Fig. 1. O:H–O segmental length cooperative relaxation.^[63,78] (a) O—O distance changes by shortening one segment and lengthening the other because of coupling interaction. The O:H always relaxes more than the H–O does because of the O:H–O segmental disparity. The primary rule for (b) O:H–O length cooperativity by compression (p), thermal excitation (t), molecular undercoordination (z), and electrostatic polarization (e).

Fig. 2. Specific-heat disparity and multiphase thermal mass density oscillation.^[79] (a) Intersection points define the quasisolid phase (QS) whose boundaries close to temperatures for homogeneous ice nucleation (T_N) and melting (T_m). Molecular undercoordination disperses the QS boundary outwardly by H–O contraction and O:H elongation through Einstein’s relation. (b) Segmental specific-heat ratio defines the thermal slope of density over all phases for water ice and the critical temperatures vary with volume size at the nanometer scale (T ≥ 273 K bulk water; T ≤ 273 K 1.4 nm sized droplet).^[21]

Fig. 3. Undercoordination resolved bond relaxation and O:H–O potentials.^[1,82,85] (a) BOLS formulation of atomic undercoordination-resolved bond contraction (z < 12), with m > 0 being the bond nature index. (b) O:H–O potential paths for the sized (H₂O)_N = 2 – 6 clusters (Δd_H < 0, ΔE_H > 0; Δd_L > 0, ΔE_L > 0). The blue dots in (b) are the initial equilibrium for N = 6.

Fig. 4. O:H–O bond segmental length cooperative relaxation. O—O repulsion dislocates O ions in the same direction by different amounts (insets) under (a) mechanical compression, cooling of (b) liquid and (c) quasisolid (QS) phase (in units of °C), and (d) undercoordination by reducing the (H₂O)_N size from N = 6 to 2. Arrows denote the master pieces and their relaxation directions. The H–O bond always relaxes less than the O:H and both of them relax contrastingly in their curvatures and slopes, irrespective of the applied stimulus or the structural order because of the O–O Coulomb repulsive coupling (reprinted with permission from Ref. [63]).

Fig. 5. Skin O:H–O segmental length and stiffness cooperative relaxation.^[53] Length and stiffness (inset) transition for the (a) H–O bond and (b) O:H nonbond from the bulk value (b) to the skin (s) and to the H–O free radicals (r). Inset (a) shows the complex unit cell denoted with bulk, skin, and a vacuum slab. The P components arise from the screening and splitting of the crystal potentials by polarization.

Fig. 6. Computational H–O stretching vibration modes in the (H₂O)_n clusters.^[88] The black dashed lines convolute the H–O vibration modes of the entire clusters. The sharp feature D corresponds to the H–O dangling bonds, C to the H–O of the O:H–O bonds between molecules at rims, features A and B to the H–O bonds inside the clusters. Reprinted with copyright permission from Ref. [88].

Fig. 7. H–O stiffness transition from the bulk to the skins of large volumes and nanodroplets.^[53,74] (a) Skins of 298 K water and (253–258) K ice^[65] share an identical ω_H of 3450 cm⁻¹ and the (b) D–O phonon^[33] transits from below 2550 to its above for the skin of droplet. Insets show ice slipperiness, water skin toughness, and the elasticity and hydrophobicity of skins of droplet and water.

Fig. 8. Site and orientation resolved SFG H–O vibration frequency. Undercoordination resolves the frequencies of the O_B1:H–O_B2 and the O_B1 – H:O_B2 bonds (inset) between outermost two sublayers of the ice I_h(0001) skin.^[35] Insets illustrate segmental bond lengths, orientations, and frequencies of the H–O stretching vibrations. The positive peak (< 3270 cm⁻¹) corresponds to the H–O_B2 vibration (shaded in green) and the valley (> 3270 cm⁻¹) to the O_B1-H (shaded in blue). The less coordinated O_B1-H is shorter and stiffer and its H:O_B2 is longer and softer than the O_B1:H–O_B2.

Fig. 9. Undercoordination induced quantum entrapment and polarization.^{[31,63,64,67,68]} (a) The O 1s energy shifts from the bulk value of 536.6 eV to 538.1 eV for the skin and to 549.7 eV for gaseous state. (b) The hydrated nonbonding electron shifts its energy from the core value of 2.4 eV and the skin of 1.2 eV to the limit of 0.4 eV for N = 5 cluster. Inset (b) shows the cluster size dependent ω_H blueshift.^[4]

Fig. 10. Ultrafast phonon and hydrated electron processes of water droplets.^[32,33] Lifetime of (a) the D–O phonons in the droplet skin compared with that in the NaBr/H₂O solutions^[33] and the lifetime of (b) electron hydrated by the sized (H₂O)_n and (D₂O)_n clusters and their skins.^[32] Insets in (a) illustrate the ionic hydration and the core-shelled water droplet. The skin supersolidity hinders the motion dynamics of both phonons and nonbonding electrons.^[55]

Fig. 11. Supersolid skin thermal stability.^[57] Temperature dependence of (a) the full-frequency Raman spectra and (b) the frequency shift of the bulk, skin, and H–O dangling bond components. The H–O dangling bond undergoes thermal expansion (redshift) at T > 300 K. Both the bulk and the skin components undergo thermal contraction, at different slopes. At T > 340 K, the skin component turns to be thermal elongation, showing the O—O repulsive weakening.

Fig. 12. Skin thermal-diffusivity elevation and specific-heat depression.^[102,55] Numerical reproduction of the measured (insets) initial-temperature dependence of (a) the θ (θ_i, t) decay and (b) the skin-bulk temperature difference Δθ (θ_i, t) of warm water. High skin thermal diffusivity due to density loss ensures the characteristic intersection. (b) The Δθ (θ_i, t) arises from the skin lower specific heat because heat flux conserves at the interface.

Fig. 13. Nano-supersolid T_N depression – supercooling.^[48,53] Skin supersolidity takes the full responsibility for (a) water skin toughness, ice slipperiness, and (b) droplet-size and surface-curvature (inset) resolved T_N depression (called supercooling).^[106,112] Droplet of slightly more-curved surface freezes 68.4 s later at 269 K than the contrast deposited on smooth Ag surface.^[111]

Table 1. Typical examples for the molecular undercoordination resolved anomalies of low-dimensional water ice.