 Welcome back and we are continuing on phase transition and after doing the Landau theory of phase transition and the general classification of phase transition, we went on to and the Landau theory, we free energy landscape, we started on a very important area of phase transition which is not routinely covered in the books and courses of statistical mechanics but which are of paramount importance. And that is the formation of a new phase and this is known as nucleation. Nucleation has become of huge importance in recent times because of emergence of nanoscience or nanomaterials as an important area of research. It was always important nucleation because of the formation of solids from liquids or droplets in the from gas liquid droplets in the atmosphere so nucleation much of the nucleation study was done by chemical engineers and who are also interested in the formation of bubbles in the or cavity inside a hot liquid because that had to do with corrosion and many other things. So nucleation always but of late last 15 years or so that it has become an exceedingly important subject. And we discussed in nucleation that how there is a barrier, a barrier comes because although the new phase is more stable that is one of the prime condition of the formation of the new phase by nucleation that the new or the new phase must be more stable thermodynamically so it will have a lower free energy than the old phase otherwise nucleation cannot take place. The reason that it cannot take place nucleation because whenever you form a new phase inside then there is a large penalty for surface tension. So bubble must grow or the new phase must grow by fluctuation the embryo of the new phase or the new phase of the new phase must grow so that the it can outweigh the energy cost or a penalty because of the surface tension. So we worked out the equation this delta the free energy change equal to minus 4 pi by 3 r cube delta G B plus 4 pi r square gamma. So as I starting that we take the derivative and find out the position of the maximum which is called the critical nucleus this was the the well known Becker during theory. So Becker during theory gave you the free energy barrier and the size of the critical nucleus in terms of just two parameters which is delta G B and the surface tension. Delta G B is the free energy per unit volume gap between the new and the old phase well and good and that has been exceedingly popular and in any any books of the nano materials or something you start you becker during theory of nutrition is the first thing. So we often see that the formation of a new phase is thermodynamic controlled or kinetic controlled those languages very much come from the becker during theory that that who is going to so many times the phase that forms is not thermodynamic it is not thermodynamic most stable phase but it is the kinetically accessible phase. This way of thinking about deformation of a new phase is particularly important in say for example like zeolites in zeolites which have a few arts number of zeolites like you start with folgersite then you have a calciumite and then many other things and going ultimately to quartz. So there are about 10 zeolite phases and if you have the high temperature the paradoxical result is that when you are starting at high temperature then it is the quartz most stable phase that forms but at low temperature it is the least stable form that my folgersite forms from the salt of alumina silicate. This wonderful diversity and then of course there are many drugs in pharmaceutical industry and then these drugs are polymorphic means that they can exist in many different solid phases and what is really interesting that some of the solid phases are biologically active or can act as a medicine but the many other polymorphic phases have no activity so when that is why when you keep a drug for a long time a tablet many times it goes to a different polymorphic form and it becomes useless. So in the presence of multiple solid phases of the same material and even of the same composition like in quartz well their number of water molecules are changing but other than that they are the same composition of alumina silicate. Another case is the ice, ice is known to form in some 18 different phases ice starting with ice 1H ice 1C then ice 2, 3, 4, 5, 6, 7, 8, 9 all the way so these different ices are formed at you know at different thermodynamic conditions but they are same they are from water H2O so how to understand all these different things and they show these what statistical mechanics allows you to do, statistical mechanics allows you to ask the questions and answer what are the phases that are will form in what condition and obtain the study of them from nucleation is kind of a difficult because you need to know the surface tension you need to know even if you know the delta GV the free energy gap between them surface tension is not always accessible like for surface tension between folgers site and calcium it is a very tough thing to ask from an experimentalist however a long time ago there was a beautiful work done by great Ostwald you know sometimes called the father of interchemistry William Ostwald who lived for in during the period which is shown here this is shown here and he did many many many things you know huge amount of work on chemical kinetics then in the phase transition Ostwald ripening Ostwald step rule both are very important in the context of nucleation and formation of the new phases today as a natural consequence to nucleation will do the Ostwald step rule and this is actually the step rule which is you put in so succinctly is the following the formation of a phase is not determined by its absolute stability and but by closeness of the growing phase is such a year statement so he is saying if I start from Sol then it is not the course that should form it should focus site which is closest then fall from focus site it should go to calciumite and many many such examples are now coming out in nature and signs of big big papers that people are and and and they are like titanate not just you are a titanate then phosphates and they form multiple solid forms and they really all of them obey Ostwald step rule when Ostwald formulate the rule and somebody would come to him and say okay this this solid does not follow your rule it does not follow he would say okay go and look carefully even his statement was that even if seems like the the closest phase