 the size of the precipitates increases with time and here I am talking about the average size of the precipitate. And the function by which it increases is that r cube average minus r 0 square r 0 cube is equal to k into t. That means, it is the cube of the average size which is related to the time and r 0 is the average radius at time t equal to 0 and this constant k is given by is directly proportional to a product of the diffusivity the interface energy and the equilibrium concentration x e which is in adjacent to the precipitate which we have seen already how to look up x e from the common tangent construction. This implies that with increasing time the average precipitate size increases and if you look at the diffusivity the diffusivity goes as exponential with temperature d equal to d 0 exponential cube by k t. That implies that at higher temperature the rate of coarsening is going to be much faster. So, if you plot the curve at a lower temperature I would get a r versus r average versus t curve which looks like this, but with increasing temperature you will see that the r versus t curve shifts upward that means that for a given time I will have an higher r average if I have a higher temperature. Now, this diffusivity I am talking about is typically the volume diffusion rate and if there are other mechanisms of diffusion in the material like interface diffusion pipe diffusion which is at course of dislocations. Then this kind of a linear r cube versus t relation may break down and these are short circuit paths even though they have a smaller cross section may tend to dominate especially in nano crystals. And if you look if you differentiate this function r cube average minus r 0 cube equal to k t then you can see that d r average by d t which is the rate of increase of the precipitate size with time is proportional to 1 by r square average. That implies that smaller precipitates coarsen more rapidly as compared to larger precipitates because if r 0 square increases then you see that this rate increases therefore, if you have a smaller collection of smaller precipitates that will coarsen faster as compared to a collection of larger precipitates. And with time you will find that you initially wanted to retain a small distribution of fine precipitates to have a higher hardness and we have already previously done a solved example wherein we have seen how the a frankereed source operates around these fine particles which is giving rise to hardening. And if these particles are coherent then in some cases if the stress is good enough these the dislocations can actually glide through the precipitates while if they are incoherent then they bow around the precipitates leaving dislocation loops around them which is called a frankereed mechanism or an equivalent mechanism known as the aura 1 mechanism. Therefore, with time if the precipitates coarsen and they also tend to go from as I pointed out from the metastable precipitates to the equilibrium precipitates then I lose my hardness. So, how can I retain the hardness the crux lies in this three terms which is related to the constant in this equation which is d gamma n x c. If I can maintain a small value of d gamma n x c then I can actually reduce amount of coarsening in the material and therefore, I can control the rate of coarsening by actually having a small value of this product d gamma n x c. So, the three strategies either I can start with a I can have a low gamma or I can have a low x c value or I can have a low d value such that overall my coarsening rate is small. And if you are looking at precipitation hardening systems which have to be employed at high temperature then we have to have these conditions satisfied. So, that I can have hardness even at high temperatures. Nemonic alloys are special alloys which are designed for high temperatures and a typical mnemonic alloy for instance a mnemonic 90 alloy would have a composition in which there is about 54 percent nickel about 18 to 20 percent chromium about 15 to 20 percent cobalt some small percentage of titanium and aluminum. And this composition retains its strength at high temperatures the basis of strength in these alloys these mnemonic alloys is the presence of fine dispersion of gamma prime precipitates. And the prime here refers to the fact that you actually have an ordered structure here which is an ordered f c c n i 3 t i a l structure which is in which case there are two sub lattices one occupied by nickel or one of the sub lattice occupied by three of the sub lattice occupied by nickel and two of them one of them occupied by titanium or aluminum. And this ordered precipitate sits in an f c c nickel rich matrix of course, as you can see that it is not pure nickel matrix, but there are other alloying elements which also sit in the matrix. And this fine dispersion of gamma prime ordered precipitate and since the precipitate is ordered we expect that it is harder. And because you know that ordered structure always has a larger burgers vector and it is going to be harder that means it is going to have an higher pulse stress and in the process it is going to give me hardness. But the important thing is that if you look at the matrix which is a nickel rich solid solution and the interface between the matrix and the gamma prime precipitate is fully coherent with a low interfacial energy of about 30 millijoule per meter square. And since the interfacial energy is very low there is low tendency for this precipitate to coarsen that means I am keeping a low gamma between the matrix in the precipitate and this is at the heart of the strategy to keep these precipitates or to avoid the precipitates from coarsening. And in the process this alloy can be used as a prestation hardened alloy at high temperatures. Now, this misfit obviously is a function of composition of the precipitate and of course, the composition of the matrix and typically it varies between 0 percent and 2 percent. And it is observed that if my misfit is large which implies that there is going to be coherency stresses then the creep rupture life actually increases when the misfit is actually small. That means if I have low interfacial energy and also a low misfit then actually I get better properties with this kind of an alloy in which my main strategy is keeping my low gamma. But as I pointed out that not only can you keep a low gamma and get a good property a good high temperature property of the material which does not deteriorate with time at high temperatures, but also I can use a low x c. This example given here is actually an extreme example where in the solubility of when I am talking about the solubility I am talking about the solubility of the solute in this beta phase is actually extremely small. Now in this case if I am talking about toria dispersed in tungsten, toria and tungsten are there is very little solubility of the metals in toria or toria in the metal. And this can be actually thought of as a dispersion hardening system rather than a prestate hardening system because if it is a prestate hardening system then actually have the prestate coming out of the normal reaction of the solid solution. But in this case it is not true the toria is actually intentionally added as a fine dispersion in the metal matrix and oxides being insoluble metals. The stability of these microstructures can be retained at high temperatures and as I pointed out this is an extreme example because there is practically no solubility of toria and tungsten therefore, you have low value of x c which gives you a low value for the term d gamma x c. And the same example if you note that in the same example you can actually cite the fact that there is a very low