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GIAN PAOLO BERETTA: OK, good morning.
So we are still here in studying cross effects that
emerge from the Onsager theory of non-equilibrium near
equilibrium as a result of linear relations between forces
and fluxes.
Just as a reminder, in this slide,
we got these expressions that summarize
the relation between energy flux and entropy flux
and constituents flux, including current, electrical charge flux.
And these are diffusive fluxes.
So they are relative to the--
if we have a motion, like in a fluid,
they're relative to the barycenter of the fluid parcels.
And this other slide also, which we have reviewed the last time,
allowed us to write this term in the expression for the rate
of entropy production in terms of independent fluxes,
or, actually, in terms of these Jij's.
These are differences between fluxes of pairs of constituents
that are present in our system.
And they go together.
So the conjugated force to these diffusive relative fluxes are--
we could call them diffusive relative forces.
And they are given by the difference
in the gradients of the chemical potentials of the same two kinds
of particles.
Notice that these by definition are antisymmetric.
So you change-- you swap the indices, you change the sign,
of course, because it's obvious from here.
And therefore, you see here we have 1/2 of the summation over
ij.
And you're supposed to sum over all the i's and all the j's.
And that will pick each of these elements twice.
So the 1/2 will go away.
And so, for example, if you have just one pair,
you will have J12 plus J21, and they will end up--
this sum will end up being just one term--
J12 times X12.
You also should remember that we have
to use the Gibbs-Duhem relation, which
relates the gradient in pressure to the gradients in chemical
potentials.
For example, for a pure substance--
so there's only one constituent--
this also holds.
And it says that the gradient in pressure
is equal to the concentration of the substance times the gradient
in the chemical potential at constant temperature
of that substance.
We will use this today in one of the examples.
So here I've included up there--
here there's still the full expression.
But now I want to focus only on the vectorial fluxes
and forces--
so the fluxes of extensive properties
that are of vectorial yeah, vectorial character.
So we exclude the scalars, chemical reactions.
We exclude the tensorial fluxes.
So we exclude the dissipation function.
You see here that we have many different terms.
And then let's suppose we have only three kinds of particles
in our system.
That summation over the i's and j's ends up
contributing only three of these products of forces and fluxes.
Only three of them are nonzero and independent.
So this will be the interdiffusion of particle 1
with respect to 2, of particle 1 with respect to 3,
of particle 2 with respect to 3.
And then we have the diffusion of electrical charge related
to a change in electrostatic potential-- gradient
in electrostatic potential.
Of course, heat is in the gradient of 1 over T.
And then there is also this term with the pressure, which
gives you a flux of particles driven
by a gradient in the pressure.
So we have six terms, six different extensive properties
that can be transported in our system,
and six different degrees of disequilibrium
which control and drive the system back towards equilibrium.
This is also related to the Le Chatelier-Braun principle.
And if you remember, the more--
see, each one of these degrees of disequilibrium
is related to a necessary condition
for mutual equilibrium.
So if you have no gradient in temperature,
you have equality of temperature.
So from that point of view, you are OK for equilibrium.
But that's not the only condition.
You also need no gradient in pressure.
That's pressure equality, if the systems can exchange volume.
This is part of the chemical potential,
but it's the electrostatic.
So if you-- so you are at equilibrium, at least
from that point of view, from the electrostatics,
if you have no gradient in the electrostatic potential.
And here also you have the equality of chemical potentials
of the various pairs.
If they are all equal to 0, then we are at equilibrium.
The more you have of them that are different from 0,
the further you are from equilibrium,
the more they adapt to the degrees of equilibrium.
And therefore, like Le Chatelier-Braun says,
your system will have a stronger tendency
to return to equilibrium.
And the way the system returns to equilibrium
is described by this big matrix.
In fact, I didn't put the tensor signs, but these are--
each of them is a tensor because the most general relation
between a vector and another vector is through--
even linear, it's through a tensor.
So you see how rich the variety of phenomena
that we can encounter and that we can model
is So, of course, if you want to include all of them,
you'll need a computer program to do it.
