 After having a broad overview on phase transformation and the important concept of a critical radius for phase transformations, let us look at melting of nano crystals. We had noted that we have a well defined melting point T m at which the entire lattice breaks and therefore, you have a molten state and this is unlike the freezing which we had talked about. On the other hand when we talked about the melting point, it was the melting point of what is known as a bulk crystal or a large crystal. The melting point of a material is an indicator of the bronze strength of the material, though the boiling point is a better reference as it is structure independent. In the solid state for instance, you have various kind of phases possible like the BCC crystal or the FCC crystal and this would affect the melting point. But boiling point is from the liquid state to the gaseous state and that is a better indicator of the bond strength, but nevertheless melting point also gives you a some kind of an indication of what is the bond strength in a material. In bulk system, surface to volume ratio is small and curvature effects can be ignored. That essentially implies we can talk about a single number known as the bulk melting point. In nano crystals, the number of surface atoms is considerable fraction of the total number of atoms and these atoms as we know have got a higher energy and higher degree of freedom, which means they have freedom of more vibration as compared to the atoms in the bulk. This implies as a first intuitive guess that surface atoms are expected to melt below the melting point of the bulk. So, now this is how much is it true, how much it does stand to experimental results will come scrutiny, we will see in a moment. But as a first intuitive guess, we would expect that surface atoms because of the higher energy they have got and the higher freedom they enjoy, they would tend to melt before the bulk atoms. And we have already noted that in 0 d and 1 d nano crystals, curvature effects are important and cannot be ignored. Now, as the size of a free standing nano crystal is reduced below about say for instance about 100 nanometer and I am talking about single crystals, which are free standing. It is noticed that its melting point is reduced with respect to the bulk melting point. In fact, experiments have shown that gold crystals about 10 nanometers in size melt about 100 degrees below the melting point of the bulk. So, there is a considerable depression in the melting point when you go to nano scale and for that we really have to go really small sizes like about 10 nanometers. And the bulk melting point of gold happens to be 1064 degrees Celsius. Now, there is a continuum formula which exists and we will though how correct is this formula or how accurate it is, we will ignore for now and we will consider a formula in which the melting point of the surface is given with respect to the melting point of the bulk material. The formula goes as T surface melting is T m into 1 minus 2 gamma solid liquid by delta h f into r, where T m is as we noted is the bulk melting point. The surface melting point that means as if we are assuming now that the surface atoms are melting first, the gamma solid liquid is the solid liquid interfacial energy and delta h f is the enthalpy of fusion of the bulk material and r is the radius of the particle. The important take home message from this formula is that if I reduce my size that means if r decreases then the surface T m surface would decrease. That means that there because this is 1 minus this fraction and therefore, you would notice that if I decrease my radius of the particle which means also that curvature is going to increase. Then I would note that the T m surface would actually decrease that means the surface atoms would show a depression in the melting point with respect to the bulk melting point. Now, if I now track my melting point with vis-a-vis the radius of the particle then of course, you have the bulk melting point and this is some kind of a schematic drawn schematic trend line which is drawn for gold nanoparticles. Then we notice that there is a severe depression in the melting point as you reduce the size and this starts to play a considerable role when the size of the particle is below about 15 nanometers and we already noted that when the size is 10 nanometers there is almost 100 degrees depression in the melting point. Now, this implies there are two points we have talked about and these two points need to be resolved and the first point we mentioned is that surface would tend to melt below the melting point of the bulk this is point number one and we have a few more things to say regarding that how true is it or how true it is not. Second thing we are noticing an experimental result wherein you are noticing that actually the melting point of a nano crystal and now I am not differentiating the surface from the interior. I am assuming that the entire nano crystal is melting and this we have seen there is a depression in the melting point of the nano crystal and this trend line which is seen for gold can also be observed for other materials like lead, copper, bismuth, silicon and the trend lines are very similar. In other words if you go below about 15 nanometer size there is a depression in the melting point. Now, the question obvious question which arises is that is the surface melting first and therefore, the next layer becomes a surface then the next layer becomes a surface or the next layer becomes then therefore, there is a continuous melting from outward to inward there are two possibilities. So, let me write down the scenario of melting. So, there is one possibility that the surface actually melts first. So, this layer gets molten which means now the effective surface is moved inward and this is the new surface and which is even smaller in size and since the temperature is already above the melting point of the surface this crystal it is definitely this is going to be above the melting point of this crystal and therefore, my melting would progress inward. This is scenario one the other possibility is that the entire crystal melts at a single temperature and now I call it of course, the T m of the nano crystal which is now obviously, a function of the radius R. Now, which of these two is true is a important question is it that the melting actually happens in the surface and progresses inward or does the whole nanoparticle melt at a single temperature and let us answer this question that where does melting start first a simple minded guess would be that the melting starts the surface of the nano crystal as the formula I pointed out in the previous slide. But in actual experiments and the one which I would like to cite now is Bevang et al which is goes back to 1999 which they did on gold nano rods heated by laser beam they found that they could not