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