 So, what we will do today is an important class which is hardly taught in chemistry departments, at least nowhere in India in chemistry department it is taught and so I insisted that this should be dealt in very elementary form and in detail. And the in the book Stratmag book there I have devoted 5 chapters of 30 to 5 chapters so that kind of tells so how much importance I give to the phase transition. As I told in the last class when we did the Mayer's theory, so Mayer's theory is one of those 5 chapters in Mayer's theory Mayer was the first to do a theory starting from interaction as I told you ideal gases do not they do not have phase transition, ideal gases do not become liquid they do not become crystal they are in good reference systems but they cannot explain the real world. The what you and me exist on the phase transition happen because of molecules and atoms interact and one of the most dramatic consequence of the molecules interact is the phase transition. And however the understanding of phase transition is very tricky because the concepts are profound and deep and many of the things that we know in physical chemistry and physics chemical even chemical kinetics and the as far field as chemical kinetics are derived from the basic idea that phase transition get to us. For example concept of all the parameter, concept of free energy landscape, the given glass transition, ruggedness all the things that we hear or multidimensional processes many of these things have their origin in this is beautiful and the very profound and fundamental field of phase transition. So this what we will be doing as though it is a land of phase transition it will be lot more so the main theory of this so in 1937 and you know this great big bank episode that when penny was seldom was teaching penny physics and the one warm summer evening in Greece I suddenly remembered what I am going to say reminded me of big bank theory if you have not seen that particular episode you should see that is really one of the most hilarious in Sheldon and penny. Now in 1937 you know much later than that Greece there are two things happened in the same year Joseph Meyer came out with the theory of condensation and which went on as I mentioned the last class the our cluster concept of cluster expansion leading representing interacting molecules as dots and lines between them which gives to graph theoretical picture of physics that ultimately developed into Feynman's integrals and all these things the representation of an integral in theoretical physics as the graphs started with Joseph Meyer then it went on to become Meyer our cell cluster expansion in quantum mechanics okay and of course over the Feynman's did in a quantum electrodynamics and we are talking of atoms and molecules and phase transitions the same year 1937 Landau came up with a theory of phase transition which is extremely simple and elementary but it captures very essence of many of the things that we will be talking today but before I can do that I will try to go through some very basics as I said because the generality of phase transition is not what it came is usually used to see we are used to specific properties of atoms and molecules that attract us. So when it comes to thinking generally in general picture this is not something that is our forte okay so I will have many elementary concept basics chapters that will be there you can just quickly read or I can read but it will be not spending too much time on that that is ubiquitous in nature it is manifested in a large expended atom main thing is that there is a my infinitesimal change in a control parameter like temperature pressure or which will be out of the parameter I will talk about it later infinitesimal change leads to a macroscopic change or thermodynamic change so the you know remember the concept of derivative you put you put the change of your dependent variable in the numerator and your independent variable in the denominator so so we always think of change in terms of a derivative so you can easily see if I have a infinitesimal change in a control parameter and I have a macroscopic thermodynamic change then that means you are talking essentially of a very peculiar phenomena which is called singularity see much of the things that you read in your calculus are the ones of the functions that are called entire functions or their first or second derivatives exist so that is how you work with them if you have a singularity you cannot do anything so that is the first difficulty we will have so it is the many branches of natural biological science like you know of late we are the protein folding that is completely used the language of glass transition and the folding funnel or which is all these things are just completely come borrowed from phase transition or nowadays in many of the things with genomic studies and all this again from phase transition so it is a discipline by its own merit with entire books devoted to phase transition like the book by Eugenie Stanley is on phase transition then there is one by Balescu there are many books just on phase transition by itself and I find people I disdain people who do not find it interesting very frankly this is such a beautiful subject okay is a strong language but I think sometimes you need the strong language at least in India so this is amazing it has different diverse types of phase transition that we encounter in nature yet there is an amazing unity in such diversity these are all lines from my my own textbook and my own writing okay among all this there is a unified property among all the refined kinds of phase transition is the nature of discontinuity so I can characterize a huge number of phase transitions by saying the where is the first order or second order as soon as I say it is