Attempts to understand the molecular structure of water were first made well over a century ago. Looking back at the various attempts, it is illuminating to see how these were conditioned by the state of knowledge of chemistry and physics at the time and the experimental and theoretical tools then available. Progress in the intervening years has been facilitated by not only conceptual and theoretical advances in physics and chemistry but also the development of experimental techniques and instrumentation. Exploitation of powerful computational methods in interpreting what at first sight may seem impenetrable experimental data has led us to the consistent and detailed picture we have today of not only the structure of liquid water itself and how it changes with temperature and pressure but also its interactions with other molecules, in particular those relevant to water’s role in important chemical and biological processes. Much remains to be done in the latter areas, but the experimental and computational techniques that now enable us to do what might reasonably be termed “liquid state crystallography” have opened the door to make possible further advances. Consequently, we now have the tools to explore further the role of water in those processes that underpin life itself—the very prospect that inspired Bernal to develop his ideas on the structure of liquids in general and of water in particular.

The first attempt to address the problem of water structure seems to be generally accepted to be that of Röntgen in 1892.1 However, there were interesting earlier attempts, and some of that earlier literature throws interesting light on how those early ideas were developed and how they related to the state of physics and chemistry at the time.

A 1928 review by Chadwell2 gives some interesting pointers. Underlying the early approaches is the concept of liquid water as an “associated” liquid mixture of two or more components. The thought that the so-called “anomalous” properties of liquid water such as the existence of a temperature of maximum density might be explained by a temperature-driven change in the relative fractions of two distinguishable entities fitted with the contemporary views on liquid mixtures. It is therefore understandable that a “mixture” approach dominated ideas of water structure in those early years.

As to the nature of the components of such a mixture, a wide range of ideas was put forward from the 1880s onward. Whiting’s 1884 thesis3 proposed the existence of what he called “solid particles” in the liquid, though it is perhaps important to note that the understanding then of the structures of solids—let alone liquids—was far from what we know about the structures of both phases today. Whiting did in fact derive formulas to explain water’s physical properties, but water did not satisfy those equations. Thomsen in 1883,4 together with Raoult5 from his studies of freezing points of solutions, proposed that liquid water molecules were twice as heavy as vapor ones. While this may seem to us a strange idea, it perhaps contains hints of there being strong interactions between water molecules in the liquid state. Armstrong in 18886 suggested that liquids in general are complexes of “fundamental” gas molecules, while Vernon in 18917 was perhaps the first to make specific proposals for the “components” making up liquid water by explaining the temperature of maximum density by a mixture of (H2O)2 “molecules” and (H2O)4 “complexes.”

Following these early forays, in 1892 Röntgen1 took the concept of water as a mixture of “species” to a new level. He postulated that water is a saturated solution of “ice molecules,” whose concentration depended on temperature, further assuming that the change from the more complex “ice” molecules to “simple” molecules resulted in a decrease in volume. Hence, by choosing his parameters appropriately, he could explain not only the temperature of maximum density but also the existence of a point of minimum compressibility, the anomalous thermal coefficient of expansion at high pressures, the lowering of the freezing point of water under pressure, and the decrease in viscosity with increasing pressure. All in all, an impressive achievement, especially considering that the knowledge of molecular structures of both liquids and solids was rudimentary.

Accepting the possibility of molecules associating in the liquid state, Ramsay and Shields in 1893–18948–10 argued for an alternative route to that chosen by Raoult to estimate the number of molecules associating in the liquid. In an interesting set of papers on what they term “the molecular complexity of liquids,” they considered the variation with temperature of molecular surface energy. From their results, they divided over 50 liquids into two classes: those that they considered had the same molecular weight in the liquid and gaseous states and those in which the molecular weight in the liquid was higher than that in the gas. At the end of one of their 1893 papers,8 they considered water as a special case, concluding that “about ordinary temperatures, it approaches the formula H8O4 = (H2O)4,” also commenting that it was “very remarkable that no substance yet investigated exhibits a higher molecular weight than that corresponding to four times its formula in the gaseous state.”

Interestingly, a paper a year later10 revises this figure down from near 4 to near 2. Nevertheless, the idea of the possibility of specific interactions between water molecules in the liquid state (and as set out in Ramsay’s papers, between molecules in many other liquids) was clearly in the collective psyche of interested chemists at the time.

Before moving on to later developments, it is perhaps worthwhile to sit back and take stock of the situation at the turn of the 19th century.

First, all the models of liquid water were essentially “chemical.” With an understanding that a binary solution of different molecules could give rise to the kind of “anomalous” behavior observed in water, it was perhaps inevitable that—not having any insight into the way molecules are arranged in a liquid—a “mixture model” approach should arise. The scientific environment was one in which chemical equilibrium was a familiar concept, and that phenomenon suggested an obvious way of explaining water’s anomalous behavior.

Second, these theories say nothing about the actual structures involved. Not only do they say nothing about molecular structures in the liquid, but also little was known about the arrangement of atoms or molecules in the solid (the molecular structure of ice was not to be solved for another three decades). Nor did they say anything about the ways in which the different assumed components might interact with each another or how those interactions would affect the conclusions drawn. Each component is considered as a noninteracting separate entity, with no consideration given to the interactions between them. So although these kinds of chemical models appeared capable of explaining elements of the anomalous behavior of water, they could do so only by (a) making assumptions regarding the selected properties of the different components and their variation with temperature and pressure and (b) ignoring the influence of the interactions between them. This latter interface problem incidentally continued to be ignored by mixture models that have continued to be proposed up to the present day.

Mixture models continued to dominate in the first two decades of the 20th century. Table I sets out a number of these.

TABLE I.

Variations on a theme: Some mixture models of the late 19th/early 20th century.

AuthorDateSpeciesComments
Van Laar11  1899 Double and single molecules Adding alcohol breaks up the doubles, leading to a contraction 
Witt12  1900 (H2O)2 the water molecule; (H2O)8 the “ice” molecule Used Röntgen’s approach to explain heats of solution and vapor pressure 
Sutherland13  1901 (H2O)3 and (H2O)2 As ice is hexagonal, the trimer is taken to be the component with the greater volume 
Bousfield and Lowry14  1905 (H2O)3 and (H2O)2 at low temperature; (H2O)2 and (H2O)1 at higher temperature Considers water as a limiting case of NaOH solutions 
Armstrong et al.15  1908 “Active” (H2O) and “inactive” isomers “Inactive” molecules “closed systems”—e.g., closed rings of three or four molecules 
Dutoit and Mojoui16  1909 Average (H2O)2.5 at 0 C; (H2O)2 at 100 C From surface tension as Ramsay10 but using different formula 
Walden17  1909 (H2O)2 at 100 C  
Guye18  1910 Equilibrium between (H2O)2 and 2(H2O)1 Relative fractions calculated by assuming association constant the same as in the gas 
Sutherland19  1910 0 C: mostly (H2O)3. 100 C: mostly (H2O)2 Trihydrol may be three water molecules interacting through their oxygens. This structure fits neatly into a hexagon, so postulates this could be the structure of water in ice 
Nernst20  1910 (H2O)2 and (H2O)1 Specific heat explainable assuming heat of dissociation is 2.519 kcal mol−1 
Tammann21  1926 “Ice-like” up to 50 C plus (H2O)9, with the latter dissociating to 9(H2O)1 or 2(H2O)3 Heats of dissociation and specific heats estimated 
AuthorDateSpeciesComments
Van Laar11  1899 Double and single molecules Adding alcohol breaks up the doubles, leading to a contraction 
Witt12  1900 (H2O)2 the water molecule; (H2O)8 the “ice” molecule Used Röntgen’s approach to explain heats of solution and vapor pressure 
Sutherland13  1901 (H2O)3 and (H2O)2 As ice is hexagonal, the trimer is taken to be the component with the greater volume 
Bousfield and Lowry14  1905 (H2O)3 and (H2O)2 at low temperature; (H2O)2 and (H2O)1 at higher temperature Considers water as a limiting case of NaOH solutions 
Armstrong et al.15  1908 “Active” (H2O) and “inactive” isomers “Inactive” molecules “closed systems”—e.g., closed rings of three or four molecules 
Dutoit and Mojoui16  1909 Average (H2O)2.5 at 0 C; (H2O)2 at 100 C From surface tension as Ramsay10 but using different formula 
Walden17  1909 (H2O)2 at 100 C  
Guye18  1910 Equilibrium between (H2O)2 and 2(H2O)1 Relative fractions calculated by assuming association constant the same as in the gas 
Sutherland19  1910 0 C: mostly (H2O)3. 100 C: mostly (H2O)2 Trihydrol may be three water molecules interacting through their oxygens. This structure fits neatly into a hexagon, so postulates this could be the structure of water in ice 
Nernst20  1910 (H2O)2 and (H2O)1 Specific heat explainable assuming heat of dissociation is 2.519 kcal mol−1 
Tammann21  1926 “Ice-like” up to 50 C plus (H2O)9, with the latter dissociating to 9(H2O)1 or 2(H2O)3 Heats of dissociation and specific heats estimated 

In considering the contents of Table I, it is interesting to note the concluding comment to a 1910 Faraday Discussion by the chairman of the meeting Walker:22 

“…perhaps the best feature of the Papers, taking them all together, is that the conclusions to which they lead seem to be very much the same all round. Often in a discussion of this nature we have a great diversity of opinion, but here there seems to be little divergence, although the experimental material which has been worked with is so exceedingly diverse in the different cases. I should think as a result of this discussion, one will soon find even in the textbooks that while ice is trihydrol, and steam monohydrol, liquid water is mostly dihydrol with some trihydrol in it near the freezing point and a little monohydrol near the boiling point.”

So, the problem of liquid water was seen by this expert as being solved. But in reality, things began to take a different turn.

The first x-ray study of a crystal by Friedrich and Knipping23 was published in 1912. This followed an approach to Max von Laue by Paul Ewald, a young theorist Ph.D. student of Sommerfeld, suggesting that x rays might be able to give information on the structures of crystals. That successful experiment opened up the possibilities of determining actual molecular structures.

However, further versions of “chemical” mixture models continued to be put forward, though even before Friedrich and Knipping’s experiment, the relevance of structure had begun to enter the discussion. This was perhaps presaged by suggestions that the water of crystallization in crystalline hydrates might be one of the components of the water “mixture.” Even before the structures of crystalline hydrates had been determined, attempts were made to calculate the density of water in them and to use those data to specify the density of one of the components. For example, Sutherland in 191019 calculated the density of water in Li2SO4 · H2O and assumed this to be the density of solid monohydrol. Not surprisingly, in the absence of the actual structures, calculated densities of the water component varied considerably, though an extensive 1911 study by Rosenstiehl24,25 of 179 hydrated salts led him to conclude from the number of water molecules lost in each step of dehydration that liquid water is a ternary mixture of (H2O)1, (H2O)2, and (H2O)3.

