 Welcome back to this course on statistical mechanics and what we are trying to do here is to develop an understanding of basic phenomena from the point of view of statistical mechanics and as I repeatedly tell that the under graduates understanding our under graduate studies of physical chemistry is wonderful going through all different phenomena which then finds use in biology, biophysical chemistry, materials, organic chemistry all kinds of things that physical chemistry. Physical chemistry is sometimes called the soul of chemistry but it is certainly the basic understanding of chemistry and also that branch of physics which interface with physics and chemistry. During my course and travel through the world I found one very interesting things that what we studied in physical chemistry in universities of United States are the that what we call physical chemistry theoretical physical chemistry liquids polymers and all other aspects of biophysics protein folding they are all studied in the physics departments in Europe like in Japan in not only Europe in Japan Europe in France and Germany they are very much so theory of liquid is the physics department and chemistry used to be the traditional chemistry colloids organic synthesis things like that the reason is that the physical understanding the physical chemistry or the things that will are in in under graduates requires lot of mathematics and the concepts which are very common and already existed in physics like Liouville equation I told you which came to be VBGKY or the VBGKY Bogliobov and Bond they are physicists, Yofan physicists, Karkut was a physical chemistry chemistry department, Grim was a physicist, Onsager was a physical chemist, Trandas chemical engineer then other great guys like Monzyk they are in chemistry department. So, Stillinger is a chemist and instead of Hormone is a physicist. So, this branch of statistical mechanics has physics and chemistry working together to unravel to understand these rather complex systems. Now, we now start one thing for next 40 minutes or so we will try to do a very very interesting things which is a big subject by itself just like in theory of liquids there are books on theory of liquids here also there are books on endless number of books on polymers but our emphasis here is to introduce to students the concept of statistical mechanics to understand polymers and there are some beautiful work has been done several Nobel Prizes have come. The picture that we have here is that of Paul Flory and Paul Flory is considered the father of polymer chemistry or polymer physics and we have this is chapter in my book in statistical mechanics chemistry and material science and we have taken this picture from his book and Paul Flory along with many other people many other big names are there in that field but he introduced the concepts of statistical mechanics concepts of probability and he modernized the polymer science that is why one has so much he got a single Nobel prize for this I think 1960s or 1970 sometime like that and so what we will do many today is many of the things that Flory did and he did pioneering contributions on the distribution of polymer size into a distribution of polymer introduce the concept of excluded volume which nobody thought will be so important and so interesting and then he also went on to do a theory of solgyl transition and so that his his his stamp on the polymer science is really amazingly diverse and he was both an experimentist and the theoretician he did many of the experiments himself and he then developed the theory. So this was probably one of the last of those kind of gentlemen who did both in physical chemistry both did theory and experiment but of course he had to be organic chemistry also in order to synthesize these large polymers. Now we continue with Paul Flory's work so with millions of connected so polymers are what we have one have to remember their millions we are not talking of 10 particle or 20 particle things we are the conventional polymers that we use are every day nylon and many other things teflon or rubber all these things coming they are millions and millions of monomers connected and this is a natural playground of statistical mechanics because of this note the term natural playground of statistical mechanics that is very very nice way to put into it because this completely many body system these are statistical system and there are some beautiful things that please talk about like critical phenomena where there is a long range correlation emerges that density fluctuation in one region gets correlated there is a fluctuation at a distance thousand angstrom apart that means 50 100 molecules apart or 500 molecules apart in polymer there is already the correlation because a polymer is a connected thing but then one monomer knows another monomer which might be 10000 monomer apart along the contour along the chain because they cannot cross. So this excluded volume introduce by Flory introduced a long range correlation into polymer which that is what later Pierre Dijon and Michael Fisher use the concept of critical phenomena to understand polymer science and polymer physics in a much much detail way. So while Flory what did what we will use the term polymer chemistry or polymer physical chemistry what Dijon and Michael Fisher did was considered polymer physics but you know they also the same thing. So it was by initiated by statistical applications were initiated by