 I'd like to explain to you that this is not actually exactly thermodynamics and neither is it exactly kinetics. So if I change to the next slide here. This is what we call stable equilibrium that if I give this ball, an infinite asthma perturbation it will come back to its original location. This clearly is not equilibrium because the ball is rolling down this non-linear here and therefore it's dissipating free energy as it rolls down. Now the difference between this and this is that this is a steady state process because we have a constant gradient here. So if I'm an observer located on this ball, I actually won't see any change happening, even though free energy is being dissipated. So the thermodynamics of irreversible processes is strictly about the steady state processes. For example, you know if we have a bar with the temperature at one end being hot and the other end being cold, and there's a constant gradient of temperature, then at any location within that bar, you will see the same temperature, even though heat is flowing down that bar. So this is a subject which is between kinetics and thermodynamics. And whereas in equilibrium, you know, we can write an equation that G alpha equals G gamma at equilibrium in both of these cases, that becomes an inequality because we are actually dissipating free energy so G alpha, maybe less than G gamma in both of these cases. And just to remind you of the meaning of a reversible process and an irreversible process, because you know the title of this talk has the term irreversible. If I have a cylinder here containing an ideal gas, and there is no friction between the piston and the gas, then if I start to increase the pressure along this curve, then the volume will decrease. And when I reverse the process if you go exactly back along that line. So, in this process of going there and coming back in this cyclic process. There is no free energy dissipated, you recover all the energy that you put in. In contrast, in this case, if I want to decrease volume, because I have friction between the piston and the cylinder. I want to increase the pressure until the friction is overcome and then follow this path here, where the volume decreases as the pressure increases. And when I start to reduce the pressure, again to overcome the friction I'll have to go to a much lower pressure than given by the blue curve, and so on until I get back to the original point, and the free energy dissipated in this process is the area within this red loop. So it's process like this is called irreversible because entropy is being created during that process. And we can express this in the following way. So, let's assume that Jay is some kind of a flux. Okay, it could be a heat flux. And X is a generalized force, which is driving that heat flow. Then the product of the flux and the force is equal to the temperature times the rate of entropy production. In other words, the rate of energy dissipation in an isothermal process. Jay here is a generalized flux. It could be a heat flux, it could be a diffusion flux or an electrical current. For example, and this is the force which drives that flux. And if I can write an equation like this where the temperature multiplied by the by the rate of entropy production, which is basically the free energy dissipation rate at an isothermal temperature. And it will equal to the product of the flux and the force, if you have expressed these relations properly. In other words, you know, in the case of a diffusion flux this would not be a concentration gradient, but a chemical potential gradient. So as that is as thought is the rate of entropy production, generalized flux and generalized force. Now, the importance of this equation is that if I can write such an equation, then we generally find that the flux is proportional to the force. Okay, if I go to the next bit. If we can express the forces and flux says, in terms of the free energy dissipation rate, then we find that the flux is proportional to the force and I'll give you examples. So here, here is the arms law representation where we have a potential difference, given by this battery. And this is the current flowing. And this is a resistor which resist the flow of the current. And, you know, it's obvious that if I multiply the current by the voltage, I get a free energy dissipation rate, because that that's the energy that's dissipated. And therefore, we find that the current is proportional to the potential difference, we. This gives us our own slow with one upon our being the proportionality constant. Okay, so this satisfies what we said in the last slide that if I can write an equation in which I express the energy dissipation rate as a product of a force and a flux, then the current will be proportional to then the flux will be proportional to the force. And in this particular case of homes law. The constant is one upon our. Now if you look at heat transfer as the second example. I've got these two chambers which are at different temperatures Th and TL, and the area here is a, and we are transferring a quantity of heat DH into this side. From an earlier lecture, you know that the entropy change on doing this transfer is one upon the lower temperature minus one upon the higher temperature. In other words, we have actually produced entropy because the higher temperature is smaller than the lower temperature. And the rate of entropy production is simply DS by DT, and this is per unit volume, which we can expand by substituting for DS here. So it's DH by DT into one upon TL minus one upon TH. And therefore the rate of entropy production is equal to a flux. The relationship between the flux going through a unit area and DH DT. So if I take a onto this side then that flux