If you are attending a class in rheology of polymers you will learn that for a given set of pressure, temperature and strain rate of shear deformation, the melt viscosity is known and calculable by formulas with tabulated parameters.
OK, simple enough.
The problem in that statement is that it is simply NOT TRUE: the storm!
This is due to the “elastic dissipative” nature of polymer melts: the rainbow!
The correct statement could read like this: for a given set of pressure, temperature and strain rate, one cannot define a unique viscosity!!!
Let’s qualify this answer:
- the viscosity is function of the stability of the state of interactions between the macromolecules , itself function of the thermal-mechanical history of the melt.
- the viscosity can remain apparently stable for long times (say 200,000 to 1 million times longer than its longest relaxation time at the corresponding T,P), yet applying even a small amount of shear energy can destabilize the melt which, upon release of that energy will remain apparently stable at that new viscosity .
Some examples
Graph 1
In this Fig. a PC melt (Makrolon 2608) is studied under N2 in a dynamic rheometer, submitted to a time sweep at constant frequency of 1 Hz and a small strain amplitude of 1%. T=300 oC. The sample was vacuum dried before the test. Its initial measured viscosity is 450 Pa-s. This viscosity is in the Newtonian range, at this high T, low w and low strain. It is stable for 17 minutes.
No one that I know would wait 17 min to check that the Newtonian viscosity of a melt is stable. Or perhaps those who study the thermal degradation process, or the effect of additives on the chemical or structural stability.
OK, these are true concerns.
BTW you might be wondering about the small fluctuation of the viscosity signal in the graph:??? Please ignore this phenomenon for the moment, which will take your focus away from the current purpose (it is due to a small T sinusoidal fluctuation which was added to understand some features of the elastic dissipative state: this will be reported another time ).
After about 17 min the complex viscosity starts to increase and after approximately 2 hours its magnitude has almost doubled: from 450 Pa-s it has reached 825 Pa-s. Note, the viscosity increases, not decreases, which eliminates all kinds of explanations based on the chemical degradation of the molecular weight.
Now for the possibility that an increase of molecular weight would be responsible for this increase of viscosity, we checked the Mw by GPC and nothing had changed (within the accuracy of this method).
I discuss other possible explanations (migration of small Mw fractions, esterification) in a paper published recently:
“Investigation of the Dynamic Rheological Properties of a Polycarbonate Melt Presenting Solid-Like Characteristics and a Departure from Pure Liquid Newtonian Behavior at Long Relaxation Times”, J. P. Ibar, Z. Zhang, Z. M. Li, and A. Santamaria, Journal of Macromolecular Science, Part B: Physics, 54 (6): 649-710, 2015.
The paper is downloadable here.
However, as explored in the paper, none of these possible explanations survive all the other experimental tests designed to understand the meaning of the viscosity changes.
Let us consider another example, Graph 2 (Fig.1 from the publication above):
Graph 2
Dynamic viscosity vs. w for Makrolon 2207 at T = 255 oC. The strain is 5%, the gap sub-millimetric. The viscosity decreases as w increases according to two mechanisms. The low w viscosity tail (w < 0.6 rad/s) corresponds to solid-like characteristics of the melt. Then we observe the classical Newtonian viscosity plateau (0.6 < w< 3.0), also depicted with the dotted line, and finally, for w > 3 rad/s. the shear-thinning mechanism observed can be fitted with the classical formula by Carreau, for instance.
This shear-thinning behavior at larger w is a classic, emerging from the Newtonian state, which is well established for w values below w ~ 3 rad/s. “Normally”, for a stable Newtonian state, the value of the Newtonian viscosity at that temperature of 255 oC would correspond to the points on the dashed line at low frequencies, down to 0.01 rad/s. What is unusual is the departure of the viscosity at low w, which we call “the low w tail”. For instance if we run a downsweep experiment (w varies from high to low), we observe that the Newtonian value starts to rise at each new measurement along the way, as frequency decreases. The final viscosity at the end of the frequency downsweep is about 3 times the value of the Newtonian viscosity.
