Critical point free scenario

Going under the assumption that the form of the excess Cp T) reported in these latter experiments differs completely from that in common glass formers, but resembles that of the classical order-disorder transition, an interesting analysis of these calorimetric data of nanoconlined water has been proposed [122]. Note that the order-disorder transition (critical point-free) scenario differs little from the second critical point scenario, which attributes all water anomalies to the existence of a second critical point. 

Below, we describe some of the scenarios that have been explored as a way of rationalizing the thermodynamics of liquids displaying anomalies, such as water. These are (i) the stability limit conjecture [46,55,58,59], (ii) the liquid-liquid critical point scenario [16], (ill) the singularity free scenario [60], and (iv) the critical point free scenario [37],... 

Critical Point Free Scenario Recently, Angell [37] has discussed a possibility, related to some of the early observation of Speedy and Angell [46], in which the high temperature liquid encounters a spinodal at positive pressure, but this is a spinodal associated with a first order transition between two liquid states. Such a first order transition, however, does not terminate in a critical point but may terminate at the liquid gas spinodal. A weaker version of this picture is that no critical point may exist at positive pressures. Analysis of a model calculation by Stokely et al. [72] indicates that such a scenario may indeed arise in the limit of extreme cooperativity of hydrogen bond formation. 

Figure 13. Schematic phase diagram of water s metastable states. Line (1) indicates the upstroke transition LDA —>HDA —>VHDA discussed in Refs. [173, 174], Line (2) indicates the standard preparation procedure of VHDA (annealing of uHDA to 160 K at 1.1 GPa) as discussed in Ref. [152]. Line (3) indicates the reverse downstroke transition VHDA—>HDA LDA as discussed in Ref. [155]. The thick gray line marked Tx represents the crystallization temperature above which rapid crystallization is observed (adapted from Mishima [153]). The metastability fields for LDA and HDA are delineated by a sharp line, which is the possible extension of a first-order liquid-liquid transition ending in a hypothesized second critical point. The metastability fields for HDA and VHDA are delineated by a broad area, which may either become broader (according to the singularity free scenario [202, 203]) or alternatively become more narrow (in case the transition is limited by kinetics) as the temperature is increased. The question marks indicate that the extrapolation of the abrupt LDA<- HDA and the smeared HDA <-> VHDA transitions at 140 K to higher temperatures are speculative. For simplicity, we average out the hysteresis effect observed during upstroke and downstroke transitions as previously done by Mishima [153], which results in a HDA <-> VHDA transition at T=140K and P 0.70 GPa, which is 0.25 GPa broad and a LDA <-> HDA transition at T = 140 K and P 0.20 GPa, which is at most 0.01 GPa broad (i.e., discontinuous) within the experimental resolution.

Although the presented scenarios are still under discussion, the existence of a first-order like transition between metastable high- and low-density supercooled water with a second critical point at negative pressures in bulk water and positive pressures in confinement is strongly suggested (29). Alternatively, singularity-free scenarios are discussed to explain the properties of supercooled water (24, 29). 

Several explanations have been proposed to account for the anomalies of water (i) the stability limit conjecture [18], (ii) the metastable liquid-liquid critical point (LLCP) hypothesis [19], and (iii) the singularity free (SF) scenario [20]. An excellent review of the properties of supercooled water and the explanations proposed can be found in Ref. [16]. We just give here a brief overview. 

We then calculated the Arrhenius activation energy (P) from the low-T slope of log T versus l/T and extrapolated the temperature T P) at which t reaches a fixed macroscopic time ta > tc, with Tg, P) smaller than the crossover temperature. For Ta = 10 Monte Carlo (MC) steps > 100 s, Ea(P) and Tg (P) decrease upon increasing P in both scenarios, providing no distinction between the two interpretations. Instead, there is a dramatic difference in the -dependence of the quantity E / k T in the two scenarios, increasing for the LL critical point and approximately constant for the singularity free. 

Figure 3. Schematic phase diagrams in the pressure-temperature (P, Ij plane illustrating three scenarios for liquids displaying anomalous thermodynamic behaviour, (a) The spinodal retracing scenario. (b) The liquid-liquid critical point scenario, (c) The singularity free scenario. The dashed line represent the liquid-gas coexistence line, the dotted line is the liquid liquid coexistence line, the thick solid line is the liquid spinodal, the long dashed lines is the locus of compressibility extrema and the dot dashed line is the locus of density extrema. The liquid-gas critical point is represented by filled circle and the liquid-liquid critical point by filled square.

The Singularity Free Hypothesis Sastry et al. [60] proposed that a minimal scenario that was consistent with the salient anomalies did not require recourse to any thermodynamic singularities, such as a critical point or a retracing spinodal. They analyzed the interrelationship between the locus of density and the compressibility extrema and showed that the change of slope of the locus of density maxima (TMD) was associated with an intersection with the locus of compressibility extrema (TEC) (Fig. 3c). The relationship between the temperature dependence of isothermal compressibility at the TMD and the slope of the TMD is given by... 

The M3 scenario. This occurs because of an upstream control, as by the sluice gate. The bed slope is not sufficient to sustain lower-stage flow, and, at a certain point determined by energy and momentum relations, the water surface will pass through a hydraulic jump to upper-stage flow unless this is made unnecessary by the existence of a free overfall before the M3 crest reaches critical depth. 

Because the recent experiments and simulations reviewed here concentrated on the universal aspects of the novel non-equilibrium transition, focus will be laid on the MCT-ITT approach. Reassuringly, however, many similarities between the MCT-ITT equations and the results by Miyazaki and Relchman exist, even though these authors used a different, field theoretic approach to derive their results. This supports the robustness of the mechanism of shear-advection in (7) entering the MCT vertices in (lid, 14), which were derived independently in [40, 41] and [43 5] from quite different theoretical routes. This mechanism had been known from earlier work on the dynamics of critical fluctuations in sheared systems close to phase transition points [61], on current fluctuations in simple liquids [62], and on incoherent density fluctuations in dilute solutions [63], Different possibilities also exist to include shear into MCT-inspired approaches, especially the one worked out by Schweizer and coworkers including strain into an effective free energy [42]. This approach does not recover the (idealized) MCT results reviewed below but starts from the extended MCT where no true glass transition exists and describes a crossover scenario without, e.g., a true dynamic yield stress as discussed below. 

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