24 Linear viscoelasticity

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24 Linear viscoelasticity
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Lecture 24 - Linear viscoelasticity. George Phillies lectures on polymer dynamics based on his book "Phenomenology of Polymer Solution Dynamics".
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glasses in polymer dynamic based on
George Phillies book phenomenology of polymer solution dynamics Cambridge University Press 2011 ends today this
lecture E is lecture 24 this
Scully lastest city the storage modulus the loss modulus thinning BEA I'm
festival is today we're going to finish off our discussion viscosity and push on to discuss this commitments this at all let me recall where we left off roughly speaking we left off in Chapter 12 discussing figures 12 . 20 to 12 . 4 blocked Twitter treatment of the pot molecular weight dependence of the viscosity of the solution now there is a bacteria actually fairly considerable number of measurements of the East and they're actually fairly consistent namely what they find what is found with considerable consistency many systems is the viscosity depends on the life of the way this pressure exponential way and unless brought out up quite emphatically by for example Figure 12 . 1 4 of 12 . 1 4 ,comma measurements they actually here in doorway and Edwards and report them as measurements by home nobody at all on the measurements are a little more complete amendments the original paper although having said that the an interesting feature here is that if you go back and literature and asked what people said about this these measurements you can find people who say they have measured the molecular weight at its viscosity and asked predicted we see how a lot depends of the molecular weight there actually are measurements that fine Powell molecular weight St opt for example as we discussed earlier how all adults measurements which pushed out to the Mets find close to melt a towel on the dependence of viscosity on molecular weight in volume a fraction of of what it was like plot where rapid clearly straight lines viscosity changing as a towel on em and if you actually saying will stay fairly close to the melts and we will explode points that seem to be deviating from the region's straight line you actually could find but how and reading however most of amendments to find which are down considerably lower concentrations do not find out power behavior they find stretched exponential interesting example of this is is provided and this is also Onassis stretching over to the next section which is in discussion of policy part of the government's 5 bracelets and gracefully look at polymers and they look at linear chains and they look at foreign stars and they look 6 hours starters and if you looking that the results they show as a function of concentration for living and earlier changes of molecular weight of on the 1 . 6 6 that had billed as a function of concentration wall-to-wall plot they find what is very clearly a straight line and look at it it's clearly a straight line it is quite clear that the offer you were seeing Powell on the other uses for 2 Armstrong linear chain to stop on the other hand for F equals and perhaps he say very clearly on the same plots Lisa chains of about the same molecular weight on this is about 190 5 Medical polymer and the other is about a what was it was about 1 . 4 were told to hold her energy for the other 2 very serious barely see stretched stretched exponential molecular weight dependents concentration fixed molecular weight the concentration is if you look at the molecular weight and where they have a considerable number of samples they were looking at ,comma concentrations of about 300 rampantly polymer extremely concentrated systems and the molecular weight and straight line for linear polymer they are was a stretched exponential To stretched exponential actually for we all stop all as a font taken as a function of concentration on the altar of the year the linear chains showed the strength of our law behavior the stars from stretched exponential here's the viscosity of the stock was for the most part unless the viscosity of linear all are however if you look carefully at the measurements against concentration you can see it .period up here and so is the old concentration was feted the kind of eventually sold the stop always as the obliged to give functional behavior they shall viscosity that was higher viscosity stars of linear chains all however for the most part start polymers comparable molecular weights all are often fairly high concentrations less viscous that linear chains and it faced concentration as a function of molecular weight against the stars are less viscous the millenium changes Monday but you could also look at results of the duration of you can also look at results of the furor and collaborators but they looked at 3 Armstrong but the stewards were halted healthful diet styrene on the they're look at a series of concentrations and molecular weights on you can see the results in all my years 12 . 6 and what they found was that again if you're about to send concentration about the same molecular weight the viscosity all the stock owners to get the restart is less than the viscosity although why is the statement that the stock holders have a lower viscosity leading chain of some major all the number of models of polymer dynamics which asserted that linear chains have extra modes of motions that are in principle not accessible the stops are in particular for example often reputation type models say that if you have a linear polymer he can creep through holds in a transient last like snake in the world but if I am a Stockholm in order to make forward progress when I have to is is retract by fluctuation 1 of my arms it's difficult push it out in new directions and now I
