The hidden structure of liquids

Here’s a Commentary written for the collection of essays curated by Nature Materials for the International Year of Crystallography. It is worth also taking a look at Nature’s Crystallography Milestones collection.

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From its earliest days, crystallography has been viewed as a means to probe order in matter. J. D. Bernal’s work on the structure of water reframed it as a means of examining the extent to which matter can be regarded as orderly.

In 1953, J. Desmond Bernal wrote that, as a result of the development of X-ray crystallography,

“science was beginning to find explanations in terms of atoms and their combinations not only of the phenomena of physics and chemistry but of the behaviour of ordinary things. The beating out of metal under the hammer, the brittleness of glass and the cleavage of mica, the plasticity of clay, the lightness of ice, the greasiness of oil, the elasticity of rubber, the contraction of muscle, the waving of hair, and the hardening of a boiled egg are among the hundreds of phenomena that had already been completely or partially explained." [1]

What is striking here is how far beyond crystalline and ordered matter Bernal perceived the technique to have gone: into soft matter, amorphous solids, and viscous liquids. For biological polymers, Bernal himself had pioneered the study of globular proteins, while William Astbury, Bernal’s one-time colleague in William Bragg’s laboratory at the Royal Institution in London, had by mutual agreement focused on the fibrous proteins that constitute hair and muscle. Of course, in the year in which Bernal was writing, the most celebrated X-ray structure of a fibrous biological macromolecule, DNA, was solved by Crick and Watson under the somewhat sceptical auspices of William’s son Lawrence Bragg, head of the Cavendish Laboratory in Cambridge.

All those macromolecular materials do form crystals. But one of Bernal’s great insights (if not his alone) was to recognize that the lack of long-ranged order in a material was no obstacle to the use of X-rays for deducing its structure. That one could meaningfully talk about a structure for the liquid state was itself something of a revelation. What is sometimes overlooked is the good fortune that the natural first choice for such investigation of liquids – water, ubiquitous and central to life and the environment – happens to have an unusually high degree of structure. Indeed, Bernal first began his studies of liquid-state structure by regarding it as a kind of defective crystal.

The liquid state is notoriously problematic precisely because it bridges other states that can, at least in ideal terms, be considered as perfectly ordered (the crystal) and perfectly disordered (the gas). Is the liquid a dense gas or an imperfect solid? It has become clear today that neither view does full justice to the issue – not least because, in liquids, structure must be considered not only as a spatial but also as a temporal property. We are still coming to terms with that fact and how best to represent it, which is one reason why there is still no consensual “structure of water” in the same way as there is a structure of ice. What is more, it is also now recognized that there is a rich middle ground between crystal and gas, of which the liquid occupies only a part: this discussion must also encompass the quasi-order or partial order of liquid crystals and quasicrystals, the ‘frozen disorder’ of glasses, and the delicate interplay of kinetic and thermodynamic stability. X-ray diffraction has been central to all of these ideas, and it offered Bernal and others the first inkling of how we might meaningfully talk about the elusive liquid state.

Mixed metaphors

One of the first attempts to provide a molecular picture of liquid water came from the discoverer of X-rays themselves, Wilhelm Röntgen. In 1891 Röntgen suggested that the liquid might be a mixture of freely diffusing water molecules and what he termed “ice molecules” – something akin to ice-like clusters dispersed in the fluid state. He suggested that such a ‘mixture model’, as it has become known, could account for many of water’s anomalous properties, such as the decrease in viscosity at high pressure. Mixture models are still proposed today [2,3], attesting to the tenacity of the idea that there is something crystal-like in water structure.

X-ray scattering was already being applied to liquids, in particular to water, by Peter Debye and others in the late 1920s. These experiments showed that there was structural information in the pattern: a few broad but clearly identifiable peaks, which Debye interpreted as coming from both intra- and intermolecular interference. In 1933 Bernal and his colleague Ralph Fowler set out to devise a structural model that might explain the diffraction pattern measured from water. It had been found only the previous year that the water molecule has a V shape, and Bernal and Fowler argued from quantum-chemical considerations that it should have positive charges at the hydrogen atoms, balanced by two lobes of negative charge on the oxygen to produce a tetrahedral motif. On electrostatic grounds, each molecule should then form hydrogen bonds with four others in a tetrahedral arrangement. Noting the similarity with the tetrahedral structure in silicates, Bernal and Fowler developed a model in which water was regarded as a kind of distorted quartz. Their calculations produced fair agreement with the X-ray data: the peaks were in the right places, even if their intensities did not match so well [4].

This work established some of the core ideas of water structure, in particular the tetrahedral coordination. It set the scene for other models that started from a crystalline viewpoint. Notably, Henry Eyring and colleagues at the University of Utah devised a general picture of the liquid state consisting of an essentially crystalline close-packing threaded with many dislocations [5]. Molecules that escape from this close-packing can, in Eyring’s picture, wander almost gas-like between the dense clusters, making it a descendent of Röntgen’s mixture model.

