ANNA MORALES MELGARES: Hi.
My name is Anna.
I'm from Barcelona, although I'm starting an EPFL, which
is in Switzerland.
It's the Polytechnical Federal School of Lausanne.
And today, I wanted to show you potential energy surfaces.
But I thought that might be a bit boring,
and since it's my video, and I do
what I want, I'm going to show you
theme parks and slides, because why not?
Going to make myself smaller here.
OK.
What do we have here?
Well, this is the Water World.
The Water World is this theme park
which is in Lloret de Mar, a city next to Barcelona,
kind of.
And as you can see, they have these amazing slides
that end up in a funnel, like you
can see this guy at the bottom of the funnel just fall
through this hole.
But what if I want to simulate this?
I would need a software that enables me to do this, right?
There is some software that can do something like this,
but I chose to use Wolfram Mathematica.
And now I will show you my little funnel.
Here it is.
My humans are these black particles here.
Now, what if I let some time pass?
What happens to my humans?
They get closer to the center of the funnel.
That makes sense, because gravity pulls them
to the lowest height, right?
This height is actually related to gravitational energy,
since gravitational energy is proportional to height.
Therefore, this is just exactly the shape
that the potential energy surface
for the gravitational energy looks like.
So we have constructed actually a potential energy surface.
But these kind of surfaces can represent
many other types of energy, not only gravitational energy,
and can be plotted with other coordinates rather than space.
Here, we have x and y, which only defined the space where
these humans and funnels sit, but I
could use other coordinates.
Like for example, the distance between two items,
or the state of the system, or things like this.
Things in which energy, the energy that employing
depends on.
In order to simulate these things-- to simulate the funnel
that I did or what I'm going to show you after,
I need statistics.
But don't run away, they're easy to understand.
So what I'm going to do in order to simulate this
is the following.
Imagine that I have this blue line, which
will be like a slide, and imagine
that the bottom of this blue line will represent the floor--
the ground.
We'll call it ground state.
And the upper part is like something
which is higher in energy.
Now, I'm going to separate this slide into some finer parts.
This is going to be the discretization of the system.
I'm going to discretize my system.
Here, I discretized it in five steps.
You can see these red steps.
What I'm going to tell my particles,
which will represent the state of the system,
is the following.
So my particle will be in the initial state.
I will tell my particle the following.
It will attempt to go either down or either up.
It will have 50% probabilities of trying to go up
and trying to go down.
Now, there is something we have to differentiate.
One thing is the attempt.
One thing is to try to go somewhere,
and another different thing is actually going somewhere.
So my particle is going to try to go up and down
with equal probability.
Now, if it chooses to go down, it
will have probability equal to one to go down.
This means that if it tries to go down,
it will actually go down.
Now, if it chooses to go up, it's
not that clear that it's going to go up because--
and I'm going to make my screen bigger so I can show you--
probability of going up will be equal to this term here.
This term depends on energy two, which
is the energy of the upper state,
and energy one, which is the energy of the initial state.
The more separated they are, this
means the bigger this barrier is.
This is actually called energy barrier.
The bigger this step is, the more difficult
it will be for my particle to actually go up.
There is another term I didn't talk
about yet, which is this one.
This is temperature.
Temperature, here, will-- if temperature increases,
the probability of my particle to go up will also increase.
This means that the more energy we give--
the more thermal energy we give the system, the more
temperature we give the system, it
will be more probable for my particle to be able to go up.
In a system where my temperature is close to zero,
in a system where my temperature is super low,
my particle will tend to go to the ground state, which
is the bottom of my curve.
If I give energy enough, then particle
will be able to go anywhere.
Now, what I'm talking about here is not a system
with only one coordinate.
I'm talking about a system with two coordinates,
therefore I'm not talking about lines.
I'm not talking about curves.
I'm talking about surfaces.
Then, the example I showed before
was basically a particle that can go up and can go down,
as you can see here with this blue row.
But now, I'm going to have a surface,
so my particle will be able to go many other places.
Therefore, my particle will attempt--
the attempts that it can do are not two.
There are going to be eight which
will describe this surface.
Now, what I wanted to show you is actually
a real case where these surfaces can
be really useful to understand the system,
and these are the transition states.
