 So, this is X plane zebra light pattern in phase phase and alloys and what we do now, we just like what we did in nucleation in Zeldovich, we write down a kind of a diffusion equation kind of scenario and then we go into that and I will now this time I will describe little bit more of what you mean by diffusion equation and how these things develops little bit more detail. Okay, now let us consider the diffusion equation this you know from your undergraduate very very well and it is actually just a part of time-dependent statistical mechanics which we will do much more later but we need a little bit of it here and it is very very easy. This gradient, this quantity gradient here is a three-dimensional vector that means d dx, d dy, d dz nothing to be a and j is the current term. So, this is called conservation of mass that I take the derivative of position dependent composition C and composition C is XA-XB then this is the just the statement of the continue we will call it continuity equation is a statement of law of conservation. That means if I have a like that then I take an area here then the particle coming here and then I want to consider change what is happening here then the change inside will be the amount coming in and the amount going out and flux is the j so dd, dj dx, d dx that will be dc dt and there is a minus because the more going out it will be less and it is more than. So, this is nothing but the continuity equation and this is the one that we just describe it. So, this is the continuity equation just the conservation but that means we essentially say particle cannot be created or particle cannot be destroyed. So, it is a it is a intimately connected law of conservation but continuity equation is a very very simple thing and you might have faced it in your undergraduate you know in electro chemistry chapter and simple and kinetic theory chapter. Now, we know one more thing this equation by itself is not complete because they have two unknowns. Now, what is done is something is extremely nice which is called Fick's law remember what is Fick's law? Fick's law is now this is an exact this is an exact but Fick's law is not exact but Fick's law is an assumption Fick's law tells that the flux is derivative of chemical potential. So, if there is more chemical potential then matter flows from large chemical potential to lower chemical potential here I use you as a chemical potential. So, Fick's law is a what you call constitutive relation is an assumption and this assumption as one case it is kind of with a linear regime that means small fluctuation regime is valid. So, this you can you can read these things in Google you can read these things little in Wikipedia but these are very very fundamental very very nice things. So, this is my chemical potential they essentially saying matter flows from large chemical potential low chemical potential large free and G2 low free however it flows with the rate and that rate is the diffusion d by KBT and KBT is put in here if you could have guessed it it is because that KBT can as well go in the chemical potential. So, this is a dimension of energy KBT is dimension of energy. So, U by KBT is dimensionless. This as you can see here is T A comes because this is concentration or concentration. So, 1 over L cube and this is L cube by T this flux is L square remember amount of matter per unit area per second right. So, flux it L square by T this is gradient. So, you get number per unit time per area. So, this is 1 over L square T so gradient comes with 1 over L. So, this is 1 over L cube T this is 1 over L cube T dimensional analysis. So, now so this is 1 over L square T flux now this takes care of that and this is energy taken care of and this is 1 over L here and D is L square by T and so D is L square by T. So, D is L square by T. So, there is some interesting thing here. So, D is diffusion L square by T we are with me for a minute and 1 over L and then it is M L square by T square and that is then L goes then L cube by T cube but I have a k B T. So, that again T square by M L square. So, that goes out and 1 over L L square by L by T L by T L by T L by T. If this is composition then it is L by T. So, C is used as a composition the difference in mole fraction. So, C is used as a dimensionless quantity. So, if C is dimensionless quantity then flux is 1 over L and flux is number of particle by 1 over L square T. So, this is 1 over L square 1 over L square L square by T and this is 1 over L. So, I think there is something a bit screwy up here but we have to continue and I will fix it up sometime later that these things. So, now we continue it now there is the issue either I am doing something wrong here likely most probably it is correct but one of the notation used here probably is different from what I have in mind but you can figure it out probably yourself. So, now this is the but the rest of the things is okay this is continuity equation this is continuity equation and this is the flux this definition is correct and the chemical potential remember chemical potential is chemical potential and Z is dA free energy by density that is the chemical potential. So, here the chemical potential is similarly exactly same thing here chemical potential is U chemical potential is functional derivative of free energy with respect to composition or concentration that because so this is that exactly same thing as that