 In understanding the distribution of phases, implicitly we have introduced one kind of a defect, which is the interface between the phase A and the phase B. To understand the properties of materials, we have to understand not only one kind of a defect, but what is called the entire defect structure in the material. So, we ask of course, the question what is meant by the defect structure. Actually, the definition that small term or the phrase defect structure hides in it a lot of details. It is actually a very what you call involved term, wherein you have to use lot of parameters in order to characterize the defect structure. And more further, we have to take it to a stage where the defect structure can be understood in terms of its influence on the properties of the material. So, we have to worry about the kinds of defects present along with their dimensionality, when we want to understand the defect structure in a material. We need to understand the nature of these defects and in terms of their origin. For instance, are these defects statistical in nature or they structure in nature. Some of these terms may be new to you like what is meant by a structural defect or a statistically stored defect. We will briefly take up these terms in the coming slides to understand these kind of classification of defects. We need to understand the nature of these defects in terms of their position. In other words, are these defects random? Are they ordered? And how would be the difference in properties with respect to a single defect? Let me take an example of a vacancy. How a random vacancy would be differed different from an ordered vacancy? I need to know the density of these defects and their spatial distribution. Not only that, I need to know what we might call we had been discussing before the spatial temporal evolution of these defects. In other words, if I am putting a component to service or a material to service or I am doing a processing in order to engineer the material to have a desired defect structure, then how the defect structure evolves with time and in the defect structure, how is the spatial distribution and terms like this density and location etcetera can be used to understand the spatial distribution of these defects? Last but not the least, I need to understand the interaction and association of these defects with each other. It is not sufficient that I talk about one kind of a defect, let it be a vacancy or an interstitial or a dislocation or may be a segregated region along the grain boundary. I need to understand that how one kind of a defect interacts with the other kind of a defect and how this kind of an interaction may lead to what we may call the association of these defects. This is going to be the theme of the understanding which we will take up in the coming slides. Needless to say, this kind of an endeavor is extremely complicated and some of this is barely being undertaken today because when I am talking about these defects like a vacancy or a precipitate, these depresents not only a single lens scale but multiple lens scales and not only that a single defect like a dislocation could have its expression in multiple lens scales and therefore, for instance at the at a single dislocation level, the effective stress field of a dislocation can be thought of as pervading about few tens of nanometers. But when I am talking about the dislocation stress structure or dislocation sitting along the grain boundaries, then I am talking about a different lens scale associated with these dislocations. Therefore, I am actually talking about a really difficult problem of material science wherein I am trying to understand the defect structure across multiple lens scales. In other words, the spatio temporal evolution of the defect structure and this problem as I pointed out is going to be one of the key steps in understanding how our material behavior is dependent on the microstructure. Now, it is worthwhile for me to now briefly revise this slide because lot of new terminology has been introduced here and but essentially what we are talking about is defects in the material and what I would like to do in the coming slides is to understand these defects in terms of their dimensionality, in terms of their origin, in terms of their positional origin, in terms of the density, in terms of their spatial distribution, last but not the least in terms of their interaction and association. So, how can I simplify this problem of understanding defects? What I can do is that initially of course, I just can take up an isolated defect and of course, when I am talking about an isolated defect, I could be talking about a single interstitial atom, I could be talking about a single edge dislocation, I could be talking about a single precipitate in a crystal, I could also associate with these defects certain quantities which are very important like the stress field associated with the defect, if there are any charges associated with these defects, what is the energy of these defects etcetera. So, when I am talking about an isolated defect, it simplifies my problem of understanding the larger picture of the defect structure and have to approach this isolated defect given all its what you might call all the facets of the problem. For instance, now if this isolated defect can talk to other defects, why are the long range stress fields with which it is associated, if this defect is going to cause the crystal energy, then such a defect could be in a meta stable state and when you anneal the crystal, the defect is going to leave the crystal. In other words, this energy associated with the defect is going to dictate if it is stable or meta stable or how much of energy is associated of course, at the core of the defect and also around it, if there are any charges associated with these defects, then this could affect the properties of the defect. For instance, we will soon see that surfaces in ionic crystal like sodium chloride can be polar such as surface could have because they have charges associated with them. The kind of segregation which occurs at these surfaces, the kind of energies of these surfaces is going to be different from a surface for instance in a copper crystal, which is non polar in nature. Then, once I am having a reasonable handle on the isolated defect, then I can consider a pair wise interaction of these defects. These pair wise interactions would essentially imply that I am talking about short range interactions. Some examples of these pair wise interactions could be a vacancy vacancy interaction leading to the formation of a die vacancy. A vacancies interaction with a cluster of vacancies leading to the formation of a larger cluster vacancy cluster. It could be the interaction between a dislocation and an interstitial solute, which finally leads to the segregation of these solutes along the core of the dislocation, typically the