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  • Correlation Effectsin

    Super-Arrhenius Diffusion

    Vanessa de SouzaTelluride 2007

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

  • Overview

    Introduction Strong and Fragile GlassesSimulation Details

    A Quantifiable Correlation EffectNon-Ergodic Diffusion Constants

    The Relationship betweenD(τ) andD(∞)

    Correlation,cos(θjk)

    Recovering Super-Arrhenius Behaviour

    Arrhenius DiffusionNon-Ergodic Arrhenius Diffusion

    Arrhenius−→ Super-Arrhenius Behaviour

    Fitting to Super-Arrhenius Behaviour

    Glassy LandscapesReversibility 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

    log10η/p

    oise

    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 particles256-atom: 204 type A and 52 type B particlesInteraction 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 GlassesSimulation Details

    A Quantifiable Correlation EffectNon-Ergodic Diffusion Constants

    The Relationship betweenD(τ) andD(∞)

    Correlation,cos(θjk)

    Recovering Super-Arrhenius Behaviour

    Arrhenius DiffusionNon-Ergodic Arrhenius Diffusion

    Arrhenius−→ Super-Arrhenius Behaviour

    Fitting to Super-Arrhenius Behaviour

    Glassy LandscapesReversibility 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 intervalsaveraged over anergodic trajectory

    1/T

    lnD

    0.50 0.75 1.00 1.25 1.50

    -2

    -4

    -6

    -8

    -10

    -12

    most mobile

    least mobile

    average τ = 25 (mσ2AA/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

    lnD

    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 = limt→∞

    〈∆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

    lnD

    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

    2.505.007.5012.525.062.5125250

    0.50 0.75 1.00 1.25

    −3

    −4

    −5

    −6

    −7

    −8

    −9

    1/T

    lnD

    0.50 0.75 1.00 1.25

    −3

    −4

    −5

    −6

    −7

    −8

    −9

    1/Tln

    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 GlassesSimulation Details

    A Quantifiable Correlation EffectNon-Ergodic Diffusion Constants

    The Relationship betweenD(τ) andD(∞)

    Correlation,cos(θjk)

    Recovering Super-Arrhenius Behaviour

    Arrhenius DiffusionNon-Ergodic Arrhenius Diffusion

    Arrhenius−→ Super-Arrhenius Behaviour

    Fitting to Super-Arrhenius Behaviour

    Glassy LandscapesReversibility in the Energy Landscape

    Cage-Breaking

    Non-Diffusive Connections

    Diffusive Connections

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

  • Non-Ergodic Arrhenius Diffusion

    2.505.007.5012.525.062.5125

    0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3

    −4

    −5

    −6

    −7

    −8

    −9

    1/T

    lnD

    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

    grad

    ient

    inte

    rcep

    t

    τ

    012

    −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:

    tergτ0

    = exp(

    hT j

    )

    − 1 lnDerg = −(

    mT

    )n− cT + lnD0

    0.6 0.8 1.0 1.21/T

    101

    102

    103

    t erg

    /τ0

    0.6 0.8 1.0 1.2

    −4

    −5

    −6

    −7

    −8

    −9

    1/T

    lnD

    (terg

    )

    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))

    〈r2(t

    )〉

    t

    -1

    -1

    -2

    1

    1 2 3

    0

    0

    10

    10

    10

    10

    1010101010

    low temperature

    Diffusive behaviour

    Ballistic motion〈r2(t)〉 ∝ τ2

    〈r2(t)〉 ∝ τ

    Correlation Effects in Super-Arrhenius Diffusion – p.17/27

  • 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))

    〈r2(t

    )〉

    t

    -1

    -1

    -2

    1

    1 2 3

    0

    0

    10

    10

    10

    10

    1010101010

    terg

    Correlation Effects in Super-Arrhenius Diffusion – p.17/27

  • Fitting to Super-Arrhenius Behaviour

    ln Derg(T ) = −(

    mT

    )n− cT + lnD0 can be used to fit ergodic diffusion

    constants.

    Arrhenius component and a correction,−(

    mT

    )n.

    0

    1.00.5 1.5 2.52.0

    −2

    −4

    −6

    −8

    −10

    1/T

    lnD

    Correlation Effects in Super-Arrhenius Diffusion – p.18/27

  • 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.

    Arrhenius Diffusion

    Non-ergodic diffusion is described by a time-dependent Arrhenius form.

    A formalism is introduced to describe super-Arrhenius behaviour based on

    Arrhenius and non-Arrhenius contributions to the diffusion constant.