has not formed that means the closest way did form and disappearance disappeared so if you are be careful you will be able to detect the closed form this became a very big industry and it is called disappearance of polymorphs is a very common thing it turns out the understanding of Ostwald step rule took us to very profound results of conditional size in randomizing model or random disorder systems and it forms a beautiful bridge from many many fascinating areas of science some of them I will just be able to just touch the surface but I want to tell you about this rule and this beautiful consequence of many many consequence that that will ask so let me tell you from the very beginning when at the very outset that the closeness of the growing phase is indeed turned out to be correct and at what extent Ostwald understood it I do not know what did that has to do with the surface tension so if I have two phases a three phases a b c and a is close to b and c is close to b and far from a then one can show the surface tension between a and b is much less than that between a and c so it is the what we learned in Euclidation that create a barrier is 16 pi by 3 16 pi by 3 and let me write it down because this so delta G star of nucleation so there are two things now we have three phases a b and c so let c a is your original phase now there is a phase in between you know but in order of form it has to be more stable than you so this is a then this is the frame b then far from that there is a phase more stable c now I have plotted against the order parameter which is actually measure of the closeness so and this is free energy now a and b are close to each other but c is more stable however if a has to go to c it has to create m brow which is very far from the a now look at this free energy barrier then gamma cube gamma is a surface tension what Ostwald Ostwald did not have Baker during theory this Baker during theory it did not be the theory so I did not know the whether he knew surface tension or not that very subtle did not have this expression but he correctly case that this is the when he said close this then a is close to v which in our language now will translate is that surface tension between a and b is less so gamma comes with a cube and delta G comes with a square this is of course not as trivial as I am telling this much more because it will depend on the value of delta G v and gamma but surface tension between a and b is much less than surface tension between a and c so now there is a balance between the two how big is the free energy gap and how different is the surface tension so this is what determines so Ostwald step rule is a routinely used in a thing so let us go to little bit more on that so Ostwald was a when people say you know who is not working here a fantastic is it really lot of any other very very smart man Ostwald offered no explanation he is just stated it from his experimental observation he offered no explanation and now comes when people used to come and tell him he said they are under undoubtedly examples of phase transformation where a metastable phase exist but does not form in such cases we can always assume that the intermediate structure does form but then immediately transforms into another phase this is amazing so this is classical nucleation theory which partly explains it it does not fully explain but moves it to the you have a low interfacial free energy I said but then comes very difficult task of relating the surface tension to the statement of closeness of Ostwald how can we now relate the surface tension to the outer parameter that is the question we will not do a full job but it will be a part job okay now so there are many others that says that the metastable liquid phase fluctuations with the local symmetry are undoubtedly favored and it crystal is not most stable then it could be a picker sir and all these things could act as a picker sir and all these things we do not need to go into that right now so here is the two examples that very much is a form of the from the you know course material and discussed in my book Alexander Mactake I do not have in the book but here comes these zeolites so from this all alumina circuit it goes to foge site in this two calcimide finally it goes to most dense phase and most stable phase that is the quartz and there is another beautiful example that came out in few ego in nature physics in lithium ferrous phosphate and it wanted now because of the enormous improvement in electron microscopy and almost time given electron microscope you can detect all the intermediate states that actually also spoke of so in lithium ferrous phosphate one could see in an something like an interval of 1 hour 2 hour or maybe a day interval that you can see that the soul of lithium ferrous phosphate it goes through all the phases and ultimately the most stable form dense form forms it is just amazing exactly at Ostwald predicted and the free energy diagram then one say the one that I just drew two pages ago that is how it goes from one this is the one which is closest to the in closeness to quote Ostwald that forms first which would be this one then forms this one which is this one then forms this one which is this one and finally the stable crystal form which is this one so one sees in real world the verification of Ostwald step rule one by one and here we in zeolite you also see what happened in zeolite if we do it low temperature then you find the foge site then calcimide however in high temperature again you do not see but you are you just see this one because that if you follow Ostwald then this indeed forms but it transforms itself to success succession of intermediate states and goes over to the courts okay so how do we do this this is kind of interesting and very rather complicated theory but we will go through that but for that let me consider the following following condition that I have a metastable state then I have two I have a melt which is metastable with this final then I have two metastable phases metastable one and metastable two and then I have the final stable solid which is very very similar to what we we saw in this so this picture on from the nature physics recent methods article and the picture that I have drawn here is very very similar and then one wants to calculate how the surface tension going to be and the way this calculation goes is the following and I will not go to the detail calculation but let me