edge dislocation which is called the Cotterl atmosphere. And of course, how this association of these interactions lead to finally, change in properties that is the goal of course. So, when I am considering pair wise interaction of defects, again I worry about some of the quantities, we worried at the stage when we are talking about an isolated defect. Like I could worry about the stress fields of this combined or the paired defect. I need to know about the energy of this paired defect, has there been a lowering of energy, has there been increase in energy of the system, because now I am having two of these defects. I need to know the same kind of an effects on the charge, is the overall charge reduced, is there a redistribution of the spatial charge, because of this interaction of these defects. And therefore, I need to worry about all these issues and this becomes an important goal or an intermediate step in understanding the larger picture, which is now considering the entire defect structure along with the external constraints. Now, why do I have to take the external constraints into effect is, because now the presence of these internal defects could influence each other. In addition, the external influence also is going to play an important role and how these defects are going to behave. So, let me take an example for instance, suppose I have an edge dislocation in a material. And now suppose I apply shear stress on this material, then there is a tendency for this dislocation to move. And therefore, finally of course, this dislocation would come out of the crystal and create the surface step of magnitude B. In other words, I have an internal defect, which is a dislocation, which is an edge dislocation I have considered. The external stresses interact with the stress field of the dislocation driving it to the surface. On the other hand, if I am considering two dislocations in the material, couple of possibilities here, the configuration A and the configuration B. In configuration A, the two dislocations are in position in such a way that they would attract each other. In configuration B, they are positioned each in such a way that they repel each other. What is the agency by which the interaction is mediated? It is obviously the long range stress fields of the dislocation. Now, if I plot the sigma x stress field of a dislocation and supposing this is my, I would have iso stress contours. So, this is be my compressive region and this will be my tensile region and I would have a tensile stress. Of course, this plot would display a certain kind of mirror symmetry down the middle. Now, if I have two dislocations in this configuration, the compressive region of this dislocation would repel the compressive region of this dislocation. Similarly, the tensile stress fields would repel each other and therefore, these dislocation would repel each other. On the other hand, such a configuration would mean that my compressive region is in connection with the tensile region and therefore, vice versa on the top and therefore, they would attract each other via the long range stress fields. If the dislocations in a future picture go towards each other, that means they come closer to each other, then this being the separation of the dislocations, then the energy of the system comes down and therefore, there is a natural tendency for the system to go from here to here. This of course, could happen even in the absence of external stresses, if the attractive force between the dislocations is greater than the pulse force, which is the inherent lattice friction. In other words, if my then these dislocations can spontaneously move towards each other and unable each other. Similarly, if the repulsive force is larger than the pulse force or the pulse stress, then automatically the dislocations can move for further apart till of course, the value of the stress is that repulsive stress becomes lower than that of the pulse stress. Therefore, when I am considering pair wise interactions, I these defects can talk to each other via the long range stress fields and additionally, we have to also worry about the external stresses or external constraints, which have been imposed on the system and its effect on the evolution of the defect structure. In other words, in this clear cut example, you can see that this is now an edge dislocation and here there is no dislocation, but only a surface step and the long range stress field associated with the dislocation has vanished when the surface step forms. Therefore, you can clearly see one defect has evolved into another defect and in the process, the energy of the system has been lowered and this is being mediated by an externally applied shear stress tau. Of course, externally what you may call an externally applied force because typically what you can apply are constraints on the surface, what evolves inside the system is a stress field. So, stresses are typically inside the material and what you apply outside we can call of the shear force. In other words, we cannot apply shear stresses, but we need to apply shear forces. Now, when I am trying to understand this in this example, I can take an isolated defect, I can see its interaction with the external constraints or the forces I am applying, then I can take a pair of defects, then I can understand how this system is going to evolve given these pair of defects and the parameters I would like to track during this is obviously the combined stress field. In other words, in this case of course, the stress overall stress region decreases because of the cancellation of the compressive and tensile regions. The energy as the dislocation structure evolves and of course, in some cases as I told you, there may be charges also involved which I need to track. And finally, I have to talk not only about these one or two dislocations, but entire gamete of dislocations and many of these dislocations will be lying on multiple slip planes, they could have defects within these dislocations which we will consider briefly later. In other words, I have a very complicated structure which is evolving under the external constraint and I have to track how this evolution of structure gives rise to what we might call the properties of the material. Some of the properties which typically we could see is for instance, the effect of strain hardening. We could be seeing actually what we might call the evolution of the structure by the formation of what is called the low angle grain boundaries which could happen during after a deformation when you try to do recovery. That is a thermal treatment called recovery which can give rise to low angle grain boundary. So, I can take a material and plastically deform it. So, when I take a deform material and heat it and call the process annealing, then the system evolves to a low energy state in which there is a formation of low angle grain boundaries. Now, this