Ice Regelation: Hydrogen-bond extraordinary recoverability and water quasisolid-phase-boundary dispersivity

Regelation, i.e., ice melts under compression and freezes again when the pressure is relieved, remains puzzling since its discovery in 1850’s by Faraday. Here we show that hydrogen bond (O:H-O) cooperativity and its extraordinary recoverability resolve this anomaly. The H-O bond and the O:H nonbond possesses each a specific heat ηx(T/ΘDx) whose Debye temperature ΘDx is proportional to its characteristic phonon frequency ωx according to Einstein’s relationship. A superposition of the ηx(T/ΘDx) curves for the H-O bond (x = H, ωH ~ 3200 cm−1) and the O:H nonbond (x = L, ωL ~ 200 cm−1, ΘDL = 198 K) yields two intersecting temperatures that define the liquid/quasisolid/solid phase boundaries. Compression shortens the O:H nonbond and stiffens its phonon but does the opposite to the H-O bond through O-O Coulomb repulsion, which closes up the intersection temperatures and hence depress the melting temperature of quasisolid ice. Reproduction of the Tm(P) profile clarifies that the H-O bond energy EH determines the Tm with derivative of EH = 3.97 eV for bulk water and ice. Oxygen atom always finds bonding partners to retain its sp3-orbital hybridization once the O:H breaks, which ensures O:H-O bond recoverability to its original state once the pressure is relieved.

of modern literature suggesting that Faraday's surmise of an anomalous ice layer may be correct but it is not actually true.
The Regelation can easily be demonstrated by looping a wire around a block of ice with a heavy weight attached to it. This loaded wire melts the local ice gradually until the wire passing through the entire block. The wire's track will refill as soon as it passes, so the ice block will remain solid even after wire passes completely through. Another example is that a glacier can exert a sufficient amount of pressure on its lower surface to lower the melting point of its ice, allowing liquid water flow from the base of a glacier to lower elevations when the temperature of the air is above the freezing point of water (258K). The regelation is exceedingly interesting, because of its relation to glacial action under nature circumstances 4 , in its bearing upon molecular action 5 , and self-repairing of damaged living cells.
It is usual in 'normal' materials that compression raises the critical temperature (T C ) at all phase transitions [6][7][8] ; however, according to the phase diagram of water and ice, the freezing temperature of liquid water is lowered to − 22 °C by applying 210 MPa pressure; stretching ice (i.e. tensile, or negative, pressure) has the opposite effect -ice melts at + 6.5 °C when subjected to − 95 MPa pressure 9 . Conversely, the T C for ice drops from 280 to 150 K at the transition from ordered ice-VIII to proton-disordered ice-VII phase when P is increased from 1 to 50 GPa [10][11][12] . A molecular-dynamics (MD) study of a nanowire cutting through ice suggests that the transition mode and the cutting rate depend on the wetting properties of the wire -hydrophobic and thicker wires cut ice faster 13 .
However, a consistent understanding with numerical reproduction of regelation has yet been achieved despite intensive investigations. It might be true that regelation can occur for substances with the property of expanding upon freezing, but mechanisms for neither freezing expansion nor regelation is clear 14 . These issues are beyond the scope of classical thermodynamics in terms of equation of states, which inspires alternative ways of thinking and approaching to unlocking these puzzles.

Principle: Hydrogen bond cooperative relaxation
General bond potential. Figure 1a shows a pairing potential u(r) for the interatomic bonding. The coordinates (d, E b ) at equilibrium are the bond length and bond energy. We are concerned how the d and E b respond to external stimulus regardless of the shape of the particular u(r). A Taylor series approximates the pairing potential u(r) as follows:  The zeroth differential is the bond energy at equilibrium E b, which can be determined from photoelectron spectrometrics 15 . Higher-order differentials corresponding to the harmonic and nonlinear vibrations determine the shape of the u(r). The vibration amplitude x is 3% or less than atomic distance d of the substance below melting.
Generally, external stimuli, such as stressing and heating modulate the length d(T, P) and energy E(T, P) of the representative bond along a path denoted f(T, P) 6 . For instance, compression stores energy into a substance by shortening and stiffening all bonds with possible plastic deformation while tension does the opposite, as illustrated in Fig. 1a, and formulated as follows 15 : where T 0 and P 0 are the ambient referential conditions. The α (t) is the thermal expansion coefficient. recover completely its initial states once the compression is relieved. Conversely, once the O:H nonbond breaks, oxygen atom finds immediately bonding partner to retain its sp 3 -orbital hybridization that occurs at 5 K 24 temperature and above even in gaseous phase 25 .
With the aid of quantum calculations, Lagrangian oscillating mechanics and Fourier fluid thermo dynamics, and phonon spectrometrics, we have been able to consistently and quantitatively resolve quantitatively a few issues such as: 1) Mpemba effect -hot water freezes quicker than its cold 16 , 2) supersolid skins for the slipperiness of ice and the hydrophobic and tough skin of water liquid 26 , 3) ice expansion and mass density oscillation over full temperatures range 14 , 4) anomalies of water molecules with fewer than four nearest neighbors in clusters and droplets 22 , 5) Hofmeister effect -NaCl mediation of O-O repulsion 23 , 6) density-geometry-dimension correlation of molecules packed in water and ice 20 , 7) low compressibility and proton centralization of ice 18 , and, 7) mapping the local potential paths for the O:H-O bond relaxing with stimulus 21 , etc. Progress made insofar has formed the subject of a recent treatise 17 .  Figure 2b shows the ω x cooperative shift of ice under compression at 80 K. Phonon frequencies relax monotonically up to 60 GPa even though the pressure is increased 30,31 . In accordance to the length relxation, compression shifts the ω H toward higher frequencies and the ω L to lower. The length and stiffness trend of O:H-O bond relxation hold for all phases of water and ice with negligible slope variation 17 .

