Figure 1. Isochore lines of water at sub- and supercritical regions obtained by molecular dynamics simulation. The density values are shown in the figure whose dimensions are all g.cm-3. Experimental value of the critical point is shown as small circle.
The convergence of the isochore lines disappeared above the pressure of 20 MPa and the temperature of 650 K. That is, the gas-liquid coexistence curve disappeared at this point. This indicates that the critical point exists at this point in our simulation. Although it is difficult to specify the critical point exactly by the simulation, the experimentally obtained value of the critical point of water ( T = 647 K, P = 22 MPa, r = 0.32 g.cm-3 ) also exists at the same position where the convergence of the density lines disappeared. Though TIP3P is a simple model of the water, it can reproduce well the critical point. Guissani and Guillot obtained the critical point by using another water model of SPCE . They also found that the critical value of their simulation corresponded well with the experimental ones. These two models are both parameterized for the water at ambient condition, but they can reproduce the critical phenomena with good accuracy for our purpose. Therefore, we next analyzed the density fluctuation from sub- to supercritical regions.
Analysis of the density fluctuation
In order to investigate the density fluctuation along the critical isochore curve, some of the state points (T, P) with the average density of 0.3 g.cm-3 were selected from Figure 1. The selected state points were shown in Table 1. The density fluctuations of these points were analyzed using bin analysis. In this analysis, the number of water molecules in each bin was calculated at each time step. Then, the time averaged percentage of the number of bins which include different numbers of water molecules were calculated. These values were plotted as ordinate and the number of water molecules in a bin was plotted as abscissa in Figure 2. In order to clear the figure, the results of the five representative state points out of ten were shown in the figure. As the simulation box was divided equally by 125 in this analysis, the average number of water molecules per a bin should be 4.09. Therefore, the histogram in Figure 2 should show a sharp peak at 4.09 if the density of the system is homogeneous. Figure 2 showed that the peak and the width of the histogram become lower and broader when the thermodynamic state approaches the critical point. Especially, at the state point of T=640K, P=23MPa, which is the closest calculated state to the critical point, the shape of the histogram distorted and showed a flattening of the peak. This indicates that the density fluctuation of the water is large near the critical point. With increasing of the pressure and the temperature, the shape of the histogram became sharp and the peak position approached the value of 4.09. These results qualitatively indicate that the density fluctuation is large near the critical point and gradually decreases with the increment of the temperature and the pressure.
Table 1. Trajectory averaged values of the temperature and pressure calculated at some selected points on the isochore curve, which were used for the bin analysis.
|Average Temperature||Average Pressure||Average Density|
|( K )||( MPa )||( g.cm-3 )|
Figure 2. Average percentage of the number of bins which include different number of water molecules were plotted against the number of water in a bin. The measured (T, P) points were on the critical isochore whose values are shown in the figure.
In order to investigate the density fluctuation quantitatively, the density fluctuation defined in the next equation  was calculated from the histograms in Figure 2.
The calculated values of the density fluctuation were plotted against temperatures in Figure 3. It can be seen that the density fluctuation gradually decreased with the increment of the temperature along the critical isochore line from the critical point. Nishikawa et al measured the density fluctuation of supercritical water near the critical point ( T = 660 ~ 690K and P = 22.5 ~ 38.8MPa ) using the small angle x-ray scattering method . In their measurement, density fluctuation took a maximum value of 31.3 at T = 659.6K, P = 25.22 MPa and r = 0.2914 g.cm-3, and the value rapidly decreased to 1.17 at their most remote state from the critical point at T=687.4K, P=23.69, r=0.1244 g.cm-3. Their value is ten times larger than ours near the critical point, though both took similar values at the peripheral points. These differences should attribute to the number of water and the size of bin used in this study. The occurrence of the large cluster and the density fluctuation near the critical point could not be represented by the size of our 512 water molecules. Nishikawa also obtained the value of 16A for the correlation length near the critical point of water . The estimated size of our bin is 7 ~ 8A in average near the critical point in our simulation. In order to investigate the fluctuation near the critical point, the simulation box with more than 3000 water molecules at least should be considered. Although the quantitative analysis of the density fluctuation near the critical point can not be achieved in this study, the rest of the wide supercritical region can be represented well by this bin analysis. Further studies with a larger scale simulation will give us a more accurate picture around the critical point.
Figure 3. Temperature dependence of the density fluctuations along the critical isochore line.
The temperature dependences of the density fluctuation along the isobaric lines were shown in Figure 4. At the pressure of 25 MPa, density fluctuation began to rapidly increase near the critical temperature, and reached a maximum value at 700K, then gradually decreased again. As isobaric lines were taken at successively higher pressures, the peak of the density fluctuation shifted toward the high temperature gradually. The temperature and the pressure values at the peak point along with each isobaric line were listed in Table 2 with its peak value of the density fluctuation.
