Coexistence density

Below T, liquid and vapour coexist and their densities approach each other along the coexistence curve in the T-Vplane until they coincide at the critical temperature T. The coexisting densities in the critical region are related to T-T by the power law... 

The coexisting densities below are detennined by the equalities of the chemical potentials and pressures of the coexisting phases, which implies that tire horizontal line joining the coexisting vapour and liquid phases obeys the condition... 

In order to demonstrate that the systems in question exhibit nonzero wetting temperature, we have displayed the results of calculations for one of the systems (with =1 at T = 0.7). Fig. 12 testifies that only a thin (monolayer) film develops even at densities extremely close to the bulk coexistence density (p/,(T — 0.7) — 0.001 664). In Fig. 13(a) we show the density profiles obtained at temperature 0.9 evaluated for = 7. Part (b) of this figure presents the fraction of nonassociated particles, x( )- We... 

Larger system sizes suffer less from this effect and can be used to obtain coexistence densities near critical points. 

A single temperature and pressure point (T - 333 K, P - 150 bar) for which experimental data have been reported (17) was selected for the calculations. The simulations proceeded in a similar manner as for the binary systems. Because of the low reduced temperature and high coexistence density of the water-rich phases, the simulations were quite long, requiring 3-5x10 Monte Carlo steps. A total number of 500 particles was used in the two regions. The number of attempted transfers was 2,000 per MC cycle of 500 attempted displacements. 

Figure 9. The coexistence densities of the full (solid lines and filled symbols) and short-range models of polar and associating fluids. The straight lines correspond to the rectilinear diameter rule.

Figure 4. Phase diagrams for the AHS fluid with a = 6 (top), a = 4 (center), and a = 3.1 (bottom) coexistence densities from GEMC simulations (open circles) critical parameters from MFFSS analyses assuming Ising criticality (filled circles) fits of the form ft] p A(T — T ) B T — T ) eff (solid fines). For a = 6 the effective order parameter exponent is /left = 0.36, for a = 4 / eff = 0.46, and for a = 3.1 /Jeff = 0.46.

Fig. 3. (a) Results from a multicanonical simulation of the 3D Lennard-Jones fluid at a point on the coexistence curve. The figure shows both the multicanonical sampling distribution Ps p) (symbols o) and the corresponding estimate of the equilibrium distribution Po p) with p = N/V the number density. The inset shows the value of the equilibrium distribution in the interfacial region [44]. (b) Coexistence density distributions for a selection of temperatures. The coexistence densities can be simply read off from the peak positions... 

The dependence on the system size is explored in Fig. 11 In the left panel we plot the chemical potential vs. density for system sizes ranging from L = 11.3a to 22.5a. The turning point of the curves shifts closer to the coexistence density of the vapor, Ap —> 0, as we increase the system size. Also the maximum slope increases with increasing L, indicating that for L —> oo a sharp transition occurs. The right panel of Fig. 11 presents the probability distribution of the energy, U, at the droplet evaporation/condensation for different system sizes. In qualitative agreement with the expectations the droplet evaporation/condensation becomes sharper and both states (the supersaturated vapor and drop) become more separated as we increase the system size. 

Generally, the density of the liquid inside the drop will also deviate from the coexistence density of the liquid. Since the compressibility of the liquid phase, however, is much smaller than that of the vapor the deviation of the density inside the drop from the coexistence value will be much smaller than the deviation in the vapor phase. 

Fig. 10 Left Mean-field equation of state for the hydrophobic A strand melt, for = —15.15 and u>aaa = 0.564375 (see (15)). The vertical green line marks, at P/kgT = 0, the coexistence density of the melt with its vapor, Pa = 20. The red line marks the tangent to the P/kgT curve at P/ksT = 0,Pa = 20. Its slope is inversely proportional to the melt compressibility. Right The density distribution of the A and B bilayer components is shown across a free edge, membrane patch (see inset) which is thin in the z direction (i.e., vertically to the graph plane). The red and the green symbols correspond to the densities of the hydrophobic and the hydrophilic segments, respectively...

Pallas suggests that the slope in Kim and Cannell s isotherms above 26.3°C is the result of a leaching out of surface-active impurities from the Teflon trough in which the measurements were made. The differences in the coexisting densities are attributed to the effect of residual impurities in the sample of pentadecanoic acid used by Kim and Cannell this does not seem likely. 

The sensitivity of the pressure measurement technique used by Pallas and Pethica was 0.2 Nm The precision of the measurements is much poorer, probably because they were made mostly by successive additions, and Pallas and Pethica include in the two-phase regions points that differ by as much as 3 /iNm from the mean pressure. In constructing the isotherms, they have chosen to have the one-phase and two-phase portions meet at an acute angle. This choice leads to coexisting densities markedly different from those reported by Kim and Canned. Which of the interpretations in correct As we shad see, other types of experiments allow us to choose between them. 

Figure 2.7. Reduced coexistence densities = pd, where d is the diameter of the sphere with the same volume as the dumbbell) for hard dumbbells versus reduced bondlength L = L/a). The filled circles and lines are the results of Vega et al. [60,240] and the open squares are the results of Singer and Mumaugh [2 9]. The dashed lines give estimates of the plastic crystal to orientationally ordered solid.

Iteration for Coexisting Densities. Orthobaric densities near the critical point generally cannot be obtained accurately from isochoric PpT data by extrapolation to the vapor-pressure curve because the isochore curvatures become extremely large near the critical point. The present, nonanalytic equation of state, however, can be used to estimate these densities by a simple, iterative procedure. Assume that nonlinear parameters in the equation of state have been estimated in preliminary work. For data along a given experimental isochore (density), it is necessary merely to find the coexistence temperature, Ta(p), by trial (iteration) for a best, least-squares fit of these data. 

These data are replotted in a different form in Figure 12, on the assumption that the order parameter (the coexistence density gap) for the LJ system should behave in an Ising-like manner. This is reflected in the nearly straight-line behavior of much of the data very close to the critical points the data deviate from linearity, becoming mean-field-like because of the limitation on fluctuations in a finite system. The precision of the results puts us in position to study this finite-size crossover and also other nonuniversal properties of the critical behavior of fluid phase transitions. 

Figure 10. A portion of one subcritical isotherm. The dashed line shows the double tangent, obtained numerically after local interpolation in the vicinity of the coexisting densities the squares mark the resulting liquid and gas phases. The slope of the double tangent gives the vapor pressure in view of the high precision of the free energies, the vapor pressure is obtained very precisely.

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