is an important concept and we will explore it a little further later perhaps during the course, but essentially we are seeing that how the system is evolving during plastic deformation and how the system is evolving afterwards during the heat treatment. And this when you are talking about system evolution I am tracking the what you might call the spatiotemporal defect structure in the material. And just to reiterate the problem involved that this consideration of multiple defects or interactions their evolution is actually one of the difficult problems in material science. And when I am talking about the collective motion of dislocations as in the example we are talking here leading to plastic deformation and work hardening. We need to not only consider millions of dislocations or millions and billions of dislocations, but millions and billions of dislocations in a very complicated structure like a cell walls or along the grain boundaries. And their interaction with other defects in the material which makes this problem extremely difficult. And this is already only a single phase material I am talking about a multi phase material like the example we had considered before for instance here we talked about a two phase material. In such a case in a two phase material things would obviously become much more complicated, but nevertheless it is extremely important for us to understand the defect structure in order that can I have an handle on the properties of the material. Now we have said that it is important to understand the defects in the material. And here in this course since this is a course on nanomaterials and nanoscience and nanotechnology I am we will not have the time to go into details of all the aspects of defect structures. But nevertheless a student can keep in mind the overview of the subject and whenever he is faced with a problem or a difficulty on a certain topic he can go and refer to some of this terminology and get more details on the subject from other sources. Now why this broad overview is required is because in nanomaterials often the defect structure is highly altered. And this alteration of defect structure is what is makes the nanomaterials special in terms of its properties. So, how can I understand and classify what we might call a large number of defects and what important ways of classification can actually highlight the role of a defect in terms of its property. So, we can classify defects based on dimensionality. I can classify defects based on association with symmetry and symmetry breaking. I can classify defects based or on their origin we will I will clarify more in a coming slides what those these things mean. I can classify defects based on their position and I can also classify defects if the defect has been defined with respect to a geometrical entity or a physical property. We had seen in the definition of a crystal that the very definition of a crystal hinges on my consideration of merely a geometrical entity like an atom or a molecule or a cluster of atoms etcetera or if it is based on a physical property. And we had also seen that a combination of both is also possible. Therefore, similarly here the defects also in a similar structure is going to be either of geometrical origin or they could have a origin in the physical property. As just to reiterate once more that in this possibility we are not going into too much detail, but we are having a broad overview of the subject. And a student can actually go and refer to some detail text when he wants lot of knowledge on these and other allied topics. So, let us see how we can actually classify defects based on these methods. And why is it important to classify these defects because this classification is going to simplify our job. And when I am talking about a defect I can also understand it from these multiple angles which helps me to straight away pinpoint its role in terms of a properties of a material. So, the usual method of classification of defects is based on the dimensionality of the defect on band. And I am talking about dimensionality here we mean not in the strict geometrical sense, but in a more physical sense. We can have defects which are 0 dimensional which are otherwise called the point defects. We can have 1 dimensional defects otherwise called the line defects. We can have 2 dimensional defects which are the surface or interface defects. And we can have 3 dimensional defects which are the volume defects. It is it must be obvious to any student of material science that is a couple of defects are absolutely unavoidable whenever you have a material. For instance when you have a material it is going to be finite you are never going to have an infinite material. Therefore, surface is one of the unavoidable defects in a material. So, you need to have a surface if you have any material. The second unavoidable defect is what you might call thermal vibration that is because we always have materials at a finite temperature. That means that atoms are vibrating about their mean positions they are vibrating about their mean lattice positions. And therefore, if I am drawing a picture of a lattice here. And if I take a snapshot at any point of time this could be a picture which I might see. In other words atoms are not located right at their atomic mean positions and they are actually vibrating around them. So, this is of course, a crude schematic in reality things could be little more ordered. So, therefore, atoms are vibrating allowed their mean position typically in all three dimensions. Therefore, I can see that this perfect concept of a perfect order or a perfect lattice being a seat of a atom is actually violated. And because of this becomes an unavoidable defect and as we know that we always have materials above 0 Kelvin. Therefore, this would be another unavoidable defect in a material. Now, to have a brief overview what are the point effects which we need to consider? What are the important line defects which we need to consider? And what are the volume and surface and interface defects we need to consider? One other unavoidable defect in the thermodynamic sense is the 0 dimensional defect of vacancy. So, what is a vacancy? A vacancy is for instance a missing atom from a lattice position. For instance a copper atom or a copper ion could be missing from its lattice position and this is called a vacancy. This atom going missing is actually going to cost internal energy in terms of the broken bonds to the material. In other words the internal energy of the material is going to increase in the presence of a vacancy. But it is not internal energy which is going to determine my stability of my substance at constant temperature and pressure it is usually the Gibbs free energy. So, the stability of my system under at constant temperature and pressure is determined by the Gibbs free energy which is given by u being the internal