 Welcome back. So in the last class we started on a beautiful subject of which goes on the kinetics of phase transition and but it is essentially a thermodynamic treatment and we have included these in this course because of the generality. In fact phase transition itself if you have noticed is quite general. It does not talk of any given system. It is applicable to all systems. That means I can I am talking of nucleation not just of gas liquid. At the same breath I am talking of freezing, formation of ice nuclei, melting, formation of water inside ice. I am talking of magnetic transition in the presence of a magnetic field when it is first order in the absence of magnetic field is second order. Why the grand generality appeals to some people? This grand generality at the same time makes many people uncomfortable because people many times like to think in terms of specific systems, specific examples. But there are things where generality is extremely helpful and useful because you learn one thing you can apply to all different systems. As I was telling my ex-student in Japan you know he learned these things in a class you know this nucleation and spin order composition he found it is an experimental physical chemist not my own student. But he was just taking the course and he found it extremely helpful. Even now he is so grateful whenever he comes to the professor now in IIT Madras or IIT Chennai whenever he comes to Bangalore to make sure that he comes and sees me such is the reach of such a problem that you can apply to almost many, many disciplines. So now we are trying to get this thing going. So basic idea then is that we have this kind of a scenario and this is delta G R and this is R and this is the critical nucleus and we have to cross this barrier. I describe the analogy with chemical kinetics where it is we know how to do it. Here we know per day because we know the probability of being in the top has to be much more into the bar minus delta G star that we know. But you do not know how to calculate the rate of going over. In chemical kinetics is a bond breaking and or twisting there we can write the detail motion of that molecule rotating or bond breaking. But here we do not have that. So how do we get it? This was the one which was done by Zeldo which as I told you is a nucleophysist. So the way it is now done is there are two factors is the probability or number of critical if you consider rate as a flow like water going through a pipe the rate of water we talk of rate of water flow. Similarly, a new phase is forming and from the old phase then if we can see that the rate as a time independent process then you know then how do I show. I will say okay number of nucleus at the top a critical nucleus number of critical nucleonc and that is I can find on the top of the barrier and then at the rate it will go over. So this I said is the same thing that we do in chemical kinetics this part and this part is the one that is not trivial and I need to calculate that. So if I now have a nucleus sitting here at the top then what would let it if I am sitting on the top a nucleus sitting in the top what will make it go over it will make it go over if another particle comes and impringes on that. So I need a practice for another more particles actually not one several but this is the radius R. So this is already a pretty macroscopic coordinate. So by changing small r means several monomers is coming. So these monomers are going to impringes on this. They are very interesting thing you know in many of examples many problems we get this quantity. I remember I passed a comprehensive at my in Brown University because I could argue this one this was the problem given in a comprehensive exam in PhD exam without telling us and I did that time I did norm nucleation and but I could figure it out by using kinetic theory of gases that okay the number of molecules that can pass I realized that what could be then I said okay we all done in Maxwell Boltzmann kinetic theory of gases that there is unit area of a surface and the pressure is nothing but the molecules coming and colliding with it. What are the number of molecules colliding with it that is within certain parallel pipe remember we draw the parallel pipe and okay these are the number of molecules in these and which has the velocity towards the wall. So the rate of impringement of monomers is very similar logic and on the surface but then also the surface area of the nucleus larger the surface area more now you may argue why I am taking different surface area because a different temperature the size of the chemical nucleus is different remember R star is 2 gamma by delta GV and delta GV is temperature dependent. So as temperature becomes larger and larger and I will come to this little bit more detail now or maybe I should just do now and then come back because it is very important thing I had it in my last I have maybe forgot it. So if I now plot this delta G R against R at different temperature then I know gamma is also temperature dependent surface tension but much weaker than this thing delta GV however when they are at equilibrium two phases old and new then these 0 delta GV is 0 so R star is infinity. So when I go little bit below then R star it is start decreasing but still can be very large. So I can consider the following situation that I can say okay this is when it is large very close to coexistence then it so as I lower the temperature