STRUCTURE AND PROPERTIES OF WATER D. J. G. Ives, D.Sc., A.R.C.S., F.R.I.C. Birkbeck College, Malet St., London W C I and T. H. Lemon, B.Sc., M.Phil. Post Office Research Station, Brook Rd., Dollis Hill, London N W 2 Introduction Bases of comparison A general view of related hydrides, 65 General comment on solid-liquid-gas transitions, 66 A structural theory of liquids, 68 Liquids over a wide temperature range, 71 . . .. . . . . . . .. . . . . . . . . .. The abnormal hydrides . . .. . . . . . . The hydrogen bond Infrared and Raman spectra Theories of the structure of water . . . . . . .. . . . . .. .. .. .. .. .. The water molecule, 87 Ice, 87 x-Ray scattering, 9 1 The Bernal and Fowler theory, 91 The distorted bond model, 92 Interstitial models, 92 Cluster models, 94 Thermal anomalies Hydrophobic bonding and co-solvent behaviour * ... .. .. * . . . . . 100 Conclusion . . .. .. .. .. .. .. .. . . 102 Water is familiar to us from our earliest years. We get it by turning a tap, and what could be less remarkable? We see that ice floats on water, and we are shown at school that water has a maximum density at 4°C. We may be told that these are happy circumstances because, if ice did not act as an insulating upper layer and, if slightly warm water did not remain at the bottom, lakes and rivers might freeze solid, with fatal effects to fish. Mildly interesting this, but natural, i.e. ‘according to, or provided by, nature’.l We are not by experience inclined to think of water as anything but normal, and there is certainly a lot more of it than of any other liquid.Little may happen during a chemist’s education to disturb this phlegmatic attitude to water. It was the solvent predominantly used in the early develop- ment of the physical chemistry of solutions (with, oddly, sucrose as the most normal of solutes). It is, no doubt, gradually borne in upon us as students that water has outstandingly convenient properties. It is fortunate that it can remain in the liquid state over a curiously wide and high range of tempera- 62 62 65 . . .. .. .. .. .. .. . . .. 75 .. 81 . . 78 .. 86 .. 99 .. .. RIC Reviews tures, and that it is the nearest approach we have to the universal solvent- especially that it is an ‘ionizing solvent’, ostensibly because of its high dielec- tric constant.It is also fortunate that water has a broad domain of thermodynamic stability2 and can participate in acid-base equilibria over a range of 16 pH units, and sustain redox equilibria over a potential range of more than two volts. A great deal of chemistry depends on these providential properties of water, but it is the chemistry rather than the water which receives attention. There is little change in emphasis when we come to more recent fundamental studies of equilibria and reaction rates in aqueous solutions, or even to work leading to present knowledge of electrolytic solution^.^ Interionic attraction theory has reached a sophisticated level by treating the solvent-water or another-as if it were merely a ‘dielectric continuum’.The physico-organic chemists have founded an unassailable structure of theoretical organic chemistry very largely on studies of reactions in solution.4 They have used a variety of solvents, including water and aqueous-organic mixtures, and have recognized the profound importance of solvent effects.5 Until quite recently,6 however, they have not been forced to give detailed consideration to the problems of solvent structure and of solvation pheno- mena in their own right. It is as if there were some general, simplifying prin- ciple which has shielded chemists from the effects of the more complex phenomena of nature, allowing them often to get away with treating solvents just as media, of status little more than that of reaction vessel.It will be suggested later that there is indeed such a principle. In these circumstances, students are not to blame if they graduate knowing no more about water than that it is an abnormal, hydrogen-bonded liquid. Further, the growth of science imposes a progressive squeeze on syllabuses, leaving little time for overly-debatable topics. Not even normal liquids are well understood, so water, as a mystery within a mystery, gets little attention. The same applies to the hydration of ions-a messy and contentious subject, fortunately not obtrusive-and still more to the hydration of molecules. The study of binary liquid systems is an associated field, formally intimidating and difficult of interpretation without exhaustive mustering of diverse lines of evidence,7 so usually this also gets short shrift.If it is agreed that solutions are important-aqueous solutions above all others-must it not be admitted that solute-solute, solute-solvent and solvent- solvent interactions should all be studied in a reasonably balanced way? Would it be an overstatement to say that, in general and for many years past, only the first and least important of these interactions has been studied seriously by the average student ? Activity coefficients, Debye-Huckel, and all that-thermodynamic formalism and a theory of simple electrolytes admittedly based on a grossly oversimplified model. Perhaps the progress of physical chemistry in a vitally important direction has been retarded by the constraints and limitations of undergraduate courses-students can hardly become interested in a subject if they barely know of its existence.Water is the keystone of this argument, because of its ubiquitous and supremely important role in natural processes. All living structures are composed mostly of water. Many lines of enquiry are now converging on Zves and Lemon 63 5 0- Y n 0 60, ,3 49 - 1 44 5.58 I O! TM 600 500 - 400 + 2 0 300 F E 2oc I oc 0.43 C 3 Ne L 0 079 the problem of the structure of water, how this structure varies with tempera- ature and is modified in the vicinity of charged and uncharged solute particles and of charged and uncharged interfaces, and how water may play a specific part in life-processes.Over the last several years there has been rapidly growing awareness of this situation, and at the present time there is intense activity directed to these problems; there is also as little agreement as there is much debate. The special difficulty of reviewing a subject in this state has been met by a plan discernible from the table of contents-first to establish a broad foundation to support consideration of the more special problems. It might be described as a conscious effort to avoid not being able to see the wood for the trees. If this could be well done it might turn out that much had been learned about water ‘in advance’, but whether the reviewers succeed or fail in this, the main objective may still be achieved-to demonstrate the importance, difficulty and fascination of the challenge that this subject presents.BASES OF COMPARISON Abnormality must be considered by reference to standards of normality. Water is a volatile hydride and it is a liquid. It is reasonable first to give a general view of how water is abnormal by reference to other hydrides and to other liquids, first deciding what may be considered normal for each. A general view of related hydrides Water, as a hydride, is to be compared with its congeners in the periodic table. The basis for such a comparison is provided in Fig. 1, which displays melting and boiling points, critical temperatures (TM, TB, Tc) and appropriate latent heats (AHm, AHE, kcal mole-l) for hydrides of elements in the first, second and third periods.A noble gas as ‘zeroth hydride’ is included in each of the three isoelectronic groups. There is a close resemblance between the second (Ar-SiH4) and third (Kr-GeH4) groups; they hardly differ except for an expected displacement on the temperature scale. This must be due to similarities in the cohesional forces responsible for the existence of the liquid and solid phases of the hydrides in these groups. These forces are accounted for by dispersion, dipole-dipole and dipole-induced dipole attractions*- collectively, van der Waals interactions. In general, except for highly polar molecules, the dispersion or London forces predominate; they have the important characteristic of additivity, i.e.the total interaction between a number of molecules is the sum of the interactions of all the pairs of mole- cules.* * These forces, acting between a pair of molecules, give rise to the following interaction energy in which the ternis can be easily identified : cy is polarizability, v is the zero-point frequency of oscillation responsible for the dispersion effect, p is dipole moment, r is intermolecular distance and other symbols have their usual meanings. It may be useful to note the trends of dipole moments and polarizabilities shown in Fig. 1 . Ives a d Lemon 65 It is very significant that the resemblance between these two groups of hydrides extends, broadly, to the differences between boiling points and melting points (TB - TM): to explain why, and to provide a background for discussion of the behaviour of the abnormal hydrides, calls for general comment on transitions between solid, liquid and gaseous states.9 General comment on solid-liquid-gas transitions At sufficiently low temperatures, the ordered structure of a crystalline solid is such as to minimize potential energy. With rising temperature, entropy must increase and potential energy must follow suit.The entropy requirement increasingly militates against the tendency to minimization of energy-in other words, rising temperature compels a shift of the competitive balance between the influence of order-producing forces and the randomizing effect of thermal agitation in favour of the latter. Although melting marks a defeat for the ordering forces, it may not be a major defeat, since order is relaxed in stages, of which melting is not the last,g or, in some cases, the first.Melting may be viewed thermodynamically in terms of the relation AGM = AHM - TASM representing the Gibbs free energy, enthalpy and entropy changes accompany- ing the process at temperature T. For a hypothetical solid-to-liquid transition below TM, AGM is positive-the process is unnatural and does not occur. But AHm and ASM are different functions of temperature,* and, as tempera- ture rises, TASM overtakes AHM, and brings AGM to zero at TM, where solid and liquid phases are in equilibrium. Figure 2a presents approximate data for the melting of ice. A broader view of phase transitions is given by Fig.2b, consisting of formal plots of free energy G (on no absolute scale), against temperature. The lines (for simplicity, straight) have relative slopes, (aG/aT), = - S, broadly consistent with relative entropies of solid, liquid and gaseous phases at 1 atm pressure. The intersections of the lines determine TM and TB. At some prevailing pressure much lower than 1 atm-below the triple-point pressures of normal substances-a very steep ‘gas line’ would cut the ‘solid line’ directly. A native observer on the moon might be unaware of the existence of liquids. Under more normal conditions, a ‘liquid line’ cuts off the corner between the other two, representing the fact that a liquid, with assistance from external pressure, can retain effective cohesion despite the absence of long-range order in the liquid state; the cohesion is always less, however, than that of the ordered solid state from which the liquid was derived by melting.Inspection of Fig. 2b indicates that if a liquid is of high entropy, so that the ‘liquid line’ is steeper, then, other things being equal, the intersection with the ‘gas line’ would be at a higher temperature than for a liquid of low entropy, i.e. TB would be raised. It is generally true that the necessity for jumping into the next state of greater randomness is deferred to a higher RZC Reviews Solid - Gas 1 atrn L G\ u ' - 5 - I I I I I I I I I I I I Temperature ("Cl Fig. 2. (a) Thermodynamic functions for ice-+water; (b) formal G(T) plot embracing solid- liquid-gas transitions.temperature the higher the entropy of a phase, or the greater its capacity for accommodating entropy. Similar thought can be applied to other features of the basic diagram. At the boiling point, TB, cohesion of the liquid fails, but it is to be noted that TB is not normally far ahead on the temperature scale of the intersection of the gas line and the extrapolated solid line, suggesting that cohesion in the liquid is related to, if less than, cohesion in the solid. An approximate assessment of either can be made because ASB, the large entropy increment accompanying evaporation at 1 atm pressure does not vary greatly (except in special cases) from one substance to another ; its near-constancy is expressed in Trouton's rule.Since TB = AHB/ASB for reversible evaporation, TB serves as a rough measure of the energy (- AHB) required to overcome the cohesional forces.