 Next, we consider the defect which is the twin and the associated interface defect which is the twin boundary. Like before, we can now think of a twin either based on a geometrical entity which is the usual kind of a definition or we can think of a twin in terms of a physical property. There are examples on these slide which show you both these kind of twins. The one on the top is twinning with respect to the magnetization vectors and let me point out at this stage that these are schematics and in real systems the actual orientation of these spin vectors could be slightly different and that depends on the specific system under consideration or for instance even the property under consideration. The figure on the bottom shows the right bottom first shows twin as seen at an optical microscope. The sample here is actually annealed copper and in annealed copper you find these straight lines which can be seen here. For instance these lines which are marked in red here these two lines which can be thought which are actually the twin boundaries. So, let me extend this straight line to show you where the twin boundaries are and as you can see their signature is that they are extremely straight. There are other twin boundaries right here in the same picture this another twin boundary and this twin boundaries reside within a grain. Now, what happens at this twin boundary the other twin boundaries you can see that the atomic planes are reflected and since we are here talking about the reflection. Therefore, these twins are called the mirror twins as the slide title shows you these are the mirror twins and on one side of this mirror plane which I am marking with an arrow. You can see that atoms atomic planes go in this orientation and on the other side you can see that they are reflected by the twin boundary which acts like a mirror plane. This kind of a twin boundary of course is called a coherent twin boundary, but at this stage we will not go into the details of what is the difference between a coherent twin and an incoherent twin. But the important point to note here is that the twin region itself is a volume defect if you talk about this defect with respect to the remaining of the grain. So, this is my grain and in this grain the twin boundary introduces a misorientation of the plane. The second twin boundary restores the orientation of the plane and if I have had to look at my atomic planes and just drawing some schematics here they will be on the same orientation across the twin boundary. This implies that the twin interface which is otherwise called the twin boundary is one of the atomically sharp interfaces as compared to a grain boundary or other kind of interfaces typically which are somewhat diffuse. Now, a twin need not only be a mirror twin there are other kinds of twins which are possible and just to cite such an example there is one here which is called the rotation twin and in a rotation twin there are many variants and for instance this is one variant. There is a second variant right here this is the third variant and the fourth variant and the fifth variant put together you can see that on an average these are misoriented by an angle of about 72 degrees of course it is not accurately 72 degrees in real systems, but it is approximately 72 degrees and I can go from region 1 to region 2 by a rotation of 72 degrees and therefore this is called a rotation twin. In other words the mirror twin is associated with reflection symmetry the rotation twin with a rotation symmetry. The general principle in this area is that the mirror twin mirror boundary or the twin boundary or the symmetry operation I am considering cannot be the inherent symmetry operation of the crystal if it were so then it will be a single crystal and they will not be two variants. Of course when I am talking about variant 1 and 2 I can also think of this boundary between the two which I am marking with an arrow as sort of a mirror twin as well. So, you can see that this can also be thought of as a reflection of planes, but on the whole this is a rotation twin and the importance of this rotation twin in this context is that suppose one where to perform a diffraction experiment say in a transmission electron microscope by putting a selected area diffraction of purchase in a region encompassing these 5 twins then the person would see a symmetry higher than that available for a single variant. In other words this twin configuration mimics a symmetry which is higher than that available for the crystal. So, in this case you will observe a 5 volt symmetry which as you know is disallowed in crystals. So, let us have a look of a totally different kind of a schematic or a or let us put it this way a different kind of understanding of a real crystal and this actually happens to be the from the nano world. This is an example of a porous alumina which can be thought of as a two dimensional crystal and this two dimensional crystal has crystalline regions and this is not a single crystal. So, there are crystalline regions, but you can see that a region like this is nicely single crystalline. So, this is some sort of a polycrystalline alumina and, but what is the motif here? The motifs are these pores along with the region encompassing this. Therefore, I can think of a motif which is a pore plus some material which is actually decorating the two dimensional lattice. And if I try to identify defects in such a material then I can think of a region like this as a vacancy as marked here. So, this is not a vacancy where in atom has gone missing. This is not a vacancy where in a set of ions have gone missing, but this is a vacancy in a two dimensional crystal where in the motif is actually some pore along with some alumina. So, this is a totally different kind of an example and if you look at the dimensions of these pores they are in the scale of nanometer as the scale bar on the left bottom of the micro graph shows. So, this is a nice example of a nanomaterial and we will come back to it when we start to classify nanomaterials and try to understand them better, but this happens to be a nice example. And you can see that even in this nanomaterial we can think of a region as if it were a low angle grain boundary. And this low angle grain boundary as shown here has an extra half plate in other word it is a dislocation. In a certain different region you can also visualize certain kind of a defect where in instead of having this region having can be also thought of as a disclimation. Therefore, we can visualize many defects even in crystals which are not the conventional crystals or there are crystals in a different length scale all together. So, let us briefly revise the various defects a collage of the way which we have considered before. So, we had said that there can be 0 dimensional defects like the interstitial the substitutional atoms the vacancy or the short key defect. We can have one dimensional defects like the dislocation and we consider two kinds of dislocation one which is localized to a interface as shown in the example of the G S i 1 S i. So, this interfacial dislocation is called the interfacial misfit dislocation and is actually playing a structural role. And therefore, can be classified as a structural dislocation. The other inevitable defect we said was a surface in a crystal and we considered one important kind of a surface which is polar in nature which is which I mean circling here. We also briefly talked about this, but we will consider little more detail that the surface may have further defects which are kings and ledges. And therefore, the surface itself may have a complicated structure unlike a very simple structure which normally we consider. The grain boundary is another important structural unit and it is the boundary which is between two grains in a poly crystalline material. And we even noted that as sometimes the grain boundary can be pretty straight. We also took a very specific example of another kind of a structural dislocation which accommodates the misfit the tilt misfit between two