so if it is T3 T2 T1 then T1 is greater than T2 greater than T3. So barrier comes down size becomes small as delta GV increases. So this is the reason why the surface area depends very strongly on temperature then also depends on system to system because these temperature dependence and the free energy landscape completely varies like if I talking of melting it is different from that of freezing or more than that you know more from gas liquid condensation that is what I wanted to tell. So this nucleus with the surface S and C sitting so then I would be the rate would be proportional to the rate of infringement let me call it beta I and S and C. So this should be proportional to that and then I need a rate to go over that I call this is Z then I can go now and write my so this is the expression. So this is the probability of having a number of critical size nuclei that is NC then this is the rate of infringement on the surface and then the probability there is also probability that during the unit time this beta is the rate of infringement per unit time per unit time it can also disappear that means it can also come down here. So there is a non-equilibrium process and this is what it brings it very close to chemical kinetics that I will discuss a little bit later that effect that it can come down is included through these factors Z which is called the Eldovich factor. Now we will go through the calculation and the calculation is a little complicated and but we have to live with it this is one of the probably after you our canonical ensemble calculation is the most detailed calculation that we will be doing but we have to do that anyway and we will go through that with the heroism it is so this is the thing we have to calculate Z how do I calculate Z the way I calculate Z now is that I consider as if a train is moving so the growth of cluster is going like this this can also so I add one it goes there then I add one more a n minus one is becoming a then a n plus n plus it of course disappears that I will consider later right now I am trying to set up a train and my aim is that just as you are in front of it if you have today in front of a passing train then this compartment after compartment going in front of you when the train moves in a steady state velocity then what happens each compartment comes to you at a regular interval okay and that is a steady state steady state means just the flow of water in a steady state that means it is a time independent process. So I want to find the time independent rate rate is always when rate is a constant is the time independent it gives a steady flow complete independent time independence is equilibrium when nothing is happening okay that is a if I take a derivative of all the quantities then at equilibrium then though they are all 0 concentration time derivative of concentration is 0 but here it is not this is the next time independent thing that is steady state but it is not equilibrium. So we have to make a distinction between equilibrium time independent process and steady state time independent process I am going to do here that steady state because I want to study the rate whenever I talk of a rate it is a it is a time independent is a steady state otherwise you do not have it then we say time in time dependent rate you know that is a different thing all together okay. So now I am going to calculate the flow as if a train so however this quantity that I call c n c that the this going over I can write as Boltzmann this is equilibrium c n is an equilibrium concentration I now change my notation I might change my notation from r dependence to n dependence that means because I want to talk of adding of infringement of particles on these things I and then growth I want to do in terms of number plane the number of particles in the system then I know there is one to one correspondence between radius and if I assume a sphere then then volume into 4 pi by 3 r cube is 4 pi by 4 pi by 3 r cube is n number of particles and then volume v so that means r scales as n to the power one third because is volume of a one molecule volume acquired by one molecule into number of molecules in the cluster nv that has to be 4 pi by 3 r cube both are same dimension of volume okay this is a trivial if that is so then I know this is 4 pi by 3 r cube per unit volume then that is nothing but proportional to n and that one volume I absorb in n so that because per volume and that v I multiply so these become dimensionless is just but I am sorry dimension of energy energy per particle and now there is a surface term which is r to the power r square r square is nothing but n to the power 2 by 2 okay so my earlier expression minus 4 pi by 3 r cube plus 4 pi r square that becomes minus n delta g n plus gamma n to the power 2. Why I am going to end representation because I want to add the molecular picture r does not allow me a molecular picture it allows a thermodynamic picture because I can I could do the surface tension but now I add a molecular picture because I am adding molecules one by one and it is flowing and then I have the following ingredients I have the Boltzmann distribution and I have conversion of delta g gamma to the power n to the power 3 okay. Now I want to see that how the now I introduce a very important quantity that is FNT. FNT is time dependent this is the probability of having a particle a cluster a nucleus of n particles at a time t okay. Now that