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“Strain Hardening” in Simple Shear of Branched Polystyrene Solutions
Gengxin Liu, Shi-Qing Wang Department of Polymer Science, The University of Akron, Akron, Ohio, USA 44325-3909
February 11, 2013, Monday, 10:50 AM
84th Society of Rheology annual meeting, Pasadena, CA 10
2
103
104
105
106
102
103
104
105
106
107
108
109
1010
10-2
10-1
100
101
102
103
104
+
E
(Pa.s)
+
(Pa.s)
t (s)
shear rates s-1
extension Hencky rates s-1
LCB-PS 4M TCP 7% -15 oC
LCBPS 4M melt 190 oC
0.1 at 25 oC
0.01
0.1
0.30.6
1.52
3
5
1
0.0030.03
0.01
0.1
1
0.3
310
2 Read, D. J.; Auhl, D.; Das, C.; den Doelder, J.; Kapnistos, M.; Vittorias, I.; McLeish, T. C. B., Linking Models of Polymerization and Dynamics to Predict Branched Polymer Structure and Flow. Science 2011, 333 (6051), 1871-1874
Extension – hardening? Shear – softening? Long chain branching
Low rates Extension Shear
Long chain Branched
hardening softening
Linear softening softening
Low Density Polyethylene
Meissner, J., Modifications of the weissenberg rheogoniometer for measurement of transient rheological properties of molten polyethylene under shear. Comparison with tensile data. J. Appl. Polym. Sci. 1972, 16 (11), 2877-2899. Laun, H. M.; Munstedt, H., Elongational behavior of a low-density polyethylene melt .1. strain rate and stress dependence of viscosity and recoverable strain in steady-state - comparison with shear data - influence of interfacial-tension. Rheol. Acta 1978, 17 (4), 415-425.
hardening
softening
3
Extension – hardening? Shear – softening? Pom-Pom Model (Dr. McLeish and Dr. Larson )
Bishko, G.; McLeish, T. C. B.; Harlen, O. G.; Larson, R. G. Phys. Rev. Lett. 1997, 79, 2352.
“Gaussian chain statistics … curvilinear tension of kT/a.”
“The backbone can readily be stretched, until their tension is sufficient to withdraw. “The macroscopic consequence of chain stretch is elongation hardening”
q=3 pom-pom polymer
Malmberg, A.; Gabriel, C.; Steffl, T.; Münstedt, H.; Löfgren, B. Macromolecules 2001, 35, 1038.
Auhl, D.; Chambon, P.; McLeish, T. C. B.; Read, D. J. Phys. Rev. Lett. 2009, 103.
In start-up of shear … backbone stretches temporarily, and eventually collapses as the molecule is aligned, producing strain softening.”
4
s g
t
Extension – hardening, Shear – softening Our Understanding: Yielding
Yield point
"New experiments for improved theoretical description of nonlinear rheology of entangled polymers", S. Q. Wang, Y. wang, S. Cheng, X. Lin,X. Zhu and H. Sun submitted to Macromolecules
gtotal
gmax
grecov
Yield point
5
Extension – hardening, Shear – softening Our Understanding: Geometry
sengr
e
t
Yield point
0
2 104
4 104
6 104
8 104
1 105
0 5 10 15 20lamda
sen
grP
a
HDPE(linear)
LDPE(branch)
irrecoverable Yield point Elastic recoverable
105
106
0.1 1 10
+
E
(Pa.s)
t (s)
+
E /
+
E /
HDPE(linear)
0.3 s-1
180 oC
LDPE(branch)
1 s-1
150 oC
sengr = F/A0= s/
s = F/A(t)=F/A0
Branch: Yields later
Liu, G.; Sun, H.; Rangou, S.; Ntetsikas, K.; Avgeropoulos, A.; Wang, S. Q., Studying the origin of "strain hardening": Basic difference between extension and shear. J. Rheol. 2013, 57 (1), 89-104. Dealy, J. M., DO POLYMERIC LIQUIDS EXHIBIT STRAIN-HARDENING. J. Rheol. 1990, 34 (7), 1133-1147.
6
Extension – hardening, Shear – softening Our Understanding: Geometry, Yielding
Liu, G.; Sun, H.; Rangou, S.; Ntetsikas, K.; Avgeropoulos, A.; Wang, S. Q., Studying the origin of "strain hardening": Basic difference between extension and shear. J. Rheol. 2013, 57 (1), 89-104.
0
2 104
4 104
6 104
8 104
1 105
0 5 10 15 20lamda
sen
grP
a
HDPE(linear)
LDPE(branch)
Yielding of entanglement
Geometrical shrinkage of cross-section area.
