 Okay so welcome everybody to this second lecture on thermodynamics and before I go into explaining the concept of equilibrium you see a picture of an engine which I took at IIT Kharagpur in 2002 and the interesting thing about this engine is that it's a steam driven engine. Now of course it was attempts to improve the efficiency of steam engines, steam powered facilities that led to the Carnot cycle and the concept of entropy itself. So this picture is highly relevant to what we've been talking about but I want to show you another engine which also illustrates the concept of entropy and this particular engine that I'm going to show you was constructed by my secretary Jean Pompryt who is about 70 years old and the idea that I'm going to explain I have to acknowledge a really delightful character called Steve Mold who used this particular engine to explain the concept of entropy. So this is a sterling engine and you can see there are two plates over here and this is my Tata steel cup of tea. You can see the Tata steel logo and there's a hot tea in there. What happens is that this plate becomes hot and therefore the air inside the chamber expands and pushes the piston upwards. That causes the wheel to turn and the momentum of the wheel will then push the piston down and the air above it will then cool in this second plate. So effectively we are transferring heat from the tea to these plates here and this is exactly the same as the diagram I used yesterday where we have these two systems at different temperatures where T2 is greater than T1 and we transfer heat into this blue part and the change in entropy then has to be positive because T2 is greater than T1. So what Steve Mold was explaining is that this engine is driven by a difference in temperature. Let me just show you how effective this is. The difference in temperature here is about 20 degrees centigrade and yet it's driving this engine rapidly and the idea is that you need energy to be clumped in order to be useful. That means we have a hot cup of tea and we have the environment which is at room temperature and it's a difference in temperature which drives this engine. If these two plates were at the same temperature no matter what the temperature there would be no motion. So you need to think about the fact that we are using clumped energy in the form of hot tea and eventually it cools down as we do work and what we are doing is we are degrading energy. We are taking a concentrated form of energy and spreading it out and it's that spreading of energy that is associated with an increase in entropy and eventually as Steve Mold said the entire universe will be at a uniform temperature. There will be no useful way of using the energy but that might be a long way away. So this is a beautiful device to illustrate the concept that the spreading of energy and that is equivalent to an increase in disorder results in an increase in entropy. So organized energy is more ordered than disorganized energy spread out over the universe. Okay going on to equilibrium. So here I'm illustrating different forms of mechanical equilibrium. So in this case we have stable equilibrium and stable equilibrium is defined by saying that look if I give that ball an infinitesimal perturbation then it will go back to its original position. So I'm not talking about huge perturbation etc just has to be an infinitesimal perturbation. If it goes back to its original position that's what we call stable equilibrium. Now of course there may be other minima in the system. So this here is known as metastable equilibrium but really from the point of view of thermodynamics we don't differentiate between these two because here if I give an infinitesimal perturbation to this ball it will return to its original position and we never know what the lowest minimum is. Okay so there's no difference from our point of view between metastable equilibrium and stable equilibrium. We cannot actually say that something is a stable equilibrium because you might be able to find somewhere a lower minimum. Unstable equilibrium is where this ball is on top of a hill and perfectly located and not moving but if I give it an infinitesimal perturbation it will tend to roll down the hill. So this is known as unstable equilibrium. It's like taking a needle and balancing it on its point. In principle it should stand vertically if the center of gravity is precisely in that point but the slightest of perturbation and the needle will fall. Now in this case this is not a systematic equilibrium it's like the heat transfer that we were doing from the temperature T2 to T1. This ball is rolling down this non-linear hill and is dissipating energy just as we were dissipating energy from the hot cup of tea to the engine. So this is the realm of kinetics. Okay a very very difficult realm to deal with because there are various factors which are unexpected and uncontrolled. Okay now if you are dealing with a pure substance for example pure iron and it can adopt different crystal structures at different temperatures then the free energy is simply a function of temperature or pressure and we can plot these curves here showing the free energy of the body-centered cubic form of iron and the cubic closed packed form of iron and where they intersect these two phases are in equilibrium because they have exactly the same free energy. So it's easy to define equilibrium for a case where there is no composition change all right. So it need not be pure iron it could be say an iron manganese or a copper aluminum alloy but none of the atoms are able to diffuse and therefore the parent and product lattices have exactly the same compositions and exactly the same atomic positions. So I'll come back to this later but this point here is we