Morphometric approach to many-body correlations in hard spheres

Supercooled liquids become more and more viscous as their temperature is reduced. The increased viscosity corresponds to an enormous increase in the characteristic time for the relaxation of density fluctuations. What is often puzzling is that, differently from many other physical phenomena, this dramatic change in the correlations in time appears to be weakly reflected in conventional measures of spatial correlations. These are typically so-called pair or two-body correlations, measuring how likely it is to find randomly chosen pairs of particles at particular characteristic distances.

The lack of strong correlations between two-body spatial correlation and the emergent, enormous relaxation times of supercooled liquids suggests that more complex, eventually multi-body correlations may be at play.

Thanks to the work of a very gifted PhD student in Paddy Royall‘s group, Joshua F. Robinson, we have obtained a first theoretical insight on the origin of such emergent correlations in a reference model for supercooled liquids, i.e. hard spheres, which are often employed to understand the behaviour of colloidal particles and as a basis to develop approximate theories of liquids.

We rooted our work in a geometric approach to treat the free energy of thermal hard spheres developed by Roland Roth (a co-author of our work) termed morphometric theory and this has allowed us to study the free energy of a certain number of thermal structural motifs of hard spheres immersed in an effective medium and predict with a high degree of precision their respective populations. Furthermore, the approach that we have used has revealed that it is possible to follow local deformations of the motifs and compute the free energy barriers between them.

The work appeared as an Editor’s Suggestion in Physical Review Letters:

Joshua F. Robinson, Francesco Turci, Roland Roth, C. Patrick Royall, Phys. Rev. Lett. 122, 068004 (2019) doi: 10.1103/PhysRevLett.122.068004

and it has also been paired by a Viewpoint in Physics by Thomas Speck.

Coupling of sedimentation and liquid structure: Influence on hard sphere nucleation

Colloidal hard spheres at high volume fractions (beyond ~0.49) can crystallise: up to to 0.54, they coexist with a fluid phase, and at even higher densities they completely crystallise. The way the hard sphere fluid transforms into the solid is called crystal nucleation: due to thermal fluctuations, every now and then locally denser and more ordered regions appear and disappear; occasionally, these are large enough to further grow irreversibly, and form a crystalline region.

Nucleation is a generic process: in hard spheres, it should present its simplest traits, as it can be driven only by entropic forces. Yet, nucleation rates in colloidal hard spheres are rather different from what can be predicted from theory and numerical simulations. In particular, the discrepancy between the two increases of many orders of magnitude with decreasing the volume fraction from, for example, 0.55 to 0.53. Simulations normally consider idealised hard spheres in an ideal solvent. What could possibly go wrong?

Nick Wood, a talented PhD student here in Bristol, has taken care of some potential origins of the discrepancy analysing the effect of sedimentation on local order. Together with John Russo and Paddy Royall, we have considered in our recent paper how the flow induced by sedimentation may, via hydrodynamics interactions, transform the structure of the liquid, compared to the case in absence of sedimentation, and how such changes would impact on the nucleation barriers. The result is that some structural signatures clearly vary as a function of the density mismatch between colloids and solvent and that this leads to an estimated correction of the rates in the right direction, but by an amount that is not sufficent to address the entire discrepancy.

The complete article can be found here:
Nicholas Wood, John Russo, Francesco Turci and C. Patrick Royall J. Chem. Phys. 149, 204506 (2018); https://doi.org/10.1063/1.5050397

Visualisation
Structure of the colloidal hard-sphere fluid with (left) and without (right) sedimentation. In green are highlighted motifs that hinder the crystal formation and anti-correlate with it, as demonstrated here.

Local structure in deeply supercooled liquids exhibits growing lengthscales and dynamical correlations

Computer simulations are very powerful: in the case of molecular dynamics, we can model the positions and velocities of atoms or molecules and observe the emergence of pattern and structures in situ, following each individual atom in its trajectory.

However, when we study supercooled liquids or glasses, it is hard to probe in computer simulations very low temperatures or very tightly packed systems, unless we opt for indirect and clever routes to glean some information on the low temperature behaviour. It would be great if one could directly take a very cold (or, similarly, very dense) liquid at equilibrium and see how the constituent particles are arranged.

This is precisely what James Hallett has managed to do during his stay in Bristol using super-resolution microscopy: this method can access the coordinates of equilibrium themal packings so dense that a direct simulation would never do. Thanks to James’s clever imaging, we have then carefully analysed the individual coordinates and trajectories of dense repulsive colloids and managed to clearly show how the local environment of these densely packed equilibrium systems changes as the density is increased.

