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NICOLA MARZARI: As you have
heard that this is actually

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the last class for the course,
it will be a split lecture.

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I'll start for half an hour.

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Then Professor Ceder
will continue and wrap up

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the course, and
then at the end, I

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will also spend 15 minutes
doing course evaluations.

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So we'll just leave you here
with the forms and filling up.

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On my side here, I
think I'll end up

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discussing a topic that's
actually very dear to me.

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I think it's been a very
intriguing and very fruitful

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application of modeling
and, in particular,

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modeling based on quantum
mechanical energy schemes

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to an interesting set
of scientific problems.

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We have said it
over and over again.

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A set of simulations
allow you to do

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sort of computational
experiments

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in areas where maybe things are
happening too fast to follow

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them with real
experiments, or maybe they

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happen on time scales--

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sorry, on length scales that
are too small to really follow.

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Or as in this case,
they take place

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under certain
thermodynamic conditions

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that are very difficult or
even impossible to reproduce

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with an experimental apparatus.

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Let's look, actually,
at the first case,

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and this is research
which a lot of work

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has been done at University
College in London

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by Dario Alfe and Mike Gillan.

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So this is actually a cut
out of the Earth interior,

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and in case you're not familiar
with this, the way it works

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is that we are actually sitting
on a large liquid region.

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So you see, on top here, we
have sort of the Earth crust.

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And what is very important
for us is actually

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this region, dark
yellow and light yellow,

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in which we have the so-called
core of the planet that

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is mainly constituted by iron.

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Iron actually is a
very stable nucleus.

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So instead of the early
formation of stars,

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it's one of the elements
that sort of, once created,

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sort of was more
stable than the other.

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And what happens
in our planet is

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that we have a sort of
increasing pressure going

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inside, and the inner
core, so-called,

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again, sort of mostly
iron is actually solid.

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And we know very well
what is the pressure here

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at the boundary between the
outer core and the inner core.

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This part is liquid.

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Pressure is increasing here.

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So this part is liquid.

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This part is solid.

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It's mostly iron.

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And for this sort of
case, we'll assume

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that is 100% iron,
although there

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is another problem of
really how much impurities

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they are diluted.

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And we know very well what
is the pressure there,

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because we can sort of
look at seismic waves

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and how seismic
waves get deflected

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by that discontinuities.

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And the pressure is
roughly 330 gigapascal,

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but we actually don't know what
is the phase diagram of iron

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at that pressure.

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And you see, if we knew what
the phase diagram of iron

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at that pressure was, we
could pinpoint exactly

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the temperature here,
because these points here

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are sort of points
of coexistence

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at 330 gigapascal
between the liquid

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and the solid phase
at that boundary.

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So if you want, by
knowing the pressure,

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we would be able via the
phase diagram of iron

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to actually pinpoint what is the
temperature inside the Earth.

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And before sort of the set of
experimental and theoretical

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work that I'll show, the
estimates, actually, for this

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were really sort of ranging
in a very wide spectrum

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of possibilities from 3,000
Kelvin to 8,000 Kelvin.

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And this is where sort of
our initial simulations

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become very useful,
because really,

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once you believe your energy
model, there is no reason

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to sort of think
that it would fail,

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either because the
pressure is too large

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or because the
temperature is too high.

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In all of this current
experimental capabilities

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to study the high
pressure system,

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the best instrument is what is
called a diamond anvil cell.

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That is basically
made by 2 diamonds.

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So the more perfect they
are, the better that they

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squeeze whatever
you're putting inside.

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And you know diamond
is transparent to a lot

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of wavelengths, so they are
very useful to probe them

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with spectroscopic techniques.

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But basically, now,
the best you can do--

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and you still sort
of break up hundreds

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of diamonds in the process--

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is that you can get
to 300 gigapascal

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if you are doing this
at room temperature.

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But as soon as you want to study
things at a higher temperature,

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really 200 gigapascal is the
limit that you can reach.

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Well, we want to figure
out what's happening here.

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There are other techniques--
experimental techniques

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to study matter under conditions
of very high pressure and very

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high temperatures done with
what is called a gunshot

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cannon in which you really
sort of shoot pellets

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and try to establish
what's going

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on in the instant in which
they hit their target.

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And so they find themselves
at very high condition

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of pressure, but
it's very difficult

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to get accurate
data out of this.

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And so we'll see
here actually sort

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of how ab initio simulation
and, in particular,

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thermodynamic integration has
been used to figure out what

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is going on in this region--

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in this region here.

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And as usual, you
need to make sure

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that you believe
your energy model.

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So these are a couple
of validation curves.

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This is for an HCP iron.

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That is one of the
high pressure phases,

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and the comparison between the
sort of experimental results

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and simulation
where the simulation

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are the sort of solid lines.

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And you see it agrees
very well in describing

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the equation of state, pressure
versus volume in this case.

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And then in particular,
this is a calculation

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of the phonon modes of
the vibrational frequency.

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This is something
that actually you

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could very easily do with
the quantum mechanical tools

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that you have done as a
sort of post-processing

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of your laboratory three,
but again, it works.

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It works.

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It works very well.

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So once you believe
iron, now the issue

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is trying to figure out what
is its phase diagram at, say,

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which temperature
you have coexistence

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between the solid
and the liquid.

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And you have actually done
this in your laboratory

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for a sort of explicit method
to find out the transition

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temperature by just doing
a simulation of a liquid

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and a solid phase, and figuring
out at which temperature

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the boundary between
the two phases

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doesn't really grow
into each other.

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So the solid doesn't
keep growing,

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or the liquid
doesn't keep growing.

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And you see an example of
this in sort of what follows.

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The caveat that in doing
this kind of simulation

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is that they are still sort of
very expensive to do ab initio.

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You usually need at
least 400 or 500 atoms

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to do such kind of
simulation and sort

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of to kill the
final size effects.

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And this is, even
now, borderline

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for what we can do even
for the simplest elements.

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There is another approach that
I wanted to introduce first,

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and this is the one in which
we actually sort of determine

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explicitly the
Gibbs free energy.

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So in this case, what
you want to find out

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is thermodynamic stability
at a given pressure

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and at a given temperature.

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So the Gibbs free energy is
your correct thermodynamic

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potential.

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And so your goal,
if you want, is

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being able to calculate
the Gibbs free energy

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of your solid system as
a function of temperature

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and the Gibbs free energy
of your liquid system.

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And of course, the point where
these two free energy cross

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is really your
coexistence point.

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Above that critical
temperature, the liquid phase

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is more stable.

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And below that critical
temperature, the solid--

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that solid phase is more stable.

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And sort of one possible way
of calculating, actually,

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explicitly this free
energy or, in particular,

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calculating the free energy
difference with respect

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to a known phase is to use
thermodynamic integration.

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That is a technique that
sort of has been discussed

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in some of the previous
lecture by Professor Ceder,

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so I've just put
up a reminder here.

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And the nomenclature is
that we are using, actually,

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a Hamiltonian or an
energy model that

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depends on a parameter lambda.

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And the parameter lambda
is something that, say,

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brings us from the solid to the
liquid, because ultimately what

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we want to sort of
figure out is, what

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is the difference in
Gibbs free energy,

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say, between the
liquid and the solid,

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or maybe between two solids,
between one reference

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phase of the solid and another
reference phase of the solid?

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And again, the idea behind
thermodynamic integration

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is fairly simple once you
skip all the math here.

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Again, one can write out
explicitly the partition,

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or the partition
function is written here.

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And then it just requires
a little bit of math,

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sort of figuring out
that the derivative

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of the, in this case,
Gibbs free energy--

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well, sorry, this is
actually a constant volume.

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00:10:00,850 --> 00:10:04,970
So it would be a sort of
[? isentropic ?] simulation.

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00:10:04,970 --> 00:10:05,820
No, sorry.

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00:10:05,820 --> 00:10:09,180
It would be constant
volume, constant temperature

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00:10:09,180 --> 00:10:10,140
simulation.

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00:10:10,140 --> 00:10:12,600
So [INAUDIBLE] energy.

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00:10:12,600 --> 00:10:14,820
So the derivative
of the free energy

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00:10:14,820 --> 00:10:19,200
with respect to this
parameter lambda--

201
00:10:19,200 --> 00:10:22,470
you see once you sort make the
appropriate substitution-- that

202
00:10:22,470 --> 00:10:27,060
is, you put the logarithm
of the partition function,

203
00:10:27,060 --> 00:10:28,950
and you work out
explicitly what is

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00:10:28,950 --> 00:10:31,990
that logarithm of the
partition function in there,

205
00:10:31,990 --> 00:10:36,225
you have 1 over q times the q
over the lambda, and 1 over q

206
00:10:36,225 --> 00:10:38,220
is the partition function here.

207
00:10:38,220 --> 00:10:40,830
And this is the derivative
of the partition function

208
00:10:40,830 --> 00:10:41,910
with respect lambda.

209
00:10:41,910 --> 00:10:45,480
You basically get
an expression that

210
00:10:45,480 --> 00:10:51,210
is nothing else than the average
in your ensemble normalized

211
00:10:51,210 --> 00:10:55,000
by the partition function
of the u over the lambda.

212
00:10:55,000 --> 00:10:58,620
So in order to calculate
what is the derivative

213
00:10:58,620 --> 00:11:01,740
of the free energy
with respect to lambda,

214
00:11:01,740 --> 00:11:06,000
you need to calculate what
is the thermodynamic average

215
00:11:06,000 --> 00:11:09,360
in your ensemble of the
derivative of the energy

216
00:11:09,360 --> 00:11:11,640
with respect to lambda.

217
00:11:11,640 --> 00:11:15,900
And so this is a quantity
that you can always calculate,

218
00:11:15,900 --> 00:11:20,860
and this is just an
average over your ensemble.

219
00:11:20,860 --> 00:11:23,050
So let's look at
sort of how this

220
00:11:23,050 --> 00:11:26,150
is done in a specific case.

221
00:11:26,150 --> 00:11:30,100
So again, I just returned
here the partition function,

222
00:11:30,100 --> 00:11:31,720
the logarithm--

223
00:11:31,720 --> 00:11:34,390
the free energy, sorry, as
the logarithm of the partition

224
00:11:34,390 --> 00:11:35,620
function.

225
00:11:35,620 --> 00:11:38,620
u will be our internal
energy, and this

226
00:11:38,620 --> 00:11:43,090
is what you calculate with
your quantum mechanical model.

227
00:11:43,090 --> 00:11:46,330
And for simplicity,
I've actually

228
00:11:46,330 --> 00:11:50,050
broken this in three
parts, and that is I've

229
00:11:50,050 --> 00:11:55,180
written out what is the
energy at the equilibrium

230
00:11:55,180 --> 00:11:59,920
and then the energy for
any possible configuration

231
00:11:59,920 --> 00:12:03,040
of the ions can be
written as the energy

232
00:12:03,040 --> 00:12:07,900
at equilibrium plus what is
called the harmonic term.

233
00:12:07,900 --> 00:12:12,310
So when you expand the energy
of a system around equilibrium,

234
00:12:12,310 --> 00:12:15,200
the first derivatives
are 0 by definition.

235
00:12:15,200 --> 00:12:18,310
And so here you have
the quadratic terms

236
00:12:18,310 --> 00:12:20,650
in the displacement
of the position

237
00:12:20,650 --> 00:12:22,210
around that
equilibrium position.

238
00:12:22,210 --> 00:12:25,780
And then you have all
the nonlinear term that

239
00:12:25,780 --> 00:12:28,360
will call sort of anharmonic.

240
00:12:28,360 --> 00:12:31,510
And it's actually, say,
in the case of a solid,

241
00:12:31,510 --> 00:12:33,920
this makes a lot of
sense, because it's

242
00:12:33,920 --> 00:12:38,500
sort of fairly easy to
calculate explicitly

243
00:12:38,500 --> 00:12:43,780
the integral of the partition
function for this term

244
00:12:43,780 --> 00:12:46,870
in the overall energy function.

245
00:12:46,870 --> 00:12:48,670
And then this sort
of is reserved

246
00:12:48,670 --> 00:12:53,860
for an explicit integration
than using molecular dynamics.