Force in shear and extension t
7
Meissner, J., Modifications of the weissenberg rheogoniometer for measurement of transient rheological properties of molten polyethylene under shear. Comparison with tensile data. J. Appl. Polym. Sci. 1972, 16 (11), 2877-2899.
Extension – hardening, Shear – softening
Effect of rates? Low rates Extension Shear
Long chain Branched
hardening softening
Linear softening softening
8
Extension – hardening, Shear – softening
At high rates-Linear Extension High rates Extension Shear
Linear hardening
Wang, Y.; Wang, S.-Q., From elastic deformation to terminal flow of a monodisperse entangled melt in uniaxial extension. J. Rheol. 2008, 52 (6), 1275-1290.
Auhl, D.; Chambon, P.; McLeish, T. C. B.; Read, D. J., Elongational flow of blends of long and short polymers: effective stretch relaxation time. Phys. Rev. Lett. 2009, 103).
9
Meissner, J., Modifications of the weissenberg rheogoniometer for measurement of transient rheological properties of molten polyethylene under shear. Comparison with tensile data. J. Appl. Polym. Sci. 1972, 16 (11), 2877-2899.
Extension – hardening, Shear – softening
At high rates-Linear Shear High rates Extension Shear
Linear hardening Softening
105
106
107
0.4 0.8 1.2 1.6 2 2.4 2.8
Crosslinked SBR 160K
sliding plate shear at rate 1.1 s-1
|*|+
+
, |
*| (
Pa
.s)
t (s), 1/
SUN Hao, W. S.-Q., Shear and extensional rheology of entangled polymer melts: Similarities and differences. SCIENCE CHINA Chemistry 2012, 55 (5), 779-786.
What if slightly crosslinked (1%)?
In shear, chains sliding by each other.
No overshoot, no yielding.
tmax << t, Wi>>1
V
s
elastic irrecoverable
Yield point
0
Meissner, J., Modifications of the weissenberg rheogoniometer for measurement of transient rheological properties of molten polyethylene under shear. Comparison with tensile data. J. Appl. Polym. Sci. 1972, 16 (11), 2877-2899.
Extension – hardening, Shear – softening
High rates-Branches in shear??? High rates Shear
Long chain Branched
Softening Hardening?
Linear softening
11
McLeish, T. C. B.; Allgaier, J.; Bick, D. K.; Bishko, G.; Biswas, P.; Blackwell, R.; Blottière, B.; Clarke, N.; Gibbs, B.; Groves, D. J.; Hakiki, A.; Heenan, R. K.; Johnson, J. M.; Kant, R.; Read, D. J.; Young, R. N. Macromolecules 1999, 32, (20), 6734-6758.
Model branch samples in literatures:
11
Archer, L. A.; Juliani, Linear and Nonlinear Viscoelasticity of Entangled Multiarm (Pom-Pom) Polymer Liquids. Macromolecules 2004, 37 (3), 1076-1088.
Polyisoprene: Arm (21k)2.9-backbone 89k-A2.9
Polybutadiene: Arm (20k)2-backbone 110k-A2
12
Nielsen, J. K.; Rasmussen, H. K.; Denberg, M.; Almdal, K.; Hassager, O., Nonlinear branch-point dynamics of multiarm polystyrene. Macromolecules 2006, 39 (25), 8844-8853.
Model branch samples in literatures:
Polystyrene: Arm (27k)2.5-backbone 140k-A2.5
Grafted Comb PS
Hepperle, J.; Münstedt, H.; Haug, K., Peter ; Eisenbach, D. C., Rheological properties of branched polystyrenes: linear viscoelastic behavior. Rheol. Acta 2005, 45, 151-163. Rheological properties of branched polystyrenes: nonlinear shear and extensional behavior. Rheol. Acta 2005, 45, 717-727.
13
Only shear softening is observed
Outline:
Current information:
• Synthesis Long-chain branching (LCB)
• Different regions of dynamics from SAOS
• Simple shear of LCB polystyrene solutions
-> Strain hardening
-> Non-Gaussian chain stretching
->Extraordinary elastic recovery
• Summary
Liu, G., S. Cheng, H. Lee, H. Ma, H. Xu, T. Chang, R. P. Quirk and S. Q. Wang, "Strain Hardening in Startup Shear of Long-Chain Branched Polymer Solutions," Phys. Rev. Lett. 111, 068302 (2013).
