 Let us continue with the discussion of self-assembled ordered nanostructures with assembling particles with multiple sizes or phases. It has been shown that for micron size particles to size spheres which are labeled A and B, the following was predicted. It was seen that for a size ratio between R A by R B in the range of 0.48 to 0.62, the mixture will form a stable structure. Out of this, if the size is particularly close to 0.58, then you will form an A B 2 phase. And we are noting here that these sizes we are talking about is in the range of microns and not at in the nano scale. And it was also predicted that if the size range is between 0.458 and 0.48, then or the size ratio range is in between 0.458 and 0.48 phase separation where occur. So, it is important to test that if can we form binary mixtures of the type A B 2 with spheres in the nano scale. For that people have tried gold particles which have been thiol stabilized and soon we will see why we need to stabilize these molecules with thiol. And if the size ratio was taken to be close to 0.58, then the prediction which was seen for micron size particles was confirmed for the case of nano crystals as well. That is in even for the nano crystals, if the size ratio is close to 0.58, then we can actually form a stable A B 2 kind of a phase which can be thought of as an binary intermediate phase. The arrangement of the two particles in this phase is as shown in the figure below. The larger particle is the one which is shown in blue color. So, this is the larger particle and now these particles are in the nano scale regime. And the smaller particle is the one which is shown by the sphere here and I will shade these smaller particles as follows. So, surrounding each one of those larger particles, you have six smaller particles and surrounding each one of the smaller particles, you actually have three larger particles. Now therefore, I can actually draw an unit cell for the structure which is now a two dimensional crystal with a stoichiometry A B 2. This is my unit cell, but to calculate the stoichiometry to be A B 2, it is better that I consider a hexagonal kind of a cell which is shown on the right hand side here. Here you have the larger particle and these are parts of the smaller particle, six of which are surrounding the larger particle. And now I can calculate the stoichiometry because now one third of each of the particles is contributing to the cell, there are six of them. So, that makes it two of those for each that means two of B for each one of the A particles which is the larger particles thus giving the stoichiometry A B 2. And of course, in this schematic diagram the thiol molecules have not been shown, but it is essential that if I am using gold nano particles then it should be stabilized with thiol otherwise these particles may center and therefore, I will not get a structure which is described as in the schematic below. So, this is now a two dimensional crystal with a unit cell as shown here which is now in the nano scale regime. Now the question arises we have micron says size crystals which can form with a stoichiometry A B 2, there are nano scale crystals which can form with the stoichiometry A B 2, but are there analogs of this in the normal atomic crystals. And obviously, there are and the normal atomic crystals are called the lavais phases which are atomic binary compounds. And if you look at the largest scheme of size factor compounds there are two kinds of disguise factor compounds, one is the lavais phases another is the fan transfer phases. And if you look at the lavais phases they form with the stoichiometry A B 2, which is the stoichiometry we are talking about in the case of micron sized or the nano sized particles forming the precipitates. But here the clear cut difference is that these A and B are atoms in the case of the lavais phases. And there are well known examples of such kind of lavais phases like M G is an N 2 and M G N I 2 both of which have hexagonal crystal structure or belong to the hexagonal lattice, M G C U 2 which is a cubic close or has the FCC lattice that is underline lattice. And therefore, we also have analogs of these kind of A B 2 binary compounds which are basically size factor compounds. That means size is the determining factor for the formation of these compounds unlike for instance some of the humero 3 compounds wherein E by A ratio stabilizes the compounds. These lavais phases are regarded as tetrahedral close pack structures with an ideal radius ratio R A by R B turning out to be root of 3 by 2 which is 1.225. Obviously, this 1.225 the inverse of which is 0.186 is clearly a different number compared to the 0.58 which we have seen for the case of the largest phase spheres which form an A B 2 stoichiometry. But, nevertheless these are stabilized for this size ratios 1 close to 1.225 and typically they form not just at 1.225, but for a radius ratio in the range of about 1.1 to 1.6. And if the R A by R B is 1.225 then a high packing density is achieved with the chemical formula A by 2 with an average coordination number which is 13.3. Of course, you might say that the maximum coordination possible in the case of for instance FCC or HCP is 12, but that is for a mono dispersed sizes. But, mono dispersed that means you have particles of the same size or spheres of the same size, but when you consider two sizes then you can have an higher coordination number which can be 13.3 in the case of the R A by R B being about 1.225. There are about more than 1400 members belonging to this Lavais family that means it is very abundant and well known kind of a structure which is can be thought of as an analog of the nano crystal which we just saw which forms an A B 2 kind of phase. Of course, these phases are three dimensional crystals these Lavais phases unlike the two dimensional crystal which was shown to be forming with an A B 2 in the size ratio giving 0.58. So, this is the important difference and many of the ternary and multinary representation of the Lavais is also been reported with excess of A or B elements some ternary Lavais phases are known in systems with no corresponding binary Lavais phases. So, there are some important members in the ternary class as well when it comes to Lavais phases. Therefore, to summarize this part when I take nano crystals of two sizes and then I can form a stable structure I may even see phase separation or you may form a very well defined ordered structure with two which might call a super lattice structure with an A B 2 kind of stoichiometry. This A B 2 kind of stoichiometry can be observed when R A by R B is in the range of about 0.58. Now, we have considered a few types of the self assemble structures for instance we talked about the nano crystalline solid. We also talked about and this was one example of nano crystalline solid where silver nano crystal was used to form a crystal using the FCC lattice. Now, we have seen a binary nano crystal or nano crystalline solid, but many properties and applications have also been some envisage for these self assemble nano crystals. The properties of self assemble nano crystals are different from either the nano either the nano crystals or the bulk material. In some sense because we