Effect of strain%: if we increase the strain from 5% to 35% the lower viscosity tail decreases in magnitude to become almost flat.
Two observations; two conclusions:
-1st: the time lapsed to do the measurements from w=1 rad/s down to w=0.01 rad/s is very long: 6113 sec, i.e. 102 min (out of a total time of 105 min for the overall test): this means that the melt is annealed for 102 min while it is in the Newtonian state, a situation very similar to what is presented in the 1st graph of this blog and with the same results (an increase of the Newtonian viscosity).
-2nd: the test was immediately followed by a frequency upsweep using the same parameters (10 points per decade, same T). We also run the frequency upsweep first, followed by the downsweep: same results: the reversibility of the results obtained for upsweeps and downsweeps eliminates all explanations based on artifacts due to the type of instrument used, chain segregation, chain growth and mutation by polycondensation and chain degradation.
[47] Noirez et al. solid-like results : Download their paper: Here
The increase of viscosity at vanishing frequency w seen in Graph 2 has the same solid-like appearance as that mentioned by Noirez et al.[47] for very narrow gaps for PBD and polybutylacrylate. This is shown in Graph 3.
Graph 3
Graph 3 is a plot of h*(w) vs w for the polybutylacrylate M=47,500 data used by Noirez et al [47]. The gap is 25 m and the temperature 25 oC , i.e. 89 o above the Tg of this polymer (Note: the data of Noirez et al. were kindly provided by these authors to permit this new analysis of their Fig.4). The curve shown corresponds to a frequency sweep done at 30% strain, in the linear visco-elastic region. Noirez et al. also varied the strain, from 0.2% to 200% to show that increasing strain decreases the magnitude of the low viscosity tail: see Graph 4 below (note: the original data by Noirez et al. include lower strain % values, down to 0.2%, not shown because they show some scatter).
Graph 4
The axes are linear in Graphs 3 and 4, suggesting a hyperbolic variation of h*(w), converging towards the Newtonian viscosity h*o (dashed line). It is clear that increasing strain reduces the amplitude of the viscosity tail observed, an experimental fact also observed for Makrolon 2207, as mentioned above. The most obvious correspondence between the solid-like data of Noirez et al. (PBuA) and the results of Graph 2 on PC is best demonstrated by using a log-log plot for the 200% strain data (bottom curve of Graph 4):
Graph 5
In this plot, we observe the same features as in Graph 2: the low w tail below w~1 and the change of curvature to initiate the classical shear-thinning at w>3 rad/s. Clearly, one could curvefit the 10 data points located beyond w~10 by a Carreau’s equation leading to the dashed line for the Newtonian viscosity.
To understand what phenomenon creates this instability of the liquid Newtonian state, it is interesting to consider how (G’/G*)2 varies with w. Graph 6 is such a plot for the data of Graph 5, i.e for the 200% strain. Graph 7 is the same plot for all the strains corresponding to Graph 4.
Graph 6
This graph shows that the stored elasticity in the melt is at its maximum at low w (thus the solid-like character of the melt), that it decreases as w increases (the red curve), reaches a minimum and then increases (the blue curve). The overall variation of c=(G’/G*)2 can be fitted by the sum of two functions, c1 and c2, drawn as the red and the blue line, respectively, in Graph 6:
c=c1+c2
with c1=p1 (1-tanh(p2 Logw+p3)) and
with c1=p1 (1-tanh(p2 Logw+p3)) and
c2= 0.5 (1+tanh(p5 Logw+p6))
with p1,p2,…,p6 fitting parameters.
It really does not matter (for the present discussion) how the expressions of c1(w) and c2(w) look like, what is important is that c1 decreases sigmoidally from a maximum which decreases as strain increases (see Graph 7), and that c2 increases sigmoidally itself (although we only observe the lower portion of the sigmoid as the blue curve).