grains which have a small tilt misfit. And this is called the low angle grain boundary which can be thought of as an array of dislocations according to a model which is called the Reed-Shortley model of a low angle grain boundary. We also talked about mirror twins and we noticed that the twin boundary is a two dimensional defect while the twin region itself can be thought of as a three dimensional defect residing typically within a grain. As in the example of a copper which we saw right here. We can also think of rotation twins and we noted the important point that the rotation twin can actually mimic certain symmetry which is disallowed normally within the single crystal which we normally consider. And we also noted that when a defect becomes part of the crystal structure itself like for instance a vacancy getting ordered or for instance a stacking fault getting ordered then the symmetry not only that it can be thought of as a defect in the original parent structure but it becomes part of the structure itself. And therefore, we get a new crystal structure altogether which may have a different unit cell and also a different kind of a symmetry translational and other kind of symmetry is possible. Therefore, defects can not only dictate the properties but they can at a fundamental level even alter the crystal structure and that is why defects are very very important. Next let us take up an example wherein we are talking about interaction of two kinds of defects and here this interaction of two kinds of defects serves as an engineering application to in fact strengthen the material. I take an aluminum crystal an aluminum as an FCC lattice where in aluminum I on sits and it this kind of a crystal typically is very ductile but on the other hand is very weak and therefore, it is not suitable directly as a structural application. Such a crystal can actually be strengthened by adding an alloying element like in the case here you can make an aluminum copper solid solution. And this solid solution of copper and aluminum actually serves as to strengthen the parent crystal but further to that we can have further strengthening by actually precipitating certain precipitates in the aluminum copper system. And we had briefly considered how we do what is called a solutionizing treatment followed by an aging treatment wherein we actually precipitate certain fine distribution of precipitates which gives us an enhanced strength. What is the origin of this strength? It is we had noted that what is weakening the crystal is actually the presence of dislocation. Suppose I had a single crystal without any dislocations and the strength of the crystal would be all the order of giga Pascal the shear strength but in the presence of dislocations the crystal is severely weakened by a couple of orders few orders of magnitude. And now we have a much weaker crystal in the presence of a dislocation or a large number of dislocation as the usually the case is. Now if I want to strengthen the crystal back again I have two strategies at my disposal and let me briefly write down those two strategies. So, there are two strategies available for me to strengthen the crystal which has weakened by the presence of dislocation either I can make it totally dislocation free to obtain something known as a whisker. These have been produced with a small thin cylindrical specimens which are totally dislocation free and such as material will actually have an enhanced strength. But the problem with this strategy is that under actual service such a whisker may actually nucleate a dislocation first of all it is very difficult to produce these whiskers which are totally dislocation free. But in service again dislocation may nucleate thus weakening the crystal therefore, this is in most cases not a viable option to actually strengthen a crystal. So, what is the better option or the usual engineering option is to make it difficult for dislocations to move. And in this respect we have many strategies how do we make it we can do something known as a solid solution strengthening we can do precipitation hardening. So, some of the strategies we typically used for increasing the strength of a weekend material is to use solid solution hardening which in some sense actually increases in your inherent lattice friction for the dislocation to move. You can use precipitation hardening in the words you put a fine distribution of precipitates which can impede the motion of the dislocations. You can actually do what is called a half pitch hardening in which case you decrease the green size and the green boundaries act as impediments to the motion of dislocations which is thus increasing your strength. We can do cold working and this cold working essentially increases your dislocation density and thus you have an increase in the strength of the material. Of course, it is a curious question that actually dislocations are weakening then how an increase in dislocation density can lead to a strengthening effect. This is because when you think of a single dislocation moving in a forest of other dislocations there are interactions between these dislocations and these interactions lead to need to lead to increase in dislocation lens in other words increase in dislocation density and therefore, you have an increase strength. We are going to solve this example for the case wherein we are precipitating certain second phase into the matrix which is going to give a strengthening. And the case we are specifically talking about here is the aluminum copper system in which a second phase precipitate has been put and we will make certain simplifying assumptions to understand how this distribution of precipitates actually gives rise to hardening. We will also dwell into little detail of regarding the mechanism which is actually giving rise to this hardening. And when we go to nano materials we will see that some of these mechanisms can be effectively used and some of them will actually cease to exist when we go to nano materials. And that is why it is important to consider what is one example in detail with regard to the hardening of a material which is one of the important mechanical properties. Now, let us consider a fine distribution of precipitates for instance a 0.5 micron size and for now we will consider what are called incoherent precipitates that means lattice planes are not continuous across from the precipitate to the matrix. And we will assume an average inter particle spacing of about 5 microns which is shown in the figure below. So, for now we will assume some spherical precipitates and we should note that in actually in the aluminum copper system the precipitates may not be spherical or in fact are not spherical. But, we just for understanding of the concept we will consider that and we will assume that the diameter of these particles about 0.5 microns and the inter particle spacing is about 5 microns. The distance from surface to surface thus turns out to be about 4.5 microns or 4.5 micrometers. The lattice parameter of aluminum is about 4.05 angstroms and in the presence of a solid solution this will be altered. But, we will for now assume that this is the lattice parameter we need to consider. The burgers vector in the FCC lattice is root 2 a by 2 in other words it is a by root 2. And for this case it will turn out to be about 2.86 angstroms. The shear modulus of aluminum is 25 megapascals and we will now make a calculation as to how what is the increase in strength up and above the normal yield strength or the normal shear strength of the material. Now, what is going to give rise to the hardening in this case? The hardening is going to come from a fact that when the dislocation moves there are precipitates and these precipitates being incoherent. Therefore, the dislocation cannot actually glide through the precipitate. In fact it will bow around the precipitate by a mechanism known as the Frank Reed mechanism. And for now we will assume that the entire plastic deformation is being dictated by this process. In other words we will not