* No similat statement can be made about TM because of the very wide variation of ASM, from one substance to another. All that can be said, in terms of TM = AHM/ASM, is that the melting point will depend on the magnitude of the cohesional forces to be overcome in melting and on the disparity of entropy between liquid and solid phases. No ad hoc conclusion can be drawn from whether TM is high or low, because four factors are involved-energies and entropies in two phases. This uncertainty is reflected in the great variation of (TB - TM) shown by substances, even those of not dissimilar TB.For instance, boron and tin have boiling points of 2823 and 2533°K respectively, but (TB - TM) values of 250 and 2028". Boron is a * It could be argued that TC would be better than TB as such a criterion. It can be replied that Tc has less fundamental significance than is usually assigned to it because there is no similar temperature known to limit the existence of solid phases, i.e. above which a tion is low and compressibility approaches infinity. Since, however, TB - +Tc (Guldberg's normally given encountered because Moreover, VC - solid 12 Nw, cannot where exist. N is Avogadro's often TC number TB S- may TB and and be w more conditions is the suited volume near to of discussion TC a are molecule, far from of liquids coordina- normal as : rule), approximate argument can be conducted on either basis.Ives and Lemon 67 hard solid built of interlinked BIZ icosahedra-a structure of great geometrical specialization formed by strong, directional bonds; tin is a metal, not close- packed in the solid state, owing its cohesion when molten to strong, non- directional metallic bonding. No doubt this is an extreme example, but it is true to say in general that TIM and (TB - TM) depend critically on the nature of structures and forces in solid and liquid states. It is therefore evident that the previously noted resemblance between the second and third groups of hydrides (Ar-SiH4, Kr-GeH4) in Fig.1 is more remarkable than was at first apparent. This may strengthen the justi- fication for using their common pattern of behaviour to assess abnormalities in the first group, but, before making such comparisons, it is well to reflect that there is nothing absolute about such a criterion. The considerable variation of (TB - TM) between the members of the ‘normal’ groups suggests that they have their own structural problems. Since interest centres on the liquid phase, there is a need to define a normal liquid and to know something about its behaviour, particularly from the structural angle. It would hardly be sensible to approach the problem of the peculiarities of water as a liquid without attempting to satisfy this need.A structural theory of liquids Comparatively recently it was said that ‘we are completely in the dark as to the degree of order in a liquid’.lO This is disconcerting, since the problem is of such wide interest and also because the generally high specific heats of liquids suggest that they do have considerable order to lose and entropy to gain. In these circumstances, it is reasonable to turn in the first instance to the noble gases. Their monatomic molecules attract each other only by short- range, non-directional dispersion forces. Within close limits they have the same reduced melting point TM/Tc (0.556 & 0.002) and the same entropy of fusion, ASM (3.29 & 0.05 cal OK-lmole-1). This uniformity in properties (which could be illustrated further) indicates that they are likely to form the simplest kind of liquid phase, suitable as a standard of normal behaviour.ll Considering the noble gases together, we note that low TB and AH, (Fig. 1) denote feeble cohesion.But (TB - TM) is small so that, relatively, TM is high. Noting that triple-point pressures are high (Ar, 0.68 atm), that RIC Reviews TM is proportional to AHA[ because ASM is constant and that AH, and AHB vary with atomic number in a manner to be expected from dispersion forces alone (Eqn. (1); Fig. l), we see that ‘high TM’ is not due to order- promoting, stabilizing forces of attraction characteristic of the solid state. On the contrary, the solid state is such as to be able to gain entropy with the least restriction from such forces.The noble gas elements form close-packed solid phases, each atom having 12 equivalent nearest neighbours: this is a structure determined by forces of repulsion rather than attraction, i.e. the simplest regular structure to be expected from the most space-economizing packing together of hard spheres. When the solid noble gas elements melt, they expand by 15 per cent-an additional uniformity in behaviour which suggests that they provide us also with the simplest melting process, adequately defined as a transition from regular to irregular closest packing. This occurs 68 when the constancy in number, and the symmetry in arrangement of nearest neighbours essential to preservation of long-range order can no longer be sustained against the increasing attack of thermal agitation.This is essentially the picture used by Bernal in his ‘polyhedral hole theory’ of these simple liquids,l2 which is considered to present a new structural viewpoint basic to the understanding of liquids in general. In this theory, Bernal views a liquid neither as a ‘blurred solid’ nor as a condensed gas, but directly, in terms of how a homogeneous, coherent, densely but irregularly packed assembly of spherical molecules can be described structurally. The essential feature of the liquid state is variation of coordination-a continuous change in identity and number of nearest neighbours on which the self-diffusion and fluidity of liquids must depend. It may seem odd to look for structure under circumstances which doubly forbid the existence of long-range order, but it is to be remembered that the yield and shear involved in fluid flow are slower by orders of magnitude than the mainly vibrational thermal motions of the molecules.On a short enough time-scale, liquids show solid-like properties (as in ultrasonic and neutron diffraction studies); at any instant they are geometric- ally, rather than physically, dissimilar to crystalline solids, and the dis- similarity is that of irregularity as compared to regularity. A liquid may be regarded as passing through a random sequence of irregular molecular arrangements evolving continuously from each other. There is a vast number of such energetically equivalent arrangements, compared with the very few regular ones, and a constant flux between them.It is the ready transition from one to another under the slightest stress which is the basis of fluidity. The question arises whether these irregular arrangements have any common structural features; whether liquids present a static as well as a dynamic problem. Bernal studied the static, structural aspect by a method briefly explicable in the following way. Consider four spheres of identical radius in mutual contact. Imaginary straight lines between their centres trace the equal edges of a regular tetra- hedron enclosing the gap or ‘hole’ between the four spheres. The hole may be called a ‘tetrahedral hole’. In any regular assembly of many such spheres each making the maximum number of contacts with others, only two kinds of ‘polyhedral hole’ are possible-tetrahedral and octahedral.Such a ‘spheres- in-contact’ method can give only a model of a regular crystal, but it may be noted that the structure can be described in terms of an assembly of poly- hedra (containing holes) just as well as in the more conventional way. In building a model of a liquid, there must be no geometrical constraint on each added sphere to occupy a position precisely defined by its previously built-in neighbours. The model must be looser to represent the higher energy, less cohesion, great volume and, above all, the irregularity of the liquid. This representation becomes possible if nearest-neighbour distances are allowed latitude to increase by up to 15 per cent above the minimum corresponding to spheres-in-contact.With no longer any repeating unit of pattern, the model must be large. Each sphere must be placed randomly within the general requirement that, when all have been placed, the best economy of Ives and Lemon 69 space in irregular packing is achieved, with the least possible stretching of nearest-neighbour distances (maximization of entropy, minimization of energy). To build such a model is an exercise in statistical geometry-ideally the model should be allowed to ‘build itself’, without human interference. The most successful attempt to solve this problem objectively was experi- mental. Conducted by Bernal and his team, it was based on a random heap of a large number of ball-bearings, consolidated by dousing with paint- subsequently allowed to drain and harden.Clearly-marked ball-to-ball contacts could ultimately be counted, and confirmed the essential variability of coordination. By an ingenious method, the spatial coordinates of each ball in the heap were determined, distances and directions were computed, and a ‘transparent’, much enlarged, ball-and-wire replica of the heap was constructed. Examination of this open model showed that it contained (apart from a few significant ‘accidents’) holes of only five polyhedral types : tetrahedron, octahedron, and three somewhat larger ‘deltahedra’ (poly- hedra with triangular faces) lacking the symmetry for regular packing. These are the only polyhedra, generated by sphere packing, small enough not to admit an extra sphere.It was shown independently that these five poly- hedra can be packed together, with but little distortion from equality of edge-length, in an indefinitely great number of space-filling ways, but the minimum volume for such packing is always 15 per cent greater than that for regular cl ose-packing. That this model successfully simulated the simplest liquid was supported by calculation of a radial distribution curve* for the model-it was in agree- ment (with appropriate change of distance scale) with that derived for liquid argon by neutron scattering. The following less-expected features of the model therefore merit close attention. The irregularity of the model was not absolute.Tetrahedral holes, which predominated, tended to occur in company, orderly arranged. Tetrahedra can pack together economically of space, forming a number of structures, some closed (e.g. 20 tetrahedra each sharing three faces, with a common vertex at the centre), others not (e.g. a ‘triple helix’ formed by face-sharing tetra- hedra). The density of packing in such local regions is higher than can be attained in regular close-packing, but all the ordered structures formed in this way must remain local, because they incorporate an element of five-fold symmetry which forbids their indefinite extension in space. Bernal calls them ‘pseudonuclei’ because they are like crystal nuclei which can never grow, and can never form the basis of long-range order.In considering whether these features of the model indeed reflect what happens in the simplest liquid, we recall the standing conflict between the interests of energy and entropy, and ask why, in the liquid, energy-minimizing forces should not promote a kind of order forbidden to the solid simply because it is of the wrong symmetry? When the solid has melted, the prohi- bition of five-fold symmetry is lifted, and local order based upon it could conceivably be more probable than any other. At least the thought should be retained that order in a liquid need not be related to that of the solid from * RadiaI distribution curves show the probability, as a function of distance from any molecule, of encountering another molecule. RIC Reviews 70 which the liquid was derived by melting.The occurrence in liquids of local regions of less than average energy and entropy is acceptable thermodynami- cally ; constant free energy accommodates such fluctuations,* which are, indeed, to be expected.13 If it is accepted that simple liquids contain evanescent, super-dense pseudo- nuclei, it must be remembered that the liquid as a whole is less dense and of higher energy than the corresponding regular solid. There must be other local regions with density lower and energy higher than the average. Reference to the model indicates a higher than average proportion of the larger, more irregular, polyhedral holes in the immediate vicinity of the pseudonuclei, i.e. the denser, more ordered regions are surrounded by emptier, more disordered zones.Several of Bernal’s experiments suggest that in assemblies in which there is an order-disorder balance, any locally ordered region may be adjoined by a misfit zone of greater than average disorder. If this is generally so, it may have implications in relation to solvation and to special effects at interfaces. The model could, of course, represent only one possible configuration of an assembly of molecules loosened just enough to be able to acquire the irregularity of the liquid state, or of a just-melted solid. Valuable as it has proved to be, it could not be expected to give guidance on the changes occurring in liquids with rise of temperature towards Tc-with one possible exception. The ‘accidents’ previously mentioned were holes large enough to admit an extra sphere; they correspond with vacancies-the kind of accident to be expected particularly in liquids.Liquids over a wide temperature range The minor expansion normally associated with melting is just adequate to establish the irregularity of the liquid state. With rising temperature, it is followed by a gradual but major expansion-commonly, volume is more than trebled between triple and critical points. A general theory must say how liquids use this excess volume. It is not unreasonable to consider that multiplication of volume between TM and TC may involve profound change in the state of a liquid. At temperatures little above TM, liquids can be viewed as melts, best to be understood in terms of a lattice theory, and to be compared with the solid phases from which they came.Towards the upper end of the temperature range, comparison with the gas phase might be more appropriate -the classical ‘continuity of state theorem’ certainly suggests this. Both approaches have been used in discussing liquids. Liquids, however, are very diverse-even in the restricted field of molecular liquids-because of wide variation in kind and symmetry of intermolecular force and of molecular size and geometry. Perhaps, therefore, no more than a general theory of broad principles is to be looked for. Particular liquids may require their own theories within the general framework-perhaps one for temperatures near TIM, and another for higher temperatures where entropy requirements have prevailed more or less decisively. For the first kind of particular theory it may be relevant to consider that structure in crystalline solids has always * G = H - T S .Ives and Lemon 71 been revealed by external form; structure in liquids has been concealed by lack of it. Eyring’s ‘significant structures theory of liquids’l* encourages this attitude. It is based on a generalized model of liquids which ascribes both solid-like and gas-like behaviour to the molecules they contain, and is de- signed to accord with two main observations. First, x-ray scattering studies of simple liquids such as argon over the temperature range between TM and Tc show that the average coordination- 10 or 11 just above TM-steadily decreases with rising temperature to about four at about 5” below Tc, but the distance between nearest neighbours remains almost constant.Expansion is therefore largely confined to the holes between the molecules. Nearer to Tc attractions begin to fail, nearest-neigh- bour distances increase, and coordination rises to six. The theory describes the holes as ‘fluidized vacancies’ and suggests that they move about in the liquid as freely as molecules in a gas. This involves the second main observation, which is the classical ‘law of rectilinear dia- meters’-to the effect that the average of the densities of a liquid and its equilibrium vapour is nearly independent of temperature, but decreases slowly and linearly from TM to Tc.The liquid seems to bear some kind of reciprocal relationship with the vapour. It is adopted as a cardinal principle of the theory that the liquid indeed ‘mirrors’ the gas or vapour with which it is in equilibrium. To quote:14 ‘individual molecules translating in the vapour are mirrored as vacancies translating in the liquid. Rotating molecules are mirrored as rotating vacancies. Association of vapour molecules is inatched in the liquid by association of the corresponding vacancies which translate and rotate like their molecular counterparts.’ The translation of vacancies is of course due to the contrary movement of molecules (just as the migration of positive holes is due to movement of electrons), but there is only one phenomenon and it is immaterial which description is applied to it.The theory requires some of the molecules some of the time to have translational freed om. The solid-like behaviour of molecules has nothing to do with any crystal- linity in the liquid; it depends only on the vibrational motion of a molecule temporarily trapped in the potential well created by its nearest neighbours. The motion is adequately represented by an Einstein oscillator* of the same characteristic temperature, 8, as that of the parent solid. Occasionally, such a molecule will acquire enough energy to push its neighbours aside, and a vacancy is generated. The energy required to create a vacancy is directly related to the sublimation energy of the solid. The theory gives a very simple answer to the problem of what proportions of the molecules are gas-like or solid-like (i.e.having translational or vibra- tional motions) at any temperature. It is that (VL - VS)/VL is the fraction * The Einstein theory of the heat capacity of a crystal, considered as N independent oscillators gives ,o co = 3Nk ( ;)2 J 1‘ - 1)2 where 8 = hv/k; v is a characteristic frequency, and other symbols have their usual mean- ings. When T 9 8, Cu -+ ca 6 cal°K-lmole-l. RIC Reviews 72 of gas-like, and VS/VL is the fraction of solid-like molecules, where VL and VS are molar volumes of liquid and solid respectively. It is remarkable how well the simple expression fits the heat capacity data for liquid argon from TM to Tc. The theory is tested by setting up a partition function for the liquid as a product of partition functions for solid- and gas-like states (the ‘significant structures’), weighted according to their contributions, i.e.where N is the Avogadro number. For the boundary conditions VL = VS and VL > VS, f L assumes the limiting valuesfs and fG respectively. Appropriately expanded with feeding-in of molecular weight, Einstein 8, sublimation energy and molar volumes, this partition function can be used to calculate the thermodynamic properties of a liquid from TM to Tc. The results are then compared with the experi- mental data. In general, tests of this kind have been remarkably successful for a wide variety of liquids; the model on which the theory is based must therefore be given weight.On the grounds that there is no majority group of liquids that can safely be described as normal, it is expedient to abandon the attempt to define a normal liquid in favour of tracing the events that may occur and the sequence of states that may be traversed on the way from solid to gas. To do this adequately would require much closer attention to the phenomenon of melting15 than is at present permissible. The brief resumd which is alone practicable must start with the statement that crystalline solids in their lowest free energy states contain defects-the cost in energy to produce defects is met by entropy profit-at concentrations which increase exponenti- ally with rising temperature. There is, of course, no reason to suppose that most solids are not in a state of frozen-in disequilibrium but this is likely to be relieved as TM is approached from below, and, particularly if co- operative positional defects develop, ‘pre-melting’ phenomena may be observed-they are in any case to be expected from fluctuation theory.It is of interest that some melts show pre-freezing effects as TIM is ap- proached from above, but they do not normally ‘mirror’ pre-melting as might be expected. The general inference is that melting may not be as sharp as unsophisticated thermodynamic theory suggests. In the present context, however, the main point is that the expansion which normally accompanies melting so greatly lowers the energy to produce defects that positional disordering becomes almost, if not quite, complete and all long-range order is lost.In this order-disorder transformation the distinction between right and wrong positions vanishes, so that ‘positional defect’ loses meaning. Nevertheless, in the fluid state attained by melting, it is fundamental to expect the fluctuating balance of the ‘energy-entropy conflict’ to be even more significant, if less externally apparent. Whereas in the ordered solid lves and Lemon 73 state fluctuations had to be towards disorder, in the liquid we should expect them to be of both kinds-toward local order in the interests of energy minimization, and towards even greater disorder as entropy prevails. We look for an even greater prevalence of ‘defects’, but they now need redefining by reference to a system not considered defective.It is reasonable to adopt Bernal’s model of the ideal ‘monatomic liquid’ as the new basis of comparison-or an averaged-out version of the assemblies represented by alternative packings of the five ‘canonical holes’. The less regular of these polyhedral holes cannot be considered as defects, since they are formed with insignificant energy effect. What is to be looked for is something of enhanced energy and entropy, increasing in concentration with rising temperature until cohesion is undermined and the system is ready to take the major entropy jump of transition to the gas phase. Bernal’s ‘acci- dents’-holes large enough to admit an extra molecule-or Eyring’s ‘fluidized vacancies’ answer this description.The ‘hole theory of liquids’ in this sense is generally acceptable and is basic to mass-transfer phenomena in liquids- self-diffusion and fluid flow. Bernal’s ‘pseudonuclei’ represent the opposite kind of fluctuation, but a greater width of possibilities needs consideration. The relaxation of symmetry restrictions imposed by the long-range order of crystals, and the greater volume and freer movement of molecules make possible the formation of various types of more or less evanescent clusters with a variety of structures. If crystal-type symmetries are not excluded, but must merely take their chance, the following types of cluster may be envisaged in liquids, probably confined to temperatures not too far above TIM. ( 1) Quasi-crystalline, or ‘crystallizable’ clusters, with structure related to that of the solid phase in equilibrium with the liquid at TIM.The adjective ‘quasi-crystalline’ is often used loosely; it should be confined to the result of incomplete ‘mucking-up’ of a crystal, by expansion and introduction of defects of various kinds in number insufficient to complete the destruction of order. This creates a picture of a tattered remnant of a crystal, but there is the possibility of a tighter, less imperfect cluster too small to serve as a nucleus for growth. (2) Quasi-crystalline clusters with structures related to some other real or conceivable crystalline phase the substance might form, other than that normally existing at TM-still, in principle ‘crystallizable’ and perhaps either ‘tattered’ or ‘tight’.(3) Anti-crystalline or non-crystallizable clusters-pseudo-nuclei, of growth-forbidding symmetry. If these are small ‘closed’ units, they might tend to associate together in a quasi-crystalline way. They may, as Bernal showed, be capable of extension in one dimension. Disordered clusters, distinguished only by lowered energy and changed density, cannot be ex- cluded; they might, indeed, be the most probable of all, because of the lesser penalty in entropy. There are no grounds to deny that more than one-perhaps several-of these possibilities might be open to a single liquid. From one liquid to another, preferred types of clustering would clearly depend on the directional nature of forces and the shapes of molecules.There are but two further RIC Reviews 74 comments to be made on this largely unexplored because nearly inaccessible problem. It is believed that attention should be paid to Bernal’s finding that, adjacent to local order, special misfit disorder is to be looked for. The second comment is that all clusters must sooner or later vanish as the temperature is raised and it might be thought that this would happen but little above Tm,, but undoubtedly there are exceptions. For these exceptions, it is of particular interest how quasi- or anti-crystalline clusters would vanish. The order of these structures does not extend throughout a phase, so their ‘melting’ into disorder is not constrained by the phase rule to take place at a single characteristic temperature for each.On the other hand, their order, if not long-range, is co-operative in nature, and its failure and dissolution might be expected to show the general feature of order-disorder transformation-that once the rot sets in, complete collapse is not long delayed. The ‘melting’ of a cluster would then occur over quite a narrow temperature range and give rise to an effect on the properties of the liquid concerned-not expected to be very marked or easily discernible. If, in a given liquid, more than one kind of cluster is formed each will have its own energy-entropy balance, to swing over at its own ‘submerged’melting point. A sequence of second or higher order transitions in the properties of the liquid might then be detectable along the rising temperature scale.This is discussed later (p. 99). THE ABNORMAL HYDRIDES Comparison can now be made between normal and abnormal hydrides. The previously noted similarity between the second and third groups of hydrides in Fig. 1 does not extend to the first group; in particular HF, H2O and NH3 show enormously greater cohesion in condensed phases than would be expected. This is, of course, due to hydrogen bonding. The extent of the abnormalities can be assessed, first by means of the thermal data given in the first three columns of Table 1 and secondly by inventing a ‘normal’ first group of hydrides by extrapolation from the other two. Neon and methane should help in this, but are abnormal-the former because of low Table I : Some thermal data for HF.HzO and N k I 7 1 8 1 9 HF 1 1 9 0 * 1 = g & I 293*1=- I790 -- 226 0.24 0-57 0*140* 503 54 129 184 108 647 311 0-35 0.59 0.148 HzO 5.258 6 - 10 9717 373. I = ~ 26 * 04 79 128 0-62 0.62 0.242 NH3 I436 273. I =-- Qy;4 239*7= __ 558 I 23 * 28 l 9 5 * 4 = 7 * AHB calcd. for evap. to monomolecular vapour = 7.80 kcal mole-1. Units are O K , cal mole-1, cal O K - 1 mole-’. Ives and Lemon 75 mass, the latter because of rotational freedom in the solid. Nevertheless, the rough constancy of TM, TB and TC differences between corresponding members of the third and second groups (23 & 6, 24 5 4,47 3 6", respec- tively) can be used for a linear extrapolation of each which may have a little sigrdicance.It leads to 'abnormality increments', ATM, ATB and ATc, probably underestimated and relatively uncertain to about 10 ", entered in columns 4 to 6 of Table 1. Somewhat unusual 'reduced increments' appear in columns 7 and 8 and column 9 contains ratios of latent heats. It is obvious that water shows the greatest anomaly; very high AHB and high ASB indicate strong cohesion and residual organization in the liquid at TB. High TC and normal TB/Tc (0.577) show that cohesion does not fall away abnormally with further substantial rise of temperature- weak hydrogen bonding could be expected to be more vulnerable. This is all the more striking because water vapour at 1 atm pressure does not depart greatly from ideal gas behaviour.It seems that pair-wise hydrogen bonding between water molecules is weak, but is somehow much strengthened in condensed phases. The behaviour of HF stands in strong contrast to that of H2O. Very low AHB and ASB are due to persistence of association in the saturated vapour which consists of polymeric, zig-zag chains of average length (HF)3.5. Evi- dently, polymeric molecules in the liquid at TB need little excess energy (1.8 kcal mole-1) to slip into the vapour state, but a further 6.0 kcal mole-1 are required to break them up into monomers. This is in line with superior stability of F-H ..... F bonds, but not with considerable stabilization of them in the liquid as compared with the gaseous state. Cohesion in liquid HF could be said to be strong in one direction-along the chains-but weak in others; this is a situation conducive to ordering in one dimension but not in three.The rather high ASM perhaps reflects an appropriate disparity of order between liquid state and solid (an ordered assembly of infinite zig- zag chains). Hydrogen bonds between like atoms decrease in strength in the electro- negativity sequence F > 0 > N. Accordingly, NH3 shows least non-ideality as a gas and abnormality as a liquid, although still relatively high AHB and ASB indicate appreciable hydrogen-bonding