are now in making these nano crystals with some passivating agent in many cases. As we saw in the case of gold we have to use a passivating agent and therefore, the property is some kind of outcome of the as a composite of the nano crystal and the passivating agent. So, it is not a merely a property of the nano particle which is going on to form the crystal it is not merely a property of the passivating molecule, but it is some kind of an average or some kind of an additional property which can come because of this kind of a composite. For instance some applications have been envisaged for these kind of self assemble nano structures like heterostructure diodes have been made with metallic self assemble nano crystals connected by a molecular nano wires. The idea here being to reduce the dimension of the diode below about 10 nano meters. So, this is one of the important applications which has been envisaged for metallic self assemble nano crystals connected by a molecular nano wires. Super lattice of gold nano crystals about 3.7 nano meter in diameter was assembled by organic interconnects on a silica substrate and this showed non-linear coulomb charging behavior. We will be talking about more about this coulomb charging behavior and coulomb blockade later on when we talk about electrical properties of nano materials, but this is one of the interesting applications wherein we have a gold nano clusters which have been assembled by a organic interconnects and which shows some kind of a non-linear coulomb charging behavior. Like we have ordered crystals, we can also have ordered pores and these pores can exist in multiple lens scales. They can be in fact macroscopic which are visible to the naked eye. They can be mesoporous in which case we are talking about the pore size in the range of about 25 nano meters and we can have narrow porous materials where the pore size is extremely small. So, we can in other words we are now going to describe ordered self-assembled mesoporous materials and though we using the word mesoporous we are going to take a picture at the largest picture wherein we are also going to briefly talk about these other kind of porous materials like the nano porous and the macro porous materials. Porous materials have a low density because now we do not have a homogenous monolithic material, but we have pores part of the end structure and therefore, there is could be a reduction in the density and the density of the porous material could be about 10 to 30 percent smaller than the bulk density. Such materials obviously, have large surface area or some of which is internal of course and some of which is external and therefore, this surface area and low electric low dielectric contents gives it lot of applications specially in catalysis, filtration, sensors, structural members etcetera. When you are talking about structural members for instance this lower density also implies that now we can other can be a considerable weight saving and there are could be other structural members which are taking the maximum load and this structural member for instance can be used for space filling. Additionally, if you look at these porous materials you can fill the pores with a secondary material which can form an integrated component in the functioning of the whole component. For instance now I can fill the pores if they are small enough with some kind of a grease material. Now, this secondary material would function additionally to the main material in giving some kind of a lubrication while in service. Now as we said these porous materials can exist in three length scales the nano porous materials and nice examples of these are zeolites and metal organic frameworks and here the pore itself can be considered part of the crystal structure. That is why the pore size is extremely small it is of the order of 1.5 nanometers. So, these are pores within in some sense the crystal structure itself. In piece mesoporous materials which are typically inorganic in nature the pore size is much larger and these have to be specially made using some artificial methods like we shall see like templating methods. Macro porous materials like we just said given example which are the pores are visible to the naked eye and one example would be aluminum foam are also finding widespread application nowadays and we shall not discuss these macro porous materials in detail here. Now how do we how can I make these mesoporous materials a template assisted method has emerged the most versatile and common method to form ordered mesoporous materials. And here typically we take a mono dispersed silica or polystyrene nano diameter size spheres and typically this means that we are using a single size sphere. And we form a colloidal suspension ordered structure of form when actually the suspension is dry. Therefore, I take a mono dispersed silica or polystyrene that means fixed size spheres and in colloidal suspension I allowed the suspension the particles to order when actually dry the substance. And using this method ordered porous oxides graphite organic materials etcetera have been synthesized. There are other techniques also which have been used by people to assemble or self assemble mesoporous materials and these include super molecular templating and self assembling mechanism etcetera. Now as I was mentioning that you may leave the porous material as a pore and use it like for instance in filtration or catalysis where you have a lot of surface area internal surface area especially which can be used advantageously in a mechanism like catalysis where you really need lot of surface area. But additionally as I pointed out that you can actually fill these pores with another material and this material could actually be an active component in the properties. And one example which I am just randomly taking from literature is in which people have incorporated high dispersed gold nano particles into the pore channels of mesoporous silica thin films and they have studied the optical response behavior which has found to be non-linear. Therefore, now when I make a mesoporous solid or even a nano porous solid I need not work just with the porous material which I could do of course I can directly use it for direct application. But I can use it as a medium for actually putting in a second member. One nice example is in the case of the metal organic framework wherein actually and a one nice example of a metal organic framework is the what is called the MO of phi. Where in you have for instance I will just draw a schematic of this MO of phi on the board. So, you have these Z n O 4 tetrahedra and each one of these Z n O 4 tetrahedra which is of course there are four of them in each one of these corners. And I have schematically shown by dotted lines and each one of this is connected to the neighboring 4 tetrahedra by dichorboxylic acid group which have schematically shown by these double red lines these double red lines. And now this structure implies that these sort of dense regions of matter are separated from each other by these dichorboxylic acid group. And there is lot of volume within the unit cell and in fact you can schematically sometime draw this volume to be a sphere which is a large sphere which exists