 With this third lecture, we continue our study of polymer solutions. So now, Flory said okay, there are two very interesting things there. One is, keep the polymer in a good solvent. If the polymers, monomer of the polymers don't like each other, then we have a effect, repulsion between them. However, it cannot go on swelling because when you increase, remember the distribution is e to the power minus r square, where the r square comes because all the many configurations. What is the configurational degree of freedom tells you? It tells you the entropy. Now, you can see e to the power minus r square has a maximum, you have r equal to 0. But this remember this is an average because many different though orientations, that's why r0 comes. However, it also tells you cannot stretch it too far. This is called elasticity of the polymer. So polymer loses lots of entropy if you stretch it. So there is these two forces acting opposite to each other. One is that repulsion energy which tries to push it out but elasticity of the polymer which wants to keep it at certain size. This is what we realized and it is a beautiful calculation. Okay, say internal concentration of the N by RT as we discussed already. Then I can consider repulsion energy with this equation that every repulsive is half k by 18 nu Pc square. So this nu T is the effective parameter and as I said this is essentially nothing but Van der Waal's theory that we are doing. And so see now here the important thing to know two terms nu T and let us know VT, not nu it is VT, which is a temperature dependent excluded volume parameter which is like very much like a parameter in Van der Waal's theory. That means it is a volume parameter but it essentially takes care of the interaction energy. Now and so see is the concentration in the number within a volume. So volume is R to the power D number is there. So this is repulsion energy. And Florie introduced this parameter 1 minus 2 XB to the power D and we will learn a lot about it later. But it is just to show that it can go either from plus to minus. And then integration of the volume element if I integrate with this volume element of this C to N to the power RT and then I get this quantity volume element then I will get the total repulsion energy is N square because there is coming with N here RT and then I take all the all the all the monomers then I get N square. So integration of volume element essentially meaning taking all this. So the important thing is that repulsion energy scales as N square by RT. Now if I take the elastic term which again is coming. Now it is nothing but what is in the exponent of Gaussian distribution. So R square by N square that's what I had in the Gaussian distribution. So since e to the power volumetric distribution gives me a value of a quantity X e to the power minus X by KBT. So I just rearrange terms I get the elastic terms trivially KBT R square by NB square. So I have the repulsive term of N number of monomers connected in a polymer in a chain is N square by RT KBT V and V is essentially Vendol's A and then it plays a very similar level of A and comes out in a very similar way that we have already done Mayer's theory. When you do that then we see we add the two free energy repulsion and repulsion and the elastic term which is an effective attraction which is kind of putting it closer together. When I add them I get these things sorry it's a little bit but we will give you the idea. I have screwed up equation by V by N square by RT that is where just two terms are added. Now the important thing is that I minimize it with respect to R and if I minimize R with respect to R then I get a beautiful result. This is because in a DVF DR equal to 0 then there is a R to the power T plus 1 it becomes and these terms come R. So D plus 1 and R but they are in opposite side one is numerator and denominator. So when I rearrange the terms then I get there is a negative comes from R square by RT and positive comes R square by RT. So they have worked out right. So you get then RT D plus 2 scales as the N cube and then you get the beautiful result. RF is for flow rate that N to the power nu the radius of flow rate in the nu is called flow rate parameter 3 by D plus 2. That means so when I get D equal to 3 it becomes 3 by 5. Very important result D equal to 3 I get 3 by 5. So suddenly you see in ideal polymer chain these N to N distance scales as N to the power half square root in half nu is half. However in when you take attraction and impulses all these things into account and in a good polymer solder then you get this huge difference. That means N to the power half because N to the power 3 by 5. But when you go to D equal to 4 another surprising interesting result D equal to 4 it is 3 by 6 go to half. So in 4th dimension that is very important or theoretically not much relevant in the real world but you get ideal chain behavior partly because when you have 4th dimension then the concentration of polymer there is so much more volume around each polymer than this repulsive interactions and all these things are not that important. That's why you get back the ideal polymer chain but 3 dimension and 2 dimension things are very different. Now it is we have discussed this before but I would like to emphasize it RF is N to the power nu for ideal chain nu is half but it is not so in A. So in the real chain it is very very different. Tharman N is a polymer solution we already did quite a bit because at the end of the day it is the same thing that we will be having this take care of these 2 forces one is the interaction with the solvent monomer solvent interaction that's one term which is I already said 40 parameter chi that will come here and will be a little bit more quantified here and then we will do the mixing of polymer in a solvent. So solvent has to come in. So in addition to the entropy that we talked about and polymer interaction that at all you will have one more entropy that entropy is a mixing entropy with the solvent than the polymer. So basically the entropy term here is essentially the arrangement of polymer chains that exist. So what one considers