Crossing the Widom line

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. 

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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... 

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). 

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]. 

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. 

Figure 4.9 shows the temperature dependence of the isobaric thermal expansion coefficient, Up, below, at, and above the critical pressure. The dashed line is the ideal gas result. Below the critical pressure a jump occurs when the gas-liquid saturation line (cf. Fig. 4.4 right panel) is crossed. At the critical point we observe a divergence. Above the critical point a maximum marks the smooth continuation of the gas-liquid saturation line, which sometimes is called Widom line. 

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