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  • Hierarchy of models for entangled polymers

    Alexei E. Likhtman

    Department of Mathematics, University of Reading

    With Sathish K. Sukumaran, Ponmurugan Mani, Jing Cao

  • Hierarchical modelling in polymer dynamics

    • Constitutive equations

    – Tube theories

    • Single chain models

    ?),(σ σ

    f Dt

    D =

    Tube

    Model? jm slip-spring

    • Single chain models

    – Coarse-grained

    many-chains models

    » Atomistic simulations

    > Quantum mechanics simulations

    bead-spring

    molecular dynamics

    Well established coarse-

    graining procedures,

    force-fields,

    commercial packages

    CR

    Model? ja

    slip-spring

    model

  • 40 years of the Tube theory 1967-2007

    • Original assumptions

    – Independent motion of each chain

    – along the primitive path like a 1-D Rouse chain

    – purely entropic stress (Kramer’s)

    • Predictions

    – Stress plateau

    – Two stage relaxation and damping function

    – Mean square displacement

    – for stars

    4.3

    wd M∝∝τη

    )exp( Mνη ∝

    • Problems

    – constraint release vs. tube dilution

    – branch point motion

    – tube diameter under deformation

    – tube field and its stress contribution

    – how to find the tube in MD?

    – for stars

    – Contour length fluctuations

    )exp( wMνη ∝

    •many contradicting theories coexist

    •many proven results remain ignored

    •one theory postdoc per

    new experiment

  • 20 instantaneous trajectories

  • 20 mean paths

  • 20 mean paths

  • Construction of the slip-spring model

    ja

    jm

    timeelementary

    size coil

    re temperatu

    parameters model Rouse

    0

    2

    τ

    gR

    T

    chain thealonglink -slip offriction -

    chain) anchoring in the monomers ofnumber effective(or link -slip ofstrength -

    links-slipbetween beads ofnumber average -

    parameters New

    s

    s

    e

    N

    N

    ζ

    slip_links19.exe A.L., Macromolecules, 2005, 38 (14), 6128

    Rubinstein, Panyukov, Macromolecules 2002, 35, 6670-6686

  • Construction of the slip-spring model

    ja

    jm

    timeelementary

    size coil

    re temperatu

    parameters model Rouse

    0

    2

    τ

    gR

    T

    chain thealonglink -slip offriction -

    chain) anchoring in the monomers ofnumber effective(or link -slip ofstrength -

    links-slipbetween beads ofnumber average -

    parameters New

    s

    s

    e

    N

    N

    ζ

    slip_links19.exe A.L., Macromolecules, 2005, 38 (14), 6128

    Rubinstein, Panyukov, Macromolecules 2002, 35, 6670-6686

  • Constraint release

    Hua and Schieber 1998

    Shanbhag, Larson, Takimoto, Doi 2001

  • Outline

    • Molecular dynamics as an experimental tool

    • Molecular dynamics fitted by slip-springs• Molecular dynamics fitted by slip-springs

    • Microscopic definition of entanglements

  • Molecular Dynamics -- Kremer-Grest model

    • Polymers – Bead-FENE

    spring chains

    0

    2 2

    2

    0

    ( ) ln 1 2

    FENE

    kR r U r

    R

      = − − 

     

    • With excluded volume – Purely

    • k = 30ε/σ2

    • R0=1.5σ

    Density, ρ = 0.85• With excluded volume – Purely

    repulsive Lennard-Jones

    interaction between beads

    otherwise 0

    2 r 4

    1 4)( 61

    612

    =

    <  

     

     +

      

     −

      

     =

    rr rUrLJ

    σσ ε

    Density, ρ = 0.85

    Friction coefficent, ζ = 0.5

    Time step, dt = 0.012

    Temperature, T = ε/k

    K.Kremer, G. S. Grest

    JCP 92 5057 (1990)

  • Stress relaxation for N=350

    0

    30

    60

    1

    10

    100

    G (t

    )

    G (t

    )

    bond length

    relaxation 0.1 1 10

    0.01 0.1 1 10 100 1000 10000 100000 1E-3

    0.01

    0.1 t

    G (t

    )

    t

    G N

    (0)

    τ e

    relaxation

    collisions

    "Rouse" dynamics

    entangled

  • Multiple tau correlator

    G (t

    )

    1e-3

    1e-2

    1e-1

    1e0

    1e1

    G (t

    )