Graph 7
The increase of c2 with log w is a characteristic of the blinking mechanism describing shear-thinning in the Dual-Phase model of polymers rheology, which other models fit with the Carreau’s equation. This corresponds to the high w end viscosity decline in Graphs 2 and 5. It is a network property. The deformation occurs by a sequential stretch/relax mechanism involving parts of the network channel characterizing the cross-dual-phases (the entanglement phase-line).
The variation of c1 has a different origin. It has two characteristics: it is a function of the b-grain population of the melt due to the [b/F <-->(c,g,t)] statistics, but it is also the Newtonian state for the blinking mechanism, corresponding to the relaxed stage of the stretch/relax sequence. The population of b-grains (the b-conformers) determines the degree of “glassy-like” character of the melt. By “glassy” I don’t mean to qualify the kinetics, which, in a melt, are orders of magnitude faster than in a glass, I relate to the population of b-grains that dominates to a very large extent in a glass (95% or more). Graph 7 shows that the low frequencies and low strains induce the highest elasticity in the melt, the highest value of c1, c1max=2 p1= 0.975 (extrapolated to w=0 and strain=0), hence the highest concentrations of b-grains. But the thermal/mechanical stability of these b-grains is totally different below Tg and in a melt at Tg+89 oC.
For the PBuA sample of Noirez et al. which I re-analyze here, the magnitude of c1 and the separation of c1 and c2 are remarkable; it is due to the small gap used (25 m) and we offer an explanation why below. At larger gaps, say > 1mm, or at large strain, c1 does not take on such large values, as will be explained below, yet the effect of the shear deformation on the b/F structuring, although low, is still measurable even for classical 2mm thick samples but one needs to look for it and extract it from the value of c (see “The Great Myths of Polymer Rheology. Part III: Elasticity of the network of entanglement”, J. Macromol. Sci., Part B, Phys. 52:222-308, 2013).
Download this paper here
The effect of strain on the fitting parameters describing c1(w), c2(w) can be quantified to tell us what are the favorable conditions to observe such solid-state behavior in the melt, and what to expect for c1 when the gap is not micrometric.
Graph 8
The maximum value of c1 , corresponding to w=0, is equal to 2p1 which is shown to decrease with g
Graph 9
Graph 10
This graph shows that the log w for the inflection point of the sigmoid of c1(w), i.e. (-p3/p2), decreases with strain. At high strain, c1(w) drops to 0 for very low w values. This is why the melt looks liquid-like.
Graphs 8 -10 show how p1, p3 and (–p3/p2) vary with strain: one can use very simple fitting functions to express these variations quantitatively. For instance, the variation of p1 in the expression of c1, extrapolated to large strains, tells us that p1 non only becomes very small but also negative ( -0.086). We observed such negative values for c1 for the melts we studied in the paper quoted above (The Great Myths of Rheology, part III) which used classical gaps and classical surface for the plates. But, as we already pointed out, this is the same phenomenon: the proof of the existence of the b/F duality.
Shear deformation decreases c1 because the entropy defined by the fluctuations of the phase-line channel is compensating for the thermal agitation of the b/F conformers, creating a fluctuating standing wave propagating constantly through the medium to homogenize (average out) the differences between the phases. This is the equivalent of an elastic wave, which I call ‘phase-wave’, sweeping through the entire network of conformers in interactions. In the dual-phase model, this ‘sweeping’ mode of deformation of a melt is not simply due to thermal energy (kT) but also to the mechanical energy input. This is the mode of deformation at low wg, occurring for a melt ‘at rest’, i.e. in Newtonian conditions. w’o is the frequency for the sweeping wave. Graphs 2-6 show that w’o is not simply function of T, but also depends on wg, external parameters: this is a very different view than pure thermodynamic equilibrium!
