Transcriber: Robert Tucker Reviewer: Ariana Bleau Lugo
I'm here to talk to you about entropy.
Entropy can order stuff.
Entropy cares about shape, and entropy cares about shape in such a way
that it's become surprising to scientists and we're just beginning to appreciate
what the ramifications are.
Designing new building blocks of matter with concepts of entropy and shape in mind
is giving rise to a new generation of very exciting ideas and new materials
with really cool properties.
Well, let's start at the beginning.
We all have learned back in kindergarten that matter uses all kinds of forces
to hold stuff together.
So we've learned in basic chemistry and physics classes about covalent bonds
and ionic bonds and hydrogen bonds that hold water molecules together
and many of you have maybe heard about van der Waals forces
that are responsible for helping geckos to stick
onto walls and ceilings and not fall off when they walk.
And these kinds of forces are what holds stuff together
and we've all seen the periodic table and we know that these elements
either in combination or on their own
can form all sorts of interesting crystal structures
because of the forces that we just talked about.
For example sodium and chlorine can combine together
into the familiar rock salt structure.
And there's even more complicated kinds of crystal structures.
One of the most complicated for a single element to form on its own
is the one that manganese forms, which has 20 atoms in a repeat unit.
Another concept that is critically important
to how stuff is held together is the concept of entropy.
And entropy is something that's not a force like these other forces
but it can be thought loosely as a statistical force.
So, what is entropy?
Entropy is the reason why we don't all fall asleep right now
due to oxygen deprivation.
And why is this?
Entropy is the reason why all the oxygen molecules in the room
prefer to be scattered randomly throughout the room
and are not instead piled up under my hand to form a big block letter “M”.
And the reason for this is that there are so, so many more ways
for oxygen molecules to randomly be disordered throughout the room
than to organize in one small space or in any kind of a pattern,
say in an “M” in my hand.
And the fact that entropy is related to the number of ways
of organizing stuff was recognized by Mr Boltzmann
who lived in the latter part of the 19th century.
And this is a picture of his tombstone in Vienna,
which at the top, it may be hard to see,
but circled in yellow is his expression
that I wrote up at the top that says that entropy,
which he called the letter “S”, is equal to a constant times the logarithm
of the number of ways of arranging stuff.
And this expression to a statistical physicist
or a thermodynamicist is as revered as Newton's F = m a,
or Einstein's E = m c^2.
And it's exactly this example of oxygen molecules preferring
to be randomly distributed throughout this room
that contributes to our everyday or colloquial notion that entropy
is synonymous with disorder.
But that's only sometimes correct and if you get one thing out of this talk today,
it's that instead of thinking of entropy as being disorder,
we should be thinking of entropy as being synonymous with options.
Entropy is about ways of arranging stuff and that's important because entropy
can order stuff if there are more ways
of being ordered than being disordered.
And that may sound counter-intuitive to many of you,
but let me give you some examples.
So here are two computer simulations that I'm going to show you.
There are two boxes of hard particles in them.
The one on the right has prolate ellipsoids
that are like long rice grains and the one on the left
are oblate ellipsoids like Spree candies.
And think of them as being hard particles almost like billiard balls
if they were suspended in some kind of solution
and could jiggle around. There are no other interactions,
no attractive forces, no covalent bonds, ionic bonds, none of that stuff,
just banging together due to thermal collisions.
And now we'll run these computer simulations
and we start off initially with all of the particles randomly arranged
and the color red denotes that the particles
have a random orientation or position relative to their neighbors
and as I start this simulation you'll see by the color green,
which we use to denote particles
that are ordered locally relative to their neighbors,
that they start to align, they start to order with each other.
And if we run it long enough, we see that this order
starts to pervade the whole box.
And this is because there are more ways for these particles
to be aligned and ordered with each other than for them to stay disordered.
Now this may not seem so counter-intuitive,
after all we know that if we had a bunch of pencils on a table
and we started to corral them together and they started to pack more and more densely,
then pretty soon the pencils would run into one another
and they'd have to line up if we wanted to get them even denser.
But what is counter-intuitive to many people
is that entropy can even order spheres.
So here on the left is a computer simulation
just like I the ones I showed a minute ago,
only the particles are spherical now instead.
They start off initially completely randomly
and now we run the simulation
and the particles are jiggling with thermal motion
and they're banging against each other and pretty soon they start to order,
and that's denoted by these green regions, into crystals.
So this box of particles is crystallizing, right before our eyes.
And what's happening is that there are more ways
for the particles to be ordered than there are to be [un]ordered
and so entropy is driving the crystallization of these spheres.
