Theoretical and Applied Phase-Field: Glimpses of the activities in India
ICR_2012
1. Andreas Hess1 and Nuri Aksel2
1 TU Bergakademie Freiberg, 2 Universität Bayreuth
Evaluation of Strain-Rate
Frequency Superposition (SRFS) Data by
Stress Decomposition Method (SDM)
August 6, 2012
The XVIth International Congress on Rheology
August 5-10, 2012 – Lisbon, Portugal
2. Outline
Basics of stress decomposition method (SDM)
Basics of strain-rate frequency superposition (SRFS)
Combining stress decomposition method (SDM) with strain-rate
frequency superposition (SRFS) to characterize the mechanical and
relaxation behavior of a nonlinear viscoelastic material
Comparing the new approach with Fourier-transform (FT) rheology
TU Bergakademie Freiberg 1
3. Nonlinear Viscoelasticity
Viscoelastic model material:
Carbopol microgel
Yield stress
Network microstructure1
1
Cryo-TEM image from J.-Y. Kim et al., Colloid
Polym. Sci. 128, 614 (2003)
Visualization of rheological behavior:
Strainamplitudeγ0
Deborah number λω
Pipkin diagram
Newtonianfluid
?
Linear viscoelasticity
Nonlinear viscoelasticity
Linearelasticity
Rheometric test: oscillatory shear!
TU Bergakademie Freiberg 2
4. Nonlinear Viscoelasticity
Viscoelastic model material:
Carbopol microgel
Yield stress
Network microstructure1
1
Cryo-TEM image from J.-Y. Kim et al., Colloid
Polym. Sci. 128, 614 (2003)
Visualization of rheological behavior:
Strainamplitudeγ0
Deborah number λω
Pipkin diagram
Newtonianfluid
?
Linear viscoelasticity
Nonlinear viscoelasticity
Linearelasticity
Rheometric test: oscillatory shear!
TU Bergakademie Freiberg 2
5. Waveform Analysis
Time domain:
time
γ(t)
σ(t)
δ
Deformation domain:
-1
0
1
-1 0 1
σ(t)/σ0
γ(t)/γ0
-1
0
1
-1 0 1
σ(t)/σ0
˙γ(t)/˙γ0
Storage, G′, and loss, G′′,
modulus
Fourier-transform (FT)
rheology
Stress decomposition in elastic, σ′(γ), and
viscous, σ′′( ˙γ), stress
We follow this route!
TU Bergakademie Freiberg 3
6. Waveform Analysis
Time domain:
time
γ(t)
σ(t)
δ
Deformation domain:
-1
0
1
-1 0 1
σ(t)/σ0
γ(t)/γ0
-1
0
1
-1 0 1
σ(t)/σ0
˙γ(t)/˙γ0
Storage, G′, and loss, G′′,
modulus
Fourier-transform (FT)
rheology
Stress decomposition in elastic, σ′(γ), and
viscous, σ′′( ˙γ), stress
We follow this route!
TU Bergakademie Freiberg 3
7. Stress Decomposition Method (SDM)
Approximating σ′ and σ′′ by Chebyshev polynomials, Tn (1th kind)1:
σ′(γ(t)/γ0) = γ0 ∑
n:odd
enTn(γ(t)/γ0), σ′′( ˙γ(t)/ ˙γ0) = ˙γ0 ∑
n:odd
vnTn( ˙γ(t)/ ˙γ0)
-1
0
1
-1 0 1
σ(t)/σ0
γ(t)/γ0
Elastic stress σ′
-1
0
1
-1 0 1
σ(t)/σ0
˙γ(t)/˙γ0
Viscous stress σ′′
Changing the closed-loop Lissajous representation to the single valued functions
σ′ and σ′′!