call the T0 temperature where the two phases of the same composition and same free energy are in equilibrium at this point they're exactly the same free energy more relevant mostly is solutions where we have more than one component and therefore there's a possibility that the parent and product phases will not have the same chemical composition. So I'm going to introduce the free energy of solutions but before I do that I want to introduce the concept of a mechanical mixture mechanical mixture if I take a chunk of zinc and a chunk of copper and I just put them into contact that's a mechanical mixture there's it's not a solution of copper and zinc. So here is a mechanical mixture where we have a component A and another chunk of material of component B and if I work out the mole fraction of A then it's one minus x and the mole fraction of B is x and it's very easy to calculate the free energy of a mechanical mixture it's simply the weighted sum of the free energy of pure A and the free energy of pure B. So this is the symbol that I use for the free energy of pure A and the free energy of pure B and we simply take a weighted average according to the concentrations and that gives us the free energy of a mechanical mixture. So this is not a solution you can separate out the two two bits just by picking them apart. Now we talked about entropy earlier and you know we expressed it in a reversible process as the change in entropy here infinitesimal change in entropy is an infinitesimal change in the heat that you put into the body divided by the temperature of that body and we can also integrate over a range of temperature from heat capacity data here okay and we noted that entropy is a capacitive property like mass that means we can add things together all right and it fits in with the idea of probabilities because of the Boltzmann equation which I'll go into later and the reason why I'm coming back to entropy is that when I mix these two up there is a change in entropy okay even if they're separable because you have different arrangements that you can make depending on how many particles of air we have and how many particles of B. So going back to this I'll explain configurational entropy slightly differently from what I did in the previous lecture but the same principle here I have a cylinder of gas a cylinder and all the gas atoms are on one side there's no wall over here but nevertheless they've all moved in the same direction and ended up in half the chamber with a resulting pressure in that region of p and no pressure here okay so there is only one arrangement of this configuration where all the atoms have moved simultaneously into one half of that cylinder if you now allow the atoms to fill all space then actually the number of arrangements here would be very large if I identified a sort of a lattice inside this cylinder and the pressure would be a half and the probability of this is much greater than the probability of this you know of all the atoms moving simultaneously into it's possible but the probability is incredibly small so this is a disordered system and this is an ordered system and because the number of configurations is not a capacity property that means we we can't add the configuration and work out the total entropy uh Bozeman concluded that we need to use the log of the number of configurations because the logarithms you can add up and not the total number of configurations to measure as a measure of disorder okay so what I want to do is I want to create a solution right I've got large chunks of material and I want to create a solution by mixing up the atoms so here's a large chunk of a and a large chunk of b in the proportion uh one minus x and x respectively and somehow or other we make them form an atomic solution an intimate mixture of a and b atoms now the free energy of this mechanical mixture is given by the rated sum of pure a and pure b rated by the concentrations so g star here is mu naught a times one minus x plus mu b times x so it simply is along this straight line joining the pure elements so at this stage the atoms really haven't perceived the presence of the b atoms when we mix them up into an intimate solution there is a reduction in free energy which is given by this delta g m and that reduction depends on the configurational entropy of mixing and various other terms but let's assume for the moment that this is an ideal solution and therefore we can work out the configurational entropy as normal and just to remind you because entropy is a difficult concept i'm going to go through it once again so here's a binary solution and the total number of atoms we have is capital n and this is a little bit of revision okay but it's important to repeat total number of atoms is capital n and there are small n atoms of type a and therefore capital n minus n atoms of type b and i'm going to place them onto a lattice so here is my lattice and the first atom i can put on any one of these positions right so i have n capital n number of locations where i can put that atom the second atom i can put on n minus capital n minus one locations but you know we can't distinguish two atom being here or two atom being here and one atom being here so we have to divide that by two these two possibilities are equivalent because the atoms are identical similarly with with this third atom we have n minus two locations but these are all identical so we have to divide by three factorial which is six and so on so that's n for the first one n over n into n minus one for the two atoms but we have to divide that by two because one two and two one are not distinguishable and similarly in the third case we have n into n minus one into n minus two over three factorial and in general we come up with this as our