The Source of Molecular Motion and Flow in Polymers

I have just posted a new Video Clip Lecture (#21) which explains how interactive coupling of Dual Phases can be used to describe thermally stimulated molecular motions and flow (rheology).

This subject is closely related to my previous post, yet the information is presented differently and the format (video lecture) is more casual.

This lecture is available on the WIZIQ teaching platform:

http://www.wiziq.com/tutorial/194924-The-Source-of-Molecular-Motion-and-Flow-in-Polymers

Dual Phase Visco-Elasticity

Polar plot of Modulus vs Phase Angle for the dissipative and Elastic components of the two Dual Phases (red and blue).

I have finished part III of my series ¨ The Great Myths of Rheology¨. The paper is going to be published momentarily. This paper talks about the melt network elasticity, diffusion, shear-thinning and strain softening, in other words melt deformation and flow.

It is a rather long paper, which no one likes to consider reading, yet I write it like a novel, introducing my parameters like an author would introduce his actors. I also make flash backs and replace intuitive images by equations, which themselves produce new conclusions, more intuitive images. I am never far from my subject, which is to describe the experimental data, but I try to innovate not just on the ideas advanced, but also on the style to present them.

Well, the objective was not really to entertain, but to launch new ideas. The style may help the reader swallow this whopper.

My real objective was to lay down the background for the Grain Field statistics, knowing that it would take two or three more lectures before I am ready to expose the mathematical model.

It took me a while (several years) to develop the connection between molecular motion and flow within the framework of the dual-phase concept. Flow induces transport of matter and it was a real challenge to integrate this with interactive local motions, which I thought I had described well in my work on thermal stimulated relaxation (see one of the previous posts of this blog).

I reproduce below the Introduction of my new paper.


Abstract

The dynamic data of polymer melts are analyzed in a novel way, presenting new correlations between the viscosity, G’ and G” (the elastic and loss moduli), and strain rate and the implications of the new formulas on our understanding of melt entanglement network elasticity are discussed. In the two previous articles of this series [1,2], we showed that the existing models valid in the linear visco-elastic deformation range were not adequate to extrapolate to the non-linear regime, suggesting that the stability of the network of entanglements was at the center of the discrepancies.

In this article, we introduce new tools for the analysis of the dynamic data and suggest new ideas for the understanding of melt deformation based on this different focus. In particular, we express classical concepts, such as shear-thinning, melt diffusion or melt elasticity and viscosity, in a different context, that of the existence of a Dual-Phase interaction, essential to our treatment of the statistics of interaction of the bonds responsible for the system coherence and cohesion. It is within this framework that visco-elasticity parameters emerge and the new view of the deformation of a polymer melt results in a different definition of the entanglement network.

INTRODUCTION BACKGROUND
The mathematical treatment serves as a way to support the concepts, but the reverse is also true, the concepts of dual-phase naturally led to the search for these mathematical tools. Thus, the concepts are introduced early on, in a qualitative and intuitive way, and refined as the results emerge giving support or challenging the initial ideas. For instance, thermal diffusion in polymer melts is imaged, in our views, by a continuous coherent sweeping motion of “the phase-lines”, defining the boundaries between the dual phases, organized as a continuous network. These phase-lines are constantly in motion, with natural frequency w’o, to insure melt isotropicity and homogeneity despite the free volume difference between the dual-phases. At one stage of melt deformation, the orientation of the phase-lines occurs and creates anisotropicity which is compensated, at least partially, by an increase of the sweeping wave frequency to maintain the homogeneity of the cohesion between the interactive bonds. We describe this mechanism (and other competing ones) mathematically in this article.

We consider a new parameter, wR= w /(G'/G*)^2, where w is the radial frequency, G' is the elastic modulus and G* the amplitude of the complex modulus and study how it correlates to viscosity, suggesting that shear-thinning can be simply expressed in terms of w and (G’/G*)^2. We show that (G’/G*)^2 can be split into two terms, k1 and k2 , i.e. (G’/G*)^2 = k1 + k2 , the variation of k1 and k2 with w and temperature being fundamentally related to the mechanisms of deformation of the network of interactions (inter-and intra-molecular in nature, working coherently and defining the viscous cohesion). We show that the k2 term is related to the energy stored by the network of activated phase-lines (“entanglements”) which may lead to its entropic modification (orientation) resulting in a further increase of the sweep wave frequency, so k2 is a characteristic of the deformation mechanism occurring in the “strand-channel-phase” of the two dual-phases. By contrast, we show that the k1 term is related to the core-phase, the other dual-phase, which participates in the response to deformation by way of compensation with k2, either by diffusion (at low strain) or by a stretch-relax mechanism (at higher strain) similar to what is observed for the k2-phase when shear-thinning is active.

