 Welcome back we just finished studies of several chapters on phase transition which are all described in my book but what I am trying to do here take you through several aspects and give you more of the physics and the physical picture. Many cases like we did in Zellovich and on the partition functions and many other cases beat on the Landau theory we did it mathematics but we will do now again some mathematics but that is not the real aim because details of mathematics you can get in my book the statistical mechanics this book you can get it in all the chapters are there though what I am teaching not in the same chapter because I want to make it interesting I want to make it engaging to students and I want to make it worthwhile for them they should know why they are studying you know I do not believe that one should study because it is a course material one must have some amount of enjoyment I am not saying fully you have to extract you can jump up and out but you should have some amount of involvement and some liking for the subject otherwise it will not stay with you and not only equations do not stay with you but physical picture will stay with you. Now coming we are so we were doing a Landau theory phase transition free energy functional Landau free energy wrote down the beautiful free energy functional that if it is F naught which is equal to 0 then E dot of L eta then D like that and then since free energy has to be minimum then this D F D eta has to be 0 at D F D eta at has to be 0 at eta equal to 0 that immediately means this term disappears and then we are left with simpler things and that is F eta F naught plus D eta square plus C eta cube plus D eta 4 and then this thing I can this thing I can just remove for the time being and make this delta F so then I have B eta square and C eta cube and this is important in the dependent term and for first order phase transition I need the asymmetry of metastability and hysteresis I need the C term but for critical phenomena where I have a flat like that then it becomes like that I do not need this term. So because of parity because free energy must be same here and there and they are equally displaced from origin one is minus and the plus so no odd term can be there so then the free energy just eta square eta plus this sometimes called phi 4 theory and very very common in field theory where these things are. So this was the Landau beautiful then one can go ahead and can get the entropy from here and one can get the I will do it again when I do the critical phenomena little bit more we can get the entropy from here we can get the specific if you come here and the similar things but that I will do little bit more when I do critical phenomena right now I am doing the first order kind of phase transition and similar things which are more we need it much more in finger chemistry materials character the nucleation and the austral step rule and similar things. Now we also did but now many of the phase transitions require the heterogeneity so there must be a homogeneous system must become heterogeneous because this we create a an embryo in a so there is heterogeneity any fluctuation of the new phase is a heterogeneous fluctuation. So there is a density fluctuation gas liquids when gas goes to liquid suddenly a hugely more dense region comes and but when in chemical engineers do the cavity bubble forms inside the liquid boiling liquid and that is how it evaporates then you suddenly have a very low density region in a density so this is heterogeneity because this huge density change takes place at the surface as I go through okay. So now so how do I describe that heterogeneity Landau theory cannot describe the heterogeneity it has the macroscopic density in it that is why Ginsberg Landau comes in Ginsberg Landau now tells me okay if I have a order parameter or density as a function of z then the way to get the free energy is you get the free energy as a function of okay this let me then a as eta as a function of z this F is this F this F is this Landau free energy so this is Landau however when I create the heterogeneity I have to describe that and that is done by using a harmonic and this is which allows me to vary order parameter like this an order parameter in a gas liquid liquid solid interface or at nucleation then this is called the famous Ginsberg Landau free energy function of in nucleation in nucleation what we do we said okay I have something like that and this is the new one has to integrate from the old one and then what do we do we do breaker during and this is goes over a barrier comes like that okay so this is the nucleation r star we did it r and this delta g r okay that is delta g as a function of order parameter this order parameter and but this is a function of r so one has to be careful okay but now I am going to so we have done nucleation we have done surface tension I described how one talks of surface tension because you create a surface a create a surface and then you go through the surface and creation of surface means you go through the an interface and this region is neither here not here surface tension always the equilibrium so free energy is the same so this region is rather in this unfavorable domain so it seems it has to go through there is no other option any one face to other face has to create this interface and creation of interface has to put matter in the free energy large free energy barrier now the system tries to minimize the free energy and the way it minimize the free energy it plays around with the profile so now you have to put in this see that is what means what land off does that there is a variation with cost to energy and there is the this putting in the unfavorable sort of cost to energy both costing energy but you have to minimize that and that gives the surface tension so that is why this is a statement you have always entered a why raindrops are spherical because it it it minimizes the surface okay and so it is the same principle that is working there so now what you have done till now we are going from one minima to another minima however there is a large class of phenomena where something else happens like when a falcon reacts and there is many many pattern in nature we see formation of patterns they go through completely different setup and this is the phenomena of spinodal decomposition this is again is a very famous and well-known subject in phase transition and in material science you know because this is something always taught in materials metallurgy departments so here what happens we consider the following thing we have a homogeneous space at high temperature homogeneous mixture of a and b homogeneous high temperature now we suddenly quench it for example say we have