This is done in simulations.
But even though nowadays we have computers
and we have softwares that are very--
they're able to model very complex phenomena in fluids--
and not only fluids, because, of course,
I'm talking with fluids, but also in solids,
we have the same ideas, very seldom they
are implemented to include all these off-diagonal terms.
Most of the times, you just keep the diagonal ones.
For example, here I've already sort of shaded out these
off-diagonal ones because most of the times,
you assume that each pair--
so the relative flux of pair 12 is
driven by the relative difference
in gradient of chemical potentials of that pair and not
the other ones.
But these are possible cross effects.
And these possible cross effects may result in,
for example, the possibility that concentration gradients--
you have two different--
you could have a flow upward of the concentration gradient
of one constituent driven by the flow downward
of two other constituents.
And part of it could be captured by this.
But I think that if you want the most general description
of that, you would need to include also
these off-diagonal terms.
OK, now, we have already done thermoelectric effects.
And from this point of view, the thermoelectric effects
cover these degrees of disequilibrium--
so a material in which you have a gradient in temperature
and a gradient in electrostatic field.
And so our system becomes just 2 by 2, two quantities,
extensive quantities transported and two degrees
of disequilibrium.
And since we are at it, I think I should--
there was something wrong in the viewgraph-- in the slide where I
did this calculation last time.
So I corrected it now because I was using the wrong expression
for the work, the work triple prime.
So here I've taken a slightly different approach
to correct that.
First of all, I write what the reversible work is.
And you remember in the situation
like we have done when we did heat transfer,
we had here, this is x.
Here is T, and here is T plus dT.
And then we put in an engine that
receives a heat flux, like a Carnot engine with the Carnot
efficiency, because it's a reversible engine,
produces a w--
let's call it th for thermal--
reversible, which is given by that minus q gradient in T
over T.
Now, here, we also have a gradient
in the electrostatic potential.
So we can use that, by extracting,
with an electric motor, for example,
some work also this way.
And that will add an electrical work
reversible, which is given by--
since the gradient in the electrostatic potential
is driven by the Seebeck effect, this
is given by minus I times the gradient in phi.
And that's minus I times epsilon gradient in T. Or maybe--
sorry, there is no minus here.
I drive a current down from high potential to low potential,
and the power of the engine is current per voltage.
The power is current per voltage.
And so there is no I here.
The minus sign comes from the relation for the Seebeck effect.
So that's what I've done here.
And next, I say, well, the actual--
if I put my thermoelectric generator,
and it does have some entropy produced by irreversibility,
and that entropy produced by irreversibility
is given by the Fourier term and the Joule effect term,
of course, the actual work of the generator will be less.
It will be the reversible minus T times sigma.
And since I was there, I just decided to keep also all
the tensors that I know complicates the story,
but since I haven't seen it done anywhere--
not that I looked everywhere, but--
so I said, well, why not working it out?
And I hope I did the right thing.
So using that expression and substituting
the thermoelectric relations, these ones,
you find an expression for the actual electrical work.
You can maximize it, so you find at which current density
it has the maximum power.
So maximum power doesn't mean that the engine is reversible.
It's a particular point of operation
of the engine or generator.
And so if you compute this--
if you substitute this current density at maximum power
into the expression for the work,
you get this expression here, where there begins to appear--
now I highlight this parameter Z,
which for an isotropic material takes this form.
And it's what is called the parameter of merit,
the figure of merit for the thermoelectric material.
And there are some materials that
have that parameter of the order of 1,
but there are also some nanostructured materials
that go well above 1, maybe, I think, even 2, or even 3.
And this is why the fact that you can achieve,
and that nowadays, you are actually engineering materials
by layering.
Exactly like we did for the nonisotropic heat transfer
example, here you can cook up layered materials
that have properties, alternate properties
with high conductivity or low--
I mean, yeah, high Seebeck coefficient
and low thermal conductivity.