247
00:12:53,860 --> 00:12:57,580
And because basically you have
a logarithm of an exponential,

248
00:12:57,580 --> 00:13:00,310
if you sort of break
out of this thing

249
00:13:00,310 --> 00:13:04,360
in the sum of different
terms, all the terms that

250
00:13:04,360 --> 00:13:11,060
contribute separate free energy
pieces to your overall system.

251
00:13:11,060 --> 00:13:14,500
And so we could
calculate, let's say,

252
00:13:14,500 --> 00:13:20,410
for a solid system what is
the additional vibrational--

253
00:13:20,410 --> 00:13:22,210
in this case, free energy--

254
00:13:22,210 --> 00:13:25,510
that comes on top of the
equilibrium ground state

255
00:13:25,510 --> 00:13:30,040
energy, and coming just
from the harmonic term.

256
00:13:30,040 --> 00:13:34,090
And I haven't worked out
the algebra explicitly,

257
00:13:34,090 --> 00:13:37,480
but it's actually fairly
easy to do these integrals

258
00:13:37,480 --> 00:13:43,540
by writing the harmonic term in
terms of the interatomic force

259
00:13:43,540 --> 00:13:44,590
constants.

260
00:13:44,590 --> 00:13:50,380
So if you sort of figure out
with electronic structure

261
00:13:50,380 --> 00:13:54,070
codes what are the normal
modes of your system--

262
00:13:54,070 --> 00:13:57,580
that is, what are the
modes that give you

263
00:13:57,580 --> 00:14:01,240
an expression for the
harmonic term that

264
00:14:01,240 --> 00:14:03,610
is diagonal in the
displacement-- and you find out

265
00:14:03,610 --> 00:14:05,650
the frequencies,
well, you can actually

266
00:14:05,650 --> 00:14:10,720
substitute that expression in
here, work out the integral,

267
00:14:10,720 --> 00:14:16,820
and basically what you obtain is
that your harmonic vibrational

268
00:14:16,820 --> 00:14:24,700
free energy is just given by a
sum over the logarithm of all

269
00:14:24,700 --> 00:14:26,470
your vibrational frequency.

270
00:14:26,470 --> 00:14:29,890
So this omega here would
be your phonon frequency

271
00:14:29,890 --> 00:14:33,850
if you go back to what
we had seen before

272
00:14:33,850 --> 00:14:40,450
for the case of iron when we
have calculated the phonon

273
00:14:40,450 --> 00:14:41,140
spectrum.

274
00:14:41,140 --> 00:14:42,250
Here it is.

275
00:14:42,250 --> 00:14:47,680
So here, what we have calculated
is all the possible frequency

276
00:14:47,680 --> 00:14:51,860
for iron, depending
on the wave vector.

277
00:14:51,860 --> 00:14:55,520
So in principle, we have
a continuum of frequency,

278
00:14:55,520 --> 00:14:58,300
and we just need to basically
integrate these curves

279
00:14:58,300 --> 00:15:03,040
or sum over our course
representation of them

280
00:15:03,040 --> 00:15:05,440
to get the vibrational
free energy--

281
00:15:05,440 --> 00:15:07,810
at least the harmonic term.

282
00:15:07,810 --> 00:15:08,740
And this is it.

283
00:15:08,740 --> 00:15:11,500
And so this is sort
of the first step.

284
00:15:11,500 --> 00:15:16,050
This can give me, say,
what is the change

285
00:15:16,050 --> 00:15:19,770
in the vibrational
free energy in going,

286
00:15:19,770 --> 00:15:22,570
say, from one
phase to the other.

287
00:15:22,570 --> 00:15:24,960
And I just calculate
the phonon frequency

288
00:15:24,960 --> 00:15:28,020
in sort of two different phases.

289
00:15:28,020 --> 00:15:32,250
What is more intriguing
is the calculation

290
00:15:32,250 --> 00:15:35,340
of all the nonlinear
terms that can be very

291
00:15:35,340 --> 00:15:39,210
important at high temperature.

292
00:15:39,210 --> 00:15:42,780
And this is sort of the
slightly more difficult task,

293
00:15:42,780 --> 00:15:47,190
because especially in a
sort of ab initio framework,

294
00:15:47,190 --> 00:15:51,930
it requires very
extensive integration

295
00:15:51,930 --> 00:15:53,910
of your thermodynamic ensemble.

296
00:15:53,910 --> 00:15:57,330
So you would do something like
molecular dynamic simulations

297
00:15:57,330 --> 00:15:59,700
or Monte Carlo
simulation to sample

298
00:15:59,700 --> 00:16:04,470
as much as possible a set of
representative microstates,

299
00:16:04,470 --> 00:16:08,020
but it tends to
be very expensive.

300
00:16:08,020 --> 00:16:10,920
So there is actually
sort of a very useful,

301
00:16:10,920 --> 00:16:16,920
if you want trick that is often
used in order to calculate

302
00:16:16,920 --> 00:16:21,540
this anharmonic term, that
remember the anharmonic term is

303
00:16:21,540 --> 00:16:26,070
nothing else than the overall
vibration of free energy

304
00:16:26,070 --> 00:16:28,620
minus the harmonic term.

305
00:16:28,620 --> 00:16:34,800
And the trick are consistently
used in introducing a reference

306
00:16:34,800 --> 00:16:41,220
classical potential so that a
lot of your expensive sampling

307
00:16:41,220 --> 00:16:45,420
of the phase space can be done
with classical simulations

308
00:16:45,420 --> 00:16:48,900
and a very expansive ab
initio simulation are only

309
00:16:48,900 --> 00:16:52,530
used to figure out what
is the sort of difference

310
00:16:52,530 --> 00:16:56,250
in thermodynamical terms
between your quantum

311
00:16:56,250 --> 00:16:58,960
potential and your
classical potential.

312
00:16:58,960 --> 00:17:01,810
So again, sort of just
a mathematical term.

313
00:17:01,810 --> 00:17:09,510
I mean, we can sort of
think at our overall energy.

314
00:17:09,510 --> 00:17:13,200
So this would be the
overall, apart from this sort

315
00:17:13,200 --> 00:17:14,829
of equilibrium term.

316
00:17:14,829 --> 00:17:17,310
So what we are
trying to calculate

317
00:17:17,310 --> 00:17:20,190
is the difference
between the whole term

318
00:17:20,190 --> 00:17:22,859
minus the harmonic
term, but we actually

319
00:17:22,859 --> 00:17:26,609
introduce a classical potential
that sort of hopefully

320
00:17:26,609 --> 00:17:29,640
reproduces as
closely as possible

321
00:17:29,640 --> 00:17:33,900
the energies, the interactions
between all our atoms.

322
00:17:33,900 --> 00:17:38,610
And so we can actually break
down this term in two parts--

323
00:17:38,610 --> 00:17:40,560
one that has to do
with the difference

324
00:17:40,560 --> 00:17:43,050
between the harmonic crystal
and the reference system,

325
00:17:43,050 --> 00:17:45,660
and one that has to do with
the difference between sort

326
00:17:45,660 --> 00:17:50,700
of our overall energy term
here, or the overall free energy

327
00:17:50,700 --> 00:17:54,870
term, including non-linear
fx minus their reference--

328
00:17:54,870 --> 00:17:56,610
minus the reference system.

329
00:17:56,610 --> 00:18:00,360
And now, this here
is a calculation

330
00:18:00,360 --> 00:18:04,560
that involves only sort of
classical potential calculation

331
00:18:04,560 --> 00:18:05,910
of your phonon frequency.

332
00:18:05,910 --> 00:18:09,720
So this can be done
very extensively

333
00:18:09,720 --> 00:18:12,750
over longer classical
simulation, and in a way,

334
00:18:12,750 --> 00:18:16,830
it captures all the
complexity of the phase space.

335
00:18:16,830 --> 00:18:18,660
While this, that
is very expensive,

336
00:18:18,660 --> 00:18:22,140
because you need to calculate
the internal energy with sort

337
00:18:22,140 --> 00:18:24,720
of the quantum
mechanical model--

338
00:18:24,720 --> 00:18:28,320
is sort of you are
trying to sample

339
00:18:28,320 --> 00:18:31,500
the difference between
two quantities that

340
00:18:31,500 --> 00:18:33,870
are as similar as possible.

341
00:18:33,870 --> 00:18:37,770
And in the ideal case in
which your classical potential

342
00:18:37,770 --> 00:18:41,160
is able actually to reproduce
your quantum potential,

343
00:18:41,160 --> 00:18:43,140
this goes towards 0.

344
00:18:43,140 --> 00:18:45,780
So it's something that is
very simple to integrate.

345
00:18:45,780 --> 00:18:48,030
So by introducing basically
a reference potential,

346
00:18:48,030 --> 00:18:51,780
you are shifting a lot of
the thermodynamic integration

347
00:18:51,780 --> 00:18:53,520
to the classical simulation.

348
00:18:53,520 --> 00:18:57,030
And you are just using the
quantum mechanical simulations

349
00:18:57,030 --> 00:18:59,680
to sort of sample
the difference.

350
00:18:59,680 --> 00:19:01,170
And so this can
be done, actually,

351
00:19:01,170 --> 00:19:05,520
with much shorter and
much smaller simulations.

352
00:19:05,520 --> 00:19:08,220
And I want to go actually into
the details of what you need

353
00:19:08,220 --> 00:19:12,270
to do for the case of iron that
is what is a good potential,

354
00:19:12,270 --> 00:19:15,480
but people have figured out
that something as simple

355
00:19:15,480 --> 00:19:17,440
as basically a sort
of Lennard-Jones

356
00:19:17,440 --> 00:19:21,600
just like kind of potential
on top of anharmonic term

357
00:19:21,600 --> 00:19:26,980
is actually good enough to
provide a reference potential.

358
00:19:26,980 --> 00:19:30,090
So basically with
all this formalism,

359
00:19:30,090 --> 00:19:32,490
what one is able
to do at the end

360
00:19:32,490 --> 00:19:36,570
is calculate the free
energy differences in which,

361
00:19:36,570 --> 00:19:40,710
if you want the fundamental
components are calculating

362
00:19:40,710 --> 00:19:43,740
the phonon frequency
of your system--

363
00:19:43,740 --> 00:19:47,190
in this case, specifically
for the case of a solid--

364
00:19:47,190 --> 00:19:50,250
that gives you the first step
in the vibrational free energy.

365
00:19:50,250 --> 00:19:53,400
And if nothing else, then
this case, a density function

366
00:19:53,400 --> 00:19:55,080
or perturbational
theory calculation.

367
00:19:55,080 --> 00:19:57,060
You want the phonon
spectrum, and then you

368
00:19:57,060 --> 00:20:00,750
need to do a number of
molecular dynamic simulations

369
00:20:00,750 --> 00:20:04,050
on very large and
sort of very extensive

370
00:20:04,050 --> 00:20:05,670
for classical potential.

371
00:20:05,670 --> 00:20:08,400
And then if you want,
you just correct

372
00:20:08,400 --> 00:20:11,710
your classical simulations
with the difference

373
00:20:11,710 --> 00:20:15,370
between the classical and
the quantum potential.

374
00:20:15,370 --> 00:20:17,460
And once you have
all of data, you

375
00:20:17,460 --> 00:20:21,780
are basically able to calculate
all sort of the free energies

376
00:20:21,780 --> 00:20:22,840
that you need.

377
00:20:22,840 --> 00:20:24,840
And I guess this is an example.

378
00:20:24,840 --> 00:20:27,510
I'll show in a moment
a more telling example

379
00:20:27,510 --> 00:20:30,720
in which, again, sort
of we compare actually

380
00:20:30,720 --> 00:20:34,020
sort of what our prediction
are for the pressure,

381
00:20:34,020 --> 00:20:38,040
volume, curve coming from
the ab initio simulation

382
00:20:38,040 --> 00:20:39,840
and experimental results.

383
00:20:39,840 --> 00:20:42,060
But more importantly,
if you want,

384
00:20:42,060 --> 00:20:44,190
we are actually
able to figure out

385
00:20:44,190 --> 00:20:46,290
when the free
energy of the solid

386
00:20:46,290 --> 00:20:49,020
and the free energy
of the liquid cross.

387
00:20:49,020 --> 00:20:52,840
And we can do this as
a function of pressure.