can take a very modest step forward on the Net result is that 1 would expect that is harder for star polymers to move with respect to each other than at least if they're very concentrated than it is for linear chains that does not appear to be be quite near certain question it can also be a possibility the hearing the the all that could be an issue in after however In the solution the amount of hacking is determined by by the concentration and the answer is that the of constant work we make comparisons that people concentration for linear struck you know they thing could call With respect to let us have changed to know is that I have to start all over I have 1 point where there were 3 chains coming together there is an equals 3 stock there is a narrow region around this where there were 3 of my chains close together however ,comma large stop Oliver over most of my life and look exactly like the lineage therefore that this sort of infected you might find interesting effects with small chains with a large ones you would not expect that OK we advanced the year the whole discussion around figures 12 months and then the issue around figure 12 . 28 configured 12 . 1 per cent 2 years from the polymer and we changed the solvent all what happens if we change the soul car while the expectations but if we are in favor of solvent closer to which means we also have the right here we have a policy change in solutions although you would rather see some as its neighbor than see something else to come and therefore we increased concentration at nothing terribly interesting will have to change rates they didn't say nothing at all because don't it you can get perfect cancelation effects to approximately on the other hand over the goods sold are we see what we saw in the chapter on dielectric relaxation which is what has been used for most of the namely as increased concentration look at charities the chain radius falls with increasing concentration :colon so there'd be these however stacked properties the hobby is the chain of command takes a snapshot we're talking about viscosity here viscosity is a dynamic property and we ask what happens to the dollar the behavior of the system if we change solvent quality and such but the 1st answer is if we look at a series of changes different molecular weight the disaster as in the case of most of systems not all the systems and not sufficiently high molecular weight concentration but the concentration independence is stretched exponential and what we happen find as we increase polymer molecular weight and our measurements we find but the parameter alpha increases and the parameters so there is also and there is new and both parameters ,comma molecular weight and the 2nd thing we can do is to compare these parameters In a good solvent In latest and what we find is that in a state of solvent patterns discussed here a parameter in new orders from the whole point of the 120 5 work and if we go however take the same material over to the insolvent the parameters movies Poland . 6 of a point so great that is we're going in to head can we change the solvent quality knowing that you're going to get better solving the parameters new falls but the model that derives this equation in fact predicts this behavior we will eventually get to it later but if you look at the figures especially 12 . 28 this change of new was physically quite visible if you look at the son-in-law "quotation mark a a lot of viscosity versus concentration for invader solvent B on concentration and the data is fairly close to a straight line because firm boast was stretched exponential but if you look at the measurements and good solvent you typically see a more tiles curvature because you were looking at a stretched exponential red purists but but now having said this let us push ahead and consider the general features of this gossip military losses specific measurements and what we do cost find viscosity depends on polymer concentration polymer molecular weight not to mention some of 1 would the apology and 1 could put insolvent quality following With respect to the apology but there was a very long dispute this goes back a considerable piece about the viscosity of ring followers all in there that was mostly belts which the book is not covered on the issues there was well you can try to make it melts you have asked party in trying to make rains and put them in melts here a ring is a nice open or during years since synthesis process didn't get itself all tied in knots so that its ball Paul under modern conditions if you want to answer that question you can resort to using DNA tests since home you can synthesize DNA is with absolute molecular weight control but absolutely mean we have something that's 180 thousand base pairs meaning its molecular weight is tens of millions and the molecular weight is well the fluctuations because several isotopes of carbon material nature and part and interior hydrogen saying nature all but the molecular weights are totally exactly control no fluctuations you can also synthesize things synthetic DNA not stuff that occurs in nature that is a star are completely 1 of his 1st you could also synthesized it happens naturally some viral DNA students on the energies the brink polymers there again we on Spurs electron micrograph those readings indicate the readings are a completely open they are tied up in knots on
