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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. 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

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  1. A Computational Approach To Mesoscopic Polymer Modelling C.P. Lowe, A. Berkenbos University of Amsterdam

  2. 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

  3. The Problem • Consequences: • Their large size makes their dynamics slow and complex • Their slow dynamics makes their effect on the fluid complex

  4. 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

  5. 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.

  6. 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

  7. A Tractable Simulation Model [II] Modelling The Solvent Ingredients are: hydrodynamics (fluid like behaviour) and fluctuations (that jiggle the polymer around)

  8. 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

  9. 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

  10. 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.

  11. Time Scales time it takes momentum to diffuse l time it takes sound to travel l time it takes a polymer to diffuse l

  12. Time Scales Reality: τsonic <τvisc << τpoly Model (N = 2):τsonic~τvisc<τpoly Gets better with increasing N

  13. Hydrodynamics of polymer diffusion b a a is the hydrodynamic radius b is the kuhn length

  14. Hydrodynamics of polymer diffusion For a short chain: hydrodynamic bead For a long chain (N →∞) :

  15. Dynamic scaling Choosing the Kuhn length b: For a value a/b ~ ¼ the scaling holds for small N

  16. 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

  17. 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

  18. Centre of mass motion Convergence excellent. Not exponential decay. (Time dependence effect)

  19. Surprise, it’s algebraic

  20. Movies N = 16 (?) N = 32 (?)

  21. Stress-stress (short) τb = time to diffuse b

  22. Stress-stress (long) τp = τpoly

  23. Solves a more relevant (and testing) problem… viscosity Time dependent polymer contribution to the viscosity For polyethylene τp ~ 0.1 s

  24. 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

  25. Solid-Fluid Boundary Conditions Solution: introduce a buffer lay with an external slip boundary

  26. Solid-Fluid Boundary Conditions Result: Poiseuille flow between two plates

  27. 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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