diffusivity d and in the process actually I can have no coarsening taking place because the matrix does not dissolve into the prestate the prestate does not dissolve into the matrix. And therefore, there is very little diffusivity of toria into tungsten, but if you look at cementite dispersions in tempered steel they coarsen due to high d of the interstitial carbon. If you look at as you know there are two kinds of diffusion interstitial diffusion and substitution diffusion if you look at interstitial diffusion then there is only one term in the exponential and d equal to d 0 exponential minus h m delta h m which is the migration enthalpy of the interstitial carbon from one interstitial side to the other and as expected this goes exponentially with temperature. And if you see that when you temper steel actually you get a fine dispersion of many of these cementite or equivalent carbides and these carbides in the fine scale actually give you very high hardness, but they tend to coarsen because now the carbon can easily diffuse through the matrix. Now, if you add a substitution alloying element which specially segregates the carbide in other words you just do not have an iron carbide, but you also have an iron alloying element combination carbide the rate of coarsening actually decreases due to low d for the substitution element. In other words now I can design my alloy in such a way now that there is an additional alloying element which segregates the carbide and this implies that when the carbide has to grow not only has to carbon to diffuse, but also the alloying element has to diffuse. And since that is going to be a substitution alloying element its diffusivity is going to be smaller than that of the carbon which is an interstitial solid solution. Because if you look at the diffusivity of substitution elements you have an additional term delta h f which means that substitution elements diffuse by vacancy mechanism. And that implies that the exponential term has two terms one related to the enthalpy of formation of vacancy that means the chance of finding a vacant neighboring side. So, that the substitution element can actually jump into that site and the second which is a usual migration enthalpy for the substitution element this implies that the diffusion of the alloying element is going to be which is a substitution alloying element is going to be difficult as compared to carbon. And overall you find that the coarsening rate actually comes down. So, we have seen so far that you have you can produce a fine dispersion of a straight to increase the hardness typically say of an aluminum copper alloy. This prestation harden system tends to not only change the character of the precipitate the characteristic of the interface with time, but also the prestate coarsens with time which leads to a loss in hardness. And therefore, I would like to keep the fine particle distribution we had also seen the reason behind the actually the coarsening one of the heart of the reason is the Gibbs Thomson effect. Where in there is a curvature dependent solute concentration around the prestate followed by a diffusion down the concentration gradient. And we have seen the strategies so that I can actually avoid the coarsening of precipitates. So, that I can retain a fine dispersion of a states at high temperatures. And of course, when I am talking about high temperatures I mean a temperature which is above the recrystallization temperature which is typically about 0.5 the melting point of the material. Now, let us take up some more studies where in we can see the importance of curvature effects. That means we had already pointed out that small particles are curved more and therefore, there the curvature effects tend to dominate. And here we take up the specific example of vacancy concentration and sintering of nanoparticles. We already seen and we have said that suppose I want to retain a gold nanoparticle the nano scale then I have to isolate one particle from the other using some kind of a passivating molecule like thiol. Otherwise these particle will tend to sinter with each other and would coarsen and become large green. Therefore, we will see that what is the underlying mechanism that you actually have which is giving rise to this kind of a sintering in nanoparticles at a temperature much lower than what you would expect for bulk particles. Now, we know that the surface atoms have a higher internal energy and then higher entropy because there is a larger freedom for the surface atoms. Now, they are in a different state as of energy as an entropy as compared to the bulk of the material. Now, we also know that the broken bonds cost energy to put in a crystal. That means if you have a vacancy in a crystal then it cost you excess additional energy you need to put. But, there is a configurational benefit which you get and therefore, there is an increase in configurational entropy and at any given temperature any positive Kelvin temperature there is an equilibrium concentration of vacancies which is given by the formula n bulk by n where n is the total number of atoms in this crystal n bulk is a number of vacancies in the bulk crystal is given by exponential minus delta h f by k t where delta h f is the enthalpy of formation of a vacancy. That means joules per atom and if I am talking about I can replace this k with r if I am talking about joules per mole of vacancies. That means at any given temperature t positive Kelvin temperature there is an equilibrium concentration of vacancies in a bulk crystal. Now, if you heat a metal and take it close to the melting point you would notice that the fraction of vacancies about 10 power minus 4 that is 1 in about 10000 is vacant that means this is a concentration or the mole fraction of vacancies when you actually heat the metal close to the melting point though this is a small number. But, there are actually theories which are called the critical vacancy concentration theories which are related that they which tell you the importance of these kind of vacancies in the melting process. So, approximately 1 in about 10000 is the concentration of vacancies close to the melting point of metals like gold, silver and copper. These vacancies otherwise called thermal vacancies because they are coming from temperature alone have an equilibrium concentration of the bulk. And we have just now seen that they are responsible for substitutional or self diffusion. And for instance the diffusivity of nickel in ion at 1000 degrees is 2 into 10 power minus 16 meter square per second which is purely coming from the vacancies present in the crystal. Now, curvature effects as we are known is important in nano crystals and the vacancy concentration below a curved surface is altered from the value which we assume for a bulk crystal. That means if you have a flat crystal the vacancy concentration below it can be given by a formula like this which is n bulk by n. But, suppose now I take into account the curvature effects then the vacancy concentration can be altered with respect to the bulk. An extreme example of this is the case where when you make the crystal very small then the crystal can spontaneously become free of vacancies. But, before we go to that we will now deal with the aspect of curvature dependence on vacancies. The sign of curvature determines if there is an excess or depletion of vacancies with respect to the value for a flat surface. So, now I have three kind of surfaces which I can talk about a flat surface, a positively curved surface and a negatively curved surface. So, this I can call a convex surface and this is a concave surface and for now I will assume that this is the concentration of vacancies. Now, I