Rheological study of a simulated polymeric gel: shear banding
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Rheological study of a simulated polymeric gel: shear banding
J. Billen, J. Stegen+, M. Wilson, A. Rabinovitch°, A.R.C. Baljon
+ Eindhoven University of Technology (The Netherlands)
° Ben Gurion University of the Negev (Be’er Sheva, Israel)
Funded by:
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Polymers
• Long-chain molecules of high molecular weight
[Introduction to Physical Polymer Science,L. Sperling (2006)]
polyethylene
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Motivation of research
Polymer science
Polymer chemistry (synthesis) Polymer physics
Polymer rheology
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Introduction: polymeric gels
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Polymeric gelsReversible junctions between endgroups
(telechelic polymers)
TemperatureSol Gel
Concentration
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• Examples– PEO (polyethylene glycol) chains terminated by
hydrophobic moieties
– Poly-(N-isopropylacrylamide) (PNIPAM)
• Importance:– laxatives, skin creams, tooth paste, paintball fill,
preservative for objects salvaged from underwater, eye drops, print heads, spandex, foam cushions,…
– cytoskeleton
Polymeric gels
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Visco-elastic properties
Shear rate
Vis
cosi
ty
[J.Sprakel et al., Soft Matter (2009)]
stress
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Hybrid MD/MC simulation of a polymeric gel
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Molecular dynamics simulationITERATE
• Give initial positions, choose short time t
• Get forces and acceleration a=F/m
• Move atoms
• Move time t = t + t
rd
dUF
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Bead-spring model
•Temperature control through coupling with heat bath
[K. Kremer and G. S. Krest.J. Chem. Phys 1990]
1
Distance
U
12.1,4612612
c
ccijijLJ rr
rrrrU
2
0
2 1ln2
10 R
rkRU ij
FENE
Attraction beads in chain
Repulsion all beads
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Associating polymer • Junctions between end groups : FENE + Association energy
• Dynamics …
[A. Baljon et al., J. Chem. Phys., 044907 2007]
LJnobond
LJFENEassocbond
UU
UUUU
U bond
Unobond
U
Distance
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Dynamics of associating polymer (I)•Monte Carlo: attempt to form junction
)exp(~Tk
UP
b
FENEassoc
oldwpossiblene
UU
UUU
P<1possibleform
P=1form
Distance
Uassoc
U
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Dynamics of associating polymer (II)•Monte Carlo: attempt to break junction
)exp(~Tk
UP
b
FENEassoc
oldwpossiblene
UU
UUU
P=1break
P<1possible break
Distance
U
Uassoc
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Simulation details• 1000 polymeric chains, 8 beads/chain
• Units: (length), (energy&temperature), m (mass), (m/ (time);
• Box size: (23.5 x 20.5 x 27.4) with periodic boundary conditions
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Simulated polymeric gel
T=1.0only endgroupsshown
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Shearing the systemSome chains grafted to wall;
move wall with constant shear rate
fixed wall
moving wall
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Shear banding in polymeric gel
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Shear-Banding in Associating Polymers
Plateau in stress-shear curve two shear bandsfixed w
all
moving w
all
distanceshear rate
stre
ss
velo
city
• PEO in Taylor-Couette system
[J.Sprakel et al., Phys Rev. E 79, 056306 (2009)]
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Shear-banding in viscoelastic fluids
• Interface instabilities in worm-like micelles
[Lerouge et al.,PRL 96,088301 (2006).]
time
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Stress under constant shear
/1059.3 2
stress yield peak
plateau
dist
ance
fro
m w
all
0
30
All results T=0.35 (< micelle transition T=0.5 )
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Before yield peak:
homogeneous
After yield peak:
2 shear bands
Velocity profilesdi
stan
ce f
rom
wal
l
0
30
/1059.3 2
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Velocity profile over time
• Fluctuations of interface
fixed wall
moving wall
velo
city
time
dis
tan
ce fr
om
wa
ll
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Chain Orientation
ij
ji
r
rrQij
3
1
2
32
Shear direction
x
z
y
rij
Qxx=1 Qzz=-0.5
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Chain orientation
ij
ji
r
rrQij
3
1
2
32
• Effects more outspoken in high shear band
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Aggregate sizes• Sheared: more smaller and larger
aggregates
• High shear band: largest aggregates as likely
size=4
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• MD/MC simulation reproduces experiments– Plateau in shear-stress curve
– Shear banding observed
– Temporal fluctuations in velocity profile
• Microscopic differences between sheared/ unsheared system– Chain orientation
– Aggregate size distribution
• Small differences between shear bands
• Current work: local stresses, positional order, secondary flow, network structure
Conclusions
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Equation of Motion
)(tWrUr iiij
iji
FENEij
LJijij UUU
K. Kremer and G. S. Grest. Dynamics of entangled linear polymer melts: Amolecular-dynamics simulation. Journal of Chemical Physics, 92:5057, 1990.
W
•Interaction energy
•Friction constant;
•Heat bath coupling – all complicated interactions
•Gaussian white noise
•<Wi2>=6 kB T (fluctuation dissipation theorem)
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2) From calculate forces and acceleration at t+t
Predictor-corrector algorithm
t=0.005
1)Predictor: Taylor: estimate at t+t
3) Estimate size of error in prediction step:
4) Corrector step:
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TemperatureSol Gel
Polymeric gelsAssociating: reversible junctions between
endgroups
Concentration
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Simulation details• 1000 polymeric chains, 8 beads/chain• Units: (length), (energy&temperature),
m (mass), (m/ (time);• Box size: (23.5 x 20.5 x 27.4) with
periodic boundary conditions• Concentration = 0.6/(in overlap regime)• Radius of gyration:
• Bond life time > 1 / shear rate
22
1
2 69.21
N
kmeankg rr
NR