Building liquids by hand

But Bernal was not happy with this view of the liquid as a defective solid, saying that it postulates “a greater degree of order…in the liquid than actually exists there” [6]. In the 1950s he started again, this time by considering a ‘simple liquid’ in which the molecules are spheres that clump together in an unstructured (and presumably dynamic) heap. Bernal needed physical models to guide his intuition, and during this period he constructed many of them, some now sadly lost. He used ball bearings to build dense random packings, or to see the internal structure better he would prop apart ping-pong balls or rubber balls with wires or rods, sometimes trying to turn himself into the required randomizing influence by selecting rods of different length without thinking. He was able to construct models of water that respected the local tetrahedral arrangement while producing no long- or medium-range order among molecules: a random hydrogen-bonded network in which the molecules are connected in rings with between four and seven members, as opposed to the uniformly six-membered rings of ordinary ice. Not only did this structure produce a good fit to the X-ray data (he counted out the interatomic distances by hand and plotted them as histograms), but the model liquid proved to have a higher density than ice, just as is the case for water [7].

This ‘mixed-ring’ random network supplies the basis for most subsequent molecular models of water [8, 9], although it is now clear that the network is highly dynamic – hydrogen bonds have a lifetime of typically 1 ps – and permeated with defects such as bifurcated and distorted hydrogen bonds [9, 10].

But although the tetrahedron seems to fit the local structure of water, that liquid is unusual in this regard, having a local geometry that is dictated by the high directionality of the hydrogen bonds. At the same time as Bernal was developing these ideas in the 1950s, Charles Frank at the University of Bristol proposed that for simple liquids, such as monatomic liquids and molten metals, a very common motif of short-ranged structure is instead the icosahedron [11]. This structure, Frank argued, provides the closest packing for a small number of atoms. But as one adds successive layers to an icosahedral cluster, the close-packing breaks down. What is more, the clusters have fivefold symmetry, which is incompatible with any crystalline arrangement. It is because of this incommensurability, Frank said, that liquid metals can be deeply supercooled without nucleating the solid phase. It was after hearing Frank speak on these ideas about polyhedral packings with local fivefold symmetry – very much within the context of the solid-state physics that was Frank’s speciality – that Bernal was prompted to revisit his model of the liquid state in the 1950s.

Forbidden order

Both X-ray scattering [12, 13] and X-ray spectroscopy [14] now offer some support for Frank’s picture of liquid metals, showing that something like icosahedral structures do form in metastable supercooled melts. Frank’s hypothesis was already recalled in 1984, however, when the first discovery was reported of a material that seemed to have a crystalline icosahedral order: a quasicrystal [15]. X-ray diffraction from an alloy of aluminium and manganese produced a pattern of sharp peaks with tenfold symmetry, which is now rationalized in terms of a solid structure that has local five- and tenfold symmetry but no perfect long-range translational order. Such materials are now recognized by the International Union of Crystallography as formally crystalline, according to the definition that they produce sharp, regular diffraction peaks. Frank’s icosahedral liquid clusters could provide the nuclei from which these quasicrystalline phases form, and indeed synchrotron X-ray crystallography of a supercooled melt of a Ti-Zr-Ni alloy shows that their formation does indeed precede the activated formation first of a metastable quasicrystalline phase and then of a stable crystal [12].

It seems fitting that Linus Pauling, whose work helped to explain the structures of water and ice [16], should have entered the early debate over the interpretation of quasicrystal diffraction. Pauling was on the wrong side, insisting dismissively that this was probably just a case of crystal twinning. But he, Bernal, Frank, and indeed William Bragg himself (a pioneer of X-ray studies of liquid crystals), all grappled with the question of determining how far ideas from crystalline matter can be imported into the study of the liquid state. Or to put it another way, they showed that X-ray crystallography is better viewed not as a method of probing order in matter, but as a means of examining the extent to which matter can be regarded as orderly. With the advent of high-intensity synchrotron sources that reduce the exposure times sufficiently to study ultrafast dynamic processes by X-ray diffraction [17, 18], it is now possible to explore that question as a function of the timescale being probed. It has been suggested that recent ddebate about the structure of water – a discussion that has oscillated between the poles of Bernal’s random tetrahedral network and mixture models – is itself all a matter of defining the notion of ‘structure’ on an appropriate timescale [19].

Such studies also seem to be confirming that Bernal asked the right question about the liquid state in 1959 (even if he phrased it as a statement): “it is not the fluidity of the liquid that gives rise to its irregularity. It is its irregularity that gives rise to its fluidity.” [7] Which is it, really? Do defects such as bifurcated hydrogen-bonds give water its fluidity [10]? Or is it dynamical making and breaking of hydrogen-bonds that undermines the clathrate-like regularity proposed by Pauling [20]? Whatever the case, there is little doubt now that Bernal’s perception that extending X-ray diffraction to biomolecules and liquids – and now to quasicrystals and all manner of soft matter – has led to a broader view of what crystallography is:

“And so there are no rules, or the old rules are enormously changed… We are seeing now a generalized crystallography, although it hasn’t been written up as such… [These materials] have their own inner logic, the same kind of logic but a different chapter of the logic that applies to the three-dimensional regular lattice crystals.” [21]

References
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