[MUSIC] Dear students in today's lecture. We will talk about the grain grows after recrystallization. So, previously we have ended other point where the completion of the recrystallization takes place, which means your old deform microstructure is completely consumed by your new grains with very low defect densities. So, what will happen after this completion of recrystallization, so if you continue to heat up your sample at elevated temperature afterwards. A phenomenon called green growth will take place because by intuition, you have a polycrystalline material with a lot of. Grain boundaries. And then you will have a total free energy of your system equals the free energy of the interior of the grains which is essentially the bulk free energy plus your free energy associated with your grain boundaries. So essentially the system wants to reduce the energy associated with the green boundaries and this is to reduce the total green boundary area. And this is to increase the grain size, so you have fewer green boundaries. So that's the logic why you have grain growth process. This is to reduce and minimize the total free energy of your system. And then by intuition you can come into directly into these two conclusions. The first one is the growth of larger grains and the shrinkage of smaller grains because you want to minimise the grain boundary area. So you want to maximise your grain size. And second is to reduce the curvature and green boundaries also minimize the green boundary area. This is because if you have so in this two dimensional case, if you have a curved green boundary relative to this flat green boundary, you can easily imagine that the flat one will have a lower green boundary area, smaller green boundary area and a lower green boundary energy. So, essentially, the thermodynamics and kinetics of your materials really really determines the the geometry sighs and the shape of your new grains after recrystallization in the grain growth process, and particularly, your larger grains will grow on the expense of these smaller grains, right? So the larger ones will eat up the smaller ones, and curved grain boundaries they will become flat ones. They are both to minimize the grain boundary area and to minimize the grain boundary energy. This phenomenal is called normal grain growth. And here I just talked about the size of the grams, of the grains growth. So to maximize the size of the grain to reduce grain boundary area. And in case of the shape of the grains, after the grain grows. We will start from this discussion of simple two dimensional cases. Suppose we have a triple junction where you have 1, 2, 3, 3 grains they meet each other. So this is a triple so called triple junction and this will correspond to your 1, 2, 3 different or identical green boundary energies one, two, three green boundary angles. So from simple geometry and force balance, you can come up with this equation. Where the interfacial or the green boundary energy divided by your sin theta 1 into one green boundary energy divided by sin theta 2 and gamma three divided by sine theta 3, this should be Identical. What does that mean? So, in the differ chapter, we talked about the grain boundary energy as a function of misorientation angle at a grain boundaries. So for high angle grain boundaries, you have a fairly constant grain boundary energy as a function of Green boundary misorientation angle so that you have gamma equals gamma 2 equals gamma 3. So if you go back to the previous equation, you can come to the conclusion that your three angles and the triple junction are all approximately 120 degrees. So this means. Your your stable green boundary shape in the two dimensional case will be this 120 degrees and your green boundary will be a flatline right? So you will have this case which guarantees you have this 120 degree green boundary angle and with a flat green boundary in the two dimensional case. So if you have more size, for example, you have eight sides or 10 sides to meet the requirement of this 120 degree requirement. You have to have a concave, so concave. Concave, a concave the grain boundary. So with concave the grain boundary you will immediately know that these bigger grains they will grow and similarly if you have fewer size that six to maintain this 120 degree green boundary angle, these green boundaries have to have a convex have to have a convex shape. So in that case you will immediately know that these green boundaries, they will grow inwards and these brains will shrink so essentially you will end up with so ideally you will end up with always six eyes green shape with 120 degree and with flat green boundaries. So that's for two-dimensional case. And the stable shape of your brains are six size with flat boundaries 120 degree green boundary angles, okay? In a more complex three dimensional case it can be verified in computer simulations that the stable, the stable shape of your grains is 14 sides, okay. That's much, much more complex. And next we are going to talk about the kinetics of grain growth. So in the atomic scale what happens during green grows is the atomic drum from one grain to another grain. Because it's a kinetic process, you can easily imagine that it follows the Boltzmann Maxwell statistics, and there should be a green boundary migration activation energy there and indeed. The so called brain boundary mobility is calculated by a constant term m0 times an exponential term. Where's the green boundary migration activation energy divided by kT. Okay? So that's a general form of green boundary mobility. So then, the velocity of the green boundaries can be calculated by the green boundary mobility times the driving force delta p and