Liquid-solid coexistence curve

Fig. 11.1 (a) Theoretical phase diagrams of temperature versus polymer volume fraction in polymer solutions with the chain length 32 monomers, (b) Molecular simulation results under parallel conditions. The reduced interaction parameter sets are labeled as (Ep/Ec, B/E and the liquid-solid coexistence curve T EpIEc, B/Ec) (Hu et al. 2003a) (Reprinted with permission)... 

In Fig. 11.3, we made a comparison between the binodals obtained from dynamic Monte Carlo simulations (data points) and from mean-field statistical thermodynamics (solid lines). First, one can see that even with zero mixing interactions B = 0, due to the contribution of Ep, the binodal curve is still located above the liquid-solid coexistence curve (dashed lines). This result implies that the phase separation of polymer blends occurs prior to the crystallization on cooUng. This is exactly the component-selective crystallizability-driven phase separation, as discussed above. Second, one can see that, far away from the liquid-solid coexistence curves (dashed lines), the simulated binodals (data points) are well consistent... 

Unlike the liquid-solid coexistence curve, the Uquid-vapor coexistence curve is not infinite. A maximum has been observed on this curve. Consequently, the liquid and vapor states cannot be considered as two separate states, but as the extremes of the same continuum (the dotted arrow) [2]. 

Figure 13.5 shows the parallel results of liquid-liquid demixing curves and liquid-solid coexistence curves in polymer solutions with different energy parameter sets, obtained from the lattice mean-field theory and Monte Carlo simulations. Theoretical calculations and Monte Carlo simulations agree well with each other, validating the mean-field assumption in theoretical approaches. 

Figure 7.1 Schematic phase diagram of water (not to scale), showing phase boundaries (heavy solid lines), triple point (triangle), critical point (circle-x), and a representative point (circle, dotted lines) at 25°C on the liquid-vapor coexistence curve.

FIGURE 5.1. Liquid/vapour coexistence curves of SPC water (solid line) and OPLS methanol (dashed line) following hrom the RISM-KH theory. Molecular simulation results for water [45] and methanol [46] (open circles and squares, respectively) and critical point extrapolations (filled symbols). 

FIGURE 5.8. Liquid/vapour coexistence curve of the Lennard-Jones fluid predicted by the lOZ/LMBW theory in the KHM, KH and VM approximation (solid, short-dash and long-dash lines, respectively). Monte Carlo (MC) simulations are shown by the open circles. 

Figure 7 The vapor pressure and liquid-vapor coexistence curves (including estimated critical points) for methanethiol from simulation and experiment. The square symbols are obtained using the quantum-mechanical-based potential in simulation, the triangles from using the OPLS-UA potential, and the solid line shows the smoothed experimental data. The filled symbols are measured and estimated critical points (ref 22)...

FIGURE 5.2. Liquid/vapour coexistence curves of hydrogen fluoride (solid line), methanol (short-dashed line), water (long-dashed line) and dimethylsulfoxide (dash-dotted line) following from theRISM-KH theory. Experimental data for their critical points (filled triangle, square, circle and rhomb, respectively), and at ambient conditions (corresponding open symbols). 

Fig. 3. Equation of state obtained from the energy equation. The curves are isotherms and are labeled with appropriate t. The broken portion of the isotherms occur in the two-phase region which bounded by the liquid vapor coexistence curve. The solid dot is the critical point. The quantity Po =...

Figure 2.3. The liquid-vapor coexistence curves, T vs p, as predicted by Guggenheim (solid line) and van der Waals.

The simplest class of binary phase diagram is class I as shown in Figure 1.2-3. The component with the lower critical temperature is designated as component 1. The solid lines in Figure 1.2-3(b) represent the pure component liquid-vapor coexistence curves which terminate at the pure component critical points (Cj and C2). The feature of importance in this phase diagram is that the mixture critical line (dashed line in Figure 1.2-3(b)) is continuous between the two critical points. The mixture critical line represents the locus of critical points for all mixtures of the two components. The area bounded by the solid and dashed lines represents the two-phase, liquid-vapor (LV) region. The mixture-critical... 

