 So, now let me briefly tell you the theoretical strategies that one of the thing first phenological models may are came later, but before may are around the same time, but after Vantof and Ostwald we started doing a we put this solute and solvent molecules on a lattice. Because that could be solved because I could now give epsilon A A epsilon B B and epsilon A B just like we did in binary alloy and I can count now how many ways I can put A and B. So, putting on a lattice has its advantage and that done in a in a way to a in polymer by Freud-Hurgin's theory very famous theory which still plays a very important role in all studies of polymer solution even in ordinary solution. This is very similar to Bragg Williams approximation with it in quasi chemical approximation with it in a model. Then we have a external mechanism with the lattice for the lattice model which we will discuss briefly of lattice model done in a model we will describe it and decodal description base which will stress theory these are the famous things on stanzanic equation which we will discuss little bit and then many other things we do not need to know now, but everywhere remember everywhere the main central issue remains the issue of the force fields that is where osmotic pressure plays a very important role that gives way and that is what Freud realized and used it in the polymer and now we go to Freud-Hurgin's theory. So, as I said when you put A and B together in a mixture A and B together then one thing I get is an entropy of mixing and that is very well known is the entropy of mixing per particle I would rather take it out from here and put it here by N is KB XAL and XA KB is Boltzmann constant. So, XA is mole fraction of A, B is mole fraction of B. So, this is say B and this is this is my A ok. So, I have XAL and XA plus XPL and XB these things by MAM mixture you know from come from simple combinatorics you calculate the Boltzmann omega number of ways from N factorial by N, N factorial by NB factorial that is where lattice finds N number of lattice sides and you are putting A and B together it is very very useful to get that. Now, this is the entropy of mixing now comes in extremely important which we have been talking, but we have not quantified that how do I talk of A, B and I need something to say A and B different from each other I need a quantification that is done by this guy is discriminant. So, epsilon delta E I introduce a term delta epsilon AB in terms of now if epsilon AB is epsilon A plus epsilon B by 2 what is this called? Suvam what is this called? This is Barthel rule there is one more called geometric rule ok. So, if A and B interaction is just some of the average of do then you get the ideal solution then you get Raoult's law, but that is not like what are DMSO you cannot say what are DMSO interaction by whatever interaction the DMSO-DMSO interaction at the because then the hydrogen bonding between A and B does not appears ok. So, the interaction between A and B brings a very essential new ingredient and that is captured by this term delta epsilon AB. So, we called it discriminant that how different A and B how different A and B are in their properties and in their interactions that what captures by this is flowy still we are doing all along flowy. So, flowy then said ok this is something we discussed a little bit, but if Z which is in gamma in my isent model things is a number of your coordination number in a lattice site if this is my lattice site then my coordination number is 4 into D simple key will lattice in 6 D would be 6 3 D will be 6. So, Z is now I want to know that number of contacts within A and B then I can say ok I pick up number of A there then I can say ok the coordination number and this is a very mean field means average and the mole fraction. So, that gives you number of contacts AB these also give they are these two are exactly same quantities. Now I have the I have the entropy I am doing all these things in order to do the enthalpy and sure enough once I know the contact then I say ok I can also write in terms of any it makes sense to focus on N B little bit more because they are in our mind they are less in number because B is this or A is this or that is the kind of picture we have. So, then N B Z X A this will be the enthalpy because these are the these are the contact this is the number of contacts and each contact brings a discriminant. So, the enthalpy of mixing is the change delta H mixing is the change like entropy of mixing is the change in entropy this is a new thing that comes in ok which was not there because of mixing and that is delta H mixing. So, now I add the two enthalpy I get the free energy now define flow redefine the following parameter which is famous as a flow parameter which is this quantity chi F is number of coordination number delta epsilon AB by dimensionless dimensionless quantity this is no dimension then we get delta H mixing now this term we are going to rewrite in terms of just introduce the Z Z into delta AB combine that divide and multiply by KBT and then we get this thing the delta H mixture remember now we have explicitly the flow interaction parameter flow parameter. Now, I get the free energy of mixing I get the free energy of mixing by doing entropy term and the enthalpy term and low and the odd I have this beautiful thing. Now, I have a elementary theory elementary I am not yet connected with my osmotic pressure that is my goal I will do that, but now I can start saying some little bit more little bit more teeth into my understanding of chemical chemistry the phenomenology that we did in the other graduate and that you already I already told that now I want to say if they are structure making or structure breaking if they are structure making and structure breaking I will be able to tell them in terms of flow parameter because if flow parameter is there negative then is structure making because they like delta H they like each other more they are more negative, but structure breaking is the flow parameter is positive. So, suddenly I am beginning to have little bit of sense of structure making and structure breaking and I can now begin to see if I can have some idea of structure making and structure breaking then I can put them into a statistical mechanical theory. So, from these structure making delta H epsilon AB if I can estimate of that then I can take