inside this unit cell. So, there is a large volume which is resides in the unit cell and this volume can actually be used not to not only transport molecules small molecules like hydrogen, but also store hydrogen. These interaction of hydrogen with this MO of phi is more like a physics option that means there is no strong covalent or other kind of bonding. This implies that if I really have to store put put hydrogen and store hydrogen then I have to work at low temperatures where in the physics of the hydrogen does not absorb because of entropic effects. Therefore, because of these free pores and you can visualize that these free pores are connected to each other that means now I have unit cells which are beside each other. And there is this interconnectivity of these pores and therefore hydrogen can be transported across these unit cells via the phase center phase centers. And in the end of course this volume inside this unit cell itself can be used to actually store hydrogen albedad slightly lower temperature. So, that now the physics option forces are not broken by thermal effects. So, we can clearly see that these mesoporous materials they as you said as synthesized condition itself they offer certain benefits where in you have a high surface to volume ratio which can be used for catalysis or these pores can actually be used for filtration. That means now certain only certain molecules will be allowed through the pores depending on the pore size and other molecules will be kept out because of this filtration process. But additionally you can actually use this as an active medium like we saw in the case of the MOFI for storage of hydrogen. And even if you are and we definitely saw another effect that suppose I put in additional particles like the case of the gold nano particles in the channels of mesoporous silica then I can have important effects like non-linear optical response. So ordered self-assembled mesoporous materials which includes now I am talking about nanoporous materials additionally definitely are finding very important applications and there are important class of materials in their own right. Additionally we can have hierarchically structured nano materials. In an early example we had seen that how when you use when you see the structure of lotus leaf it is hierarchically ordered and this hierarchical construction can actually give rise to the important property which is super hydrophobicity. We had also seen certain kind of hierarchical constructions like in the abnon shell which gives it extreme impact toughness. So hierarchical structures are abound in nature and these hierarchical structures we had even schematically drawn earlier that how we can actually make a hierarchical structure. Now in the present case and we had noted that this hierarchical ordered structure has ordering in more than one length scale. And the subunit which is formed at one level can actually be used to construct a higher level structure which itself forms a subunit for even a higher level structure. Now similarly similar concepts can be applied for the case of porous materials where in spheres of mesoporous material can be ordered with voids between the spheres to produce a material with pores in two length scales. That means let me now think of this as pores at two length scales. Let us see one of these spheres which is drawn below here. So in this sphere you can clearly see that there are ordered pores and these pores are small pores. Now I can take each one of these particles and arrange them in for instance this example shown here in the form of an hexagonal unit. And as we know in this hexagonal unit actually your unit cell will be this rhombus. Now when I do this two order assembly first level one being the particle itself and level two being this assembly into this hexagonal kind of an array. Then I see that there are pores here. So I can call this pore one which is at the smaller length scale. But additionally you also form these larger voids between the particles. Therefore you can clearly see this is an example of a case where there are two length scales of pores. One length scale of the order of tens of nanometers which could be the smaller pore which is shown here. And between the pores which could be hundreds of nanometers. Now therefore and such structures have formed examples of this would be the mesoporous silica spheres. And this concept of hierarchical organization can actually apply to other structures as well. Just to summarize here we have already seen that how we can use hierarchical construction to make materials. Where in you take a sub unit this sub unit forms a part of a larger unit. And that unit itself forms a part of a becomes a sub unit for a even larger length scale construction. And even we had even considered a very nice schematic example at that stage. Where in we can think of for instance suppose I have an ordering at this length scale which is shown by say for instance a sine wave. Then I can have a smaller length scale associated with which is overlaid on top of this. And finally you could also visualize even smaller length scale on top of this. So, let me draw that with red color. So, you could have for instance this smaller length scale on top of this. So, these are for instance can be thought of as roughness as at one length scale the second length scale. And of course they could be higher and higher length scales. Therefore, this can be thought of as hierarchical construction. And in this case of course this could be for example be a example of surface roughness. Similarly, we said that there can be hierarchical construction of materials not only of surfaces, but also materials. And finally here we see that there can be hierarchical construction of pores in materials. Where in I can incorporate more than one length scale of pores of course finally intended for some kind of an application which would make this construction useful. So, in this particular example we saw here we are actually using silica nanospheres which have pores. And these pores decide within silica nanospheres which themselves are ordered to give pores at two length scales the smaller pores and the larger pores. Therefore, this is a nice example where we have we can not only have we have an example of hierarchically structured porous materials. The next class of materials we take up briefly here and we will describe some important properties of these materials later is the example of what you might call the core shell nanostructures. A typical core shell nanostructure consists of a material which is inside this which I call the core. And around this I have a layer which I call the shell. One typical example would be where this for instance could be a cobalt particle. And this could be a cobalt oxide. That means it is a natural oxide which forms on top of the core which you expect if you leave a cobalt particle outside in air. It is not necessary of course that this shell need be very thin. You can actually visualize cases where you can have thin cores and large shells around these cores is also possible. And it is also not necessary that these cores need to be spherical. You can actually visualize cylindrical cores also. Another shapes of course around