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Rheology with Application to Polyolefins
Teresa Karjala, Ph.D. and Dr. Ir. Sylvie VervoortThe Dow Chemical Company, Packaging and Specialty Plastics
Lake Jackson, Texas and Terneuzen, The Netherlands Thoi Ho and Terry Vermass Tutorial
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Outline
• Introduction• Steady Shear Flow• Oscillatory Shear Flow• Extensional Rheology• Concluding Remarks
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Rheology
• Science dealing with deformation and flow of materials.
• Includes both molten and solid state behavior.
• Most commonly measured for materials such as polymers, polymer solutions,paint, food, and blood.
• Requires measuring the deformation resulting from a given force or measuring aforce required to produce a given deformation.
• Importance for polymers: Processing: extrusion, gear pumps, flow through pipes, pressure drops, etc. Relation to molecular structure such as:
o Molecular weight (Mw)o Molecular weight distribution (MWD)o Long chain branching (LCB)
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Deformation
4
• Measurement and analysis• Single deformation modes• Straight-forward description• Measure material properties
• Flow in applications• Mixed deformation modes• Complex analysis
Lo
F
Lo
Lo s
LoLt
FF
• Shear
• Elongation
4
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Viscosity (Newtonian Fluid)
• Viscosity relates to the resistance of a material to flow.
• Linear relationship between shear stress and shear rate: 𝜎 = h 𝛾.
For simple shear, the constant of proportionality is the viscosity, h .
• A material that behaves in this way is a Newtonian fluid. The viscosity does not depend on the shear rate.
Stre
ss(s
)
Slope is viscosity, h
Vis
cosi
ty(h
)
Shear rate ( 𝛾) Shear rate ( 𝛾)
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Viscoelasticity and Non-Newtonian Behavior
• The majority of materials are neither purely elastic or viscous, but are consideredviscoelastic (exhibit viscous resistance and elasticity).
• In this case, the relationship between the stress and strain rate is no longer linearand cannot be described in terms of a single constant, h.
• Generalized equation for steady simple shear: h ( 𝛾) = s / 𝛾in which the h is function of 𝛾.
Shear rate ( 𝛾)
Vis
cosi
ty (
h)
Newtonian
Shear thinning/Pseudoplastic
Shear thickening/Dilatant
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7
10
100
1000
10000
0.01 1 100 10000
ho
r h
* (P
a.s
)
shear rate (s-1) or frequency (rad.s-1)
Polypropylene at 190 °C
steady shear
dynamic
capillary
Plateau regionViscosity constant Zero-shear viscosity, h0
Shear thinning regionViscosity with shear rate
Non-Newtonian Behavior: Typical Polymer Flow Curve
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Non-Newtonian Behavior: Viscosity Models
8
Power law
𝜎 = K 𝛾𝑛
𝜂 = K 𝛾𝑛−1
Cross: 𝜂
𝜂0=
1
1+ λ 𝛾1−𝑛
n-1 < 0: shear-thinning
n-1 > 0: shear-thickening
n = 1: Newtonian fluid
h = viscosity, 𝛾 = shear rate
ho = zero shear viscosity
λ = relaxation time
n = high shear rate fitting parameter
𝐶𝑎𝑟𝑟𝑒𝑎𝑢:𝜂
𝜂𝑜=
1
1 + (𝜆 𝛾 2 (1−𝑛)/2
[T.A. Plumley et al., SPE ANTEC Proceedings, p. 1221 (1994)]
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Temperature Dependency of Viscosity (Activation Energy)
9
• Viscosity with temperature
• G’ and G” with temperature
Overlay
Temperature
Non-Newtonian polymerIsothermal frequency sweeps
Arrhenius: 𝑎𝑇 =h0(𝑇)
h0(𝑇𝑅𝑒𝑓)
= 𝑒𝑥𝑝𝐸𝑎
𝑅
1
𝑇−
1
𝑇𝑅𝑒𝑓,
Ea flow activation energy
TTS
Overlay
Master curve at TRef = 190 oC
Temperature
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Viscometric Flows • With these flow geometries, the viscosity, as well as fluid velocities, shear rates,
and pressure distributions can be determined analytically.