We define w’ as the dynamic frequency of the entanglement network, w’ = w/ k2 , and show that w’ correlates simply with the total stress generated by the flow mechanism in the shear-thinning regime at low strain. At vanishing w, w’ converges to a finite value, w’o, that we associate, as already said, with the fundamental static diffusion of the network of entanglements, i.e. with the natural sweeping wave frequency of the entanglement phase to interpenetrate the core phase, delimiting the contours of the boundaries between the dual-phases. We correlate w’o with the onset of non-Newtonian viscous flow behavior. Subtle differences of the variation of k1 and k2 emerge for various thermo-mechanical treatments of the melt or by varying temperature or the magnitude of the strain applied.

The analysis of the split of (G’/G*)^2 into k1 and k2 suggests to assign a physical dynamic attribute to the elastic entanglement network, whose deformation occurs by an activated mechanism of stretch-relax, and the need to characterize its stability under stress. We also define the elastic cohesive energy of the dynamic network, DELTAw, which varies with both frequency, w, and strain since it directly correlates with the number of activated strands of the dynamic network, k2 . We study the influence of the Talpha transition, the mechanical manifestation of Tg, which varies with w and strain ,and which we write Tg(w,strain), on the visco-elastic behavior, showing that it plays a significant role in the mechanism of shear-thinning and strain softening, and propose a way to evaluate its impact on k1 and k2. Multiple examples are given comparing k1 and k2 for linear low density polyethylene (LLDPE), polymethylmethacrylate (PMMA), polycarbonate (PC), polystyrene (PS), polyethylene terephtalate glycol (PETG) and polypropylene (PP) melts. The influence of temperature on the elasticity of the dynamic network of entanglements suggests a change of the characteristics of the elastic network in the melt above Tg, an observation already foreseen in a previous communication [1].

The effect of strain is an important section of this paper. We show that the essential role of strain is to activate the k1-phase to participate actively (by shear-thinning) in the deformation process. In linear viscoelastic conditions, the conformers[1] in the k1 dual-phase do not deform, their motion is through diffusional reorganization, i.e. delocalization in the structure triggered by the stretch-relax deformation mechanism (shear-thinning) of the k2-phase conformers. When the k1-phase is activated by an increase of the strain, strain softening occurs. In the discussion, we present a new understanding of “the network of entanglement” and show how its orientation and gradual instability gives rise to the mechanisms of deformation observed from very low w to high w, at various strains. We suggest that the network character of deformation is not due to topological considerations but, instead, due to the cooperative coupling nature of the interactions between the macromolecules conformers which organize according to a Dual-Grain Field-Statistics. In this model, the duality aspect comes twice: it comes at the local level of interactions between the conformers, and this duality is dealt with by the introduction of the Grain-Field Statistics applicable to macro-coil systems. The equations of the Grain-Field Statistics predict the dynamic aspect of the interactions between conformers. But the interaction between macro-coils introduces a second level of duality, above a certain size for the macro-coils (which we consider to be the onset of entanglements), responsible for the molecular characteristics of the dynamic network.

In summary, we introduce in this article new methods of analysis of the rheological results which appear to confirm an essential aspect of the cohesion of the interactions between the conformers and the existence of the “entanglements”, the existence of a Dual-Phase structure. The question of the stability of the network of interactions, which was an essential focus of experimental investigation in part II of this series [2] is reviewed here in terms of the Dual-Phase model.

[1] Conformers are defined in refs. 33-35. Also see Fig. 12a.

Dependence of Viscosity on Molecular Weight at constant Free Volume

I already expressed my interest in determining if the famous 3.4 exponent that characterizes the melt viscosity dependence on molecular weight (for entangled polymers) would be different when the rheological data are determined at constant free volume.

