Hierarchy of models for entangled polymers

Alexei E. Likhtman

Department of Mathematics, University of Reading

With Sathish K. Sukumaran, Ponmurugan Mani, Jing Cao

Hierarchical modelling in polymer dynamics

• Constitutive equations

– Tube theories

• Single chain models

?),(σσ

fDt

D=

Tube

Model?jm

slip-spring

• Single chain models

– Coarse-grained

many-chains models

» Atomistic simulations

> Quantum mechanics simulations

bead-spring

molecular dynamics

Well established coarse-

graining procedures,

force-fields,

commercial packages

CR

Model?ja

slip-spring

model

40 years of the Tube theory 1967-2007

• Original assumptions

– Independent motion of each chain

– along the primitive path like a 1-D Rouse chain

– purely entropic stress (Kramer’s)

• Predictions

– Stress plateau

–

– Two stage relaxation and damping function

– Mean square displacement

– for stars

4.3

wd M∝∝τη

)exp( Mνη ∝

• Problems

– constraint release vs. tube dilution

– branch point motion

– tube diameter under deformation

– tube field and its stress contribution

– how to find the tube in MD?

– for stars

– Contour length fluctuations

)exp( wMνη ∝

•many contradicting theories coexist

•many proven results remain ignored

•one theory postdoc per

new experiment

20 instantaneous trajectories

20 mean paths

20 mean paths

Construction of the slip-spring model

ja

jm

timeelementary

size coil

re temperatu

parameters model Rouse

0

2

−

−

−

τ

gR

T

chain thealonglink -slip offriction -

chain) anchoring in the monomers ofnumber effective(or link -slip ofstrength -

links-slipbetween beads ofnumber average -

parameters New

s

s

e

N

N

ζ

slip_links19.exeA.L., Macromolecules, 2005, 38 (14), 6128

Rubinstein, Panyukov, Macromolecules 2002, 35, 6670-6686

Construction of the slip-spring model

ja

jm

timeelementary

size coil

re temperatu

parameters model Rouse

0

2

−

−

−

τ

gR

T

chain thealonglink -slip offriction -

chain) anchoring in the monomers ofnumber effective(or link -slip ofstrength -

links-slipbetween beads ofnumber average -

parameters New

s

s

e

N

N

ζ

slip_links19.exeA.L., Macromolecules, 2005, 38 (14), 6128

Rubinstein, Panyukov, Macromolecules 2002, 35, 6670-6686

Constraint release

Hua and Schieber 1998

Shanbhag, Larson, Takimoto, Doi 2001

Outline

• Molecular dynamics as an experimental tool

• Molecular dynamics fitted by slip-springs• Molecular dynamics fitted by slip-springs

• Microscopic definition of entanglements

Molecular Dynamics -- Kremer-Grest model

• Polymers – Bead-FENE

spring chains

0

2 2

2

0

( ) ln 12

FENE

kR rU r

R

= − −

• With excluded volume – Purely

• k = 30ε/σ2

• R0=1.5σ

Density, ρ = 0.85• With excluded volume – Purely

repulsive Lennard-Jones

interaction between beads

otherwise 0

2 r 4

14)( 61

612

=

<

+

−

=

rrrUrLJ

σσε

Density, ρ = 0.85

Friction coefficent, ζ = 0.5

Time step, dt = 0.012

Temperature, T = ε/k

K.Kremer, G. S. Grest

JCP 92 5057 (1990)

Stress relaxation for N=350

0

30

60

1

10

100

G(t

)

G(t

)

bond length

relaxation0.1 1 10

0.01 0.1 1 10 100 1000 10000 1000001E-3

0.01

0.1 t

G(t

)

t

GN

(0)

τe

relaxation

collisions

"Rouse" dynamics

entangled

Multiple tau correlator

G(t

)

1e-3

1e-2

1e-1

1e0

1e1

G(t

)

1e-3

1e-2

1e-1

1e0

1e1

Magatti and Ferri, Applied optics, 40, 4011 (2000)

t0.1 1 10 100 1,000 10,000 100,000

t0.1 1 10 100 1,000 10,000 100,000

Stress autocorrelation

N=1,2,5,10,25,50,100,200,350

1

10 N=350

N=200

N=100

N=50

N=25

N=10

N=5

N=2

G(t

)