And depending on how you structure them,
you may obtain good aspects of this tensor, which,
of course, relates to the properties of how
much the directions come in, because you see
the gradient in temperature dotted into the tensor means
that the directions are important,
the relative direction of your material
and the direction in which you put the gradient in.
Then, once you have these, you can also
substitute all the rest in all the rest
and compute also the second-law efficiency.
Second-law efficiency is the ratio
between the work at maximum power and the reversible work
that you could get with that current
that you have at the maximum power.
And this is the generalized expression
for that efficiency, which, for the isotropic material,
becomes this one--
the figure of merit divided by 1 plus 2Z.
And if you look at this function--
l mean, Z and--
what is it-- Z divided by 1 plus 2Z.
Well, it's a function that, of course, is 0 at 0,
and then it goes up and eventually, it could reach 1.
Does it reach 1?
Or does it reach 1/2?
Ah, that's a good question, eh?
Yeah, it looks 1/2, yeah.
Yeah.
OK, all right.
So this fixes that error that I made a while ago.
But errors are instructive and stimulating so
that you try to correct them.
So that's good.
Here is another example somewhat similar to that one.
Think of having, instead of the electrostatic potential, having
the pressure.
So you have two reservoirs, one at T and p
and the other at T plus dT and p plus dp.
So here you could--
since you have-- suppose this is a fluid.
Well, then, in addition to the thermal work,
you could use the difference in pressure
to drive a flow through a hydraulic turbine.
And the work of the hydraulic turbine
is given by the difference in pressure
times the volumetric flow rate, like in a pump,
so that it's very analogous to the thermoelectric effect.
And so with this pair of degrees of disequilibrium--
OK, so-- you treat exactly-- you take those two elements
that survive in the expression for the entropy production.
You assume linear relations.
And now the flows are heat and, instead of charge here,
now you have just particles.
And if you divide the flow of particles by the concentration,
you get a volumetric flow rate.
So this is cubic meters per second per square meter flowing.
So you can rewrite, because, usually,
what we do is rewrite these expressions where we have been
careful in using the entropic degrees of disequilibrium
so that we know that that's where the Onsager
reciprocity applies.
And so I can safely say that these coefficients, LQP,
is equal to LPQ.
Here Q stands for heat and P stands for pressure, of course.
And so now at this level, I can apply Onsager.
But then we like to work with more familiar variables, which,
in this case, are the volumetric flow rate
and the gradient in pressure is already
there, and the gradient in temperature instead
of the inverse temperature.
So we introduce parameters, like the thermal conductivity,
the other diagonal element is usually written
as k over nu (I meant, eta), where nu (eta) is the viscosity,
the shear viscosity of your fluid.
And k (kappa) is the Darcy permeability,
because if you had only this diagonal term here,
this is simply Darcy's law of flow in a capillary
or in a porous medium driven by the gradient in pressure, which
is what you would use, for example, to describe
the flow in a porous material or a permeable membrane,
like here, or in a capillary tube
or in a microfluidic device or things like that.
But the off-diagonal terms are also there.
And they give rise to two cross phenomena
that are related to one another That sometimes
go under the name of thermo-osmotic effect
or mechano-caloric effect.
So the thermo-osmotic is this one.
No, I'm sorry, thermo-osmotic is this one.
You have a flow of material in the absence
of a gradient in pressure driven by a gradient in temperature.
We could actually have this flow go--
if the gradient in pressure is small but opposite
to what the Darcy's law would drive the flow,
you could go against that pressure gradient
by using a thermal gradient, appropriate thermal gradient.
So that is one way in which you could somehow
drive your material in a capillary or microfluidic device
with a gradient in temperature against a gradient in pressure.
And the opposite or the conjugated cross effect
is this one, where, without a gradient in temperature,
you would have a gradient in pressure that
drives a heat flux.
But also, here, you could have--
depending on the relative differences,
you could have a gradient in pressure drive heat against--
up a temperature gradient, so in the opposite direction
of what you would expect if you just had Fourier.
So now let's move into the realm of diffusion of particles.