It appears that the phase diagram for very small q <0.3) is much simpler than for larger q [144—145, 213, 214]. For small q the only effect of adding polymer chains to the pure hard-sphere dispersion is the widening of the fluid-solid coexistence region. A gas-liquid phase transition occurs at larger polymer concentrations above the fluid-solid phase line and is metastable, see Sect. 3.3.4. Only above a certain range of attraction, the (colloidal) gas-liquid phase transition shifts below the fluid-solid coexistence curve. For q close to 1/3 the critical point hits the fluid-solid coexistence curve. This critical point is the critical endpoint, which is rather insensitive to the shape of the interaction potential used [215]. 

Fig. 4.20 Schematic equilibrium phase diagram of a colloid-polymer mixture for small q. Full curve is the fluid-solid coexistence curve, dashed curve is the metastable gas-liquid coexistence region...

At high pressures, a substance may have additional triple points for two solid phases and the liquid, or for three solid phases. This is illustrated by the pressure-temperature phase diagram of H2O in Fig. 8.4 on the next page, which extends to pressures up to 30 kbar. (On this scale, the liquid-gas coexistence curve lies too close to the horizontal axis to be visible.) The diagram shows seven different solid phases of H2O differing in crystal structure and... 

Since all three of these transition enthalpies are positive, it follows that Asub-f is greater than Ayap f at the triple point. Therefore, according to Eq. 8.4.10, the slope of the solid-gas coexistence curve at the triple point is slightly greater than the slope of the liquid-gas coexistence curve. 

Fig. 3.2. Electronic, paramagnetic volume susceptibility of liquid cesium (derived from data of Preyland, 1979) as a function of reduced density p/p. Total susceptibility data are corrected for ionic diamagnetism and, for the liquid state, are corrected for conduction electron diamagnetism using theories of I nazawa and Matsudawa (1960) solid line) and Vignale et al. (1988) dot-dash line). Dashed line represents Curie law susceptibility along liquid-vapor coexistence curve, calculated for monovalent, atomic cesium. Note the deviation from Curie law behavior of the vapor for p/p. 2.

Fig. 3.16. Test of Eq. (3.27) comparing the density- solid line) and wave nnmber- dashed line) derivatives of the liquid structure factor for cesium at three temperatures along the liquid-vapor coexistence curve (Winter et al., 1988).

Experimental data for the DC conductivity pressure dependences of the conductivity at constant temperature near the liquid-vapor critical point. Comparison with the equation-of-state data displayed in Fig. 2.3(a) clearly shows a qualitative relationship between rapid variation in the conductivity and density. Conductivity data obtained along the liquid-vapor coexistence curve, shown in Fig. 3.20, demonstrate, furthermore that the electronic structures of the liquid and vapor are fundamentally different. The liquid structure of cesium just above the melting point is characterized by a high degree of correlation in the atomic positions (see Section 3.4) and, as we have noted previously, cesium is a normal liquid metal with physical properties very similar to those of the corresponding solid. The electrical conductivity, in particular, is ical for a material with metallic electron concentration, that is, an electron density comparable with the atomic density. 

Fig. 4.12. The isochoric thermal pressure cx>efiicient, yy, of fluid mercury versus density along the liquid-vapor coexistence curve (open circles) derived from isochores of Fig. 4.10. Solid lines represent predictions of the soft-sphere equation of state, Eq. (4.11) with = 12, appropriate for rare gases, and n = 7. Points (x) denote predictions of Eq. (4.11) for n = 15.5.

When two phases of a single substance are at equilibrium, the pressure is a function only of the temperature. A phase diagram for a pure substance contains three curves representing this dependence for the solid-liquid, solid-gas, and liquid-gas equilibria. These three curves meet at a point called the triple point. The liquid-vapor coexistence curve terminates at the critical point. Above the critical temperature, no gas-liquid phase transition occurs and there is only one fluid phase. The law of corresponding states was introduced, according to which all substances obey the same equation of state in terms of reduced variables... 

Figure 1 Experimental liquid-vapor coexistence curve of water [3] (thick solid line). Liquid-vapor coexistence curves of several water models ST2 [6,10],ST2RF[6],TIP4P[6],SPCE [6],andSPCE [11]. Fit of the data for the TIP 4P model to the extended scaling equation with leading asymptotic behavior described by eq. (1) is shown by thin solid line.

Figure 4 Liquid-vapor coexistence curve of water (thick solid line) and various specific lines emanating from the critical point maximum heat capacity Cp [3] (dashed line), minimum speed of sound [22] (circles), and percolation transition of water clusters [24] (star).

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