Liquid-gas coexistence

McConnell et al. [196] and Andelman and co-workers have predicted [197,198] an ordered array of liquid domains in the gas-liquid coexistence regime caused by the dipole moment difference between the phases. These superstructures were observed in monolayers of dipalmitoyl phosphatidylcholine monolayers [170]. 

Fig. 27. Phase diagram of an adsorbed film in- the simple cubic lattice from mean-fleld calculations (full curves - flrst-order transitions, broken curves -second-order transitions) and from a Monte Carlo calculation (dash-dotted curve - only the transition of the first layer is shown). Phases shown are the lattice gas (G), the ordered (2x1) phase in the first layer, lattice fluid in the first layer F(l) and in the bulk F(a>). For the sake of clarity, layering transitions in layers higher than the second layer (which nearly coincide with the layering of the second layer and merge at 7 (2), are not shown. The chemical potential at gas-liquid coexistence is denoted as ttg, and 7 / is the mean-field bulk critical temperature. While the layering transition of the second layer ends in a critical point Tj(2), mean-field theory predicts two tricritical points 7 (1), 7 (1) in the first layer. Parameters of this calculation are R = —0.75, e = 2.5p, 112 = Mi/ = d/2, D = 20, and L varied from 6 to 24. (From Wagner and Binder .)...

Fig. 40 Schematic phase diagram of the disordered phases for particles interacting with isotropic potentials (upper panel) and limited valence potentials (lower panel). In the first, standard case, the glass line hits the gas-liquid spinodal at large densities. In the limited valence case, the shrunk gas-liquid coexistence region leaves a new region in which a stable network with saturated bonding can develop. Reproduced with permission from [167]...

The gas—liquid coexistence curve is known as the boiling curve. If one moves upwards along the boiling curve, increasing both temperature and pressure, the liquid then becomes less dense due to thermal expansion, and as the pressure rises the gas becomes denser. Eventually, the densities of the two phases converge and become identical, eliminating the distinction between gas and liquid. For example, the critical point of C02 occurs at a pressure of 73.8 bar and a temperature of 31.1 °C. 

In 1822, Cagniard de la Tour showed the existence of a critical temperature for each individual substance above which such a substance can only occur as a fluid and not as either a liquid or a gas. This critical point is reached as one moves upward along the gas-liquid coexistence curve, where both temperature and pressure increase. The original liquid becomes less dense through thermal expansion and the gas becomes more dense as the pressure rises. At the critical point, the densities of the two phases are equal, the distinction between the gas and liquid vanishes and the coexistence curve comes to an end at the critical point, where the substance is described as a fluid. Supercritical fluids exhibit key features such as compressibility, homogeneity and a continuous change from gas-like to liquid-like properties. 

In addition to the importance of the M41S materials for size- and shape-selective applications, these materials have been also regarded as a suitable mesoporous model adsorbent for testing theoretical predictions of pore condensation. Pore condensation represents a first order phase transition from a gas-like state to a liquid-like state of a pore fluid in presence of a bulk fluid reservoir, which occurs at a pressure p less than the saturation pressure po at gas-liquid coexistence of the bulk fluid [6,7]. In this sense pore condensation can be regarded as a shifted gas-liquid bulk phase transition due to confinement of a fluid to a pore. Recent work has shown that in fact the complete phase diagram of the confined fluid is shifted to lower temperature and higher mean density as compared with the bulk coexistence curve [e.g., 8,9]. 

Figure 2 P-T mixture critical curve (solid line) for ethanol-C02 modeled from the Peng-Robinson equation of state, also shown in relation to the extension of the gas-liquid coexistence curve for pure CO2 (dashed line) (8).

If, on the other hand, the substrates are sufficiently attractive, one notices from the plots in Fig. 4.7 that F (T, pb) may either vary continuously or discontinuously depending on whether the (bulk) isochoric path is super-or subcritical, respectively, with regard to the critical point of the confined fluid. Hence, discontinuities in the plots in Fig. 4.7 indicate capillary condensation (evaporation) in the model pore prior to condensation in the bulk, which would, of course, occur at bulk gas-liquid coexistence, i.e., at T - Tzb) /Tzb = 0. 

