Transcript Construction of a toolbox for analysis of dummy toy models

A computational study of shear
banding in reversible associating
polymers
J. Billen, J. Stegen+, A.R.C. Baljon
San Diego State University
+ Eindhoven University of Technology (The Netherlands)
Funded by:
Associating polymers: reversible networks:
Outline
•
•
•
Molecular dynamics/Monte Carlo simulation
Shear banding
Shear-induced differences
– Shear vs Unshear
– High vs Low shear band
Sol
Temperature
Gel
Hybrid MD / MC simulation (I)
Molecular dynamics simulation:
Bead-spring model (Kremer-Grest)
interactions within chain
Monte Carlo: Junctions
between end groups
•Lennard-Jones interaction between all beads
U LJ
  12    6   12    6 
 4            , r  rc = 21/6
r 
rc 
rc  
 rij 
ij 





•FENE: between beads in chain and junctions
2

 rij  
1 2
U FENE   kR0 ln 1    
2
  R0  
•Temperature control (coupled to heat bath)
•Units: (length),(energy),t=(m/)1/2 (time)
[A. Baljon et al., J. Chem.
Phys., 044907 2007]
Hybrid MD / MC simulation (II)
•Monte Carlo: junctions formed / destroyed
with probability: P ~ exp(U  / kbT )
U bond  U assoc  U FENE  U LJ

U nobond  U LJ
Uassoc= -22 
Simulation details
• 1000 polymeric chains, 8 beads/chain
• Box size: (23.5 x 20.5 x 27.9) 3 with
periodic boundary conditions in x/y direction
• Concentration = 0.6 beads / 3 ( in overlap
regime )
• Radius of gyration:
N
2
1
Rg = 2.69
r2k  rmean   2.69

N k 1
2
Numerical study of associating
polymers
f
d2f/dT2=0
Order parameter:
Number of junctions f
[Baljon et al.;
J Chem. Phys. 126, 044907 2007]
Temperature
• Gel transition: T=0.5
• Percolation transition T=1.5
Topological changes at the gel
transition
Billen et al.
Europhys. Lett. 87
(2009) 68003.
Node
Temperature
T=0.5
Gel transition
T=1.5
Percolation
transition
Experiment: Shear-Banding in
Associating Polymers
• Polyethylene Oxide (with octadecyl alkane
(hydrophobic) groups at chain ends) under shear
Plateau in stress-shear curve
stress
velocity
two shear bands
moving wall
[J.Sprakel et al.,
Phys Rev. E 79, 056306 (2009)]
fixed wall
shear rate
distance
Theory: Visco-elastic fluid
t relaxation time

Shear banding
[S. Fielding, Soft Matter 3,1262 (2007)]
Simulation: Constant shear
5% chains grafted to wall
Moving wall
Shear velocity:
v
h
Moving wall
h
Fixed wall
Fixed wall

shear rate:

= v/h
measure: stress 
Stress under constant shear
• All results T=0.35 below gel transition; t0 = 3.9 104 t
stress yield peak

  3.59102-4 / t
plateau
Velocity profiles

-4
  3.59102 / t
Before yield peak:
homogeneous
After yield peak:
2 shear bands
Microscopic differences
• Endgroup concentration 2.5% larger in high
shear band
• Average # aggregates larger in high shear
band than in low shear band
• Junctions live longer in low shear band
• Radius of gyration:
High shear band
low shear band
unsheared
Chain Orientation
3  ri rj 1 
 2   ij 
Qij 
2 r
3 
Qxx=1 Qzz=-0.5
z
Shear direction
y
rij
x
Aggregate sizes
• Sheared/Unsheared
– Increase of smaller and larger
aggregates during shear
– Lower preference for average
size <s> under shear
• High / low shear band
– Average aggregate size <s>
lower in high shear band
<s>
unsheared
19.1
high shear
13.0
low shear
19.4
unsheared
low shear band
high shear band
Topological differences
Nomenclature
Node =
3 endgroups
Single
bridge
Double
bridge
Link
Loop
• Links can consist of more than 1 bridging molecule
Influence of shear on network structure
unsheared
unsheared
sheared
sheared
unsheared sheared
• no change # loops/ bridging
molecules
• drop # links
Influence of shear on network structure
average
unsheared
average unsheared
average
unsheared
average unsheared
Spatial distribution of bridges
unsheared
• High/low shear band:
– Dip # links near
interface
– More multiple bridges
in high shear band
sheared
sheared
unsheared
Conclusions
• MD / MC simulation associating polymers
• Shear banding observed in simulations
• Study of differences
– Sheared vs Unsheared
•
•
•
•
Chains orient and Rg increases
No change in # loops / # bridging molecules
Number of links decreases
Formation of links with high # bridging molecules
– High vs Low shear band
•
•
•
•
More aggregates in high shear band
Larger endgroup concentration in high shear band
Smaller aggregate size in high shear band
More links with high # bridging molecules
Poster 49 on Wednesday
Spatial distribution of bridges (II)
Bead-spring model
[K. Kremer and G. S. Krest.
J. Chem. Phys 1990]
Attraction beads in chain
U []
U FENE
2


r


1 2
ij
  kR0 ln 1    
2
  R0  
Repulsion all beads
U LJ
  12    6   12    6 
 4            , r  rc  1.12
r 
 rij 
 rc  
 ij   rc 

Distance []
•Temperature control through coupling with heat bath
1
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 []
U nobond  U LJ
Unobond
Distance []
Dynamics of associating polymer (I)
•Monte Carlo: attempt to form junction
 U []
 U
P ~ exp(
)
kbT
U  Upossiblenew  Uold
 U assoc  U FENE
P=1
form
P<1
possible
form
Uassoc
Distance []
Dynamics of associating polymer (II)
•Monte Carlo: attempt to break junction
 U []
 U
P ~ exp(
)
kbT
U  Upossiblenew  Uold
 U assoc  U FENE 
P<1
possible break
P=1
break
Uassoc
Distance []
Velocity profile over time
• Fluctuations of interface
fixed
wall
velocity [/t]
distance from wall []
moving
wall
time [t]
Endgroup distributions
• unsheared
• sheared