which you can have actually a shell which would typically follow the contour of the particle shape. In core shell structures the core of one material is covered with a shell of another material and this can be thought of as a hybrid. Therefore, this is now like a composite of the nanoscale where in or a hybrid of the nanoscale where in there are two materials involved. The shell of course may be intentionally designed to impart special properties like exchange anisotropy. This exchange anisotropy is an important magnetic property which we will deal in detail when we talk about magnetic properties of nanoparticles and nano crystals. But essentially we want to say that it has intensely been designed. It can enhance a preexisting property. For instance it need not impart special new properties to the particle but may actually perform the role of enhancing a preexisting property. It may be introduced to protect the core from the environment like suppose you are talking about super paramagnetic iron particles which are used for drug delivery. It is typically coated with the polymer and then introduced in the blood stream. So, that it actually has bio compatibility with the blood. So, iron particles if bar iron particles are introduced they will be reactive and they could be actually not have be bio compatible and not only that the system actually remove these particles before the drug delivery is done. And therefore, this shell actually performs an important role in the intended application which is targeted drug delivery. In which case it protects it from the environment which happens to be the blood of a person if you are talking about targeted drug delivery. This could actually arise from an unintentional process like it could be a natural consequence of oxidation or reaction. As we saw in the case of cobalt oxide over cobalt if you leave the cobalt particle outside but nevertheless once this combination of the shell over the core has been achieved. Typically it is intended for a certain purpose though it is unintentionally appearing but typically we expect that certain additional beneficial properties are coming from this core shell structure which makes it useful to actually synthesize this core shell kind of a structure. Even in the case of thiol nano gold nano particles we had seen that we actually coated with thiol particles. So, that we passivate the gold surface. So, that we avoid sintering at room temperature even and the thiol particle can be thought of as a shell around the gold core. But we will see more examples of these kind of core shell particles wherein intentionally the shell has been designed to as we saw just now to impart certain special properties. The shape of the nano particle is usually spherical but can be irregular rod like wire like or any other shape which is important in a certain application. Silver an example of an alternate geometry rather than the sphere is the silver nano wires have been coated with amorphous silica and these nano wires have are more than 1000 nano meters in length and have a diameter of about 50 nano meters. So, that means there are nano wires with an aspect ratio of about 20 and these have a shell of amorphous silica over silver which is a as you know conducting material. Typically the convention or the notation used to describe these core shell particles is written as core at shell. Though I might actually prefer the opposite notation which is shell at core because shell actually resides on the core. But typically in literature you note the notation that people use the notation that it is core at shell. That means the shell is the one which is the one which is outside which is shown in the notation here. Just to summarize this slide we have seen that there are hierarchical constructions possible there are hybrids possible. And there is one special class of hybrids which is called the core shell nano structures when in a core of one material is covered with a shell of another material. When you say other material it should be noted that it could actually be just the oxide of the same material itself. And we have noted that there are four reasons why we actually have a shell around the core. One could for instance we saw the least of the trivial is that it is a natural consequence of oxidation or reaction when you leave the nano particles outside. But more in more usually it is intentionally designed to import special properties like for instance we said it could have actually something like an exchange anisotropy in magnetic particles. You could actually enhance a preexisting property and we will take up examples of fluorescence etcetera. When we want to actually enhance a preexisting property that means the shell itself core itself has this property. But by putting a shell around it this property is enhanced. And we more often than not actually introduce a shell to protect the core from the environment. And one nice example of this we saw was when we are using the nano particle like an ion super paramagnetic ion which is used for targeted drug delivery. And in this case what happens why we need super paramagnetic ion is because the magnetic field is actually used for transporting the particles to near the region where the affected region or the region which is been damaged. And we want to transport the drug to that region and therefore we are using super paramagnetic ion. And we will of course later on see what is super paramagnetic ion and why how this super paramagnetic property come in ion. This we will see later. But nevertheless we want to passivate this molecule and typically a polymer is used to protect this ion nano particles from the blood stream. So the core usually shows the relevant property while the shell may actually stabilize the core as we had seen before make the core compatible with the environment or change the charge reactivity or behavior of the core surface. So there are important functions of the shell performs and that is why this core shell composite actually gives some beneficial effects. And the synthesis of typically of these core shell nano structures is done either by formation of the core for followed by the shell means you first synthesize the core then you have a shell synthesis around it or you can actually use an in situ method where in the core and the shell actually are manufactured or form together. Some common methods of synthesis we are listing here without actually going into the detail. The shell can be formed by a surface chemical reaction by simple absorption of molecules or small nano particles or the whole core shell nano particle. They can also be formed by self assembly and cross linking of macro molecules that means there is a method we can use more than one method actually to put a core around the shell around the core. And this could be a simple reaction adsorption of molecules or they can actually be some kind of a cross linking or assembly process wherein you put a shell around the core. Here there are listed below some examples of shells around cores and typically the core is can be metallic a semi conducting core. It can be metal oxide it can be a cross linked polymer or even an in organic core and the shell could consist of for instance could be a