in Tharman N is a very famous theory that one considers that the gel lattice and the lattice in polymer because polymer is so big much bigger than solvent that one can consider as I told you as a random work. And that since the size of the monomer is not important one can as well take the advantage of the lattice. So now we have to calculate that how many ways or make a estimate how many ways I can put monomer the polymers, monomers of the polymer and the solvent molecules on a lattice and that will give an entropic term. And there is an enthalpy term because interactions between solvent molecules and solvent and polymer and polymer monomer monomer. So enthalpy we have three terms and we now we already considered that effective interaction of these kind of good solvent but the good solvent here is the same thing and what we are doing is very similar. But we are doing a little bit more justification of the polymer or polymer solvent interaction that is a little bit done better than quantify that. That is why otherwise nothing different from the what we did earlier or say minimization of free energy and everything will come. So when you do that we said how many ways I can put the polymer then you can see the first term and the second term on the right hand side. This is the free energy of mixing and free energy of mixing as the entropy term. That means how many ways I can place a polymer N of size N, polymer size N let's say Np, N is the total number of sites and then and calculate that is entropy of mixing because I am mixing solvent molecules and the polymer and I am laying down the polymer in the lattice. And how many ways I can do that essentially same as binary mixture entropy of mixing. When you do entropy of mixing in your thermodynamics in binary mixture what do you do? You do N factorial divided by N a factorial N b factorial and then that gives you the omega the total number of ways and then you take log of that get the entropy by using Boltzmann formula. And that gives you X1 ln X1 plus X2 ln X2 you can also write X1 ln X1 plus 1 minus X1 ln 1 minus X1 because X1 are X2 mole fractions together they become what? It is a very similar thing here in polymer here however one has to take care of the fact that you have a polymer is a monomer. So the first term and by polymer constitute of many monomers the first term takes care of that fact. Second part is just the solvent part and phi is the volume fraction the size of the through phi enters the size of the of the polymer. Now the important thing the chi I said is the floating parameter chi that has exactly the same thing we have in binary mixtures and we did the binary mixtures or you are not even looking my book by many mixtures as you done in great detail. That is the if I a and b then a interaction and a b interaction and b b interaction when a and b like each other in this parameter that means monomer and of the polymer and the solvent they like each other. Then the chi term that you see on the first term on the right hand side will be negative and chi will become negative on the other hand even. a and b here monomer of the polymer and the solvent they don't like each other then and a and b they like each other and a like each other b like each other then this term is become because then chi and chi are negative then the chi becomes positive. So a good solvent chi is negative and the bad solvent chi is positive. So through this parameter we essentially what flow we did use the ideas of entropy of mixing and the entropy of mixing and the binary mixture theory the ideality non-ideality parameter that we know in the breakdown of Raoult's law. So the idea of this classical physical chemistry is combined with statistical mechanics or statistical concepts to develop this statistical thermodynamic theory of polymer solution in a very elegant and very simple way which has took the test of time. So this again the same thing this is side because I divide the total number of sides and then you get the entropy of the solvent as I said an entropy of the things I am not going to derivations you will find the derivations in many places including my book and also in give video. So now I want to calculate the entropy is the entropy of laying down the polymer into a lattice side polymer of volume fraction phi lattice of insides that's why the first n in the denominator comes. Now monomer-monomer interaction as we just discussed is the Van der Waalsen this KBT I put KB equal to 1 KBT by the pro-reparabiter chi for monomer-monomer phi square the monomer solvent because monomer and solvent it is phi into 1 minus phi because 1 minus phi is the solvent and solvent solvents 1 minus phi square. There are three terms what I do now I next add all these terms together and I keep okay I keep in the term which is linear so I find out in that sense there is the maximum contribution comes from the one that I need is the monomer solvent interaction is phi into 1 minus phi. So this becomes the important term the first term becomes important term for me in this understanding the thermodynamics of polymer because solvent solvent polymer interaction is chi phi into 1 minus phi. Chi gives the relative interaction and phi into 1 minus phi gives the probability that a monomer is interacting with the solvent molecule. Then I add this all these things together and then I again minimize and when I minimize the free energy then I get the theory theory of the polymer solution and that theory when I do with the value of chi it is essentially Van der Waals theory that then explains these good solvent and bad solvent collapse in terms of these things. So the chi in the good solvent chi is negative but when it crosses over to poor solvent then chi is positive. And this is exactly the thing that you get the free energy and then you find out what is the side and you find there is a crossover in the size from depending on the value of chi. Okay this is the essentially this free energy embodies the thermodynamics of polymer solution and proteagin theory. More details are worked out in my book. I am sorry we don't have but we will try to put a slide there on these