Glassy Dynamics in the Potential Energy Landscape 10 AA Vanessa - PowerPoint PPT Presentation

Glassy Dynamics in the Potential Energy Landscape 10 ǫ AA Vanessa de Souza University of Granada, Spain University of Cambridge Glassy Dynamics in the Potential Energy Landscape – p. 1/

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 Glassy Dynamics in the Potential Energy Landscape – p. 2/

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 = η 0 exp[ A/T ] η = η 0 exp[ A/ ( T − T 0 )] Angell’s classification Strong Fragile Glassy Dynamics in the Potential Energy Landscape – p. 3/

Strong and Fragile Glasses 12 8 log 10 η /poise Strong 4 0 Fragile − 4 0 0 . 2 0 . 4 0 . 6 0 . 8 1 T g / T Arrhenius Super-Arrhenius Temperature dependence Arrhenius Law VTF equation η = η 0 exp[ A/T ] η = η 0 exp[ A/ ( T − T 0 )] Angell’s classification Strong Fragile Glassy Dynamics in the Potential Energy Landscape – p. 3/

The Loch Ness Monster Glassy Dynamics in the Potential Energy Landscape – p. 4/

The Loch Ness Monster Glassy Dynamics in the Potential Energy Landscape – p. 4/

The Loch Ness Monster Glassy Dynamics in the Potential Energy Landscape – p. 4/

Potential Energy Landscapes Potential Energy Landscape (PEL): the potential energy as a function of all the relevant particle coordinates. transition state minimum Any structure can be minimised to find its inherent structure, a minimum on the PEL. minimum Discretisation and simplification of configuration space. Dynamics requires information about transition states, the highest point on the lowest-energy pathway between two minima. Glassy Dynamics in the Potential Energy Landscape – p. 5/

Visualising the Landscape - Crystal Landscapes Disconnectivity Graphs 10 ǫ AA Calvo, Bogdan, de Souza and Wales, JCP 127, 044508 (2007) Glassy Dynamics in the Potential Energy Landscape – p. 6/

Visualising the Landscape - Glassy Landscapes Disconnectivity Graphs 10 ǫ AA de Souza and Wales, JCP 129, 164507 (2008) Glassy Dynamics in the Potential Energy Landscape – p. 7/

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 Glassy Dynamics in the Potential Energy Landscape – p. 8/

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. Glassy Dynamics in the Potential Energy Landscape – p. 9/

Fitting to Super-Arrhenius Behaviour � n − c � m ln D erg ( T ) = − T + ln D 0 T de Souza and Wales Arrhenius component: − c T + ln D 0 PRB 74, 134202 (2006) � m PRL 96, 057802 (2006) � n Correction: − T 0 − 2 − 4 ln D − 6 space here − 8 − 10 0.5 1.0 1.5 2.0 2.5 1 /T space here Glassy Dynamics in the Potential Energy Landscape – p. 10/

Levels of Coarse-Graining 10 ǫ AA Negative correlation in Minima-to-Minima Transitions ⇓ Negatively correlated Diffusive Processes ⇓ Random Walk between Metabasins Glassy Dynamics in the Potential Energy Landscape – p. 11/

Mean square displacement → Diffusion Einstein relation: D = lim t →∞ 1 6 t � ∆ r 2 ( t ) � Diffusive behaviour 1 � r 2 ( t ) � ∝ t 10 � r 2 ( t ) � 0 10 low temperature -1 10 Ballistic motion -2 10 � r 2 ( t ) � ∝ t 2 -1 0 1 2 3 10 10 10 10 10 t Glassy Dynamics in the Potential Energy Landscape – p. 12/

Nearest Neighbours g AA ( r ) 8 AA interaction end of first-neighbour shell 4 0 1 . 8 1 . 6 0.8 1.0 1.2 1.4 2.0 r g AB ( r ) AB interaction 8 4 0 1 . 6 1 . 8 0.8 1.0 1.2 1.4 2.0 r g BB ( r ) 8 BB interaction 4 0 1 . 6 1 . 8 0.8 1.0 1.2 1.4 2.0 r Glassy Dynamics in the Potential Energy Landscape – p. 13/

Nearest Neighbours g AA ( r ) 8 AA interaction 4 0 1 . 8 2 . 2 2 . 4 1 . 6 2 . 6 0.8 1.0 1.2 1.4 2.0 g AB ( r ) AB interaction 8 4 0 1 . 6 1 . 8 2 . 2 2 . 4 2 . 6 0.8 1.0 1.2 1.4 2.0 g BB ( r ) 8 BB interaction 4 0 1 . 6 1 . 8 2 . 2 2 . 4 2 . 6 0.8 1.0 1.2 1.4 2.0 r Glassy Dynamics in the Potential Energy Landscape – p. 13/

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) Glassy Dynamics in the Potential Energy Landscape – p. 14/

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. 3 cage-breaks 3 space 2 reversals 2 here Last cage-break is Productive 1 Glassy Dynamics in the Potential Energy Landscape – p. 15/

Diffusion from Productive Cage-Breaks M � Productive Cage-breaks follow a random walk, � r 2 ( t ) � = L 2 j j =1 − 3 − 4 − 5 − 6 ln D − 7 − 8 − 9 − 10 0.5 1.0 1.5 2.0 2.5 3.0 3.5 1 /T 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 Glassy Dynamics in the Potential Energy Landscape – p. 16/

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 ) . � � 1 + 2 − n (1) + n (2) − n (3) + · · · � r 2 ( t ) � = ML 2 M Reversal chain, z=2. space Two reversal chains, z=1. here n (1) = 2 and n (2) = 1 Glassy Dynamics in the Potential Energy Landscape – p. 17/

Diffusion from All Cage-Breaks M � � 1 + 2 − n (1) + n (2) − n (3) + · · · All � � r 2 ( t ) � = L 2 j × M Cage-Breaks j =1 − 3 − 4 − 5 − 6 ln D − 7 − 8 − 9 − 10 0.5 1.0 1.5 2.0 2.5 3.0 3.5 1 /T Glassy Dynamics in the Potential Energy Landscape – p. 18/

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 Glassy Dynamics in the Potential Energy Landscape – p. 19/

Levels of Coarse-Graining 10 ǫ AA Negative correlation in Minima-to-Minima Transitions ⇓ Negatively correlated Diffusive Processes ⇓ Random Walk between Metabasins Glassy Dynamics in the Potential Energy Landscape – p. 20/

Levels of Coarse-Graining 10 ǫ AA Negative correlation in Minima-to-Minima Transitions ⇓ Correlated Random Walk of Cage-Breaking events ⇓ Random Walk between Metabasins Glassy Dynamics in the Potential Energy Landscape – p. 20/

Levels of Coarse-Graining 10 ǫ AA Negative correlation in Minima-to-Minima Transitions ⇓ Correlated Random Walk of Cage-Breaking events ⇓ Random Walk of Productive Cage-Breaking events Glassy Dynamics in the Potential Energy Landscape – p. 20/

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. Glassy Dynamics in the Potential Energy Landscape – p. 21/