Correlation between 2D Square Ice and 3D Bulk Ice by Critical Crystallization Pressure

Low-dimensional ice trapped in nanocapillaries is a fascinating phenomenon and is ubiquitous in our daily lives. As a decisive factor of the confinement effect, the size of nanocapillary significantly affects the critical crystallization pressure and crystalline structure, especially for multi-layered ices. By choosing square ice as a typical two-dimensional (2D) multi-layered ice pattern and using all-atom molecular dynamics simulations, we further unveil the variation mechanism of critical crystallization pressure with the nanocapillary size. The results show a strong dependence of the critical crystallization pressure on the size of the graphene sheet for monolayer, bilayer, and trilayer square ice. The quasi-macroscopic crystallization pressure, the actual pressure of water molecules, and the freezable region between them are all strongly dependent on the nanocapillary width. As the size of the capillary becomes larger in all three directions, the critical crystallization pressure converges to the true macroscopic crystallization pressure, which is very close to the value of the crystallization pressure for bulk ice. A direct correlation is established between 2D square ice and three-dimensional (3D) bulk ice by the critical crystallization pressure. There is an unfreezable threshold for crystallizing spontaneously in practice when the quasi-macroscopic crystallization pressure is equal to the actual pressure, which can explain the limit of nanocapillary width for multi-layered ice.

💡 Research Summary

The paper investigates how the critical crystallization pressure governs the formation of two‑dimensional (2D) square ice confined within graphene nanocapillaries and how this pressure connects 2D ice to three‑dimensional (3D) bulk ice. Using all‑atom molecular dynamics (MD) simulations, the authors construct a model in which two parallel graphene sheets form a slit‑like capillary. The graphene sheets are varied in size (Dx × Dz) from 26.6 Å × 21.9 Å up to 108.8 Å × 89.6 Å, while the slit width (h) is set to 6.5 Å, 9.0 Å, 11.5 Å, and 14.0 Å to accommodate one, two, three, and four layers of water, respectively. Water is represented by the SPC/E model, and interactions with graphene are described by a 12‑6 Lennard‑Jones potential (ε = 0.114 kcal mol⁻¹, σ = 0.328 nm). Simulations are performed in the NPT ensemble at 298 K, with lateral pressure (Pz) ramped from 0 to 4 GPa at rates of 0.02–0.12 GPa ns⁻¹.

Three independent criteria are used to locate the phase transition: (1) a sudden drop in potential energy, (2) changes in the square‑icing order parameters MCV1 and MCV2, and (3) visual inspection of simulation snapshots. For each graphene size, six independent runs with different initial velocities are carried out, yielding a distribution of critical crystallization pressures. The key findings are:

1. Size Dependence – The average critical pressure increases as the graphene sheet becomes smaller, and the spread of values narrows for larger sheets. This reflects the need for higher external pressure to overcome confinement when the available lateral area is limited.

2. Convergence to Bulk – As the graphene dimensions grow, the critical pressure approaches a quasi‑macroscopic value (Pm) of roughly 1 GPa, which matches the experimentally known crystallization pressure of bulk ice.

3. Actual vs. Quasi‑Macroscopic Pressure – The internal hydrostatic pressure exerted by the confined water (Pa) is estimated from the adhesion‑energy difference (≈30 meV Å⁻²) and the increase in inter‑graphene spacing (≈5.6 Å), giving Pa ≈ 1 GPa. When Pa equals Pm, the system reaches an “unfreezable threshold.” Below this threshold, the internal pressure is insufficient to trigger spontaneous crystallization, explaining why very narrow or short capillaries do not produce square ice in practice.

4. Layer‑Number Effects – Single‑layer ice (h = 6.5 Å) exhibits the highest critical pressures, while bilayer, trilayer, and four‑layer ice show progressively lower values, indicating that inter‑layer hydrogen‑bond networks relieve part of the confinement stress.

5. Rate Independence – Varying the pressure‑ramp rate does not materially affect the critical pressure, confirming that the observed size dependence is not an artifact of finite‑rate dynamics.

The authors therefore establish a direct, pressure‑based correlation between 2D square ice and 3D bulk ice. By identifying the unfreezable threshold, they provide a quantitative explanation for the experimentally observed maximum number of layers that can be stabilized in nanocapillaries. This insight offers practical guidelines for engineering nanoconfined water systems—such as in filtration membranes, nano‑thermal management, and cryogenic storage—where control over ice phase and thickness is crucial.