The structural symmetry and molecular separation in water and ice remain uncertain. We present herewith a solution to unifying the density, the structure order and symmetry, the size (H-O length dH) and the separation (dOO = dL + dH or the O:H length dL) of molecules packing in water and ice in terms of statistic mean. This solution reconciles: i) the dL and the dH symmetrization of the O:H-O bond in compressed ice, ii) the dOO relaxation of cooling water and ice and, iii) the dOO expansion of a dimer and between molecules at water surface. With any one of the dOO, the density ρ(g·cm−3), the dL and the dH, as a known input, one can resolve the rest quantities using this solution that is probing conditions or methods independent. We clarified that: i) liquid water prefers statistically the mono-phase of tetrahedrally-coordinated structure with fluctuation, ii) the low-density phase (supersolid phase as it is strongly polarized with even lower density) exists only in regions consisting molecules with fewer than four neighbors and, iii) repulsion between electron pairs on adjacent oxygen atoms dictates the cooperative relaxation of the segmented O:H-O bond, which is responsible for the performance of water and ice.
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.
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.
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.
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.
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.
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:
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.
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.
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.
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.
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.
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.
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.
Medcraft, C. et al. Water ice nanoparticles: size and temperature effects on the mid-infrared spectrum. PCCP 15, 3630–3639 (2013).
Petkov, V., Ren, Y. & Suchomel, M. Molecular arrangement in water: random but not quite. J. Phys.: Condens. Matter 24, 155102 (2012).
Tsai, M. K., Kuo, J. L. & Lu, J. M. The dynamics and spectroscopic fingerprint of hydroxyl radical generation through water dimer ionization: ab initio molecular dynamic simulation study. PCCP 14, 13402–13408 (2012).
Liu, Y. & Wu, J. Communication: Long-range angular correlations in liquid water. J. Chem. Phys. 139, 041103 (2013).
Ostmeyer, J. et al. Recovery from slow inactivation in K channels is controlled by water molecules. Nature 501, 121–124 (2013).
Gierszal, K. P. et al. π-Hydrogen Bonding in Liquid Water. J. Phys. Chem. Lett. 2, 2930–2933 (2011).
Medcraft, C. et al. Size and temperature dependence in the far-Ir spectra of water ice particles. Astrophys. J. 758, 17 (2012).
Skinner, L. B. et al. Benchmark oxygen-oxygen pair-distribution function of ambient water from x-ray diffraction measurements with a wide Q-range. J. Chem. Phys. 138, 074506 (2013).
Wikfeldt, K. T. et al. Oxygen-oxygen correlations in liquid water: Addressing the discrepancy between diffraction and extended x-ray absorption fine-structure using a novel multiple-data set fitting technique. J. Chem. Phys. 132, 104513 (2010).
Bergmann, U. et al. Nearest-neighbor oxygen distances in liquid water and ice observed by x-ray Raman based extended x-ray absorption fine structure. J. Chem. Phys. 127, 174504 (2007).
Morgan, J. & Warren, B. E. X-ray analysis of the structure of water. J. Chem. Phys. 6, 666–673 (1938).
Naslund, L. A. et al. X-ray absorption spectroscopy study of the hydrogen bond network in the bulk water of aqueous solutions. J. Phys. Chem. A 109, 5995–6002 (2005).
Orgel, L. The hydrogen bond. Rev. Mod. Phys. 31, 100–102 (1959).
Wilson, K. R. et al. X-ray spectroscopy of liquid water microjets. J. Phys. Chem. B 105, 3346–3349 (2001).
Narten, A. H., Thiessen, W. E. & Blum, L. Atom pair distribution functions of liquid water at 25°C from neutron diffraction. Science 217, 1033–1034 (1982).
Fu, L., Bienenstock, A. & Brennan, S. X-ray study of the structure of liquid water. J. Chem. Phys. 131, 234702 (2009).
Kuo, J. L., Klein, M. L. & Kuhs, W. F. The effect of proton disorder on the structure of ice-Ih: A theoretical study. J. Chem. Phys. 123, 134505 (2005).