want to compare this with the concentration of vacancies for convex and concave. So, I have three kinds of configurations which I need to talk about and I need to see how the concentration of vacancies changes. Below a concave surface there is an excess concentration of vacancies and the opposite is true for convex surfaces. That means if I am in concave surface there is an excess vacancy concentration and this automatically implies that below concave surfaces there is an enhanced diffusivity or there is an enhanced diffusion. So, if I that means that if I am talking about diffusion below a flat surface or diffusion below a convex surface or diffusion below a concave surface the value I would obtain is not going to be the same. For a concave surface there is an enhanced diffusion because now there is an excess vacancy concentration in a concave surface and with respect to the flat surface the convex surface would show a reduced diffusivity. This effect obviously is dependent on the curvature and as the curvature increases the reduction that means by the reduction of size this effects become more pronounced and which is going to be all the more so in smaller particles. Now, the concentration of vacancies in a nano crystal can be written as n nano crystal vacancy vis-a-vis the fraction in a bulk crystal can be given as exponential minus 1 by r omega gamma by k t. Gamma is of course the surface energy omega is the atomic volume and r is the radius of curvature. For a convex surface r is positive this means that x nano crystal is going to be smaller than x bulk. For a concave surface r is negative on the other hand this implies that x nano crystal is going to be greater than the x bulk and x of course the fraction of vacancies either below a convex surface or a concave surface. Therefore, now my vacancy concentration is altered in a convex surface or a concave surface and this has important consequences when we are talking about sintering of nano particles. Now, atoms will tend to diffuse from convex region to concave regions leading to a sintering of particles and this sintering will occur at lower temperatures than that employed for large size particles because now it is enhanced by the presence of additional vacancies. And this is the reason we have seen that typically in metals have to be kept isolated from each other by a passive weighting layer that will be nano particles and nano crystals have to be kept even at room temperature. So, now what is happening here in the case of nano crystals is that suppose I have to now so when the sintering process starts you have a neck formation and here I have and this region is convex and I would have diffusion of mass from here to here leading to the sintering of the particle. Of course, this diffusion can be aided bulk diffusion and other kind of interface diffusion which can finally lead to the sintering of the particle which means that the neck is going to grow and the particles are going to the material is going to sinter and join with each other. Therefore, we are seeing that purely from curvature effects we have important phenomena coming like coarsening and enhanced or reduced diffusivity in selected regions of the nano particle which is going to give you actually an enhanced sintering. So, just to go through the sintering part again we have a bulk concentration of vacancies which we call the equilibrium concentration of vacancies which is coming from a thermodynamic stabilization a configurational stabilization of vacancies in the bulk. But when we go to nano scale materials that means where there is a curvature effect there the vacancy concentration is either elevated or depressed with respect to the normal bulk material a flat material where there is no geometrical contribution. And below concave surfaces there is an excess of vacancies and therefore, an enhanced diffusion and below concave convex surface on their hand there is a depression of a number of vacancies or fraction of vacancies. And therefore, there is reduced diffusion and this implies that when you have a connection of particles between two particles these particles can be sintered because of mass transport from regions of convexity to a region where the neck is formed. And this implies to me that one of the thermodynamically nano particles are very much unstable as compared bulk particles and they have to be kept in an isolator environment otherwise they will they will be sinter and of other of course, these grain size of these crystallites may grow and finally, I will end up with a system which is not there are no nano particles anymore. Now, let us move on to the next topic which is again closely related to surface effects which is the lattice parameter of nano crystals. Now, when we are talking about lattice parameter of bulk crystals we know that if you talk about a cubic crystal then you have three lattice parameters the a b and c of course, the alpha equal to beta is equal to gamma is 90 degree is a typical lattice parameter which we used to describe a cubic crystal. And so forth there are many if you are talking about a lower symmetry system then you need more number of lattice parameters to describe the crystal. And one important quantity which we can come from all these lattice parameters is the volume of the unit cell. So, in of course, we can track each track each one of these lattice parameters as you reduce a crystallite size or we can track the atomic or the unit cell volume. And in this current discussions we will stick to the unit cell volume because that makes consideration easy we will have to deal with a single number rather than independent lattice parameters which may actually show increase or decrease with when the size is reduced. It is obvious that as the size of the nano crystal is reduced surface tension effects tend to dominate because now we have more number of atoms on the surface for a given volume of material. Additionally you have more unsaturated material that means the now because of increase curvature we already pointed out in the some of the previous lectures that your surface atoms are less bonded as you go down to smaller and smaller side. This implies that surface tension effects tend to dominate in nano crystals. This surface tension effect can lead to a reduction in lattice parameter of the crystal. Now, how is this happening is that now I have atoms sitting on the surface. So, I will assume a spherical particle and there is a layer which I can call the surface. And we have already pointed out that when I am talking about the surface actually it need not be a mono layer it could be one more than one layer which we together call the surface. Now, these atoms on the surface are in a state of higher energy and therefore, the surface would want to shrink itself it wants to pull itself on itself and in the process it will tend to exert a pressure into the crystal. Because now the surface does not the atoms on the surface want to get bonded more they want to get closer to each other and in the process they actually employ as some side of a compression on the particle. Now, this effect actually lead leads to a decrease in the lattice parameter of the crystal and the typical decrease when you decrease the size to say about 10 nanometer is about 1.5 percent decrease in the lattice parameter when you go from a bulk crystal bulk single crystal particle to particle which is of the size of about 10 nanometer. So, this is a decrease you are talking about in the value of the lattice parameter. Now, we can understand this by actually looking at what is known as the Gauss Laplace formula which is related to the change in pressure delta p with the surface energy and the