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Water's Two-State Cluster History

'Because of the importance of water in the theoretical speculations as well as the practical routine of chemists,

a review of the theories concerning its structure is pertinent.'

Harris Marshal Chadwell 1927    


V Two state theories

V Are the icosahedral clustering model and the outer structure two-state mixture models related?

Two-state theories

The anomalous properties of water have excited chemists and physicists in their attempts to find explanations for many years. Frank concluded in 1970 that physicists were mostly drawn to 'uniformist' models in which all molecules must be regarded as equivalent whereas chemists generally preferred 'mixture' models [767j]. This conflict has continued to the present day with entrenched positions on both sides. However, a slow conversion towards the acceptance of the mixture models is becoming increasingly noticeable of late. The mixture hypotheses that have put forward have mainly involved consideration of the existence of different water clusters within liquid water [205c, 765, 767g, 767i, 767k, 992]. Generally, two states are hypothesized although more states may be used. Such clusters are either low-density (similar in density to that of hexagonal ice), low energy or of higher density, high energy; their changing relative concentrations determining the changes in the physical properties. In order to fit the anomalous properties of water, these structural forms must exhibit a relatively large difference in volume between the low-density open and the high-density close-packed structures. Thus the properties of water are shown to follow those of a more typical liquid at higher temperatures (> ~50 °C) based on the higher density form and follow those of a very different liquid at lower temperatures (~ 0 °C) based on the lower density form, with about equal amounts of the two forms present at about 0 °C. Such two-state theories have been proposed for well over 100 years Throughout that time they have been successfully used to explain the anomalous properties of liquid water. The main steps in the progress of two-state proposals for water are given below.

Ice Ih cell cluster

Low-density ice-like particles

 Dense liquid water

Higher-density liquid matrix


The first reported suggestion for clusters being responsible for water's anomalous density maximum was by Whiting in 1884 and Röntgen in 1892 [764]. Melting ice was proposed to release solid low-density ice crystals (left) which remained within the higher density matrix (right). Whiting proposed that the concentrations of these solid particles changed with temperature and pressure. Chadwell [765] reviewed the first 40 years in the development of this idea.


Bousfield and Lowry [767f] described water as a mixture of less-dense single-molecules (H2O, like steam) and higher-density dimers, (H2O)2 (called dihydrol) to explain water's interaction with hydroxide ions. Later, trimers (H2O)3 (called trihydrol) and ice crystallites were added to the proposed system to help describe pure water and the changes towards ice formation. Although this system was successful for explaining some properties of water, its popularity dropped when more information concerning the structuring of water became available and the additional clusters needed to better fit the density data. These extensions were described in the Meeting of the Faraday Society held in 1910 [767g].

Ice Ih cell cluster


Low-density ice-like clusters







 Walrafen pentamer;  'quartz-like' cluster


Higher-density clusters


In 1933, Bernal and Fowler developed this model in two important respects [1177]. Firstly the denser matrix was proposed to be made up of 'quartz-like' clusters (right) that may close-pack together. Secondly, and significantly, they introduced the idea of equilibration of water molecules interconverting between the clusters [766], but as the temperature is raised rather than always coexisting. The less-dense structure remained ice-like (see far left) and was only significantly present below 4 °C. Above 200 °C, the water behaved like a normal close-packed liquid.

Ice Ih cell cluster


Low-density ice-like clusters

Ice Ih cell cluster, with an interstitial water molecule


Cluster plus an interstitial molecule


An interesting idea, based on interstitial water molecules within the ice hexagonal box (right) was put forward by Samoilov in 1946 [767a] and Narten and Levy in 1967 [767d] based on their X-ray data. Such clusters are not now generally thought to be present in significant quantities. However, some researchers still make use of this hypothesis [2009], as the interstitial structure approximates to a close-packed water structure. It has 1.5 times the density of ice as there are twice as many molecules as cavities.

8-molecule open water cluster

Low-density (H2O)8, clusters

H2O, (H2O)2 and (H2O)4

High-density structures


Eucken's theory [767d] was that the molecules in liquid water form an equilibrium between monomers, dimers, tetramers, and octamers. The octamers take up a larger per-molecule volume and occur importantly in many other models including the icosahedral water cluster model. Eucken suggested
that about 30 % of the water molecules are in the octamers. This theory was simplified by Hall in 1948 [767e], where the high-density structures are less specifically defined.