differentiate the melting of the surface from the interior and both the inside and of the nano crystal and the surface seem to actually melt as a whole though more work has to be done on various systems and more careful experimentation has to reveal really does actually the surface melt before the interior. But as it stands today the understanding seems to be that it is not a scenario one which is operative but a scenario two wherein actually the whole nano crystal melts at a single temperature which is what you expect for the bulk crystal in a bulk crystal you define a melting point which is not surface interior etcetera dependent. So, you expect that it is melting at a single temperature though as I pointed out more work has to be actually done to understand this phenomena. Now it is not always necessary that this nano crystal need be a free standing nano crystal often this nano crystal it could be embedded in a matrix like for instance to the way to produce lead nano crystal is to actually melt spin melt spinning as a technique in which you actually take you melt your lead and aluminum together and spin it in the form of a ribbon and often you will get lead nano particles which are embedded in a matrix and a schematic of such a system is shown here wherein for instance this lead is the this lead could be this phase which is the black phase or and this could be the matrix which could be aluminum. So, another in the system the nano particles or nano crystals are not free standing but they are embedded in a matrix in such a scenario there is an extension of the concept we just know saw about melting that it is now not purely we cannot say that there is going to be a depression or an elevation in the melting point both of which are possible we will have to take it system by system. So, if the crystal is embedded in a matrix its melting point may decrease or increase with respect to the bulk depending on the relative magnitudes of the particle matrix that means a crystal matrix and liquid matrix interfacial energies. So, now if I know the relative magnitudes of these two the crystal matrix and the liquid matrix interfacial energies I can predict will I have a depression in the melting point or will there be something known as super heating which is means an elevation in the melting point. Now, if gamma particle matrix is greater than the gamma liquid matrix in other words the gamma liquid matrix is the lower of the two interfacial energies then formation of the interfacial liquid lowers the energy it is obvious. Therefore, now suppose originally there was an unmolten state in which case I had this particle in a matrix and now the interfacial relevant interfacial energy is for the solid with the liquid which is gamma solid or particle matrix. Now, if instead of this this thing melts this is region of space melts then I will have to replace it with this quantity which is now my this is a molten state in which case I can replace this by gamma liquid matrix and since gamma liquid matrix is lower in energy it would rather melt at a lower. So, overall reduce the energy of the system and that means there will be a depression in the melting point the opposite would be true if the gamma particle matrix is lower than the gamma liquid matrix in such a case you would observe an elevation in melting point which is called super heating and embedded crystal will meet above its bulk melting point and this is observed in example of aluminum indium alloy matrix. So, suppose you had aluminum indium particles in an aluminum alloy matrix then you would actually observe an elevation in melting point which is called super heating. So, just to summarize the slides related to melting we observe that if you have free standing nano crystals then there is a depression in the melting point which could be a large fraction of the overall melting point. For instance this is 100 degrees depression would imply that there is about 10 percent reduction in melting point if you reduce the crystal size to about 10 nanometer. Second thing we notice is that as the understanding stands today that the entire nano particle melts at a single temperature rather than the surface inward melting which could also be a possibility which needs to be explored. Now, and the third thing we notice was that if I have this nano crystal not as a free nano crystal, but as embedded in a matrix then there could be an elevation in the melting point or they could be a depression in the melting point that depends on the relative interfacial energies between the solid and the liquid or the crystal and the matrix. Now, let us take up the example of solid to solid phase transformation in nano crystals and we already notice that in when you are talking about for instance liquid to solid or a solid to liquid phase transformation then I can ignore something known as the strain energy, but when you talk about solid to solid phase transformation still the concept of an r star is valid, but in the calculation of r star I need to invoke the strain energy term as well that is what we had noted before. Phase transformations in nano crystals is expected to be different from bulk materials. So, this is very obvious and we will see why it is obvious very soon. Heterogen is nucleation at the surface is expected to play a dominant role due to its proximity. So, far we said that suppose I am doing a melt solidification experiment and suppose I do this experiment for instance in a container less method that means I have no container there is no wall then I can reasonably expect that homogeneous nucleation is going to take place, but in a solid to solid phase transformation it is heterogeneous nucleation which is going to be a dominant form of nucleation, because you cannot avoid defects in a material like there are going to be green boundaries there are going to be stacking faults there are going to be dislocations and even enriched regions of vacancies have known to be plain have known to play a role in heterogeneous nucleation. So, heterogeneous nucleation and in the case of nano particle there is one defect which is very close to the bulk of the material which is the surface and therefore, surface is expected to play an important role in heterogeneous nucleation in nano crystals or heterogeneous nucleation leading to a phase transformation in nano crystals. Additionally, we also expect that the activation energies for phase transformation is expected to be different from the bulk which is not an unreasonable expectation. Additionally, apart from the thermodynamic aspects we also would notice that the kinetics of phase transformation is expected to be highly enhanced with respect to the bulk and the reason obvious or we list the three reason important reasons that smaller length scale for diffusion involved and suppose I am talking about the diffusional transformation for now like we have been