Figure 4. Temperature dependence of the density fluctuations along the isobaric curves. The pressures are 25MPa ( ), 30MPa ( ), 40MPa ( ), and 50MPa ( ) , respectively.
Table 2. The peak values of the density fluctuation and their (T, P) value along the isobaric lines which were obtained from Figure 4.
|Peak Temperature||Peak Pressure||Peak Density Fluctuation|
|( K )||( MPa )|
By combining the data along the isochore and isobaric lines, the density fluctuations were represented as a contour map and superimposed on the isochore lines on the P-T phase diagram in Figure 5. The contour line shows that the highest point of the density fluctuation occurs in a little high temperature region from the critical point. However, the highest density fluctuation should be at the critical point. As the simulation of this work does not have a sufficient accuracy for the fluctuation analysis at the critical point, the highest area of the fluctuation should include the critical point and the area may spread from critical point to a little high temperature region.
Figure 5. Contour plot of the density fluctuation ( - ) of supercritical water superimposed on the isochore lines ( ... ). The peak points of the density fluctuation along the isobaric lines shown in Table 2 were plotted as ( × ). Experimental value of the critical point is shown as small circle.
Compared to the isochore lines, the contour lines of density fluctuation below the value of 0.8 behave almost parallel to the isochore lines of the average density above 0.4 g.cm-3. However, the contour lines of density fluctuation above the value of 1.0 are not parallel to the isochore lines, and instead, it forms a maximum near the critical point. The threshold line locates around the isochore line with critical density of 0.3 g.cm-3. From the density fluctuation point of view, this threshold line may be regarded as the boundary of the sub- and supercritical regions.
The maxima points of the density fluctuation along the isobaric lines shown in Table 2 are also plotted in the same figure. The connected line of these peak values of the density fluctuation, that is, the ridge of the density fluctuation, roughly lies along the isochore line at density of 0.2 g.cm-3. This indicates that the ridge of the density fluctuation deviates to a little lower density side from the critical isochore line of 0.3 g.cm-3. This deviation becomes large with the increase of the pressure. Nishikawa et al  showed that the ridge of the density fluctuation exists and deviates to a little lower density region than the critical isochore line. The result of our simulation is in accord with their results. However, the ( P, T ) region investigated by their experiment was limited because of the experimental difficulty. They investigated the temperature ranging from 660K to 690K and pressure ranging from 22.5 MPa to 38.8 MPa. Compared to the experimental method, simulation has an advantage in that it can investigate at the extreme conditions. The results obtained in this simulation could predict that the ridge of the density fluctuation spreads to a wide area of the supercritical field. Futhermore, the ridge line deviates from the critical isochore line and lies along the isochore line at 0.2 g.cm-3.
In order to investigate the water structure around the ridge line of the density fluctuation, the coordination number of water in the first hydration shell was calculated. The coordination number, <N>, was evaluated by integrating the oxygen-oxygen radial distribution function until it takes the first minimum at 4.0 A. The results were shown in Figure 6(a). It can be seen that the coordination number decreases with increasing temperature along each isobaric line. Along the isobaric line at 25MPa, the coordination number rapidly decreases at around the critical temperature and reaches a constant value at high temperature. With the elevation of the pressure, the coordination number decreases monotonically. As the bulk density changes according to the elevation of the temperature along the isobaric line, the coordination number should be affected by the bulk density. Therefore, in order to analyze the coordination behavior in more detail, the ratio of <N>/r was plotted in Figure 6(b). It can be seen that all the values of ratio <N>/r take almost constant value below the critical temperature. This means that the coordination number changes proportionally to the bulk density. However, the ratio <N>/r increases above the critical temperature in all cases. Especially, the increment is largest at 25 MPa among the isobaric lines. This could show that the preferential aggregation of water molecules occurred above the critical temperature. This behavior is in accord with that reported by Yoshii et al . Although the curve greatly waves at 25MPa due to the large fluctuation, the peak exists in all cases of the isobaric lines and their values shift toward the higher temperature with the increment of the pressure. This behavior is similar to that of density fluctuation, which was shown in Figure 4. These results may suggest that the preferential aggregation of water becomes maximum along the ridge line of the density fluctuation.
Figure 6. Temperature dependence of the coordination number of water, <N>, in the first hydration shell within the radius of 4.0 A along the isobaric curves (a). The ratio of the coordination number of water and the bulk density, <N>/r, was also plotted in (b). The pressures are 25MPa ( ), 30MPa ( ), 40MPa ( ), and 50MPa ( ) , respectively.