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H. Meyer, J. Wittmer
A. Johner, A. N. Semenov
J. Baschnagel
Institut Charles Sadron
Université de Strasbourg
Strasbourg, France
Molecular simulations in polymer physics
Introduction● neutral chain
● good solvent
Polymer := chain molecule with N monomers
Example: polyethylene (monomer = CH2)
end-to-end distance:
N = 1 04 R~ 1 02n m
bond length ~ 0.1 nm
connectivity
repulsive interaction
Introduction● neutral chain
● good solvent
Polymer := chain molecule with N monomers
local properties
depend on chemistry
Example: polyethylene (monomer = CH2)
end-to-end distance:
N = 1 04 R~ 1 02n m
bond length ~ 0.1 nm
connectivity
repulsive interaction
Introduction● neutral chain
● good solvent
Polymer := chain molecule with N monomers
local properties
depend on chemistry
global properties = universal
1 /N = T−T c/T c
polymer critical system
P.-G. de Gennes
1972
Example: polyethylene (monomer = CH2)
end-to-end distance:
N = 1 04 R~ 1 02n m
R = be N0.588
bond length ~ 0.1 nm
connectivity
repulsive interaction
Introduction● neutral chain
● good solvent
Polymer := chain molecule with N monomers
all atom models
polymeric materials
mesoscopic models polymer melt
many chain
systems
Example: polyethylene (monomer = CH2)
end-to-end distance:
N = 1 04 R~ 1 02n m
R = be N0.588
bond length ~ 0.1 nm
connectivity
repulsive interaction
Introduction● neutral chain
● good solvent
Polymer := chain molecule with N monomers
all atom models
polymeric materials
mesoscopic models
1. generic models
2. derived models
polymer melt
many chain
systems
Example: polyethylene (monomer = CH2)
end-to-end distance:
N = 1 04 R~ 1 02n m
R = be N0.588
bond length ~ 0.1 nm
connectivity
repulsive interaction
Introduction● neutral chain
● good solvent
Polymer := chain molecule with N monomers
all atom models
polymeric materials
mesoscopic models
1. generic models
2. derived models
polymer melt
many chain
systems
Example: polyethylene (monomer = CH2)
end-to-end distance:
N = 1 04 R~ 1 02n m
R = be N0.588
bond length ~ 0.1 nm
connectivity
repulsive interaction
Polymer melts: universal properties
Polymer melt : many-chain system
Polymer melts: universal properties
Polymer melt : many-chain system
solution melt
isolated chains chain overlap
connectivity
repulsion
Polymer melts: universal properties
Polymer melt : many-chain system
solution melt
isolated chains chain overlap UEr = weak
UEr Edwards potential
1975
S. F. EdwardsS. F. Edwardsconnectivity
repulsion
Polymer melts: universal properties
Polymer melt : many-chain system
Flory ideality hypothesis (1949) :
“Chains in a melt are (nearly) ideal.”
solution melt
isolated chains chain overlap ideal chainUEr = weak
UEr Edwards potential
connectivity
repulsion
connectivity
Bond-bond correlations
n
m = n + s
s monomers
b n
bm
Ps ∝ ⟨b n⋅bns ⟩ ∝ exp− sl p ideal chains
Bond-bond correlations
n
m = n + s
s monomers
b n
bm
Ps ∝ ⟨b n⋅bns ⟩ ∝ exp− sl p ideal chains
interaction
Bond-fluctuation model
[ J. Wittmer, H. Meyer, et al., PRE 76, 011803 (2007)]
Bond-bond correlations
n
m = n + s
s monomers
b n
bm
Ps ∝ ⟨b n⋅bns ⟩ ∝ exp− sl p ideal chains
interaction
Bond-fluctuation model
[ J. Wittmer, H. Meyer, et al., PRE 76, 011803 (2007)]
Bond-bond correlations
n
m = n + s
s monomers
b n
