 Good morning or good afternoon everybody depending wherever, whatever is your time frame and welcome back to this course on statistical mechanics for chemistry and material scientists. This is a shared course for equilibrium statistical mechanics and we have been studying of late polymer, the physical chemistry of polymer and the picture that you see on the slide in front of you is that of Paul Florey who played an enormous role in fashioning this subject. Before I proceed further let me tell you this course will be little different from the earlier courses which were recorded in Bombay IIT studio. This one is because of the coronavirus epidemic I am recording it from home so you won't be able to see me but rest of the things will be okay. You will be able to see the slides and you will be able to follow and what I will do go through revise the course first 5-10 minutes of what I did in the last class and then we will go into do some new things. This might be a bit of a long East class but and there will be some expressions and derivations but we will not derive the we will not derive the lengthy equations but I will sketch the derivation and you will find the derivations in my book on the same title of the course and there are two other books I want to bring to your attention. One is the book by Paul Florey on principles of polymer chemistry and there is book by Dijan on scaling concepts in polymer science. These two books together just will be more than enough and as I was telling you about Paul Florey who created almost a much of what you know about polymer science today and he is rightfully called the father of polymer chemistry he got Nobel Prize in 1974 and there are many many things that he did in polymer and he did both theory and experiment which is remarkable because he did theory at a very high level. Paul Florey was born in 1910 in a place in Illinois near Chicago and he graduated from Vice State University and then joined industry and from industry he then moved to Cornell University and at the Cornell where he did much of his pioneer work and he also published his book from the Cornell University. Now the next slide I give the plan of the lecture. So, this is something we did in the last lecture polymer size end to end distribution and with an random work analogy it takes you to Gaussian distribution and then we started polymer thermodynamics we will do little bit more of polymer thermodynamics and we will talk two kinds of transitions in polymer, two kinds of phase transition one is a solgial transition while the reason of doing it is that it is it bears close similarity to Mayer's theory of condensation gas to liquid transition. It is just is a kind of a clustering transition here sol means solution of monomers. The monomers connect together to form a branched polymer and which is then become a highly viscous solution. It is a very important industrial process and it is used widely as I have told you already that polymer is one of the most important part of our daily life and from textile to rubber to many many things that we use every day. For example, the your diode in many things is organic semiconductor is a polymer MEHPPV. So, and these many of these things are made by some kind of a solgial transition. So, it is an important thing. So, we will just go through this in a rather simple fashion without the blackboard I will be more qualitative, but I will show the equations. So, in the third slide I briefly do the overview that a polymer that we think that millions and millions of connected monomers. Monomers themselves are somewhat complex molecules and we polymerize them means we add these monomers together and many of the first applications of. So, polymer is since is a many is a large system with a large number of monomers it is a natural playground of statistical mechanics and one of the important property of these polymer is end to end distribution function of a polymer which is expressed in terms of number of monomers and which is end and the but in the monomers. We will talk of effective interaction solvent effect and we will talk of several other remarkable properties and some observations that we have got. In the next slide we shoot the random coil polymer chain and this is a kind of a ribbon diagram and you can see the two open ends of the polymer and in between that there is a long contour where the connected chain connected of monomers is kind of it takes a random zigzag fashion and that is why it is called a random walk. One of the reason that it is so random lies in the chemistry you know in a basic chemistry and that is when I start with a 0 to a 1 and the beginning to the first bond between a 0 to a 1 that bond many of this is cc single bond carbon carbon single bond and they can take visited to a total geometry that means that that means it can rotate in three dimensions. So, that bond can take many many directions and then the next one so you join to one bond which already can rotate in many reactions to another bond which again can take many reactions. So, between if by the time I have 10 some bonds connected and each of them take many configurations we already have huge number of configuration space that means rotational configuration space that this polymer can enjoy and that is a very important property of the polymer. There is huge number of configuration space the configuration degrees of freedom that a polymer enjoys because of this rotational flexibility or rotational degrees of freedom. So, these different polymer that arise because of the rotational degrees of freedom sometimes there is a preference of certain