Frank and Wen [97] described water as a mixture of more-dense single-molecules (H2O) and lower-density hydrogen-bonded clusters, (H2O)n. They introduced the term 'flickering structures' to describe the short-lived structures, as opposed to the more stable structures previously envisioned. This term lives on to the present day. They suggested the half-life of such clusters to be 10-10 -10-11 s, to fit with the dielectric and relaxation time of water, but long enough (102 - 103 times the molecular vibrations) to give it a meaningful existence.

Dodecahedral water cluster

Low-density clathrate clusters

Dodecahedral water cluster, with an interstitial water molecule

Cluster plus an interstitial molecule


Pauling suggested another interstitial arrangement in 1959, making use of his interest in clathrate structures [8b]. These clusters were quickly dismissed, however, for being unable to explain the available diffraction data if the main constituents in liquid water.


Wada [767h] developed a simplified Eucken model [767d] and described water as a simple two-state model that water is equilibrated between the so-called "iceberg" having locally the low-density ice structure and a more closely-packed high-density structure.

(H2O)icy =(H2O)packed

The model gave a 42% composition of the (H2O)icy state at 0 °C. It was used to explain a number of the properties of liquid water.


Nemethy and Scheraga [767g] described water as a mixture of more-dense single-molecules (H2O) and lower-density hydrogen-bonded clusters, (H2O)n. It was based on Frank and Wen's model [97] and used to help the understanding the thermodynamic properties of water.

Ice Ih cell cluster

Low-density ice-like particles

Dense water shown as 2 water tetramers

Close-packed hexagons


Davis and Litovitz proposed a variation on Whiting's hypothesis [2160]. The high-density state consists of almost close-packed water hexagon rings. The fraction of ice-like particles was about 0.6 at 0 °C.

Ice Ih cell cluster

Low-density ice-like particles

Dense liquid water


Higher-density liquid of

unbonded monomers


Arakawa and Sasaki also proposed a variation on Whiting's hypothesis [2161]. The high-density state consists of close-packed unbonded molecules. The fraction of ice-like particles was 0.52 at 0 °C.

   4-membered cyclic water cluster     5-membered cyclic water cluster    6-membered cyclic water cluster      7-membered cyclic water cluster  8-membered cyclic water cluster


A random network model of water, also introduced by Bernal [1177] and published in 1975 [19], contained a mixture of water clusters including 4-, 5-, 6-, 7- and 8-membered rings. Some success has been had using this model but its homogeneous nature is not universally applicable or productive.

Ice Ih cell cluster

An ice Ih cluster


Wilse Robinson's research group introduced the outer structure two-state mixture model in 1987, b which involved a mixture of water clusters related to ices 1h, II and III. The group went on to produce a number of papers using the model to successfully and quantitatively explain many of water's anomalies (for example, [23, 56, 57, 60, 69, 73, 148, 826, 1354]).



A cluster from ice-two


An ice II cluster

A cluster from ice-three

An ice III cluster



The difficulty with this model concerned whether such clusters could exist for significant lengths of time in the liquid water. Therefore, the clusters were more realistically considered as rapidly fluctuating 'indicative' (or representative) structures [1354c].

Dodecahedral water cluster






5-membered cyclic water cluster6-membered cyclic water cluster


In 1998 Dougherty and Howard proposed an equilibrium model for water [15] involving dodecahedra, 5- and 6-membered clusters based on several of water's anomalous properties.

Dense water shown as 2 water tetramers

High-density cluster




8-molecule open water cluster

Low-density cluster


These water cluster models lead logically on to the icosahedral water cluster model; published in 2000 [55] and described at this site. This model is based on dense and less dense clusters equivalent to an equilibrium opposite (see animated gif). The less-dense bicyclo(2,2,2) structures (right) occur when the hydrogen bonding is strongest and the dense structures (far left) occur when the weaker but more numerous van der Waals interactions






  8-molecule open water cluster


Low-density cluster









280-molecule icosahedral open water cluster

Icosahedral cluster



When conditions arise (such as on supercooling at low temperatures) when there is a high concentration of the expanded 8-membered bicyclo-clusters, (H2O)8, partial to complete icosahedral clusters (right) may arise. Partial structures may involve cyclo-fragments including (H2O)5 , (H2O)10, (H2O)12, (H2O)20, and (H2O)100. A full description of this is given elsewhere on this site. These icosahedral clusters contain the dodecahedral, ice Ih cell, 5-, 6-membered and 8-membered bicyclo(2,2,2) subclusters included in the historical survey of models, above. This model fits well with and extends the general theory of water two-state clustering.