however having said the rating :colon murders are Mom's 1st stop polymers are truly modest 1st insisted that the comes their way up there is the significant practical issue that people have not done studies of DNA viscosity there is extensive it as the viscosity studies at the onset of fall OK having said it let us look at the question the 1st quake which are going to emphasize significantly more than was emphasized this area is not universal functions folksy that is if you think that you can take your viscosity measurements and yes there will be some chemical dependence Constance and the way things that depend on the concentration of molecular weight or maybe even 1 strands some more like the way some representation of amount of material and all sizes chain although if you think that all of the chains are the same because of that and therefore there should be a universal functions into to viscosity is a function concentration on you're going to be disappointed instead it is quite clear from literature the viscosity is is not universal function of and instead there are 2 distinct phenomenology In 1 of the terminology you see is simply a stretched exponential which in measurements of data goes a data from other people go out more was and another set of systems using stretched exponential all afternoon .period and then beyond .period you see Powell copy it is worthwhile to recall but the same phenomenology this class is occupied by satirical call satirical colleagues show the same behavior the the hourlong the hands are on the move exact material for spherical Collins slope is considered is appreciably steeper than from any other chains and if you look at the many arms stock polymers the slope creams off and makes the transition from linear change behavior of very nearly parts security what it should be be clear that any model that explains this phenomenon is subject to the constraints that it must also be on workforce Terkel Polaroid's skiers can do not on offer things but on Form entanglement lattices is not really 1 of and former of wrapped tape is certainly not what so we have the issue discussed behavior of life not quite what you would have expected from some of your reading from some years OK during war if you were shocked to viscosity of volatile and the the rest Asia over it's your own I don't have any and most of those measurements ,comma literature studies to prove all that there's some nice work holding off however if you look at that it doesn't appear that if you get about a here instead of continuing along on power law there is words deviation the nature of the words deviation is not quite certain all what might however for example proposed to do since were extremely high concentrations that some sort of a jailing transition in which things actually mechanically get in each other's way and the chains can't slide across each other because they can't after all the rigid over short distances they can't move sideways as easily enough to get through however that is a fairly specific that's a fairly rare exception was not a lot of studies look on the articular if you look for a predictions that the conditions on the next what all that prediction is very generally not true polymer solutions there are some exceptions to this this family but predictions that you can simply assumes scaling do her not sustain other than if you wanna say have meetings playing was sometimes you scale and sometimes you don't all otherwise you can't just OK so that is the sort of behavior we get all we might then asked Well what others generalizations and and what the answer is for a very large family is if you have we have created over a 0 is the song constant golf see the new and now I'm home all of molecular native independence of Alpha Tau both actually seagrass had said you but we actually look at the measurements and asked OK we get this thing to fear what do we see well beyond what is usually In a range of that I can't be more precise than their best reasonable approximation called in .period affair is I should add something to go from faded with salt you go from various solvents solvents the parameters of and what I call health care because it's helpful him laughter what I call on all the animals and the amount of it's not seen increases as we from there solvents to resolve generalization context so what behaviors Ruiz get any answer is seen In Figure 12 . territory where we plot helpful to which the terms of this he way his health and you know coefficient C the plot health against all molecular weight now I just shrugged .period Suri action would be Gee that was a nice piece of graph paper professor abilities at least it was very nice before for Eastern way back and started firing the shot gun and points are all over the place however if you look carefully at this point I have labeled the points by chemicals family all of the polymer so we have a serious I identified separately each set of homologous Talbot followers that are chemically identical differently and if you do that you see that through each single-family family ophthalmologist polymers alkalis variable interest rate .period on the straight line has slowed so there is a lot lot plot so we say Alpha goes into power in March and what can we