equation and we demonstrated that when we take the logarithm of this using sterling's approximation we end up with an entropy of mixing and when i multiply this by minus t that gives us the free energy of mixing as a function of concentration assuming that we are mixing things at random and that's another way of saying that we are dealing with an ideal solution and an ideal solution means that there is no change in entropy when we mix up all these atoms so what does this plot look like well the entropy of mixing does that okay it's maximum when the concentrations of a and b are identical and remember when I talked about the high entropy alloys I explained that you know when you have five components nickel cobalt manganese iron and something else okay five components if they are of equal concentration then you get a maximum entropy of mixing okay so that's logical that we reach a maximum entropy of mixing when the concentration is a half if I take this curve and I now multiply it by minus t then I end up with the free energy of mixing and this is the ideal free energy of mixing and what it says is that these atoms like to be next to unlike neighbors right because the free energy is at a minimum when the concentration is a half if they do not like to be next to each other then there would be a tendency for clustering which I'll describe later so this is the free energy of mixing for an ideal solution and as I said it's obtained by multiplying the entropy of mixing by minus t because we don't have any enthalpy term as yet so that's the equation for the free energy of mixing that's avocados constant and that is the Boltzmann constant now of course it's it's very rare that we get a solution which is ideal okay so there will be enthalpy terms associated with breaking a bonds breaking bb bonds and creating ab bonds so we we identify that by writing this term here which is an excess enthalpy uh excess above that of the ideal solution where it is zero okay and there may be other entropy terms for example due to lattice vibrations changing with concentration and and so on thermal entropy terms which are not accounted for in the configurational term so this is now what we call a regular solution and think about it is there an approximation here I'll just give you a moment all right so we have here an enthalpy change when we mix the atoms and this is the ideal entropy of mixing that means the atoms are at random okay so the answer is obviously the entropy of mixing here is an approximation because we are assuming that the atoms are disposed at random but if there's a tendency for air atoms to like each other more than they like b atoms then we will have an enthalpy term which will make the solution non-random but as an approximation we will use this as the configurational entropy later on in the very last lecture I will show you how to take account of the fact that this is not actually a random mixture when we have a finite enthalpy of mixing except at very high temperatures at very high temperatures no matter what the bonding is you will tend to get a mixing of all the atoms okay now how do we define the binding energy between two atoms and at the moment we are only looking at pairwise binding energies well if you take a pair of atoms really far apart okay infinite distance apart and you bring them together there may be an attractive force and as they come closer and closer there will be a strong repulsion because their electron clouds begin to overlap in a big way so this is the curve that I was describing where I take a pair of atoms with a large separation bring them closer they start interacting attractively because we are talking about an air bond and then you go through a minimum when you start to push the atoms too close together the energy rises into a repulsion and you know this curve is actually important because this is an anharmonic curve and what that means is that if if my pair of atoms is located at the energy minimum and I raise the temperature to this point yeah then the center point of this and this is no longer along this line along this line it moves like so and that means that you get thermal expansion okay so from this curve you could estimate the thermal expansion coefficient if you had this curve similarly you know the the force required to push or pull the atoms is you know the derivative of the energy as a function of separation so the slopes here also define the elastic moduli of your material so this is an important important function here which represents the force between represents the binding between two atoms and by convention we take the binding energy as being minus two epsilon AA because there are two atoms involved and you know okay so if I have a random mixture of A and B atoms then the probability of finding an A atom is one minus x the concentration of A and the probability of finding another A atom next to it is also one minus x so the product of these gives you the probability of finding an AA bond inside your random mixture and we've got a factor of half in front because the two A atoms cannot be distinguished right similarly the chance of finding a B atom is x and therefore the chance of finding a BB pair is x squared and again we have this factor half here z is simply a coordination number okay that depends on the sort of lattice you have so if it's a primitive cubic lattice then you have a coordination number of six that means so you have six atoms surrounding a particular atom and those are the BB bonds or the AA bonds now to find AB bonds we take the probability of finding an A atom multiplied by the probability of finding a B atom and also a BA bond which can be distinguished from an AB bond and therefore there's no no factor of half over here so all we have