Arrest, Crystallization and Surfaces Properties of ...

Arrest, Crystallization and Surfaces Properties of Amorphous Materials

Computer Simulations and Future Challenges in Amorphous Materials

Peter HarrowellSchool of Chemistry

University of Sydney

2019 International Graduate Summer School on “Frontiers of Soft Matter and Amorphous Materials”

Outline of Lectures

Lecture 1. Structure in Liquids and its Consequences

Lecture 2. Dynamics: Spatial Heterogeneities and its Link to Structure

Lecture 3. Amorphous Surfaces: Kinetic Enhancement and Surface-Mediated Glass Fabrication

The Kinetic Arrest of a Supercooled Liquid is Called the Glass Transition

Hawaii volcano fills sky with acid plumes and glass shards as lava hits sea 22 May 2018

“Pele’s Hair” – glass fibres formed during rapid stretching and cooling of lava

obsidian

pumice outcrop, Kamchatka Peninsula

Dynamics: Spatial Heterogeneity and the Link to Structure

Tm

Glass Supercooled Liquid

Glass transition

halpy

Temperature

Crystal

fast cooling

slow cooling

The glass state is a consequence of the liquid slowing down sufficiently to fall out of equilibrium

Structural Explanations of Dynamics

These are generally difficult since dynamics disrupts the very structure that we would use to explain it.

Defect mediated dynamics is one case in which most of the structural features are persistent.

1. Diffusing defect theory

2. ‘Zipper’ motion of defects in a random tiling phase

3. Grain boundary mobility …….

Diffusing Defect Model of Relaxation

Relaxation at a site occurs once a diffusing defect makes contact

JCP 33, 639 (1960)



1/2

( ) exp tC tτ

= −

a ‘stretched’ exponential1D

JCP 33, 639 (1960)



1/2

( ) exp tC tτ

= −


Chem. Phys. Lett 32, 592 (1975)3D( ) exp tC t

τ = −

stretching lost !

JCP 33, 639 (1960)



1/2

( ) exp tC tτ

= −


Chem. Phys. Lett 32, 592 (1975)3D( ) exp tC t

τ = −

stretching lost !

3D ( ) exp tC tα

τ = −

stretching recovered in 3Dusing Levy walks

Diffusing Defects in a Random Tiling Phase

A binary mixture of soft discs can form a random tiling phase in 2D

Allowed local structures




Defects (disclinations)




Defects (disclinations)

Elementary ‘flip’

Passage of Defect Through the Random Tiling

Transitions from one random tiling to another are achieved by the passage of these defects.

Defects are possible in random arrangements

Passage of Defect Through the Random Tiling

Transitions from one random tiling to another are achieved by the passage of these defects.

Defects are possible in random arrangements

In these examples fluctuations are crucial to dynamics.

A mean field treatment would produce no motion since the disorder averaged over the sample would be locally insignificant

Diffusing Defect Models can be Characterised by Dynamic Heterogeneities

Instead of monitoring structure, we could just monitor dynamics.

In the case of diffusing defects, this would be sufficient to determine concentration and mobility of the objects – even if we could not explain why

they moved and other regions did not.

Introducing Dynamic Heterogeneities

T= 0.6T= 0.5T= 0.4T= 0.3

Models of Heterogeneous Dynamics in Amorphous Materials

Progress in understanding the relation between structure and relaxation dynamics has (and continues) to relay heavily on models.

It is important to understand the limits of each model

1. Facilitated Kinetic Ising Model2. Random Bond (Constraint) Model3. Molecular Dynamics

Fredrickson and Andersen, PRL 53, 1244 (1984)

Butler and Harrowell, JCP 95, 4454 (1991)

Simple thermodynamics of uncoupled spins

Complicated dynamics

A spin can’t flip unless n or more of it neighbours are up

- and the field h favours down spins

Dynamics and Structure:Facilitated Kinetic Ising Model

‘Facilitated kinetics ’ is, in retrospect, an misleading name – ‘structure determined kinetics’ is more accurate.


Relaxation (white) propagates out in time from ‘seeds’ whose concentration decreases with decreasing T

Unflipped spin (black)

Flipped spin (white)


High Temperature


Relaxation (white) propagates out in time from ‘seeds’ whose concentration decreases with decreasing T

Note trapping of relaxation in small domains

Unflipped spin (black)

Flipped spin (white)


Low Temperature


Relaxation ‘seeds’ are those that avoid being trapped by ‘walls’ of down spins


As the concentration of up spins decreases, the probability of complete walls of larger extent increases


This model introduced the notions that dynamic heterogeneities might

a) exist as a generic feature of glass relaxation and

b) that these heterogeneities provided a real space signature of the ‘hidden’ structural fluctuations responsible for dynamics (i.e. the analogues if ‘defects’)


Are these notions generally true?