different metal on one metal. It could be a semi conductor on a metal or it could be an insulator on a metal. Similarly, on a semi conducting core you could actually put a metal semi conductor or an insulator. And on a metal or a metal oxide you could actually put an organic which is could be a polymer or a series of molecules around it. And if you have a polymeric core then you could actually put which is typically done is you is typically put polymeric molecules around polymeric cores. Typically there are fewer examples of metallic or semi conducting shells around polymeric cores. And you can actually put bio molecules around like for instance DNA around in organic cores. So, there are wide variety of choices when it comes to the core and shell that means I have now a process by which not only I am modifying as my particle and the modification either as we saw can lead to stability or actually make the core more compatible with the environmental or actually modify something on the surface of the core such that I can now tailor my properties of the core. And additionally we saw could actually introduce newer properties which did not exist in the core. And now since I have a gamete of possibilities of not only the core kind of core particle that I can choose, but also the shell particle. Therefore, there are lot of interesting properties and explorations which are going on where in the explore a wide variety of cores and shells to impart very special properties. We will take up some examples of these core shell and the properties which are rising them when we especially talk about for instance optical properties and magnetic properties. Next we take up the important topic of curvature effects in nano materials. Now when you are talking about a bulk material we typically assume that a bulk material is infinite. In other words I can schematically draw a bulk material like having no boundaries and this is infinite in all directions. And typically when I use for instance a property like Gibbs free energy G then I am talking about a Gibbs free energy of an infinite material. Where in there is no geometry there is no description of any kind of a shape in the property. But it is obvious then when I am talking about nanoparticles and nanomaterials then there is a certain geometry like just now we talked about spherical nanoparticles. Now we can have a spherical nanoparticle of small size we can have a spherical nanoparticle of a larger size. It is obvious that the properties which are related to the surface are going to be different for these two kind of nanoparticles. For instance this could be a fine nanometer particle and this could be for instance a much larger particle like a 15 nanometer particle. Unlike a bulk material which we described with a parameter like G whenever I am talking about a real material in which case I have to deal with the concept of a surface. That means I have something which is the termination for the bulk. And typically if you want to consider what you might call a semi infinite space then you can have a flat interface between the material and the air or vacuum. Unlike this kind of a flat interface it is clear that these particles actually have a curved interface. And now suppose I am talking about a cylindrical particle then you can clearly see that you have one phase which is curved cylindrical side while the other surface is flat. So it is in some sense a combination of a semi infinite bulk material and a spherical nanoparticle. So in these kind of materials it is obvious that the surface effects are going to come into play. And therefore I cannot use a value like G which is typically meant to say that it is G bulk. Important effects and properties come into play when I am actually using a nanomaterial nanoparticle because of the surface or interface effects. Therefore in bulk materials curvature effects can be ignored and the Gibbs free energy has no dependence on any geometrical aspect of the material. So we are talking about a bulk material I really do not consider any geometry or a surface or an interface and I just assume that there is a single parameter G which describes my Gibbs free energy. In the case of nanomaterials and typically I am here talking about nanoparticles or nanocrystalline materials where the grain size in the scale of nano then we are actually dealing with certain interfaces like it could be a grain boundary or a free surface. Then for example now I will talk about a particular example which will be used to exemplify the importance of curvature in nanomaterials is the case of nanocrystalline precipitates or in a matrix. Actually I am talking about fine nanocrystalline precipitates in a matrix and so this requires a little correction. So this is fine nanocrystalline precipitates in a matrix. The free energy of the precipitate is now not a single number but is dependent on the size of the precipitate. And now for instance I am assuming that the precipitate is spherical though actually we will see in real systems the precipitate could be disk shape it could be an oblate spheroid it could even be a sprolate spheroid. In other words suppose I need not the precipitate need not be of this kind of a shape it can be a squashed it can be a long plurate spheroid that means it is like a spheroid like this it has a circular cross section but it is elongated which is sometimes is called a needle shape it could typically be in the form of even of a disk or it could be in the same form of a plate. So there are various geometries of these precipitates possible but for now I will assume that the precipitate is spherical. And this free energy of this precipitate is now going to be curvature dependent in other words if I am talking about a spherical particle I am assuming that it is size dependent therefore gives free energy is not a constant number now anymore and it is dependent on the size. And smaller the size of the spherical precipitate higher is going to be my curvature and as we shall see soon that therefore the energy of the system is going to be higher. Now this is what we consider for precipitate is also going to be true for free standing nano crystals. And the reason is not difficult to understand why this free energy of this nano particles has to or nano crystals has to be higher than that of the bulk material this is because the atoms on the surface are less bonded as compared to the bulk atoms this leads to an increase in the energy of the system. Now of course when I am talking about an increased energy or lowering of energy we have already noted that before that the reference state for the surface is the crystalline state and not the isolated atoms state. So when I am saying it is higher energy I am assuming that I am comparing my free energy of this particle with the free energy of a bulk crystalline solid and not with the unbonded state wherein the atoms are far apart. Now why should curvature actually determine your bondedness we will have draw a schematic to understand this. So let us take a flat surface like this vis-a-vis a highly curved surface like for instance the particle drawn here. So in a flat surface for instance the atoms sitting on the surface might have neighbors like this. So each one of them is going