a first order then there is a huge number of phase transitions which immediately become a you know that their certain characteristics which are common or universal when I say second order phase transition again a huge number of phase transitions can be grouped together so this characterization of first or second order phase transition you know is possible and that brings this the unity in diversity that I have been talking about okay so depending if this transition is categorized in different order namely first order second order and classification is the grouping of phenomena in similar characteristics together and a few of them large number of phase transition observed you know so what it one is trying to do now characterization and it's a very powerful for example when it's a first order phase transition gas liquid phase transition liquid to crystal phase transition many many other phase transitions or even magnetization magnetic phase transition in the presence of a magnetic field electrical transition is basically a huge number of isotopic nematic phase transition nematic smectic phase transition liquid crystals the huge number of things in polymer as soon as I say it's a first order phase transition they are all very similar so I know what does mean by first order phase transition similarly when I say second order phase transition the order disorder type of phase transition gas liquid phase transition critical point and huge number of transitions like that which this now so we started off saying the order now before that I tell what is an order let me tell you what we need to know something to characterize the phase transition this is done through a commonsent or order parameter it is a physical quantity so chosen that it is 0 before the phase transition but becomes non-zero and acquires a finite value after the phase transition that the definition of order parameter which is selected which is chosen for example if I have a gas to liquid transition then so the canonical gas liquid will have we will see it many time pressure versus density then this is the so this is gas this is liquid this is crystal solid and this is the so this is the coexistence now the order parameter at a given temperature will be this change in density that means density here rho L minus rho gas that will be order parameter in a magnetic transition it is a magnetization that will be the order parameter because that is 0 before the magnetic transition ferromagnetic transition and non-zero afterwards so these kind of identification is extremely important and that was the one that was implemented by Landau because this change that I will be seeing here is which characterize so the whole idea is that if I want to understand how a phase transition takes place and why a phase transition takes place this why and how will depend I see if I understanding is proper or not whether I can explain this change in density for example typically let us say as I described earlier that one of the very another unifying thing we do is we introduce rho star that is n by v cube of molecular diameter the universality come because all the liquids have very narrow range of rho star they may have very different number density like 10 to the power 22 per cc then they 10 to a 23 per cc but in that when you take the molecular diameter the number density and multiply you find they are all between 0.8 and 0.9 this is wonderful then most of the gases they are widely different but at the condensation typically not to quite a bit lower than critical temperature but not too low this is typically will be between 0.05 and 0.1 it is a 0.1 so now I suddenly have a fantastic whether carbon dioxide or methane or water I have a fantastic or methanol a fantastic unification when I go to the dimension this quantity now as I was saying I am just giving a number this rho star is 0.1 and then liquid typically goes to 0.7 or 0.8 so I have to if I understand properly I should be able to say why gas to liquid transition density increases by factor of 7 ok and what pressure it happens and I have to be able to say what are the other characteristics of the change like latent heat so these are the kind of properties that one would like to know the holds the a starts hold this understanding of phase transition starts with the identifying an order parameter which is as I said 0 so if I did well mine as rho g so it is 0 order parameter 0 just before there then it jumps by a large number ok. Now this was the very famous classification by Aaron first Aaron first introduced this and this is the beginning order parameter and Aaron first classification is the beginning of the our formal study of phase transition so Aaron first defined the following way he noticed that like gas liquid liquid crystal these two here and you know as I said magnetic transition in the presence of external magnetic field all are characterized by finite change in a first order property first order property mean first derivative of free energy like entropy and then pressure or other idea you can get the density so the first order properties change then he notice that there are cases where it is a like superconductivity it is the second order property that is changing order disorder transition metallic alloys with the second order property like this specific heat the susceptibility specifically second derivative free energy susceptibility second derivative with respect to following ok. So, so there is a class of phase transitions which are characterized by first derivative discontinuity or singularity or anomaly in the first derivative of free energy and their properties which are second derivative and this is suddenly you begin to say ok then there are characteristics between liquid liquid and nematic and spectic or gas and liquid liquid