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Transcript of Correlation Effects in Super-Arrhenius Diffusion · PDF file The correct super-Arrhenius...

  • Correlation Effects in

    Super-Arrhenius Diffusion

    Vanessa de Souza Telluride 2007

    Correlation Effects in Super-Arrhenius Diffusion – p.1/27

  • Overview

    Introduction Strong and Fragile Glasses Simulation Details

    A Quantifiable Correlation Effect Non-Ergodic Diffusion Constants

    The Relationship betweenD(τ) andD(∞)

    Correlation,cos(θjk)

    Recovering Super-Arrhenius Behaviour

    Arrhenius Diffusion Non-Ergodic Arrhenius Diffusion

    Arrhenius−→ Super-Arrhenius Behaviour

    Fitting to Super-Arrhenius Behaviour

    Glassy Landscapes Reversibility in the Energy Landscape

    Cage-Breaking

    Non-Diffusive Connections

    Diffusive Connections

    Correlation Effects in Super-Arrhenius Diffusion – p.2/27

  • Strong and Fragile Glasses

    ‘Super-Arrhenius’ behaviour

    For some supercooled liquids, the temperature

    dependence of relaxation times or transport

    properties such as the diffusion constant,D, is

    stronger than predicted by the Arrhenius law.

    Arrhenius Super-Arrhenius

    Temperature dependence Arrhenius Law VTF equation

    η = η0 exp[A/T ] η = η0 exp[A/(T − T0)]

    Angell’s classification Strong Fragile

    Correlation Effects in Super-Arrhenius Diffusion – p.3/27

  • Strong and Fragile Glasses

    Strong

    Fragile

    lo g 1 0 η /p

    oi se

    Tg/T

    0

    12

    4

    -4

    8

    0.2 0.4 0.6 0.8 10

    Arrhenius Super-Arrhenius

    Temperature dependence Arrhenius Law VTF equation

    η = η0 exp[A/T ] η = η0 exp[A/(T − T0)]

    Angell’s classification Strong Fragile

    Correlation Effects in Super-Arrhenius Diffusion – p.3/27

  • Simulation Details

    Binary Lennard-Jones mixtures

    Number densities of 1.1, 1.2 and 1.3σ−3AA

    60-atom: 48 type A and 12 type B particles 256-atom: 204 type A and 52 type B particles Interaction parameters:σAA = 1.0, σAB = 0.8, σBB = 0.88

    ǫAA = 1.0, ǫAB = 1.5, ǫBB = 0.5

    used with theStoddard-Ford quadratic cutoff

    Microcanonical Molecular Dynamics simulations with a timestep of

    0.005(mσ2AA/ǫAA) 1/2

    The Mountain and Thirumalai energy fluctuation metric for the average

    energy was used to diagnose broken ergodicity.

    Correlation Effects in Super-Arrhenius Diffusion – p.4/27

  • Overview

    Introduction Strong and Fragile Glasses Simulation Details

    A Quantifiable Correlation Effect Non-Ergodic Diffusion Constants

    The Relationship betweenD(τ) andD(∞)

    Correlation,cos(θjk)

    Recovering Super-Arrhenius Behaviour

    Arrhenius Diffusion Non-Ergodic Arrhenius Diffusion

    Arrhenius−→ Super-Arrhenius Behaviour

    Fitting to Super-Arrhenius Behaviour

    Glassy Landscapes Reversibility in the Energy Landscape

    Cage-Breaking

    Non-Diffusive Connections

    Diffusive Connections

    Correlation Effects in Super-Arrhenius Diffusion – p.5/27

  • Non-Ergodic Diffusion Constants

    Einstein’s relation for diffusion should be used in the limit t → ∞

    Non-ergodic diffusion constants,D(τ): short,non-ergodictime intervals averaged over anergodic trajectory

    1/T

    ln D

    0.50 0.75 1.00 1.25 1.50

    -2

    -4

    -6

    -8

    -10

    -12

    most mobile

    least mobile

    average τ = 25 (mσ 2 AA/ǫAA)

    1/2

    Correlation Effects in Super-Arrhenius Diffusion – p.6/27

  • Non-Ergodic Diffusion Constants

    As τ increases, expectD(τ) → D(∞)

    Super-Arrhenius curvature is recovered as expected

    τ = 25, 250 and 2500

    (mσ2AA/ǫAA) 1/2

    1/T

    ln D

    0.50 0.75 1.00 1.25 1.50

    -2

    -4

    -6

    -8

    -10

    -12

    -14

    most mobile

    least mobile

    average

    Correlation Effects in Super-Arrhenius Diffusion – p.7/27

  • The Relationship betweenD(τ) and D(∞)

    D = lim t→∞

    〈∆ri(t) 2〉

    6t

    If the total trajectory is divided intom time intervals of lengthτ and

    ∆ri(j) = ri(jτ) − ri(jτ − τ),

    ∆ri(t) 2 =

    m ∑

    j=1

    ∆ri(j) 2 + 2

    j

  • Correlation, cos(θjk)

    k = j + 1 k = j + 2

    cos θjk

    low temperature

    −0.5−1.0

    0.5

    0.5

    1.0

    1.00.0

    1.5

    2.0

    2.5

    cos θjk

    low temperature

    −0.5−1.0

    0.5

    0.5

    1.0

    1.00.0

    1.5

    2.0

    2.5

    〈cos θjk〉 6= 0 for adjacent time intervals only.

