Rheology

the science of flow and deformation of matter

{Force}MaterialElement

(m){Deformation}

Material Functionor

Rheological Equation of state

)( ,, tijijfij

Stress

ij, , ijij

Strain

Difference bt. Rheology and Fluid Mechanics

Equation of motion for the

system+

Measuredpressure dropand flow ratefor the fluid

in the system

Rheologicalproperties of

the fluid

Fluid Mechanics

Rheology

Equation of motion for the

system+

Measuredpressure dropand flow ratefor the fluid

in the system

Rheologicalproperties of

the fluid

Applied Fields of Rheology

• Polymer Rheology (Solution, Melt, Solid) • Suspension and Emulsion Rheology • Electro- and Magneto- Rheology• Food Rheology• Bio-rheology, Hemo-rheology, Psyco-rheology• Chemorheology• Lubricant Rheology• Surface Rheology

Why Rheology ?

Rheology is sensitive to material structure => characterization

Rheology describes the flow behaviour => processing behaviour

Rheology correlates with end use performance => material performance

Molecular Structure• MW & MWD• Chain Branching and Cross-linking• Interaction of Fillers with Matrix Polymer• Single or Multi-Phase Structure

Viscoelastic PropertiesAs a function of :• Strain Rate(frequency)• Strain Amplitude• Temperature

Processability & Product Performance

Interrelationship bt. Structure-Property and Processing

1. 시간 의존성을 갖는 완화 탄성율 G(t) :

점탄성

2. 전단담화 점도 거동 η( )

3. 정상상태의 단순 전단장 하에서의 수직

응력 τ11-τ22 > 0

4. 연신심화 점도 거동 ηE( )

Viscoelastic Behaviors

Rod Climbing (Weissenberg) Effect

Newtonian Viscoelastic

Free surface shape for a rotating rod in a reservior

Viscoelastic Fluid Flow in a Sudden Contraction Tube

Streakline photographs illustrating the changing vortex growth as a function of λ for a viscoelastic liquid flowing in a 4.08 to 1 circular contraction (from Mackay and Boger, 1988).

Die Swell

• Die swell is related to the elastic properties of materials: result of a disorientation of macromolecules which have been

oriented within the die by the high shear field.: result of the recovery of the elastic deformation imposed in the die.

• Die swell ratio depends on molecular parameters: increase with MW and MWD : increase with long chain branching

• Die swell ratio depends on process parameters

tT, ,DLf swellDie

Rheological Explanation on Die swell

Die Design

smooth

sharkskin

Slip-stick

slip

Gloss

Melt fracture

Fig. Apparent wall shear stress vs. apparent shear rate of the metallocene based LLPDE resin at T=120oC. ( L/D=30, D=1mm )

Melt Fracture

Influence of Long Chain Branching on Melt Fracture

Fig. Photographs of the extruded strands for three resins at four apparent shear rates. (150oC, Tungsten carbide die, D=1mm, L/D =30; (a) 40.3 s-1, (b) 115.4 s-1, (c) 639.6 s-1, (d) 1246.4 s-1)

smooth surface melt fracturesharkskin gross melt fracture

Apparent shear rate (s-1)

101 102 103

Resin A

Resin B

Resin C

Fig. Photographs of the extruded strands for Resin Aat various processing conditions. ( Powder A; (a) 115.4 s-1,(b) 224.1 s-1, (c) 851.3 s-1, (d) 2007.4 s-1)

Temperature : 2000oCPressure : 14 MPaBinder : B2O3(boric acid ;2-5 wt%)

Hot-pressed Boron Nitride die

Diameter : 1 mmL/D : 30 Entry angle : 180o

Die Surface Effect on Melt Fracture

Ultrasonic Improvement of the Productivity of Extrusion

PS extrudates at 200 oCEffect of ultrasonic vibrations on the pressure drop

Fig. Stained extrudate cross sections of Nylon6,12/HDPE blends from the 1.5” extruder (Nylon appears black)

220oC

240oC

Polymer Migration:

The lower viscosity component tends to migrate to the region of higher shear rate

