A Computational Approach To Mesoscopic Polymer Modelling

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A Computational Approach To Mesoscopic Polymer
Modelling

• C.P. Lowe, A. Berkenbos
• University of Amsterdam

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The Problem
Polymers are very large molecules, typically
there are millions of repeat units.
This makes them mesoscopic Large by atomic
standards but still invisible
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The Problem

• Consequences
• Their large size makes their dynamics slow and
  complex
• Their slow dynamics makes their effect on the
  fluid complex

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A Tractable Simulation Model
I Modelling The Polymer Step 1 Simplify the
polymer to a bead-spring model that still
reproduces the statistics of a real polymer
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A Tractable Simulation Model
I Modelling The Polymer We still need to
simplify the problem because simulating even this
at the atomic level needs ?t 10-9 s. We
need to simulate for t gt 1 s.
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A Tractable Simulation Model
I Modelling The Polymer Step 2 Simplify the
bead-spring model further to a model with a few
beads keeping the essential (?) feature of the
original long polymer
Rg0 , Dp0
Rg Rg0 Dp Dp0
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A Tractable Simulation Model
II Modelling The Solvent Ingredients
are hydrodynamics (fluid like behaviour)
and fluctuations (that jiggle the polymer
around)
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A Tractable Simulation Model
II Modelling The Solvent The solvent is
modelled explicitly as an ideal gas couple to a
Lowe-Andersen thermostat - Gallilean
invariant - Conservation of momentum -
Isotropic fluctuations fluctuating
hydrodynamics
Hydrodynamics
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A Tractable Simulation Model
II Modelling The Solvent We use an ideal gas
coupled to a Lowe-Andersen thermostat
(1) For all particles identify neighbours within
a distance rc (using cell and neighbour lists)
(2) Decide with some probability if a pair will
undergo a bath collision
(3) If yes, take a new relative velocity from a
Maxwellian, and give the particles the new
velocity such that momentum is conserved
(4) Advect particles
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A Tractable Simulation Model
III Modelling Bead-Solvent interactions Therm
ostat interactions between the beads and the
solvent are the same as the solvent-solvent
interactions. There are no bead-bead
interactions.
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Time Scales
time it takes momentum to diffuse l time it
takes sound to travel l time it takes a
polymer to diffuse l
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Time Scales
Reality tsonic lt tvisc ltlt tpoly Model (N 2)
tsonic tvisc lt tpoly Gets better with
increasing N
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Hydrodynamics of polymer diffusion
b
a
a is the hydrodynamic radius b is the kuhn length
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Hydrodynamics of polymer diffusion
For a short chain
hydrodynamic
bead
For a long chain (N ?8)
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Dynamic scaling
Choosing the Kuhn length b For a value a/b ¼
the scaling
holds for small N
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Dynamic scaling

• Dynamic scaling requires only one time-scale to
  enter the system
• For the motion of the centre of mass this choice
  enforces this for small N
• Hope it rapidly converges to the large N results

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Does It Work?
b 4a requires b solvent particle separation
so
Hydrodynamic contribution to the diffusion
coefficient for model chains with varying bead
number N
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Centre of mass motion
Convergence excellent. Not exponential decay.
(Time dependence effect)
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Surprise, its algebraic
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Movies
N 16 (?)
N 32 (?)
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Stress-stress (short)
tb time to diffuse b
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Stress-stress (long)
tp tpoly
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Solves a more relevant (and testing) problem
viscosity
Time dependent polymer contribution to the
viscosity For polyethylene tp 0.1 s
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Solid-Fluid Boundary Conditions
We can impose solid/fluid boundary conditions
using a bounce back rule
But near the boundary a particle has less
neighbours ? less thermostat collisions ?
lower viscosity, thus creating a massive boundary
artefact
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Solid-Fluid Boundary Conditions
Solution introduce a buffer lay with an
external slip boundary
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Solid-Fluid Boundary Conditions
Result Poiseuille flow between two plates
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Conclusions

• (1) The method works
• (2) It takes 16 beads to simulate the long time
  viscoelastic response of an infinitely long
  polymer