As the glassification of the melt occurs (F-->b), favored by anything which permits the conformers to reach the b-conformer state- which is of less energy thus more stable- the b-grain population increases, the diffusional sweeping phase-wave requires more energy to diffuse through the melt since there are more b-grains to carry through, which means, in rheological terms, that viscosity increases as the concentration of b-grains increases (also equivalent to a decrease of free volume).
This is a well-known result, that the Newtonian viscosity can be described in terms of free volume (Doolittle, Ferry). In other words, the momentum of the sweeping phase-wave, its density times its speed of propagation, is expected to vary with the local density of the melt, i.e. with the number of b-grains/free volume, and this is the reason we observe the increase of viscosity in Graph 1 (for a melt initially out of equilibrium), or the instability of the Newtonian state as w decreases in Graphs 2 and 3.
One can express the variation of the free volume in Graph 3 by making the T2 of the Vogel-Fulcher’s formulation of the Newtonian viscosity VARIABLE with w:
Graph 11
T2(w) is the value which makes h*(w)= h*o calculated by the Vogel-Fulcher. T2c is the classical value.As w increases, b-->F, and the system returns to its T2c value which remains constant during blinking.
In simple terms, the decrease of c1, due to b--> F, is equivalent to an increase of free-volume, which can be interpreted as the lowering of T2 in the Vogel-Fulcher’s equation of the Newtonian viscosity, or an increase of w’o, the phase-wave sweeping frequency. As soon as blinking starts to operate and c2 controls the viscosity variations, T2 remains constant.
Of course, one could also express the free volume variation due to the b/F structuring in terms of Tg instead of T2. This is expressed by the WLF equation:
(C2g ~ 50 oC according to the WLF equation)
This is what I mean by “melt glassification”, the increase of the b-grain population makes the melt more glassy-like, a melt for which the fluctuation of local density favors the clusters of b-conformers, either increases their number or their size, and this translates into an increase of Tg and log hog , not just superficially, at the interface between the plates and the melt, but in the bulk.
Graph 12
The lower level is the b-conformer level, the same for all b-conformers regardless of their cis-gauche-trans conformation and whether they belong to the cross-dual-phase e or c (the blue and black colors). The F-levels are differentiable according to the conformer’s rotational isomeric energies. What differentiates the e and c states is their population statistics (in the F and b levels) which we symbolize by [b/F ß>(c,g,t)]e ↔ [b/F ß>(c,g,t)]c . The fact that this overall statistics remains stable is due to the elastic dissipative nature of the interactions. The elastic dissipative phase-wave propagates through the melt to homogenize in time the population statistics differences. For an un-entangled melt, the blue lines in the graph are not existent. For a melt which can crystallize, the tF level of the F-state can stabilize into another level, st-t, which is more stable than the b-level, and which gives rise to nuclei.
Why is narrowing the gap makes c1 increase (making the melt more “glassy-like”)?
Noirez et al. report that working at thinner gaps favors the appearance of the solid-like character (increases c1 in Graph 7), and that, conversely, increasing the gap makes the melt go back to its classical behavior of a pure liquid (in the Newtonian regime). This observation made many scientists believe that Noirez et al. only observed surface effects, probably due to capillary forces, in any case nothing which could justify the claim by these authors that the bulk of the melt exhibited solid-like characteristics (see my previous blog #23).
As I said, the phenomenon of b-grain glassification in the molten state is a property of the bulk, but it is expected to be influenced by anything which favors the creation of b-conformers, such as the influence of melt confinement which is known to affect the Tg near the surface. But is this the reason for the Newtonian viscosity increase seen in Graphs 2-5?
No, not according to the dual-phase view of the Newtonian state as a standing sweeping elastic wave propagating through the medium to keep it homogeneous.
Strain is defined as the product of a geometrical factor (R/e) and of the angle of oscillation, q. If the gap e decreases, the strain increases, so to keep in line with the type of strain which produces linear visco-elasticity, one can work at lower oscillation amplitude, which decreases the velocity of deformation, thus of the momentum of the standing elastic dissipative wave. The need to homogenize the melt requires an increase of the density to balance the loss of velocity, and this is the reason for the increase of b-grains. The increase of strain at constant gap is done by increasing q, which increases the momentum of the elastic dissipative phase-wave and, to compensate, the b-grain population now starts to melt out.
