 So, there are some simple calculations now we will be doing which will be telling us certain scaling relations which plays a very important role in because the size is so big and since there are so much you know correlations between these polymer chain it makes sense to talk in terms of certain very simple to find out how different properties of the polymer scales with the size n like we already found that the size Rg scales as n to the power half but now that is when the polymers the monomers do not interact at all however in a real world the polymers in monomers interact with each other particularly when they kind of a collapse state like in protein or in many other polymers in a polymer in a pore solvent they collapse and that means it brings little bit of the structure of liquids it brings to us the I have to talk of local correlations I have to talk of the local density and local volume and but however at the same time I do not want to give up the beauty or that this scaling relation that polymer is famous for and that polymer really that really works so the large size of the polymer at the same time can be again exploited to get these certain ideas about the size so so in the Gaussian chain polymer the freely jointed chain polymer that flow de polymer so these are all the same name that flow de polymer then Gaussian chain then freely jointed chain these are all Gaussian polymer they are all the same things that we talk about that so it is essentially means that polymers monomers do not interact with each other so one of the major thing of that that that so this kind of thing so this is allowed so this is allowed this is allowed so as a result in a freely jointed chain all these crossings are allowed that and that means no volume is excluded so no volume is excluded but really world of course you cannot do that in real world you cannot do that you is not allowed this is not allowed okay this is not allowed it might seems that is a small thing but no these excluded volume interactions that the some volumes excluded plays a very important role but how do I talk about it at the same time I do not want to talk up completely the probabilistic description because this is a many body system which are lots of randomness that brings us very close to the concept of liquids that so the so the concept that we developed in theory of liquids like distribution function is already used which is like the distribution function of liquids but also the concept of density and the arrangement here is never arrangement these kind of things can be used in polymer and that is what we are going to do next so and that is done by saying okay and that we know this let us see Gaussian chain but number of monomer-monomer contact we are now going to talk of monomer-monomer interaction number of monomer-monomer contacts between two things are can be shown I had a DAA of that proof of that I do not know where that disappeared okay so number of contact 5 star is the number of contact so number of contacts is 5 star number of contacts is 5 star and it can be shown that number of okay so 5 star each is the 5 star where each monomer the number of contacts made by each monomer so then n 5 star so number of contacts made by this was in my notes but I do not know what happened by okay each polymer each monomer is 5 star so total number of contacts the n monomer is n 5 star this is very very important and I will just show you how important it is okay okay yeah this is not should be here so probability of a given monomer probability is 5 star that a given monomer encounter another monomer so this is not the number but probability but we are going to the number 2 from that so if how would that come it will come because it will be just like in liquid state that there is a volume of a monomer just like in liquid state I have a volume of a monomer just in liquid is a volume of a monomer and then the volume of the monomer which I can take like that with a student volume then then the number of molecules there the contact within the same in liquid state in a shell will be the density so then density of my polymer average density of the polymer which is same is the density of the liquid analogous quantity is total number of monomers divided by the volume of the monomer so volume of the monomer are 3 by D so in D is the dimension this is something which is in polymer is more common now not in flow is time flow you would have put D equal to 3 everywhere but in a physics literature D when is very popular and favorite is D equal to 4 because at D equal to 4 we get a Gaussian polymer so one can do the calculations about dimension 4 by making introduce a constant constant called epsilon and which is the comes from clinical phenomena and the motion group and then epsilon is D minus 4 and so or 4 minus D and D is 3 so epsilon becomes 1 and you would do a calculation where there is a lot of fun one does but we do not need to worry about it so if the monomer volume B is for bead the bead that is introduced because in physics kind of language is nothing but a bead many times the chemists do not like to consider monomer as a because we deal with monomers all the time and but phase is nothing but a bead and neglect model so the bead B so B cube B is the R size of the bead R cube but the R is kept for the end to end distance so this is the volume of the monomer then volume of the monomer into the density and density is total number of monomers divided by the volume which is R cube once we have first we show we have the first and then we can show since R equal to n to the power half I can combine it before and it becomes n to the power 1 by d by 2 and n phi study total number of contacts in my entire polymer chain is n to the power 2 by now this is very interesting for three dimension then I have n to the power 2 minus 3 by 2 is root over n so number of contacts is enormous so again in my if I take 100 million which is a very good number and for polymers then 100 million means 10 to the power 8 so I have 10 to the power 4 contacts there are 10,000 contacts and contacts cannot be allowed so you immediately see that if I exclude the contacts then size of the polymer will increase so the simple analysis tells me the Gaussian polymer chain is inadequate the random work model so what is a polymer chain are doing is nothing but doing the random walk that is why this is the result that we get into size is same as that of a random walk that how much a random