to be bonded say 1, 2, 3, 4. So there are 4 bonded atoms around this. Now suppose I consider a highly curved surface in which case I have a particle like this then I can see that an atom on the surface may actually be bonded only to 3 atoms. So this is of course a crude schematic which I am drawing for you to for illustrate the point but if I had a flat surface which is like more like a low curvature system then I would typically find my bonding is better I need to go this is 4 bonds you can see here and while in this case there are 3 bonds. That means that there is one bond lower here and that means that this crystal in is in a higher state of energy with respect to the system which is having a lower curvature or a high radius of curvature. So this can be thought of a system with high radius of curvature and this is a small particle and high curvature automatically implies that it is got a small radius of curvature. Now having this kind of an effect that means an curvature effect has important consequences on the properties of materials and one of these properties we will take up in a little more detail now which is the phenomena of coarsening for instance of precipitates. Now this there are 2 important aspects when you are talking about coarsening one is the reduction in the overall energy which is a global effect and additionally we are also bothered about some very local effect which is coming from the curvature and both these effects we will try to understand and how the Gibbs free energy is going to determine what you might call the solubility of a solute and the equilibrium concentration around the particle. So just to summarize this slide curvature plays a very important role in nano materials and nano crystals and nano particles unlike the case of the bulk materials where I can describe the system by a quantity which is the Gibbs free energy which is not geometry or size dependent. When you come to smaller and smaller particles because of this effect of actually unsatisfied bonds or there is an excess energy up and above the bulk energy and therefore these particles are in some sense more unstable as compared to the bulk material and they are and we have already seen the consequence of that they are very reactive and we will take up one specific example of these kind of curvature effects in a phenomena known as coarsening of precipitates. Now when does coarsening take place of course the first thing and as an example now I will take the aluminum copper system or the aluminum 4 percent copper system which is shown here and this is a typical system known for what is called an age hardening system. And in this age hardening system we follow four steps to actually give rise to a fine dispersion of precipitates and these fine dispersion of precipitates could be actually in the nano scale. So the four steps we follow are and this is actually not the first step we first solutionize at high temperatures is for solutionize at high temperatures followed by quenching from high temperatures then we age then we cool to room temperature. Now when we actually take this alloy which already consists of stable precipitates and stable precipitates in the aluminum copper system actually happens to be the C u Al 2 Al 2 C u precipitates. Now we take it to high temperature dissolve the precipitates and we take it to a region in the phase diagram where you have a solid solution that means uniform solid solution then we quench from high temperature to room temperature to actually obtain a super saturated solid solution that means now all the copper is in the solid solution form in the aluminum matrix then we actually heat it again to a higher temperature which is about 180 degree Celsius even actually if the solution is left at room temperature at a over period of long periods of time this precipitates will tend to form and therefore, prestation does take place even at room temperature. But to accelerate the process we can actually heat it to high temperatures about 180 degrees and therefore, we will actually have a prestation process the choice of 180 degrees we will explain very soon why that is required and finally, we cool to room temperature to obtain a fine dispersion of precipitates. As we shall see later this fine dispersion of precipitates actually required to strengthen this material if you take an aluminum copper solid solution it is actually already stronger than the pure aluminum because of solid solution strengthening. But this strengthening alone is not enough because aluminum is a beautiful material because of its corrosion resistance it is light weight etcetera. And therefore, we would want to make lot of structural applications with aluminum, but because of its lack of strength being an FCC crystal or what you might call a cubic close pack crystal it is not very strong and we want to strengthen it and this age hardening process can actually increases strength from about 100 MPa to about 400 or 500 MPa. And therefore, we want to have a fine dispersion of precipitates which will now impede my dislocation motion which will give me my strengthening effect. So, the important thing in the whole process is to get a fine dispersion of precipitates. So, in the initial stages as we shall see that whatever was a super saturated solid solution will actually precipitate. And this prestation process is not a state forward process that actually the equilibrium precipitate theta which is a tetragonal phase is not directly obtained from the cause of the reason that if a theta precipitate comes out then the activation barrier for such a process is higher. And therefore, it a series of meta stable precipitates are obtained and therefore, these series involves actually the prestation of GP zones first followed by theta double prime precipitate which is a meta stable phase the theta prime phase which is another meta stable phase. And finally, the theta phase which is a stable phase now which is actually found in the equilibrium phase diagram. Now, suppose I took a I did a slow cooling and did not use a process which I have described here then what happens is that you will actually obtain very coarse theta precipitates which will not be effective in the impeding the motion of dislocation which means the strength obtained will be lower. But by now by actually doing this process and getting a fine dispersion of precipitates we can actually obtain a higher hardness. In the initial stages the precipitate size is in the nanoscale for instance GP zones are a couple of atomic layers thick and aging can be controlled to get COL to precipitates in the size of about 100 nanometers. That means these precipitates are already in the nanoscale and they are can be called if this whole system can be called in some sense an nano crystalline material. But remembering that it is not the grain size we are talking about which is in the nanoscale it is actually the precipitate size which is in the nanoscale. Now, the region we actually heat to get this uniform solid solution is suppose the aluminum 4 percent copper alloy somewhere here let me draw some schematically. So, this is now my aluminum copper alloy I heat it to this alpha region which will give me an solid solution then I quench it to room