consider other kind of competing processes in this system. And we will try to find out what is the additional strengthening effect up and above that due to the solid solution strengthening. And for that we need to know what is the stress required to operate a Frank Reed source? The stress required to operate a Frank Reed source is given by this formula which is given by tau Frank Reed which is the maximum of the stress which we require to actually operate a Frank Reed mechanism. And in the coming slides I will show you what is this Frank Reed mechanism and how it actually works. But the tau max is given for now by the formula G B by L where G is the shear modulus of the material B is the burgers vector and L is the distance between the particles which happens to be 4.5 microns. So, I have all my parameters in place if you plug it into this formula and calculate my tau max it turns out to be above 1.59 megapascals. Now I need to compare this number with the critical result shear stress or what is known as the tau CRSS. Tau CRSS is very similar to the pearl stress which we had talked about before. The only difference being tau CRSS is usually experimentally determined while pearl stress is theoretically determined. And the second thing is that the tau CRSS or the critical resolved shear stress is usually at some finite temperature like it could be the room temperature as the example considered here. So, we are talking about the resolved shear stress which needs to act on the slip plane to activate slip and that is called the critical resolved shear stress. And critical resolved shear stress like as I told you the pearl stress is an inherent material property at the atomic level or the slip plane level. Unlike a gross macroscopic property like the yield stress which is an average of this over an entire system or entire material. Now if you take the critical resolved shear stress value for aluminum it turns out to be having a value of 0.75 megapascal and I can clearly see that by adding these incoherent precipitates I obtain a strengthening effect above this 0.75 to value of 1.59 megapascals. Now if I go to look at this formula it tells me that the L is sitting in the denominator. So, how do I increase my strength either I have a material with a larger value of b a larger value of g which are not under our control because we have considered a certain specific system. The second possibility is to actually have a smaller value of l because as I told out at the slip plane level if I am able to impede the motion of dislocations. Then automatically the strength of the material is going to increase because this is what is actually weakening the crystal the motion of dislocations leaving the crystal is what is weakening the crystal. Now that means if I am able to distribute the precipitates in a very fine manner by actually decreasing l which is sitting in the denominator of this formula then I am going to have an increased strength. In fact this is the whole strategy behind what is called precipitation hardening in an precipitation hardening system like aluminum copper and though I have mentioned this before may be briefly I will talk about the steps involved in this mechanism. The step one is to solutionize the step two is to quench and the quench substance produces and super saturated solid solution after quenching I age it to produce fine precipitates. Of course, one may ask the system why do I have to adopt such a complicated procedure of three steps of solutionizing quenching aging to produce instead of actually slowly cooling the system because anyhow at room temperature the theta phase is made as a the amount of solubility is very less for a copper and aluminum. Therefore, automatically the theta phase would precipitate out the answer lies in the fact that if I do a slow cooling then actually the system would be growth dominant and nucleation restricted. In other words when I slowly cool it you will have a few nuclei which will grow large and therefore, I will have a very coarse precipitate that means the theta phases which will be produced would have a microstructure something like this. So, this is a slow cooled system and we are already considered a microstructure like this before this will be slow cooled vis a vis suppose I consider a system which has gone through this kind of a treatment which will have a fine distribution of precipitates. Of course, the morphology of the precipitate will not be what I am showing here this is just a crude schematic. In fact, if you look at a transition electron micrographs the GP zones have precipitated out they will look something like this they will have a morphology like this. Now, that is the detail the important thing is that there will be a fine distribution of precipitates and the reason is amply clear now as to what this fine distribution of precipitates can do in terms of the strength additional strength which I obtain. Now, we have talked about this Frank Reed source let us understand what is the implication of having a source like a Frank Reed source and what are the other competing mechanisms which are possible because at the heart of this strengthening is this double ended Frank Reed source which has been shown here on the right hand side. Now, in the double ended Frank Reed source the starting point is that we are having a dislocation which is moving on a slip plane on the slip plane there are these incoherent precipitate particles which hinder the motion of the dislocation. So, let us consider a configuration like this and therefore, these are my precipitates and this is my dislocation line segment and for now we will understand that this dislocation line segment has been pinned at these two points in the words wherever the precipitate is it cannot move. There are other sources of pinning possible like for instance at points A and B the actually the dislocation may leave the slip plane or they could be other kind of impediments to the motion of the dislocation. But, for now we are considering precipitates which are sitting there and for now they are incoherent precipitates and in a aging schematic like this when you have a solutionizing a quenching and an aging this kind of incoherent precipitates are produced either by aging at higher temperature or aging for a long time at a lower temperature. Now, what happens what is the sequence of events in increasing stress when you actually try to press this dislocation against this two pinned points. So, let us think about it the system. So, you have two variables in the system one is an external applied shear stress which translates your externally applied shear force which translates into a shear stress at the level of the slip plane. On the other hand if the dislocation line is not straight then you have a line tension which is trying to make the dislocation straight. So, therefore, there are two competing forces and if these forces at the level of the dislocation are equal then the dislocation will be in a stable configuration. Now, this is my initial configuration wherein the dislocation has been pinned at points A and B please do not look at this region carefully because this is just a schematic there will be more it is not actually on the precipitate, but it will be stuck around the precipitate or slightly away from it. Now, on the application of the shear stress at the dislocation line level there will be a force which is tau B which is acting on this this is coming from the externally applied shear force on the material. And as you increases externally applied force the dislocation would tend to get into a curved geometry. The force on the dislocation line can be written as tau B d s d s being the unit length along the dislocation line. The dislocation energy per unit length is proportional to G B square and approximately it is half G B square wherein we have actually ignored something known as a core energy of the dislocation. The line tension force which is actually trying to contract the dislocation line can be written as 2 gamma sin B theta by 2. And if you