The simplest is when we have two constituents,
a pair of particles.
So we have interdiffusion of one into the other.
And so this is a direct law case, which gives rise
to Fick's law of diffusion.
Or also, since we're talking about a pair,
it belongs to-- instead of Fick, it's
called Maxwell-Stefan law of diffusion.
And here is the story.
Actually, see, on the left-hand side here
I've just repeated all the various steps that took us
to the relation I just said.
So let's just look at what we have
here on the right-hand side, which is the new aspect.
So we assume a linear force flux law
and, as we have seen in the previous, just a direct effect.
The degree of disequilibrium X12 drives a relative flux, J12,
through a coefficient that-- yeah,
on the previous slide we call it L1212, here,
just drop the double 12.
Usually, there are a couple of ways in which this is typically
rewritten.
So there is no single familiar way of rewriting this.
We like to rewrite it in terms of diffusion coefficients, which
are the things that we measure in the laboratory.
Or we measure also, nowadays, with Monte Carlo
or molecular dynamics simulations.
And so we rewrite--
we call this L12 this way, identify--
so these are the concentrations of the two constituents.
And, of course, they are related to one another,
because the mole fraction of the second species
is 1 minus the mole fraction of the other one.
So also between the concentrations,
you have a similar relation because--
all right.
And the other aspect is that you somewhere--
yeah, here, yeah, here, you notice
that you can prove that the X12 can also be written by virtue
of the Gibbs-Duhem relation.
It can be written also in terms of either just one gradient--
X1 is just the gradient in chemical potential
of substance 1.
So it's just is this one--
minus 1 over T times gradient of mu 1.
And that's X1.
But you can also define an X2, and these are all
related to X12.
So by substituting and expressing
these in terms of the flux of particles 1, proportional
to the gradient in chemical potential of that particle,
you can rewrite it this way.
And then the gradient in chemical potential of particle 1
can be related, if you do properly
the various derivatives, since the chemical potential is
a function of the concentration.
You remember our usual expression here,
for example, for an ideal mixture.
For an ideal mixture, you can write this.
And therefore, the gradient in the chemical potential done here
at fixed temperature and pressure.
So this doesn't enter into the gradient.
And so then you get that gradient in mu
is RT times the gradient of the logarithm.
So that's the gradient of y over y.
And, of course, since the mole fraction
are related to the concentration,
you can also rewrite it in terms of concentration.
So that is--
I mean, this expression is for the ideal.
If you have a more general situation,
you would have to do the partial derivative of mu with respect
to n, which is written here.
And that's what appears here.
So if you include all these, then you
have a relation between the flux of particle 1
and the concentration of particle 1 times the gradient
in the mole fraction.
And then you can call this group the Fick diffusion coefficient,
which you see is related to the Maxwell-Stefan diffusion
coefficient.
And so it all depends on giving names to these diffusion
coefficients.
Typically, people measure Fick diffusion coefficients.
Notice also that you can also define a self-diffusion case.
This is done, for example--
suppose you have just water in your box.
But suppose that some of the oxygens,
some of the water molecules, some percentage
of water molecules, are doped.
Well, actually, they're made with isotopes of oxygen.
So instead of using oxygen-16, you use oxygen-18.
So you have water molecules that have an isotope for the oxygen.
The isotope doesn't change the chemical properties.
But that allows you to, with proper detection,
to follow the diffusion of the heavy--
it's not really heavy water, because the heavy water
is with deuterium, not with the oxygen-18, but in this case,
it's a heavier water than the other one.
So you can see the interdiffusion.
And, of course, this diffusion is
related to the Brownian motion, the random motion generated
by collisions and the kinetic energy
that the particles always have at finite temperatures.
So in that case, you measure the self-diffusion
of water, which is an interesting subcase of this.
Now, if in addition to having gradients in concentrations,
you have also gradient in temperature,
this opens up to diffusion or thermo-diffusion effects.
And so the transport-- thermophoresis means--
remember, "phoresis" means "transport."
