Lecture 30 : Part 02 - Materials Design at Different Length Scale-IV

Students, welcome back to the part 2 of the lecture number 30 and here we are talking about how we utilize CALFAD, a calculation of phase diagram, CL stands for calculation, PA stands for phase diagram page and D stands for diagram. So we are going to discuss more details about that. So, you know, as I said you, so any kind of calculation of phase diagram require you first to capture data. Once you capture data, data can be captured either from experiments or from calculation, other type of calculations also. And once you capture the data, what you need to do is to make a good assessment of the initial assessment of a data. Sometime it requires you to do preprocessing and then after that we will be doing optimization of your data. Finally, once it is optimized, we will be storing data. Do not worry about it. We will discuss about this thing in detail manner. So what do you do? A data capture can be obtained either obtained from experiments or even calculated, right, like in Ibanezio calculations one can obtain that. So that is what it is. So it has to be captured. You can take the data either from experiments or from calculations and you can assess the data using analytics and modeling. Many of the preprocessing of data will be done by modeling, right, that is very important. And then you know there will be lot of free parameters in your model and you have to do super fitting of that model, free parameter needs to be optimized. So that once you predict something, it will not predict wrongly. That is what is done by optimization. So optimization allows you to determine these free parameters precisely or probably range of values for the free parameters. And finally once this optimization is done, you know you are happy with your predictions which are coming from experiments or may be some calculations, then you can store in terms of certain data files. And then it can be used for validations or any further application, that is what is done. So thermodynamic databases which are basically nothing but things which will come after the stage of storage, you can see here databases, they actually allow us all kinds of things. This allow us to do assessments of your phase, microstructure, many other things or even you know kinetic aspects like diffusion. And then it can also allow us to do some literature analysis of the whole thing, okay. If I have to solve a problem from which has been taking place for whether let us say 20 years or 30 years duration, many times we need to look at literature and do a thorough analysis what are the gaps, research gaps happened in a particular research aspects. So nonetheless it is very important to have this as input data and most of the cases input data is basically Gibbs free energy models of the phases. So let us talk about in details, what is it? First is the data capture, right. Because as you know in a Calvert data is the key thing or are the key thing, without data you cannot do anything, that is like a machine learning also. So data capture, these are all taken from Thermocal, as you can see here thermocal.com you can easily find out this. So data capture is what? Collection of high quality experimental data on any material systems, it can be on phase equilibria, it can be thermomechanical properties, like enthalpy of mixing or formation energies of the phases or compounds or even crystal structures, that is the thing we need to collect and we need to collect high quality data. But in many of you will find such a data either not available or partially available. So in that case you have to use ab initio calculations like DFT or something which will be talking some of the later lectures or you can do it using machine learning or you can do it empirical relationship also. So important aspect to make the data available to everybody, so that that can be used for very genuine purposes, that is the critical aspects of that. So now in a critical assessment of a preprocessing of data stage, after you capture the data you have to critically assess the data, why? For a phase diagram assessment of any given system, let us say you have two elements C and D, you want to do a phase diagram assessment of the system. There may exist large number of phases like let us talk about aluminum nickel system, there are large number of phases present, but it is used for super alloys, right. So for each of these phases we need to have a model for Gibbs free energy, Gibbs free energy is what is important and this has to be assigned by human expert depending on the crystal structure, depending on the chemistry. Gibbs free energy is in general is a polynomial, polynomial of chemical compositions or thermodynamic temperatures or pressure of the material and this is the key thing in computational thermodynamics as or the calphor adapter. Therefore this data once you capture from experiments or calculations or we obtain from empirical relationships or machine learning that needs to be properly processed and this processing is done for critical assessment of the things, okay. That is what this stage is required. So now once you have done that, you know once you have preprocessing of data everything is done, you may not know that model which you are going to develop will be properly done. So any kind of free energy model, free energy equations which you see will have different kinds of phases and microstructures, okay. And free energy, free parameters will be there in any model and this free parameter needs to be fitted with the input data and that is what is called