measuring polymers using a rotational rheometer in ... polymers using a rotational rheometer in...
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Measuring Polymers using a Rotational Rheometer in Oscillatory Mode
Steve GoodyerProduct Manager for Rheology
Anton Paar Ltd.
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←←←← Extrusion (e.g. Polystyrene, PS):extrudate swellingand melt fracture
IntroductionViscoelastic Behavior
Polymer melts
Blow moulding →→→→(e.g. Polyethylene, PE):
orange peelor
shark skin
polymers
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Agenda
� (A little bit of theory – G’ and G’’)Measuring Polymers with a rheometer�Amplitude sweeps – lvr�Frequency Sweeps
- molecular interactions- fingerprints- degree of cross linking- molecular weight from zsv- relaxation time from x-over ����mmd
�Temperature-DMTA
�Time Sweeps- cure profile- reaction kinetics
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Introduction Viscoelastic Behavior
ideally viscousliquids
like water, oilsLaw of Newton
ideally elastic(rigid) solids
like stone, steelLaw of Hooke
viscoelasticliquids
like glues, shampoos
viscoelasticsolids
like pastes, gels, rubbers
←←←← rotational tests →→→→ |←←←← ←←←← ←←←← ←←←← ←←←← ←←←← oscillatory tests →→→→ →→→→ →→→→ →→→→ →→→→ →→→→
Using a simple illustrative picture:
„The Rheology Road“
viscous viscoelastic elastic←←←← →→→→
→ e-learning (Eiffel tower)
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DefinitionsShear Stress, shear deformation or shear strain
shear stress
shear deformationor shear strain
The Two-Plates Model
ττττ =AF
unit: 1 N / m 2 = 1 Pa (Pascal)
γγγγ =hs
unit: 1 m / m = 1 = 100 %
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DefinitionsElasticity Law
γγγγττττ
====GRobert Hooke (1635 to 1703)
unit of the shear modulus: (1 Pa / 1 = ) 1 Pa
further units:1 GPa = 1000 MPa = 106 kPa = 109 Pa(Giga-pascal, Mega-pascal, kilo-pascal)
Spring Law: F / s = C
spring force Fdeflection path sspring constant C (stiffness)
definition of theshear modulus
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Rheometry
Oscillatory Tests: Basics (1)
Two-Plates Model
Ideally elastic behaviorof a totally stiff sample(e.g. a stone, or steel):
There is no shift betweenthe sine curves of shear strain (deformation) and shear stress :the curves of γγγγ and ττττare “in phase “
→ Movie(2-plates-model,ideal-elastic behavior)
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RheometryOscillatory Tests: Basics (2)
Preset: constant frequency and constant amplitude
Result: Most samples are showing viscoelastic behaviorwith the phase shift δδδδ between the sine curves of the test preset (e.g. strain) and the measuring result (then: stress),as a retardation of the measuring response to the preset oscillation.
It counts: 0° ≤≤≤≤ δδδδ ≤≤≤≤ 90°ideally elastic ideally viscous behavior
→ Movie(2-plates-model,visco-elastic behavior)
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Physically:G' for the stored and G'' for the lost (dissipated) deformation energy
tan δδδδ [1][1][1][1] = G''/ G' Loss Factor or Damping Factor as the ratio between the viscous and the elastic portion
RheometryOscillatory Tests: Basics (3)
Elasticity Law of Hooke(for oscillation):
Index A for „Amplitude“
G* [Pa] Complex Shear Modulus
G' [Pa] Storage Modulus, elastic portionG'' [Pa] Loss Modulus, viscous portion
of the viscoelastic behavior
A
A*
γτ=G
Vector Diagram
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Preset:constant frequency
(e.g. the angular frequencyωωωω = 10 rad/s or s-1)
andvariable strain (deformation)
Frequency Conversion: ωωωω = 2ππππ ⋅⋅⋅⋅ f with angular frequency ωωωω [s-1] and frequency f [Hz]
(since Hz is not an SI unit !)