Notice also that, at temperatures higher than that corresponding to capillary condensation, F (T. pb) increases with decreasing T. Tliis is indicative of a regime where one would observe growth of a wetting film, which at a mean-field level, manifests itself as an increase in overall density of an otherwise homogeneous low-density phase adsorbed all across the slit pore. For temperatures lower than that at which capillary condensation sets in, F (pb) turns out to be nearly independent of T the lower T becomes. This reflects that, when sufficiently close to bulk gas-liquid coexistence, the pore is com-... 

However, as. So = 0,1, no new morphologies arise. The onty effect of confinement by hard, repulsive substrates is an upward shift in the chemical potential at gas liquid coexistence. By solving the analog of Eq. (4.99) we obtain... 

The two satnration fines—c-a representing satnrated gas and c-b representing saturated liquid— meet at c, which is the highest point of the L+G two-phase area. Point c is the gas-liquid critical state. As temperature is increased from below to approach the critical, the flat gas-liquid coexistence segment of the isotherm shrinks to the point c at the critical temperature. The critical point is a point of inflection of the isotherm. At c. 

We start with gas-liquid equilibrium (gle). For simple mixtures of similar components, gas-liquid coexistence states are observed at a wide range of T and p that are bounded at the high-temperature side by the critical states and at the low-temperature side by the formation of solids. In mixtures of large difference in molecular attractive forces, liquid immiscibility takes place. On the high-temperature side, liquid-liquid equilibrium (lie) either merges into gas-liquid equilibrium (gle) or becomes bounded by separate critical states. With extremely dissimilar molecules, the liquid-liquid coexistence phenomenon persists without limit to high temperature, where it turns into gas-gas equilibrium (gge). 

It is now established both theoretically and experimentally that many thermodynamic variables assume a simple power-law behaviour at or near critical points in both pure and mixed fluids. The actual functional dependence of one variable on another can be characterized by the so-called critical indices a, 5, etc. The critical index j8, for example, defines both the shape of the gas-liquid coexistence curve for a pure fluid and the liquid-liquid coexistence curve of a binary mixture in the vicinity of either an upper or a lower critical solution temperature. The correspondence between critical phenomena in one-, two-,... 

Fig. 2 Overview of the (2r, rn)-parameter plane containing the location of typical charged colloids with monovalent counterions in aqueous solution at room temperature and the location of Systems I-IV (squares). The directions of increased value in (i) counterion charge Z, (ii) macroion charge Zm, and (iii) the combined parameter are shown by arrows. Also shown are other simulated systems (dots) and the gas-liquid coexistence curve (dashed curve with triangles) [22]...

Fig. 1.22 Monte Carlo computer simulation results for the gas-liquid coexistences of hard spheres mixed with excluded volume polymers for = 3.86 (open circles), 5.58 (crosses) and 7.78 (filled diamonds), redrawn from Bolhuis et al. [177]. The binodal curves are drawn to guide the eye...

Fig. 3.10 The dimensionless semi-grand potential 12 as a function of volume fraction (j>. Schematic view of the common tangent construction (straight lines) to determine the phase coexistence in mixtures of colloidal hard spheres and phs. (i) gas-4iquid coexistence, (ii) fluid-solid coexistence, (iii) gas-liquid-solid triple coexistence, and (iv) fluid-solid coexistence near a metastable (dashed lines represent the common tangent construction for this case) gas-liquid coexistence...

When two compositions can be connected through the common tangent (the thin straight lines in the figures connecting these compositions), binodal points are found the intercepts of the extrapolated lines correspond to the total pressure —P. Scenario (i) in Fig. 3.10 corresponds to gas-liquid coexistence. In situation (ii) Q(( ) are given for both the fluid state and for the solid state and the common tangent shows the compositions where fluid and solid coexist. A combination of (i)... 

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