but this is the functional because C this C is a which is C is a let me write C R is X A R and minus X B R and that is I think is my confusion. So, this is a mole fraction and this is a mole fraction of A this mole fraction of B. So, if I have the particle like that then I divide into grid I am saying something very very interesting. So, please listen carefully. So, then I choose one of them and I say this is my location R and in that location now I find out where in this location R I find out what is the number of A molecules and what is number of B molecules. So, what is then that will give me the mole fraction. So, A molecules minus B molecules. So, that is the composition C. So, the way to consider the position dependent composition is to consider your system to vary into grid and then each grid you go you can make the grid very small but do not make it too small. So, it does not contain they let it contain about 8 to 10 molecules then you can see. So, when they are 50 50 equal number of A and B then you will have composition 0 but it varies from position to position and because of the inhomogeneity that is being created and also because of fluctuations. So, now this is the definition of composition. So, when I take derivative of free and due to desperate composition that gives me chemical potential which is just like in thermodynamics. So, Fieg's law is driven Fieg's law is diffusion gradient of chemical potential. This cavity might be the problem of the dimensional thing but we can now exactly what I said before the free energy I am going to create a inhomogeneity now I am going to create the free energy really lot of fun now. I am going to create from here the homogenous where in every grid my C is 0 from there now I am going to place where C is 1 and here and minus 1 here. So, I am going to create through this the heterogeneity and I already told you the way to describe heterogeneity is Ginsburg Landau that I already told you that Ginsburg Landau allows you to get this free energy this is free energy as a fc. So, this fc this quantity is this is really lot of fun this quantity is this thing. So, this is this is the thing fc okay now and this is the energy you have to pay in order to create a heterogeneity. So, this is the surface tension term. So, this is the Landau term which is the free energy and this is the surface tension term because I have to create surface to create surface I have to pay energy and so combination of the two is a. So, now if I want to see the WL if I have to create the WL and I know how to create the WL and I know that is just Landau now the beauty of this thing that I can create single WL to WL by saying if A is greater than 0 then I have a single WL that is the high temperature then I cohench to low temperature when I cohench to low temperature A becomes negative A becomes negative and this changing then this single one like that becomes something like that. So, that is a very clever way of doing. So, I can now use Landau free energy into Ginsberg Landau and I can I can do something very very clever okay. So, I combine the diffusion equation dc dt I had two things one flux term and the A term one flux term remember one dc dt equal to minus del dot j and I have a j term that is in terms of d by KBT delta u and then u is in terms of del f delta c these three things three things are combined here. So, there is one del here and that del here that gives a del square d by KBT comes out and the third part delta f del c or the chemical potential dc dc dc so three ingredients. So, I get the diffusion equation dc dt d by del square now I can consider this equation in terms of fluctuation but that can be we can we do not need to do that we can work with this equation this beautiful equation now what we do is the following we are almost done that we consider just like we did in nucleation exactly the gel dovish and all these things that we consider in infinitesimal fluctuations or position dependent composition. So, I said okay there is a fluctuation delta c then I go back I said okay fc plus delta c is fc and I write fc plus as a result of that fluctuation as a result of this fluctuation my three energies undergone at a position r some small change how do I do that now okay I go back my Ginsberg landup and put in place of c c plus del c and here c plus del c now this quantity I can do a Taylor expansion fc plus del c fc plus df this is just a first order expression. Now and this quantity del c plus del c square so I now going to do that grad c plus delta c square so now I multiply that means this is a vector this is a great delta dot delta c grad dot delta c when I do that I get delta c dot delta grad c grad c grad c square and then del grad delta c grad delta c square so grad delta c square is a small term so that is neglected because this is a non-linear or quadratic in fluctuation then two terms remain to delta c delta c so I can now go back I will be and put all these things together and that is given here you know you put this term that then then you realize that this term and this term these two can be combined to get a far this quantity and I am left with this I am left with okay this combine I am left with this term and this term which I collect here and then I integrate these by parts delta c delta c and I can get this term I integrate by parts I will get delta square c delta I brought the linear thing out and then I