to do now is to see how many AA bonds and AB bonds we had to break in order to create this many unlike atom bonds and that gives us our entropy of mixing here which is the Avogadro's number coordination number concentration of A concentration of B times this term omega which is the difference in the binding energies of the original configurations of AA and BB atoms BB bonds and our final result which gives us the AB bonds so this term here is taking account of the fact that when you break an AA bond and a BB bond to create two AB bonds this is the change in energy and that gives us the enthalpy of mixing here now this product of the Avogadro's number the coordination number and omega is known as the regular solution constant okay so if you look at papers on thermodynamics and so on they might give you values of a regular solution constant and now if this is the case that by breaking air bonds and BB bonds the energy of an AB bond is less than that that means that unlike atoms like to be next to each other okay so this would be a system which favors some sort of ordering at a low temperature low temperature because at high temperatures entropy dominates you know because you're multiplying the entropy of mixing by a temperature so obviously that term becomes larger at high temperatures and dominates anything about enthalpy you had complete mixing so at low temperatures this would be a system which favors ordering in the solution this on the other hand would favor clustering that means that in the solution you would be able to find many clusters of A atoms and clusters of B atoms with AB bonds not being the preferred option okay so let's see what the enthalpy function looks like and this is the case here for an ideal solution right where there's no enthalpy of mixing and this is when the enthalpy of mixing is less than zero so we are favoring favoring ordering and you can see a deeper curve but the shape is identical with a minimum at at a half concentration and I explained to you that if you go to a high enough temperature then the system will become disordered now this is again our our reference for the enthalpy of mixing at a thousand kelvin and if this is favoring clustering of atoms then the free energy curve changes like this and has two minima in this case it's a gentle minimum but here's here you can see clearly there is a minimum here and minimum here and a maximum in the middle indicating that you know the A rich and B rich regions are favored and supposing I start at a high temperature where everything is mixed so this is three thousand kelvin and again clustering is favored but at three thousand kelvin you know everything will be mixed because temperature knocks everything into a uniform mixture random mixture when I call that the solution will tend to separate into A rich and B rich regions so this is known as spinodal decomposition I'll explain that in another lecture but basically it means that you know if you had a homogeneous solution which you created at a high temperature and supercooled it it would tend to separate out into A rich and B rich regions and that separation can be spontaneous okay now here is a free energy curve for a solution of A and B and if I fix the composition at this value x here then the free energy of that solution is given by g of x if I now draw a tangent to the free energy curve at this concentration then the intercept on the vertical axis is mu A and the intercept on the other vertical axis of pure B is mu B I can write gx as the weighted mean of mu A and mu B as follows where this is the concentration of A times mu A and this is the concentration of B times mu B and that is equal to the net free energy of a solution of composition x so we call these quantities mu A and mu B which are per mole as the chemical potential of A in a solution of a specific composition and this is of course the mole fraction of A so the mole fraction of A multiplied by the chemical potential of A gives me the contribution of only the A atoms to the total free energy of the solution and similarly this is the contribution of only the B atoms to a solution of composition x so we have effectively separated out the free energy of the solution into two components one a contribution due to A atoms and another contribution due to B atoms and of course these chemical potentials vary with compositions if my tangent had been drawn at a different point than the chemical potential of B atoms in another solution would be different useful well if you look at this graph where we have alpha and gamma the free energy curves of alpha and gamma and if they share a common tangent then the chemical potential of carbon in gamma is exactly the same as the chemical potential of carbon in alpha and of course the chemical potential of iron in gamma is the same as the chemical potential of iron in alpha so even if these two phases have a different chemical composition okay so these are the equilibrium compositions of ferrite and of austenite even if they have a different chemical composition there is no tendency whatsoever if they have these compositions for the composition to homogenize the reason being that the free energy of the carbon atoms in the alpha is the same as that of carbon atoms in the gamma even though their concentrations are different and similarly the free energy of iron atoms in alpha is the same as free energy of iron atoms in gamma even though they're different concentrations so the common tangent allows us to obtain the equilibrium composition once we know the free energy function and this is an example I used in a previous lecture where we are looking at ice which freezes from seawater but turns out to be pure whereas this is rich in salt and the reason why that remains the case is