Deformation of polyethylene by Monte Carlo simulation From virial theorem to stressstrain behaviour

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Deformation of polyethylene by Monte Carlo
simulationFrom virial theorem to stress-strain
behaviour

• Jing Li

Group Polymer Physics, Eindhoven Polymer
Laboratories and Dutch Polymer Institute,
Technische Universiteit Eindhoven, The
Netherlands
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Typical stress-strain curves
H.G.H. v. Melick, Deformation and failure of
polymer glasses, PhD thesis, Eindhoven
university of Technology (2002)
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Virial theorem
Monte Carlo simulation
Lennard-Jones liquid
Diatomic molecule
Linear Polyethylene
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Outline

• Brief introduction of Monte Carlo method and
  virial theorem
• Polyethylene
• Pressure calculation
• Simulation procedure
• Conclusions
• Outlooks

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Monte Carlo method

• A stochastic method first developed by
  Metropolis in 1953.

• A procedure for evaluating configuration space
  equilibrium averages

• Many algorithms are developed to generate a
  sequence of configurations.

• The average of any property over this sequence
  is an approximation to the measured value of that
  property for the same thermodynamic state (e.g.
  N, P, T).

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Virial theorem
In standard thermodynamics the pressure is given
by
where the Hamiltonian
The pressure
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Polyethylene model
Linear Low mol.wt Polydisperse Coarse-grained
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Polybead model
Methyl group is considered as beads.
Chemical bond is replaced by rod or spring.
OR
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Pressure calculation
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Pressure calculation
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Pressure calculation
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Stress tensor
In fact, pressure is a tensor with the more
general form
For isotropic case, the scalar pressure is the
average of three components on the diagonal of
matrix.
Stress tensor
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Simulation procedure
Geom. builder
Equilibration (MC)
Deformation (MC)
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Force fields
1. Harmonic bond stretching
2. Harmonic angle bending
3. Dihedral angle torsion
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Force fields
4. Non-bonded interaction

• Intra-molecular interaction begins from beads
  separated by at least 4 segments.
• Potential is cut off at certain distance.
• Long-range tail interaction is replaced by a
  quintic spline.

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Geometry builder

• Random walk process.
• Include bonded interaction.
• Reject hard-sphere overlap.

5000 monomers in 100 chains. The sample is
prepared under T 450K with experimental density
corresponding to this temperature. The box size
is about 50Å.
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Monte Carlo Local moves
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Monte Carlo Global moves
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Equilibration of structure

• Temperature is gradually reduced to target value.
• The system is then relaxed under compressive
  pressure to remove excess volume.
• The pressure is then switched back to atmosphere
  pressure until it reaches a steady state in the
  energy and volume behaviour.

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Deformation

• Constant velocity deformation
• increase box length by ?L.
• Keep length of this direction fixed and relax for
  m steps of MC moves.

3. End-bridging is switched off during
deformation.
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Evolution of density
The densities goes down at small strain
corresponding to elastic deformation. After
yielding molecules start to flow which keeps
volume constant.
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Stress-strain behaviour
Strain rate dependence
Temperature dependence
Slower rate and lower temperature give a higher
stress behaviour.
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Snapshots of deformation
Strain 0.4
Strain 0
Strain 0.6
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Conclusions

• We implement the virial theorem in Monte Carlo
  simulation.
• The pressure for the model with constraints can
  also be well calculated.
• Many advanced Monte Carlo algorithms are applied
  to amorphous polyethylene system.
• Stress-strain behaviour during deformation of
  polyethylene reveals the natural dependence of
  temperature and strain rate.

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Outlooks

• Calculate pressure for model of polyethylene
  with rigid bonds.
• Compare deformations of polyethylene with rigid
  and flexible bonds.
• Youngs modulus, yielding and strain hardening
  etc. can be studied under different temperatures
  and strain rates.
• Introduce the Monte Carlo method to complicated
  systems (PS, PC).

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Acknowledgment
ir. Tim Mulder ir. Bart Vorselaars dr. Alexey
Lyulin prof.dr. Thijs Michels