In particular, I wanted to know if de Gennes had been right in the first place, coining the exponent at 3.0 in his famous 1971 paper.

The analytical technique to analyze rheological data at constant free volume is presented in a WIZIQ lecture found at http://www.wiziq.com/NewSchoolPolymerPhysics977161

(class#2: On the incidence of Tg and Talpha on the formulation of rheolgical equations)

I have now addressed and completed a series of monodispersed polystyrene grades and the result is shown in the figure above:

THE EXPONENT IS NOT 3.0, NOR 3.4, IT IS 5.3

Besides, in the Vogel-Fulcher's expression of the temperature dependence of the friction factor, the famous T2 - for which viscosity becomes infinity- is raised from 55 oC to 123 oC, when the free volume is accounted for.

These results show how important it is to correctly describe the effect of the free volume on molecular mobility when analyzing dynamic rheological data. The mythical constants, 3.4 for the viscosity exponent, T2=Tg-52.5 for the WLF equation, are the results of cooperative contributions from free volume and conformer rotations. The influence of free volume is not separable the way it is traditionally presented: in our analysis, free volume influences both the T and M factors in the viscosity expression.

Interestingly, the temperature of 123 oC is 23 oC above the Tg of polystyrene, determined by DSC, for instance. And this temperature is precisely the temperature of compensation for this polymer for all the relaxation modes occuring below Tg. Refer to a previous blog page on "Interactive Coupling between Relaxation Modes". One knows that the coupling between the molecular motions below Tg, resulting in compensation, occurs in a very restricted free volume environment, compared to what is assumed to occur above Tg. It is, therefore, somewhat satisfying to find that the T2 obtained after removing the effect of free volume is the same as the compensation temperature found from a study of motions in the solid state.

It is also remarkable to see how the free volume is intrinsically coupled with the effect of molecular chain length above Tg: mobility is much more reduced (by a factor 100) than what one thought was only due to molecular weight alone. The influence of molecular weight is described by the exponent 5.3, an extraordinary large number. This is only because the free volume is interactively coupled with the configurational effect that we observe the 3.4 exponent. As said before, viscosity does not separate into a term that varies with T only and a term that is function of M only. This formulation is only a convenient approximate representation.

So, unfortunately, the brilliant demonstration by Prof. de Gennes to explain the restrictive mobility in polymer melts will remain, in my mind, incomplete, perhaps even "un faux pas".

The Optimist: the little man at instant t

The Optimist by Baptiste Ibar (2001)

I am looking at a painting across the room, above the fireplace, “the Optimist” it is called. It is a large painting, showing just a few elements, a black dead tree with branches, a red ladder that folds up in the air in a bizarre way, leaves that fly like butterflies and seem to pass by, disappearing towards the horizon, and a little man suspended from a rope, posing his feet at the intersection of the ladder and the tree. The whole scene is overlaid on top of a background color divide between just two colors, gold and burgundy. The flying leaves are all colored in aqua green.

It is breathtaking, mysterious, deep but simple, naïve but profound, it brings me peace and a sense of existing, as if time was passing in front of me, like fishes in an aquarium...

There are several basic concepts I relate to in this painting. The structure of the painting is this basic split between two colors, although note that these two colors are not complementary. What makes it work, it seems, is the presence of the aqua green stains which spread on boths sides, ignoring the divide, giving the appearance that the two colors are much more compatible than they actually are. The aqua green color turns out to be beautifully contrasted with each background color, gold and burgundy, giving that appearance. This may be a basic principle: get a compatibilizer, a diffuser, it will blur a divide. A divide is a split, like a discrete quantum event, either 0 or 1, not a smooth variation between 0 and 1. The leaves are also a set of discrete objects, yet their presence, as a transversal spread, smoothes out the discontinuity, renders the divide more continuous.

Perhaps this illusion also comes from the fact that the leaves are independent objects, present in the picture but without belonging, since they seem to ignore the main story of the painting, the little man, the rope, the tree and the ladder, they are parallel to the event, yet they play an essential role in the overall aspect of the painting. The size of the butterfly leaves decreases when they approach the horizon divide line, which gives them the apparence of being in motion across the scenery towards infinity. This adds to the picture a dynamic dimension, introduces time as a reference, making the event of the suspended man a space event, at instant t. Perhaps the little man on his tree is watching the passing of the butterflies and, during his time of observation, his position in space is stable.

