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

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The problem The large size of polymers makes their dynamics slow and complex

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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

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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.

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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

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

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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

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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

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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.

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

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

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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.

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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.

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

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Hydrodynamics of polymer diffusion For a short chain: For a long chain (n → ∞) a/b is irrelevant: bead hydrodynamic

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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.

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

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

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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:

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

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Surprise, it’s algebraic

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

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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

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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)?

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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.

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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.

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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)

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

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Interacting chains Plot one wayPlot another way

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

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Interacting chains This works for depressing large N.

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

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