Transcript Construction of a toolbox for analysis of dummy toy models

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:
Polymers
• Long-chain molecules of high molecular
weight
polyethylene
[Introduction to Physical
Polymer Science,
L. Sperling (2006)]
Motivation of research
Polymer science
Polymer chemistry
(synthesis)
Polymer physics
Polymer rheology
Introduction: polymeric gels
Polymeric gels
Reversible junctions between endgroups
(telechelic polymers)
Concentration
Sol
Temperature
Gel
Polymeric gels
• 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

Visco-elastic properties
   stress
Viscosity

Shear rate
[J.Sprakel et al.,
Soft Matter (2009)]


Hybrid MD/MC simulation of a
polymeric gel
Molecular dynamics simulation
ITERATE
• Give initial positions, choose short time Dt
dU
• Get forces F    and acceleration a=F/m
dr
• Move atoms
• Move time t = t + Dt
Bead-spring model
[K. Kremer and G. S. Krest.
J. Chem. Phys 1990]
Attraction beads in chain
U [e]
U FENE
2

r
 ij  
1 2
  kR0 ln 1    
2
  R0  
Repulsion all beads
U LJ
 s 12  s 6  s 12  s  6 
 4e            , r  rc  1.12
r 
 rij 
 rc  
 ij   rc 

Distance [s]
•Temperature control through coupling with heat bath
1s
Associating polymer
[A. Baljon et al., J. Chem.
Phys., 044907 2007]
• Junctions between end groups : FENE + Association energy
• Dynamics …
U bond  U assoc  U FENE  U LJ
U [e]
U nobond  U LJ
Unobond
Distance [s]
Dynamics of associating polymer (I)
D U [e]
•Monte Carlo: attempt to form junction
 DU
P ~ exp(
)
kbT
DU  Upossiblenew  Uold
 U assoc  U FENE
P=1
form
P<1
possible
form
Uassoc
Distance [s]
Dynamics of associating polymer (II)
D U [e]
•Monte Carlo: attempt to break junction
DU  Upossiblenew  Uold
 DU
P ~ exp(
)
kbT
 U assoc  U FENE 
P<1
possible break
P=1
break
Uassoc
Distance [s]
Simulation details
• 1000 polymeric chains, 8 beads/chain
• Units: s (length), e (energy&temperature),
m (mass), ts(m/e1/2 (time);
• Box size: (23.5 x 20.5 x 27.4) s3 with
periodic boundary conditions
Simulated polymeric gel
T=1.0
only
endgroups
shown
Shearing the system
Some chains grafted to wall;
move wall with constant shear rate
moving wall
fixed wall
Shear banding in polymeric gel
Shear-Banding in Associating
Polymers


• PEO in Taylor-Couette
system
two shear bands
stress
velocity
Plateau in stress-shear curve
moving wall
[J.Sprakel et al.,
Phys Rev. E 79, 056306 (2009)]
fixed wall
shear rate
distance
Shear-banding in viscoelastic fluids
• Interface instabilities in worm-like micelles
time
[Lerouge et al.,PRL 96,088301 (2006).]
distance from wall [s]


30
Stress under constant shear
0
All results T=0.35 e (< micelle transition T=0.5 e)
stress yield peak
plateau

  3.59 102 / t
distance from wall [s]


30
Velocity profiles

  3.59 102 / t
0
Before yield peak:
homogeneous
After yield peak:
2 shear bands
Velocity profile over time
• Fluctuations of interface
fixed
wall
velocity [s/t]
distance from wall [s]
moving
wall
time [t]
Chain Orientation
3  ri rj 1 
Qij   2   ij 
2 r
3 
Qxx=1 Qzz=-0.5
z
Shear direction
y
rij
x
Chain orientation
Qij 
3  ri rj 1 
 2   ij 
2 r
3 
• Effects more outspoken in high shear band
Aggregate sizes
• Sheared: more smaller and larger
aggregates
size=4
• High shear band: largest
aggregates as likely
Conclusions
• 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
Equation of Motion
K. Kremer and G. S. Grest. Dynamics of entangled linear polymer melts: A
molecular-dynamics simulation. Journal of Chemical Physics, 92:5057, 1990.
ri   U ij  ri  Wi (t )
j i
Uij  UijLJ  UijFENE

•Interaction energy
•Friction constant;
•Heat bath coupling – all complicated interactions
W
•Gaussian white noise
•<Wi2>=6 kB T

(fluctuation dissipation theorem)
Predictor-corrector algorithm
1)Predictor: Taylor: estimate at t+t
4) Corrector step:
2) From
calculate forces and
acceleration
at t+t
3) Estimate size of error in prediction step:
Dt=0.005 t
Polymeric gels
Associating: reversible junctions between
endgroups
Concentration
Sol
Temperature
Gel
Simulation details
• 1000 polymeric chains, 8 beads/chain
• Units: s (length), e (energy&temperature),
m (mass), ts(m/e1/2 (time);
• Box size: (23.5 x 20.5 x 27.4) s3 with
periodic boundary conditions
• Concentration = 0.6/s3 (in overlap regime)
• Radius of gyration:
Rg
2
1

N
N
 r
k 1
k
2
 rmean   2.69s
2
• Bond life time > 1 / shear rate