Rheometry (Oscillation)Amplitude Sweeps, preset
→ Movie(amplitude sweep)
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Result:storage modulus G' (elastic behavior),loss modulus G'' (viscous behavior),limiting value of the linear viscoelastic (LVE- ) r ange when reaching γγγγL
- at the given test conditions, i.e., at the preset (angular) frequency -
left side: G‘ > G‘‘ (“gel - like structure“) in the LVE - range right side: G‘‘ > G‘ (“liquid - like structure“) in the LVE - range
Limiting valueof the LVE - range
Viscoelastic Behavior Amplitude Sweeps
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Pa
lg G'
lg G''
10-2
10-1
100
101
102
103
%strain lg γγγγ
Polymer Melt
ω = 10 rad/sT = +180°C
Viscoelastic Behavior Amplitude Sweeps
↑↑↑↑
↓↓↓↓
limit of the LVE range at
γγγγ = 10% = 0.1
viscoelastic liquid,
liquid-like character since G‘‘ > G‘
polymers105
103
104
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0.1
1
10
100
1000
kPa
0.001 0.01 0.1 1 10 100%
Sealant
ω = 10 rad/sT = +25°C
paste-like, viscoelasticgel-like character
in the LVE rangesince G‘ > G‘‘
Viscoelastic Behavior Amplitude Sweeps
↑↑↑↑
↓↓↓↓
Limit of the LVE-range at
γγγγ = 0.026% = 2.6 ⋅ 10-4
(with 10% tolerance deviation)
polymers dispersions
lg G'
lg G''
strain lg γγγγ
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Frequency Sweeps
�Measure time dependency of viscoelastic properties.
�Frequency of applied strain with constant amplitude is logarithmically varied.
�Results typically plotted as G’, G’’ vs frequency, f (or ωωωω= 2ππππf).
�Generally speaking the shorter the timescale the more elastic a material behaves.
�Consider as viscoelastic spectrum.
�Results relate to molecular structure of the sample.t=0 t= 5 min
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Frequency SweepVisco-elastic liquid (no gel, unlinked, no filler)
� Long term: newtonian behaviour� Short term: viscoelastic behaviour
Angular frequency ω
Complex viscosity
G‘‘ G‘1
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1
� No network structure
� No links between macro-molecules
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γγγγ = 10 %T = +23°C
101
102
103
104
105
Pa
103
104
Pas
10-3
10-2
10-1
100
101
102
103
rad/sangular frequency lg ωωωω
Typical behavior of an unlinked polymer
Viscoelastic Behavior Frequency Sweeps
G′′′′′′′′ > G′′′′ ←←←← crossover →→→→ G′′′′ > G′′′′′′′′
ηηηη0 = 35 kPas
polymers
lg G'
lg G''
PDMS(poly - di - methyl
- siloxane)
• lg |ηηηη*|
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Frequency SweepVisco-elastic, partially linked
� No long term relaxation� Gel stability due to 3D-network structure
Angular frequency ω
Complex viscosity
G‘
G‘‘Slope:
� Strength of structure at rest
Absolute value:� Stiffness of gel
Damping G‘‘/G‘� Damping
behaviour
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Frequency Sweep – the Regions
1 Unlinked polymer with narrow MWD2 Unlinked polymer with broad MWD3 Lowly cross-linked polymer, soft gel or dispersion with weak structure4 Highly cross-linked polymer. stiff gel or dispersion with strong structure
lg G'4
3
2 1
lg ωωωω
G1
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Frequency SweepExample for Polymersolutions & MeltsThe most important parameter
100
101
102
103
104
Pa·s
|η*|
100
101
102
103
104
105
107
Pa
G'
G''
10-1
100
101
102
103
104
105
106
1/sAngular Frequency
Physica Messtechnik GmbH
Polystyrol 200°C]
|η*| Complex Viskosity
G' Storage Modulus
G'' Loss Modulus
4,3w0 kM=η
Zero Shear Viscosity η0(Direct Relation MW)
Cross Over Point
( ) ( )( )( )
∞
−
∞ η+⋅λ+⋅η−η=η a
1na
0 x1
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Frequency Sweep Molar Mass MW & Distribution MMD
G'
G''
Angular Frequency ω
> narrow < MMD
< wide MMD >
lower average molar mass (<MW)
shorter / less branched molecules
higher average molar mass (>MW)
longer / branched molecules
GX, ωX
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Frequency Sweep – Master CurveHorizontal shift towards the reference temperature T0
� TTS example: shift of storage modulus G‘� The range abover the transition region is called glassy region
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Frequency Sweep – Master CurveHorizontal shift towards the reference temperature T0
� TTS example: horizontal shift of storage modulus G‘
Angular frequency ω
Storage modulus G‘
260°C
160°C
180°C
200°C
230°C
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From Frequency Sweep to Master Curve (I)
101
102
103
104
105
107
Pa
G'
G''
-2
142 °C
10 10-1
100
101
102
103
1/sAngular Frequency ωωωω
170 °C
Shift Factor a T
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From Frequency Sweep to Master Curve (II)
100
101
102
103
1/sAngular Frequency ωωωω
142 °C
101
102
103
104
105
107
Pa
G'
G''
10-2
10-1
170 °C
Shift Factor a T
142 °C shifted
Enlarged frequency range
at170°C
RESULT
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From Frequency Sweep to Master Curve (III)
101
102
103
104
105
107
Pa
G'
G''
10-2
10-1
100
101
102
103
104
105
1/sAngular Frequency ωωωω
LONGMOLECULES
dominant
Long term behavior
SHORTMOLECULES
dominant
short term behavior
Interactionsbetween
MOLECULES
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Frequency Sweep – Molar Mass M W, MMDMMD in just 2 analysis steps
Frequency Sweep [1/s]or
Master Curve
Additional parameterfrom literature!