    1e-3

    1e-2

    1e-1

    1e0

    1e1

    Magatti and Ferri, Applied optics, 40, 4011 (2000)

    t 0.1 1 10 100 1,000 10,000 100,000

    t 0.1 1 10 100 1,000 10,000 100,000

  • Stress autocorrelation

    N=1,2,5,10,25,50,100,200,350

    1

    10 N=350

    N=200

    N=100

    N=50

    N=25

    N=10

    N=5

    N=2

    G (t

    )

    0.1 1 10 100 1000 10000 100000

    0.01

    0.1

    N=2

    N=1

    G (t

    )

    t A.E.Likhtman, S.K.Sukumaran, J.Ramirez, Macromolecules, 2007, 40, 6748-6757

  • 1.0

    1.2

    1.4

    1.6

    1.8

    Normalized G(t)

    1 10 100 1000 10000 100000 0.0

    0.2

    0.4

    0.6

    0.8

    G (t

    )t 1

    /2

    t

    Rouse

    Entanglements

  • Fit with slip-springs model as an extrapolation tool

    A.E.Likhtman, S.K.Sukumaran, J.Ramirez, Macromolecules, 2007, 40, 6748-6757

  • Varying chain stiffness

    1E-3

    0.01

    0.1

    1

    10

    50 100

    20 30

    50 70

    50 100

    200

    k b =3, ev2, x10

    -2

    k =1.1, x10 -3

    k b =1.5, x0.1

    k b =3, x10

    k b =5,ev4

    G (t

    )

    200

    0.003

    0.004

    0.005

    0.006

    1.5 1.1 3odd

    0

    -3 rho06

    rho0.4

    1 10 100 1000 10000 100000

    1E-8

    1E-7

    1E-6

    1E-5

    1E-4

    300 100

    50 200100

    100

    350

    50

    100 150

    200

    200

    ρ=0.6, x10 -5

    k b =-3, x10

    -4

    k b =1.1, x10

    -3

    t

    0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.11.2 1E-3

    0.002

    0.003

    3

    5ev4

    G~p -3

    (Fetters)

    G N

    (0 ) p

    3

    p

    slip-spring fit with free G0 and M0

    same with fixed M0=rho/G0

    PPA

    G~p -7/5

    (Semenov)

    A.E.Likhtman, S.K.Sukumaran, J.Ramirez, Macromolecules, 2007, 40, 6748-6757

  • g1(t) – monomer mean square displacement

    100

    1000

    (t )

    N=50,100,200,350

    10 100 1000 10000 100000 1000000 1

    10

    g 1 (t

    )

    t

  • g1(i,t) – monomer mean square displacement

    0.2

    0.4

    0.6

    0.8

    1.0

    1.2

    1.4

    0.2

    0.4

    0.6

    0.8

    1.0

    1.2

    1.4 g

    1 i/t

    1 /2

    g 1 i/t

    1 /2

    N=50 N=100

    10 100 1000 10000 100000 1000000 0.2

    10 100 1000 10000 100000 1000000 0.2

    10 100 1000 10000 100000 1000000 0.2

    0.4

    0.6

    0.8

    1.0

    10 100 1000 10000 100000 1000000 0.2

    0.4

    0.6

    0.8

    1.0 t t

    g 1 i/t

    1 /2

    t

    g 1 i/t

    1 /2

    t

    N=200 N=350

  • Slip-links vs MD

    10 1

    10 2

    10 3

    10 4

    10 5

    10 6

    0.4

    0.6

    0.8

    1

    (g 1 (t

    )/ σ

    2 )/

    (t /τ

    )0 .5

    10 1

    10 2

    10 3

    10 4

    0.4

    0.6

    0.8

    1

    1.2

    (a) (c)

    Central

    Peripheral

    N=200 N=50

    10 1

    10 2

    10 3

    10 4

    10 5

    10 6

    t/τ

    10 -1

    10 0

    (G (t

    )/ ε/

    σ 3 )(

    t/ τ)

    0 .5

    10 1

    10 2

    10 3

    10 4

    t/τ

    10 -1

    10 0

    (b) (d)

    τ e

    S.K. Sukumaran and AEL, Macromolecules 2009, 42, 4300–4309

  • Mismatch between static properties in MD and slip-springs

    )( 2RR ji >−<

    ||

    )(

    ji

    RR ji

    >−

  • Additional interactions in slip-spring model

  • Static properties of the melt in MD

    ||

    )( 2

    ji

    RR ji

    >−<

    >−−=< ++++ ))(()( 11 sisiii RRRRsP

  • N=50, slip-spring model with additional potential

  • Simulations of Langevin equations with memory

  • N=200, slip-spring model with additional potential and with memory

    S.K. Sukumaran and AEL, Macromolecul

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