Also drawn upon in trying to understand the structure of water was the work of Bridgman26 and Tammann27 on the phase diagram of ice. Although their structures were unknown, five different phases were known at that time, some of which are denser and others lighter than liquid water. So, it was perhaps natural to suggest, as Tammann did, that liquid water consisted of a mixture of two of these forms of ice—one of which is (a) heavier and the other (b) lighter than liquid water. A mixture of the two forms would be the actual liquid. To quote Tammann27 (my translation of the German):

“In fact, water exists in two groups of crystal forms, which are very different from each other in terms of their volume.”

So, we are still in the realm of mixture models, but ones for which we now have structural content—although still assuming the liquid can be considered as having a crystalline-like structure.

With respect to using x rays to determine crystal structures, techniques for interpreting diffraction data were in the very early stages of development. Nevertheless, information on unit cell symmetry and relative dimensions could, with care, be obtained. Early x-ray studies on ice by Rinne in 191728 and Dennison in 192129,30 came up with unit cells of hexagonal symmetry with respective axial ratios c/a of 1.68 and 1.62. From his results for the density and the lattice parameters, Dennison concluded that “the formula for the ice molecule is (H2O)2 or H4O2.” He was also perhaps the first to suggest how the water molecules were arranged in the crystal, proposing that they were arranged at the vertices of equilateral triangles.

It was, however, Bragg who in 1922 came up with the first realistic molecular structure of ice by a theoretical argument.31 Referring initially to the x-ray data of both Dennison30 and St. John,32 he made “an attempt to arrive at the structure of ice by an independent method, which, though not depending on general information gained by x-ray analysis, does not require special application of the analysis to ice itself.” Knowing the density of ice and the hexagonal structure that could be generated from that of diamond, he argued impressively that each individual water molecule interacts with four essentially tetrahedrally distributed neighbors in the arrangement shown in Fig. 1, reproduced from his original paper.31 Each molecule could be considered as a single molecule interacting equally with four neighbors, with no need to postulate larger molecular structures. Though his proposed structure was the result of a theoretical argument, he showed that this structure was consistent with the lattice parameters determined by Dennison, and largely consistent with the x-ray intensities reported in Dennison’s paper.

FIG. 1.

The structure of ice as proposed by Bragg.31 Black circles represent hydrogen atoms, white circles represent oxygen atoms. Each water molecule interacts with four neighbors arranged tetrahedrally, a motif that became central to the early structural work on water. Reproduced with permission from Bragg, Proc. Phys. Soc. London 34, 98 (1922). Copyright 1922 IoP Publishing.

FIG. 1.

The structure of ice as proposed by Bragg.31 Black circles represent hydrogen atoms, white circles represent oxygen atoms. Each water molecule interacts with four neighbors arranged tetrahedrally, a motif that became central to the early structural work on water. Reproduced with permission from Bragg, Proc. Phys. Soc. London 34, 98 (1922). Copyright 1922 IoP Publishing.

Close modal

This local tetrahedral structure might be justifiably considered to be the major step forward in understanding the essential geometrical consequence of the water–water intermolecular interaction in its condensed phases—a local geometry that was to have a major influence on subsequent approaches to the structure of water. Seven years later in 1929, Barnes’s x-ray measurements finally confirmed the structure of ice,33 in a paper that also gives an extensive discussion of previous work on its structure.

Bragg, in his deduction of the geometrical structure of ice,31 treated the water molecule as a negative oxygen ion and two positive hydrogen ions. However, prior to the work of both Bragg and Barnes, the “chemical” nature of the interaction between individual water molecules was being addressed. This work led in the early 1920s to the concept of the hydrogen bond as the interaction between water molecules in its condensed states.

The origin of the idea of the hydrogen bond is usually attributed to a 1920 paper by Latimer and Rodebush,34 which discusses polarity and ionization (though Pauling has argued35 that preference should be given to a 1912 study on amines in aqueous solution by Moore and Winmill36). Noting the footnote on page 1431 of the Latimer and Rodebush paper34 that “Mr. Huggins of this laboratory in some work as yet unpublished, has used the idea of a hydrogen kernel held between two atoms as a theory in regard to certain organic compounds,” perhaps the credit should at least equally be given to Maurice Huggins, who many of us working on water in the second half of the 20th century will remember from discussions especially at Water Gordon Conferences. Huggins’s paper celebrating 50 years of hydrogen bond theory37 is well worth reading and makes his case for priority of the concept he arrived at in 191938,39—though it was 1922 before his work was published.39–41 

Wherever the priority for the concept of the hydrogen bond lies, its central relevance to understanding the structure of water has been clear for pretty well the last century. And Latimer and Rodebush’s comment34 that the result of this interaction between water molecules “need not be limited to the formation of double or triple molecules” could be read as a realization that theories based on mixture models, e.g., of (H2O)2 and (H2O)3, were going down the wrong track. And perhaps their following comment is prescient of what was to come thirty years later in the form of continuum models: “Indeed the liquid may be made up of large aggregates of molecules, continually breaking up and reforming under the influence of thermal agitation.”

With x-ray work on crystalline solids growing in the 1920s, it would seem natural to apply the same experimental technique to liquids. In fact, what is probably the first attempt to understand x-ray scattering from a general array of scattering centers was that of Debye in 191542 and applied by Debye and Scherrer in 1916 to the structure of liquid benzene,43 only four years after Friedrich and Knipping first directed x rays onto a crystal. However, they were only able to obtain an estimate of the diameter of the molecule. Six years later, Keesom and de Smedt44 reported results on a series of experiments on a number of liquids from which they estimated average intermolecular separations—including that for water. However, a direct relationship between the diffraction pattern of a noncrystalline system and the atomic density at a given separation had to wait until 1927 when Zernike and Prins45 connected the two by applying the Fourier integral theorem. This work laid the foundation for liquid structure analysis through the radial distribution function (RDF) or pair correlation function. It seems to have been first used to interpret diffraction from a liquid in an experiment on liquid mercury in 1930 by Debye and Menke,46 who concluded that a “quasi-crystalline structure” exists in the liquid.

Although diffraction data on water was taken at about the same time,47–49 the results were not interpreted by inverting the diffracted intensity to obtain the radial distribution function. Rather, comparisons were made between the scattering angles of the main peak maxima and those obtained from powder patterns of ice. As the three strongest ice peaks occurred at similar angles to the maxima of the three broad peaks in the liquid pattern, Stewart48 drew the conclusion that “such periodicities as are found in the liquid state have strong quantitative similarity to the most important of those in the solid state”—and although a “detailed description of the molecular arrangement in water from x-ray data is not at present possible yet it simulates the crystal arrangement in ice.” However, the fact that the liquid peaks did not correspond exactly with the powder lines of the crystalline solid caused him to comment that this correspondence was only approximate, and he gave no indication of the nature of this structural approximation. Even so, taken together with his assertion that his data implied that models involving complexes of small numbers of molecules [e.g., (H2O)2; (H2O)3] should be abandoned, this work could be seen as an early, significant advance on the road to a structural model of liquid water.

Making use of what appears to be Amaldi’s diffraction data,49 in 1933 Bernal and Fowler,50 in the first year of this journal, did make use of the relationship between the measured x-ray intensity and the radial distribution function set out by Zernike and Prins.45 However, rather than try to Fourier invert the measured diffraction patterns, Bernal reversed the procedure by setting up models of possible structures, calculating their expected diffraction patterns and then comparing those with the measured experimental diffraction.

Before he did this, he realized he needed additional information to that in the x-ray pattern. The 1933 paper50 sets out how he made use of this additional information in developing his ideas on the liquid structure, and it is illuminating to try to follow his lines of argument in the context of ideas that were around at the time.

First of all, he wanted an idea of the size and shape of the water molecule itself. With respect to the size, guided by the nearest approach of 2.76 Å between water molecules in the then recently solved structure of ice33 he used a radius of 1.4 Å. Regarding the shape, on the basis of spectroscopic measurements, Mecke and Baumann51 had proposed in the previous year that the water molecule was essentially V-shaped. They proposed that the oxygen connected to two hydrogens in the shape of a V, with an HOH angle between 103° and 106°, which Bernal and Fowler interestingly noted50 was close to the tetrahedral angle of 109°.

Bernal and Fowler then argued what this shape might mean for the way electrical charge was distributed around the molecule, an important question to answer if they wanted to understand how water molecules were likely to interact with each other. Using his understanding of quantum mechanics (another relatively new scientific concept at the time), Bernal argued that the charge distribution in the molecule could be represented by positive charges on the hydrogen atoms and negative charges “behind” the oxygens in the regions shown in Fig. 2. This deduced charge distribution was a key factor in his next suggestion: that “The net electronic density distribution will therefore resemble a tetrahedron with two corners of positive charge and two of negative charge.”50 With this kind of near-tetrahedral arrangement of (two positive and two negative) charges, he argued that, since unlike charges attract, a single water molecule would tend to interact with four neighbors to form a tetrahedral motif of Fig. 3, as had been found in ice.33 This four-coordinated local arrangement was to be a central concept in his models of the liquid water structure.

FIG. 2.

The electron distribution on the water molecule as proposed by Bernal and Fowler. Reproduced from Bernal and Fowler, J. Chem. Phys. 1, 515 (1933), with the permission of AIP Publishing.

FIG. 2.

The electron distribution on the water molecule as proposed by Bernal and Fowler. Reproduced from Bernal and Fowler, J. Chem. Phys. 1, 515 (1933), with the permission of AIP Publishing.

Close modal
FIG. 3.

The local structure of first neighbor water molecules as envisaged by Bernal and Fowler. Reproduced from Bernal and Fowler, J. Chem. Phys. 1, 515 (1933), with the permission of AIP Publishing.

FIG. 3.

The local structure of first neighbor water molecules as envisaged by Bernal and Fowler. Reproduced from Bernal and Fowler, J. Chem. Phys. 1, 515 (1933), with the permission of AIP Publishing.

Close modal

So much for the molecule itself and how it is likely to interact with its nearest neighbors. However, to understand the liquid structure, we need to know how these kind of local arrangements connect to one another—information that should be in the x-ray data that were available at the time. In contemplating how Bernal, as a crystallographer, approached this issue, one can imagine this tetrahedral geometry structure setting a bell ringing, for a similar motif is found in the structures of silicates. Perhaps, therefore, something could be learned about the structure of water from the known structures of silicates?

Two silicate structures that were of particular interest to him in this context were quartz and tridymite. Though these two structures have different densities (quartz is denser than tridymite), like other silicates, they both have the same underlying tetrahedral motif, with each silicon connected to four neighbors via intervening oxygens—similar to the way one water molecule is linked to four neighbors through intervening hydrogens in ice. With this similarity in mind, he then calculated low resolution radial distribution functions of quartz and tridymite (scaled to the assumed size of the water molecule) and then used the approach of Zernike and Prins45 to calculate the diffraction pattern to compare with the experimental data.