Introducing a Non-Local Model of Heterogeneous Dynamics:

Constraint Model of Philips and Thorpe

Philips (1979), Thorpe (1983) provide a generic model of disordered solids and the non-local action of the constraints on particle motion.

Jacobs and Thorpe, PRE (1996)

A floppy structure consisting of 4 rigid clusters

(each linking bond is a rigid cluster of two particles)

Addition of one more bond results in single rigid cluster

The Dynamic Heterogeneity of Floppy Modes in a Constraint Network

Jacobs and Thorpe, PRE (1996)

overconstrained

marginally constrained

(= isostatic)

pivots associated with floppy modes

There is a clear but nontrivial connection between the spatial distribution of bonds and the distribution of floppy modes

Unconstrained Motion vs Floppy Modes

While we can rigorously count the number of floppy modes remaining in any bond network we cannot uniquely identify the particles associated with them. (A problem similar to uniquely identifying degenerate vibrational modes.)

De Souza and Harrowell PNAS 106, 15136, (2009)

The pentagonal ring has two internal floppy modes

We could nominate these two modes but the choice is not unique.

Unconstrained Motion vs Floppy Modes

A unique assignment can be made to each unconstrained motion. Unlike floppy modes, unconstrained motions are not necessarily independent. This is not a problem if what we want is a map of the possible individual motions.

De Souza and Harrowell PNAS, (2009)

Particles in an unconstrained motion are those which become rigid by the addition of a bond across a pivot.

The ratio of the number of unconstrained motions to floppy motions

Thermal motion of rigid clusters linked by pivots

Motion consists of the translation and rotation of free clusters and the coupled rotation of clusters joined by flexible pivots.

Consider some rigid clusters linked by floppy pivot points

Thermal motion of rigid clusters linked by pivotsMotion consists of the translation and rotation of free clusters and the coupled rotation of clusters joined by flexible pivots.

Rigid clusters constrained by a single pivot can still rotate independently


Rotation of rigid clusters that are constrained by 2 or more pivots requires the mutual rotation of all the connected clusters

similar to random gear model Zwanzig JCP 1987


Rotation of rigid clusters that are constrained by 2 or more pivots requires the mutual rotation of all the connected clusters

similar to random gear model Zwanzig JCP 1987

Relaxation rate ωi of particles in cluster i

ωi ~ (kBT/Ii)1/2

where Ii = total moment of inertia of the clusters coupled to the motion of cluster i

Idea: modes slow down as the size of the coupled cluster grows

Moment of inertia of rigid clusters resembles that of compact disks

Despite the random shapes of clusters

Ii ~ si2.05

similar to that expected for compact 2D object

si = number of particles in cluster i

ln(moment of inertia of cluster i)

Distribution of Frequencies of Unconstrained Motions

bond density increasing

rigidity percolation

The width of frequency distribution has a maximum at the rigidity percolation density

Spatial Distribution of the Time Scale of the Fastest Unconstrained Motion each Particle can Access

increasing bond density


How Sensitive is the Spatial Distribution of Relaxation Times to the Change of a Single Bond?



Maps of the average variation in relaxation times due to random changes of a single bond

How Sensitive is the Spatial Distribution of Relaxation Times to the Change of a Single Bond?



Maps of the average variation in relaxation times due to random changes of a single bond

While the variation is localised at high bond density, we find massive nonlocal sensitivity close to the rigidity percolation. This feature was not found in the Facilitated Kinetic Ising model along with the system size dependence of dynamic length scale

Structural relaxation of a Permanent Network

bond densities above percolation

bond density below percolation

at rigidity percolation

When the bond configuration is fixed, structure can only completely relax below the rigidity percolation

Bertheir 2011

Dynamic Heterogeneities: Maps and Measures

Non-Gaussian parameter

Molecular dynamics of binary mixtures in 2D

Dynamic Heterogeneities: Maps and Measures

Non-Gaussian parameter

Molecular dynamics of binary mixtures in 2D

Model

Particles diffuse via a diffusion coefficient that is, itself, undergoing a stochastic variation that obeys

To give

Comment on the Interpretation of Non-Gaussian Dynamics

Heterogeneous dynamics implies non-Gaussian dynamics - True

But non-Gaussian dynamics does not automatically imply heterogeneity

Trap-diffusion models

i.e. the transition from incremental continuous random walks to dynamics characterised by rare large jumps

will also generate non-Gaussian behaviour – even when homogeneously distributed in space.

Role of Collective Strain in Relaxation Dynamics: Irreversibility and Unrecoverable Strain

Structural relaxation typically involves particles moving a distance of ~ 0.3 x diameter - smaller than the distances typically involved in establish a diffusion coefficient.

These structural relaxation distances are small enough to overlap with collective vibrations and so we have to separate reversible from irreversible relaxation.




