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

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

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

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 ~ s. We need to simulate for t > 1 s.

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 R g 0, D p 0 R g = R g 0 D p = D p 0

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

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

A 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 r c (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 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 Scales time it takes momentum to diffuse l time it takes sound to travel l time it takes a polymer to diffuse l

Time Scales Reality: τ sonic < τ visc << τ poly Model (N = 2): τ sonic ~ τ visc < τ poly Gets better with increasing N

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

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

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

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

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 motion Convergence excellent. Not exponential decay. (Time dependence effect)

Surprise, it’s algebraic

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

Stress-stress (short) τ b = time to diffuse b

Stress-stress (long) τ p = τ poly

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

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

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

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

Conclusions (1) (1) The method works (2) (2) It takes 16 beads to simulate the long time viscoelastic response of an infinitely long polymer