Overview of Polymer Physics Concepts

Shall We [Music] Begin okay greetings to the participants in this class and uh thank you for your enthusiasm to take this course introductory course and the title is Introduction to polymer physics and we have uh uh many Concepts to follow to understand and Implement so basically what I like to do is to begin with uh the scope what are we going to do the general picture well all all of us know uh your homopolymer say for example carbon based polymers right would be a connection of many of these monomers with different chemical groups and there are n such repeat units and as we know the richness of polymers arises from these chemical groups R1 or two or three or four of course instead of carbon we could have silicon based but or right oxygen based a carbon and oxygen alternating like polye oxy say and in general this is very rich with the chemical details given the variety of chemical details we can have what are the things we like to understand what are the things that we like to measure and understand and use our knowledge the first thing is this quantity namely the degree of polymerization how do we measure this which is related to molar mass what is the molecular weight of a particular molecule one polymer so this m mass is dependent on the degree of polymerization of course M mass is proportional to degree of polymerization namely mass of one repeat unit times number of those repeat units that's something that we want to do the second thing that we want to know is what is the size of such a molecule the polymer could be depending upon chemical details it could be like that so it will look like a ball of wool if you wish you know some rough sphere and there is some radius we can imagine what is that radius that's what weze or it could be more open like this and again what is that radius or you could be highly branched and from a linear polymer it could be Branch like this again and what is radius of that then the other question is shape what is the shape of the molecule one is I have already drawn it could be roughly spherical or it could be semi flexible Rod like this is more this is semi flexible so these are the basic quantities that we want to describe the molecule first we our chemical chemistry friends make these molecules and our task is to go and measure the molar mass or understand how the molar mass can be measured and then based upon this what is the size of this molecule and how does that depend on on molar mass what's the relationship between the size and the degree of polymerization and then what is the shape of this molecule what's the relationship between the shape and size and chemical architecture like a branch polymer versus a linear polymer etc etc so that is our primary goal and once we characterize them we will have their properties uh thermodynamic properties mechanical properties and how they respond depending upon the conditions so what do we want to do what are the what is the scope once we know uh how to characterize individual molecules we want to know structure structure alone and also there assembles how do they assemble alone and together and we want to know thermodynamical properties and we also want to know how do they move around right with respect to temperature how do they move around Dynamics then we also want to know how do they undergo phase Transformations from liquid to solid you know and solid to vapor and the Blends Etc so phe transitions these are the properties that we like to understand once we know how to CPR the polymer okay after we do these things we need to understand how do we Tinker with experiments to harness the desired properties that we like to have in order to do that we need to understand these things we need to understand these things in terms of various experimental variables what are the experimental variables first of all chemistry of the polymer namely the one I told you about R1 R2 R3 R4 whether it's a single Bond or double bond here right the chemical nature of the polymer then we need to we are going to make this in a liquid in a solution in some solvent so solvent quality this is also chemistry like a dealine is different from T it's different from cyoh heex these are the solvents so we need to know how to understand the role of solent then of course the natural variables are temperature and polymer concentration how much polymer we put in a solution or is it a melt that is there is no solvent at all and of course there are also driving forces like a flow so these are the experimental handles okay so this is basically the scope of of this course first we will discuss how to characterize a polymer in terms of molar mass size and shape and also in terms of architecture then we will try to understand the structure of these molecules and their assemblies thonic properties Dynamics phase transitions Etc and they handle the experimental variables are chemistry of the polymer solvent quality temperature polymer concentration for okay so that's the whole outline of this course now what I like to do next is to address this issue of chemistry because this is a problem of Physics course but we must understand how to account for the chemistry because the chemistry is essential because all the the properties that we are going to discuss will depend upon chemical nature so how do we want to treat the chemical properties first to chemistry of a single polymer chain how do you do that let me consider the simplest example okay this is uh I want to distinguish between uh short length scale properties versus the lar large length of scale properties so I'm going to talk about a global property is versus local properties so let me consider the simplistic case of polyethylene as an example so here R1 is some R Group in this particular case is proton hogen atom men here is R here is r and here is R same group like this if you do this here I have an example can you see me probably not can you see me Jun FR can you see my yeah what can I see what do you see do you see a model in my hand yeah I I can see the model oh good cannot see right perhaps I should stop this sharing only then you can see it yeah now now all us can see you can see you can see my model yeah we can see your model oh good okay so if you look at look at this polymer chain right for example going back to the screen so this is part of the chain I'm looking at let's look at for example right the the white one which is me that's a big Strand and the red one is another strand I'm looking at the middle carbon carbon Bond so what do you see is if I have something like this there's a very bulky group another bulky group they don't like each there's going to be steric interaction right therefore what will happen is I can draw this in a way like this there is one carbon here another carbon here there is one bulky group towards me right and then this is carbon is tetrahedral as you know then I have another one another carbon there's a bulky group there like that this is expensive in energy so if I plant energy of this particular Bond around this Bond what you will see is the energy is going to be high in this configuration right in this configuration which I'm going to call this to be CIS because these two groups are in alignment then what I will do is I will rotate it around when I rotate it say towards me I go like this this will be lowest in energy so I'll go into a configuration where I will go like that like that and there is another carbon and like that this will be lower in energy because there's no repulsion so I'll have lower energy somewhere here so when I rotate it from here when I rotate like this all right the energy will come down like this then if I keep on rotating right then I'll get into a state where this bulky group is exactly the opposite of this so I will get into another state here which I'm going to call Trans because they are opposite then what I have is this carbon there here is carbon here like that like that this would be TR and inside there's a barrier for this right corresponding to this there is a barrier like this then I rotate even more keep going then I'll have something like this this is carbon like that like that then that energy since the R1 r r r r since they are same then I'll have again something like this and then keep rotating more and then come back to my original state of a Cy and that will be higher energy so this is CIS this is trans and this is also CIS and the angle that I'm rotating is the dihal angle and in the intermediate States they are called go this is go this is also go to distinguish these two let me put plus and minus there so this would be G+ and this would be G minus this is a typical landscape energy landscape whenever you go to a particular bond in the polymer when you rotate it around you get a landscape like this and the feature of this land landscape depends on the chemistry because whether this barrier is high or small or how deep this one is how stable thisr is all of them depend upon the particular nature of RS here R1 R2 R3 R4 that's where the chemistry enters along the backbone of the chain but there are two important quantities in this landscape first quantity is the difference between the goost and and the TR let me write that to be Delta Epsilon and another important quantity is the barrier height for a confirmation to transition to make a transition from Trans State to go state so there is going to be a barrier here that is that barrier let me call that be capital E Delta e so the given any chemistry generically there will be two important quantities one is the energy difference between energy difference between the go and trans the other one is the barrier energy barrier for the bond to rotate you know the groups to rotate around a bond from a Trans State to a go State either this state or that state okay this is a genetic feature now naturally this Delta Epsilon if it's very small if it very small then Trans State is is equally accessible as the go state or go state is equally accessible to Trans State because the energy difference is very small they can equally make either this or that on the other hand if Delta Epsilon is Big then a trans state is preferred that mean the chain will be stiffer if Delta is very large right then trans is more favorable when I say trans it will go zigzag right one group is like this always it's a trance then next Bond will be also trance etc etc but at the same time if a Delta e the barrier is small then they are dynamically flexible if the barrier is small then they can equally easily go back and forth so the chain becomes on a Time framework they become very flexible but if the barrier is very large then you have to work very hard to make to access other states so this Delta e has something to do with the Dynamics and Delta Epsilon has something to do with the flexibility in equilibrium situation and that's the way it works so therefore what I'm going to say is I'm going to make a statement that Delta Epsilon by KT K is Vol constant T is the temperature and absolute deg celvin right Delta epon by KT this is a measure of chain stiffness right if Delta Epsilon again Epsilon small then trans and G are equally possible therefore it's flexible there is no preferred orientation flexible although if a Delta Epsilon small suppose I reduce a temperature when I go to very low temperature then although Delta Epsilon small at low temperature what happens is this quantity becomes large because temperature is in the denominator it becomes very large if it's very large then only trans will be preferable right therefore the combination of the chemistry and temperature they work together right therefore this is also giving me a persistence length that is how long along the backround of the chain the bond orientation will persist and we'll come back to this in couple of lectures later on how to describe the persistance length later on we'll talk about it later but right now qualitatively you can imagine this ratio Delta Epsilon by KT T is a measure of pris length which we work it out in a similar way if you look at the other quantity that is is a barrier height Delta e again once again we have to worry about this ratio this is a measure of dynamical flexibility we will this becomes important when you this is how we get into glassy Behavior Uh to some extent a crystallization Behavior we'll talk about it there is a dynamical flexibility which is dictated by chemistry right which is a Delta e or KT this also means this gives me your characteristic Time character time which I'll write as a to KN or to you can choose any symbol you like characteristic time for your monomer for a monomer to jump around to flip uh from a trans confirmation into a g confirmation what is the time involved for a local movement of your monomer right and that is given by the is a t which is related to Delta e by KT we will work it out later on when we talk about glassy Behavior we'll come back to this right in fact this T is exponential of Delta e by KT as we are going to work out later on so now I'm going to Define our goal for this particular course so here I go back and look at what I said I talked about local properties right that is how long would it take for a bond from the trans confirmation to flip into your G confirmation and that's a local similarly what is the partitioning of a particular Bond into trans confirmation versus G confirmation a bond that is again local now what I like to talk about I want to mention is I want to Define what do I mean by global properties so I want to Define what I mean by global properties I Define it this way Global properties deal with length scales scales larger than persistence length and times longer than to that to is this t for a monomer so this is what we mean by global properties okay so through our course we are going to be interested in global properties because that's the problem of physics right