Figure 4 compares his calculated diffraction patterns for a “quartz-like” structure (curve No. 3), a “tridymite-like” one (curve No. 4), and an irregular close packing (curve No. 1)52 with the experimental data (curve No. 2).53 The calculated pattern from the close-packed structure is clearly in disagreement with the experimental data and so was immediately rejected. With respect to the other two calculated patterns, he argued that the quartz-like one, adjusted to “diminish the regularity of the packing particularly at greater distances than 4 Å from any molecule” (or as stated in a paper published in the same year:54 “introducing a slight general disorder to account for heat motion”), fitted the data best.

FIG. 4.

The experimental x-ray scattering curve for water (No. 2) compared with the calculated curves for models based on irregular close packing (No. 1), quartz-like (No. 3), and tridymite–ice-like (No. 4) structures. Reproduced from Bernal and Fowler, J. Chem. Phys. 1, 515 (1933), with the permission of AIP Publishing.

FIG. 4.

The experimental x-ray scattering curve for water (No. 2) compared with the calculated curves for models based on irregular close packing (No. 1), quartz-like (No. 3), and tridymite–ice-like (No. 4) structures. Reproduced from Bernal and Fowler, J. Chem. Phys. 1, 515 (1933), with the permission of AIP Publishing.

Close modal

The comparison with experimental data is shown in Fig. 5. The broad peaks are all in the right places. Their average intensities are not quite right, but for a first try the agreement is encouraging. By present-day standards, it may be far from perfect, but it is pretty good considering the state of the field at the time.

FIG. 5.

A comparison of the calculated scattering of Bernal and Fowler’s water structure model with the experimental data. Reproduced from Bernal and Fowler, J. Chem. Phys. 1, 515 (1933), with the permission of AIP Publishing.

FIG. 5.

A comparison of the calculated scattering of Bernal and Fowler’s water structure model with the experimental data. Reproduced from Bernal and Fowler, J. Chem. Phys. 1, 515 (1933), with the permission of AIP Publishing.

Close modal

With respect to changes in temperature, his observations on the changes in the experimental diffraction pattern on reducing temperature (in particular a deepening of first minimum) led him to suggest a molecular distribution in which the “ice–tridymite-like” arrangement was beginning to occur. Noting the fact that quartz has a higher density than tridymite, he argued50 that its disappearance in favor of the quartz-like structure as temperature is increased is consistent with the anomalous contraction of water on heating above 0 C. However, he was cautious: “…we must not say more than that cold water is on the whole quartz-like with appreciable tridymite–ice-like tendencies.”

The final structural conclusions of the paper were that there were three main forms of arrangements of water molecules in the liquid. What Bernal called “water I” was tridymite–ice-like. It was “rather rare” but present to some degree below 4 C. “Water II” was quartz-like and dominated at ordinary temperatures, while “water III,” which predominated at high temperatures, was similar to a close-packed liquid—though it is not clear either from the 1933 paper or his other publications about that time what he understood to be the structure of a close-packed liquid.52 

With respect to contemporary theories of water as mixtures, he stressed50 that trihydrol and dihydrol had no direct structural analogy with water I, II, and III but that his scheme explained, “by means of the geometrical internal structure of the liquid, physical properties which the former attempted to explain in terms of hypothetical molecules.” So in this rejection of what he described as theories that “have been conceived too much in the manner of molecular chemistry,”50 he echoed the conclusions that Stewart had previously drawn48 from his work on his (Stewart’s) x-ray diffraction data.

At first sight, these conclusions might lead us to think that Bernal was proposing a mixture model based on the mixing of different structures. However, he stressed that these forms would pass continuously into each other with change of temperature. He insisted50 that “there is no question of a mixture of volumes with different structures; at all temperatures the liquid is homogeneous but the average mutual arrangements of the molecules resemble water I, II and III in more or less degree”—a view that was also being advanced by Stewart.48 One can perhaps feel here that he is trying to develop a concept relating to structural disorder that did not, at the time, exist but would follow later as his understanding of simple liquids developed.55,56 In fact, as he said later,56 the approach of the 1933 paper “was ultimately to prove rather a delusive approach, postulating a greater degree of order, particularly long-range order, in the liquid than actually exists there. A liquid, in my present opinion, is not simply a blurred solid.”

In addition to the structure of water, Bernal’s 1933 paper50 addressed a wide range of water-related issues. Before leaving this paper, it is perhaps worth looking at one of those issues specifically—namely how the charge distribution in the water molecule might best be described. As set out above, in developing his ideas on the structure of the liquid, he placed two negative charges “behind” the oxygen as denoted in Fig. 2. When later in the paper he addresses the internal energy of water and of ice, and the dielectric properties, he replaces the two centers of negative charge by a single charge, not on or behind the oxygen center, but displaced along the bisector of the HOH angle (Fig. 6). This creates no problem as far as approximate tetrahedrality of the water–water interaction is concerned and, interestingly, is very similar to some of the most successful three point charge potential functions that were to become used in computer simulation calculations some 40–50 years later, such as SPC,57 TIPS,58 and RWK1/2.59 

FIG. 6.

A three-point charge model of the water molecule according to Bernal. Reproduced from Bernal and Fowler, J. Chem. Phys. 1, 515 (1933), with the permission of AIP Publishing.

FIG. 6.

A three-point charge model of the water molecule according to Bernal. Reproduced from Bernal and Fowler, J. Chem. Phys. 1, 515 (1933), with the permission of AIP Publishing.

Close modal

Judging from the published literature from the 1930s to the 1960s, Bernal’s structural approach, though representing a major step forward in understanding the properties of water and some aqueous solutions in terms of their structures, appears to have made little impact on the field. Rather than focus on molecular structure, work seemed to follow the following approach:

  1. First, postulate a model based on some experimental evidence (combined with some intuition).

  2. Then, translate the model into a simple function with variable parameters.

  3. Fit those parameters to chosen experimental properties.

Models that were developed on this kind of basis included the idea of hydrogen-bonded “flickering clusters” in a sea of “free” molecules,60 structural mixture models by various authors from 1948 to 1966 invoking “dense” (possibly “closely packed”) and “bulky” (often “ice-like”) local structures,61–64 “interstitial” models where single water molecules are considered to occupy cavities in a hydrogen-bonded framework,65–67 and other mixture models based on “species” defined, for example, by the number of hydrogen bonds made by each molecule. As examples of the latter, Vand and Senior in 196568 proposed water molecules formed 0, 1, or 2 hydrogen bonds, while Nemethy and Scheraga69,70 considered water to be a mixture of water molecules forming 0, 1, 2, 3, or 4 hydrogen bonds. Other models were produced by Walrafen71 on the basis of intensity changes in the Raman spectrum and by the authors of the work of Haggis et al.72 from dielectric and latent heat data.

Clearly, a good fit would suggest a model may be plausible, but it does not prove it is correct. For us to have some confidence in a particular model, it would need to predict other properties that were not used in the fitting. So, to the list above should perhaps be added the following:

  1. Use the model to predict other properties that were not used in the parameter fitting.

On the surface, each of these models might be considered successful in that, by varying parameters, all of them could fit some data. However, they fail when used to predict other properties. And the different “broken bond” models give wide variations in the estimates of the fractions of broken bonds.

Taking a sample of these estimates from different models, Falk and Ford73 listed the predicted fraction of intact hydrogen bonds. This varied between different models from 0.02 to 0.72 at 0 C, with a similar spread ranging from 0 ∼0.8 at 100 C.74 Even among spectroscopists, there was significant disagreement, with Walrafen75 estimating over 80% of hydrogen bonds are broken at 65 C, while Wall and Hornig76 put the figure at less than 5%. These models also gave estimates of the energy of a hydrogen bond in liquid water that varied by a factor of more than three: between 1.3 and 4.5 kcal/mol.74 Another unencouraging sign!

In a paper given at a 1965 symposium on desalination, Bernal77 summarized the water structure situation at the time as follows:

“In general, practically all the theories of structure of liquids, particularly those of liquids such as water, are remarkably theory-insensitive. The same results can be obtained from different theoretical approaches. It is not that we lack theories of the structure of water, but rather that we have too many theories, not all of which can be true, and we lack the criteria to distinguish between them.”

As to how to proceed, he proposed the following:

“A new attempt should dispense with such arbitrary models or hypotheses and try to build up a more detailed model based on what we can deduce from the nature of the water molecule and from the general picture of the theories of liquid structure. Such a model should contain a minimum of auxiliary hypotheses and especially of disposable parameters….”

He interestingly noted that his 1933 theory would not satisfy these criteria—by postulating variable amounts of quartz-like and tridymite-like structures to account for the volume–temperature variation, he pointed out that just such arbitrariness was introduced.

From examining the details of these kinds of models and their predictions, Eisenberg and Kauzmann in their 1969 book74—a masterwork of critical examination of the water structure field up to that time—seemed to come to a similar view, concluding that the basic premise of water being a mixture of a small number of distinctly different species “is not in accord with experimental data.” In addition to the problems of the wide variations in broken bond populations and hydrogen bond energies mentioned above, they argued also that the uncoupled (IR, Raman) stretching bands show that the liquid contains a variety of molecular environments, that the liquid is characterized by a wide distribution of environments, and that models that take no account of this cannot be accurate representations of liquid water. It seems to me that these statements hold as true today as they did in 1969, though it does not seem to have prevented the periodic re-emergence of models based on a mixture of a small number of species.

In arguing we should move on from mixture models, Eisenberg and Kauzmann concluded74 that “distorted h-bond models seem to be in accord with most of what is known…from experiment.” Moreover, they asserted that extensive hydrogen bonding can account for many properties, such as the large dielectric constant, the energy of vaporization, the abnormal proton mobility, and the low quadrupole coupling constant of deuterons in D2O. Hence, they proposed that distorted hydrogen bond models merited further investigation.

Here, they were specifically drawing attention to two separate but conceptually related theories. First, Pople’s 1951 theoretical approach considered hydrogen bonds to be distorted rather than broken.78 Second, Bernal’s four-coordinated random network model,56 which was developed according to the principles he set out and are quoted above—a “model based on what we can deduce from the nature of the water molecule and from the general picture of the theories of liquid structure.”

Before looking at how both these models fared on comparing with the experimental diffraction data, it is interesting to look at how Bernal’s thinking had developed in getting to this model.

With respect to the nature of the water molecule (the first criterion he set out for developing a model, as quoted above), he remained with the basic four-coordinated local motif of a water molecule hydrogen bonding to four neighbors. With respect to “a general picture of the theories of liquid structure” that he could embed this motif in, he had developed a structural model of a simple liquid such as argon based upon a random packing of spheres (Fig. 7).

FIG. 7.

Bernal’s random packing of equal spheres as a structural model of a simple liquid (top) compared to the equivalent ideal crystal (bottom). Finney, personal collection.

FIG. 7.

Bernal’s random packing of equal spheres as a structural model of a simple liquid (top) compared to the equivalent ideal crystal (bottom). Finney, personal collection.