stabilization of the liquid state. Liquid ammonia is probably the most normal of the three liquids; this view is supported by their dielectric constants (NH3, 22 at - 34°C; HF, 83.6 at 0°C; HzO, 87-7 at OOC), which broadly reflect the extent of co-operative association.On the other hand, AT, and, especially, ATM/ATc place NH3 differently, suggesting that it is particularly well stabilized in the solid state. Disparity between solid and liquid in this respect is confirmed by high AS,* and AHM/AHB and is supported by relatively low (TB - TIM). In solid NH3, each nitrogen atom has six equivalent neighbours at 3.38 A and six more at the greater distance of 3.95 A. This suggests that the nitrogen atoms form six weak hydrogen bonds-they would have to be weak, but would contribute to cohesion in three dimensions. If this picture is correct, it is a remarkable example of the fortification of weak interactions by * The situation is, however, complicated by the ready inversion of the pyramidal NH3 molecule in liquid and gaseous, but not solid, states.RIC Reviews 76 co-operation depending on precise geometrical requirements-positional, orientational-to be attained in the regularity of a crystal but not conceivably in the irregularity of a liquid. Water comes into comparison again because ASM is relatively low, despite the strong hydrogen bonding in ice and its comparatively uncomplicated structure. Although allowance must be made for positional disorder of hydro- gen atoms in ice (equivalent to orientational disorder of molecules), this is consistent with retention of considerable order after melting, for which there appears to be a reserve of ‘hydrogen-bonding power’. Some inferences may be drawn from this intercomparison of the three abnormal hydrides. Their properties are dominated by hydrogen bonding.The hydrogen bond, normally formed between a hydrogen atom of one molecule and a lone pair of electrons of another, can be regarded as a Brarn- sted acid-base interaction, and it is convenient to refer to one participating molecule as the proton donor and to the other as the proton acceptor. All the three hydride molecules can act in both capacities, but the numbers of protons and lone pairs of electrons available per molecule are, respectively : HF, 1 and 3; HzO, 2 and 2; NH3, 3 and 1. In very many crystalline substances hydrogen bonding is a contributory factor determining the solid state structures that are adopted. In the regularity of the crystal, with participating molecules held in optimum positional and orientational relations, even very weak bonding is co-operatively protected and cumulative in effect.This appears to be so in solid NH3. In liquids, weak bonding has no such privileged position and must sustain the knocks of its hostile, uncoordinated surroundings. This may be the basis of the general rule that, except in the solid state, hydrogen bonding in one-com- ponent systems is confined to equal numbers per molecule of protons donated and accepted. Whereas HF and NH3 molecules form but two hydrogen bonds each under such conditions (acting once as donor and once as acceptor), H20 can give two protons and accept two. It is thus unique in its capability of promoting three-dimensional order, and is the only molecule which, from a single atomic centre, can give rise to four hydrogen bonds directed in space.Whatever the ways in which water may exploit this facility in the liquid state, there can be no doubt that co-operation is involved. Two kinds of co-operative effect must be distinguished. The first is a function of the nature of hydrogen bonding between molecules which can both donate and accept protons. It is co-operative in the sense that a molecule that has acted as donor is more ready than before to act as acceptor, and vice versa. Donation and acceptance are mutually supporting and hydrogen bonding between like molecules is autocatalytic. This is no doubt why, in the association of monohydric alcohols, the dimer stage is very nearly skipped.7 The second effect is the crystal-like kind in which co-operation depends on the satisfaction of precise, three-dimensional geometrical requirements.The problem is, whether hydrogen bonding in liquid water brings this second kind of co-operation into play, so that it is significantly fortified by the formation of evanescent quasi-crystalline or anti-crystalline clusters, or both. If so, does this occur over the whole temperature span between TIM and Tc, Ives and Lemon 77 or only over restricted lower ranges of temperature? On the other hand, is the first kind of co-operation, intrinsic to hydrogen bonding, sufficient, leading to amorphous clusters of no distinguishable symmetry, representing no more than energy and density fluctuations? These are some of the main questions in debate.THEHYDROGENBOND It is mandatory to consider a bond so basic to the subject under review. Strictly, the name ‘bond’ is unsuitable in connexion with the highly variable ‘lone pair interactions’ concerned but ‘hydrogen bond’ is conventionally, if ambiguously, applied to the three atom system X-H *.... Y, where X and Y are covalently bound atoms, rarely other than F, 0 or N, in separate molecules or in the same molecule. In this formulation, the dotted line represents the interaction between a lone pair of electrons, provided by Y , and the hydrogen atom covalently bound to X. In so far as the electrons of this bond are with- drawn towards X (of higher electronegativity), Y may ‘see’ the hydrogen atom as an imperfectly screened proton of radius lO-l3cm-unique as a centre of attraction for its lone pair electrons. It is common to speak of ‘hydrogen bonding between molecules’, i.e.one molecule carrying the group X-H and the other, the atom Y, implying that it is the H - . . . . Y interaction which is being defined as the hydrogen bond. Seemingly logical, this is not acceptable, because the bond X-H is profoundly affected by the interaction. Even to consider X--H--.*Y as a hydrogen bond between X and Y may be satisfactory only as a first approximation because so much depends on what else is attached to X and Y. This is implicit in the reference already made to the co-operative nature of hydrogen bonding.Energetically, the hydrogen bond can be understood only by assessment and algebraic summation of a number of interaction energies, no one term of the summation being of decisively predominant weight. Theoretically based calculation of small, net bond energies (1-6 kcal mole-1) appearing as differences between larger quantities, turns out to be of almost impossible difficulty. This is one reason why the present state of knowledge on hydrogen bonding is regrettably inadequate. Difficulties are enhanced by wide variation in hydrogen-bond geometry. Thus, 0-H ..... 0 bonds vary in length (0 *..-- 0 internuclear distance) from 2.44 to 3.36 A-to be compared with 3.5 A, the normal distance of closest approach between non-bonded atoms.Although colinearity of the three atoms favours maximum stability, some intramolecular hydrogen bonds (e.g. that in salicylic acid) must be ‘bent’, with the hydrogen atom displaced from the line of centres of the terminal (oxygen) atoms. Presumably, then, intermolecular hydrogen bonds can also bend, but how this would affect the bond energy depends on the nature of the bond. The simplest assumption, not inconsistent with known bond energies, is that only electro- static interaction is involved-dipole-dipole attraction between the X-H bond and the ‘atomic dipole’ (lone pair-nuclear charge) of Y. Energy would then bear a simple cosine relation to angle of bend, and considerable de- parture from colinearity would have little effect.If, on the other hand, the bonding were covalent in nature, there would be a much more critical RIC Reviews 78 dependence of energy on angle. It is clearly fundamental to the description of the hydrogen bond to decide which of these views is correct. There is strong evidence that the electrostatic model is inadequate. It does not satisfactorily explain the increase in intensity of infrared absorption due to 0-H vibration which accompanies hydrogen bonding, nor the shortening by nearly 1 A of the H.--*O distance below the sum of the normal van der Waals radii of hydrogen and oxygen atoms. Purely electro- static attraction implies that the attracted species suffer no mutual deforma- tion. At the close approach involved this is impossible-they are bound to polarize each other, and the distortions of the charge clouds must give rise to delocalization of electrons and perhaps change in hybridization.Coordi- nated motions of electrons in atoms brought close together will give rise to dispersion forces, and overlapping of charge clouds will bring repulsion into play. Accordingly, electrostatic, delocalization, dispersion and repulsion energies have been taken into account in attempts to calculate hydrogen-bond energy. Coulsonl6 considered delocalization in terms of contributions from ionic - 4 - - + (e.g. 0 H *.... 0) and covalent (e.g. 0 H-0) valence-bond structures to the complete molecular wave function of the three-atom system and found, in agreement with other calculations, that these contributions are significant.Additional difficulties in estimating dispersion and repulsion terms augment the uncertainty of his assessment of the four main contributions to hydrogen bond energy; these, for the hydrogen bond in ice, are shown in Table 2.l6 I kcal mole-1 I Table 2: Contributions to hydrogen bond energy in ice16 Electrostatic Delocalization Dispersion Repulsion Experimental (from sublimation energy) +6 $8 -8.4 + 3 +8.6 +6*1 It might be remarked that a better result is obtained if the bonding is assumed to be entirely electrostatic, and perhaps this is the reason why this assumption has not been universally discarded. It is unfortunate if trivial coincidence between two energy terms obscures other indications of the theoretical treatment.The most important of these is that decrease of 0 **.** 0 distance is accompanied by increase in delocalization and increase in 0-H bond length. As the 0 ***.. 0 separation shortens towards 2.45 A, the proton moves towards a central position, and probably attains it in a number of cases (e.g. the acid maleate ion). There is no doubt that the [FHFJ- ion is symmetrical, with equal F-H distances of 1.13 (the normal HF bond length is 0.92 A). Electrostatic attraction will not do for such cases. Ives and Lemon 79 6 It is appropriate to interpolate a general comment on hydrogen-bond energy. The heat of formation of the HF, ion from HF and F- is 58 kcal mole-l ; this provides a reason for modifying any impression that the hydrogen bond is always characteristically weak.Another reason comes from an alternative theoretical modell7 which views the hydrogen atom in 0-H ..... 0 as common to two 0-H bonds, one strong and slightly stretched, the other weak and highly stretched. Energies for each are calculated from semi- empirical interatomic potentials and with repulsion and electrostatic attrac- tion terms, provide hydrogen-bond energies as a function of 0 **... 0 distance with considerable success. Particular interest lies in the equivalent treatment of the two halves of the hydrogen bond, and the indication provided that the ‘lone pair attraction-long covalent bond’ half is a good deal stronger than the over-all bond energy might suggest.Examples, in terms of inter- oxygen distance, ‘weak bond’ and total bond energies, quoted in sequence are: 2-70 A, 11.0 and 5.6 kcal mole-1; 2.50 A, 35.5 and 14-4 kcal mole-1. Even if these figures are challenged as derived on too simple a basis, they illustrate a valid point relevant to the vulnerability of hydrogen bonds to thermal disruption. In this connexion, there is evidence that the volume of a system, rather than its temperature, is the decisive independent variable. The theoretical conclusions about delocalization have strong experimental support from the infrared absorption spectra of hydrogen-bonded systems. The increase in intensity of absorption in the 3 p region which accompanies the formation of O-H....*O bonds requires the 0-H vibration to be accompanied by a fluctuating charge separation.This implies considerable and mobile charge migration within the three-atom system in response to the motions of the proton. There is also a frequency displacement (approxi- mately in proportion to the increase in intensity) which reflects the stretching of the 0-H bond-longer bonds have lower frequencies than shorter ones. The third general effect is an increase in the width of absorption bands, less for intra- than for intermolecular hydrogen bonding because in any macro- scopic molecular system there must be considerable statistical variation of 0 ....