Lecture 31 : Part 02 - Materials Design at Different Length Scale-VI

Students, welcome back to the lecture part number 2 or second part of the lecture 31. We have been discussing about the different length scale issues as well as the calculations required for obtaining data. So, in the last part or the first part of the lecture rather, we talked about CALFORD. I showed you different kinds of mathematical formalism as well as plots for both bulk scale as well as nano scale systems. Now you can also use CALFORD provides you phases and their quantities in the microstructure, but does not provide you real microstructure. How these phases are distributed, how they look like, what are their sizes, shapes, morphologies, it does not provide because it is a thermodynamic calculations. Secondly it also provides you mostly the microstructure or the stable phases, thermodynamically stable phases. It is not easy to obtain the microstructure phase diagrams of the metastable phases. Well, it is possible to do nowadays, but not easy. So nonetheless, it only provides different phases and their amounts of volume percentage or volume fractions, whatever you can say. It does not provide you how the phases will look like, how they will be distributed and also it does not provide also the phases microstructure, where there are lot of strain energy effects and you can also have issues of transformations. So in a nutshell, we can get this information using phase field simulations. That is what we will discuss in this part of the lecture. So as you know, we have to connect thermodynamics to kinetics to the microstructure. That is the key things in getting that. Again, this particular slide is obtained from PANDAT and THERMOCAL both. That is what you see these two features. The picture of a PANDAT tells you PANDAT and picture of this kind of ternary things called is taken from CALFED. So we can actually get the kinetic parameters like diffusivities, atomistic, whatever calculations. Once you get that, you can calculate the mobility databases. That can be used for DICTA diffusion build calculations and we can also use experimental data and optimize. Once you optimize that, you can get a much better understanding of how the rate of transformation can happen. Like here, the chemical diffusivities of titanium in an alloy is shown. But this is okay, nice, but does not provide you the microstructure of the titanium alloy. As you know, the microstructure will be controlled by many things, not only diffusivities, mobilities, free energies, but also strain energies. When this kind of phases are present, you can see here in this picture, which is a basket with microstructure we call in titanium alloys. So you can have strain energies playing a very important role between the phases. So let us talk about phase field simulations. What is the main idea? Well, main idea is to solve a single set of partial differential equations over the entire domain. So before that, let me tell you that let us imagine the growth of precipitates in aluminum copper alloy and these precipitates are growing in a matrix, right? As you have seen, let us talk about this picture, which I have shown you, precipitates here. This is the picture you see, precipitates are going in a matrix. So these are the precipitates, which are shown here, these ones and this blue color thing is called matrix. So if that is the case, we are thinking about it. So there are three distinct quantities to consider. One is the precipitate obviously, second is the matrix on which you are forming and third is the precipitous matrix interface. These three has to be considered. So if I draw it here again, that picture, let me explain to you. So this is the microstructure, these are the precipitates. So this is the matrix. So we have three pieces here, matrix, precipitate and you have an interface between precipitates and matrix, that is the interface. That is constitute our system, which is where I have drawn it here. So there are three, these three distinct quantities. Interfaces can be described in different ways, like this actually evolving surfaces, whose motion is controlled according to the boundary conditions exist. This precipitates can grow, then this motion will happen or the boundaries will also move. So that is what is the thing, which allows you to develop the microstructure. Initially precipitates may start forming a small particles, then they grow and become bigger. We are talking about that, because phase precipitates allows you understand how the microstructure actually evolve as a function of time, as a function of temperature, as a function of chemistry, all kinds of things, that is what. So that is why the interface in this sign of systems require a mathematical descriptions. CALFED provides you free energy descriptions of the matrix and the precipitates. There are two different phases, we can always obtain free energies or g or precipitate and the matrix, but it does not provide you the information about the interface. Well you may say it is provide, but when the interface moves, it does not provide much information about that. So we need to do that. So phase pre-model, there are two main issues are used. One is the bulk thermodynamics, which you already discussed in CALFED. Like here you have a system, I shown you my phase diagram FCC L1 to FCC L2 phases. So it will give you free energy functions for each of these phases, but as you told like in the last slide about the precipitate matrix microstructure, you have a interface between a precipitate and the matrix. So therefore, there is a interface that requires you to know surface properties also. So in a phase pre-model, these two are dealt together, but they are two separate issues or separate quantities. But we have to develop, we have to solve a single PDE over the entire domain. That is the main idea of this phase pre-build. How do you do that? So phase pre-build incorporates both the bulk thermodynamics of the multiphase systems. Just now I talked about precipitates and the matrix, as well as also it provides the information of the surface thermodynamics. Sometime people call it Gibbs surface excess quantities, they are nothing but a surface properties. That is what they do. So now how do you characterize the interface? We will talk about these equations now. So in a phase pre-model, state of the entire microstructure is represented by continuously one single variable. The state of the whole, because we have to develop one single PDE, partial