walker will go in n step is exactly this distance is root over n which is the size of the polymer however if you now make the random walker no you cannot visit the sites you have already visited once you are excluded from that then the random walker has to take new and new sites as a result the rain 20 seconds the distance the random walker will go now will become larger and that has a tremendous implication in these things and that will do now so I go back again so I have considered a phi star probability of a monomer is encountering another monomer and that is monomer phi is b to the power d and density is n by rd then n phi star is total number so then they become n square and then n square and these are comes on the up so n phi n square and rd root over n so I get d by 2 okay so now then the next let us go ahead that the things that we have done so is if there is one more thing that comes in addition to this excluded volume interaction there is another term that is very popular so the I want to add here so excluded volume because contacts not allowed you might think it is a small thing but it is not because they are large number of then another thing is a different thing is a effective interaction that is used and that is this thing that means in polymer you put polymer in liquid then this is a very good example of saying that there is a chain polymer chain in kind of reduced kind of description but it captures these things and they say these are water molecules or ethanol molecules because polymers are not solid water so maybe ethanol molecules or methanol ethanol is a very good solvent for polymers now I want to but I am not interested in the water molecules or ethanol molecules I am interested in the polymer configuration however this polymer is solvated by the water molecule but I want to have a description where these not there because too difficult to bring all this because the polymer is so big as I told you 100 million monomers a huge guy if I have to put it in a solvent even what I do want to do theory or simulation I will have billions and billions of water molecule I need trillion water molecules that is not possible and also does not make sense because they seem to polymer seem to behave in certain way so the one the concept that came in that one say that what happened if I could average an implicit solvent implicit solvent means I will remove the water molecules behave as if it is a continuum and then I will have interaction these two guys who otherwise are not interacting but they will be interacting because of the interaction just like we discussed in liquid state interact interaction they are proceeding through intervening in molecules just as in liquid state and actually the all the concepts of liquid state comes to polymer very handy then I remove the water molecules instead I have an effective interaction between these two which is somewhat longer distance and there is of course this heart sphere interaction it also interacts with shortness but in long distance there could be an attractive interaction so this effective interaction is the thing to find from arm but we can again look into a liquid state theory if we can get a radial distribution function between them g r between these monomer polymer then I know I have cannot write beta is 1 over KBWR then WR is the potential of mean force and is the effective interaction okay so that is the same kind of game we play that we are removing the water molecules and we are replacing that all the complexity are taken into account by potential of mean force or an effective interaction between them or the indirect interaction in the theory of the liquid state we remember what we did in the derivation of percussive equation or direct interaction like direct correlation function and indirect one which are again goes through this potential of mean force so we are essentially introducing a potential of mean force then so effective interaction is a very important role played it is introduced by Flory so if the polymer is such that if the polymer is such the solvent is such that it does not like the polymer then the water molecules from where we will go away and the polymer will collapse however if they like then they will be in have it inside and then the polymer will swell so in a good solvent good good for the polymer the Githanol is good for many things then they will polymer swell then they do not want to cross each other so they will almost mimic what is the excluded volume interaction much stronger excluded volume interaction and size of polymer goes up and that is a very important thing so good solvent is a is where polymer is large size is large however if you put for example lot of water which does not like the hydropoic interaction does not like the polymer then polymer will collapse and this is shown here that good solvent this is a good solvent then what all everywhere water molecules comes in and then you get a swelling of polymer very important phenomena however if the solvent is out if they do not like each other for solvent then it collapse protein folding which is this very very essential hydropoic collapse so these collapse collapse of polymer is like is hydropoic collapse that is so important in protein folding except in protein folding we do not have a homo polymer what we are discussing here all the monomers are the same that is called homo polymer but protein is a hetero polymer and protein collapse of protein in a pore solvent is so because protein folding we do in water much of the studies or at least large water and there are other parts also but that is what makes so when protein folds all the inside all the things are the hydropoic code like alanine valine phenylalanine and outside here all the hydropoic things that is essentially architecture of protein which is the polymer of intermediate sized polymer but many of the things that we learned in polymer goes are used in understanding protein folding and many other things so polymer is a very important for because of the biopolymers also but I want to emphasize that this the concept of liquid state the densities b by n by r r to the power d and then monomer monomer contact and all these things actually are very useful in the polymer so then the the is a important thing that this is a good solvent at the pore solvent that effective interaction is repulsive good solvent this interaction is attractive so we now have to think of how to describe interaction