Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers...

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Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran, Ponmurugan Mani, Jing Cao

Transcript of Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers...

Page 1: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

Hierarchy of models for entangled polymers

Alexei E. Likhtman

Department of Mathematics, University of Reading

With Sathish K. Sukumaran, Ponmurugan Mani, Jing Cao

Page 2: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,
Page 3: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

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

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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

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20 instantaneous trajectories

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20 mean paths

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20 mean paths

Page 8: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,
Page 9: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

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

Page 10: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

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

Page 11: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

Constraint release

Hua and Schieber 1998

Shanbhag, Larson, Takimoto, Doi 2001

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Outline

• Molecular dynamics as an experimental tool

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

• Microscopic definition of entanglements

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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)

Page 14: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

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

Page 15: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

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

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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

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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

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Fit with slip-springs model as an extrapolation tool

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

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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

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g1(t) – monomer mean square displacement

100

1000

(t)

N=50,100,200,350

10 100 1000 10000 100000 10000001

10

g1(t

)

t

Page 21: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

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

Page 22: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

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

Page 23: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

Mismatch between static properties in MD and slip-springs

)( 2RR ji >−<

||

)(

ji

RR ji

>−<

Page 24: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

Additional interactions in slip-spring model

Page 25: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

Static properties of the melt in MD

||

)( 2

ji

RR ji

>−<

>−−=< ++++ ))(()( 11 sisiii RRRRsP

Page 26: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

N=50, slip-spring model with additional potential

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Simulations of Langevin equations with memory

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

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

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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

Page 30: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

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

Page 31: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

Theory of mean paths

Free energy of the mean path

Page 32: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

010=avτ

Page 33: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

110=avτ

Page 34: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

210=avτ

Page 35: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

310=avτ

Page 36: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

410=avτ

Page 37: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

510=avτ

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Page 39: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

Minimal distance between mean paths

sq

ua

re m

inim

al d

ista

nce

time

sq

ua

re m

inim

al d

ista

nce

Page 40: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

E. W. Dijkstra, Numerische

Mathematik, 1, 269 (1959).

Page 41: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

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

Page 42: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

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

Page 43: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,
Page 44: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

Entangled mean paths

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Page 46: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

Distribution of entanglement strengths

<d> is the average distance fluctuations

during entanglement

sNd ~><

Cut-off

Page 47: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

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)

Page 48: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

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)

Page 49: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

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

Page 50: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

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|

Page 51: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

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

Page 52: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

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

Page 53: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

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

Page 54: Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers Alexei E. Likhtman Department of Mathematics, University of Reading With Sathish K. Sukumaran,

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.