Introduction

Water and ice has attracted much attention because of its anomalies pertaining to issues from galaxy to geology, astrophysics, biology, climate and to our daily lives1,2,3,4,5,6,7. However, the structure order, the geometric symmetry, the size and the separation between molecules packing in water and ice (H2O) and their correlation remain yet highly disputed, independent issues despite decades-long intensive investigation. For instances, the separation between adjacent oxygen atoms (dOO) was measured to vary from 2.70 to 3.00 Å8,9,10,11,12,13,14,15,16,17,18,19,20 and the molecular size (the H-O bond length dH) changes from 0.970 to 1.001 Å21. A H2O molecule demonstrates high instantaneous asymmetry with coordination numbers varying from two22 to four or even greater23. The geometric structure of the weekly-ordered H2O liquid was interpreted in terms of either the monomial-phase of tetrahedrally-coordinated structures with thermal fluctuation2,24,25,26 or the mixed-phase of low- and high-density fragmentation27,28,29. However, uncertainties in these seemingly independent issues determine jointly the density of water and ice that is macroscopically detectable but the correlation among these quantities is often ignored in consideration. This fact serves as one essential constraint for the solution to the uniqueness of structure order and molecular separation, in terms of statistic expectation, that water molecules prefer. Therefore, these structural and dimensional discrepancies can be resolved simultaneously based on the framework reported in this Letter without needing any assumption or approximation.

Results

Firstly, the sp3-orbital hybridization is the unique choice of oxygen upon reacting with atoms of relatively lower electronegativity, irrespective of the structural phase30. As shown in Figure 1a, an oxygen atom (2s22p4) catches two electrons from neighboring atoms such as hydrogen (H) and metals and then hybridizes its sp orbits with tetrahedrally directional orbits26. In the case of H2O, one O forms two intramolecular H-O bonds with shared electron pairs and ~ 4.0 eV binding energy26 and fills up the rest two orbits with its nonbonding electron lone pairs “:” to form the intermolecular O:H non-covalent bonds of < 0.1 eV binding energy31. The inhomogeneous distribution of charge and energy around the central oxygen atom entitles a H2O molecule only Cv2 group symmetry except for the rotation and vibration of the molecule. Therefore, an oxygen atom always tends to find four neighbors to form a stable tetrahedron but the nonequivalent bond angles (∠H-O-H < 104.5° and ∠H:O:H > 109.5°) and the repulsion between electron pairs on oxygen26,32 refrain the steady tetrahedron from being formed in the liquid phase. The strong fluctuation proceeds more like the motion of a complex pendulum surrounded by four non-bonding lone pairs, because of the O:H bond switching on and off restlessly in a period of sub-picosecond2,25,28,29. Therefore, it would be more realistic and meaningful to consider the statistic expectation of the coordination number, the structure order and the molecular separation in all phases at question for a long time span rather than seeking for the instantaneous accuracy of a certain independent quantity by taking the snapshot at a quick flash25 for the highly correlated and fluctuating system.

Figure 1
figure 1

Sampling procedure for the structure of water and the segmented O:H-O bond.

(a) An extension of (a) the sp3-hybridized oxygen (red) motif with two nonbonding electron lone pairs (blue) and two bonding electron pairs (yellow) results in (b) an ideal tetrahedron that contains two equivalent H2O molecules connected by four identical O:H-O bonds of different orientations. The packing of the basic building blocks (b) forms (c) a diamond structure, which ensures the tetrahedral coordination of the central oxygen atom in the coordination origin. Therefore, only four of the eight cubes in (c) are occupied by (b) and the rest four remain empty. (d) The O:H-O bond forms an asymmetric, coupled, H-bridged oscillators whose relaxation in length and energy and the associated local charge distribution determine the physical properties of water and ice26,32,35. Small pairing dots on oxygen represent the electron pairs.

