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The Great Myths of Rheology Part III: Elasticity of the Network of Entanglements

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Journal of Macromolecular Science
R
, Part B: Physics
, 52:222–308, 2013Copyright
©
Taylor & Francis Group, LLCISSN: 0022-2348 print / 1525-609X onlineDOI: 10.1080/00222348.2012.693338
The Great Myths of Rheology Part III:Elasticity of the Network of Entanglements
J. P. IBAR
IPREM-EPCP, Universit`e de Pau et Pays de l’Adour, Pau, FrancePOLYMAT Institute, The University of the Basque Country, Donostia SanSebastian, Spain
The dynamic data of polymer melts are analyzed in a novel way, presenting new corre-lations between the viscosity, G
and G
(the elastic and loss moduli), and strain rateand the implications of the new formulas on our understanding of melt entanglement network elasticity are discussed. In the two previous articles of this series, Part I and Part II, we showed that the existing models valid in the linear viscoelastic deformationrange were not adequate to extrapolate to the nonlinear regime, suggesting that thestability of the network of entanglements was at the center of the discrepancies. Inthis 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 inter-action, 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 viscoelasticity parameters emerge and the new view of the deformation of a polymer melt results in adifferent deﬁnition of the entanglement network.
Keywords
cross dual-phase model, entanglement, grain-ﬁeld-statistics, new polymerphysics, visco-elasticity theory
Introductory Background
The mathematical treatment serves as a way to support the concepts, but the reverse is alsotrue, 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 reﬁnedas the results emerge giving support or challenging the initial ideas. For instance, thermaldiffusion in polymer melts is imaged, in our views, by a continuous coherent sweepingmotion of “the phase-lines,” deﬁning the boundaries between the dual phases, organized asa continuous network. These phase-lines are constantly in motion, with natural frequency
ω
0
, to insure melt isotropicity and homogeneity despite the free volume difference betweenthe dual phases. At one stage of melt deformation, the orientation of the phase-linesoccurs and creates anisotropicity which is compensated, at least partially, by an increaseof the sweeping wave frequency to maintain the homogeneity of the cohesion between theinteractive bonds. We describe this mechanism (and other competing ones) mathematicallyin this article.
Received 11 November 2011; accepted 12 April 2012.Address correspondence to J. P. Ibar, POLYMAT Institute, University of the Basque Country(UPV/EHU), Av. de Tolosa, 72 Donostia 20018, Spain. E-mail: jpibar@alum.mit.edu
222
The Great Myths of Rheology Part III 223
We consider a new parameter,
ω
R
=
ω
/(
G
/
G
∗
)
2
, where
ω
is the radial frequency,
G
is the elastic modulus and
G
∗
the amplitude of the complex modulus and study how itcorrelates to viscosity, suggesting that shear-thinning can be simply expressed in terms of
ω
and(
G
/
G
∗
)
2
.Weshowthat(
G
/
G
∗
)
2
canbesplitintotwoterms,
κ
1
and
κ
2
,i.e.,(
G
/
G
∗
)
2
=
κ
1
+
κ
2
, the variation of
κ
1
and
κ
2
with
ω
and temperature being fundamentally related tothe mechanisms of deformation of the network of interactions (inter- and intramolecular innature, working coherently and deﬁning the viscous cohesion). We show that the
κ
2
termis 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 thesweepwavefrequency,so
κ
2
isacharacteristicofthedeformationmechanismoccurringin the “strand-channel-phase” of the two dual phases. In contrast, we show that the
κ
1
termis related to the core phase, the other dual phase, which participates in the response todeformation by way of compensation with
κ
2
, either by diffusion (at low strain) or by astretch–relax mechanism (at higher strain) similar to what is observed for the
κ
2
-phasewhen shear-thinning is active.We deﬁne
ω
as the dynamic frequency of the entanglement network,
ω
=
ω
/
κ
2
, andshow that
ω
correlates simply with the total stress generated by the ﬂow mechanism inthe shear-thinning regime at low strain. At vanishing
ω
,
ω
converges to a ﬁnite value,
ω
0
,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 phaseto interpenetrate the core phase, delimiting the contours of the boundaries between the dualphases. We correlate
ω
0
with the onset of non-Newtonian viscous ﬂow behavior. Subtledifferences of the variation of
κ
1
and
κ
2
emerge for various thermo-mechanical treatmentsof the melt or by varying temperature or the magnitude of the strain applied.The analysis of the split of (
G
/
G
∗
)
2
into
κ
1
and
κ
2
suggests to assign a physicaldynamic attribute to the elastic entanglement network, whose deformation occurs by anactivated mechanism of stretch–relax, and the need to characterize its stability under stress.We also deﬁne the elastic cohesive energy of the dynamic network,
ω
,
γ
, which varieswith both frequency,
ω
, and strain,
γ
, because it directly correlates with the number of ac-tivated strands of the dynamic network,
κ
2
. We study the inﬂuence of the
T
α
transition, themechanical manifestation of
T
g
, which varies with
ω
and
γ
, and which we write
T
g
(
ω
,
γ
),on the viscoelastic 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
κ
1
and
κ
2
. Multiple examples are given comparing
κ
1
and
κ
2
for linear low-density polyethy-lene (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 changeof the characteristics of the elastic network in the melt above
T
g
, an observation alreadyforeseen in a previous communication.
