Polymer Rheology - Department of Chemical Engineering - Indian

Polymer Rheology

P Sunthar

Department of Chemical EngineeringIndian Institute of Technology, Bombay

Mumbai 400076, [email protected]

05 Jan 2010

[email protected]

Introduction Phenomenology Modelling

Outline of the Lecture

1 Introduction

2 Phenomenology

3 Modelling

P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 2 / 44

Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology

Outline of this Section

1 IntroductionNature of Polymeric LiquidsPolymer Rheology

2 Phenomenology

3 Modelling



Questions to Ask for a New Phenomena

Fundamental QuestionsWhat 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 theuniverse ?

Application oriented questionsCan it be employed for betterment of quality of life?Consequences to processes that manipulate the material ?



Polymeric Liquids

DefinitionLiquids that contain Polymers

Liquids: Materials that flowSimple Liquids

Definition: Material that does not support shear stress atrest

Complex fluidsLiquid (viscous) and Solid (elastic) like behaviourDynamic properties are not thermodynamic constantsEg: Viscosity η = f (γ̇), η = f (t).



Chemical Nature

Long chain monomers joined by chemical bondsLarge molecular weights: 1000 to 109

Linear or branchedNatural (DNA, Proteins) or Synthetic

Linear

Branched



Physical Nature

Linearity of large portions: L� dFlexibility: Not rigid long rodsIs NOT: Suspension ofpolystyrene beads



States of Polymeric Liquids

Polymer MeltsT > Tg. Eg HDPEConcentratedSolutionSemi-dilute solutionDilute Solution, Eg:Polystyrene incyclohexane

Polymer Melt

Semi-DiluteSolution

ConcentratedSolution

Dilute Solution



Role of Temperature

Noodle Soup

What is the difference between apolymeric liquid to us and a hugebowl of noodles to a Giant?

Noodles are linear, Soup is like asolvent.Difference Random lineartranslating motionNoodles is a zero temperature(Frozen) systemPolymeric liquid is a finitetemperature system



Role of Temperature

Noodle Soup





Role of Temperature

Noodle Soup





Role of Temperature

Noodle Soup





Need for Study of Polymeric Liquids

Polymer ProcessingReactors and MixersExtrusion MouldingFilmsFibre Spinning

Consumer ProductsShampooPastesPrinting InksPaintsLamination and Coating

Food AdditivesGumsGlycerine



Nobel in Physics

Pierre-Gilles de Gennes \dU-zhen\1932–2007Nobel in Physics: 1991Nobel for generalising theory of phasetransitions to polymers and liquidcrystals.Scaling Theory in Polymeric liquidsReptation in Polymer MeltsCoil-stretch transitions in ExtensionalflowsPolymer induced Turbulent dragreduction



Polymer Rheology

Industrial Flows are ComplexGeometryPolydisperse and Multi-component

Understand Response to Simple flows (Viscometric)ShearElongational

Understand Response of Simple Materials (reproducible)Single or two component systemsMonodisperse molecular weightDilute SystemsMelts (Pure polymer)

Rheology

Science of Deformation and Flow



Polymer Rheology

Industrial Flows are ComplexGeometryPolydisperse and Multi-component

Understand Response to Simple flows (Viscometric)ShearElongational

Understand Response of Simple Materials (reproducible)Single or two component systemsMonodisperse molecular weightDilute SystemsMelts (Pure polymer)

Rheology

Science of Deformation and Flow



Rheology Core: Viscosity and Elasticity

What is Deformation?Relative displacements withinmaterialMeasured by Deformation(Strain): γResisted by Elasticity

G =σxy

γ

What is Flow?Continuous Relative motionMeasured by rate ofDeformation (Strain rate): γ̇Resisted by viscosity

η =σxy

γ̇

Deformation

Flow

Shear

Elongation

Elongation

Shear



Polymers, Soft Matter, Complex Fluids

Liquid Viscosity Modulusη (Pa.s) G (Pa)

Water 10−3 109

An Oil 0.1 108

A polymer solution 1 10A polymer melt 105 104

A glass > 1015 > 1010

Soft MaterialsElasticity has Entropic Origin (Not Energetic origin as forsolids)G proportional to kBT times number concentration offlexible unitsPhysical feel of softness, intermediate GComplex mechanical response and microstructure


Introduction Phenomenology Modelling Visual Linear Nonlinear


1 Introduction

2 PhenomenologyVisual PhenomenaLinear viscoelasticityNonlinear Phenomena

3 Modelling



Weissenberg Rod Climbing Effect

Rod rotating in a polymericliquidFluid “climbs” the rodCommon fluids that show

Gum solutionsBatter (with egg white)

