a computational approach to mesoscopic modelling a computational approach to mesoscopic polymer...

A Computational Approach To A Computational Approach To Mesoscopic Mesoscopic Polymer Modelling Modelling

C.P. Lowe, A. BerkenbosUniversity of Amsterdam

The ProblemThe Problem

This makes them ”mesoscopic”:

Large by atomic standards but still invisible

Polymers are very large molecules,

typically there are millions of repeat units.

The ProblemThe Problem

Consequences:

• Their large size makes their dynamics slow and complex

• Their slow dynamics makes their effect on the fluid complex

A Tractable Simulation ModelA 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 ModelA 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 ModelA 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 , Dp

0

Rg = Rg0

Dp = Dp0

A Tractable Simulation ModelA Tractable Simulation Model

[II] Modelling The Solvent

Ingredients are:

hydrodynamics (fluid like behaviour)

and

fluctuations (that jiggle the polymer around)

A Tractable Simulation ModelA 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 ModelA Tractable Simulation Model

[II] Modelling The Solvent

We use an ideal gas coupled to a Lowe-Andersen thermostat:

(1)(1) For all particles identify neighbours within a distance rc (using cell and neighbour lists)

(2)(2) Decide with some probability if a pair will undergo a bath collision

(3)(3) If yes, take a new relative velocity from a Maxwellian, and give the particles the new velocity such that momentum is conserved

(4)(4) Advect particles

A Tractable Simulation ModelA 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 ScalesTime Scales

D

l

C

l

l

poly

ssonic

visc

2

2

time it takes momentum to diffuse l

time it takes sound to travel l

time it takes a polymer to diffuse l

Time ScalesTime Scales

Reality: τsonic < τvisc << τpoly

Model (N = 2): τsonic ~ τvisc < τpoly

Gets better with increasing N

Hydrodynamics of polymer diffusionHydrodynamics of polymer diffusion

a is the hydrodynamic radius

b is the kuhn length

b a

beadD

kTa

6

Hydrodynamics of polymer diffusionHydrodynamics of polymer diffusion

)(1

nfb

a

ND

D

monmon

poly

N

constnf )(

For a short chain:

For a long chain (N →∞) :

NDpoly

1

bead

hydrodynamic

Dynamic scalingDynamic scaling

Choosing the Kuhn length b:

For a value a/b ~ ¼ the scaling

ND

D

mon

poly 1~

holds for small N

Dynamic scalingDynamic 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?Does It Work?

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

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

Centre of mass motionCentre of mass motion

Convergence excellent.

Not exponential decay. (Time dependence effect)

Surprise, it’s algebraicSurprise, it’s algebraic

MoviesMovies

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

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

Solid-Fluid Boundary ConditionsSolid-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 ConditionsSolid-Fluid Boundary Conditions

Solution: introduce a buffer lay with an external slip boundary

cR

Result: Poiseuille flow between two plates

Solid-Fluid Boundary ConditionsSolid-Fluid Boundary Conditions

ConclusionsConclusions

(1) (1) The method works

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