Close modal

In trying to see how to construct a model that was consistent with both the local coordination and the noncrystallinity of the extended structure that was implicit in his simple liquid model, he appears to have had an inspiration from the discovery of keatite,79 a crystalline form of silica that, unlike the quartz structure he used in the 1933 paper, had the same density ratio to tridymite as water has with ice. In that structure (keatite), he noted that, rather than being connected through sixfold rings, some of the silicons connected through fivefold rings—a point that suggested that in developing a liquid water model, he did not need to be confined to the sixfold ring structures found in many silicates, but that the ring structures could range from 4 to 7 without serious deformation of the local four coordination. Accordingly, he built such a physical structure in the laboratory (Fig. 8), commenting56 that the irregularity of the liquid structure “is not in the co-ordination itself, but in the rings that are formed by co-ordinated molecules. Mostly on account of the internal structure of the molecule, these rings usually contain five members.”

FIG. 8.

Bernal’s laboratory realization of his random network water model. Finney, personal collection.

FIG. 8.

Bernal’s laboratory realization of his random network water model. Finney, personal collection.

Close modal

How did these two approaches fare in comparison with X-ray diffraction?

First, from measurements taken by Katzoff,80 work that was contemporaneous with the Bernal and Fowler paper but published in the following year, the author concluded that the local four-coordinated motif was consistent with the measured data. With respect to the way in which these local structures connected with one another, the paper argues for a “continuous”—and by implication noncrystalline—arrangement. No evidence was found for a “quartz-like” arrangement postulated by Bernal, and Katzoff suggests that Bernal and Fowler’s choice of wording “was perhaps unfortunate in its emphasis of the analogy to the crystalline structures,” a criticism which Bernal himself would subsequently make.56 Katzoff also interestingly suggests that the decrease in water density on approaching the freezing point is due to the increasing tendency of the O–H⋯O angles to straighten toward 180° “as the heat motion becomes less violent.” So, perhaps a case can be made for Katzoff being the first to suggest the structural explanation of the volume–temperature anomaly, and perhaps also the need to move away from models based on structures related to those in crystals.80 

Second, following Warren’s 1937 measurements on vitreous silica and liquid sodium,81 the 1938 experiments of Morgan and Warren82 had produced improved quality x-ray data on the liquid against which structural models could be directly tested. As shown in Fig. 9, Pople’s distorted hydrogen bond model fitted Morgan and Warren’s data reasonably well.78 And a comparison of the predicted scattering from Bernal’s laboratory-built random network model with x-ray data was also encouraging (Fig. 10). However, the availability of the even higher quality x-ray data presented in the work of Narten et al.83 did not kill off the mixture model concept, as they preferred to fit their data to a model of a (distorted) ice-I lattice with interstitial non-hydrogen-bonded molecules within the cages. The interstitials were considered necessary to explain the existence of a small peak in the data at a distance of about 3.6 Å, though this was later agreed by Narten84 to have been an artifact resulting from the truncation of Fourier inversion of the x-ray data that is performed to obtain the radial distribution function. However, the Narten and Levy data were sufficient to invalidate Pauling’s 1959 hypothesis65 of water as a “self-clathrate” structure similar to that of chlorine hydrate.

FIG. 9.

The radial density function predicted by Pople’s distorted hydrogen bond model78 (dotted-dashed line) compared to that obtained from the experimental data (solid line) of Morgan and Warren.82 Dashed lines show contributions of separate shells. Reproduced from Pople, Proc. R. Soc. London, Ser. A 205, 163 (1951). Article out of copyright.

FIG. 9.

The radial density function predicted by Pople’s distorted hydrogen bond model78 (dotted-dashed line) compared to that obtained from the experimental data (solid line) of Morgan and Warren.82 Dashed lines show contributions of separate shells. Reproduced from Pople, Proc. R. Soc. London, Ser. A 205, 163 (1951). Article out of copyright.

Close modal
FIG. 10.

Bernal’s comparison of the RDF of water from his random network model (dashed line) compared with contemporary experimental data (solid line). Reproduced with permission from Bernal, Proc. R. Soc. London, Ser. A 280, 299 (1964). Copyright 1964 The Royal Society (U.K.). Permission conveyed through Copyright Clearance Center, Inc.

FIG. 10.

Bernal’s comparison of the RDF of water from his random network model (dashed line) compared with contemporary experimental data (solid line). Reproduced with permission from Bernal, Proc. R. Soc. London, Ser. A 280, 299 (1964). Copyright 1964 The Royal Society (U.K.). Permission conveyed through Copyright Clearance Center, Inc.

Close modal

Eisenberg and Kauzmann’s critical synthesis summarized the state of the field in the 1960s and pointed the way forward—away from chemical and mixture models and toward an exploration of homogeneous structural models of the kind that had been pioneered by Pople and Bernal. It was to take the advent of computer simulation, allied to the development and exploitation of more powerful neutron and synchrotron radiation sources, to provide experimental evidence that would justify Eisenberg and Kauzmann’s forward look.

Computing facilities in the 1960s had developed to the stage where they could begin to simulate the structures of simple liquids, incidentally confirming Bernal’s random sphere packing model as an ideal model of such liquids.85–87 The first attempt to simulate liquid water seems to have been a Monte Carlo simulation by Barker and Watts in 196988 using a potential function derived by Rowlinson89 and limited to a periodic cubic cell containing 64 molecules. The authors commented that agreement with experiment was “not outstanding”—the radial distribution function looked more like that from a simple liquid with the second peak closer to two molecular diameters rather than that at about 1.5 that is characteristic of water. On the positive side, this pioneering work did show results that were encouraging enough to establish the feasibility of this approach to understand water structure and suggested more needed to be done to refine the potential functions used in simulations.

Exploiting the technique realistically required two main conditions to be satisfied: first, a realistic potential function that could be used in the simulation. Although it was suspected by many chemists that non-pair-additivity could be involved in water hydrogen bonding interactions, “effective” pair additivity was, of computational necessity, initially assumed. The potential functions were generally different varieties of point charge models that were usually fitted to known experimental data. As has been commented earlier, some of the most successful pair potential functions used bore a strong resemblance to the three-point charge model proposed in the Bernal and Fowler 1933 paper50—see Fig. 6. Second, adequate computer power was necessary to enable large enough assemblies of molecules to be followed long enough for equilibrium to be reached. And as computational power increased year on year, larger assemblies and more complex potential functions (including non-pair-additive ones and ones derived from first principles calculations) could be explored.

Consequently, the early 1970s saw major advances in simulations of water structure. Using a four-point-charge potential function designed by Ben-Naim and Stillinger,90 and a periodic cell of 216 molecules, Rahman and Stillinger91 concluded that “agreement obtains only in modest degree.” However, they particularly noted a number of important issues that pointed to the inadequacy of popular models that had been current up to that time, many of which have been discussed above. First, the structures sampled showed a high degree of structural (i.e., noncrystalline) disorder—their results “lead to a picture of liquid water as a random, defective, and highly strained network of hydrogen bonds that fill space.” There was general tetrahedrality in the interactions between first neighbor water molecules. The structures were homogeneous—there were no clusters of anomalous density. There were no signs of local ice or clathrate structures. The molecules could not be divided into ones that were “network” or “interstitial”—they were all part of the network. And there was no interpenetration of hydrogen bonds through rings of hydrogen-bonded water molecules, such as is found in some of the high pressure ices, e.g., ices IV92 and V.93 

This one set of “experiments,” provided one accepted that the potential function used was sufficiently realistic, thus had serious implications for most of the water structure models extant at the time. On the positive side, the observations from the simulations were consistent with the distorted hydrogen bond type of model (Pople and Bernal) favored by Eisenberg and Kauzmann. In one fell swoop, microcrystalline and mixture models were seen to be inconsistent with the simulation results.

In considering how to improve their initial simulations, Rahman and Stillinger concluded that the potential function they used was probably too strongly tetrahedral, a conclusion strengthened by a following simulation.94 Consequently, they shifted the negative charges representing the lone pair electrons in from 1 Å from the oxygen center to 0.8 Å, a change that not only resulted in better agreement with scattering data but also95 demonstrated that the existence of a temperature of maximum density could be explained without having to resort to a mixture model.

There followed a mushrooming industry of simulations using a range of potential function models, the earlier ones of which represented the water molecule with three or four point charges.96 Many of these produced results that agreed to some degree with the experimental data, though as the structural data were mostly x-ray data, comparisons largely concentrated on the oxygen–oxygen RDF. There were a few attempts to develop non-pair-additive potential functions that it was thought might be more capable of reproducing structural changes with temperature and pressure,97,98 but none of those proved satisfactory. And with further increase in computer power, attention increasingly focused on exploring quantum mechanical approaches to the water–water interaction, such as the early MCY potential.99 

Where models of the water molecule included a representation of the hydrogen atoms, in addition to the oxygen–oxygen RDF, computer simulations could also access the oxygen–hydrogen and hydrogen–hydrogen correlations, and hence throw light on the orientations of water molecules in the liquid predicted by the modeling. However, as the x-ray scattering power of hydrogen is much smaller than that of oxygen, the diffraction measurements available at the time of the early simulations were unable to yield good information on the hydrogens that was needed to test the simulated orientational structure.

In contrast, neutrons are scattered strongly by—especially—deuterium, and so neutron diffraction entered the stage in the late 1950s and 1960s. Probably the first neutron diffraction experiments to probe the structure were those on D2O by Brockhouse at Chalk River in 1958100 and Page on the DIDO reactor at Harwell in 1967.101 However, neither was able to extract specific structural information, although Forte and Menardi,102 from their 1966 small angle neutron scattering data, concluded there was no evidence for significant aggregates in the liquid. That neutrons are generally very penetrating also gives them an advantage in structural studies requiring bulky sample environment containment, for example water under pressure and over a wide temperature range, including in the supercritical region.103–107 

In dealing with raw neutron diffraction data, a number of corrections need to be applied to account for absorption, multiple and incoherent scattering, and the so-called Placzek correction.108 All except the latter are relatively straightforward. However, the Placzek correction is small if the energy transfers are small compared to the energy of the impinging radiation. This condition can be satisfied using high energy neutrons such as are produced by pulsed spallation neutron sources, and it is the exploitation of these—particularly the ISIS pulsed neutron and muon source in the UK—that has led to the detailed picture of the structure of liquid water that we now have. This has been further helped by exploiting the fact that the scattering lengths of hydrogen and deuterium are different, making possible the exploitation of isotope substitution pioneered by the work of Enderby et al. originally on liquid metals109 to separate the three partial RDFs: oxygen–oxygen, oxygen–hydrogen, and hydrogen–hydrogen. The early landmark work of Soper110,111 is of particular note in kick-starting the exploitation of isotope substitution and pulsed neutron sources in tackling water structure.

There are, however, other problems in getting at direct structural information from the scattering. Formally, as originally set out by Zernike and Prins,45 the measured scattering factor is related to the corresponding radial distribution function through a Fourier transformation. However, the measured data are limited in the range of scattering vector that can be accessed, so the resulting transform is likely to contain truncation errors that are in danger of being interpreted as real features. An example here is the 3.6 Å “hump” in Narten’s x-ray RDF83 referred to above, which led to their erroneous interpretation in terms of interstitial molecules in surrounding water cages.84 As the accessible scattering vector range in a pulsed source experiment is much greater than that at a reactor source, this problem is less for pulsed source measurements, but it is still there to be dealt with. So, rather than try to correct for the problem, work from the 1990s onward has increasingly taken a different approach, taking advantage of the continually increasing computational power available.

