Ole Hassager

Extensional viscosity for polymer melts measured in the filament stretching rheometer

A new filament stretching rheometer has been constructed to measure the elongational viscosity of polymer melts at high temperatures. Two polymer melts, a LDPE and a LLDPE, were investigated with this rheometer. A constant elongational rate has been obtained by an iterative application of the Orr–Sridhar method for specification of the end-plate movement. Agreement has been found with linear viscoelastic measurements performed in shear. Hencky strains up to about 6 have been reached. Steady values of the viscosity have been sustained in some cases for about two Hencky strain units.

Analytical solutions for squeeze flow with partial wall slip

Abstract Squeeze flow between parallel plates of a purely viscous material is considered for small gaps both for a Newtonian and power law fluid with partial wall slip. The results for the squeeze force as a function of the squeezing speed reduce to the Stefan and Scott equations in the no slip limit, respectively. The slip velocity at the plate increases linearly with the radius up to the rim slip velocity v s . For small gaps H , the resulting apparent Newtonian rim shear rate—measured for a constant rim shear stress, i.e. an imposed force increasing proportional to 1/ H —yields a straight line if plotted versus 1/ H . The slope of the straight line is equal to 6 v s whereas the intersect with the ordinate yields the effective Newtonian rim shear rate to be converted into the true rim shear rate by means of the power law exponent. The advantage of the new technique is the separation of bulk shear and wall slip from a single test. A more general derivation for the Newtonian case being also valid for full lubrication and large gaps is used to explain the gap dependence of the squeeze modulus of an elastic material.

The Considère condition and rapid stretching of linear and branched polymer melts

We analyze the onset of “necking” and subsequent filament failure during the transient uniaxial elongation of viscoelastic fluid samples in extensional rheometers. In the limit of rapid elongation (such that no molecular relaxation occurs), the external work applied is all stored elastically and the Considere criterion originally developed in solid mechanics can be used to quantitatively predict the critical Hencky strain to failure. By comparing the predictions of the Doi–Edwards model for linear homopolymer melts with those of the “Pom-Pom” model recently proposed by McLeish and Larson [J. Rheol. 42, 81–110 (1998)] for prototypical branched melts we show that the critical strain to failure in rapid elongation of a rubbery material is intimately linked to the molecular topology of the chain, especially the degree of chain branching. The onset of necking instability is monotonically shifted to larger Hencky strains as the number of branches is increased. Numerical computations at finite Deborah numbers al...

Transient filament stretching rheometer II: Numerical simulation

The Lagrangian specification is used to simulate the transient filament stretching rheometer. Simulations are performed for a dilute PIB-solution modeled as a four mode Oldroyd-B fluid and a semidilute PIB-solution modeled as a non-linear single integral equation. The simulations are used to investigate a sequence of filament stretching rheometers. The questions of special and temporal homogeneity of the experiment are investigated. The simulations are compared with data from a specific rheometer by Tirtaadmadja and Sridhar.

Quantitative prediction of transient and steady-state elongational viscosity of nearly monodisperse polystyrene melts

Elongational behavior of four narrow molar mass distribution polystyrene melts of masses 50 000, 100 000, 200 000, and 390 000, g∕mol, respectively was investigated up to Hencky strains of 5. All melts show strain hardening behavior. For the two highest molar mass polystyrenes, strain hardening starts at elongation rates larger than the inverse reptation time, and the steady-state elongational viscosities decrease with increasing elongation rate according to a power law with a power-law index of approximately −1∕2 instead of −1 as predicted by the original Doi–Edwards tube model. Marrucci and Ianniruberto [Macromolecules 37, 3934 (2004)] have introduced an interchain pressure term arising from lateral forces between the chain and the tube wall into the Doi–Edwards model to account for the latter effect. Based on the molecular stress function theory allowing for a strain-dependent tube diameter, we show that the transient and steady-state elongational viscosities of the nearly monodisperse polystyrene melt...

Elongational Viscosity of Monodisperse and Bidisperse Polystyrene Melts

The start-up and steady uniaxial elongational viscosity have been measured for two monodisperse polystyrene melts with molecular weights of 52 and 103kg∕mole, and for three bidisperse polystyrene melts. The monodisperse melts show a maximum in the steady elongational viscosity vs. the elongational rate, ϵ, of about two times the limiting value of 3η0 expected for a Newtonian fluid, whereas the bidisperse melts have a maximum of up to a factor of seven times the Trouton limit of 3η0. The Wiest model which incorporates anisotropic drag and finite extensibility may be used to interpret the results in molecular terms.

Viscosity overshoot in the start-up of uniaxial elongation of low density polyethylene melts

The transient uniaxial elongational viscosity of BASF Lupolen 1840D and 3020D melts has been measured on a filament stretch rheometer up to Hencky strains of 6–7. The elongational viscosity of both melts was measured at 130°C within a broad range of elongational rates. At high elongation rates, an overshoot or maximum in the transient elongational viscosity followed by a steady viscosity was observed. The steady elongation viscosity was about 40%–50% less than the maximum at high strain rates. The steady elongational viscosity as a function of the elongation rate, ϵ, decreases approximately as ϵ−0.6 in both melts at high strain rates. The transient elongational viscosity, measured at a specific elongation rate at 170°C on the BASF Lupolen 3020D melt, did not follow the time temperature superposition principle based on linear viscoelasticity during the decrease in the transient elongational viscosity towards the steady state.

Failure and nonfailure of fluid filaments in extension

The phenomenon of ductile failure of Newtonian and viscoelastic fluid filaments without surface tension is studied by a 2D finite element method and by 1D non-linear analysis. The viscoelastic fluids are described by single integral constitutive equations. The main conclusions are: (1) Newtonian fluid filaments do not exhibit ductile failure without surface tension; (2) some viscoelastic fluids form stable filaments while other fluids exhibit ductile failure as a result of an elastic instability; (3) for large Deborah numbers, the Considere condition may be used to predict the Hencky strain of the elastic instability.

A Lagrangian finite element method for the simulation of flow of non-newtonian liquids

Abstract A Lagrangian method for the simulation of flow of non-Newtonian liquids is implemented. The fluid mechanical equations are formulated in the form of a variational principle, and a discretization is performed by finite elements. The method is applied to the slow of a contravariant convected Maxwell liquid around a sphere moving axially in a cylinder. The simulations show that the friction factor for a sphere in a narrow cylinder is a rapidly decreasing function of the Deborah number, while the friction factor for a sphere in a very wide cylinder is not significantly affected by fluid elasticity. It is demostrated that the simulated wall effect on the motion of the sphere may be utilized in an experimental identification of a time constant for a given liquid.

Flow of viscoplastic fluids in eccentric annular geometries

Abstract A classification of flowfields for the flow of a Bingham fluid in general eccentric annular geometries is presented. Simple arguments show that a singularity can exist in the stress gradient on boundaries between zones with yielded and un-yielded fluid respectively. A Finite Element code is used to verify this property of the Bingham fluid. An analytical solution for the flowfield in case of small eccentricities is derived.