27/11/2007 Bonn, Institut fuer Numerische Simulation 1 A century of viscoelastic modelling: from Maxwell to the eXtended Pom- Pom molecule Giancarlo Russo Cardiff School of Mathematics

227/11/2007 Bonn, Institut fuer Numerische Simulation Outline Different models for different problems Different models for different problems - Disperse polymer solutions -Upper Convective Maxwell -Oldroyd-B -Finite Extensible Nonlinear Elasticity A few words about my research project - Motivation - Results achieved - Coming next (hopefully…) Viscoelastic matter: a brief description and main features Viscoelastic matter: a brief description and main features - Concentrated polymer solutions - Phan Tien – Tanner - Tube model - Pom-Pom model - eXtended Pom-Pom model

327/11/2007 Bonn, Institut fuer Numerische Simulation Stress-Strain relation (constitutive equation) Modelling continuum mechanics Field Equation of Momentum Continuity Equation Newtonian Fluids Non-Newtonian Fluids Viscoelastic Fluids ?

427/11/2007 Bonn, Institut fuer Numerische Simulation The relaxation time, aka “will I run out of patience before the flow ?” About 10d-12 for water; more than a day for glass. Relaxation Time and Deborah Number The Deborah number, aka “I’ve been waiting enough !!!” Setting the time of the experiment will determine the behaviour of the matter. 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 Maxwell Model: Young modulus and viscosity gathered together “Memory” of the fluid

527/11/2007 Bonn, Institut fuer Numerische Simulation Some (not so obvious…) effects of elasticity High extensional viscosity/shear viscosity ratio: the open siphon effect Releasing tension along the streamlines: the Die Swell Supporting tension along the SL: Rod Climbing Visualizing different Deborah numbers: 9 to 1 mixture of cornstarch and water

627/11/2007 Bonn, Institut fuer Numerische Simulation …leads to the Upper Convective Maxwell model for the constitutive equation.T is total extra-stress and Gamma-dot is the rate of strain. Modelling disperse polymer solutions I: UCM and Kelvin (late XIXth century) The Maxwell model for shear flows… 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. 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 Let’s combine them…

727/11/2007 Bonn, Institut fuer Numerische Simulation 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. 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. Solvent vs Total viscosity ratio Weissenberg number, measure of elasticity of the fluid

827/11/2007 Bonn, Institut fuer Numerische Simulation 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. Modelling disperse polymer solutions III: Oldroyd-B at the microscope m2m2 m1m1 r1r1 r2r2 F1F1 Fluid Bulk R = r 2 - r 1 F2F2 A common expression relating stress and the product RF is due to Kramers: 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.

927/11/2007 Bonn, Institut fuer Numerische Simulation Modelling disperse polymer solutions IV: FENE models FENE FENE – P (Peterlin) FENE – CR (Chilcott - Rallison) Limiting the dumbbells extensibility through bounding functions; overcoming the discontinuity in the extensional viscosity. Mainly suitable for extensional flows. Combining the Kramers, Giesekus and the FENE-P we obtain The quantity R 0 is the maximum extension the spring can reach. It predicts constant shear flow, a non- zero 1 st 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. The FENE-CR replaces the expression above by the following: It presents the same features as Old-B but the extensibility is bounded.

1027/11/2007 Bonn, Institut fuer Numerische Simulation 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: Shear thinning; extensional bounded; 1 st 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 2 nd NSD is the main limit for extensional simulations. Multiplying both sides by rho and averaging, we obtain the Const. Eq. 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.

1127/11/2007 Bonn, Institut fuer Numerische Simulation 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 Disentanglement time comparison Stress-orientation tensor Q(E) relation Psi is the key; the reptation dynamics is responsible for the change of conformation when elastic effects are taken into account..

1227/11/2007 Bonn, Institut fuer Numerische Simulation 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) 2 nd NSD is not predicted

1327/11/2007 Bonn, Institut fuer Numerische Simulation Modelling concentrated polymer solutions IV: the XPP model (2001) Viscoelastic Stress Backbone stretch and stretch relaxation time Extra Function Relaxation time tensor Blackwell modification: a smoother approach to the maximum BB stretch by mean of withdrawing arms before such maximum stretch is achieved Tackling issue # 1 Tackling issue # 2 High strain rates means the 1 st term totally outweighs the 2 nd 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 The Giesekus - like 2 nd order term is responsible for predicting 2 nd NSD. ARE WE HAPPY ??? SADLY NOT…

1427/11/2007 Bonn, Institut fuer Numerische SimulationModel Fluids described FeaturesDrawbacksViscosityOldroyd-B Boger fluids; Polymer Solutions (molecules are far enough from each other so that their interactions can be neglected) Analytical Solutions Available for both transient and steady flow (Channel); direct derivation from F - P eq.; 1 st NSDiff predicted Infinite extensibility of dumbbells allowed (unphysical); no shear thinning predicted; no 2 nd NSD Shear : constant over a wide range of shear rates; Ext.: possibly unbounded. FENE models Polymer solutions General agreement with experiments in extensional flows. Maximum extension bounded. No 2 nd NSD predicted. Poor representation of shear flows. FENE-P is shear thinning. The others reflects much better extensional properties. PTT (Phan T hien-Tanner) Polymer Melts (molecules are not so dispersed anymore) Extension of dumbells bounded; shear thinning predicted; 1 st NSDiff predicted. No 2 nd NSD predicted Shear thinning, Extensional bounded. Pom Pom Polymer Melts (molecules are not so dispersed anymore) Vortex prediction in contraction flows due to branches effect; shear thinning and strain hardening predicted. Derivatives of ext. viscosity discontinuous; no 2 nd NSD. Orientation tensor possibly unbounded. Shear thinning, constant plateau without Blackwell’s mod.; Tension Thickening at low strain rates. XPP (eXtended Pom Pom) Polymer Melts (molecules are not so dispersed anymore) Derivatives of ext. viscosity are smooth; 2 nd NSD also predicted by Giesekus term. Ext. unbounded and shear approaching constant plateau if ETA is switched off; possible unbounded stress and imaginary stretch. Shear thinning. Tension Thickening at low strain rates. Review

27/11/2007 Bonn, Institut fuer Numerische Simulation 15 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. EXTRUDATE SWELL A viscoelastic fluid is extruded from a pipe. The 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 FILAMENT STRETCHING 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. Velocity extrapolated on the FS along the vertical gridlines. Nodes shifted. New transfinite mapping built. Moving forward in time. Pressure HorizontalVelocityVerticalVelocity My research project in few words 1) Discretization in space: the Spectral Element Methods + transfinite techniques 1) Extending the code to the real problems 2) Discretization in time: usually 1st order + OIFS for the material derivative RESULTS ACHIEVED 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. Ongoing and doming next (hopefully…) 2) Improve the MATRIX-LOG version ( problem with some stress profiles) and make it model independent (so far just Oldroyd-B) 3) Gather the two approaches

1627/11/2007 Bonn, Institut fuer Numerische Simulation [1] PHILLIPS T.N., OWENS. R., Computational Rheology, Imperial College Press, [2] KEUNINGS R, On the Peterlin approximation for finitely extensible dumbbells, Journal of Non-Newtonian Fluid Mechanics, 68-1:85-100, [3] PHAN-TIEN N., TANNER R.I., A new constitutive equation derived from network theory, Journal of Non-Newtonian Fluid Mechanics, 2: , [4] DOI M.,EDWARDS S.F., The theory of polymer dynamics, Oxford University Press, [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: , [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: , 2001 [7] PHAN-TIEN N., Understanding Viscoelasticity, Springer, Berlin, References