Tuning the Liquid-Liquid Transition by Modulating the Hydrogen-Bond Angular Flexibility in a Model for Water

We propose a simple extension of the well known ST2 model for water [F. H. Stillinger and A. Rahman, J. Chem. Phys. 60, 1545 (1974)] that allows for a continuous modification of the hydrogen-bond angular flexibility. We show that the bond flexibility affects the relative thermodynamic stability of the liquid and of the hexagonal (or cubic) ice. On increasing the flexibility, the liquid-liquid critical point, which in the original ST2 model is located in the no-man’s land (i.e., the region where ice is the thermodynamically stable phase) progressively moves to a temperature where the liquid is more stable than ice. Our study definitively proves that the liquid-liquid transition in the ST2 model is a genuine phenomenon, of high relevance in all tetrahedral network-forming liquids, including water.

Figure 1

Schematic plot of the ST2 water model and of the proposed extension to modulate hydrogen-bond flexibility. Solid lines indicate the position of the H and LP sites in the rigid original ST2 model. The cones have an angular amplitude equal to ${\theta }_{\max }$ and define the volume limiting the position of the same sites in the flexible model (dashed lines).

Figure 2

(a) Probability distribution of the $\angle (\mathrm{O},\mathrm{O},\mathrm{O})$ angle for bonded triplets at $\rho =0.90\mathrm{g}/{\mathrm{cm}}^3$ for three different values of the flexibility. Note the increase of the variance of the distribution on increasing flexibility. Two adjacent water molecules are considered bonded if the O-O distance is less than 3.2 Å. (b) Oxygen-oxygen structure factors $S(q)$ for the same state points, displaying the effect of the flexibility on the prepeak.

Figure 3

Distribution $P(N)$ in the number $N$ of particles populating a volume of $8{\mathrm{nm}}^3$ at fixed temperature and chemical potential $\mu$, for three different bond flexibilities ${\theta }_{\max }=0^{\circ }$ (a), ${\theta }_{\max }=11.5^{\circ }$ (b) and ${\theta }_{\max }=14^{\circ }$ (c). This last quantity controls the average density and it is selected to provide equal area below the low-density and high-density liquid phases. $P(N)$ evolves from a single-peak to a double peak shape on crossing $T_c^{\mathrm{LL}}$. The data for (a) are redrawn from Ref. [19].

Figure 4

Dependence of the liquid-liquid critical point temperature $T_c^{\mathrm{LL}}$ on bond flexibility calculated from the free energy estimate based on successive umbrella sampling simulations. These grand-canonical simulations are performed for a volume of $8{\mathrm{nm}}^3$. The dashed red line indicates $T_m$, the melting $T$ for the liquid to $I_h$ and $I_c$ transformation at the critical pressure. The two insets show, respectively, the critical pressure $P_c$ and the critical density ${\rho }_c$ as a function of ${\theta }_{\max }$.

Figure 5

Pressure $P$ dependence of the reduced chemical potential $\beta \mu$ for $I_h$ and $I_c$ and for the low- and high-density liquid phases at $T_c^{\mathrm{LL}}$ for (a) ${\theta }_{\max }=0°$ and (b) ${\theta }_{\max }=11.5°$. Note that while for ${\theta }_{\max }=0°$ (e.g., for the original ST2 model) $\beta \mu$ of the liquid phases is higher than the one of $I_h$ and $I_c$; the opposite is found already for ${\theta }_{\max }=11.5°$. Under these conditions, the liquid is thermodynamically more stable than $I_h$ and $I_c$.