a temperature scale temperature just to one I want quench it very far I want to point and that happens see when you take the volcanic rocks you find that there are many of the volcanic rocks a beautiful pattern okay many of the patterns are because of this iron bearing sol and you find the stripes same stripes you find in zebra you find the same stripes many many places in nature and the reason is that these are non equilibrium processes so now I suddenly quench it but what happened at low temperature low temperature these not homogeneous low temperature a and b phase separated so a is with a and b is with b like you can do aluminum and manganese and that alloy it is it is it is a it is a homogeneous mixture high temperature quench it if they are phase separated now these are happening maybe happening all in solid state so in high temperature they are disordered but when you go to low temperature they separate however I do it suddenly so then suddenly the system finds system finds that it is put in this region so these are free energy and this is the maybe composition which is the order parameter here so define that the composition is 0 when they are homogeneous so you can say it is x a minus x b is the composition order parameter 50-50 mixture so completely mixed so minimum here is where this quantity difference is 0 and is there a harmonic as we are describing again and again fluctuations are harmonic in the first order because simply because the first derivative is 0 because it has to be minimum with respect to free energy free energy much minimum with respect to fluctuation that is a thing the first term goes to 0 so first term this the real the relevant term comes this harmonic term that means z x square okay then delta x cube could be there but for small fluctuation delta x cube is also not important because delta x cube is small so the initially it is always harmonic like that okay now they are low temperature they are phase separated so they are just like the coefficients I have been drawing whole morning or previous lectures so this is a phase and this will be phase they are phase separated so a and b are phase separated and indeed now when I coin something like that is very interesting thing going to happen the system now has to go into this phase and has to go into this phase however it is a homogeneous system at at high temperature they are homogeneous a and b are mixed with each other now I am suddenly telling them to phase separate and it starts to face but it is a nucleation there is a mass bulk phase separation but phase separation in a which it was it finds it was in free energy surface like the upper free energy surface it then suddenly finds in lower free energy surface but mass is conserved I cannot just destroy one a particle in place and that if placed by b particles similarly I cannot take a b particle from here and put indian a particle I cannot do that number is conserved so I must do it in a progressive fashion moving around how do they move in such a dense thing they move by diffusion that is the only thing they must move it so however this is a very strange diffusion diffusion usually diffusion makes diffusion makes inhomogeneous inhomogeneous system homogeneous that is what diffusion is actually diffusion is very closely connected with entropy there is a relation between diffusion and entropy there are actually several relations diffusion increases entropy increases so diffusion favors entropy so you have a inhomogeneous system which is lower entropy then diffusion makes them homogeneous that because entropy increases this is very easily understandable very pictorial and very although profound but very simple to understand however now we are in a different situation we are now in a homogeneous system that homogeneous system has to become inhomogeneous but with the diffusion so this that is why this is sometimes called spinodal decomposition is called aphyl aphyl diffusion let me give you some example now because we are talking a lot so this is an example of computer generated spinodal decomposition and the evolution of structure through dynamics so at the high temperature a and b I made one a may be white b may be black so the black and white white is just empty and blacks are these small squares so here is near almost 50 percent 50 percent and then I suddenly quench it then it starts to phase separate it is different from equation you can see is a large scale large scale large amplitude phase separation scenario so now it goes intermediate you can see nascent phase separation how does it do now a likes a now a wants to be with a that is what the free energy dictates it b wants to be with b that is what free energy dictates it so now the nearby black ones they start forming chains and white ones they form chains so you start seeing the initial pattern formation then they move around more and blacks form and but they have to avoid each other and in the process they form this beautiful pattern which is called spinodal pattern and this is what one sees in many many many many cases in nature in a metallurgy in in physics chemistry in in in any kind of things and in also in basic theory goes over almost to the animal world a very similar equations that we are going to use that is used so so the formation of this pattern and that can be experimentally studied through x-ray and x-ray will be x-ray scattering will pick pick out this pattern formation will pick out the emergence of the length scale and this length scale then shows as a pick in the x-ray structure factor there are many many things like that these multi strong stronger attraction between similar space component that a a like a b likes a then homogeneous mixture high temperature entropy dominates but suddenly coincidental temperature is enthalpy that is a driving force and the non equilibrium conditions are in temperature drop etc phase separation mechanism is difference from nucleation and growth what we studied in the last lecture or so there is a nucleation or here before last two lectures nucleation and growth now it is different but it is nevertheless is a very well known of phase separation and so there is the showing that the volcano the volcanic rocks that form and when the volcano takes place then after that it is just go out and suddenly finds itself from something like 5000 degree centigrade or something like that to room temperature you know to it 0 degree centigrade for 5 degree centigrade and then these beautiful patterns in that form because a volcanic melt has many different components ferrous sulphate sulphide one of them and some zinc and other things are there and they like to be among themselves and so that is beautiful so you can imagine always that here out of a