One way of avoiding the Fourier truncation problem would be to (a) simulate a system using either Monte Carlo or molecular dynamics and an assumed potential function and then (b) calculate the scattering that simulated assembly would produce and compare that with the experimental results. However, making that comparison between the experimental scattering and that predicted by the simulation would not be able to throw light on the structural reasons for disagreement between the real system and the simulated one.

An alternative approach would be to reverse the process, as is done in the Reverse Monte Carlo (RMC) procedure of McGreevy.112 This begins by setting up a starting configuration of atoms at the required density, from which the radial distribution function, and then the predicted (x-ray or neutron) structure factor, is calculated. This is then compared with the experimental structure factor and a random atom moved with a probability that depends on the difference between the measured and simulated structure factor. The new structure factor difference is then calculated and the procedure continued until the difference settles down to an equilibrium value. The resulting configuration should be a three-dimensional structure that is consistent with the experimental data.

An advance on the RMC method, which it is argued gets over the problem of uniqueness that has been raised concerning the process,113 is provided by also allowing perturbations to the assumed starting potential function—the Empirical Potential Structure Refinement (EPSR) approach developed by Soper.114 As with RMC, the extent to which a resulting structure can be considered reliable will depend upon the extent and quality of the experimental data that are fed into the modeling, and here the use of isotope substitution (in the case of water, taking advantage of the different neutron scattering factors of hydrogen and deuterium) provides additional constraints on the EPSR-derived structure. Further constraints can also be imposed, for example the x-ray scattering from the same system.

With continual improvement in data from both synchrotron x-ray and neutron sources, the structure of water at normal temperature and pressure in terms of the partial radial distribution functions is pretty well understood.115,116 Moreover, scattering experiments as functions of temperature and pressure are extending our structural understanding throughout the phase diagram (see, for example, Refs. 107 and 116124). No longer do we need to speculate on what the different “species” are in the liquid—the experimental results from neutron and x-ray scattering are consistent with the average structure of liquid water under normal conditions being a homogeneous, disordered, partially defective, hydrogen-bonded network of molecules. Moreover, we can follow how the degree of hydrogen bond bending is reduced as we go into the supercooled region121—resulting in less distortion of the average first neighbor tetrahedral motif—as well as the increased distortion and breakdown of the hydrogen bonding as we go in the other direction toward the supercritical region.120 

Liquid structure as described by a radial distribution function tells us nothing about molecular orientations: To describe the detailed local molecular structure, we need to look deeper. The procedures now available indeed do allow us to go beyond partial pair correlation functions and interrogate experimentally consistent local three-dimensional structures through spatial distribution functions (SDFs).125 These enable us to look in detail at the average three-dimensional local structure surrounding a particular type of atom—for example, Fig. 11 shows the average first neighbor arrangement of water molecules around a typical “central” molecule in liquid water at 283 K.

FIG. 11.

The average arrangement at 283 K of first neighbor molecules around a central water molecule that is consistent with neutron scattering data taken by Soper and reproduced with his permission.

FIG. 11.

The average arrangement at 283 K of first neighbor molecules around a central water molecule that is consistent with neutron scattering data taken by Soper and reproduced with his permission.

Close modal

The basic tetrahedral organization is evident, though the fact that the neighboring water hydrogens interacting with the negative region (the “underside” here) of the central molecule are shown by a single probability distribution underlines the less strict control on directional structure of the negative “lone pair” region. This less rigid orientational control is consistent with not only the “soft” tetrahedrality of many popular potential functions but also those three-point-charge models where a single charge is used to represent the oxygen lone pairs.96 It is perhaps also worth noting that the lack of separation of charge in the lone pair region was indicated many years earlier (for example, see Refs. 126 and 127; note also Bernal’s 1933 model in Fig. 6), though this was not noticed by the early simulations that favored strongly tetrahedral models until their over-tetrahedrality was demonstrated by the disagreement between their predicted RDFs and the experimental data.

This way of looking at the structural data can be extended to neighbors that are further out and can also be exploited to examine changes in local structure with, e.g., temperature and pressure. For example, Fig. 12 shows how the first and second neighbor distribution changes as pressure is increased to a few thousand atmospheres. Although they are partly hidden by the second neighbor lobes, those representing the first neighbor distribution are similar in both images. So the first neighbor average near-tetrahedral structure remains relatively undisturbed. However, when looking at the response of the second neighbor lobes to pressure, we see both something the same and something different. Though they look basically similar in both cases, at the higher pressure these lobes have been pushed in closer to the central molecule. Whether there has been some restructuring of the water network as a result of some bond breaking and reforming is not clear, though there are perhaps indications that this might be the case. However, it might just be that the structure has adapted to the reduced volume by a uniform compression. It is perhaps interesting to note that the SDF of high-density amorphous ice looks very similar,128 suggesting that its structure might bear a resemblance to that of liquid water under pressure.

FIG. 12.

The spatial density function showing average first and second neighbor distributions at (left) ambient and (right) high pressure. Rerun and smoothed from Ref. 129 and reproduced with permission from Soper.

FIG. 12.

The spatial density function showing average first and second neighbor distributions at (left) ambient and (right) high pressure. Rerun and smoothed from Ref. 129 and reproduced with permission from Soper.

Close modal

Although mixture models seemed to have been put to bed in the late 1960s by Eisenberg and Kauzmann,74 supported by later computer simulation and x-ray and neutron scattering experiments, the idea that there might be two liquid phases—generally labeled high density (HDL) and low density (LDL) liquids—has again raised its head130–135 and has become an issue of very active controversy in the last 30 years or so.

As in the earlier mixture model discussions, this idea has been put forward as an explanation of liquid water’s anomalous properties. However, as both theory and simulation have demonstrated, the anomalies can be explained in terms of two opposing tendencies rather than a change in relative populations of two different local structures. As temperature is increased from zero centigrade, two opposing tendencies come into play: (1) thermal expansion to increase volume and (2) increased bond bending that facilitates a volume reduction. These processes can be looked at in slightly different ways. For example, Debenedetti and Stanley have pointed out136 the anticorrelation between volume and entropy fluctuations in water, the microscopic cause of which is the local tetrahedrality, while Errington and Debenedetti137 consider the issue in terms of the temperature and pressure dependencies of translational and orientational order parameters. So the anomalies can be explained in terms of the local tetrahedrality, with no need to invoke any kind of two structures concept.

The origin of this re-emergence of a two liquid concept is, however, somewhat different—as a rationalization of the low temperature behavior of supercooled water.138 As the temperature is reduced to below the freezing point, the isothermal compressibility, isobaric heat capacity, and thermal expansion coefficient all change rapidly. As crystallization at the homogeneous nucleation temperature of 231 K would seem to prevent further measurement of these quantities to lower temperatures, a power-law fit139—though the use of a power law is disputed136—suggests all three quantities diverge at 228 K.

Of the four main theories put forward to explain this behavior,138,140–142 two138,141 invoke the idea of there being two liquid structures, the first of these proposing the existence of a liquid–liquid critical point located in a region of the phase diagram to which access is denied by crystallization. This “two liquids” scenario has, in addition to extensive computer simulation, stimulated imaginative experimentation. For example, small angle x-ray scattering measurements, together with results from x-ray emission spectroscopy and x-ray Raman scattering have been interpreted in terms of fluctuations between low and high density regions.130,143,144 Furthermore, ingenious and challenging experiments145–147 have exploited x-ray free electron lasers and ultrafast cooling of micron-sized droplets to determine thermodynamic response functions such as isothermal compressibility and heat capacity, as well as correlation functions, before crystallization can occur. The results from these experiments have provided support for a liquid–liquid critical point and the observation of a liquid–liquid transition under pressure. These and other studies—both experimental and simulations—have generated an extensive literature (for examples see Refs. 119, 129, 130, and 143163). There is vigorous discussion both in the literature and at scientific meetings, but the water community has yet to come to a consensus on this issue.

While accepting that this issue is clearly of interest with respect to the behavior of supercooled water, one might question if this is really a matter of particular importance to our understanding of water and its interactions in “real-life” situations. In saying this, I am not suggesting that water may not be able to exist in two distinct structural forms under different external conditions. Indeed, Stell has argued164 that such a scenario is theoretically possible given certain conditions on the potential function, and there is experimental evidence for liquid–liquid transitions in both phosphorus165 and silica.166 

Whether the effective water potential fulfills these conditions is unclear. Perhaps, more effort trying to establish this or otherwise might resolve the issue, but whatever the outcome of future work, there would seem to be no need to employ the concept to explain the anomalous behavior of water in its equilibrium states at biological temperatures and normal pressures.

Our attempts to understand the structure of liquid water have developed in the context of the contemporary intellectual environments and scientific concepts and the experimental and computational techniques available at the time. In the years before x rays opened up a route to experimentally determining even crystal structures, models involving an equilibrium between two—or more—different components could be parametrized to explain some of water’s so-called anomalous behavior. The early x-ray crystallographers brought molecular structure onto the scene, with Bragg’s31 early insight suggesting the tetrahedral first neighbor interaction well before the structure of ice Ih was determined being, in the context of the day, a tour de force. However, early structural models still depended on the idea of mixtures—but now on mixtures of particular structures that were known at the molecular level. This is in fact how Bernal initially approached the problem,50 interpreting the then-available very limited x-ray data on liquid water in terms of disordered “quartz-like” and “tridymite-like” structural units—though the nature of the disorder was not specified, nor was the relevance of the interfaces between these purported structures to the x-ray pattern addressed. Though three-dimensional structure had now entered the field, mixture approaches still continued to dominate the scene.

Despite the improved x-ray data in the 1930s,82 further improved on in the 1960s,83 and increased understanding of how to extract average structural information on liquids from diffraction experiments, mixture models continued to flourish in the 1950s and 60s. These were of increasingly imaginative forms, defined by their “nature:” for example “dense” and “bulky” species; number of “broken bonds.” However, none of these was able to sit comfortably with the experimental data. Eisenberg and Kauzmann’s 1969 in-depth analysis74 should have been the death knell of these kinds of theories. However, some bulky/dense mixture models have continued to raise their heads from time to time since then. Although imaginative experiments have been performed to address the issue, their conclusions remain highly controversial. Moreover, like the early “chemical” models, later mixture models that propose distinct structures also fail to consider the interface between the two components. They leave out a critical interaction: that between molecules in one component and molecules in the other.

As computer power increased significantly from the 1960s, computer simulation results justified Eisenberg and Kauzmann’s judgment that continuum models were the way forward, citing Pople’s 1953 distorted hydrogen bond approach78 and Bernal’s random network model56 that he developed in the late 1950s/early 1960s as an adaptation of his simple liquid structural model. However, it was not until neutron scattering could be effectively exploited with pulsed spallation sources, allied to further advanced computer-based interpretation techniques such as EPSR,114 that a consistent picture of water structure would emerge.