0.1 1 10 100 1000 10000 100000

0.01

0.1

N=2

N=1

G(t

)

tA.E.Likhtman, S.K.Sukumaran, J.Ramirez, Macromolecules, 2007, 40, 6748-6757

1.0

1.2

1.4

1.6

1.8

Normalized G(t)

1 10 100 1000 10000 1000000.0

0.2

0.4

0.6

0.8

G(t

)t1

/2

t

Rouse

Entanglements

Fit with slip-springs model as an extrapolation tool

A.E.Likhtman, S.K.Sukumaran, J.Ramirez, Macromolecules, 2007, 40, 6748-6757

Varying chain stiffness

1E-3

0.01

0.1

1

10

50100

2030

50 70

50100

200

kb=3, ev2, x10

-2

k =1.1, x10-3

kb=1.5, x0.1

kb=3, x10

kb=5,ev4

G(t

)

200

0.003

0.004

0.005

0.006

1.5 1.13odd

0

-3rho06

rho0.4

1 10 100 1000 10000 100000

1E-8

1E-7

1E-6

1E-5

1E-4

300100

50200100

100

350

50

100 150

200

200

ρ=0.6, x10-5

kb=-3, x10

-4

kb=1.1, x10

-3

t

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.11.21E-3

0.002

0.003

3

5ev4

G~p-3

(Fetters)

GN

(0) p

3

p

slip-spring fit with free G0 and M0

same with fixed M0=rho/G0

PPA

G~p-7/5

(Semenov)

A.E.Likhtman, S.K.Sukumaran, J.Ramirez, Macromolecules, 2007, 40, 6748-6757

g1(t) – monomer mean square displacement

100

1000

(t)

N=50,100,200,350

10 100 1000 10000 100000 10000001

10

g1(t

)

t

g1(i,t) – monomer mean square displacement

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.2

0.4

0.6

0.8

1.0

1.2

1.4g

1i/t

1/2

g1i/t

1/2

N=50 N=100

10 100 1000 10000 100000 10000000.2

10 100 1000 10000 100000 10000000.2

10 100 1000 10000 100000 10000000.2

0.4

0.6

0.8

1.0

10 100 1000 10000 100000 10000000.2

0.4

0.6

0.8

1.0t t

g1i/t

1/2

t

g1i/t

1/2

t

N=200N=350

Slip-links vs MD

101

102

103

104

105

106

0.4

0.6

0.8

1

(g1(t

)/σ

2)/

(t/τ

)0.5

101

102

103

104

0.4

0.6

0.8

1

1.2

(a) (c)

Central

Peripheral

N=200 N=50

101

102

103

104

105

106

t/τ

10-1

100

(G(t

)/ε/

σ3)(

t/τ)

0.5

101

102

103

104

t/τ

10-1

100

(b) (d)

τe

S.K. Sukumaran and AEL, Macromolecules 2009, 42, 4300–4309

Mismatch between static properties in MD and slip-springs

)( 2RR ji >−<

||

)(

ji

RR ji

−

>−<

Additional interactions in slip-spring model

Static properties of the melt in MD

||

)( 2

ji

RR ji

−

>−<

>−−=< ++++ ))(()( 11 sisiii RRRRsP

N=50, slip-spring model with additional potential

Simulations of Langevin equations with memory

N=200, slip-spring model with additional potential and with memory

S.K. Sukumaran and AEL, Macromolecules 2009, 42, 4300–4309

Stress relaxation for N=200, slip-spring model with additional potential and with memory

)(tGt

S.K. Sukumaran and AEL, Macromolecules 2009, 42, 4300–4309

Slip-springs model summary

•Large deformation / fast flows

•Slip-spring strength should change accordingly

(at least in networks: Panyukov-Rubinstein Macromolecules 2002)

Advantages comparing to the tube model:•one simple consistent model with specified assumptions and few parameters

to describe many experiments simultaneously.