[1]
Theeffect ofstrainisanimportant sectionofthispaper. Weshow that theessentialroleofstrainistoactivatethe
κ
1
-phasetoparticipateactively(byshear-thinning)inthedeforma-tion process. In linear viscoelastic conditions, the conformers
1
in the
κ
1
dual phase do notdeform, their motion is through diffusional reorganization, i.e., delocalization in the struc-ture triggered by the stretch–relax deformation mechanism (shear-thinning) of the
κ
2
-phaseconformers. When the
κ
1
-phase is activated by an increase of the strain, strain softeningoccurs. In the discussion, we present a new understanding of “the network of entangle-ment” and show how its orientation and gradual instability gives rise to the mechanisms of
1
Conformers are deﬁned in refs.33–35. Also see Fig. 12a.
224 J. P. Ibar
deformationobservedfromverylow
ω
tohigh
ω
,atvariousstrains.Wesuggestthatthenet-work character of deformation is not due to topological considerations but, instead, due tothecooperativecouplingnatureoftheinteractionsbetweenthemacromoleculesconformerswhich organize according to a
Dual-Grain Field-Statistics
. In this model, the duality aspectcomes twice: it comes at the local level of interactions between the conformers, and thisdualityisdealtwithbytheintroductionoftheGrain-FieldStatisticsapplicabletomacrocoilsystems. The equations of the Grain-Field Statistics predict the dynamic aspect of the inter-actions between conformers. But the interaction between macrocoils introduces a secondlevel of duality, above a certain size for the macrocoils (which we consider to be the onsetof entanglements), responsible for the molecular characteristics of the dynamic network.In summary, we introduce in this article new methods of analysis of the rheologicalresults, which seem to conﬁrm an essential aspect of the cohesion of the interactionsbetween 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 anessential focus of experimental investigation in Part II of this series
[2]
is reviewed here interms of the dual-phase model.
Introduction
The viscosity of polymers is key to their behavior in the molten state and thus to theirprocessing. Viscosity is a scalar equal to the stress divided by the strain rate, which, in thecase of a dynamic deformation, can be rewritten as
η
∗
=
τ
˙
γ
=
G
∗
·
γ ω
·
γ
=
G
∗
ω,
(1)where
η
∗
is the dynamic viscosity,
τ
is the shear stress, and ˙
γ
is the strain rate, the otherparameters having been deﬁned above. The viscosity is known to remain constant at lowstrain rate of ﬂow, in the so-called Newtonian region, and its value is the “Newtonianviscosity,”
η
0
. Polymer melts are not as simple as Newtonian ﬂuids, and, as the strain rateincreases, the viscosity becomes strain rate dependent, a phenomenon described as “shear-thinning” if the viscosity decreases with increasing shear rate. It is important to realizethat the stress continues to grow with strain rate, as shear-thinning occurs, it is simply notgrowing as fast as it would if the Newtonian stress still applied (in the Newtonian regimeof deformation, stress is proportional to the shear rate).The phenomenon of shear-thinning of polymeric melts has been analyzed and mathe-matically modeled “satisfactorily” for more than 60 years.
[1]
The power-law equation wasoneoftheﬁrstequationsusedtoquantifythestrainratedependenceofviscosity,andapplieswell, at high strain rate, over a short range of strain rate. The power law is useful whenmodelingﬂowincomplexgeometryatthestrainrateusuallyappliedinindustrialprocesses.The power-law equation is not applicable at low strain rate: it predicts much higher valuesfor the viscosity than what is observed at low strain rate, and does not provide the Newto-nian value. Shear-thinning is classically described by another equation, the Cross-Carreauequation, also called the Carreau-Yasuda equation, that has the advantage to converge tothe power-law formula at high strain rate, and to predict the Newtonian viscosity value atlow strain rate. Critical issues related to the validity of the Cross-Carreau’s formula are notdiscussed in this paper; they were presented in Part I of this series.
