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