energy. So, when I put a vacancy in a material because of certain broken bonds we can have an increase in internal energy of the system. But at any finite Kelvin temperature the configurational entropy of the system given by s is equal to k ln omega the famous Boltzmann equation in which omega is the number of configurations possible and k is the Boltzmann constant. Therefore, because when I put a vacancy in a material this vacancy could actually sit in one of the many lattice positions. For instance if this is my vacancy this vacancy could be present here, here, here, here, here or in one of the many possible lattice positions and this implies that this increases the configurational richness of the system and that hence provides an entropy benefit. Therefore, a system with a vacancy has an higher entropy as compared to a system without a vacancy. When we put more and more vacancies it is going to cause the crystal more and more in terms of the internal energy and hence enthalpy. But there is going to be a certain number of vacancies which are going to be thermodynamically stable. In other words if I put a certain number of vacancies it will give me the maximum entropy benefits such that h minus t s term actually turns out to be negative. And if the state it happens to be negative that implies that vacancies are going to be stable. Of course, we are not talking about a any number of vacancies we are talking about a certain fixed number of vacancies which will lead to a minima in the Gibbs free energy function. Hence, because the fact that now in the presence of vacancy the Gibbs free energy can actually be reduced compared to a state where there are no vacancies. And now we are talking about certain positive Kelvin temperature like we could be talking about room temperature we could be talking about 0 Kelvin or we could talk about 200 degree Celsius. In such a situation we see that there are a certain equilibrium concentration of vacancy which are stable in a material. And therefore, vacancy becomes a thermodynamically stable defect unlike the many other defects which will be dealing with like the example of a dislocation here or the example of a grain boundary which are thermodynamically metastable defects. Now, what is the implication of having a defect which is stable or metastable or as the case may be even unstable. Of course, unstable defects will spontaneously leave the crystal and therefore, the crystal would become free of these defects. But a stable defect how much ever you heat the substance or anneal the substance would not go away. In other words just because I have a vacancy I cannot heat the system to get rid of vacancies because there is an equilibrium concentration of vacancies which will always remain in a substance. Of course, if I heat it to higher temperature more number of vacancies would become stable because this is the weighing term for the entropy. When of course, I slowly cool down then the number of vacancies if given an opportunity to leave the crystal and which you reach a new equilibrium at lower temperature which would be a lesser number of vacancies. On the other hand suppose I had a system with dislocations or a system with grain boundaries and I heat such a system there will be a tendency for dislocations and grain boundaries to leave the crystal and in the process actually reduce the Gibbs free energy of the crystal. So, this is an important point to note that some defects like surface and thermal vibration are unavoidable in the crystal. Some defects like vacancy though in spite of being defects are actually thermodynamically stable. The other important 0 dimensional defects which we need to consider are actually what we may call the term impurity of course, is a casual term. We can think of a substitutional or interstitial atoms and that is a very important kind of defects. Though here we are talking in terms of defects, but these could intentionally be added to increase the properties. For instance in steel we add carbon to actually increase the properties. In other words, if you want to have a harder material then I may add carbon to it. Similarly, I could have at substitutional impurity atoms to increase the what you might again the strength of the material. So, in such cases the second atom which has been added would actually be beneficial and should be called an alloying element instead of an impurity. So, but since we are talking in terms of a perfect lattice the presence of these additional elements actually may actually cause distortion to the lattice and therefore we classify them as defects as with respect to the perfect crystal which is a crystal in which there is only one species of atoms. There are other defects which come in ionic materials like the Frankl defect and the Schottky defect. We may briefly consider some of these as we go along. The important 1 dimensional defect is the dislocation and we have been briefly considering the concept of a dislocation and the stress field etcetera. There are other possible one line defects like the disclination and desperation which of course we will not take up during this course in any kind of detail. The important 2 dimensional defects are the surface which we have already talked about and surface represents a region of higher energy in the material and surface is also associated not only with surface energy, but also with a concept of surface tension. And this is very very important to note here. The term often associated with the surface is sometime it is also called the free surface in the absence of any constraints. And there is a possibility that in certain nano materials the surface especially in polar materials that the surface actually can be under compression which is normally not found in bulk materials. So, we can have a surface associated with 2 important quantities the surface tension and the surface energy. And we are already exposed to quite a few important aspects of surface tension especially in liquids like the capillarity effect or insects walking on water this is all possible because of surface tension. And later on when we talk about phenomena like super hydrophobicity we will see that how surface tension is at the heart of those kind of phenomena which we are very very important for us. We can also talk about an interface boundary which also we briefly had considered sometime back. An interface boundary is the boundary between 2 phases. Suppose I am taking my example which we had considered before now this is my interface the one I am highlighting in red. So, this is my interface so this is my interface boundary. Now, as I had pointed out when we are discussing this aspect that the interface properties is very very interface properties is very very important with regard to the material as a whole. In other words if we have interfacial debonding or any one of those phenomena then or nucleation of cracks at the interface