Introduction

The physical origin and theoretical reproduction of the measured anomalies of water and ice remains a great challenge1,2,3,4,5,6,7,8,9. For example, under compression of the O:H-O bond, the H-O covalent bond becomes longer while the total O–O distance is shortened, leading to the proton centring in the O–O of ice VIII-X phase transition2,10,11,12,13,14. The behaviour of the proton is so strange that thermal fluctuation15 and quantum effect of nuclei16 were both considered contributing to the ambiguous behaviour. Unlike other materials whose phonons were harden by external pressure, ice-VIII demonstrated the anomalous softening of the H-O vibration mode at frequency greater than 3000 cm−1 while stiffening of the O:H vibration mode at frequency lower than 400 cm−117. Ice melts under compression and freezes again when the pressure is relieved, evidencing extraordinary recoverability of O:H-O bond21. Hydrogen bond interaction potential, typical double-well potentials of the symmetrical22 and the asymmetrical23 forms, is still under debate.

An accurate description of the H-bonding between water-water molecules is widely recognised as the crucial factor in the understanding of water anomalies9,24,25,26. Progresses has been made by considering the flexibility of the covalent bonding in water/ice in models such as the flexible-SPC27 and TIP4PF28. Kumagai et al. have proposed a flexible KKY potential which has three separate terms: VHO, VHH and VOO in order to account the intra- and inter-molecular interactions separately11 and adopted by our previous MD calculation29. However, since those models have not distinguished O–O linked by O:H-O or not, the reproduction of the proton centring of ice VIII-X has seldom been reported. Ab initio MD (AIMD) enabled people to use the ab initio-derived force field, though time-expensive. Bernasconi et al.15,30 reproduced the ice VIII/VII/X phase transition by AIMD-derived lattice relaxation and infrared spectra. Sugimura et al.9,11 determined the intermediate structure during the ice VII/X phase transition using AIMD. Katoh et al.31,32 confirmed the fast protonic diffusion coefficient at high temperature molecular phase of ice VII. However, previous researches mostly focused on the structure determination or proton transferring, while the mechanism of the proton centring and the cooperative relaxation of the O:H and H-O are not fully understood. A potential model for time-saving calculations of the proton centring in the ice VIII/X phase transition is still lacking.

Recent progresses find that the hidden force33, i.e. the repulsive force between the H-O bonding electrons and the O:H nonbonding pair within a O:H-O bond, should play a significant role in various anomalies of water and ice, such as extraordinary recoverability, skin lubricity, etc.33,34,35,36. For example, in the temperature range from 250 K to 277 K, H-O covalent bond exhibit the normal thermal expansion, while O:H nonbond shrinks due to the reducing repulsive force, leading to the contraction of the total O–O distance34,36. At the skin of ice, due to the decrease of the molecular coordination, H-O covalence bond contracts spontaneously to lower the cohesive energy, and hence O:H nonbond is polarized, leading to the surface lubricity35,37. This force is beyond the conventional intra- and inter-molecular interactions but depending on the existence of the O:H-O link, i.e. if there is no O:H-O bond like between two O atoms in difference sublattice in ice VIII, hidden force (3 times stronger than Van de Vaals force) interaction would not show38. However, it has still seldom reported to address the hidden force of O:H-O bond in a H-bond model, although the force plays a significant role in altering the properties of water and ice.

In this paper, in order to investigate the strange behaviour of proton centring in phase VIII-X transition, we clarified a hidden force model considering the quantum interactions between electron clouds of covalent bond and nonbond and reproduced the anomalous behaviour of proton in ice VIII-X transition.

Potential Model Proposed

Based on our extensive investigation of water anomalies21,34,35,36, we proposed hidden force model for dynamics of the “O: H-O” bond under external pressures, as shown in Fig. 1a. In the model, “O: H-O” bond is similar to be connected by three “non-harmonic converted to anharmonic springs”. Black spring represents covalent bonding of H-O (denoted as subscript C); Grey spring represents van der Waals interaction between lone pair (blue circles) and proton of O:H (denoted as subscript H); Blue spring represents repulsive interaction between bonding electron pair and lone pair (denoted as subscript OO, but the interaction is not the conventional O–O coulomb repulsion). A water molecule is surrounded by four nearest neighbours (Pauling’s ice rule). Four identical O: H-O forms a tetrahedron as the basic unit of water structure (Supplementary Fig. S1). The bond length (x), forces (f) and potential(V) of O:H, H-O and O–O need to reach to a balance under external force (fp). Figure 1b shows two stages of O:H-O relaxation under compression.

