Isotope Effects and the Negative Thermal Expansion Phenomena in Ice and Water

TL;DR

The paper investigates why ice and water exhibit negative thermal expansion (NTE) and an abnormal volume isotope effect (VIE). It introduces a Born-Oppenheimer–like separation between fast high-energy intramolecular phonons and slow low-energy intermolecular modes, assigning the zero-point energy of the fast modes to an effective potential that modulates the slow modes, and uses the Lindemann criterion to link vibrational amplitudes to melting. The authors show that zero-point phonons, thermal phonons, and the hydrogen-bond network compete to produce NTE and VIE, with heavier isotopes (e.g., D$_2$O) experiencing different ZP contributions that shift equilibrium volumes and transition temperatures. This QM-centric framework explains isotope-dependent shifts in $T_m$, $T_{MD}$, and the low-temperature NTE in ice-Ih, extending to water near the freezing transition, and emphasizes the quantum nature of phenomena traditionally treated classically in H$_2$O systems.

Abstract

H2O is a unique substance with exceptional thermal properties arising from the subtle interplay between its electronic, phononic, and structural degrees of freedom. Of particular interest in H2O are the negative thermal expansion (NTE) phenomena, observed in its solid phase (ice) at low temperature, and in its liquid phase (water) near the freezing temperature. Furthermore, ice and water exhibit the abnormal volume isotope effect (VIE), where volume expansions occur when replacing H with its heavier isotope, deuterium (D). In order to capture more conceptual and intuitive understanding of intriguing NTE and VIE phenomena in ice and water, we have explored isotope effects in their NTE and melting properties by employing a type of Born-Oppenheimer-approximation approach and the Lindemann criterion. Our findings demonstrate that unusual isotope effects in these phenomena stem from competition between zero-point-energy phonons, thermal phonons, and the hydrogen bonding in H2O. All these components originate from nuclear quantum mechanical (QM) processes, revealing that QM physics plays a crucial role in the seemingly classical ice/water systems.

Figures (4)

• Figure 1: (a) Phase diagram of H$_2$O PD-h2o. A, B, C, and D correspond to the triple point, freezing point, boiling point, and critical point, respectively. (b) Thermal expansion data near the freezing point Vol-T. Note the NTE (abrupt volume jump) at the water to ice transition at $T=0~^{\circ}$C (273 K), and the NTE of water between $0~^{\circ}$C (273 K) and $4~^{\circ}$C (277 K) shown in the inset. (c) The thermal expansion data of ice-Ih phases of H$_2$O and D$_2$O. The NTEs are observed below $\sim$ 70 K for both. D$_2$O has the larger volume than H$_2$O, manifesting the abnormal VIE. (d) Volume variations of supercooled-water isotopes as a function of $T$Water-websiteTmd. (e) Open hexagonal crystal structure of ice-Ih (viewed along the $c$ axis). (f) Hydrogen-bonding (H:O-bonding) in tetrahedral building block of ice-Ih. Each oxygen forms two-short H-O covalent bonds, and two-long hydrogen bonds (H:O-bonds) with neighboring H ions.
• Figure 2: (a) Typical volume-dependent behaviors of high-energy phonon modes $\omega_s(V)$. The upper one exhibiting the pressure-induced hardening produces the positive Grüneisen parameter and the PTE, while the lower one exhibiting the pressure-induced softening produces the negative Grüneisen parameter and the NTE. (b) Effective intermolecular potential energies $U_{eff}$'s ($U_{eff}=U_{0}+U_{ZP}$) for the PTE-anharmonic (top) and the NTE-anharmonic (bottom) systems, respectively. $U_{0}$ is assumed to be harmonic (middle). Red dotted lines in the top and bottom illustrate the volume expansion and contraction, respectively, with increasing $T$. (c) The intermolecular potential energy, $U_{0}(V)$ (red curve), which is intrinsically PTE-anharmonic, can be approximated by the harmonic potential (blue curve) at low $T$. (d) For the NTE case with negative $\gamma_s$, the heavier-mass system of D$_2$O ice has the larger volume (red dot) than the lighter-mass system of H$_2$O ice (blue dot) ($\Delta V_0 ~(\equiv V_0^{D}-V_0^{H}) > 0$). Due to the mass-dependent QM zero-point energy $U_{ZP}$, D$_2$O ice would have larger binding energy than H$_2$O ice by $\Delta U_{ZP}$ ($\Delta U_{ZP}$ value taken from Refs. Rasti22Pamuk15).
• Figure 3: (a) Schematic plot of $T$-dependent variations of the kinetic molecular vibration energies ($E_k$'s) and the ($-1\times$) cohesive energies ($-E_{coh}$'s) of H$_2$O (in blue) and D$_2$O (in red) ices. The variation of $E_k$ is somewhat exaggerated. Melting occurs when $E_k$ overcomes $E_{coh}$. It is shown that $T_{m}$ of D$_2$O is higher than that of H$_2$O. $\Delta E_k$ and $\Delta E_{HB}$ values are taken from Ref. Romanelli24 and Ref. Pamuk15, respectively. (b) Thermal-average amplitudes of ionic displacements ${\langle u^2 \rangle}^{1/2}$ of H$_2$O (blue) and D$_2$O (red) ices. Melting occurs when ${\langle u^2 \rangle}^{1/2} \geq fa$ ($a$: intermolecular distance, $f$: fractional number ($\leq 0.2)$).
• Figure 4: (a) Energy configurations of H$_2$O and D$_2$O waters and ice-Ih's. The nuclear QM effects are still retained in the liquid phase near $T_m$, and so D$_2$O water has the larger H:O-bonding energy $E^{W}_{HB}$ and the larger volume than H$_2$O water, as in ice-Ih phase. The value of $\Delta E$ is taken from Refs. Ramirez10Nemethy62, and that of $E^{W}_{HB}$ is taken from Ref. Hakem07. (b) Thermal expansion behaviors of H$_2$O and D$_2$O ice-Ih's and waters expected from the energy configurations of (a).