we are going to be talking about large length scale Behavior large time scale Behavior very rarely we'll go and describe local properties like how does a particular monomer move etc etc but in general the rest of the course we are going to be talking about global properties okay okay so if that's the case I have a challenge the challenge is how do I build a bridge between the local properties and Global properties because the chemistry is operating at local level but I'm interested in physical properties at Global level how do I build a bridge between these two uh descriptions how am I going to do and that's what I'm going to do right now in another 20 minutes okay I'm going to build a bridge or begin to build a bridge right let's do that so let me take what I would call a a skeletal chain okay so a skeletal chain as I said before is right so so there is a carbon one hydrogen there another hydrogen there the carbon and this is hydrogen hydrogen there and on and on it goes like this so what I'm going to do is I'm going to combine them right combine that that group into uh what I would call a skeletal group I would call this to be skeletal sorry skeletal group some people call this to be United atom that's that's my nor clure if that's the case then what I can do is I can do the following okay I have I start with the one uh uh United atom at one end and then another one and uh another one another one and on and on right another one this each each circle that I'm drawing is one methylene group right and it keeps going keep going so let me call this to be a0 that's a position Vector of the first one and let me call this a bond this one that is the bond Bond Vector okay and then of course this would be this would be A1 this is A2 etc etc eventually I'll have this to be a n and this is a bond Vector you know one point 5 an right that's magnitude but in that direction and finally this would be a n where little N is a number degree of polymerization right as I pointed out n is this little n is the degree of polymerization so I have converted into the a confirmation of the chain into a skeletal chain now let's look at how it's going to be at a local level so let me go and draw uh say let me choose in the middle say e Bond so let me choose this to be the E bond for example say for example I choose this to be ice Bond right when I choose ice Bond so let's say the direction of that iond is like this so this is my a i this is my Ai and then next B comes right next one is going to be say something like this okay and this is the tetral angle 190° right that's a tetral angle but it is a convention in polymer Community all right that the complimentary angle is called the bond angle 18 80° is minus that one is Bond angle okay and then let me call that to be Theta I and of course this Vector is a i + 1 okay now next one next Bond is going to be say something like this okay and with respect to that one uh this angle would be Theta I + 1 but because of the the dial angle I mentioned to you about not only be in a trans or it could be go or something like this so what will happen is it can be right this could be anywhere along an ellipse or a circle and then with respect to this there is going to be an angle five the five that I told you before this is the dial angle that's what will happen in here there's some angle there so I need to specify right I need to specify the thetas and F and corresponding to that confirmation the energy will be different okay this five is actually for 5 I + 1 this is 5 I + 1 okay that gives a description of your confirmation of the chain okay and what are the quantities once I have a confirmation and for this confirmation I have an energy because I go and compute right I go and compute the energy landscape depending upon chemical details and let's let's do that let's do this calculation if you do this calculation there are two important quantities that I like to look at the first quantity I like to look at in general is mean Square n n distance because that is your largest scale quantity and I'm going to write this to be R squ where R is n distance what does that R mean R means I'm going to add this A1 A2 correspond little A2 little A3 the red terms I add all these vectors together and then that will give me n to end distance Vector right this will be n to n distance vector and then vectorial addition will give me that n distance Vector so therefore if I want to know R squ what I have to do is this is equal to summation over I from 0 to n a i Square One Is The End into distance and square is that then I average over all average over all confirmations all allowed confirmations that probability is going to depend on this input which in turn is going to depend on R1 R2 R3 R4 okay so that's how it works that's R squ and you can simplify the calculation I can simply say this sum and that is equal to number of bonds times the bond length squared plus right two times summation I summation over J but larger than I then I have this quantity a i dotted with AJ average average over confirmation this is a very general result because this n a square is coming from when I take the square same term right uh yeah same I when I have double sum here when I take the square it will be summation I and summation J when I is equal to J I get this term when I is not equal to J I get this one so this is a very general result the other quantity that I like to do is radius of right so let me write down mean Square radius of gation okay and before I Define this let me take this uh let me draw it once again suppose I have yeah skeletal chain and goes on and on suppose I had skeletal chain like this for this confirmation there is going to be a center of mass somewhere let me say the center of mass is going to be here this is Center of mass of of this confirmation then what I can do is I Define say the position of the I skeletal this is i s i that's the radial distance of the I skeletal atom or United atom skeletal group from the center of mass similarly let me draw say for example J let me draw the radial distance of SJ and of course the difference between them I'm going to there's going to be inter segment distance which I'm going to call say r i j the distance between I United atom and J United atom once I have this I can write down the radius of gation square to be mean Square radius of gation which I'm going to write this symbol RG is radius of duration and and square right that is RG squ and mean because I'm going to take an average so this is equal to I can write down to be this is equal to 1 / n + 1 sum I from 0 to n SI I squ average where I have defined what SI is that's my SI and I run over SI here SI i s i + 1 s i - 1 is I + 2 ET ET some all of them Square it and all of them divide 1 n + 1 I get radius generation Square you can also show you can also show that this is equal to 1/ n + 1 squ right summation over I summation over J J is larger than I then I have r i j s as I have defined here r j is this one R is that Vector Square this and then do the double sum for practical purposes it turns out that this is better to to calculate for computational purposes so people usually use this definition to get AIO SC they are identical you can prove it exactly okay these are the two important quantities that uh that comes out from from considering the chemical structure so the road map is right the road map is then uh account for chemical details account for chemical details from Computing this right and then you take that into into this average and then construct a various quantities like mean Square NN distance mean Square ration and any other quantity that we are interested in and that's the road map now what we are going to do is we are going to consider some simple examples right because this is this is a big picture this is how we have to do but it's difficult to do right because the chemical details are so much and the potentials are are quite complex so therefore we like to come with some simple models particularly in view of the global versus local properties we are look we are looking for some big pictures what are the most important aspects where we can understand the physical properties of this polymers at the same time parameterizing the chemical details okay that's all go so the first thing I'm going to do is you might think that I'm being silly but I'm going to do this in a very simple way all right and I want to talk about U what is called a freely jointed chain freely join your chain what does that mean I hope you can see my picture here um Jun you had to help me are you able to see my picture here my my demonstration here yeah okay so so here is taken plenty of paper clips I hook them together and this is the chain but this model is kind of baby like model very simple model right I take one Bond another Bond and the bond can rotate completely completely freely jointed it can bend no torsion whatever it's completely flexible you know not only flexible it's freely jointed so this is the simplest model that we can think of let's think about that let's take this freely jointed model what does that mean right what that means is I go here right that means the bond I bond is not correlated with the J Bond even if they neighbors because AI right is completely freely jointed there is no correlation at all that means this average goes to zero in this model so in a very simple way right this simply goes to zero right because they are not correlated completely freely jointed therefore one of the quantities that we are interested in is going to be r² is equal to n a² okay because this is zero this term is zero okay now what I want to emphasize here is I like to put a subscript here I'm going to put a subscrip to zero Okay the reason I do that is when I take a chain like this right and there is going to be always interactions there's going to be some interaction between a monomer and monomer because they are coming very close there will be Vandal interactions that's why you know the gases when you lower the temperature make a liquid because there are attractions and if they when they come very close there is a repulsion right so we are ignoring that I'm looking only at the skeletal chain at this moment to emphasize that I'm not looking at these quantities I'm putting this subscript to zero here and as we we shall see later on these are collectively called excluded volume interactions we will come back in the next lecture exra volume interaction right and we are ignoring that for the time being okay so ignore that we we are not taking that into account right so we don't do that yet ignore maybe I should say ignore ignore this for now and that's the reason why I'm putting that I'm sorry I'm put ping that zero there because I'm ignoring that all right if that's the case then I can also do other calculations like uh r i j s is equal to a² IUS J and in fact n to I is one I United atom J is another United atom if it's zero and a little n that will be n same as that one and then again to be uh consistent I'll put a subscript zero and what happens to radius of gation RG squar and I'll put emphasize that and that turns out to be equal to n a² by 6 if n is large enough because what I have to do is I have to go back you know I have this double sum I know r a square that is little a square modulus of IUS J that in do the double s if you do this large end limit you get that one and the important thing that I want you to recognize in all these things is for a freely jointed chain I want to Simply say right I want you to recognize R which is a measure of size whether root mean Square n to distance or radius of dation this is essentially proportional to n for freely jointed chain this is the degree of polymerization as soon as we see this as soon as we recognize this uh our imagination should be triggered this is fantastic as we we are going to see later on okay that's for a freely jointed chain now let's do one more one more model well two more models very quickly okay now instead of this I want to do freely jointed sorry freely rotating chain freely rotating chain if you do this that is right if it keeps on rotating right then uh then you can show r² without any extra volume effect is equal to n a² time let me Define Alpha is equal to cosine Theta Theta is our bond angle right that's my definition right so this Bec * 1 + alpha 1 - Alpha - 2 Alpha by n 1 - Alpha power n n is number of De polymerization and 1 - Alpha squ this is a Formula I you don't have to copy this because I sent a PDF where these formulas are given all right and here I I wish that you could follow the logic how we compose the arguments how we are building the picture okay and of course if for n much larger than one right look at this 1 / n here right 1 / n uh is very small when n is very L then R square average goes like n a s 1 + Alpha there be 1us Alpha once again you you see a remarkable thing R is proportional to oh did I make a mistake no this is square root of it right R squ goes like n r squ goes like n once again okay there's a there's a picture that's coming in right if it's a freely jointed I get r s going like n and then if um freely rotating I'm adding a little more information yes the number is different right number is different for a tetrahedral case like a carbon Alpha is equal to 1/3 tetrahedral coordination tetral coordination Alpha isal 1/3 that means r² goes like 2 n a squ okay but still the important thing is r² goes like n the preactor that's different so they when I bring a little more chemistry the preactor changes but yet R squ is proportional to okay um shall we can I add one more or maybe I should stop here I think what do you think or shall we go ahead for another 5 minutes June fine yeah what's a what do you want me to do we go and complete this part extra 5 minutes okay now let me go to uh another situation right uh now I have a barrier for rotation I have a barrier right but there