Additional relaxation at low frequency is a result of the spherical domain relaxation

1E-031E-02

1E-011E+00

1E+011E+02

1E+031E+04

Frequency (rad/s)

1E+00

1E+01

1E+02

1E+03

1E+04

1E+05

1E+06

1E+07G

’; G

’’ (P

a˜G’ B02 G’’ B10G’ B20 G’’’ B02G’ B10 G’’ B20

PS/PMMA Blend

LDPE/SEBS/PS Blend

Temperature

-65 -50 -35 01E+05

1E+06

1E+07

1E+08

1E+09

1E+10M

odul

us E

˜ [P

a]

0.001

0.01

0.1

1

10

tan

E˜ (SBR˜ tan delta (SBR˜E˜(SBR +CB˜ tan delta (SBR+CB˜E˜ (NR˜ tan delta (NR˜

Rubber

SBR has a Tg at 44oC. Adding carbon black increases the modulus.

If the SBR is replaced with polyisoprene (natural rubber*, the transition shifts to lower temperature (56oC).

25 30 35 40Time t [min]

1E˜01

1E+00

1E+01

1E+02

1E+03

1E+04

1E+05

Mod

uli G

˜, G

" [P

a]

1E˜01

1E+00

1E+01

1E+02

tan

Reactive Systems

The time dependence of the moduli allows to follow the cure.

The crossover point of G‘ and G“ can be correlated with the gel point

0.1 1 10 100frequency [rad/s]

1E+00

1E+01

1E+02

1E+03

1E+04

G',G

'' [P

a]; V

isco

sity

[Pas

]

G'

G''*

tan = G''/G' >1 to 1.5good stability

The frequency dependence of the modulus below the yield characterizes the internal structure:

• tan must be between 1 - 1.5 for best stability

• tan <1: elasticity too high, interparticle forces cause aggregation

• tan >1.5: purely viscous behaviour, no interparticle forcesprevent coagulation

Storage Stability of Complex Fluids

Category of Rheometerwhat material functions they can measure

An instrument that measures both stress and deformation history on a material to determine material functions

Rheometer

Kinematics shear rheometerextension rheometer

Type of strainingsmall strainlarge strainsteady straining

Type of geometryhomogeneousnon-homogeneousindexer

Type of Rheometer

Bubble collapseRotating clamps,inflation methodsSimple extension,lubricated compression

Sliding platesConcentric cylindersCone and plateEccentric rotating disksShear surfaceParallel disksCapillary

SlitAnnulusN

onho

mog

eneo

usH

omog

eneo

us

EXTE

NSI

ON

SHEA

R

RHEOMETERS

),( t

),( tG

)( t)( tG

),( tu

),( tE

Shear

Extension

largestrain

small strain

Steadystraining

)(),( 1

)(2

)(),( 21

Fiber spinningStagnation flows

Falling ballRotating diskExtrudate swellPressure holeSqueezing flows

Entrance flowsINDEXERS

Spectrum of Material Classification in simple shear deformation

Rigid Solid(Euclidean)

Linear Elastic Solid(Hookean)

Nonlinear Elastic Solid

Nonlinear Viscous Fluid(Non-Newtonian)

Linear Viscous Fluid(Newtonian)

Inviscid Fluid(Pascalian)

Viscoelastic

0

0

)(

G

)(G

),,( tF

Solid

Fluid

Stress

333231

232221

131211

ij

F~

1F

3F

2F

X1

X2

X3

etc. 3333

2121

1212

AFAFAF

1

3

13

12

11

221

22

23

33

32

31

Classical Strain

Displacementgradient at point 1 1

021

21 limlim121

sd

udsu

ssuu

sss

Strain : a quantitative measure of the deformation of a material element

Deformation occurs whenever any twopoints in a material are displaced from their initial position such that a change in the separation between them results

The magnitude of the deformation isdetermined by the relative displacements of the points.