So you generate flux of particles
with a gradient in temperature, even
if you don't have the driving force of the gradient
of the chemical potential.
So even if you don't have gradients in concentrations,
just the gradient in temperature drives particles.
So that is-- this is written explicitly X12, which is minus 1
over T times the gradient in mu 1 minus mu 2.
We can write it also in terms like I've said here
with the partial derivatives starting from the chemical
potentials.
You can write it this way, in terms of the concentration.
Yeah, sorry, this should be a c1 over c1, not i.
I'll correct that.
OK, so this is the setup.
So that's the usual formula for the entropy production
from which you infer what are the entropic degrees
of disequilibrium, which are the ones that
enter in the linear force-flux relations.
And so the force-flux relations are these.
You therefore put in a matrix of coefficients,
and the matrix has to be symmetric.
So we will have that LQ12 is equal to L12Q.
And we rewrite it in so-called familiar variables,
or the traditional ways, so as to single out the gradients
in concentrations instead of the gradients in chemical
potentials.
However, when you do that, your system of equations
loses the apparent symmetry.
So you see these off-diagonal terms are not identical.
But I have used, in order to put this DT here--
you see, DT is called the thermo-diffusive coefficient.
And whereas D12 is the Fick--
so this is-- the Fick diffusion coefficient is the direct effect
between--
so it drives a flow--
a flux of particles, the diffusion
of particles, by the concentration gradient of one
of them, whereas this drives the flow of particles,
the flux of particles, the diffusive flux of particles,
with a gradient in temperature.
And you can enter these equations
in many different ways.
For example, if you assume no gradient in concentration,
then you have the heat flux that is simply given by Fourier law.
Or if you choose to work at zero gradient in temperature,
the second equation gives you Fick's law.
You could also consider, though, that when
you work at no gradient in concentration,
you don't have only the first equation.
You also have the second equation.
And that gives you a flux of particles
driven by just the gradient in temperature.
That is the Soret effect.
This is the effect whereby, for example, you
have dust particles in the air that collect.
You may have noticed, or you probably
know that in your house, you often
find dust near the cold walls.
It's the Soret effect, because, from the mechanistic point
of view, suppose this is a dust particle.
So it's a large particle with respect to the molecules of air.
And let's say these are molecules of oxygen, nitrogen.
And so suppose you have a gradient in temperature
this way.
So you get cold here and hotter there.
So the idea is simply that the particles on this side
have a higher kinetic energy.
And when they bounce on the particle, of course,
they exchange--
they exert a force on one side that
is slightly less than the one exerted by the collisions
with the particles that come from the cold side, which
are a bit slower.
So this pushes your dust particle towards the cold.
Notice also, yeah, and the opposite effect,
the conjugate effect, is the Dufour effect.
So when I have a Fick law at no gradient in temperature,
so I have a flux of particles driven by concentration,
but I also have a heat flux driven by the concentration.
So it's like the flux of particles that carries
with it also a heat flux.
And that's the Dufour effect.
These effects were discovered separately,
then only later they were linked by the Onsager
theory of reciprocity.
Also, it's interesting to note this, that, for example, suppose
you have no flux of particles.
So you set this equal to 0.
OK, so if you set that equal to 0,
you see that there comes a relation between gradient
and concentration and a gradient in temperature.
So if you impose a gradient in temperature, then--
and there is no way that--
and you don't allow this flux of particles--
then the gradient in temperature builds up
a gradient in concentration, and that gradient in concentration
has a play in the first equation up here.
And it reduces the effective conductivity.
So the thermal conductivity at no concentration (gradient)
is higher than the thermal conductivity
at no diffusive flux.
Now, that simple picture that I have shown here
on the blackboard is one possibility.
And it's one mechanistic possibility
that explains the Soret effect, or the thermophoresis
for the transport of dust particles towards the cold side.
In part, that is due to the size of the particles.
And so it belongs--
this nice article that summarizes
thermophoretic mechanisms shows that there
are many other mechanisms that drive the Soret effect, that
control the Soret effect.