are not correlated every rotation is independent okay so change CH with chains with independent rotation independent rotational barriers that means dial angle right there will be a preferred dial angle rotational potentials if you do this you do the calculation you know you know flow Paul FL did this uh and there's a book written by him second book and I have given reference in my handout but the answer is r² average is equal to once again it's n a² of course 1 + Alpha by 1 - Alpha where Alpha is equal to cosine Theta and then with the dial angle I'm going to write cosine of f p is dial angle and average this going to be 1 + e 1us still it's simply a number Alpha is a number right 1/3 and it roughly 1/3 if it's a silicon you'll have different number and if you have double bond you'll have a different number for Alpha but here ITA is again some number okay but once again the important thing is r² goes like n but this is fantastic right this is really fantastic because go back to the very difficult task we had this is a difficult task we have and independent of the details r one or two or three or four independent of the details right independent of everything in general if you do not worry about exal volume interaction if you ignore that then in general it turns to be that R square is proportional to number of bonds R square is proportional to De of polymerization this is a very profound uh observation why is that the reason is remember I if I take a walk right if I take a walk then if I go on a straight line like a rod like polymer if I just take a walk a straight line then R square n to n distance Square would be proportional to number of bonds square right because length length is number of steps I'm making it will be length will be proportion number of steps therefore length to square would be number of Step squared but in here what we have is mean square and into distance that is the square of the distance I travel is proportional to time this is a brown in motion this is Einstein's Brown in Motion in terms of time space so this means uh there is an inspiration here once you do these calculations you realize in the simplis model you realize that maybe we can use all our knowledge about classical Brownian motion all theorems developed by Einstein and various other people and we could probably copy some of those answers and get stimulated and try to understand polymer physics polymer physics very complicated but in terms of captured in the chemical details maybe there is a chance but I have to convince you that this chance is actually a good chance okay and before I do that I want to make one quick summary The quick summary is this in the literature you will see the capturing of chemical details with what is called a characteristic ratio and usually the symbol used is C Infinity in general for a polymer of n bonds CN that's defined to be the r² once again I want to put without extra volume interaction divided by the freely jointed chain result for freely jointed chain result it's n a square because we are always saying R square is proportional to n right and R square is a length Square so n a square is a bond length squar so that's the ratio this is CN C Infinity is when n goes to Infinity okay again for freely jointed chain right CN is equal to one obviously right and let's look at the freely rotating chain freely rotating chain and we can calculate CN CN is equal to exactly what I did before here this is the formula right r² / n s that is the term in in this square bracket just for completion I WR down to be 1 + Alpha be 1- Alpha right - 2 Alpha by n 1 - Alp power n / 1 - alpha s okay let me write it for completeness and then for tetrahedral tal coordination for tal coordination I told you Alpha is equal to 13 so let me plot this let me plot CN versus n take this function when I plot it what you will see is for short end small NS it will be linear it will go like this and eventually it will curve into something like this for this particular case right when n goes to Infinity C Infinity is 1 + alpha 1us alpha which we know it's two so that value is two when Alpha is equal to one you can work it out 1 minus Alpha this n is equal to 1 when you work it out that's going to be one so CN goes from one to two and this is around 20 n is around 20 number of ons is 20 so the point that I'm trying to make here is once again once again R squar is n and the different n a square if you right and the difference is really within a factor of unity right 1.2 1.3 maybe six for some polymers maybe the inste of two you'll get six but the preactor here is still only a number a small number of the order of unity okay it of the order of unity so that's is the punch line of all the models we have done so far this is a critical thing so I could also write equivalently I can write R the typical size goes like square root of n according to this model remember this was one of our tasks going back here one of our task is a size right we are going to discuss maybe in the next lecture how we are going to determine Mass we going to do that but before that given n how does the size the r we have already gotten it by accounting for chemical details all these chemical details come where do they come they come they come here here let me write some symbol here okay that's my chemical details this is a remarkable observation right because if you want to take every polymer as a special polymer and we work it out it will be a daunting task right but here what you see is in general if you don't worry about extra volume interactions right uh normally what we get is R squ is proportional to n and all those numbers are representation the other numbers are representations of chemical details and also when I look look at this characteristic ratio what you see is this here for a short n it's a rod like for large n it's a flexible when I say for n large for n large I get this result when I go back in here right for n large here when n is large I get this result what is large that depends upon the chemical details right so so this is measured this indicates persistence length how stiff the polymer is locally the way the character ratio changes from this linear line to this plate value the rate with which it does gives a measure of Prive length it gives Prive length how many carbons you have to go along the backbone so that it becomes coil like so in this region it's a rod like locally in this region it's a coil like when I say coil like it's fors back and forth therefore is not R square is not going like right is not going like n that's coil like inside it goes like n Square because if it's linear CN is proportional to n CN is proportional n therefore R square goes like n squ that's more like a NE case Newtonian walks just you take an automobile you go on a straight line the distance is proportional to right the time time in the case of Einstein of bur in motion the distance is proportional to square root of time so we have Newtonian analog Einstein analog here Rod like analog like situation here coil like situation all these things come from uh CN by Computing CN and then in in principle we are going to be in large and limit when you are large and limit we are here okay that's a global limit and degree of polymerization we are noted 10 20 things like that we are going to be for these polymers most of the polymers we going to be end to be 100 thousand things like that okay I believe that this is a good place to stop uh and then we can take a break what do you think Jun [Music] bre okay let's do that [Music] Jun fun Jun fun are you there [Music] [Music] unform you [Music] for for [Music] open so moment D are you there yes I cannot hear say that again we can start yeah okay okay all right so so what uh we have come to the conclusion is basically here and this conclusion is an important one because now what this allows us is to imagine as I have several times alluded to that I can make an analogy with a random walk Brown in motion of a particle undergoing motion time so we can make an analogy this was done by Kon that takes me to chain model so what happens with this suppose you take a a confirmation at some temperature what you will see is okay locally it will be trans and then there will be a go like that and then that will go like a trans like that and then your go again trans like this then go that's a that's a goost right trans like this then again maybe right like that R again and on and on that means okay this is a typical confirmation since I do know this that comes very easily right that I know that right I know that because because of the the very first thing that we did right freely jointed chain right because we did this right because this is a general definition and then if these are not correlated AI AJ is zero then I will get this result right when they get correlated I get some other number but still I'm getting only this result n based upon the chemistry but the important thing is that these vectors are not getting correlated if they not correlated then I will be able to get this one so whaton imagined is let me imagine that there is a some local step okay because I already told you here for when n is short number of bonds is small in number four five something like this it is going to be Rod like so look right we know that only when we go to very large values of n it's going to be coil like therefore I would imagine this is Rod like here and then this is Rod like locally this is another segment another segment another segment Etc right then I would say these are freely jointed because if it's freely jointed I do not have to worry about the bond Vector the net Bond Vector for this segment and the net Bond Vector for this segment they are not correlated anymore right so he imagined this an imagination this is a beautiful imagination right imagination is Imagine This is the critical step in the life of a scientist or an engineer or a poet imagine right imagine we imagine he imagined that a confirmation can be considered to be made up of freely jointed segments freely jointed segments and you don't need anything more I need only only this paper clip chain of paper clips right it's freely jointed except now every paper clip now is a segment previously it was a bond in the freely jointed chain and here I'm imagine in the real chain with all the chemical potential chemical details right is Can Be Imagined to be a series of segments which would call this to be segments named after him segment of average length l so let me call this to be L average length L right and there are now I'm going to use capital N there are n segments per chain in other words what I'm doing is I take this confirmation and I write this to be something like this I write to be okay for this one I write this one that is a CO segment and then for this one I see another segment for this one I write another segment for this one I write another segment for this one I write another segment and it continues this way so I'm saying this is segment length L and there are n there are n such segments there are n such okay there are n such con segments this n capital N is not equal to little n but it's proportional to little n okay so this is the model in fact I can tell you the result for the relation between Co chain model and the real units the connection maybe this is for your notes the connection between capital n and little n capital N is equal to little n CR squ cosine squ Theta by 2 Theta is a bond angle in the way we defined divid C Infinity similarly the length is equal to C Infinity of course the bond length a actual Bond a divided by cosine of theta by 2 so this is a connection between real chemistry right real chemistry and the parameterized chemistry although we have rotational angle right uh Bond angle and then Bond length right and C Infinity which depends on chemistry once we know these things then we know that capital N is proportional to little n number of bonds and length L is proportional to the bond length real Bond length a and there are three factors the chemistry is converted into these two parameters so that's is our model but nice thing about this is is freely jointed that means right this is freely jointed freely because if it freely jointed we know how to do the calculations because that's exactly so you could also imagine that okay I have a a brownan particle here makes a step of length L without any memory of where the particle was before freely jointed right it could make move by another distance L to a new place it could be here it could be there it could be there it could be there right then again without any memory whatsoever without any Persistence of the orientation it can make another move in a random Direction and on and on and on so this is a trajectory of your random Walker the co chain model here right a confirmation G confirmation yeah chain confirmation is equivalent to a trajectory trajectory of yeah random Walker random Walker so we could call this chain model as a random walk chain why not you can call this Co chain model as a random walk chain now since it's a freely jointed chain we can write down the answers exactly with paper and pencil we can just look at it copy for freely jointed chain we can write the answer so what I'm going to do is I'm going to write the various important quantities okay so let's do that first of all all right basically what I'm trying to say is all right since it's a random walking chain I could imagine like this that's my confirmation for large enough n okay and the kind of quantity that we are interested in is what is RG and then suppose I had one end somewhere there another end in here here so what is um n to n distance then also we can look at what is RG squ for example things like that this is this is RG things of that type we can calculate this so it can be shown for this a random walk chain or chain it can be shown if capital N is large enough because