things. The next slide is missing. This is the theory that you know we know experimentally when we are doing a transition we are doing a we find a transition. That means when I add many important things with the gel many of the applications of material. What do we do? We go on adding the concentration in these monomers and then we sometime add something to gel them together. But what happened when the monomer concentration goes beyond certain thing concentration or temperature is lowered. Then we find there is a dramatic transition a phase transition that takes place from salt to gel. And that is called the soldier transition is a very well known phenomena which is essentially very close to what we do in is very close to what we do in in mayas theory. So soldier transition in very similar to gas liquid transition and the reason of my including in this course and also in the book is that to show you how mayas theory is used. Mayas theory of 1937 and Florin did almost at the same time and Stockman did later in 1941 the same theory Stockman was a student of Joseph Mayer. So basic idea that instead of mayas clusters we have the real polymer clusters. And you remember the mayas theory two guys come close to each other and we have a maya function here we have the maya function is placed by something called functionality f and the polymers like nylon they never form they are trade like they never form a gel but they become very long and it is important that they don't form a gel for all our textiles. But in many other case like rubber we want them to form a gel that is when functionality has to be catered. So I briefly tell you now this method of soldier transition and there is a basic idea then is that I have a solution phase where viscosity is low and maneuvers are kind of dispersed and then this underway polymerization and they form a giant molecule and they precipitated out the reason it is important to understand it because the soldier transition if we can control we can control the property of the gel. And so this exactly it follows like mayas theory system in identical monomers and carrying identical functional groups. So monomers react with each other to form a polymer and the functionality is f chosen very similar to mayas f function also function with f. Now we have to say at the after that some stage of the polymerization we have mn gives the number of monomers of size n and then some of mn the last equation in the bottom is the total number of polymers. And nmn is the total number of monomers is exactly what we did in mayas theory the nmn is the number of n clusters of maya nmn is the total number of molecules or atoms the system name and the nmn is there also the capital m was the size of the polymer. Okay so the total number of ways now we do exactly what maya did so we can remember what we did the mayas partition function when here that is the number of ways I can form monomers timer. And so this is our essentially our partition omega is partition function q and in this whole transition we do not we interaction is already taken into account by forming monomers bond as I told you that is essentially maya function. So we all we need to find out how many ways I can form this and then that is that exactly maya theory omega n to the power omega n by n factorial omega n is what you call the n or maya cluster integrals so it is exactly maya partition function. And then we need to know what is the maya here remember in mayas theory we did omega n or bl in terms of the irreducible cluster integrals by time so that is now done in this case the omega n is evaluated and can be evaluated directly and that is I will give it here. The story goes in the following maya theory I told you the relation between cluster integrals the maya did the expression of the maya cluster integral q in terms of reducible cluster integral bl but the derivation of bl in terms of beta k was highly non trivial and that was done by his wife. And the maya gopard maya got the Nobel Prize later in nuclear shell theory who was didn't have a job because those days women it was very difficult for women to get a job so and she was she came from America and then was in Colombia for some time they moved to Chicago in Chicago she was not doing the job in the beginning. But it was the maya gopard maya who helped stock maya which is in stock mayas 1941 paper that is that is acknowledged that but she didn't want to become a co-author of the paper. So omega n neither was she co-author of the maya's paper. So those days the women many times did not get the due credit the dessert that is something which one of our famous scientist Dan Frank repeatedly brings out these days that many of the things in old days were done actually by women and that were not given much credit but whatever. So this is the expression in terms of the f and n and when you do that put this do that should be f n mistake f n minus one factorial so there are two factorial actually one factorial sign is missing. But when you do that now you have to do exactly like in maya still you know why now maximize the omega that's this thing you know in statistical mechanics same as many magicians of free energy. Maximization omega is same as maximization of entropy and that is same as many magicians of free energy and when you do that I get a beautiful relation which is in terms of the most probable distribution which is actually parallel to the one maya's case. Maya's ML maya's theory that has now that where there comes a fugacity in maya's theory here that quantity is by zeta and beauty of that I can now evaluate e and zeta and the a is the following expression what you have. If n one minus alpha square by alpha and alpha is the extent of reaction very common in chemistry that means how many the number of function. So if I don't bring that's why absent problem polymer cyclization comes on and as an hypothesis so then alpha is to you can easily convince yourself this is the fraction of because each each reaction consumes two functional books that's why you see the factor of f total number of function so extent of reaction to n minus m by f n that's the extent