Due to Normal stress differences

psidot, Youtube:npZzlgKjs0I


http://www.youtube.com/watch?v=npZzlgKjs0I

http://www.youtube.com/watch?v=npZzlgKjs0I


Extrudate or Die Swell

POLYOXTM(PEO, PEG) SolutionEjected from a syringeSignificant increased diameterupon exitAlso known as Barus EffectNewtonian fluids diameter doesnot change significantlyDue to Normal stress differences

psidot, Youtube:KcNWLIpv8g


http://www.youtube.com/watch?v=KcNWLIpv8g

http://www.youtube.com/watch?v=KcNWLIpv8g


Tubeless Syphon

Elongational flowStresses hold up against gravityand surface tensionAfter initial pouring (suction) afree-surface syphon ismaintained.Also known as Fano Flow

psidot, Youtube:aY7xiGQ-7iw


http://www.youtube.com/watch?v=aY7xiGQ-7iw

http://www.youtube.com/watch?v=aY7xiGQ-7iw


Drop Formation

Jet and Drop breakupElongational flowDilute PEO solutionElongational stresses holdagainst surface tension andgravity driven breakupSatellite drop



Turbulent Drag Reduction

Small amounts of polymers (ppm) to waterFluid drag in pipelines reduced significantlyTransportation of liquids.2Firefighting: Farther throw



Contraction Flow

Sudden contraction low Re FlowElongational flowLip-vorticesCorner Vortices

Newtonian

Polymeric



Relaxation Times

Observable microscopic time scale, λSimple liquids λ ∼ 10−15 secTime for large scale changes in polymer configurationsMicroseconds to minutesSimilar order of macroscopic observation period andprocessing ratesConfigurations altered by thermal energy⇔ Elasticity⇒ λis an Elastic time scale



Dimensionless Numbers

Macroscopic time scalesKinematic (rate of deformation)time scale

γ̇ for shear flowsε̇ for extensional flows

Dynamic time scale, tdTime to traverse a geometry orsectionPulsatile flowMay not be known apriori

Weissenberg Number

For Viscometric flows(with kinematictimescale)

Wi = λ γ̇ or λ ε̇ (1)

Deborah NumberFor complex flows (withdynamic timescale)

De =λ

td(2)



Molecular Weight Dependence of Relaxation Time

Large scale motion depends on MScaling dependence for a class of liquids

Class Scaling

Dilute solution in poor solvent λ ∼M1.0

Dilute solution in θ-conditions λ ∼M1.5

Dilute solution in good solvent λ ∼M1.8

Semi dilute solution λchain ∼M2

Entangled Melts λrep ∼M3.4



Linear Response

Response to small imposed deformationLinearity means additive responseLinearity of Response in

Viscous propertiesElastic properties

Linear Viscoelastic PropertiesMainly “Polymer physics”

Liquid Viscosity Relaxation time Modulusη (Pa.s) λ (s) G (Pa)

Water 10−3 10−12 109

An Oil 0.1 10−9 108

A polymer solution 1 0.1 10A polymer melt 105 10 104

A glass > 1015 105 > 1010



Rheological Tests

OscillatoryControlled StressControlled Strain

Stress RelaxationAfter step strainAfter cessation of shear flow

Creep (Constant stress applied)



Zero-shear rate viscosity

Linear response (γ̇ → 0)Micro-structural informationDilute: c < c∗

Intrinsic Viscosity (inverseconcentration)

[η]0 ≡ limγ̇→0

[η] ≡ limγ̇→0

limc→0

η − ηs

c ηs

[η]0 ∼λ

M

Semi-dilute: c∗ < c < c∗∗

ηsp0 = η0 − ηs

Entangled: c > c∗∗

1

2

14/3

Semi−

Dilute

Entangled

Dilute

log c

logη

sp0

c∗

c∗∗



Small Amplitude Oscillatory Tests

G′: Elastic Modulus; G′′: Viscous

Rubbery/PlateauGlassy

Viscous Transition to Flow

log(ω)

log(

G′ )

log(

G′′ )

∼ λ−1

G′

G′′



Plateau Modulus with Molecular Weight

Increased M⇒ IncreasedEntanglementsRubber like networkEntanglements are likecross-links

Crosslinked Polymer

Entangled Melt

Unentangled Melt

log(ω)

log(

G′ )

G0N

log(ω)

log(

G′ )

M



Characteristic Relaxation Time

Low Frequency response always Viscous

G′′ > G′

Wait long enough, even Mountains will flow!Low frequency scaling for all polymeric liquids (Maxwellmodel)