    〈cos θjk〉 is negative, an atom is most likely to move backwards relative to

    the displacement vector in the previous time window.

    Correlation Effects in Super-Arrhenius Diffusion – p.9/27

  • Recovering Super-Arrhenius Behaviour

    If only the average behaviour of〈cos θj,j−1〉 is important:

    ∆ri(t) 2 =

    m ∑

    j=1

    ∆ri(j) 2 × (1 + 2〈cos θj,j−1〉)

    D(τ)∗ = D(τ) × (1 + 2〈cos θj,j−1〉)

    Correlation Effects in Super-Arrhenius Diffusion – p.10/27

  • Recovering Super-Arrhenius Behaviour

    If only the average behaviour of〈cos θj,j−1〉 is important:

    ∆ri(t) 2 =

    m ∑

    j=1

    ∆ri(j) 2 × (1 + 2〈cos θj,j−1〉)

    D(τ)∗ = D(τ) × (1 + 2〈cos θj,j−1〉)

    Super-Arrhenius

    behaviour is recovered

    1/T

    ln D

    0.50 0.75 1.00 1.25 1.50

    -2

    -4

    -6

    -8

    -10

    -12

    -14

    Correlation Effects in Super-Arrhenius Diffusion – p.10/27

  • Recovering Super-Arrhenius Behaviour

    2.50 5.00 7.50 12.5 25.0 62.5 125 250

    0.50 0.75 1.00 1.25

    −3

    −4

    −5

    −6

    −7

    −8

    −9

    1/T

    ln D

    0.50 0.75 1.00 1.25

    −3

    −4

    −5

    −6

    −7

    −8

    −9

    1/T ln

    D (τ

    )∗

    The effect extends to larger system sizes

    Finite size effects are small

    The correction is sufficient except at the shortest time scales

    Correlation Effects in Super-Arrhenius Diffusion – p.11/27

  • Summary

    A Quantifiable Correlation Effect

    The diffusion constant is overestimated for time windows that are too short

    to register negative correlations in atomic positions.

    The correct super-Arrhenius behaviour can be recovered:

    a) in thelimit of long time intervals

    b) with acorrection containing the average anglebetween steps

    We can directly link negative correlation with an increase in effective

    activation energy at low temperature.

    Correlation Effects in Super-Arrhenius Diffusion – p.12/27

  • Overview

    Introduction Strong and Fragile Glasses Simulation Details

    A Quantifiable Correlation Effect Non-Ergodic Diffusion Constants

    The Relationship betweenD(τ) andD(∞)

    Correlation,cos(θjk)

    Recovering Super-Arrhenius Behaviour

    Arrhenius Diffusion Non-Ergodic Arrhenius Diffusion

    Arrhenius−→ Super-Arrhenius Behaviour

    Fitting to Super-Arrhenius Behaviour

    Glassy Landscapes Reversibility in the Energy Landscape

    Cage-Breaking

    Non-Diffusive Connections

    Diffusive Connections

    Correlation Effects in Super-Arrhenius Diffusion – p.13/27

  • Non-Ergodic Arrhenius Diffusion

    2.50 5.00 7.50 12.5 25.0 62.5 125

    0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3

    −4

    −5

    −6

    −7

    −8

    −9

    1/T

    ln D

    Arrhenius forms can be found over a range of timescales

    Exclude ergodic and ‘caged’ results

    Correlation Effects in Super-Arrhenius Diffusion – p.14/27

  • Arrhenius −→ Super-Arrhenius Diffusion

    gradient/intercept = a(ln(τ/τ0 + 1)) b + c

    0.05 cutoff 0.1 cutoff 0.2 cutoff

    10

    10

    100

    100

    gr ad

    ie nt

    in te

    rc ep

    t

    τ

    0 1 2

    −1

    −2

    −2

    −4

    −6

    −8

    lnD(T, τ) = −a(ln(τ/τ0 + 1))

    b − c

    T + d(ln(τ/τ0 + 1))

    e + lnD0

    Correlation Effects in Super-Arrhenius Diffusion – p.15/27

  • Arrhenius −→ Super-Arrhenius Diffusion

    Minimisation gives an ergodic time and an ergodic diffusionconstant:

    terg τ0

    = exp (

    h T j

    )

    − 1 lnDerg = − (

    m T

    )n − cT + lnD0

    0.6 0.8 1.0 1.2 1/T

    101

    102

    103

    t e rg

    /τ 0

    0.6 0.8 1.0 1.2

    −4

    −5

    −6

    −7

    −8

    −9

    1/T

    ln D

    (t e rg

    )

    Correlation Effects in Super-Arrhenius Diffusion – p.16/27

  • Arrhenius −→ Super-Arrhenius Diffusion

    D(T, τ) is linked to the local behaviour of the mean square displacement.

    〈r2(T, τ)〉 = 6τ exp(ln D(T, τ))

    〈r 2 (t

    )〉

    t

    -1

    -1

    -2

    1

    1 2 3

    0

    0

    10