Polymer Rheology. P Sunthar. Department of Chemical Engineering Indian Institute of Technology, Bombay Mumbai , India
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1 Polymer Rheology P Sunthar Department of Chemical Engineering Indian Institute of Technology, Bombay Mumbai , India P.Sunthar@iitb.ac.in 05 Jan 2010
2 Introduction Phenomenology Modelling Outline of the Lecture 1 Introduction 2 Phenomenology 3 Modelling P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
3 Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology Outline of this Section 1 Introduction Nature of Polymeric Liquids Polymer Rheology 2 Phenomenology 3 Modelling P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
4 Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology Questions to Ask for a New Phenomena Fundamental Questions What makes the phenomena different? How to represent in terms of a mathematical model? Are there distinct laws or rules for the behaviour? Are there other known phenomena that obey similar laws? What role has this played in the current state of the universe? Application oriented questions Can it be employed for betterment of quality of life? Consequences to processes that manipulate the material? P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
5 Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology Polymeric Liquids Definition Liquids that contain Polymers Liquids: Materials that flow Simple Liquids Definition: Material that does not support shear stress at rest Complex fluids Liquid (viscous) and Solid (elastic) like behaviour Dynamic properties are not thermodynamic constants Eg: Viscosity η = f ( γ), η = f (t). P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
6 Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology Chemical Nature Long chain monomers joined by chemical bonds Large molecular weights: 1000 to 10 9 Linear or branched Natural (DNA, Proteins) or Synthetic Linear Branched P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
7 Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology Physical Nature Linearity of large portions: L d Flexibility: Not rigid long rods Is NOT: Suspension of polystyrene beads P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
8 Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology States of Polymeric Liquids Polymer Melts T > T g. Eg HDPE Concentrated Solution Semi-dilute solution Dilute Solution, Eg: Polystyrene in cyclohexane Polymer Melt Concentrated Solution Semi-Dilute Solution Dilute Solution P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
9 Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology Role of Temperature Noodle Soup What is the difference between a polymeric liquid to us and a huge bowl of noodles to a Giant? Noodles are linear, Soup is like a solvent. Difference Random linear translating motion Noodles is a zero temperature (Frozen) system Polymeric liquid is a finite temperature system P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
10 Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology Role of Temperature Noodle Soup What is the difference between a polymeric liquid to us and a huge bowl of noodles to a Giant? Noodles are linear, Soup is like a solvent. Difference Random linear translating motion Noodles is a zero temperature (Frozen) system Polymeric liquid is a finite temperature system P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
11 Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology Role of Temperature Noodle Soup What is the difference between a polymeric liquid to us and a huge bowl of noodles to a Giant? Noodles are linear, Soup is like a solvent. Difference Random linear translating motion Noodles is a zero temperature (Frozen) system Polymeric liquid is a finite temperature system P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
12 Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology Role of Temperature Noodle Soup What is the difference between a polymeric liquid to us and a huge bowl of noodles to a Giant? Noodles are linear, Soup is like a solvent. Difference Random linear translating motion Noodles is a zero temperature (Frozen) system Polymeric liquid is a finite temperature system P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
13 Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology Need for Study of Polymeric Liquids Polymer Processing Reactors and Mixers Extrusion Moulding Films Fibre Spinning Consumer Products Shampoo Pastes Printing Inks Paints Lamination and Coating Food Additives Gums Glycerine P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