diameter of the droplet. So, this was originally developed for liquids, but we will use it for solids. So, the delta p is the pressure difference between the inside and outside of the droplet d is the diameter of the droplet and gamma is the surface energy. That means, if I decrease my droplet size the delta p is actually going to increase. Now, for the bulk I can write down the definition of compressibility k as 1 by v naught dou v by dou p at constant temperature. Now, this v 0 is the volume of the unit cell when the particle size is large. That means, it is in a crystal where surface tension effects are negligible and this view v can be written as a cube and in fact, when I am saying a cube I actually mean an a 0 cube which is the where a 0 is the bulk lattice parameter. Now, I can actually substitute I can take now I can differentiate this d v is equal to 3 a 0 square d a and I can actually write instead of d v I can write it in terms of delta v and I can make this a delta a and I can now substitute this relation into this relation here. And additionally I can substitute this Gauss Laplace formula into this relation here and therefore, I can get a relation between my gamma by d and delta a by a where delta a is the change in lattice parameter as I reduce my crystal size from infinity. That means, if d goes to infinity then delta a goes to 0 that means, delta a is the change in or reduction in lattice parameter from a bulk crystal to a small nano crystal. Now, therefore, if you look at this function which is gives that gamma by d is equal to 3 by 4 k where k as we notice is the bulk compressibility and to delta a by a where a as I pointed out is the lattice parameter of the bulk crystal which can otherwise be written as a 0. Now, this formula implies that as I decrease my d that means, as I decrease my particle diameter then my delta a increases that means, asymptotically goes with my dereduction in my lattice parameter the size. Therefore, you can see here in this reduction of lattice parameter with particle size this is schematic trend line arrived at looking at many systems and you can see that if you reduce the particle size and the effect seems to dominate only when the particle diameter is below about 15 nanometers. So, if you can safely say that if you take a nano particle or a nano crystal which is above 15 nanometer in size then that behaves like a bulk even though that is a nano scale with respect to lattice parameter it is as good as a bulk crystal. And this is an important point we mentioned when we try to understand what is bulk and what is nano with respect to this specific parameter lattice parameter or the atomic volume then you see that it is as good as a bulk. Now, but on the other hand if you take very small particles like 5 nanometer particles or 10 nanometer particles there is depression about lattice parameter of 1.5 percent. And this schematic trend line is specifically for aluminum, but other metals also show similar kind of schematic trend lines. Therefore, we have now come across an important effect that purely because of surface tension effects the lattice parameter of nano crystals is actually reducing and this decrease could be as large as about 1.5 to 2 percent. Next, we take up the another aspect with wherein certain interesting phenomena interesting effects are seen and that is related to thermal expansion of nano materials. Thermal expansion of nano materials as a first expectation we may think is going to be dominated by surface tension effects and reduced geometrical constraint on particles. So, when I mean reduced geometrical constraint I would expect that because it is a nano particle and there is very little of material around each atom to constrain it from actually expanding on heating that this I would expect would play an important role in thermal expansion of material apart from the fact which I mentioned just now which is the surface tension effects. And obviously, we do expect some shape dependence in this regard, but we are in store for some surprise regarding this that there are other effects which also come in which is going to actually alter the scenario and we are going to obtain some very interesting and startling observations which we will actually consider here. If you talk about normal materials then when you heat the normal material bulk material it actually expands of because of two reasons and the two reasons we already know. So, if you have two contributions here the dominant contribution is the thermal vibration contribution and we know that the potential energy versus the distance plot actually is asymmetric and the average distance actually moves out to larger distances when there is a the potential well that means therefore, when there is a thermal vibration the material would expand. The increased vacancy concentration is a much smaller effect one in about one in ten thousand effect nevertheless when vacancies are created within the crystal these atoms go and sit on the surface and this as we know that is because we think a higher temperature there is an higher equilibrium concentration of vacancies. And now this is the thermodynamic equilibrium concentration we are talking about and this moment of creation of vacancies and the moment of atoms would lead to an increasingly size of the material in other words there would be an expansion with temperature. Now even though we noted again that the second effect is actually a smaller effect. Now it has been observed and this is a very startling example that in 4 nanometer gold nanoparticles that below 125 degrees 125 Kelvin the thermal expansion is positive that means it is a normal kind of a thermal expansion and the thermal expansion coefficient is about 3.5 3.2 into 10 power minus 5 per Kelvin. And this is not very different from what you expect for bulk materials. But if you look at the expansion in the regime beyond about 125 Kelvin on heating the thermal expansion actually becomes negative. That means the material tends to contract after you heat above 125 Kelvin and the negative coefficient of expansion turns out to be about minus 2.5 into 10 power minus 5 per Kelvin. That means that the lattice parameter of the particle is about 0.2 percent smaller than bulk gold. And this is a very startling effect and the physics of this effect is not fully understood yet. And the important contributions which go into giving rise to this what you might call the negative thermal expansion which is often written in acronym as NTE. So, this effect can be called negative thermal expansion effect is thought to arise from valence electron potential and equilibrium lattice separation. So, these are thought of as two important mechanisms which is supposed to play an important role towards contributing to this negative thermal expansion in gold nanoparticles. And we are talking about extremely small gold nanoparticles of the size of about 4 nanometers. And we are talking about temperatures which are also pretty cold which is below the room temperature. And this happens as we note in only a certain temperature regime and not in all temperature regimes. And perhaps with time we will come to know what is really the mechanism behind such an effect. Nevertheless, it is very interesting to note that in nanoparticles and nanocrystals you can actually have negative coefficient of thermal expansion. And such that too of the same order of magnitude as you notice here as the positive coefficient of thermal expansion. It is both of the order of 10 power minus 5 and even the magnitude is pretty much similar. So, it is a definitely a very significant effect. Now, there is another reason why you could actually have negative thermal