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Secondly, the packing order of H2O molecules follows Pauling's Ice Rule33 in all phases except for water under extremely high temperature and high pressure34. Despite thermal fluctuation in the O:H non-covalent bond lengths and the ∠O:H-O bond angles, the average separation and the size of molecules will change when the H2O transits from the strongly-ordered solid phase, to the weakly-ordered liquid phase and to the disordered amorphous or vapor phase, as the Ice Rule retains. An extension of the Ice Rule results in an ideal tetrahedron, shown in Figure 1b, with higher C3 group symmetry. This tetrahedron containing two equivalent H2O molecules and four identical O:H-O bonds at different orientations forms the basic block building up the bulk water and ice despite fluctuations.

Thirdly, as illustrated in Figure 1c, four of the eight cubes are occupied by the basic 2H2O block tetrahedrally and the rest four cubes are empty, which means that each cube of a3 volume accommodates only one H2O molecule on average. With the known mass of a H2O molecule consisting 8 neutrons, 10 protons and 10 electrons, M = (10 × 1.672621 + 8 × 1.674927 + 10 × 0.000911) × 10−27 kg and the known density ρ = M/a3 = 1 (gcm−3) at 4°C under the atmospheric pressure, this structural order defines immediately and unambiguously the density-dependent molecular separation, dOO and the next-nearest neighboring distance √2a (unit in Å),

Finally, the O:H-O bond, in Figure 1d, consists of the longer-and-softer part of the O:H van der Waals bond (dL) and the shorter-and-stiffer part of the H-O polar-covalent bond (dH) rather than either of them alone. The O:H-O bond approximates a pair of asymmetric and H-bridged oscillators coupled by Coulomb-repulsion, whose relaxation in length and energy and the associated local charge distribution determine the anomalies of water ice under various stimuli such as compression32, coordination number reduction26 and cooling1,7,35. Under excitation, oxygen atoms dislocate along the O:H-O bond in the same direction but by different amounts with H atom as the coordination origin. The O:H-O interaction in Figure 1d holds statistically true in any phase including amorphous despite the strong fluctuations whose extent is subject to the thermal conditions due to the switching on and off the O:H interactions.

A molecular dynamics (MD) computation has enabled us to decompose the measured volume-pressure V(P) profile of compressed ice36,37 into the dH(P) and the dL(P) cooperative curves32, see Figure 2. The dx(P) curves meet at dL = dH = 1.12 Å under ~ 59 GPa pressure of ice, which is exactly the measured proton symmetrization of hydrogen bond in ice38,39.

Figure 2
figure 2

(a) MD calculation reproduction of the V(P) profile of ice36 with derivatives of the O:H and H–O lengths meeting at dH = dL = 1.12 Å under 58.6 GPa compression, which agrees with the measurements of dH = dL = 1.12 Å at 59 GPa38,39.

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This coincidence indicates that the MD derived dx(P) relation represents the true cooperativity of the dL and the dH bond relaxation. Plotting the dL(P) against the dH(P) yields immediately the (projection along the O—O) length cooperativity that is free from probing conditions or probing methods,

The dx (x = L and H) value approaches the true bond length with ~ 1.5% deviation (1-cos(10°) = 0.015) as the O:H-O angle remains 160° in liquid and greater in solid35. Combining eqs (1) and (2), one is able to scale the size dH and the separation dOO of H2O molecules with the given packing order in Figure 1c and the measured density under various conditions. If the dOO or the dH matches those of direct measurement, the structure order in Figure 1c and eqs (1) and (2) are justified true and unique.