388
00:20:52,840 --> 00:20:57,990
And so this is, at this
stage, our best prediction

389
00:20:57,990 --> 00:21:02,310
for the phase diagram of iron
as a function of pressure.

390
00:21:02,310 --> 00:21:04,920
In particular, the
black line here

391
00:21:04,920 --> 00:21:08,820
is sort of the set of
simulation that I've described.

392
00:21:08,820 --> 00:21:11,220
And there starts to be
a reasonable agreement

393
00:21:11,220 --> 00:21:13,950
between a number of
Shockley-Gunn experiments

394
00:21:13,950 --> 00:21:18,120
and a number of simulation that
puts roughly the temperature

395
00:21:18,120 --> 00:21:23,770
at 330 gigapascal, just
sort of above 6,000 Kelvin.

396
00:21:23,770 --> 00:21:24,630
And so this was--

397
00:21:24,630 --> 00:21:27,390
when it came out, the
scientific literature,

398
00:21:27,390 --> 00:21:30,660
it was sort of hailed as
taking the temperature

399
00:21:30,660 --> 00:21:32,280
of the core of the Earth.

400
00:21:32,280 --> 00:21:33,840
And at this stage,
it's, again, sort

401
00:21:33,840 --> 00:21:39,840
of our most accurate prediction
of what is going on inside.

402
00:21:39,840 --> 00:21:44,640
Just sort of to make the point
that this is not the only way,

403
00:21:44,640 --> 00:21:47,790
remember that one can
also do the simulation

404
00:21:47,790 --> 00:21:49,830
of the coexistence
between the solid

405
00:21:49,830 --> 00:21:55,560
and the liquid exactly as you
have done in your laboratory,

406
00:21:55,560 --> 00:21:57,300
in your lab number four.

407
00:21:57,300 --> 00:22:01,050
This is actually a paper,
again, sort of Science 2000,

408
00:22:01,050 --> 00:22:05,340
really sort of state
of the art research.

409
00:22:05,340 --> 00:22:07,550
And in these
particular simulations,

410
00:22:07,550 --> 00:22:09,730
they used classical potential.

411
00:22:09,730 --> 00:22:12,050
So again, not very
far from what you

412
00:22:12,050 --> 00:22:16,010
have done with the only caveat
that the classical potential

413
00:22:16,010 --> 00:22:19,460
had been extensively
optimized to describe

414
00:22:19,460 --> 00:22:21,650
solid and liquid iron.

415
00:22:21,650 --> 00:22:25,960
And the way that
optimization was done,

416
00:22:25,960 --> 00:22:29,420
it used what is called
a force matching

417
00:22:29,420 --> 00:22:32,660
method, in which
basically you do, again,

418
00:22:32,660 --> 00:22:37,400
extensive but small ab initio
molecular dynamic simulation.

419
00:22:37,400 --> 00:22:40,760
And you sort of tune
your classical potential

420
00:22:40,760 --> 00:22:43,340
in order to have
this more or less

421
00:22:43,340 --> 00:22:46,160
the mean square
error with respect

422
00:22:46,160 --> 00:22:48,050
to the ab initio simulation.

423
00:22:48,050 --> 00:22:50,180
And that often, because
you have basically so

424
00:22:50,180 --> 00:22:53,060
many configurations,
and so you sample

425
00:22:53,060 --> 00:22:56,180
so many possible different
environments for your atoms,

426
00:22:56,180 --> 00:22:59,660
it can somehow give
you a reasonably robust

427
00:22:59,660 --> 00:23:04,130
classical potentials, again,
in sort of very simple cases.

428
00:23:04,130 --> 00:23:06,860
This case is just
an elemental system,

429
00:23:06,860 --> 00:23:12,420
and sort of in a range of
temperature and pressure.

430
00:23:12,420 --> 00:23:14,720
So this was sort of
an example that I

431
00:23:14,720 --> 00:23:18,500
wanted to discuss a
little bit more in detail.

432
00:23:18,500 --> 00:23:22,370
I think all this sort of
research area of studying

433
00:23:22,370 --> 00:23:25,460
the properties of matter
at extreme condition

434
00:23:25,460 --> 00:23:28,910
has been extremely fruitful from
the point of view of ab initio

435
00:23:28,910 --> 00:23:30,170
simulations.

436
00:23:30,170 --> 00:23:33,350
And sort of there
are other cases

437
00:23:33,350 --> 00:23:39,260
in which actually the phase
diagram of, say, methane

438
00:23:39,260 --> 00:23:41,630
under high pressure
has sort of led

439
00:23:41,630 --> 00:23:43,790
to a number of
interesting discoveries.

440
00:23:43,790 --> 00:23:46,880
There are all the planets--

441
00:23:46,880 --> 00:23:53,820
Jupiter and beyond often are
made by a mixture of methane,

442
00:23:53,820 --> 00:23:57,290
water, or often, say,
in the case of Jupiter,

443
00:23:57,290 --> 00:24:00,080
a large amount of hydrogen.

444
00:24:00,080 --> 00:24:02,810
And the phase diagram of this
system, again, at very high

445
00:24:02,810 --> 00:24:05,150
pressure is largely unknown.

446
00:24:05,150 --> 00:24:07,430
One of the most interesting
sort of prediction

447
00:24:07,430 --> 00:24:10,970
that was made already
in the '30s by Wigner

448
00:24:10,970 --> 00:24:16,120
was that hydrogen,
under high pressure,

449
00:24:16,120 --> 00:24:21,130
should really start to behave
as all the other alkali metals

450
00:24:21,130 --> 00:24:22,060
in the first group.

451
00:24:22,060 --> 00:24:24,500
If you think
hydrogen is in there,

452
00:24:24,500 --> 00:24:28,090
and there's a sort of molecular
gas at ambient condition,

453
00:24:28,090 --> 00:24:30,190
but everything below it--

454
00:24:30,190 --> 00:24:32,560
it's really a simple metal--

455
00:24:32,560 --> 00:24:36,140
lithium, sodium, potassium,
and so on and so forth.

456
00:24:36,140 --> 00:24:39,640
And at ambient condition,
hydrogen, if you want,

457
00:24:39,640 --> 00:24:41,890
would belong much
more to group seven

458
00:24:41,890 --> 00:24:44,740
with the halogens
that really form

459
00:24:44,740 --> 00:24:49,960
dimer and gases, like a chlorine
or bromine, than on group one.

460
00:24:49,960 --> 00:24:52,240
But if you keep
compressing, compressing,

461
00:24:52,240 --> 00:24:56,110
there should be a point in
which it becomes metallic.

462
00:24:56,110 --> 00:24:58,960
And we haven't really
reached the state.

463
00:24:58,960 --> 00:25:01,480
There are indications from
the shotgun experiments

464
00:25:01,480 --> 00:25:06,970
that it could reach the metallic
state at 350, 400 gigapascal,

465
00:25:06,970 --> 00:25:09,010
but this is extremely
important, because, say,

466
00:25:09,010 --> 00:25:11,410
for a planet like
Jupiter, there is

467
00:25:11,410 --> 00:25:13,330
a huge difference
if we are going

468
00:25:13,330 --> 00:25:17,890
to have in the inside of the
planet, say, a molecular solid,

469
00:25:17,890 --> 00:25:21,130
or a metallic solid,
or a metallic fluid.

470
00:25:21,130 --> 00:25:24,610
If we have a hydrogen that
all of a sudden inside

471
00:25:24,610 --> 00:25:28,480
is a metallic fluid,
well, we can have there

472
00:25:28,480 --> 00:25:31,990
right away, like for the case
of Earth, a significant source

473
00:25:31,990 --> 00:25:35,980
of magnetic field if we start
to have a sort of circulation

474
00:25:35,980 --> 00:25:37,090
in that fluid.

475
00:25:37,090 --> 00:25:39,310
So all of this is
very important,

476
00:25:39,310 --> 00:25:43,380
and I think one of the most
intriguing prediction that

477
00:25:43,380 --> 00:25:46,270
actually sort of
had originally come

478
00:25:46,270 --> 00:25:50,860
just as a hypothesis
by Martin Ross in 1981

479
00:25:50,860 --> 00:25:53,290
was that, in some
of this planet,

480
00:25:53,290 --> 00:25:57,100
there could be an interesting
decomposition of methane.

481
00:25:57,100 --> 00:25:59,440
So basically, what
could be going on

482
00:25:59,440 --> 00:26:05,200
is that inside, say, Neptune,
as you increase the pressure,

483
00:26:05,200 --> 00:26:09,760
your methane molecule under
the effective pressure

484
00:26:09,760 --> 00:26:12,250
starts decomposing.

485
00:26:12,250 --> 00:26:15,340
And this actually
has been seen in

486
00:26:15,340 --> 00:26:17,560
quantum mechanical simulations.

487
00:26:17,560 --> 00:26:20,410
And so what you have
is that-- on one side

488
00:26:20,410 --> 00:26:23,620
is that the carbon atoms--
and you have a breaking

489
00:26:23,620 --> 00:26:25,670
of the carbon-hydrogen bond.

490
00:26:25,670 --> 00:26:31,150
And so you have formation
of pure carbon, of diamonds,

491
00:26:31,150 --> 00:26:34,480
and you have formation
of higher hydrocarbons.

492
00:26:34,480 --> 00:26:35,950
And actually, there
has always been

493
00:26:35,950 --> 00:26:38,260
a sort of experimental
observation

494
00:26:38,260 --> 00:26:41,950
of an anomalous presence
of high order hydrocarbons

495
00:26:41,950 --> 00:26:44,000
on the surface of this planet.

496
00:26:44,000 --> 00:26:46,330
And so this simulation
for the first time

497
00:26:46,330 --> 00:26:48,760
provided a reason
why there should

498
00:26:48,760 --> 00:26:51,400
be higher order hydrocarbons.

499
00:26:51,400 --> 00:26:55,330
They are basically created
when methane collapses

500
00:26:55,330 --> 00:26:59,500
under pressure, and then somehow
from deep inside the planet,

501
00:26:59,500 --> 00:27:01,570
these sort of
convective currents

502
00:27:01,570 --> 00:27:03,910
bring these hydrocarbons up.

503
00:27:03,910 --> 00:27:06,520
But the other prediction
that, I think,

504
00:27:06,520 --> 00:27:09,820
even if not confirmed at the
time is the most appealing,

505
00:27:09,820 --> 00:27:13,330
is that at the same time you
have nucleation of diamond,

506
00:27:13,330 --> 00:27:16,990
and so there is this beautiful
picture of basically a rain

507
00:27:16,990 --> 00:27:20,770
of diamonds taking
place inside the planet

508
00:27:20,770 --> 00:27:24,320
and basically converging
to the inside.

509
00:27:24,320 --> 00:27:26,980
And again, this was
just a prediction,

510
00:27:26,980 --> 00:27:32,680
but I think in 2002 or 2003--
maybe it's mentioned here.

511
00:27:32,680 --> 00:27:38,530
No, 1999, so just two or three
years after the previous paper,

512
00:27:38,530 --> 00:27:41,950
finally there were sort of
diamond anvil cell experiments

513
00:27:41,950 --> 00:27:45,970
made by Raymond
Jeanloz at Berkeley.

514
00:27:45,970 --> 00:27:50,410
And all of a sudden, methane
under pressure sort of

515
00:27:50,410 --> 00:27:53,110
gave rise both to
higher hydrocarbons,

516
00:27:53,110 --> 00:27:55,740
but also to the clear
signature of diamonds.

517
00:27:55,740 --> 00:27:59,650
So they saw basically the
nucleation of pure diamonds

518
00:27:59,650 --> 00:28:04,180
just under pressure, and I
guess this brought in general,

519
00:28:04,180 --> 00:28:08,470
both in the geophysical and in
the planetary community, a lot

520
00:28:08,470 --> 00:28:12,400
of resonance for the importance
of this ab initio simulation

521
00:28:12,400 --> 00:28:14,690
in the field.

522
00:28:14,690 --> 00:28:17,410
And the last example
that I wanted

523
00:28:17,410 --> 00:28:21,280
to show in terms of sort
of geophysical or planetary

524
00:28:21,280 --> 00:28:24,070
importance is actually
the one of water

525
00:28:24,070 --> 00:28:28,540
that, as a very intriguing
phase that was sort of

526
00:28:28,540 --> 00:28:32,750
suggested for the first
time, again, in 1999--

527
00:28:32,750 --> 00:28:37,510
and what I'm showing here is
just a network of solid oxygen.