s

21

1X

2X

3X

u

1s

2u

2s

1u 2u

u

xu

xu

xu

xu

xu

xu

xu

xu

xu

xu

sdud

j

i ~

3

3

2

3

1

3

3

2

2

2

1

2

3

1

2

1

1

1

TT

i

j

j

i

i

j

j

i

j

i uuuuxu

xu

xu

xu

xu ~~

21~~

21

21

21

Pure deformation Pure rotation

Deformation Tensor

jiT

i

j

j

i

j

i

i

ji

j

j

iijij

vvxv

xv

tu

xtu

xxu

xu

tte

~~

T

i

j

j

iij uu

xu

xue ~~

Strain Tensor

Rate of Strain Tensor

Strain and Rate of Strain Tensor

-p

Isotropic and Anisotropic Stress and Strain

Isotropic (volumetric) stress and strain

Anisotropic (shear) stress and strain

)()( 31 , 332211 ijijijij tr

v3

3

2

2

1

1332211 3

2)( 32)(

31

exv

xv

xveee

ijijij

For an incompressible (isochoric) material

0 , , ijijoijijijoij pbriumat equilip

)( ,0)tr( 31 ij

i

j

j

iijij x

vxve

Total Stress and Strain

Constitutive Equation

ijij f

ijij f

t ,, ijijij f

Purely Viscous Fluid

Elastic Solid

Viscoelastic Fluid

Criterion of Viscoelasticity

Fig. Schematic diagram showing the behavior of viscoelastic fluids.

Deborah No.flow

fluid

tDe

- flow instabilities- slip-stick- extrudate roughness

flow timethe inverse of the typical deformation rate

1

The inverse of the amplitude of the oscillatory strain times its frequency 1

0

relaxation time

GG

fluid

Pipkin-Diagram

Maximum relaxation time

What is a maximum relaxation time?

in transient: G0e-t/ for t== max = t(0.367G0)

in dynamic: G' = G'' = Gc

=> max=1/

G0

tGc

0 0 zyx vvyv ,,

222111

233222

Viscosity Coefficient

First Normal Stress Difference Coefficient

Second Normal Stress Difference Coefficient

12

Simple Shear Flow

x2

x1x3

V

2211x xvv xx

x2

x1x3

x2

x1x3

V

2211x xvv xx

Fig. Master curves for the viscosity and first normal stress difference coefficient as functions of shear rate for the LDPEmelt. Reference temperature = 423 K.

Viscosity and 1st normal stress difference coefficient as a function of shear rate

1st and 2nd normal stress difference coefficient as a function of shear rate

Fig. Dependence of the first and second normal stress coefficients on shear rate for two polymer solutions and a soap solution

2% PIB in Primol

7% aluminium lauratein decalin and m-cresol

1.5% PAA in a water-glycerin mixture

2.5% PAAin a 50/50water-glycerin mixture

3% PEO in a 57/38/5Water-glycerin-isopropylalcohol mixture

y

x

A. steadyshearflow

yv x

y

x

B. small-ampitudeoscillatoryshear

ytv ox ))cos((

y

x

C. stress growthupon inceptionof steady shearflow

Fluid at rest

0xv yv ox

Steady shear flow

Stressgrowth

t < 0 t > 0

Various Types of Simple Shear Flow

y

x

D. Stress relaxationafter cessationof steady shearflow t > 0

yv ox

Steady shear flow

t < 0

0xv

Motion suddenly stopped

Stressrelaxation

y

x

E. Stress relaxationafter a suddenshearingdisplacement

t > 0

Fluid is rest

t < 0

0xv yv ox

Fluid is rest

Stressrelaxation

y

x

F. Creep

t > 0

Constant shear stress appliedFluid is rest

t < 0

0xv ytv x )( Creep

y

xrecoil

G. Constrainedrecoil aftersteady shearflow

t > 0

Shear stress suddenly removedSteady shear flow

t < 0

yv ox ytv x )(

Various Types of Simple Shear Flow(continued)

Material Functions in Simple Shear Flows

Flow Material Function

Steady shear flow

Small-amplitude oscillatory shear

stress growth upon inception of steady shear flow

Stress relaxation after cessation of steady shear flow

Stress relaxation after a suddenshearing displacement

Creep

Constrained recoil aftersteady shear flow

constantyx

tcoso

000 t , t o

00 ,0 t t yxoyx

toyx )(

0 ,00 t t oyxyx

000 t , t yxoyx

2 1 ,,

GG

,

,

02010 ,,,,, ttt

0 0 0 21 ,,,,, ttt

0G 0 1 ,,, ttG

0,tJ

0 0 00 0 er Jt ,,,,

Deformation that involves stretching along streamlines.