We have found some notable features: as we take denser samples, the liquid becomes gradually richer in regions where particles are arranged into five-fold symmetric structures; those regions display reduced mobility compared to other regions of the sample; randomly selected domains of the system become more and more “similar to each other” as the density is increased, accompanied by the decrease in the number of distinct configurations the liquid can take, a quantity related to its so-called configurational entropy.

This work has just appeared in Nature Communications. Full text here:

James E. Hallett, Francesco Turci, and C. Patrick Royall. Nature Communications 9(1), 3272 (2018).

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Measuring the similarity between randomly sampled regions: (a) picking at random a blue and yellow group of particles from the (beautiful) experimental coordinates we can use methods based on singular value decomposition (SVD) to find the best-fitting rotation and translation. The overlap Q between blue and yellow particles is then calculated. In (b) (c) and (d) we see how the probability distribution of the overalap changes as (b) we change the volume fraction φ; (c) when we look at 5-fold symmetric regions (LFS-rich) at low φ and high φ. Only at high φ LFS-rich regions are significantly more similar.

The race to the bottom: approaching the ideal glass?

Together with CP Royall, S Tatsumi, J Russo and PhD student J Robinson we have just published on the Journal of Physics: Condensed Matter a handy review on recent approaches to the exploration of very stable glasses in experiments and simulations.

We cover a variety of topics, including vapor deposited glasses in experiments, importance sampling in trajectory space, random pinning, representing distinct attempts to address the following question: is glassiness linked to some kind of novel thermodynamic transition in very low temperature liquids?

Find out more here:

CP Royall, F Turci, S Tatsumi, J Russo, J Robinson, Journal of Physics: Condensed Matter, Volume 30, Number 36 (2018)

Structural-dynamical transition in the Wahnström mixture

Supercooled liquids present dynamical heterogeneities at low temperatures: on a certain length and timescale, some areas are very mobile (active) while others are much more solid-like (inactive). This feature is often interpreted as the signature of the fact that the liquid, when supercooled, starts exploring different metastable regions of the free energy landscape.

A possible route to illustrate this effect is through large deviations of structural-dynamical obserables, as we first did in the case of a canonical atomistic model for glassformers, the Kob-Andersen mixture. A main observation of that work was that dynamical heterogeneities correspond to a first order phase transition in a (reweighted) space of possible steady states between high energy trajectories that are rich in structure and low energy trajectories that are poor in structure. Moreover, such a transition has a strong temperature dependence, so that the structure-rich trajectories become more and more likely to be observed as the temperature decreases.

Now, we have published a follow-up work on the European Physical Journal E, where we show that the same mechanism is at play in another model glass-former (the Wahnström mixture), showing that while the overall qualitative picture may be general, the details depend on the nature of the interactions between the constituents. Moreover, we also show that configurations extracted from the structure-rich trajectories have much larger yield stresses than the normal supercooled liquid: the emerging rigidity of glasses appears to be strongly related to the structural-dynamical transition that we have highlighted.

More information in

F Turci, T Speck and C P Royall, Structural-dynamical transition in the Wahnström mixture Eur. Phys. J. E , 41: 54 (2018) https://doi.org/10.1140/epje/i2018-11662-3

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A structure-rich configuration of the Wahnström binary mixture under shear: red areas are icosahedral motifs while flashing yellow particles are those that manifest the largest non-affine square displacement

Structural Covariance in the Hard Sphere Fluid

Hard spheres are one of the simplest model of fluid. When the density is increased, the motion is slowed down and non-trivial local arrangements of particles characteristic of the liquid emerge. We call these local structural motifs.

In hard spheres, special role is played by structures with pentagonal rings, such as icosahedra or pentagonal pyramids.

In collaboration with Pierre Ronceray (Princeton), to appear on the Journal of Chemical Physics, we have shown that if we study the cross-correlations between the variation of populations of different motifs we can understand more of the phase behaviour of this model liquid: for example, we can predict its propensity to form more motifs of a given type if an external field is applied; or, we can understand and classify different motifs that coexist in the liquid phase.

More details can be found in the JCP Editor’s Pick:

B. Carter, F. Turci, P. Ronceray, C.P. Royall, Structural Covariance in the Hard Sphere Fluid, The Journal of Chemical Physics 148, 204511 (2018); https://doi.org/10.1063/1.5024462

Experimental determination of configurational entropy in a two-dimensional liquid under random pinning

A glass is (broadly speaking) mostly composed of particles that very slowly move due to surrounding cages formed by the disordered structure of the neighbouring particles. A way to study the glass transition is to actually freeze-in a subset of the particles and observe how this induces changes to the slow relaxation and how this relates to the emergence of local order.