say about gambling if we look at all the measurements that all of measurements were the same way out of a large number of different polymer molecular weights on all those bad things are consistent and the answer is that the answer is taking the lead in the range of people . 6 0 . 6 cent to 1 can find a cases where there is no way that people directly people . 9 4 words quite large you can find the words hold .period 1 on if you look at this he knows there are different slopes the Nunavut materials and therefore were not universe the slope is not quite universal constant quest for a given set of samples the power a lot of friends is actually fairly sharp on this result is also this result is also this the heart of the systematic review of all of the collaborators and all the wonderful Russian data piled in 1 place for us because we also says is the is a good reducing that is if you take all of these measurements and plot was functions of times the intrinsic viscosity rayon so we have a lot of him see here you find pursued whole series different polymer molecular weights the measurements the file on the same person downward for the expansion alliance the same problem when you are doing this lot must be please say that some of the cost of city in which means that all of the molecular weight dependence of viscosity Gaston built into the molecular weight maintenance the intrinsic viscosity you can't get any of the molecular weight dependents out years of boys because if you didn't want plotted viscosity for a series of polymers of different molecular weights they on on the same line by line along the same lines because viscosity is a function of sea times the intrinsic viscosity that statement universal functions goes all the way up to now however it is important the viscosity the intrinsic viscosity depends on however therefore this equation says viscosity use of function of 1 of the function of the times news conference on the day and the example of a function that has a property that viscosity has this tendency is given by this fall now you would have to say this is the health see them in the back of the power and therefore the are Diskin the interest the Times New yes formed the next following came by multiplying the expert said and therefore this equation and their wages only consistent so long as gamma a new or related like this novel last year false results are entirely consistent with everything he said at the I would be delighted to say that I protection of movies movies and molecular weight solve all the event really is quite different from the 1st obstacle is that if you look this formula for the viscosity data from the article the office seems to new was the exponent of the which means it's a bit hard furthermore new doesn't change a whole lot if we send who is almost always in the range of 1 1 but yeah that's correct all and therefore there's not a lot of range and if there's any difficulty at all measuring you accurately you run into problems with Brossard scale there are some other recent parameters however that have been sort of hearing that I had weakened .period has fueled the 1 we can talk about we know have to the call telling is the transition concentrations the key transition concentration yes we're not limiting ourselves to systems that show the the solution like smell like transition this transition they have charge of like that and there is a concentration of substrate scenes solvent he to would translate time for spot leaders and discuss status of key when we discussed this call a well 1st fears there's it's always usually only 1 per cent volume fraction the period and he was out of the water toward the higher and his staff that the rate decisions divided by mistrust insolvent about and also minus 5 that's very clearly not trend concentration and viscosity which start getting states that an question is that of a linear polymer is how does the transition concentration and the answer given in 12 . 3 3 in on points above the lives of more or less on the straight lines and the transition concentration of and my hope 1 of . and 1 resource that is as you increase the polymer molecular weight the concentration of the transition faults yeah quality of course you need a set of measurements where the same material the chose the transition has been studied for a lot of different polymer molecular weights otherwise you should be hesitant about leaving the Fed and the results that do exist are on and fossilized sigh and they discussed for and on you look at those and G you get a roughly this behavior change to you OK that's the transition concentration I you might also said Well gee that's the transition concentration and maybe we should really express the transition concentration in natural and when I say in the actual concentration for polymer solution with people generally need to see the stars know there are people who say the old cease-fire were the stars the overlap concentration but in fact the overlap concentration is usually inferred from the intrinsic viscosity namely the stock he holds solid numbers range from what I've seen things small moments Florida the war literature the distinction but not difference OK so we will look and see and will ask out loud transition concentrating natural movements received transition and if