So, typically, the heavy particles
like to get moved to the cold side, or the skewed particles,
or the large particles.
But then there are also more sophisticated actions,
including some double layer effects
on the surface of particles that have
some charge on their surface.
So there is a whole realm of possibilities with which you
can use Soret effect to drive things towards the hot
or towards the cold, depending on how
you engineer your materials and/or the properties
of your materials.
This is another paper that shows how
you can use a gradient in temperature
to move particles against friction force in a small pipe.
This is a carbon nanotube in which you have water molecules.
And so you manage, with a gradient in temperature,
to win the friction forces, because one
of the problems in moving stuff in small pipes
is that you would need huge pressure drops if you
wanted to do it with a pump.
And the device will break because of the high pressures,
whereas the thermophoresis or electrophoresis are
more effective and more doable.
And this is also in a thermofluidic device,
the idea that the gradient in temperature-- so you have
a hot flow which keeps a hot side
and a cold flow which keeps a cold side.
And then you have your actual device with particles in it,
and you want to separate the particles.
And that's what happens by the Soret effect,
let's say, the particles go to the cold side,
and so you reach a separation in your device.
Now, it is also important to realize
that what we have covered here is
only a very minor aspect of the phenomenology that
goes with diffusing particles.
And so what I'm doing with these slides is--
I don't pretend I can describe what happens,
but I want just to suggest that the variety of phenomena that
needs to be discussed, if we had, let's
say, another course, but also by another teacher,
because I'm not prepared on this topic.
But maybe the next teacher of this course will be.
And so there are--
for example, we talked about the Knudsen number,
the relative size of the mean free path
relative to the dimension of your device.
So if the Knudsen number is very small, then your particles--
let's say that if they make at least 100 collisions
before traveling a distance of the order of magnitude
of your device, which could be the diameter of your pipe, then
we are safe, and you can make the continuum assumption
that we have made--
in what we have done.
But if the Knudsen number becomes larger,
you may reach the other extreme, which
is the ballistic extreme, in which a particle may travel all
the way from one side to the other of your device
without even having a collision.
So this is not dominated by collisions between particles,
but it's dominated by collisions between particles
and the walls of your device.
Plus, there are surface effects.
And so you see there is a variety of things
that need to be considered.
Now, thermophoresis can also be--
is also used as a way to move particles around.
This is called self-propelling particles.
For example, here we have a PhD thesis and an article
that talks about using a phenomenon that we have
discussed, which is spinodal decomposition, in order
to move around particles.
And there are two kinds of particles.
This one, which has a double face,
is called a Janus particles because Janus
is the god with two faces, the Greek god with two faces.
So this particle is gold-plated on one side
and not on the other side.
And the gold surface layer guarantees--
since it has a high conductivity,
guarantees a uniform temperature.
And so then what they do is to shine some laser light
to heat up that particle.
And therefore, a temperature profile builds up.
And then they find a solution.
So the particle is in a solution which
is nonmiscible at high temperatures.
So we have studied spinodal decomposition
in substances that have an upper critical point.
Remember?
And this is the spinodal curve.
This is the unstable, and this is the metastable region.
So you could do that phenomenon, that thing if you
could cool down the particle.
But it's easier to heat up than cool.
So they found it easier to develop mixtures
that work the other way around.
From what I remember also, water and nicotine has a lower bound--
has a lower critical point like this.
I don't know exactly what substances they use here--
water and 2.6-lutidine, OK, whatever that is.
In any case, the idea is that you
start with the blue mixture, which is surrounding
the particle at this point.
So it's fully mixed.
And then, since you heat up the surface
of the particle in this boundary layer around it,
the temperature goes up, and you enter into the unstable region.
And the unstable region creates--
let's say, frees the free energy.
And that free energy, if properly directed,
can be used to propel the particles in two ways.
You need asymmetries.
So for the Janus particles, the asymmetry is on the fact
that you have more heating on one side than on the other.
And therefore, you generate a concentration gradient
from this side to the other side.