look at the next picture so, D theta being along the radius of the what you might call the dislocation line the angle it subtends. And therefore, this is approximately equal to gamma D theta for equilibrium in a curved configuration. We need to have gamma D theta is equal to tau B d s which gives us by putting all these equations together we get the formula which we had used in the case of finding these strengthening effect of the precipitates which is tau which is given by 2 G B by 2 R. Obviously, when R is the maximum you get the R is the minimum you get the maximum and tau max. And let us see when does actually you have the maxima in the shear stress which needs to be shear force which needs to be applied. So, in this figure you can see that you have configuration 1 which is a starting configuration the configuration 2 3 4 and 5. So, we start with an pinning points at a distance of length L which is shown here and we increase the external shear stress. And therefore, in under the presence of the shear stress the dislocation length increases and also it gets into a curved configuration. On further increasing of shear stress we see that it can actually get into a semi circular configuration. This semi circular configuration is important because still we obtain a semi circular configuration we the stress has to be increased. And beyond that point actually the system goes down hill in stress as from configuration 3 to configuration 4 where in actually the system you do not have to apply extra shear stress. But the system itself goes down hill in stress and it up takes the dislocation takes a very convoluted appearance as you can see here in the form of a certain kind of a curve. Now, when I look at a curve like this and initially suppose I was talking about a pure screw dislocation suppose this segment was a pure screw then these segments would constitute positive and negative edge segments. And as we know that positive and negative edge segments will actually attract themselves and therefore, they will tend to come together as this loop expands further. And in the end you would notice that so this segment will tend to come towards this segment in an expanded configuration. And in the end you will see that the positive and negative segments under each other. And finally, we are left with one circular loop and the original segment which we started off with. So, what is the implication of such a mechanism the implications are twofold number one that actually the dislocation density has increased. Because originally we had a dislocation length which is between points a and b suppose I were to label these points a and b. But now we have an additional dislocation loop which is sitting around these two precipitates. The second implication is that suppose I were to drive want to operate the same source again in other words suppose I start with a new dislocation line which is approaching my loop. Then to operate the same source I need because now this dislocation loop is actually going to repel my oncoming loop or oncoming dislocation line. And therefore, I need to apply additional stresses to operate the same or actually to try to move the dislocation which is coming in. Therefore, this will automatically lead to a strengthening effect. And since I have done work on the system to actually produces extra dislocation length. This means that the system is actually going to harden as I am going to do the deformation. So, let us summarize the Frank Reed mechanism. The Frank Reed mechanism can operate a single resource without depletion can keep on operating to produce more loops each time. The loop produces a slip of 1 b on the slip plane each time it operates. The maximum stress corresponds to the semicircle configuration. And at that configuration the stress required is tau max which is given by G B by L its order of G B by L. If I have various spinning segments in the material like for instance a dislocation various dislocations may be pinned at various precipitates which are at not at a constant distance L, but different lengths. Then the ones with the largest distance will operate first and the ones with which are separated by the smallest distance will actually operate later. And since with time I am going to exhaust all my longer segments and I will only left with shorter segments which I need to operate. This will tell me that there is work hardening or in other words I need to apply higher and higher stress to actually deform the material plastically. Now, when the back stress is greater than tau max the applied or the applied force which translates into shear stress at the slip plane level the source will cease to operate. One of course, beside point which we need to note here is that typically in normal experiments they have found that double ended fracture resources are not that common, but it is one of the possible mechanisms by which actually you can have 2 fold things. One is increase in dislocation density and the second being an increased stress required for further deformation which is otherwise called work hardening. In this whole analysis we are assuming that a single mechanism which is the double ended franque rate source mechanism is operating, but we could have other competing mechanisms like the oro on bowing mechanism or certain other mechanism of plastic deformation which may be competing with the mechanism I am considering like for instance in the figure below the mechanism shown is the oro on bowing mechanism. And for now I have ignored these other mechanism which are competing with the franque rate mechanism or the franque rate source mechanism for plastic deformation. Therefore, I made many of the simplifying assumptions, but the heart of this whole analysis lies the fact that you have defects these defects may interact with each other these defects may grow in density. That means as we saw that the dislocation density increasing I need to know a lot of details about these defects like for instance is the boundary of this prostate with the matrix coherent is it semi coherent is it incoherent. Because if the prostate happen to be coherent and the dislocation actually glide through the prostate and you will not be operating a franque rate mechanism. Therefore, all these details I need to know about the defect structure the material. And it is this evolving scenario of defect interactions which is actually giving rise to many other properties like in the case we considered the yield strength of the material. Before we leave the concept of defects and the details regarding a defect structure let us explore couple of interesting view points which are which is the concept of a defect in a defect the concept of a defect association and what you might call hierarchy of defects. Now, often these are not found in elementary textbooks, but these are very important view points when we want to understand the properties of a material. Now, to explore this concept let us consider this defect in a defect concept in 0 dimension in 1 dimension and in 2 dimensions. Now, these are just examples we are considering here, but on the in the detail these are subjects of study in themselves. Now, we had noted that vacancy is a point defect in a material in a crystal vacancies can get ordered to form vacancy ordered phases. Now, once you have a sub lattice which is being occupied only by vacancies. Therefore, our vacancy ordered system in which one of the sub lattices is actually occupied purely by vacancies then there could be a scenario in which the vacancy the awakened sub lattice side could actually an atom could be present. Now, since this is a vacancy sub lattice the presence of an atom in this sub lattice is actually a defect what is the defect here? It is an atom. So, in the vacancy sub lattice an atom is a defect therefore, this is a defect in a defect concept because vacancy is a defect in a material the atom in the vacancy sub lattice is a defect in the vacancy ordered sub lattice. Therefore, this is a concept which is a defect of the concept of a defect in a