This we not only now have, but the techniques that have led us to this point have opened up tremendous possibilities for examining molecular-level structure and processes in more complex systems.

These techniques have not only enabled us to clarify—and in some cases change our understanding of—the structural basis of a number of ideas in solution chemistry. For example, the ability neutron scattering with isotope substitution has given us to look at local environments in liquids (“liquid state crystallography”167) has, through the work initiated by Enderby and Neilson,168 completely changed our understanding of hydration of ions in aqueous electrolyte solutions. More recently, we have been able to open up the study of structure in relatively complex—including biologically relevant—aqueous systems. For example, aqueous solutions of alcohols such as methanol that were previously thought to be homogeneous regular solutions have been demonstrated to show clustering of the alcohol molecules169 (see Fig. 13). Moreover, the standard model of the hydrophobic interaction first proposed by Frank and Evans170—that the anomalous negative excess entropy of mixing is caused by the formation of ordered “iceberg-like” structures around the hydrophobic entity in water—is not borne out by the structural evidence:171 Rather, there is a much simpler explanation in terms of the observed tendency of the apolar solute molecules to segregate at the molecular level.172 Furthermore, although the solute clustering observed in the self-assembly of aqueous amphiphiles may qualitatively appear to be consistent with being driven by hydrophobic interactions, the effects observed at the molecular level imply a rather more complex interplay between the nonpolar and polar interactions operational in such systems.173 

FIG. 13.

Snapshot of a mixture of methanol (black spheres) and water (white spheres) at a methanol mole fraction of 0.54. Obtained from EPSR simulation of neutron diffraction data. Reprinted with permission from Soper et al., J. Phys. Chem. B 110, 3472 (2006). Copyright 2006 American Chemical Society.

FIG. 13.

Snapshot of a mixture of methanol (black spheres) and water (white spheres) at a methanol mole fraction of 0.54. Obtained from EPSR simulation of neutron diffraction data. Reprinted with permission from Soper et al., J. Phys. Chem. B 110, 3472 (2006). Copyright 2006 American Chemical Society.

Close modal

Furthermore, the molecular interactions that are operational in salting out have been examined,174 where the system of nine different kind of atoms has required the extraction of 45 partial radial distribution functions and the construction of specific SDFs to demonstrate the local structures (Figs. 14 and 15). With respect to understanding the role of water in biological systems, the interactions between water and a number of biologically relevant molecules have begun to be explored, for example, amino acids175 and neurotransmitters.176 

FIG. 14.

Spatial density functions showing the more favored regions of molecular distribution of (from left to right) t-butanol around t-butanol (a), (b), and (c); water around t-butanol (d), (e), and (f); water around water (g), (h), and (i). For comparative purposes, (a), (d), and (g) relate to the SDFs calculated for a 0.06-mol-fraction solution of tertiary butanol in water.177 Panels (b), (e), and (h) refer to a 0.02-mol-fraction solution of tertiary butanol in water without salt and panels (c), (f), and (i) to a 0.02-mol-fraction solution with salt. Reproduced from Bowron and Finney, J. Chem. Phys. 118, 8357 (2003), with the permission of AIP Publishing.

FIG. 14.

Spatial density functions showing the more favored regions of molecular distribution of (from left to right) t-butanol around t-butanol (a), (b), and (c); water around t-butanol (d), (e), and (f); water around water (g), (h), and (i). For comparative purposes, (a), (d), and (g) relate to the SDFs calculated for a 0.06-mol-fraction solution of tertiary butanol in water.177 Panels (b), (e), and (h) refer to a 0.02-mol-fraction solution of tertiary butanol in water without salt and panels (c), (f), and (i) to a 0.02-mol-fraction solution with salt. Reproduced from Bowron and Finney, J. Chem. Phys. 118, 8357 (2003), with the permission of AIP Publishing.

Close modal
FIG. 15.

Spatial density functions showing the distribution of the sodium cation (a) and chloride anion (b) around t-butanol. Reproduced from Bowron and Finney, J. Chem. Phys. 118, 8357 (2003), with the permission of AIP Publishing.

FIG. 15.

Spatial density functions showing the distribution of the sodium cation (a) and chloride anion (b) around t-butanol. Reproduced from Bowron and Finney, J. Chem. Phys. 118, 8357 (2003), with the permission of AIP Publishing.

Close modal

No doubt more work on water structure exploiting recent developments such as machine learning and neural networks is likely. It also seems inevitable that efforts will continue to address the two liquids issue—the apparent divergence of the specific heat near 228 K means there must be some kind of hiatus in the entropy near this value that needs to be explained.149,178–180 However, some (though, I would strongly emphasize, not all) of this work gives me the impression of being predisposed to interpret new results based on, or in favor of, a two liquid model, rather than look at the data in an unbiased and critical way. This—as well as the hype that sometimes accompanies some publications and their associated publicity—I find worrying. Worrying also are attempts that seem to want to explain all the “anomalous” behavior of liquid water in terms of a two liquid scenario. As discussed above, these models were put forward originally to explain the behavior of supercooled water, not the behavior of water elsewhere in its phase diagram—behavior that has a perfectly straightforward structural explanation as discussed above. Moreover, the ballooning literature promulgating preferentially the two liquid scenario may be leading to an increasing problem of potentially misleading publications—including in the nonscientific press181,182—and an associated feedback loop.148 While a two liquid scenario may turn out to be a reasonable way of understanding water’s low temperature behavior, a skeptical approach should be the way to approach the issue, not one that biases data interpretation in terms of the theory and tries to explain everything about water in terms of the scenario. And for the sake of the integrity of scientific research, we should avoid the sensationalism that is sometimes attached to this kind of work.

In practice, settling the two liquid question is not necessary for understanding liquid water’s properties under the conditions where they are mostly relevant to us—as has been stressed above, its behavior above its melting point can be perfectly well explained otherwise. However, there are some fairly detailed structural issues that I would find of interest to be explored further—for example, the detailed hydrogen-bonded structure as pressure is increased is unclear. As suggested in regard to Fig. 12, does increasing pressure cause some restructuring of the water network as a result of bond breaking and reforming? Might the system be accommodating the reduced volume available to it by some “threading” of hydrogen bonds through ring structures as, for example, in ices IV and V? Does the structure of water as the temperature is reduced through the supercooled region move continuously toward that of low density amorphous ice? And can we develop useful ways of quantifying degree of order/disorder in a system that itself is inherently disordered—progressing Bernal’s concept of statistical geometry perhaps,56 or some other imaginative approach.183 In this context, it is worth noting that the distribution of density fluctuations in (neutron and x-ray) diffraction-consistent EPSR models119,149 has been found to be unimodal rather than bimodal: The latter would be expected if distinct domains of low and high density water were present.

While these questions may be relevant to those of us interested in the details of water structure, the wider—nonscientific as well as scientific—world might be justified in suggesting there might be better returns, both with respect to the “pure” science as well as its applicability, to explore the structure and behavior of water in its interaction with other systems. How it is, or is not, perturbed at biological, chemical, and physical interfaces is likely to be relevant to the properties and the functioning of such systems.

As experimental techniques and computational capabilities continue to develop, we should expect to be able to tackle increasingly complex aqueous systems and explore how water may be perturbed in such interfacial systems. For example, the NIMROD instrument184 at the UK’s pulsed spallation neutron source ISIS, through being able to take neutron scattering data over a very wide range of scattering vector in the same measurement, has already enabled us to look at both the intermolecular and intermediate range structures in complex aqueous systems, for example the self-assembly of colloids185 and counterion binding and water structuring in micelles.186 And computing developments under way promise the ability to interpret the molecular-level structures and processes in even larger systems of potentially over 106 atoms.187 

With continuing developments such as these on both the experimental techniques and computing fronts, the future looks bright for beginning to satisfy the reason Bernal originally became interested in the structure of water—to understand its role in biological systems.56 We now have a pretty good understanding of the structure of water itself over a wide range of temperatures and pressures. What is more, with the experimental and computational techniques that have led us to this understanding, we can now, in effect, do crystallography in the liquid state.167 And using these tools, we now have the ability to mine—and hopefully to move to understand—water’s behavior in not only the biological systems that stimulated Bernal’s interest, but also in other complex systems of physical and chemical importance.

The author has no conflicts to disclose.

John L. Finney: Writing – original draft (lead); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