14 Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology Nobel in Physics Pierre-Gilles de Gennes \d -zhen\ Nobel in Physics: 1991 Nobel for generalising theory of phase transitions to polymers and liquid crystals. Scaling Theory in Polymeric liquids Reptation in Polymer Melts Coil-stretch transitions in Extensional flows Polymer induced Turbulent drag reduction P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
15 Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology Polymer Rheology Industrial Flows are Complex Geometry Polydisperse and Multi-component Understand Response to Simple flows (Viscometric) Shear Elongational Understand Response of Simple Materials (reproducible) Single or two component systems Monodisperse molecular weight Dilute Systems Melts (Pure polymer) Rheology Science of Deformation and Flow P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
16 Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology Polymer Rheology Industrial Flows are Complex Geometry Polydisperse and Multi-component Understand Response to Simple flows (Viscometric) Shear Elongational Understand Response of Simple Materials (reproducible) Single or two component systems Monodisperse molecular weight Dilute Systems Melts (Pure polymer) Rheology Science of Deformation and Flow P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
17 Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology Rheology Core: Viscosity and Elasticity What is Deformation? Relative displacements within material Measured by Deformation (Strain): γ Resisted by Elasticity Deformation Shear Elongation G = σ xy γ What is Flow? Continuous Relative motion Measured by rate of Deformation (Strain rate): γ Resisted by viscosity η = σ xy γ Flow Shear Elongation P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
18 Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology Polymers, Soft Matter, Complex Fluids Liquid Viscosity Modulus η (Pa.s) G (Pa) Water An Oil A polymer solution 1 10 A polymer melt A glass > > Soft Materials Elasticity has Entropic Origin (Not Energetic origin as for solids) G proportional to k B T times number concentration of flexible units Physical feel of softness, intermediate G Complex mechanical response and microstructure P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
19 Introduction Phenomenology Modelling Visual Linear Nonlinear Outline of this Section 1 Introduction 2 Phenomenology Visual Phenomena Linear viscoelasticity Nonlinear Phenomena 3 Modelling P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
20 Introduction Phenomenology Modelling Visual Linear Nonlinear Weissenberg Rod Climbing Effect Rod rotating in a polymeric liquid Fluid climbs the rod Common fluids that show Gum solutions Batter (with egg white) Due to Normal stress differences psidot, Youtube: npzzlgkjs0i P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
21 Introduction Phenomenology Modelling Visual Linear Nonlinear Extrudate or Die Swell POLYOX TM (PEO, PEG) Solution Ejected from a syringe Significant increased diameter upon exit Also known as Barus Effect Newtonian fluids diameter does not change significantly Due to Normal stress differences psidot, Youtube: KcNWLIpv8g P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
22 Introduction Phenomenology Modelling Visual Linear Nonlinear Tubeless Syphon Elongational flow Stresses hold up against gravity and surface tension After initial pouring (suction) a free-surface syphon is maintained. Also known as Fano Flow psidot, Youtube: ay7xigq-7iw P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
23 Introduction Phenomenology Modelling Visual Linear Nonlinear Drop Formation Jet and Drop breakup Elongational flow Dilute PEO solution Elongational stresses hold against surface tension and gravity driven breakup Satellite drop P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
24 Introduction Phenomenology Modelling Visual Linear Nonlinear Turbulent Drag Reduction Small amounts of polymers (ppm) to water Fluid drag in pipelines reduced significantly Transportation of liquids.2 Firefighting: Farther throw P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
25 Introduction Phenomenology Modelling Visual Linear Nonlinear Contraction Flow Sudden contraction low Re Flow Elongational flow Lip-vortices Corner Vortices Newtonian Polymeric P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
