Entanglements and stress correlations in coarsegrained

molecular dynamics

Alexei E. Likhtman, Sathish K. Sukumuran,

Jorge RamirezDepartment of Applied Mathematics,

University of Leeds, Leeds LS2 9JT, UKA.Likhtman@leeds.ac.uk

Hierarchical modelling in polymer dynamicsHierarchical modelling in polymer dynamics

• Constitutive equations

– Tube theories

• Single chain models

– Coarse-grained many-chains models

» Atomistic simulations

> Quantum mechanics simulations

?),(f

Dt

D

Kremer-Grest MD, Padding-Briels Twentanglemets,

NAPLES

Well established coarse-graining procedures,

force-fields, commercial packages

Traditional rheology

Traditional physics

CR

TubeModel?

The weakestlink

The missing linkThe missing link

Many chains system

+ self-consistent field + self-consistent field

One chain model

The ultimate goal: Stochastic equation of motion

for the chain in self-consistent entanglement field

Is there a tube model?Is there a tube model?

Best definition of the tube model:one-dimensional Rouse chain projected onto three-dimensional random walk tube.

Open questions:

•Can I have expression for the tube field, please?•How to “measure” tube in MD?•Is the tube semiflexible?•Diameter = persistence length?•Branch point motion•How does the contour length changes with deformation?•Tube parameters for different polymers?•Tube parameters for different concentrations?

Rubinstein-Panyukov network modelRubinstein-Panyukov network model

Rubinstein and Panyukov, Macromolecules 2002, 6670

Construction of the modelConstruction of the model

timeelementary

size coil

re temperatu

parameters model Rouse

0

2

gR

T

timeelementary

size coil

re temperatu

parameters model Rouse

0

2

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

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

ja

jm

Constraint releaseConstraint release

Hua and Schieber 1998Shanbhag, Larson, Takimoto, Doi 2001

1 10 1000.2

0.4

0.6

0.8

1.0

0.1 1 10 100 1,000103

104

105

1 10 1000.2

0.4

0.6

0.8

1.0

1k 10k 100k

6x10-5

1.2x10-4

1.8x10-4

1k 10k 100k

1E-12

1E-11

101 102 103 104 105 106 107 108 109

104

105

106

1k 10k 100k 1M

1E-11

1E-10

1k 10k 100k 1M

2x10-5

4x10-5

6x10-58x10-510-4

10k 100k 1M

5E-12

1E-11

1.5E-11

2E-11

1k 10k 100k 1M 10M

1E-10

1E-9

1k 10k

4x10-5

6x10-5

8x10-5

102 103 104 105 106 107

104

105

106

10-2 10-1 100 101 102 103 104 105 106104

105

106

1k 10k 100k 1M

1E-11

1E-10

1k 10k 100k

5x10-5

10-4

1.5x10-42x10-4

G(0)

N=2.2MPa

by extrapolation

too slow

too unstable?

experiments needed

experiments needed

q=0.05A-1

q=0.077A-1

12.4K 24.7K 190K q=0.115A-1

125K 61K 34K

s-1

q=0.03A-1

q=0.05A-1

q=0.068A-1

q=0.076A-1

q=0.096A-1

q=0.115A-1

/M

3 w

PS

PBd

PI

PEP

PE

DiffusionDM2

w (m2

/s)(g/mol)2

ViscosityG'/G''(Pa vs (s-1

))

NSES(q,t)/S(q,0) vs t (ns)

/M3

w (Pa*s/(g/mol)

3)

/ M

3 w

A.E.Likhtman, Macromolecules 2005

t, ns0,1 1 10 100

S(q

,t)/S

(q,0

)

1

0,95

0,9

0,85

0,8

0,75

0,7

0,65

0,6

0,55

0,5

0,45

0,4

0,35

0,3

0,25

0,2

0,15

0,1

0,05

2k

6k

12k

Mwmat

Rouse

Relaxation of dilute long chains (36K) in a short matrix: constraint release

Relaxation of dilute long chains (36K) in a short matrix: constraint release

M.Zamponi et al, PRL 2006

labeled

Molecular Dynamics -- Kremer-GrestMolecular Dynamics -- Kremer-Grest

• Polymers – Bead-FENE spring chains

0

2 2

20

( ) ln 12FENE

kR rU r

R

• With excluded volume – Purely repulsive Lennard-Jones

interaction between beads

otherwise 0

2 r 4

14)( 61

612

rrrU rLJ

• k = 30/2

• R0=1.5

Density, = 0.85

Friction coefficent, = 0.5

Time step, dt = 0.012

Temperature, T = /k

K.Kremer, G. S. Grest

JCP 92 5057 (1990)

g1(t) from MD for N=100,350g1(t) from MD for N=100,350

t10 100 1,000 10,000 100,000

g1(t)

1e0

1e1

1e2

1e3 1

0.5 1/4

0.5

21 , ( , ) ( ,0)g i t i t i r r

1

11( ) 1 ,

N

i

g t g i tN

e

d

R

t10 100 1,000 10,000 100,000

g1(t)

1.1e0

1e0

9e-1

8e-1

7e-1

6e-1

5e-1

4e-1

3e-1

2e-1

g1(i,t)/t0.5 from MD for N=350g1(i,t)/t0.5 from MD for N=350g

1(i,t

)/t0

.5

ends

middle

t

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

G(t)

1e-4

1e-3

1e-2

1e-1

1e0

1e1

G(t) from MD for N=50,100,200,350 (Ne~50)G(t) from MD for N=50,100,200,350 (Ne~50)

e

( ) ( ) (0)V

G t tkT

t1 10 100 1,000 10,000 100,000

G(t)

1e0

G(t) from MD for N=50,100,200,350 (Ne~70)

G(t) from MD for N=50,100,200,350 (Ne~70)

e

ttG )(

G(t) from MD for N=50,100,200,350 (Ne~50)G(t) from MD for N=50,100,200,350 (Ne~50)

t

10 100 1,000 10,000 100,000

g1(t)/t^0.5

1e0

9.5e-1

9e-1

8.5e-1

8e-1

7.5e-17e-1

6.5e-1

6e-1

5.5e-1

5e-1

4.5e-1

4e-1

3.5e-13e-1

g1(i,t) -- MD vs sliplinks mapping 1:1 (N=200)g1(i,t) -- MD vs sliplinks mapping 1:1 (N=200)g

1(i,t

)/t0

.5

t

1 1

0

e

d

Lines - MDPoints - slip-links

Lines - MDPoints - slip-links

t

10 100 1,000 10,000 100,000

G(t)*t^0.5

1e0

G(t) -- MD vs sliplinks mapping 1:1 (N=200)G(t) -- MD vs sliplinks mapping 1:1 (N=200)G

(t)*

t1/2

t

1 50

e

d

Lines - MDPoints - slip-links

Lines - MDPoints - slip-links

)0()()0()( virtualchainchainchain tt

)0()()0()(

)0()()0()(

virtualvirtualchainvirtual

virtualchainchainchain

tt

tt

)0()( chainchain t

Questions for discussionQuestions for discussion

• Binary nature of entanglements?– Can one propose an experiment which contradicts

this?

• Non-linear flows: – do entanglements appear in the middle of the

chain?

• Is there an instability in monodisperse linear polymers?

Log(gamma)210-1-2

Log(Sxy) 5e0

4e0