Micro and Macro in the Dynamics of Dilute Polymer Solutions

Micro and Macro in the Dynamics of Dilute Polymer Solutions Ravi Prakash Jagadeeshan

Complex Fluid Mechanics Closed form equations Constitutive Equations Stress Calculator Simulations (BDS etc) Homogeneous Flows Kinetic Theory (Microscopic) Coarse-grained Models Microscopic Physics Phenomenology Continuum Mechanics (Macroscopic) Conservation Laws Complex Flows Simulations (FEM etc) Rheological Properties in Shear and Extensional Flows Experimental Validation Velocity and Stress fields

Oldroyd-B Model for Polymer Solutions In polymer solutions, solvent contribution to stress is significant. Total stress tensor : Solvent contribution to stress: Polymer contribution to stress (UCM Model):

The Non-dimensional Oldroyd-B Model Weissenberg Number Viscosity Ratio =

The High Weissenberg Number Problem Computations with the Oldroyd-B model break down at Weissenberg number Accompanied by large stresses and stress gradients in narrow regions of the flow domain Driving force for the development of Numerical techniques - EEME (1989), EVSS (1990), DEVSS (1995), DEVSS-G (1995), AVSS (1996), DAVSS (1999), DEVSS-TG (2002) Viscoelastic flow computations are not yet safe and routine!

HWNP in Benchmark Flows Most computations of benchmark flow break down at Flow around a cylinder 4:1 Contraction flow Lid driven cavity

The Molecular Approach Microstructure cannot be ignored for complex fluids. Aim of molecular rheology is to predict properties from a molecular point of view. Chemical structure of different polymer molecules is not important because of the existence of universal behavior.

Origin of Universal Behavior The large scale features of a polymer molecule, such as: its stretchability its orientability and the large number of degrees of freedom are responsible for its universal behavior

Elements of a Molecular Theory The polymer molecule can be replaced with a coarse-grained model There is a vast separation of time-scales between the motion of the polymer and the solvent molecules The solvent molecules are replaced by the influence they have on the polymer molecules. They exert: a Brownian force a drag force

The Hookean Dumbbell Model Dumbbell model F b F d Equation of motion SPRING FORCE F s Q F s NO INERTIA HYDRODYNAMIC FORCE BROWNIAN FORCE F b F d

The Stress Tensor n The stress tensor describes the forces transmitted across an arbitrary plane in the fluid The polymer molecules contribute to the stress by carrying momentum across the surface by straddling the plane Stress is given by Kramers expression: where is the stress tensor

Conformation Tensor

Hookean Dumbbell and Oldroyd-B are the same! Polymer stress is given by: The conformation tensor is given by: Equivalent to the Oldroyd-B model with

Shear Flows Surface area, A Velocity, V Force, F y h z x Shear stress : Shear rate : Viscosity :

Shear Thinning Zero-shearrate viscosity, η 0 Power law regime The Hookean Dumbbell model fails to predict shear thinning!

Extensional Flows z 1 Λ Extensional viscosity x y

Hookean Dumbbell Model in Extensional Flows Polymer molecules unravel close to a critical Stress becomes unbounded at The polymer molecules undergo a coil-stretch transition!

Finite Size Effects The use of linear springs leads to the prediction of an unbounded stretch Bead-rod model Most realistic Difficult to simulate N k Kuhn steps

Nonlinear Springs The solution lies in using non-linear springs that account for the finite contour length of the chain The flexibility of the polymer leads to different non-linear force laws: Finitely Extensible Nonlinear Elastic (FENE) force law F Worm-like-chain (WLC) force law Q/Q max

FENE Models in Extensional Flows b : dimensionless maximum stretchable length of the molecules Bounded stresses and extensional viscosity The Hookean Dumbbell model is recovered as

Non-Linear Phenomenon need BDS Non-linear microscopic effects cannot be accounted for at the macroscopic scale without closure approximations Example of a closed form equation is the FENE-P model Brownian dynamics simulations lead to exact predictions of models with non-linear physics on the microscopic scale