bm
Ps ∝ ⟨b n⋅bns ⟩ ∝ exp− sl p ideal chains
interaction
Bond-fluctuation model
[ J. Wittmer, H. Meyer, et al., PRE 76, 011803 (2007)]
Bond-bond correlations
n
m = n + s
s monomers
b n
bm
Ps ∝ ⟨b n⋅bns ⟩ ∝ exp− sl p ideal chains
interaction
Bond-fluctuation model
[ J. Wittmer, H. Meyer, et al., PRE 76, 011803 (2007)]
Bond-bond correlations
n
m = n + s
s monomers
b n
bm
Ps ∝ ⟨b n⋅bns ⟩ ∝ exp− sl p ideal chains
interaction
Bond-fluctuation model
[ J. Wittmer, H. Meyer, et al., PRE 76, 011803 (2007)]
Bond-bond correlations
n
m = n + s
s monomers
b n
bm
Ps ∝ ⟨b n⋅bns ⟩ ∝ exp− sl p ideal chains
interaction
Bond-fluctuation model
[ J. Wittmer, H. Meyer, et al., PRE 76, 011803 (2007)]
Bond-bond correlations
n
m = n + s
s monomers
b n
bm
Ps ∝ ⟨b n⋅bns ⟩ ∝ exp− sl p ideal chains
interaction
Bond-fluctuation model
[ J. Wittmer, H. Meyer, et al., PRE 76, 011803 (2007)]
Bond-bond correlations
n
m = n + s
s monomers
b n
bm
Ps ∝ ⟨b n⋅bns ⟩ ∝ exp− sl p
Perturbation theory N ∞
UEr = weakEdwards potential
ideal chains
[ J. Wittmer, H. Meyer, et al., PRE 76, 011803 (2007)]
Bond-bond correlations
n
m = n + s
s monomers
b n
bm
Ps ∝ ⟨b n⋅bns ⟩ ∝ exp− sl p
Perturbation theory N ∞
UEr = weakEdwards potential
Ps =cP
s3 /2, cP ∝
1
be3
ideal chains
monomer density
[ J. Wittmer, H. Meyer, et al., PRE 76, 011803 (2007)]
Bond-bond correlations
n
m = n + s
s monomers
b n
bm
Ps ∝ ⟨b n⋅bns ⟩ ∝ exp− sl p ideal chains
interaction
Bond-fluctuation model
N = 8192 (128 chains): ~ 1000 000 monomers
[ J. Wittmer, H. Meyer, et al., PRE 76, 011803 (2007)]
Simulating long chains
displacement: r i r i '= r i r
⇔ x x ' ⇒ U x '
min 1,exp{−[U x '−U x ] / kBT }
accept according to Metropolis criterion:
configuration x : U xBond-fluctuation model:
interaction
local move
[J. Baschnagel, et al., NIC series 23, 83 (2004); http://www.fz-juelich.de/nic-series]
Simulating long chains
displacement: r i r i '= r i r
⇔ x x ' ⇒ U x '
min 1,exp{−[U x '−U x ] / kBT }
accept according to Metropolis criterion:
configuration x : U xBond-fluctuation model:
interaction
local move
slithering snake
nonlocal moves
[J. Baschnagel, et al., NIC series 23, 83 (2004); http://www.fz-juelich.de/nic-series]
Simulating long chains configuration x : U x
displacement: r i r i '= r i r
⇔ x x ' ⇒ U x '
min 1,exp{−[U x '−U x ] / kBT }
accept according to Metropolis criterion:
Bond-fluctuation model:
interaction
local move
slithering snake
nonlocal moves
N-conserving connectivity altering move
[J. Baschnagel, et al., NIC series 23, 83 (2004); http://www.fz-juelich.de/nic-series]
10-2 10-1 100 101 102 103 104 10510-3
10-2
10-1
100
∝ t-0.25
Ne = 36
PB18000 N=324PB11400 N=205PB9470 N=169PB4600 N=83PB2020 N=36 (= M
e)
PB466 N=8
Kreer et al. N=512 N=128 N=64 N=32 N=16
∝ t-0.5
t / τs
(Rouse)
φ b(t
/τs)
Bond-bond correlations: dynamics
b t =⟨ bn t ⋅
bn 0 ⟩
⟨ b n20 ⟩
Bond-fluctuation model:local moves
Polybutadiene:field cycling NMR
b n 0 b n t
n n
time
increases
b s =1e
[A. Herrmann, et al., Macromolecules 42, 2063 (2009)]
1
bt /s
Summary
end-to-end distance:
N = 1 04 R~ 1 02n m
bond length ~ 0.1 nm
connectivity
repulsive interaction
all atom models
polymeric materials
mesoscopic models
● conformation,
structure
● dynamics
polymer melt
many chain
systems
chemical specificity large degree of universal behavior