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Polymer Rheology
P Sunthar
Department of Chemical EngineeringIndian Institute of Technology, Bombay
Mumbai 400076, [email protected]
05 Jan 2010
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
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 3 / 44
Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology
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 ?
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 4 / 44
Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology
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).
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 5 / 44
Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology
Chemical Nature
Long chain monomers joined by chemical bondsLarge molecular weights: 1000 to 109
Linear or branchedNatural (DNA, Proteins) or Synthetic
Linear
Branched
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 6 / 44
Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology
Physical Nature
Linearity of large portions: L� dFlexibility: Not rigid long rodsIs NOT: Suspension ofpolystyrene beads
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 7 / 44
Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology
States of Polymeric Liquids
Polymer MeltsT > Tg. Eg HDPEConcentratedSolutionSemi-dilute solutionDilute Solution, Eg:Polystyrene incyclohexane
Polymer Melt
Semi-DiluteSolution
ConcentratedSolution
Dilute Solution
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 8 / 44
Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology
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
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 9 / 44
Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology
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
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 9 / 44
Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology
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
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 9 / 44
Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology
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
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 9 / 44
Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology
Need for Study of Polymeric Liquids
Polymer ProcessingReactors and MixersExtrusion MouldingFilmsFibre Spinning
Consumer ProductsShampooPastesPrinting InksPaintsLamination and Coating
Food AdditivesGumsGlycerine
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 10 / 44
Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology
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
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 11 / 44
Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology
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
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 12 / 44
Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology
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
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 12 / 44
Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology
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
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 13 / 44
Introduction Phenomenology Modelling Nature of Polymeric Liquids Polymer Rheology
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
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 14 / 44
Introduction Phenomenology Modelling Visual Linear Nonlinear
Outline of this Section
1 Introduction
2 PhenomenologyVisual PhenomenaLinear viscoelasticityNonlinear Phenomena
3 Modelling
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 15 / 44
Introduction Phenomenology Modelling Visual Linear Nonlinear
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
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 16 / 44
Introduction Phenomenology Modelling Visual Linear Nonlinear
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
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 17 / 44
Introduction Phenomenology Modelling Visual Linear Nonlinear
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
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 18 / 44
Introduction Phenomenology Modelling Visual Linear Nonlinear
Drop Formation
Jet and Drop breakupElongational flowDilute PEO solutionElongational stresses holdagainst surface tension andgravity driven breakupSatellite drop
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 19 / 44
Introduction Phenomenology Modelling Visual Linear Nonlinear
Turbulent Drag Reduction
Small amounts of polymers (ppm) to waterFluid drag in pipelines reduced significantlyTransportation of liquids.2Firefighting: Farther throw
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 20 / 44
Introduction Phenomenology Modelling Visual Linear Nonlinear
Contraction Flow
Sudden contraction low Re FlowElongational flowLip-vorticesCorner Vortices
Newtonian
Polymeric
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 21 / 44
Introduction Phenomenology Modelling Visual Linear Nonlinear
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
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 22 / 44
Introduction Phenomenology Modelling Visual Linear Nonlinear
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)
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 23 / 44
Introduction Phenomenology Modelling Visual Linear Nonlinear
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
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 24 / 44
Introduction Phenomenology Modelling Visual Linear Nonlinear
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
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 25 / 44
Introduction Phenomenology Modelling Visual Linear Nonlinear
Rheological Tests
OscillatoryControlled StressControlled Strain
Stress RelaxationAfter step strainAfter cessation of shear flow
Creep (Constant stress applied)
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 26 / 44
Introduction Phenomenology Modelling Visual Linear Nonlinear
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∗∗
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 27 / 44
Introduction Phenomenology Modelling Visual Linear Nonlinear
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′′
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 28 / 44
Introduction Phenomenology Modelling Visual Linear Nonlinear
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
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 29 / 44
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 (Maxwellmodel)
G′ ≈ Gλ2ω2
G′′ ≈ η0 ω
Cross over frequency or Characteristic relaxation time
λ =G′
G′′ ωZero-shear rate viscosity estimate
η0 ≈G′′
ω
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 30 / 44
Introduction Phenomenology Modelling Visual Linear Nonlinear
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
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 31 / 44
Introduction Phenomenology Modelling Visual Linear Nonlinear
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
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 32 / 44
Introduction Phenomenology Modelling Visual Linear Nonlinear
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
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 33 / 44
Introduction Phenomenology Modelling Visual Linear Nonlinear
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
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 34 / 44
Introduction Phenomenology Modelling Visual Linear Nonlinear
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
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 35 / 44
Introduction Phenomenology Modelling Solution Viscosity Normal Stresses Extensional Viscosity
Outline of this Section
1 Introduction
2 Phenomenology
3 ModellingBasicsShear ThinningNormal StressesExtensional Viscosity
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 36 / 44
Introduction Phenomenology Modelling Solution Viscosity Normal Stresses Extensional 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
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 37 / 44
Introduction Phenomenology Modelling Solution Viscosity Normal Stresses Extensional Viscosity
Tube Model
Chains cannot cross each otherEntanglement is like a crosslinkMotion between entanglementsPervaded volume: Tube [SamEdwards, 1967]Primitive path
Melt
Entanglement
Tube
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 38 / 44
Introduction Phenomenology Modelling Solution Viscosity Normal Stresses Extensional Viscosity
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
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 39 / 44
Introduction Phenomenology Modelling Solution Viscosity Normal Stresses Extensional Viscosity
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
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 40 / 44
Introduction Phenomenology Modelling Solution Viscosity Normal Stresses Extensional Viscosity
Shear Thinning in Melts
Entangled state (rubber like) high viscosityEntanglements are constraints for motionShear flow releases some constraintsHigh shear rate chains align along flow
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 41 / 44
Introduction Phenomenology Modelling Solution Viscosity Normal Stresses Extensional Viscosity
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
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 42 / 44
Introduction Phenomenology Modelling Solution Viscosity Normal Stresses Extensional Viscosity
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.
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 43 / 44
Introduction Phenomenology Modelling Solution Viscosity Normal Stresses Extensional Viscosity
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
P Sunthar (IIT Bombay) Polymer Rheology ComFlu 2009 44 / 44