Later publications have described two-state systems with a low-density state and a high-density state but without using explicit structures. Partially, this is due to the lack of water monomers found by using vibration spectroscopy. Thus Maréchal describes them entirely from their infrared spectra [1738] (with the low-density form having peaks at 744 cm-1, 1669 cm-1, and 3356 cm-1 whereas the high-density form possesses peaks at 518 cm-1, 1639 cm-1 and 3519 cm-1), Taschin et al describes them as tetrahedrally hydrogen bonded (associated with a Raman peak at 225 cm-1) and disordered (associated with a Raman peak at 180 cm-1) [2019], and Nilsson et al describes them as tetrahedrally hydrogen bonded (low-density liquid-like) and closely packed disordered with distorted hydrogen bonds (high-density liquid-like) [1899].


Shows how cluster lifetime is independent of hydrogen bond lifetimeThe meaning of the term 'cluster' has evolved during this time as well. In the beginning, water clusters were thought of as discrete molecular entities, like crystals. These were proposed to have long (e.g. > seconds) individual lifetimes where the same molecules were involved, within the cluster, throughout the cluster lifetime. Nowadays, we know that molecules may leave or add to clusters a with frequencies that depend on their situation and obey statistical laws with clusters appearing, evolving, and disappearing with the involvement of physically different molecular constituents (that is, the water molecules come and go, see the cartoon, right). The aqueous environment is heterogeneous with more than one type of environment present and with the relative preponderance of these environments changing with temperature, pressure, solutes, and surfaces. Clusters are now thought of as dynamic entities offering a simplified view into a complex, broken and rapidly shifting environment. As such they reveal water's underlying, if elusive, nature.


a   Some authors prefer the term 'dynamic heterogeneities' to 'clusters', but this site does not. The same type of structuring is meant in both cases although 'dynamic heterogeneities' may include the scenario where there is a gradual change in cluster type with temperature (the type being a single cluster arrangement under a set of conditions) whereas 'cluster' is more easily understood in two state systems as a mixture of cluster arrangements in equilibrium with each other). [Back]

b Are the icosahedral clustering model and the outer structure two-state mixture models related?

The outer structure two-state mixture model has been successfully used to explain many of the anomalous properties of water (for example, [23, 56, 57, 60, 69, 73, 148, 826, 1354]). Although this model explicitly includes ice Ih (hexagonal ice) and ice-two (ice II) substructures, these may be thought of as representing fully tetrahedral open low-density structures (ice Ih equivalent in the icosahedral network model to ES) and tetrahedral

14-molecule tetrahedron of water molecules structures with close second neighbors, giving a higher density 'collapsed' structure (ice-two equivalent in the icosahedral network model to CS). This equivalence can best be envisioned by examining the behavior of a 'c'-type water molecule. They are the most numerous water molecules in icosahedral water clusters (120/280 = 43%) and they are the least affected by the movements due to the ESreversible arrowCS equilibrium. These water molecules are tetrahedrally hydrogen-bonded to the surrounding four (2 'b' and 2 'c' type) water molecules in both ES and CS. In ES the next 'outer' structure (including 2 'a' type) water molecules are also tetrahedrally positioned but in CS the two 'a' type water molecules in this 'outer' structure are flexible and collapsed, allowing close second neighbors. It may be noted that the outer structure two-state mixture model does not depend on the explicit structures of ice Ih and ice-two in order to explain water's anomalous properties; in fact, the existence of such explicit structures is unlikely as shown by many of the other properties of water (for example, see earlier). This means that arguments using this model can also be used in support of the icosahedral cluster model (but not explicitly vice versa). [Back]




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