26 Introduction Phenomenology Modelling Visual Linear Nonlinear Relaxation Times Observable microscopic time scale, λ Simple liquids λ sec Time for large scale changes in polymer configurations Microseconds to minutes Similar order of macroscopic observation period and processing rates Configurations altered by thermal energy Elasticity λ is an Elastic time scale P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
27 Introduction Phenomenology Modelling Visual Linear Nonlinear Dimensionless Numbers Macroscopic time scales Kinematic (rate of deformation) time scale γ for shear flows ɛ for extensional flows Dynamic time scale, t d Time to traverse a geometry or section Pulsatile flow May not be known apriori Weissenberg Number For Viscometric flows (with kinematic timescale) Wi = λ γ or λ ɛ (1) Deborah Number For complex flows (with dynamic timescale) De = λ t d (2) P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
28 Introduction Phenomenology Modelling Visual Linear Nonlinear Molecular Weight Dependence of Relaxation Time Large scale motion depends on M Scaling dependence for a class of liquids Class Scaling Dilute solution in poor solvent λ M 1.0 Dilute solution in θ-conditions λ M 1.5 Dilute solution in good solvent λ M 1.8 Semi dilute solution λ chain M 2 Entangled Melts λ rep M 3.4 P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
29 Introduction Phenomenology Modelling Visual Linear Nonlinear Linear Response Response to small imposed deformation Linearity means additive response Linearity of Response in Viscous properties Elastic properties Linear Viscoelastic Properties Mainly Polymer physics Liquid Viscosity Relaxation time Modulus η (Pa.s) λ (s) G (Pa) Water An Oil A polymer solution A polymer melt A glass > > P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
30 Introduction Phenomenology Modelling Visual Linear Nonlinear Rheological Tests Oscillatory Controlled Stress Controlled Strain Stress Relaxation After step strain After cessation of shear flow Creep (Constant stress applied) P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
31 Introduction Phenomenology Modelling Visual Linear Nonlinear Zero-shear rate viscosity Linear response ( γ 0) Micro-structural information Dilute: c < c Intrinsic Viscosity (inverse concentration) [η] 0 lim γ 0 [η] lim γ 0 lim c 0 η η s c η s [η] 0 λ M Semi-dilute: c < c < c log ηsp0 Dilute Entangled Semi Dilute c 2 1 c log c 14/3 η sp0 = η 0 η s Entangled: c > c P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
32 Introduction Phenomenology Modelling Visual Linear Nonlinear Small Amplitude Oscillatory Tests G : Elastic Modulus; G : Viscous log(g ) log(g ) Viscous Transition to Flow Rubbery/Plateau G G λ 1 log(ω) Glassy P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
33 Introduction Phenomenology Modelling Visual Linear Nonlinear Plateau Modulus with Molecular Weight Increased M Increased Entanglements Rubber like network Entanglements are like cross-links log(g ) M Crosslinked Polymer log(g ) Entangled Melt G 0 N log(ω) Unentangled Melt log(ω) P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
34 Introduction Phenomenology Modelling Visual Linear Nonlinear Characteristic Relaxation Time Low Frequency response always Viscous G > G Wait long enough, even Mountains will flow! Low frequency scaling for all polymeric liquids (Maxwell model) G G λ 2 ω 2 G η 0 ω Cross over frequency or Characteristic relaxation time λ = G G ω Zero-shear rate viscosity estimate η 0 G ω P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
35 Introduction Phenomenology Modelling Visual Linear Nonlinear Stress Relaxation Small step strain γ is linear Response G(t) = σ xy /γ G(t) Fourier Transform G (ω) Small t large ω: Elastic Large t small ω: Viscous (flow) η 0 = Area under the G(t) curve G(t) log G(t) Rouse τe G 0 N G 0 N τrep Reptation t η 0 λ G(0) for exponentially decaying tail: exp t/λ Monomer τ0 log t Rouse τe τrep Reptation P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
36 Introduction Phenomenology Modelling Visual Linear Nonlinear Shear Thinning Decrease in viscosity upon shear More pronounced in concentrated solutions than dilute Intermediate shear rates: Power Law Fluid Worm-like Micelles Living Polymers abrupt changes Cylindrical micelles Breaking and forming Large shear rates most are small fragments log η Dilute Solution Worm like Micelle 1 log γ Concentrated solution P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