doing so far the first order diffusional transformation which produced by nucleation which progress by nucleation and growth. Then I would notice that this is a overall particle size is very very small the overall length scale for diffusion for phase transformation to be completed is going to be small it is all the going to be order of nano scale. Then there are going to be other effects like surface diffusion which are going to play a prominent role because now the entire material is dominated by surface. Then the third important point is that there is going to be lesser constraint on the system. We have noted that previously by drawing some schematics that how if I take a region out of a material and it transforms to a different volume then I have to refit that material into the bulk of the material which is going to give rise to strain energy. But suppose the system is very small then the free surfaces can actually relax and therefore the system is less constrained that implies that phase transformation typically involve volume changes and since the amount of constraining material is less the transformed volume stresses caused by the transformation are lower and that means that there is going to be less impediment to the phase transformation. And we had noted earlier that actually the out of the three terms energy is involved in a phase transformation it the interfacial energy and the strain energy term tends to oppose the phase transformation and since now the system is less constrained that means if there is a volume change or a shape change this can easily be accommodated because there is a free surface in proximity and that implies that typically the strain energy term is going to be a lower term in a nano crystal transformation as compared to a bulk transformation. So, let us understand this now suppose I have a bulk crystal and I have a certain region which is transforming and assume that after transformation this volume becomes like this. That means there is a shape change and there is a volume change. Now in the case of the bulk this transformed volume has to be reinserted into this volume and therefore there is going to be lot of strain energy term. In the case of a nano crystal there is no surrounding medium therefore this transformation can proceed with lesser constrained and therefore and additionally we have already noted that suppose there can be surface diffusion dominance there can also be the overall length scale for diffusion for this could be small and therefore if even if it is a diffusional transformation or a diffusion less shear transformation the overall expectation is that the strain energy term is going to be a small term and many a times you may like to ignore the strain energy term in the phase transformation. Now we have already noted many times before that in bulk materials multiple nucleation events are necessary and we all characterize nucleation as nucleation rate that means number of nuclei forming per unit time per unit volume and that is what leads to phase transformation. So, suppose and the schematic is shown here for that. So, you have a region of material wherein there is certain nuclei which form the black dots and with time these nuclei grow and as these nuclei grow and fresh nuclei formed in different regions and finally the entire volume is transformed. So, you the new transformed volume. So, it is actually should be shaded black. So, maybe it is a good idea to shade the whole region black because now this is the transformed volume which is now the black material. Now in the case of nano crystals the size of the transformation nucleus may become comparable to the volume of material and when we talked about this r star the critical size for nucleation nucleus we noted that it is of the order of 1 to 10 nanometer is of the order of the nano scale and therefore a single nucleation event may lead to a solid to solid phase transformation in a nano crystal which is very very different from the case of a bulk crystal. And the schematic is shown here that you have a nano crystal at the bottom which is shown by A and I assume that the nano crystal is of the order of nanometers then a single nucleation event is as good as transforming the material entire material and therefore the transformed material is shown on the right. There have been some good research work and this is typically been done on materials like CDSC and CDS which are semiconductors and the particle size range of 2.3 to 4.3 nanometers shows that for pressure induced transformation and now this is not that usual heating transformation, but pressure induced transformation. It is seen that for a transformation from the wood side to the rock salt structure wood side happens to be a Z n o type of structure which is an hexagonal kind of a structure which has an A and C parameters as given here. On the other hand the rock salt structure is a familiar N A T L N A C L type structure which is cubic and you can notice here that the wood side structure has a higher volume per motif as compared to the rock salt structure that means if I increase the pressure the tendency would be to actually convert this wood side to rock salt. And this pressure typically is of the order of gigapascals and if you apply 3 gigapascals it is seen that a single nucleation event can lead to the transformation of the whole volume and that implies that this situation is very very different from that of a bulk crystal wherein nucleation alone does not lead to transformation there has to be nucleation followed by growth. Now, of course there is has to be a transition from the nucleation event to single nucleation event to a bulk system wherein there is multiple nucleation events and the transition occurs via what is known as a single nucleation followed by growth regime. So, now I can divide my size regimes into three parts one is this very small regime of the order of 2 nanometers, one a intermediate size regime of the order of 10 nanometers or may be 10 to 15 of course these depends on system to system and in this case of course I am taking the example of CDSE or CDS crystals and therefore, if I change the system these numbers would be different. And the third is the bulk which is now I am when I am talking about bulk it could be more than about 100 nanometers or 500 nanometers which I call a bulk system. In the small nanoscale regime I notice that this 2 nanometer is of the same order of magnitude as the R star my critical nucleation radius. So, this of the order of critical nucleation radius and therefore, a single nucleation event can lead to phase transformation. In this intermediate size particles like about 10 nanometer particles we notice that single nucleation can lead to a phase transformation, but there needs to be a certain growth because the nucleus size as you can see here as indicate by this black is actually smaller than