Viscometric flow Flow example
Steady tube flow Capillary flow
Steady slit flow Cast die
Annular pressure flow Blown film die
Steady concentric flow Brookfield viscosity
Steady parallel disk flow DMS
Steady cone and plate flow DMS
Steady sliding cylinder flow Wire coating
Steady helical flow Spiral die
Combined drag/P flow Extrusion
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Useful Equations for Steady Flow Through a Capillary and Slit
Flow Geometry Shear Rate, 𝛾
Shear Stress,t or s
Depiction of Flow
Capillary 𝛾 = 32𝑄
𝜋𝐷3=
4𝑄
𝜋 𝑅3𝜏 =
∆𝑃
4 (𝐿
𝐷)
=∆𝑃
2 (𝐿
𝑅)
Slit 𝛾 = 6𝑄
𝑊𝐻2𝜏 =
∆𝑃
2 (𝐿
𝐻)
𝛾 = shear rate
Q = volumetric flow rate
DP = pressure drop
V = velocity
R = radius of capillary
L = length of capillary
D = diameter of capillary
W = width of slit
H = height of slit
11
R VzQ D
HW Q Vz H
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Typical Shear Rates in Common Processes
Process Shear rate (s-1) Application
Extrusion 100 - 103 Polymer melts, food
Mixing 101 - 103 Liquid manufacturing
Spraying, brushing 103 - 104 Spray-drying, paints
Rubbing 104 - 105 Creams & lotions
Injection molding 102 - 105 Polymer melts
Coating flows 105 - 106 Paper
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Steady Shear Flow
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Shear Flow: Measurement MethodsRotational rheometer Capillary rheometer
• Uniform simple shear flow• Steady, oscillatory or creep flow• Rates: low (creep) to moderate• Variety of tools
• Steady tube flow in capillary die• Steady shear flow• Entry/exit effects Apparent viscosity• Rates: moderate to high• Variety of dies
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Melt Index (ASTM D-1238, ISO 1133)
• Industry standard.• Similar to capillary rheology except use load instead of instrumented
crosshead and short die.• Single-point measurement of flow rate:
Melt index: g/10 minMelt index , viscosity
• For Polyethylene:T = 190 oCLoad = 2.16 kg (I2); 10 kg (I10); 21.16 (I21)2Rb=0.376”2R=0.0825”L/D=3.818”
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Combining Different Methods Gives Broad Flow Curves
EPDM at 190 oC
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Melt Index Correlations
• Based upon modeling a melt indexer as capillary flow:• Thus, the shear rate at which higher melt indexes are measured is greater. The
stress correspondingly increases with the MI load.
25.2~
.I
100
1000
104
105
0.1 1 10 100 1000 104
1 MI30 MI
Vis
cosity (
pois
e)
Shear Rate (s-1
)
I2 I
10
I2 I
10
Melt Index
Weight (kg)
Shear Stress (dyn/cm2)
Pressure (Psi)
Pressure (MPa)
I2 2.16 1.93 x 105 43 0.296
I5 5 4.47 x 105 99 0.683
I10 10 8.94 x 105 198 1.37
I21 21.6 1.93 x 106 429 2.96
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Shear Flow: Melt Fracture at High Shear Rates, Capillary Flow
Smooth Surface Melt Fracture Gross Melt Fracture
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Effect of Molecular Weight on Zero Shear Viscosity
Mw dependency
• h0 as Mw
• Above Mw,cr: 𝜂0 = 𝑘𝑀𝑤3.4
k = f(T)
Mw,cr depends on polymer type
• Below Mw,cr: 𝜂0 = 𝑘𝑀𝑤1
10
100
1,000
10,000
100,000
1,000 10,000 100,000
Vis
cosi
ty (
cP)
14
9 o
C
Weight Average Molecular Weight, Mw (g/mol)
Polyethylene
Slope ~1
Slope ~3.4
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Shear Flow: Effect of Polymer Properties
h wM0
= 2.29•10-15
a = 3.65 ± 0.08
10-1
100
101
102
103
104
105
106
107
h0(1
90
°C),
Pa•s
4 5 6
104
2 3 4 5 6