then the material as a whole would actually perform poorly. The third defect which in actually engineering or practical terms is we might even think of it as an unavoidable defect is the grain boundary. Because unless we take extreme good care to produce a material like we use a technique like a Bridgman technique or a Zocralski technique we cannot or we normally do not obtain a single crystalline material. When I take for instance a rod of copper or a rod of aluminum or any one of the metals we normally see in day to day life typically they are polycrystalline. Of course, there are important engineering applications if single crystal like we know the silicon chip which goes in the formation of a computer arithmetic and logic unit. Therefore, there is there are specific examples where single crystals are specially grown and used one of them being with respect to computer. The other thing could be for instance single crystalline turbine blades have been used in gas turbine engines. But these require extremely special care to make and therefore, are usually costly. But if I take a normal material typically it will be polycrystalline. In other words, it will consist of single crystalline regions and these single crystalline regions would be separated by what is known as a grain boundary. We will have a look at example grain boundary very soon. The other kind of a two dimensional defect is what is called a twin boundary and we will also talk about a twin boundary very soon. In close packed crystals there is another kind of a defect which comes which is called a stacking fault and a special kind of a stacking fault is what is called an anti phase boundary which is found in ordered materials. So, we can have many kinds of two dimensional defects and of course, when I mean two dimensional they need not be a flat two dimensional defect they could be curved and in other words they could have a varying curvature from place to place. Three dimensional or volume defects for instance is can be thought of a twin which is bounded by twin boundaries can be thought of as a three dimensional or volume defect. A precipitate can be thought of as a region wherein there is a second phase sitting in a matrix. So, the second phase is actually a disruption in the order of the perfect crystal the parent crystal or the matrix crystal and therefore, can be thought of as a defect in the parent or the perfect crystal. And we will take up an example sometime soon wherein we will see that how this precipitate itself can be engineered to obtain important properties. In other words far from being a defect in the sense of being unwanted these precipitates actually can play a very important role in engineering the properties of a material. The fault and region which is bounded by the stacking faults is also a region which can be thought of as three dimensional defect. So, I can make these connections in other words a precipitate is bounded by an interface boundary twins are bounded by twin boundaries a faulted region is bounded by stacking faults. Voids and cracks are again typically though how much ever we want to avoid them they are typically unavoidable in a material. And therefore, at least to in its typical engineering process like casting or welding you would notice that to some extent we would actually have some amount of these defects like voids cracks etcetera. And we shall see that these voids and cracks actually play a very important role in sometime most of the times actually in ways of deteriorating the properties of a crystal. But under special circumstances like we have seen in porous materials voids and cracks can actually be beneficial and this can or more not really cracks, but voids can be beneficial in giving special properties in a material like having allowing it to have lesser weight. So, we have considered here a broad overview of classification of defects based on dimensionality. So, let us now consider other kinds of classification or defects apart from based on dimensionality. We can think of defects being associated with symmetry like for instance the dislocation can be thought of as being associated with the translational symmetry of a crystal. Similarly, a discoloration can be thought of as being associated with the rotational symmetry of a crystal and a discredition being associated with these crew symmetry of a crystal. And this is of course, thinking of symmetry associated defects at an atomic level at a certain larger level or the macroscopic multi atom level. We can think of as twins being associated with certain kind of trimetries like we can have mirror twins, we can have rotational twins and also we can have inversion twins. We will see certain schematic examples incoming slides of mirror and rotation twins. In other words suppose I have a dislocation in a material that in some sense is going to be also breaking my translational symmetry of the crystal. So, this is the reason why we call them symmetry associated defects. Defects based on truly symmetry breaking concept can be called topological defects or non-topological defects. Though we are not going into details at this stage is what is a topological or non-topological defects, but these are genuine fields of interest and study at least in specific circles. This is we will spend a little more time over the understanding of what is called a defect classification based on origin. There can be statistically stored defects in other words which is present in non for non specific reason in the crystal and what we may call structural defects. And of course, we will I will show you by what is one example that how a statistically stored defect is different from a structural defect. The importance of this classification is that a statistically stored or a random statistical defect has a different role in material behavior as compared to what you might call a structural defect. A single defect can also go from being a statistically stored defect to a structural defect and in fact we have already considered one such example, but we will return to that example once again in the from a view point of classifying defects as statistically stored or structural. Structural defects typically make certain configurations possible in a crystalline material and therefore, they are localized to a certain region in the material. For instance the angular misorientation at a green boundary can be produced by an array of dislocations. So, this is a very important concept that the dislocations which are present along for instance a low angle green boundary and we will see actually a beautiful high resolution picture in a coming slide. Such a dislocation which is making possible the misorientation typically a low angle misorientation possible between two regions of a or a two crystals is what we can thought of as a structural array of dislocations. Now, let us consider one example which will may exemplify