This is the other concept that I see coming out of this painting. Consider the chances for this exceptional event consisting of this moment of coincidence where the branch of the tree, the ladder step and the feet of the little man suspended from his rope exactly meet. This event is like an interaction in space and time. The painting is a snapshot of this coincidence. The tree has many branches, filling space, increasing the chances that events of this sort will occur. Branching permits to duplicate the possibilities, expanding the area for the reaches with little expansion demands from the tree itself, a very efficient way to increase the chances to reach out (create interactions), but the search is done randomly, at irregular intervals, it symbolizes unpredictability. The ladder is like a one dimension space filling, it explores space by changing direction, the whole ladder is redirected in the new direction, this is a different approach than branching. The presence of rungs, transversal to the main direction, provides a second degree of space filling, it symbolizes regularity, periodicity, the order dimension. And our little man is there, in equilibrium between the tree, the last bar of the ladder and he holds himself from the rope above him. What a coincidence! None of the other tree branches could do that, none of the other ladder configurations, no other rungs could do that, no other position of the little man on the rope would work. The occurrence of the event has to be very sparse, almost unique. How long will it last?

This occurrence reminds me of the way nature works, it tries and tries, fails a million times, continues to try until a coincidence occurs, a node, an interaction, and the important question is its stability, the durability of the interaction. Matter emerges from that mechanism, which I call chrono-condensation. Structuring can occur at different scales, depending on the stability of past solutions, increasing the inertia to create new coupled interactions. The ladder of solutions interlocks like a chinese lantern,from small to large, from 1 small to 1 large, yet still through the same mechanism, repeated over and over again. We are part of those interlocked cycles, we are part of why it stands. We are hanging to our rope, just for the time to let the butterfly leaves pass by.

All of these ideas emerge from this beautiful painting. I assure you, I see all these things: physics is everywhere!

This is what the artist said about his painting:
“I decided to call it "The optimist" after thinking about our talk and how all of the elements and events meet up at a certain point in space/time.... the optimist believes in these events happening perfectly in the present moment. Therefore feeling fearless of the future and free from the past.”

Here is a portrait of the artist in his studio in Brooklyn, NY

On the Incidence of Tg and Talpha on the Formulation of Rheological Equations

One often uses Frequency sweeps at constant temperature to analyse the rheological characteristics of polymer melts. But, I suggest, it would be preferable to popularize Temperature sweeps at constant frequency w instead.

Why?

Because frequency plays two roles and it would be easier to deconvolute their respective influence:

- Frequency increases the value of Tg (or Talpha) and therefore decreases free volume when T remains constant; this increases viscosity.

- Frequency plays a significant shear-thinning role in reducing viscosity.

By working at (T-Tg(w)) constant one can study how the rheological parameters vary at constant free volume. There are surprising and interesting results from such an analysis, but it requires to correct the data obtained at constant T and to know the variation of Tg(w), i.e. the frequency map for the Talpha relaxation.

By working at constant w, Tg(w) is fixed, and one can obtain shear-thinning results by simply sweeping the temperature (cooling or heating) at a given rate.

In the video-lecture available from the following link, we study the incidence of working at constant free volume on the classical relationships found in rheology, such as the Maxwell fits at low w of G'(w) and G"(w), or the evaluation of the terminal time from the maximum of G'/w vs logw etc.


http://www.eknetcampus.com/videopresentation/On the incidence of Tg and Talpha on the formulation of rheological equations2.avi


This work suggests the importance of incorporating the influence of several parameters: w Frequency, T temperature, P Pressure, M molecular weight, L lambda, the elongational stretch ratio, on the value of Tg, i.e.

Talpha (w,T,P,M,lamda)

A New Understanding of Entanglement -The Quizz

I launched a series of 12 classes on January 13th 2011 on the WIZIQ teaching platform.