Problems

• but what about newly created slip-springs,

• and how do existing ones relax back to equilibrium

•In flows much faster than reptation time all slip-springs will disappear,

but the chains still do not obey Rouse – creation algorithm should

change in non-linear regime

•Branch point motion, especially in non-linear regime

Reason: slip-springs is still an empirical model

Theory of mean paths

Free energy of the mean path

010=avτ

110=avτ

210=avτ

310=avτ

410=avτ

510=avτ

Minimal distance between mean paths

sq

ua

re m

inim

al d

ista

nce

time

sq

ua

re m

inim

al d

ista

nce

E. W. Dijkstra, Numerische

Mathematik, 1, 269 (1959).

Period of Entanglement for

pair of chain

MD time step

Number of Entanglem

ents

300280260240220200180160140120100806040200

3.0E+0

2.0E+0

1.0E+0

0.0E+0

3

2

1

Period of Entanglement for

pair of chainNumber of Entanglem

ents

Number of Entanglem

ents

300280260240220200180160140120100806040200

3.0E+0

2.0E+0

1.0E+0

0.0E+0

3

2

1

MD time step

Entangled mean paths

Distribution of entanglement strengths

<d> is the average distance fluctuations

during entanglement

sNd ~><

Cut-off

Entanglement survival probability

1

G(t

) an

d P

(t)

G(t)

d2<1

100,3 == Nkb

100 1000 10000 100000

0.1

G(t

) an

d P

(t)

t

d <1

1<d2<1.5

1.5<d2<2

2<d2<2.5

2.5<d2<3

d2>3

(arbitrary vertical shift applied)

0.1

1G

(t)

an

d P

(t)

G(t)

Entanglement survival probability

350,0 == Nkb

1000 10000 100000

0.01

G(t

) a

nd P

(t)

t

G(t)

d2<9

d2<6

d2<5

d2<4

d2<3

(arbitrary vertical shift applied)

Entanglement density along the chain

1.0

1.1

1.2

1.3

1.4

1.5norm

aliz

ed d

ensity o

f enta

ngle

ments 150,3 == Nkb

0 20 40 60 80 100 120 1400.5

0.6

0.7

0.8

0.9

1.0

norm

aliz

ed d

ensity o

f enta

ngle

ments

monomer number

d2<2

2<d2<3

3<d2<4

Segment survival function

0.6

0.8

1.0(s

,t)

t0=1; (psi0)

t1=16; (psi1)

t2=32; (psi2)

t3=64; (psi3)

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.2

0.4ψ(s

,t)

s=|i/n-0.5|

Fit with Doi-Edwards solution

6.5E-1

6.0E-1

5.5E-1

5.0E-1

4.5E-1

4.0E-1

3.5E-1

0.0E+0 2.7E-1

psi

4.5E-14.0E-13.5E-13.0E-12.5E-12.0E-11.5E-11.0E-15.0E-26.9E-17

psi

3.5E-1

3.0E-1

2.5E-1

2.0E-1

1.5E-1

1.0E-1

5.0E-2

1.4E-17

-5.0E-2

0.0E+0 2.7E-1

No constraint release

0.50

0.45

0.40

0.35

0.30

5% of free chains

in the 95% fixed ends environment

psi

0.450.400.350.300.250.200.150.100.050.00

psi

0.25

0.20

0.15

0.10

0.05

0.00

s

Mean-square displacementof entanglement point

100

mea

n-s

qua

re d

ispla

cem

ent

100<t<120, da<1.5

100<t<120, any da

200<t<220, da<1.5

- slope 1/4

- msd of the middle monomer

- (msd of the middle monomer)/3

- g1 of difference between r2 and r1

order of magnitude for the

100 1000 10000 100000 1000000

1

10

τe

mea

n-s

qua

re d

ispla

cem

ent

t

order of magnitude for the

square of tube diameter

τd

Conclusions.

•Slip-springs model agrees with

molecular dynamics very well

•BUT to describe early time you

probably need excluded volume

and memory functions in slip-springsand memory functions in slip-springs

•Long lived tight contacts definitely

exist – experimental fact.

•One needs to study their dynamics

and feed the results to slip-links or

tube model.