Paul Flory as already so what Paul Flory did and there are some other gentlemen also did the end to end distribution that means if I have a polymer long chain then it is very important to know what is the distribution end to end because from that then I can get the size and size of a polymer is very important it is not only measured in light scattering that also plays a role in giving rise to the viscosity of the polymer solution and many other properties of the polymer. Okay and then this effective interaction or excluded volume interaction was introduced by Paul Flory and these remarkable many, many theories it is one of the place where theory and experiment agree extremely well. Here is on us is a polymer is called random coil this is random coil because they are connected and even if the saturated polymer means they are tetrahedral one C connected to another C but then because they are tetrahedral. So beyond that point it becomes again another tetrahedral but then this guy can rotate around this guy. So by the time you have come 1, 2, 1, 2, 3, 4, 4th monomer that it already has a very large number of configuration space. So because of that this distribution end to end distribution has huge number of configuration space it can access to that is why you have to talk of a distribution like in statistical mechanics. So this is so basically this is what I said is described here that if we start at one of them like this with this guy then how you go around like that and come back okay we start at A then go low like that from here going like that then you get essentially beyond the point a random coil. So if I want to know the position of a monomer nth of a specific monomer which n contued the stand apart then all I have to do is to sum over all the vectors that is shown here that I sum over all the vectors and then I then I will add this vector this vector with this vector with this vector with this vector and then if I have then I this vector. So this if I add this vector this vector this vector this vector then I get that. So if I want to get these distance then I have to sum over all the vectors that is very very nice these vectors are clearly random because of the tetrahedrality because of the rotation around the bond we called it backboard and that gives rise to a wonderful thing is that so I end to end distribution is sum over these things and these are projections and they can take many values so they are random numbers. Now that is now we have if this distribution end to end distribution is a beautiful things from central limit theorem and polymer size distribution. So that should be the title the central limit theorem and polymer size distribution or polymer size distribution and central limit theorem. So central limit theorem is something we discussed a little bit before but let me state again that if I have a n number of random if value I have one variable x and x takes n number of distributions or I have x n number of random variables which are taking all values within a certain range and if I add them up I sum then if these my random variables which are weekly correlated or not correlated random number then the sum becomes a Gaussian function. This is one of the most powerful theorem in the theory of probability and that is the name central limit theorem as I always tell in my class that mathematicians are not given to unlike sometimes physicists the theory of everything and many other things that we do mathematicians are very conservative people they are not used to given very big big names they very rarely say central limit theorem or fundamental like they have a fundamental theorem of algebra and that is the most important theorem of algebra you know that which gives you that polynomial of the n as we have n roots and that that depending on the coefficients of a signs of A and D the complex conjugates and complex roots have to be in equivalent that is the fundamental theorem which is used every whole complex analysis is based on that fundamental theorem. Similarly central limit theorem is a hugely important in the theory of probability it can be derived in many different ways has been derived in many different ways but we do not need that here. So this is what is that that if sum is all these numbers then probability of A is this is the Gaussian. So that may have values between plus 1 and minus 1 randomly picked up for example in a way we develop a we generate a Gaussian distribution in computer that we call a random number and random number between this is minus 1 plus 1 and then we add them up and it is about just about 12 such random numbers is enough that mean n can be 12 below 12 it does not work that well sometimes we take 20 but n equal to 12 will give you fairly good Gaussian distribution and this is something which one should have it is uniformly distributed between plus 1 and minus 1 then this average goes to 0 then you just have e to the power minus s square by 2 sigma square and the normalization constant in salt. So now we are coming back to polymer in the polymer we all have this random we are adding up so one bond this one is really not that correlated with that one because it can rotate. So if I think of distance between my central and the final one then it is adding up these projections which are really random. So then my aid that my central limit theorem I can surmise that the end to end distribution is a Gaussian and that exactly what holds on that this one of the most most