Structure and Dynamics of Polymer Introduction to Polymer Physics Generic Polymer Models Molecular...

download Structure and Dynamics of Polymer Introduction to Polymer Physics Generic Polymer Models Molecular Simulation

of 40

  • date post

    10-Jul-2020
  • Category

    Documents

  • view

    1
  • download

    0

Embed Size (px)

Transcript of Structure and Dynamics of Polymer Introduction to Polymer Physics Generic Polymer Models Molecular...

  • Introduction to Polymer Physics

    Generic Polymer Models

    Molecular Simulation Methods

    Polymer Films I: Chain Extension

    Polymer Films II: Glass Transition

    Summary

    Structure and Dynamics of Polymer Films

    J. Baschnagel, S. Peter, H. Meyer, J. P. Wittmer

    Institut Charles Sadron Université Louis Pasteur

    Strasbourg, France

    NanoSoft Nanomatériaux, Surfaces et Objets FoncTionnalisés

    21–25 May 2007 / Roscoff

  • Introduction to Polymer Physics

    Generic Polymer Models

    Molecular Simulation Methods

    Polymer Films I: Chain Extension

    Polymer Films II: Glass Transition

    Summary

    Outline

    Introduction to Polymer Physics

    Generic Polymer Models

    Molecular Simulation Methods

    Polymer Films I: Chain Extension

    Polymer Films II: Glass Transition

  • Introduction to Polymer Physics

    Generic Polymer Models

    Molecular Simulation Methods

    Polymer Films I: Chain Extension

    Polymer Films II: Glass Transition

    Summary

    Polymer: Definition and Conformation Polymer := macromolecule of N monomers

    conformation: x = (~r1, . . . ,~rN) (monomer positions) x = (~r1, ~b1 . . . , ~bN−1) (bond vectors)

    Example: polyethylene (monomer = CH2)

    persistence length `p ∼ 5 Å

    bond length `0 ∼ 1 Å

    θ

    φ

    end-to-end distance

    radius of gyration Rg

    N =104: Re∼103Å

  • Introduction to Polymer Physics

    Generic Polymer Models

    Molecular Simulation Methods

    Polymer Films I: Chain Extension

    Polymer Films II: Glass Transition

    Summary

    Polymer Physics Polymer physics = calculate macroscopic properties

    A(x ) from microscopic interactions U(x )

    〈A〉 = 1 Z

    ∫ dx A(x ) exp

    [ − βU(x )

    ] , β =

    1 kBT

    Assumption about U:

    U(x ) = N−1∑ i=1

    U0(~bi , ~bi+1, . . . , ~bi+imax)︸ ︷︷ ︸ “short range”: `,θ,φ

    +U1(x , solvent)︸ ︷︷ ︸ “long range”

    `p

    `0

    θ

    φ Rg

    Re

  • Introduction to Polymer Physics

    Generic Polymer Models

    Molecular Simulation Methods

    Polymer Films I: Chain Extension

    Polymer Films II: Glass Transition

    Summary

    Ideal Chains Ideal polymer := U1 = 0

    `p := “memory” of orientation along the chain

    `p = `0

    ∞∑ k=0

    〈b̂i · b̂i+k 〉 (b̂i = unit vector)

    R2e = N−1∑ i=1

    N−1∑ j=1

    〈~bi · ~bj〉 = 2`20 N−1∑ i=1

    N−1∑ j=i

    〈b̂i · b̂j〉 − (N − 1)`20

    ' 2`20 N∑

    i=1

    ∞∑ k=0

    〈b̂i · b̂i+k 〉︸ ︷︷ ︸ =`p/`0

    −N`20 = N [ 2`0`p − `20

    ] ︸ ︷︷ ︸

    = b2e (effective bond length)

    Result: N ↔ t and Re =̂ distance covered in time t

    Re ∝ N1/2 (Brownian motion or random walk)

  • Introduction to Polymer Physics

    Generic Polymer Models

    Molecular Simulation Methods

    Polymer Films I: Chain Extension

    Polymer Films II: Glass Transition

    Summary

    Real Chains Real polymer := U0 + U1, good solvent: U1 = repulsive

    Re ∝ Nν 1 2

    < ν < 1 (ν = 0.588)

    `p

    `0

    θ

    φ

    local properties depend on chemistry

    Rg Re

    global properties = universal:

    polymer ↔ critical system 1/N ↔ (T − Tc)/Tc = τ

    Re ∝ Rg ∼ Nν ↔ ξ ∼ τ−ν

  • Introduction to Polymer Physics

    Generic Polymer Models

    Molecular Simulation Methods

    Polymer Films I: Chain Extension

    Polymer Films II: Glass Transition

    Summary

    Coarse-Graining the Model

    b

    Θ

    I spatial continuum

    I `, θ, φ, . . . = continuous variables

    I realistic potentials

    self-avoiding walk

    b

    Θ

    I lattice (e.g. simple cubic)

    I b = lattice constant Θ = 90◦, 180◦

    I connectivity, excluded volume

    When can this be a viable model?

