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. 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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A computational approach to mesoscopic polymer modelling

A Computational Approach To Mesoscopic Polymer Modelling

C.P. Lowe, A. Berkenbos

University of Amsterdam


The problem
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


The problem1
The Problem

  • Consequences:

  • Their large size makes their dynamics slow and complex

  • Their slow dynamics makes their effect on the fluid complex


A tractable simulation model
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


A tractable simulation model1
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 > 1 s.


A tractable simulation model2
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


A tractable simulation model3
A Tractable Simulation Model

[II] Modelling The Solvent

Ingredients are:

hydrodynamics (fluid like behaviour)

and

fluctuations (that jiggle the polymer around)


A tractable simulation model4
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


A tractable simulation model5
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


A tractable simulation model6
A Tractable Simulation Model

[III] Modelling Bead-Solvent interactions

Thermostat interactions between the beads and the solvent are the same as the solvent-solvent interactions.

There are no bead-bead interactions.


Time scales
Time Scales

time it takes momentum to diffuse l

time it takes sound to travel l

time it takes a polymer to diffuse l


Time scales1
Time Scales

Reality: τsonic <τvisc << τpoly

Model (N = 2):τsonic~τvisc<τpoly

Gets better with increasing N


Hydrodynamics of polymer diffusion
Hydrodynamics of polymer diffusion

b

a

a is the hydrodynamic radius

b is the kuhn length


Hydrodynamics of polymer diffusion1
Hydrodynamics of polymer diffusion

For a short chain:

hydrodynamic

bead

For a long chain (N →∞) :


Dynamic scaling
Dynamic scaling

Choosing the Kuhn length b:

For a value a/b ~ ¼ the scaling

holds for small N


Dynamic scaling1
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


Does it work

b = 4a requires b ~ solvent particle separation so:

Does It Work?

Hydrodynamic contribution to the diffusion coefficient for model chains with varying bead number N


Centre of mass motion
Centre of mass motion

Convergence excellent.

Not exponential decay. (Time dependence effect)



Movies
Movies

N = 16 (?)

N = 32 (?)


Stress stress short
Stress-stress (short)

τb = time to diffuse b


Stress stress long
Stress-stress (long)

τp = τpoly


Solves a more relevant and testing problem viscosity
Solves a more relevant (and testing) problem… viscosity

Time dependent polymer contribution to the viscosity For polyethylene τp ~ 0.1 s


Solid fluid boundary conditions
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


Solid fluid boundary conditions1
Solid-Fluid Boundary Conditions

Solution: introduce a buffer lay with an external slip boundary


Solid fluid boundary conditions2
Solid-Fluid Boundary Conditions

Result: Poiseuille flow between two plates


Conclusions
Conclusions

  • (1) The method works

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


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