the particle, but then there is not sufficient region around it for a second nucleus to form with that means given a small size and therefore, this nucleus grows leading to a complete phase transformation. And finally, of course you have the bulk wherein which is I told you could be of the order 100 nanometers or more multiple nucleation events followed by growth which is a usual mechanism for bulk crystals. Now, further if you look at closely at the 2 nanometer size crystals which is you can see was a reason for or in the same region as the experimental study which is from 2.3 to 4.3 nanometers you notice that you can subdivide this region into very small sizes which wherein the nucleation event is interface control that means, surface control and you can talk about what you might call a volume control regime which is slightly bigger than those very small volumes. So, let me summarize this very important slide since r star which is a critical nucleation radius is all the order of nanometers usually typically 1 nanometer or 2 say may be 10 nanometers in some systems not assume a system like CDSE or CDS wherein it is all the order of 1 nanometer then a single nucleation event can lead to a phase transformation which is unlike the bulk crystals. Now, the important work on CDSE has been done on this pressure induced transformation and the pressure for transformation is pretty high of the order of gigapascals. We and we are talking about the phase transformation from the wood side structure to the rock salt structure that means, from an hexagonal structure it is going to a cubic structure and that it since its pressure induced transformation we expect that there is going to be a reduction in volume of the unit cell which is what is seen here from the volume of the unit cells and in such a system we see that in small scales a single nucleation event is enough to cause the entire transformation of the volume. Now, nucleation could occur in the bulk or on the surface and the activation energy for the transformation is seen to increase with size regime size construct. So, if you are talking about the experiments conducted in CDSE and CDS and you are talking about these nanoscale particles then the activation energy for the transformation is seen to increase with size. At larger crystallized sizes single nucleation event is followed by growth of transformation to be complete which we noted before. Additionally, we note that smaller crystallites require higher pressure for transformation and this for this particular case we will see more examples and we will actually we will see one more example wherein we will actually note the pressure at which smaller crystals transform as compared to larger crystals both being in the nanoscale. In other example we consider here and here we are taking up some illustrative examples, but much of this can the concepts can actually be applied across many very many systems including metallic systems and other systems. In silicon and now we are talking about Si O 2 coated silicon pressure induced transformation from diamond cubic to primitive hexagonal structure it is seen that crystals as large as 50 nanometers transform by a single nucleation event. So, here we noted that for instance that it is about of the order of 2 nanometers that be a single about 10 nanometers sorry of the order of 10 nanometers that a single nucleation event is responsible, but in other systems this size can as long as be as long as 15 nanometers and this also tells us that the concept is applicable to broader class of systems and in this case we have to especially note that is not a free sanding silicon crystal, but it is some kind of a core shell structure where silicon is coated with Si O 2 and this is a natural can be also thought of as a natural consequence of the oxidation of the silicon crystal. And as expected the shape of the crystals change on phase transformation and this is of course, a homogenous deformation we are talking about and bulk behavior in this case of the silicon is expected to take over after about 100 nanometers. So, it is clear that when you study more than one size system that up to about 15 nanometers seems to be the regime where you can actually have a single nucleation event of course, it could be nucleation followed by growth, but a single nucleation event is responsible for the entire phase transformation. And when you go to for instance about 100 nanometer or more we can assume that it is bulk like behavior with respect to what you might call and then phase transformation by nucleation and growth. So, now we are taking more and more examples we are seeing more and more examples and we had asked and very early in the course we had asked a question that what is bulk. So, now with respect to nucleation and phase transformation we can see that 100 nanometer crystal can be called bulk even though we know it is a nano scale crystal because in terms of the mechanism for phase transformation it is no different from bulk crystal or it is not very different from bulk crystal because now you have multiple nucleation events followed by growth. So, I can call it bulk like even though it is still in the nano scale. Another example of a solid to solid phase transformation which has been studied is the transformation from gamma Fe 2 O 3 to alpha Fe 2 O 3 again this is pressure induced phase transformation and this gamma Fe 2 O 3 is actually a meta stable form at room temperature with respect to the alpha form. This is a cubic form and it is volume of unit cell is about 0.58. The alpha form and which is a smaller unit cell is actually a rhombohedral crystal and it has got A and C parameters as A equal to 0.504 and C is 1.37 is an alumina form of a crystal. So, you have two forms one with a larger unit cell one with a smaller unit cell and this transformation as you can as perhaps you can guess that can be driven by pressure and the important thing to note is that as you decrease the size and now we are really in the very small scale regime 7 nanometer to 5 nanometer to 3 nanometer. You can see that the transformation pressure actually increases that means for smaller and smaller particles actually you need larger and larger pressures for phase transformation to take place. And of course these pressures are employed by using a diamond anvil and so there are inherent complications in actually studying the in situ phase transformation and getting the signal of how the transformation has taken place. So, these are very careful experiments which have to be conducted, but the general trend line seems to be that as you reduce the crystallite size you need to apply more and more pressure for phase transformation to take place. Next we take up a couple of more physical properties and both of these are actually vast areas of research where in lot of work has been done, but we will take a small sample of