differential equation. And you have free energy of the phases and surface energy of the interfaces. So what do you do? You cannot have different sets of equation, then you have problem of tracking the interface. So what do you do is that we define a single variable. By interface, it is represented by a single variable or rather, do not interpret the entire microstructure is represented by a single variable. That variable is always called as a phi and as you see here, this is 1, this is 0. So there are two phases here, one phase is more denser atomically than other phase. So we have seen that this phi is 1 for the less dense phase, 0 for the more dense phase. And the interface, the phi varies from 1 to 0. So phi is called an order parameter or indicator function. I hope you got it. And this phi is, as I told you, phi is 1 and 0. So therefore, phi between 0 and 1, 0, phi, 1, it indicates the interface or that is why it is changing, the values are changing. For the less dense phase, it is 1, more dense phase, it is 0, close to 0. In this regime, interface is diffused and we call this is the width of the interface as you can see here, width of the interface. So this interface is kind of defined by the change of phi in this regime, 0 to 1. I hope it is clear. The range for which it changes, this is the width of the interface and remember this interface is diffused. It is not sharp, right. If it is sharp interface, then it will have a jump. Sharp interface will look like this, correct. But here we are showing the interface in a different fashion. So this is a, this kind of interface. So there is a finite width, correct. This finite width is defined the diffused interface. So this is the main aspect and this variable is phi is called a phase field variable, order parameter or phase field. This is what it is called. Now how do you understand the evolution of microstructure? So let us go back to the board. We will come back to this equation little later. So evolution of microstructure, so this is basically, it has two parts. One is the thermodynamics, thermodynamical description, description of phases and second one is surface properties or interface properties, these two are the things. So this is defined by the phi energies, this is defined by the surface energies as well as width of that. So let me draw a microstructure again. So you have precipitates in a matrix. These are precipitates, this is matrix, there is a interface. I write EI between the phases. That is why we need a surface interface, surface properties of that interface. Well that is, but now the question is how do we define the interface? That is what we are going to do. We basically want to develop a single equation. So we define interface by this way. So we see the value, how this variable phi is varying. So you can see that this is 0 and 1. So phi is varying. This is the width of the interface. That is what it is. This is width of the interface. So we can define that interface by this width, also define the interface by the value of the phi. So phi is 1 for one phase and phi is 0 for another phase, phase 1, phase 2 and phi varies from 0 to 1 for interface. Now it is easy, very easy to simulate the microstructure evolution because we need to formulate equation how phi is varying as a function of time. So how do you do? We write this equation or basically like that. What is that? Evolution of microstructure. Evolution of microstructure will be given by this equation del phi by del t equal to m m del g by del phi. So this is the first term here. This term is telling you variation of phi or phase field variable. So phi is what? Phi is your phase field variable. So this is the variation of phi with time. This one is called mobility related to diffusivity. This is related to diffusivity. And then what is g? g is nothing but free energy functional. g describes how free energy will change or free energy varies as a function of phase field variable or phi. So this is your master equation which you need to solve. I told you we need to have one single PDE, partial differential equation. This is t by the way, do not get confused. One partial equation which will provide you everything. That is what will give you evolution of microstructure as a function of time. Now g has to be written. g as you know has two parts. One is g is from this phases, precipitation matrix, other one is the interface. You can always write down g for the whole volume V. So first term is your g 0 phi time temperature plus a constant epsilon into grad phi square. Let me explain. So this is the total free energy which is integrated over the whole volume. That is what it is called. It is integrated over the whole volume. So here T is temperature, V is volume and phi is your phase field variable. So g 0 for solidification let us take can be written as h into g s plus 1 minus h into g l plus q into f. So h is the fraction of solid multiplied by the free energy of solid in solidification. When liquid solid we are talking about it, not precipitates. And then this 1 minus h is the fraction of liquid multiple plus a function q into f. That is what can be written. And this q is f. Basically q is the height of the activation and barrier at the interface. So we can write down for the precipitates as like this h into this matrix and q f. So this is the volume fraction of the precipitates multiplied by free energy of precipitates plus volume fraction of the matrix multiplied by the free energy of the matrix plus q f. And q is basically the barrier height of the interface. What does it mean? It means how strong the interface is because interface will grow. The precipitates will become bigger and bigger. Then interface will become grow. That is what. So a barrier height is very important then as precipitates become thicker and thicker. So now this function h can be also written as a function of phi or f also can be written. So as you can see everything you can get down in terms of the phi, the phase field variables. Everything here you can get down. G is this. So we should know the free energy of the precipitate and we should know free energy of the matrix. That is okay. That can be obtained from Calford and q is the barrier height. That also can be obtained from a diffusion calculations and h and f can be obtained like this. I will talk about why you are using this kind of functions. So finally we can find out g as a function of phi. So now we can take the first derivative, partial derivative of phi and then we can solve this