G′ ≈ Gλ2ω2

G′′ ≈ η0 ω

Cross over frequency or Characteristic relaxation time

λ =G′

G′′ ωZero-shear rate viscosity estimate

η0 ≈G′′

ω



Stress Relaxation

Small step strain γ is linearResponse G(t) = σxy/γ

G(t)→ Fourier Transform→ G∗(ω)Small t⇒ large ω: ElasticLarge t⇒ small ω: Viscous (flow)η0 = Area under the G(t) curve

η0 ∼ λG(0)

for exponentially decaying tail:exp−t/λ

Reptation

Rouse

t

G(t

)

G0N

τe

τrep

Reptation

RouseMonomer

log t

log

G(t

)

G0N

τ0 τe

τrep



Shear Thinning

Decrease in viscosity upon shearMore pronounced inconcentrated solutions thandiluteIntermediate shear rates: PowerLaw FluidWorm-like Micelles “LivingPolymers” abrupt changes

Cylindrical micellesBreaking and formingLarge shear rates most aresmall fragments

−2

2

−5 −1 3

4

0Dilute Solution

Concentrated solution

Worm−like Micelle

log γ̇

logη



Normal Stresses

Simple liquids: Normal stress isthe pressureComplex fluids: Microstructureleads to flow induced anisotropyNormal Stresses:

N1 = τxx − τyy

N2 = τyy − τzz

Shear thinning for ψ1 = N1/γ̇2

N2 is usually 0 for polymericliquids

log γ̇

logσ,

N1

N1

σ



Extensional Viscosity

Stretching and Compressing flowfield

Contraction flowStagnation pointsSpinning of fibresBreak up of jets to dropsBlow moulding

Elongational viscosity ηE

Experiments: Transient (notSteady) η+

E “Tensile StressGrowth Coefficient”Strain (ε̇ t) hardening

log t

logη+ E ǫ̇



Trouton Ratio

Ratio of extensional to shearviscosity

TR =ηE(ε)η(√

3 ε̇)

Newtonian Liquids: TR = 3

SolutionsBranched Melts

Linear Melts

log ǫ̇, log γ̇

logη

logη

E

η

ηE

×3

3

100

1000

Melts

Inelastic liquid

Dilute Solution

log ǫ̇, log γ̇lo

gT

R

1/2λ


Introduction Phenomenology Modelling Solution Viscosity Normal Stresses Extensional Viscosity


1 Introduction

2 Phenomenology

3 ModellingBasicsShear ThinningNormal StressesExtensional Viscosity



Dilute Solution and Colloidal Suspensions

Spherical particles only on theaverageLike Porous particles (fluid canpass through)Suspension viscosity (Einstein)

η = ηs (1 + 2.5φ)

Dilute polymer solution

η = ηs(1 + UηR φ

)UηR = 1.66 Zimm theoryUηR ≈ 1.5 Molecular simulationsand Experiments



Tube Model

Chains cannot cross each otherEntanglement is like a crosslinkMotion between entanglementsPervaded volume: Tube [SamEdwards, 1967]Primitive path

Melt

Entanglement

Tube



Reptation and other Relaxation Times

Smallest time τ0: MonomerrelaxationIntermediate τe: Rouserelaxation betweenentanglementsLargest τrep: Reptation orrelaxation along the lengthof the tube [P G de Gennes,1971]Diffusion time of polymeris reptation time

Monomer

Rouse

Reptation



Relaxation Modulus and Reptation

Relaxation after step strainInitial monomer relaxation τ0

Plateau region, relaxationbetween entaglements τe

Terminal region, reptation τrep

Viscosity related to reptation time

η0 ∼ τrep G(0)

Reptation

Rouse

t

G(t

)

G0N

τe

τrep

Reptation

RouseMonomer

log t

log

G(t

)

G0N

τ0 τe

τrep



Shear Thinning in Melts

Entangled state (rubber like) high viscosityEntanglements are constraints for motionShear flow releases some constraintsHigh shear rate chains align along flow



Understanding Normal Stress Difference

Anisotropy in microstructureEquilibrium: spherical pervadedvolumeShear Flow: Stretch and TumbleShear pervaded volume: inclinedellipsoidalRestoring force in normal planesare different⇒ Normal stress difference

Shear

Equilibrium

yy

xx



Extensional Viscosity in Dilute Solutions

Equilibrium: Spherical pervadedvolumeSmall extension rates ε̇ λ < 0.5,small deformationLarge extension rates: stretchingof chain, larger stress

Equilibrium

Small Extn.

Large Extn.



Extensional Viscosity in Melts

ReptationEntanglements and Confiningtube

Tube orientationRouse time: Chain Stretching

Reptation Orientation Stretching

Ful

ly S

tret

ched

log ǫ̇

logη

E

τ−1rep τ−1

e


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Polymer Rheology - Department of Chemical Engineering - Indian