Here is the link: http://www.wiziq.com/online-class/430878-the-need-for-a-new-understanding-of-entanglement-in-polymer-physics

The video recording of the class is available to people who access the above link. I post (below) another direct link to view the class, without the need to register with WIZIQ. It appears, though, that viewers need to be online in order to view the recording, saved or not, meaning that if the file is saved on a computer, it still needs to be watched on-line!

http://www.eknetcampus.com/WIZIQ/Class1_video_recording/634305601542868750.exe

In any case, I like to communicate, before, during, and after the class, and I have enjoyed the many exchanges and discussions by email.

I preceeded the announcement of the class by a quizz, THE ENTANGLEMENT QUIZZ, with 22 questions (http://www.wiziq.com/online-tests/22446-the-entanglement-quizz).

I had 56 students who tried the test, with an average of 8 out of 22 good answers. Of course, "good answer" depends where one stands with respect to the entanglement concept. What it really shows is how different I stand on the topic.

Like Boris Vian once said (about something else): "My views are right since I made them up".

I have posted a pdf file with my answers and comments to the entanglement quizz:

http://www.eknetcampus.com/WIZIQ/ANSWERS%20to%20the%20Entanglement%20Quizz.pdf

The main idea to remember from my first class is that "entanglement" manifests the reality of the physics of dual-phases and how it changes scales with the number of interactive units involved. It is a "slow motion" example of how energy of interaction and population size interact to define the scale of a statistics. It's the chance of a lifetime for statisticians, who normally deal with very fast relaxation processes, therefore solutions for steady state problems. With polymers, motions are considerably slower, giving the opportunity to conceive and adapt simpler models to describe "scaling" , hence to illustrate complex mathematical abstract concepts by using common language: Ah... writing a Renormalization Group Theory for Dummies... using polymer entanglements!

It is true that de Gennes initially thought of that idea, but he stayed too far from Prigogine to set up his initial statistical frame, and got stuck with single chain dynamics in steady state.

Here is a picture of entangled dual-phases. The duality is at two levels for M > Me. This is why there is a critical molecular weight. Below it, there is only one duality, the one responsible for the compensation phenomena below Tg and for the Boson peak observed for glasses.




My next class is scheduled for February 17th, 10:30 am Eatern US standard Time.

check it out: http://www.wiziq.com/online-class/454992-molecular-weight-and-frequency-dependence-of-tg-on-rheological-data

The Great Myths Part II: Abstract, Summary, Conclusions

Drawing by Baptiste Ibar (2009)

Some of the students have asked me to put a link to the Abstract, a short Summary of the paper and the Conclusions. Here it is:

ABSTRACT
https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B1EGViD3w4U0NzU5NTQ2NDEtODQzNS00NTk4LTkzOTItMDU0NTI0ZWM5ZjVm&hl=en

SUMMARY

To summarize some of the findings and thoughts expressed in this paper:

- Transients and steady states must be described by a unique theory of the deformation of interactive conformers. We suggest it is necessary to understand non-linear effects first and to have linear viscoelasticity derived by extrapolation to infinite time. In other words, time, frequency (or strain rate) and strain should be involved in the mathematical description of the deformation process (in the quantification of the moduli).

- A melt can be brought out of equilibrium with respect to its entanglement state. The return to equilibrium explains the transient properties. New entanglement states can be made quasi-stable, even at high temperature in the melt, by coupling entropic and enthalpic effects produced under specific conditions of melt processing.

- The currently accepted descriptions of rheology only apply to a stable entanglement state, which is not general enough. For instance, the WLF-Carreau equation of viscosity-strain rate does not correctly describe the rheology of an unstable entanglement network. The modelization of the influence of a network of entanglement on the melt deformation mechanism in terms of parameters introduced in linear viscoelasticity (tauo, GoN, Me) provides the wrong answers when the entanglement network has become transient.

- The influence of strain on the rheological equations is currently not addressing the issue of its influence on the stability of the network of entanglement, and therefore is incomplete.

