A century of viscoelastic modelling: from Maxwell to the eXtended Pom-Pom molecule

1. A century of viscoelastic modelling: from Maxwell to the eXtended Pom-Pom molecule Giancarlo Russo Cardiff School of Mathematics Bonn, Institut fuer Numerische Simulation

2. Outline • Viscoelastic matter: a brief description and main features • Different models for different problems-Disperse polymer solutions-Upper Convective Maxwell -Oldroyd-B -Finite Extensible Nonlinear Elasticity - Concentrated polymer solutions - Phan Tien – Tanner - Tube model - Pom-Pom model - eXtended Pom-Pom model • A few words about my research project - Motivation - Results achieved - Coming next (hopefully…) Bonn, Institut fuer Numerische Simulation

3. Modelling continuum mechanics Field Equation of Momentum Continuity Equation Newtonian Fluids Non-Newtonian Fluids Stress-Strain relation (constitutive equation) ? Viscoelastic Fluids Bonn, Institut fuer Numerische Simulation

4. Relaxation Time and Deborah Number Maxwell Model: Young modulus and viscosity gathered together The relaxation time, aka “will I run out of patience before the flow ?” About 10d-12 for water; more than a day for glass. Solution relating shear stress to rate of strain: the stress at any time depends on the whole strain history; the further back in time, the more memory fades The Deborah number, aka “I’ve been waiting enough !!!” Setting the time of the experiment will determine the behaviour of the matter. “Memory” of the fluid Bonn, Institut fuer Numerische Simulation

5. Some (not so obvious…) effects of elasticity Releasing tension along the streamlines: the Die Swell Supporting tension along the SL: Rod Climbing High extensional viscosity/shear viscosity ratio: the open siphon effect Visualizing different Deborah numbers: 9 to 1 mixture of cornstarch and water Bonn, Institut fuer Numerische Simulation

6. Modelling disperse polymer solutions I: UCM and Kelvin (late XIXth century) The Maxwell model for shear flows… …leads to the Upper Convective Maxwell model for the constitutive equation.T is total extra-stress and Gamma-dot is the rate of strain. The Maxwell model above is one of two basic ways of “mixing” the Young modulus with the Newtonian viscosity. Another one is the Kelvin model The upper convective derivative takes into account the deformation induced by the rate of strain, a feature which is typical of elastic fluids, and adds to the usual material derivative describing the flux. Let’s combine them… Bonn, Institut fuer Numerische Simulation

7. Modelling disperse polymer solutions II: Oldroyd-B (1950) …and we find the Oldroyd-B model Physically the pure elastic molecules are replaced by bulks of particles and fluid, and these will be the clusters swimming in the solvent. Solvent vs Total viscosity ratioWeissenberg number, measure of elasticity of the fluid This replacement leads to the splitting of the extra-stress into its solvent and polymeric contributions. The former is plugged as diffusive term into the balance of momentum; the latter is what is practically computed in the constitutive equation, here in its dimensionless form. Bonn, Institut fuer Numerische Simulation

8. Modelling disperse polymer solutions III: Oldroyd-B at the microscope The equations of motions for the bulks include some Brownian forces; these Brownian forces depends on a probability density function, say q;A Fokker-Planck equation (diffusion equation for q) is derived. F1 Fluid m1 F2 r1 m2 r2 Bulk A common expression relating stress and the product RF is due to Kramers: R = r2 - r1 Combining this and the FP equation we find the so called Giesekus expr. The whole point is the choice for F :F hookean means Oldroyd-B. Unfortunately this leads to an infinite extensibility of the dumbbells!!!Another problem is a discontinuity in the extensional viscosity. Bonn, Institut fuer Numerische Simulation

9. Modelling disperse polymer solutions IV: FENE models Limiting the dumbbells extensibility through bounding functions; overcoming the discontinuity in the extensional viscosity. Mainly suitable for extensional flows. FENE – P (Peterlin) FENE FENE – CR (Chilcott - Rallison) Combining the Kramers, Giesekus and the FENE-P we obtain The FENE-CR replaces the expression above by the following: The quantity R0 is the maximum extension the spring can reach. It predicts constant shear flow, a non-zero 1st normal stress difference and the extension is bounded. The model is shear-thinning; the extensional viscosity exhibit continuous dependence on the extensional rate. Being shear thinning is suitable for shear flows. It presents the same features as Old-B but the extensibility is bounded. Bonn, Institut fuer Numerische Simulation

10. Modelling concentrated polymer solutions: the PTT model (1977) In describing these melts, is essentials to represent in a proper way the entanglement between molecules. The PTT looks at the polymer molecules and their interaction as a network. Strands are linked through rigid junctions. Slip is the cause of dynamics of the strands, modelled as follows: f is the probability distribution of the junctions; here is its rate of change balance, with g and h rate of creation and destruction of the junctions. Multiplying bothsides by rho and averaging, we obtain the Const. Eq. Shear thinning; extensional bounded; 1st normal stress predicted. This makes PTT a suitable model for shear flows of polymer melts. Stress overshoots at high strain rates in elongational flows are also fairly reproduced. The absence of 2nd NSD is the main limit for extensional simulations. Bonn, Institut fuer Numerische Simulation