you go through all the accumulated evidence behind transitions the highest in you can find things that occurred about 35 now is more typical but you can also find a transition concentrations as below for example hydroxy prompted cellulose which Carroll's philosophy and I and others about the war and so if you look at the transition concentration and natural units in covers a very wide range of values you should realize this is a significant part of the way now often meld concentrations in revolves men analysis were not only about 300 all on the transition concentration were on hydroxy probable cellulose all the intrinsic viscosity was 1 over all wanted to grant leader in the transition occurred around 6 grams per liter so the if we say put the concentration of natural units you don't see any tendencies measurements flaps on there is however something you can which gets the dimensions make not perfectly but actually pretty well and the answer is if you look at all those curves where you see something like this for a linear polymer not years call for the wrong measurements for people's looked at a lot of different molecular weights you look at the graph you realize that the crossovers occurred about the same viscosity and fat the key is absolutely not in fact Bloomberg aided studios in some long relied too much on the phone that is there is not a concentration of which used cross over but there doesn't appear to be a viscosity universal viscosity which the crossover says in September's fiercest be different and the crossover and viscosity occurs when data over a 0 reduced viscosity is few the cave produce something that's fairly viscous not completely impossible work Oh yeah we want set up the power of the behavior of year you might legitimately get Westernized bodies expert and the answer for referral systems is in excess of 5 3 in the however in confinement systems in which more to Florida of well I know much larger somewhat larger for example and hydroxy probable cellulose answers about 4 . 4 and therefore there is a power locks from making even tell you how a let us give back From the systems that do show transition which well it is universally universal transition a function transition viscosity it's not universal transition this function concentration all of you but it wasn't even here that song system so let's go back and let us look at the system management I'm not seeing transition which is 1 of going out in some cases very heightened constantly to use discussing itself about him 7 in cases in which no transitions OK so there is courage and 1 of the things you might legitimately ask is well she couldn't we managed describe this measurement perhaps coupled Powell as you go through the literature you find the broad Colin Walker a here which is sort of tangential goes points over some distance and you find them here typically a linear curve and between the song transition what sort of talked over as Jeter's Transitions Abroad and therefore you might propose yes the measurements actually show a crossover from 1 side the the other but the crossover is a little hard to see because there are other things going on it's not there's not a clear transition away there would be worst phase transition I don't think anyone has proposed that the letters White almost no 1 who goes through the system so there's there's a transition but it's clearly not a phase transition not the only question is is this a reasonable distractions and will answer I think is that if we look the the office he knew all be stretched exponential behavior marches out health and and the concentration of worship C 0 rate and extra long the whole purpose and therefore viewed from this perspective there is no transition there is a single smooth firm just shrugs in your version OK I have just run us out of sheer we have now come the of this discussion starts I am not going to advance discussed this last December the worried that they have brought picture the idealized fisheries Europe infinitely long the plates now that's a little dangerous because plates :colon infinitely long and there are several ways around this for example bent plates in circles and have sent cylinders for you to have a flat plate we have a rotating columns you may say what you want rotating color and the answer is the velocity of here he is always up for that loss to the parallel play this distance distance out from the center increases heart therefore this distance because of this distance is is proportional and therefore violated the velocity of plane over the distance from the late father and there is a velocity gradient here and the velocity gradient is uniform this season now you have to be a little careful with this because is all going to be some question of whether the uniforms he this way the potential complication nonetheless we should persevere and I will draw this spring and despite the bottom plate stationary and the top players is being moved back and forth and the top place has a displacement of local and remote and displacement is something I don't would do it "quotation mark thank that is we have displacement and the extent of the
displacement oscillations of the harmonic I you can do that the with either the displacements were fairly small now you might say Well gee could you get information out of you made displacements Florida and the answer is yes and there's an object and experimental method called Todd English possible to every year and we shall reach in the next chapter Chapter however just read Chapter 13 which talks about linear discovery last year OK so we have this displacement and the results reported that this placement the have force on the upper plate to move back forth now if you think about this for a moment you realize that if the player has inertia you're also accelerating the slated for sale time anyway have to deal with this Texas basically an experimental quest and the inertia of of the place has to be made small in some sense that you can measure the current forces the fact liquidating time waiting for all this a 0 the amplitude of oscillation comeback other notation for Poland