that the free energy of sodium chloride in the sea is the same as in ice and similarly for water molecules in ice the same as in the sea even though the ice is pure and the seawater is not so there's no tendency for salt to transfer from the water to the ice the ice can grow while rejecting the salt into the water if the temperature drops but there if there are at equilibrium there's no tendency for a redistribution of solute between these two species okay so this is the equation that the graph that I showed earlier where we have a free energy of a solution of composition x this is the chemical potential of a for this particular concentration of a so I'm not multiplying mu a by one minus x it's just saying this is the chemical potential at this concentration of a and this is the chemical potential of b at this concentration of b and we had an equation where we were able to partition this free energy into a component due to a and a component due to b and these are the free energies of pure a and pure b now what's happening here why do we have this enormous difference here okay well that difference is what we write as rt log of the activity of a so this distance here we write as rt log activity of a and why is that well you know we had the free energy of mixing as given by you know rt into x log x plus one minus x log one minus x so that x log x term comes from the atoms and if you just take the log x because we are not considering concentration we are not multiplying by concentration we can do that later as we did when we partitioned this free energy into components so this is pure a and this is the contribution from rt log x okay which which we dealt with in the configuration and for mixing but of course I haven't got x in that equation I've got something called activity so what does activity actually mean why isn't this a concentration as we would expect from the equation we derived earlier why is this a peculiar thing which we call activity well let me give you a physical meaning for activity and notice of course that this this intercept is a function of the composition if I had a different tangent then the chemical potential of air would change and the distance between these two would also change and the activity would change okay so I want to give you a physical explanation of what activity means supposing I have a mixture of 50 percent of alcohol in water and it's a completely ideal solution that means a mix at random and I look at the vapor pressure of water above this mixture then that's an indication of the activity of the water in this case it will simply reflect the concentration of water inside that mixture on the other hand if the alcohol repels the water okay it's no longer an ideal solution the water molecules want to be next to each other and the alcohol molecules want to be next to each other then the activity of water about this solution will increase the vapor pressure of water will increase so that's what we really mean by activity the concentrations in these two cases are exactly identical but the effective concentration of water is larger in the second case so activity should be thought about as an effective concentration if the activity is large then the particular element or molecule is behaving as if it has a larger concentration and we write the relationship between activity and concentration using an activity coefficient here which is this capital gamma now it's it's interesting some of you might recognize him but this is a mixture of alcohol and water and if you use a particular technique known as soft x-ray fluorescence then you can investigate how the molecular structure of that liquid is it turns out that alcohol and water don't mix on a very small scale so the water molecules tend to form clusters with the alcohol molecules forming sort of like not quite polymer rings but rings around them rings around them so the entropy of this alcohol water mixture is actually lower than what you would expect if it was a completely mixed system so you think that alcohol and water mix but when you look at it on a molecular level they do not and it is the it is a fact that the entropy is smaller than you expect and it's also a fact that the activity of water is larger than its concentration in this mixture okay if I now look at this relationship here it appears that if this capital gamma is a constant but that can depend right for an ideal solution the activity scales exactly with the mole fraction and we have a straight line here of slope one and yes the activity coefficient is constant in this case but for a solution which in which the like atoms unlike atoms attract you get a deviation from this line so the activity coefficient is no longer constant okay and this region is known as the Raoultian region where you know the concentration of one of the species is very low and this is the Henry's law region where the concentration of the solute is very low but that doesn't matter okay and similarly this is the case where you know the atoms tend to cluster so you clearly are not following the ideal solution now it turns out that there is I can find an example of an ideal solution when we look at oxides and this was work done by the very famous Darken and Carrick. Darken did sort of seminal work on on diffusion and many other aspects of thermodynamics so here is a graph that I've taken from his paper from dear paper in this journal Geo-Chemika and Cosmo-Chemika actor and I'm plotting here two lines because this represents the line for magnesium oxide in a solution of magnesium oxide and nickel oxide and this is the activity of nickel oxide in a solution of magnesium oxide and nickel oxide and these two exhibit ideal behavior right I before I decided to write this lecture I did a search