Experimental validation of a tube based constitutive equation for linear polymer melts with inter-chain tube pressure effect

Rudy Valette,J.-F Agassant,A Durin,G Boukellal

doi:10.1051/meca/2013048

Abstract

A polydisperse case of an entangled linear polymer melts constitutive equation was studied. This constitutive equation, proposed by S. Dhole et al. [J. Non-Newtonian Fluid Mech. 161 (2009) 10–18], based on the reptation theory and the tube model, was tested on a polystyrene in shear (capillary rheometry) and planar extension in a complex flow (fieldwise measurements in a contraction flow) for different level of strain rates. A good quantitative prediction of all the set of experiments was obtained, using no adjustable nonlinear parameters.

Highlights

IntroductionConvective constraint release parameter Shear rate (s−1) Source wavelength (m) Cauchy stress tensor (Pa) First principal stress difference (Pa) Reptation time of mode i (s) Inter-chain pressure relaxation time of mode i (s) Rouse time of mode i (s) Finite extensibility parameter Conformation tensor Stress-optical coefficient (Pa−1) Finite extensibility function Elastic modulus of mode i (Pa) Identity matrix Polydispersity index Fringe order Weight-averaged molecular mass (kg.mol−1) Pressure (Pa) Velocity vector (m.s−1) Velocity gradient tensor (s−1) Slit width (m) Number of entangled segments Number of entanglements per chain
Where p is the pressure term, G = 4ρRT /5Me is the elastic modulus (ρ and Me being respectively polymer density and molecular mass between entanglements), b is the finite extensibility coefficient and C is the conformation tensor
It is important to notice that no nonlinear parameter were adjusted, the only difference between this polystyrene and the one used by Bach et al [16] and identified by Dhole et al [1] being its molecular mass distribution, from which depends only the viscoelastic spectrum

Summary

Introduction

Convective constraint release parameter Shear rate (s−1) Source wavelength (m) Cauchy stress tensor (Pa) First principal stress difference (Pa) Reptation time of mode i (s) Inter-chain pressure relaxation time of mode i (s) Rouse time of mode i (s) Finite extensibility parameter Conformation tensor Stress-optical coefficient (Pa−1) Finite extensibility function Elastic modulus of mode i (Pa) Identity matrix Polydispersity index Fringe order Weight-averaged molecular mass (kg.mol−1) Pressure (Pa) Velocity vector (m.s−1) Velocity gradient tensor (s−1) Slit width (m) Number of entangled segments Number of entanglements per chain. The parameter β controls the rate of chains disentanglement in strong flows (i.e. large strain rates) This model predicts a shear-thinning effect in stationary shear beyond a shear rate value of γ = 1/θ, where β controls the value of the power law exponent on γfor moderate values of chain stretch (characterized by the trace of the tensor C). Elongational flows, θR controls the short time non linear response In stationary elongation, this model predicts a strain hardening for strain rates larger than 1/θR and a final saturation due to the finite extensibility (controlled by the parameter b). This model predicts a strain hardening for strain rates larger than 1/θR and a final saturation due to the finite extensibility (controlled by the parameter b) Between these two regimes, inter-chain pressure, controlled by θp, limits elongation softening, as observed by Bach et al [16]. A multimode extension of this model was considered in order to take into account the polydispersity of the studied polymer

Studied polymer: polystyrene

Set of model parameters

Fieldwise measurements in a contraction flow

Conclusion