J. Stegen + , J. Billen ° , M. Wilson ° , A.R.C. Baljon ° . A.V. Lyulin +
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Transcript of J. Stegen + , J. Billen ° , M. Wilson ° , A.R.C. Baljon ° . A.V. Lyulin +
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Structural origin of non-Newtonian rheologyComputer simulations on a solution of telechelic associating polymers
J. Stegen+, J. Billen°, M. Wilson °, A.R.C. Baljon °. A.V. Lyulin+
+ Eindhoven University of Technology (The Netherlands)
° San Diego State University (USA)
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Introduction
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Polymeric gels
Reversible junctions between end groups (telechelic associating polymers)
Temperature
Sol Gel
Concentration
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Constitutive relation for gel
Stress Shear rateViscosity
Constitutive relation for gelRegime where stress decreases with increasing shear due to shear induced structure:•decrease in number of elastic junctions•increased orientation in shear direction
/ /
/ , /x
F A F xy
x z v z
shear ratest
ress
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Hybrid MD/MC simulation of a polymeric gel
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Molecular dynamics simulation
Molecular dynamics:
Grest-Kremer bead-spring model
Equations of motion:
(Langevin equation, coupling to heat bath through fluctuation dissipation theorem)
i i i ir U rm r R t ��������������
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Bead-spring model [K. Kremer and G. S. Krest.J. Chem. Phys 1990]
1
Distance
U
2
0
2 1ln2
10 R
rkRU ij
FENE
Repulsion all beads
Attraction beads in chain
12 6 12 6
4 ,
1.12
LJij ij c c
c
Ur r r r
r r
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Associating polymer
Junction between end groups : LJ + FENE + Association energy
[A. Baljon et al., J. Chem. Phys., 044907 2007]
LJnobond
LJFENEassocbond
UU
UUUU
U bo
nd
Unobond
U
Distance
22assocU ò
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Dynamics of associating polymer
Monte Carlo: attempt to form or destroy junction
~ exp( )B
UP
k T
new old
assoc FENE
U U U
U U
P<1possibleform
P=1form
Distance
Uassoc=-22
U
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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 in x,y
direction.• Fixed walls in z-direction
• Average volume density in system: 0.32
• NVT simulation
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Shearing the system
Move wall with constant shear rate.
Some chains grafted to wall to minimise wall slip (50 per wall)
fixed wall
moving wall
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Nomenclature
Bead (8 per chain) • Chain bead (6 per chain, white/gray)• End group (2 per chain)
• Dangler (blue)• Loop (orange)• Aggregate (red & orange)
Single chain
Network structure of 4 chains
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Structural properties in equilibrium
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Structural properties in mechanical equilibrium I
phase # aggregates # loops # danglers
T=1.0 Solution 390 ± 11 67 ± 8 593 ± 23
T=0.55 Gel transition
198 ± 7 184 ± 12 151 ± 11
T=0.35 Gel 107 ± 4 257 ± 4 62 ± 4
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Structural properties in mechanical equilibrium II
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Structural properties in mechanical equilibrium II
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Structural properties in mechanical equilibrium III
T=1.0
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Structural properties in mechanical equilibrium III
T=0.55
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Structural properties in mechanical equilibrium III
T=0.35
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Structural properties in mechanical equilibrium IV: Conclusions
• Aggregates increase in size with decreasing temperature
• Gel network immobile, macroscopic lifetime
• Spatial ordering of aggregates observed in gel phase
• Boundary effects visible at all temperatures, induces structure and ordering at lower temperature
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Shear Banding
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Shear banding: theory
Instable region in constitutiverelation (striped)
Stable configuration throughtwo shear bands coexisting ata stress σ
Lever rule: 3 1 1 2 2
1 2
· · · ,d
d
d d
d d
Plateau in shear-stress curve
Difference in mesoscopicstructure between bands
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Shear banding: force and velocity profile
Simulation details: T=0.35εwall velocity 0.01 σ/τshear rate 3.6*10-4 τ -1
total wall displacement ~700 σ
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Shear banding: aggregate size distribution
• More small and large aggregates in shear banding state• Large aggregates strong influence on velocity profile?
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Shear banding: orientation function
ij
ji
r
rrQij
3
1
2
32
Orientation in xx-direction, xz-direction and perpendicular to zz-direction: effects of applied shear on chains decrease
No significant differences between shear bands
xx
zzxz
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Shear banding: spatial distribution
High shear band very small (~5σ), too small to contain mesoscopic structure?
Fluctuations in density of ~10% at bottom of high shear band. No stationary flow but hopping like behaviour of end groups at interface?
Shear direction
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Conclusion
• Shear bands in velocity profile observed.
• High shear band too small to accommodate a mesoscopic structure different from the low shear band.No significant differences in structure observed between bands.
• More large aggregates in a sheared system, these could be responsible for the observed shear banding.
• Fluctuations in end-group density at interface, no steady flow.
• Validity of lever rule has not been checked. Uncertain if observed shear banding corresponds to the shear banding observed in experiment.
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Other work…
Jammed system at constant stress & fluctuation relation• Elastic behaviour visible • Two types of behaviour observed in time• Deviations from fluctuation relation observed
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Questions?