results to understand that how nanomaterials can actually be different from bulk materials and as usual we take up those examples where the demarcation is very clear cut and often there is a startling or a very different kind of a variation as compared to the bulk. So, that we can understand that at the nano scale there are some very interesting effects which we do not observe in the bulk materials. So, let us one of those physical properties we talk about now next is that thermal conductivity. We know that thermal conduction occurs due to electrons and phonons in normal metallic materials electrons are the dominant form of conduction, but suppose I am talking about a diamond we know that diamond has no free electrons and therefore, very little free electrons at room temperature. Therefore, it is phonons which are responsible for thermal conduction actually diamond happens to a very good thermal conductor. In fact, if one holds a diamond blade and actually cuts through eyes the heat of the hand can actually conducted very nicely through the diamond and it will actually cut eyes very nicely and this phononic mode of conduction as you know phonons are quantized normal modes of lattice vibrations. And later on we will talk about plasmon in the context of optical properties plasmon are collective oscillations of electrons while phonons are collective oscillations of quantized oscillations of elastic oscillations of atoms or we may call lattice vibrations. In nano materials the phonon wave length can be comparable to the lens scale of the microstructure or in the case a nano particle itself. So, this is an important point that means phonons can get confined in the material because now it cannot travel through the medium. So, that is an important point to note and therefore, you do expect that when phononic conduction is the dominant form of conduction that in nano scale materials things are going to be very different. Even in cases where electronic conduction is the important mechanism you do expect that it is going to be different from the normal materials. Now, if you talk about phonon wave length you can write down e phonon is H v by lambda v is the velocity of sound and the momentum of photon H by lambda putting together we can see the lambda phonon is of the order of 10 power minus 10 or in some cases you can talk about it of the being in the nano scale. Now, in experiments there are two at least two experiments we will take up and or two results we will summarize here and both of these results are you might say opposite sides of the whole spectrum. One is the case of carbon nanotubes where you see a highly enhanced thermal conductivity which is occurring because of phonons and in fact you observe something like about 3000 watt per meter per Kelvin which is 7 times or more than that for copper and this is observed in carbon nanotubes along the length and this is expected to be because of the phononic mechanism. Now, we later on when we talk about conduction electrical conduction we will also notice that in carbon nanotubes you can actually have electrical transport which is ballistic again that means unimpeded transport of electrons across the length of the carbon nanotube which means that you cannot have ballistic conduction. So, you can see the thermal conductivities and electronic conductivities along carbon nanotubes is very high and the result which is contrary to that we are also talking about which we will take up now is the case of nano size platinum films and we are talking about in plane conductivity and its dependence on thickness. So, we have two examples in front of us one is the case of the carbon nanotube where in phononic conductivity is giving us very high conductivity as compared to even that of copper along the length of the tube. The other case is the case of platinum where you expect free electrons to be present and we notice that the in plane conductivity of these thin films these are platinum thin films actually decreased with size. Now, suppose I am talking about thermal conductivity and its plot with temperature then you notice that for platinum bulk that the thermal conductivity actually decreases with temperature which is the normal behavior you would observe because now you are going to have more and more collisions of electrons with phonons and therefore, you actually would have a decreased mean free path and therefore, you expect that the resistance of the material is going to actually decrease increase which is case of electrical conductivity and similarly you would also notice that the thermal conductivity is actually going to degrade with temperature. So, thermal conductivity and electric conductivity both you expect to degrade with temperature and which is what you see in the case of the platinum bulk. Now, if you take about talk about in plane thermal conductivity of platinum thin films there are two important things to be noticed which are both of which are unexpected and startling in some sense. Number one is that the overall conductivity value for any given temperature actually is much much lower. So, you can see that a 28 nanometer film the conductivity for instance you take at temperature like 120 Kelvin you notice that it is about 20 and this is slightly less than 80. So, there is a considerable reduction in the thermal conductivity at any given temperature and even at a higher temperature at 300 Kelvin you notice this is about 30 and this is about 70. So, it is more than double or two and half times reduced with respect to the bulk value. Further you also noticed if the film size is made thinner then the thermal conductivity actually decreases even more. That means a 15 nanometer film is actually having a lower thermal conductivity as compared to a 28 nanometer film which is having a lower thermal conductivity as compared to the bulk. So, therefore, we keep on increasing the thickness at some point of time I would perhaps have a transition from what you might call the nano scale regime to the bulk regime. Now, there is another important effect which we see from the curve which is very different from that of the bulk and this is seen in the regime of 70 to about 340 Kelvin which in which is the regime in which the experimentation was done in the platinum films. You can notice that in this regime actually the thermal conductivity increases with temperature in the case of the bulk semiconductor you already noticed that the thermal conductivity decreases with temperature, but unlike the bulk for the nano size films the thermal conductivity actually increased with temperature. So, the through startling effects in the case of materials having electronic conductivity number one is that the thermal conductivity actually increased with temperature in a nano scale materials and number two the thermal conductivity of the nano scale material and you are talking about in plain