And that movement of particle in one direction
propels the big particle in the other direction.
And so you get the particle move.
Or here, they have still a gold-plated particle
but all around.
So it's not Janus.
But it's not completely spherical.
So it's a nonspherical particle.
And the nonspherical aspect here makes it
so that the gradient in the concentration and the gradient
in temperature cross.
So that crossing means that these two gradients are not
directed in the same--
so there is a component of the gradient in concentration
that goes orthogonal to the gradient in temperature.
And that will cause the movement of particles
that will push your big particle, I think, downward.
So you can play also.
You can play-- people are playing
because this is recent work.
There's a lot that people are doing using these methods.
Or another way to exploit or deploy nonequilibrium is--
what they are called--
molecular motors, essentially, are
nanoparticles that develop motion
at the expense of chemical reactions.
And here, for example, the particle, simply
has a catalyst, or an enzymatic site which catalyzes
the chemical reaction.
So it lowers the Arrhenius activation energy
only locally here.
So it promotes chemical reactions
in this side of the particles.
Of course, the chemical reactions
change the composition.
The change in composition creates
a gradient in concentration.
And that's how you get a flux of particles.
In fact, we said that the Curie symmetry principle prevents
in an isotropic medium to couple chemical reactions, which
are scalar transport, with vectorial transports.
But on the other hand, diffusion is sometimes the limiting aspect
in chemically reacting system because it is by diffusion
that particles go to the proper place
where they're used up for reacting
with the other reactants.
So indirectly, diffusion processes
are very important in controlling chemical reactions.
So they are effectively not coupled
in the way of the Onsager kind, but they are coupled overall
in order to feed the chemical reactions with the reactants
where they need to be.
And then, if you have simultaneously diffusion,
so many constituents, and also the electrostatic field,
the charges play a role.
They may play a role just locally
like this kind of double layers around particles
that have charges inside that can move to the surface.
And these effects are things that
happen, for example, at the electrodes in a fuel
cell or an electrolytic device.
Or simply you have a thermoelectric effect
coupled to diffusion.
So this is what you--
So, electrochemical systems belong to this kind of class.
And these are the kind of equations
that you need to consider if you model
an electrochemical system with, say, three different particles.
And three different particles is very common
because you may have the solvent and you have two solutes,
but often you have many more.
And these solutes could be the ions
that travel in your electrolytic solution.
The typical models are done this way,
with this interdiffusion taking up only the diagonal ones
and often forgetting these other more
complicated or second-order, maybe, cross effects.
But if you are interested in modeling electrochemical
systems, then these are the references
that you should not overlook.
This book, Non-Equilibrium Thermodynamics for Heterogeneous
Systems by Kjelstrup and Bedeaux,
is an excellent starting point for doing properly things
the way we've done, and also for realizing that--
so you can compute distributions in the heat.
They use the De Groot and Mazur notation.
So what we call the q double prime,
they call measurable heat flux.
But other than that, they essentially
use the same notation that we've done.
And also, a very important aspect
here is the interfaces at the electrodes,
where you have jumps from one substance to the other.
And, yes, you could define continuous or diffuse interface
theories, which is another aspect that needs to be covered.
But they do it with sharp interfaces, which is the Gibbs
traditional method.
But the reason why I urge you to go to their work
is that they do it correctly and carefully.
And in particular, also, regarding this role
of interfaces and nonequilibrium thermodynamics at surfaces,
there is this paper, again, by one of them, Dick Bedeaux,
in 1986, which is really wonderful.
And you should study that paper if you are in this business.
Now, of course, we are at the end of the course,
so I'm not covering anything more.
But I want to mention that there are
many other directions in which you could
go in the field of thermodynamics, which is related
to the general principles that we have developed,
but perhaps not the detailed models.
So, of course, the queen is statistical mechanics,
which is behind--
we've used also some of it when we did the single particle
in a box.
But then you have also kinetic theory
to deal with rarefied gas dynamics
if your Knudsen number is not very small.