orientations and then one talk of rotational isomer this is a term which was coined by Russian scientist Volkanstein and which Florie also use later. So, in this slide that we are showing that we are starting one then we have these different bonds and then bonds are going and last the other ball at an is a red ball. So, this is the ping pong diagram and in order to get into a distance that we from a 1 to a n from one into another end that is called R the universal notation R and that is what we that is what we will calculate now and that into end vector R can be it can be easily seen is nothing, but some or all these small small vectors and these are small small bond vectors and each of the bond vector has lot of independence from other because of the rotational degree of freedom. So, these bonds arise have the because the vectorial nature the length is could be the same that is the bond length L, but they are vector. So, because of the vectorial nature of the bond and the rotation degrees of freedom R is a random number and that probability distribution of R is a subject of great interest and that is the one which is a Gaussian distribution as we discussed last class and also we will discuss today. Okay now this is we discussed but let us discuss again this is essentially a rough caricature of a polymer which is a rebound we call this as a rebound diagram and you can see there is an end to end distribution going on here and these end to end distribution is you can see from one end to the other end that that because of the large intervening contude going from one end to the other end which can take many configurations in a certain way it reminds you of rope or is dangling or dancing rope or snake and this ability to take many many configurations determine the end to end distribution. So, roughly the size of the polymer when you say size of a polymer we essentially talk of two things how long the average how far is my end it's average quantity what is the average distance and since it's a random quantity that average is a mean square no and this is very close close analogy is random work so as if it has our the monomer to monomer distance is almost as a random work and we will discuss this now immediately little bit extremely important to realize that the end to end distribution along with the end to end distribution to give you the size of a polymer there is one more quantity is the radius of guidance radius of guidance is nothing but the kind of momentum inertia that means you take the position and m i r i square and sum over and that gives you also a called r g that also gives you a measure of the size and these two quantities the mean square end to end distribution size and the radius of guidance are very closely connected to each other they have the same size number of same dependence or number of monomers but they have some numerical factor which make them different from each other so we can use either radius of guidance or mean square root mean square distribution end to end distribution as the end this is what I was telling that we will be discussing little bit of the random work and then go to a1 a2 a3 and then go over and go to end point that two end points are given by two red spheres and one important thing to know here that when I go from a0 to a1 and a1 to a2 then a0 to a1 one filter can rotate and can take a large number of orientations this this is not quite 360 degrees because there is some steric hindrance posed by a molecular molecular architecture but it is fairly flexible that means it can take many many orientations so now you consider that you are orienting one from the other and then to other so by the time you have just 8 or 10 monomers connected there's a 10 mark already you can see a huge number of configurations that can this polymer can take is extremely important to realize the inherent huge flexibility of a polymer the huge number of internal degrees of freedom the configurational degree of freedom that a polymer enjoys and that is very important part of its properties which flurry realized very early along with volkenstein and volkenstein coined the name rotational isomer these multiple configurations that polymer gates was recognized by both these two giants of polymer chemistry so now if I want to go into a distance we call it r then you realize that this is nothing but the sum of the vector just like random work I want to go one place to another place to another place to finally place then where I go finally I if I number of steps then I add up the distance the vector and that's why vector is important depending on orientation that's why orientation is so important so you can now see the end to end distance end to end distance both the vector and the scalar is a dynamic quantity and is also random hugely random because and you see it now in one position then if you are in a solution polymer in a little later you will see in a different position so it is completely moving around coiling and uncoiling and things like that so the way to say that we do try to have an idea of this is what appears to be extremely important extremely complicated but it turns out to be not complicated not only not complicated it is a beautiful universality that prolifolkenstein realized and later was I told you has been enormously exploited in physics community where polymer physics is a big subject now so this is the sum we get and now we are to get a distribution of this r and r vector as I told you this r is a stochastic quantity it is a random quantity now the way to do that then is one many ways it can be done it can be just apply you don't have to do