Measuring Polymers using a Rotational
Rheometer in Oscillatory Mode
Steve Goodyer
Product Manager for Rheology
Anton Paar Ltd.
www.anton-paar.com
Introduction
Viscoelastic Behavior
polymers
Polymer melts
←
Extrusion
(e.g. Polystyrene, PS):
extrudate swelling
and
melt fracture
Blow moulding →
(e.g. Polyethylene, PE):
orange peel
or
shark skin
2
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
3
Introduction
Viscoelastic Behavior
Using a simple illustrative picture:
„The Rheology Road“
viscous
ideally viscous
liquids
like water, oils
Law of Newton
←
viscoelastic
viscoelastic
liquids
like glues,
shampoos
←
rotational tests
←←←←←←
→
viscoelastic
solids
like pastes,
gels, rubbers
→
|
oscillatory tests
elastic
ideally elastic
(rigid) solids
like stone, steel
Law of Hooke
→→→→→→
→ e-learning
4
(Eiffel tower)
Definitions
Shear Stress, shear deformation or shear strain
The Two-Plates Model
shear stress
τ
unit:
1 N / m2 = 1 Pa (Pascal)
shear deformation γ
or shear strain
unit:
5
=
=
F
A
s
h
1 m / m = 1 = 100 %
Definitions
Elasticity Law
Spring Law:
F/s=C
spring force F
deflection path s
spring constant C (stiffness)
Robert Hooke (1635 to 1703)
τ
G=
γ
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)
6
definition of the
shear modulus
Rheometry
Oscillatory Tests: Basics (1)
Two-Plates Model
Ideally elastic behavior
of a totally stiff sample
(e.g. a stone, or steel):
There is no shift between
the sine curves of
shear strain (deformation)
and shear stress :
the curves of γ and τ
are “in phase“
7
→ Movie
(2-plates-model,
ideal-elastic
behavior)
Rheometry
Oscillatory Tests: Basics (2)
Preset: constant frequency and constant amplitude
Result: Most samples are showing
viscoelastic behavior
with the
phase shift δ
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:
8
0° ≤ δ ≤ 90°
ideally elastic
ideally viscous behavior
→ Movie
(2-plates-model,
visco-elastic behavior)
Rheometry
Oscillatory Tests: Basics (3)
Vector Diagram
Elasticity Law of Hooke
(for oscillation):
Index A for „Amplitude“
G* [Pa]
G' [Pa]
G'' [Pa]
Physically:
G' for the stored and
G'' for the lost (dissipated)
G
τ
=
γ
A
A
Complex Shear Modulus
Storage Modulus, elastic portion
Loss Modulus, viscous portion
of the viscoelastic behavior
deformation energy
tan δ [1] = G''/ G'
Loss Factor or Damping Factor
as the ratio between the viscous and the elastic portion
9
*
Rheometry (Oscillation)
Amplitude Sweeps, preset
Preset:
constant frequency
(e.g. the angular frequency
ω = 10 rad/s or s-1)
and
variable strain (deformation)
Frequency Conversion: ω = 2π
π⋅f
with
angular frequency ω [s-1]
(since Hz is not an SI unit !)
and
frequency f [Hz]
→ Movie
(amplitude sweep)
10
Viscoelastic Behavior
Amplitude Sweeps
Limiting value
of the LVE - range
Result:
storage modulus
G'
(elastic behavior),
loss modulus
G''
(viscous behavior),
limiting value of the linear viscoelastic (LVE- ) range when reaching γL
- at the given test conditions, i.e., at the preset (angular) frequency left side:
right side:
11
G‘ > G‘‘ (“gel - like structure“)
in the LVE - range
G‘‘ > G‘ (“liquid - like structure“) in the LVE - range
Viscoelastic Behavior
Amplitude Sweeps
polymers
105
Pa
Polymer Melt
lg G'
viscoelastic liquid,
104
lg G''
liquid-like character
since G‘‘ > G‘
↑
limit of the LVE range at
γ = 10% = 0.1
ω = 10 rad/s
T = +180°C
↓
103
-2
10
-1
10
0
10
strain lg γ
1
10
2
10
%
3
10
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Viscoelastic Behavior
Amplitude Sweeps
polymers
dispersions
1000
kPa
Sealant
↑
100
paste-like, viscoelastic
gel-like character
in the LVE range
since G‘ > G‘‘
lg G'
10
lg G''
1
0.1
0.001
Limit of the LVE-range at
γ = 0.026% = 2.6 ⋅ 10-4
(with 10% tolerance deviation)
ω = 10 rad/s
T = +25°C
↓
0.01
0.1
1
strain lg γ
10
% 100
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Frequency Sweeps
Measure time dependency
t=0
14
t= 5 min
of viscoelastic properties.