1.
W. C.
Röntgen
,
Ann. Phys.
281
,
91
(
1892
).
2.
H. M.
Chadwell
,
Chem. Rev.
4
,
375
(
1928
).
3.
H.
Whiting
, Thesis,
Harvard University
,
1884
.
4.
J.
Thomsen
,
Thermochemische untersuchungen
(
J. A. Barth, Leipzig
,
1883
), Vol.
3
, p.
181
.
5.
F.-M.
Raoult
,
Ann. Chim. Phys.
4
,
401
(
1885
).
6.
H. E.
Armstrong
,
J. Chem. Soc., Trans.
53
,
125
(
1888
).
7.
H. M.
Vernon
,
London, Edinburgh Dublin Philos. Mag. J. Sci.
31
,
387
(
1891
).
8.
W.
Ramsay
and
J.
Shields
,
J. Chem. Soc., Trans.
63
,
1089
(
1893
).
9.
W.
Ramsay
and
J.
Shields
,
Philos. Trans. R. Soc. A
184
,
647
(
1893
).
10.
W.
Ramsay
,
Proc. R. Soc. London
56
,
171
(
1894
).
11.
J. J.
Van Laar
,
Z. Phys. Chem.
31U
,
1
(
1899
).
12.
O. N.
Witt
,
Vetens Akad. Forhandl.
57
,
63
(
1900
).
13.
W.
Sutherland
,
Philos. Mag.
50
,
460
(
1901
).
14.
W. R.
Bousfield
and
T. M.
Lowry
,
Philos. Trans. R. Soc. London, Ser. A
204
,
253
(
1905
).
15.
H. E.
Armstrong
,
E.
Wheeler
,
D.
Crothers
,
R. J.
Caldwell
, and
R.
Whymper
,
Proc. R. Soc. London, Ser. A
81
,
80
(
1908
).
16.
P.
Dutoit
and
P.
Mojoiu
,
J. Chim. Phys.
7
,
169
(
1909
).
17.
P.
Walden
,
Z. Phys. Chem.
65U
,
129
(
1909
).
18.
P. A.
Guye
,
Trans. Faraday Soc.
6
,
78
(
1910
).
19.
W.
Sutherland
,
Trans. Faraday Soc.
6
,
105
(
1910
).
20.
W.
Nernst
,
Trans. Faraday Soc.
6
,
117
(
1910
).
21.
G.
Tammann
,
Z. Anorg. Allg. Chem.
158
,
1
(
1926
).
22.
J.
Walker
,
Trans. Faraday Soc.
6
,
123
(
1910
).
23.
W.
Friedrich
,
P.
Knipping
, and
M.
von Laue
,
Sitzungsberichte der Math. Phys. Klasse (Kgl.)
,
Bayerische Akademie der Wissenschaften
,
1912
, p.
303
.
24.
M. A.
Rosenstiehl
,
C. R.
152
,
598
(
1911
).
25.
M. A.
Rosenstiehl
,
Bull. Soc. Chim.
9
,
281
(
1911
).
26.
P. W.
Bridgman
,
Proc. Am. Acad. Arts Sci.
47
,
441
(
1912
).
27.
G.
Tammann
,
Z. Phys. Chem.
72U
,
609
(
1910
).
28.
F.
Rinne
,
Ber. K. Sachs. Ges. Wiss. Mat.- Phys. Klasse
69
,
57
(
1917
).
29.
D. M.
Dennison
,
Science
52
,
296
(
1920
).
30.
D. M.
Dennison
,
Phys. Rev.
17
,
20
(
1921
).
31.
W. H.
Bragg
,
Proc. Phys. Soc. London
34
,
98
(
1921
).
32.
A.
St. John
,
Proc. Natl. Acad. Sci. U. S. A.
4
,
193
(
1918
).
33.
W. H.
Barnes
,
Proc. R. Soc. London, Ser. A
125
,
670
(
1929
).
34.
W. M.
Latimer
and
W. H.
Rodebush
,
J. Am. Chem. Soc.
42
,
1419
(
1920
).
35.
L.
Pauling
, in
The Nature of the Chemical Bond
, 3rd ed. (
Cornell University Press
,
Ithaca, New York
,
1960
), p.
450
.
36.
T. S.
Moore
and
T. F.
Winmill
,
J. Chem. Soc.
101
,
1635
(
1912
).
37.
M. L.
Huggins
,
Angew. Chem., Int. Ed. Engl.
10
,
147
(
1971
).
38.
M. L.
Huggins
, Thesis in Advanced Inorganic Chemistry Course,
University of California
,
1919
.
39.
40.
M. L.
Huggins
,
J. Phys. Chem.
26
,
601
(
1922
).
41.
M. L.
Huggins
,
Phys. Rev.
19
,
346
(
1922
).
42.
P. J. W.
Debye
,
Ann. Phys.
351
,
809
(
1915
).
43.
P. J. W.
Debye
and
P.
Scherrer
,
Phys. Z.
17
,
277
(
1916
).
44.
W. H.
Keesom
and
J.
de Smedt
,
Proc. Sect. Sci. K. Akad. Wet. Amsterdam
25
,
118
(
1922
).
45.
F.
Zernike
and
J. A.
Prins
,
Z. Phys. A: Hadrons Nucl.
41
,
184
(
1927
).
46.
P.
Debye
and
H.
Menke
,
Phys. Z.
31
,
797
(
1930
).
47.
H. H.
Meyer
,
Ann. Phys.
397
,
701
(
1930
).
48.
G. W.
Stewart
,
Phys. Rev.
37
,
9
(
1931
).
49.
E.
Amaldi
,
Phys. Z.
32
,
914
(
1931
).
50.
J. D.
Bernal
and
R. H.
Fowler
,
J. Chem. Phys.
1
,
515
(
1933
).
51.
R.
Mecke
and
W.
Baumann
,
Phys. Z.
33
,
833
(
1932
).
52.

It is unclear where this curve for irregular close packing came from as his model of a simple liquid as an irregular close packing of spheres was not put forward until many years later. The fact that he did produce a radial distribution function for a close-packed structure (Fig. 3 of Ref. 50) might suggest that these ideas were in his mind at the time—much earlier than the literature suggests. They were to be developed more fully later and would eventually feed back into his later ideas of the structure of liquid water itself (see later discussion). However, his comments in Ref. 56, page 300, suggest he may have constructed his close-packed radial distribution based on the spacings found in a close-packed crystal.

53.

Though Bernal refers to the x-ray measurements of Meyer, Stewart, and Amaldi (Refs. 47–49), he does not say which experimental data he used. However, in a paper the same year (Ref. 54), he shows a figure similar to Fig. 5 (qv. below) in which the experimental x-ray data are that of Amaldi (Ref. 49).

