J. Stegen + , J. Billen ° , M. Wilson ° , A.R.C. Baljon ° . A.V. Lyulin +

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)

Introduction

Polymeric gels

Reversible junctions between end groups (telechelic associating polymers)

Temperature

Sol Gel

Concentration

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

Hybrid MD/MC simulation of a polymeric gel

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 ��������������

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

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

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

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

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

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

Structural properties in equilibrium

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

Structural properties in mechanical equilibrium II

Structural properties in mechanical equilibrium II

Structural properties in mechanical equilibrium III

T=1.0

Structural properties in mechanical equilibrium III

T=0.55

Structural properties in mechanical equilibrium III

T=0.35

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

Shear Banding

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

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

Shear banding: aggregate size distribution

• More small and large aggregates in shear banding state• Large aggregates strong influence on velocity profile?

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

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

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.

Other work…

Jammed system at constant stress & fluctuation relation• Elastic behaviour visible • Two types of behaviour observed in time• Deviations from fluctuation relation observed

Questions?