528
00:28:37,510 --> 00:28:40,510
So here I'm just showing the
oxygen for a high pressure

529
00:28:40,510 --> 00:28:41,200
phase.

530
00:28:41,200 --> 00:28:43,540
In this simulation,
we were looking

531
00:28:43,540 --> 00:28:46,660
at 20 gigapascal of pressure.

532
00:28:46,660 --> 00:28:49,180
And what happens is
that, if you increase

533
00:28:49,180 --> 00:28:52,060
the temperature
enough, actually water

534
00:28:52,060 --> 00:28:55,720
becomes superprotonic--
that is a very exotic phase.

535
00:28:55,720 --> 00:28:59,380
So the covalent bonds
between hydrogen and oxygen

536
00:28:59,380 --> 00:29:04,000
breaks apart, and the
system is still a solid.

537
00:29:04,000 --> 00:29:07,180
Again, high pressure even if
there is high temperature,

538
00:29:07,180 --> 00:29:11,920
but the hydrogens start
moving around like a liquid.

539
00:29:11,920 --> 00:29:14,350
So you have a
coexistence of phases

540
00:29:14,350 --> 00:29:17,200
between a solid phase
of one sublattice

541
00:29:17,200 --> 00:29:20,560
and the other sublattice
that has gone liquid.

542
00:29:20,560 --> 00:29:22,360
There are a number
of sort of materials

543
00:29:22,360 --> 00:29:25,180
that, even at ordinary
conditions, do this.

544
00:29:25,180 --> 00:29:29,380
There are some salts silver
iodide that does this,

545
00:29:29,380 --> 00:29:32,830
and there are interesting
fuel cell materials

546
00:29:32,830 --> 00:29:35,570
that do this, because,
again, in this case,

547
00:29:35,570 --> 00:29:38,710
say, when you have a
superprotonic, in this case,

548
00:29:38,710 --> 00:29:41,770
phase-- that is, when you
have that the protons behave

549
00:29:41,770 --> 00:29:44,320
as a liquid-- well this
system becomes a very

550
00:29:44,320 --> 00:29:46,900
intriguing ionic conductor.

551
00:29:46,900 --> 00:29:50,090
But again, this was a
prediction from 1999.

552
00:29:50,090 --> 00:29:52,330
And actually, just
a few weeks ago

553
00:29:52,330 --> 00:29:54,820
came the experimental
confirmation

554
00:29:54,820 --> 00:29:58,750
from Lawrence Livermore,
and lo and behold,

555
00:29:58,750 --> 00:30:01,970
the superprotonic phase
of water has been found.

556
00:30:01,970 --> 00:30:05,030
It has been found that 47
gigapascal instead of 20

557
00:30:05,030 --> 00:30:08,260
gigapascal as usual,
I mean, because of all

558
00:30:08,260 --> 00:30:11,230
the errors involved in
density function theory,

559
00:30:11,230 --> 00:30:13,900
and in this specific case,
also, the the errors involved

560
00:30:13,900 --> 00:30:16,780
with the sort of quantum
mechanical description

561
00:30:16,780 --> 00:30:19,300
of the vibrational
excitation that, remember,

562
00:30:19,300 --> 00:30:21,790
is a Bose-Einstein gas.

563
00:30:21,790 --> 00:30:26,110
And I'll conclude
with the last slide

564
00:30:26,110 --> 00:30:30,950
that, again, it's not any
more geophysical or planetary,

565
00:30:30,950 --> 00:30:34,810
but is, again, another very
intriguing example of what

566
00:30:34,810 --> 00:30:39,160
was discovered in the
high pressure regime.

567
00:30:39,160 --> 00:30:43,060
And this is, again, some
work from the late '90s that

568
00:30:43,060 --> 00:30:47,560
has to do, actually, with
the behavior of alkali,

569
00:30:47,560 --> 00:30:51,520
like sodium or lithium
under pressure.

570
00:30:51,520 --> 00:30:56,650
These are metals that we really
sort of describe and consider--

571
00:30:56,650 --> 00:31:00,340
simple metals, where
the valence electron

572
00:31:00,340 --> 00:31:04,750
behave as almost nearly
free electron gas.

573
00:31:04,750 --> 00:31:06,670
Think at lithium for a moment.

574
00:31:06,670 --> 00:31:10,330
You have a core with two
electron in the 1s orbital,

575
00:31:10,330 --> 00:31:13,930
and then you have a single
electron on the 2s orbital that

576
00:31:13,930 --> 00:31:15,910
becomes very delocalized.

577
00:31:15,910 --> 00:31:18,630
And now this is what was
discovered that, I think,

578
00:31:18,630 --> 00:31:20,020
it was very appealing.

579
00:31:20,020 --> 00:31:22,720
And at some level,
it was also confirmed

580
00:31:22,720 --> 00:31:25,330
recently by the
Carnegie group that does

581
00:31:25,330 --> 00:31:27,640
a lot of these high
pressure experiments,

582
00:31:27,640 --> 00:31:31,930
but basically, when you start
compressing your system, what

583
00:31:31,930 --> 00:31:35,560
you have is that the
2s electrons start

584
00:31:35,560 --> 00:31:37,100
overlapping more and more.

585
00:31:37,100 --> 00:31:38,800
If you want that
very high pressure,

586
00:31:38,800 --> 00:31:43,990
even the core of your lithium
atoms start overlapping.

587
00:31:43,990 --> 00:31:47,410
And what's happening
is that Pauli principle

588
00:31:47,410 --> 00:31:49,270
becomes exceedingly relevant.

589
00:31:49,270 --> 00:31:53,560
That is, you start to
have 2s electrons that,

590
00:31:53,560 --> 00:31:57,590
by pressure, would be forced to
be almost in the same quantum

591
00:31:57,590 --> 00:31:58,090
state.

592
00:31:58,090 --> 00:32:01,000
They start to overlap
very significantly.

593
00:32:01,000 --> 00:32:04,780
And what happens is that,
in order to escape this,

594
00:32:04,780 --> 00:32:07,270
the electrons
actually self localize

595
00:32:07,270 --> 00:32:09,190
in the interstitial position.

596
00:32:09,190 --> 00:32:12,970
So the 2s electrons, from sort
of being around the lithium

597
00:32:12,970 --> 00:32:17,170
atom, sort of moves
around and jumps

598
00:32:17,170 --> 00:32:19,090
into the interstitial position.

599
00:32:19,090 --> 00:32:23,040
It localizes, and it actually
pairs with another electron.

600
00:32:23,040 --> 00:32:27,250
So you have that your unit
cell becomes double or sort of

601
00:32:27,250 --> 00:32:28,720
becomes more complex.

602
00:32:28,720 --> 00:32:32,260
And actually, your system
is a phase transition

603
00:32:32,260 --> 00:32:37,090
from the metallic case of the
sort of ordinary simple metals

604
00:32:37,090 --> 00:32:39,850
to an insulating case
in which, actually,

605
00:32:39,850 --> 00:32:44,230
really a pair of electrons
localize in the interstitials.

606
00:32:44,230 --> 00:32:47,980
And again, these are actually
sort of cutting edge research

607
00:32:47,980 --> 00:32:51,160
that one nowadays can
do on a simple computer,

608
00:32:51,160 --> 00:32:55,090
because studying something
like the equation of state

609
00:32:55,090 --> 00:32:59,500
of a simple metal in a variety
of [INAUDIBLE] lattices

610
00:32:59,500 --> 00:33:03,070
and a variety of
crystallographic incarnation

611
00:33:03,070 --> 00:33:04,990
is actually very simple to do.

612
00:33:04,990 --> 00:33:06,490
It's nothing different
from what you

613
00:33:06,490 --> 00:33:09,910
have done for, say, silicon
or the semiconductors

614
00:33:09,910 --> 00:33:12,040
in your first laboratory.

615
00:33:12,040 --> 00:33:14,170
With this, I'll
conclude my part,

616
00:33:14,170 --> 00:33:17,630
and Professor Ceder
will continue from here.

617
00:33:17,630 --> 00:33:19,480
And again, as a
summary, just wanted

618
00:33:19,480 --> 00:33:24,100
to give you an example of our
whole line of research that

619
00:33:24,100 --> 00:33:28,810
has come out from modeling
and very accurate modeling,

620
00:33:28,810 --> 00:33:33,407
and really going in directions
where experiments would be--

621
00:33:33,407 --> 00:33:38,410
or still are exceedingly
difficult, or even impossible

622
00:33:38,410 --> 00:33:40,570
to do.

623
00:33:40,570 --> 00:33:44,710
And with this, I'll switch
on to the next lecture.

624
00:33:44,710 --> 00:33:45,630
Let me--

625
00:33:45,630 --> 00:33:49,570
GERBRAND CEDER: So let me hammer
in again the sort of overview,

626
00:33:49,570 --> 00:33:53,260
the kind of guideline by
which we set up the course.

627
00:33:53,260 --> 00:33:56,590
I think by now hopefully
you'll see the structure.

628
00:33:56,590 --> 00:33:58,120
We covered energy methods.

629
00:33:58,120 --> 00:34:00,940
We covered simulation
methods, and then

630
00:34:00,940 --> 00:34:02,810
the last third of
the course was really

631
00:34:02,810 --> 00:34:07,390
a kind of showing you how
these are combined together,

632
00:34:07,390 --> 00:34:09,280
and that's often still
a very hard part,

633
00:34:09,280 --> 00:34:16,070
how you combine all these things
to make impact on something.

634
00:34:16,070 --> 00:34:18,830
And this is-- being
at MIT, impact

635
00:34:18,830 --> 00:34:22,550
is the one thing we
worry a lot about.

636
00:34:22,550 --> 00:34:25,610
And if you see the
kind of things that--

637
00:34:25,610 --> 00:34:29,810
to integrate modeling more
with materials research,

638
00:34:29,810 --> 00:34:32,639
there are really these
four poles that you need.

639
00:34:32,639 --> 00:34:35,030
There's obviously
method development,

640
00:34:35,030 --> 00:34:36,739
and we told you
a lot about that.

641
00:34:36,739 --> 00:34:38,510
We told you what
goes into DFT, what

642
00:34:38,510 --> 00:34:43,120
goes into molecular
dynamics, Monte Carlo,

643
00:34:43,120 --> 00:34:46,010
there's an important
factor of dissemination,

644
00:34:46,010 --> 00:34:48,469
which is getting better
and better done, I think,

645
00:34:48,469 --> 00:34:49,250
in modeling.

646
00:34:49,250 --> 00:34:51,800
There's now a lot of stuff
where you really just

647
00:34:51,800 --> 00:34:53,929
can fairly easily
get access to codes.

648
00:34:53,929 --> 00:34:57,230
You get quantum codes, good MD
codes-- sometimes for money,

649
00:34:57,230 --> 00:34:59,240
but in many cases, for free.

650
00:34:59,240 --> 00:35:01,280
And then there's, of
course, the education part,

651
00:35:01,280 --> 00:35:06,380
which we try to address here.

652
00:35:06,380 --> 00:35:10,780
This is actually a picture from
last year, or two years ago.

653
00:35:10,780 --> 00:35:15,330
I actually think I have one
from longer ago where, see,

654
00:35:15,330 --> 00:35:18,572
Professor Marzari still has
a beard, which is how you

655
00:35:18,572 --> 00:35:19,780
know how old that picture is.

656
00:35:19,780 --> 00:35:23,327
But anyway, but I think,
then, the fourth pole

657
00:35:23,327 --> 00:35:24,660
is the one you shouldn't forget.

658
00:35:27,630 --> 00:35:30,000
You think that for modeling
you only need modeling,

659
00:35:30,000 --> 00:35:35,280
but I think what you need more
sometimes than modeling is

660
00:35:35,280 --> 00:35:37,140
the basic signs of your field.

