[MUSIC] Dear students, in today's lecture, we are going to discuss the kinetics of recovery and recrystallization. So, the lecture today is just for part one. So in previous lectures, we have talked about the energy stored inside a plastically deformed sample. So essentially after plastic deformation, there will be the energy storation into your material, so the total energy of your system goes up. And your system is under a high energy state with a lot of crystalline defects and residual stresses. And at a relatively low temperature, the thermal energy from the environment are not sufficient for the kinetic process of recovery and recrystallization to take place. However if you raise up the temperature to sufficiently high levels, that external thermal energy will be sufficient for the system to relax and going through this recovery and recrystallization process. So the exact meaning of these two kinetic processes we'll explain shortly. So these figures show schematically what happened during recovery and recrystallization. So, suppose you have a heavily deformed microstructure with a lot of residual stresses and defects. And if you heat up this heavily deformed sample to an intermediate temperature which is sufficient for recovery to take place, you will have an annihilation of your point effects as well as a rearrangement of dislocations. However in the recovery regime, your thermal energy input from your external environment is not sufficient for any major microstructural change. And if you continue to heat up your material at higher temperatures, you will enter into a regime called recrystallization. Where just inside your heavily deformed microstructure, you will have the generation of new defect free, dislocation free, residual stress free grains. And then if you continue to heat up, these new grains will grow on the expense of the older grains and eventually you will generate a completely new microstructure with grains, with very low defect and very low residual stress. And if you continue to heat up, you enter into a process called grain growth, where your bigger grains will grow on the expense of smaller ones. And then eventually, you will have an increase in grain size and usually the grain size at the completion of this grain growth process, they will remain similar to each other. So this is the complete scenario of your recovery, recrystallization and grain growth process. So we will look at each of these steps one by one. The first one is the recovery process where you have an annealing or heating temperature usually smaller than one third of your melting temperature. So here I need to remind you that the important temperature here is actually not the absolute temperature but the so called normalize the temperature which is essentially the real temperature normalized by your melting temperature. Because that's a measure of how much external thermal energy input your system relative to the chemical bonding strengths of your material. So and this analysis applies to so called cold-worked metals and the term cold here doesn't mean absolutely cold, it means a low normalized temperature. So the cold-worked metal means the deformation or the work of your material happened below the recovery recrystallization temperatures. So first, we are going to see what happens in recovery process. So in this relatively low annealing temperature, you have a recovery and because you have an annihilation of point effects. So you know that there are equilibrium concentration of point defects. So if you enter into this recovery regime with relatively high temperature, there will be an annihilation of extra point defects generated during plastic deformation. And very limited microstructural change will happen in this recovery regime because you have relatively low temperature. And the only thing happening regarding microstructural change is the dislocation rearrangement. Okay, so it's something like this. So Initially in your heavily deformed sample, you have a kind of a random and complex dislocation arrangement. And then in the recovery regime, the energy is not enough to get rid of these dislocations but they are sufficient to rearrange these dislocation into some ordered situation where you have a relatively reduced stress and strain field. So that's what happened during recovery. And then we move on to talk about what happened in recrystallization which is much more important. So we are now talking about this regime. So smaller grains with very low defect density and with very low residual stress they generate inside this heavily deformed microstructure. And eventually they grow and consume the entire old structure. So this is called recrystallization. And this happens with a relatively high annealing temperature between one sort of melting temperature to about half the melting temperature. So the thermal energy input from your outside is sufficient enough for microstructural change. And two comments before we go into the details of recrystallization, because there is no crystal structure and chemical composition change before and after recrystallization, recrystallization is not a phase transformation. Second, most of the stored energy as result of plastic deformation will be released in the recrystallization process, so your total energy of your system is greatly reduced. So the recrystallization takes place by nucleation and growth process. Nucleation means the generation of small nuclei of new grains from your deformed old microstructure and the driving force for the nucleation is the elastic strain energy stored inside your system associated with the plastic deformation. And after you have nucleation, the nuclei will grow and this is your growth process. And usually the nucleation is more difficult than growth and is thus more critical than growth. And preferred nucleation size includes high energy locations such as high angle grain boundaries, phase boundaries, twin boundaries, all kinds of surfaces and interfaces. So for better understanding of the nucleation and growth kinetics of recrystallisation, I think we have to go through the basic theory or elements of nucleation. So the most easy simple example is the homogeneous nucleation of a pure solid in a pure liquid. But this kind of logic will apply for any nucleation and growth problems. For example, you can consider the generation of a new grain inside an old microstructure. So in the case of a solid nucleation in a pure liquid, if you consider the formation of this spherical solid nuclei inside this liquid upon solidification. So what is the energy associated with the process? First, you will have a driving force because you are cooling the sample below its melting temperature. So there will be a driving force, and suppose you have this nuclei with a sphere shape, then the driving force will be something like this, right? So the 4/3 pi r square is just the Wallum of this nuclei and delta Gv is the free energy reduction per unit Wallum of this liquid solid transformation. So essentially this is the reduction of free energy as a result of the solidification