Idea

a rearrangement re-sets the minimum of the potential so strain does not recover




Idea

a rearrangement re-sets the minimum of the potential so strain does not recover

Note the particle movement was achieved as a vibration – it was ‘converted’ to a relaxation via the shift in the minima

The Role of Collective Strain in Structural Relaxation of a Supercooled Liquid

Filled circles are particles that relaxed via reorganization (i.e. changing nearest neighbours)

Open circles are particles that relaxed while losing no more than one neighbour (i.e. strains)

At low T, ~ 70% of structural relaxation is achieved by unrecovered strain – not rearrangement

So What Exactly, is Heterogeneous about Dynamic Heterogeneities?

Consider 2 nearest neighbour separating – we will call this a ‘bond break’

Let Pij be the probability that bond ij breaks in some time interval

and Fij be the probability that the bond ij is broken after some longer time

bond length

‘other’coordinates

threshold determined by geometry

Pij

Fij

Note that Fij can differ from Pij due to reversals

Pij F ij relaxation

Results: The Spatial Distribution of Pij and Fij

Dynamic heterogeneities are determined by the probability of reversals, not of the breaks themselves

bond breaking

What part of dynamic heterogeneity can be attributed to a configuration?

-15 -10 -5 0 5 10 15

-15

-10

-5

0

5

10

15

00.20.40.60.81.01.21.41.61.82.0

Three different runs from the same starting configuration but different momenta.

Defining the Propensity for Motion of a Configuration

Average squared displacements over many (>100) runs from a single initial configuration.

Persistent structure must be due to that configuration.

We call these ensemble averaged mean squared displacements the dynamic propensity.

ionconfiguratisoirr −>−< 2)()0(( ατ

Phys.Rev.Lett 93, 135701 (2004)

τα = 103 τ at T = 0.4

48

Maps of the participation fractions for the 11 lowest frequency modes in an IS of the 2D binary disk mixture. Localised modes (participation ratio < 0.34) are coloured red

Heterogeneity and the Normal Modes of the Inherent Structures

Perry, Widmer-Cooper, Harrowell & Reichman, JCP 131, 194508 (2009)

localized

plane wave

49

Participation fraction of the 30 lowest frequency modes of 6 different inherent structures of the 2D binary soft disk mixture

Mapping the Local Susceptibility of an Inherent Structure

Perry, Widmer-Cooper, Harrowell & Reichman, Nature Physics 4, 711 (2008)

50

The low frequency mode participation overlaid by those particles whose probability for irreversible relaxation within 200τ exceeds 0.01

Perry, Widmer-Cooper, Harrowell & Reichman, Nature Physics 4, 711 (2008)

Irreversible Relaxation Occurs at Soft Localised Modes

Circles correspond to particles with a high probability of losing > 3 initial neighbours

And the Connection Between Structure and Dynamics?

The correlation between dynamics and the localized modes of the inherent structure suggests that degree of local constraint is more important than the specific geometries by which it was achieved.

But how does the inherent structure (a T = 0 structure) influence the dynamics at non-zero temperatures and for displacements well beyond the harmonic limit?

That remains an unanswered question.

Fluctuations and the Shear Modulus

][ 22 ><−><−= ∞ σσβVGGeq

The zero frequency shear modulus Geq can be written in terms of the fluctuations of the shear stress σ

Squire, Holt and Hoover, Physica 42, 388 (1969)


][ 22 ><−><−= ∞ σσβVGGeq

Born modulus- response to affine strain

reduction of modulus due to stress fluctuations. Associated with non-affine strains



Note: G∞ not the same as the plateau modulus Gp , the experimental high frequency modulus.


][ 22 ><−><−= ∞ σσβVGGeq

Born modulus- response to affine strain



When the averages are unconstrained <σ> = 0 and

><=∞2σβVG Zwanzig and Mountain, JCP. 43, 4464 (1965).

so that Geq = 0

reduction of modulus due to stress fluctuations. Associated with non-affine strains

On The Relation Between Rigidity and Constraint

It follows that the averages appearing in the expression for the modulus Geq must include some form of constraint, let’s imagine that it can be quantified by κ, where constraint increases with κ.

][ 22 ><−><−= ∞ σσβVGGeq κ κ

Then

1/κ

Geq

0

G∞

a non-trivial relation must relate the shear modulus to the degree of constraint

We shall use finite time trajectories as a means of applying an implicitconstraint.Monte Carlo averaging with explicit constraints is an alternative approach.

Two Step Relaxation with Random Bond Changes

t t

below percolation at percolation above percolation

Green line indicates the faction of particles in a percolating cluster

Relaxation in a single trajectory shows significant fluctuations close to rigidity transition due to the appearance and disappearance of percolating clusters

χ4 and the Length Scale of Dynamic Heterogeneities

increasing system size

Going from the marginal solid at percolation to the totally rigid solid:

• χ4 decreases with increasing bond density

• the system size dependence of χ4 decreases

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