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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 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
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
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
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
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)
Adapted from Laun et. al. (1992)
Latex Suspensions with Yield Stress
The yield stress is best measured with a stress controlled rotational rheometer