105
2 3 4 5 6
Mw, g/mol
HDS set
HD set
h0 = 2.29•10-15
(Mw)3.65±0.08
95% confidence limits
Molecular Weight and h0
[Karjala et al., J. Appl. Polym. Sci., 119, 636-646 (2011)]
Polymer architecture and h
1. h0 increases with LCB2. Shear thinning increases with LCB
104
105
0.01 0.1 1 10 100
Co
mp
lex V
isco
sity (
po
ise
)
Frequency (rad/s)
Linear
CGCLLDPE w/ LCB
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Shear Flow: Effect of Polymer Properties
LCB as low as 0.005 is easily detected for resins with
Mw=110,000 g/mol
NMR GPC GPC h 0 (creep),
Resin LCB M w M w/M n Pa•s ZSVR
/1000C g/mol 190 oC
B0P2 0 121,200 2.03 9,569 1.16
B1P2 0.005 121,900 2.11 13,670 1.63
B2P2 0.01 120,200 2.17 13,720 1.72
B3P2 0.056 101,700 2.58 10,640 2.46
B4P2 0.109 87,800 2.18 16,040 6.33
B0P4 0 122,100 4.14 10,790 1.28
B1P4 0.014 112,300 4.47 11,040 1.78
B2P4 0.052 104,900 3.96 15,640 3.23
B3P4 0.106 85,800 3.81 17,190 7.38
B0P6 0 126,900 6.75 9,597 0.99
B1P6 0.013 102,300 6.09 10,040 2.27
B2P6 0.052 102,600 6.37 14,190 3.17
B3P6 0.106 83,200 5.55 11,540 5.55
B0P8 0 131,300 8.64 9,844 0.89
B1P8 0.014 104,400 8.85 10,050 2.11
B2P8 0.052 100,900 8.2 14,310 3.40
B0P10 0 131,300 11.03 10,920 0.99
B1P10 0.014 102,200 10.02 9,918 2.25
B2P10 0.07 93,400 10.12 12,510 3.94
ZSVR Sensitive to identify LCB
[Karjala et al., J. Appl. Polym. Sci., 119, 636-646 (2011)]
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Oscillatory Shear Flow
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Parallel plates
23
Shear Flow Rheometers
• One part rotating at rad/s, torque M• Steady or oscillatory shear flow• Small amplitude oscillatory flow
h(r)
Cone and plate
V
h(r)
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Small Amplitude Oscillatory Shear Flow
• Oscillatory (dynamic, sinusoidal) deformation (0, ) 0: maximum amplitude, typically a small deformation : angular frequency
• Response: sinusoidal stress (s0, ) at a radial distance : phase angle Measure for viscoelasticity
• Use: to study structured materials without disturbing the structure proxy for steady shear flow (Cox-Merz rule)
90°: viscous liquid0°: elastic solid
Stress, s*
Strain,
time, t
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Oscillatory Shear Flow
• Properties measured: h*: complex viscosity G’: Storage modulus (elastic) G”: Loss modulus (viscous) tan = G”/G: damping factor
• Typical use: Polymer melts Viscoelastic materials Structured materials
10-1
100
101
102
102
103
104
40.0
45.0
50.0
55.0
60.0
65.0
70.0
75.0
Freq [rad/s]
E
ta*
()
[P
a-s
]
PhaseA
ngle
()
[°]
LDPE 621 (2.3 MI) 190°C Ares-2 5% strain
10-1
100
101
102
102
103
104
105
Freq [rad/s]
G
' (
)
[P
a]
G
" (
)
[P
a]
LDPE 621 (2.3 MI) 190°C Ares-2 5% strain
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Oscillatory Shear Flow
logG’logG”
log
G’
G”
logG’logG”
log
G’
G”
Typical liquid- Frequency dependent- G’ < G”
Typical solid- Frequency independent- G’ > G”
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Oscillatory Shear Flow
Polymer behaviorSmall scale motions
Typically measured range
Mo
du
lus
(lo
g)
Angular frequency (log)
12
G’
G”
G
GN0
Terminal Plateau Transition Glassy
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van Gurp - Palmen (vGP) Plot
• (G*) Impression of sample topology
/ morphology Verify TTS
• Reduced vGP plot (G*/GN0)
Polymer topology Across chemistries
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Extensional Rheology