will be per unit area and this is DH DT. I've substituted now for DH DT by the product of the flux and the area. And that is, if you make these temperature differences small enough than that is the flux into minus one upon T squared DT by the Z, where Z is coordinating this direction. And therefore the temperature, if I take one of these temperature terms onto this side, then the temperature times the rate of entropy production gives us the heat flux. And this is the correctly expressed force that is driving the heat flux. Now, you might, you might be a little bit confused here because normally we say heat flux depends on the temperature gradient. But that would not be consistent with this equation. It actually depends on one upon minus one upon T times the temperature gradient. So if we go back to the heat flow equation of Fourier, where, you know, the flux is proportional to the temperature gradient and this is a thermal conductivity, K really is a function of temperature. Not a constant. Okay. So this equation here tells us that J is proportional to this entire product here. And therefore, you know, if you take Fourier's law, then the conductivity will have a temperature dependence. Now, the temperature dependence actually depends on the mechanism by which the heat is transferred, but it should be temperature dependent. And that comes directly from that equation where we said that if I can write a product of T times s dot equals J times x, then J will be proportional to x. As long as x is expressed correctly so that the product of J and x gives us a rate of energy dissipation. Okay, this is a story which I might already have explained. That fixes law of diffusion where, you know, the flux is proportional to a concentration gradient and proportionality constant is a diffusion coefficient D. We know that it shouldn't be the case that it's proportional to the concentration gradient in general, because when we look at circumstances like these where we have two phases in equilibrium but with different compositions, there is a concentration gradient, but we will not get a flux because the chemical potentials of the species are uniform, even though the chemical compositions are different. So diffusion is actually driven by a chemical potential gradient so if you look at this free energy curve here, and we have in the same sample, we have two regions with different chemical potentials, then the chemical potentials are clearly different in those two regions, the free energies are clearly different. And therefore, there will be a driving force for diffusion, and which will tend to homogenize this material given, given sufficient time and thermal activation. So we demonstrated that by writing fixes law and by writing another law which says J is proportional to the gradient of chemical potential. And because this is per unit concentration we also multiply by the concentration here. And this is now a proportionality constant. So let's find that a little bit to express it like this where it's d mu by DC and DC by the Z equals to the mu by the set. Then we have the analogy with the diffusion coefficient that the diffusion coefficient actually is a function of how the chemical potential changes with concentration, and M is what we call a mobility and we have to use a symbol capital L for for mobility and sometimes capital M doesn't matter. Okay, so diffusion is driven by a chemical potential gradient, not by, not in general by a concentration gradient. And just to summarize what we've done so far. In the case of electrical current we had the voltage which was driving the current and that's basically an expression for the electromotive force heat flux minus one upon T into the gradient of temperature and diffusion flux, the gradient of free energy. And of course if you are deforming something plastically, then the product of stress and strain rate is actually the free energy dissipation rate. And therefore, you know, you will find the strain rate to be proportional to the stress. Of course there will be complications for example work hardening and so forth but let's assume we have a perfectly plastic solid. So I haven't actually explained to you. So far, why, you know, if you write an equation which is D times s dot equals j times x, then why does j turn out to be proportional to x. So I'm going to explain that empirically now. So, this is the flux and when I use this curly brackets I mean that j is a function of x this is not a product of j and x. And I'm going to expand this J in terms of x as a polynomial. So the first term here, second term, third term where here there is a direct dependence on the force and on the square of the force and so on. Now, if you look at this equation carefully, this term has to be zero because there's no flux when there is no force. Okay, so we can delete that. And as an approximation I'm going to ignore all the terms which are in higher order of x. So if you do that, then your flux becomes proportional to the force. Okay. And this also emphasizes another point that this theory can only apply when there aren't large forces. Okay, so we're talking about relatively small forces. Now, the obvious question to ask is how large does the force have to be before this stops applying. And the answer is that you simply have to do an experiment to see whether your original hypothesis that J is proportional to x works or not. There's no rigorous way of telling you that look when axis of this magnitude, the proportionality between the force and flux will disappear. And I once asked a physicist whether ohm's law is satisfied when the potential difference is very large. The answer is very difficult in that case because you also get heating of the sample if you have a very