Frequency of applied strain
with constant amplitude is
logarithmically varied.
Results typically plotted as
G’, G’’ vs frequency, f (or ω
= 2π
πf).
Generally speaking the
shorter the timescale the
more elastic a material
behaves.
Consider as viscoelastic
spectrum.
Results relate to molecular
structure of the sample.
Frequency Sweep
Visco-elastic liquid (no gel, unlinked, no filler)
Long term: newtonian behaviour
Short term: viscoelastic behaviour
No network
structure
No links between
macro-molecules
Complex viscosity
G‘‘ G‘
1
2
1
1
15
Angular frequency ω
Viscoelastic Behavior
Frequency Sweeps
polymers
Typical behavior of an unlinked polymer
η0 = 35 kPas
10
PDMS
(poly - di - methyl
- siloxane)
5
Pa
10
4
10
Pas
lg G'
lg G''
4
10
10
10
3
• lg |ηη*|
2
10
3
1
G′′
′′ > G′′ ← crossover → G′′ > G′′
′′
γ = 10 %
T = +23°C
10
-3
10
-2
10
-1
10
0
angular frequency lg ω
10
1
10
2
rad/s
10
3
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Frequency Sweep
Visco-elastic, partially linked
No long term relaxation
Gel stability due to 3D-network structure
G‘
G‘‘
Slope:
Strength of
structure at rest
Absolute value:
Stiffness of gel
Complex viscosity
Angular frequency ω
17
Damping G‘‘/G‘
Damping
behaviour
lg G'
Frequency Sweep
– the Regions
4
G1
3
2
1
lg ω
1
2
3
4
18
Unlinked polymer with narrow MWD
Unlinked polymer with broad MWD
Lowly cross-linked polymer, soft gel or dispersion with weak structure
Highly cross-linked polymer. stiff gel or dispersion with strong structure
Frequency Sweep
Example for Polymersolutions & Melts
The most important parameter
Zero Shear Viscosity η0
(Direct Relation MW)
4
10
η0 = kMw
(
η = (η0 − η∞ ) ⋅ 1 + (λ ⋅ x )
3, 4
)
a
+ η∞
7
Pa
3
10
5
10
Polystyrol 200°C]
4
10
2
|η*| 10
3
G'
10 G''
2
10
1
10
Cross Over Point
1
10
0
0
0
10
1
2
3
10
10
10
Angular Frequency
4
10
Physica Messtechnik GmbH
19
(n −1)
10
Pa·s
10
-1
10
a
5
10
6
10 1/s 10
|η*| Complex Viskosity
G'
Storage Modulus
G'' Loss Modulus
Frequency Sweep
Molar Mass MW & Distribution MMD
> narrow <
MMD
higher average
molar mass (>MW )
G'
G''
GX, ωX
longer / branched
molecules
lower average
molar mass (<MW )
shorter / less
branched molecules
< wide MMD >
Angular Frequency ω
20
Frequency Sweep – Master Curve
Horizontal shift towards the reference temperature T0
TTS example: shift of storage modulus G‘
The range abover the transition region is called glassy region
21
Frequency Sweep – Master Curve
Horizontal shift towards the reference temperature T0
TTS example: horizontal shift of storage modulus G‘
Storage modulus G‘
160°C
180°C
200°C
230°C
260°C
Angular frequency ω
22
From
Frequency Sweep to Master Curve (I)
7
10
142 °C
Pa
5
10
G'
4
Shift Factor aT
10
G''
3
10
2
10
170 °C
1
10
-2
10
23
-1
0
10
10
10
Angular Frequency ω
1
2
10
3
1/s
10
From
Frequency Sweep to Master Curve (II)
7
10
142 °C shifted
142 °C
Pa
RESULT
5
Enlarged
frequency range
at
170°C
10
G'
4
Shift Factor aT
10
G''
3