- The interpretation of the phase angle between stress and strain in terms of a dissipative and an elastic component represents an over-simplification of the mechanism of deformation which, we believe, mischaracterizes the relative influence of a network of strands on the elasticity / relaxation process versus the influence of the local bond orientation (the conformer statistics). The difference between the two permits to define the amount of interactive coupling reorganization due to entropic vs enthalpic drives and under what conditions of strain rate and strain they occur. An entropic driven coupling mechanism of deformation can be viewed as an activation, then orientation process of the active network of strands. We have made the suggestion, in this paper, that the active number of system strands (defining the EKNET network) is proportional to (G'/G*)^2. In fact, the active number of strands is not exactly proportional to (G'/G*)^2 but can be calculated from (G’/G*)^2 , and is almost exactly equal to (G’/G*)^2 shifted by a constant when its value is approximately less than 80% of the maximum of (G'/G*)2 .The enthalpic contribution starts beyond that.point and corresponds to the orientation of the network. We suggest that only certain compensations of enthalpic and entropic contributions result in stable “sustained oriented entanglement states”. This set of conditions would be the equivalent of “plastic yielding” and implies highly anisotropic samples.

- An increase or decrease of G* (t) and thus of viscosity can be produced when the network of strands is unchanged (Figs 1a to d of the paper) and local orientation/relaxation is responsible for the transient behavior, and the relaxation times relate to the properties of this network. In order to obtain a modification of the network, one needs to add energy to it until it yields. Strain rate or frequency are capable of reaching that point for any strain % deformation, but the value of the strain % allows to decrease the frequency or strain rate at which the network starts to deform.

CONCLUSION

The deformation of a polymer melt in shear mode is the main subject of interest in the science of rheology of such materials. It is a crucial topic for successfully processing these materials. As illustrated in part I of this series and in the above examples, it is a complex and rich subject which is far from being fully understood.

In part I of this series, we suggested that even the linear visco-elastic behavior of polymer melts (at low strain rate and low strain) was not satisfactorily described by the accepted theoretical models, when carefully comparing experiments and theoretical predictions. In the non-linear range, at high strain rate and strain, the subject of this part II, it is generally admitted that the current theoretical developments that successfully predict the main characteristics of polymer melts in the linear range come short but merely need improvements. The improvements proposed generally consist in tweaking certain assumptions of the linear viscoelastic model to be able to extrapolate to the non-linear behavior. There is no current theoretical challenge to the dominant reptation model of melt deformation in polymer physics. The aura this model has reached among polymer scientists makes it more difficult to search for other explanations for visco-elasticity and rubber elasticity. Yet, as we suggest, it is possible that the experiments described in this paper challenge the reptation school to its limits, to the edge of usefulness.

As already concluded in part I of this series of papers dedicated to flow, the theory seems to be fine in the linear range in appearance only. The “devil is in the details” says the old saying. The present understanding of the physics of macromolecules is based on an analysis of the properties of a single chain. The presence of the other chains is perceived as a mean field influence on the properties of that chain. The reptation school considers that this mean-field can be described as a topology, an homogeneous field of obstacles restricting the motion of the single chain and explaining the molecular weight dependence of viscosity. The mobility is constrained within an imaginary tube and the chain “reptates” within that tube. The shortcomings of the predictions of that model made the initial static tube evolve into a more dynamic tube, capable of evolution, in time and as a consequence of the various modes of deformation of the melt. The tube was therefore thought to have a stability of its own, it could fluctuate in length, and, to address some of the non-linear issues, it could get thinner and elongate in length. In other words, the tube itself had evolved into a “super macromolecule” capable of deformation very similar to what early polymer scientists would assign to macromolecular chains themselves. Perhaps, at the horizon of the reptation school, also lies the concept of entanglement of the tubes themselves!. We are not suggesting this idea totally ironically, because it illustrates another concept that we will develop in a follow up article of this series, that of the need to not only define the scale of the basic unit that participates in the deformation process, but also to determine the link and the modulation between cooperative scales.

In explaining several figures of this paper, we made reference to a “network of strands” to describe the cooperative interactive process resulting from the macroscopic deformation. We obviously referred to a basic unit of deformation that involved the cooperative motion of a group of bonds responding as a set. We must define what cooperation means, how many bonds cooperate in an active strand and where they are located, on a single chain or on several chains?. The physics of dealing with all the chains at once is the model that we have adopted to describe the deformation of polymer melts and solids, above Tg and below Tg. The theory not only addresses the interaction between the conformers of a single chain to assume the shape of a macro-coil (which can be deformed), but also defines why entangled macro-coils exhibit the response of a network of active strands when all the chains participate cooperatively in the deformation process. The link between the deformation of a conformer, of a macro-coil and of a network of strands must be fully described.