do that I am going to do I get the following equation delta fr is this quantity this is the one now I am going to use to find out my chemical potential and when I do that put it together this is the equation one gets the small goes equation now the important thing is that with these equations are given you can this is not difficult at all is just simple linear algebraic same thing you did in calculus but the important thing is now I have an equation for small fluctuation and that is of the following form there is a diffusion second order term and the term BFDC so in the next stage one go and say okay I go back to original thing that this equation is a very powerful equation and lot of things can be done with the equation but we are not going to do that now we are doing something very simple we are going to do just here so my cr my fluctuation I consider just around c little fluctuation here this little fluctuation then I say okay my DFDC term there I can write like that DFDC 0 and then second derivative and these of course we know is DFDC here maximum is 0 so that goes to 0 I am left with this term if I do that then I go back and put it back here DC a du du C is u plus C C0 plus u so it become du dt so then I get du dt d del square minus k square d x so this is the final equation what you call linearized equation and then something very very nice to so with this equation now one very very simple thing one can do is that we can say okay if I have the small fluctuations what happens to the fluctuations so this is my small fluctuation now is this fluctuation going to grow or the fluctuation going to decay if fluctuation going to decay I am not going to get the phase separation of the composition but the fluctuation grows then there is an instability and that indeed happens one introduce a Fourier analysis and does a Fourier space analysis here and one can show that if that because this equation is fully linear equation I can do a Fourier transform these become d k square these become k small k square this k square and so k square plus k4 and then you get this equation k square and k square k4 I take k square out and I just take the surface tension term out so in this mineral decomposition then we are talking of a small fluctuation in this initial say we call a linearized regime because u is very small and the way one of the way to study this kind of small fluctuations through a linearized equation like that is to go to Fourier space we call this a stability analysis so the stability analysis that we are doing here is a wonderful thing for stability of fluctuation even is the same thing as the stability of that we do in the free energy with the that means whenever we do a fluctuation if the energy increases as a result of fluctuation then the system is stable like if I here for example I give a small fluctuation small fluctuation if it stays then it's stable but if it is in a in a in a region like that marginal stability if it falls like that then that's that that's the free energy decreases here in this case gravity energy if by inferential fluctuation free energy decreases then the system is unstable so this is the way we talk even in in thermodynamics the stability and instability just because we look into the first period even second period the same thing we are doing here we already have a we already have a beautiful beautiful equation for the evolution of the small fluctuation here and we the small fluctuation and I want to know what happens to the small fluctuation and this equation here is going to tell me what will happen to the small fluctuation that equation already there is a great character it has a surface tension term which resist building of heterogeneity and if we here energy term which which is held in the first operation okay because it is this term is coming from this term that help but surface tension term as I said by diffusion they don't like it okay they don't don't like creation of surface that's why it comes with the negative term and these two are opposite in sign they fight against each other and in that fight who wins that is called stability analysis and the way to go to stability analysis is just introduce a Fourier why we do Fourier it is simply because this equation is linear equation it points beautifully so if I do Fourier transform in K square will term another K square term so K4 term so this case go with this and K4 go with that and that exactly what happens you know this K square and K square K4 term and this is the K square term so now if I say okay my this quantity my fluctuation grows it grows or decays with this omega K so now I solve that I said okay Ukt is equal to e to the power minus Wkt so this is my equation the from this equation I did nothing I said okay I have and introduce this definition then I will go back and do the Fourier transform I find the UK DT becomes omega K UK and then that omega K is given by this term and this equation means I have this solution now if on a fluctuation the fluctuation has to grow then omega K has to be negative if the fluctuation decays the very simple stability analysis that you probably have done in the stability analysis of the differential equations in your undergraduate those were little mathematics course they will have the stability analysis of the diffusion of differential equation is a very very a thing important thing to for a differential equation whether it