Size, separation, structural order, and mass density of molecules packing in water and ice

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.

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 d H ), and the separation (d OO  the d OO relaxation of cooling water and ice and, iii) the d OO expansion of a dimer and between molecules at water surface. With any one of the d OO , the density r(g?cm 23 ), the d L , and the d H , 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. W ater and ice has attracted much attention because of its anomalies pertaining to issues from galaxy to geology, astrophysics, biology, climate, and to our daily lives [1][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 (H 2 O) and their correlation remain yet highly disputed, independent issues despite decades-long intensive investigation. For instances, the separation between adjacent oxygen atoms (d OO ) was measured to vary from 2.70 to 3.00 Å [8][9][10][11][12][13][14][15][16][17][18][19][20] and the molecular size (the H-O bond length d H ) changes from 0.970 to 1.001 Å 21 . A H 2 O molecule demonstrates high instantaneous asymmetry with coordination numbers varying from two 22 to four or even greater 23 . The geometric structure of the weekly-ordered H 2 O liquid was interpreted in terms of either the monomial-phase of tetrahedrally-coordinated structures with thermal fluctuation 2,24-26 or the mixed-phase of low-and high-density fragmentation [27][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 sp 3 -orbital hybridization is the unique choice of oxygen upon reacting with atoms of relatively lower electronegativity, irrespective of the structural phase 30 . As shown in Figure 1a, an oxygen atom (2s 2 2p 4 ) catches two electrons from neighboring atoms such as hydrogen (H) and metals and then hybridizes its sp orbits with tetrahedrally directional orbits 26  109.5u) and the repulsion between electron pairs on oxygen 26,32 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 subpicosecond 2,25,28,29 . 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 flash 25 for the highly correlated and fluctuating system.
Secondly, the packing order of H 2 O molecules follows Pauling's Ice Rule 33 in all phases except for water under extremely high temperature and high pressure 34  Thirdly, as illustrated in Figure 1c, four of the eight cubes are occupied by the basic 2H 2 O block tetrahedrally and the rest four cubes are empty, which means that each cube of a 3 volume accommodates only one H 2 O molecule on average. With the known mass of a H 2 O molecule consisting 8 neutrons, 10 protons, and 10 electrons, M 5 (10 3 1.672621 1 8 3 1.674927 1 10 3 0.000911) 3 10 227 kg and the known density r 5 M/a 3 5 1 (gcm 23 ) at 4uC under the atmospheric pressure, this structural order defines immediately and unambiguously the density-dependent molecular separation, d OO , and the next-nearest neighboring distance !2a (unit in Å ),  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 ice 36,37 into the d H (P) and the d L (P) cooperative curves 32 , see Figure 2. The d x (P) curves meet at d L 5 d H 5 1.12 Å under , 59 GPa pressure of ice, which is exactly the measured proton symmetrization of hydrogen bond in ice 38,39 .
This coincidence indicates that the MD derived d x (P) relation represents the true cooperativity of the d L and the d H bond relaxation. Plotting the d L (P) against the d H (P) yields immediately the (projection along the O-O) length cooperativity that is free from probing conditions or probing methods, The d  (1) and (2), one is able to scale the size d H and the separation d OO of H 2 O molecules with the given packing order in Figure 1c and the measured density under various conditions. If the d OO or the d H 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 r(T) profiles of water droplets of different sizes  26 , which shifts the cross points of the two specific heat to temperatures outwardly away from that of the bulk (refer to Ref. 35). The d OO in a water droplet expands additionally in the skin region 42 but one can only measure its aver-   Figure 1c describe the true situations in both water and ice. Furthermore, the data reported in Ref. 10 is essentially accurate and correct.

Discussion
The non-covalent bond length d L , molecular size d H , molecular separation d OO , and the mass density r can be obtained by solving the equation with any one of these parameters as a known input,  Figure 4 shows the solution consistency to the measured molecular size d H , molecular separation d L (or d OO ), mass density r, and structural order of: i) compressed ice 36 , ii) cooling water and ice 40,41 , and, iii) water surface and dimer 10,19 . The d H of 1.0004 Å at unity density is within the measured values ranging from 0.970 to 1.001 Å 21 . The d OO values greater than the ideal value of 2.6950 Å at r 5 1 (g?cm 23 ) correspond to the supersolid phase (low-density, LDP) that exists indeed [27][28][29] but only presents in the skins of water ice composed of molecules with fewer than four neighbors (Figure 4b) 26 .
Wilson et al 19 have discovered that the surface d OO expands by 5.9% from 2.801 to 2.965 Å at room temperature. If one considers the shortest distance of 2.70 Å 10 and the longest 2.965 Å 19 of measurements, the surface d OO expands by up to 10%. Furthermore, the volume of water molecules confined in 5.1 and 2.8 nm TiO 2 pores increase by 4 and 7.5%, respectively, with respect to that in the bulk 43 . With a 5-10 Å thick air gap existing in between molecules and the hydrophobic surface 44 , water molecules at the interface exhibit skin vibration attributes 45 of 3400 cm 21 compared to that of 3200 cm 21 for the bulk water. The separation d OO 5 2.980 Å for a dimer is even greater.
In these supersolid regions, molecular under-coordination shortens the d H and lengthens the d L , resulting in d OO expansion and polarization because of the inter electron-pair repulsion 26 . The least density of ice is 0.92, which corresponds to d OO 5 2.695(0.92) 21/3 5 2.7710 Å . However, the density of the supersolid phase is r 5 (2.695/ 2.965) 3 5 0.7509 g?cm 23 , 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 d OO 5 d L 1 d H grows and molecular size d H shrinks simultaneously at the skins because of the molecular undercoordination 26 . The H-O bond contraction follows Goldschmidt-Pauling's rule of ''atomic coordination number-radius'' correlation 46,47 ; the d OO expansion results from the Coulomb repulsion between electron pairs on adjacent oxygen atoms 26,32 . The skin region, consisting molecules with fewer than four neighbors, forms such an amazing supersolid phase that possesses the attributes of low-density 19 , high elasticity 48 , polarized 49,50 , dielectric instability 51 , thermally stable 52 and hydrophobic 53,54 with densely entrapped bonding electrons [55][56][57][58] . The timescale for hydrogen-bond switching dynamics at the surface is about three times slower than that in the bulk 59 because of the strong polarization and high viscosity.
The findings apply to any situations including solid-liquid (waterice) 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 stable 26 . An air gap of 0.5 , 1.0 nm thick presents between the superhydrophobic substrate and water 44 .
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. (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.

Methods
The MD calculations were performed using Forcite's package with ab initio optimized forcefield Compass27 60 . The Compass27 has been widely used in dealing with the electronic structures and the hydrogen bond network of water and amorphous ices 61 as well as water chains in hydrophobic crystal channels 62 . Figure 4 | Accordance of (a) molecular size d H , molecular separation d L (or d OO 5 d H 1 d L ), (b) mass density r, and packing order (see Figure 1c)