 Welcome back and good morning wherever you are or good afternoon again. So we discussed in the last class the review little bit of phase transition then I talked about two things I left out and not in great detail but students should know about them is the solutel transition which is a very common chemical industrial phenomena and percolation which is again very very common in electrical engineering and these things percolation is very common in natural as I said in sand bed and many many other places flow of water is dictated by percolation and then we talked about that the Landau literally revised Landau theory the Landau's great free energy landscape picture of phenomena and then we talked about that first order phase transition is where you have a metastability as given by hysteresis or given by the fact that you know we all call that we all say that water freezes into high is at 0 degree centigrade or go below 0 degree centigrade but that does not quite happen if you have a very pure water you can make in a droplets that no impurities there then that remains in the liquid water till minus 40 degree centigrade mercury and gallium you can super cool we call them super cool to about minus you know below they are melting where the freezing point 50 degree centigrade 60 degree centigrade even in upstairs in clouds the water the temperature has to go down before the really clouds to the droplets form and this is what is already in Landau's theory in a kind of Landau's loop and we know in a metastability in magnetic systems you you switch the field in the other direction but nothing happens the spin still point magnetic moment still point in up in the old direction is that to turn up your field quite a bit before they turn around so metastability means that system is trapped in a minimum and that can be explained very easily from Landau's picture that there the in a first order of extension the old minima remains so even though new minima deeper minima deeper minimum has appeared the system does not know how to go around it does not know even that probably knows that there is a new way of doing it but it does not know but what happened system always undergoing fluctuations and this fluctuations is a way of exploring the thing and as we discussed in the thermodynamics that these fluctuations must be positive for a system to be stable that is why specific heat stability specific we go greater than 0 is a condition of thermodynamic stability ish thermal compressibility positive is a condition of mechanical stability essentially these fluctuations are we consider in a stable system in a in a the fluctuations around a harmonic surface so free energy can be that's why the how does it come because if the probability of the fluctuation of energy that is Gaussian e to the power minus delta e square and what do you have in the denominator of delta e square you have the specific heat so so if I talk of a fluctuation delta e then that goes as a normalization constant e to the power minus delta e square by CV so if I have a harmonic thing then this CV is fluctuating then what is the probability of fluctuation delta x this is x then it is again e to the power minus omega square delta x square so omega square now you compare the two the frequency is CV 1 over CV so omega square is 1 over CV so this a system a stable system is in a bound in a harmonic surface so right now what we are having then we are having this kind of situation this is harmonic but this barrier is large the big barrier is a macroscopic so barrier is microscopic very important so how does the system go then from that question in second order of estimation it does not arise second order of estimation does not have metastability it has just large scale giant fluctuations have here with the divergence of compressibility and speak it and you suddenly get these systems split into two okay ordered and this order like in magnetic transition you have order state and this order state and a system choose between one of the two system choose between sorry three sense up spin and down spin and that says beautiful phenomena called spontaneous symmetry making but we are not going to go into that right now we are going to do with first order of estimation so this is the first order of estimation we will do second order more detail later but something extremely important to a physical chemist and to nanomaterial synthesis of more importance of late now into material scientist and to physical chemist or in the department I am in first order of structural chemistry or the next my department chemical engineering is very very important that one understands in inclusion phenomena and this is something I can tell you a small story once I was visiting Japan and I get talk and I was the first time and I was not comfortable with Japanese food and I was very hungry after the my talk lecture so there is an Indian student who was my ex-student I was not my own PhD student but a student I remember he was very eager send me many mails he wants to see me so I remember it because after the talk I was very tired and he took me to his I do a long talk in Japan sometimes you give two hours lecture I went to at 6 o'clock I went to his office and I told him first before you talk you give me some food so he gave me couple of banana but then he was so told me sir I wanted to see you because you saved my life I said how well you taught that course or nucleation and spin order composition and here when I came they were doing something on a surface which was actually nucleation and they didn't know anything about it and so but I knew from your class so I could explain the experimental phenomena to them the formation of islands and then they and they were exc and percolation and all these things and they are extremely extremely happy and the first paper that I could read in right within a short time in Japan it took a long time sometimes to write a paper especially by foreign students but this guy was so so then I realized that this guy was doing a surface scientist so all the critical phenomena critical exponent realization group that you teach them is not useful to bulk majority of material scientist and the physical chemist or organic chemist but nucleation is very important because nucleation is more important than critical phenomena to this group of people because they face that all the time a physical chemist is an experimental physical chemist so so that I always the preempt I before critical phenomena any other things I teach nucleation and I want students to learn nucleation is a beautiful phenomena so now how does nucleation they know what happens actually in this in the system when you have this phase my old phase in that old phase the way it goes over to new phase is that a small embryo forms so these embryo is a nuclei that nucleus of the new phase and that tries to that tries to grow so in these landscape this is somewhere here so these guy tries to grow by fluctuation is a spontaneous fluctuation