homogeneous state we are getting an homogeneous state so you immediately one thing that comes to your mind okay so this is the free energy minimum okay too free energy minimum but I have to separate them out so in diffusion yes but there is also surface tension will play an important role because I have to create the surface there was no surface before now there is a surface so again so so surface is being created and many many surface being created okay so surface tension going to play a very important role so system tries to avoid formation of surface but in this case there is no way but you can you can imagine what will happen in the load very long time this patterns will become thicker and thicker so slowly some islands will disappear like this island here this will disappear they will go into here and there and then they next these island will disappear these island will disappear so there will be thickening or called coarsening ultimately in a very long time there will be phase separation just like one is one but that may take a very long time that is why even volcanic rocks from the million years before you still have the pattern and so this evolution takes place time dependent and we will discuss them how to understand that okay so this is what I just said a sudden temperature quench then a and b and these three much greater than homogeneous sinks then a plus b here and then then we can also say that very important phase diagram in a binary mixture is a very common phase diagram you study in your undergraduate physical chemistry that if an a and b then homogeneous phase where a plus b are here and then it is like gas liquid then this is a phase that can equals the boundary and then inside you remember the lever rule and all these things then the phase separate like that but there is you can coin if you coin sheet usually you see only this one but if you coin sheet then you get this is the spin order line and that you form somewhere I will see the somewhere here it forms and then there is an unstable region that we will we will discuss little bit now okay so important thing of spin order composition very important in absence of an activation barrier it is controlled by diffusion it is all through it is very important occurs through all through the bulk it is not unlike nucleation which occurs in a local region so it is a that is what we call as large amplitude phenomena it is almost everywhere in the system it is happening system is falling out of equilibrium and falling out of equilibrium in a real dramatic fashion then free energy gradient but it is it is driven by free energy gradient but resisted by diffusion so many many system this thing is seen I am repeating here but I just show you one of the computer simulation results that you know is a binary mixture two species the A and B blue and red and then you coin sheet and then in the long term and the coin sheet and then you remove some vibration then you find this beautiful pattern so these guys are connected one and blue is connected below that like that so it is just like the pattern I showed you that they form this beautiful pattern intertwined in a beautiful way and this is a pattern formation in a spin order decomposition so this is again the same thing different stages of pattern formation we showed here so now let us do the theory little bit of theory I will do not to a great deal but some amount of understanding with evolve again these are nothing but based on Landau theory and the very similar kind of thing we did Becker during thing but now the in Becker during all the way to go up now we are coming down so free energy and all these beautiful things happen one of the reason that I like to do this spin order decomposition and nucleation because this is take you to a deeper understanding but they at the same time without too much difficult calculations but at the same time it is a beautiful beautiful physical inside that the student gets and it is something which is a as I told you this is the same thing used in the very large number of cases and very very nice okay so enough of that now let us see it is out front and a little bit of equations that I said nothing but Landau again so I am now in the initially at time t equal to 0 I am here I from here I have dropped here I have dropped the red one is the guy I have dropped now that thing now is going to going to go into this direction and he is going to go this direction and I want to do the evolution okay now so let me start with a free energy at the this is C0 this is my C0 now I want to say okay I want a small fluctuation in composition around this region so I write a C0 plus delta C by this small fluctuation little bit on this direction or little bit on the direction any of the direction I have chosen the symmetric so it does not matter so now I expand it in a Landau expansion retailer expansion f C0 f dot C0 this then half delta C so it is a second derivative that comes in and already as I have described many times this is derivative first derivative relative C0 here that things maximum so this term goes to 0 and I again take to this part to here and call it delta f this quantity is delta f and then I have only this term left on the right hand side so this is my simple equation so if I do a small fluctuation now this is a maximum so secondary very this quantity is negative f double is less than 0 then I am coming down so they are 2 region in this region I a small fluctuation decreases free energy but if I ultimately reach here then small any small fluctuation increases energy so here small fluctuations increases energy free energy here small fluctuation decreases free energy so it goes through f double prime greater than 0 which is the condition of the minimum less than 0 so in between there somewhere it must be 0 so somewhere here f double prime C must be equal to 0 so this time where the f double prime C changes sign from negative to positive is an inflection point this is the point which is called the spinodal point that is where the character nature of force coming from free energy on the system changes and this is the called spinodal line fairly well known terminology we use but in the in the in the level of free energy it is described in terms of second derivative free energy at the function of the composition okay so when we did nucleation we went up like that and now in the spinodal decommission we are coming down from the free energy landscape how how a simple free energy landscape is used introduced by Landau he describes such huge phenomena and that is why in physics community in on this matter science in general these kind of pictures are so so admired because they give us we we invest so little and we get back so much in in in return