Long chain branched Polystyrenes
Sample Molecular weight Polydispersity index
LCB-PS 1M 1.5* 106 g/mole 2.6
LCB-PS 4M 4.7* 106 g/mole 1.5
Polymer Solvent Volume Fraction
Relaxation time
LCB-PS 1M TCP 11% 0.9s (-15 oC)
LCB-PS 4M TCP 7% 20s (-15 oC)
LCB-PS 4M TCP 14% 40s ( oC)
LCB-PS 4M TCP 21% 1780s (25 oC)
LCB-PS 4M DEP 22% 2000s (25 oC)
TCP
DEP
Fixtures: Anton Paar Physica MCR 301 D=15mm, Cone-Plate 4o
D=25mm, Cone-Plate 2o
Solvents:
15
Shear: LCB-PS 4M in TCP 21 %
Dynamics of long chain branching in different regions
tbackbone tarm
101
102
103
104
105
100
101
102
103
104
105
106
10-1
101
103
105
107
10-4
10-3
10-2
10-1
100
101
102
103
104
De = t
(rad/s)
LCB-PS 4M
(21%-TCP)
t = 1780s
Ref. T= 25 oC
|*| (P
a.s
)
G' ,
G''
(Pa)
0
50000
1 105
1.5 105
2 105
0 1000 2000 3000 4000 5000 6000
Strain(%)
10s-1
6s-1
3s-1
1s-1
0.6s-1
0.3s-1
shear rates at -10 oC
aT to 25
oC= 273
LCB-PS 4M
(21%-TCP)
s
Pa
𝝈 = 𝑮 × 𝜸
102
103
104
105
106
102
103
104
105
106
107
108
109
1010
10-2
10-1
100
101
102
103
104
+
E
(Pa.s)
+
(Pa.s)
t (s)
shear rates s-1
extension Hencky rates s-1
LCB-PS 4M TCP 7% -15 oC
LCBPS 4M melt 190 oC
0.1 at 25 oC
0.01
0.1
0.30.6
1.52
3
5
1
0.0030.03
0.01
0.1
1
0.3
310
Shear: LCB-PS 4M in TCP 7wt %
Shear: LCB-PS 1M in TCP 11 %
102
103
104
0.1 1 10 100 1,000
Startup simple shear
30 s-1
20 s-1
10 s-1
7 s-1
1 s-1
3 s-1
LCB-PS 1M in TCP 11 %
t (s)
+ (
Pa.
s)
T = -15oC
|*|
Transient viscosity higher than the envelope:
“hardening”
17
5 103
1 104
1.5 104
2 104
2.5 104
3 104
3.5 104
4 104
0 20 40 60 80 100 120Strain (1)
5
3
21.5
1
0.6
LCB-PS 4M TCP
7% -15 oC
s
Pa
shear rates: s-1
neo Hookean: s=G*g
Shear hardening: Non-Gaussian stretching
18
Strain recovery
Straighten an entanglement strand of Me g*≈lent/(b(C+1))=lent/lKuhn=4 Yielding of linear Chain Straighten between two branch point of 22.7Me
g* ≈ √22.7 ×lent/lKuhn = 20 Straighten between two end of 152Me
g* ≈ = √152 ×lent/lKuhn= 50
s g
t
gtotal grecov
19
0
1000
2000
3000
4000
5000
6000
0 5000 10000 15000 20000 25000 30000
reco
ver
Str
ain
(%)
Strain(%)
100%
recovery
line
-15 oC rate 3 s
-1
LCB2PS-TCP 7%
0 oC rate 1 s
-1
Strain recovery
Straighten an entanglement strand of Me g*≈lent/(b(C+1))=lent/lKuhn=4 Yielding of linear Chain Straighten between two branch point of 22.7Me
g* ≈ √22.7 ×lent/lKuhn = 20 Straighten between two end of 152Me
g* ≈ = √152 ×lent/lKuhn= 50
20
Summary, long chain branching (LCB) can:
1. Disentanglement (sliding from chain end) is easy; 2. Unless at high rates in extension;
Linear chains
Chains with long branching
LCB will not easily retracts/ pull out: 1. Non-Gaussian stretching,
strain hardening in shear; 2. Breakdown is delayed; postpones stress overshoot huge strain recovery in shear; strain hardening in extension;
Thank you for your time This work is supported by NSF DMR-1105135
High rates Extension Shear
Long chain Branched
hardening hardening
Linear hardening softening
Low rates Extension Shear
Long chain Branched
hardening softening
Linear softening softening
102
103
104
105
106
102
103
104
105
106
107
108
109
1010
10-2
10-1
100
101
102
103
104
+
E
(Pa.s)
+
(Pa.s)
t (s)
shear rates s-1
extension Hencky rates s-1
LCB-PS 4M TCP 7% -15 oC
LCBPS 4M melt 190 oC
0.1 at 25 oC
0.01
0.1
0.30.6
1.52
3
5
1
0.0030.03
0.01
0.1
1
0.3
310