optimization, optimization of these free parameters for the best possible use and this determines, this requires extensive human adjustment. Here you cannot use you know computers for do that and that is because the free parameter all phases should be consistent with each other, otherwise it will give you wrong results. So in a multi, this is basically a multi-objective optimization process. We can use higher order machine learning approaches but that is for optimizations like bias optimization or something. But it will have lot of constraint and that is why it is called training the multi-model, multiple model machine learning approach type which is analogous to that, it is not same. So this is a very key stage in a calcified approach, this optimizations, okay, where you are going to optimize this free parameters from the input data and that is why you need to use judgment of the human, extensive judgment of human. So once it is done, it will be stored for all the phases to the experimental data and gives free energy function then can be used as a database and that one everybody can use as a calculation. Nowadays, you know, free energy, this database is available to everybody. You can buy it, you can even use it, some of the cases free access is available. So the final step, you need to also do a validation because validation is important for, you know, previous experimental results, whether your optimized data are going to satisfy the previous experimental results or not, that is what is the step of validation. So in a, for a multi-component database or specific system, validation against data for the commercial or other multi-component is very critical because most of the cases, databases are not available. Even if they are available, they are not be perfectly optimized, so that is why validation is very important before you use it. So the agreement with the real multi-component alloys is not good for some key points. We have to go back to the optimization for the one or more loader systems, that is the only way to rectify this. So you understand there are five steps, first is data capture, then you critically assess the data and also preprocess the data if something is missing. Many cases you have developed a model for this free energy of the phases, how do you do that? And then you have to do optimization of the few parameters, so that you can actually use it for different kind of purposes and this is a multi-objective optimization with various kind of constraints in it and after that you will store some database and then finally you will validate these kind of databases. So this is what is shown again taken from the Thermocol website. First we capture data, then we do that, we will come back to Gibbs free energy functions very soon in the next couple of slides. Once you have done that optimization, then you have to do predictions, so predictions can be done. So this is putting them together, both the assessment and the optimization together once the data is captured. Then you can do the predictions and prediction allows to validate that your things, prediction can be simply plotting, this is the aluminum magnesium phase diagram, so many phases are present there and one can also use this diffusion calculation using DICTA, diffusion calculations that is what is called DICTA and one can also use a software like a Prisma which can allow you to understand the various kinds of phase transformations also, so that is it. So now let us go back to the board and let us see what is the free energy and what are these things. As you know free energy is the critical aspects in the Calphat calculation, so free energy is always described as a two things, so what is called Gibbs free energy and we marked as a G and G is written as H-GS, where H is known as a enthalpy and S is known as a entropy right and T is temperature, this is the fundamental relationship which allow us to estimate Gibbs free energy but remember we are always dealing with alloys. So then alloys H and S will also depends on chemistry, temperature and many other processing conditions right, as we have talked about it. So normally as you can see here free energy of any system is written as like this, G is let us talk about summation of you know I will explain to it all these things okay. So these are the three important parameters, three important parameters of free energy expression for any alloy system for any system actually. So first one is the free energy of components, in a binary systems you will have two components let us say A and B, so this will be corresponding to G A of pure A or G of pure B okay. For ternary system there will be three, for higher order system there will be many more is called as a mole fraction okay, so Xi is called the mole fraction that is the chemistry, how much of the particular element is present in the material okay. So I will give an example also and Gi 0 is free energy, sorry free energy of the pure phase and that is why I use 0 okay, that is why I use 0. Now this is this first part okay or I would say one, let us talk about the second part, what about the second part? Second part is once you mixed two or three elements there will be change of entropy right that is what is coming from S and as you can clearly see this is RT, so T is there and this is the logarithmic summation of the logarithmic term, this is coming from Boltzmann law, delta S mix is equal to minus R okay, Xi or for the two component system or let us write for summation only Xi ln I okay, and you know Xi ln I comes from Boltzmann statistics okay, because entropy is given as SK ln