And, in fact, to cover that, the phenomenology
of essentially the entire fluid mechanics
is redone at the non- very small Knudsen number.
People have redeveloped.
There's been a couple of decades of very active work
on small-scale hydrodynamics--
still ongoing.
Then, on the side of developing the foundations even
for continuum, we have this issue.
Remember the local equilibrium, where
we assume that we are at this state, but we're actually there.
And so there is this difference in entropy
that we are neglecting if you do local equilibrium, which is not
neglected in these other theories,
like rational extended thermodynamics
or other nonequilibrium thermodynamics theories.
And there are not only papers but books
about these approaches.
And you've seen that by using these,
we've seen one example, the Cattaneo heat transfer equation.
Well, in that same idea, you can develop,
and you should develop also, all the things that we've done, even
including what we've done today.
So there is a lot to be done also in that direction.
There is also--
OK, then there is quantum aspects.
We talked already about that.
And there is also these diffuse interface methods
that, again, if you have two phases,
you could consider them as sharp,
like, for example, liquid and vapor.
You could have-- this is the density of the vapor,
and this is the density of the liquid.
You could consider it sharp.
But if you enlarge this region, it's
actually something like this.
And by assuming the fundamental relation for your states,
for example, by assuming that s is not just a function of u, v,
n, but it's also a function of the gradients in concentration,
and, actually, since you need a scalar,
you can take the square of the gradient.
And you put it in as an additional variable,
which determines equilibrium.
And now you're talking about equilibrium
because that's a stable equilibrium
state for equilibrium.
That term there, which was present also in a paper
by Van der Waals, which actually is a very nice paper--
it's been translated by Rowlinson later on.
And I suggest that you go and see
that paper if you are studying this.
Well, this theory here has been developed by Landau.
It has now become known as Cahn and Hilliard models or diffuse
interface methods.
With that thing, you can predict the exact shape of this curve
at equilibrium.
And actually, there is a very nice paper
in The Physical Review by Antanovskii
that I suggest that you read if you
are interested in that kind of development.
Plus, of course, you have a list of--
an infinite list of other approaches,
from statistical to numerical simulations--
the Boltzmann equation, approximations
of the Boltzmann equation.
This is a famous one, which allows
you to compute the hydrodynamic quantities like heat flux,
the fluxes of particles, but also
the momentum and diffusive--
as moments of the Boltzmann distribution off equilibrium.
So we are talking about models that
can cover also nonequilibrium and far nonequilibrium.
Another approximation is one which
derives the equivalent of the Navier-Stokes equations,
but more complicated with more moments.
Then there's the Fokker-Planck, the stochastic equations.
These are equations that have some random noise terms in it.
And then you're certainly familiar
with molecular dynamics, Monte Carlo, and various types
of molecular dynamics, various flavors of molecular dynamics
that have been developed over the years
to overcome some of the limitations.
So, with this, I think I'd like to end the course remembering
that we have celebrated Carnot's 200th birthday with--
and I also, yes, I want to remind you
of the Prigogine Prize.
So if you ever end up with a PhD thesis
in the field of thermodynamics--
all aspects of thermodynamics, from applied to experimental
to very theoretical--
don't forget to apply for this prize.
It is done every two years.
The next deadline is February 1 next year.
And for this call, it will cover PhD defended between January 1
of last year, 2023, and December 31 of this year.
It's a pretty international prize.
In the last eight editions, we had 134 candidates
from 28 different countries.
You can click here for all the details.
And it is sponsored by a series of conferences called the Joint
European Thermodynamics Conference
but has become an international one.
And the next one will be in Belgrade in Serbia next year,
in May or June.
The date has not been decided yet.
As soon as it will be out, I'll put in a link
here so that you can see what it is about.
All right, so I'd like to conclude,
and I'd like to thank you for attending the lectures, which
I dedicate to my mentors.
But also, I thank you very much for the quality
of your video homeworks, because they
have been stimulating me for--
at least giving me some feedback that some of the messages I
wanted to pass have been passed.
So thank you very much.