mathematical partial differential equations. So this is the main theme of phase field evolution. So we track how phi varies as a function of time. So you can see here free energy is given this. Again same thing. This is this. Here is k is corresponding to epsilon there. I told you and this phi square is basically related to the surface energy or interfacial energy and this is the free energy functions. Now why we take such a kind of free energy functions like h and f is shown in the board. That is because taking the corollary of liquid and solids because we want to have two minima each for liquid and solid and this functions tells you that okay. You can clearly see h as a minima at 3 by 2 and f as a minima as phi equal to 1 and both as a minima as phi equal to 0 right. Both the function goes to 0 when phi is 0 and h goes to also 0 when phi become 3 by 2, f goes to 0 and when phi becomes 1 right. So that is why you need this kind of double well potentials. The double well potential helps you to do that okay. So total free energy is a kind of it is a functional that is why it is called free energy function. g is considered as a free energy functional there and that is why g is to be written like that. So in a nutshell if I have to write the final equation for this okay, final equation for this del phi by del t. So del phi by del t will look like this that is equal to m mobility multiplied by epsilon phi square epsilon sorry not phi grad phi square that phi tells you the interpersonal energy. Epsilon is a constant we multiplied to take care of that then h l sorry h g l minus g s or rather g matrix minus g precipitate minus q f. So that will be the final equation if I have to do mathematics properly and this is what you are going to solve. So something is shown here also you can see here on the right side they have taken m out m 1 by m del phi by del t and this is your the surface energy terms okay, capillary terms and this is what is your free energy terms this is called canal and equations in a way. So let us not talk about much of these too much in details. So we can always define free energy functional like this also epsilon square del phi the square the gradient phi square and sometime people take half also that is okay, but finally you are solving this fundamental equation. So let us come back to the some of the examples of that I will not go back into all these equations they are they will talk many more physics into that, but that will make your life more difficult. So once you solve the equations one of these are shown here. So you can get different kinds of microstructures let us talk about that. So phase field simulations on the growth of dendrites very important. So as you see here once you solve that this is taken from NIST okay, National Institute of Science and Technology US. This is showing nicely grown dendrites not only that you can one can see when the secondary arm then this dendrite signed arms are bridging can you see that they are bridging they are forming bridge right here also they are bridging correct that can also be tracked very easy because it is basically solid and liquid for dendritic growth. So it will be GL and GS instead of GP and G matrix GM equations will remain same. So one can easily track how the dendrites grows into the liquid. Here it is showing even how actually a one single dendrite showing you can see that actually one can easily get this kind of steps and dendrites this is taken from Mullis and Rossum Mullis et al basically and from the University of Leeds and you can see that dendrites are going typically in the liquid and the concentration profile across the liquid from the solid to liquid can be also tracked or can be seen that. So this is a liquid composition this is a solid composition and around that solid there is a segregation of these solids or the extent of the solid that is why they segregated and slowly decreases. One can also look at the dendrites you know morphology of dendrites much better way also this is what is shown again here you can see that how the dendrites started growing from initial as a spherical nucleus then the spherical nucleus get broken down and finally, finally, finally four pieces starts forming from the center. You can also do a phase-based simulation of eutectics this is taken from our eutectic grows like two-phase microstructures you can see there that one is white other one is grey. So we can start with liquid and two phases and then you can see that how the two phases will start developing for the different conditions C1 and C2 and one can not only that one can also look at if you have a seed crystal one phase the other phase will form you can see that other phase forming another phase is going on the top of the second phase that is also possible the red color phase as was growing it grows and finally, the second phase started forming. So it is also shown here on this picture you can see that the two phases they directly go in the liquid and there finally, becomes a full scale microstructure of that. So this is very clear aspects of that we can do. So now I will also like to give you one information the lecture number 32 and 33 will be delivered by a guest faculty Professor Tarun Kumar Kundu he is from IIT Kharagpur he is from the department of metallurgical materials engineering and this is a details his email address is written here see he will be taking two lectures DFT for lecture number 32 and MC and MD molecular dynamics and molecular simulations will be taught in lecture number 33. So therefore, for this part of the lecture as I told you for these materials at different length scale there are five lectures. So I spent three lectures describing about different length scale issues phase diagram Calford as well as I talked about today phase field simulations by very brief way I understand phase field simulation itself will be a one course but I did not want to talk about it. Now if you want to go down from the microstructure to the atomic level or even the electrons then you need to talk about density functional theories molecular dynamics multicolor simulations that is what will be dealt by Professor Tarun Kumar Kundu he will be delivering these lectures for subsequent two lectures and I hope you will be getting same flavor or when better flavor than what I am providing in different lectures. So let me stop here for this lecture number 31 and I will come back for lecture number 34 where we will talk about the advanced deep learning techniques thank you.