and the reason is that if look at the system if you make plays a bigger and bigger amount goes up but it's basically a linear process and therefore do interesting things worse for you in areas and the forced relocation area is known as the In the 1st forestry unit area which by the way is going to be oscillating as a function of time the stress is determined by the amount of oscillation the displacement divided by distance Allen that is defined this whole apparatus make twice as big vertically the amount of oscillations seen locally by all there is small and it starts out as neighbors along this line and we increased the lead to its maximum uses oscillating the neighbors displaced sideways 1 way or the other in losses back-and-forth banter and the amount of local displacement is determined by the ratio of the Reiher displacement of flight to the distance between the plates and this is known as the spread so far so good now suppose we actually do this experiment there are a couple of different ways we could do 1 thing we could do used to apply an ocelot Tory foresaw here and there the motion and so we have force which is very precisely controlled Houston's back forward and we asked how they played more we can also have a driver and feedback electronics and remove the played back and forth through a fixed distance and we instrumentally determine how much force were having to preserve so the place actually does so those 2 experiments ,comma level that is are they add that measure series of curves were certain about cost per unit area certain stressed and a certain strain on match with each other 1 from 1 boat whether you were actually experimentally creating stress measuring spring vise versa doesn't matter if there is 1 curve and the distinction there is purely how you build your machine OK no having said all this we now it when measure this question is this the all we would basically say that suggests only because the militants this is all about it seems to me the answer is no that statement is more general than its linear response that is you have stressed you have a strain identified doubles stress what happens to the strike well-illustrated may or may not bubble in many years responsible adults but however the statement is there's a stress strain relationship would be true whether it was when you're not the only places where you would get into trouble with this is that you could have a history cessation where on if the will apply different for example different strains and the stress changes someplace or vise versa you might not be able to to reach every point on her instrumentally very easily by doing 1 now this is linear response in the sense that the theoretical analysis that is going to be near his going last city assumes that if I apply a series of forces at different times the food has a response at later times and I can just 2 linear response that is add up all the responses of fluid the worst fire fighting forced several times and served time delays and added get the answer to get too many years so what is the net result that is all that is needed for the actually look at the spring stressed the area and we compare it with the strain we find itself and what we can find that because this is a year we can get tossed can provide out the EU 0 we have forced per unit area per unit of strain and that forced turns out have to pieces there is 1 piece of this tournament functions in the crime of all murder and crime of is now multiplying the displacement traditionally Steinbrenner postponing all right "quotation mark .period foes there is 1 component where the response has been response that is being phased With this place so you have displacement and there is a restoring force that's what areas that grows linearly and displaced the oscillating misplaced and then we have another moment and other components is 90 degrees out of their lives and so there is a response that is 90 degrees out of phase 2 the displacement meaning this is a force that is the largest displacement is 0
local very odd but you in phases out of phase which sponsors you might ask what does this all mean why is the forced him out of phase with displacement is liquidate after all if you made the displacement stopped and sat there after a while force would disappear because the just flop that's a little more complicated the answer is the low shear viscosity is it will also low treatments in the heat prime momentum over these 2 objects have been used by the way he pralines storage model In GW opponents I'm lost want In light of legitimately want gene that seems like a two-year wage represents things and I will now show something that makes much clearer for physicists pointed to what you were looking I'm going to multiply each of these objects by what and of course where a physicist says I'm going to multiplied by 1 you know it's going to be some huge object that is in fact 1 of the books but not to look and multiply this despite warned wormhole all that we're this and I got this 1 by 1 in the form of major overhaul OK that's perfectly generally notices that looks exactly like Wisconsin but now I OK I have something that goes as Omega times the the displacement In his 19 reads out of phase with displacement and I have something that goes where times displacement what my looking at your role if you think about harmonic oscillator you realized In this object if the velocity and I have determined the forest is determined by the velocity that's a lost tourism and here I have determined that goes Omega squared times the displacement nasty acceleration and the