In other words, I suggest that the gap effect is the same as the strain effect or the increase of w effect (which also decreases c1), and this is actually what we observe.
These observations are actual illustrations of what an elastic dissipative melt is like (Graph 12).
Now, is there a surface effect in the experiments reported by Noirez et al?
I suggest that there is one, and it is observed for the 3 lowest w values in Graph 7. What is seen for these 3 points is a reversal of the b-grain formation (c1 ↘) as w decreases, as if the stored elasticity now declined (and-although more difficult to assess- as if the viscosity had started to nose down). But this makes sense. Noirez told me that she used special low surface energy materials for the plates so that the melt could perfectly wet the surface: this situation creates the opposite of a high surface energy surface which raises Tg (with respect to the bulk Tg value).
The paper that studies the effect of the surface energy of surfaces and the thickness on the Tg of ultrathin polymer films is by David S. Fryer, Richard D. Peters, Eui Jun Kim, Jeanne E. Tomaszewski, Juan J. de Pablo, Paul F. Nealey, Chris C. White and Wen-li Wu, Macromolecules 2001, 34, 5627-5634 “Dependence of the Glass Transition Temperature of Polymer Films on Interfacial Energy and Thickness”. This is what the Abstract says:
The glass transition temperatures (Tg’s) of ultrathin films (thickness 80-18 nm) of
polystyrene (PS) and poly(methyl methacrylate) (PMMA) were measured on surfaces with interfacial energies (gSL) ranging from 0.50 to 6.48 mJ/m2. The surfaces consisted of self-assembled films of octadecyltrichlorosilane (OTS) that were exposed to X-rays in the presence of air. Exposure to X-ray radiation systematically modified the OTS by incorporating oxygen-containing groups on the surface. The interfacial energy for PS and PMMA on the OTS surface was quantified as a function of X-ray dose using the Fowkes-van Oss-Chaudhury-Good model of surface tension. The Tg values of the films were characterized by three complementary techniques: local thermal analysis, ellipsometry, and X-ray reflectivity. Within the resolution of the techniques, the results were in agreement. At low values of gSL, the Tg values of PS and PMMA films were below the respective bulk values of the polymers. At high values of gSL, the Tg values of PS and PMMA films were higher than the bulk values and increased monotonically with increasing gSL. The deviation of the Tg values of the films compared to the bulk values
increased with decreasing film thickness. For a specific film thickness of PS and PMMA, the difference between the Tg of the film and Tg of the bulk polymer (DTg = (Tg film – Tg bulk) scaled linearly with gSL irrespective of the chemistry of the polymer.
According to this study, although it concerns ultrathin samples, the melt surface interface could play a role on the Tg of the layers located just below the surface, which I would explain in terms of an increase or decrease of the b/F population, with respect to the bulk value. The paper gives examples of a rise or a decrease of Tg by ±30 oC, a substantial amount, indeed, the thinner the sample the more the DTg varies. So, it is possible that the surface effect plays a small role in the samples by Noirez, with a decrease of Tg effect for the first 3 w, as we suggested, which could explain the systematic drop of c1 observed for almost all strains in Graph 7 for the first 3 w.
Even if the above explanation can be validated, the surface energy effect would remain very small on thicker samples, since 25m is a thick sample compared to the 18-80 nanometer thickness of the films studied by these authors. This is why, perhaps, it can only be perceived at the lowest w values for which the sweeping wave has its lowest momentum, being over-ridden by the opposite effect (b-->F) at higher w. In summary, the solid-like character observed is not a surface effect, it is an elastic dissipative effect in the bulk.
The question of the nature of the elasticity in the solid-like melt.
This question raises, perhaps, one issue where my explanation differs from that of Noirez et al.
