Mesoscopic simulations of entangled polymers, blends, copolymers, and

branched structures

F. Greco, G. Ianniruberto, and G. Marrucci Naples, ITALY

Y. MasubuchiTokyo, JAPAN

Network of entangled polymers

Actual chains have slackPrimitive chains are shortest path

Microscopic simulations:

• Atomistic molecular dynamics (Theodorou, Mavrantzas, etc.)• Coarse-grained molecular dynamics (Kremer, Grest, Everaers et al.; Briels et al.)• Lattice Monte Carlo methods (Evans-Edwards, Binder, Shaffer, Larson et al.)

Mesoscopic simulations:

• Brownian dynamics of primitive chains (Takimoto and Doi, Schieber et al.)• Brownian dynamics of the primitive chain network (NAPLES)

Brownian dynamics of primitive chains along their contour

Sliplinks move affinelySliplinks are renewed at chain endsEach sliplink couples the test chain to a virtual companion

3D sliplink model

Simulation box typically contains ca. 2 x 104 chain segments

Nodes of the rubberlike network are sliplinks (entanglements) instead of crosslinks

Crucial difference: Monomers can slide through the sliplink

Primitive Chain Network ModelJ. Chem. Phys. 2001

+

3D motion of nodes

1D monomer sliding along primitive path

Dynamic variables: node positions R monomer number in each segment n number of segments in each chain Z

Node motion

Elastic springs Brownian force

Chemical potential

Fr

RκR

23 4

12

i i

i

nb

kT

Relative velocity of node

Monomer sliding

fn

r

n

r

b

kT

d

n

i

i

i

i

1

12

3

2

= local linear density of monomers

1

1

2

1

i

i

i

i

r

n

r

nd

n = rate of change of monomers in i-th segment due to arrival from segment i-1

d

n= sliding velocity of monomers from i-1 to i

Network topological rearrangement

ni monomers at the end

End

21

0n

niif Unhooking (constraint release)

else if211

0n

ni Hooking (constraint creation)

n0: average equilibrium value of n

if 0

if 1

2

kT

E

Chemical potential of chain segment from free energy E

The numerical parameter was fixed at 0.5, which appears sufficient to avoid unphysical clustering. The average segment density <> is not a relevant parameter. We adopted a value of 10 chain segments in the volume a3, where a is the entanglement distance.

Non-dimensional equations(units: length = a=bno , time = a2/6kT = , energy= kT)

n=n/no

Fr

RκR 3

1

2

1 4

i i

i

n f

n

r

n

r

d

n

i

i

i

i

3

1

1

1

n

rr

b

kT

2

3T Stress tensor:

n

rr

kT

3

T

Relevant parameters:

Nondimensional simulation: equilibrium value of <Z> (slightly different from initial value Z0)

Comparison with dimensional data: modulus G = kT = RT/Me

elementary time

LVE prediction of linear polymer melts

104

2

4

68105

2

4

6810

6

2

4

6810

7

101 102 103 104 105 106 107

(sec-1)

6810

-2

2

4

6810-1

2

4

6810

0

2

4

10-3 10-2 10-1 100 101 102

(simulation)

PBWang et al (2003)

M=43.9kSimulation

<Z>=27.9

Polybutadiene melt at 313K from Wang et al., Macromolecules 2003

Polyisoprene melt at 313K from Matsumiya et al., Macromolecules 2000

104

2

4

6810

5

2

4

6810

6

2

4

6810

7

10-3

10-2

10-1

100

101

102

103

(sec-1)

810-2

2

4

6810

-1

2

4

6810

0

2

4

68

10-4

10-3

10-2

10-1

100

101

(simulation)

PMMAFuchs et al. (1996)

M=46k, 71kSimulation

<Z>=11.2, 18.6

Polymethylmethacrylate melts at 463K from Fuchs et al., Macromolecules 1996

Polymers G (MPa) Me (kDa) Me literature

Me

(s)

PS (453K) 0.33 11 1.7 0.002

PB (313K) 1.8 1.6 1.6 7x10-6

PI (313K) 0.63 3.5 1.4 5x10-5

PMMA (463K) 1.25 3.9 1.6 0.6

G = kT = RT/Me <Z> = M/Me

Polystyrene solution by Inoue et al., Macromolecules 2002

Simulations by Yaoita with the NAPLES code

Step strain relaxation modulus G(t,)

Viscosity growth. Shear rates (s-1) are: 0.0113, 0.049, 0.129, 0.392, 0.97, 4.9

Primary normal stress coefficient. Shear rates as before.

