= Modulus = Viscosity TAINSTRUMENTS.COM - - PDF document

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• Sarah Cotts

TA Instruments Rubber Testing Seminar CUICAR, Greenville SC

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• = Viscosity
• = Modulus

Rheology: The study of stress-deformation relationships Rheology: The study of stress-deformation relationships

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• =

!Ω

Ω Ω Ω =

!θ

θ θ θ " =

# $% !M

Stress (σ σ σ σ) Strain (γ γ γ γ) Strain rate ()

r = plate radius h = distance between plates M = torque (µN.m) θ = Angular motor deflection (radians) Ω= Motor angular velocity (rad/s)

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• Controlled Strain

ARES G2

Applied Strain or Rotation Measured Torque (Stress) Direct Drive Motor Transducer Sample

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Applied Strain or Rotation Measured Torque (Stress) Direct Drive Motor Transducer

RPA Elite

Controlled Strain

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!

• Viscoelasticity: Having both Viscous and Elastic properties.
• Elastic Behavior: Stress is dependent on Strain

Hooke’s Law of Elasticity: Modulus = Stress/Strain

• Viscous Behavior: Stress is dependent on Strain Rate

Newton’s Law of Viscosity: Viscosity = Stress/ Strain Rate

• Viscoelastic materials cannot be fully understood using only one of

these relationships. Oscillation measurements are able to measure stress as a function of both strain and strain rate.

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• For an Elastic Solid, Stress and

Strain have a constant proportionality = E*

• If the material follows Hooke’s

Law, the deformation will be reversible when the stress is removed

• The modulus of a Hookean solid will not show any

time dependence- the stress depends on the strain, but not the strain rate

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• For a Viscous Liquid, Stress is proportional to

Strain Rate d/dt by a coefficient of Viscosity = * d/dt

• The deformation of a liquid is non-reversible

Stress Strain Rate

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!

σ = E*ε + η*dε/dt

Kelvin-Voigt Model (Creep) Maxwell Model (Stress Relaxation)

Viscoelastic Materials: Force depends on both Deformation and Rate of Deformation and vice versa.

L1 L2

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T is short [< 1s] T is long [24 hours]

Deborah Number [De] = τ/T

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&*

Long deformation time: pitch behaves like a highly viscous liquid

• 9th drop fell July 2013

Short deformation time: pitch behaves like a solid

http://www.theatlantic.com/technology/archive/2013/07/the-3-most-exciting-words-in-science-right-now-the-pitch-dropped/277919/

Started in 1927 by Thomas Parnell in Queensland, Australia

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• Silly Putties have different characteristic relaxation times
• Dynamic (oscillatory) testing can measure time-dependent

viscoelastic properties more accurately and efficiently by varying frequency (deformation time)

()!

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Frequency of modulus crossover correlates with Relaxation Time

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RUBBER BALL TENNIS BALL

X

STORAGE (G’) LOSS (G”)

X

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• The motor applies an oscillating

deformation to the sample at a set Amplitude and Frequency.

Amplitude: Degree of Arc, or % Strain Frequency: Hz (cycles per second) or CPM (cycles per minute)

• The torque transducer measures the

response of the sample.

Amplitude: Torque (S*), or Stress

• The phase lag between the deformation

and the torque is used to determine the Elastic and Viscous response.

Deformation Torque, S* Phase Angle

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In a purely Elastic material, Stress is in phase with the Strain Viscoelastic material In a purely Viscous material, Stress is out of phase with the Strain

Phase angle= 0° Phase angle= 90° 0˚ < Phase angle< 90°

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() %-)

Elastic Material

Viscoelastic material Viscous Material

Strain Strain Strain Stress Stress Stress

Raw oscillation data can also be plotted as stress vs. strain. An elliptical shape represents a viscoelastic response.

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The Phase Angle can be used to separate the stress signal into the elastic and viscous components G’: Storage Modulus

Measure of elasticity, or the ability to store energy G’ = (Stress/Strain)*cos()

G”: Loss Modulus

Measure of viscosity, or the ability to lose energy G” = (Stress/Strain)*sin()

Tan Delta

Measure of dampening properties Tan () = G”/G’

G*: Complex Modulus

Measure of resistance to deformation G*= Stress/Strain

*: Complex Viscosity

Measure of resistance to flow * = Stress/Strain Rate

Storage Modulus Loss Modulus

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,+

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10-1

100 101

102 105

106 107

108 10-2

10-1

100 Freq [rad/s] E ' ( ) [ d y n / c m ² ] t a n _ d e l t a ( ) [ ]

Unhappy ball Happy ball

Modulus is the same tan δ is very different Happy & Unhappy Balls

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• TAINSTRUMENTS.COM

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Apart from chemical nature, tacticity and microstructure, polymers have 3 major characteristics which affect processability

• 1. Average molecular weight
• 2. Molecular weight distribution
• 3. Branching (Type and architecture)

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% .

• Long Chain Branching (LCB) provides Shear Thinning, Strain

Hardening and Melt Strength

• Very effective only one per 10000C needed.
• Difficult to fully characterize length of branches, number per chain

and distribution.

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Chromatography (SEC) NMR FTIR

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The G’ and G” curves are shifted to lower frequency (longer time) with increasing molecular weight.

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Modulus G', G'' [Pa]

SBR Mw [g/mol] G' 130 000 G'' 130 000 G' 430 000 G'' 430 000 G' 230 000 G'' 230 000

Freq ω [rad/s]

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The zero shear viscosity increases with increasing molecular weight. TTS is applied to obtain the extended frequency range.