661
00:35:37,140 --> 00:35:40,920
I think, without understanding
the field in which you work--

662
00:35:40,920 --> 00:35:42,960
the theory, the
material science of it--

663
00:35:42,960 --> 00:35:44,370
there's really
very little impact

664
00:35:44,370 --> 00:35:45,912
you can do with
modeling, because you

665
00:35:45,912 --> 00:35:48,780
can be a good modeler, but if
you don't know what to model,

666
00:35:48,780 --> 00:35:50,730
you're never going
to have impact,

667
00:35:50,730 --> 00:35:53,880
unless you choose to remain
on the pure theory and method

668
00:35:53,880 --> 00:35:55,030
development side.

669
00:35:55,030 --> 00:35:59,000
But I would argue that, even
then, you need to understand.

670
00:35:59,000 --> 00:36:01,500
You have to have field specific
knowledge, because otherwise

671
00:36:01,500 --> 00:36:06,220
you never know what is going
to be important to develop.

672
00:36:06,220 --> 00:36:10,740
So I sort of hope you
don't forget that.

673
00:36:10,740 --> 00:36:14,970
So this being the
end of the course,

674
00:36:14,970 --> 00:36:20,230
I'll give you a few more maybe
sort of loaded perspective--

675
00:36:20,230 --> 00:36:23,740
personal perspectives on
the field, and what's good,

676
00:36:23,740 --> 00:36:26,500
and what are the potential
directions about it.

677
00:36:29,550 --> 00:36:32,330
This is what people
want from you.

678
00:36:32,330 --> 00:36:34,190
They want properties.

679
00:36:34,190 --> 00:36:36,230
They want behavior.

680
00:36:36,230 --> 00:36:38,900
And this is where you
essentially start,

681
00:36:38,900 --> 00:36:41,630
and typically you see people
draw these kind of slides

682
00:36:41,630 --> 00:36:43,370
of the multi scale perspective.

683
00:36:43,370 --> 00:36:46,940
You go from atoms to
microstructure to continuum.

684
00:36:46,940 --> 00:36:50,720
This is essentially--
has almost never worked.

685
00:36:50,720 --> 00:36:52,830
This is great for raising money.

686
00:36:52,830 --> 00:36:56,030
It's a great sort of cartoon
view grab of what you do.

687
00:36:56,030 --> 00:36:59,720
It's rarely that you
actually ever do this.

688
00:36:59,720 --> 00:37:02,120
There are very few
properties where you make it

689
00:37:02,120 --> 00:37:04,760
across all the length scales.

690
00:37:04,760 --> 00:37:12,000
And so what you really do is you
kind of connect at some level.

691
00:37:12,000 --> 00:37:13,880
You sort of connect
these through your brain.

692
00:37:13,880 --> 00:37:16,670
In many cases,
there are properties

693
00:37:16,670 --> 00:37:21,140
we determine just by doing
stuff at the electronic level.

694
00:37:21,140 --> 00:37:24,110
And we don't barter
coarse-graining that explicitly

695
00:37:24,110 --> 00:37:25,407
through the length scales.

696
00:37:25,407 --> 00:37:26,990
And what you're
really doing is you're

697
00:37:26,990 --> 00:37:30,840
using your basic materials
theory knowledge-- for example,

698
00:37:30,840 --> 00:37:36,530
that maybe when you work
on color of materials.

699
00:37:36,530 --> 00:37:39,380
I mean, I had this one--
jewelers once came to me

700
00:37:39,380 --> 00:37:41,690
and asked if I
could predict color

701
00:37:41,690 --> 00:37:47,090
for gemstones and for thin
layer-- thin surface layers

702
00:37:47,090 --> 00:37:49,050
that they could use to coat.

703
00:37:49,050 --> 00:37:52,540
So in many cases, those are
fairly local properties,

704
00:37:52,540 --> 00:37:54,510
especially in oxides.

705
00:37:54,510 --> 00:37:57,720
So there you sort of decide
yourself, gee, there's

706
00:37:57,720 --> 00:37:59,910
kind of one length
scale that's important--

707
00:37:59,910 --> 00:38:03,310
this one-- and I'm going to
directly calculate my property.

708
00:38:03,310 --> 00:38:06,520
So this is you.

709
00:38:06,520 --> 00:38:09,010
This is, I think, why your
field specific knowledge is,

710
00:38:09,010 --> 00:38:13,810
in some sense, more important
than all these multi scale

711
00:38:13,810 --> 00:38:16,560
techniques.

712
00:38:16,560 --> 00:38:21,830
This is my pet topic that--

713
00:38:21,830 --> 00:38:25,750
it may seem obvious
that you can--

714
00:38:25,750 --> 00:38:28,310
an obvious goal to try
to simulate all the way

715
00:38:28,310 --> 00:38:31,120
from electrons to properties
by coarse-graining the length

716
00:38:31,120 --> 00:38:31,650
scale.

717
00:38:31,650 --> 00:38:34,670
But you have so much more
impact if you just substitute

718
00:38:34,670 --> 00:38:37,670
that idea by knowledge.

719
00:38:37,670 --> 00:38:39,260
I think, if all
the money had been

720
00:38:39,260 --> 00:38:42,410
spent on trying to integrate
length scale had been spent

721
00:38:42,410 --> 00:38:44,810
on actually trying to
integrate materials

722
00:38:44,810 --> 00:38:48,950
modeling with material science
by connecting your field

723
00:38:48,950 --> 00:38:52,020
specific knowledge, by trying
to figure out what to compute,

724
00:38:52,020 --> 00:38:57,720
which is often the hard part,
we'd actually done much better.

725
00:38:57,720 --> 00:39:03,570
So I'm going to give you an
example, which I really like.

726
00:39:03,570 --> 00:39:04,580
It's not my work.

727
00:39:04,580 --> 00:39:07,040
It's done by people
at Northwestern.

728
00:39:07,040 --> 00:39:10,505
It's on intergranular
embrittlement of iron.

729
00:39:10,505 --> 00:39:12,130
And this is sort of
one of these things

730
00:39:12,130 --> 00:39:14,350
that you really think
you need a multi scale

731
00:39:14,350 --> 00:39:17,700
sort of big kind of
modeling approach on.

732
00:39:17,700 --> 00:39:20,590
Essentially, the
observation is that there's

733
00:39:20,590 --> 00:39:23,740
quite a few impurities in
steel that embrittle steel,

734
00:39:23,740 --> 00:39:28,220
and there's some that actually
enhance the internal cohesion.

735
00:39:28,220 --> 00:39:31,120
Sort of the typical
examples are phosphorous,

736
00:39:31,120 --> 00:39:32,110
which is really bad.

737
00:39:32,110 --> 00:39:34,780
Phosphorus is a really bad
embrittler of high strength

738
00:39:34,780 --> 00:39:39,910
steel, whereas boron tends to
enhance intergranular cohesion.

739
00:39:39,910 --> 00:39:42,650
So when you sort of hear
about this for the first time,

740
00:39:42,650 --> 00:39:44,750
you think, well, how am
I going to study this?

741
00:39:44,750 --> 00:39:48,340
Maybe I'll set up a big
simulation with a grain

742
00:39:48,340 --> 00:39:50,920
boundary in, and I'll
put phosphorus there,

743
00:39:50,920 --> 00:39:53,770
or I'll put born there, and
I'll try to pull it apart.

744
00:39:53,770 --> 00:39:57,190
And that'd sort of be
the brute force, maybe

745
00:39:57,190 --> 00:39:58,360
multiscale approach.

746
00:39:58,360 --> 00:40:01,550
You could embed that
in a continuum theory,

747
00:40:01,550 --> 00:40:03,910
but there's a sort
of very simpler--

748
00:40:03,910 --> 00:40:05,920
a very basic theory
that tells you there's

749
00:40:05,920 --> 00:40:07,970
a much simpler way
to understand this.

750
00:40:07,970 --> 00:40:10,810
And that's the Rice-Wang
theory, which essentially

751
00:40:10,810 --> 00:40:14,770
states that the embrittling
tendency of a solute

752
00:40:14,770 --> 00:40:17,170
depends on the
difference in segregation

753
00:40:17,170 --> 00:40:20,110
energy between the segregation
energy for that element

754
00:40:20,110 --> 00:40:23,810
to a grain boundary
and to a free surface.

755
00:40:23,810 --> 00:40:26,150
And you can sort of
intuitively-- this is actually

756
00:40:26,150 --> 00:40:27,420
mathematically justified.

757
00:40:27,420 --> 00:40:29,060
It's actually a quite
elaborate theory

758
00:40:29,060 --> 00:40:32,750
that you can catch
in two sentences,

759
00:40:32,750 --> 00:40:36,020
but you can sort of see why this
is true from a driving force

760
00:40:36,020 --> 00:40:37,340
perspective.

761
00:40:37,340 --> 00:40:40,670
If you have an impurity
that has a large segregation

762
00:40:40,670 --> 00:40:44,390
energy to a surface rather
than a grain boundary,

763
00:40:44,390 --> 00:40:47,588
then its presence is
going to enhance surfaces,

764
00:40:47,588 --> 00:40:49,130
because then it can
have that benefit

765
00:40:49,130 --> 00:40:52,230
from that large
segregation energy.

766
00:40:52,230 --> 00:40:55,430
So those are going to be
embrittlers, because they will

767
00:40:55,430 --> 00:40:57,650
tend to promote decohesion.

768
00:40:57,650 --> 00:41:00,060
Whereas the ones that have
a very large segregation

769
00:41:00,060 --> 00:41:02,870
energy at the grain boundary,
rather than the free surface,

770
00:41:02,870 --> 00:41:06,210
will tend to promote cohesion.

771
00:41:06,210 --> 00:41:08,270
And so if you believe
Rice-Wang theory,

772
00:41:08,270 --> 00:41:09,780
then this problem
is very simple.

773
00:41:09,780 --> 00:41:13,410
All you have to do is
calculate segregation energies,

774
00:41:13,410 --> 00:41:15,070
and that you can do.

775
00:41:15,070 --> 00:41:17,580
See, now you're back
to the atomistic level.

776
00:41:17,580 --> 00:41:20,190
You could set up a supercell
with a grain boundary

777
00:41:20,190 --> 00:41:22,790
and really calculate
the energy there.

778
00:41:22,790 --> 00:41:24,540
Do the same on the
surface, and you really

779
00:41:24,540 --> 00:41:26,865
don't have to do any dynamics.

780
00:41:31,977 --> 00:41:33,310
And that's what these folks did.

781
00:41:33,310 --> 00:41:34,320
This is our work.

782
00:41:34,320 --> 00:41:37,240
This is work that was published
in Science about 10 years ago

783
00:41:37,240 --> 00:41:41,020
and already by Art Freeman's
group and Greg Olsen, who

784
00:41:41,020 --> 00:41:44,200
sort of runs a company that
does atomistic modeling

785
00:41:44,200 --> 00:41:47,200
and other modeling
for steel development.

786
00:41:47,200 --> 00:41:51,490
And essentially, you see
that this theory works out

787
00:41:51,490 --> 00:41:52,870
very well.

788
00:41:52,870 --> 00:41:57,760
On one side, you have the
strong embrittlers, phosphorus

789
00:41:57,760 --> 00:41:58,480
and sulfur.

790
00:41:58,480 --> 00:42:00,740
And they essentially
have much higher--

791
00:42:00,740 --> 00:42:03,070
this is the difference
in segregation energy

792
00:42:03,070 --> 00:42:06,550
between the surface
and the grain boundary.

793
00:42:06,550 --> 00:42:10,490
These very much want to
segregate to the surface.

794
00:42:10,490 --> 00:42:13,390
And that's why they tend to
create internal surfaces,

795
00:42:13,390 --> 00:42:16,150
whereas carbon, to
a lesser extent,

796
00:42:16,150 --> 00:42:19,720
but boron to a large
extent promotes decohesion.

797
00:42:19,720 --> 00:42:22,990
So you can get a fairly
easily good insight

798
00:42:22,990 --> 00:42:27,730
of what are embrittlers
without having to coarse-grain

799
00:42:27,730 --> 00:42:31,856
across all the length scales.

800
00:42:31,856 --> 00:42:35,017
AUDIENCE: Why is the
separation different?

801
00:42:35,017 --> 00:42:36,100
GERBRAND CEDER: I'm sorry?

802
00:42:36,100 --> 00:42:39,284
AUDIENCE: Why is the separation
different in both boron

803
00:42:39,284 --> 00:42:42,480
and [INAUDIBLE].