to find an ideal solution because I did not know of any and what these people say is that look if the difference in molar volume between these two oxides is small then they tend to form an ideal solution and their explanation was in terms of the hume-rothery rules you may or may not know of the hume-rothery rules but there's a factor in there which takes account of the misfit when a solute atom is put into a solution so the idea is that in these oxide solutions when the difference in the molar volume is small they tend to form an ideal solution and remember these are ionic molecules but when the difference in molar volume is large and again many examples I've just taken a selected two then you get a highly non-ideal solution so this is magnesium oxide the activity of magnesium oxide in a solution of magnesium oxide and manganese oxide and this of manganese oxide in the same solution so it is very rare I think to find an ideal solution but it is possible to pick a few examples okay so that deals with activity activity coefficients which you'll encounter frequently when you deal with thermodynamics now I've got here the two free energy curves of austenite and of ferrite at a particular temperature and we find the equilibrium compositions of the two phases by constructing a common tension so that the chemical potentials of iron and carbon are uniform everywhere even though their chemical compositions are different now if we extrapolate these points as a function of temperature onto a phase diagram then of course we obtain these equilibrium phase boundaries this is the a3 phase boundary and this is the a1 phase boundary which defines the equilibrium between ferrite and austenite normally this is the iron carbon phase diagram without these extrapolations of the phase boundaries now supposing we extrapolate these phase boundaries because we have suppressed the decomposition reaction of gamma into pearlite to a lower temperature then we need to know the equilibrium compositions at the suppressed temperature and that can be obtained by extrapolating these phase boundaries using thermodynamic calculations the interesting thing is that within this extrapolated region it is possible for both cementite and ferrite to grow simultaneously from austenite and that means that in order to obtain a fully pearlitic structure the composition need not be exactly the eutectoid composition supposing we have a hyper eutectoid steel with a composition here and we supercool it so that we avoid the precipitation of cementite into this region then it's possible to get a fully pearlitic microstructure albeit with a larger fraction of cementite in the pearlite similarly if i have a hypo eutectoid steel and i avoid the precipitation of ferrite until i get to a temperature within this extrapolated region then it's possible to get a fully pearlitic microstructure in a hypo eutectoid steel the volume fraction of ferrite would be greater than you would expect from just a eutectoid composition this region is known as the hulkron extrapolation region and it helps us to design pearlite which is much stronger than the pearlite that you get from the reaction at the eutectoid temperature and the eutectoid composition so here for example is a rail steel with carbon concentration of 0.95 weight percent so it's a hyper eutectoid steel and it is cooled into a region where it can become fully pearlitic without any precipitation of pro eutectoid cementite which is not good for the properties the interesting thing is that even even with this fully pearlitic structure because we have formed it at a relatively low temperature the inter lamella spacing is really fine so this is of the order of 200 nanometers here and this is one of the modern rail steels if you go back to 1970 then for that rail steel the inter lamella spacing was about five micrometers so this is a reduction by a factor of about 20 in the inter lamella spacing achieving a much higher hardness and wear resistance for railway tracks so sometimes we can look at the equilibrium between two phases even in domains two or more phases even in domains where other reactions should happen if those reactions have been suppressed because of kinetic effects so here is that diagram again where this is the point where ferrite and austenite of the same composition at a particular temperature have the same free energy and if i plot the locus of these points onto my phase diagram then i get a curve which is called the t zero curve okay and these boundaries here are simply obtained from the common tangent to the alpha and gamma surfaces free energy curves now what is the importance of this this t zero point well if i have austenite of this chemical composition and i try to transform it into ferrite of the same chemical composition then i would get an increasing free energy and that thermodynamically is not possible okay on the other hand if i'm to the left of that point and i take austenite of that composition and transform it into ferrite without any change in composition then it is thermodynamically possible you get a smaller free energy change than if the material was transforming to its equilibrium composition which would be here that would be the free energy change if it's separated out into alpha and gamma of these compositions but that is when there is no change in composition so what this means is that martensitic transformation is only possible if the carbon concentration of austenite is in this region of the phase diagram to the left of the t zero curve there is no possibility of getting martensitic transformation if the composition of the austenite falls in this region so bear in mind that martensite is not an equilibrium transformation and here we are using equilibrium thermodynamics to