Understanding the Density Maximum of Water with Machine Learned Potentials

Abstract

After melting, at ambient pressure, the density of water continues to increase with temperature until it reaches a maximum around 4 °C. For nearly a century, this phenomenon has been qualitatively attributed to a mixture of ordered and disordered structures. Herein, we employ a deep neural network to train a machine learned (ML) interatomic potential for water using electronic structure data from advanced density functional theory. Notably, molecular dynamics simulations with the ML potential reproduce both the experimental water density anomaly and the thermal expansion coefficient. Detailed structural analysis of the computed hydrogen-bond network reveals that the density anomaly arises from an emergent liquid structure that retains nearly ideal tetrahedral coordination at short range but collapses at intermediate range. Our findings point to a more delicate mechanism causing the density maximum than the conventional picture, emphasizing the collective roles of structural orderings at different length scales.

Figures (11)

• Figure 1: Structural decomposition of water’s density anomaly. (A) Temperature-dependent water density and thermal expansion coefficient (inset) from theory and experiment kell1967hare1987. Red circles indicate the TMD. All curves are horizontally shifted by the melting temperature (314.0 K). (B) Snapshot of liquid water at 330 K (approximately 16 K above the melting point), with Voronoi cells color-coded by local density. The color scale ranges from low local density (orange) to high local density (blue). A representative Voronoi cell is highlighted with a black outline. (C) Oxygen–oxygen PDFs, $g_{\mathrm{OO}}(r)$ (dashed lines), and the corresponding partial contributions from Voronoi neighbors, $g_{\mathrm{OO}_{\mathrm{VN}}}(r)$ (solid lines). Colors indicate different simulation temperatures. Inset shows a representative Voronoi cell (gray polyhedron) of a central water molecule (red); neighbors forming H bonds with the center molecule are marked in blue, others in gray. The two gray circles indicate the first and second coordination shells, respectively. (D) Decomposition of the total density change $\Delta \rho_{\mathrm{total}}$ into short-range ($\Delta \rho_{\mathrm{SR}}$) and intermediate-range ($\Delta \rho_{\mathrm{IR}}$) contributions. The intermediate-range component is further divided into contributions from tetrahedral ($\Delta \rho_{\mathrm{IR}}^{\mathrm{t}}$) and disrupted tetrahedral ($\Delta \rho_{\mathrm{IR}}^{\mathrm{d}}$) structures, as defined in Eq. (3).
• Figure 2: Short-range ordering in the H-bond network. (A) Temperature-dependent local densities of tetrahedral ($\langle \rho_t \rangle$) and disrupted tetrahedral ($\langle \rho_d \rangle$) structures. Insets show the representative molecular structures for each structural type. The disrupted tetrahedral structure exhibits a larger Voronoi cell volume. (B) Temperature-dependent population fractions of tetrahedral ($n_\mathrm{t}$) and disrupted tetrahedral structures ($1-n_\mathrm{t}$).
• Figure 3: Intermediate-range ordering in the H-bond network. (A) Temperature dependent PDFs between a central water molecule forming a tetrahedral structure and its Voronoi neighbors. Each PDF is decomposed into contributions from Voronoi neighbors in the first coordination shell and those beyond it, with circles and diamonds indicating the average radial positions of each group, respectively. Average radial positions and corresponding standard deviation values for each temperature are provided in Table \ref{['tab:variance']}. Insets illustrate how the accumulation of interstitial water molecules (between the two gray circles) leads to a reduction in the Voronoi cell volume of the central molecule at higher temperatures. Arrows indicate the inward movement of water molecules at IR. (B) Decomposition of IR density change of tetrahedral structures $\Delta \rho_{\mathrm{IR}}^{\mathrm{t}}$ into $\Delta \rho_{\mathrm{IR}}^{\mathrm{t}\,^{(-)}}$ and $\Delta \rho_{\mathrm{IR}}^{\mathrm{t}\,^{(+)}}$. The corresponding LSI is shown on the right axis in red.
• Figure S1: Comparison between the actual and estimated mean local densities of water molecules adopting tetrahedral structures, as a function of the average H-bond neighbor distance.
• Figure S2: Temperature dependence of the prefactors $A$ and $k$.