things a little more. Now, we had seen the process wherein we had taken a deformed crystal and we are annealed it and the process which occurred during this anneal is called polygonization which led to the formation of in a deformed crystal. Actually, if I now consider a polycrystalline material and I am just going to draw some crude schematics here there are going to be lot of dislocations and of course, though I am using a symbol for an edge dislocation a symbol a general dislocation in a crystal could actually have a mixed character it could be partially screw it could be partially edge and therefore, it could have any general character. So, I am just drawing some schematic of showing these dislocations and if I am looking at a single crystal single grain within this this single grain actually could get bent because of the presence of these dislocations. Now, during the annealing process these dislocations can come together to actually form an array one below the other and we already seen that the reason for formation of these arrays is because of lowering of energy. Now, this is my compressive region of the dislocation and this is my tensile region and the tensile region of this dislocation partially annals the compressive region of an dislocation below it and therefore, they would rather tend to align in this format rather than being randomly positioned in the crystal this kind of a distribution of dislocations I would call as a and in the formation of in the formation process where you actually end up forming what is this called a low angle grain boundary and we will see a picture of this in a coming soon and as you can see this low angle grain boundary actually consists of an array of dislocations. The specific low angle grain boundary being considered here is what is called a low angle tilt boundary and suppose I were to construct a low angle twist boundary such a low angle twist boundary would actually consists of an array of screw dislocations. But, what we have seen here is that this statistically stored dislocations in the process of this annealing actually go on to form what is called as structural dislocation. Now, this structural dislocation and of course, I do not always have to start with a statistically stored dislocation to form a structural dislocation this is not a necessity I am just taking one beautiful example where such a transformation actually takes place. Now, in this structural dislocation or this low angle grain boundary the structural dislocation has been localized to the grain boundary. So, it is now and it is providing a structural feature here in other words it is accommodating the tilt between the two regions of the crystal which I can call crystal 1 and crystal 2. So, the misorientation between these two regions which is now I have just pointed out is a small misorientation of something less than 10 or 15 degrees has been accommodated by this array of dislocations. Now, suppose if I try to plastically deform this specimen we saw the this specimen it is clear that these dislocations are free to move because they play no structural role in the material. On the other hand if a dislocation this dislocation leaves this boundary then the misorientation angle is going to change and therefore, it is energetically not feasible or not that easy to drive a dislocation away from this boundary. Also this is already in a low energy configuration and when that this dislocation leaves the energy of the system is going to be higher. Therefore, more work has to be done to drive this dislocation away from this boundary as compared to a dislocation moving from here to a neighboring position like this. Therefore, you can clearly see that a structural dislocation though in terms of its character is this is also an edge dislocation that is also an edge dislocation is far more localized the structural dislocation and its response to an external stress is going to be different as compared to a statistically stored dislocation. So, for other kind of structural vis-a-vis statistically stored effects like a vacancy. For instance suppose in when I am talking about vacancies again I can have structurally stored vacancies or I can have I mean statistically stored vacancies or structural vacancies. As I had pointed out that vacancies are typically unavoidable in a material because we are always at positive Kelvin temperature and therefore, there is an equilibrium concentration of vacancies. But, there is a possibility that vacancies arise because of other reasons as well and why it is considered one such example. Let me consider a b 2 phase which is a simple cubic phase having a composition a 50 b 50. Now, it is possible that some of these a b kind of phases one example could be n i a l phase for instance they have certain or they tolerate certain of stoichiometry. For instance I when I want to make not a 50 b 50 phase, but there could be a possibility that this is not actually a b is not a line compound and it can actually tolerate a little amount of deviation from stoichiometry and let me now assume that this can become a 51 b 49. In other words this is now an a rich composition with respect to the perfect stoichiometry of a 50 b 50. Now, when I am trying to make this a rich composition there are two possibilities which represent myself one is I can actually put an anti side defect or I can put a vacancy. In other words suppose I start putting a atoms in the b sub lattice then I can get actually a rich composition as you might be aware that this kind of a structure is called an ordered structure and this ordered structure is sometimes also referred to as a super lattice and the super lattice can be thought of as two interpenetrating simple cubic sub lattices. One simple cubic sub lattice having its origin at 0 0 0 another sub lattice having its origin at half half half and therefore, this simple cubic super lattice can be thought of as having two sub lattices one completely occupied by a another completely occupied by b. Now, if I start putting a atoms in b sub lattice then I can make this composition a rich another possibility of making an a rich composition is by actually putting vacancies in the b sub lattice. So, of course, the system depending on the system specificity and in which direction are we actually considering the of stoichiometry I am trying to make an a rich composition or a b rich composition it certain systems will actually show as a preference for vacancies in one of the sub lattices. Now, this vacancy which have introduced purely because of of stoichiometry with respect to a ordered compound. So, this vacancy is to very different from the thermal vacancies which arose because of thermodynamic reasons right. So, this vacancy is there because of of stoichiometry and now if this vacancy has been put in the b sub lattice and I am talking about vacancy diffusion or atomic diffusion as a specific what you