54.
R. H.
Fowler
and
J. D.
Bernal
,
Trans. Faraday Soc.
29
,
1049
(
1933
).
55.
J. D.
Bernal
,
Trans. Faraday Soc.
33
,
27
(
1937
).
56.
J. D.
Bernal
,
Proc. R. Soc. London, Ser. A
280
,
299
(
1964
).
57.
H. J. C.
Berendsen
,
J. P. M.
Postma
,
W. F.
van Gunsteren
, and
J.
Hermans
, in
Intermolecular Forces
, edited by
B.
Pullman
(
Reidel
,
Dordrecht
,
1981
), p.
331
.
58.
W. L.
Jorgensen
,
J. Am. Chem. Soc.
103
,
335
(
1981
).
59.
J. R.
Reimers
,
R. O.
Watts
, and
M. L.
Klein
,
Chem. Phys.
64
,
95
(
1982
).
60.
H. S.
Frank
and
W. Y.
Wen
,
Discuss. Faraday Soc.
24
,
133
(
1957
).
61.
L.
Hall
,
Phys. Rev.
73
,
775
(
1948
).
62.
K.
Grjotheim
,
J.
Krogh-Moe
et al,
Acta Chem. Scand.
8
,
1193
(
1954
).
63.
G.
Wada
,
Bull. Chem. Soc. Jpn.
34
,
955
(
1961
).
64.
C. M.
Davis
, Jr.
and
T. A.
Litovitz
,
J. Chem. Phys.
42
,
2563
(
1965
).
65.
L.
Pauling
, in
Hydrogen Bonding
, edited by
D.
Hadzi
(
Pergamon
,
London
,
1959
).
66.
H. S.
Frank
and
A. S.
Quist
,
J. Chem. Phys.
34
,
604
(
1961
).
67.
R. P.
Marchi
and
H.
Eyring
,
J. Phys. Chem.
68
,
221
(
1964
).
68.
V.
Vand
and
W. A.
Senior
,
J. Chem. Phys.
43
,
1878
(
1965
).
69.
G.
Nemethy
and
H.
Scheraga
,
J. Chem. Phys.
36
,
3382
(
1962
).
70.
G.
Nemethy
and
H.
Scheraga
,
J. Chem. Phys.
41
,
680
(
1964
).
71.
G. E.
Walrafen
,
J. Chem. Phys.
47
,
114
(
1967
).
72.
G. H.
Haggis
,
J. B.
Hasted
, and
T. J.
Buchanan
,
J. Chem. Phys.
20
,
1452
(
1952
).
73.
M.
Falk
and
T. A.
Ford
,
Can. J. Chem.
44
,
1699
(
1966
).
74.
D.
Eisenberg
and
W.
Kauzmann
,
The Structure and Properties of Water
(
Oxford University Press
,
London
,
1969
), The predictions of broken hydrogen bond models are discussed on pages 176–179.
75.
G. E.
Walrafen
,
J. Chem. Phys.
44
,
1546
(
1966
).
76.
T. T.
Wall
and
D. F.
Hornig
,
J. Chem. Phys.
43
,
2079
(
1965
).
77.
J. D.
Bernal
, in
Proceedings of the First International Symposium on Water Desalination
(
U.S. Department of the Interior
,
Washington, D.C.
,
1965
),
Vol. 1
, p.
371
.
78.
J. A.
Pople
,
Proc. R. Soc. A
205
,
163
(
1951
).
79.
J.
Shropshire
,
P. P.
Keat
, and
P. A.
Vaughan
,
Z. Kristallogr.
112
,
409
(
1959
).
80.
S.
Katzoff
,
J. Chem. Phys.
2
,
841
(
1934
).
81.
B. E.
Warren
,
J. Appl. Phys.
8
,
645
(
1937
), In the context of the idea of a liquid as a distorted crystal, it is interesting to note Warren’s comment in this paper on the early attempts to interpret liquid diffraction patterns in terms of Bragg reflections from planar layers as in crystalline structures: “…there is no excuse for calling a material crystalline when one is merely trying to express the idea that it has roughly the same coordination number, or interatomic distances, or density as found in the crystalline form.”
82.
J.
Morgan
and
B. E.
Warren
,
J. Chem. Phys.
6
,
666
(
1938
).
83.
A. H.
Narten
,
M. D.
Danford
, and
H. A.
Levy
,
Discuss. Faraday Soc.
43
,
97
(
1967
).
84.
A. H.
Narten
, personal communication,
1969
.
85.
J. L.
Finney
,
Proc. R. Soc. A
319
,
495
(
1970
).
86.
J. L.
Finney
and
L. V.
Woodcock
,
J. Phys.: Condens. Matter
26
,
463102
(
2014
).
87.
For a discussion of the history of random packing as a model of non-crystalline systems, see
J. L.
Finney
,
Philos. Mag.
93
,
3940
(
2013
).
88.
J. A.
Barker
and
R. O.
Watts
,
Chem. Phys. Lett.
3
,
144
(
1969
).
89.
J. S.
Rowlinson
,
Trans. Faraday Soc.
47
,
120
(
1951
).
90.
A.
Ben-Nairn
and
F. H.
Stillinger
, “
Aspects of the statistical mechanical theory of water
,” in
Structure and Transport Processes in Water and Aqueous Solutions
, edited by
R. A.
Horne
(
Wiley-Interscience
,
New York
,
1972
).
91.
A.
Rahman
and
F. H.
Stillinger
,
J. Chem. Phys.
55
,
3336
(
1971
).
92.
H.
Engelhardt
and
B.
Kamb
,
J. Chem. Phys.
75
,
5887
(
1981
).
93.
B.
Kamb
,
A.
Prakash
, and
C.
Knobler
,
Acta Crystallogr.
22
,
706
(
1967
).
94.
F. H.
Stillinger
and
A.
Rahman
,
J. Chem. Phys.
57
,
1281
(
1972
).
95.
F. H.
Stillinger
and
A.
Rahman
,
J. Chem. Phys.
60
,
1545
(
1974
).
96.
W. L.
Jorgensen
,
J.
Chandrasekhar
,
J. D.
Madura
,
R. W.
Impey
, and
M. L.
Klein
,
J. Chem. Phys.
79
,
926
(
1983
).
97.
P.
Barnes
,
J. L.
Finney
,
J. D.
Nicholas
, and
J. E.
Quinn
,
Nature
282
,
459
(
1979
).
98.
H. J. C.
Berendsen
,
J. R.
Grigera
, and
T. P.
Straatsma
,
J. Phys. Chem.
91
,
6269
(
1987
).
99.
O.
Matsuoka
,
E.
Clementi
, and
M.
Yoshimine
,
J. Chem. Phys.
64
,
1351
(
1976
).
100.
B. N.
Brockhouse
,
Il Nuovo Cimento
9
(
S1
),
45
(
1958
).
101.
D. I.
Page
,
Discuss. Faraday Soc.
43
,
130
(
1967
).
102.
M.
Forte
and
S.
Menardi
,
Il Nuovo Cimento B
46
,
7
(
1966
).
103.
A.
Wu
,
E.
Whalley
, and
G.
Dolling
,
Mol. Phys.
47
,
603
(
1982
).
104.
P.
Postorino
et al,
Nature
366
,
668
(
1993
).
105.
M. C.
Bellissent-Funel
and
L.
Bosio
,
J. Chem. Phys.
102
,
3727
(
1995
).
106.
T.
Strässle
et al,
Phys. Rev. Lett.
96
,
067801
(
2006
).
107.
A. K.
Soper
,
F.
Bruni
, and
M. A.
Ricci
,
J. Chem. Phys.
106
,
247
(
1997
).
108.
G.
Placzek
,
Phys. Rev.
86
,
377
(
1952
).
109.
J. E.
Enderby
,
D. M.
North
, and
P. A.
Egelstaff
,
Philos. Mag.
14
,
961
(
1966
).
110.
A. K.
Soper
and
R. N.
Silver
,
Phys. Rev. Lett.
49
,
471
(
1982
).
111.
A. K.
Soper
,
Chem. Phys.
88
,
187
(
1984
).
112.
R. L.
McGreevy
,
J. Phys.: Condens. Matter
13
,
R877
(
2001
).
113.
R. L.
McGreevy
,
Nucl. Instrum. Methods Phys. Res., Sect. A
354
,
1
(
1995
).
114.
A. K.
Soper
,
Chem. Phys.
202
,
295
(
1996
).
115.
A. K.
Soper
,
J. Phys.: Condens. Matter
19
,
335206
(
2007
).
116.
A. K.
Soper
,
Int. Scholarly Res. Not.
2013
,
279463
.
117.
A. K.
Soper
,
Chem. Phys.
258
,
121
(
2000
).
118.
A. K.
Soper
and
M. A.
Ricci
,
Phys. Rev. Lett.
84
,
2881
(
2000
).
119.
A. K.
Soper
,
Pure Appl. Chem.
82
,
1855
(
2010
).
120.
A.
Botti
,
F.
Bruni
,
M. A.
Ricci
, and
A. K.
Soper
,
J. Chem. Phys.
109
,
3180
(
1998
).
121.
A.
Botti
,
F.
Bruni
,
A.
Isopo
,
M. A.
Ricci
, and
A. K.
Soper
,
J. Chem. Phys.
117
,
6196
(
2002
).
122.
L. B.
Skinner
et al,
J. Chem. Phys.
138
,
074506
(
2013
).
123.
L. B.
Skinner
et al,
J. Chem. Phys.
141
,
214507
(
2014
).
124.
D.
Schlesinger
et al,
J. Chem. Phys.
145
,
084503
(
2016
).
125.
I. M.
Svishchev
and
P. G.
Kusalik
,
Chem. Phys. Lett.
239
,
349
(
1995
).
126.
R. F. W.
Bader
and
G. A.
Jones
,
Can. J. Chem.
41
,
586
(
1963
).
127.
G. H. F.
Diercksen
,
Theor. Chim. Acta
21
,
335
(
1971
).
128.
J. L.
Finney
,
A.
Hallbrucker
,
I.
Kohl
,
A. K.
Soper
, and
D. T.
Bowron
,
Phys. Rev. Lett.
88
,
225503
(
2002
).
129.
G. N. I.
Clark
,
G. L.
Hura
,
J.
Teixeira
,
A. K.
Soper
, and
T.
Head-Gordon
,
Proc. Natl. Acad. Sci. U. S. A.
107
,
14003
(
2010
).
130.
F.
Huang
et al,
Proc. Natl. Acad. Sci. U. S. A.
106
,
15214
(
2009
).
131.
V.
Holten
and
M. A.
Anisimov
,
Sci. Rep.
2
,
713
(
2012
).
132.
J.
Russo
and
H.
Tanaka
,
Nat. Commun.
5
,
3556
(
2014
).
133.
P.
Wernet
et al,
Science
304
,
995
(
2004
).
134.
G.
Robinson
,
C.
Cho
, and
J.
Urquidi
,
J. Chem. Phys.
111
,
698
(
1999
).
135.
F.
Rull
,
Pure Appl. Chem.
74
,
1859
(
2002
).
136.
P. G.
Debenedetti
and
H.
Stanley
,
Phys. Today
56
(
6
),
40
(
2003
).
137.
J. R.
Errington
and
P. G.
Debenedetti
,
Nature
409
,
318
(
2001
).
138.
P. H.
Poole
,
F.
Sciortino
,
U.
Essmann
, and
H.
Stanley
,
Nature
360
,
324
(
1992
).
139.
R.
Speedy
and
C.
Angell
,
J. Chem. Phys.
65
,
851
(
1976
).
140.
R. J.
Speedy
,
J. Phys. Chem.
86
,
982
(
1982
).
141.
P. H.
Poole
,
F.
Sciortino
,
T.
Grande
,
H. E.
Stanley
, and
C. A.
Angell
,
Phys. Rev. Lett.
73
,
1632
(
1994
).
142.
S.
Sastry
,
P. G.
Debenedetti
,
F.
Sciortino
, and
H. E.
Stanley
,
Phys. Rev. E
53
,
6144
(
1996
).
143.
F.
Perakis
et al,
Proc. Nat. Acad. Sci. U. S. A.
114
,
8193
(
2017
).
144.
C.
Huang
et al,
J. Chem. Phys.
133
,
134504
(
2010
).
145.
J. A.
Sellberg
,
C.
Huang
,
T.
McQueen
et al,
Nature
510
,
381
(
2014
).
146.
K. H.
Kim
et al,
Science
358
,
1589
(
2017
).
147.
K. H.
Kim
et al,
Science
370
,
978
(
2020
).
148.
J.
Bachler
, Doctor of Science Dissertation (
University of Innsbruck
,
2023
), Chap. 2.6.
149.
A. K.
Soper
,
J. Chem. Phys.
150
,
234503
(
2019
).
150.
A. K.
Soper
,
Nat. Mater.
13
,
671
(
2014
).
151.
C. A.
Angell
,
Nat. Mater.
13
,
673
(
2014
).
152.
P.
Gallo
,
K.
Amann-Winkel
,
C. A.
Angell
et al,
Chem. Rev.
116
,
7463
(
2016
).
153.
P. M.
Piaggi
,
T. E.
Gartner
III
,
R.
Car
, and
P. G.
Debenedetti
,
J. Chem. Phys.
159
,
054502
(
2023
).
154.
P.
Gallo
,
J.
Bachler
,
L. E.
Bove
et al,
Eur. Phys. J. E
44
,
143
(
2021
).
155.
J. C.
Palmer
,
P. H.
Poole
,
F.
Sciortino
, and
P. G.
Debenedetti
,
Chem. Rev.
118
,
9129
(
2018
).
156.
P. G.
Debenedetti
,
F.
Sciortino
, and
G. H.
Zerze
,
Science
369
,
289
(
2020
).
157.
A.
Nilsson
and
L. G. M.
Pettersson
,
Nat. Commun.
6
,
8998
(
2015
).
158.
J.
Niskanen
et al,
Proc. Natl. Acad. Sci. U. S. A.
116
,
4058
(
2019
).
159.
L. G. M.
Pettersson
,
Y.
Harada
, and
A.
Nilsson
,
Proc. Natl. Acad. Sci. U. S. A.
116
,
17156
(
2019
).
160.
P.
Geissler
,
J. Am. Chem. Soc.
127
,
14930
(
2005
).
161.
J. D.
Smith
et al,
Proc. Natl. Acad. Sci. U. S. A.
102
,
14171
(
2005
).
162.
A. K.
Soper
,
Mol. Phys.
106
,
2053
(
2008
).
163.
J.
Niskanen
et al,
Proc. Natl. Acad. Sci. U. S. A.
116
,
17158
(
2019
).
164.
P. C.
Hemmer
and
G.
Stell
,
Phys. Rev. Lett.
24
,
1284
(
1970
).
165.
Y.
Katayama
et al,
Nature
403
,
170
(
2000
).
166.
L.
Henry
et al,
Nature
584
,
382
(
2020
).
167.
J. L.
Finney
,
Struct. Chem.
13
,
231
(
2002
).
168.
J. E.
Enderby
and
G. W.
Neilson
, in
Water: A Comprehensive Treatise
, edited by
F.
Franks
(
Plenum
,
New York
,
1979
), Vol.
6
, p.
1
.
169.
S.
Dixit
,
A. K.
Soper
,
J. L.
Finney
, and
J.
Crain
,
Europhys. Lett.
59
,
377
(
2002
).
170.
H. S.
Frank
and
M. W.
Evans
,
J. Chem. Phys.
13
,
507
(
1945
).
171.
A. K.
Soper
and
J. L.
Finney
,
Phys. Rev. Lett.
71
,
4346
(
1993
).
172.
A. K.
Soper
,
L.
Dougan
,
J.
Crain
, and
J. L.
Finney
,
J. Phys. Chem. B
110
,
3472
(
2006
).
173.
D. T.
Bowron
and
J. L.
Finney
,
J. Phys. Chem. B
111
,
9838
(
2007
).
174.
D. T.
Bowron
and
J. L.
Finney
,
J. Chem. Phys.
118
,
8357
(
2003
).
175.
S. E.
McLain
,
A. K.
Soper
, and
A.
Watts
,
Eur. Biophys. J.
37
,
647
(
2008
).
176.
E. C.
Hulme
,
A. K.
Soper
,
S. E.
McLain
, and
J. L.
Finney
,
Biophys. J.
91
,
2371
(
2006
).
177.
D. T.
Bowron
,
A. K.
Soper
, and
J. L.
Finney
,
J. Chem. Phys.
114
,
6203
(
2001
).
178.
A.
Angell
,
W. J.
Sichina
, and
M.
Oguni
,
J. Phys. Chem.
86
,
998
(
1982
).
179.
P.
Johari
,
G.
Fleissner
,
A.
Hallbrucker
, and
E.
Mayer
,
J. Phys. Chem.
98
,
4719
(
1994
).
180.
E.
Tombari
,
C.
Ferrari
, and
G.
Salvetti
,
Chem. Phys. Lett.
300
,
749
(
1999
).
181.
K.
Sanderson
,
New Sci.
238
(
3180
),
26
(
2018
).
182.
R.
Brazil
,
Chemistry World
(
Royal Society of Chemistry
,
Cambridge, UK
,
2020
), p.
26
, https://www.chemistryworld.com/features/the-weirdness-of-water/4011260.article.
183.
From Semiconductors to Proteins: Beyond the Average Structure
, edited by
S. F.
Billinge
and
M. F.
Thorpe
(
Kluwer Academic/Plenum
,
Dordrect
,
2002
).
184.
See https://www.isis.stfc.ac.uk/Pages/Nimrod.aspx for more information about the instrument specification and some science highlights.
185.
K. J.
Edler
and
D. T.
Bowron
,
Curr. Opin. Colloid Interface Sci.
20
,
227
(
2015
).
186.
D. T.
Bowron
and
K. J.
Edler
,
Langmuir
33
,
262
(
2017
).
187.
Published open access through an agreement with JISC Collections