Water is one of the most well-studied substances in the physical sciences. This is not surprising in the light of its importance to our everyday lives, and indeed to life itself. Water is the only common substance that appears in all three basic states of matter – vapour, liquid and solid – under everyday conditions. Solid water has an especially large number of polymorphs, although some only appear at very high pressures that are not easily accessible to experiment. There are many other physical properties of water that make it especially unusual and therefore interesting to study.
In order to understand better what causes some of water's unusual properties, we have focussed on determining its phase behaviour in two ways: first, we need to understand what the stable phase of water is under various conditions, which entails computing the phase diagram; second, we need to understand how transitions between phases can occur.
Of course the phase diagram of water is – to some extent at least – well known experimentally. However, predicting it from first principles would have been a task well beyond our reach even a few years ago: quantum calculations such as density functional theory are computationally demanding, so calculations are of necessity limited to small system sizes and short simulation times. Determining a system's free energy, by contrast, requires us to sample the configuration space of atom positions thoroughly and is sensitive to simulation artefacts introduced by small system sizes. As a result, phase diagrams could largely only be computed using (semi-)empirical force fields, which inevitably entail severe approximations. In collaboration with Bingqing Cheng, we have recently combined machine-learning methods, which enable us to obtain a good approximation to the underlying quantum-mechanical behaviour of water, with advanced free-energy techniques to compute ab initio phase diagrams of water. Since phase diagrams are very sensitive to the underlying potential, our calculations show not only that machine-learned potentials can reproduce most of the thermodynamic properties of water, but, by implication, that they can prove to be a very useful proxy for studying other properties of water, including those that are less easily accessible to experiment, such as the behaviour of water on other planets of the solar system.
Thermodynamic stability is the first step in understanding what the likely behaviour of a substance is. However, substances cooled below their thermodynamic freezing point do not necessarily freeze, especially when they are very pure. According to classical nucleation theory, a critical cluster must spontaneously form in the supercooled liquid before crystallisation can proceed: a free-energy barrier thus arises from a competition between the (favourable) bulk free-energy difference and the (unfavourable) formation of an interface between the phases. The presence of a free-energy barrier to nucleation makes homogeneous nucleation a rare event; this means that it happens infrequently, but when it does, it tends to proceed rapidly. This makes it challenging to study in computer simulation, since it is seldom possible to simulate in a brute-force manner. Advanced methods must be used to drive the process, and for this we generally need to identify a suitable order parameter to act as a ‘reaction co-ordinate’. We have used such approaches to investigate the nucleation of ice with various empirical potentials.