We investigate, for two water models displaying a liquid-liquid critical point, the relation between changes in dynamic and thermodynamic anomalies arising from the presence of the liquidliquid critical point. We find a correlation between the dynamic crossover and the locus of specific heat maxima C P max (''Widom line'') emanating from the critical point. Our findings are consistent with a possible relation between the previously hypothesized liquid-liquid phase transition and the transition in the dynamics recently observed in neutron scattering experiments on confined water. More generally, we argue that this connection between C P max and dynamic crossover is not limited to the case of water, a hydrogen bond network-forming liquid, but is a more general feature of crossing the Widom line. Specifically, we also study the Jagla potential, a spherically symmetric two-scale potential known to possess a liquid-liquid critical point, in which the competition between two liquid structures is generated by repulsive and attractive ramp interactions. By definition, in a first-order phase transition, thermodynamic state functions such as density and enthalpy H change discontinuously as we cool the system along a path crossing the equilibrium coexistence line (Fig. 1a, path ␤). However, in a real experiment, this discontinuous change may not occur at the coexistence line because a substance can remain in a supercooled metastable phase until a limit of stability (a spinodal) is reached (1) (Fig. 1b, path ␤).If the system is cooled isobarically along a path above the critical pressure P c (Fig. 1b, path ␣), the state functions continuously change from the values characteristic of a high-temperature phase (gas) to those characteristic of a low-temperature phase (liquid). The thermodynamic response functions, which are the derivatives of the state functions with respect to temperature (e.g., isobaric heat capacity C P ϭ (ѨH͞ѨT) P ), have maxima at temperatures denoted T max (P). Remarkably, these maxima are still prominent far above the critical pressure (2-5), and the values of the response functions at T max (P) (e.g., C P max ) diverge as the critical point is approached. The lines of the maxima for different response functions asymptotically approach one another as the critical point is approached, because all response functions become expressible in terms of the correlation length. This asymptotic line is sometimes called the ''Widom line'' and is often regarded as an extension of the coexistence line into the ''one-phase region.'' If the system is cooled at constant pressure P 0 , and P 0 is not too far from the critical pressure P c , then there are two classes of behavior possible. (i) If P 0 Ͼ P c (path ␣), then experimentally measured quantities will change dramatically but continuously in the vicinity of the Widom line (with huge fluctuations as measured by, e.g., C P ). (ii) If P 0 Ͻ P c (path ␤), experimentally measured quantities will change discontinuously if the coexistence line is actually seen. However, ...

Bulk water has three phases: solid, liquid and vapour. In addition to undergoing a phase transition (of the first order) between them, liquid and vapour can deform continuously into each other without crossing a transition line-in other words, there is no intrinsic distinction between the two phases. Hence, the first-order line of the liquid-vapour phase transition should terminate at a critical point. In contrast, the firstorder transition line between solid and liquid is believed to persist indefinitely without terminating at a critical point 1 . In recent years, however, it was reported that inside carbon nanotubes, freezing of water may occur continuously as well as discontinuously through a first-order phase transition 2 . Here we present simulation results for water in a quasi-twodimensional hydrophobic nanopore slit, which are consistent with the idea that water may freeze by means of both firstorder and continuous phase transitions. Our results lead us to hypothesize the existence of a connection point at which first-order and continuous transition lines meet 3,4 .It is widely believed that a solid-liquid phase transition line does not terminate at a critical point, although it has not been possible to prove rigorously the non-existence of the solid-liquid critical point 1 . On the other hand, after the discovery of carbon nanotubes 5 , a solid-liquid phase transition inside these structures has been reported, together with the intriguing possibility of a solid-liquid critical point, presumably because of the constraints in packing inside carbon nanotubes 6 . In recent years, therefore, there has been a great deal of interest in the simulation report that freezing of water inside carbon nanotubes may occur continuously as well as discontinuously through a first-order transition 2,6,7 .As for the solid-liquid phase transition, two-dimensional systems are of particular interest. According to the Mermin-Wagner theorem, continuous symmetry in two dimensions cannot be broken at non-zero temperatures, which means that there is no true long-ranged crystalline order in two-dimensional systems 8 . However, this does not necessarily rule out the presence of a phase transition; instead the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) theory states that melting in two dimensions may involve two continuous transitions, one from the solid phase to a hexatic phase and the other from the hexatic phase to the liquid phase 9 . Although the KTHNY theory seems rather appealing and general, it is still controversial whether the KTHNY theory provides the only mechanism of the melting in two dimensions 9,10 . Here we present simulation results supporting the possibility that a solid-liquid phase transition of water in quasi-twodimensions may occur through both first-order and continuous phase transitions, depending on the density or pressure.We carry out extensive molecular dynamics simulations of the TIP5P model 11 of N = 256 water molecules confined between two unstructured and smooth hydrophobic plates (see ref. 12

We perform molecular dynamics simulations of 512 waterlike molecules that interact via the TIP5P potential and are confined between two smooth hydrophobic plates that are separated by 1.10 nm. We find that the anomalous thermodynamic properties of water are shifted to lower temperatures relative to the bulk by Ϸ40 K. The dynamics and structure of the confined water resemble bulk water at higher temperatures, consistent with the shift of thermodynamic anomalies to lower temperature. Because of this T shift, our confined water simulations ͑down to T = 220 K͒ do not reach sufficiently low temperature to observe a liquid-liquid phase transition found for bulk water at T Ϸ 215 K using the TIP5P potential, but we see inflections in P-isotherms at lower temperatures presumably due to the presence of a liquid-liquid critical point. We find that the different crystalline structures that can form for two different separations of the plates, 0.7 and 1.10 nm, have no counterparts in the bulk system, and we discuss the relevance to experiments on confined water.