temperature as step 2. And then after quenching I actually age it at some temperature which is in this region this is my. And the reason for aging at low temperatures is now if I age at low temperatures the left hand side is my phase diagram and in the right hand side I have got my TTT diagram which is the time temperature transformation diagram. And I have separate C curves for the transformation of this phase from the alpha phase which is on the left hand side of this diagram to the various phases. So, this whole left region now let me shade this left region this is now my alpha phase field in this TTT diagram. And on the right hand side are various other phases and now suppose I age at low temperatures now this temperature I am aging is this temperature. Then I can see that I can actually go through a sequence of precipitates which is now first the G P zone then you got the theta double prime precipitate the theta prime precipitate and if I wait long enough I will actually get my equilibrium theta precipitate. But typically I do not wait that long and I actually try to control my precipitation process. So, that I can have some of these coherent precipitates which are fine scale are distributed in my matrix. Now using this kind of a process I am actually trying to make a fine scale of precipitates. But what happens if I wait long enough what happens initially of course we have seen that you have a precipitation. That means that you are going to obtain a nucleus of a phase and this nucleus will grow and I will show that schematically on the board. So, in the initial prestation you have a uniform solid solution here and initially at some point T there will be some nucleus which will form and typically the size of this nuclei itself is in the nano scale. Then at a later time you will notice that these would have grown to some size these precipitate nuclei and at the same time new nuclei would have appeared at various other places. If I wait further longer then I would notice that these would have grown to even larger size while these nuclei which appeared later would have grown a little longer and additionally there will be new nuclei which would have appeared at a time which is T 3. This process will continue till all the solute has come out in the form of the precipitate. But will the system freeze at that stage no after all the prestates have come out. Now let me draw a schematic for that stage when all almost all the solute is already in the form of precipitates that means now my system has come down to an equilibrium state microstructure level equilibrium where in now I have got precipitates in the system and of course all these prestates need not be of same size some could be smaller some could be larger and some could be of medium size. Now at this stage the concentrations are such that the matrix is an approximately given by the phase diagram and there are certain distribution of phases. The system does not freeze here and the reason the system does not freeze here goes back to two reasons one is that because there is an interface between now my matrix and the precipitate and this is now this interface is going to be and high energy region as we have just noted. So instead of having so many prestates of these sizes suppose I put larger and larger prestates that means my interface area to volume ratio would actually decrease and that means that by the system is will go downhill in energy. In other words if I do not put small many precipitates I put large precipitates few number then the system will actually go downhill in energy and this implies that the system wants to go from a high energy state which is the state with many small particles or many small precipitates to a case where there are a few number of large precipitates this is of course the process known as coarsening. Now the question arises why would such a process or how would such a process take place it obviously implies that if all the precipitates of course increase in size then the there is no mass conservation this implies that some of the smaller precipitates will have to vanish decrease in size and vanish and the larger ones will have to grow. But these particles are not actually touching each other so mass transport from one particle to another say the smaller particle to this larger particle will actually have to be mediated by the matrix why would such a mass transport actually take place is what is important to understand. In other words the global criteria is decrease in Gibbs free energy or increase in energy and therefore, we want to put fewer number of larger particles which will actually lead to a reduction in energy that is obvious. But there has to be a local reason and this local reason is what is given by curvature effects and that is what we will try to understand how curvature effects in these especially in these nano scale precipitates will actually give rise to effects of coarsening which is what we are trying to consider now. During the aging process many things are actually parallely happening and it is important to know what are those things which happen before we understand the phenomenon of coarsening. The precipitate type itself changes that mean there is not a single type of precipitate which exists as we have seen that initially you have the g p zones which goes to theta double prime which finally goes to theta prime to theta which is the equilibrium phase. The size of these precipitates will increase with time suppose I have the theta prime which will grow with time and then of course, finally it as we saw it will transform to the theta prime will transform theta double prime to theta when you actually hit the c curve in the time transformation diagram. An additional thing which actually happens parallely is there initially the interface between the precipitate like for instance the theta double prime and the matrix tends to be coherent and with time it becomes semi coherent and finally, certain in certain phase sets may become incoherent another remain coherent and therefore, the interface tendency is to go from coherency to semi coherency to incoherency as with time. The sequence of precipitates which we have seen is that initially you have uniform solid solution on aging you get a theta double prime phase which is a distorted FCC phase and typically this is in the form of a disc which is about or a plate which is about 100, 10 nanometer thick and about 100 nanometer in diameter. So, these are actually nanoscale precipitates and these precipitates form with a certain orientation relationship with the matrix and in this case of the alpha theta double prime it happens to be the 001 plane in the theta double prime is parallel to the 001 plane of the matrix alpha and the 100 direction in theta double prime phase is parallel to the 100 direction in the alpha matrix. If you wait further long enough while aging then you will actually get the theta prime phase which is a tetragonal phase with a composition approximately equal to C u a l 2 and even this theta prime phase has a certain orientation relationship with respect to the matrix. And when this theta double prime phase forms certain interface are coherent while certain others are incoherent semi coherent and