Figure 1
figure 1

(a) Hidden force model for O: H-O bond. xH and xC are the O atomic position counting from x = 0 (H atom). fOO is the hidden repulsive force. fP is the external compression force. fH and fC are the recovery force of O:H and O-H, in value of slope of the potential curves (blue dash lines) of VH and VC. Star represents the curvature turning point. (b) Two stages of O:H-O relaxation under compression. Stage I (ice VIII): A small external force compresses the O:H at first since O:H interaction is weaker than O-H. foo is then increased to extend O-H. Stage II (ice X): under a large external force, fH goes across the curvature turning point, leading to the shrinks of both O:H and O-H.

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In stage I, a small external force compresses the O:H at first since O:H interaction is weaker than O-H. The repulsive foo then is increased by the contraction of O…O, leading to H-O extension. In stage II, under a large fp, fHgoes across the curvature turning point, leading to the shrinks of both O:H and O-H. Intermediate phases exist during the transition where proton disordered symmetrization occurs with delocalized proton in a relatively broad potential well. The delocalization of proton (or proton-transfer) may attributed to quantum effect of nuclei16,39, or to thermal fluctuations obtained by AIMD15. In this work, we focus on the proton centring due to electron interactions within the O:H-O bond.

Without external force, taking centre proton as reference, three springs can reach to equilibrium with their forces along the directions as indicated in Fig. 1a. Due to the foo, both O atoms are pushed away a little from their lowest-energy positions in the potentials of isolated bonds. Thus, the initial potential recovery forces fC and fH will both point inward to center proton; while repulsive force fOO will point outward. Supposing , at equilibrium, total forces (force values) added on O atoms should both be zero:

where, xC0 and xH0 (length values in positive numbers) are the equilibrium position of O atoms counting form x = 0. Thus, in equilibrium without external force,

Considering an identical force fp at both sides, the forces satisfy:

if

Supposing the displacement is small enough to consider k as a constant, since , combining eq. (2) and eq. (3), we can get:

Eq. (4) tells us:

i) The values of and should be determined by the potential forms of VH and VC. Since O:H bond is much longer and weaker than H-O bond, VH is more flat (smaller curvature) than VC, i.e. |kC| >| kH|. Thus, the value of is always larger than .

ii)If and have opposite signs, i.e. xC and xH do not both increase or decrease together, kH and kC must have opposite sign with each other, which means the curvatures of potentials are opposite, as shown in Fig. 1a.

iii) If and have same signs, i.e. xC and xH both increase or decrease together, kH and kC must have the same sign.

Due to the charge polarization within the O:H-O bond, the curvature of the H-O potential has the opposite sign with O:H potential in the low-pressure region, leading to the contraction of O:H bond but extension of H-O bond and the proton centring. Hence, for O: H-O bond relaxation under external fp, there are two stages as shown in Fig. 1b:

i) Stage I (ice VIII): in quasi-static process, external force contracts the O:H bond but extends the H-O bond with the total O–O length shortened, when kC ∙ kH < 0. Both the work of fP and the change of VC provide energy for increase of VH and work of fOO:

ii) Stage II (ice X): external force is strong enough to make xH shrink over the curvature turning point (star in Fig. 1) and kC ∙ kH > 0. External force contract both segments, pushing O atoms back towards the energy-lowest positions, compressing all the three springs. In this case, fP and energy increases for both segments provides the sharp increase of the VOO:

Calculations and Verifications

To verify and quantify our potential model, we performed density functional theory (DFT) calculation using CASTEP. The generalized gradient approximation (GGA) function of PW91, HCTH and RPBE were used to describe the exchange–correlation effects (results show little differences among the different functions and hence the PW91 results were presented here). The van der Waals force was also examed by adopting the DFT-D, its effect to the stability of the ice structures has reported, a small effect of red-shifts (~2 meV) for the main peaks of the vibrational spectra was observed. The phonon spectra were calculated using the CASTEP module with finite displacement method. The force constants produced by the CASTEP for the phonon spectra calculations were obtained from the output files. The force for the atom i, fi (= dE/dri), is simply the derivative of the total energy E. By applying a second small displacement for the atom j, the force constant kij (= dU/dridrj) is obtained for the pair atoms i and j. A force constant matrix k for the unit cell is a 3 N×3 N matrix (where N is total number of atoms in the unit cell) and were constructed based on above procedure.