Now this particular crystal structure is known to material scientists and chemists
as a face-centered cubic crystal structure
and that's what its atomic structure looks like.
And if you go back to the periodic table of the elements you see that there are
many examples of structures that are
the same face-centered cubic structure and you can mix elements together
and also get the face-centered cubic structure.
What's really interesting is that this way of arranging spheres
has been known back in early Sanskrit writings,
back to 499 CE, they wrote about what's the best way to organize spheres
if you wanted to pack them as densely as possible.
And Johannes Kepler conjectured in the 1600's
that the densest you could possibly pack spheres together
is Pi over the square root of 18.
But it took a mathematician by the name of Hales till 1998;
he proved that Kepler was correct, but it took the mathematician
7 years to check the proof, so it wasn't published until 2005.
And so the densest,
if you have hard spheres all of identical sizes,
and you try to put them together in a box,
you can't get them any denser than 74.05%.
And why is that interesting? Well because shape matters
and any of us who has ever gone on a trip
and has to pack a suitcase with stuff knows
that the shape of the stuff that you want to put in the suitcase
matters to how densely you can pack that suitcase.
So if you have a bunch of baseballs or if you have a bunch of books,
you can pack the books much more efficiently
with less space left over in the suitcase than you can a bunch of baseballs.
And in fact spheres pack the least densely of all convex objects.
And this is really important to a number of different industries.
It's important to the candy makers who have to figure out how many M&M's
they can fit in a bag of candy, or to the people who make Toblerone
to try figure out how to shape their boxes so they can pack
as many boxes together into a pallet.
And it's also important to pharmaceutical manufacturers
who have to figure out how different-shaped tablets or different-shaped pills,
how many of them will fit into a bottle of a certain size and shape
or even how many granules of medicine will fill a single capsule.
And it's even important to fruit growers,
and this a real photograph of cubic water melon, I kid you not,
water melon that's been grown in a cubic box to give it this shape
because cubes pack more efficiently than spheres do.
So, what in the world does all of this, and water melons,
have to do with nanotechnology?
Well, it turns out that in the past 10 years
some of the most exciting advances in nanotechnology have been
the community's ability to make particles of all sorts of different shapes.
These particles can be made out of metals and ceramics and semiconductors
and plastics, polymers, and they're very small particles,
so they're particles that are smaller than a red blood cell
but slightly larger than, say, the width of a DNA double helix.
So these are really microscopic or nanoscopic objects
and the ability to make them has become so exciting and so promising
that now these kinds of nanoscopic objects
could be the atoms and molecules of a next generation of materials,
if only we could figure out how to put these particles together,
how to assemble them into some kind
of organized structure for new materials and devices.
So, we're here in Michigan, I would be remiss to not show
a picture from the original assembly line at Ford around 1920.
And if we were to think about using the concepts of assembly lines
which revolutionized all of manufacturing in this country and the world,
if we thought about applying those principles to nanotechnology,
just consider that a 2 cm cube made of nanoparticles
would have about 10^20 nanoparticles in it.
That's a 1 with 20 zeros after it. That's a lot of particles.
And even if you were to place them a million at a time sequentially,
and you placed a million of them every second,
it would take over 25000 years to build up a 2cm cube of nanoparticles.
So clearly we need different kinds of concepts
than the traditional assembly line.
And what we need is to exploit the forces that are naturally there
between nanoparticles and entropy
to get the particles to assemble themselves
into an interesting and useful structure.
And so over the past few years, my students and I
have become very excited to study the role of shape in self-assembly
and how we can use that and entropy
to make new kinds of structures and new materials.
And so we've been studying polyhedral shapes.
So polyhedra – the simplest polyhedra are the Platonic solids,
and there are 5 of them.
And they're the simplest because each of them
is made of only one kind of face and the face or the facet
is a simple polygon and they're all the same
and there's only one kind of vertex and of these simplest polyhedra
the very simplest Platonic solid is the tetrahedron
because it has four sides and that's the minimum number of sides
that you need to make a polyhedron.
And what's interesting is that the question of how tetrahedra arrange,
how they can pack, was actually discussed by Aristotle,
and this a quote translated by Guthrie from Aristotle around 330 BCE,
and he thought that tetrahedra would be able to fill space perfectly
like cubes with no space left over.
And but, of course, you can see that they don't and all you need is to get
your dungeons and dragons dice out of a drawer and stick them together
with modeling putty that you can buy at the local hardware store
and if you put 5 of then together then you see they fit together pretty well
– maybe there's a little gap in the middle.