conductivity of films and this is not nanoparticles, this is not nanotubes, but actually films and this films is much lower than the bulk counter parts. And this is in contrast with a nano structure like carbon nanotube wherein you actually observe much higher or a highly enhanced thermal conductivity along the length of a carbon nanotube. So, though we only concerned a very little part of the whole story of the thermal conductivity, but this I hope can convey the what you call important message that the thermal conductivity of nano materials is going to be very very different from that of the bulk materials. And this may have important consequences when you are actually designing devices and you are worried about the cooling of these devices, because there is lot of heat generated because of and you may want to put design various strategies for the cooling purposes. And you will notice that the cooling rate or the dissipation of heat through these some of these for instance nano structures is going to be very different from that of the bulk. And therefore, that aspect has to be kept in mind when the devices are designed. The next property which perhaps is in some sense is very much related to the other properties we have been seeing before which is related to the surface property. Because in some sense the conduction we were talking about is more dominant by the bulk and rather than the surface, but in catalysis it is clear that it is a surface effect which is very very important. And we know what is a catalyst a catalyst is something which enhances the rate of the reaction, but it itself does not get consumed by the reaction. So, that is an important thing about the catalyst that it itself does not get consumed and you know lot of industries rely upon good catalysts for industrial production. And catalysts are known to play an important role in reactions like oxidation of hydrogen, hydrogenation of hydrocarbons, oxidation of carbon monoxide, polypropylene, epi oxidation etcetera. Again like before we will view this from a slightly classical view point and also we will take up only a couple of examples which are startling. While literature on this area is again extremely vast and it as because of the industrial application this is also very very useful to study catalysis and especially role of nano particles and nano crystals in catalysis. If you look at the history of the material only about 12 metals they and these typically belong to groups 8 and 1 b which are elemental catalyst. So, there are very small representation of the whole element periodic table which are naturally catalysts. Most widely used ones of those are 3 D metals like iron, cobalt, nickel, copper, 4 D metals like rhenium, palladium, silver and 5 D metals like platinum. So, these are some of the usually used catalysts for these some of these reactions which we have seen before. Copper is used in the catalysis of methanol production by hydrogenation of carbon monoxide and silver is used in the production of ethylene oxide by the reaction of ethylene with oxygen. So, there are some examples. Now, how does this catalytic activity come in? So, again we go back a little back in time perhaps and take a classical viewpoint and try to understand this. That group 8 metals owe their catalytic activity to an optimum vacancy in the D band like nickel and palladium. They have a some kind of an optimum vacancy in the D band and therefore, these group 8 metals have you expect them to be having some catalytic activity. On the other hand, if you look at copper, silver and gold, they have a completely filled D band, but by the virtue that they have a low ionization potential and they can lose electrons from the D band. These can also function as catalysts in bulk form. So, therefore, either you need to have an vacancy in the D band or if you have a low ionization potential, then you expect that such a material would be able to perform the role of a catalyst. And this reaction we could be talking about is the example of ethylene, reaction ethylene, oxygen, production ethylene oxide or hydrogenation of carbon monoxide etcetera. Gold if you look at, gold does not satisfy any of these criteria and it has an high ionization potential like 9.23 and therefore, you expect gold to perform poorly as a catalyst in bulk form. So, the classically in bulk form if you see gold, then you would say a gold is not going to be a catalyst and therefore, I would discard it in the bulk form to play the role of a catalyst. Instead of talking about bulk crystals, suppose I am talking about nano particles, then it is obvious that nano particles are expected to have certain better catalytic property as compared to bulk materials. And these comes from two obvious effects. One is the increase in surface area per unit volume and the increase in surface activity to high degree of unsaturated bonds. So, we do expect already that when you go to the nano scale, there will be a certain benefit in using a catalyst in the nano crystalline or nano form. Now, we have just noted that gold actually does not perform good when you are talking about a bulk gold, but then it has been found that gold can actually be used as a catalyst in the nano form. So, gold has a high ionization potential thus giving it poor affinity for molecular hydrogen and oxygen and expected to perform poorly as a catalyst for hydrogenation and oxidation reaction under ambient conditions. Of course, it should be noted that the poor catalytic activity is for smooth gold surfaces at low temperature. So, this is what we are referring to and the expectation is that gold is not a very good catalyst. And it is very surprising that gold can perform the role of a catalyst in nano form. Gold displays a drastic change in behavior and reduction of size. Gold nano particles supported on substrates like Fe 2 O 3 N N I O can be used as catalyst for CO oxidation even at low temperatures like 473 degree Celsius. So, the observation is that now gold which is not a bulk catalyst can actually be used as a catalyst in nano form. And needless to say we have already noted before that these gold nano particles cannot be kept in touch with each other then they will tend to coagulate. Therefore, they are typically embedded in a substrate and these substrates have noted to be Fe 2 O 3 N N I O. And we will note shortly the relevance and the importance of the substrate in this whole process, but the important thing to note is that that gold now in nano form can be used as a catalyst and this is a very startling expectation or startling behavior. Factors which play an important role in catalytic, catalytic properties of gold are first of course, we are talking about gold nano particles size that means size and size distribution and shape of the gold nano particle. And additionally which is the important point is the substrate that means what is the type