tell you this one one finds that if I have the if I plot these multiple phases and multiple surface tension then surface tension as a function of the order parameter difference within the two goes as delta eta to the power and exponent alpha and these alpha is between 3 to 4 that means if I have one thing like that and as another thing like that then surface tension of these because the order parameter difference is delta eta is this much and order period is delta eta much later then this will have a much much larger surface tension how do you calculate that the way to calculate is actually goes through very very interesting and I will briefly discuss probably and then it will be the more quantitative way one does is the following I consider the old phase we let us see the new phase and these old phase and let us see this is the I have a planar interface so I have new here old here and surface remember surface tension equilibrium property that means surface tension is when the two phases coexist so under this condition I now try to find out how the order parameter varies and that one can do by using the other sophisticated theory but you do not need that right now but we have to understand appreciate the so this is the order parameter say delta eta and and this is the two minima this new and these old and is other way around but does not matter so this is the difference so I take the order parameter of the old phase as 0 and then it becomes just like in landout it becomes non-zero now if I plot it then what happens that this is becoming like that now what is the meaning of surface tension so this is order parameter profile I call it order parameter profile it goes from here then this region it falls from the new to the old goes to this now very interesting what is the meaning of surface tension remember surface tension and equilibrium is the equilibrium phenomena so surface tension happens in the coexistence like in the pressure temperature plane this is the critical point this is the triple point and this is liquid solid and gas so these lines are the coexistence line and surface tension is defined at coexistence along this line only and along the coexistence free energy is the same okay now let us go little bit more thinking so let me now draw a free energy free energy which are the same okay so these two phases are the same free energy old and new now they are at coexistence like here so let us old new so when the free energy when the system is here which is here the free energy or chemical potential is the same when it is here because they are at coexistence if I consider solid liquid then these are coexistence at coexistence free energy system however in order to bring old and new together in order to create a interface in order to create a interface I have to go through this region of order parameter so I am plotting against a direction z but it has the order parameter values are intermediate order parameter neither this one which is this one neither this one which is this one but it is in between so the a certain amount of phase has to be in this region if you can think in terms of spin then there is a region which is all spins are up the ordered ferromagnetic phase and then you have a which is disordered so there is a completely now in between it will be a region if I have to put a interface in between there is a region where they are partly ordered that puts it the system in this region so that is this region but this is a high energy region high free energy and as a result as a result you have to spend extra energy and that extra energy that you have to spend is the surface tension so this is the definition of surface tension which is not told in most of the under graduate text books or even in books that they do not explain that the reason of the surface tension at a molecular there is a molecular level understanding of surface tension which is because you the this a part of the material a part of the material has to be in the intermediate order parameter values now very interesting comes now what is the next thing now if I have an interface between these then I have to now understand a system always minimizes the extra energy spent so the system tries to minimize amount of material in this intermediate region so it acts to make it narrower so there is now competition between the which is in the one is in z plane and so you are looking up two dimension one is really special dimension z and the dimension is order parameter so when one wants to do that one wants to derive a free energy expression which takes into account the free energy of the system as a function of order parameter as we traverse this thing plus the extra energy cost of creating this thing and creating such a gradient and that is the thing that we are going to describe now little bit so this is a we are not going to go into very detail into it but I just want to tell you that if now density like gas liquid density is the free is the order parameter then it has the following kind of a thing one is the free energy at a given position row or and then other is the gradient term this term that means how sharp you are very if you are varying very sharp then you have to pay an energy but if you make it wider you have to pay an energy here so there are these two competitions then one wants to make it narrow the interface narrow but the other one making it a narrow is putting these two opposite phases next to each other that they also do not like and so that is the square gradient term we call you a square gradient term so there is a term this term is like a free energy of van der Waals term this term so these are the two things with the condition that this for a free energy or chemical potential that chemical potential of the two phase alpha and beta is the same and then we minimize this so then important thing to know is in an interface we are you are trying to make this guy narrow in order to minimize the amount here but in order to make it narrow if you make it too narrow then you have to pay a energy in terms of the square gradient term because you know if you vary too fast then you have to pay because you are making a density change very very sharply and that system does not like because system does not like fluctuations you know as I told you the all these density terms are harmonic they are quadratic in a density so it does not like because then what happened that the packaging is in the molecular interactions