the tendency is to go from there to incoherency as a prostate size increases. Finally, of course you form the body centered tetragonal phase which is theta which is the equilibrium phase which is found in the phase diagram. Now, we said that we need to understand the local reason why coarsening takes place. To understand this let us assume that there is an uniform distribution of precipitates with an average concentration of solute and the solute in this case happening to be copper in the example of aluminum copper system which is present in the matrix. In the setting of this average composition let me consider two precipitate particles or precipitate crystals one a smaller one with radius r 2 and a larger one with a radius r 1. And want to understand that how such a small precipitate which is the beta phase can actually dissolve and the larger precipitate which is r which has a radius r 1 can grow to a larger size. And if this I understand this mechanism I can hypothesize how as in a assembly of various sizes how smaller precipitate will tend to vanish how larger precipitates will tend to increase in size. And therefore, overall reducing my interface area and therefore, reducing my energy of the system. Now, the composition of the solute which is in equilibrium with the precipitate is now not of independent number, but actually is a function of the curvature. As the curvature increases the solute concentration in the matrix adjacent to the particle actually increases. That means, if I have a smaller particle there and the concentration of the solute is x 2 this is higher around the smaller particle as compared to a particle x 1 where in the concentration is lower. This sets up concentration gradients in the matrix and that implies that solute will diffuse from near the small particles towards the large particles. So, I have a higher concentration of solute around this smaller particle I have a lower concentration of solute around this larger particle. And this implies that I am going to have a diffusion of solute from the smaller precipitate to the larger precipitate. Now, if the and of course, we are not at explained that how I am getting at this x 2 value and the x 1 value which we will see in the next slide, but we are trying to understand the overview first. Now, if x 2 is depleted by diffusion because now the solute has left this part of the material and has actually gone to this part the copper which implies that now the value around the beat of this r 2 particle is actually smaller. That means, the particle is now going to dissolve such that the equilibrium concentration of solute is obtained. That means, some of the copper will actually come out from the r 2 particle which implies that the r 2 particle will tend to shrink. Now, when the solute reaches the particle and reaches x 1 close to the r 1 particle, then this matrix around it is richer in the solute and this first state actually can take that solute and actually grow in size. Therefore, the larger particle will tend to grow and on an average you will notice that smaller particle shrink and larger particles grow. This phenomena is known as coarsening and therefore, with increasing time at for now we will hold at constant temperature the total number of particles decrease and the mean radius increases with time. So, this is what we observe in the case of coarsening. So, the key link now which we need to understand why is that x 2 is larger than x 1. To understand that let us now plot a Gibbs free energy composition plot and now this Gibbs free energy composition plot is a special kind of a plot where in now the Gibbs free energy for the precipitate is shown here the beta phase on the right hand side which is shown by these three curves while there is only one curve for the alpha phase which is shown here. In other words these tell you now that the Gibbs free energy of the precipitate is not a constant number or a not a single curve, but is actually dependent on the curvature of the particle. And now suppose I assume a flat interface which means r is equal to infinity then I have a lower Gibbs free energy. As I make the particle curvature increase the particle curvature or in other words decrease the radius of curvature then the Gibbs free energy actually tends to increase. So, this goes up from the particle which is flat and finally, if you make the particle size even smaller then you notice that the Gibbs free energy further increases. And obviously, the Gibbs free energy is a function of the concentration of the solute. So, you get these Gibbs free energy composition curves. Now, the Gibbs free energy of the entire system can actually or can be found by what is known as a common tangent construction. Therefore, now I can draw common tangents with respect to suppose I had a large precipitate like the beta phase then I can draw a common tangent which is shown by this line here between this and this. And you would notice that the concentration of the solute which is x 1 for the radius r 1 which is an equilibrium with the precipitate of this size r 1 is x 1. That means, if you have a r 1 size precipitate by the common tangent construction I would notice that the common tangent intersects my alpha curve here. And that implies the concentration of solute around this precipitate is x 1. Now, suppose I have a smaller particle then you would notice that it is a smaller particle with radius r 2. Now, if I make a common tangent construction for this smaller particle then I would notice that the solute in equilibrium with this precipitate is x intersects at a point x 2. And you can clearly see that x 2 is larger than x 1. This is what we started off by saying in the previous case that there is a larger concentration of solute around the smaller precipitate particle as compared to a larger precipitate particle. Therefore, now I can clearly understand why there is an enrichment of solute around a smaller particle and this effect is actually called the Gibbs Thompson effect which is a result of the curvature effect on the Gibbs free energy. Just to summarize this slide we can clearly see that now we do not have a single curve representing the Gibbs free energy of a precipitate. There are multiple curves and these curves depends on the curvature of the particle or in other words the radius of curvature of the particle. As the curvature increases the radius of curvature decreases then the Gibbs free energy increases which implies that you have a common tangent construction with the alpha phase which is now going to give me the Gibbs free energy of the system. Then I see I can notice that the solute which is in equilibrium with the larger particle has a lower concentration around the larger particle as compared to the smaller particle which is in has a higher concentration in equilibrium around it. And this implies automatically that there is going to be a concentration gradient which is going to be set up and therefore, there will be diffusion of solute from the smaller particle to the larger particle which finally, of course results in the smaller particle reducing in size the larger particle increasing in size finally, as a global picture you would notice that the average particle size increases with time.