Polystyrene solution fitting parameters:

Vertical shift, G = 210 Pa

Horizontal shift, = 0.55 s

<Z> = 18.4 implying Me = 296

Blends and block copolymers

kTEmix /

Phase separation kinetics in blends

t=0 2.5

<Z> = 10 (d ~ 40), =0.5, =4.0

0/ dt 5.0

10.0 20.0 40.0

Block ratio = 0.5

= 0.5

<Z> = 40

BLOCKCOPOLYMERS

Block ratio 0.1

Block ratio 0.3

<Z> = 40 = 2

Branched polymers

Backbone-backbone entanglements cannot be renewed

two entangled H-molecules

Backbone chains have no chain ends

Sliplink

Branch point

End

A star polymer with q=5 arms

Free arm

If one of the arms happens to have no entanglements, …

it has the chance to change topology

1/q

1/q

1/q

1/q

1/q

Possible topological changesThe free arm has q options, all equally probable

(under equilbrium conditions)

Double-entanglement

It can penetrate a sliplink of another arm, thus forming a …

If later another arm becomes entanglement-free, …

the topological options are …

Enhanced probability for the double entanglement because the coherent pull of the 2 chains makes the branch point closer to double entanglement

2/q

1/q

1/q

1/q

If the multiple entanglement is “chosen”, …

the branch point is “sucked” through the multiple entanglemet

The multiple entanglement has now the chance to be “destroyed” by arm fluctuations

Similar topological changes would allow backbone-backbone entanglements in H polymers to be renewed

H-polymer simulations

Click to play

Relaxation modulus for H-polymers

1010

10

10 1020

With the topologicalchange(liquid behavior)

without(solid behavior)

Stress auto-correlation

5

6

789

10-1

2

3

4

5

6

789

100

100

101

102

103

104

t

w BPM H20 H10

w/o BPM H20 H10

2

''

2

2

''

''

''')(

tt

tt

tt

tttttC

Effect on diffusion of 3-arm star polymers

Diffusion coefficient

Arm molecular weight, Za 5 10 20

Topological change w/o w w/o w w/o w

Diffusion Coefficient 4.8 e-3 6.0 e-3 4.3 e-4 4.3 e-4 2 e-6 2 e-6

Acceleration Ratio 1.2 1.0 1.0

Code H05 H10 H20

Topological chan ge w/o w w/o w w/o w

Diffusion Coefficient 1.6e-3 2.4e-3 4e-5 1.3e-3 1e-8 6e-4

Acceleration Ratio 1.5 ~33 >1000

For 3-arm stars

For H’s having arms with Za= 5

Backbone-backbone entanglement (BBE) cluster

105

10

The largest BBE cluster for H05 including 58 molecules

Size distribution of BBE cluster

100

2

46

101

2

46

102

2

46

103

100

2 4 6 8

101

2 4 6 8

102

2 4 6 8

103

Size of BBE cluster

H05H10H20

Conclusions• Mesoscopic simulations based on the entangled

network of primitive chains describe many different aspects of the slow polymer dynamics

• For linear polymers, quantitative agreement is obtained with 2 (or at most 3) chemistry-and-temperature-dependent fitting parameters.

• More complex situations are being developed, and appear promising.

• A word of caution: Recent data by several authors (McKenna, Martinoty, Noirez) on thin films (nano or even micro) show that supramolecular structures can exist. These can hardly be captured by simulations.

Conclusion (social)http://masubuchi.jp to get the code & docs.

NAPLESNew Algorithm for Polymeric Liquids

Entangled and Strained