is all the same same yes they are all very similar and we will see they are similar even more they are all have the latent heat while the second order phase transition they have no latent heat this is very important they have no latent heat they have discontinuity second order property. So, so the end phase classification at one short allowed us to classify very different kind of phase transition which apparently so different but they are similar ok. Again motivation of the questions I have said that these I have said all the various phase of solid liquid magnetic include order disorder transition, metal insulator transition, superconductor transition, sol-gel transition, liquid crystal all these things. Now, one very important thing I should point out that it is a field simply intuitive arguments fail miserably and this is one of the reason our the theories are held so so highly regarded and I will say how it does not work like one of the most important thing is the second order phase transition is the which was solved in 1971 by K. Wilson by introducing the renovation group which is one of the most formidable thing that has been created by our mental things many human beings our intellectual feet. Now, so what we have to say I as I say always why and how why we study I just said you and what are the main characteristics of the phenomena why does it occur in the first place one would like to understand both from microscopics is not here microscopics and from macroscopics both from fundamental and phenological point of view. So, there are many examples I said let us go through them liquid so this is the one of the most common things we see everyday life and the famous paper of come from India of T.V. Ramakrishna and you saw of Kanpur IIT that one of the highly cited paper is the freezing transition the first line of that beautiful paper was that freezing is the most ubiquitous of all the phase transitions great line they started with then parametric phenomena phase transition or the disorder phase transition binary mixture large change of composition normal super fluid transition liquid helium taking to super conductivity super fluidity then metal insulator transition again change in the estivity sol-gel transition helix coil transition DNA protein folding least goes on no very good question many of them are second order like superconducting transition is second order superconducting is very unique I will talk of superconductivity helix coil transition is the first order phase transition helix coil is a first order phase transition protein folding is considered also first order phase transition and but the change is very small helix coil transition order parameter is the we use helical pitch yeah exactly exactly and this theory if you are interested take a note it was done by Ziemann Bragg Ziemann Bragg 1959 two pages next to each other both one and half page or even less than that and other by my advisor gives and debuts you and Julian Gibbs and is beautiful theory the reason it becomes phase transition the first order it is very nice see as we will discuss a second order phase transition particularly order disorder gas liquid at critical point they are characterized by large cal fluctuations and so huge density fluctuation near gas liquid transition the compressibility goes to zero and gas liquid transition order disorder transition again your composition is hugely changing that is the composition that is the order parameter that first order phase transition characterized by that you need an inflation that means it does not have it is not characterized by that kind of huge phase transition so this is so when helix coil transition takes place if you want a DNA helix to be broken in into coil then you have to first break the three hydrogen bonds to break the first hydrogen bond you do not get second to break the third one then it starts rotating you get the rotational entropy okay so that is the essential inflation phenomena in that okay we would not be able to understand everything today in this one lecture as I told you full semester course is given on phase transition but the idea here is to introduce the universality and the grandeur of the field and the concepts like order parameter then first and second order phase transition which I said that is the first and by the determined by the derivative of the free energy where it becomes discontinuous that is this is the iron press classification the order of it and given by the first derivative of the free energy with respect to order parameter or other thing order parameter that shows discontinuity so in second order phase transition first order derivative like density entropy they are not discontinuous but in the first order phase transition is the entropy that is what you call latent heat that is discontinuity so heat content entropy jumps and from crystal to liquid entropy jumps and entropy is the first derivative of free energy that is why melting freezing are first order phase transition okay these I said this continuous change microscopic variables finite change of a macroscopic variable caused by infinitesimal change the idea I said but it is very important that infinitesimal change in a variable but a macroscopic change in a variable is the hallmark of a phase transition that is the uniqueness and the beauty of the phase transition okay these are the you know many of these things this is the famous pressure temperature density plane so this is the vapor this is the liquid and this is the critical point and this is the coexistence all across coexistence we get a phase transition here we get condensation here we get boiling okay and this is the coexistence when liquid and crystal and here you get the phase transition lot of interesting things happen and lot of study going