Thermal conductivity of supercooled water

The heat capacity of supercooled water, measured down to $-37{}^{\circ }\mathrm{C}$, shows an anomalous increase as temperature decreases. The thermal diffusivity, i.e., the ratio of the thermal conductivity and the heat capacity per unit volume, shows a decrease. These anomalies may be associated with a hypothesized liquid-liquid critical point in supercooled water below the line of homogeneous nucleation. However, while the thermal conductivity is known to diverge at the vapor-liquid critical point due to critical density fluctuations, the thermal conductivity of supercooled water, calculated as the product of thermal diffusivity and heat capacity, does not show any sign of such an anomaly. We have used mode-coupling theory to investigate the possible effect of critical fluctuations on the thermal conductivity of supercooled water and found that indeed any critical thermal-conductivity enhancement would be too small to be measurable at experimentally accessible temperatures. Moreover, the behavior of thermal conductivity can be explained by the observed anomalies of the thermodynamic properties. In particular, we show that thermal conductivity should go through a minimum when temperature is decreased, as Kumar and Stanley observed in the TIP5P model of water. We discuss physical reasons for the striking difference between the behavior of thermal conductivity in water near the vapor-liquid and liquid-liquid critical points.

Figure 1

Thermal-diffusivity data of Benchikh et al. [21] and Taschin et al. [20], compared with the thermal diffusivity calculated from Bridgman's (4) and Eyring's (5) formulas. We use the TSEOS to evaluate the thermodynamic properties in these formulas [17]. The inset shows simulation results of Kumar and Stanley for the TIP5P model [12].

Figure 2

Heat capacity data of Archer and Carter [3] and Angell et al. [24]. The solid curve shows the prediction of the TSEOS [17].

Figure 3

Thermal conductivity calculated from thermal diffusivity [20, 21] and the TSEOS [17]. The formulations due to Bridgman (4) and Eyring (5) are both shown, as is the IAPWS correlation for thermal conductivity (which is only guaranteed down to 273 K) [22]. The inset shows the simulation data of Kumar and Stanley for the TIP5P model [12], and the dashed curve on the inset is a quadratic fit to the simulation data. $T_{\mathrm{W}}$ refers to the Widom temperature in both the main graph and the inset.

Figure 4

Viscosity data taken by Osipov et al. at atmospheric pressure (dots) [31]. The solid curve shows the IAPWS formulation for viscosity at atmospheric pressure [29, 30], while the dashed curve is a fit to the data of a super-Arrhenius or Vogel-Fulcher-Tamman law: $\eta =0.0885\exp [220/(T-197\left)\right]$, with $T$ in $K$. The inset shows the product of viscosity and thermal conductivity. The increase in the viscosity completely dominates the decrease in the thermal conductivity, so it is clear that the decrease in thermal conductivity is not a result of the increase in viscosity.

Figure 5

The main figure shows thermal conductivity in water, both from experimental data [20, 21] and from the Eyring's equation (5), evaluated with the TSEOS [17]. The inset shows the critical enhancement in the same units but magnified by a factor of one million (${10}^6$). The curves change from solid to dashed at the temperature of homogeneous nucleation. The critical effect is several orders of magnitude smaller than the background and will be completely undetectable.