expansion. And this is seen beautifully in the system like cobalt or copper oxide or magnesium manganese fluoride MNF2. Both these are anti ferromagnetic materials though we have noted dealt with magnetic properties of nanocrystals in detail. And we will take up some of it in lot of detail in some of the coming lectures. But it is in this one this is related to thermal expansion we are considering this system here. And copper oxide and manganese MNF2 unlike a very similarly related system like NIO. NIO is also a system wherein you actually have anti ferromagnetic ordering. But NIO does not show this effect. So, we do not consider it here and we take up the example of C U O or MNF2. We notice that these systems show magnetic ordering on cooling and this leads to an expansion in the material via the magnetostriction effect. And all of you must be familiar with magnetostriction because that is the hum created when you send an AC signal through a transformer core. The transformer core actually hums and that is that humming sound is actually coming from acoustic emissions which is coming from this magnetostriction effect wherein the magnetic field is leading to a change in the dimension of the materials which is coming from in the presence of a magnetic field which is applied. And this is caused by shift in domain boundaries and rotation of domains within the material. So, there is a shift in domain boundary there is a rotation of domains and this leads to a change in dimensions of the crystal when you are applying a magnetic field. And this effect is called magnetostriction and this magnetostriction as I pointed out gives rise to the familiar effect of a hum in a transformer when you actually apply an AC signal. And the important thing to notice that in this materials like C U O and M N F 2 there are two important criteria under which you actually see this negative thermal expansion effect. One is that the temperature has to be below the needle temperature and the needle temperature as you know is the anti ferromagnetic ordering temperature. That means below the needle temperature the material is anti ferromagnetic and above the needle temperature the material is actually paramagnetic. And above T N only the U above T N of course, the needle temperature the material you can actually see a positive thermal expansion which means it behaves like a normal material. Therefore, it is essential that I retain this the anti ferromagnetic state to observe the special negative thermal coefficient negative expansion state. Additionally the particle size has to be small it has to be in the nano scale it has to be very very small and for about 5 nanometers. And if you look at micron size particles the negative thermal expansion barely is able to come out go around or get around the thermal vibrational effects. And actually you will see that only you observe a positive coefficient of expansion. So, let me write down the two criteria for negative thermal expansion in anti ferromagnetic materials. And we take examples like copper oxide and M N F 2. Now, if I observe my volume of the crystal with respect to the temperature I would notice that for if I now take my nano size particles and compare it with micron size particles. You see that in micron size particles you see the normal coefficient of expansion though there is a pronounced alteration in the coefficient below the anti ferromagnetic ordering temperature which is shown by the vertical dotted line. So, in nano size particles though there is this hidden effect which is giving rise to an altered what you might call coefficient of expansion. But the coefficient of expansion still remains positive because it is not able to overcome the normal vibrational expansion which is what is going to dominate in micron size or larger size particles. But if you take nano size particles and I am talking about very small sizes like 5 nanometers. Then it is clear that if you look at the plot here which is shown in the orange line you can see that below the above the needle temperature the coefficient is positive. And if you go slightly below the needle temperature you see that the region marked in red box is the region where you actually see an expansion in the volume of the volume as you are reducing the size. This is the expansion in the volume of the unit cell when you reduce the size and you can see that there is this is a significant effect and this cannot be what you call accounted by other methods. And we have to invoke the fact that there is an anti ferromagnetic ordering in these particles and this anti ferromagnetic ordering is giving rise to a magnetostriction effect. And this magnetostriction effect is able to actually overcome the usual thermal vibrational effect which is normally seen for all normal materials. So, we have seen at least couple of examples straight away wherein we are noticing that there can be negative thermal expansion coefficient one of which is not fully understood yet which is in the case of gold above 125 Kelvin. But in this case there seems to be better understanding in the underlying mechanism which is giving rise to this negative thermal expansion in these nanoparticles of copper oxide or magnesium manganese difluoride. Now this is not true of course that it is only nanoparticles where you see negative thermal expansion there are some rare examples of bulk materials also which actually show a negative thermal expansion coefficient. And one such example is zirconium Z r w 2 o 8 which is called a zirconium tongue state and here the geometric flexibility in the linkages is supposed to be the underlying mechanism which is giving rise to this kind of a negative thermal expansion coefficient in the Z r w 2 o 8 structure the Z r o w linkages become increasingly linear at low temperatures. And therefore, the amplitude of this is happening as amplitude of the oxygen vibration decreases. Now this is a schematic explaining this effect now there is a molecule which is an angular form in which the oxygen is shown by this blue sphere. And what happens when you heat this actually cool this material is it the molecule tends to become progressively linear. As the thermal vibration of the oxygen molecules reduce when you reduce the temperature then the molecule goes from an angular kind of a shape to a linear kind of a shape. This implies that actually the spacing between these molecules actually increases. And therefore, actually by reducing temperature you are seeing a thermal expansion rather than a thermal contraction. And this can be thought of as a negative thermal expansion effect. What the important point to note is that in this bulk kind of a material which is itself is a very rare example of N T E effect the thermal expansion effect is negative expansion effect is only one fifth of that which is found for a material like copper oxide or M N F 2 where in we know that in nanoparticles the effect is much more pronounced. Therefore, there are very interesting effects in nanoparticles which includes this startling observation that you can actually have negative thermal expansion coefficient. And this the underlying mechanism can be more than one. And you have to take system by system and understand how this negative thermal expansion is actually occurring in nanocrystals of copper oxide or gold or are equivalent systems. The next topic we take up is phase transformations in nanocrystals. Now phase transformation of course is a broad area and it may not be possible to cover all aspects of phase transformations here. But we will take up selective examples of phase transformations in nanocrystals and important effects which are coming because now you are talking about a scale of