defect and this is of course, in 0 dimensions. Of course, we will come back to this point later where when we talk about the hierarchy of defects, but for now we will take it that when somebody says there is a vacancy we should not always think of it as a missing atom as the interesting example points out in the vacancy sub lattice the presence of an atom is actually a vacancy in some sense of understanding of the word. Let us now consider a dislocation. Now, a dislocation is a defect in a perfect crystal. Now, we can have a kink or a jog in a dislocation line which is a defect in the dislocation line a kink or a jog is not a defect directly or easier way to understand it it is not by thinking of it as a defect in the crystal, but thinking of it as a defect in the dislocation line. Therefore, a kink is a defect in the dislocation line a dislocation line is a defect in the crystal. Therefore, a kink or a jog is a defect in the defect. Now, what is a kink or a jog let me see if I got a figure here for instance that this shows you that. Now, therefore, this is now a jog and this jog takes the dislocation from one slip plane for instance the one which is above here to another slip plane and in the process serves as a defect in a defect. The kink which does not take the dislocation line out of the slip plane. In fact, a single slip plane can be thought of as you can see here as a bend in the dislocation line segment it is a bend in the dislocation line and therefore, this kink is a defect in the straight dislocation which is what we want to consider here suppose I am looking at this segment it is a straight dislocation line the straight dislocation line and a kink without taking the dislocation outside the slip plane actually produces an additional length segment. In other words it also cause energy to the crystal to put kinks into the material and actually kinks and jogs can be produced by intersection of dislocations and this itself can serve as a mechanism by which the material hardens. Therefore, I can think of a kink and a jog as a defect in a dislocation line which is a defect in the crystal. In 2D surfaces I can think of a surface as a defect in the perfect crystal in in fact it is a termination to a perfect crystal. In other words I take a single crystal I make a or a infinite crystal and make a cut then I produce 2 surfaces and on the surface of the bonds are broken and in fact the surface may undergo other kind of transformations like relaxation and reconstruction on and it will not merely be the termination of the bulk that may not be the scenario. But here we are talking about how we can think of defects in the surface steps on a surface is a defect in the surface and steps in these steps can be thought of as a defect in those steps. Let me now show you a figure which can actually exemplify the point. So, I have a surface which as I pointed out is a cut in an infinite crystal and I also pointed out that this need not be have a structure which is just the termination of the bulk. In other words it may undergo relaxation or reconstruction and typically though we talk about a 2 dimensional surface to be a 2 dimensional defect. This is as I pointed out when I talked about defects that this is in some sense a geometrical viewing of the structure. In reality the surface can be thought of as a few atomic layer starting from the true geometrical surface. In other words the extent of the surface where atomic bonding has been disturbed the bulk lattice parameter has been changed would actually be a few atomic diameters from the true geometrical surface. Now, in this surface I could have these steps which are called ledges and of course, this ledges and terraces the terraces have been label blue and the ledges have been a colored orange and this ledges and terraces can be thought of as a defect or in the surface. Further we can have a break up of these ledges into these kinks. So, you can see that if I am climbing a staircase then I can think of myself going from the step 1 to step 2 to step 3 via these ledges. Now, within a single ledge I can think of going from this 1 to 2 to 3 this is the orange to the green to the orange to the green via series of steps which are kinks in the ledge. Therefore, I have now a defect in a defect a kink is a defect in the ledge the ledge is a defect in the surface the surface is a defect in the crystal. So, this is an alternate view point which makes it very interesting to study some of these defects and. So, let me briefly summarize the concept of a defect in a defect I may have a defect belonging to 1 dimensions 0 dimension or 2 dimension. In the case of the 0 dimensional defect we considered the vacancy ordered phase and for instance in this figure I see a vacancy sub lattice which is formed by these crosses. So, I have this vacancy sub lattice and now in the vacancy sub lattice I may have a missing vacancy like in this case here where a cross should have appeared actually there is an atom sitting there and therefore, this is a defect in the vacancy sub lattice and vacancy itself is a defect in the perfect crystal or perfect crystalline order. A word of caution here we had already pointed out this aspect that actually in a vacancy ordered phase the vacancy is sometimes not or more precisely not thought of as a defect, but belonging to the part of the crystal structure itself. Because if I talk about the original unit cell which is a smaller unit cell for the vacancy ordered phase you actually have a larger unit cell which goes on to give you the entire vacancy ordered phase crystal structure. But, nevertheless this is an interesting view point which you can keep in mind same case we had talked about jogs and kings in a dislocation line and also ledges kings ledges and kings and ledges on a perfect surface which itself is a defect in the perfect crystal. Next interesting view point which we considering is a concept of hierarchy of defects. Now, again we will use the concept of a vacancy ordered phase to understand what is meant by hierarchy of defects though in the practical sense some of these may be actually rare that you may actually have some of these hierarchy of defects. And from a point of view of nano materials this is going to become important because in a totally different context we will actually consider what is called hierarchical structures. Now, where is this hierarchy of defects coming from we already seen that a vacancy can get ordered in a vacancy ordered phase. Now, we had seen that a presence of an atom is actually a defect in the vacancy ordered sub lattice. Now, these atoms in the vacancy ordered sub lattice may themselves get ordered. And therefore, you can have think of an atom sub lattice in the vacancy sub lattice which is present actually in the crystal. Therefore, now we can think of two levels of sub lattices one sub lattice where in the vacancies are ordered and sub lattice within the vacancy ordered sub lattice where in atoms are ordered. Therefore, this is in some sense a hierarchy of defects. So, we have two orders of hierarchy of defects and this is pointed out though may be intellectual exercise, but it is an interesting exercise to understand what you might call the defects not only are present as defects they can get ordered, but they can even have a hierarchy amongst themselves. Let us take up the next topic which is exceptionally or extremely important is the concept of association of defects. We already considered one example of clear cut example of interaction of defects and that interaction of defects was the interaction of a dislocation line with a precipitate in the crystal. Now, this in this example we are talking about association of defects. Defects can exist in isolation or they can be associated with each other. And as a to reiterate the important point such a defect association can actually drastically alter the properties of a material. Now, the important thing is that the defect of one dimensionality for instance a zero dimensional defect like an interstitial atom can associate with