most is not by n can be even 12 is enough here as I said in 12 is sufficient and then we can continue little bit to make a little bit more a as I said that it just angle cos theta i angle make with that. So if l is the bond length l is the bond length as it is given here theta is the individual bond vectors then this theta i and cos theta i is random just between minus 1 and plus 1 as I said then r becomes so it becomes l square and then I get this is double sum and one of them gives you 0 or 1 then this double sum become proportional to n you can do it with a random number one of them average where then the idea with respect to that and the rest one you can sum. So this should be capital N this should be capital N this should be capital N this should be capital N. So one sum is taken care to give you this random number second and that seems from distribution second sum into capital N. So r square the beautiful thing is amazing result amazing result. So root mean square now I can say root mean square r square goes as n to the power half this is one of the first result that that is kind of known though fluid systematized it there are other people who have got this result that size of the polymer scales as n to the power this if you think this is a remarkable result because I think of is a remarkable result that root over r square which is the size is n to the power half l and remember our n it can be 100 million so 10 to the power 8 then r square the size become 10 to the power 4 l l is 2 angstrom or say 5 angstrom then this 5 into 10 to the power 4 angstrom that means it is 50000 angstrom. So I now know the size of the polymer this is very important because that can that can be obtained by experiment. So we almost doing no work because of central limit theorem almost doing no work other than this little thing that we have done and that is trivial. So we get that you know if they are correlated that they are not they are equal to 0 then only i equal to this survives and that is a random number that u minus 1 to plus 1 that average there should be probably a factor of half pc here you get nl square. So we get this huge result because of the central limit theorem. And Prody did not do it by central limit theorem he just took them with a number and random numbers and he used probability theory but alternative way he arrived at the same exactly same expression. So we continue then what is the probability if I have n a p nr is I hope this small mistakes are not in the book this is the book small in capital N mistakes okay 4 pi 3 by 2 pi nl square 3 by 2 okay. So this is number as I said this is p nr capital R. So this is the full formula of the distribution of n number of n to n distribution of n number of for example if I do then I have 1 2 3 4 5 6 7 8 9 10 and then this is the 10 n equal to 10 this n equal to 10 and these distance then these distance is the end to end distance this is the R. So now this will give me if I even if I told you if I have 20 then that is enough then the end to end distribution will be given by all of us have done this calculation with the polymer and we have found it out that this this works beautifully well okay to the extent that this is a universal we consider universal formula and the details of these bonds and the rigidity all these things are not important as per so far the form is concerned there are certain changes in this rigidity and the kind of restriction on the randomness of the angles that is taken care by introducing normalization a kind of fitting parameter here and that is the end of the end distribution consequence. So now one important quantity which is the radius of guidance which is comes in the light scattering and many many theories and the radius of guidance is nothing but essentially same radius of guidance and moment of inertia that we find in any classical mechanics like if we have four particles then we find the radius of guidance is take it as a as a central point and then you want to find out what is the radius that this guy is rotating and that is mass weighted from central as I was saying from central mass. So this is the definition of the radius of guidance so when these things are we face that in classical mechanics when these guys of different these are different mass then different mass then we use this as the say if mass is the same everywhere then Rg is thus just simply sum over Ri minus Rcm where that means now I am going to take the distance I am going to take the distance from a central mass so in a polymer I can go to the center of mass of a polymer sorry so if that is such a polymer then I can go and find out a center of mass here then now I want to do about the center of mass what are the other guys are just like in the classical mechanics. Then that gives me from the center of mass how the size looks and then that is this quantity and one can show it is same as these into comes out because this is also a random number these distances that I am considering are also random number and then I again get the central limit theorem to get me this quantity and again is a factor except factor is different I get Rg is 1 over root 6 Rg square if I do not have that then I have Rg square Rg square 1 over 6 is very famous result 1 over 6 Nl square Rg square 1 over 6 Nl square. So basic thing is the same as that means Rg square so Rg square is same as R square with the except the numerical factor so the scaling remains the same so the size grows as root over