- 0 separation. Particularly because of the sensitivity of delocalization to this separation, there will be a correspondingly wide variation in the frequency of the 0-H vibration.At this point it might seem justifiable to make the inference that hydrogen bonding is not purely electrostatic; on the contrary, there must be an import- ant covalent contribution to it. It has been strongly argued that the co- operative strengthening of hydrogen bonding in systems of like molecules depends fundamentally on delocalization. If this inference is correct, it follows that hydrogen bonding is the more directional in nature, and that hydrogen bonds will have the greater tendency to be straight because of a considerable dependence of energy on angle of bend. Unfortunately, true as this may be for the ‘ideal’ hydrogen bond, the situation remains confused, and the fact must be faced that there is a wide range of interactions legitimately included under the name of ‘hydrogen bonding’. Proposals have been made for a closer classification-short, straight, long, bent, shading off into attractions which are purely electrostatic.The difficulty is, where to draw an agreed line between interactions which are, or are not, to be considered ‘true’ RIC Reviews 80 hydrogen bonds. Although electrostatic interactions are, in general, somewhat weaker they are not substantially weaker than ‘authentic’ hydrogen bonds, nor are they completely non-directional, because of the localization of elec- tronic charge in hybridized lone-pair orbitals. INFRARED AND RAMAN SPECTRA The expectation that infrared and Raman spectroscopy would be unrivalled in providing information about hydrogen bonds is realized only in the study of dilute solutions of hydrogen-bonding substances in inert solvent-evidence for the co-operative nature of the bonding has been gained in this way.18 For pure, hydrogen-bonded liquids such as water the situation is less favour- able because of greatly enhanced difficulties.Coupling of molecular motions, overlap of overtone, combination and resonance bands, added to the broaden- ing caused by the structural disorder of the liquid state, make it hard to identify peak frequencies and to assign them to specific modes of motion. The literature is correspondingly large and contentious ; the present writers have no alternative to arbitrary selection of topics with the hope of presenting an unbiased view of the present situation.Table 3 lists one selection of principal vibration frequencies, from Raman and infrared spectra, for HzO in the three states of aggregation. v2 Bend v1 Symmetrical stretching v3 Asymmetrical stretching I Ref 19 3755 3397 3434 I595 I642 164’ Table 3: Spectroscopic frequencies for H2O (cm-1) Water vapour Liquid water (3°C) Liquid water (70°C) Ice Ref 20 3650 3448 3448 3360 - The vapour assignments are not in doubt. Raman intensities, v1 9 v3; i.r. intensities, vl < v3; frequencies, v1 < v3. But in ice, the main Raman band is at 3360 cm-1, and is expected t o be vl; the main i.r. band is at 3210 cm-1 and is expected t o be v3. This puts the frequencies in the reverse order, v1 > v3.Hornig et a/. adopted this, but other workers seem either not t o have noticed the transposition, o r t o have chosen to disregard the intensity evidence, preferring t o assume that the frequencies for ice must remain in the order V I < v3. This problem i s not properly resolved-there are arguments either way, and, of course, the uncertainty extends to liquid water. The uncertainty is worsened by the asymmetry of local fields of force in the liquid; the effective symmetry of a water molecule is lowered from tetrahedral (CZ,), so that v 1 and v3 are no longer purely symmetrical and antisymmetrical stretching vibrations-they lie close to each other and to the overtone 2 ~ 2 . Assignments remain correspondingly questionable.Ives and Lemon 81 Bending frequency is little affected by phase change, but the stretching frequencies decrease in the sequence vapour > liquid > solid, with liquid water about two-thirds the way to ice. Increase of temperature has but little effect on the vibrational frequencies of liquid water and hence, it would be thought, little hydrogen-bond breaking effect Wall and Hornig21 have made precise photoelectric Raman spectroscopic studies of HDO using five mole per cent solutions of D2O in H2O and of H2O in D2O. This method of applying isotopic substitution has the advantage that the five nearest-neighbour positions of each OH oscillator are pre- dominantly deuterated, or of each OD oscillator, protonated, so that un- coupled fundamental stretching frequencies can be observed. The simplified, uncoupled bands are still broad (although very narrow in the case of deuter- ated ice), and it is asserted that their breadth directly reflects the spread of 0 *.*..0 distances in liquid water. Using a well-substantiated correlation between this distance and frequency, Wall and Hornig derived a distribution function for nearest-neighbour distances in good agreement with that from the best x-ray scattering measurements. This supported their assertion, but perhaps more important was the fact that the Raman bands and the distri- bution curves were smooth, continuous functions, indicating that allowable values for intermolecular distances in water are densely distributed within a finite range, with one most probable distance represented by a single maximum not far from the centre of the range.Frequency shift with temperature was small, confirming the earlier indication that hydrogen bond strength varies little with temperature. It is clear that these results must be taken into account in discussion of the nature of any clusters that may be postulated in liquid water. The conclusions of Wall and Hornig have been supported by Falk and Ford’s studies21a of the infrared absorption of dilute solutions of HDO in H2O and D2O between 0 and 130°C. This method has an advantage over Raman spectroscopy in greater resolution, better ‘signal to noise’ ratio, and consequent improved detectability of small shoulders on band profiles.Well separated v1 (2505 cm-l) and v3 (3400 cm-1) bands were identified, of broad, smooth, singly-peaked, nearly Gaussian contour, decreasing in intensity with rising temperature in a manner consistent with a gradual weakening of hydrogen bonds. High resolution examination of the region (3550-3750 cm-l), where absorption due to non-hydrogen-bonded OH would be expected, showed no trace even of a shoulder-and similarly for OD. This, with the parallel Raman result, was taken as conclusive evidence that there is in water a continuous distribution of hydrogen-bond strengths from weak to as strong as in ice, and that the existence of discrete species differing in the extent of hydrogen bonding is ruled out. No less weight, however, must be given to Walrafen’s equally careful broad 152-175 cm-l was assigned to hydrogen-bond stretching, A in studies22 Raman band centred of intermolecular at - 60 cm-1 vibrations to hydrogen-bond water.bending. band and These of a frequency weaker, bands decreased in intensity quite rapidly with rising temperature, and also with increasing concentration of added electrolyte. Both these effects strongly suggest extensive breaking of hydrogen bonds, and Walrafen was able to support this view with plausible thermodynamic argument. RIC Reviews a2 More recently,22a Walrafen has substantially supplemented the evidence for his viewpoint in two main ways. First, he has shown that the broad Raman band between 300 and 1000 cm-1, associated with intermolecular libration in liquid water, can be resolved into three Gaussian components which vary little in frequency, half-width or relative proportion between 0 and 90°C.All three have essentially the same negative temperature coefficient of intensity as the previously studied hydrogen-bond stretching and bending vibrations. ‘411 these frequencies (717, 538, 439, 152-175 and 60 cm-I), which are associated with water molecules in a tetrahedral, hydrogen-bonded environment of CzV symmetry, fade out together. It is relevant to later discussion of water structure theories (e.g. that of NCmethy and Scheraga) that if this fading out is attributed to the breaking of hydrogen bonds, then the constancy in relative proportions of the librational components indicates that each molecule breaks its bonds in one single step, and not in consecutive steps.This implies that it is not acceptable to assume different energy levels for water molecules forming one, two, three or four hydrogen bonds. The second way in which Walrafen has supported his case is by re-examina- tion of the intramolecular Raman bands of liquid water in the 2800-3900 cm-1 region, from 10 to 90°C. These, by analogue computer technique, have been resolved into four Gaussian components. Two, at 3247 and 3435 cm-1, are strong, decrease in intensity with rising temperature and are not found for water near Tc; two, at 3535 and 3622 cm-l, are weak, but increase in intensity with rising temperature and are found for water near TC (with but slight change in frequency).There is an intermediate frequency, 3460 cm-l, of intensity independent of temperature ; this is equivalent to an isosbestic point. This behaviour is judged by Walrafen to be irrefutable evidence of an equilibrium, shifting with temperature, between hydrogen-bonded (‘lattice’) water and non-hydrogen-bonded water. It is clear that there is disagreement in the interpretation of observations of fundamental intramolecular and intermolecular frequencies, perhaps to be resolved by the suggestion that alternative methods of study are not ‘seeing’ the same phenomenon. Walrafen himself points out that the inconsistency can be clarified only by better knowledge of so-called hydrogen bonding, and that his ‘non-hydrogen-bonded’ water molecules (with spectra unlike those of high pressure steam or of dilute solutions of water in inert solvents) are still subject to strong intermolecular forces.The situation emphasizes the quandary mentioned in the last section. It is perhaps that intermolecular Raman effects are associated with straight ‘covalent’ hydrogen bonds, and that it is the disappearance of tetrahedral symmetry of vibrating groups with rising temperature that is being observed.22b If this is so, we need some hydrogen bonds other than the straight ‘covalent’ ones, hardly less effective in promoting liquid phase cohesion and in their influence on intramolecular vibrations. Alternatively, we require some other kind of bonding altogether.A further spectral region of interest to the study of water extends from 5000 to 11 000 cni-1 in the near infrared, where there are several systems of well-marked overtone and combination bands of not universally agreed assignment. Quite different interpretations, each with important implications in relation to the structure of water, have been given to essentially identical Ives and Lemon 83 O 8t 0 31- 0.2 f I 01 - L t L U I I 2 0 I 2 5 I 3 0 115 I 05 I10 I 30 Fig. 3. Near infrared absorption of water, after Luck. E = optical density, p = density, d = optical path length. observations-notably those of Buijs and Choppin23 and of Some of Luck's spectra, which extend over the wider temperature range (> TC - TM) are reproduced in Fig.3. Buijs and Choppin assigned frequencies of 8620 cm-1 (1.16 p), 8330 cm-1 (1.20 p) and 8000 cm-1 (1.25 p) to water molecules forming zero, one and two hydrogen bonds respectively. From appropriate intensity measurements, they calculated the mole fractions of these species to be (in the same order) 0.27, 0.42 and 0.31 at 6"C, and 0.40, 0.42 and 0-18 at 72"C, these being the extremes of their experimental temperature range. Extrapolation gave 46 per cent hydrogen-bond breakage accompanying the melting of ice; further extensive breaking with rise of temperature from 0°C was clearly inferred. Luck contended that Buijs and Choppin's solution of their simultaneous equations for the mole fractions of three water species was not unique, and that, in any case, non-hydrogen-bonded molecules in liquid water cannot be detected readily by infrared absorption. Although there are nine 'hydro- gen-bonding states' of a water molecule,* it is only proton donation, not acceptance, that has an appreciable effect on vibration frequency, largely confined to that of the OH group acting as donor, and almost independent of the state of the other OH group in the same molecule. The appropriate assignment of the peak near 1.15 p (Fig.3) to free OH groups was supported by the cfose comparison afforded by the similar band system of methanol in the same region. The large proportional increase in intensity between TB and TC is to be noted-it suggests that this is the temperature range for most of the hydrogen-bond breaking.One independently determined per- *The molecule can act zero, one or two times as proton donor and, independently, zero, one or two times as proton acceptor. 84 RIC Reviews centage of free OH groups at one temperature was needed to calibrate the intensity scale. Studies of the dielectric constant of water up to Tc25 had led to an estimate of nine per cent at 0°C; adoption of this figure brought the dielectric constant and infrared data into agreement over the whole tempera- ture range. Twenty per cent of non-hydrogen-bonded OH groups at 100°C provides a very different picture of water from that presented by Buijs and Choppin, but more consistent with the high TC of water. For hydrogen-bonded OH groups, Luck assigned absorption at 1.25 p to those making linear (ice-like) bonds of maximum strength, and at 1.20 p to those involved in ‘unfavourable’ bent or long hydrogen bonds, considered collectively.This rough classification was based first on observations by the ‘matrix isolation method’ of the infrared absorption of low, varied concen- trations of H2O in solid nitrogen at 20”K26, which gave the fundamental frequencies of monomers, di-, tri- . . . n-mers. These showed the expected trend (e.g. in increasing order of complexity, higher v only: 3725, 3545, 3510, 3390 . . . 