that's what we are interested in remember we are interested in large length scale properties if capital N is large enough then we can calculate the probability that that chain will have n to n distance r a confirmation with an N to n distance r with n steps n Co steps each having length L okay if you do that this turns to be 3 2 pi n l s 3x 2 here Theus 3 r² by 2 n l s you can show this because remember I told you that this is like a trajectory of random Walker right when came up with this model there were already 30 years of experience with a random walk or more than 30 years of experience random box so it's easy to copy the answer of course you can also easily derive it but it's easy to copy it if you do this the probability of getting n to n distance are after n steps for freely jointed chain namely the chain random walk chain is given this formula so R appears in here n appears in here and capital N is proportional to little n because I told you that capital N is proportional little n that means is proportional to molecular weight so wherever you see capital N throughout the rest of the lectures capital N means is proportional molecular weight degree of polymerization and there are some pre factors if necessary you'll account for them in general for Global properties we take it for granted that we know how to deal with this so we will not be explicit about it right now the cool length is proportional to little a bond length but it is not Bond length length could be 1 nanometer whereas little a could be .15 nomer right that depends upon Thea depends on C Infinity all those things depending upon all those things length will be yes some proportional number to a some proportion constant times the bond okay so given that uh for any polymer now we are making a huge uh jump any chemistry if we ignore EX volume interactions if you ignore solvents you're just looking only skeletal properties of polymer we could write down this formula as long as n is large okay so what's the impf of the implication of this first of all I want to talk about mean square into distance r² average because R square average is using this probability we are going to calculate R square because you know this that's how you you know how to calculate the average right integrate over r² * P RN divided by normaliz interal of that's the way you do if you do this naturally you should get NL squ because freely jointed again going back to the formulas that we had right this is r² is equal to this instead of now it's the bond length is little L little n is now capital N right capital N so they are not the the Orient TS of these segments are not correlated therefore this average is going to be zero therefore I have capital N time the little L squ okay that's an squ now let's do RG squ and you know this already RG squ without any extra volume interaction is equal to n l s by six before I go further I wanted to you know right you have this distribution function probity function then you can calculate these quantities I defined what RG is before so we can calculate them but look at the shape of this equation right the shape is e to the where does r appear R appears like e to the minus r² right that is called gausian E to the minus x² where X is a variable it's called gausian so this is a gausian the probability is a gan therefore therefore the chain is equal to Gan chain so we could use these names interchangeably we can say random walking chain or a gausian chain or a chain all right this is exact if n is larger than one much larger than one in the limit it is cred if n is small of course you know that you have to worry about prist length there are corrections to that but for all practical purposes we will call this to be chain to be a gaussian chain because when capital N is large then these are the results that come about okay now what I like to say uh an important consequence of this as I have already alluded to here I said in general the realistic chains now it also happens for gusin chain R squ is proportional to n capital N now in this case that means R is proportional to square root of capital n okay so therefore I could write this to be in the following F fashion or going like n to I'm going to write half in a in a general way I'm going to write like like that where this new is equal to half same thing R is proportional to square root of n I'm writing that but except that I'm giving a symbol for or this half an exponent so new is called size exponent in general you'll see when we talk about all other effects we'll come back and look at the general formula new is size exponent there's also one more thing I can do right I could Define instead of R depending on Mass power capital N is number of Co steps number of segments in a chain it is proportional degree of polymerization therefore it's proportional to mass therefore here what I'm saying is R is proportional to mass power new new is a size exponent instead of doing this I could also think in terms of exactly the mass is proportional to R to 1 / new same thing right R for one new is proportional n this gives me kind of a different definition this means R I'm going to give a symbol DF is proportional to n where DF is called fractor dimension dfal 1/ new it's called the fractal dimension of this object this will come particularly when you do scattering measurements this comes often we we'll deal deal with uh exal uh adventures and exploring form physics and we'll come back and talk about DF DF is a fror dimension and fror dimension simply reciprocal of in our context reciprocal of the size exponent to new and the reason why we call this to be one new is the following suppose I had right suppose I had a ball of Steel then volume will be R power some number three R power three is proportional to mass because the density is fixed volume is proportional to mass analogously here lot of this not a solid ball it's not a steel ball right there lots of uh gaps in here right but in general in analogy we like to write some dimension in a three dimension volume is R cubed if you suppose you have a pancake in a phosph liid membrane suppose I have a pancake then there'll be R squ area will be proportional mass and things like that so analogously we call this to be dimension of that object geometrical dimension of that object that's the reason why we call this to be a dimension that one/ new which is DF is a dimension but it's fractal because it's not three it's not two it's not one it's some number it's a half in this particular case sorry in this particular case two which is an integer although polymer lives in threedimensional world right instead of two you could have some other number as we are going to see later on so that's the origin of this nen clature okay so what have we done here based upon what we got in here we got we got uh probable distribution function to find an N to n distance r that turns to be gausian chain gausian result therefore we call this chain and mean Square distance proportion n mean Square ration is proportional to n is actually n l s 6 in general R can be written as n to the new right and then or alternatively R to the one new is n now I want to make a connection between why are we doing this I I want make a connection with the real experimental World okay this is a model capturing all chemical details but at the same time I like to take the pleasure of talking to uh experimental systems experimental tools okay how do we do this in order to do that I construct what is called a form factor for this model chain form factor and I'm going to going to write in my notation I'm going to write s of K right I'll tell you in a minute what this means s of K maybe I'll put an arrow is a vector that's equal I'm going to Define this this is one/ number of Co segments summation or I sum or j e to the i k do r i j remember I Define r j before r i j is a distance between the I segment and J segment I defined between atoms before skeletal atoms before here these are for segments our idea is the distance between the I segment and J segment and average it out suppose I do this this is a f transform of that distance and average over all confirmations if I do this I get for the gausian chain I get a very beautiful result that turns out to be 2 n y k to the 4 and let me see let me write rg/ 4 e to the minus k s r g 0 S - 1 + k s r g 0 S that's what I get and as you know as you know r z square is equal to you know that right uh this is called this expression is called D structure Factor this is called div structure fact this is really fantastic because you know why it's fantastic let's go to our first challenge first task our first task was was to look at the m mass and size we want to know what n is M mass is and what R is here this formula immediately gives me yeah an idea I look at this one see I'm interested in N I'm interest in size this is Theory model Theory right g chain Theory right this is model but it gives me a hint about how to measure and and RG right away I can get both n m mass and size how do I do this then the way you do that is this K right this s of K can be measured experimentally this can be measured experimentally how do you do this the way do that is okay so here is my my Polymer here and then sending a beam some wavelength Lambda some instant beam of your liking we'll talk the next lecture about how to choose this experiment and then you measure scatter intensity from some angle Theta here is your detector and then in this setup scattering setup K this is scattering wave vector scattering W vector k w vector k 4 pi/ Lambda sin Theta 2 where this Theta is scattering angle this Theta is a scattering angle Lambda isong the instant B so i s a Lambda I have Theta here then I know what K is that's a k that appears in here in the form factor that's the K that appears in this formula okay for example as an example I take this function and then I look at this function s of K right and that's related to this scattering intensity I of K this is cering intensity cing intensity I of K depending upon whether you use a synchron neutron scattering light scattering depending upon that there will will be a number here which I'm not going to write down but that is proportional to this s of K that is the form factor so this is something you measure and your laboratory or a national facility will tell us what that number is here and then that is related to this s of K this s of K if I take this right and uh we'll we'll think a little more about it if you work out this this formula you get the following results you get y 1 minus k s by 3 and squ plus etc for K * RG is much smaller than one if K is small small angle that's what you will get on the other hand on the other hand if you are larger K * R is larger than 1 you get 12 by k s l s that's the answer you'll get so this gives you a clue right this gives you a fantastic clue right that is you go to your scattering Laboratory you choose the instrument technique light Neutron xrays in such a way you satisfy this condition K I have given K that is you choose a Lambda or scattering angle you could tune K either with the insent beam nature of the insent beam and also in terms of scattering angle you choose in such a way K * RG is less than one then if you measure scatter intensity you can measure n and you can measure the size and this is what we are going to do in the next lecture I'm going to augment to this and look at the consequence of this at this moment I'm looking at a microscopic details of how the chain sizes mol masses but I also want to use this gausian chain model to understand the free energy thermodynamic properties suppose I take this and I put a force on it how is that going to respond what is the force right how is that going to happen right and that's what I'm going to pick up in the beginning of the next lecture so uh at this moment what I have done is we have made a huge imagination which turned so to be a fantastic idea good idea very fruitful idea and it is the basis this imagination is the basis of poly physics All chemical details are surrogated through L and N but the key thing is capital N is proportional n number number of actual number of bonds little L is proportional to actual Bond length but there are prefactors depending on which polymer days these things will appear in making the mapping that is taking chemical details into the co chain model we do that step okay so once we do this we have without any extra volume interaction accounted for we have R is going to be proportional size is proportional to n power half molecular weight power half okay in general and then I'm also telling you right away this allows us to determine it gives me a handle experimental handle to measure molar mass and also radius Direction I think I should stop here you must be tired at this moment having listened to me a couple of lectures right now and we'll pick it up next week and what I'm going to do next week is to use this idea a little bit more and I'll show you some external data and we'll make a critique of that we'll also talk about the free energy of a polymer chain and its consequences and then we will ask about how good this model is in real world because real world we have right we have solvent and solvent molecules are not going to be idle they are going to kicking the monomers of the polymer chain and there are going to be consequences of that both the structural consequences and the thermic consequences and that's how we are going to build our Concepts in the in the next uh lectures okay and I like to stop here if there are any questions Jun F this is a good time to ask from [Music] anybody well they may think about it we can I'll entertain their questions again later on next lectures okay maybe you find the floor is yours do as you wish I'm going to stop sharing yeahor stop yeah okay I'll stop the recording all right okay thank you that