of the support, what is the interface with the support substrate which is having all these plays an important role in final outcome of the gold nano particles catalytic behavior. We already noted that gold nano particle melts at lower temperatures as compared to bulk materials. For instance, we have noted that 5 nanometer gold actually melts 200 degrees below the melting point and this enhances the coagulation tendency and nano particles are left in contact. And we have also noted that actually nano particles can sinter at temperatures much lower then and we have noted the role of vacancies in this process of sintering. And therefore, we cannot leave these gold nano particles in touch with each other, they have to be isolated and the substrate therefore, performs the first important role of isolating these gold nano particles. And but the important thing which comes out of the study which the experimental study is that there are additional benefits with respect to catalysis of using the substrate. And we have noticed two substrates we noted before Fe 2 O 3 and N I O. Now, when I embedded gold nano particle in a substrate there are important parameters which automatically come about what is of course, the shape of the gold nano particle is the embedding partial like this picture on the right or is it more to more like this where in the particle is not embedded deep into the substrate. So, we could have a range of embeddings. So, if I have a substrate so my gold nano particle could be embedded such that only a small part of it is embedded it could be such that it is almost like a semicircle or it could be even deeper embedded. And what is the difference between all these of course, it is of course, the surface area of gold which is exposed which is going to be this number this area. And you can see that the surface area of gold is actually decreasing as you go from a small embedding to a large embedding. Additionally, there is one more parameter to be noted in this embedding and that is the interface. Suppose, I look at the three dimensional figure of this I can actually notice that when I have a particle which is embedded then I would have actually a circular interface which is the line of contact between the particle and the substrate. So, this area interface or a triple point length which is the interface between the gold nano particle the substrate and air and that length would change depending on the embedding. So, in this case the length is small in this case it is the maximum possible and in this case there is somewhat intermediate length of embedding. That means, now the triple line which is passing as the interface between the three phases that is also changing depending on the embedding. Now, this has important consequence on the catalysis and we will note that by reading this phrase here the substrate used in as support in the gold nano particle plays a profound role in catalytic properties of the system. First of all the substrate determines the contact angle with which the gold nano particles make with the substrate and which is of course, size dependent. Gold and titanium 2 nanometer particles tended to wet the surface while larger particles 5 nanometer have a contact angle of 90 degrees. That means 90 degree means partial wetting and in the case of 2 nanometer particles which is extremely small particles they tend to form a uniform coverage that means, there is a wetting tendency. So, the wetting tendency itself is a function of the size of the nano particle. This wetting also is going to tell you what is the amount of interface that the gold nano particles will have with the substrate which in turn plays a profound role in the catalysis. Hemispherical particles with more interface perimeter show higher catalytic efficiency than spherical particles attached to the substrate. So, if you compare the two pictures which are drawn here schematically below one is the case of the non wetting kind of a situation where the particle is sitting virtually outside versus a case of a 90 degree contact angle which were in the perimeter areas large. Then you notice that the one which is embedded nicely is the one or a hemispherical cap kind of a embedding is actually giving you a higher catalytic activity as compared to the one which is not wetting the surface. It is understood that the reacting species adsorb on the surface and interface defects like ledges, kinks and interface lines. So, interface lines play an important role apart from ledges and kinks which anyhow exist on the gold surface in determining the catalytic activity. At room temperature and low temperatures about less than 300 Kelvin gold nano particles beat platinum palladium hands down by four orders of magnetic magnitude in their catalytic activity. If you go back and look at this slide where we talked about catalyst we noted that platinum and gold are well known catalysts. Platinum and palladium are well known catalysts, but gold we said in bulk form is not a catalyst, but when you go to nano scale it looks like the roles are reversed. And you can actually note that gold can outperform platinum palladium of course, in a certain regime of temperature certain regime of embedding that it can actually go and beat platinum and palladium with respect to the catalytic activity. And here we are talking about of course, as I said a limited regime we are where is the cobalt carbon monoxide oxidation reaction has been studied. And this is understood due to the low activation energy of about 30 kilo joule per mole for use by using gold nano crystals. Therefore, we see that when you go to the nano scale there are behaviors which are highly unexpected. And this is this continues to be so in the case of catalysis where in spite of gold not being in the list of the usual known catalysts like the 3 D metals like and the 4 D metals and 5 D metals like platinum palladium silver cobalt. But we find that when you go to the nano scale and especially when we are trying to track catalytic activity of gold nano particles in the low temperature regime. And for a particular reaction like carbon monoxide you can actually go ahead and beat some of the well known catalyst. And the important parameters of course, in this whole thing is of course, shape of the gold nano particles the size of the gold nano particles the substrate medium is it titanium or is it nickel oxide or is it one of the other materials which can be chosen as substrate material. The contact angle that gold nano particles makes with the substrate and all this put together it can be seen that there is an effective role that gold in nano scale can actually perform in the case of catalysis. And this is again one of the startling effects which you see only when you go to the nano scale. The next chapter we take up is related to the mechanical behavior of nano materials. And since defect structure plays a