In contrast to crystalline solids - for which a precise framework exists for describing structure1 - quantifying structural order in liquids and glasses has proved more difficult because even though such systems possess short-range order, they lack long-range crystalline order. Some progress has been made using model systems of hard spheres2,3, but it remains difficult to describe accurately liquids such as water, where directional attractions (hydrogen bonds) combine with short-range repulsions to determine the relative orientation of neighbouring molecules as well as their instantaneous separation. This difficulty is particularly relevant when discussing the anomalous kinetic and thermodynamic properties of water, which have long been interpreted qualitatively in terms of underlying structural causes. Here we attempt to gain a quantitative understanding of these structure-property relationships through the study of translational2,3 and orientational4 order in a model5 of water. Using molecular dynamics simulations, we identify a structurally anomalous region - bounded by loci of maximum orientational order (at low densities) and minimum translational order (at high densities) - in which order decreases on compression, and where orientational and translational order are strongly coupled. This region encloses the entire range of temperatures and densities for which the anomalous diffusivity6-9 and thermal expansion coefficient10 of water are observed, and enables us to quantify the degree of structural order needed for these anomalies to occur. We also find that these structural, kinetic and thermodynamic anomalies constitute a cascade: they occur consecutively as the degree of order is increased.
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