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Uniaxial Elongational Flow Devices
• Rotating drums on rotational rheometers
• Different vendors, different names EVF, SER, UXF,…
• Homogeneous deformation
• Only for high viscosity materials sample should not sag
• Limited to relatively low elongation rates
• Max. H = 4
• Sources of error: slip, necking, sagging
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Extensional Flow: Strain Hardening
1 MI LDPE150 oC
• Strain-hardening if hE > LVE envelope
• Strain-hardening factor 𝑆𝐻𝐹 =𝜂𝐸
3𝜂𝑆
• Strain-hardening required to withstand flow in stretching processes
• LCB contributes heavily
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Extensional Flow: Melt Strength by Rheotens
• Melt strength• Drawability• Draw resonance
• Fiber is drawn from a die at increasing pull-off velocity (increasing force) until filament breaks
• Can reach high stretch rates, ~processing• Transient, non-uniform stretching
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Extensional Flow: Melt Strength
• Effect of Mw (MI)• Effect of polymer structure
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
0 50 100 150 200 250 300 350 400 450
Me
lt S
tre
ngt
h (
cN)
Velocity (mm/s)
190 oC
LDPE-0.18MI
LDPE-0.65MI
LDPE-1.85MI
22 cN
14 cN
8 cN
[Karjala et al., SPE ANTEC Proceedings (2016)]
0
5
10
15
20
25
0 10 20 30 40 50 60 70 80 90 100
Me
lt S
tre
ngt
h (
cN)
% LDPE (in LLDPE-1 or DOWLEXTM 2045G)
190 oC
LDPE-0.65MI
LDPE-1.85MI
®™Trademark of The Dow Chemical Company (“Dow”) or an affiliated company of Dow
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Concluding Remarks
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Summary
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References
• General rheology references T.G. Mezger, Applied Rheology, Anton Paar GmbH (2015). F. A. Morrison, Understanding Rheology, Oxford (2001). R.G. Larson, The Structure and Rheology of Complex Fluids, Oxford (1998). C. W. Macosko, Rheology: Principles, Measurements, and Applications, VCH (1994). A.A. Collyer et al., Rheological Measurement, Springer (1993). J. E. Mark et al., Physical Properties of Polymers, 2nd Ed., ACS (1993). J. M. Dealy et al., Melt Rheology and Its Role in Plastics Processing, Wiley (1990). H. A Barnes et al., An Introduction to Rheology, Elsevier (1989). J. M. Dealy, Rheometers for Molten Plastics, SPE (1982).
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References Continued
• Detailed (classic references) R.B. Bird, R.C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Wiley (1987). J. D. Ferry, Viscoelastic Properties of Polymers, 3rd Ed., Wiley (1980).
• Fluid mechanics M. M. Denn, Process Fluid Mechanics, Prentice-Hall (1980). R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, Wiley (1960).
• Polymer Processing Z. Tadmor and C. G. Gogos, Principles of Polymer Processing, Wiley (1979). S. Middleman, Fundamentals of Polymer Processing, McGraw-Hill (1977).
• Journals/Proceedings J. Applied Polymer Science, J. Polymer Science, J. Rheology, Rheologica Acta,
Macromolecules, Polymer Engineering and Science, Applied Rheology Society of Plastics Engineers ANTEC Proceedings
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Acknowledgements: Teresita Kashyap of The Dow Chemical Company
Questions, feel free to contact either presenter at:Teresa Karjala ([email protected])Sylvie Vervoort ([email protected])