Widom line

R. Evans, J. R. Henderson, D. C. Hoyle, A. O. Parry, Z. A. Sabeur. Asymptotic decay of liquid structure oscillatory liquid-vapour density profiles and the Fisher-Widom line. Mol Phys 50 755-775, 1993. 

Therefore, the maximum of Cp °° occurs where the correlation length associated with the tetrahedral order is maximum, i.e. along the Widom line associated with the LL phase transition." In MF we may compare Cp calculated for the LLCP scenario J(j > 0) with Cp calculated for the SF scenario J(j = 0) [Fig. 7(b)]. We see that the sharper maximum is present only in the LLCP scenario, while the less sharp maximum occurs at the same T in both scenarios. We conclude that the sharper maximum is due to the fluctuations of the tetrahedral order, critical at the LLCP, while the less sharp maximum is due to fluctuations in bond formation. The similarity of our results with the experiments in nanopores is striking. Data in ref. [ °] show two maxima in Cp. They have been interpreted as an out-of-equilibrium dynamic effect in [ °], but more recent experiments show that they are a feature of equilibrated confined water. Therefore, our interpretation of the two maxima is of considerable interest. 

The Widom line is a line in the P-T phase diagram that starts from the eritical point and rises above the eritical pressure while remaining at the critical temperature [3]. It has been observed that when one approaches this line at constant pressure above the critical pressure by varying the temperature (all above 7c), the system experiences large density fluctuations, as reflected in the increase of compressibility. Speciflc heat also increases. Thus, the anomalous properties in the supercritical state can be correlated with the proximity to the Widom line. 

The presence of this Widom line has been proposed to explain low-temperature thermodynamic anomalies. As is evident from the above, the Widom line starts from the critical point where density fluctuation is highest and the fluctuation becomes weaker as the state point goes away from the critical point. 

The concept of the Widom line is also applicable to the density fluctuation and inhomogeneity observed for different systems beyond critical point in the supercritical region. 

Some of the anomalies of supercritical fluids can be understood by using the idea of the Widom line. One can then relate, for example, file width of a Raman tine to the temperature- and density-dependent correlation length of the fluid. As we cross the Widom line at constant density, we would expect a sharp rise in the width of the Raman... 

It is interesting to note that the useful properties of supercritical water arise from the breakdown of the extensive HB network that is at least partly responsible for many of the anomalies of liquid water. We have discussed how the use of the idea inherent in the Widom line helps in understanding the large-scale fluctuations observed in supercritical water. Because of the large separation of timescales between vibrational relaxation and density relaxation, the vibrational line widths are influenced significantly by the transient density inhomogeneity present near the critical temperature. 

G. G. Simeoni, T. Bryk, F. A. Gorelh, et al. The Widom line as flie crossover between liquid-Uke and gas-like behaviour in supercritical fluids. Nat. Phys., 6 (2010), 503-507. 

In order to understand the utility of this picture, let us now consider the gas-liquid critical point in the (P,T) plane. While there exists only a fluid state beyond the critical point, one can still find that across the line that extends straight above the Tq the density fluctuation in the system shows a maximum when this line is crossed at different pressures. Thus across this line, the response functions show a maximum. This line is widely known as the Widom line. Now if one assumes that there exists a similar Widom line in the supercooled region corresponding to the second critical point, then one expects the maximum in response function across the line (see Figure 22.3) [6]. 

As discussed in the introductory chapter, volume fluctuation is directly proportional to the isothermal compressibility (kt) of the system and therefore behaves anomalously in supercooled water. The k-y starts increasing with a decrease in temperature below T = 320 K and has a maximum on crossing the Widom line (observed in experiments on nano-confined systems). 

P. H. Poole, F. Sciortino, U. Essmann, and H. E. Stanley, Phase behavior of metastable water. Nature, 360 (1992), 324—328 L. Xu, P. Kumar, S. V. Buldyrev, et ah. Relation between the Widom line and the dynamic crossover in systems with a liquid-liquid phase transition. Proc. Natl. Acad. Sci. USA, 102 (2005), 16558-16562. 

L. Xu, et al.. Relation between the Widom line and the d3uiaimc crossover in systems with a liquid-liquid phase transition, Proc. Nad. Acad. Sci. USA 102, 16558-16562 (2005). 