it over to my linear zones parameter and I can do a statistical mechanical theory. Now, Florie now said that in case of polymer I am not going to go into details of that into polymer what Florie showed that this number that goes in here this number and this volume fractions volume fraction. So, number of contacts that we have done NA and B depends on the volume fraction if one guy is much bigger than the other then you have to talk of the volume fraction and that is then goes into this we will do derive these things when we do the polymer theory of polymers, but right now I am just going to quote it that because this is a beautiful work of Florie that the same thing goes over and a much more powerful and direct way where Florie parameter becomes a really powerful parameter to talk of ferrous solvent and many properties of the many many properties of the polymer solution because solvent is so small and polymer is so big it now it over that interaction between solvents effective interaction between solute and solvent becomes very important this remember the term effective interaction and we will come to that, but what Florie parameter is trying to do is an effective interaction and we will come back to that again. But before that I want to this beautiful theory delta in the x-sync for a polymer is you instead of you have volume fraction, but you also you have the mole fraction of solvent, but in case of solute the polymer it becomes the volume fraction this I can trivial thing, but this is not that trivial okay. So, now I want to do before going on I want to connect the that we have done little between the ising model how these two models are. So, I now say spin up is equivalent to a particle a spin down is particle b. So, in a lattice I have a spin up that means I have a particle spin down have a b particle. So, when a interaction is spin interaction that if it is favored ferromagnetic interaction then epsilon a a is more negative epsilon and b b they like each other if I can tune this thing of course and a and b do not like each other then that is up and down spin they do not like each other that is again the ferromagnetic interaction. If down spin like each other then that is called anti-ferromagnetic interaction that is also very well known thing, but the main idea is that see part of ising model you remember doing after making all the approximation after those five variables n plus n plus plus and n plus minus n minus minus n minus eliminating all these thing by three conditions of the bonds of nearest neighbors. We get n plus n plus then we did a combinatorics after making a Bragg Williams or quasi chemical or whatever mean field approximation. So, here down I do exactly same thing I write the binary mixed energy as just like I do epsilon a n a a these are the n a a are contact n b b n a b. So, just like there I have there I have n plus plus this is n minus minus this is n plus minus. So, now just like in model we have these three conditions. So, the same as five variables become equal to three variables exactly same thing goes over we will not know as the total energy can be written in the following plan by eliminating all the other variables keeping a a and a this is our n a a is n plus plus and n a is n plus. So, same as before we do not have to do an extra work now or everything has been done by Bragg and Williams we just map it into and use the free energy that exactly happens. So, here is the given the a between then connection between them n plus is n a then n plus plus is n a a here and then energy of the Ising model is this quantity and that we did in the Ising model class remember the factor of four the two and this quantity except z is gamma in Ising model I am sorry about the change of notation, but z is very common in it is very common in polymer literature and while gamma is quite common in Ising model. So, they are the we went by the kind of z also used in Ising model and lattice coordination number actually chemist like z more than however this is kind of gamma is linear it is never interactions. Partly came probably from Carson one once famous takes book of studies mechanics. But main thing is that there is a complete isomorphism between them as a result of that we can now write down the free energy. So, write down the free energy in the case of Bragg Williams approximation this was the free energy wrote down that free energy in the presence of a magnetic field H is this quantity now here the free magnetic field H of course we are doing magnetic field H equal to 0. So, this is the for H equal to 0 or our B equal to 0 this goes over and this is the expression the equation that we need to solve and epsilon here is epsilon ad. So, we get the solution of binary mixture by implicit that has the beauty is that now. So, the Ising model Ising model mapping allows us to explain very important this phase separation of binary mixture wonderful that means we need not do any extra work. But we realize that it is an isomorphic problem this wonderful realization was done by physically in 1952 two wonderful papers where these did and that is gas many other people and it has done also by Guggenheim I should not give only a name other person who do and he has a beautiful book theory of solutions Guggenheim beautiful beautiful beautiful beautiful book at that all the students would really go through I was really in condom it is amazing how these days students do not learn in I should not really start thinking about it, but I find that there is much less and because of the internet probably and because of the our everything at your desktop all the world is coming to you at your computer. But you know what time is to go and and the advantage was that we used to discover many old books many classic books like the Guggenheim books there is one book by Fowler and Guggenheim statistical mechanics wonderful book and another is Guggenheim theory of solutions where he does the quasi chemical approximation actually quasi chemical approximation was done in respective of the Ising model by Guggenheim I think I understated and underestimated the contribution of Guggenheim. So, theory of solution and then it was found by T. D. Lee and C. N. Young that they are they are exactly the same and but Guggenheim did it quite to the force in solving the quasi chemical quasi chemical name was given by Guggenheim and Brad Williams