    Correlation Effects in Super-Arrhenius Diffusion – p.19/27

  • Overview

    Introduction Strong and Fragile GlassesSimulation Details

    A Quantifiable Correlation EffectNon-Ergodic Diffusion Constants

    The Relationship betweenD(τ) andD(∞)

    Correlation,cos(θjk)

    Recovering Super-Arrhenius Behaviour

    Arrhenius DiffusionNon-Ergodic Arrhenius Diffusion

    Arrhenius−→ Super-Arrhenius Behaviour

    Fitting to Super-Arrhenius Behaviour

    Glassy LandscapesReversibility in the Energy Landscape

    Cage-Breaking

    Non-Diffusive Connections

    Diffusive Connections

    Correlation Effects in Super-Arrhenius Diffusion – p.20/27

  • Glassy Landscapes

    Quench configurations from an MD trajectory

    Connect the resulting minima via double-ended searches

    −205

    −210

    −215

    −220

    −225

    −230

    Many disordered minima at similar energies separated by high barriers

    Correlation Effects in Super-Arrhenius Diffusion – p.21/27

  • Reversibility in the Energy Landscape

    A wide spectrum of potential energy barriers exists which can be loosely divided

    into diffusive and nondiffusive processes.1,2

    NONDIFFUSIVENearest neighbour coordination shells do not change

    DIFFUSIVE One or more atoms change their nearest neighbours

    Negative correlation is present in minima to minima transitions3

    Correlated diffusive processes

    The landscape can be coarse-grained into a random walk between metabasins4

    [1] T. F. Middleton and D. J. Wales, Phys. Rev. B64, 024205 (2001).[2] T. F. Middleton and D. J. Wales, J. Chem. Phys118, 4583 (2004).[3] T. Keyes and J. Chowdhary, Phys. Rev. E64, 032201 (2001).[4] B. Doliwa and A. Heuer, Phys. Rev. E67, 031506 (2003).

    Correlation Effects in Super-Arrhenius Diffusion – p.22/27

  • Cage-Breaking

    Sequence ofminimum – transition state – minimumfor a cagebreaking

    event.

    The cagebreaking atom (shown in red) is identified by the lossand gain of

    neighbours during the cagebreaking event.

    Correlation Effects in Super-Arrhenius Diffusion – p.23/27

  • Non-Diffusive Connections

    −205

    −210

    −215

    −220

    −225

    −230

    Minima connected bynon-cagebreakingpaths

    Changes in colour show disconnected branches – highly disconnected

    Correlation Effects in Super-Arrhenius Diffusion – p.24/27

  • Non-Diffusive Connections

    −205

    −210

    −215

    −220

    −225

    −230

    −205

    −210

    −215

    −220

    −225

    −230

    Correlation Effects in Super-Arrhenius Diffusion – p.24/27

  • Diffusive Connections

    −205

    −210

    −215

    −220

    −225

    −230

    Minima connected bycage-breakingpaths

    Changes in colour show disconnected branches – highly connected

    Correlation Effects in Super-Arrhenius Diffusion – p.25/27

  • Diffusive Connections

    −205

    −210

    −215

    −220

    −225

    −230

    −205

    −210

    −215

    −220

    −225

    −230

    Correlation Effects in Super-Arrhenius Diffusion – p.25/27

  • Summary

    Glassy Landscapes

    Disconnectivity trees have a highly frustrated appearance, with a large

    number of potential energy funnels separated by relativelyhigh barriers.

    There are no clear differences between trees at different temperatures and

    for systems with differing fragility (density) suggestingthat a more subtle

    characterisation of the landscape is needed.

    We can identify cage-breaking or diffusive moves and find that these

    provide an accurate description of the exploration of the PEL.

    Reversals are evident in these diffusive moves and they increase with

    decreasing temperature as found for previous studies of ’jumps’ in a

    trajectory.

    Correlation Effects in Super-Arrhenius Diffusion – p.26/27

  • References

    V. K. de Souza and D. J. Wales, Phys. Rev. B, 74, 134202 (2006)

    Correlation Effects and Super-Arrhenius Diffusion in Binary

    Lennard-Jones Mixtures

    V. K. de Souza and D. J. Wales, Phys. Rev. Lett., 96, 057802 (2006)

    Super-Arrhenius Diffusion in a Binary Lennard-Jones Liquid Results from

    a Quantifiable Correlation Effect

    V. K. de Souza and D. J. Wales, J. Chem. Phys., 123, 134504 (2005)

    Diagnosing broken ergodicity using an energy fluctuation metric

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

    OverviewStrong and Fragile GlassesSimulation DetailsOverviewNon-Ergodic Diffusion ConstantsNon-Ergodic Diffusion ConstantsThe Relationship between $D(au )$and $D(infty )$Correlation, $�oldsymbol {cos (heta _{jk})}$Recovering Super-Arrhenius BehaviourRecovering Super-Arrhenius BehaviourSummaryOverviewNon-Ergodic Arrhenius DiffusionArrhenius $longrightarrow $ Super-Arrhenius DiffusionArrhenius $longrightarrow $ Super-Arrhenius DiffusionArrhenius $longrightarrow $ Super-Arrhenius DiffusionFitting to Super-Arrhenius BehaviourSummaryOverviewGlassy LandscapesReversibility in the Energy LandscapeCage-BreakingNon-Diffusive ConnectionsDiffusive ConnectionsSummaryReferences