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Glassy Dynamics in the Potential Energy Landscape

Vanessa de Souza

University of Granada, Spain University of Cambridge

10 ǫAA

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Overview

Introduction Strong and Fragile Glasses Potential Energy Landscape Visualising the Potential Energy Landscape Glassy Dynamics Coarse-graining the Landscape - Metabasins Cage-breaking Reversed and Productive Cagebreaks Calculating Diffusion Constants Cage-break Metabasins Random Walk Metabasins vs. Cagebreaks

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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

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Strong and Fragile Glasses

Strong Fragile log10 η/poise Tg/T 12 4 −4 8 0.2 0.4 0.6 0.8 1 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

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The Loch Ness Monster

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The Loch Ness Monster

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The Loch Ness Monster

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Potential Energy Landscapes

Potential Energy Landscape (PEL): the potential energy as a function of all the relevant particle coordinates. Any structure can be minimised to find its inherent structure, a minimum on the PEL. Discretisation and simplification of configuration space.

minimum minimum transition state

Dynamics requires information about transition states, the highest point on the lowest-energy pathway between two minima.

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Visualising the Landscape - Crystal Landscapes

Disconnectivity Graphs 10 ǫAA

Calvo, Bogdan, de Souza and Wales, JCP 127, 044508 (2007)

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Visualising the Landscape - Glassy Landscapes

Disconnectivity Graphs 10 ǫAA

de Souza and Wales, JCP 129, 164507 (2008)

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Overview

Introduction Strong and Fragile Glasses Potential Energy Landscape Visualising the Potential Energy Landscape Glassy Dynamics Coarse-graining the Landscape - Metabasins Cage-breaking Reversed and Productive Cagebreaks Calculating Diffusion Constants Cage-break Metabasins Random Walk Metabasins vs. Cagebreaks

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Coarse-graining the landscape

Transitions between metabasins follow a random walk Metabasins are well-characterised by an energy and waiting time Diffusion constants can be calculated

Doliwa and Heuer, PRE (2003)

Problems with this approach: How but not Why. No information about microscopic mechanisms, within metabasins or for transitions between metabasins. Identify minima by total system energy, the method cannot be scaled for larger system sizes, restricted to around 65 atoms.

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Fitting to Super-Arrhenius Behaviour

ln Derg(T) = − m

T

n − c

T + ln D0

Arrhenius component: − c

T + ln D0

Correction: − m

T

n

de Souza and Wales PRB 74, 134202 (2006) PRL 96, 057802 (2006)

space here 1.0 0.5 1.5 2.5 2.0 −2 −4 −6 −8 −10 1/T ln D space here

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Levels of Coarse-Graining

10 ǫAA Negative correlation in Minima-to-Minima Transitions ⇓ Negatively correlated Diffusive Processes ⇓ Random Walk between Metabasins

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Mean square displacement → Diffusion

Einstein relation: D = limt→∞ 1

6t∆r2(t)

r2(t) t

• 1
• 1
• 2

1 1 2 3 10 10 10 10 10 10 10 10 10 low temperature Diffusive behaviour Ballistic motion r2(t) ∝ t2 r2(t) ∝ t

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Nearest Neighbours

0.8 0.8 0.8 1.0 1.0 1.0 1.2 1.2 1.2 1.4 1.4 1.4 2.0 2.0 2.0 gAA(r) gBB(r) gAB(r) r r r 4 4 4 8 8 8 1.6 1.6 1.6 1.8 1.8 1.8 end of first-neighbour shell AA interaction AB interaction BB interaction

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Nearest Neighbours

0.8 0.8 0.8 1.0 1.0 1.0 1.2 1.2 1.2 1.4 1.4 1.4 2.0 2.0 2.0 gAA(r) gBB(r) gAB(r) r 4 4 4 8 8 8 1.6 1.6 1.6 1.8 1.8 1.8 2.2 2.2 2.2 2.4 2.4 2.4 2.6 2.6 2.6 AA interaction AB interaction BB interaction