between the polymers and that was also fluid so if I had the interaction between polymers I can do spatial mechanics now I can bring in the whole apparatus of liquid state and that has been done so then how do I talk of the interaction between the polymers this is the following way this probably used it again to develop the interactions and the the evolution of the coil dimension of polymer between an m now this is a bit this is not effective repulsion energy actually if effective interaction energy that we will see that it can be both positive and negative it can be both attractive so interaction or effective interaction energy that is very important effective interaction energy that we see as we just discussed before that the density or concentration I can as well call it density rho but concentration C is total number of monomers divided by the volume up to the power dd is a dimension now I can now make the following answer it is just a completely Van der Waals that attraction is the or interaction between two polymers in Van der Waals is that exactly what the free energy contribution of Van der Waals because of attraction term exactly if you carry out the free energy calculation of which was given in my book that it is half a rho square and rho is density so here C is the monomer density and so the interaction C square now Florida did something amazing the up to this is just just Van der Waals C square is rho square half k bit k bit is introduced by hand and newt is a parameter which is the Florida parameter excluded many many for you may be following now now what Florida he said okay these newt that I have can be both positive and negative if this is what I say the wrong is called repulsion interaction if newt is negative then I get attraction and that is Van der Waals but if newt is positive I get repulsion and that is just the beauty so now Florida introduced a famous parameter again called Florida parameter that is exactly that quantity 1 minus 2 chi and he derived an expression for chi later which exactly works out according to this so when chi is greater than half then this is negative and this attraction but when chi is small then it is repulsion so monomer this interaction parameter chi simulates when the monomers attract each other and monomers repel each other and we know when monomers attract each other monomers attract each other when you force all that so chi becomes large in a force solvent but chi becomes small in a good solvent then the monomers repel each other and they swell this is the beauty of this description so then one does little bit more this interaction of this is just what I wrote down this free energy is we just say total volume because that is between monomer monomer in a volume bd total volume if I do interaction then I get these are interacting over the total volume and I will get the another if total volume is proportional to the r size r square and c square as n by rd so when I put these things I get n square as I said I get these quantity as the interaction both attraction of the pulse so I get mainly n square by rd term we are done in two minutes and then effect through the consideration of then however I another interesting thing is that in the polymer I cannot just swell it because there is an elasticity that through the through the chain so it is an elastic I cannot ate too much and then they die and that is a proportional to r square there is a potential energy which is like harmonic and that term the elastic term goes like r square because elasticity and one can show it because this essentially comes from e to the power r square by nl square so this elastic part it comes completely very easily from the Gaussian distribution so Gaussian chain has a preferential size and that is an entropic effect so the entropic effect give rise to an elasticity because it is a favorable size is there if you go out from that you have to pay if you come close to that you have to pay so is the harmonic potential that harmonic potential is the coming of the center limit theorem and that leads to you then have to add so you have to add the interaction energy and the elastic energy and this is the total free energy as a size r now you minimize that and you get go low and behold one of the most beautiful result of polymer science if you minimize that you say df dr r equal to 0 then you get r d plus 2 flow size r a for r star equal to n cube and you get r is r is n to the power 3 by d by 2 so this exponent that introduce an exponent that r this flow rate so because of interaction the excluded volume the Gaussian polymer size r get modified to rf and these 3 n to the power 3 by 5 so this goes as n to the power half so because of interaction n to the power half swells to n to the power 3 by 5 if I go with my n equal to 10 to the power 8 then that is 10 to the power 4 and now if I do n to the power 3 by 5 into sorry I have to do then 10 to the power 8 3 by 5 then I see that it is 10 to the power 24 by 5 that means almost 10 to the power 5 so I see a 10 fold increase a little less 8 actually huge increase in the size of the polymer because of the interaction between the polymers so this is one of the most beautiful results of polymer science that size of the because of the interactions which is a contribution of the interaction between the polymers mediated through the solvent and the elastic part and elastic part is from the Gaussian chain that I can model as Gaussian chain as my independence goes Gaussian chain is okay but the size now changing so effective interaction renormalize the size and gets to a beautiful new result and this is for flores flores a r equal to n by this is the result flory is n to the power nu and nu is a flory parameter nu a f is 3 by d plus 2 so flory got everything correct this is amazing flory got everything correct flory exponent we got that nu equal to this ideal chain ideal chain nu half and that is underestimated this error then it goes over to nu equal to 3 by 5 for 3 dimension and this is the some thermodynamics of solution for a Young's theory we will do in the next class okay for the time we will stop and the next few more things more work of flory and and beautiful work of flory continues the saga of flory's contribution continues to into flory a thermodynamics of polymer solution for a Young's theory then you will do solider transition was also done by flory so these two will be the next class okay I hope you enjoyed this beautiful class