[1]
The Great Myths of Rheology Part III 225
The Cross-Carreau equation can be written as:
η
=
b
1
1
+
(
b
2
.ω
)
b
3
(
n
−
1)
/b
3
,
(2)where
b
1
,
b
2
,
b
3
and
n
are all curve-ﬁtting constants.
n
is between 0 and 1,
b
1
is the valuefor
ω
=
0, so it is the Newtonian viscosity
η
0
. For large
ω
, Eq. (2) simpliﬁes to a powerlaw, since the second term inside the parenthesis becomes much larger than 1:
η
=
η
0
(
b
2
ω
)
(
n
−
1)
.
(3)On a log–log plot of
η
vs.
ω
, a straight line with slope (
n
– 1) is observed for Eq. (3).Although Eq. (2) has been veriﬁed with great accuracy for many unbranched polymers,
b
3
and
n
are often found to vary with temperature, even though slightly, and thus are nottrue constants. Another problem is the multiplicity of solutions found for the constants
b
1
,
b
2
,
b
3
, and
n
by regression, all providing an apparent excellent ﬁt. Besides, for dynamicdata, which only vary over a small
ω
range (typically from 0.1 to 300 rad/s at low strainamplitude,
∼
2%), one needs to assume rheological simplicity for the melt to extend the
ω
range to larger spans by use of the frequency–temperature superposition principle. Weshowed in Part I
[1]
that truesuperposition was not veriﬁed inmost cases, even inthose caseswith “apparent” good curve overlap.As explained in Part I of this series,
[1]
our interest in using (
G
/
G
∗
) to analyze dynamicdata srcinated from our review of the claims of the time–temperature superposition tem-perature. In order to avoid addressing the need, whereas performing superposition of theviscosity–
ω
curves, to know the melt density
ρ
, which enters the expression of the verticalshift factor, one can introduce the ratio of two moduli, say (
G
/
G
∗
), which cancels out thecorrection for density and absolute temperature. The vertical shifting is thus eliminated.Furthermore, plotsof(
G
/
G
∗
)
2
vs.log .
ω
seemed even moreappropriate when suchhorizon-talshiftingwasperformed,becauseoftheinterestbroughttothespecialcase(
G
/
G
∗
)
2
=
0.5corresponding to the “cross-over point”
G
x
=
G
x
, often considered as an important char-acteristic of the molecular weight of the chains. Figure 1 displays such a plot for PS data.
Figure 1.
(
G
/
G
∗
)
2
vs.
ω
for polystyrene at 10% strain for
T
165–245
◦
C (color ﬁgure availableonline).
226 J. P. Ibar
Figure 2.
(a) (
G
/
G
∗
)
2
vs.
G
∗
for Virgin and treated polycarbonate (treatment
1
). The Virgin is thetop curve: for a given melt elasticity,
G
∗
is greater for the treated melt. (b) log(
η
∗
(
ω
)) vs. log(
ω
) forVirgin and treated PC (treatment
1
). Viscosity is higher for the melt presenting the greatest modulusfor agiven melt elasticity. (c) (
G
/
G
∗
)
2
vs.
G
∗
for Virgin and treated Polycarbonate (treatment
2
). Sameas in (a) but with a different mechanical treatment. This time,
G
∗
is smaller for the treated melt, fora given melt elasticity. (d) log(
η
∗
(
ω
)) vs. log(
ω
) for Virgin and treated PC (treatment
2
). Like in (b)viscosity is higher for the melt exhibiting the greatest
G
∗
for a given melt elasticity, here the Virginsample (color ﬁgure available online).
This failure of the time–temperature principle drove our interest over the last fewyears to analyze, in detail, plots of (
G
/
G
∗
)
2
vs. log(
ω
) and, as this article will show, morespeciﬁcally plots of (
G
/
G
∗
)
2
vs.
G
∗
. The latter pair of variables was found in Part II of this series
[2]
to be most appropriate to characterize melts “treated”
2
to exhibit variousstates of disentanglement or reentanglement. Related plots are shown in Figs. 2(a)–(d)for PC.In Figs. 2(a) and (b), a treated melt (“treatment
1
”, squares) is compared with a virginmelt (circles), and, similarly, a different disentanglement treatment (“treatment
2
”) providesthecomparativeplotsfoundinFigs.2(c)and(d).Itisremarkablethattherespectiveposition
2
The treatment was mechanically done, mostly consisting of the superposition of pressure ﬂowand cross-lateral drag ﬂow (combining rotational and vibrational shear).

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