a defect of a different dimensionality like for instance a one dimensional defect like a dislocation. Of course, there could be association with of defects of the same dimensionality for instance a vacancy could also get associated with another vacancy. Now, what is the reason that such an association should take place? The reason is that the enthalpy of the system may reduce when such association takes place. In other words the enthalpy delta H change in enthalpy is negative. In other words this change in enthalpy is helping the process. Now, we are considering those kind of associations only for now wherein the enthalpy happens to be negative that is we are assuming that the association actually does occur. And therefore, this is helping our process to take place. Now, what will happen if such an association occurs? There will be a negative delta S. In other words the entropy change will try to oppose our process. Now, I am assuming at the starting itself that the certain association for instance association of an interstitial atom with a dislocation is taking place. So, what is the criteria I would use to actually see the feasibility of the association? I had actually use delta G as the my criteria at constant temperature and pressure. And I am assuming that the delta G is negative. What I am trying to do now is further analyze this delta G in terms of a delta H and a delta S. And I am seeing actually that the delta H in this association is trying to help the process to take place while the delta S is negative and actually tries to oppose the process of occurring. So, let us see why this happens in a little more detail. Now, why should delta H be negative? We have seen that whenever there is a defect I can think of it being associated with a certain kind of defectiveness. Of course, I am using defectiveness in a rather what you might call a casual sense. But, we all have an understanding of what is meant by defectiveness. It is the amount of strain for instance it costs in the lattice, amount of free volume that is introduced by the presence of a defect, amount of excess charge locally at least which is present in the defect etcetera etcetera or in the disruption to the perfect crystalline order which the defect introduces. Now, when two defects come together then it could so happen that this defectiveness is actually reduced. So, let me take up a schematic to show how this can happen. For instance I have a single vacancy in a crystal and this is responsible for of course, this I am showing here without the relaxation of atoms which would take place when the vacancies present we crudely talk about four bonds being broken. Now, if there were two such vacancies located say at point A and point B for instance this could be point A which I have here and a point B somewhere else then I would have eight broken bonds. So, that means I am going to cost the crystal lot of energy. Now, suppose I talk about the formation of a die vacancy which means two vacancies located each two other next to each other then I would notice that this is going to cost my only one, two, three and three on the other side six broken bonds. Therefore, clearly in many of these associations of defects there is an enthalpy benefit. Therefore, enthalpy is going to help me in the association of the defects. Now, why should the entropy of the system come down when I have a defect association? This is because in the absence of the defect association let me again go back to the example of this two vacancies here. This vacancy A could exist in one of all the lattice crystal lattice positions and vacancy B again can be present in any one of the lattice positions except those adjacent to the place where A is located and therefore, I have lot of configurations at my disposal two independent possible configurations in the entire mole of substance or larger. On the hand when I have a die vacancy this pair has to configure together. This is posing a severe constraint on the number of available configurations and we had already noted that S is equal to k ln omega and since the number of available configurations is reduced a lot this implies that my entropy benefit which is associated this configuration entropy is reduced. Therefore, my delta S is going to oppose this process of defect association. Therefore, now given the fact I am already assuming that the defect association is thermodynamically feasible it is in some sense can be thought of that these are the two factors which go into it, but overall the delta G for the system is negative. Now, we had talked about the presence earlier we talked about the presence of thermodynamically stable vacancies. Now, in the in the when we introduce the concept of die vacancy the situation is little more complicated which will come to in a moment. Now, how do these association of defect take place the association of defects can take place through the long range stress fields associated with these and if kinetics permits then one or both of these defects may move and get associated. Now, that means that if these defects originally existed in isolated form and it is we are interacting actually through a long range stress field like in the example of one dislocation and other dislocation then they actually attract each other and if the kinetics permits then they would come to close to each other and may get associated with one other which is what we saw happened in the case of polygonization or the formation of a low angle grain boundary. We had originally considered. So, let me redraw that schematic for you. So, originally we had considered that in a bent crystal there are these statistically stored dislocations then these dislocations may move in the presence of thermal activation to form a low angle grain boundary which is nothing but an array of dislocations this is my low angle grain boundary and therefore the crystal 1 and crystal 2 are misoriented. So, this is some sort of a sub grain boundary. Now, this is in some sense an extreme form of defect association where in all the dislocations have been arranged one below the other and they are associated one other to form a low angle grain boundary. But, for such a kind of a motion to occur you there needs to be some kind of an activation which could be a thermal activation as in the case of the example of formation of the low angle grain boundary. And of course, I pointed out that the long range stress fields help in such a process to take place. Now, let us take these association of defect one by one first we talk about the zero dimensional one then we talk about a zero dimensional one dimensional association and also a zero dimensional two dimensional association. And finally of course, we will just revisit the example which we just now talked about which is a one dimensional one dimensional deceptive association and we will consider some more examples also. Now, so we had already seen that there is a specific reason why a die vacancy should form that is why and we had rationalize the formation of a die vacancy in terms of the enthalpy benefit which finally translates into a reduced energy in terms of the number of bonds broken. We are also said that this actually now puts a constraint on the number of configurations available for the system and therefore, it is going to decrease my configurational entropy which is going to oppose my formation of die vacancies. Therefore, if I have any particular temperature in mind for instance I could be talking about 50 degree Celsius or 200 degree Celsius I know that there is going to be a certain thermodynamic number of stable vacancies. Now, this number of vacancies not all of them would be completely associated not all of them would be completely dissociated, but the point there will be an equilibrium number of mono vacancies. There will be an certain number of die vacancies which are also present and finally, there could also there is a possibility of higher vacancies also being present. In other words though my formation of die vacancy is going to help me in terms of the