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Cage-Breaking Criteria

Nearest neighbours are within a distance of 1.25 for an AA interaction. For the loss of a neighbour, relative distance changes by more than 0.561, which corresponds to half the equilibrium pair separation. A cage-break occurs with the loss/gain of at least two neighbours. Sequence of minimum – transition state – minimum for a cagebreak.

de Souza and Wales, JCP 129, 164507 (2008)

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Reversed Cage-Breaks

Identified when the net displacement squared is less than 10−5. Chains of repeatedly reversed cage-breaks are found. Determine cage-breaks which are Productive towards long-term diffusion: The cage-break is not followed by the reverse event. The cage-break is not part of a reversal chain OR ends a chain with an even number of reversals. 1 2 3 space here 3 cage-breaks 2 reversals Last cage-break is Productive

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Diffusion from Productive Cage-Breaks

Productive Cage-breaks follow a random walk, r2(t) =

M

• j=1

L2

j

1.0 0.5 1.5 2.5 3.5 2.0 3.0 −3 −4 −5 −6 −7 −8 −9 −10 1/T ln D 60-atom binary Lennard-Jones at number densities of 1.3 and 1.1 Landscape-influenced regime (1/T): 0.78 and 1.78 Landscape-dominanced regime (1/T): 1.56 and 3.56

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Accounting for correlation

The following simplifications are suggested by our studies of diffusion using Molecular Dynamics trajectories: The displacements of cage-breaks are similar and can be represented by a constant, L. Correlation arises from direct return events. We can account for correlation effects using a count of reversal chains of length z, n(z). r2(t) = ML2

• 1 + 2−n(1) + n(2) − n(3) + · · ·

M

• space

here Reversal chain, z=2. Two reversal chains, z=1. n(1) = 2 and n(2) = 1

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Diffusion from All Cage-Breaks

r2(t) =

M

• j=1

L2

j ×

• 1 + 2−n(1) + n(2) − n(3) + · · ·

M

• All

Cage-Breaks 1.0 0.5 1.5 2.5 3.5 2.0 3.0 −3 −4 −5 −6 −7 −8 −9 −10 1/T ln D

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Overview

Introduction Strong and Fragile Glasses Potential Energy Landscape Visualising the Potential Energy Landscape Glassy Dynamics Coarse-graining the Landscape - Metabasins Cage-breaking Reversed and Productive Cagebreaks Calculating Diffusion Constants Cage-break Metabasins Random Walk Metabasins vs. Cagebreaks

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Levels of Coarse-Graining

10 ǫAA Negative correlation in Minima-to-Minima Transitions ⇓ Negatively correlated Diffusive Processes ⇓ Random Walk between Metabasins

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Levels of Coarse-Graining

10 ǫAA Negative correlation in Minima-to-Minima Transitions ⇓ Correlated Random Walk of Cage-Breaking events ⇓ Random Walk between Metabasins

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Levels of Coarse-Graining

10 ǫAA Negative correlation in Minima-to-Minima Transitions ⇓ Correlated Random Walk of Cage-Breaking events ⇓ Random Walk of Productive Cage-Breaking events

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Metabasins vs. Cagebreaks

Transitions between metabasins follow a random walk Metabasins are well-characterised by an energy and waiting time Diffusion constants can be calculated

de Souza, Rehwald and Heuer, in preparation (2013)

Advantages of this method: How and Why. Information about microscopic mechanisms, within metabasins and for transitions between metabasins. Method can be scaled for larger system sizes.

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Conclusions

The Potential Energy Landscape for glass-forming systems is extremely complex. The landscape can be coarse-grained into metabasins Important transitions such as cagebreaks can be identified We have reconciled the two approaches, providing a microscopic description for metabasins within the PEL in the form of productive cagebreaks. Microscopic mechanisms <–> Macroscopic properties

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The Loch Ness Monster

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