on now in the super cool super fluid super critical fluid and you also know these are we asked in our interview and about 80 percent students cannot answer any of them I have done statistics 80 percent of the student cannot put they are coming for doing PhD pressure versus temperature plane they cannot do you can ask this question it is very interesting they will put solid here liquid here and so it is a kind of a rubies cube we make them do okay but here is a first order phase transition a liquid solid melting here sublimation is the first order phase transition gas liquid is the first order phase transition but it is only here it is the at the critical point nature of the transition changes it becomes a second order phase transition and that second order phase transition is like this so this I have drawn the van der Waals loop also to this Maxwell construction pressure versus density of senior first year undergraduate and in the first year undergraduate books there is a great great section which is called law of corresponding states that is done in a huge way now it is time for you to ask a question what is the law of corresponding states and why it is so unique absolutely this is the our first in physical chemistry or physics the first first a with the universality that I can P by PC V by VC and T by TC and I get an equation from van der Waals remember is on the right hand side 8 by 3 R and you get that equation now describes everything suddenly you get a master car you find when you plot pressure versus density everything is different you cannot even plot them on the same graph paper because density is so different pressure is so different but suddenly when I plot P by PC and V by VC or O by OC suddenly everything collapses you get a master car when van der Waals did that you of course knew and van der Waals did enormous I as I mentioned once before that the he is one of the underrated scientists is huge amount of work he has done so this law of corresponding states first told us that there is something very interesting very fundamental going on I do not have a slide on law of corresponding states I should have had but I forgot okay now then order parameter is chosen physical quantity as it 0 before the transition becomes nonzero afterwards so and that provided by liquid oil transition a clear example that we will go to but here I have said entropy versus temperature is the first order phase transition however in a second order phase transition it is of course this is but it is specific it that diverges all of us know that only there is only one truly second order phase transition in R and sense only one that we know and that is the superconductivity so entropy entropy goes there are lot of things on that because this has to be a positive because DSDT is specific heat those are the things will do a little bit then resistivity against temperature this is the only is this superconductivity but most of the second order phase transition that we call two kinds of phase transition is a end phase sense other one where lambda transition in super fluidity so you guys have seen that these figures teachers they by thing everywhere professors or whoever teach it or write books they are all excited about this amazing thing that is here and this itself has given rise to huge amount of theories understanding papers Nobel prizes is and as I said is highly regarded I happen to do my PhD start after a few years after Kenneth Wilson's famous renunciation group calculations and there is still so much excitement going on and Leo Katanoff who started some of the things was at Brown and Leon Cooper who did the superconductivity Cooper pair was at Brown so it was a lot of fun to seeing the people who created physics around you so this is the example of a Erenfest first order phase transition this one Erenfest first order phase transition Erenfest second order phase transition this we call continuous transition or Erenfest second order phase transition of the continuous type that the little bit we do as I said some of the things will be elementary I to introduce people to the nomenclature ok now one thing as I said I always teach because this has come out from India though the first one that by Karkut and Monroe in 1941 but this that was not successful for reasons I cannot go here but the first successful theory of melting a freezing was done by Ramakrishnan and you saw in 19th written paper written in 1977 came out in 1979 in physical review what they did and I teach it because it is a wonderful example of outer parameter so this is the density of the solid which is in homogeneous position dependent density liquid is homogeneous there are some position dependence that's average density is Rho L if you have read Kittel or any solid state physics books then the density of solid is written in an expansion where G is the reciprocal that is vectors so when you do the x-ray you get reflection of this plane these are the sharp peaks however they the solid will not show there unless this factor is in front this is a phi G this is the order parameter that becomes nonzero when liquid becomes crystal and phi naught is the fractional density change between liquid and crystal like in when I go to liquid to water twice then this phi naught is negative but most of the other cases sodium and all other this is our molecular liquids this is positive because density increases phi naught and phi G are the one of the best examples of the order parameter so there are certain things given a parameter magnetization some other examples gas to liquid density as I said Rho L minus Rho G and liquid to crystal then phi naught and phi G and there is a bunch of them because you have a bunch of reciprocal that is vectors in principle you need to take all of them but in practice