Ice VIII has a cubic structure containing two sets of interpenetrated ice sub-lattices as shown in Fig. 2a. Hence, each water molecule has 8 nearest neighbours, 4 of them are H-bonded and the others are not, at almost the same O–O distance. The oxygen atoms in the same sublattice were denoted as O and in the other sublattice as O’. Blue isosurface of deformation electron density (density difference before and after bonding,) indicates the position of the bonding electron pair and the O:H nonbond pair. The charge distribution in a water molecule was polarized in the tetrahedral directions by the O:H-O bond. Figure 2b shows the pressure-dependent bond length of O:H(xH) and O-H(xC). Proton centring is obtained as pressure increases, in accord with the experimental observation2,12 and other DFT calculation33. The changing trends of xH and xC verify the two stages of O:H-O under compression. xH shrinks and xC extends in stage I. As pressure increases, xH and xC both shrink in stage II like normal material.

Figure 2
figure 2

(a) Structure of ice VIII. Two nested sublattices connected by green and purple lines respectively are shown. The oxygen atoms in the same sublattice were denoted as O and in the other sublattice as O’. Deformation electron density was plotted as blue isosurface. (b) DFT-derived bond length of O:H(xH) and O-H(xC) versus pressure. The changing trend of xH and xC falls into two stages.

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The DFT calculation can reproduce phonon spectrum for Ice VIII as shown in Fig. 3a. The simulation reproduces the main features of inelastic neutron scattering (INS) spectrum24, such as the three peaks in the translational region (<50 meV) and the two sharp peaks of the librational region at about 65 meV. The small peak in the right hand side of librational region at about 100 meV was not shown in the INS spectra due to large Debye-Waller effect in the measured spectra which smeared the spectra dramatically at higher energy transfer. Force constants kC, kH and kOO were also obtained from the calculation of phonon spectrum of the ice VIII and are plotted as a function of pressure in Fig. 3b. Apart from the trend for the kC, kB to merge at xOO = 2.3 Å the force constants between the H-bonded O–O atoms kOO also shows a rapid increase under the external pressure. For the non H-bonded O…O’ atoms between the two sub-lattices of ice VIII, a new force constant, kOO’, was obtained which remains at small values around kH. This implies that the O–O interaction only becomes relevant when the two O atoms are linked by an H atom and hence this large value of kOO is not due to the simple coulomb interaction between O atoms. As shown by the isosurface in Fig. 2a, charge distribution was polarized by O:H-O bonding, i.e. the electron clouds were accumulated in the line of O:H-O, leading to the increase of the repulsive force between bonding pair and nonbonding pair of O–O and the decrease of the interaction between O and O’.

Figure 3
figure 3

(a) Phonon spectra of ice VIII measured using inelastic neutron scattering and calculated uing DFT. (b) Force constants of O-H, O:H, O–O in O:H-O bonding and O–O’ without bonding.

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Potential Model Parametrization

Fitting from DFT-derived k-x results of stage I of P-dependent IceVIII (Fig. 3b), we get the relations among ki and xi (I = H, C, OO) according to our potential model,

where we add negative sign on kH since DFT only take the positive square root value. Based on k-x curve, we fit the expression of f-x and V-x curves (Supplementary Eq. S1 and S2). The shapes of f-x and V-x are plotted in Fig. 4.

Figure 4
figure 4

The DFT-fitting curves of bond length x-dependent (a), fC and fH, (b), fOO, (c), VC and VH, and (d), VOO according to water model. Black arrows indicate the opposite relaxation in length of xC and xH in stage I; Blue arrows indicate the both decreasing in length of xC and xH in stage II.

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In stage I, as fp increases, fH = fC should also increases according to Eq.(3). Thus, xH decreases while xC increases as indicated by black arrows in Fig. 4a. In stage II, when xH decreases to about 1.1 and reach to the curvature turning point, fH changes to decrease and xC will decreases back (as shown by blue arrows) to make fC = fH. The absolute value of repulsive force fOO always increases as distance xOO decreses. We focus on the parametrization of stage I in this work since it is more interesting than the normal compression-contraction in stage II.

Hence, for a given fP, there is only one set of solutions of xH and xC obtained in Fig. 5. Figure 5a shows that, for a given xH, fH = fC gets the xC. Then, Fig. 5b shows that, fOO can be calculated by xH + xC and fP is correlated with xH by fp = fOO−fH. Hence, we have obtained the fP-xH curve in Fig. 5b, from which a given external fP can determine the relaxation of xH and xC. Thus, the correlations among external compression force, the length relaxation of H-O and O:H bond, the forces of O-H, O:H and O:O are all determined. Figure 5b also indicates that, as fP increases, fOO rises much more significantly than fH. Thus, fOO is indeed a hidden force that should not be ignored.