Simple extension: (same streamlines)

Simple shear: (same streamlines) (different streamlines)

Extentional (Shearfree ) Flow

Strong Flow:

(Weak flow in shear flow:

Irrotational flow - deformation by stretching & aligning(rotational flow in shear flow - deformation by tumbling &stretching)

Not a viscometric flowThe nonvanishing third invariant of deformation rate tensor

Three major different types of extensional flows;uniaxial, biaxial, planar extensional flows

texpL)t(L 0 (exponential function)

tx)0(x)t(x 211 (linear function)

Characteristics of Extensional Flow

biaxial extension

uniaxial extension

planar extension

Three Major Typesof Extensional Flow

Rate of deformation (Strain) tensor:

m1000m0001

2uniaxial: m = -1/2biaxial: m = 1planar: m = 0

Three Major Typesof Extensional Flow (continued)

Unification of shearfree flows

xb121Vx

yb121Vy

zVz

uniaxial: b=0,biaxial: b=0,planar: b=1,

000

Extensional Material Functions

Uniaxial extension:

),t(,t EE 33112211E

for linear viscoelastic region: )t(3)t(),t(lim EE0

Biaxial extension: B

BBBB

),t(,t

33223311B

for linear viscoelastic region:

Planar extension:

33111P ,t

33222P ,t

)t(6)t(),t(lim BBB0B

for linear viscoelastic region: )t(41P )t(22P

Extensional Viscosity of LDPE

Method Advantages Disadvantages

Cone and plate Homogeneous 0.1 rad Best for N1 Best for G(t, )

High : low, edge failure, loading difficult

Low : inertia Evaporation Need good alignment

Parallel disks (Torsional flow)

Easy to load viscous samples Best for G’ and G” for melt, curing

Vary by h and (N1-N2)( )

Nonhomogeneous:not good for G(t,) Ok for G(t) and ( ) Edge failure Evaporation

Concentric cylinders (Couette flow)

Low , high Homogeneous if Ri/Ro0.95 Good for suspension settling

End correction N1 impractical High fluids are difficult to clean

Capillary(Poiseuille flow)

High Sealed Process simulation ext from Pent Wide range with L

Corrections for Pent time-consuming Nonhomogeneous: no G(t,) Bad for time dependence Extrudate swell only qualitative for N1

Comparison of Shear Rheometers

Comparison of Shear Rheometers

Method Advantages Disadvantages

Sliding plates Simple design Homogeneous Linear motion High , G(t, ) t 10-3 s

Edges limit <10 Gap control Loading

Slit flow No Pent with wall-mounted pressure transients

(p) Pex, Ph give N1

Edge effects with W/B>5 Similar to capillary Difficult to clean

Axial annular Flow

Slit with no edges P can give N2

Difficult construction and clean

Falling ball Very simple Neddle better Sealed rheometer High T, p

Not useful for viscoelastic fluids Nonhomogeneous Transparent fluid Need

Squeeze flow Simple Process simulation ( ) at long times

Indexer flow: mixed shear rates and shear transients

Contained bobs Sealed Process simulator

Indexers Friction limits range

(continued)

Viscosity measured by several Rheometers

Adapted from Laun (1988)

Adapted from Laun et. al. (1992)

Latex Suspensions with Yield Stress

The yield stress is best measured with a stress controlled rotational rheometer

The Cox-Merz Relation

3.5% PAA in water

PDMS melt

Closed symbols: cone and plate

Open symbols: birefringence

Comparison of Viscosity and First Normal Stress Coefficient