You put 9 of them together, 20 of them together makes an icosahedron,
we could stick tetrahedra onto that makes a hexecontahedron
and the more tetrahedra that you stick to this, the more you see
that there are these gaps and the gaps get bigger
and there's more of them as you go out and out.
And that is because tetrahedra cannot fill space
so you can never pack tetrahedra with a density or packing fraction of 1.
And this is a problem that started back with Aristotle in 330 BCE,
Johannes Müller aka Regiomontanus pointed out in the 1400's
that Aristotle was actually wrong.
Then a bunch of mathematicians got interested in it
and it's been studied a lot.
This is a timeline that my student put together for me yesterday,
and he's here, so I need to show it.
But I love it, it's a little busy, but it's beautiful because
(Laughter)
because it shows you that you start back in ancient times,
and then you go up through the millennia, and then through the centuries
and then through the decades, the years, the months
and then we get to 27 December 2009, or as I finally refer to as:
“how I spent my Christmas vacation in 2009 where we figured out
what is the densest you could possibly pack tetrahedra.”
Just in case anyone ever stops you in the street and says,
“Wait! What is the densest you could possibly pack tetrahedra?”
You can tell them that number is 85.63% and it's a world record
that is held by the University of Michigan.
(Whooping, applause)
“The New York Times” thought that all of this flurry and excitement
over how many tetrahedra you could fit into a box
was so interesting that the science writer Kenneth Chang
wrote an article about it on January 4th, 2010.
So for one week it was the science story of the decade.
So we were studying this problem and one of the things we did was,
we took a bunch of tetrahedra and we threw them into a box
and did the same kind of computer simulation
that I showed you earlier. So there's a bunch of tetrahedra,
jiggling around, banging around,
and it's hard to see, but they're actually ordering.
And if you took a slice of the box and turned it on its side,
you'd see this incredible kind of order in the system
where these particles figure out how to organize themselves
into all of these structures.
And this thing on the right is a diffraction pattern, it's hard to see,
it's what you get if you were to bounce x-rays
off of this structure and it's the kind of image
that seriously makes a materials scientist weep with joy.
Because they look at it and they say, “Oh, this is so complex.
How could the system have figured out how to do this
without any kind of attractive forces? Just entropy.
Just because there are more ways for the particles to be ordered,
more options, than if the particles stayed disordered.”
And if we changed the shape of the particles
we can get different kinds of structures.
Always starting from some initial random thing,
and comparing it, you find that they always want to be
in these crystal structures. So we can get, for example, diamond,
which is a high-pressure form of carbon,
and you can get high-pressure lithium, which is a structure that atomistically
was only discovered a few years ago.
And we can get other kinds of crystal structures,
the metallic form of tin, tungsten, manganese, and even brass,
just by changing the shape of the particles.
And one of the most exciting, one of my personal favorites,
is this super cube where 6 of these particles up here
come together into a little cube and the cubes assemble into bigger cubes,
which will assemble into bigger cubes; it's a hierarchically assembled structure.
Again there are no instructions in the particles that say where to go,
they just explore all the possible ways of organizing
and there are so many more ways for them to be ordered than disordered,
that this is what they form.
And so because we have some really fast computers
here at the University of Michigan, we are able to study
literally hundreds of different shapes and do tens of thousands of simulations.
And what we've been able to do is connect now for the first time
the shape of particles with the kinds of structures
that they want to form, the kinds of crystals that they want to form.
And so even without traditional forces, based only on particle shape,
we now have the blueprints to make materials classes
whose properties mimic their atomic and molecular counterparts.
And this is important and exciting because now we can begin to think about
making new types of materials with new kinds of properties.
So we can make shape-changing materials like the skin on this concept car
that can change shape due to the changing aerodynamic conditions
or the canonical Übershapeshifter from the Terminator or even a Transformer.
We can also think about making smart materials like this iron-man armor,
that could be strong as steel one second and lighter than air another
simply by changing environmental conditions.
And we can also think about making new kinds of functional materials,
like those that can bend light in interesting ways,
to make things look differently than they really are,
like this cloaked Klingon bird of prey. And before you get me started,
I can assure I know that this is Romulan technology
and that the Klingons only acquired it from the Romulans
(Laughter)
(Applause)
in 2267 in exchange for some D7-class battlecruisers, but –
(Laughter)
maybe we will all learn one day, that the Romulans actually acquired
that technology by licensing it from the University of Michigan.
Thank you for listening.
(Applause)