Soper, A. K. Joint structure refinement of x-ray and neutron diffraction data on disordered materials: application to liquid water. J Phys Condens Matter 19, 335206 (2007).
Wilson, K. R. et al. Surface relaxation in liquid water and methanol studied by x-ray absorption spectroscopy. J. Chem. Phys. 117, 7738–7744 (2002).
Liu, K., Cruzan, J. D. & Saykally, R. J. Water clusters. Science 271, 929–933 (1996).
Hakala, M. et al. Intra- and intermolecular effects in the Compton profile of water. Phys. Rev. B 73, 035432 (2006).
Wernet, P. et al. The Structure of the first coordination shell in liquid water. Science 304, 995–999 (2004).
Soper, A. K. Recent water myths. Pure Appl. Chem. 82, 1855–1867 (2010).
Head-Gordon, T. & Johnson, M. E. Tetrahedral structure or chains for liquid water. PNAS 103, 7973–7977 (2006).
Kuhne, T. D. & Khaliullin, R. Z. Electronic signature of the instantaneous asymmetry in the first coordination shell of liquid water. Nat. commun. 4, 1450 (2013).
Sun, C. Q. et al. Density, elasticity and stability anomalies of water molecules with fewer than four neighbors. J. Phys. Chem. Lett. 4, 2565–2570 (2013).
Huang, C. et al. The inhomogeneous structure of water at ambient conditions. PNAS 106, 15214–15218 (2009).
Wernet, P. et al. The structure of the first coordination shell in liquid water. Science 304, 995–999 (2004).
Nilsson, A., Huang, C. & Pettersson, L. G. M. Fluctuations in ambient water. J. Mol. Liq. 176, 2–16 (2012).
Sun, C. Q. Oxidation electronics: bond-band-barrier correlation and its applications. Prog. Mater Sci. 48, 521–685 (2003).
Zhao, M. et al. Atomistic origin, temperature dependence and responsibilities of surface energetics: An extended broken-bond rule. Phys. Rev. B 75, 085427 (2007).
Sun, C. Q., Zhang, X. & Zheng, W. T. Hidden force opposing ice compression. Chem. Sci. 3, 1455–1460 (2012).
Pauling, L. The structure and entropy of ice and of other crystals with some randomness of atomic arrangement. J. Am. Chem. Soc. 57, 2680–2684 (1935).
Wang, Y. et al. High pressure partially ionic phase of water ice. Nat. Commun 2, 563 (2011).
Sun, C. Q. et al. Density and phonon-stiffness anomalies of water and ice in the full temperature range. J Phys. Chem. Lett. 4, 3238–3244 (2013).
Yoshimura, Y. et al. High-pressure x-ray diffraction and Raman spectroscopy of ice VIII. J. Chem. Phys. 124, 024502 (2006).
Sugimura, E. et al. Compression of H2O ice to 126 GPa and implications for hydrogen-bond symmetrization: Synchrotron x-ray diffraction measurements and density-functional calculations. Phys. Rev. B 77, 214103 (2008).
Benoit, M., Marx, D. & Parrinello, M. Tunnelling and zero-point motion in high-pressure ice. Nature 392, 258–261 (1998).
Goncharov, A. F., Struzhkin, V. V., Mao, H.-k. & Hemley, R. J. Raman spectroscopy of dense H2O and the transition to symmetric hydrogen bonds. Phys. Rev. Lett. 83, 1998–2001 (1999).
Mallamace, F. et al. The anomalous behavior of the density of water in the range 30 K < T < 373 K. PNAS 104, 18387–18391 (2007).
Erko, M. et al. Density minimum of confined water at low temperatures: a combined study by small-angle scattering of X-rays and neutrons. PCCP 14, 3852–3858 (2012).
Sulpizi, M., Salanne, M., Sprik, M. & Gaigeot, M.-P. Vibrational sum frequency generation spectroscopy of the water liquid–vapor interface from density functional theory-based molecular dynamics simulations. J. Phys. Chem. Lett. 4, 83–87 (2013).