large current, but I will show you another example now, where clearly the proportionality between force and flux breaks down when you go to large forces. Okay, and this is an example which will be familiar to you, and it's about grain boundary motion. So, let's assume that this is one crystal here gamma. And this is just because the lattice is periodic so we have these potential wells corresponding to the periodicity. And this is the crystal alpha, which has a lower free energy so gamma is tending to transform into alpha. And this is the activation barrier in the transfer of atoms in the austenite into the ferrite. And this delta G is the driving force. Now in any any reaction, there will be a forward flux and a reverse flux. Okay, so the forward flux will be proportional to the Arrhenius term, including the activation barrier G star. And this is proportional to exponential of minus G star upon RT. And the reverse flux has a larger activation barrier, which is the sum of G star and the magnitude of the driving force so G star and the magnitude of driving force. And the velocity of your interface should be proportional to the difference between these two quantities here, the forward flux and the reverse flux. So we're going to substitute for these terms using these identities. And then you get the velocity is proportional to exponential minus G star upon RT into this, because one times that is simply that. And this times this is this. Okay, so you find the velocity is not directly proportional to the driving force. But it violates the principle that T s dot equals J x then J is proportional to x. So this is a physically derived equation, which ought to work for any magnitude of the driving force. But if you look at the maths of this expression and you can show this for yourself when delta G is small. So the term becomes magnitude of delta G over RT, and we recover the relationship between the velocity and the driving force, a direct proportionality. Okay, so how small. Well, you can try and use this equation and substitute values of delta G and see what deviation you get between these two relationships that the velocity directly proportional to this. Or using this sort of an equation. Okay, so this illustrates that you need to be careful when looking at very large forces. And this was one of the problems that I discussed when I gave the seminar at IIT Kharagpur, that when we deal with diffusion in very steep gradients, we can no longer assume that there isn't an additional cost, which is greater than just the chemical potential gradient. Okay. Okay, I'll give you another example, which may be closer to home that is in the context of steel making. Now you might recognize this person here. This is what's known as Linz Donovitz converter, and it's located in Austria. And, you know, the Basima process was able to produce steel but it contained quite a lot of phosphorus which is quite difficult to eliminate. But by blowing oxygen at high pressure in this converter, you can actually reduce the phosphorus content using also a slag of lime into which the Lyman dolomite dolomite is a magnesium oxide into which the phosphorus partitions. So this is a really good technology which made all the difference to the mechanical properties of steels in the old days. Okay, so again, if you look at the transport between the slag and the metal interface, we can use the same sort of analysis, as we did in this case, where we look at a forward reaction from the slag and the metal and the reverse from the metal to the slag. And that is the origin of this equation, where you look at the flux of an ion across the slag metal interface. This is the driving force, the Faraday times the valency times an over potential. And obviously this has units of energy, because it's the same units as our team. You know, this subscript I is actually for many components, not just phosphorus. So you have many elements in the slag and you might have many elements in the metal as well. So if you want to develop a model for this process. It's difficult using these highly non linear equations, but by applying irreversible thermodynamics you can simplify something like this, assuming that the approximation is adequate to more or less linear approximation like this where you know the rate of the reaction across the slag metal interface is directly proportional to this driving force, exactly as in irreversible thermodynamics so this paper here that I got this information from is basically about applying irreversible thermodynamics to slag metal interfaces. And that is the same as we discussed for the grain boundary where you look at the forward reaction rate and the reverse reaction rate. So this method is actually extremely powerful and extremely helpful in many circumstances where the driving forces are not very large that it's actually quite rare that we have very large driving forces because for example in the case of transformations in this case we would have to under cool a lot in order to get a large driving force, and that makes it difficult to maintain the particular mechanism of transformation that operates at a high temperature. So it's, it's very difficult to actually super cool things to provide this very large driving force. So in general, the theory will work rather nicely. It's a really interesting bit. Okay, where if I asked you how a thermocouple works. You would have to think because nowadays we take thermocouples for granted. But it's really strange, you know, you've got a temperature difference between a reference and the electrode, you're using to sense the temperature. Temperature difference creates a voltage. Okay, so it's strange we have a temperature difference creating a voltage. And there's also