Abstract
In polymeric materials subject to both polymerization reactions and flow, there can be a complex interplay between reactions and stress relaxation processes. For example, reversible scission reactions can “shuffle” stresses across the molecular weight distribution, narrowing the stress relaxation spectra and decreasing the typical stress relaxation time. In addition, flow can stretch chains and make them more likely to break apart, leading to changes in the underlying reaction kinetics. Existing strategies for modeling the coupling between reactions and flow in polymer systems are limited in their range of applicability or dubious in their underlying approximations. Here, we develop a more flexible modeling approach with coupled population balance models that move both material and stress across the molecular weight distribution. The full model, which we call the “living Rolie Poly” (LRP) model, reproduces some earlier findings on linear rheology and offers new insights into nonlinear rheology and the role of flow-induced scission. For systems that are not inclined to shear band in the absence of flow-induced scission, we predict that flow-induced scission produces an additional shear thinning effect for steady shear flow, and in steady extensional flow, we predict that flow-induced scission acts like a finite extensibility correction, preventing the divergence of the steady viscosity. The LRP model is too complex to use in spatially resolved calculations or complex flow geometries at this time, but a “simplified” model with no such limitation arises naturally in the “fast--breaking” limit and shows good agreement with the full LRP model predictions.
Highlights
AND INTRODUCTIONapply well to any system of well-entangled living polymers with a linear chain architecture.At high densities and long chain lengths, wormlike micelles can exhibit viscoelastic behaviors similar to those of wellentangled polymers
In the living Rolie Poly” (LRP) model, we have eliminated the most severe approximations of the VCM model: we have considered the entire molecular weight distribution, we have enforced a realistic length-dependence to all chain relaxation times, and we have accounted for nonlinear modes of stress relaxation known to be important in entangled polymer rheology
We have developed a constitutive model for wormlike micelles by combining a simple population balance model with the Rolie-Poly constitutive equation
Summary
Apply well to any system of well-entangled living polymers with a linear chain architecture (i.e., no branching). More recent refinements introduced chain stretching as needed for describing finite entanglement numbers [14] These constitutive models all assume an equilibrium molecular weight distribution and a system that is “fast breaking” with respect to its stress relaxation dynamics. While the VCM model does describe a coupling between the constitutive equation and the molecular weight distribution, it is not clear that its predictions bear any correspondence to the systems it purports to describe (namely, well-entangled wormlike micelles). VI develops a simplified “singlemode” picture of each stress relaxation mechanism in the “fast-breaking” limit, which we call LRP-S These results are rejoined into a final equation that preserves essential features of flow-induced scission while remaining simple enough for CFD calculations. Contrary to what has been shown for the VCM model, flow-induced scission cannot cause shear banding in the LRP model
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