1
R. H. Ewoldt et al., J. Rheol. 52, 1427 (2008)
TU Bergakademie Freiberg 4
8. Stress Decomposition Method (SDM)
Approximating σ′ and σ′′ by Chebyshev polynomials, Tn (1th kind)1:
σ′(γ(t)/γ0) = γ0 ∑
n:odd
enTn(γ(t)/γ0), σ′′( ˙γ(t)/ ˙γ0) = ˙γ0 ∑
n:odd
vnTn( ˙γ(t)/ ˙γ0)
-1
0
1
-1 0 1
σ(t)/σ0
γ(t)/γ0
Elastic stress σ′
-1
0
1
-1 0 1
σ(t)/σ0
˙γ(t)/˙γ0
Viscous stress σ′′
Changing the closed-loop Lissajous representation to the single valued functions
σ′ and σ′′!
1
R. H. Ewoldt et al., J. Rheol. 52, 1427 (2008)
TU Bergakademie Freiberg 4
9. Viscoelastic Measures
Sign of curvature of σ′ or σ′′ characterizes type of response, e.g.
d2
σ′
dx2 = γ0 (e3 ·24x+...)σ′
γ
Elastic stress
strain hardening
strain softening
G
σ′′
˙γ
Viscous stress
shear thickening
shear thinning
η
Elastic Chebyshev intensity
E =
n:odd
∑
3
en
e1
> 0, strain hardening
= 0, linear elastic
< 0, strain softening
Viscous Chebyshev intensity
V =
n:odd
∑
3
vn
v1
> 0, shear thickening
= 0, linear viscous
< 0, shear thinning
TU Bergakademie Freiberg 5
10. Strain-Rate Frequency Superposition (SRFS)
Sampling the Pipkin diagram with
constant strain-rate sweeps1,
˙γ0 = γ0ω = const..
Determining the elastic, E, (and
viscous, V) Chebyshev intensity at
each constant strain-rate sweep.
Strainamplitudeγ0
Deborah number λω
Pipkin diagram
Newtonianfluid
?
Linear viscoelasticity
Nonlinear viscoelasticity
Linearelasticity
10-2
10-1
100
101
102
103
104
10-2 10-1 100 101 102
γ0
ω (rad/s)
˙γ0=const.
−−−−−−−→
-0.2
0
0.2
0.4
0.6
0.8
1
10-2 10-1 100 101 102E
ω (rad/s)
1
H. M. Wyss et al., Phys. Rev. Lett. 98, 238303 (2007)
TU Bergakademie Freiberg 6
11. Strain-Rate Frequency Superposition (SRFS)
Shifting the elastic, E, (and viscous, V) Chebyshev intensity along the frequency
axes
Frequency shift factor
-0.2
0
0.2
0.4
0.6
0.8
1
10-2 10-1 100 101 102
E
ω (rad/s)
100
101
102
103
10-3
10-2
10-1
100
101
b
˙γ0 (s−1
)
1
−−−−−−−−−−−−−−−−−→ -0.2
0
0.2
0.4
0.6
0.8
1
10-4 10-3 10-2 10-1 100 101
E
ω/b( ˙γ0) (rad/s)
All data collapse onto a single curve!
TU Bergakademie Freiberg 7
12. Strain-Rate Frequency Superposition (SRFS)
Shifting the elastic, E, (and viscous, V) Chebyshev intensity along the frequency
axes
Frequency shift factor
100
101
102
103
10-3
10-2
10-1
100
101b
˙γ0 (s−1
)
1
-0.2
0
0.2
0.4
0.6
0.8
1
10-2 10-1 100 101 102
E
ω (rad/s)
100
101
102
103
10-3
10-2
10-1
100
101
b
˙γ0 (s−1
)
1
−−−−−−−−−−−−−−−−−→ -0.2
0
0.2
0.4
0.6
0.8
1
10-4 10-3 10-2 10-1 100 101
E
ω/b( ˙γ0) (rad/s)
All data collapse onto a single curve!
TU Bergakademie Freiberg 7
13. Strain-Rate Frequency Superposition (SRFS)
Shifting the elastic, E, (and viscous, V) Chebyshev intensity along the frequency
axes
Frequency shift factor
-0.2
0
0.2
0.4
0.6
0.8
1
10-2 10-1 100 101 102
E
ω (rad/s)
100
101
102
103
10-3
10-2
10-1
100
101
b
˙γ0 (s−1
)
1
−−−−−−−−−−−−−−−−−→ -0.2
0
0.2
0.4
0.6
0.8
1
10-4 10-3 10-2 10-1 100 101
E
ω/b( ˙γ0) (rad/s)
All data collapse onto a single curve!