enthalpy benefit this is not the final story and any given temperature I may have a certain equilibrium number of mono vacancies die vacancies and maybe even higher cluster of vacancies which can form. And therefore, if I increase the temperature further there will be more preference for dissociation of these associated vacancies and the reason is very obvious because the function which is going to dictate my stability. So, the term which is going to dominate high temperature is the entropy term because of the weighing factor of temperature and therefore, at higher temperature you will have a tendency for these die vacancies to more and more dissociate and at lower temperature you may have more and more association of these die vacancies. As die vacancies are costly in terms of the entropy. Now, let us consider the next example which is the association of a 0 dimensional defect like an interstitial solute atom with a 1 dimensional defect which is the dislocation. And the classic nice example of this is the presence of carbon solute in b c c iron which is otherwise called steel. And let us see what is the effect of this carbon solid solution and dislocations a dislocation as we had noted is associated with a tensile field and a compressive field. Now, I am talking about an edge dislocation and an edge dislocation is associated with a compressive field wherever the extra half plane is that I am there. So, this is my extra half plane and this is where the compressive regions are and the tensile regions and I am plotting sigma x x for now and these are iso stress contours. So, as you go away and away then you will notice that the stress contours decrease in magnitude. So, this is higher stress contour this is lower and lower as you go away from the core of the dislocation. Now, a vacancy is also associated locally with a stress field and it is associated with the tensile stress field. And therefore, if a vacancy is present in the compressive region it will actually be attracted to this region. On the other hand interstitial atom is associated with the compressive stress field and therefore, if I have an interstitial atom it will be attracted to the tensile stress field of the dislocation. So, this is my extra half plane. The carbon atom is attracted towards the tensile stress field of the edge dislocation and can lead to a segregation of carbon at the core region of the edge dislocation. This selective segregation of the core of an edge dislocation is called deformation of the cortral atmosphere and this is important consequences as shown in the figure below. The energy associated with the defect the dislocation interstitial is lower than the independent defect like only a dislocation or only an isolated or far away interstitial. Now, because of this as I am saying that because of this defect association in the case the interstitial being associated with the core of the edge dislocation the energy of the system decreases as this interstitial would partially anal the stress field of the dislocation or give some relief to it. Now, therefore, when I apply external shear stress it has to pull the dislocation and that means I am applying external shear force to cause plasticity which would imply in terms of the slip plane to move this dislocation on the slip plane. I have to do additional work to move this dislocation away from this cortral atmosphere and the configuration of the dislocation with the interstitial carbon is a low energy configuration. Therefore, I need to apply additional force additional shear force to pull the dislocation out of this atmosphere of carbon and this is reflected in terms of the increased yield stress which you see in the stress strain diagram. Of course, this is a nice schematic in which you have a stress strain diagram in which you have the elastic region and you notice that you need to apply an additional stress till the dislocations can break free of the cortral atmosphere. And once it does so then the yield stress drops and then of course, it oscillates till some more many of the other dislocations would do the same thing. This implies this whole yield point phenomena is related to the locking of the dislocation and locking I am using in a in the sense I told you of decreased energy state which is resulting from the association is resulting from the locking of the dislocation by the solute atmosphere. Therefore, the yield point phenomena which is now a gross macroscopic phenomena seen in a normal uniaxial tension test is a product of this microscopic or atomic level segregation of carbon atoms at the core of the dislocations in a b c c i n crystal. Now, similar as I pointed out to the interstitials vacancies can also be attracted towards the compressive regions of an edge dislocation and this phenomena actually plays an important role in the climb of an edge dislocation. And this would be very very important when you are talking about high temperature loading of a material. So, this kind of a vacancy or dislocation interaction can lead to what is called dislocation climb and therefore, plays an important role in the creep of materials. So, though this looks like a very microscopic phenomena occurring at the atomic level it has a gross macroscopic signature and therefore, I cannot ignore defect association. So, let us consider some more examples of defect association we can think of substitutional atoms which are segregating the grain boundary. Substitutional atoms which are insoluble in the grain or have a low solubility in the grain as you want to put it tend to segregate to the grain boundaries. Now, because grain boundaries associated with a certain amount of higher free volume and can better accommodate the substitutional atoms which are otherwise have a very low solubility in the grain. As in the case of an interstitial solute atom dislocation association this combination of an substitutional atom with the grain boundaries leads to a lowering of energy. And of course, when I am talking about energy here I am actually talking about the Gibbs free energy of the system. Now, this has important consequences this segregation actually may make the grain boundary very weak this segregation may actually lead to a precipitation at the grain boundary. This segregation may lead to what you might called hot shortness or cold shortness when I am doing a plastic deformation. And therefore, the materials properties could actually be drastically affected by the presence of this segregation of this solute to the grain boundary region. Further suppose I am talking about a phenomena like creep or any other phenomena like grain growth wherein I need to talk about the mobility of the grain boundaries this presence of the solute actually has a pinning effect on the grain boundary motion. Now, if the grain boundary wants to move the solute atom has to move along with it because that is the low energy configuration. In other words if the grain boundary tries to break free of the solute atoms there is a drag on the dislocation as it tries to migrate. This is similar to the pinning effect of interstitial solute atoms on the dislocation when it wants to move. This segregation of solute atoms to grain boundaries can actually have very very drastic effects. And one very drastic example of this I would like to cite here because it is a very exciting and drastic example happens to be the case of segregation of gallium to the grain boundaries of aluminum. Actually one can do a nice experiment one can take a thin aluminum foil sheet small strip of it and hang it by a small weight. After doing so you may you may want to etch away a small region of this to remove the oxide layer in the aluminum foil. So, if I am looking at this side if you look on edge on this is a thin foil this is a very thin foil. On this edge surface of course, I may want to heat the system by a for instance a hair dryer on of those things. So, that I give some hot air to the system to actually heat up and therefore, improve the kinetics. So, there the kinetics of the process which I am going to