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# A Computational Approach To Mesoscopic Polymer Modelling - PowerPoint PPT Presentation

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

C.P. Lowe, A. Berkenbos

University of Amsterdam

Polymers are very large molecules,

typically there are millions of repeat units.

This makes them ”mesoscopic”:

Large by atomic standards but still invisible

• Consequences:

• Their large size makes their dynamics slow and complex

• Their slow dynamics makes their effect on the fluid complex

[I] Modelling The Polymer

Step #1:Simplify the polymer to a bead-spring model that still reproduces the statistics of a real polymer

[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.

[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

[II] Modelling The Solvent

Ingredients are:

hydrodynamics (fluid like behaviour)

and

fluctuations (that jiggle the polymer around)

[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

[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

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

time it takes momentum to diffuse l

time it takes sound to travel l

time it takes a polymer to diffuse l

Reality: τsonic <τvisc << τpoly

Model (N = 2):τsonic~τvisc<τpoly

Gets better with increasing N

b

a

b is the kuhn length

For a short chain:

hydrodynamic

For a long chain (N →∞) :

Choosing the Kuhn length b:

For a value a/b ~ ¼ the scaling

holds for small N

• 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?

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

Convergence excellent.

Not exponential decay. (Time dependence effect)

N = 16 (?)

N = 32 (?)

τb = time to diffuse b

τp = τpoly

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

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

Solution: introduce a buffer lay with an external slip boundary

Result: Poiseuille flow between two plates

• (1) The method works

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