Figure 5
figure 5

(a) Correlation between xH and xC. (b) Correlation of xH, fP. fH, and fOO. Under external fp, the length and force relaxations inside the O:H-O bonding are all determined.

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Based on our potential model, foo, fH and fp are functions of xH and xC determined by Supplementary Eq. S1 and Fig. 4. We run the time-dependent dynamic process of external pressure on O:H-O bond in Fig. 6. The dynamic equation can be expressed as:

Figure 6
figure 6

Time-dependent relaxation dynamics of O:H-O bond with the same initial lengths and different external forces.

(a) xH relaxes with time and (b), xC relaxes with time under fp = 0.1, 0.5 and 1.5 eV/Å. The initial lengths are 1.9 and 0.98269 at 0 external force, which give initial energies for bonds to oscillate around the equbrillium point under pressures. xH and xC under fp = 0.5 and 1.5 eV/Å were taken average.

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The initial condition of xH and xC are the lengths (Å) without pressure. The partial differential equation is solved by 4th order Runge-Kutta method (Supplementary Method).

Figure 6 shows the xH and xC relaxation with the same initial lenghs and different external forces under fp = 0.1, 0.5 and 1.5 eV/Å. The initial lengths are bond lengths at fp = 0, which give energies to atoms to oscillate. If the process is slow, i.e. initial coordinates are around the stable point, the oscillation will decrease. xH and xC under fp = 0.5 and 1.5 eV/Å were taken average as . The results show that, as fp increases, xH decreases from 1.890 to 1.811 and to 1.645 Å in average, while xC increases from 0.983 to 0.992 and to 1.015 Å in average, agreeing well with proton centring obtained both in DFT calculation and experimental observation2,12.

Figure 7 a and b show the time-dependent dynamics of O:H-O bond in the quasi-static process under fp from 0.1 to 0.5 eV/Å. The initial lengths at each fp were taken as the average bond lengths of the previous fp. Hence, bond length relaxes more stably with time. Figure 7c shows the average bond lengths obtained from the quasi-static states at different fp. The dynamic curves oscillated stably raise the reliability of the average lengths. Results show that as fp increases, O:H bond length decreases and H-O bond length increases, approaching to equal at about fp  = 6.91 eV/Å. the Therefore, upon compression, in the equilibrium status of O:H-O, proton and oxygen shift both towards the other oxygen while proton shift a little to the weaker O:H part. Besides, O:H vibrates much slower than H-O bond due to its weaker force constant, as indicated by the frequency of the relaxation curves.

Figure 7
figure 7

Time-dependent dynamics of O:H-O bond in the quasi-static process, with different initial bond lengths.

(a) xH relaxes with time and (b), xC relaxes with time under fp from 0.1 to 0.5 eV/Å. The initial lengths at each fp were taken as the average bond lengths of the previous fp. Hence, the oscillations of bonds were largely reduced. (c), the average bond lengths derived at fp from 0.1 to 6.91 eV/Å. Proton centring occurs near fp = 6.91 eV/Å.

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The potential model of O:H-O can be further applied to water molecules and crystal structures considering the interactions of H—H and O–O’. VOO’ follows the weak dispersion interaction as indicated by the force constant developed by DFT in Fig. 3b, since electron cloud distributes mostly on the line of O:H-O bond as shown in Fig. 2a. Current potential model provides an efficient tool to investigate bond length relaxation, time-dependent proton dynamics, mass density and intermediate structure determination of dense ice phases.

Conclusion

DFT and neutron scattering observations have enabled a O:H-O bond potential for the ice VIII-X phase transition under compression. Before transition, H-O undergoes elongation and the O:H compression because of the O:H-O bond segmental disparity and O-O repulsion; after transition, both H-O and O:H undergoes slight contraction because of the excessive compression. The effectiveness of current potential model was illustrated when dealing with H-bonded systems under pressure. More importantly, this additional force could provide the necessary mechanism to explain a range of water anomalies, for instance the complex phase diagram, i.e. its morphism, this is probably because the hydrogen bonds become very easy to buckle under the extra strong O–O interaction if the O:H-O is not in straight line. This hidden force, fOO has often been overlooked experimentally in the past, perhaps because it is easily concealed by the fH and fC. The hidden force potential model reproduced the contraction of O:H bond and extension of H-O bond and the proton centralization under compression. The potential model to develop further can serve as a key for theoretical reproduction of the O:H-O bond asymmetric relaxation under external stimuli.

Additional Information

How to cite this article: Zhang, X. et al. Hydrogen-bond potential for ice VIII-X phase transition. Sci. Rep. 6, 37161; doi: 10.1038/srep37161 (2016).

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