Filament Stretching Rheometer Extensional Trouton Viscosity Ratio 10 3 (d) SM-1 Fluid M.I.T. De = 17 Monash De = 14 10 2 Toronto De = 12 (c) 10 1 (b) (a) Tr = 3 10 0 0 1 2 3 4 5 6 7 Hencky Strain Strain Extensional viscosity first measured accurately at Monash University

Extensional Viscosity: FENE chains and Experiment JOR, 44, 291-322, 2000 Need to adjust parameters to obtain a good fit to data

Physics in the Constitutive Equation Rallison and Hinch (1988) argued that the HWNP has a physical origin in the unrealistic behaviour of the Oldroyd-B model Rallison and Hinch suggest the use of a finitelyextensible spring Chillcott & Rallison (1988) model a dilute solution as a suspension of finitely extensible dumbbells Examine the flow of a FENE-CR fluid around a sphere and a cylinder

Flow of a FENE-CR model around a cylinder CR found that at high, polymers are most highly stretched close to the rear stagnation point in the flow around a cylinder. Stagnation Points Region of High Stretch Solutions exist for all examined for the FENE-CR model results in loss of convergence

The Continued Use of the Oldroyd-B Model The notion of self-correction (Any real flow will adapt in order to avoid infinite stresses) Solutions exist for arbitrarily large values of, it s just that current numerical techniques are unable to resolve the high stress and stress boundary layers Wapperom and Renardy (2005) used a Lagrangian technique to solve viscoelastic flow past a cylinder and showed that for an ultra-dilute solution, the governing equations for the Oldroyd-B model can be solved for arbitrarily large values of

The Log Conformation Tensor Model Fattal & Kupferman (2004) Stress is exponential in regions of high deformation rates or stagnation points Numerical instability caused by failure to balance exponential growth with convection Inappropriateness of polynomial-based approximations to represent the stress Resolution is to change variable to the matrix logarithm of the conformation tensor Still no sign of mesh convergence for Oldroyd-B!

Physics at the Mesoscale Two important long range interactions between different parts of the polymer chain must be taken into account Hydrodynamic Interactions For an accurate prediction of dynamic properties: Rheological behaviour Excluded Volume Interactions For an accurate prediction of static properties: Scaling with molecular weight Cross-over from θ to good solvent behaviour

Hydrodynamic Interactions The motion of one bead disturbs the solvent velocity field near another bead The presence of hydrodynamic interactions couples the motion of one bead to the motion of all the other beads

Consequences of HI Diffusivity ~ Diffusivity ~ 1 M 1 M without HI with HI HI ensures drag is conformation dependent

Coil-Stretch hysteresis A cross-slot cell Wi Two disparate states can exist at the same strain rate

Stress hysteresis Sridhar, Nguyen, Prabhakar, Prakash, PRL 98, 167801 (2007) Constant stress Constant strain rate The strains at quench are indicated next to the triangle symbols. Triangles of the same color have nearly the same postquench strain-rates, but different quench strains. Multiple values of stress at identical strain rates!

Glassy dynamics and hysteresis Wi 0 = 10 Wi = 0.5 Experimental observations can be reproduced by Brownian dynamics simulations

Successive Fine Graining SFG: A procedure to systematically increase the number of beads in the chain Bead-rod predictions obtained in the limit N N k (the number of Kuhn steps)

SFG & Experimental Observations " p! 10 9 10 8 10 7 10 6 10 5 (a) Wi = 9.6 0 2 4 6 8 10! Model Expt. " p! 10 9 10 8 10 7 10 6 10 5 (b) Wi = 18 0 2 4 6 8 10! Model Expt. Benchmark data from uniaxial extensional flow experiments, carried out in a filament stretching rheometer, by Sridhar and co-workers " p! 10 9 10 8 10 7 (c) Wi = 47 " p! 10 9 10 8 10 7 (d) Wi = 85 Results are for 2 million molecular weight polystyrene 10 6 10 5 Model Expt. 10 6 10 5 Model Expt. Parameter free predictions 0 2 4 6 8 10! 0 2 4 6 8 10 Prabhakar, R., Prakash, J. R. and Sridhar, T. (2004) J. Rheol., 116, 163-182.!

