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
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