10
2
10
170 °C
1
10
-2
10
24
-1
0
10
10
10
Angular Frequency ω
1
2
10
3
1/s
10
From
Frequency Sweep to Master Curve (III)
7
10
Pa
5
10
G'
4
10
G''
3
10
2
Long term behavior
short term behavior
LONG
MOLECULES
dominant
SHORT
MOLECULES
dominant
10
1
10
-2
10
25
-1
10
0
1
10
10
10
Angular Frequency ω
Interactions
between
MOLECULES
2
3
10
4
10 1/s 10
5
Frequency Sweep – Molar Mass MW, MMD
MMD in just 2 analysis steps
4
6
10
1
Frequency Sweep [1/s]
or
Master Curve
10
Pa
Pa·s
4
10
3
10
3
|η*| 10
G'
2
10
1
G''
10
0
10
2
10
0,001
-1
0,01
0,1
1
Kreisfrequenz ω
10
100 1/s 1.000
10
Relaxtation Time Spectrum
5
2
10
Pa
Relaxation Time
Spectrum [s]
1.000
3
Pa·s
10
600
1
10
H(λ
λ)
400
-1
10
200
-3
10
0
-4
10
Additional parameter
from literature!
3
26
MMD:
MMD Bimodal or MMD Kernels (for the experts)
[g/mol]
0,5
0,45
0,4
0,35
0,3
0,25
wi
0,2
0,15
0,1
0,05
0
10.000
-3
10
-2
-1
0
10
10
10
Relaxationszeit λ
100.000
Molmasse M i
1
10
2
10
3
s 10
1.000.000 g/mol 10.000.000
H(λ
λ )·λ
λ
Frequency Sweep – The Regions
10
5
10
Terminal Flow
Pas
10
Glassy State
Tg Region
Rubbery Plateau
Pa
Slope in the region
of η0
4
10
|η
η *| 10
10
7
3
10
1:1
10
2
2:1
10
5
4
G'
3
G''
2
measured
10
1
10
27
-2
10
-1
0
1
10
10
10
Angular Frequency ω
2
10
3
4
10 1/s 10
10
5
1
Viscoelastic Behavior
Temperature - dependent Behavior
amorphous
partially crystalline
Tg...glass transition temperature
28
cross-linked
Tm...melting temperature
Partially Crystalline Polymer - DMTA
29
Rheometry
Fixtures for Solids
Torsion Bar
Fixture
typical
bar dimensions:
50 x 10 x 1
(in mm)
30
Viscoelastic Behavior of Solids
Dynamic Mechanical Thermo-Analysis (DMTA)
Temperature - dependent Behavior
polymers
10
reinforced Laminate
GPa
↓
1
unmodified Laminate
lg G'
↑
0.1
lg G''
softening →
0.01
ω = 10 rad/s
γ = 0.01 %
Summary: shift of Tg from +132 to +152°C
+20
50
100
temperature
T
150 °C 180
dimensions
of the solid bar :
50x10x1 (in mm)
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Viscoelastic Behavior
Time - dependent Curing
coatings
106
→ melting
200
curing →
180
Pa
°C
160
105
140
polymers
Comparison
of two
Powder
Coatings
PC 1
lg G'
104
lg G''
Analysis:
1) Minimum of G' or G''
2) crossover G' = G''
3) end of curing
103
102 0
PC 2
T
200
400
600
800
s
1000
γ
= 0.1 %
ω = 10 rad/s
preset: T = T(t)
disposable
measuring
plates
time t
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Reaction Kinetics for Thermosetting
Polymer
33
EC-TwistTM
Modular Dynamic Mechanical Analyzer
Melts
Material
characterization
DMTA
Sealants,
Adhesives
Mechanical
properties
Elastomers
34
Time, temperature, frequency
http://www.anton-paar.com/DE/de/Web/Document/download/11158?clng=en
Curing
Learn More about Rheology ?
Rheology Workshop
University of Nottingham
2 day course at £399
Contact steve.goodyer@anton-paar.com
Free Sample Work / Rheology Audit
Rheology Handbook, Thomas Mezger
British Society of Rheology
http://www.bsr.org.uk/
35