a system which is one nanometer or five nanometer or very small system. Now if you want to classify phase transformations and we understand that phase transformation means we are going from one phase to another phase and this phase could be defined based on for instance a structure like you could be going from a cubic phase to a tetragonal phase or it could also be based on a property like you could be going from a paramagnetic phase to a ferromagnetic phase or anti ferromagnetic phase to a paramagnetic phase. So, we could actually base it on a property you could also be talking about a transition from a super conducting state to a normal state and etcetera. Here we restrict ourselves to what we call normal structural transformations and we will take up couple of examples of melting and also crystal structure change to illustrate the point that how actually nanocrystals or nanomaterials can actually be very different from bulk materials and what are the important effects which we need to take into account when you are talking about very small size crystals. Now phase transformations can be classified based on mass transport as diffusional or diffusionless and in diffusional transport we imply that there is actually long range diffusion and in an example of a diffusion transport like martensitic transformation there is actually a shear of the crystal and there is no long range mass transport involved and therefore, diffusionless transformations can occur at a very fast speed. In fact martensitic transformation the transformation front can move almost at the shear wave velocity of the of sound in the material. Based on order of the transformation typically their class phase transformations are classified as first order which involve nucleation and growth and we also have second order transformations where in the entire volume transforms. In second order transformation there I also included for instance lambda transformation other transformations, but here in this brief lecture series we will worry about diffusional transformations which are of the first order and which implies that we are going to be talking about nucleation and growth which is what is going to give rise to our phase transformation. Now since we are talking about this first order nucleation and growth kind of transformation and I am talking about the phase transformation from the alpha phase to the beta phase. First of course in the alpha matrix I need to nucleate a beta phase and finally and as a second step the alpha phase has to be exhausted that means the beta phase has to grow to its final state such that all the alpha is consumed. If of course it is going to be a transformation where in both alpha and beta can coexist then you will at the end of the transformation you will have a certain volume fraction of the beta phase and you will have a certain volume fraction of the alpha phase. Now the important thing in this whole transformation is about nucleation. Nucleation is actually a technical term and we will soon see why what is meant by nucleation and why nucleation is actually a barrier to phase transformation. And we will also consider the importance of nucleation especially nano scale systems where we note that heterogeneous nucleation can dominate over homogeneous nucleation. In heterogeneous nucleation every point in the medium suppose I am talking about a volume of alpha. Now this is my volume alpha and I am talking about some region of it transforming to beta then I would note that any point in this material I can randomly take any point is as favorable to be a site for nucleation of the beta phase as any other. That means there is no preferred site for nucleation. On the other hand when I am talking about heterogeneous nucleation suppose this where a polycrystalline material then I would observe that this polycrystalline material would actually have green boundaries inside. And it could so be that this green boundary would be a preferred site for the nucleation of the second phase which is now my beta phase. And if there are preferred sites which includes the surface the green boundaries, stacking faults, dislocations etcetera these would then be called heterogeneous nucleation. And the reason that heterogeneous nucleation takes place is that because it gives you a benefit in terms of the activation energy for the phase nucleation to take place. Now what are the energies involved when I am talking about a phase transformation. The driving force comes from the fact that below the transformation temperature the beta phase is stable as compared to the alpha phase. And suppose I am talking about a solid to solid phase transformation then I have to consider three kinds of energies. And if I am talking about a liquid to solid phase transformation for instance solidification then I will note that one of these energies can be neglected. Now the reason that the phase transformation is taking place is because the Gibbs free energy of the system reduces if and that means the beta phase is stable as compared to the alpha phase. But when a nucleus for instance now the green region of the beta phase forms in a matrix of alpha. So, I have a matrix of alpha in which there is a small region which is the beta phase which forms then clearly the red region is the interface which is forming between the two materials. So, I can see that there is an interface region which has to be created if I have to nucleate beta within the alpha phase. And this implies that my interfacial energy is actually going to increase. And in other words this interfacial energy which I call gamma is going to oppose my phase transformation that is why it is marked in red. Also the beta phase is obviously not going to occupy the same volume as the alpha phase. Now suppose I make a construction which is known as a shall be construction in which case I take a small region of material cut it out. And now I allow the phase transformation to take place and for a simplification I will assume that only there is an volume change involved in the transformation then the transformed volume for now I will take it to be larger. Therefore, this is my transformed volume as compared to the volume which is the pre existing volume here. That means now my precipitate beta is going to occupy a larger volume as compared to the region from which it was cut out. Now I obviously this beta phase is not going to exist in free space, but has to be accommodated in this hole from which I had cut it out. That means that the beta phase is going to push out on the matrix. The matrix is going to push in on the precipitate and there will be an equilibrium configuration. That means there will be strain energy associated with the phase transformation. And this strain energy is obviously going to oppose my transformation. Needless to say this kind of a strain energy is only involved in solid to solid phase transformation. Of course, this could play a role wherein we have a bulk of a material in a small liquid forms inside. But typically this is an important consideration in a solid to solid phase transformation and I am talking about for instance solidification then I can actually ignore this strain energy contribution. So, to summarize this slide we are talking about first order phase transitions in which a second phase beta is forming from the first phase alpha. And as a further simplification we assume that the entire alpha is turning into beta. This can happen only if I have a thing known as a nucleus forming of beta in various regions of the alpha phase. And then all these nuclei grow till all the alpha is exhausted. So, this is the nucleation and growth process. And we said that nucleation is a technical term which