These authors suggest that the melt solid-like elasticity is that of the rubbery plateau elasticity. I propose a different interpretation. Take the extrapolation in Graph 11 of (T2-T2c) vs w for w=0: it is 50 oC, i.e the value of C2g in the WLF equation. This means that T2=Tg i.e. the viscosity is infinity at Tg. I conclude that the extrapolated elastic state of the melt at w=0 is that of the glass, not the rubbery state.
There is another reason why I believe the melt solid-like elasticity is fundamentally not related to the modulus of the rubbery plateau: rubber elasticity is characterized by the entropic deformation of the entanglement network, which is controlled by c2, unlike the increase of the elastic energy at low w (Graph 7) which is controlled by c1. c2 verifies the linearity of a MXPLOT, c1 does not. This is shown in Graph 13 below applied to the 200% strain data for which c2 is not negligible.
Graph 13
An MXPLOT establishes a relationship between the stretched strands, Ds c, and the relaxing strands, (1-c)Do, for a blinking mechanism of deformation, by studying the variation of the cohesive energy of interactions under shear, Dw against the number of activated strands for a given (w,T), c. One can find Dw from the variation of h*(w). See The Great Myths of Rheology, part III referenced above. (Do-Ds) relates to the average isomeric state of the F-conformers; it is positive for a stretched system because the trans conformation, tF, is more stable than the cis and gauche conformations, cg1, cgi (Graph 12), and because the stretching stage of blinking involves an increase of the trans conformers in the direction of flow. This is, indeed, observed for line 2 in Graph 13 (the red arrow) for which we determine Do= 712.40 and Ds=593.36 (thus (Do-Ds)=119 >0), but it is not true for the upper line in Graph 13 (the cyan arrow) with a slope higher than Do and Ds. c decreases in region 1, at low w, because c1 decreases towards 0, whereas c increases in region 2, for the highest w values, because c2 increases in this region (see Graph 6). We conclude that region 1 corresponds to the b/F structuring within the sweeping elastic dissipative wave characteristic of the Newtonian state, and that region 2 corresponds to the stretch/relax mechanism of the blinking process, characteristics of the classical shear-thinning.
I show in the Great Myths of Rheology, part III that as w continues to increase in region 2, c2 increases at first then decreases, which I associate with the orientation of the cross-dual-phase network for an entangled (M>Me) melt. This network entropy effect may be coupled with a change of the b/F structure which modifies c1, but it is a compensating effect and has nothing to do with the low w instability of the sweeping (diffusion) wave.
CONCLUSION
The solid-like character of polymeric melts is due to the elastic dissipative nature of the interactions which favors the b-state (Graph 12). The Grain-Field Statistics applied to all the conformers belonging to all the macromolecules provides the population of the [b/Fß-->(c,g,t)]e and [b/Fß-->(c,g,t)]c. When the mechanical energy input produced by the deformation is very low, which is the condition set by working with a thin gap in linear viscoelastic mode, b-grain glassification of the Newtonian melt produces the increase of the viscosity and of the elasticity observed. This is predicted by the Cross-Dual-Phase model of polymer interactions which validates the experimental findings of Noirez et al.
In several of my previous blog posts, I insisted on the problems encountered in polymer physics by using such provocative titles as: “the great myths of rheology”, “trouble with polymer physics” etc. OK, this was my finger pointing to the storm, with thunder and dark clouds depicted in the first image of this blog post.
But now, I am focusing on teaching a different view, a different rheology to explain classical and unclassical experimental facts. I have now recorded (and will make available) about 100 hours of video lectures (which I call VCL) that details the new polymer physics, the emergence of a new understanding of the interactions. Is this the emergence of the rainbow chasing the storm away in my first image?
In any case, expect to see in my coming lectures and publications an extensive investigation of this “elastic dissipative” character of the interactions to explain in a different way than the classical interpretations the properties of polymeric glasses, melts and rubbers.