center limit theorem we can apply to random walk but center limit theorem gives a very easy way access it's a very well known theorem and I discussed last time that if I have a quantity s which is the sum of n number of random quantities and then this when I sum it up that sum is a is a Gaussian distribution that's called central limit theorem I don't want to spend too much time on that but it's a very robust theorem as I told you last time mathematicians are not given to give this kind of name fundamental theorem central limit theorem that's not there is more infirm physics and chemistry we give grand noise name like theory of everything or or very great thing but not mathematicians they're very conservative people so when the central limit theorem it is really central limit central theorem or probability theory that if I have n number of random variables I add it up then that goes over to a Gaussian distribution with a mean you and the mean is very easy to understand in this case because the it is rotating and so you can easily convince yourself the mean would be zero and but then then everything holds on to the width of the distribution which is the standard deviation which gives the size because if I now want to get the size then I have to get the average r square second moment of this distribution okay so this is one little bit on central limit theorem you can find it on do it do it do it you know uh uh internet you'll find huge amount of things on central limit theorem but main thing is that in to be large but as I told you by 10 12 you already have fairly good Gaussian distribution this is one of the things we always use in computer to generate a Gaussian distribution we generate 12 random numbers we add them up and the sum becomes the Gaussian and that we use one of the thing is that this is to be mutually unrelated that means from one step to another from one over one to one over two and one over two to three so one to two and two to three and they have to be independent and that is guaranteed by the rotational degree of freedom so then one can further quantify these things by taking the r lm z axis then you project it on the r and you get l cos theta i and uh and then uh and you take our ambient distance as z axis and then project on that then l cos l is the length l is the monomer length or bond length and theta is the one that orientation gives you that's what gives you the randomness so since theta is random cos theta is random and now we can do r squared and I can evaluate the cos theta with zero to pi and I can evaluate the r square average and that's exactly done here that we do that averaging because uh i and j is uh these are on bonds i and j are on bonds so one bond and another bond is uncorrelated so then I have cos theta ij is just one sum because otherwise this way so then we get n there and so r square is nl square so it's a beautiful result really beautiful result so it tells you mean square is nl square uh so for that I don't need the Gaussian distribution for this I don't need this entire limit theorem but I need this entire limit theorem to get the distribution of r and I can get that they are also through random work so there are two points here that mean square r square is scaling as root over n which is extremely important result because that gives you the size of the polymer and then comes the distribution so the distribution is Gaussian as I said here there is a r square factor here here I am asking the what is the probability that there is a little bit problem here that n should be capital n on both this side or small n on both the two sides but you will see these mistakes are not in the book but there are little bit mistakes here and there in these in the slides but which you go back and look so the distribution the end to end distance r is for capital n r should be there then this is distribution the the probability that you have the end within a distance r and because of the spherical symmetry of the system so at least in a in a volume element in a shell that shell is 4 pi r square so at a position r it is exponential minus 3 r square by 2 n l square plus the normalization but you have to multiply by 4 by r square okay so this is this quantity that this end to the way that this result end to the power half and that that size then you can do the radius of guidance this way and exactly same way you can derive it is also the second moment you just change your change your central of your orient system and you again see that it is r g is root over n so but basic central result is the following that probability distribution is Gaussian with 3 exponential minus 3 r square 2 n l square as a standard deviation of the width of the distribution scale as root over n and that is what we call size so as I told you before radius radius of the integration which gives a good measure of the size of the polymer and the mean square end to end distribution they are essentially the same thing and so as the num as I told you already a polymer is millions and millions of monomers that means n is upward of 10 to the power 7 say or 10 to the power 8 now if l is couple of angstrom then you can imagine that n to the power n to the power half 10 to the power 8 it is 10 to the power 4 so if this is 2 angstrom then 10,000 into 2 angstrom is essentially 20,000 angstrom so radius of guidance is 20,000 angstrom that is essentially roughly a huge huge square so you can understand this monster monster polymer is sitting in solution and they are not one polymer like that there are many polymers so these monsters 10,000 20,000 angstrom with radius yearly spherical