804
00:42:42,480 --> 00:42:45,420
GERBRAND CEDER: Yeah,
why it's different.

805
00:42:45,420 --> 00:42:48,450
That's actually what that paper
is about, why it's different,

806
00:42:48,450 --> 00:42:49,590
essentially.

807
00:42:49,590 --> 00:42:51,910
And I don't remember
anymore, but I know

808
00:42:51,910 --> 00:42:53,160
it had to do with the bonding.

809
00:42:53,160 --> 00:42:55,560
This is why these
pictures are here.

810
00:42:55,560 --> 00:42:59,130
There's a very-- I think it
probably has to do with the--

811
00:42:59,130 --> 00:43:01,080
well, first of all,
boron is smaller.

812
00:43:01,080 --> 00:43:04,170
There's definitely a size
effect, boron and carbon.

813
00:43:04,170 --> 00:43:07,470
These two are much
smaller than these two,

814
00:43:07,470 --> 00:43:09,450
but there's another
effect, I think,

815
00:43:09,450 --> 00:43:11,910
which has to do with their--

816
00:43:11,910 --> 00:43:14,970
these, I think, tend to prefer--

817
00:43:14,970 --> 00:43:19,020
in some sense, have much
more bonding potency.

818
00:43:19,020 --> 00:43:21,540
So in a grain boundary,
they have a lot--

819
00:43:21,540 --> 00:43:23,460
they have higher coordination.

820
00:43:23,460 --> 00:43:26,370
So they gain a lot more
from covalent bonding

821
00:43:26,370 --> 00:43:28,100
than these two.

822
00:43:28,100 --> 00:43:29,850
So these two don't
gain as much from being

823
00:43:29,850 --> 00:43:32,010
coordinated by iron atoms.

824
00:43:32,010 --> 00:43:34,690
I think that's what I seem
to remember, the gist of it.

825
00:43:34,690 --> 00:43:37,110
But that's exactly what
that paper is about.

826
00:43:42,770 --> 00:43:46,357
To sort of end, I wanted
to mention one thing

827
00:43:46,357 --> 00:43:48,440
that we actually haven't
talked about that I think

828
00:43:48,440 --> 00:43:50,150
that is becoming
a fairly big issue

829
00:43:50,150 --> 00:43:53,750
in computational modeling.

830
00:43:53,750 --> 00:43:57,200
There's sort of
two ways that I've

831
00:43:57,200 --> 00:43:59,750
mentioned to go from all
this stuff you can calculate

832
00:43:59,750 --> 00:44:02,960
at the atomistic and electronic
level to the macroscopic level.

833
00:44:02,960 --> 00:44:04,520
One is to just sort
of think your way

834
00:44:04,520 --> 00:44:07,303
through it, what's important,
and I'll just calculate that.

835
00:44:07,303 --> 00:44:09,470
And then one is to sort of
do this sort of much more

836
00:44:09,470 --> 00:44:13,640
explicit coarse-graining or
simply handing off information.

837
00:44:13,640 --> 00:44:15,980
With diffusion, you can
just hand off information.

838
00:44:15,980 --> 00:44:18,140
You can calculate
activation barriers.

839
00:44:18,140 --> 00:44:19,910
You can put that in
a kinetic Monte Carlo

840
00:44:19,910 --> 00:44:21,140
to get diffusivities.

841
00:44:21,140 --> 00:44:24,380
And then you can hand that off
to macroscopic diffusion theory

842
00:44:24,380 --> 00:44:26,120
and do simulations.

843
00:44:26,120 --> 00:44:28,820
There's really a third
one that's emerging,

844
00:44:28,820 --> 00:44:31,130
and that's sort of
essentially giving up

845
00:44:31,130 --> 00:44:33,980
on trying to
understand the relation

846
00:44:33,980 --> 00:44:36,740
between macroscopic
and microscopic,

847
00:44:36,740 --> 00:44:40,410
but rather try to
determine it statistically.

848
00:44:40,410 --> 00:44:43,120
And this has been done in
chemistry for a long time.

849
00:44:43,120 --> 00:44:47,580
It's really only now, I think,
reaching material science.

850
00:44:47,580 --> 00:44:49,530
Chemists have used
things like QSAR,

851
00:44:49,530 --> 00:44:52,911
which are a quantum structure--

852
00:44:52,911 --> 00:44:57,130
and I forget what the A
is-- relations or something.

853
00:44:57,130 --> 00:45:01,515
But essentially, their
idea is that maybe there

854
00:45:01,515 --> 00:45:03,140
are certain macroscopic
properties that

855
00:45:03,140 --> 00:45:06,770
are so hard to link to the
electronic structure-- that is,

856
00:45:06,770 --> 00:45:09,680
you just try to correlate them.

857
00:45:09,680 --> 00:45:11,190
And I'll give you an idea--

858
00:45:11,190 --> 00:45:12,553
an example.

859
00:45:12,553 --> 00:45:13,970
One where this is
done extensively

860
00:45:13,970 --> 00:45:17,130
is toxicity of molecules.

861
00:45:17,130 --> 00:45:21,800
How do you predict
toxicity of molecules?

862
00:45:21,800 --> 00:45:23,700
This is a very
difficult question.

863
00:45:23,700 --> 00:45:25,190
I mean, how do chemists do it?

864
00:45:25,190 --> 00:45:28,070
Well, gee, they got a lot
of experience, essentially.

865
00:45:28,070 --> 00:45:34,360
All these groups probably react
with this part of your body,

866
00:45:34,360 --> 00:45:37,222
but how do you even
coarse-grain towards toxicity.

867
00:45:37,222 --> 00:45:38,680
It's essentially,
what is toxicity?

868
00:45:38,680 --> 00:45:43,690
It's the reactivity would have
an enormous large potential

869
00:45:43,690 --> 00:45:47,990
of molecules, what your
molecule can do with it.

870
00:45:47,990 --> 00:45:49,870
So people have tried
to do that with QSARs.

871
00:45:49,870 --> 00:45:52,930
Essentially what they say
is, for a lot of molecules

872
00:45:52,930 --> 00:45:56,110
for which I know that they're
either toxic or not toxic,

873
00:45:56,110 --> 00:45:58,990
I'm going to calculate
with quantum mechanics,

874
00:45:58,990 --> 00:46:02,830
as much as I can, bond
lengths, electron densities,

875
00:46:02,830 --> 00:46:04,360
electron negativities.

876
00:46:04,360 --> 00:46:06,700
I mean, people literally
have parameterized

877
00:46:06,700 --> 00:46:09,940
charged density
surfaces for molecules.

878
00:46:09,940 --> 00:46:11,680
And then they do a
correlation study.

879
00:46:11,680 --> 00:46:14,590
They do something like
principal component analysis

880
00:46:14,590 --> 00:46:16,180
or linear regression,
and they see

881
00:46:16,180 --> 00:46:19,150
which factors at
the atomic scale

882
00:46:19,150 --> 00:46:21,980
correlate most to the output.

883
00:46:21,980 --> 00:46:24,950
And there are powerful
data mining techniques

884
00:46:24,950 --> 00:46:26,150
that do exactly this.

885
00:46:26,150 --> 00:46:27,890
This is where the
field of science

886
00:46:27,890 --> 00:46:30,470
really intersects with
a lot of other fields.

887
00:46:30,470 --> 00:46:34,010
When you apply
for a credit card,

888
00:46:34,010 --> 00:46:36,320
the first thing that people
do on the information you

889
00:46:36,320 --> 00:46:38,900
provide-- that the companies
do is they data mine you,

890
00:46:38,900 --> 00:46:42,330
and they predict essentially
your risk of default.

891
00:46:42,330 --> 00:46:46,430
And that's all done with
correlation studies.

892
00:46:46,430 --> 00:46:48,410
They have enormous
large databases

893
00:46:48,410 --> 00:46:50,060
of people who have
been in default,

894
00:46:50,060 --> 00:46:52,850
and they know that, well,
if you're 23 years old

895
00:46:52,850 --> 00:46:55,917
you're probably more likely to
default than when you're 47.

896
00:46:55,917 --> 00:46:57,500
And so all that comes
out of the data.

897
00:46:57,500 --> 00:46:58,730
And people have
essentially tried

898
00:46:58,730 --> 00:47:00,320
to do the same thing
with the relation

899
00:47:00,320 --> 00:47:03,458
between macroscopic behavior
and electronic structure.

900
00:47:03,458 --> 00:47:05,750
And in chemistry, it's very
well advanced, I would say.

901
00:47:05,750 --> 00:47:07,530
You can buy QSAR software.

902
00:47:07,530 --> 00:47:08,990
You do a search on Google.

903
00:47:08,990 --> 00:47:11,000
You'll find it everywhere.

904
00:47:11,000 --> 00:47:13,830
In sort of solid
condensed matter,

905
00:47:13,830 --> 00:47:15,860
it's really sort
of in its infancy,

906
00:47:15,860 --> 00:47:17,640
but you're starting to see it.

907
00:47:17,640 --> 00:47:19,860
And one of the reasons I
think it's in its infancy,

908
00:47:19,860 --> 00:47:21,980
it's harder--

909
00:47:21,980 --> 00:47:23,540
the link between
those properties

910
00:47:23,540 --> 00:47:26,660
and electronic
structure is harder.

911
00:47:26,660 --> 00:47:31,430
There are intermediate scales
which can mess up the relation.

912
00:47:31,430 --> 00:47:33,470
This would never be
successful probably

913
00:47:33,470 --> 00:47:36,140
for mechanical
properties because we

914
00:47:36,140 --> 00:47:38,030
know that things that--

915
00:47:38,030 --> 00:47:40,340
two materials that
look almost identical

916
00:47:40,340 --> 00:47:41,990
on the microscopic
scale can have

917
00:47:41,990 --> 00:47:43,880
very different
mechanical behavior

918
00:47:43,880 --> 00:47:45,980
depending on the
impurity level, depending

919
00:47:45,980 --> 00:47:47,570
on the microstructure.

920
00:47:47,570 --> 00:47:49,940
So you kind of already
know ahead of time

921
00:47:49,940 --> 00:47:51,770
you're lost if you
try to correlate just

922
00:47:51,770 --> 00:47:56,853
to atomistic properties, but
it may work for other cases.

923
00:47:56,853 --> 00:47:58,520
The other thing that
makes this possible

924
00:47:58,520 --> 00:48:01,100
is that essentially we get
enormous amounts of computing

925
00:48:01,100 --> 00:48:05,000
power, and we can do very high
throughput data generation

926
00:48:05,000 --> 00:48:05,990
at the atomistic level.

927
00:48:05,990 --> 00:48:07,865
And I was going to show
you just one example.

928
00:48:12,440 --> 00:48:15,940
So this is the idea that
you essentially correlate

929
00:48:15,940 --> 00:48:19,025
these things, but let
me skip through that--

930
00:48:19,025 --> 00:48:22,410
that you use some
kind of learning

931
00:48:22,410 --> 00:48:25,650
method like neural networks,
or Bayesian statistics,

932
00:48:25,650 --> 00:48:28,230
or some kind of
correlation method

933
00:48:28,230 --> 00:48:34,330
to correlate output
to calculable input.

934
00:48:34,330 --> 00:48:37,830
So let me show you one example.

935
00:48:37,830 --> 00:48:40,260
This is something I
had gotten into myself,

936
00:48:40,260 --> 00:48:44,250
and still I mean
since a few years--

937
00:48:44,250 --> 00:48:46,440
sort of a very old
problem, and it

938
00:48:46,440 --> 00:48:48,900
was essentially
trying to predict

939
00:48:48,900 --> 00:48:50,947
crystal structure of materials.

940
00:48:50,947 --> 00:48:52,530
It's a really old
premise, essentially

941
00:48:52,530 --> 00:48:55,290
an unsolved problem, and
it's an intriguing one,

942
00:48:55,290 --> 00:48:58,920
because you can actually
show quite dramatically

943
00:48:58,920 --> 00:49:01,440
that the energetics of
quantum mechanics, the way we

944
00:49:01,440 --> 00:49:03,270
do it in our density
functional theory--

945
00:49:03,270 --> 00:49:06,330
is actually pretty much most
of the time accurate enough

946
00:49:06,330 --> 00:49:10,030
to predict the stable
crystal structure.