But in order to do this, I'm going
to talk to you about the Lennard-Jones potential first.
The Lennard-Jones potential is a very well known potential
energy curve that describes the interaction--
describes the behavior of energy of the system as two atoms
get closer, and closer, and closer together.
Imagine, here, two atoms, which could be two hydrogen atoms
and are teared apart.
They're super far from each other,
and then the energy is zero.
When they are getting closer, and closer, and closer,
there is a point where energy will be at its minimum.
This minimum energy is the most stable state of the system.
This distance will be the optimal distance--
will be the equilibrium state.
Now, if we put these atoms even closer, and closer, and closer,
the energy of the system will increase.
An increase in energy means that the state is less happy.
The state is less stable.
Therefore, the most stable part of the system
will be the minimum.
The state will always try to be at it's minimum.
Here, this minimum would be in a distance between these two
atoms.
What I wanted to talk to you about,
actually, is triatomic hypothetic molecules.
This molecule, especially HHF.
HHF is a molecule which is--
it cannot be isolated.
There are some experiments that show its existence
for nano or femtoseconds.
This exists for a super, super short amount of time.
We could even say they are virtual,
but it's a state where the system has to pass through
in order to change.
I'll explain after.
The thing that we have here is one hydrogen atom which has--
we will plot.
One of the coordinates will be--
the distance between this hydrogen atom,
and the other hydrogen atom, and the other coordinate
that we will be plotting is the distance between this hydrogen
atom and fluorine.
This means that we will have two Lennard-Jones potentials.
One between HH and one between HF.
This will describe a surface.
First of all, I'm going to explain you what
this transition state means.
We will have different states of our system.
First of all, we will have a molecule
which is H2, and some fluorine somewhere.
This particles-- the molecule and fluorine--
will collide, and we will end up with an HF
molecule and a hydrogen atom somewhere else.
In order for this to happen, the system
has to undergo through another state, which
is called a transition state, which is actually our HHF
virtual or hypothetic molecule.
If a system wants to go from H2 plus F
to HF plus H or visa versa, it will preferentially
go through this transition state.
How can we actually see this in a potential energy surface?
Here, I show you how the potential energy
surface of HHF system is.
[INAUDIBLE] described by the HF distance,
where this minimum represents the state of the system where
the molecule HF exists, and the other hydrogen atom
is somewhere else.
Here, we have another Lennard-Jones potential,
described by the HH distance.
So in this minimum, the state of the system
is an H2 molecule and some fluorine somewhere else.
Now, the preferential path for this change
to happen from HH to HF, or vise versa,
will be through the transition state, which is a subtle point.
What is a subtle point?
A subtle point is somewhere in the surface
where the derivative of energy is zero,
but it's not a minimum, and it's not a maximum.
It's actually a minimum in one direction,
and a maximum in another direction,
and this will be the HHF state of the system,
as you can see in this contour plot on the left.
To illustrate this, I'm going to show you.
Now, my particles-- I'm going to show
some particles in the surface.
Now, this particle means it represents the system, not
atoms.
It's the system.
Here, I have an HF molecule because the system
is in this minimum.
Now, if the particle for some reason,
when given enough temperature, it goes through this state--
this is the transition state.
The system is HHF, but if this particle falls
into this minimum, then the system is HH--
this hydrogen molecule and the fluorine somewhere else.
Knowing this, I'm going to show you
what happens when I have lots of particles which
are lots of representations of the same system
because I want to see the probability of the system to be
in one state or the other.
I'm going to give [INAUDIBLE].
This is an histogram.
An histogram is this plot with bars
that shows me the population.
As you can see here, the particles
are just running all over the place
because I gave energy enough for them to overcome any energy
barriers.
But if I decrease temperature, [INAUDIBLE]..
OK, now, here you can see that we actually
have some preferential places for these particles--
these states of the system to be.
They are in the minimums because they cannot jump in the minimum
because they cannot jump over the energy barriers.
Now, as you can see, they are also pretty--
like, in the histogram, you can see
that if I increase temperature a little bit,
the preferential path they take, more or less,
is the transition state, which is done at this point.
This is one of the things that we can use Mathematica for,
but it's much more powerful.
It can actually do many other things,
and it can solve many other science related problems.
And not even science related, you
can even do art with Mathematica.