Shall We [Music] Begin okay greetings to the participants in this class and uh thank you for your enthusiasm to take this course introductory course and the title is Introduction to polymer physics and we have uh uh many Concepts to follow to understand and Implement so basically what I like to do is to begin with uh the scope what are we going to do the general picture well all all of us know uh your homopolymer say for example carbon based polymers right would be a connection of many of these monomers with different chemical groups and there are n such repeat units and as we know the richness of polymers arises from these chemical groups R1 or two or three or four of course instead of carbon we could have silicon based but or right oxygen based a carbon and oxygen alternating like polye oxy say and in general this is very rich with the chemical details given the variety of chemical details we can have what are the things we like to understand what are the things that we like to measure and understand and use our knowledge the first thing is this quantity namely the degree of polymerization how do we measure this which is related to molar mass what is the molecular weight of a particular molecule one polymer so this m mass is dependent on the degree of polymerization of course M mass is proportional to degree of polymerization namely mass of one repeat unit times number of those repeat units that's something that we want to do the second thing that we want to know is what is the size of such a molecule the polymer could be depending upon chemical details it could be like that so it will look like a ball of wool if you wish you know some rough sphere and there is some radius we can imagine what is that radius that's what weze or it could be more open like this and again what is that radius or you could be highly branched and from a linear polymer it could be Branch like this again and what is radius of that then the other question is shape what is the shape of the molecule one is I have already drawn it could be roughly spherical or it could be semi flexible Rod like this is more this is semi flexible so these are the basic quantities that we want to describe the molecule first we our chemical chemistry friends make these molecules and our task is to go and measure the molar mass or understand how the molar mass can be measured and then based upon this what is the size of this molecule and how does that depend on on molar mass what's the relationship between the size and the degree of polymerization and then what is the shape of this molecule what's the relationship between the shape and size and chemical architecture like a branch polymer versus a linear polymer etc etc so that is our primary goal and once we characterize them we will have their properties uh thermodynamic properties mechanical properties and how they respond depending upon the conditions so what do we want to do what are the what is the scope once we know uh how to characterize individual molecules we want to know structure structure alone and also there assembles how do they assemble alone and together and we want to know thermodynamical properties and we also want to know how do they move around right with respect to temperature how do they move around Dynamics then we also want to know how do they undergo phase Transformations from liquid to solid you know and solid to vapor and the Blends Etc so phe transitions these are the properties that we like to understand once we know how to CPR the polymer okay after we do these things we need to understand how do we Tinker with experiments to harness the desired properties that we like to have in order to do that we need to understand these things we need to understand these things in terms of various experimental variables what are the experimental variables first of all chemistry of the polymer namely the one I told you about R1 R2 R3 R4 whether it's a single Bond or double bond here right the chemical nature of the polymer then we need to we are going to make this in a liquid in a solution in some solvent so solvent quality this is also chemistry like a dealine is different from T it's different from cyoh heex these are the solvents so we need to know how to understand the role of solent then of course the natural variables are temperature and polymer concentration how much polymer we put in a solution or is it a melt that is there is no solvent at all and of course there are also driving forces like a flow so these are the experimental handles okay so this is basically the scope of of this course first we will discuss how to characterize a polymer in terms of molar mass size and shape and also in terms of architecture then we will try to understand the structure of these molecules and their assemblies thonic properties Dynamics phase transitions Etc and they handle the experimental variables are chemistry of the polymer solvent quality temperature polymer concentration for okay so that's the whole outline of this course now what I like to do next is to address this issue of chemistry because this is a problem of Physics course but we must understand how to account for the chemistry because the chemistry is essential because all the the properties that we are going to discuss will depend upon chemical nature so how do we want to treat the chemical properties first to chemistry of a single polymer chain how do you do that let me consider the simplest example okay this is uh I want to distinguish between uh short length scale properties versus the lar large length of scale properties so I'm going to talk about a global property is versus local properties so let me consider the simplistic case of polyethylene as an example so here R1 is some R Group in this particular case is proton hogen atom men here is R here is r and here is R same group like this if you do this here I have an example can you see me probably not can you see me Jun FR can you see my yeah what can I see what do you see do you see a model in my hand yeah I I can see the model oh good cannot see right perhaps I should stop this sharing only then you can see it yeah now now all us can see you can see you can see my model yeah we can see your model oh good okay so if you look at look at this polymer chain right for example going back to the screen so this is part of the chain I'm looking at let's look at for example right the the white one which is me that's a big Strand and the red one is another strand I'm looking at the middle carbon carbon Bond so what do you see is if I have something like this there's a very bulky group another bulky group they don't like each there's going to be steric interaction right therefore what will happen is I can draw this in a way like this there is one carbon here another carbon here there is one bulky group towards me right and then this is carbon is tetrahedral as you know then I have another one another carbon there's a bulky group there like that this is expensive in energy so if I plant energy of this particular Bond around this Bond what you will see is the energy is going to be high in this configuration right in this configuration which I'm going to call this to be CIS because these two groups are in alignment then what I will do is I will rotate it around when I rotate it say towards me I go like this this will be lowest in energy so I'll go into a configuration where I will go like that like that and there is another carbon and like that this will be lower in energy because there's no repulsion so I'll have lower energy somewhere here so when I rotate it from here when I rotate like this all right the energy will come down like this then if I keep on rotating right then I'll get into a state where this bulky group is exactly the opposite of this so I will get into another state here which I'm going to call Trans because they are opposite then what I have is this carbon there here is carbon here like that like that this would be TR and inside there's a barrier for this right corresponding to this there is a barrier like this then I rotate even more keep going then I'll have something like this this is carbon like that like that then that energy since the R1 r r r r since they are same then I'll have again something like this and then keep rotating more and then come back to my original state of a Cy and that will be higher energy so this is CIS this is trans and this is also CIS and the angle that I'm rotating is the dihal angle and in the intermediate States they are called go this is go this is also go to distinguish these two let me put plus and minus there so this would be G+ and this would be G minus this is a typical landscape energy landscape whenever you go to a particular bond in the polymer when you rotate it around you get a landscape like this and the feature of this land landscape depends on the chemistry because whether this barrier is high or small or how deep this one is how stable thisr is all of them depend upon the particular nature of RS here R1 R2 R3 R4 that's where the chemistry enters along the backbone of the chain but there are two important quantities in this landscape first quantity is the difference between the goost and and the TR let me write that to be Delta Epsilon and another important quantity is the barrier height for a confirmation to transition to make a transition from Trans State to go state so there is going to be a barrier here that is that barrier let me call that be capital E Delta e so the given any chemistry generically there will be two important quantities one is the energy difference between energy difference between the go and trans the other one is the barrier energy barrier for the bond to rotate you know the groups to rotate around a bond from a Trans State to a go State either this state or that state okay this is a genetic feature now naturally this Delta Epsilon if it's very small if it very small then Trans State is is equally accessible as the go state or go state is equally accessible to Trans State because the energy difference is very small they can equally make either this or that on the other hand if Delta Epsilon is Big then a trans state is preferred that mean the chain will be stiffer if Delta is very large right then trans is more favorable when I say trans it will go zigzag right one group is like this always it's a trance then next Bond will be also trance etc etc but at the same time if a Delta e the barrier is small then they are dynamically flexible if the barrier is small then they can equally easily go back and forth so the chain becomes on a Time framework they become very flexible but if the barrier is very large then you have to work very hard to make to access other states so this Delta e has something to do with the Dynamics and Delta Epsilon has something to do with the flexibility in equilibrium situation and that's the way it works so therefore what I'm going to say is I'm going to make a statement that Delta Epsilon by KT K is Vol constant T is the temperature and absolute deg celvin right Delta epon by KT this is a measure of chain stiffness right if Delta Epsilon again Epsilon small then trans and G are equally possible therefore it's flexible there is no preferred orientation flexible although if a Delta Epsilon small suppose I reduce a temperature when I go to very low temperature then although Delta Epsilon small at low temperature what happens is this quantity becomes large because temperature is in the denominator it becomes very large if it's very large then only trans will be preferable right therefore the combination of the chemistry and temperature they work together right therefore this is also giving me a persistence length that is how long along the backround of the chain the bond orientation will persist and we'll come back to this in couple of lectures later on how to describe the persistance length later on we'll talk about it later but right now qualitatively you can imagine this ratio Delta Epsilon by KT T is a measure of pris length which we work it out in a similar way if you look at the other quantity that is is a barrier height Delta e again once again we have to worry about this ratio this is a measure of dynamical flexibility we will this becomes important when you this is how we get into glassy Behavior Uh to some extent a crystallization Behavior we'll talk about it there is a dynamical flexibility which is dictated by chemistry right which is a Delta e or KT this also means this gives me your characteristic Time character time which I'll write as a to KN or to you can choose any symbol you like characteristic time for your monomer for a monomer to jump around to flip uh from a trans confirmation into a g confirmation what is the time involved for a local movement of your monomer right and that is given by the is a t which is related to Delta e by KT we will work it out later on when we talk about glassy Behavior we'll come back to this right in fact this T is exponential of Delta e by KT as we are going to work out later on so now I'm going to Define our goal for this particular course so here I go back and look at what I said I talked about local properties right that is how long would it take for a bond from the trans confirmation to flip into your G confirmation and that's a local similarly what is the partitioning of a particular Bond into trans confirmation versus G confirmation a bond that is again local now what I like to talk about I want to mention is I want to Define what do I mean by global properties so I want to Define what I mean by global properties I Define it this way Global properties deal with length scales scales larger than persistence length and times longer than to that to is this t for a monomer so this is what we mean by global properties okay so through our course we are going to be interested in global properties because that's the problem of physics right we are going to be talking about large length scale Behavior large time