are putting molecule next to each other that becomes very difficult so this extra term so in addition so you have basically saying okay free energy of these kind of a term has two terms why integrate over the z I have a term which is a free energy as a function of rho z this is at every point what is the free energy I call that a Van der Waals term and then you have to do so you have a one term which is Van der Waals term which is just picks up what is the value of free energy at that z so this is this comes from this comes from this thing so this rho as z is changing rho is changing and as rho is changing it is going from this climbing this up and then so this is a Van der Waals term but then how sharp it is varying that I said the kind of a spring constant of the system that is to so this is the quantity now what you do you listen carefully because this is something very neat and very clean and very simple it is just very new or quite new to students so now what one does one minimizes one minimizes the free energy cost with respect to the so this is the functional derivative means the derivative of a function of function when you do that then you get the solve this equation with these free energy functional then you output is this rho z and once you have the rosette you put it back in the free energy and you get the excess free energy and excess free energy is the surface tension so now we know how to do these things we know how to think about it that when so there is a very interesting thing why if you make this very huge difference in auto parameter then you have to make this region rather broad because there is a very large variation and then you are facing regions which are very high region here so as a result of that nucleation barrier goes up so while Baker during theory with Zelda which correction tells you that how to tells you that the nucleation barrier and how to understand nucleation it does not really tell you the basic essence why surface tension bf 0 it bf it does not connect to Ostrich step but these logic which Ostrich had probably had in the back of his mind because he offered no explanation but these required lot of work in the last 10 15 years that people begin to put things together work out this beautiful theory this theory goes as a name of this particular free energy functional I have written down is goes by the name of Ginsberg land of free energy functional it is such a noble thing let me write down that it is the one on a whole critical phenomena is based on this free energy functional so this is that is what I wrote down here actually we do more in terms of general just one dimensional you can have three-dimensional also and actually three-dimensional is you want that is you commonly see but I just catered it to my own need here and this is to be understood as a free energy varying so you pick a Z then that Z give you value out of a meter then you go and put it in the land of free energy function of and also energy function and then you have this is the Ginsberg land of free energy functional in many different forms it comes in many different areas of the natural science condensed matter science okay so there are certain says the nucleation stands in the boundary because I think it is a very it is a very nice slide let us go through that this is really very interesting the nucleation stands at the boundary between thermodynamics and kinetics that because we are calculating the rate but we are going from thermodynamics we really did not do much of kinetics as such except in Zeldovich the theory gives simple nucleation by in terms of energy difference of the parent and daughter phase and surface tension this is the Becker during theory then we know we had a we say it and rule of surface tension in the Ostwald step rule that is a very nice thing so Ostwald step rule then tells you these Ostwald step rule 10 called at OSR that tells you that yes when I have a situation like that then if this is the melt or solid in a stable 1 meter stable 2 and the stable solid phase then how it forms depends very much not just on the depth not just on the gap not just on this gap but also because the locations of the surface tension so when people try to talk of the stability as if the phase formation of a phase is given by the stability they talk of this free energy gap but what people do not talk of a county to a dot Ostwald pointed out as this one so this is the kind of thing one routinely sees in experiments that you see the stable solid is surrounded by coating of the metastable 2 which is this one and then metastable 1 which is this one and finally outside is the melt so this is like for example when you form that what I did not go along Alexander and Mac takes paper that often we see in FCC forming solid is coated by BCC so and that because BCC phase is closer in density and order to the liquid remember that if I have a hard spheres then the liquid typical density is in a dimension is even it just the number density is multiplied by cube of the diameter that you call rho sigma cube that is about you know about 0.5 per liquid then it becomes 0.68 for BCC and cross pack and 0.74 for FCC this is the cross pack so when forming from liquid then it goes to BCC first then within the BCC embryo this is not a universally varied picture but reasonably close to it then within the BCC FCC forms that is what Alexander Mac did say and this is again connected Rostrold's step rule that the from the melt the least stable that is what Rostrold said least stable but which is closest to the in symmetry in order parameter in density that is closest to the melt will first form from that something else grows so this is a particular advice since this is a course not just to the alchemist but also we hopefully for material scientists this is a bread and butter in material science where people first gave is the formation or synthetics of a new material and one still struggles one still struggles to crystallize one phase and at least having some understanding would help to figure out what is going on into this rather fascinating and emerging new field you know which is kind of borderline between in between between physics and chemistry and material science and we take a great great importance into this thing of okay I think we will stop here for Rostrold's step rule and we will do something else very similar to this now in a in a next lecture that is in spinodal decomposition thank you.