Download - Mesoscopic simulations of entangled polymers, blends, copolymers, and branched structures
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Mesoscopic simulations of entangled polymers, blends, copolymers, and
branched structures
F. Greco, G. Ianniruberto, and G. Marrucci Naples, ITALY
Y. MasubuchiTokyo, JAPAN
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Network of entangled polymers
Actual chains have slackPrimitive chains are shortest path
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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)
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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
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3D sliplink model
Simulation box typically contains ca. 2 x 104 chain segments
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Nodes of the rubberlike network are sliplinks (entanglements) instead of crosslinks
Crucial difference: Monomers can slide through the sliplink
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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
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Node motion
Elastic springs Brownian force
Chemical potential
Fr
RκR
23 4
12
i i
i
nb
kT
Relative velocity of node
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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
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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
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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.
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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
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LVE prediction of linear polymer melts
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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
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Polyisoprene melt at 313K from Matsumiya et al., Macromolecules 2000
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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
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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
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Polystyrene solution by Inoue et al., Macromolecules 2002
Simulations by Yaoita with the NAPLES code
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Step strain relaxation modulus G(t,)
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Viscosity growth. Shear rates (s-1) are: 0.0113, 0.049, 0.129, 0.392, 0.97, 4.9
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Primary normal stress coefficient. Shear rates as before.
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Polystyrene solution fitting parameters:
Vertical shift, G = 210 Pa
Horizontal shift, = 0.55 s
<Z> = 18.4 implying Me = 296
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Blends and block copolymers
kTEmix /
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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
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Block ratio = 0.5
= 0.5
<Z> = 40
BLOCKCOPOLYMERS
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Block ratio 0.1
Block ratio 0.3
<Z> = 40 = 2
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Branched polymers
Backbone-backbone entanglements cannot be renewed
two entangled H-molecules
Backbone chains have no chain ends
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Sliplink
Branch point
End
A star polymer with q=5 arms
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Free arm
If one of the arms happens to have no entanglements, …
it has the chance to change topology
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1/q
1/q
1/q
1/q
1/q
Possible topological changesThe free arm has q options, all equally probable
(under equilbrium conditions)
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Double-entanglement
It can penetrate a sliplink of another arm, thus forming a …
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If later another arm becomes entanglement-free, …
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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
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If the multiple entanglement is “chosen”, …
the branch point is “sucked” through the multiple entanglemet
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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
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H-polymer simulations
Click to play
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Relaxation modulus for H-polymers
1010
10
10 1020
With the topologicalchange(liquid behavior)
without(solid behavior)
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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
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Effect on diffusion of 3-arm star polymers
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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
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Backbone-backbone entanglement (BBE) cluster
105
10
The largest BBE cluster for H05 including 58 molecules
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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
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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.
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Conclusion (social)http://masubuchi.jp to get the code & docs.
NAPLESNew Algorithm for Polymeric Liquids
Entangled and Strained
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