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

[Introduction to Physical Polymer Science,L. Sperling (2006)]

polyethylene

Motivation of research

Polymer science

Polymer chemistry (synthesis) Polymer physics

Polymer rheology

Introduction: polymeric gels

Polymeric gelsReversible junctions between endgroups

(telechelic polymers)

TemperatureSol Gel

Concentration

• 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

Visco-elastic properties

Shear rate

Vis

cosi

ty

[J.Sprakel et al., Soft Matter (2009)]

stress

Hybrid MD/MC simulation of a polymeric gel

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

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

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

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

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

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

Simulated polymeric gel

T=1.0only endgroupsshown

Shearing the systemSome chains grafted to wall;

move wall with constant shear rate

fixed wall

moving wall

Shear banding in polymeric gel

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)]

Shear-banding in viscoelastic fluids

• Interface instabilities in worm-like micelles

[Lerouge et al.,PRL 96,088301 (2006).]

time

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 )

Before yield peak:

homogeneous

After yield peak:

2 shear bands

Velocity profilesdi

stan

ce f

rom

wal

l

0

30

/1059.3 2

Velocity profile over time

• Fluctuations of interface

fixed wall

moving wall

velo

city

time

dis

tan

ce fr

om

wa

ll

Chain Orientation

ij

ji

r

rrQij

3

1

2

32

Shear direction

x

z

y

rij

Qxx=1 Qzz=-0.5

Chain orientation

ij

ji

r

rrQij

3

1

2

32

• Effects more outspoken in high shear band

Aggregate sizes• Sheared: more smaller and larger

aggregates

• High shear band: largest aggregates as likely

size=4

• 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

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)

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:

TemperatureSol Gel

Polymeric gelsAssociating: reversible junctions between

endgroups

Concentration

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