The freezing-in procedure (also called pinning) has a secondary important effect: the pinned liquid has a “simplified” configurational space, as many configurations become forbidden. The number of available configurations (and hence the configurational entropy of the liquid) is therefore reduced by the simple pinning procedure and if a thermodynamic origin of dynamical arrest is to be surmised, such a reduction would be a necessity.

Some years ago, Ian Williams designed a novel, clever way to pin large two-dimensional colloidal supercooled liquids. In an article recently accepted in the Journal of Physics: Condensed Matter we have shown that the technique allows us to observe the crossover from a free-flowing liquid to a pinned glass and that this is accompanied by very limited structural changes. However, if we map the configurational entropy of the experiments with the entropy measured from model numerical simulations, we do observe that accounting for the fraction of pinned particles leads to a reduction in the estimated configurational entropy.

Full reference:

I. Williams, F. Turci et al. (2018) Experimental determination of configurational entropy in a two-dimensional liquid under random pinning, J. Phys.: Condens. Matter in press https://doi.org/10.1088/1361-648X/aaa869

Long-lived non-equilibrium interstitial solid solutions in binary mixtures

We just published on the Journal of Chemical Physics the experimental and computer simulation work of a Ioatzin Rios de Anda (PhD student in the Royall group) on kinetically arrested crystalline phases in colloidal binary mixtures.

Despite slightly different conditions (presence/absence of confinement or polydispersity in the particle sizes) the experiments and simulations match in the fundamental message of the work: when we have two species rather different in sizes, crystallisation of the bigger species can occur before the reordering of the smaller species, forming long-lived kinetically arrested structures (interstitial solid solutions), characterised by a high density of imperfections and vacancies. This shows how challenging the formation of a well ordered binary crystal is and, on the other hand, how they can potentially be considered for the realisation of partially ordered porous matrices.

Full reference:
Rios de Anda, I., Turci, F., Sear, R., & Royall, P.  Journal of Chemical Physics, 147, 124504 (2017).

Weak temperature dependence of ageing of structural properties in atomistic model glassformers

We have just published in the Journal of Chemical Physics the simulation work of a brilliant former master student of the School of Physics at the University of Bristol (Thomas Jenkinson) on the local structural changes occurring during ageing in two atomistic glass formers with Lennard-Jones interactions (the Kob-Andersen and Wahnström mixtures).

We find some expected results (local order steadily increases as the out-of-equilibrium liquid ages) and some more surprising ones (for example, the rate at which such increase occurs changes weakly for temperatures above the apparent dynamical divergence of viscosities, T0). We also investigate the effect of transient deep quenches, finding very moderate traces of so-called rejuvenation.

The reference to the work is J. Chem. Phys. 147, 054501, (2017).

Nonequilibrium Phase Transition in an Atomistic Glassformer: the Connection to Thermodynamics

One central piece of the problem of dynamic arrest is whether the phenomenology of slow relaxation, increasing dynamical length scales, mild (or dramatic) structural changes are somewhat related to the existence of a zero entropy amorphous state emerging at a non-zero temperature.

A comprehensive theory would need on the one hand to take into account of the well established phenomenon of dynamical heterogeneitiesi.e. the non-homogenous patterns of diffusion that emerge together with the glassy dynamics itself; on the other hand, it should also rationalise the many findings that point (for several model systems) to a dramatic reduction of the so-called configurational entropy as one approaches a finite temperature (sometimes termed Kauzmann temperature) at which also the relaxation times appear to diverge.

In our recent work (Physical Review X 7, 031028) Thomas Speck,  C. Patrick Royall and I discuss a unified scenario that combines dynamical aspects to structural ones in order to sample very low energy and entropy states, employing dynamical large deviations.

We find that the equilibrium supercooled liquid competes with a secondary metastable  amorphous liquid rich in long-lived structural motifs, hidden in the tails of probability distributions in trajectory space. We also show that sampling the tails of such probabilities at a single moderate temperature allows us to retrieve the thermodynamic properties of the ordinary liquid in much wider range of temperatures, down to very low temperatures. We can then draw a diagram for the stable and metastable phase, pointing towards critical-like fluctuations in the region where the Kauzmann temperature is normally located, and allowing us to review currently proposed scenarios from an alternative point of view, rooted in the large deviation theory of metastability.

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