3355 cm-1-cf Table 3 ) and further indicated that the dimer, under these conditions, was cyclic (Fig. 4)-two weak, bent bonds doing the work of one straight bond. H \ 0-H ‘ \ , , ‘H -0 H \ Fig.4. Water dimer in solid NZ at 75°K. By assuming that all the lower polymers were cyclic, Luck obtained a relation between frequency and angle of bend. Secondly, he studied the hydrogen- bonding equilibria of numerous organic compounds in inert solvents, relating by stereochemical argument bond energy with angle of bend ; use of the Badger- Bauer rule on the dependence of frequency on energy then gave results agreeing with his other frequency-angle relationship. Luck completed his detailed assignments with 1.143 ,u to OH vibration in free HzO molecules, and 1.150 p to free OH groups sharing a molecule with a hydrogen-bonded OH group. These assignments, and intensities, led by a Buijs and Choppin type of calculation to the results illustrated in Fig.5 -stressed to be a possible solution, but not the only one. Substantial evidence for the persistence of molecular association well above TC will be mentioned later. If these calculations are considered to be overly conjectural, they are hardly more so than any others. The extent of disagreement on matters fundamental to a theory of water is well illustrated by running down a listZ1a of estimates by 19 authorities of the percentage of hydrogen bonds broken whenice melts: 71.5,66, 60, 57.5, 56,47,46,46, 38, 32, 30, 18, 18, 15, 10-5,9,9, 2.5 and 0. On the whole, opinion seems to be moving away from wholesale breaking of hydrogen bonds and considerable concentrations of ‘monomers’ Ives and Lemon 85 Temperature ("C) Fig. 5.Possible concentrations of named entities in water, after in liquid water. Vacuum ultraviolet spectroscopy of water, vapour and liquid, has led to an estimate of less than one per cent monomers up to 100°C.27 Recent studies by Glew and his collaborators27a of the fundamental valence- stretching frequencies in infrared absorption shown by water in dilute solution in a wide range of solvents demonstrate that the monomer bands (unequivocally identifiable with the vapour or nitrogen matrix bands) are to be found for solutions in one group of solvents including halogenated hydrocarbons and some alcohols. In a second group-of proton-accepting solvents (e.g. ethylene oxide, acetone, tetrahydrofuran)-the spectra show the presence of a single type of dissolved water species hydrogen bonded to the solvent, with no sign of the presence of non-hydrogen-bonded monomers.Extrapolation strongly suggests that for water in water (the best proton acceptor) there can be no significant concentration of non-bonded monomers. THEORIES OF THE STRUCTURE OF WATER It is because there is no generally agreed theory of the structure of water that the plan stated in the Introduction was adopted, and the main concern has been with factual terms of reference (e.g. that water is a liquid) which ultimately must be satisfied, and with some difficulties of inadequate know- ledge, and of interpretation of experiments, that continue to vex the theory makers. In this section, further evidence and theories are non-exhaustively presented in whatever order best suits a continuing narrative.It is suitable first to ask, what is to be expected of a theory? Two of the oldest (Rowland, 1880; Rontgen, 1892) can be combined and summarized in a statement that 'water is a mixed solution of ice and steam that varies in proportion with temperature'-very close to some modern views, and perhaps qualitatively (or better) satisfactory. Eucken's theory2* that there is in water an equilibrium between monomers, dimers, tetramers and octomers (the latter necessary for Hohlraum) gave an excellent account of many proper- ties of water. It had three equilibrium constants available for adjustment. Some recent theories (mostly as well equipped with adjustable parameters) RIC Reviews 86 satisfy more exacting thermodynamic tests, but such tests are not discriminat- ing and can be inadequate to prove a theory wrong-particularly a structural theory. More is now required than success in calculating smoothed-out macroscopic properties, because of the virtual certainty that the behaviour of water to solutes and surfaces and in biological systems presents a whole range of microscopic, structurally-based problems.It is in this direction that progress is needed but is most difficult-inordinately so on the theoretical front because a realistic model is unlikely to be simple or mathematically tractable. More structure-sensitive experiments are needed, as well as recon- sideration of things we are reasonably sure about.The narrative begins with the latter need in mind. ...... . . . . . . Fig. 6. (a) The water molecule; (b) primary unit of water association. The water molecule The structure of the water molecule (Fig. 6a) lies at the root of the properties and behaviour of water. Two protons and two lone pairs of electrons occupy approximately tetrahedrally-disposed, hybridized orbitals. The vacuum dipole moment, 1-87D, is not large, and can hardly be relevant to close-range interactions ; it results from opposed components-that due to the unscreened lone pairs and the oxygen nuclear charge has been estimated as 3~38D.~9 Then, if dipole-dipole association between water molecules should be in- voked, the question arises, what dipoles? Also, would such association lead to something distinguishable from what is generally included under the heading of hydrogen bonding? This bonding leads to the formation of a five-molecule, approximately tetrahedral, primary structural unit, somewhat crudely represented by Fig.6b. It can be seen that the net moments of the molecules, lying symmetrically between the OH bonds, neither fully support nor fully oppose each other. The tetrahedral unit is not consistent with a simple dipolar model for the water molecule, which will, accordingly not be suitable for consideration of the behaviour of water in condensed systems. Lone pair hybrids Ice Tetrahedral association of water molecules and sp3 hybridization of oxygen orbitals are perfected in ordinary ice-ice Ih-the structure of which is represented in Fig. 7, where hydrogen-bond distances have been stretched for clarity.Each oxygen atom is surrounded by four others at a distance of 2.75& three in the same puckered layer of six-membered rings, and the Ives and Lemon 87 n n n Fig. 7. Structure of ice I, Ih, plan and elevation of model. (Reprinted with permission. Copyright 0 1956 by Scientific American, Inc. All rights reserved.) fourth in an adjacent layer. Adjacent layers form between them rather large, polyhedral cavities, each with 12 vertices, which link up to form channels threading their way through the hexagonal rings. The structure is that of a form of silica-tridymite-formed by the sharing of oxygen atoms between SiO4 tetrahedra.Apart from normal crystal dislocations and defects, this representation is ideaIized. The protons are disordered. This can be understood by consider- ing that, radiating from the central oxygen atom of a five-molecule, tetra- hedral assembly (see Fig. 6b) there are four 0 . . - - * 0 directions, each accom- modating one proton; two of the four protons must be near the central oxygen, being covalently bound to it. Since there are six ways of taking two out of four things, there are six possible arrangements of the protons, giving six alternative orientations to the central water molecule. But each of these 88 RIC Reviews O C Fig. 8. Phase diagrams of water-substance. alternatives requires two of the peripheral molecules to have favourable, as opposed to equally likely unfavourable orientations.This divides the absolute probability of any of the six arrangements of protons by four. The total number of configurations in an ice-like assembly of N molecules is, accord- ingly, (6/4)N. This statistical disorder of protons, frozen-in at low tempera- tures, confers on ice a residual entropy at 0°K of S,O = R In 3/2 = 0.81 cal "K-1 mole-1, which is precisely the value required to bring 'third law' and spectroscopic entropies of water vapour into coincidence, and has been otherwise confirmed. The electrical properties of ice show that the protons are mobile at higher temperatures. The dielectric constant of ice, although small (-4) at very low temperatures, is greater than that of water at 0°C (91.2, 88*2), and proton mobility (hydrogen ion equivalent conductance) is much greater in ice than in liquid water.These facts require the presence in ice of two kinds of intrinsic defect-ionic and orientational. The former arise from proton transfer. HzO + HzO = H3Of + OH- The latter arise by rotation of a water molecule through 120" about an ..... 0 axis, whereby empty (leer) and doubly-occupied (doppelt) 0-H oxygen-oxygen 'bonds' are generated (i.e. 0 *.... 0 and 0-H ....- H-0). These are called L- and D-Bjerrum defects30 and, by rotation of adjacent water molecules, they can become separated and migrate through the ice crystal. Ives and Lemon On the grounds that one cannot afford to ignore any aspect of the be- haviour of water-substance in seeking guidance on the problem of liquid water, attention should be given to the other forms of ice.Figure 8 presents recent phase diagrams.31 Ices Ic and IV are metastable phases occupying areas in the stability regions of ices Ih and V respectively. There is a recently 89 Fig. 9. Radial distribution curves for water, after Morgan and Warren.35 reported ice V I I P of the same structure as ice VII, but orientationally immobile, and probably ‘proton-ordered’ (as is ice 11). It is of interest that ordinary water-substance is crystalline at 400°C under a pressure of 200 kbar, and also that ice VII, which can exist metastably at - 175°C and 1 atm pressure, has a density of 1.50 g cm-3, still substantially less than would be expected for close-packed water molecules (- 1.84 g cm-3).Progress has been made in determining the structures of these ices, notably by Kamb.33 In ice VII, each water molecule has eight nearest neighbours, but is bonded to only four of them-it probably consists of two interpenetrating lattices of cubic ice Ic (which has a very similar open lattice of that of Ih), each lattice being fully hydrogen bonded within itself. Ice VI also consists of two interpenetrating but not interconnected lattices. These two then can be described as ‘self-clathrates’, and this appears to be the way in which relatively high density is achieved by tetrahedrally-linked structures. It seems that all RIC Reviews 90 the ices retain substantially tetrahedral disposition of water molecules-if in some cases with considerable distortion, i.e.bending of hydrogen bonds. Conclusions might be that water molecules persist in preferring four- fold bonded coordination, and that the hydrogen bond is highly adaptable to extreme conditions. A report34 that water-substance at 1000°C and 98 kbar has a conductance of 0.7 ohm-1 cm-1 suggests an ionic bond. x-Ray scattering When ice melts, there is a contraction, but this abnormality is less remarkable than the supplementary contraction which accompanies the warming of the melted ice towards 4°C. This is very like the continuation of a melting process, and a negative coefficient of thermal expansion needs must be based on structural change. There is a compulsion to admit that, at least in this temperature range, liquid water must have a structure.The retention of structure in water is supported by radial distribution curves derived from x-ray scattering. The curves obtained by Morgan and Warren35 are reproduced in Fig. 9. They established that the preferred first and second nearest neighbour distances are 2.9 A and 4.5-4.9 A-somewhat greater than those in ice. Average coordination increased from 4.4 at 195°C to 4.9 at 83°C-to be compared with four in ice, and a number decreasing from about 11 with rising temperature for a normal liquid. There cannot be much doubt about the ice-likeness of water. If water were a normal liquid, its density would be in the region of 1.84 g CITL-~, so that x-rays would hardly be needed to prove its open structure, but, in a sense, clarify the questions to be answered.Since x-rays indicate a structural expansion accompanying the melting of ice, why is water denser than ice? Why does rising temperature seem to close up the molecular packing, rather than loosen it ? How is it that forces of intermolecular attraction succeed in holding molecules apart? I The Bernal and Fowler theory On the basis of earlier x-ray scattering studies, and of wider consideration of the properties of water, Bernal and Fowler in 193336 proposed the first, and now very well known structural theory of water. It involved preferred molecular arrangements in liquid water simulating three structures. These were described as ‘water I, tridymite-ice-like, rather rare, present to a certain degree at low temperatures below 4°C ; water 11, quartz-like, predominating at ordinary temperatures ; water 111, close-packed ideal liquid, ammonia- like, predominating at high temperatures for some distance below the critical point at 374°C.These forms pass continuously into each other with change of temperature. Throughout . . . . there is no question of a mixture of volumes with different structures: at all temperatures the liquid is homo- geneous but the average mutual arrangements of the molecules resemble water I, I1 and 111 in more or less degree.’ This passage has been quoted verbatim because the theory has been misinterpreted-it has always been a ‘uniformist theory’, never postulating the existence of water molecules in alternative, sharply distinguishable situations, nor any long range order.Ives and Lemon 91 that of quartz, - 1.08 g cm-3, although adequately explaining the maximum scope. arrangements There Thus, was if a which the difficulty density water in of molecules that tridymite the use is tend adjusted of to forms assume to of that silica did of ice, not as models - give 0.91 adequate g for cm-3, the densityphenomenon in terms of the shift water I + water TI with rising tempera- ture, was too high. It would also lead to a second nearest-neighbour distance discovery obviated The less than that of a new observed form by of x-rays. silica, keatite difficulty (density has on the been same scale - by 1.01 the g cm-3) incorporating five-membered, non-planar rings.Models of dis- ordered keatite give a calculated radial distribution function to a first approxi- mation the same as that of water. More recently,12 Bernal has generalized the theory, seeing the preferred molecular arrangements in water as simulating a network of linked four-coordinated molecules, forming rings of four, five, six, seven, or even more, molecules (perhaps with five preferred) arranged in sets of random order. The distorted bond model This model is the basis of the uniformist theory par excellence. It was proposed by Pople37 that the key to the structure of water is that energy can be absorbed, entropy increased, and long-range order destroyed by the bending of hydrogen bonds.The melting of ice removes the restriction of synchronization of these bending motions, so that, in water, the increased amplitude and randomness of bond-bending accounts for the greater heat capacity of water than ice (about double) and by bringing more molecules into the first and second nearest-neighbour zones, for the greater density and average coordination. No hydrogen-bond breaking is envisaged. It is proposed that there is in water ‘a network of bonds extending throughout the whole liquid, which is, in a sense, one large molecule.’ Radial distribution curves in agreement with those for x-ray scattering are calculable on the basis of this theory which has also given a good, quantitative account of the dielectric constant of water over a range of temperatures.38 It is supported by Wall and Hornig21 and by Falk and Fordzla on grounds already explained.Interstitial models Morgan and Warren35 extended their analysis of the radial distribution curves derived from x-ray scattering by comparing the curve of water at 1-5 “C with that to be expected from a statistically randomized, ‘softened-up’ ice lattice, with the result shown in Fig. 10. In this diagram A is the experi- mental distribution curve, B is the curve for softened-up ice, and C is the difference, A - B. It shows an excess density of water molecules at a distance of 3.5 A from the arbitrary reference molecule at the origin. Such a density could be provided by water molecules, dislodged from ice-like lattice sites, occupying the centres of the large cavities in the ice Ih structure. The idea that ice ‘melts internally’, self-stabilizing residual structure, well explains the anomaly of structural expansion accompanied by over-all contraction, and also provides a means of understanding the maximum density pheno- menon.RIC Reviews 92 I 0 20 3 0 4 0 5 0 6 0 7 0 r [HI- Fig. 10. Calculated and experimental radial distribution curves for water at I - 5 " C after Morgan and Warren.35 A = experimental, B = 'softened-up' ice, C = A - B. This was the basis of the first interstitial model of liquid water proposed by Samoilov.39 F ~ r s l i n d , ~ ~ considering the progressive generation of defects in the structure of ice by rise of temperature, arrived at a similar model.Later, the refined x-ray studies of Danford and Levy41 were shown to be consistent with the existence in liquid water of an expanded tridymite-like lattice, with partial occupation of the cavities by non-lattice molecules. Similar work by the same has recently indicated that water retains average coordination of 4.4 to 4-5 and changes but little in mean nearest-neighbour distance (2.82-2.94 A) between 4" and 200°C. The tridymite lattice arrange- ment that fits the radial. distribution curves is, however, anisotropically expanded, and the non-lattice molecules are not at the centres of the cavities- instead of six nearest lattice neighbours, they have three. This is a puzzling result, but it is not advanced as an unique solution to the distribution curves, which cannot decisively discriminate between alternative models.Not dissimilar proposals came from two aspects of the behaviour of water with non-polar solutes, including the noble gases and hydrocarbons. First, the formation of 'gas hydrates'-crystalline hydrates, such as C3Hs,17HzO, by substances noted for their lack of affinity for water. The structures of many of these hydrates are now well e~tablished~~; they are clathrates, with guest molecules held in the cavities of hydrogen bonded cages. The simplest such cage is the pentagonal dodecahedron of 20 water molecules illustrated in Fig. 11, where it is represented as enclosing a methyl halide m0lecule.~3 This unit cannot grow further in three dimensions because of five-fold sym- metry, but it has hydrogen-bonding sites left over for spatial linking of permissible symmetry with other such units.Secondly, non-polar solutes, although very sparingly soluble, dissolve in water with surprisingly great loss of enthalpy and entropy (e.g. CH4; AH" = 3.19 kcal mole-1; ASo = - 31-8 cal OK-1mole-1 at 25"C), only explicable in terms of their effect on the solvent, and water is unique as a solvent in this respect. The suggestion by Frank and Evans44 that each solute particle becomes embedded in an 'iceberg' has become celebrated, but is now 93 Ives and Lemon Fig. I I Pentagonal dodecahedra1 water cage. less graphically described in terms of structure-promotion in the ambience of the non-polar solute particle, with no implication that it is a normal ice- structure being promoted.There is no doubt about: the generality of this effect, now referred to as ‘hydrophobic hydration’. Since it is particularly typical of the gas-hydrate formers, the view is held45 that the structure promoted is that of the appropriate hydrate-but not in the sense that crystalline solute entities exist in solution. It was Pauling46 who proposed the ‘water hydrate’ theory-that water is, in effect, a clathrate hydrate of itself. The structure envisaged is shown in Fig. 12. Despite a certain advantage in maximizing hydrogen-bond-forming possibilities, the general view seems to be that this model involves more long-range order than is appropriate for a liquid.It is said not to be consistent with x-ray scattering mea~urements.41~ Interstitial models, generalized as ‘any quasi-crystalline framework with single molecules occupying interstitial sites and making no contribution to the total volume’ were shown by Frank and Q u i ~ t ~ ~ to be capable of giving a good account of the thermodynamic properties of water-except perhaps for the heat capacity, but to call for free rotation of the interstitial monomers, as if they were situated in a field-free region. Cluster models Frank and Wen’s ‘suggested picture of water stru~ture’~83 49 laid emphasis on the dynamic nature of any possible structure in water, and on the co- operative nature of hydrogen bonding.To quote49: ‘Liquid water is thus pictured as consisting of flickering clusters of bonded molecules mixed with and alternating roles with non-bonded fluid which encloses them, and con- RIC Reviews 94 Fig. 12. Pauling’s water structure model. stitutes the rest of the sample. A (larger or smaller) cluster is pictured as leaping to attention, so to speak, when the stage is set by an energy fluctuation which creates a suitably “cold” region, and relaxing “at ease” when the necessary energy of “melting” becomes available.’ Emphasis is placed on a closer definition of the hydrogen bonding con- cerned in cluster formation : that there is considerable mutual polarization of bonded molecules, resulting in charge separation and rehybridization ; a considerable non-linear dependence of such charge separation on 0-H and 0 a * .- . 0 distances ; a considerable mutual neutralization of net effective charge between each proton and lone pair involved, so that in a fully-bonded system-ice or a cluster-these charges will be largely suppressed, their interaction with other charges or dipoles virtually disappearing. This provides a basis for the integrity, or distinguishability of a cluster, and leaves open possibilities for interactions responsible for cohesion of the denser fluid (hardly to be regarded as a normal ‘monomer liquid’) surrounding it. Voluminous clusters and denser fluid of higher energy (perhaps not sub- stantially higher) provide the means to interpret the maximum density and the high heat capacity.Hydrophobic hydration appears in a new guise. Non-polar solute molecules, lacking strong external fields of force, are relatively incapable of transmitting disruptive influences. Replacing the higher energy more-disordered water molecules from which clusters normally receive their ‘heat of melting’, the inert molecules increase the half-life of clusters bordering on them, and add to the statistical ice-likeness of the system. This theory of water is based on a ‘mixture model’, which, like the inter- stitial or clathrate models, is also ‘bistructural’ in that it envisages water Ivzs and Lemon 95 7 Fig. 13. Nemethy and Scheraga’s water structure model. molecules in two main roles or situations, no doubt with rapid exchange between them.There is a sharp contrast with ‘uniformist’ theories. Evidence is available however, to support two states for water molecules in liquid water. A mixture, or ‘two-fluid7, model for water was found to be necessary by Hall50 to explain its excessive absorption of sound (which does, however, fall away quite rapidly with rising temperature) in terms of a shift of equili- brium between bulky and dense states imposed by ultrasonic compression waves. Inelastic scattering of cold neutrons51 has also indicated that water can support lattice-like vibrations, and also, with less certainty, that it contains non-translating (presumably trapped) monomers. Clusters have to be viewed on an appropriate time-scale. They are not, in the ordinary sense, polymeric entities in equilibrium with each other and with monomers, since relaxation methods do not find them.Water (unlike alcohols) has a single dielectric relaxation-time of about 10-11 sec52-not a whole family of relaxation times as might be expected-and a common activation energy, 4.6 kcal mole-1, for dielectric relaxation, self-diffusion, shear viscosity and bulk viscosity. The implication is that the same funda- mental process is concerned, and this, in the theory, is regarded as the forma- tion and dissolution of clusters. This assigns to the clusters a half-life of the order of 10-11sec, about 100 to 1000 times the period of a molecular vibration and considered adequate to confer meaningful existence to the inside and clusters.NCmethy a cluster Scheraga53 (containing have - 25 adopted molecules this model, within requiring the temperature the molecules range RIC Reviews 96 0-70°C) to be fully hydrogen bonded, and excluding the cavities within a cluster from occupation by water or non-polar solute molecules. The surfaces of the clusters contain water molecules forming three or two bonds, or one bond, and the clusters are separated by water molecules having no hydrogen bonds. This model is illustrated in Fig. 13. Water molecules were ascribed equally spaced energy states* according to the number of hydrogen bonds in which they participate: four, three, two, one or zero. Given suitable assumptions, a partition function set up for this five-state model allowed thermodynamic properties of water to be calculated by the ‘significant structures’ method with considerable success. Vand and Senior54 have modified this model, replacing the inter-cluster monomeric fluid by an equilibrium mixture containing dimers and straight or branched polymers not forming ring- or cage-systems.They also replaced the concept of discrete energy levels by energy bands, centred on the three levels assumed by Buijs and Choppin,23 and were able to calculate thermodynamics functions from 0-100°C, assuming those at 50°C, with remarkable accuracy. The dense, non-hydrogen-bonded fluid, required by cluster theories to act as a flotation medium for the ice-like clusters, presents a difficulty which has clearly caused discomfort.The molecules in this fluid cannot seriously be considered as ‘free monomers’- they must be in some strongly interacting state to hold the whole system together in a manner consistent with the high internal pressure (20 000 atm) and strong cohesion of water. Are dipole-dipole and dispersion forces, as suggested by Nkmethy and Scheraga, adequate ? Comments previously made on the basis of Fig. 1 show that this assumption is debatable. It has however been maintained55 that we misunderstand the strength of such forces, and evidence has been advanced that something other than hydrogen bonding can provide a high cohesive pressure between water molecules. For instance, DzO, normally less volatile and with greater ‘hydrogen bondedness’ than water, becomes more volatile than water above 220°C and has a lower critical temperature.This is thought to mean that the molecules have more than one way of pulling each other together, and that at low temperatures one of them preponderates, and at high temperature another. This may be so, but it is not comfortable to suggest that there is another area of ignorance which lies between us and the problem to be solved. A two-state model not involving non-hydrogen-bonded water has been proposed by Davis and Litovitz.56 Briefly, it invokes the maximal strength of linear ‘covalent’ hydrogen bonds, which, because of the bonding geometry of the water molecule, will strongly favour the formation of closed, six- membered, hydrogen-bonded rings. Ring closure would strengthen all the bonds by co-operative effect.These rings would be identical with those in the puckered layers of normal ice, so that the first step in the formation of an ice-like cluster would be accomplished. But, in liquid water, rings could conceivably come together in an alternative way providing another, less deep, potential minimum. In normal ice (Ih), adjacent layers are in mirror- image relation. Consider a hexagonal, non-planar ring in one layer, situated vertically above the corresponding ring in the layer beneath. Three oxygen * This equal spacing, is hardly consistent with the co-operative nature of hydrogen bonding, on which the formation of ‘flickering clusters’ would seem to depend. Ives and Lemon 97 7§ Fig.14. Luck’s cluster model for water just above 0°C. atoms in one ring are hydrogen bonded to their ‘image’ oxygen atoms in the other, completing delineation of a cavity. If these three inter-ring bonds break, the upper ring is free to rotate in its own plane, and a twist of 60” will bring it into ‘stacking chair relation’ with the lower ring. Collectively, such rearrangements lead to a nearly close-packed body-centred cubic structure in which each oxygen has two hydrogen-bonded nearest neigh bours and five non-bonded nearest neighbours with which it can interact otherwise. This model can be reconciled with the radial distribution curves and gives a good account of thermal expansion and heat capacity. Luck’s cluster model5’ also excludes ‘free’ water molecules to an extent consistent with Fig.5. An ice-like cluster form is adopted, and on the basis of the co-operative nature of hydrogen bonding, it is assumed that broken bonds and free OH groups will not be randomly