of reaction and when I do the minimization with Lagrangian multiplier then I get the f n one minus alpha square by alpha so one does exactly maya's theory one does the Lagrangian multiplier parallel all these things then now I got the even star now I can do some science what we measure in experiment like in viscosity is the weight average molecular weight that viscosity diverges so I now do first I define okay I know f n so I do what is the weight fraction of an n minus m what is the weight of gigantic cluster that is going to appear here what is the weight of n mark that is m n by n and then I put this m n expression here last equation and I use the a here and then I get I get made a small transformation of variable zeta the main expression here last equation and I use the a here and then I get made a small transformation of variable zeta to the power n there is a f to the power n comes that becomes x to the power n on the extreme right of my right hand side equation and fn minus n factor so that is wn okay so I go by I want to correct little bit this is f to the power n the n that is subscript here should be superscript is a power okay then I get this equation then now whatever is molecular weight this is the omega wn is a distribution so I know to get the normalized distribution I have to multiply by n and sum over wn over all n multiply by wn sum over all n when I do that this sum can be done and this is one of the another two the force and the beautiful result of statistical mechanics that I get this beautiful expression of weight average molecular weight which is observed by light scattering observed by viscosity and all other experiments now look at this beautiful equation there is something very nice about this equation and some unusual when alpha into f minus one equal to one I repeat when in the denominator on the right hand side extent of reaction increases such as that alpha into f minus one become one then the denominator diverges so when the extent of reaction becomes one over f minus one then f minus one alpha equal to one so that extent of reaction when polymonomers react to that extent increase as it is going on reacting reaches one over f minus one that value then it diverges and that is exactly what is shown here that that weight average molecular weight is undergoing a divergence so plot of a system of tri-functional unit where this becomes this extent it diverges and extent of reaction in tri-functional unit f is equal to three so three minus one is two so two into alpha when alpha become half then for a tri-functional unit this diverges this is where you find this divergence into the A things it is exactly like similar density theory now so summary of this so this is essentially same as appear in some of the liquid phase and this is one of the beautiful theory where Flory used it first then the stock mayor did it and as so we need to I just briefly summarize you polymer that we done we have done the end grain distribution and polymer size then we have done theory of polymer solution which has been remarkably successful and it can explain polymer swelling it can explain polymer collapse it can say this crossover from swelling to collapse at the theta temperature and Flory stock mayor theory of soldier transition essentially the application of a mayor's theory to polymerization and stock mayor was a student of mayor but also worked with Flory and he but Flory already did that soldier transition Flory did that but stock mayor used to make it more rational by using the mayor's theory and that is the very nice and they told you Flory was given Nobel Prize single Nobel Prize in 1974 people say Folkenstein should have shared it but but given the Flory's contributions to gigantic because he not only did the theory of polymer size distribution the theory of non-ideal polymer that is into the new new country by five the theory of polymer solution which is Flory against the soldier transition that is also Flory so that is what he was justified to give Flory the single Nobel Prize to summarize I want to tell you that this theory essentially almost like the one that we did in the course is essentially is a celebration of Flory called Flory who as I told you in the beginning was a remarkably human being who not only just did this wonderful work he did work in many other areas biology in many many places the Flory theory Flory is that's why he is also very important and respected name in biology and he is a humanitarian with a lot of workforce scientists in underdeveloped and condition countries and under difficult conditions so look up the Flory's theory of polymer look up the Jan's book of scaling concepts in polymer physics which is takes many of the Flory concepts to a much higher level and last I say the one of the reason probably Flory was given in 1974 Nobel Prize though many of his work were done before 1940s and 1950s reason was that in 1970s just after critical film and a theory was developed by Kenneth Wilson, Benjamin Michael Fisher and all these people and that is the time 1972 Dijan did a wonderful calculation of polymer excluded volume he mapped it into a magnetic problem and he could essentially do renovation calculation of the polymer problem Michael Fisher extended that and they discovered all the Flory equations all of them they discovered that yes it means square size scales as into require u and u is 3 by 5 for three dimension and half for three dimension then they discovered the Keta solvent polymer collapse so whole of Flory theory was developed and derived by using much sophisticated methods of polymer physics by Dijan Michael Fisher and some other people in France, in Germany and that probably can be realization that this what Flory need goes far beyond just the polymer it is a beautiful theory of statistical mechanics a contribution to statistical mechanics multiple comes to be contribution to statistical mechanics of interacting system okay I stop here today this is the last lecture on polymer and I strongly recommend you to read our book my book and the other two books also okay I hope you enjoyed and thank you for your attention bye