the opposite effect I think it's called the Peltier effect where if I apply a potential difference I also create. If I pass a current between a potential difference I also create a temperature difference. Obviously, you know, there is a link between the force that's driving the diffusion of heat. Okay, in your thermocouple and the creation of a potential difference. And this is very easy to handle in the framework of irreversible thermodynamics because the equation we had where we multiply temperature by the rate of entropy production equal to a flux times a force we can generalize to any number of combinations of forces and fluxes. Okay, so this remains the same. This is the rate of entropy production. I'll just use a different symbol here. I should correct that. So, this could be, you know, heat flux and the force driving the heat flux. And then you could have another term which is a flux of matter driving driven by chemical potential difference. And if you can write this equation that the net free energy dissipation rate is due to a variety of forces and fluxes. Then you can also say just as we said J is proportional to x we can say that J will be proportional to this where you know repeated indices imply a summation. So, I could, for example, have the flux of one, depending not only on the force of one, but also on the force of something else, for example, the temperature gradient, and this could be the potential difference. And similarly J2 will be a function not only of x one, but also x two. So a force can affect the flux of something else. Exactly as in a thermocouple or in the peltier effect. Now supposing the system is at equilibrium. There is an interface between the slag and the metal and the system is at equilibrium. Equilibrium is not static. You know you get items jumping in both directions at an equal rate. And that is a dynamic equilibrium equilibrium. The principle tells us that, you know, at equilibrium the rates of forward and reverse reactions must be same. So these two terms here must be identical. M12 must be equal to M21 and that applies, you know, however long these equations are that Mij will be equal to Mji. And this is called the principle of microscopic reversibility. Exceptions would be, for example, you know, they have a magnetic field because there is actually a sign of the magnetic field which comes into play, in which case Mij will be equal to minus Mji. We don't normally, well, I mean physicists use this a lot, but in materials we are mostly interested in this part where we have microscopic reversibility and M21 is really equal to M12. If there's an M31 it will be equal to M13 and so on. So that simplifies things a little bit. And I mentioned to you that we have diffusion coefficient strictly depending on how the chemical potential varies with the concentration. And this is for a binary system. Okay. We have a ternary system where I have the host, we'll call it with a substitute three and two solids, X1 and X2. Then without going into detail, the detail is in this review that I did some time ago. And also in many publications of the people who actually did this work, all the references are in there. But I don't want you to worry about that at all. I think it's slightly different than the diffusion coefficient as a function of how the chemical potential of one varies with its concentration. And yeah, that's, this is the cross diffusion coefficient, but this is the solvent. Okay. So M12 is the interaction between the first solute and the second solute. So the diffusion coefficient of one will also depend on how the chemical potential of solute two varies with concentration. So if I have a bar, which has a uniform carbon concentration, but it has a gradient of manganese, then given a chance the carbon will also develop a gradient driven by the gradient of manganese, the free energy gradient of manganese. Now, these coefficients are incorporated in these are mobility coefficients like M, okay, and they are incorporated in databases which are commercially available. So for example, when you do a calculation using a program like dictra. You're accessing matrices of mobility coefficients, which have been worked out by many, many people and compiled into a database. So in many cases, these mobility coefficients are actually available, including the cross terms. Now I'm going to show you an experiment which was conducted in 1982, in which you take a bar of steel which is about, I think, three centimeters long, water cooled at both ends, but heated resistively. So you develop a temperature peak in the middle. And the bar to begin with is completely homogeneous. It has a homogeneous distribution of carbon. But when you do the experiment, because there is this temperature gradient and heat flow, you get carbon building up in the middle. So these are direct experimental measurements showing the effect of a heat flux on carbon diffusion. Okay, we started off with a homogeneous concentration, but by building this temperature gradient water cooled at the ends and resistively heated so the temperature peaks in the middle, the carbon redistributes driven by the heat flux. Okay. So this is these are real effects, and they do influence the processing of steels, etc. If you have gradients in your material. We often think of a force and flux as uniquely related. But if you have several forces that can influence that particular flux then they will all come into play. And that's how a thermocouple works. And that's how the various examples that I've given you can be formalized in terms of the theory of irreversible thermodynamics, where clearly there are some approximations, which you can only demonstrate whether they are correct or not using experimental methods.