thing is changing it is changing because I have a flow and as a result of flow that this guy this guy is decaying by I am just saying the one way flow now if I do the equilibrium I would did both the two sides that I will do later. So these guys changing because of that flow the one way flow very important the one way flow this is decreasing because molecules are moving out. So then Jn minus Jn minus that is the way this is changing JNT I already wrote that is my impringement verb time dependent probability surface minus that is very interesting. Now this is this n plus 1 becoming n also and that is so this is this JNT it is going to n to n plus 1 okay. So it is going to n to n plus 1 but it is on the other hand n plus 1 becoming n and n plus 1 becoming n by evaporating that is n plus 1. So if I consider the this guy then this guy is going on that but in the in the previous it is going to this but on the same time a part of that is in the flow is going at that rate but at the part of that is coming back. So I am just considering this process as I said I am going to consider only this process and in this process it is going by impringement but it is also disappearing by operation. So if amount going a some amount is coming back here. Now I can do also this case but doing one enough for to get me the flow. So this is a forward and this is the backward of that I have an estimate of the rate of impringement on a surface from kinetic theory of gases as I told you the Maxwell Boltzmann FNT is the time period a very difficult thing right now I will see that we do not need that we do not we need that if it is more difficult calculation. So and I have the surface area of n and n particle this is disappearing of the evaporation also depends on the surface area clearly but how do I so this I have a control SN I know SN plus 1 I know surface the surface of the sphere of the DSR is the n to the power 2 third I know that I do not know what is Wn plus 1 this is a very very interesting condition here how do I find Wn plus 1 now there comes a very very important a quantity let me get a page to explain that. So Jnt beta I SN and FNT minus Wn plus 1 Fn plus 1 T and SN plus 1. Now we use something I go back I have to calculate this quantity and this is another beautiful thing that we use a lot in your chemical kinetics and I get your views we should principle of detail balance one of the most honour most respectable the principle called principle of detail balance on Sagar used it and analyzed it in microscopic reversibility which in his famous 1932 paper or irreversible thermodynamics and he got Nobel Prize for that now principle of detail balance says that if I have now at equilibrium I have to consider back and forth steady state I got a flow you have to make a distinction. So I have principle of detail balance is a very strong statement it says that if you are at equilibrium then it is not enough to have a time steady time independent but each state must be individually balanced that means these state they have number of particles going up in towards B to A and must be same as A to B. So individual states these the principle of detail balance of microscopic reversibility extremely important that is the only condition that guarantees equilibrium. So equilibrium means flux is 0 no flux it is everything is balanced but we are ultimately calculating a steady state we were playing out flow but see how we are using equilibrium at the same time this is exactly done many many branch as I told you if you enzyme catalysis conducting Michael's Menten then you are the minimum famous chemical kind of first order kinetics in unimolecule dissociation reaction everywhere this is used but it was essentially first done by Zeldovich okay now all of maybe all of them I have done independently but it is the same thing at least I am just mentioning three very famous theories but probably 10 theories uses this kind of logic that is why I like nucleation because it is being so much beautiful so okay now Jn equal to 0 if I put Jn equal to 0 then what that is the equilibrium but I have already defined that Cn is the equilibrium distribution so at equilibrium fnt becomes time independent but at equilibrium fnt equilibrium is nothing but my Cn so now I put 0 equal to beta i Sn Cn minus Wn plus 1 Cn plus 1 Sn plus 1 so what does it mean now that means I get Wn plus 1 so Wn plus 1 this quantity now become equal to beta i Sn I take it on that side and then Cn divided by Cn plus 1 so this is my Wn plus 1 so I have now by using the real man is I got an expression for Wn plus 1 let me write this down clearly because I need it so Wn now what are you going to do now I am going to put this quantity into this equation okay so I am going to this expression is going to be substituted in this one once I do that what do I get we are with me for a minute okay so I got this now now I notice the following thing Sn plus Sn get cancelled and I get beta i common so I take beta i Sn out I also take Cn out common then I get fnt by Cn minus fn plus 1 t by Cn plus 1 so I get for Jn this expression okay now if you look at this expression what do you see you see a beautiful thing you see that these quantity these fnt and they are now I can write it in the limit of large n I can call it a time derivative so these thing can be written as because this fn plus 1 is minus there will be minus beta i Sn Cn and d dn d sorry sorry d dn so Jn this beautiful thing