very important role in the mechanical behavior we first talk about the defect structure followed by the mechanical behavior of nano materials. The defect structure in nano materials can be highly altered with respect to bulk materials. And this actually plays a direct role in terms of the properties and these properties could be conductivity diffusion could be you could be even talking about compressive strength or fracture toughness any one of those properties. And therefore, I need to know the defect structure of nano materials and we have already previously defined what is it what is the term defect structure means it term it includes as we have noted before entities like distribution of defects its size its overall volume fraction etcetera or length per unit volume etcetera. So, we need to know the defect structure in a material. So, that I can understand the profound differences in mechanical behavior of nano materials as compared to bulk materials. So, the first of these defects I take up is vacancies. We have already talked about vacancies in the context of diffusion in the context of sintering of nano particles. And we have noted that depending on the curvature of the nano particle is it going to be convex or concave we have noted that there can be actually an enhancement in the vacancies or they can be a depression in the equilibrium concentration of vacancies. And we have noted this we will briefly revise what we mean by the equilibrium concentration of vacancies we have noted that they are equilibrium thermodynamic defects. And suppose I am for now I will restrict myself to metals wherein there is no charge involved and the arguments can be simplistic made. We know that it cause enthalpy for a for you for because there are broken bonds when you have a vacancy in a crystal that it cause enthalpy in terms of putting a vacancy. But we also know that there is a configurational benefit when you have a vacancy that means that when you are trying to minimize Gibbs free energy which is given by delta G is equal to delta H minus T delta S. And for now I am restricting myself to configurational entropy then I can note that there will be an offset because of this configurational entropy a certain temperature at each temperature there will be an equilibrium concentration of vacancies. That means that if I am talking about the Gibbs free energy versus vacancy concentration then I would notice that there initially will be a depression when I increase my temperature. But there will be an at a given temperature of course there will be when I increase the number of vacancies then there will be a depression in the Gibbs free energy. But after the equilibrium concentration then there will be an increase in the Gibbs free energy. That means the energy minima appears for only a certain number of vacancies. Now of course when you increase the temperature then you will have a increased concentration of vacancies and you go even further high temperature there will be increased concentration of vacancies. And close to the melting point in FCC metals like gold, silver and copper the fraction of vacancies which is given by this formula n v by n is which is depends goes as exponential of minus theta h by k t is about 10 power minus 4 1 in 10,000 of the lattice sites go missing close to the melting point in these metals. And as I pointed out that as you increase the temperature the fraction of vacancies keeps on increasing. And therefore, if I take any bulk material and I am talking about any finite temperature like about 300 Kelvin then I expect an equilibrium concentration of vacancies. Now, what happens in nano crystals that is the question we are asking in free standing nano crystals below a critical size the benefit in configurational entropy does not offset the energy cost of introduction of vacancy because now it is a nano crystal the number of sites over which this vacancy can configure are smaller. And this implies below a critical size you expect that vacancies may become thermodynamically unstable. Now, this critical size we can call DC the crystal can become free of vacancies of course, if you start from a high temperature and go to low temperature kinetics have to permit for these vacancies to escape. Now, for aluminum at 900 degree Celsius with an activation energy of 0.66 EV the critical size of the order of about 6 nanometers that means, aluminum crystals below 6 nanometers will be free of vacancies that means, such a crystal will have no equilibrium concentration of thermal vacancies will not be in thermodynamic equilibrium if put in such a crystal. Copper with a higher energy for formation of vacancy of about 1.29 electron volts the critical size happens to be higher of about 86 nanometers. Now, the important point to note in this whole argument is that normally when you talk about configurational entropy you are talking about Gibbs free energy we are talking about what is known as bulk thermodynamics. That means, and these configurational entropy comes from statistical physics which implies that I am talking about a large ensemble which means about a mole of atoms or more which are being sites which are being configured. But in nano crystals obviously, then sizes system is limited and the important question which arises can I use bulk thermodynamics for calculation of quantities in these nano crystals. The answer is obviously no, but it has been found that if you apply bulk statistical thermodynamics to even 100 nanometer crystals. So, it is found approximately true that the results are reliable, but if you go down to very small sizes like about 6 nanometers then the classical formulae cannot be used and they need to be modified for nano crystals. Since, we are not dealing in detail about the thermodynamics of nano scale systems, but this important effect though the calculation based on this may be 6 nanometer could be erroneous, but the important thing is that when you go down to small scale system we do expect that the system will become spontaneously free of vacancies. In other words the system will not support vacancies as a stable thermodynamic defect which is what is the case for the case of bulk crystals. Later on we will see when we talk about dislocations also that when you go to nano scale the system can actually become spontaneously free of dislocations. But in the case the important difference between a dislocation and a vacancy being that in the case of a vacancy, a vacancy can be a thermodynamic defect in bulk which or is a thermodynamic defect in bulk I mean it is thermodynamically stable defect, but dislocation is not a thermodynamic defect that means it is not stable in bulk. But in nano scale the kinetics becomes such that it can actually move out of the crystal. So, there are important differences when you are talking about nano crystals vis-a-vis a bulk crystal when it comes to defects.