Closure Approximations HI can be incorporated in a constitutive equation: Zimm theory Consistent Averaging Gaussian Approximation

Hysteresis prediction Prabhakar, Prakash and Sridhar, JOR, 506, 925, 2006 The Gaussian approximation assumes the distribution to be Gaussian Accounts for fluctuations in hydrodynamic interactions

CaBER Newtonian Filament Viscoelastic Filament Anna & McKinley, J. Rheol., 45, 115 (2001) Measurements of filament radius can be used to determine relaxation times Basis for the development of the Capillary Breakup Extensional Rheometer CaBER

Concentration dependent λ? Recent experiments by Clasen et al. show a puzzling dependence of relaxation time on concentration even when c << c *! What is the source of this concentration dependence? Clasen, Verani, Plog, McKinley, Kulicke, Proc. XIV Int. Congr. On Rheology, 2004, Korea

Comparison With Experiment Prabhakar, Prakash and Sridhar, JOR, 506, 925, 2006 Inclusion of HI though the twofold normal approximation Agreement over a wide range of concentrations All curve fits are obtained with the same linear-viscoelastic relaxation time

Flow Around a Confined Cylinder Geometry and Boundary Conditions

Region of Interest: Wake of the Cylinder Region of Interest (wake of the cylinder)

Symmetry Line Simplifications is the non-dimensional component of conformation tensor non-dimensionalized using At steady state, this can be re-arranged to: The Conformation Tensor Equation is a linear ODE along the Symmetry Line can be integrated numerically

Nature of the Maxima At the maxima along the symmetry line, denoted by, As a result, If, then,

Coil-stretch transition is responsible for the HWNP! Downstream in Dilute Solutions as Computations beyond a threshold Wi are not possible using FEM as simulations break down

Upper bound on Wi? Bajaj, Pasquali, Prakash, To appear in J. Rheol., 2007 and Is Wi 0.7 the maximum computable Wi for Oldroyd-B fluids?

Complex Fluid Mechanics Coarse-grained Models Kinetic Theory (Microscopic) Microscopic Physics Micro-Macro Simulations Experimental Validation Conservation Laws Continuum Mechanics (Macroscopic) Velocity and Stress fields

Viscoelastic Free Surface Flows The image part with relationship ID rid5 was not found in the file. DIE The image part with relationship ID rid6 was not found in the file. AIR COATING BEAD The image part with relationship ID rid4 was not found in the file. OUTFLOW Micro-Macro Simulations Finite Elements/Brownian Configuration Fields Method Collaboration with Matteo Pasquali (Rice) Wall WEB Slot-Coating Flow Inflow Finite Element Mesh Ensemble of Polymer Molecules Moving Substrate Outflow

Stress, Velocity & Conformations Bajaj, Bhat, Prakash, Pasquali, J. Non-Newtonian Fluid Mech. 140, 87, 2006 Stress Velocity RELAXED Re = 0.0, Ca=0.1, Q=0.3, β=0.75 Conformations STRONGLY EXTENDED Micro-Macro scheme is stable and appears free of the High Weissenberg Number Problem

BCF with Non-Linear Dumbbells Bajaj, Bhat, Prakash, Pasquali, J. Non-Newtonian Fluid Mech. 140, 87, 2006 HI and FENE have a significant influence Need to extend to bead-spring chains

Acknowledgents Students & Postdocs: Dr Satheesh Kumar Dr Prabhakar Ranganathan Dr P Sunthar Dr Mohit Bajaj Funding Sources: ARC Discovery Monash University VPAC Expertise program

Conclusions With kinetic theory, nearly quantitative agreement with experimental results can be obtained for both macroscopic and microscopic quantities It is essential to include non-linear microscopic phenomenon to obtain accurate predictions at the macroscopic scale and to overcome computational difficulties The future lies in closure approximations and Multi-scale simulations