W right, that is why this log term comes into picture and this minus and this minus will be plus, can you see that? So this is nothing but a mixing term, once you mix together you will have a change of entropy and that is what is given by this change in entropy right, change in S, so that is the second term. So first term is very easily understandable, this is coming from the pure components that is nothing but the pure A and B whatever is the fraction presence, like say in aluminum silicon alloy as you have seen the phase diagram aluminum silicon I can draw it here, this is the temperature, this is the atom percentage or silicon chemistry, you have seen that this is there is a phase diagram like this, silicon is like that right, this is there. So if I take an alloy composition of this which is our let us say about 10 percentage then we can calculate silicon is 10 atom percentage and your aluminum will be, this is aluminum, this is silicon, aluminum will be 90 atom percentage, in a fraction this is 0.1, this is 0.9 right, that is what it is. So Xi can be calculated easily from this kind of informations and then you can easily calculate, you can easily calculate this because Gi0 it can be obtained from that, I will talk about it also. So Gi0 is basically for the pure phase as I told you, so this will be function of temperature and pressure right, because pure, so these functions will be available for each of these component like aluminum, silicon, depend on crystal structure also okay. So one can have different crystal structure possible like titanium has BCC and HCP crystal structure, then you have a different kind of microstructure. So let us write that also, so temperature, pressure, crystal structure all these things will play a very important role, so that is about it. Now next one is very easy, once you know Xi you can easily calculate that, R is nothing but the gas constant, universal gas constant okay. So this is the second part, what is the third part, third part is the excess. So G excess is called excess free energy, very, very confusing right, why excess? This is nothing but difference between actual G and these two. So that is what thermodynamic model comes in the picture, I can use simply as simple model where the free energy of a system is given by pure as well as entropic contributions. But you will see there are many systems where the actual free energy obtained from experiments or calculations will not match with these two factors. So then you have to put an extra term which is nothing but excess free energy and that is what is called excess that is why. So there are lot of models available to obtain what is the excess free energy. So fundamentally this is our fundamental equation, remember that, to obtain free energies. So now I go back to the slides, so for binaries, binary means two component systems, so Xi 0, a Gi 0 and entropic part is very easy to measure, the two component systems, you know the alloy compositions, you can easily do that, temperature also known. So you have to find out what is the excess, that is also possible, not very difficult because experiments has been done for all binary systems. I have given you the handbook right, ASM handbook for metals, you can find out that. There is a handbook on phase diagrams of all the binary systems. But you know when you talk about higher order system, ternary, quaternary, quinary, you cannot do it, measure for everything. So what you do is you simply take to the extrapolation of the binary data to the higher order data and when you do that you need to also do the assessment of your data, that is the key aspects. So this extrapolation is what is model, what is the model you do. So then from binary you can do the extrapolation of the ternary, again it will be followed by assessment, it will be followed by your assessments of the data. So once you have done G for your particular systems for all the phases, remember this. In a system you have to measure G for all the phases, G of all phases in a system needs to be known. Then you can do actually a minimization, you can do the minimization of the G, you can see this and whichever system will have minimum free energy, that is what going to be the stable systems. So this is the key aspects of the CalPHART. Now one thing which you probably need to know here is what is this extrapolation. By the way extrapolation is a mathematical exploration, extrapolation. This is not like a we do anything. There are three important exploration, extrapolation equations I will talk about it have. First one is widely used, it is called Muganyam equations. So you can clearly see here, here the excess free energy is given like a equation. And for a three order, three systems, three component system A, B, C, if I know the free energy of this, free energy of this, free energy of this, we can obtain this. So if I know the free energy of binaries, A, B binary, A, C binary, B, C binary we can obtain the free energy of a ternary system. How it is obtained? These are the chemistry. XA is mole fraction of A, XB is the mole fraction of B and rest is basically interaction parameters. Ls are actually interaction parameters, okay. That means they are interaction parameters of 0 and 1, okay. 0 is basically pure and 1 is when it is a system in the alloy, right. That is what it is done. So these equations are built in in today's software, whether you are using CalPHART or using another calculation, they are built in. But many cases this Muganiya-U will not work. So we have to use other like Kohler or Tube's equations, which are much more complex. You can