scale Behavior very rarely we'll go and describe local properties like how does a particular monomer move etc etc but in general the rest of the course we are going to be talking about global properties okay okay so if that's the case I have a challenge the challenge is how do I build a bridge between the local properties and Global properties because the chemistry is operating at local level but I'm interested in physical properties at Global level how do I build a bridge between these two uh descriptions how am I going to do and that's what I'm going to do right now in another 20 minutes okay I'm going to build a bridge or begin to build a bridge right let's do that so let me take what I would call a a skeletal chain okay so a skeletal chain as I said before is right so so there is a carbon one hydrogen there another hydrogen there the carbon and this is hydrogen hydrogen there and on and on it goes like this so what I'm going to do is I'm going to combine them right combine that that group into uh what I would call a skeletal group I would call this to be skeletal sorry skeletal group some people call this to be United atom that's that's my nor clure if that's the case then what I can do is I can do the following okay I have I start with the one uh uh United atom at one end and then another one and uh another one another one and on and on right another one this each each circle that I'm drawing is one methylene group right and it keeps going keep going so let me call this to be a0 that's a position Vector of the first one and let me call this a bond this one that is the bond Bond Vector okay and then of course this would be this would be A1 this is A2 etc etc eventually I'll have this to be a n and this is a bond Vector you know one point 5 an right that's magnitude but in that direction and finally this would be a n where little N is a number degree of polymerization right as I pointed out n is this little n is the degree of polymerization so I have converted into the a confirmation of the chain into a skeletal chain now let's look at how it's going to be at a local level so let me go and draw uh say let me choose in the middle say e Bond so let me choose this to be the E bond for example say for example I choose this to be ice Bond right when I choose ice Bond so let's say the direction of that iond is like this so this is my a i this is my Ai and then next B comes right next one is going to be say something like this okay and this is the tetral angle 190° right that's a tetral angle but it is a convention in polymer Community all right that the complimentary angle is called the bond angle 18 80° is minus that one is Bond angle okay and then let me call that to be Theta I and of course this Vector is a i + 1 okay now next one next Bond is going to be say something like this okay and with respect to that one uh this angle would be Theta I + 1 but because of the the dial angle I mentioned to you about not only be in a trans or it could be go or something like this so what will happen is it can be right this could be anywhere along an ellipse or a circle and then with respect to this there is going to be an angle five the five that I told you before this is the dial angle that's what will happen in here there's some angle there so I need to specify right I need to specify the thetas and F and corresponding to that confirmation the energy will be different okay this five is actually for 5 I + 1 this is 5 I + 1 okay that gives a description of your confirmation of the chain okay and what are the quantities once I have a confirmation and for this confirmation I have an energy because I go and compute right I go and compute the energy landscape depending upon chemical details and let's let's do that let's do this calculation if you do this calculation there are two important quantities that I like to look at the first quantity I like to look at in general is mean Square n n distance because that is your largest scale quantity and I'm going to write this to be R squ where R is n distance what does that R mean R means I'm going to add this A1 A2 correspond little A2 little A3 the red terms I add all these vectors together and then that will give me n to end distance Vector right this will be n to n distance vector and then vectorial addition will give me that n distance Vector so therefore if I want to know R squ what I have to do is this is equal to summation over I from 0 to n a i Square One Is The End into distance and square is that then I average over all average over all confirmations all allowed confirmations that probability is going to depend on this input which in turn is going to depend on R1 R2 R3 R4 okay so that's how it works that's R squ and you can simplify the calculation I can simply say this sum and that is equal to number of bonds times the bond length squared plus right two times summation I summation over J but larger than I then I have this quantity a i dotted with AJ average average over confirmation this is a very general result because this n a square is coming from when I take the square same term right uh yeah same I when I have double sum here when I take the square it will be summation I and summation J when I is equal to J I get this term when I is not equal to J I get this one so this is a very general result the other quantity that I like to do is radius of right so let me write down mean Square radius of gation okay and before I Define this let me take this uh let me draw it once again suppose I have yeah skeletal chain and goes on and on suppose I had skeletal chain like this for this confirmation there is going to be a center of mass somewhere let me say the center of mass is going to be here this is Center of mass of of this confirmation then what I can do is I Define say the position of the I skeletal this is i s i that's the radial distance of the I skeletal atom or United atom skeletal group from the center of mass similarly let me draw say for example J let me draw the radial distance of SJ and of course the difference between them I'm going to there's going to be inter segment distance which I'm going to call say r i j the distance between I United atom and J United atom once I have this I can write down the radius of gation square to be mean Square radius of gation which I'm going to write this symbol RG is radius of duration and and square right that is RG squ and mean because I'm going to take an average so this is equal to I can write down to be this is equal to 1 / n + 1 sum I from 0 to n SI I squ average where I have defined what SI is that's my SI and I run over SI here SI i s i + 1 s i - 1 is I + 2 ET ET some all of them Square it and all of them divide 1 n + 1 I get radius generation Square you can also show you can also show that this is equal to 1/ n + 1 squ right summation over I summation over J J is larger than I then I have r i j s as I have defined here r j is this one R is that Vector Square this and then do the double sum for practical purposes it turns out that this is better to to calculate for computational purposes so people usually use this definition to get AIO SC they are identical you can prove it exactly okay these are the two important quantities that uh that comes out from from considering the chemical structure so the road map is right the road map is then uh account for chemical details account for chemical details from Computing this right and then you take that into into this average and then construct a various quantities like mean Square NN distance mean Square ration and any other quantity that we are interested in and that's the road map now what we are going to do is we are going to consider some simple examples right because this is this is a big picture this is how we have to do but it's difficult to do right because the chemical details are so much and the potentials are are quite complex so therefore we like to come with some simple models particularly in view of the global versus local properties we are look we are looking for some big pictures what are the most important aspects where we can understand the physical properties of this polymers at the same time parameterizing the chemical details okay that's all go so the first thing I'm going to do is you might think that I'm being silly but I'm going to do this in a very simple way all right and I want to talk about U what is called a freely jointed chain freely join your chain what does that mean I hope you can see my picture here um Jun you had to help me are you able to see my picture here my my demonstration here yeah okay so so here is taken plenty of paper clips I hook them together and this is the chain but this model is kind of baby like model very simple model right I take one Bond another Bond and the bond can rotate completely completely freely jointed it can bend no torsion whatever it's completely flexible you know not only flexible it's freely jointed so this is the simplest model that we can think of let's think about that let's take this freely jointed model what does that mean right what that means is I go here right that means the bond I bond is not correlated with the J Bond even if they neighbors because AI right is completely freely jointed there is no correlation at all that means this average goes to zero in this model so in a very simple way right this simply goes to zero right because they are not correlated completely freely jointed therefore one of the quantities that we are interested in is going to be r² is equal to n a² okay because this is zero this term is zero okay now what I want to emphasize here is I like to put a subscript here I'm going to put a subscrip to zero Okay the reason I do that is when I take a chain like this right and there is going to be always interactions there's going to be some interaction between a monomer and monomer because they are coming very close there will be Vandal interactions that's why you know the gases when you lower the temperature make a liquid because there are attractions and if they when they come very close there is a repulsion right so we are ignoring that I'm looking only at the skeletal chain at this moment to emphasize that I'm not looking at these quantities I'm putting this subscript to zero here and as we we shall see later on these are collectively called excluded volume interactions we will come back in the next lecture exra volume interaction right and we are ignoring that for the time being okay so ignore that we we are not taking that into account right so we don't do that yet ignore maybe I should say ignore ignore this for now and that's the reason why I'm putting that I'm sorry I'm put ping that zero there because I'm ignoring that all right if that's the case then I can also do other calculations like uh r i j s is equal to a² IUS J and in fact n to I is one I United atom J is another United atom if it's zero and a little n that will be n same as that one and then again to be uh consistent I'll put a subscript zero and what happens to radius of gation RG squar and I'll put emphasize that and that turns out to be equal to n a² by 6 if n is large enough because what I have to do is I have to go back you know I have this double sum I know r a square that is little a square modulus of IUS J that in do the double s if you do this large end limit you get that one and the important thing that I want you to recognize in all these things is for a freely jointed chain I want to Simply say right I want you to recognize R which is a measure of size whether root mean Square n to distance or radius of dation this is essentially proportional to n for freely jointed chain this is the degree of polymerization as soon as we see this as soon as we recognize this uh our imagination should be triggered this is fantastic as we we are going to see later on okay that's for a freely jointed chain now let's do one more one more model well two more models very quickly okay now instead of this I want to do freely jointed sorry freely rotating chain freely rotating chain if you do this that is right if it keeps on rotating right then uh then you can show r² without any extra volume effect is equal to n a² time let me Define Alpha is equal to cosine Theta Theta is our bond angle right that's my definition right so this Bec * 1 + alpha 1 - Alpha - 2 Alpha by n 1 - Alpha power n n is number of De polymerization and 1 - Alpha squ this is a Formula I you don't have to copy this because I sent a PDF where these formulas are given all right and here I I wish that you could follow the logic how we compose the arguments how we are building the picture okay and of course if for n much larger than one right look at this 1 / n here right 1 / n uh is very small when n is very L then R square average goes like n a s 1 + Alpha there be 1us Alpha once again you you see a remarkable thing R is proportional to oh did I make a mistake no this is square root of it right R squ goes like n r squ goes like n once again okay there's a there's a picture that's coming in right if it's a freely jointed I get r s going like n and then if um freely rotating I'm adding a little more information yes the number is different right number is different for a tetrahedral case like a carbon Alpha is equal to 1/3 tetrahedral coordination tetral coordination Alpha isal 1/3 that means r² goes like 2 n a squ okay but still the important thing is r² goes like n the preactor that's different so they when I bring a little more chemistry the preactor changes but yet R squ is proportional to okay um shall we can I add one more or maybe I should stop here I think what do you think or shall we go ahead for another 5 minutes June fine yeah what's a what do you want me to do we go and complete this part extra 5 minutes okay now let me go to uh another situation right uh now I have a barrier for rotation I have a barrier right but there are not correlated every rotation is independent okay so change CH with