Re-Formulation of the Rheological Equations of Polymer Melts Pursuant to the Dual-Phase and Cross-Dual-Phase Model of Dissipative Interactions.
Description of the Dual-Phase Rheology Protocol (DPRP).
This graph compares the
shear-thinning rheological curves for the data (black line), for Dual-Phase 1
(DP1), Dual-Phase 2 (DP2), their sum (DP1+DP2), and their sum with the addition
of the inactive Dual-Phase portion of the network of interacting
Dual-conformers: (DP1+DP2+dc), the green triangles pointing down. The viscosity
of each Dual-Phase is calculated from its c
structure, c = c1+c2, with c = (G’/G”)2 ; c1 and c2 are the respective contributions of the free
volume and of the rotation isomeric state (RIS) of the Dual-conformers to the active
elasticity c.
ABSTRACT
The current equations describing the dynamic rheology
of polymer melts provide the correlations between the modulus of
visco-elasticity, G*(w),
splitting into its elastic and viscous moduli, G’(w) and G”(w), respectively, and the frequency of
the shear deformation applied to the melt, w. The dynamic viscosity of the melt, h* = G*/w is usually used to compare melts of different
chemical nature or to study the effect of molecular weight, M, its dispersity,
and the presence or the absence of branches, short or long. The equations also
describe the influence of external variables on the rheological state of the
melt: temperature and pressure. The terms and constants that enter the
equations of rheology have themselves been interpreted, to a large extent, by
molecular mechanisms triggered by the changes occurring to the statistical
description of the individual chains: the dynamic molecular models of Rouse
(for un-entangled melts) and of de Gennes (reptation) for entangled melts
illustrate the current formulation and understanding of the rheology of polymer
melts. On the other hand, we have pointed out in several publications the
problems surfacing in the classical molecular interpretation of the rheological
data when revisited by our analysis, and the contradictions arising from these
models when they are applied to theunderstanding of new experimental work on“the
strain-induced time dependence of the melt viscosity” leading to the technologies
of “Rheo-Fluidification” of melts and of their “sustained-Orientation”.
In short, we have concluded that the current comprehension of the
visco-elasticity of polymers, in particular of the entanglement concept,is not
compatible with the reality of new experimental facts and, therefore, that the
current accepted theories should be debated and revised; more generally, that a
different approach to the physics of interactions in polymers should be
considered. We have already introduced the principles of such a theory of
interactions of Dual-conformers in which the statistical system is no longer a
single macromolecule, but a self-generated collective network of
Dual-Conformers belongingto the macromolecules. The local deformation of the
Dual-conformers and the collective network enthalpy and entropy are
interactively emerging from the solution of the Grain-Field Statistics applied
to the interactions between the Dual-conformers, demonstrating the dissipative
character of the solutions: this is our new foundation to explain the observed
experimental facts, old and new, regarding the visco-elastic effects of melts
under shear-deformation, linear or non-linear. This paper presents the
Dual-Phase Rheology Protocol (DPRP) i.e. how to derive the parameters of the
new statistical approach from the visco-elastic moduli G’(w,T) and G”(w,T) in dynamic rheology. In
particular, it is shown that a completely different empirical presentation of
the rheological results can be expressed in terms revealingthe relevance of the
Dual-Phase and Cross-Dual-Phase concepts. We introduce, for the 1st
time, an empirical analysis of the data that permits to relate the rheological
properties of an entangled melt (M > Mc, the
entanglement molecular weight) to the stability of the solution presented by
the split of Dual-Phases into Cross-Dual-Phases. In other words, the
“entanglement” of the macromolecules can be quantified in terms of the
viscosity, h, and the intrinsic elasticity, c=(G’/G*)2 of the individual interactive Dual-Phases
emerging from the split to re-establish a new rheological stability of the melt
as M increases. This paper also introduces a roaster of new rheological markers
that can be traced to analyze phenomena visible in rheology, widening the panel
of analytical tools available to characterize difficult situations (blends,
branches etc.).
