Encyclopedia of Environmental Science and Engineering, Volume I and II

1294 WATER: PROPERTIES, STRUCTURE, AND OCCURRENCE IN NATURE

The model must also be consistent with the interpreta-

tions from infrared, nuclear magnetic resonance, Raman,

and X-ray spectra of liquid water.

Eisenberg has outlined the sequence by which most of

the current theories have been developed. Experimental evi-

dence and some intuition support the postulation of a model

after which it is translated into mathematical terms, often by

expressing as a simple partition function containing several

variable parameters. The number of adjustable parameters has

typically ranged from 9 to 18. The thermodynamic expres-

sions derived from the partition function are fitted to the

empirical measurements by varying the parameters. That the

model furnishes a good correlation is not necessarily proof of

its validity, due to the inductive nature of the method.

The different water models may be roughly divided into

two classes: the continuum models (also called homogeneous

or uniform models) and the mixture models (also called solu-

tion models). Models in both classes achieve partial success

in explaining the properties of water.

The models of People and Frank—Wen, representative of

the continuum and the mixture classes, respectively, are sum-

marized here. Eisenberg and Kauzmann give a comprehensive

review of the various models.

The People Distorted Bond Model

The continuum theories ascribe essentially complete hydro-

gen bonding to water, at least at low temperature. Associated

with the hydrogen bonds is a distribution of angles, distance,

and energies. The average bond energy varies with tempera-

ture due to the changes in the distributions of bond lengths

and bond angle distortions.

People’s model (1951) postulates that when ice melts,

the hydrogen bonds become more flexible and disorder is

introduced into the structure by the bending and stretching

of the bonds rather than by breaking them. Thus the water

molecules in the liquid phase, like those in the ice, retain a

coordination number of four. All the molecules have essen-

tially the same environments, thereby giving rise to the

continuum. The model allows for a continuous spectrum

of induced hydrogen bond disorder. This model is strongly

supported by Raman and infrared spectral studies of liquid

water. As Fletcher (1970) points out, “the uniform model

of Pople stands or falls on the energy required to bend the

hydrogen bond joining two molecules which are already

participating in hydrogen bonds to other molecules.” Pople

and Lennard-Jones propounded that a significant percentage

of broken hydrogen bonds in liquid water in thermodynami-

cally impossible since the hydrogen bond energy between

two water molecules is 4.5kcalmole whereas the latent hat

of fusion is 1.4kcalmole and the average thermal energy,

RT, 0 C is 0.5 kcalmole.

The Frank–Wen Flickering Cluster Model

The flickering cluster model described water as a mixture of

hydrogen bonded clusters surrounded by monomeric water.

Frank and Wen (1957) treat the formation of hydrogen bonds

in water as a cooperative phenomenon, with a sizeable cova-

lent contribution to the energy of the bond. The existence of

a hydrogen bond on one of the hydrogen atoms of water pro-

motes the tendency of the other hydrogen to form a similar

bond to another neighboring molecule. According to Krindel

and Eliezer (1971), “Frank and Wen considered that hydro-

gen bonded molecules are mutually polarized, the resulting

charge separation being such that the free ends of the associ-

ated molecules have a higher affinity for further hydrogen

bonding than the ends of non-hydrogen bonded molecules.”

This susceptibility to further hydrogen bonding gives the

process an element of positive feedback until the disag-

gregating thermal energies from the additional molecules

supply negative compensation. Thus the formation of one

TABLE 2

Structural characteristics of ice polymorphs a

Ice I Ic II III V VI VII VIII

Number of nearest

neighbors

4444 4 48b 8 b

Distance of nearest

neighbors (Å)

2.74 2.75c 2.752.84 2.762.80 2.762.87 2.81 2.86d 2.86d

Distance of closest

non-H-bonded

neighbour (Å)

4.49 4.50c 3.24 3.47 3.28, 3.46 3.51 2.86d 2.86d

O–O–O angles (deg) 109.5±0.2 109.5 80  128  87  141  84  135  76  128  109.5 109.5

Hydrogen positions Disordered Disordered Ordered Disordered

above  40 C

Disordered Disordered Disordered Ordered

a Entries, except where noted, refer to  175 C and 1 atm pressure. Data for ices I and Ic are from Lonsdale (1958); data for the high-pressure polymorphs

are from Kamb and Datta (1960), Kamb (1964), Kamb (1965 a, b), Kamb (1967), and Kamb et al. (1967).

b 4 are hydrogen-bonded to central molecule.

c At  130 C

d At 25 kbar. In quenched ice VII at atmospheric pressure the nearest-neighbor distance is 2.95 Å (Bertie et al.,1964)

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