has a stable solution or not many many solutions are neglected because they are not stable solutions okay the same thing here we are doing nothing really fancy so now I at the end of the day I get this solution and omega K now is given by this thing so this is the constant term not important so it is K square so important thing is that K plus 1 by K double that is all now this quantity remember is negative because this is falling down this quantity is positive at function of K now there will be a K there will be a K when so small K this is positive but when so these quantity that is coming from here falling down so maximum second derivative is negative so sorry this is negative but this is positive okay so this is negative and this is positive so very small K very small K this quantity is negative but large K this become positive so when I go to large K remember I have a negative sign in front of it so when a large K it becomes positive that means the fluctuations decay but at a very small K very small K this quantity this quantity wins and then omega K becomes negative at omega K becomes negative I have a negative sign then these quantity become positive that means once in function of fluctuations grow and there is a crossover this is the crossover so below that O F number below that O F number fluctuations it is really interesting as I said here that below that O F number fluctuations grow but then large O F number fluctuations stable and this exactly exactly like our nucleation kind of thing there is a surface term is a diffusion term and there is a free energy term and so below that K C systems unstable but above that it is stable because this quantity is negative so this is the stability analysis and that explains the spin order decomposition that that why why infinitesimal fluctuations can give rise to growth of patterns and the pattern that grows grows with the O F number K C because this is the where first time because you are not going to go very large scale fluctuations you are doing local small scale fluctuations and you are trying to do local small fluctuations and you find your fluctuations just decay however suddenly you can have a fluctuation you have particles B and B particles have brought together such that there is a fluctuation a certain size of A certain size of B up there very much like nucleation so these A and B are now can be stable now so that particular is the K C that magical wave number is the K C and in K C that pattern now suddenly stabilized and then we know that is the pattern that can evolve how do you know that all these things we are telling is right or wrong well people go and can calculate this fluctuation fluctuation correlation function by it is scattering this is called structure factor just like we did wave number in a Fourier transform this is the static structure factor that one gets by a x-ray scattering or neutral scattering so the maximum scattering occurs towards then what happens that one when do you get scattering you do not get scattering from a linear system you get scattering point there is a no and because in order to scatter you have to different regions of different density just like in light scattering you need to have different refractive index and that exactly one sees that in x-ray scattering that this peak and as this fresh separation focuses the peak is the manifestation of there is this regions and this regions move to a lower and lower K value this is K with time lower value that this is the longest time t3 so and that means what happens my pattern becoming thicker and thicker and phase separation is going on more and more so the scattering shifts towards lower values this is known as course training very important thing in the phase transition like course training course training so system is course training itself evolves towards the phase separation state so I think this is the end of this mineral decomposition to summarize that it is a wonderful subject to end the phase transition for the time being we will back with the critical phenomena later and this is the kind of situation we are describing and this is the kind of course training behavior so this is very different from nucleation but they are at the same time exceedingly common and popular and is not really taught in statistical mechanics courses but as soon as you go into the realm of research this is not the first thing that you face just like equation and I think students should be prepared for this kind of things rather than knowing lot of equations and lot of lot of lot of lot of partition functions they are fine and they get us started but they do not really lead you to a very interesting and important areas of research and do not allow you for example to insert papers or nanomaterials and other things where people you will see they are always talking about nucleation they are thermally controlled or kinetic controlled they are talking about they are talking of spinodal composition but this is it because I think students should lead at the level of the SC or MSC this kind of things so this is where we stop now and we will return soon with another dose of phase transition and then we go over to more mundane and but more detailed things of liquid state theory and interacting systemizing models and phase transition and many many other things are there like by how to do binary mixtures how to talk of surface tension many other