do is going to be fast. Then I can rub a little bit of gallium on the edge surface on the surf or the pickle surface. Now, what happens when you put gallium onto this impose gallium on the surface is that suppose I am talking about now on the microscopic scale that this is my grain boundaries. The gallium selectively diffuses along the grain boundary because now it has got a very high diffusivity along the grain boundaries and a very poor solubility. When a layer of gallium segregation takes place at the grain boundary what can happen is that. So, on this is of course, multi-layer segregation which I am showing here by red color. Then gallium being almost liquid like at the temperature we are talking here the grain boundary debonds and therefore, this material will fail by these two parts of the material actually separating out here. So, these two parts of the material will actually separate out. So, this is a very simple and nice experiment to do, but it has drastic consequences as you can see because of segregation of gallium which is taking place along the grain boundaries. And now this is not we might think of as a very macro effect, but really macro effect along the grain boundaries. Since we have already talked a lot about this example of a defect association I will briefly just summarize this example which we have dealt with in detail before that dislocations which are randomly present in a crystal can come together to form a low angle grain boundary as shown in this figure which is this low angle grain boundary can be thought of as an array of dislocations. And this array gives rise to a lowered energy state because the tensile region of this dislocation or let me. So, this is my tensile region of this dislocation partially and as the compressive region of this dislocation. And therefore, have a lower energy state and such a low angle grain boundary does not have long grain stress fields unlike the case of a isolated edge dislocation. This is another interesting example wherein I am talking about an association of a 1 d defect with a 2 d defect that means a dislocation as the example is with an interface. The interface I am considering here is an epitaxial interface between for instance this is the in the example considered this is my niobium substrate in which there is an epitaxial layer which has been grown. When I mean epitaxial layer it means that there is a 1 to 1 matching of atomic rows from the substrate to the fill. And typically these are these epitaxial films are grown in a very special manner by techniques like molecular beam epitaxy etcetera. Now, this is also called a hetero epitaxial interface because this sorry this is I have labeled this wrongly may I stand corrected. So, this is actually the niobium fill which is on a sapphire substrate. So, the sapphire substrate and you can see a schematic of such a system on the left hand side. Now, typically in such systems what would happen when I grow the fill a lot thick then the misfit strains or the coherency stresses generated between the epitaxial fill and the substrate cannot hold the coherency. And you would actually observe misfit dislocation segments forming which or misfit dislocation forming which would lead to partial element of the coherency stresses. Now, this system can be thought of as an association of the hetero interface between the niobium and sapphire as this example is and a dislocation. Now, why do I need to think of it that way is because if I put a dislocation at any other position apart from the interface the system will actually have an higher energy. And at the interface it has typically unless of course, the modulus difference between the substrate and the interface is a substrate and the film is too high which we are not considering here. We will consider example where typically the moduli of the two material are comparable in that case you would note that the interface is the most stable position for the dislocation and it is localized to the interface. And as we had pointed out before such a dislocation is actually a structural dislocation. So, this is an example of an inter association of a 1 D defect with a 2 D defect. So, let me briefly summarize this once again. So, we have a substrate like sapphire in this case a single crystalline substrate on which we epitaxially grew a film. And when I am using the word epitaxial that means there is an atomic matching on a lattice parameter which are very similar. Therefore, there is an lattice spacing matching between the substrate and the film. But since they are not identical the lattice parameters or the relevant length scale there is some stresses which are which may call the misfit stresses or the epitaxial stresses. But when the film grows thicker and thicker the value of the stress becomes so much that it can no longer accommodate a completely coherent interface. That means an interface where atomic planes if you see this is now a coherent atomically matched plane. And this example itself can be thought of as a G E S I film on a silicon substrate. Now it cannot maintain that kind of a coherency and at that stage there is a formation of misfit dislocation segments. Of course, we are not going into the detail here as to how this misfit segment forms there are many mechanism by which they can form. But this interfacial misfit dislocation is localized to the interface and can be thought of as an whole system can be thought of as an association of a 1 D defect with a 2 D defect. Of course, there will be an array of such misfit dislocation decorating the semi coherent interface with the word semi coherent implies there are misfit dislocations present. And if I am talking about a film on a substrate typically there will be a biaxial strain. And therefore, there will be more arrays at an angle to the first array of dislocations which you are seeing here. But the important energetic reason the whole process is that you can clearly see that you have a substrate which is typically compressive and a film which is tensile here. And locally if you look around the dislocation this leads to a partial stress relief you can see that locally in fact the sign is reversed the tensile region becomes compressive. And this relaxation which the dislocation stress field gives to the film stress field of course, film stress field also interacts with the dislocation stress field. This energetic relaxation is what is making the system feasible energetically and this would be a nice configuration which would evolve on growth of a film. Now, let us talk about one last example of defect association this is the example of a 2 D defect being associated with a 3 D defect. And the 2 D defect we are talking about is a is a green boundary and its association of 3 D defect like a precipitate. Typically this takes place by the nucleation of a precipitate at the green boundary. Now, the reason why such a nucleation takes place the green boundary is green boundary is a high energy region and therefore, serves as what you might call a preferred heterogeneous nucleation side for the formation of a precipitate. Now, once the precipitate has formed along the green boundary green boundary diffusion itself can lead to the growth of this precipitate. Therefore, it actually helps in the process of this precipitate growing and further if you have a precipitate at the green boundary it can have an effect which is similar to the effect of a solute pin it a green boundary. That means, a precipitate can pin a green boundary and can actually retard its migration. So, when I am talking about green growth the presence of the precipitate at the green boundary can actually retard its migration. Therefore, you can see that the properties at the microscopic level which definitely reflects in terms of the properties of the macroscopic level is altered severely when defect association takes place. And therefore, it is not only important for me to understand isolated defects, but their association in the scheme of the microscopic scale and also the scheme of the macroscopic scale.