nucleation having this nuclei with a sphere shape, with a radius of r. However, there will be an energy cost as well because you are creating a solid liquid interface with an interfacial area of 4 pi r square and this is going to be multiplied by this interfacial energy gamma surface liquid. So you'll have a driving force term here and you have a energy cost term here. So you need to consider the combined behavior of the two energy contributions. So the interfacial energy has r square dependence. So it's something like this and the free energy reduction or the driving force term has r cube correlation so that you have this shape of free energy versus our curve. So essentially you have a bigger order of your radius here. So eventually if you combine these two together, you will end up with a combined curve of this. Okay, so what does it mean? It means you will have a local maximum. You will have a local maximum of your free energy and you will have a critical radius of your nuclei as well. And you will have, essentially this is your energy barrier. So if your nuclei is smaller than this, so if you're in this regime, your nuclei is unstable, right? Because increasing the nuclei size will end up with higher free energy. So the nuclei will shrink, okay, will shrink back to 0. However, if you have a nuclei equal to or larger than this critical radius, your nuclei will grow because it will directly reduce your total free energy of your system. So essentially from mathematics, you can easily derive the expressions for this critical size of your nuclei which can be calculated as the two times interfacial energy. So two times the energy cost Divided by your driving force, okay? And then your critical free energy or the free energy barrier here can be calculated by this equation. So essentially you really need a large enough sphere in order to have a successful nucleation, okay? So this is the so called homogeneous nucleation. So, we just talked about homogeneous nucleation. Now we are going to talk about hydrogeneous nucleation. So previously, I have said that for the nucleation of recrystallisation, the nucleation is easier when you have high energy surfaces or grain boundaries. So this is the case, the mathematical expression for that. So suppose you have some surface for the solid nuclei to stand on. And here is your solid nuclei with a radius of curvature r again, so with a radius of curvature r here. And so depending on the interfacial energies between your surface and your liquid, between your solid and liquid, between your solid and surface, you will have a particular geometry and shape of this solid nuclei. Okay, and then you can revise your expressions for the critical radius of the nuclei and the energy barrier for nucleation. And the result is that the mathematical expression for the critical radius of your nucleus stays the same. However, your energy barrier for nucleation is greatly reduced by this factor which is a function of this theta value. And this S function is a function of the relative magnitudes of your three interphasial energies. So essentially, you will have a much, much easier nucleation in your hydrogenous nucleation situation than your homogeneous nucleation situation because your S function is usually much smaller than one, okay. So eventually, you can consider your recrystallisation nucleation and growth problem similar to the solidification problem I have just explained. So because there are a lot of mathematical details, I will just show the results. So essentially, nucleation and growth are two correlated terms but they are different. So essentially, you will have a henuse typed nucleation rate dependence on temperature as well as a henuse typed dependence of growth rate as a function of temperature. And your nucleation and growth are having different activation energies or energy barriers. And as we have discussed, your nucleation energy barrier is usually much larger than your gross energy barrier because nucleation is more difficult, okay? So if you combine these two expressions, and if you refer to some basic theories on kinetics, you will eventually end up with some so called Johnson-Mehl-Avrami-typed recrystallization kinetics equations. And depending on the particular assumption for nucleation and growth for example, if you assume constant nucleation rate and gross rate, sphere shaped nuclei and your fraction of recrystallization is much smaller than 1. You will eventually end up with this equation where the fraction that you have, recrystallization is 1 minus this exponential term where your fraction of recrystallization is depending on the nucleation growth rate, recrystallization type, okay? And this is the isothermal recrystallization kinetics, where you have put your material under a constant high temperature and just hold there. So that's called isothermal recrystallization. And more generally, your Johnson-Mehl-Avrami equation will have this form depending on the details of your nucleation and growth kinetics. So usually you will have a 1-, again 1- term here and the exponential term of -k time to the order of n. So k is temperature dependent, and n is temperature independent. And again, depending on the details of nucleation and growth, you may have different numbers of this k and n. And you have to be reminded that this equation only applies to a relatively small recrystallization fractions, so only applies to here. Not too small and not too large, okay? So why is that? Because you need to have nucleation, right. So you need to have nucleation which means your items must vibrate and they are going to form a nucleus larger than or equal to the critical radius size, right? So this takes time and this time is called incubation time. Okay, and that's the reason why you will have a relatively low recrystallization rate at the beginning. And then once you have sufficient nuclei of recrystallization, your recrystallization rate goes up. And in this regime, the recrystallization rate can be described by the Johnson-Mehl-Avrami equations you just saw in the previous page. However, upon the completion of recrystallization, you are actually having a lot of recrystallized grains trying to touch each other. And you can easily imagine that the recrystallization rate will decrease again. So you'll have a decrease in the slope of this recrystallization fraction as a function of time. So you will have a low recrystallization rate at the beginning because of incubation of nuclei, and you will have a relatively large recrystallization rate described by your Johnson-Mehl-Avrami equations. And you'll have a relatively low recrystallisation rate again because of your impingement of your new recrystallized grains. So essentially this is what I have just a said. So essentially you will have an S shaped recrystallization fraction as a function of time. And a final comment here is that recrystallization is not a phase transformation, but is a nucleation and growth process. So in today's lecture, we have talked about the kinetics of recovery and recrystallization. Thank you very much. [MUSIC]