TU Bergakademie Freiberg 7
14. Nonlinear Mechanical and Relaxation behavior in Terms of Stress Decomposition
Method (SDM)
-0.2
0
0.2
0.4
0.6
0.8
1
10-5 10-4 10-3 10-2 10-1 100 101 102
E,V
ω b´ ˙γ0µ ´rad/sµ
SAOSyield regimeLAOS
ω0
10-1
100
101
102
103
104
10-310-2 10-1 100 101
b´˙γ0µ
˙γ0 s 1
E : open symbols
V : closed symbols
Structural relaxation frequency: ω0
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15. Elastic Chebyshev Intensity vs. Generalized Storage Modulus
Elastic Chebyshev intensity:
E =
9
∑
n=3
en
e1
Generalized storage modulus:
˜G′ =
9
∑
n=1
G′
n +
9
∑
n=3
G′′
n
-0.2
0
0.2
0.4
0.6
0.8
1
10-5 10-4 10-3 10-2 10-1 100 101
E
ω/b( ˙γ0) (rad/s)
10-1
100
101
102
103
10-5 10-4 10-310-2 10-1 100 101
˜G′/a(˙γ0)(Pa)
ω/b( ˙γ0) (rad/s)
Increasing strain hardening, E > 0
E diverges at ω/b ≈ 4·10−4
Loss in elastic energy, ˜G′ → 0
˜G′ diverges at ω/b ≈ 4·10−4
TU Bergakademie Freiberg 9
16. Viscous Chebyshev Intensity vs. First-Order Loss Modulus
Viscous Chebyshev intensity:
V =
9
∑
n=3
vn
v1
First-order loss modulus:
G′′
1
-0.2
-0.1
0
0.1
0.2
10-5 10-4 10-3 10-2 10-1 100 101
V
ω/b( ˙γ0) (rad/s)
100
101
102
10-5 10-4 10-3 10-2 10-1 100 101
G′′
1/a(˙γ0)(Pa)
ω/b( ˙γ0) (rad/s)
ν′′
Constant shear thinning,
V = const. < 0
Relaxation peak at ω/b ≈ 10−2
Constant energy dissipation,
G′′
1 = const., ν′′ = 1
Relaxation peak at ω/b ≈ 10−2
TU Bergakademie Freiberg 10
17. Conclusion
The stress decomposition method (SDM) works well in combination with
the strain rate frequency superposition principle (SRFS).
The combination of SDM with SRFS is a promising approach to
characterize nonlinear viscoelastic materials with respect to their mechanical
and relaxation behavior.
The proposed procedure is also applicable to local nonlinear viscoelastic
measures1.
1
A. Hess and N. Aksel, Phys. Rev. E 84, 01502 (2011)
TU Bergakademie Freiberg 11
18. Fourier Transform Rheology
0
0.1
0.2
0.3
0.4
0.5
10-2 10-1 100 101 102
I3/I1
ω ( ˙γ0) (rad/s)
˙γ0
γ0
0
0.1
0.2
0.3
0.4
0.5
10-2 10-1 100 101 102
9
∑
n=3
(In/I1)
ω ( ˙γ0) (rad/s)
I3/I1 contains about 25% of the signal
9
∑
n=3
In/I1 contains about 50% of the
signal
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19. Extending the Frequency Range
Limited ω-range of the rheometer
device
Idea: shifting the data along the
ω-axis by factor b( ˙γ0)
0
0.1
0.2
0.3
0.4
0.5
10-2 10-1 100 101 102
9
∑
n=3
(In/I1)
ω (rad/s)
˙γ0
γ0
0
0.1
0.2
0.3
0.4
0.5
10-5 10-4 10-3 10-2 10-1 100 101
9
∑
n=3
(In/I1)
ω/b( ˙γ0) (rad/s)
10-1
100
101
102
103
104
10-3
10-2
10-1
100
101
b(˙γ0)
˙γ0 / s−1
1
TU Bergakademie Freiberg 13