Keywords
Dual-Phase
Rheology Protocol, DPRP, Dual-Phase model, Cross-Dual-Phase model,
Grain-Field-Statistics, Dual Conformer, Cross-Duality, Entanglement,
Disentanglement, Rheo-Fluidification, Sustained-Orientation, entanglement
instability, polymer melt rheology, molecular dynamics, reptation, Newtonian
viscosity, Maxwell’s rheology equations, Thermo-Vogel-Fulcher equation of
viscosity, MX-PLOT, interactive coupling, TLL transition.
TABLE OF CONTENT
ABSRACT
TABLE OF CONTENT
INTRODUCTION p.4
GLOSSARY
OF THE TERMS AND BACKGROUND DEFINITIONS. p.6
1. Dual-Splitting. p.6
2. Scaling of the
Dual-Split Terms. Scales Correlation Maps. p.7
3. The Dynamic
Thermo-Vogel-Fulcher Equation (DTVF). p.8
4. MX-PLOTS. p.10
5. Visco-Elastic Range Dynamic Fragmentation.
p.11
6. Classical Entanglement
Rheological Criteria. Dual-Phase
Rheological Proto col (DPRP): Single-Dual-Phase vs. Two Dual-Phases Criteria. p.12
7. Dual-Phase Dynamic
Frequencies. p.13
ILLUSTRATION OF THE DPRP DEFINITIONS AND PARAMETERS. p.14
8. Data Assessment.
p.14
9. The TVF equation.
p.20
10. The MX-PLOT.p.20
11. The Structure of c into c1 and c2 . p.30
12. Discontinuities. Asymmetrical
roles of c1 and c2. p.39
SCALING OF THE DUAL-PHASE TERMS.
p.48
13. Equations for the structure of c by splitting c vs G* using Eq.
2. p.48
14. Analysis of
the X-scale and Y-scale. p.50
15.
Traditional Analysis of the X-scale and Y-scale. p.50
16. Scale Correlations Maps: The
“Esoteric” Approach. p.60
17. Single
Dual-Phase (DP) or Cross (Entangled) Dual-Phases (DP1, DP2). p.70
18. Construction of
the Cross-Dual-Phases: the Linear approach. p.73
19. Correlations
between the Active terms of the Structure of c1(w) and c2(w): B1, B2, R1 and R2. Expression of the Cross-Duality. p.83
20. Construction of
the Cross-Dual-Phases: The Non-Linear Approach. p.92
21. Conversion of the c structure
results to Viscosity. p.100
22. The Question of
the “dc” terms (Fig.11i). p.106
23. Dynamic Network
Frequencies. p.107
24. Strain Induced Time dependence
of viscoelasticity. p.110
25. Orientation of the Network of Dissipative
Interactions. p.115
SUMMARY
OF THE DUAL-PHASE RHEOLOGICAL PROTOCOL (DPRP). p.117
DISCUSSION.
p.121
CONCLUSIONS
p.132
ACKNOWLEDGMENTS.
p.134
REFERENCES.
p.135
EXCERPT from the DPRP paper:
“We have accumulated empirical knowledge through experimentation [1, 3-5] that enables the determination of the dynamic parameters (w, strain) that trigger the time dependence of the rheological parameters in polymer melts; in other words, we know empirically how to generate the rheological evidence that is denied by the current paradigm of rheology [6-8], but we had not been able to predict it mathematically until now. Our challenging model of entanglements as Cross-Dual-Phases can now address mathematically the time dependence of the rheological state under specific non-linear rheological conditions [1,38,43]; it can also explain the difference between “Rheo-Fluidification” and “Sustained-Orientation”, in particular why it was more difficult, retrospectively, to obtain the “Sustained-Orientation” benefits than the Rheo-Fluidification ones”.
FULL PAPER (downloadable):
https://doi.org/10.5281/zenodo.20645765
Jean Pierre Ibar
jpibar@alum.mit.edu