see here, I do not want to go in details of each of these terms, but they are much more complex than the Muganiya-U. Muganiya-U is very simple. It is used XA, XB, XB, XC, XA, XC and then standard things LA0, LAb0 plus LAb1, LAbC, LAb0, LAbC0 and LA1bC, LAac0, LAac1, right, standard. That is why it is widely used, which is very easy to use also. But Kohler and Tube equations are much more complex. So why they are used? Because Muganiya-U is sometimes not successful in predicting the thermodynamical values of these equations. That is why we need to use this. So remember these extrapolations, they are very important. I do not want to write in the blackboard because this will take a lot of time. But these are there for you if you want to know. So this is the key things. So now there is something else also there. So that is called how to construct a non-equilibrium free energy from equilibrium properties. That is very difficult questions to answer this moment, but I just try to tell you that free energy in a solid phase can be functions of composition as well as something known as order parameter. You know, because there are many phases like in super alloys, some phases are order. Order means what? Atomic elements are specifically given to the different sides, right, compared to disorder phases. So there how to predict this order, this, you know, phases free energy compared to the disorder phase. That is why you need to talk about the order parameter. So now if you simply take a differentiation of the free energy with respect to order parameter if it goes to 0, then you will get a value of equilibrium value of the order parameter, right. That is what it is, the minimum free energy for the order parameter, okay. And then you can actually get a equilibrium free energy equations, correct. That is what it is. So that will allow us to know the free energy of different kinds of equilibrium and non-equilibrium phases, correct. So you know, I do not want to go in details of this. know there are many various ways and means to, you know, find out the free energy of the different order solid phases, okay. And these are built-in nowadays in these, or the non-equilibrium phases actually built-in nowadays in the various kinds of softwares. Let us take some example of the model systems, then we wind up today's lecture. So as you know free energy of a ternary system like A, B, C is given like, standard one is given like this. So as you can see, this is the delta G actually, this is not G. So this is the mixing free energy. So that means this is nothing but G minus summation of xi gi naught, that is what is gm, okay. So now you can see here this is the entropic term, xi ln xa plus xp ln xp plus xc ln xc, that is nothing but, I can write down rt summation of xi ln xi, I have written in the board also. And this is the term which is xs. So you can clearly see, right, that is the term for xs. These are the terms for, you know, this. So if I have to write total G, so this will be nothing but xi gi 0 summation i123 plus rt xi ln xi i123 plus this one, okay, this one is a term l xa xc plus xp xc, right, this is the xs term. The three terms I have written. So let us take that. This is taken from Dulu et al., from this chapter. Now if I take a system like that, different kinds of multi compound system like, you know, you can see that. I do not want to go up that. Iron, manganese, nickel, chromium, cobalt is a standard one. And we know that these are the expected phases, fcc and bcc. And this is the delta H mix or enthalpy of mixing. And there are different parameter like valence selection concentrations or atomic size differences called delta. So one can actually do a thermodynamic calculations using the same approach which I discussed now and find out different phases like this is a mole fraction of phases versus temperature. You can see the three phases are fcc, liquid and bcc. Clearly for this alloy, for this first alloy, okay, sorry, for the second alloy not first alloy. So first alloy is well known in the literature that is why we have not done that, the second alloy. And that is what fcc1 is called. And then you can do it for fcc2. You can see the compositions, mole fraction of, there are more phases, sigma phase is also present. Sigma is another phase and keep on doing different kinds of things. Not only that, one can actually even do, let us take this alloy, manganese, iron, cobalt, nickel, chromper. And we want to vary the chemistry from 0.05 percent to 0.35, that is 5 percent to 35 percent for each of the elements and plot it. So what you do is, so we are, see you are plotting in a ternary triangles like, you know, here you are talking about manganese plus copper, cobalt and iron plus nickel. And you can clearly see where this fcc phase will form. I can calculate. So for the red regions it is maximum about close to 1, for the black region is close to 0 and intermediates are there. So we can do at 1000 degrees, we can do at a different, different temperature for the same systems or different alloy systems. So these are the kind of calculations one can easily do using the CalPERS nowadays. Routinely you can be doing. model calculations in front of you, show how these calculations are done and how their different kind of phase diagrams they can obtain. So this is very key aspects in getting lot of data out of thermodynamic calculations or generate phase diagrams out of thermodynamic data. So let us stop here. We will get back to you in the next lecture on more of thermodynamics, especially on the phase field modeling to predict the microstructure. Thank you.