explaining that goes with the acceleration is a restorer enforceable system looks like it has little Sprinzen and in fact there earlier Arizona math models blamed on home if I recall correctly Maxwell and this whole picture can be modeled as is a system and it has a sprained and calls to this spring is something known as -dash hot -dash parts various works Japan's people had held types the desk clerk at the playing of the what we're moving objects in the teletype comes back very fast and you don't want to wear the foreign cars because the dollar's the vibration damage that is so you have the school system like object and moving objects coming this way Hassan Kazemi ,comma coming out his erupted this year because of the robber disputes this saying and there is a little here and the this is a test but it's a leaky pistol result you compress the air slows things down and then when it's done compressing the air here blows out the the end and control the opening sizzling sliding our and so eventually Africa's small fractions of a 2nd the moving object on the teletype flocked to the interview it's brought to a stop at exactly the right location and with no unpleasant shock acceleration damages although well yeah because they have and it doesn't at high quality victory engineering the no essentially no parts of this the object of trying to sell there's nowhere to the advances that no electron it's sold on him and there is national and you can feel the dance partners in solemnly was so you have this hanging objects in liquid and the hanging objects in the Lakewood it's just a really good with the viscosity and no frequencies and some you or you frequency bands and this object represents storage model was Tennessee's represents loss so that is a picture of the this fall unless the parameters linear in terms of storage model was wasn't lost wireless and you notice that I have read parameterize them starting here doing this consistency basically and once I've done this we lost lot looks exactly like the viscosity there is however dislocation qualitative for theories however the consequences of retrenchment arises if I thought crimes were achieved all crimes himself you remembered that contains something I divide out momentum which I can safely and therefore forging prodigy knows from both go to 0 this is a little hard do want lot lot plot they both got there and low frequencies more typically what plots and never really get there is a link their behavior which is contaminated with power and then the things that happened over here and perhaps the Kurds across the divide making Omega marjoram a dramatic herbs Ramon again I'm not going to drop this in any detail the final standard texts the important issue is that if you talk about cheap prime on prime cells themselves all you find that they have heard that looks sort of like this and there is a region where things are reasonably flat low-frequency Regional high-frequency region introduced terminology will get that next time all the main issue is that there is a shape perhaps if I instead saying I'm going to go in and I waited plot to keep trying to overcome this where she novel prime over that if I that I have divided out leaving slope and fight to divide up leading solve that says that low-frequency the behavior that I'm looking at this frequency independence which is sort of what you would naturally expect curves although as you will see from a book about the same leaders last year there's a rollover and then down here there is at least at 1st at Howard and then there's something interesting happens and that is the kind of question we asked given that I have drawn these pictures is well how unwise derived from rationalized curve explains its and so
I will do you think last based on here I the theoretical while we have not yet discussed in detail and renormalization group apart and so we're going to actually involved in the life of 1 of the hydrodynamic scaling the wall of his mind on and we are going to be involved renormalization group partly through normalization argument being involved is much more general and qualitative and many renormalization group argument although it will be clear what it is we're going to start out here is a straight line representing concentrate we will match and viscosity is plotted perpendicular black war of words and what we find is the viscosity increases his idea of a higher concentration and eventually at least in some systems we reached the transition concentrations scenes and down here is the solution like reading and the solution like regimes he stretched exponential in concentration and hot here is melt like reading them through CC the expertise that's secure phenomenology now what I say is I don't use something called all positive functions renormalization and what I and II apply this in the context of the hydrodynamics scaling walls and I said I calculate a certain behavior at lower concentrations and the Altenburg a dollar renormalization group that lets turned lower concentration calculation into the stretch exponentially and we're going to assume that's true now I'm going for another dent but not the FRG Altenburg coloring or positive function normalization method actually predicts 2 sorts of functional behavior the functional behaviors predicts the depends on where water called fixed points of the renormalization group of In particular if you have a fixed point at 0 then near-zero currently stretched exponential behavior as his fun on the other hand how years of places if you have a fixed point that is way out there and that is the dominant