might call mechanism of the property which is under my consideration then in a completely of course, disordered system a and b sub lattices would be equivalent and some of these would be vacant giving rise to for the reason of thermal vacancies. And if jump from and of course, I can schematically represent a disordered unit cell as follows and so forth all the lattice points are being occupied. And this disordered structure is a b c c has a b c c lattice at its heart. Now, of course, all the lattice points are occupied by 50 percent probability of occupation of the b atom and 50 percent occupation probability with the a atom. And if some of these are vacant sites then any of the a or b atoms could jump to its neighboring site without any problem. Because now this is an ordered structure the if an a atom jumps to the b site because it is vacant and this vacancy is being a structural vacancy the energetic feasibility it is not energetically feasible to do so that implies that the diffusion arising from the structural vacancies is going to be different from that arising from the thermal vacancies. The thermal vacancies can reside in any one of the lattice points on the other hand this structural vacancies localized to one of the sub lattices it is not randomly positioned anywhere in any of the sub lattices. And that the very fact that now this vacancy is a structural vacancy makes it makes its role in diffusion different from that of a thermal vacancy in diffusion. So, it is clear from at least couple of these examples that when I classify these defects into either structure or statistical it gives me a handle on the properties and gives me the ability to recognize the role of the defect in determining a specific property under consideration. Now, another important classification which we have implicitly already taken up in the example here or at least set the stone for understanding those kind of defects in the example we considered is based on the position a defect can be random or it can be ordered. And the important difference between the two is that an ordered defect once it has been ordered becomes part of the structure and this becoming part of the structure can actually lead to a change in symmetry and hence lead to a change in the crystal structure. In principle any of the defects we have considered can actually get ordered, but some of the important defects which whose ordering we need to consider are the vacancies and the stacking fault. For an example of course, we will consider just ordering of the vacancies for now, but other defects can also get ordered and when they get ordered obviously, their overall collective behavior is going to change. In some sense we already noted that this dislocations along the grain boundary in some sense is ordered because now the average spacing of all these dislocations is fixed by the misorientation angle between the two crystals. In some sense they are ordered defects, but here we are still talking about a three dimensional crystal and a defect actually at the interface. If suppose in the case example of vacancies and stacking faults these are defects within the crystal these have a in some sense have a more profound influence on the very definition of the crystal itself. So, let us perhaps consider this because this is an important consideration as far as the symmetry of the crystal goes. Now, let me consider as a schematic the A B kind of a structure we are already noted because now I am considering of stoichiometry the vacancies in the B sub lattice are localized to the B sub lattice. So, in some sense they have been localized to the sub lattice. So, they are ordered at least within the sub lattice, but to go from there to perfectly ordered vacancy which we can call a vacancy ordered phases it has to be positionally ordered even within the sub lattice. So, let us think of how we can do that. So, suppose for simplicity let me consider a few unit cells of a square kind of a lattice. Now assuming that the vacancy can be present anywhere in other words any one of the atoms can go missing randomly. In such a structure we can think of as consisting of random vacancies, but instead of being present randomly anywhere let me for simplicity think of this structure having vacancy here. And of course correspondingly if I am talking about an extended crystal and let me try to extend this crystal would have a vacancy here would have a vacancy here and would have a vacancy here. And of course the all the other points I can fill with atoms. Now this is what we call vacancy ordered phase see that my original unit in the crystal would have been a unit cell like this a small unit cell. Now when the vacancies get ordered you can see that now my unit cell has expanded and this whole big is my unit cell sorry this is not my unit cell let me correctly draw the unit cell the unit cell would be this is my unit cell. So therefore the lattice parameter and the symmetry at the fundamental level has been altered because of the ordering of vacancies. And that is why these ordered structure ordered defects have a different role to play in the crystal as compared to a random defect. In other words an ordered defect can actually become part of the structure. And therefore it can alter the very definition of the crystal last but not the least let us consider how a defect can be classified based on a geometrical entity or a physical property. Now when I am talking about for example a domain wall in a ferromagnetic material then the domain wall is not defined with respect to the crystal structure it is defined with respect to the magnetization of the spin vector in the magnetic domain. Since I am talking about a ferromagnetic material in domain 1 all the spins are going to be aligned in the same direction. Similarly in domain 2 all the spins are going to be aligned in the same direction and this direction has been shown schematically as an arrow mark in this direction. In this for domain 1 and for domain 2 it has been shown as a vector in the other direction. Now a domain wall represents a defect in the domain structure exactly analogous to a grain boundary in a crystal structure. But here we have to use the definition based on a physical property like as we have done here the magnetic movement of the spin vector rather than using a structural definition like an atom or an ion or a molecule. Therefore when I am talking about defect classification based on a physical property we come across other kind of defects which are normally not dealt with merely when I am talking about the atomic or the geometrical entity and the example I have considered here is what is called a magnetic domain wall and as you can see here in the magnetic domain wall the magnetization vector actually rotates out of plane and goes into a new orientation when you reach the domain which is adjacent to domain 1 which is has been labeled as domain 