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