1
2
3
102
103
104
Pa·s
|ηηηη*|
10-1
100
101
102
103
104
106
Pa
G'
G''
0,001 0,01 0,1 1 10 100 1.0001/sKreisfrequenz ωωωω
MMD: MMD Bimodal or MMD Kernels (for the experts)
[g/mol]0
0,050,1
0,150,2
0,250,3
0,350,4
0,450,5
w i
10.000 100.000 1.000.000 10.000.000g/molMolmasse M i
10-3
10-1
101
103
105
Pa
H(λλλλ)
0
200
400
600
1.000
Pa·s
H(λλλλ)·λλλλ
10-4
10-3
10-2
10-1
100
101
102
103
sRelaxationszeit λλλλ
Relaxtation Time Spectrum
Relaxation TimeSpectrum [s]
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Rubbery Plateau
Glassy State
Frequency Sweep – The Regions
101
102
103
104
105
Pas
|ηηηη*|
101
102
103
104
105
107
Pa
G'
G''
10-2
10-1
100
101
102
103
104
105
1/sAngular Frequency ωωωω
Tg Region
measured
Terminal Flow
Slope in the region of η0
1:1
2:1
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amorphous partially crystalline cross-linked
Tg...glass transition temperature Tm...melting temperature
Viscoelastic BehaviorTemperature - dependent Behavior
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Partially Crystalline Polymer - DMTA
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Torsion Bar Fixture
RheometryFixtures for Solids
typicalbar dimensions:
50 x 10 x 1 (in mm)
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Viscoelastic Behavior of Solids
Temperature - dependent Behavior
0.01
0.1
1
10
GPa
lg G'
lg G''
50 100 150 °Ctemperature T
reinforced Laminate
unmodified Laminate
ω = 10 rad/s
γγγγ = 0.01 %
dimensions of the solid bar :50x10x1 (in mm)
Dynamic Mechanical Thermo-Analysis (DMTA)
↑↑↑↑
↓↓↓↓
Summary: shift of T g from +132 to +152°C
+20 180
polymers
softening →→→→
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Viscoelastic Behavior
Time - dependent Curing
102
103
104
105
106
Pa
140
160
180
200
°C
T
0 200 400 600 800 1000stime t
γγγγ = 0.1 % ω = 10 rad/s preset: T = T(t)
disposable measuringplates
Comparisonof twoPowderCoatings
PC 1
PC 2
Analysis:1) Minimum of G' or G''2) crossover G' = G''3) end of curing
polymerscoatings
→→→→ melting curing →→→→
lg G'
lg G''
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Reaction Kinetics for Thermosetting Polymer
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EC-Twist TM
Modular Dynamic Mechanical Analyzer
Mechanical properties
Material characterization
Time, temperature, frequency
Melts DMTA
Curing
http://www.anton-paar.com/DE/de/Web/Document/download/11158?clng=en
Sealants, Adhesives
Elastomers
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Learn More about Rheology ?
�Rheology Workshop University of Nottingham2 day course at £399
Contact [email protected]
�Free Sample Work / Rheology Audit
� Rheology Handbook, Thomas Mezger
�British Society of Rheologyhttp://www.bsr.org.uk/