The high frequency behavior (slope -1) is independent of the molecular weight 10

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100000 10

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Slope 3.08 +/- 0.39

Zero Shear Viscosity Zero Shear Viscosity

ηo [Pa s] Molecilar weight Mw [Daltons]

Viscosity η* [Pa s] Frequency ω aT [rad/s]

SBR Mw [g/mol] 130 000 230 000 320 000 430 000

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POE Co-Monomer Density (g/cm3) Long Chain Branching EB-1 Butene 0.901 High EB-2 Butene 0.870 Medium EB-3 Butene 0.870 High EO-1 Octene 0.870 Medium

• The level of Long Chain Branching (LCB) is difficult to

directly measure in POEs Amount and length of short chain branching interferes with NMR

• Characterize differences in LCB using rheology

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1,000 10,000 100,000 1,000,000 0.01 0.1 1 10 100 1000

α α ω ω (rad/s) (bT/α α α αT)η η η η (Poise)

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EB-1: High LCB EB-2: Med LCB EB-3: High LCB EO-1: Med LCB Viscosity (poise) Angular Frequency (rad/sec)

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EB-1: High LCB EB-2: Med LCB EB-3: High LCB EO-1: Med LCB Tan Delta Angular Frequency (rad/sec)

0.1 1 10 0.01 0.1 1 10 100 1000

α α ω ω (rad/s)

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500000 1000000 1500000 2000000 2500000 3000000 2 4 6 8 10 12 14 16 18 20

Time (s) η η η ηe (Poise)

1

• s

1 . = ε

• 190 °

C

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EB-1: High LCB EB-2: Med LCB EB-3: High LCB EO-1: Med LCB

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In the Linear Viscoelastic Region, the Stress signal is a sine wave, and meaningful rheology measurements can be made.

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At higher strains, stress becomes non-sinusoidal. Modulus is an approximation.

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%!

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SAOS LAOS

τ(t) = τ1 sin(ω1t+ϕ1) + τ3 sin(3ω1t+ϕ3) + τ5 sin(5ω1t+ϕ5) + ... = τn sin(nω1+ϕn) γ(t) = γ0 sin(ωt) τ(t) = τ0 sin(ωt+δ) γ(t) = γ0 sin(ωt)

Σ

n=1

• dd

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Sample Response Fundamental 3rd Harmonic 5th Harmonic

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0.0π 0.5π 1.0π 1.5π 2.0π

• 8000
• 6000
• 4000
• 2000

2000 4000 6000 8000

• 1200
• 1000
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• 600
• 400
• 200

200 400 600 800 1000 1200 PIB, ω=0.1 Hz, γ0=1000%, T=140°C τ [Pa] t×ω [Pa] γ [%]

• 1200
• 600

600 1200

• 10000
• 8000
• 6000
• 4000
• 2000

2000 4000 6000 8000 10000 PIB, ω=0.1 Hz, γ0=1000%, T=140° C τ [Pa] γ [%]

“The potential of large amplitude oscillatory shear to gain an insight into the long-chain branching structure of polymers” ACS Meeting 2008, Florian J. Stadler, Sunil Dhole, Adrien Leygue, Christian Bailly – Université Catholique de Louvain (UCL)

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%3%4

• 300000
• 200000
• 100000

100000 200000 300000

• 8
• 6
• 4
• 2

2 4 6 8

Shear stress (Pa) Strain rate (s-1)

Branched Linear 5% Branched in Linear

“The potential of large amplitude oscillatory shear to gain an insight into the long-chain branching structure of polymers” ACS Meeting 2008, Florian J. Stadler, Sunil Dhole, Adrien Leygue, Christian Bailly – Université Catholique de Louvain (UCL)

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“The potential of large amplitude oscillatory shear to gain an insight into the long-chain branching structure of polymers” ACS Meeting 2008, Florian J. Stadler, Sunil Dhole, Adrien Leygue, Christian Bailly – Université Catholique de Louvain (UCL)

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%

• 10
• 5

5 10

Shear rate (s-1)

• 75000
• 50000
• 25000

25000 50000 75000

Shear stress (Pa)

High visc. PP Low visc. PP 95% Low visc. PP+5% high visc. PP 50% Low visc. PP+50% high visc. PP

Secondary loops not affected by AMW and MWD

• Secondary loops can be associated

with linear polymer architecture

• There has to be a mathematical

condition to account for loops

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Ewoldt, R.H; McKinley, G. H. “On secondary loops in LAOS via self- intersection of Lissajous-Bowditch Curves.” RheologicaActa 49, no 2. February 12th, 2010. 213-219.

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/&,

ML(1+4) at 125 C

Burhin, Henri G. Polymer Process Consult

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Burhin, Henri G. Polymer Process Consult

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Burhin, Henri G. Polymer Process Consult

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Burhin, Henri G. Polymer Process Consult

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,

• Is DMA Rheology, or Thermal Analysis?
• Oscillation measurements- E’, E”, Tan Delta
• Characterize viscoelastic materials as a function of time,

temperature, stress and strain.

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120 160 300 1500 9000 30,000

M = MW between crosslinks

c

Temperature

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&&

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• Closed-Die RPA rheometer can match the viscoelastic

measurements of the ARES

• Rheological measurements within the linear visocelastic

region can be used to compare MW for non-branched polymers

• Effects of branching are seen in linear, small-amplitude

rheology measurements, especially at low frequency

• Large Amplitude Oscillatory Shear (LAOS) clearly

differentiates linear and branched polymers.

• DMA measurements also show differences in viscoelastic
• properties. These can indicate differences in chemistry or

molecular structure, but also relate to bulk properties.

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