Solveyra, E. G. et al. Structure, Dynamics and Phase Behavior of Water in TiO2 Nanopores. J. Chem. Phys. C 117, 3330–3342 (2013).
Uysal, A. et al. What x rays can tell us about the interfacial profile of water near hydrophobic surfaces. Phys. Rev. B 88, 035431 (2013).
Kasuya, M. et al. Characterization of Water confined between Silica surfaces using the resonance shear measurement. J. Phys. Chem. C 117, 13540–13546 (2013).
Goldschmidt, V. M. Crystal structure and chemical correlation. Ber Deut Chem Ges 60, 1263–1296 (1927).
Pauling, L. Atomic radii and interatomic distances in metals. J. Am. Chem. Soc. 69, 542–553 (1947).
Kahan, T. F., Reid, J. P. & Donaldson, D. J. Spectroscopic probes of the quasi-liquid layer on ice. J. Phys. Chem. A 111, 11006–11012 (2007).
Siefermann, K. R. et al. Binding energies, lifetimes and implications of bulk and interface solvated electrons in water. Nat. Chem. 2, 274–279 (2010).
Ishiyama, T., Takahashi, H. & Morita, A. Origin of vibrational spectroscopic response at ice surface. J. Phys. Chem. Lett. 3, 3001–3006 (2012).
Zhang, C., Gygi, F. & Galli, G. Strongly anisotropic dielectric relaxation of water at the nanoscale. J. Phys. Chem. Lett. 4, 2477–2481 (2013).
Qiu, H. & Guo, W. Electromelting of confined monolayer ice. Phys. Rev. Lett. 110, 195701 (2013).
Wang, C. et al. Stable liquid water droplet on a water monolayer formed at room temperature on ionic model substrates. Phys. Rev. Lett. 103, 137801–137804 (2009).
James, M. et al. Nanoscale condensation of water on self-assembled monolayers. Soft Matter 7, 5309–5318 (2011).
Winter, B. et al. Hydrogen bonds in liquid water studied by photoelectron spectroscopy. J. Chem. Phys. 126, 124504 (2007).
Abu-Samha, M. et al. The local structure of small water clusters: imprints on the core-level photoelectron spectrum. J. Phys. B 42, 055201 (2009).
Nishizawa, K. et al. High-resolution soft X-ray photoelectron spectroscopy of liquid water. PCCP 13, 413–417 (2011).
Vacha, R. et al. Charge transfer between water molecules as the possible origin of the observed charging at the surface of pure water. J. Phys. Chem. Lett. 3, 107–111 (2012).
Ni, Y., Gruenbaum, S. M. & Skinner, J. L. Slow hydrogen-bond switching dynamics at the water surface revealed by theoretical two-dimensional sum-frequency spectroscopy. PNAS 110, 1992–1998 (2013).
Sun, H. COMPASS: An ab initio force-field optimized for condensed-phase applications: Overview with details on alkane and benzene compounds. J. Phys. Chem. B 102, 7338–7364 (1998).
He, C., Lian, J. S. & Jiang, Q. Electronic structures and hydrogen bond network of ambient water and amorphous ices. Chem. Phys. Lett. 437, 45–49 (2007).
Natarajan, R., Charmant, J. P. H., Orpen, A. G. & Davis, A. P. Water chains in hydrophobic crystal channels: Nanoporous materials as supramolecular analogues of carbon nanotubes. Angew. Chem. Int. Ed. 49, 5125–5129 (2010).
Critical reading by Yi Sun and financial support received from NSF (Nos.: 21273191, 1033003 and 90922025) China are gratefully acknowledged.
The authors declare no competing financial interests.
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Huang, Y., Zhang, X., Ma, Z. et al. Size, separation, structural order and mass density of molecules packing in water and ice. Sci Rep 3, 3005 (2013). https://doi.org/10.1038/srep03005
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