Using eq 1, one can convert, as shown Figure 3a for instance, the measured density ρ(T) profiles of water droplets of different sizes (1.4 and 4.4 nm)40,41 as input into the dOO as an output for water at different temperatures. The density transition points change with water droplet size. For droplet of 1.4 nm, the transition is at 205 K, it is at 242 K for 4.4 nm droplet and 258 K for the bulk water35. The droplet size discriminated density transition arises from the specific heat disparity of the O:H- and the H-O within the O:H-O bond. As the droplet size is reduced, the H-O bond becomes shorter and stiffer yet the O:H bond the otherwise26, which shifts the cross points of the two specific heat to temperatures outwardly away from that of the bulk (refer to Ref. 35). The dOO in a water droplet expands additionally in the skin region42 but one can only measure its average26. The dOO values of 2.70 Å measured at 25°C and 2.71 Å at −16.8°C10 match exactly the conversion of 2.6950 Å that is a projection along the O—O at 4°C. This consistency justifies sufficiently that both eq 1 and the packing order in Figure 1c describe the true situations in both water and ice. Furthermore, the data reported in Ref. 10 is essentially accurate and correct.

Figure 3
figure 3

(a) The dOO ~ ρ(T) profiles (eq 1) of water droplets of different sizes (1.4 and 4.4 nm)40,41 match the dOO values measured at 25°C and −16.8°C (Figure S1 and S2 of Supporting Information)10. (b) The dH and the dL (eq 2) agrees with results of MD calculations35. Inset (b) illustrates the cooperative relaxation of the segmented O:H-O bond. One part becomes longer; the other part will be shorter by different amounts due to the inter-electron pair repulsion.

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Discussion

The non-covalent bond length dL, molecular size dH, molecular separation dOO and the mass density ρ can be obtained by solving the equation with any one of these parameters as a known input,

Figure 3b shows the decomposition of the dOO into the dx of water and ice at cooling40,41. The dx(T) profiles follow the rules of O:H-O bond relaxation26,32,35: i) both oxygen atoms dislocate in the same direction (see inset) along the O:H-O bond by different amounts with respect to the H atom; ii) the longer-and-softer O:H part always relaxes more than the shorter-and-stiffer H-O part does. The cooperativity of the dx relaxation confirms further that35: i) cooling contraction happens only to the O:H bond in the solid (T < 205 K (Data 1) or 241 K (Data 2)) and in the liquid phase (T > 277 K), which lengthens the H-O bond slightly by inter-electron-pair repulsion, resulting volume contraction; ii) in the freezing transition phase, the process of length relaxation reverses, leading to the O—O length gain and volume expansion at freezing.

Figure 4 shows the solution consistency to the measured molecular size dH, molecular separation dL (or dOO), mass density ρ and structural order of: i) compressed ice36, ii) cooling water and ice40,41, and, iii) water surface and dimer10,19. The dH of 1.0004 Å at unity density is within the measured values ranging from 0.970 to 1.001 Å21. The dOO values greater than the ideal value of 2.6950 Å at ρ = 1 (g·cm−3) correspond to the supersolid phase (low-density, LDP) that exists indeed27,28,29 but only presents in the skins of water ice composed of molecules with fewer than four neighbors (Figure 4b)26.

Figure 4
figure 4

Accordance of (a) molecular size dH, molecular separation dL(or dOO = dH + dL), (b) mass density ρ and packing order (see Figure 1c) of H2O molecules in the situations: (i) ice under compression (dH > 1.00 Å)36, (ii) water ice at cooling (0.96 < dH < 1.00 Å)40,41 and (iii) liquid surface and dimer (dH < 1.00 Å)8,9,10,14,15,16,17,18.

The derived dH = 1.0004 Å at ρ = 1 is within the measurements ranging from 0.970 to 1.001 Å21. The dH shorter than 0.96 Å corresponds to the supersolid phase in regions of molecules having fewer than four coordination neighbors (CN)19,20,26. In such regions, a H2O molecule shrinks in size and expands in separation because of inter electron-pair repulsion26.