    I no long-range (e.g., electrostatic) or specific (e.g., H-bonds) interactions

    I generic properties

  • Introduction to Polymer Physics

    Generic Polymer Models

    Molecular Simulation Methods

    Polymer Films I: Chain Extension

    Polymer Films II: Glass Transition

    Summary

    A Lattice and a Continuum Model

    Bond-Fluctuation Model

    BFM = lattice model I ~b ∈ B with:

    b = 2, √

    5, √

    6, 3, √

    10 I hard-core interaction

    b

    Bead-Spring Model

    BSM = continuum model

    ULJ(r) = 4� [(σ

    r

    )12 −

    (σ r

    )6]

    LJ � b0

    bond

    Ubond(b) = k 2

    ( b − b0

    )2

  • Introduction to Polymer Physics

    Generic Polymer Models

    Molecular Simulation Methods

    Polymer Films I: Chain Extension

    Polymer Films II: Glass Transition

    Summary

    A Lattice and a Continuum Model

    Bond-Fluctuation Model

    BFM = lattice model I ~b ∈ B with:

    b = 2, √

    5, √

    6, 3, √

    10 I hard-core interaction

    b

    Monte Carlo simulation

    Bead-Spring Model

    BSM = continuum model

    ULJ(r) = 4� [(σ

    r

    )12 −

    (σ r

    )6]

    LJ � b0

    bond

    Ubond(b) = k 2

    ( b − b0

    )2

  • Introduction to Polymer Physics

    Generic Polymer Models

    Molecular Simulation Methods

    Polymer Films I: Chain Extension

    Polymer Films II: Glass Transition

    Summary

    A Lattice and a Continuum Model

    Bond-Fluctuation Model

    BFM = lattice model I ~b ∈ B with:

    b = 2, √

    5, √

    6, 3, √

    10 I hard-core interaction

    b

    Monte Carlo simulation

    Bead-Spring Model

    BSM = continuum model

    ULJ(r) = 4� [(σ

    r

    )12 −

    (σ r

    )6]

    LJ � b0

    bond

    Ubond(b) = k 2

    ( b − b0

    )2 Monte Carlo or

    Molecular Dynamics

  • Introduction to Polymer Physics

    Generic Polymer Models

    Molecular Simulation Methods

    Polymer Films I: Chain Extension

    Polymer Films II: Glass Transition

    Summary

    Molecular Dynamics (MD) Simulations MD: numerical solution of Newton’s equations of motion

    d2~ri dt2

    = 1 m

    ~Fi ⇐⇒

     d~ri dt

    = 1 m

    ~pi = ∂H ∂~pi

    d~pi dt

    = ~Fi = − ∂H ∂~ri

    discretization−→ ~ri(tµ + h) = ~ri(tµ) +

    ~pi(tµ) m

    h + ~Fi(tµ) 2m

    h2

    ~pi(tµ + h) = ~pi(tµ) + ~Fi(tµ) h

    Observables:

    A != lim t→∞

    1 t

    ∫ t 0

    dt A(x (t)) M�1 ≈ 1

    M

    M∑ µ=1

    A(x (tµ))

  • Introduction to Polymer Physics

    Generic Polymer Models

    Molecular Simulation Methods

    Polymer Films I: Chain Extension

    Polymer Films II: Glass Transition

    Summary

    From MD to Monte Carlo Simulations . . . MD = deterministic dynamics in phase space

    x = (~r1, . . . ,~rN ; ~p1, . . . , ~pN)

    d~ri dt

    = 1 m

    ~pi

    d~pi dt

    = ~Fi

     ⇒ microcanonical ensemble

    %(x ) ∝ δ ( H(x )− E

    ) MD with noise := introduce a weak stochastic damping

    and a random force =̂ “Langevin thermostat”

    d~ri dt

    = 1 m

    ~pi

    d~pi dt

    =

    [ ~Fi −

    ζ

    m ~pi

    ] +~fi(t)

     ⇒ canonical ensemble

    %(x ) ∝ e−βH(x )

    〈~f (t)〉 = 0 〈fα(t)fβ(t ′)〉 = 2kBT ζ δαβδ

    ( t − t ′

    ) random force

    ←→ − ζ m

    ~pi

    friction

  • Introduction to Polymer Physics

    Generic Polymer Models

    Molecular Simulation Methods

    Polymer Films I: Chain Extension

    Polymer Films II: Glass Transition

    Summary

    . . . From MD to Monte Carlo Simulations

    Brownian dynamics := | �

    ~pi | � |ζ~pi/m| ⇒ stochastic dynamics in configuration space x = (~r1, . . . ,~rN)

    ζ d~ri dt

    = ζ

    m ~pi = ~Fi +~fi = −

    ∂U(x ) ∂~ri

    +~fi

    stationary distribution = canonical distribution

    %(x ) ∝ e−βU(x )

    MC := generation of a sequence of correlated configurations x via a Markov process

    · · · x W (x→x ′)−→ x ′ · · ·

    present state future state

    accept transition according to the Metropolis criterion

    W (x → x ′) = min (

    1, e−β[U(x ′)−U(x )]

    )

  • Introduction to Polymer Physics

    Generic Polymer Models

    Molecular Simulation Methods

    Polymer Films I: Chain Extension

    Polymer Films II: Glass Transition

    Summary

    Monte Carlo (MC) Simulations

    Example (local moves BFM)

    I choose a monomer and jump direction at random

    I accept jump if I excluded volume is satisfied I ~b ∈ B

    b

    Example (local moves BSM)

    ~ri

    ~r ′i

    x = (~r1, . . . ,~rN) : U(x )

    displacement: ~ri → ~r ′i = ~ri + ∆~r ⇔ x → x ′ ⇒ U(x ′)

    accept according to

    min (