fixed .period can several 6 points but as no go along here there's always a dominant fixed point is the dominant feature points out here you would get power or behavior as sir trick the viscosity is the low-frequency live all you don't try all that overall the staff that he is a function of concentrated and so is the dynamic lost loyalists and therefore when we say we are looking along this curve that discusses the I can also say put another exercise in handling vertical exercise skipped over I called vertical axis holding frequency and I'm going to say this Garcetti's what we measure along the Omega moralist maybe not quite 0 so we better at all at low frequencies and we get the viscosity Societe 0 frequency limits however it's 0 frequency limits and wife wrote as Ada might just as well that Britain has cheated on you Linda by the way concentration Oklahoma and therefore I now have a 2 variable function and I have used the renormalization group argument to work along with many since Europe now comes the somewhat I will start here but I march sideways and at 1st I didn't machine when the station .period zeros presumably still there and and then at some point I move sideways and often these other fixed points about which I know very little become dominant therefore I moved from the solution lies like however I am emerging along and see the please whole matter as the variables that I sideways along with France and I am looking at something that really is really only a function of of all that even this picture is correct and there is a great deal of extrapolation of this it's obvious sorts on the evasion 2 1 2 2 2 2 2 at lower frequencies this explains is novelist and therefore I should have a stretched exponential infrequency however eventually might cross this line and once I'm across this line I am in a region where the no life the explains out there and I shouldn't have only managed to hold this doesn't really calculate what power is at all In order to do do the wrong extrapolation even low-frequency I would need some something previously viscosity is is 0 plus presumably some intrinsic viscosity in they told me that is I would need some way to generate a model gives me the low-frequency frequency bands so I can move or lost 0 at all and after I have moved off 0 at all I can then use the old Berger dollar renormalization group to say what happens frequency however I have not found in working on things I have not yet found the of frequency but it against and therefore it is the assertion is that you find this linear staff landfill the low-frequency behavior that has not yet done and cross over the fixed points out there not knowing concentration OK come so that is the rationale ended any notice was predicts 4 days stretched exponential and frequencies at lower frequencies end and predicts towel on frequencies higher however there is an interesting if you look carefully there's a little bit scientists you and
1 that I have skipped around is what it it says about frequency dependence way out here that is if you are way out there all of this Europe like regime all the way and therefore you see simply Powell along frequency behavior could actually waited data that looks like this and the question is how long are you really going to see that the worst case that there is a very low frequency stretched exponential frequency range and correspondence that is possible this heard that over and very very close to 0 frequencies get the sufficiently of use the CDs behavior this year Idaho and answered on the other .period but if we go off to a very high frequencies he is also going to be the case but we have got into the old extra gene variant will concentrate on is this :colon planned over so they're very very low concentrations were we can just barely see particle interactions all hot we are still in the on concentration to next that stretched exponential beneath the
model does not have this very does not discuss this the models about his around there covers this large area now we could save costs GE this picture explains why some systems show all the solution like not like transition and himself not namely depending on exactly where the fixed points out here or there it might be case of the soul of the system to along the concentration access all the way to now this
purpose not have totally invented stops them and even at the mouth of the details numerical parameters going into the calculation the saw the Galway out here in the faced pointed 0 is still dominant but the other systems and all you needed differences in Canada please because parameters such that these 6 wastes the long-dominant and some concentration out here see Powell and therefore want universal physical model plus chemically dependent parameters that determine where which fixed .period his dominance predates both behaviors will last finance through the on hydrodynamic scaling model which actually allows you to have to behaviors in 1 general model is not also good for models that assume scaling because they assume that you always get scaling out here which it seems that you not OK so that is the UN's Aaron of having given you the answer us we have now reached experiment we have also reached approximately in the hour and therefore we have reached a natural .period stopped therefore I am going to stop so this is the finishing off my treatment of viscosity and advancing to my treatment of the hydrodynamic scaling model and its cons for predicting linear viscoelasticity I'm George Phillies This is the end of the lecture


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