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Wilson et al19 have discovered that the surface dOO expands by 5.9% from 2.801 to 2.965 Å at room temperature. If one considers the shortest distance of 2.70 Å10 and the longest 2.965 Å19 of measurements, the surface dOO expands by up to 10%. Furthermore, the volume of water molecules confined in 5.1 and 2.8 nm TiO2 pores increase by 4 and 7.5%, respectively, with respect to that in the bulk43. With a 5–10 Å thick air gap existing in between molecules and the hydrophobic surface44, water molecules at the interface exhibit skin vibration attributes45 of 3400 cm−1 compared to that of 3200 cm−1 for the bulk water. The separation dOO = 2.980 Å for a dimer is even greater.

In these supersolid regions, molecular under-coordination shortens the dH and lengthens the dL, resulting in dOO expansion and polarization because of the inter electron-pair repulsion26. The least density of ice is 0.92, which corresponds to dOO = 2.695(0.92)−1/3 = 2.7710 Å. However, the density of the supersolid phase is ρ = (2.695/2.965)3 = 0.7509 g·cm−3, which is far lower than the least density of the bulk ice or the maximal density of water (0.75/0.92/1.0), according to eq 1. Considering the limitation of penetration depth in the optical reflection measurements of water and ice, all the reported data for the skin are reasonably correct.

The molecular separation dOO = dL + dH grows and molecular size dH shrinks simultaneously at the skins because of the molecular under-coordination26. The H-O bond contraction follows Goldschmidt-Pauling's rule of “atomic coordination number-radius” correlation46,47; the dOO expansion results from the Coulomb repulsion between electron pairs on adjacent oxygen atoms26,32. The skin region, consisting molecules with fewer than four neighbors, forms such an amazing supersolid phase that possesses the attributes of low-density19, high elasticity48, polarized49,50, dielectric instability51, thermally stable52 and hydrophobic53,54 with densely entrapped bonding electrons55,56,57,58. The timescale for hydrogen-bond switching dynamics at the surface is about three times slower than that in the bulk59 because of the strong polarization and high viscosity.

The findings apply to any situations including solid-liquid (water-ice) interface skin as only mass and volume are involved. At the water-hydrophobic surface of different materials, this findings are only valid to the water skin that forms the low-density supersolid state of polarized, depleted, elastic and thermally stable26. An air gap of 0.5 ~ 1.0 nm thick presents between the superhydrophobic substrate and water44.

The straightforward yet simple solution presented herewith has thus resolved the seemingly independent geometry and dimension uncertainties of water and ice. We may conclude:

  1. i

    One should focus on the statistic mean of all the factors and their cooperativity involved rather than the instantaneous accuracy of the individual parameter once at a point of time for the strongly fluctuated water system.

  2. ii

    The size, separation, structural order and mass density of molecules packing in water and ice are correlated, which is independent of the structural phases of water and ice or the probing conditions.

  3. iii

    Constrained by the Ice Rue, the dH and dL cooperativity, the solution has reconciled measurements of hydrogen-bond length symmetrization of ice under compression, dOO relaxation of water and ice at cooling and dOO expansion of a water dimer and molecules at water surface.

  4. iv

    With any one of the molecular separation, mass density, O:H bond length and H-O distance as a known input, one can determine using this solution unambiguously the rest three parameters and their change with external conditions such as pressure, temperature and coordination environment.

  5. v

    The tetrahedrally-coordinated structure could be the unique choice of water and ice despite fluctuations in the dL and the ∠O:H-O angle due to the non-equivalent ∠H:O:H and ∠H-O-H bond angles and the inter-electron-pair repulsion.

  6. vi

    The supersolid (low-density) phase indeed exists but only in regions consisting water molecules with fewer than four neighbors. The supersolidity phase forms because of the Goldschmidt-Pauling's rule of H-O bond contraction due to molecular under-coordination and the inter-electron-pair repulsion pertaining to the O:H-O bond.

Methods

The MD calculations were performed using Forcite's package with ab initio optimized forcefield Compass2760. The Compass27 has been widely used in dealing with the electronic structures and the hydrogen bond network of water and amorphous ices61 as well as water chains in hydrophobic crystal channels62.