was done in Ising model quasi chemical was done by Guggenheim and these two are the same that was shown later by T. D. Lee and C. N. Young I think it is important to know this history. Now, we will the last bit we will try to connect now the importance of the osmotic pressure and so I said that the osmotic pressure is so important something not realized in the physical chemistry books just give the applications of osmotic pressure osmosis then there are all other biological applications and chemical applications. But first that my guy who is trying to explain binary mixtures and non-ideality and phase separation and binary mixtures, spin order decomposition and many other things we need the force field and this a measure of the force field comes from the osmotic that is the absolutely gold mine and so we already did osmotic pressure now we kind of connect it together to do and so the basic idea now is that I want an effective interaction is so important polymer in every possible way because it is called implicit solvent model that means we remove the solvent we are left with only if a and b and there a is numerous and small b is little larger then I can remove the a and do take on the bb but now the bb interaction is not the pure bb interaction because it is mediated by a and b it is mediated by a and it is actually very different this how do we do that then and how do we get an estimate this is what was done by this great great theory of MacMillan and we are really wonderful theory and a very very sophisticated so what may MacMillan may have did MacMillan was a student of Joseph Mayer and they mapped this problem in a very beautiful and very formidable theory very difficult theory in our book the stat mac book by Bakchi we have this is what a major thing that was done to really bring home the point okay so mayer's theory we did that interaction potential as in a binary mixture if I do a mayer's theory then it would be ua r u bb r and ua b r this would be done MacMillan mayer now said okay I will use mayer theory but I will eliminate out a and I want to get that if I can get that then I have an effective interaction now if I can have a virial series of osmotic pressure my second virial coefficient will have the interaction between b and b I hope but that means I am talking about flow charts and in the flow chart how to get the effective interaction so the basic MacMillan theory with osmotic pressure gives me this quantity which is very important quantity that explains the structure making structure making all non-ideality and this kind of thing so this is a flow chart I hope you can read it that integrate out solvent molecules to obtain solid on the system solids interact with an effective potential WR perform mayer's cluster expansion and then once you do you get these beautiful see is the concentration now or the mole fraction you get the second virial so the osmotic pressure you have a virial expansion of osmotic pressure so what you have is a virial osmotic pressure pi which is the concentration dependence so now I can vary the concentration I can vary of course temperature and I can now get these b to t and then I can measure again with an effective interaction and some form I can use and I can get that so the isomorphism continues MacMillan theory is very difficult theory it is again given in my book in quite a detail asm by Bach she will have quite a bit of that but I you are welcome and invited to do this is very important I write 19 now and they just shows that what where I spend much of my life in the last century so this is the mayer's virial expansion and this is the MacMillan mayer except these are the virial coefficient we know and this BAMM as the MacMillan mayer virial coefficient and then this was very important then what this did what MacMillan mayer did derive an expression for BAMM and so that it is exactly same thing so it is exactly the it can be explained virial coefficient can be expressed in terms of this GDD so interaction between so these quantities are the radial distribution functions they kind of tell you A and B how A and B so now I want to know if this is my B and this is A then how many how B's and A's are connected with these are placed with respect to each other in a random system in a liquid how A and B are arranged with respect to each other is of course fluctuating this continuously changing so we will have an average distribution and that average description is provided by average means is still microscopic time average description and equilibrium structure so this G is a very famous thing in the theory of liquids or theory of disorder system one of the most important equilibrium property these are called the radial distribution functions or the partial radial distribution functions and they form a very important thing of the structure of liquids so before you go any further we will now go into do the structure of liquids little bit and then we will come back so now as I said that in a very low temperature A and B phase separated if A and B do not like each other how do I know that they are phase separated microscopically I can say okay how many A is around B and how many A around A then I find in a in a locally phase separated if A and B do not like each other even at normal temperature when they appear to be homogeneous they might be locally micro heterogeneity is there they will be locally phase separated how do I know that I can calculate how many radial distribution function which it gives me how many A around B and how many B around A how many A around A how many B these are the radial distribution functions these are these quantities how A is the joint power distribution how many A is around A how many B is around B and how many A is around B they give the time average structure of the liquid and now if they are then I would know okay A and B will be lowest and that is here A and B this is A and B this is the lowest and A and A is maximum number that is A and A next is B and B so this A and A A is around A then B is around B and A and B are depleted so this is the way one would try to describe now A phase separation structure making liquid or structure making liquid is this the way we will go into it. So now we will go to the next class and try to construct and describe these radial distribution functions or how these theory of liquids are built to do these things.