37 Introduction Phenomenology Modelling Visual Linear Nonlinear Normal Stresses Simple liquids: Normal stress is the pressure Complex fluids: Microstructure leads to flow induced anisotropy Normal Stresses: log σ, N1 N 1 N 1 = τ xx τ yy σ N 2 = τ yy τ zz Shear thinning for ψ 1 = N 1 / γ 2 N 2 is usually 0 for polymeric liquids log γ P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
38 Introduction Phenomenology Modelling Visual Linear Nonlinear Extensional Viscosity Stretching and Compressing flow field Contraction flow Stagnation points Spinning of fibres Break up of jets to drops Blow moulding Elongational viscosity η E Experiments: Transient (not Steady) η + E Tensile Stress Growth Coefficient Strain ( ɛ t) hardening log η + E ǫ log t P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
39 Introduction Phenomenology Modelling Visual Linear Nonlinear Trouton Ratio Ratio of extensional to shear viscosity T R = η E(ɛ) η( 3 ɛ) Newtonian Liquids: T R = 3 log η log ηe 1000 Solutions η E 3 η log ǫ, Branched Melts log γ Dilute Solution Linear Melts log TR 100 Melts 3 Inelastic liquid 1/2 λ log ǫ, log γ P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
40 Introduction Phenomenology Modelling Solution Viscosity Normal Stresses Extensional Viscosity Outline of this Section 1 Introduction 2 Phenomenology 3 Modelling Basics Shear Thinning Normal Stresses Extensional Viscosity P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
41 Introduction Phenomenology Modelling Solution Viscosity Normal Stresses Extensional Viscosity Dilute Solution and Colloidal Suspensions Spherical particles only on the average Like Porous particles (fluid can pass through) Suspension viscosity (Einstein) η = η s ( φ) Dilute polymer solution η = η s ( 1 + UηR φ ) U ηr = 1.66 Zimm theory U ηr 1.5 Molecular simulations and Experiments P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
42 Introduction Phenomenology Modelling Solution Viscosity Normal Stresses Extensional Viscosity Tube Model Chains cannot cross each other Entanglement is like a crosslink Motion between entanglements Pervaded volume: Tube [Sam Edwards, 1967] Primitive path Melt Entanglement Tube P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
43 Introduction Phenomenology Modelling Solution Viscosity Normal Stresses Extensional Viscosity Reptation and other Relaxation Times Smallest time τ 0 : Monomer relaxation Intermediate τ e : Rouse relaxation between entanglements Largest τ rep : Reptation or relaxation along the length of the tube [P G de Gennes, 1971] Diffusion time of polymer is reptation time Monomer Rouse Reptation P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
44 Introduction Phenomenology Modelling Solution Viscosity Normal Stresses Extensional Viscosity Relaxation Modulus and Reptation Relaxation after step strain G 0 N Initial monomer relaxation τ 0 Plateau region, relaxation between entaglements τ e Terminal region, reptation τ rep Viscosity related to reptation time G(t) Rouse τe τrep Reptation t η 0 τ rep G(0) log G(t) G 0 N Monomer Rouse τ0 τe log t τrep Reptation P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
45 Introduction Phenomenology Modelling Solution Viscosity Normal Stresses Extensional Viscosity Shear Thinning in Melts Entangled state (rubber like) high viscosity Entanglements are constraints for motion Shear flow releases some constraints High shear rate chains align along flow P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
46 Introduction Phenomenology Modelling Solution Viscosity Normal Stresses Extensional Viscosity Understanding Normal Stress Difference Anisotropy in microstructure Equilibrium: spherical pervaded volume Shear Flow: Stretch and Tumble Shear pervaded volume: inclined ellipsoidal Restoring force in normal planes are different Normal stress difference Equilibrium Shear yy xx P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
47 Introduction Phenomenology Modelling Solution Viscosity Normal Stresses Extensional Viscosity Extensional Viscosity in Dilute Solutions Equilibrium: Spherical pervaded volume Small extension rates ɛ λ < 0.5, small deformation Large extension rates: stretching of chain, larger stress Equilibrium Small Extn. Large Extn. P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
48 Introduction Phenomenology Modelling Solution Viscosity Normal Stresses Extensional Viscosity Extensional Viscosity in Melts Reptation Entanglements and Confining tube Tube orientation Rouse time: Chain Stretching log ηe Reptation Orientation Stretching τ 1 rep log ǫ τ 1 e Fully Stretched P Sunthar (IIT Bombay) Polymer Rheology ComFlu / 44
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