distributed, but will be situated on fissure surfaces, or ‘defect planes’ such as Frenke158 has proposed to be characteristic of liquids. This provides a snapshot model of the kind shown in Fig. 1G‘flickering’ with a period of sec, with closing and opening of the fissure surfaces, where the excess density is located. Account is taken of ‘unfavourable’ hydrogen bonds, and the liquid is thought of as possessing a range of different states in accordance with the Vand and Senior picture.54 RIC Reviews This theory, based on an interpretation of spectroscopic observations, gives, in terms of calculations which the author describes as ‘needing re- finement’, an impressive account of the behaviour of water.Very interesting observations in the supercritical region lend support to the postulation of such extensive clustering. In effect, liquid persists, and, without a meniscus, occupies the lower part of the infrared cell at temperatures 30-50” above 98 Tc, confirming many older observations, and suggesting that the critical phenomenon is a matter for the surface chemists. The latent heat of evapora- tion of the invisible, supercritical liquid appears as a set of maxima in heat capacity which allows the normal vapour pressure curve to be smoothly extended on P(T) diagram up to T/Tc ri 1.7.F r a n ~ k , ~ ~ studying supercritical solutions of water in argon (allowing density to be widely adjusted as an independent variable), has used excess proton mobility (proton jumps along a hydrogen-bonded path) in diagnosis of cluster formation. Considerable molecular aggregation has been found on this basis at 4OO0C, at densities greater than 0.5 g cm-3. It is impracticable to follow the changes which continue to be rung on the not very clear classification of water-structure models, each supported by its own evidence, and each encountering its own difficulties. Pending new and decisive diagnostic experiments, the present need seems to be a widening of the contexts of thought and investigation-first (as has been attempted) to consolidate bases of comparison and terms of reference that are reasonably sure; secondly by studying ‘tangential’ problems-flank attacks.There is, indeed, intense activity in the latter kind of study, particularly in relation to systems of which water is one component. This is tantamount to using solutes (ionic or molecular) and surfaces as probes to find out more about the inwardness of the behaviour of water. The remaining sections are briefly concerned with some tangential studies. THERMAL ANOMALIES These are transitions of higher order in the properties of water, occurring within a restricted temperature range, and the problem is, whether or not there is acceptable evidence for their existence.This has been in debate for a long time, notably since Magat’s 1935 paper.60 Certainly not all the reported ‘kinks’ in these properties plotted against temperature can be easily swallowed. There is a company of witch-hunters maintaining that authors reporting anything but perfectly smooth and continuous functions have under-estimated their experimental errors.6l The importance of this problem to the theory of water is apparent from the previous discussion (p. 75) of the possibilities of quasi- or anti-crystalline clusters in liquids, but there is also a possible biological significance in that one of the best supported ‘submerged transi- tions’, at 30-35”C, may have something to do with the slightly higher tempera- ture that evolution has selected as the optimum body temperature of mam- mals.This view is held by Drost-Hansen, who has carried out the most exhaustive assessments of data on water, aqueous solutions, and many kinds of systems with behaviour determined by their properties.62 His firm belief in the reality of these transitions is shared by the present writers63 and new evidence of them continues to come to hand. Two items may be mentioned. Luck64 finds that the wavelength of the maximum infrared absorption in the 1-14-1.25 p region is displaced in the direction of shorter wavelength with rising temperature, but between 36” and 38°C the displacement is greater by an order of magnitude than anywhere else on the temperature scale. Ives and Lemon 99 I I 1 1 I I I I I I I 1 I I 3 0 10 I 20 30 40 50 60 70 Temperature ("C) Fig.15. Differences in i.r. absorptions of water at 2100 and 1900cm-1. Figure 15 shows, as a function of temperature, differences in infrared absorptions of water at 2100 and 1900 cm--l found and plotted realistically by Salama and Goring.65 The method of moving quadratics confirmed the already obvious inflexion between 30" and 40°C. The band at 2100cm-1 is a combination of v2 (1646 cm-l) and librational modes at 500-700 cm-l. The result is associated with a fading-out of the latter with rise of temperature, observed earlier by Magat and attributed to the onset of rotation of water molecules about their axes of symmetry. HYDROPHOBIC BONDING AND CO-SOLVENT BEHAVIOUR The hydrophobic bond, brought into prominence by NCmethy and Scheraga,66 depends for its existence on the peculiarities of water.It can be understood by considering that the dissolution of hydrocarbons in water is disfavoured by the large entropy loss involved in hydrophobic hydration, but is favoured by the heat evolution and the entropy of mixing. Solute particles are randomly distributed and move freely about the solution. The solubility at any given temperature is determined by the balance of these factors; solute particles do not aggregate together because their loss of kinetic freedom would disturb this balance. If, however, the hydrocarbon is not in the form of separate molecules, but is attached as a side-chain to some very large molecule, such as that of a protein, this factor unfavourable to aggregation is obviated.Side-chains of proteins (e.g. of leucine or valine units), normally hydrophobi- cally hydrated in an aqueous medium, do seek their own kind. Contact established between two such side-chains, possibly situated on consecutive RIC Reviews 100 6 i Fig. 16. Sound absorption of t-butyl alcohol-water mixtures as functions of mole fraction of alcohol, xe, after Blandamer et 01.67 turns of a helix, allows some hydrophobic hydration to be sloughed off. The increase of entropy arising from the release of water molecules makes a contribution to the stability of the contact, and dispersion forces do the rest. This is the hydrophobic bond which, with hydrogen bonding between functional groups, is instrumental in determining conformation.Not dissimilar effects occur in ostensibly simpler systems-such as mixtures of water with an organic ‘co-solvent’. The example of t-butyl alcohol in water may be q ~ o t e d . ~ At the limit of low concentration, the difference between the partial molar volume of the alcohol and its molar volume in the pure state (V2 - V i ) is negative-in line with many non-polar solutes. Use is being made of existing cavities in water. But with increasing mole fraction of the alcohol, 7 2 decreases to a minimum which, at 15”C, lies at x2 = 0.04, a mole fraction at which other properties show extrema or inflexions. Quite the most dramatic of these is ultrasonic absorption,67 illustrated in Fig. 16. This can be interpreted in terms of an initial acceptance by the water, as host, of the t-butyl alcohol molecules, the hydrocarbon part of the latter being hydrophobically hydrated, and the hydroxyl group participating in the structure of the hydrogen-bonded clathrate cage.With rising concentra- tion of solute, minor adjustment occurs, in the sense of the water becoming more accommodating for the guest molecules. Then there is a ‘switch’ which is suggested to be hydrophobic dimerization-in effect, hydrophobic bonding between the t-butyl groups of partnered molecules. There can be no doubt that the dimerized solute forms its own, new, larger cage of hydrophobic hydration. Other examples of such dimerization have been reported, and it no doubt plays a part in the pre-micellar behaviour of long-chain soaps9 The general comment on this is that there seems to be practically no limit to the versatility of water in forming hydrogen-bonded cages.Ives and Lemon 101 CONCLUSION An attempt at summing up can take only the form of questions, such as the following : Do current theories of the structure of water take enough guidance from theories of simpler liquids? In the Bernal theory, pseudonuclei are ‘cold’ fluctuations ; in them, the strongest intermolecular attraction available succeeds in producing evanescent, ordered regions of diminished energy and enhanced density. In water, the strongest intermolecular attraction available is almost certainly that of hydrogen bonding, producing evanescent, ordered regions of diminished energy and diminished density-provided ‘clusters’ are accepted.Are these clusters to be considered as pseudonuclei? If the strength of hydrogen bonding depends much on linearity, should the clusters not be expected to be quasi- or anti-crystalline? If so, should not thermal anomalies be a natural consequence? How strong are the grounds for excluding the trapping of ‘molecules-in-transit’ in the cavities of clusters- formed and dissipated with immense rapidity-thus rejecting interstitial models ? In view of the solid-like behaviour of liquids in general, how convincing is the evidence of bistructural or mixture models, to the exclusion of uni- formist theories? Clusters, with their cavities, do not serve to explain the increase in volume between TM and Tc, nor, because of their enhanced rigidity, the facts of fluidity.Are not vacancies of a different order from ice-like cavities required, oppositely dependent in number on temperature ? If so, should they not be built-in to a proper structural theory? One view of hydro- phobic hydration is that an inert solute particle does no more than create a cavity in the water, and that the internal, closed surface of such a cavity is as naturally ‘icy’ as the free surface of water is held to be. Other reviews would be needed to survey the theories of the behaviour of water with respect to a second component. Does a solid surface have a long- range effect on the structure of water adjacent to it? If so, is this imposed order subtended by a misfit region of enhanced disorder, so that we need a theory of a ‘structural double layer’-perhaps before we can fully under- stand the electrical double layer so fundamental to colloid systems ? What is the state of water held in clays and finely porous materials? Is there really, close to some surfaces, a radically different kind of water with inter alia, 12-15 times the viscosity of normal water?69 Similar sets of questions are posed by the reactions of water to ionic solutes-the structure-makers and structure-breakers-but at least in this area we have a theory hard to resist44: that order imposed by ionic fields is subtended by a misfit region of disorder.It is clear that the time has come for an apologia.The writers are conscious of all the evidence ignored or partially and selectively presented, but make the plea that the subject is such that they had no option. There is, however, one final question that may arise in the reader’s mind. How is it that water, being such an infernally complicated thing, so little attention seems to have been paid to it until quite recent times? This returns us to the protective, simplifying principle mentioned in the Introduction (p. 62). It is the Compen- sation Law on which ‘linear free energy relation^'^ depend. In terms of the RIC Reviews 102 standard thermodynamics functions controlling an equilibrium or a reaction rate, AGt = AH$ - TAS$ there is a natural tendency for the A H and TAS terms to bs of the same sign (e.g.if forces of attraction prevail, producing order from disorder, both AH and A S are negative). Their effects on the AG term therefore partly cancel, so that AG is a more stolid, well behaved function than AH or AS. For the reactions in solution (particularly aqueous solution) on the study of which so much physico-chemical progress has depended, there can be no doubt at all that changes in hydration accompany the transformation of reactants into products and contribute largely to enthalpies and entropies of reaction qr activation. It is not however the AH and A S terms that have been predominantly studied, but rather the AG terms-or, what amounts to much the same thing, the equilibrium constants and the velocity constants. It is the well-behaved nature of AG that has afforded the protection from complication. The argument is this70: for an equilibrium in aqueous solution, we can, in principle, split the AGO of reaction into two parts-a ‘reaction- proper’ part (with its intramolecular energy and entropy contributions) and a hydration part-to include all the changing effects of the solute reaction system on the water used as solvent, i.e.AGO = AGr + AGh But if it is assumed that all the hydrational systems remain in equilibrum with one and the same bulk solvent all the time, all the hydrationalprocesses occur at equilibrium, so that AGh = 0. On the other hand AHh and ASh are certainly not equal to zero; it is just that, as in the more familiar freezing and melting processes, AHh = TASh.This seems to be the basis of the special application of the Compensation Law-which can no longer keep us in blink- ers. AGO = AH0 - TASO; ACKNOWLEDGEMENTS The authors record thanks to Dr L. J. Bellamy and Dr D. N. Glew for advice and discussion. The authors also wish to thank the following for permission to reproduce copy- right material: Heinemann Educational Books Ltd for Figs. 3, 5 and 14; Scientific American Inc. for Fig. 7; Verlag Chemie for Fig. 8; Plenum Publishing Corporation for Figs. 9 and 10; The Faraday Society for Fig. 11 ; Pergamon Press for Fig. 12: the American Institue of Physics for Fig. 13; the American Chemical Society for Fig. 15; and The Chemical Society for Fig. 16. REFERENCES Pergamon, 1966.4 E.g. J. E. Leffler and E. Grunwald, Rates and Equilibria of Organic Reactigns. 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