Results and Discussion
A Lagrangian-Laplace transformation of the measured d x and ω x turns out the force constant k x and segmental energy E x , which maps the potential paths of the O:H-O bond under compression 21 . As shown in Table 1, compression increase the E L from 0.046 to 0.250 eV up to 40 GPa and then decrease to 0.16 eV at 60 GPa; the E H decreases monotonically from 3.97 eV to 1.16 eV at 60 GPa. Different from situation of 'normal' substance, compression lowers the total energy of the O:H-O bond rather than raise it. The O:H-O bond will fully recover its initial states once the compression is relieved without any plastic deformation.
As expected, compression shortens the d L , increases the ω L and E L of the O:H nonbond; the H-O bond responses oppositely to compression, resulting in d H elongation, ω H and bond energy E H reduction, which can be formulated in the reduced forms as follows (E x valids at P < 30 GPa; d x /d x0 = 1 + β x1 P + β x2 P 2 for instance): Generally, pressure raises the T m but ice responses to pressure in the opposite -T m drops when the pressure is increased. Reproduction of the measured P-dependent T m for melting (Fig. 3a) 32  In order to clarify this paradox, let us look at the specific heat of water 14 . Generally, the specific heat of a 'normal' substance is regarded as a macroscopic quantity integrated over all bonds of the specimen, which is also the amount of energy required to raise the temperature of the substance by 1 K degree. However, in dealing with the representative for all bonds of the entire specimen, it is necessary to consider the specific heat per bond that is obtained by dividing the bulk specific heat by the total number of bonds 6 . For a specimen of other usual materials, one bond represents all on average; therefore the thermal response is the same for all the bonds, without any discrimination among all bonds in cooling contraction and thermal expansion 36 . For water ice, however, the representative O:H-O bond is composed of two segments with strong disparity in the specific heat of the Debye approximation, η x (T, Θ Dx ) 14 . These two segments response to a thermal excitation differently. Two parameters characterize the specific heat curves each. One is the Debye temperature Θ Dx and the other is the thermal integral of the η x (T, Θ Dx ) from 0 K to the T mx . The Θ Dx determines the rate at which the specific-heat curve reaches its saturation. The η x (T, Θ Dx ) curve of a segment with a relatively lower Θ Dx value reaches saturation more rapidly than the other segment, since the Θ Dx , which is lower than T mx , is proportional to the characteristic vibration frequency ω x of the respective segment, kΘ Dx = ћω x , according to Einstein's relation 37 , where k and ћ are constants.
Conversely, the integral of η x (T, Θ Dx ) from 0 K to the T mx determines the cohesive energy per bond E x 6 . The T mx is the temperature at which the vibration amplitude of an atom or a molecule expands abruptly to more than 3% of its diameter irrespective of the environment or the size of a molecular cluster 37,38 . Thus we have: The superposition of these two η x (T, Θ Dx ) curves implies that the heat capacity of water ice differs from that of other, 'normal' , materials. Such a η x (T, Θ Dx ) disparity yields temperature regions with different η L /η H ratios over the full temperature range; see Fig. 3b. These regions correspond to phases of liquid and solid (η L /η H < 1), and quasisolid (η L /η H > 1). The intersecting temperatures (η L /η H = 1) correspond to extreme densities at boundaries of the quasisolid phase (viscose and jelly like). The high-temperature boundary corresponds to the maximal density at 4 °C and the lower to the crystallization of bulk water.
Numerical and experimental observations 14,17,20   One can imagine what will happen to the crossing temperatures if one depresses the Θ DH (ω H ) and E H , and meanwhile, elevates the Θ DL (ω L ) and E L by compression or the inverse. Compression (Δ P > 0) raises the Θ DL and E L by stiffening ω L , and meanwhile, lowers the Θ DH and E H by stiffening ω L ; however, tension (Δ P < 0) does the opposite. Figure 3b illustrates how the positive P squeezes the quasisolid phase boundaries. The E H determines approximately the T m through dispersing the upper phase boundary. The Θ Dx (ω x ) always relax simultaneously in opposite direction under a given stimulus, which will disperse the quasisolid phase boundaries resulting in the observed 'superheating/supercooling' , as one often refers. In fact, external stimulus can raise/depress the melting/freezing point by phonon relaxation, which is different from the effect of superheating/supercooling 39 .
Once the O:H bond breaks, oxygen atoms will find new partners to retain the sp 3 -orbital hybridization, which is the same to diamond oxidation and metal corrosion -oxygen atoms penetrate into the bulk when corrosion occurs 15,25