Numerical evidence now supports the idea that a liquid–liquid transition forms a generic feature of tetrahedrally coordinated liquids. This result establishes the physical validity of such a transition and provides a possible explanation for the anomalous behaviour of liquid water. One of the most controversial hypotheses for explaining the origin of the thermodynamic anomalies characterizing liquid water postulates the presence of a metastable second-order liquid–liquid critical point1 located in the ‘no-man’s land’2. In this scenario, two liquids with distinct local structure emerge near the critical temperature. Unfortunately, as spontaneous crystallization is rapid in this region, experimental support for this hypothesis relies on significant extrapolations, either from the metastable liquid or from amorphous solid water3,4. Although the liquid–liquid transition is expected to feature in many tetrahedrally coordinated liquids, including silicon5, carbon6 and silica, even numerical studies of atomic and molecular models have been unable to conclusively prove the existence of this transition. Here we provide such evidence for a model in which it is possible to continuously tune the softness of the interparticle interaction and the flexibility of the bonds, the key ingredients controlling the existence of the critical point. We show that conditions exist where the full coexistence is thermodynamically stable with respect to crystallization. Our work offers a basis for designing colloidal analogues of water exhibiting liquid–liquid transitions in equilibrium, opening the way for experimental confirmation of the original hypothesis.

The simplicity of hard spheres as a model system is deceptive. Although the particles interact solely through volume exclusion, that nevertheless suffices for a wealth of static and dynamical phenomena to emerge, making the model an important target for achieving a comprehensive understanding of matter. In addition, while real colloidal suspensions are typically governed by complex interactions, Pusey and Van Megen [Nature 320, 340 (1986)] demonstrated that suitably tuned suspensions result in hard-sphere-like behavior, thus bringing a valuable experimental complement to the renowned theoretical model. Colloidal hard spheres are thus both a material in their own right and a platform upon which phenomena exhibited by simple materials can be explored in great detail. The various purposes enable a particular synergy between experiment, theory, and computer simulation. The extensive body of work on colloidal hard spheres, which ranges from their equilibrium properties, such as phase behavior, interfaces, and confinement, to some of the nonequilibrium phenomena they exhibit, such as sedimentation, glass formation, and nucleation, is reviewed here.

Predicting the local dynamics of supercooled liquids based purely on local structure is a key challenge in our quest for understanding glassy materials. Recent years have seen an explosion of methods for making such a prediction, often via the application of increasingly complex machine learning techniques. The best predictions so far have involved so-called Graph Neural Networks (GNNs) whose accuracy comes at a cost of models that involve on the order of ${10}^5$ fit parameters. In this Letter, we propose that the key structural ingredient to the GNN method is its ability to consider not only the local structure around a central particle, but also averaged structural features centered around nearby particles. We demonstrate that this insight can be exploited to design a significantly more efficient model that provides essentially the same predictive power at a fraction of the computational complexity (approximately 1000 fit parameters), and demonstrate its success by fitting the dynamic propensity of Kob-Andersen and binary hard-sphere mixtures. We then use this to make predictions regarding the importance of radial and angular descriptors in the dynamics of both models.

In the quest to understand how structure and dynamics are connected in glasses, a number of machine learning based methods have been developed that predict dynamics in supercooled liquids. These methods include both increasingly complex machine learning techniques and increasingly sophisticated descriptors used to describe the environment around particles. In many cases, both the chosen machine learning technique and choice of structural descriptors are varied simultaneously, making it hard to quantitatively compare the performance of different machine learning approaches. Here, we use three different machine learning algorithms—linear regression, neural networks, and graph neural networks—to predict the dynamic propensity of a glassy binary hard-sphere mixture using as structural input a recursive set of order parameters recently introduced by Boattini et al. [Phys. Rev. Lett. 127, 088007 (2021)]. As we show, when these advanced descriptors are used, all three methods predict the dynamics with nearly equal accuracy. However, the linear regression is orders of magnitude faster to train, making it by far the method of choice.