Mesoscopic simulations of entangled polymers, blends, copolymers, and branched structures
description
Transcript of Mesoscopic simulations of entangled polymers, blends, copolymers, and branched structures
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