11. Modelling concentrated polymer solutions II: the tube model (1978) Developed by Doi and Edwards, translates the interactions between molecules as topological constraints. The presence of other chains surrounding a test molecule will confine the allowed configurations within a tube of a certain diameter. The primitive chain AB reptates; part of it leaves the original tube for another. This is measured by a probability distribution function, say theta: Averaged (in space) solution Stress-orientation tensor Q(E) relation Disentanglement time comparison Psi is the key; the reptation dynamics is responsible for the change of conformation when elastic effects are taken into account.. Bonn, Institut fuer Numerische Simulation

12. Modelling concentrated polymer solutions III: the Pom-Pom model (1998) McLeish and Larson “mounted” arms at the end of primitive chain, which became the “backbone” of the “Pom-Pom molecule”. These arms will be then “released” and “withdrawn” by the backbone, but only when the BB is fully stretched. It describes very accurately Low Density PolyEthylenes dynamics, whose irregular branches give raise to high level of shear thinning and strain hardening. Evolution equation for the orientation tensor Evolution equation for the backbone stretch Evolution equation for the arms motion Derivation of the stress tensor Three main problems affect this model:1) discontinuities in Grad u in steady flows; 2) the orientation tensor is unbounded at high strain rates; 3) 2nd NSD is not predicted Bonn, Institut fuer Numerische Simulation

13. Modelling concentrated polymer solutions IV: the XPP model (2001) Backbone stretch and stretch relaxation time Tackling issue # 1 Blackwell modification: a smoother approach to the maximum BB stretch by mean of withdrawing arms before such maximum stretch is achieved Extra Function Tackling issue # 2 High strain rates means the 1st term totally outweighs the 2nd one; the result is that the stretching effect is predominant enough to avoid unbounded orientation.The macroscopic dependence of the slip tensor on the stress is the physical reason. Tackling issue # 3 Relaxation time tensor The Giesekus - like 2nd order term is responsible for predicting 2nd NSD. ARE WE HAPPY ??? Viscoelastic Stress SADLY NOT… Bonn, Institut fuer Numerische Simulation

14. Review Bonn, Institut fuer Numerische Simulation

15. My research project in few words RESULTS ACHIEVED FILAMENT STRETCHING A viscoelastic fluid is confined between two plates. When these plates are pulled apart, the fluid stretches. Tracking the free surface and describing the necking effect at the centre are the main challenges. This phenomenon is common in fibre spinning and industrial processes involving thin films. EXTRUDATE SWELL A viscoelastic fluid is extruded from a pipe. The Stress gradient at the die is responsible for the extensional swelling. Capturing this gradient, as well as the free surface’s behaviour, is crucial. Extrusion processes in food and manufacturing industries are the main industrial applications THEORETICAL 1) Stability estimate for the stress tensor in the 3 fields Stokes problem; 2) Existence and uniqueness of a steady state weak solution for the 3fields die swell Stokes problem. NUMERICAL 1) A code on the shelf has been extended to include almost all the models mentioned; tests are positive 2) The MATRIX-LOGARITHM formulation has been implemented and tested for the flow past a cylinder. Higher We calculation achieved for OLD-B. Horizontal Velocity Vertical Velocity Pressure 1) Discretization in space: the Spectral Element Methods + transfinite techniques Ongoing and doming next (hopefully…) 1) Extending the code to the real problems Velocity extrapolated on the FS along the vertical gridlines. Nodes shifted. New transfinite mapping built. Moving forward in time. 2) Improve the MATRIX-LOG version ( problem with some stress profiles) and make it model independent (so far just Oldroyd-B) 2) Discretization in time: usually 1st order + OIFS for the material derivative 3) Gather the two approaches Bonn, Institut fuer Numerische Simulation

16. References [1] PHILLIPS T.N., OWENS. R., Computational Rheology, Imperial College Press, 2002. [2] KEUNINGS R, On the Peterlin approximation for finitely extensible dumbbells , Journal of Non-Newtonian Fluid Mechanics, 68-1:85-100, 1998. [3] PHAN-TIEN N., TANNER R.I., A new constitutive equation derived from network theory, Journal of Non-Newtonian Fluid Mechanics, 2:353-365, 1977. [4] DOI M.,EDWARDS S.F., The theory of polymer dynamics , Oxford University Press, 1988. [5] MCLEISH T.C.B., LARSON R.G., Molecular constitutive equations for a class of branched polymers: the pom-pom polymer , Journal of Rheology, 42: 81-110, 1998. [6] VERBEETEN W.M.H., PETERS G.W.M., BAAJIENS F.P.T., Differential constitutive equations for polymer melts: the eXtended Pom Pom model, Journal Rheology, 45-4: 823-843, 2001 [7] PHAN-TIEN N., Understanding Viscoelasticity, Springer, Berlin, 2002. Bonn, Institut fuer Numerische Simulation