947
00:49:10,030 --> 00:49:11,790
So what do I mean with predict?

948
00:49:11,790 --> 00:49:14,580
What it means is
that the true crystal

949
00:49:14,580 --> 00:49:16,550
structure will have
the lowest energy,

950
00:49:16,550 --> 00:49:19,440
but the problem is finding it.

951
00:49:19,440 --> 00:49:22,260
So what I'm saying is, if
I gave you a list of 20

952
00:49:22,260 --> 00:49:24,570
and I say the winner
is among here,

953
00:49:24,570 --> 00:49:27,030
you're done, because you
calculate the energy of all 20.

954
00:49:27,030 --> 00:49:28,710
And the lowest one
will be the real one,

955
00:49:28,710 --> 00:49:30,510
but you can't make
that list of 20.

956
00:49:30,510 --> 00:49:31,710
That's the problem.

957
00:49:31,710 --> 00:49:33,000
It's a search problem.

958
00:49:33,000 --> 00:49:35,340
It's an optimization
problem, rather than

959
00:49:35,340 --> 00:49:37,910
a sort of energy problem.

960
00:49:37,910 --> 00:49:41,760
So the idea we have
is that, rather

961
00:49:41,760 --> 00:49:45,360
than trying to calculate
the energy of all crystal

962
00:49:45,360 --> 00:49:50,100
structures, can you guess
them by calculating a few

963
00:49:50,100 --> 00:49:54,510
and correlating their relation
with a whole lot of other ones?

964
00:49:54,510 --> 00:49:55,750
So it's the same idea.

965
00:49:55,750 --> 00:49:58,140
Let's say you have
a really big vector.

966
00:49:58,140 --> 00:50:00,895
This is like the energy
of all possible crystal

967
00:50:00,895 --> 00:50:02,020
structures, whatever it is.

968
00:50:02,020 --> 00:50:04,440
In the end, we have to use
finite numbers, you'll see.

969
00:50:04,440 --> 00:50:08,520
But can I just calculate
a few that goes fast--

970
00:50:08,520 --> 00:50:09,720
that's the one in blue--

971
00:50:09,720 --> 00:50:12,240
and then predict
all the ones in red

972
00:50:12,240 --> 00:50:15,390
without calculating
them so that I get

973
00:50:15,390 --> 00:50:17,700
to sort of a complete vector?

974
00:50:17,700 --> 00:50:21,708
And the way you want to do
this is by correlation methods.

975
00:50:21,708 --> 00:50:23,250
The philosophical
question is really,

976
00:50:23,250 --> 00:50:27,540
if you calculate a few pieces
of energetics of materials,

977
00:50:27,540 --> 00:50:29,850
do you have enough
information in there

978
00:50:29,850 --> 00:50:32,790
to say a lot more
about the material?

979
00:50:32,790 --> 00:50:34,467
And you can do that explicitly.

980
00:50:34,467 --> 00:50:36,300
Some people would say
I'm going to calculate

981
00:50:36,300 --> 00:50:39,397
a few pieces of energetics--
fit potentials to it, say.

982
00:50:39,397 --> 00:50:41,730
And then with those potentials,
calculate a large amount

983
00:50:41,730 --> 00:50:42,450
of other things.

984
00:50:42,450 --> 00:50:44,183
That's the explicit approach.

985
00:50:44,183 --> 00:50:45,600
The question you
can ask yourself,

986
00:50:45,600 --> 00:50:48,090
can you do this implicitly?

987
00:50:48,090 --> 00:50:50,880
And that's what we did
by building correlations.

988
00:50:50,880 --> 00:50:54,450
Essentially, we build a
sort of a smart algorithm--

989
00:50:54,450 --> 00:50:57,030
call it a neural net or
whatever-- that essentially

990
00:50:57,030 --> 00:50:58,690
builds this relation.

991
00:50:58,690 --> 00:51:01,290
So you train it on large
amounts of data sets

992
00:51:01,290 --> 00:51:03,340
for which you have
the full vector.

993
00:51:03,340 --> 00:51:04,840
Essentially, you're
asking yourself,

994
00:51:04,840 --> 00:51:08,580
is there a correlation
between the pink data

995
00:51:08,580 --> 00:51:10,740
variables and the blue ones?

996
00:51:10,740 --> 00:51:12,990
And it turns out that there is.

997
00:51:17,390 --> 00:51:18,440
Gee, OK.

998
00:51:18,440 --> 00:51:21,950
Well, this is the
picture I forgot.

999
00:51:21,950 --> 00:51:24,980
You can actually, with
a few calculations,

1000
00:51:24,980 --> 00:51:29,310
predict very well the
energy of other things.

1001
00:51:29,310 --> 00:51:32,420
And this is sort of the summary.

1002
00:51:32,420 --> 00:51:35,960
This is out of a very large
collection of binary metals

1003
00:51:35,960 --> 00:51:39,050
where we did the test,
and this is essentially

1004
00:51:39,050 --> 00:51:41,060
out of how many
calculations do you

1005
00:51:41,060 --> 00:51:44,570
need to do to get all the
ground structures in the system

1006
00:51:44,570 --> 00:51:46,540
with a given accuracy?

1007
00:51:46,540 --> 00:51:48,560
Because of course, this
is a statistical method,

1008
00:51:48,560 --> 00:51:53,590
so you're never going to get
100% accuracy, unless you

1009
00:51:53,590 --> 00:51:55,120
calculate everything.

1010
00:51:55,120 --> 00:51:59,560
But what you see is that with
something like 15 calculations,

1011
00:51:59,560 --> 00:52:02,890
you get 85% to 90%
accuracy with--

1012
00:52:02,890 --> 00:52:05,300
this is with the data
mining technique.

1013
00:52:05,300 --> 00:52:10,210
So this is an example of you
build a predictive method

1014
00:52:10,210 --> 00:52:15,840
without a theory, purely
based on statistics.

1015
00:52:15,840 --> 00:52:19,050
And you see more and
more of that done.

1016
00:52:19,050 --> 00:52:22,170
There are people who
built correlation methods

1017
00:52:22,170 --> 00:52:26,700
between very basic electronic
structure input data

1018
00:52:26,700 --> 00:52:29,850
and finite temperature
data-- for example,

1019
00:52:29,850 --> 00:52:34,040
like melting point, which
I think, as you've seen,

1020
00:52:34,040 --> 00:52:36,407
is a non-trivial thing
to actually calculate,

1021
00:52:36,407 --> 00:52:37,990
because melting point
you sort of need

1022
00:52:37,990 --> 00:52:41,790
the intersection of the liquid
and the bulk free energy.

1023
00:52:41,790 --> 00:52:46,750
So you could argue, how is that
melting point already hidden

1024
00:52:46,750 --> 00:52:49,962
in the energetics, and can
I extract that correlation?

1025
00:52:52,580 --> 00:52:55,910
So I'm going to skip
through a bunch of slides

1026
00:52:55,910 --> 00:52:58,490
because I want to
give you enough time

1027
00:52:58,490 --> 00:53:01,220
to do the evaluations.

1028
00:53:01,220 --> 00:53:04,530
These were actually slides on
examples of materials design.

1029
00:53:04,530 --> 00:53:06,440
Well, maybe I should--
a few more minutes--

1030
00:53:06,440 --> 00:53:08,240
say a few things about it.

1031
00:53:08,240 --> 00:53:10,820
One was band gap engineering.

1032
00:53:10,820 --> 00:53:13,910
The reason I put it in there
is because bandgap is the one

1033
00:53:13,910 --> 00:53:15,440
thing people always
think with DFT,

1034
00:53:15,440 --> 00:53:17,030
well, I can't do
anything useful.

1035
00:53:17,030 --> 00:53:20,060
The bandgap is always like
a factor of 2, sometimes

1036
00:53:20,060 --> 00:53:21,420
more off.

1037
00:53:21,420 --> 00:53:23,390
But you can correct for that.

1038
00:53:23,390 --> 00:53:25,870
People have made
empirical pseudopotentials

1039
00:53:25,870 --> 00:53:26,780
that correct for it.

1040
00:53:26,780 --> 00:53:28,400
There are actually
now methods out

1041
00:53:28,400 --> 00:53:32,550
there that will give you
the bandgap more accurately.

1042
00:53:32,550 --> 00:53:35,960
But the whole idea
is that you still

1043
00:53:35,960 --> 00:53:38,660
get a fairly systematic
ordering of the bandgaps,

1044
00:53:38,660 --> 00:53:41,870
and you look at-- can look
at bunches of semiconductors.

1045
00:53:41,870 --> 00:53:43,730
And what I want
to show you here--

1046
00:53:43,730 --> 00:53:46,340
this was actually a
Monte Carlo optimization

1047
00:53:46,340 --> 00:53:48,680
of the bandgap of
super lattices.

1048
00:53:48,680 --> 00:53:52,070
These were aluminum gallium
arsenide super lattices,

1049
00:53:52,070 --> 00:53:57,560
and essentially the
configuration space

1050
00:53:57,560 --> 00:54:00,740
over which they
searched was, how do you

1051
00:54:00,740 --> 00:54:02,540
arrange the aluminum
and the gallium

1052
00:54:02,540 --> 00:54:04,430
on the cation sublattice?

1053
00:54:04,430 --> 00:54:08,360
And so essentially
this was a scheme--

1054
00:54:08,360 --> 00:54:10,280
a Monte Carlo
scheme to sort of go

1055
00:54:10,280 --> 00:54:14,590
to the largest possible
bandgap in that system.

1056
00:54:14,590 --> 00:54:18,140
So it's a fairly well known
example of materials designed

1057
00:54:18,140 --> 00:54:20,870
purely by computation,
and you find

1058
00:54:20,870 --> 00:54:23,030
that, after about 10,000
Monte Carlo moves,

1059
00:54:23,030 --> 00:54:28,190
you seem to find pretty much
what the maximal bandgap is

1060
00:54:28,190 --> 00:54:29,210
in that system.

1061
00:54:29,210 --> 00:54:32,030
And then you can try to
make that will then be--

1062
00:54:32,030 --> 00:54:34,080
a thermoelectric is
one where there's

1063
00:54:34,080 --> 00:54:38,300
been several ab initio
predictions for very high

1064
00:54:38,300 --> 00:54:39,440
thermoelectric coefficient.

1065
00:54:39,440 --> 00:54:45,450
Then I'll show you one was
this lanthanum antimony,

1066
00:54:45,450 --> 00:54:46,555
essentially.

1067
00:54:51,670 --> 00:54:54,790
So my last two slides are
on the future of modeling.

1068
00:54:54,790 --> 00:54:58,400
I can't imagine a
better feel to be in.

1069
00:54:58,400 --> 00:55:01,750
And I think there's a few
reasons of-- the two things

1070
00:55:01,750 --> 00:55:04,060
that we essentially
benefit from tremendously--

1071
00:55:04,060 --> 00:55:07,480
one is computing power.

1072
00:55:07,480 --> 00:55:10,330
I mean, when I started in
this field in the early '90s,

1073
00:55:10,330 --> 00:55:13,030
I mean, I can't even
thinking back anymore

1074
00:55:13,030 --> 00:55:14,980
how slow and expensive
computers were.

1075
00:55:14,980 --> 00:55:18,350
And that's a good 10 years ago.

1076
00:55:18,350 --> 00:55:22,690
So we ride this curve
with an anomalous benefit.

1077
00:55:22,690 --> 00:55:25,780
In my department talks,
I usually show numbers.

1078
00:55:25,780 --> 00:55:27,670
I think, in the last
20 years, you've

1079
00:55:27,670 --> 00:55:29,830
had an improvement of
sort of performance

1080
00:55:29,830 --> 00:55:34,876
versus price of something
like 10 to 50 million.

1081
00:55:34,876 --> 00:55:37,510
That's how much more computing
power you get for your buck

1082
00:55:37,510 --> 00:55:40,653
now than actually
about 15, 20 years ago.

1083
00:55:40,653 --> 00:55:41,320
That's enormous.

1084
00:55:41,320 --> 00:55:43,362
There are no other techniques
in material science

1085
00:55:43,362 --> 00:55:45,020
that do that, period.

1086
00:55:45,020 --> 00:55:47,000
So you ride this enormous curve.

