simulating mesoscopic polymer dynamics c.p. lowe, m.w. dreischor university of amsterdam

Simulating Mesoscopic Simulating Mesoscopic Polymer DynamicsPolymer Dynamics

C.P. Lowe, M.W. DreischorUniversity of Amsterdam

The problemThe problem

The large size of polymers makes their dynamics slow and 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



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.



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


[II] Modelling The Solvent

Ingredients are:

hydrodynamics (fluid like behaviour)

and

fluctuations (that jiggle the polymer around)



The solvent is modelled explicitly as an ideal gas coupled to a Lowe-Andersen thermostat:

- Detailed balance

- Gallilean invariant

- Conservation of momentum

- Isotropic

+ fluctuations = fluctuating hydrodynamics

Hydrodynamics

Statics



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


[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

AlternativesAlternatives

[I] DPD

Very similar but harder to integrate the equations of motion.

[II] Malevanets-Kapral method

Not shown to work in the correct parameter regime. Gallilean invariance must be ‘forced’ in. Boxes.

AlternativesAlternatives

[III] Lattice-Boltzmann

Beter control of the parameters. No rigorous thermodynamics (fluctuation dissipation).

[IV] Stokesian Dynamics

Neglects the time-dependence of the HI. Poor scaling with number of polymers. External geometries are difficult.

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 → ∞) a/b is irrelevant:

NDpoly

1

bead

hydrodynamic

Dynamic scalingDynamic scaling

Choosing the Kuhn length b:

For a value a/b ~ ¼ the long polymer scaling

Holds for small N. Look for behaviour independent of N.

ND

D

mon

poly 1~

Dynamic scalingDynamic scaling

Or

‘Pure’ renormalization also gives 1/√N scaling for small N to a good approximation. Either are okay.

658.3

1

)(

1

nfb

a

MoviesMovies

N = 16 (?) N = 32 (?)

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

Solves a more relevant problem… viscositySolves a more relevant problem… viscosity

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

Conclusions so farConclusions so far

(1) The method is simple but works

(2) Encouragingly, it takes 16 beads to simulate the long time viscoelastic response of an infinitely long ideal polymer

Interacting chainsInteracting chains

Really we want to simulate interacting chains. We start with one ‘excluded volume chain’.

Question:

How do we renormalize the static properties (interactions between ‘blobs’ of polymer)?


Problem [I]: Flory: excluded volume parameter υ (effective monomer volume).

υ = lattice volume x (1/2 – χ )

What is υ off-lattice (i.e. in reality)? This is solved.


Problem [I]: Flory: excluded volume parameter υ (effective monomer volume).

υ = lattice volume x (1/2 – χ )

What is υ off-lattice (i.e. in reality)? This is solved. For details ask Menno.


Problem [II]:

What do we need to reproduce with an effective monomer/monomer potential?

- Ideal chain size (easy)

- Same degree of expansion independent of N (hard)


Is this problem already solved1 ?

1. R.M. Jendrejack et al., J. Chem.

Phys 116, 7752 (2002)


Plot one way Plot another way


2/13

235 2N

b

B

Alternative:

Flory’s result:

So keep constant.

(B2 = second virial coefficient)

2/13

22N

b

B


This works for depressing large N.

ConclusionsConclusions

(1) Ideal polymers are doable

(2) Interacting polymers might be possible but much still to do

simulating mesoscopic polymer dynamics c.p. lowe, m.w. dreischor university of amsterdam

Documents

tractable simulation

real polymer slide

solvent solvent interactions

complex slide

beadspring model

polymer step

beadbead interactions

solvent ingredients