things are suspended in solution so that's what makes the polymer solution thermodynamic so interesting it is not a binary mixture it is a binary mixture but it's a binary mixture where one guy is 10,000 times larger than the other guy it is very important to get a sense of number that what makes polymer so unique that polymer solution you have this big thing floating around and many of them will precipitate if they are not preferably solvated and we see these situations in proteins you know in a folded protein is almost again a globular protein is nearly spherical but they are of course much smaller they are a few hundred many of them and so they are much smaller but even then in solution is a very interesting area so polymer solution stands out from so many times we say they are the same it is not quite the same polymer solution is the polymers are much much bigger quantities though we do map the protein into a polymer and use the language of polymer science or polymer chemistry to explain proteins okay now this was done by Wolkenstein and protein now another beautiful calculation that floating and this is just a chain for it i have said okay i wanted to find it in d dimension he of course did it in 3 dimension but he also did it for d dimension he said how do i find the size of a polymer in a d dimension he said okay he has to now say okay as i told you that this polymer is a monster it's a big thing so there is a end to end distribution there is an interaction between monomers so interaction between monomers depends on the concentration of the monomers so what is the size of the size of the monomer but d in d dimension are given d dimension r square in two dimensions now if that is so and if what a monomer has a volume b then i can consider that that a concentration of monomer why i need the concentration of monomer because i need the density why i need the density i need the density so that i can conceive of interaction why i need density to concentrate interaction because when they monomers are next to each other they are close to each other they are They are interacting and they are far from each other, they are not interacting. So interaction energy is proportional to square of the concentration. Now, where did you see such logic before? That where the interaction energy is square of the density of concentration. In current class, I would have asked you and then I would have disturbed you to come up with the answer. Ultimately, I would have got the answer out of you but since I cannot do it now, let me tell you the answer in Van der Waals theory. In Van der Waals theory, there is a repulsive term P plus V minus NB plus A by V square, N square by V square. That's where exactly same logic Van der Waals seems that in order to interact, both the two molecules should be in a small volume element. And in this, what is the probability of the volume element? If they are independent of each other, then one monomer to be there is the concentration or density. Second monomer is also concentration, so it should be productive. So that's why concentration is important which is shown here in N by RT, which is the concentration. Now, in case of chain obeying, we know in this case in this RT, if it is fluorine polymer or Gaussian polymer or that is called ideal chain, then R goes to be to the power N3. In that case, the number of monomer-monomer contacts can be very easily found out to be N5 by N. Because they are N number of monomer, so total number of monomer. So you have a contact, one polymer, one monomer in contact with other, this N1 minus d by 2 we can find for the earlier one. And however, N is this 2 minus d by 2. So this is now what we are going to... So this is, we will continue with the logic in a minute. Before that, let me consider a little bit now. I want to understand polymer-polymer interaction in solution. As I told you, it is a suspended solution. On many occasions, if the solvent is considered good solvent, it even likes the solvent, then the polymer gets so well and then a lot of solvent molecules come inside. So it is kind of a porous thing. So polymer has, within itself, in its size, I put the body in that volume element. It has all these polymer molecules. So the way we sometimes talk of that, if the polymer likes each other, then there will be more monomer inside. Then effectively, polymer and polymer is not going to like each other. However, if the polymer monomer, monomer of the polymer doesn't like a solvent molecule, then they will push these water molecules out. I will find, effectively, because the solvent molecules will give effective force, then the two monomers attract each other. So I have a situation, it is all devised by the story, but depending on the quality of the solvent, good solvent or bad solvent, two monomers effectively, there is an effective interaction like each other or they don't like each other. So this effective interaction is the one which plays a very important role because if I don't want to talk of this solvent, I will talk later, but not in the beginning, so now I will do something without that. So this is the good solvent, the solvent molecules come in and the polymer gets so well in a bad solvent, however, collapses. And there is actually a transition between the, you can change the quality of the solvent by changing temperature and at certain temperature, this transition takes place, which is called, which is called theta temperature. So this is the effect of solvent is very important and you have it in the, you will get these, you will get the slides along with the talk and you can look at it and the solution.