Simulating Mesoscopic Polymer Dynamics C.P. Lowe, M.W. Dreischor University of Amsterdam

The problem The large size of polymers makes their dynamics slow and 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 coupled to a Lowe-Andersen thermostat: - Detailed balance - Gallilean invariant - Conservation of momentum - Isotropic + fluctuations = fluctuating hydrodynamics Hydrodynamics Statics

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 r c (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

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

Alternatives [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.

Alternatives [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 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 → ∞) a/b is irrelevant: bead hydrodynamic

Dynamic 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.

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

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

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

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

Conclusions 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 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)?

Interacting chains 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.

Interacting chains 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.

Interacting chains 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)

Interacting chains Is this problem already solved 1 ? 1. R.M. Jendrejack et al., J. Chem. Phys 116, 7752 (2002)

Interacting chains Plot one wayPlot another way

Interacting chains Alternative: Flory’s result: So keep constant. (B 2 = second virial coefficient)

Interacting chains This works for depressing large N.

Conclusions (1) Ideal polymers are doable (2) Interacting polymers might be possible but much still to do