1087
00:55:47,000 --> 00:55:48,760
And for whatever
people say, this

1088
00:55:48,760 --> 00:55:51,970
is likely to continue at
least for quite a while.

1089
00:55:51,970 --> 00:55:54,430
I think another one is that
we benefit tremendously

1090
00:55:54,430 --> 00:55:57,310
from all the developments
in condensed matter theory,

1091
00:55:57,310 --> 00:55:59,620
because this allows
us to compute.

1092
00:55:59,620 --> 00:56:02,650
Those allow us-- the
benefits in condensed matter

1093
00:56:02,650 --> 00:56:05,860
theory allow us to compute
things much more accurately.

1094
00:56:05,860 --> 00:56:09,490
And the third leg is the basic
materials theory development,

1095
00:56:09,490 --> 00:56:10,990
which is probably
the sort of weaker

1096
00:56:10,990 --> 00:56:13,570
leg of the three-legged stool.

1097
00:56:13,570 --> 00:56:16,750
But so this is
one of the reasons

1098
00:56:16,750 --> 00:56:20,030
this field is still very good.

1099
00:56:20,030 --> 00:56:22,150
You should keep into
account scaling,

1100
00:56:22,150 --> 00:56:26,250
which we've only sort of
cursorily said something about.

1101
00:56:26,250 --> 00:56:29,410
But here's some examples of
why scaling can kill you.

1102
00:56:32,440 --> 00:56:35,380
I put out some
hypothetical examples

1103
00:56:35,380 --> 00:56:40,630
of linear scaling examples, cube
scaling examples, and worse.

1104
00:56:40,630 --> 00:56:43,060
If you take an example
of linear scaling,

1105
00:56:43,060 --> 00:56:46,960
I think molecular dynamics
with real space potentials

1106
00:56:46,960 --> 00:56:48,940
is probably pretty well
up to linear scaling

1107
00:56:48,940 --> 00:56:51,610
these days, because you
can partition space.

1108
00:56:51,610 --> 00:56:53,620
As soon as you start
partitioning space

1109
00:56:53,620 --> 00:56:55,120
with finite range
potential, you can

1110
00:56:55,120 --> 00:56:58,120
get basically linear scaling.

1111
00:56:58,120 --> 00:57:01,680
So I figured out, what
can you do in 40 years,

1112
00:57:01,680 --> 00:57:04,320
assuming we get about a
factor of 10 to the sixth.

1113
00:57:04,320 --> 00:57:06,360
That's what you would
get from Morse's law.

1114
00:57:06,360 --> 00:57:08,410
40 years is about
when you retire.

1115
00:57:08,410 --> 00:57:10,200
So I mean, I'm
long gone by then,

1116
00:57:10,200 --> 00:57:11,840
but you guys are
going to retire.

1117
00:57:11,840 --> 00:57:14,010
So where will you be?

1118
00:57:14,010 --> 00:57:16,440
Well, let's say we can
do 10 to the 8 atoms.

1119
00:57:16,440 --> 00:57:18,360
So what is that, 100 million?

1120
00:57:18,360 --> 00:57:20,490
The record stands
more at a billion now,

1121
00:57:20,490 --> 00:57:23,792
but nobody kind of performs
at the record all the time.

1122
00:57:23,792 --> 00:57:25,500
You have to essentially
take what you can

1123
00:57:25,500 --> 00:57:27,640
do on an almost daily basis.

1124
00:57:27,640 --> 00:57:29,700
So if you do 100 million
atoms with potentials,

1125
00:57:29,700 --> 00:57:33,510
you'll be able to do
10 to the 14 atoms.

1126
00:57:33,510 --> 00:57:34,560
What does that buy you?

1127
00:57:34,560 --> 00:57:36,668
I'm not totally sure.

1128
00:57:36,668 --> 00:57:39,210
You have to ask yourself, what
can you do with 10 to 14 atoms

1129
00:57:39,210 --> 00:57:41,810
that you can't do
with 10 to the 8?

1130
00:57:41,810 --> 00:57:44,360
Because a lot of things don't
depend on the number of atoms.

1131
00:57:44,360 --> 00:57:46,130
They depend on the length scale.

1132
00:57:46,130 --> 00:57:47,840
And so of course,
the length scale

1133
00:57:47,840 --> 00:57:50,460
is the cube root of
the number of atoms.

1134
00:57:50,460 --> 00:57:52,340
So there, you've
only gained about--

1135
00:57:52,340 --> 00:57:55,530
well, you've gained about a
factor of 100 in length scale.

1136
00:57:55,530 --> 00:57:58,460
But if you go to something
that scales like the cube.

1137
00:57:58,460 --> 00:58:00,950
And I know this is a
little overdoing it--

1138
00:58:00,950 --> 00:58:04,498
LDA scales in large problems is
much more like n squared log n.

1139
00:58:04,498 --> 00:58:06,290
But let's say on smaller
problems it's just

1140
00:58:06,290 --> 00:58:07,730
like the cube of n.

1141
00:58:07,730 --> 00:58:11,330
So if you can do 1,000
atoms now, when you retire,

1142
00:58:11,330 --> 00:58:13,530
you'll be able to
do 100,000 atoms.

1143
00:58:13,530 --> 00:58:17,037
So where is that
going to take us?

1144
00:58:17,037 --> 00:58:19,370
I think it's going to allow
us to do some things that we

1145
00:58:19,370 --> 00:58:20,480
have a hard time with--

1146
00:58:20,480 --> 00:58:25,910
biological species, for example,
where even this is really

1147
00:58:25,910 --> 00:58:27,770
cutting it close.

1148
00:58:27,770 --> 00:58:31,010
You want to do proteins and DNA.

1149
00:58:31,010 --> 00:58:33,840
Yeah, people do dabble in
it now, and you can do it,

1150
00:58:33,840 --> 00:58:37,310
but it's fighting
against the frontier.

1151
00:58:37,310 --> 00:58:39,470
You've got to do solvent,
and so people now

1152
00:58:39,470 --> 00:58:43,040
make more ad hoc models
for including the solvent.

1153
00:58:43,040 --> 00:58:45,890
I think, when you
do 100,000, you

1154
00:58:45,890 --> 00:58:48,170
are going to do
biology much more.

1155
00:58:48,170 --> 00:58:50,930
Are you going to do
mechanical behavior?

1156
00:58:50,930 --> 00:58:51,950
I don't think so.

1157
00:58:51,950 --> 00:58:53,960
Not in the direct
ab initio scale.

1158
00:58:53,960 --> 00:58:57,140
This is probably by no
means enough to sort of span

1159
00:58:57,140 --> 00:59:01,010
the length scale from
atoms to microstructure.

1160
00:59:01,010 --> 00:59:03,570
Of course, if you believe that
we're all going to be doing

1161
00:59:03,570 --> 00:59:06,630
[? LDA ?] [INAUDIBLE],,
now you're up to--

1162
00:59:06,630 --> 00:59:07,880
what is this, a billion atoms?

1163
00:59:07,880 --> 00:59:09,972
So starting to count.

1164
00:59:09,972 --> 00:59:11,680
Of course, when you
have a billion atoms,

1165
00:59:11,680 --> 00:59:13,930
you're going to have to
figure out how to look at them

1166
00:59:13,930 --> 00:59:17,110
and how to extract data from
them, which is non-trivial

1167
00:59:17,110 --> 00:59:19,080
at these large simulations.

1168
00:59:19,080 --> 00:59:21,953
The people who sort of
broke these frontiers--

1169
00:59:21,953 --> 00:59:24,370
there was a group at Louisiana
State that, for the longest

1170
00:59:24,370 --> 00:59:25,472
time, drove this forward.

1171
00:59:25,472 --> 00:59:27,680
And I think they may have
done the first billion atom

1172
00:59:27,680 --> 00:59:29,260
simulation.

1173
00:59:29,260 --> 00:59:34,060
A lot of the research is in
data transfer, data analysis,

1174
00:59:34,060 --> 00:59:36,040
visualization, because
think about it.

1175
00:59:36,040 --> 00:59:38,110
When you have a
billion atoms, how

1176
00:59:38,110 --> 00:59:39,550
do you decide what's going on?

1177
00:59:39,550 --> 00:59:41,910
You just look at it, or like--

1178
00:59:41,910 --> 00:59:45,480
so that's actually a big
chunk of that research.

1179
00:59:45,480 --> 00:59:48,060
And then I took one
that's kind of excessive,

1180
00:59:48,060 --> 00:59:51,600
but let's say you do [INAUDIBLE]
configuration interactions,

1181
00:59:51,600 --> 00:59:54,690
which is as close
to-- essentially as

1182
00:59:54,690 --> 00:59:57,000
close to the exact
solution maybe

1183
00:59:57,000 --> 00:59:58,950
that you could get
for quantum mechanics,

1184
00:59:58,950 --> 01:00:00,870
except maybe quantum
Monte Carlo now.

1185
01:00:00,870 --> 01:00:04,740
But that scaling
is depending on who

1186
01:00:04,740 --> 01:00:07,050
you believe is somewhere
from n to the fifth

1187
01:00:07,050 --> 01:00:09,800
to n to the seventh.

1188
01:00:09,800 --> 01:00:11,960
It's such small
numbers that it's not

1189
01:00:11,960 --> 01:00:14,750
even really sure you can
quite calibrate the scaling.

1190
01:00:14,750 --> 01:00:16,880
But so if you can
do 10 atoms now,

1191
01:00:16,880 --> 01:00:20,290
you can only do 100 by
the time you retire.

1192
01:00:20,290 --> 01:00:22,930
So these things are
useful for calibration,

1193
01:00:22,930 --> 01:00:27,490
but they're not going to allow
us to, say, do DNA and do it

1194
01:00:27,490 --> 01:00:28,880
more accurately.

1195
01:00:28,880 --> 01:00:36,470
So the only reason I'm saying
is that you will still not

1196
01:00:36,470 --> 01:00:39,125
be able to rely on Morse's
law for all your problems.

1197
01:00:39,125 --> 01:00:40,500
You still have to
figure out ways

1198
01:00:40,500 --> 01:00:44,360
to, in a sort of smart
way, use computations.

1199
01:00:44,360 --> 01:00:48,540
I always find it funny
the proposals I read of we

1200
01:00:48,540 --> 01:00:51,390
can't quite compute this,
but next year we can.

1201
01:00:51,390 --> 01:00:53,580
Well, most stuff
you can't compute

1202
01:00:53,580 --> 01:00:56,590
today you won't be able to
compute next year either.

1203
01:00:56,590 --> 01:00:59,940
And so you'll still
have to find shortcuts.

1204
01:00:59,940 --> 01:01:03,810
If you count on being able to
sort of put materials in a box,

1205
01:01:03,810 --> 01:01:07,260
and simulate the hell out
of them, and sort of figure

1206
01:01:07,260 --> 01:01:10,050
out what comes up, I
think it's unlikely

1207
01:01:10,050 --> 01:01:13,230
that that will happen
in any reasonable time

1208
01:01:13,230 --> 01:01:14,525
for a lot of properties.

1209
01:01:14,525 --> 01:01:16,150
There are properties
you can do it for.

1210
01:01:16,150 --> 01:01:19,110
But for most things,
I don't think you can.

1211
01:01:19,110 --> 01:01:21,942
So it was a pleasure
teaching you.

1212
01:01:21,942 --> 01:01:23,650
Professor Marzari and
I had a lot of fun.

1213
01:01:23,650 --> 01:01:26,222
We always have a lot of
fun with this course.

1214
01:01:26,222 --> 01:01:28,680
We're going to leave you with
the course evaluations, which

1215
01:01:28,680 --> 01:01:29,850
have to be done.

1216
01:01:29,850 --> 01:01:32,040
And can I count on you?

1217
01:01:32,040 --> 01:01:34,650
We're not allowed to
touch these, so one of you

1218
01:01:34,650 --> 01:01:37,740
has to return this to
department headquarters,

1219
01:01:37,740 --> 01:01:40,050
or to Kathy Farrel.

1220
01:01:40,050 --> 01:01:46,920
Can I count on one of you just
to pick them up and like--

1221
01:01:46,920 --> 01:01:48,520
OK, thanks, everyone.

1222
01:01:48,520 --> 01:01:50,860
OK, thank you.