the driving force is simply this two times green boundary energy divided by the radius of curvature of the green boundary. And this is kind of intuitive as well, because the higher the higher the green boundary energy, the higher energy costs of having green boundaries and the higher the driving force to reduce the grain boundaries through green boundary migration, right? Hear the same if you have a strongly curved green boundary with a very small radius of curvature are and then you will be able to have a larger driving force and you will have a larger kinetics or the velocity of green boundary migration. Okay, and then you can do a, a simple mathematical trick, right? So you, you write the velocity of green boundary migration and this is just the average diameter or the average size of your grains divided by the tide right dt. Divided by dt, right? And consider equally as the grains, which means you have a relatively similar size in different directions of your grain. And then the two times your radius of curvature r, will be approximately the diameter of your grains. And then with mathematics and solving this equation you can eventually end up with this equation where the, final grain size, the square of the final grain size minus the square of the initial grain size will be proportional, to the green growth time, okay. And then by some simple mathematical tricks it can be simplified to this right. So the final final green size after green growth for for for a certain time t is proportional to the square root of the green gross time. Okay, so in most general cases, this equation can be written as this one, where your final grain size after grain growth is correlated with the grain growth time t with some exponential here. And this exponential or the exact value of this exponential really depends on the green growth mechanisms. Okay, so this is just a summary of what I have discussed. So for green grows larger greens will grow smaller ones will shrink in unto all the grains have similar sizes, and then the curved grain boundaries will become flat, and the grains with in the two dimensional case, the grains with fewer sizes than six will shrink, and the grains with more size than six, they will grow. And then we will talk about the factors that may affect grain growth. The primary one is again the temperature, right? The higher the temperature, the faster the kinetics, right? And you have seen this trick for many times in this course. So if you do a log log treatment on the two sides of your grain growth equations, You can then end up with a straight line. And with the grain size up, the grain grows at a different temperatures, you will be able to calculate the grain boundary migration activation energy. And then another factor that may affect grain growth kinetics is the impurity and precipitates. And this effect is often called solute drag, where the solute items are inclined to occupy occupy these these open spaces at a grain boundaries, right and these occupying salt. Solute items will suppress green. Boundary migration because they are there and the green boundary migration has to overcome these energy barriers. And that is the reason why the presence of impurities and solid items will usually restrict green growth or slow down the green growth and the exact magnitude of this solid drag effect will really depend on the the nature of the solids. So there are all kinds of complexities associated with this solid drag effects. And finally, we will talk a little bit about an abnormal grain growth. So, sometimes especially in thin film processing, you will have an unusually large grains at the surface. And the reason is that in addition to your. In the interior Green Energy free energy or G bulk plus G green boundary or green boundary free energy, you have an additional term of surface energy. [COUGH]. Okay, so essentially you have an additional contribution of surface energy to your total energy landscape. And for some reason, some orientation will have a really low surface surface energies. And that is the reason why sometimes, especially for thin film processing, you will observe unusually large grains on a surface. This is to minimize your surface free energy. So in those cases, you really need to consider the overall contributions from the interior of your brains, your grain boundary energies and your surface energies. And to wrap up this recovery and recrystallization part. So, we will see the property evolution as a function of annealing temperature. So essentially a lot of properties will change over annealing temperature during recovery and recrystallization and because recovery only in recovery there is only point defect annihilation and limited dislocation rearrangement. So in recovery, you will just have a moderate reduction in residual stress and a moderate reduction in your yield strings. However, you will have a significant reduction in electrical resistivity because your point defects are much fewer and you will also have a moderate increase in your grain size and your density of your material and your energy release is also a relatively low level. However significant change will take place during recrystallization where you will have a significant increase in grain size, right and the density of the sample will increase in significantly as well because of the annihilation of all kinds of crystalline defects. And energy release will be most significant during recrystallization, okay? And during grain growth, you mainly have this grain size increase. However, all the other property change becomes relatively moderate. So in today's lecture, we have talked about the grain growth kinetics. Thank you very much. Thanks [MUSIC]