and it go then it comes down again then again goes finally it goes here and it goes over and then there is an explosive growth so the process of the phenomena of an embryo growing into a metastable phase is nucleation so nucleation is the process let me write down by a an embryo of the old phase undergoes growth by activation and this is spontaneous it has activation energy and we will now calculate the activation energy so this is the nucleation it so when first order phase transition happens not by any kind of big large scale fluctuation but the small fluctuation that goes so the system is destroyed from within from inside the system is destroyed and new phase comes up you know that is how ice melts that is how water goes into ice or water droplets form all the first order phase transitions or if you have a magnetic transition but in the presence of magnetic field then you this again this some localized things we get ordered from the disorder system and if you have spilled on the upside then up spins will get plastered together and they form a nucleus or an embryo and then that membrane melt and go but finally it will grow in they are growing in many places but one of them will grow and reach to the critical this critical size and then it goes over to the other phase so that is the phenomena we will describe now in a little bit more detail okay so we will now do the following thing that we will do nucleation I described already thermodynamics from nucleation and the Becker-Doring theory then we will do some things very very nice things which is the Lovic treatment of the collection of Becker-Doring by the way the Becker-Doring is a extremely important thing this is the thing which is described in many cases physical chemist might know Lenderman thing then they know enzyme kinetics Michael S. Menten and all these things they are all very similar things they are connected to each other so someday if I can get time I will talk about that okay so then plastic prediction limitations and all these things we will do so let us start with then the nucleation okay is a phenomenon observing in large kinetics and phase transitions and it is from within so there it must proceed by an equation because microscope changes the barrier it cannot this I already told this barrier is macroscopic it cannot go otherwise so it must from inside so nucleated for new phase has to be nucleated within the old phase that I said and this there however this process itself has a barrier and that is a barrier and that is a beautiful physics and that is called nucleation barrier and that is the one we will now consider okay and then an example as I already said that water should freeze at 0 degree centigrade does not it is can be super cool to great great much below much below minus 40 degree centigrade water remains water water does not pure water does not freeze at 0 degree centigrade that because the embryo needs sufficient driving force to grow it does not grow okay and this is the existence of metastability is due to the this is a beautiful beautiful sentence actually this is all taken from our book that the existence of metastability due to the presence of barrier forced by nucleation now while pure water pure water does not freeze below 40 degree minus 40 degree centigrade helium and I am sorry gallium and mercury does not with a minor they go 50 60 degree centigrade below they are freezing however you know in refrigerator ice forms you know minus 5 degree centigrade minus 3 degree centigrade 10 degree centigrade the reason is that they are surfaces and there are a lot of impurity so that is called heterogeneous nucleation heterogeneous nucleation does not face that barrier and there is the way you do the heterogeneous nucleation of course you have to do homogenous nucleation first then you include the surfaces and show that how heterogeneous nucleation takes place okay so this is a completely smooth way and you and one interesting thing is that you are having a very super you might have done in physical chemistry experiments that you have super cooled it and you have the container below 0 degree centigrade now you scratch it with a and we have many of us have seen that or in liquid you scratch it with a rod glass rod and you can see crystallites appear so the heterogeneous nucleation and this is because this is a one of the telling and great example of the of the role of surface tension and surface tension is something we want to study in this lecture later but right now we will define surface tension as the energy required to create a surface so if I have to create a surface of ice in water that is a different surface that requires extraneous energy that is energy surface tension quantitatively is defined that the energy required to create an unit area that you are creating your in your high school even that surface tension but that same high surface tension plays an extremely important role how do now how do we go to do the theory and we are now going to go out that development of the theory okay so the basic idea then that we have an embryo growing in in this in this old phase embryo of the new phase is trying to grow and we have to find out what is the energy okay when I grow something then it gains energy because these the free energy landscape is like that so this is the new phase the embryo phase so when it is here it gets free energy it gets considerable of free energy say the free energy per unit volume I define as delta gv so it gains that energy however it creates the surface and it has to pay for the surface so total delta g is then some of bulk energy which is gained this is minus negative but then surface energy which is it has to pay so the these things and some of you might already bearing to see what is that so this is the metal stable gas and in this case this is metal stability in temperature percent density and this is the critical point okay now so as we again and again saying if gas going to liquid then these the coexistence but it in order the equation to take place it must go down and I am now going to tell you why is that okay so basic equation is this is the equation okay the basic equation is that if I want to create a new clears of radius r you know old phase then I give my volume is 4 pi by 3 by r cube this is 4 pi by 3 by r cube and then delta gv delta gv these are great and delta gv is the difference this is the delta gv so this is again so this negative in front again this much again however I have to pay energy so how much is the surface we know surface 4 pi r square and gamma is the energy per unit area so 4 pi so this is the total amount of so in order to create a fluctuation of size r this is the energy that I have to pay this is the total energy cost so energy gain and energy loss now what do I do okay I go here and I ask for a new phase okay so let me now see delta gr now one is energy gain and then how does do the loop let us look at these these r cube and these are square so if I plot this delta gr against r