chains with independent rotation independent rotational barriers that means dial angle right there will be a preferred dial angle rotational potentials if you do this you do the calculation you know you know flow Paul FL did this uh and there's a book written by him second book and I have given reference in my handout but the answer is r² average is equal to once again it's n a² of course 1 + Alpha by 1 - Alpha where Alpha is equal to cosine Theta and then with the dial angle I'm going to write cosine of f p is dial angle and average this going to be 1 + e 1us still it's simply a number Alpha is a number right 1/3 and it roughly 1/3 if it's a silicon you'll have different number and if you have double bond you'll have a different number for Alpha but here ITA is again some number okay but once again the important thing is r² goes like n but this is fantastic right this is really fantastic because go back to the very difficult task we had this is a difficult task we have and independent of the details r one or two or three or four independent of the details right independent of everything in general if you do not worry about exal volume interaction if you ignore that then in general it turns to be that R square is proportional to number of bonds R square is proportional to De of polymerization this is a very profound uh observation why is that the reason is remember I if I take a walk right if I take a walk then if I go on a straight line like a rod like polymer if I just take a walk a straight line then R square n to n distance Square would be proportional to number of bonds square right because length length is number of steps I'm making it will be length will be proportion number of steps therefore length to square would be number of Step squared but in here what we have is mean square and into distance that is the square of the distance I travel is proportional to time this is a brown in motion this is Einstein's Brown in Motion in terms of time space so this means uh there is an inspiration here once you do these calculations you realize in the simplis model you realize that maybe we can use all our knowledge about classical Brownian motion all theorems developed by Einstein and various other people and we could probably copy some of those answers and get stimulated and try to understand polymer physics polymer physics very complicated but in terms of captured in the chemical details maybe there is a chance but I have to convince you that this chance is actually a good chance okay and before I do that I want to make one quick summary The quick summary is this in the literature you will see the capturing of chemical details with what is called a characteristic ratio and usually the symbol used is C Infinity in general for a polymer of n bonds CN that's defined to be the r² once again I want to put without extra volume interaction divided by the freely jointed chain result for freely jointed chain result it's n a square because we are always saying R square is proportional to n right and R square is a length Square so n a square is a bond length squar so that's the ratio this is CN C Infinity is when n goes to Infinity okay again for freely jointed chain right CN is equal to one obviously right and let's look at the freely rotating chain freely rotating chain and we can calculate CN CN is equal to exactly what I did before here this is the formula right r² / n s that is the term in in this square bracket just for completion I WR down to be 1 + Alpha be 1- Alpha right - 2 Alpha by n 1 - Alp power n / 1 - alpha s okay let me write it for completeness and then for tetrahedral tal coordination for tal coordination I told you Alpha is equal to 13 so let me plot this let me plot CN versus n take this function when I plot it what you will see is for short end small NS it will be linear it will go like this and eventually it will curve into something like this for this particular case right when n goes to Infinity C Infinity is 1 + alpha 1us alpha which we know it's two so that value is two when Alpha is equal to one you can work it out 1 minus Alpha this n is equal to 1 when you work it out that's going to be one so CN goes from one to two and this is around 20 n is around 20 number of ons is 20 so the point that I'm trying to make here is once again once again R squar is n and the different n a square if you right and the difference is really within a factor of unity right 1.2 1.3 maybe six for some polymers maybe the inste of two you'll get six but the preactor here is still only a number a small number of the order of unity okay it of the order of unity so that's is the punch line of all the models we have done so far this is a critical thing so I could also write equivalently I can write R the typical size goes like square root of n according to this model remember this was one of our tasks going back here one of our task is a size right we are going to discuss maybe in the next lecture how we are going to determine Mass we going to do that but before that given n how does the size the r we have already gotten it by accounting for chemical details all these chemical details come where do they come they come they come here here let me write some symbol here okay that's my chemical details this is a remarkable observation right because if you want to take every polymer as a special polymer and we work it out it will be a daunting task right but here what you see is in general if you don't worry about extra volume interactions right uh normally what we get is R squ is proportional to n and all those numbers are representation the other numbers are representations of chemical details and also when I look look at this characteristic ratio what you see is this here for a short n it's a rod like for large n it's a flexible when I say for n large for n large I get this result when I go back in here right for n large here when n is large I get this result what is large that depends upon the chemical details right so so this is measured this indicates persistence length how stiff the polymer is locally the way the character ratio changes from this linear line to this plate value the rate with which it does gives a measure of Prive length it gives Prive length how many carbons you have to go along the backbone so that it becomes coil like so in this region it's a rod like locally in this region it's a coil like when I say coil like it's fors back and forth therefore is not R square is not going like right is not going like n that's coil like inside it goes like n Square because if it's linear CN is proportional to n CN is proportional n therefore R square goes like n squ that's more like a NE case Newtonian walks just you take an automobile you go on a straight line the distance is proportional to right the time time in the case of Einstein of bur in motion the distance is proportional to square root of time so we have Newtonian analog Einstein analog here Rod like analog like situation here coil like situation all these things come from uh CN by Computing CN and then in in principle we are going to be in large and limit when you are large and limit we are here okay that's a global limit and degree of polymerization we are noted 10 20 things like that we are going to be for these polymers most of the polymers we going to be end to be 100 thousand things like that okay I believe that this is a good place to stop uh and then we can take a break what do you think Jun [Music] bre okay let's do that [Music] Jun fun Jun fun are you there [Music] [Music] unform you [Music] for for [Music] open so moment D are you there yes I cannot hear say that again we can start yeah okay okay all right so so what uh we have come to the conclusion is basically here and this conclusion is an important one because now what this allows us is to imagine as I have several times alluded to that I can make an analogy with a random walk Brown in motion of a particle undergoing motion time so we can make an analogy this was done by Kon that takes me to chain model so what happens with this suppose you take a a confirmation at some temperature what you will see is okay locally it will be trans and then there will be a go like that and then that will go like a trans like that and then your go again trans like this then go that's a that's a goost right trans like this then again maybe right like that R again and on and on that means okay this is a typical confirmation since I do know this that comes very easily right that I know that right I know that because because of the the very first thing that we did right freely jointed chain right because we did this right because this is a general definition and then if these are not correlated AI AJ is zero then I will get this result right when they get correlated I get some other number but still I'm getting only this result n based upon the chemistry but the important thing is that these vectors are not getting correlated if they not correlated then I will be able to get this one so whaton imagined is let me imagine that there is a some local step okay because I already told you here for when n is short number of bonds is small in number four five something like this it is going to be Rod like so look right we know that only when we go to very large values of n it's going to be coil like therefore I would imagine this is Rod like here and then this is Rod like locally this is another segment another segment another segment Etc right then I would say these are freely jointed because if it's freely jointed I do not have to worry about the bond Vector the net Bond Vector for this segment and the net Bond Vector for this segment they are not correlated anymore right so he imagined this an imagination this is a beautiful imagination right imagination is Imagine This is the critical step in the life of a scientist or an engineer or a poet imagine right imagine we imagine he imagined that a confirmation can be considered to be made up of freely jointed segments freely jointed segments and you don't need anything more I need only only this paper clip chain of paper clips right it's freely jointed except now every paper clip now is a segment previously it was a bond in the freely jointed chain and here I'm imagine in the real chain with all the chemical potential chemical details right is Can Be Imagined to be a series of segments which would call this to be  segments named after him segment of average length l so let me call this to be L average length L right and there are now I'm going to use capital N there are n segments per chain in other words what I'm doing is I take this confirmation and I write this to be something like this I write to be okay for this one I write this one that is a CO segment and then for this one I see another segment for this one I write another segment for this one I write another segment for this one I write another segment and it continues this way so I'm saying this is segment length L and there are n there are n such segments there are n such okay there are n such con segments this n capital N is not equal to little n but it's proportional to little n okay so this is the model in fact I can tell you the result for the relation between Co chain model and the real units the connection maybe this is for your notes the connection between capital n and little n capital N is equal to little n CR squ cosine squ Theta by 2 Theta is a bond angle in the way we defined divid C Infinity similarly the length is equal to C Infinity of course the bond length a actual Bond a divided by cosine of theta by 2 so this is a connection between real chemistry right real chemistry and the parameterized chemistry although we have rotational angle right uh Bond angle and then Bond length right and C Infinity which depends on chemistry once we know these things then we know that capital N is proportional to little n number of bonds and length L is proportional to the bond length real Bond length a and there are three factors the chemistry is converted into these two parameters so that's is our model but nice thing about this is is freely jointed that means right this is freely jointed freely because if it freely jointed we know how to do the calculations because that's exactly so you could also imagine that okay I have a a brownan particle here makes a step of length L without any memory of where the particle was before freely jointed right it could make move by another distance L to a new place it could be here it could be there it could be there it could be there right then again without any memory whatsoever without any Persistence of the orientation it can make another move in a random Direction and on and on and on so this is a trajectory of your random Walker the co chain model here right a confirmation G confirmation yeah chain confirmation is equivalent to a trajectory trajectory of yeah random Walker random Walker so we could call this chain model as a random walk chain why not you can call this Co chain model as a random walk chain now since it's a freely jointed chain we can write down the answers exactly with paper and pencil we can just look at it copy for freely jointed chain we can write the answer so what I'm going to do is I'm going to write the various important quantities okay so let's do that first of all all right basically what I'm trying to say is all right since it's a random walking chain I could imagine like this that's my confirmation for large enough n okay and the kind of quantity that we are interested in is what is RG and then suppose I had one end somewhere there another end in here here so what is um n to n distance then also we can look at what is RG squ for example things like that this is this is RG things of that type we can calculate this so it can be shown for this a random walk chain or chain it can be shown if capital N is large enough because that's what we are interested in remember we are interested in large