we will try to understand very soon. But before that we also noted that nucleation can be homogeneous which means that there are no preferred site for the nucleation of the beta phase. Or it can be heterogeneous in which case there are preferred sites within the medium alpha wherein nucleation can take place. We also notice that the reason of course the beta transformation is taking place because now we are below the transition temperature from alpha to beta. That means now the beta phase is thermodynamically stable as compared to the alpha phase. But as the nucleus of beta phase forms there are two energies which will tend to oppose my phase transformation. One is of course interface energy between alpha and beta and the other is the strain energy. And now whatever argument I make is what is called a continuum normal argument. And if you look at very small nuclei certain other atomistic arguments will have to come in. But we will take up the standard classically accepted argument to understand this phase transformation. Now suppose as I pointed out suppose I am talking about a solid to liquid phase transformation. Then I can ignore the strain energy contribution and this would lead to my simplification in understanding the process of nucleation. So, suppose I consider liquid to solid phase transformation that means I have an uniform liquid as shown in the figure on the right hand side. And then at some point of time when I under cool the system under cool implies I am below the melting point of the material. And when I am talking about melting point I usually define melting point in heating. So, this is an important point to be noted there never I refer to melting point it is always melting point in heating. And this is a very surprising phenomena that you have actually heat a say for example a copper metal. Then at suddenly at some temperature T m the entire lattice of copper collapses and therefore you have the liquid state which is very surprising. But never the less that is what happens now suppose you are doing the reverse process actually of cooling the metal. Then solidification does not start at the melting point that means solidification freezing point is not the same as the melting point. And we observe a phenomena which is known as under cooling and the reason we observe under cooling is the effect I told you is related to nucleation barrier. So, we will understand that now as you cool the material below the melting point then at some under cooling delta T which is shown here by you notice that there might be some nuclei which form which we had noted before. And these nuclei actually these nuclei would tend to grow to regions wherein have crystals at some point of time all this crystal of course would grow and occupy the entire volume. And all the liquid would have been transformed into a solid of course we should note that these are just schematic diagrams. Now if I track the free energy of the liquid and the solid I would notice that this my T m which have drawn here the vertical line here. Above the melting point you see that the free energy of the liquid now the blue curve is the free energy of the liquid curve and the red curve is the free energy of the solid curve. And I am plotting my temperature versus give free energy these means these are negatively curve these are curve the way they are. Now above the melting point the blue curve is lower than the red curve which means the liquid is stable as compared to the solid. And therefore, this system would prefer to remain in the liquid state. But below the melting point you can see that the red curve is below the blue curve and I am talking about melting point I am talking about the melting point of a bulk crystal. And still we are not talking about the melting point of a nano crystal which we will take up very soon. Now you can see that below the melting point and suppose I am at under cooling say for instance delta T then I would notice that at this under cooling there is a benefit for the system if it goes from the liquid state to the solid state. In other words this is my bulk gives free energy driving force for the system to solidify. But we noted as we noted before that even though there is a driving force but the system actually does not solidify and we will understand it soon. That means on cooling just below T m the solid becomes thermodynamically stable but solidification does not start. And because of this nickel actually can be under cooled about 250 kelvin below the bulk melting point. So, there is under cooling possible and of course this under cooling is only possible if you avoid heterogeneous nucleation and we have so far this kind of a large under cooling. And we have for now we will we have assumed that all the nucleation you are observing is homogeneous nucleation. To summarize this slide I am talking about the liquid to solid phase transformation which implies that strain energy effects can be ignored. The liquid is stable about the melting point the bulk melting point the solid is stable about the below the melting point. And here again I am once again to emphasize the bulk melting point I am talking about here even though I have a melting point defined in heating. In other words if I am heating then the metal melts at its melting point when I cool it the melting point is not the same as a freezing point. In other words solidification does not start and you need to under cool the system before solidification starts. Now if to understand this effect of nucleation barrier let me write down the free energy is involved. That means there is a reduction in bulk gives free energy there is an increase in surface energy and there is also an increase in strain energy. But for now we will ignore this increase in strain energy coming from because we are considering a liquid to solid transformation. So, the delta G for this transformation can be written as the volume of the material being transformed into delta G and the surface area into gamma. And suppose I assume that the nucleus or the second phase which is forming is spherical. Then I can write down the volume as 4 by 3 pi r cube and the give free energy is delta G v and the surface of the sphere will be 4 pi r square into gamma which is the interface energy between the solid and the liquid. Now this delta G function is two terms one is going as r cube and one is going as r square. We know that r cube terms will tend to dominate for larger values of r while the lower power would dominate at the lower values of r. That means now if I am suppose I am just plotting f of r versus r and I am tracking two functions r square and r cube then r cube will be the dominant function above 1. But r square is a dominant function similarly r will also dominate over r square when I am below 1. That means the second term r square is going to dominate at the smaller sizes and at larger sizes the bulk free energy r cube term is going to dominate. Now suppose I plot this curve the delta G curve then you would notice that delta G curve goes up and down with r and this is now r versus delta G curve. So I am plotting r versus delta G and you see that the curve goes up and down. And now this implies that if I form a small nucleus and now I am talking about a phase transformation from liquid to solid and I form a small nucleus of small radius r then I see that the energy of the system actually increases over the liquid state. If I form even larger nucleus then or an even larger solid phase then there is an even larger increase in the energy. If I go to even larger sizes then I notice that there is though a reduction it is similar to this and finally of course beyond the value r 0 there is actually a if I grow