Differential constitutive equations for polymer melts: The extended Pom–Pom model

Abstract

SynopsisThe Pom-Pom model, recently introduced by McLeish and Larson ͓J. Rheol. 42, 81-110 ͑1998͔͒, is a breakthrough in the field of viscoelastic constitutive equations. With this model, a correct nonlinear behavior in both elongation and shear is accomplished. The original differential equations, improved with local branch-point displacement, are modified to overcome three drawbacks: solutions in steady state elongation show discontinuities, the equation for orientation is unbounded for high strain rates, the model does not have a second normal stress difference in shear. The modified extended Pom-Pom model does not show the three problems and is easy for implementation in finite element packages, because it is written as a single equation. Quantitative agreement is shown with experimental data in uniaxial, planar, equibiaxial elongation as well as shear, reversed flow and step-strain for two commercial low density polyethylene ͑LDPE͒ melts and one high density polyethylene ͑HDPE͒ melt. Such a good agreement over a full range of well defined rheometric experiments, i.e., shear, including reversed flow for one LDPE melt, and different elongational flows, is exceptional.

Modeling Flow-Induced Crystallization

“…The former is calculated from stretch of a mode representative of the longest chains in the material and the latter from stretch of an average mode. The extended pom-pom (XPP) model [83] is used to calculate the backbone stretch, with parameters for the XPP model as given in Sect. 3.…”

“…The relation λ s,i % λ b,i /4 has been observed for several other polydisperse linear polymer melts [83]. Therefore, we took λ s,i % λ b,i /4 and q ¼ 1 for all modes, except for the mode with the longest relaxation time.…”

“…This model was originally developed for branched polymer melts, but was later found to also accurately describe the behavior of linear polydisperse melts [83,101]. The conformation tensor is given by…”

“…Here, c ∇ i denotes the upper convected derivative of the conformation tensor of mode i, λ b,i denotes the relaxation time for backbone tube orientation of mode i, λ s,i denotes backbone stretch relaxation time of mode i, and the parameter v i depends on the number of arms of the molecule q i following v i ¼ 0.1/q i [83]. The Giesekus α-parameter is set to 0.…”

“…Here, Λ hmw is the backbone stretch calculated using the XPP constitutive model [83], and μ n and g n are scaling parameters, the latter of which depends on temperature and pressure following…”