we find couple of them it describes the theory and they get very good agreement with these one of the most successful theories of phase transition it is actually also called Landau theory for reasons I will explain little bit so this is a very important slide where one gets examples of the order parameter now I want to tell you and if you can tell me it will be even better why we give so much importance on order parameter so this is again the same thing at phase transition level by always very very free energy with classifying but then first order phase transition second order phase transition, true phase transition is superconducting transition is the true phase transition is exceedingly rare we know only one case ok so now some of the questions we need to ask as I said why and how why does a system undergo such a sharp changer phase transition point what is the origin of discontinuity can we evaluate the transition parameters such as these are the things I wrote to the morning to make it really simple to bring home the point now I have to ask you a question like last question corresponding state was very good now Koila so somebody again tell me why we need order parameter well one is of course the unification one is order of a transition then unity among diversity all this beautiful stuff but there is one technical reason why Landau made it or introduced it and made it so famous there is a practical view you know the scientists are very pragmatic practical people feautism are even more practical they may go around with the air they are not practical but feautism are often very practical people because we do what we can is actually control parameter correct but I am asking more yeah that's a practical side I am saying theoretically Landau was a theoretician he built this theory and he is the one who actually introduced all these things the order parameters and all these things ok the reason is that we do all these things the reason is that we need to have a theory now let us think we need to have a theory and what do we really need for the theory what is that this thing or what is the question what is the question the question is that it reaches here it goes over to liquid or reaches 0 degree centigrade water in your refrigerator becomes ice and so when I ask why what exactly is the question why means why means that there are certain characteristics in this branch certain characteristics which manifest itself which kind of initiates or motivates the transition ok one very simple answer you could give and I expect you to give what is the thermodynamic answer why water becomes ice absolutely so free energy becomes minimum so if the free energy is like that then that is the coexistence but when free energy goes down then this will be T below 0 degree centigrade and this will be ice and this will be liquid water so thermodynamic sense we know the answer that it is it is the the free energy becomes low and so that was one of the first thing that people know and people knew it I think even in 19th century or 20th century so this part is well understood that a phase transition takes place when the free energy the system finds that it is more profitable free energy voice to go over to the new phase ok that is a valid and good answer now as I said here why liquid sodium goes to BCC while argon goes to FCC these are the kind of answer if I want to know then I would be I have to the characteristics here characteristics here in this region that holds the key that it goes to BCC sodium similarly here is a characteristics that tells you how big the jump will be at a given pressure from gas to liquid how do I describe that now ok one of the thing is that I calculate the free energy here I calculate the free energy here then I say ok I just what we did there we we we compare the free energy and both will have minimum and then I when this minimum goes down then it system goes over to the this place that as is a is an explanation at a level which is macroscopic level but we want to know little bit more than that and we also want to know why certain universal characteristics emerge in this kind of things and what allows us to calculate the transition parameters ok so here certain things are there I am not going to go too much into it this is just what I said free energy stabilization so free energy against an outer parameter x or can be like density or temperature then you here solid and melt M is the melt then this here this is stable phase or S is stable and is metastable is stable and that lower free energy then at coexistence they are both same free energy both are stable but then when you lower the temperature or increase the density then goes to the other things and there are certain stability conditions of convexity and concavity which I do not think I will have time for that but basically once one says that if there is a maximum there if I draw a line between this minima and this minima then every point will lie above that chord ok so these kind of things are maximum or if it is a minima here I draw a line from here and there then every point intervening is lies below that chord so that condition simple condition is kind of stated here and that has certain consequences and then another is that for example Gibbs potential is a concave function and that means specificity is positive concave function again means the compressibility is positive there is a negative sign in front of it so on the other hand Hemels potential is a concave function of temperature and convex function of volume which is given here these are the slides I live with it and I will not have time to go through these details of that but these are essentially fundamental things telling you stability of a system but that does not fully answer the questions that I am after here