a very large precipitate like this then I very large format called a solid then actually there is a reduction in the free energy. That means if I have a liquid and I there is a small r which forms the energy of the system does not decrease but actually increases. So even though we are talking about the we are below the bulk freezing point or the bulk melting point to be more precise the system does not benefit if this small volume or small r actually becomes a solid the energy of the system actually increases. Now there are critical points in this curve the maxima the curve point r 0 below which actually the curve goes down below 0 and we can actually find these numbers by taking this function of delta g which is shown here and actually differentiating this function d delta g by d r and if you set d delta g by d r is equal to 0 then I would find this critical value and which is the maxima in the curve and I can actually set this function delta this delta g equal to 0 and I can find the value of r 0. Now by doing so we find that the by setting d delta g by d r equal to 0 and I find that r star which is the value of r at the maxima or which is also called the critical nucleus size turns out to be 2 gamma by delta g that means larger the interface energy larger will be the r star and what is this r star r star is the maxima in the curve what is the physical meaning of this r star it implies for some reason suppose I suddenly form a solid of size r 1 what will happen is that this r 1 will shrink and in other words this solid will actually remelt it will not because that is the direction of decreasing free energy but by some other chance suddenly some region of space you put a solid of radius which is larger than r star here and I call this r 2 then the system actually can grow the radius can grow by decreasing its free energy and now I am not comparing my r 1 and r 2 the energies corresponding to that with the overall energy at the origin I am just saying by chance suppose I put I take a crystal of size r 2 and put it into the melt or equivalently by some chance and r 2 forms in the melt and I am not explaining how it is forming I just assume it forms then such an r 2 can actually grow downhill in Gibbs free energy while if I form an r 1 it will actually shrink and vanish. So r star is that critical radius above which the actually the crystal would tend to grow and we will note very soon that this r star itself actually lies in the nano scale we will note that very soon now and this r star scales as gamma that means larger the interfacial energy larger will be the size of that r star which actually can grow downhill in Gibbs free energy. Now this and how does this r star actually arise and how does a solidification take place is that the liquid is exploring all kind of configurations because now this is a liquid it atoms are mobile and all kind of configurations are being explored and some region of the liquid you would actually observe that atoms are coming in a precise configuration suppose I am talking about aluminum crystal which looks like an FCC aluminum crystal right this is occurring by what you may call a random statistical fluctuation. Now this if that region wherein you form then crystal of aluminum which look like a crystal of aluminum in terms of its atomic arrangement if it is below the value of r star then such a crystal would actually shrink away and if it is above r star then such a crystal will actually grow and lead to solidification that was it will grow to a stable regime and actually will lead to solidification. Now in some sense we will note later on that this is actually not very surprising and this is what you would expect but additionally we can actually calculate the value which is known as r 0 which is set got by setting delta g equal to 0 and we can actually obtain the value beyond which of course the whole system is actually lowering its energy but needless to say r 0 is a much larger value than r star. So, in realistic systems you would not expect crystallite sizes of r 0 to form would rather expect a crystallite size of slightly larger than r star to form by random fluctuations. Now the r's which are smaller than r star are called embryos in other they are the unstable things which will just vanish and those which are larger than r star are called super critical nuclei and the r star it is all is called the critical nucleus size. Therefore, I need to at least put an r star before which my system would grow and give me my phase transformation. Now obvious it is needless to say that the delta g v which I have used as a function in this function here delta g v which is the reduction in free energy per unit volume is a function of the under cooling and larger the under cooling. Now suppose I plot my delta g versus r curves and now this is my the previous curve which I showed you delta g versus r curve but now for different under coolings. Now if I start with a normal under cooling like this I notice that I get a curve wherein I have a certain value of r star. Now suppose I under cool the system even further which means I am now somewhere here in the curve instead of being somewhere here in the curve I am under cool my system even further and I am somewhere here in the curve. So, under cool my system even larger that means I have a larger delta g v supporting my phase transformation now this is my new delta g v at a larger under cooling delta t 2 if this is my t 2 and this is my t 1 and therefore, at a larger under cooling you would notice that the delta g versus r plot actually sees has a maximum at a lower value of r star. That means that with increasing delta t which is the under cooling my r star decreases. So, r star decreases with increasing under cooling therefore, what happens r star decreases implies that I need to form those r star kind of nucleus which is smaller and smaller. That means if I under cool only by about 5 degree Celsius then I need to assemble a crystal which is this large while suppose I cool it under cool it even further then I need to assemble a much smaller crystal and remember these crystals are arising out of random fluctuations. And therefore, there is a better chance of hitting on a smaller crystal rather than Roger crystal arising out of random fluctuation that implies that if I under cool more there is a better chance that my solidification would actually take place. Now, this r star which is the critical nucleus size is typically in the range of 1 to 10 nanometer and therefore, it is own right actually a nano crystal. Now, we will see the implications of this very soon when you talking about melting of nano crystals. So, in other words to summarize this whole process we are talking about transformations which are the first order transformation which involve nucleation and growth. And we are also talking about the fact that melting point is only in heating and cooling actually we need to under cool the system. The reason we need to under cool the system is because the interface energy plays an important role and is going to oppose your phase transformation. And in addition to what you might call the strain energy which is involved in solid to solid phase transformations. And because I have to under cool the reason I have to under cool the system is that only then actually I get a benefit in terms of my forming a smaller nucleus which actually can grow down hill in gives free energy. And this process as I told you is arising out of a random fluctuation. Therefore, we understood some of the essential aspects of what you may call a phase transformation and especially melting in bulk crystals. And we can now turn our attention to melting point of nano crystals.