Thermodynamic geometry of supercooled water

The thermodynamic curvature scalar $R$ is evaluated for supercooled water with a two-state equation of state correlated with the most recent available experimental data. This model assumes a liquid-liquid critical point. Our investigation extends the understanding of the thermodynamic behavior of $R$ considerably. We show that $R$ diverges to $-\infty$ when approaching the assumed liquid-liquid critical point. This limit is consistent with all of the fluid critical point models known so far. In addition, we demonstrate a sign change of $R$ along the liquid-liquid line from negative near the critical point to positive on moving away from the critical point in the low density “ice-like” liquid phase. We also trace out the Widom line in phase space. In addition, we investigate increasing correlation length in supercooled water and compare our results with recent published small angle x-ray scattering measurements.

Figure 1

A schematic diagram of the metastable, supercooled liquid phase extending down to the limiting homogeneous nucleation temperature $T_H$, where an amorphous solid must form. We show the first-order liquid-liquid phase transition curve between a high-density phase (HDL) and a low-density phase (LDL). This curve terminates in a second critical point ${\text{C}}^{\prime }$. This phase transition is not directly accessible to measurements in the bulk, since it falls below $T_H$. However, the thermodynamic properties in the liquid phase appear to be strongly affected by the critical properties near ${\text{C}}^{\prime }$. Further metastable and regular cooling brings us to the cubic ice crystallization curve $T_x$, with amorphous high-density and low-density amorphous phases HDA and LDA corresponding to the HDL and LDL liquid phases. Also shown is the regular liquid-vapor coexistence line, terminating in the regular critical point C.

Figure 2

$R$ diagram ($R$ contours measured in ${\text{nm}}^3$) of supercooled water for temperatures ranging from $T=220$ to 300 K and for pressures ranging from $p=-100$ to 400 MPa. In the negative pressure region we omit a colored contour field in order to indicate the extrapolated character of this region. The LLT line is marked by a red solid curve terminating at the LLCP marked by a red solid circle (${\text{C}}^{\prime }$). Two lines $R=0$ develop in the diagram, one in the stable cold water region close to the freezing line, and the other in the LDL region below the LLT line. The triple point is marked by a blue solid box, and the solid-liquid coexistence line by a blue solid curve showing the melting lines of ice I, III, and V. The liquid-vapor coexistence curve starting from the triple point is indicated as a dark red line.

Figure 3

Cube root of the thermodynamic curvature $R^{1/3}$ along the LDL side (red curve) and the HDL side (blue curve) of the LLT line as a function of temperature (a) and of density (b). On the LDL side a point $R=0$ occurs approximately at $T=223$ K and $\rho =900\text{kg}/{\text{m}}^3$ (marked by arrows) indicating a change to organized “ice-like” structures.

Figure 4

Comparison between the thermodynamic curvature $R$ (measured in ${\text{nm}}^3$) obtained from the HAS-TSEOS (red lines) [23] and the IAPWS-95 formulation (green lines) [36, 37] for temperatures ranging from $T=240$ to 330 K at different constant pressures of $p=300,200,100,60,20$, and 0.1 MPa. The blue dotted lines indicate the melting temperatures.

Figure 5

Development of the negative curvature $R$ near the LLCP along isobars (a) in a range from $-15$ to 10 MPa and along isotherms (b) ranging from 227 to 230 K. LLCP coordinates are $p_C=0$ MPa, $T_C=228$ K, and ${\rho }_C=914.844\text{kg}/{\text{m}}^3$.

Figure 6

Points with maximum $\left|R\right|$ along curves with constant temperature or pressure determining the Widom line. The slope of lines ${\left|R\right|}_{\max }$ is shown at constant temperature and constant pressure in the $\rho -T$ (a) and the $p-T$ (b) phase projections. The lines of ${\left|R\right|}_{\max }$ start at the LLCP with $p_C=0$ MPa, $T_C=228$ K, and ${\rho }_C=914.844\text{kg}/{\text{m}}^3$.

Figure 7

Temperature-dependent behavior of the correlation length $\xi$, at ambient pressure in the supercooled region of water. Circles indicate small angle x-ray scattering measurements [30] down to a temperature of 252 K. The brown curve represents their power law [30] extrapolated to values close to the LLCP. The green and the red curve show $\xi$ calculated using the relation $\left|R\right|/2={\xi }^3$ obtained from the HAS-TSEOS [23] and the IAPWS-95 formula [36], respectively.