treat that case it's perfectly reasonable to do that if you assume that there is no atomic mobility all right in other words the ferrite must be forced to accept the composition of the austenite and that of course is the case when we do transformation at temperature as much below 600 degrees centigrade so this defines the limit of diffusion less transformation okay here we have a system where we have iron carbon and manganese okay but it could be any three elements so this is no longer a curve but it is a surface in three dimensions so if you think about half of football and the lower half of the football roughly like that okay so this is a surface in three dimensions a free energy surface of ferrite as a function of manganese carbon and iron and similarly this is a surface in three dimensions now instead of a common tangent we have to use a tangent plane here and where that tangent plane touches these two surfaces defines the equilibrium composition of gamma and of alpha because the chemical potentials of manganese carbon and iron are uniform in all of the phases so it's straightforward you know the condition for equilibrium is simply that the chemical potential of all the solids should be identical everywhere and of the solvent so if i'm dealing with a 20 component alloy i simply need to write these equations with 20 terms i don't i can't actually plot plot a diagram then because it's hard to conceive of dimensions more than three but these are the equations which define equilibrium whether it's a 20 component system or a two component system so from the point of view of calculations using you know commercially available or free software we don't need to draw these diagrams all we are interested in is the thermodynamic quantities to find the circumstances where the chemical potential is uniform everywhere for all the elements to find the volume fractions of the equilibrium phases if we are talking about a particular alloy and that's that's straightforward because we know the equilibrium composition then the alloy compositions then we can work out the volume fractions and of course the chemical compositions of the phases which are given by the condition here that this point extrapolates onto our phase diagram and this point extrapolates here now there is an additional complication is that because we've added manganese here this is not a unique tie line i can take that tangent plane and rock it while still maintaining contact with the two free energy surfaces so there's an infinite number of tie lines possible at this particular temperature and this defines the alpha plus gamma phase field at a constant temperature because we have an extra degree of freedom here and this phase field alpha plus gamma simply consists of the thousands of tie lines generated by rocking that tangent plane and you know obviously if you have more elements there are greater degrees of freedom okay so that defines the case for equilibrium both in a single component system binary ternary and higher order because we don't need to draw diagrams we can simply do calculations to ensure that these kinds of equations are satisfied and we get all the information that you actually get from a phase diagram okay now let me show you a couple of examples of how you could use use this knowledge now there was a time when intermetallic compounds were quite prominent as a research activity and one of the advantages of some of the intermetallic compounds like ni3al is that their strength increases with temperature because the dislocation structure is such that as the temperature increases the dislocations tend to move onto a plane where they're less glissal and therefore the strength is maintained as a function of temperature until around 800 degrees centigrade so this ni3al or ni3ti have that property but if you just look at ni3al or ni3ti or nial or other intermetallics they were found to be extremely brittle okay now obviously if you're going to make a high temperature alloy which is going to operate in let's say an aircraft engine or something you can't afford it to be brittle so our idea was to create a metal metal composite okay consisting of three different intermetallic compounds some of which will be more ductile than others and therefore they would support the entire system and of course this is going to operate at high temperatures so we need to choose our intermetallic compounds such that they don't react with each other how can you do that well there will have different chemical compositions different crystal structures but nevertheless if we choose their compositions so that the chemical potentials of all the solids are uniform in all the intermetallics there will be no tendency for any reaction or diffusion these are the three intermetallic compounds nickel aluminium titanium mixed on the sub lattice ni2alti and ni3alti and this just shows you the final composite so these are obtained as powders and then extruded to form our composite material and this is to show that some intermetallics are brittle but the cracks are arrested by the others which are not so brittle now to ensure that they don't start reacting with each other the compositions were selected to be these three points here this this and this because that is a tie triangle which defines equilibrium between these three compounds so if you have chosen the composition so that they already are in equilibrium before you have consolidated them by extrusion then there will be no tendency for redistribution of elements or reactions between the intermetallic compounds okay so you can use phase diagrams and the condition for equilibrium as a means for designing high temperature alloys if you do not want them to be changing during service then they have to be at equilibrium okay