Figure 3. Different phase diagrams of water generated by changing parameters in the cell model of Stokely et al. [32]. (a) Singularity free (SF). (b) Liquid-liquid critical point (LLCP). (c) Liquid-liquid critical point at negative pressure, (d) Critical point free with reentrant stability limit (CPF/SL). HDL and LDL refer to high- and low-density liquids, respectively. The L-L Widom line is the locus of maxima in the correlation length emanating from the LLCP. Reprinted with permission from Ref. [32].

Measurements of transport properties would provide another means to explore the metastable regime. In particular, studies based on simulation focused on the supercooled regime [104] correlate the breakdown of the Stokes-Einstein relation Dv = constant) with the Widom line w P), locus of the correlation length maxima emanating down from the proposed liquid-liquid critical point toward lower pressures (Fig. 3b). Measurements of diffusivity could be performed at negative pressure by NMR on static samples (e.g., via MVLE or inclusions) and viscosity could be measured by capillary rheometry with the MVLE method. 

J. L. F. Abascal and C. Vega, Widom line and the liquid-liquid critical point for the TIP4P/2005 water model, J. Chem. Phys. 133(23), 234502 (2010). 

Figure 5 shows the Kj and Cp lines obtained from MD simulations of two different polymorphic liquids. The and lines are extensions of the LLPT line (T < Tc ) into the supercritical region (T > Tc) and are born at the LLCP. Moving along these maxima lines towardihe LLCP results in an increase of kt T, P) and Cp(T, P) and, at the LLCP, both kt(T, P) and Cp T, P) become infinite. Close to the LLCP, both lines asymptotically approach one another this asymptotic line is sometimes called the Widom line [41]. 

Figure 2. The phase diagram of water in the P-T plane [2,3]. Th denotes the homogeneous nucleation temperature line, Tx is the crystallization line of amorphous water, is the melting temperature line, and Tmd is the maximum density line. Tw indicates the Widom line locus.

From a structural point of view, all the proposed scenarios give a key role to the tetrahedral geometry of the local HB interaction pattern. In the liquid state this HB network governs the overall structure and dynamics of water. Further, the LLPT approach focuses on the so called Widom line, that is, the locus of the maximum correlation length [30,31]. Along the Widom line, the response functions show extremes and finally diverge at the critical point. 

Due to the experimental difficulty in exploring the No-Man s Land, the phase diagram shown in Fig. 2 and the physical scenario proposed by the LLPT hypoth esis (and in particular the Widom line) until very recently were only hypothesized and not completely proven. The power law approach, used for many years to explain water singularities, corresponds to the extension of a first-order transition line beyond the critical point. Thus, when experimentally approaching the Widom line, the thermodynamic response functions should behave as though they were going to diverge with critical exponents, but they do not. 

This interpretation of the MIT-Messina experiments relies on the Widom line concept, which is still not widely accepted—even though it has been widely known among experimentalists since its introduction in the 1958 Ph.D. thesis of J.M.H. Levelt (now Levelt Sengers) [31]. By definition, a Widom line arises only from a critical point and, if their experiments can be rationalized by the existence of such a Widom line, the MIT Messina results are consistent with the existence of a LL critical point in confined water. 

If the system is cooled isobarically along a path above the critical pressure Pc (Fig. 5b, path a), 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., C ), have maxima at temperatures denoted Pmax (P) Remarkably these maxima are still prominent far above the critical pressure [31], and the values of the response functions at Pmax(P) (e-g-, C 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, since 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 regime. ... 

Suppose the system is cooled at constant pressure Pq. (i) If Pq > Pc ( path a ), experimentally measured quantities will change dramatically but continuously in the vicinity of the Widom line (with huge fluctuations as measured by, e.g., Cp). (ii) If Po < Pc ( path j8 ), experimentally measured quantities will change discontinuously if the coexistence line is actually seen. However, the coexistence line can be difficult to detect in a pure system due to metastability, and changes will occur only when the spinodal is approached where the gas phase is no longer stable. 

Figure 5. Schematic illustration on the significance of the Widom line [54].

Using MD simulations [82,83], we studied three models, each of which has a LL critical point. Two (the TIP5P and the ST2) treat water as a multiple-site rigid body that interacts via electrostatic site-site interactions complemented by a Lennard-Jones potential. The third is the spherically symmetric two-scale Jagla potential with attractive and repulsive ramps. In all three models the loci of maxima of the relevant response functions, Ki and Cp, which coincide close to the critical point and give rise to the Widom line, were evaluated. The hypothesis that, for all three potentials, a dynamic crossover occurs when the Widom line is crossed, was carefully explored. 

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