length scale properties if capital N is large enough then we can calculate the probability that that chain will have n to n distance r a confirmation with an N to n distance r with n steps n Co steps each having length L okay if you do that this turns to be 3 2 pi n l s 3x 2 here Theus 3 r² by 2 n l s you can show this because remember I told you that this is like a trajectory of random Walker right when came up with this model there were already 30 years of experience with a random walk or more than 30 years of experience random box so it's easy to copy the answer of course you can also easily derive it but it's easy to copy it if you do this the probability of getting n to n distance are after n steps for freely jointed chain namely the chain random walk chain is given this formula so R appears in here n appears in here and capital N is proportional to little n because I told you that capital N is proportional little n that means is proportional to molecular weight so wherever you see capital N throughout the rest of the lectures capital N means is proportional molecular weight degree of polymerization and there are some pre factors if necessary you'll account for them in general for Global properties we take it for granted that we know how to deal with this so we will not be explicit about it right now the cool length is proportional to little a bond length but it is not Bond length length could be 1 nanometer whereas little a could be .15 nomer right that depends upon Thea depends on C Infinity all those things depending upon all those things length will be yes some proportional number to a some proportion constant times the bond okay so given that uh for any polymer now we are making a huge uh jump any chemistry if we ignore EX volume interactions if you ignore solvents you're just looking only skeletal properties of polymer we could write down this formula as long as n is large okay so what's the impf of the implication of this first of all I want to talk about mean square into distance r² average because R square average is using this probability we are going to calculate R square because you know this that's how you you know how to calculate the average right integrate over r² * P RN divided by normaliz interal of that's the way you do if you do this naturally you should get NL squ because freely jointed again going back to the formulas that we had right this is r² is equal to this instead of now it's the bond length is little L little n is now capital N right capital N so they are not the the Orient TS of these segments are not correlated therefore this average is going to be zero therefore I have capital N time the little L squ okay that's an squ now let's do RG squ and you know this already RG squ without any extra volume interaction is equal to n l s by six before I go further I wanted to you know right you have this distribution function probity function then you can calculate these quantities I defined what RG is before so we can calculate them but look at the shape of this equation right the shape is e to the where does r appear R appears like e to the minus r² right that is called gausian E to the minus x² where X is a variable it's called gausian so this is a gausian the probability is a gan therefore therefore the chain is equal to Gan chain so we could use these names interchangeably we can say random walking chain or a gausian chain or a chain all right this is exact if n is larger than one much larger than one in the limit it is cred if n is small of course you know that you have to worry about prist length there are corrections to that but for all practical purposes we will call this to be chain to be a gaussian chain because when capital N is large then these are the results that come about okay now what I like to say uh an important consequence of this as I have already alluded to here I said in general the realistic chains now it also happens for gusin chain R squ is proportional to n capital N now in this case that means R is proportional to square root of capital n okay so therefore I could write this to be in the following F fashion or going like n to I'm going to write half in a in a general way I'm going to write like like that where this new is equal to half same thing R is proportional to square root of n I'm writing that but except that I'm giving a symbol for or this half an exponent so new is called size exponent in general you'll see when we talk about all other effects we'll come back and look at the general formula new is size exponent there's also one more thing I can do right I could Define instead of R depending on Mass power capital N is number of Co steps number of segments in a chain it is proportional degree of polymerization therefore it's proportional to mass therefore here what I'm saying is R is proportional to mass power new new is a size exponent instead of doing this I could also think in terms of exactly the mass is proportional to R to 1 / new same thing right R for one new is proportional n this gives me kind of a different definition this means R I'm going to give a symbol DF is proportional to n where DF is called fractor dimension dfal 1/ new it's called the fractal dimension of this object this will come particularly when you do scattering measurements this comes often we we'll deal deal with uh exal uh adventures and exploring form physics and we'll come back and talk about DF DF is a fror dimension and fror dimension simply reciprocal of in our context reciprocal of the size exponent to new and the reason why we call this to be one new is the following suppose I had right suppose I had a ball of Steel then volume will be R power some number three R power three is proportional to mass because the density is fixed volume is proportional to mass analogously here lot of this not a solid ball it's not a steel ball right there lots of uh gaps in here right but in general in analogy we like to write some dimension in a three dimension volume is R cubed if you suppose you have a pancake in a phosph liid membrane suppose I have a pancake then there'll be R squ area will be proportional mass and things like that so analogously we call this to be dimension of that object geometrical dimension of that object that's the reason why we call this to be a dimension that one/ new which is DF is a dimension but it's fractal because it's not three it's not two it's not one it's some number it's a half in this particular case sorry in this particular case two which is an integer although polymer lives in threedimensional world right instead of two you could have some other number as we are going to see later on so that's the origin of this nen clature okay so what have we done here based upon what we got in here we got we got uh probable distribution function to find an N to n distance r that turns to be gausian chain gausian result therefore we call this chain and mean Square distance proportion n mean Square ration is proportional to n is actually n l s 6 in general R can be written as n to the new right and then or alternatively R to the one new is n now I want to make a connection between why are we doing this I I want make a connection with the real experimental World okay this is a model capturing all chemical details but at the same time I like to take the pleasure of talking to uh experimental systems experimental tools okay how do we do this in order to do that I construct what is called a form factor for this model chain form factor and I'm going to going to write in my notation I'm going to write s of K right I'll tell you in a minute what this means s of K maybe I'll put an arrow is a vector that's equal I'm going to Define this this is one/ number of Co segments summation or I sum or j e to the i k do r i j remember I Define r j before r i j is a distance between the I segment and J segment I defined between atoms before skeletal atoms before here these are for segments our idea is the distance between the I segment and J segment and average it out suppose I do this this is a f transform of that distance and average over all confirmations if I do this I get for the gausian chain I get a very beautiful result that turns out to be 2 n y k to the 4 and let me see let me write rg/ 4 e to the minus k s r g 0 S - 1 + k s r g 0 S that's what I get and as you know as you know r z square is equal to you know that right uh this is called this expression is called D structure Factor this is called div structure fact this is really fantastic because you know why it's fantastic let's go to our first challenge first task our first task was was to look at the m mass and size we want to know what n is M mass is and what R is here this formula immediately gives me yeah an idea I look at this one see I'm interested in N I'm interest in size this is Theory model Theory right g chain Theory right this is model but it gives me a hint about how to measure and and RG right away I can get both n m mass and size how do I do this then the way you do that is this K right this s of K can be measured experimentally this can be measured experimentally how do you do this the way do that is okay so here is my my Polymer here and then sending a beam some wavelength Lambda some instant beam of your liking we'll talk the next lecture about how to choose this experiment and then you measure scatter intensity from some angle Theta here is your detector and then in this setup scattering setup K this is scattering wave vector scattering W vector k w vector k 4 pi/ Lambda sin Theta 2 where this Theta is scattering angle this Theta is a scattering angle Lambda isong the instant B so i s a Lambda I have Theta here then I know what K is that's a k that appears in here in the form factor that's the K that appears in this formula okay for example as an example I take this function and then I look at this function s of K right and that's related to this scattering intensity I of K this is cering intensity cing intensity I of K depending upon whether you use a synchron neutron scattering light scattering depending upon that there will will be a number here which I'm not going to write down but that is proportional to this s of K that is the form factor so this is something you measure and your laboratory or a national facility will tell us what that number is here and then that is related to this s of K this s of K if I take this right and uh we'll we'll think a little more about it if you work out this this formula you get the following results you get y 1 minus k s by 3 and squ plus etc for K * RG is much smaller than one if K is small small angle that's what you will get on the other hand on the other hand if you are larger K * R is larger than 1 you get 12 by k s l s that's the answer you'll get so this gives you a clue right this gives you a fantastic clue right that is you go to your scattering Laboratory you choose the instrument technique light Neutron xrays in such a way you satisfy this condition K I have given K that is you choose a Lambda or scattering angle you could tune K either with the insent beam nature of the insent beam and also in terms of scattering angle you choose in such a way K * RG is less than one then if you measure scatter intensity you can measure n and you can measure the size and this is what we are going to do in the next lecture I'm going to augment to this and look at the consequence of this at this moment I'm looking at a microscopic details of how the chain sizes mol masses but I also want to use this gausian chain model to understand the free energy thermodynamic properties suppose I take this and I put a force on it how is that going to respond what is the force right how is that going to happen right and that's what I'm going to pick up in the beginning of the next lecture so uh at this moment what I have done is we have made a huge imagination which turned so to be a fantastic idea good idea very fruitful idea and it is the basis this imagination is the basis of poly physics All chemical details are surrogated through L and N but the key thing is capital N is proportional n number number of actual number of bonds little L is proportional to actual Bond length but there are prefactors depending on which polymer days these things will appear in making the mapping that is taking chemical details into the co chain model we do that step okay so once we do this we have without any extra volume interaction accounted for we have R is going to be proportional size is proportional to n power half molecular weight power half okay in general and then I'm also telling you right away this allows us to determine it gives me a handle experimental handle to measure molar mass and also radius Direction I think I should stop here you must be tired at this moment having listened to me a couple of lectures right now and we'll pick it up next week and what I'm going to do next week is to use this idea a little bit more and I'll show you some external data and we'll make a critique of that we'll also talk about the free energy of a polymer chain and its consequences and then we will ask about how good this model is in real world because real world we have right we have solvent and solvent molecules are not going to be idle they are going to kicking the monomers of the polymer chain and there are going to be consequences of that both the structural consequences and the thermic consequences and that's how we are going to build our Concepts in the in the next uh lectures okay and I like to stop here if there are any questions Jun F this is a good time to ask from [Music] anybody well they may think about it we can I'll entertain their questions again later on next lectures okay maybe you find the floor is yours do as you wish I'm going to stop sharing yeahor stop yeah okay I'll stop the recording all right okay thank you that

Transcript for:Overview of Polymer Physics Concepts

Transcript for:
Overview of Polymer Physics Concepts