00:03:53.420
that minimum distance
will probably be OK.

00:03:53.420 --> 00:03:56.240
I simplified it here to
get the-- and then I take

00:03:56.240 --> 00:03:59.060
the derivative again, just
so I can get that slope--

00:03:59.060 --> 00:04:01.590
and that'll be useful later.

00:04:01.590 --> 00:04:05.210
So here, we're modeling
our cubic lattice.

00:04:05.210 --> 00:04:07.880
Say we have something
like alpha polonium, which

00:04:07.880 --> 00:04:10.280
has a simple cubic structure--
the only metal that

00:04:10.280 --> 00:04:14.940
has a simple cubic structure
with a single atom motif.

00:04:14.940 --> 00:04:18.070
So we have our stress
over here, and our strain,

00:04:18.070 --> 00:04:20.540
and there's this
fourth ranked tensor

00:04:20.540 --> 00:04:23.270
that connects our second ranked
tensors with stress and strain.

00:04:23.270 --> 00:04:27.330
But, because of the wonders
of mathematics and matrices,

00:04:27.330 --> 00:04:32.450
we can actually break that down
into two first ranked tensors

00:04:32.450 --> 00:04:35.286
and a second ranked tensor.

00:04:35.286 --> 00:04:36.660
Which is pretty
great, because it

00:04:36.660 --> 00:04:39.090
means we don't want to do
some really early math.

00:04:39.090 --> 00:04:42.290
And so we can find that
infinitesimal strain tensor,

00:04:42.290 --> 00:04:45.310
if we're looking at a tiny,
tiny piece of a solid,

00:04:45.310 --> 00:04:46.950
will look something like this.

00:04:46.950 --> 00:04:52.460
It's this-- epsilon
IJ is equal to 1/2

00:04:52.460 --> 00:04:55.640
of the displacements
in each direction

00:04:55.640 --> 00:04:58.970
of the infinitesimal piece.

00:04:58.970 --> 00:05:01.070
And so from that, we
can build that out

00:05:01.070 --> 00:05:03.350
to the second rate
strain tensor,

00:05:03.350 --> 00:05:05.370
which is this tensor here.

00:05:05.370 --> 00:05:07.400
And it looks like this,
where u is the position,

00:05:07.400 --> 00:05:09.560
so we have all
these displacements.

00:05:09.560 --> 00:05:11.360
As you can see along
the principal axes,

00:05:11.360 --> 00:05:13.430
they're just partial
to the displacements

00:05:13.430 --> 00:05:15.824
and the directions
along the other axes,

00:05:15.824 --> 00:05:17.240
they're a little
more complicated,

00:05:17.240 --> 00:05:20.600
because you have things moving
in two directions at once.

00:05:20.600 --> 00:05:22.850
But the coolest part is that
only six of these entries

00:05:22.850 --> 00:05:23.420
are unique--

00:05:23.420 --> 00:05:25.430
because this is
the same as that.

00:05:25.430 --> 00:05:28.200
That's the same as that, and
that is the same as that.

00:05:28.200 --> 00:05:30.620
So what we can do is we
can just break it down

00:05:30.620 --> 00:05:32.630
to these six unique things.

00:05:32.630 --> 00:05:34.850
And then here is our second
ranked tensor, like I

00:05:34.850 --> 00:05:37.650
said before, of our elasticity.

00:05:37.650 --> 00:05:40.310
But, because we're using
Hooke's Law, these are springs

00:05:40.310 --> 00:05:42.410
and this is a simple
cubic lattice.

00:05:42.410 --> 00:05:46.680
So each atom is only attached
to six of its neighbors,

00:05:46.680 --> 00:05:48.680
so it's not going to
have any weird stresses.

00:05:48.680 --> 00:05:51.570
This is where I'm probably
wrong, but at this point,

00:05:51.570 --> 00:05:53.130
we are going to do this.

00:05:53.130 --> 00:06:00.150
So here are our diagonal
elasticity values.

00:06:00.150 --> 00:06:03.860
And if we do anything before and
simplify that spring potential,

00:06:03.860 --> 00:06:07.379
and grab the slope of that line,
we get our spring constant.

00:06:07.379 --> 00:06:08.920
So if this were a
spring, that's what

00:06:08.920 --> 00:06:10.650
the spring constant would be.

00:06:10.650 --> 00:06:13.757
And we just plug that in here
to our matrix, and we get--

00:06:13.757 --> 00:06:14.840
oh look, it's Hooke's Law.

00:06:14.840 --> 00:06:16.730
So if everything is
wonderful and linear,

00:06:16.730 --> 00:06:19.610
which it's probably
not, but if it were,

00:06:19.610 --> 00:06:22.070
we have the Hooke's
Law, and we can

00:06:22.070 --> 00:06:24.890
use that to determine from the
strain, the stress-- or vice

00:06:24.890 --> 00:06:25.765
versa.

00:06:25.765 --> 00:06:27.890
As you can see, it would
look something like this--

00:06:27.890 --> 00:06:30.710
so in the original position
of the spring's the black,

00:06:30.710 --> 00:06:33.530
and then when you drag
the atom over here,

00:06:33.530 --> 00:06:35.600
you get this red,
deformed spring.

00:06:35.600 --> 00:06:38.750
This one here's squished,
these are stretched.

00:06:38.750 --> 00:06:41.710
And yeah, that's about it.