then this thing is negative this r cube this thing is positive okay now surface tension is actually pretty large so what happens this part wins in the beginning so that is why you go up the hill and come down but if by some fluctuation it becomes large then it starts getting the advantage that this is larger because these are cube okay so then what happens as a combination of these two a maximum appears and this maximum is the this maximum is very important thing in the language these are star critical radius this is the critical barrier so in the folklore of phase transition this is a very important diagram that is telling you how nutrition takes place this is very important for nanomaterial synthesis in every case this one diagram called Becker doing diagram that when I write the energy of creation that is like embryo of size r then this has this structure now how do so again combination of two factors one is negative rapidly becoming negative other is positive and then you get that then how do I do that then next thing of course and this is the critical radius r star I have to find the expression for r star now and critical barrier I have to find an expression of critical barrier okay let me do that how do I do that now we have done it many many times in this class already we do d dr delta dr equal to 0 okay and what do I do now I get I do this so I writing little big I should write little small so when I do that I get minus 3 3 cancels this 3 and I get 4 I get minus 4 pi r square delta gv and I get that plus 8 pi r gamma equal to 0 right now one of them are equal to 0 is the solution part of the reason because this is the minimum so r equal to 0 is the solution that is not the solution we want so we then divide by r so this goes out and this become r okay now these are now become my r star I know the other second solution this r star and that is now given by r star by 2 gamma by delta gv so these are star expression so this is the radius expression of the critical radius now what I want to do I want to put these values of r star into this so now I want to do find this barrier so now I put it made it r star okay and I put r star 2 gamma by delta gv cube here I will use the same position excuse me but that will work out beautifully so 2 gamma cube and now I will put okay so now you get 8 and 32 by 3 here you get 16 by 3 and delta gv these delta gv cancels one of the delta gv here and you get gamma cube by delta gv square back in the so now I get the following expression so these are the two very important expressions this theory of nucleation is called becker during let me tell you this also the theory was used in nuclear fission and many many across the condensed matter physics and chemistry the amazing important and beautiful thing so basically what do I we have done we have done a considered a combination of surface term and the bulk term and I just wrote down a very simple expression surface tension is known so is delta gv is known because I can calculate the free energy of the old and the new phase I can come from the old phase and I can come from the new phase and I can calculate the the free energy things my standard free energy calculations I can do that by integrating one specific heat that is the way we set up the problems that I give you enthalpy I give you specific heat and you calculate the free energy you know that can be done so free energy of these phases are fairly well known and because there are many many fittings also available to these things so we know delta gv we know the surface tension and there are certain approximations and we will talk about that but since this is that my course for aim to for beginners essentially we will probably minimize the criticism of a theory of the scale and we will discuss of course now but the I want you to appreciate the generality of the concepts that are involved it is a very common generality and at the end of the day we come out with certain beautiful beautiful expressions and which are very widely used very very widely used so when you say you hear the term nucleation is thermo thermodynamically controlled or kinetically controlled all these kind of a thing then you know many times what is talking about nucleation that that if it follows with no barrier pathway even though the state is thermodynamically more stable that we will we will do now another very beautiful thing we will do just amazing the important things not usually covered in textbooks but we will do is the austral step rule and extremely important in present day scenario of solid state synthesis so so those which the thermodynamic control all these things are essentially we are talking about nucleation and so these are the two equations I just derived so this is the size of the critical nucleus and this is the barrier of the you know the beautiful equation to come of delta gv we all almost know this by by memory okay so now how do I do that how do I do now so I have a situation like that and this is my free anti-barrier delta g star my r star now you asked me what is the what is the rate how do I calculate the rate of nucleation Rn how do I calculate that ask a chemist that I have a barrier I have to go at the barrier a chemist will immediately say I know there is a back activation barrier I know what is activation barrier it is delta g star by KBT I know that okay well and good then what is the p factor so this part chemist will be then the chemist will scratch his head I am used to do transition state theory if I have a barrier I have a situation like that I write rate K equal to K Boltzmann factor by both H Boltzmann factor times temperature by Planck constant e to the power minus delta e star by star e star by KBT I know that this has been taught to me from school days that this is the rate of barrier crossing can I do it here okay so this is a Boltzmann factor the weight of the probability that the particle is at top that gives you this term that kind of problem this has to be correct then but how do I get this term I will do a chemical kind is a little bit somewhat dying towards the end of our road because we take a bond and we say okay the bond breaks and bond is a frequency a nu and h nu and that is about thermal energy KBT and then say okay nu is the frequency of breaking is KBT by h that is how we do it then how do I do it here how do I do it here well we do in a way not too different we will do in a way which is essentially follows these logic but in the process we do something really extremely beautiful which is very much like chemical kinetics and it turns out what we do is a chemical kinetics is a form of chemical kinetics and again the guy who did it is a nuclear physicist very famous nucleus and what he did here was essentially in the atomic bomb that same thing used in chain reaction and we will do a very chain reaction and in fission and this beautiful stuff so we will stop it here and in the next class we will be doing the result of its correction and finish the nucleation theory.