2. This kind of a wall of course is called an block wall and in thin films there are other kind of walls possible like the kneel wall etcetera wherein actually the rotation of in the kneel wall the rotation of magnetization spins occurs in plane rather than out of plane as shown in this figure. But nevertheless the domain wall represents a region of higher energy it represents a region of disturbance with respect to the perfect ordering which is present in the either in the domain 1 or domain 2 and therefore can be considered as a defect in the domain structure. So, when I am trying to understand a physical property based definition then additional kind of defects can arise in a material like a magnetic domain wall or of course when I am talking about a ferroelectric material then there will be domain walls in ferroelectric materials where the dipoles on either side would be oriented differently and therefore the ferroelectric domain wall would represent again a region of high energy and this hence becomes an important basis for definition of defects in materials which is which takes us to additional kind of defects which are present. Suppose I have a polycrystalline material which is also ferromagnetic then I would notice that the the grain boundary is automatically also a domain wall boundary, but domain wall domain boundaries themselves often have been found to be continuous across grain boundary. So, this is an additional observation, but the important thing is that within a single grain which is a single crystal based on crystal structure the single crystal could be divided into multiple domains and the reason for doing so is actually a lowering of energy of the material which is otherwise in a single crystal would be leading to high magnetostatic energy. So, domain walls in spite of having high energy are tolerated in materials in order to overall reduce the energy of the system which involves magnetostatic energy which throws out external magnetic fields. So, before we close this section let us consider some of what you might call gallery of some of the defects some of which we have already talked about some of which perhaps we have briefly mentioned, but not gone into any kind of detail and some new ones. So, let us start with point defects like the interstitial which has been shown here and a vacancy which we have talked about lot of detail right here. And this is two kinds of point defects are shown here one is the interstitial which is present in the interstitial void of a material or if one of the atoms has been replaced in the lattice by an atom which is been colored blue here then this would be called a substitutional defect or substitutional alloying element. We can have a pair of defects like we can have a missing anion in conjunction with the missing cation in an ionic substance and of course, a pair of them go missing because then the overall charge neutrality is maintained. Therefore, such a combination is called a Schottky defect which is a combination of a missing anion and a missing cation typically which are positioned close to each other. We already seen the example of an edge dislocation and we also such an edge dislocation can actually present at an interface between two materials and in this example this there is material A on this side which actually is a solid solution for example, this could be a solid solution between germanium and silicon and this could be pure silicon for instance. And this dislocation which is present between the two materials can be thought of as an interfacial misfit dislocation and in some sense an interfacial misfit dislocation belongs to the class of structural defects and not to a random defect because now it is localized to this interface between G E S I and S I. So, this is an interfacial misfit dislocation is a structural dislocation unlike the random dislocation which is the one which is shown on the left hand diagram. We can think of surfaces in ordered structures like for instance this is now the 1 1 1 surface in an ordered structure and of course, this is a nice ball model and this is a super lattice structure. And suppose I am talking about ionic material I can notice that if I make the same 1 1 1 surface it can either consists of purely of sodium atoms as this surfaces or it can consist purely of chlorine ions sodium ions or chlorine ions. And therefore, these surfaces in polar materials can in ionic materials can be polar. So, these two are polar surfaces cuts along other surfaces like for instance the this is now the 1 1 1 surface which has been shown here. Suppose, I make a 1 0 0 kind of a cut in such a material then I would notice that that surface would be non polar. So, surfaces can be polar or non polar even in a single ionic material then we are also seen the example of a grain boundary and one such example is in the case of the grain boundary in staunchim titanate which is a ceramic. And you can see that some grain boundaries are randomly curved and some of them can be reasonably straight as well. We had also said that there are defects in materials which are unavoidable and one of them was the surface. And now this figure here shows not only a surface, but certain other defects within the surface which we call ledges and kinks and we will come to it perhaps later during the series of lectures. Therefore, we have the surface here these are examples of surfaces here here which are two dimensional defects a grain boundary which is again a two dimensional defect an edge dislocation which can be a random statistically stored edge dislocation or a structural edge dislocation localized to an interface both these are one dimensional defects. Then of course, we have point defects here which is 0 dimensional and the top. So, your collage of many kind of defects which are shown here, but what we have to remember is that many of these defects are what are going to play a very important role in the properties of the material. And to show on some of the examples which you have been talking about quite a bit today is the case of the low angle grain boundary. So, this is an actual transmission electron micrograph in a strontium titanate poly crystal in which this region is a low angle grain boundary. So, this is my grain one and this is my grain two and between that you have a low angle grain boundary and when I have a Fourier filter image actually you can see the dislocation course nicely in such a low angle grain boundary. In other words, I can clearly see an array of dislocation sitting along the grain boundary which is now has a small misorientation of about 8 degrees it is an 8 degree tilt boundary. And therefore, it is leading to a small misorientation which I pointed out this characteristic of the low angle grain boundary. And clearly therefore, this dislocation I am describing here is leading to this misorientation and therefore, it is a structural dislocation.