Phase transitions liquid-vapour

The arguments relating a porous structure and its morphology to mechanisms of phase transitions - specially vapour-liquid transitions - reveal to be highly consistent. A previous clasification of porous structures within five types was improved by considering recent resuits of Monte Carlo estimations of porous morphology and the role of the adsorption potential. 

We discuss classical non-ideal liquids before treating solids. The strongly interacting fluid systems of interest are hard spheres characterized by their harsh repulsions, atoms and molecules with dispersion interactions responsible for the liquid-vapour transitions of the rare gases, ionic systems including strong and weak electrolytes, simple and not quite so simple polar fluids like water. The solid phase systems discussed are ferroniagnets and alloys. 

Statistical mechanical theory and computer simulations provide a link between the equation of state and the interatomic potential energy functions. A fluid-solid transition at high density has been inferred from computer simulations of hard spheres. A vapour-liquid phase transition also appears when an attractive component is present hr the interatomic potential (e.g. atoms interacting tlirough a Leimard-Jones potential) provided the temperature lies below T, the critical temperature for this transition. This is illustrated in figure A2.3.2 where the critical point is a point of inflexion of tire critical isothemr in the P - Vplane. 

The parameter /r tunes the stiffness of the potential. It is chosen such that the repulsive part of the Leimard-Jones potential makes a crossing of bonds highly improbable (e.g., k= 30). This off-lattice model has a rather realistic equation of state and reproduces many experimental features of polymer solutions. Due to the attractive interactions the model exhibits a liquid-vapour coexistence, and an isolated chain undergoes a transition from a self-avoiding walk at high temperatures to a collapsed globule at low temperatures. Since all interactions are continuous, the model is tractable by Monte Carlo simulations as well as by molecular dynamics. Generalizations of the Leimard-Jones potential to anisotropic pair interactions are available e.g., the Gay-Beme potential [29]. This latter potential has been employed to study non-spherical particles that possibly fomi liquid crystalline phases. 

The variation of enthalpy for binary mixtures is conveniently represented on a diagram. An example is shown in Figure 3.3. The diagram shows the enthalpy of mixtures of ammonia and water versus concentration with pressure and temperature as parameters. It covers the phase changes from solid to liquid to vapour, and the enthalpy values given include the latent heats for the phase transitions. 

The best-known examples of phase transition are the liquid-vapour transition (evaporation), the solid-liquid transition (melting) and the solid-vapour transition (sublimation). The relationships between the phases, expressed as a function of P, V and T consitute an equation of state that may be represented graphically in the form of a phase diagram. An idealized example, shown in figure 1, is based on the phase relationships of argon [126]. 

During each phase transition of the type illustrated here, both of the intensive parameters P and T remain constant. Because of the difference in density however, when a certain mass of liquid is converted into vapour, the total volume (extensive parameter) expands. From the Gibbs-Duhem equation (8.8) for one mole of each pure phase,... 

Phase equilibrium requires that A2 = Al and hence that the integral vanish. All conditions are satisfied if the points 1 and 2 are located such that the areas A = B. This geometry defines the Maxwell construction. It shows that stable liquid and vapour states correspond to minima in free energy and that AL = Ay when the external pressure line cuts off equal areas in the loops of the Van der Waals isotherm. At this pressure that corresponds to the saturated vapour pressure, a first-order phase transition occurs. 

In the region of a first-order transition ip has equal minima at volumes V and V2, in line with the Maxwell construction. The mixed phase is the preferred state in the volume range between V and V2. It follows that the transition from vapour to liquid does not occur by an unlikely fluctuation in which the system contracts from vapour to liquid at uniform density, as would be required by the maximum in the Van der Waals function. Maxwell construction allows the nudeation of a liquid droplet by local fluctuation within the vapour, and subsequent growth of the liquid phase. 

The explosive phenomena produced by contact of liquefied gases with water were studied. Chlorodifluoromethane produced explosions when the liquid-water temperature differential exceeded 92°C, and propene did so at differentials of 96-109°C. Liquid propane did, but ethylene did not, produce explosions under the conditions studied [1], The previous literature on superheated vapour explosions has been critically reviewed, and new experimental work shows the phenomenon to be more widespread than had been thought previously. The explosions may be quite violent, and mixtures of liquefied gases may produce overpressures above 7 bar [2], Alternative explanations involve detonation driven by phase changes [3,4] and do not involve chemical reactions. Explosive phase transitions from superheated liquid to vapour have also been induced in chlorodifluoromethane by 1.0 J pulsed ruby laser irradiation. Metastable superheated states (of 25°C) achieved lasted some 50 ms, the expected detonation pressure being 4-5 bar [5], See LIQUEFIED NATURAL GAS, SUPERHEATED LIQUIDS, VAPOUR EXPLOSIONS... 

A vapour explosion can occur on contact between 2 liquids of differing temperatures if the temperature of the hotter liquid is above the b.p. of the cooler, and the explosion is due to extremely rapid vapour generation (phase transition) of the cooler liquid. Vapour generation must be preceeded by very good heat transfer by a fragmentation... 

The thermodjmamlcs presented in the previous subsection is easily extended to account for the coexistence of different phases in pores. In particular, the liquid-vapour transition is relevant here. We shall briefly consider three aspects of this topic. 

Over the years, vapour adsorption and condensation in porous materials continue to attract a great deal of attention because of (i) the fundamental physics of low-dimension systems due to confinement and (ii) the practical applications in the field of porous solids characterisation. Particularly, the specific surface area, as in the well-known BET model [I], is obtained from an adsorbed amount of fluid that is assumed to cover uniformly the pore wall of the porous material. From a more fundamental viewpoint, the interest in studying the thickness of the adsorbed film as a function of the pressure (i.e. t = f (P/Po) the so-called t-plot) is linked to the effort in describing the capillary condensation phenomenon i.e. the gas-Fadsorbed film to liquid transition of the confined fluid. Indeed, microscopic and mesoscopic approaches underline the importance of the stability of such a film on the thermodynamical equilibrium of the confined fluid [2-3], In simple pore geometry (slit or cylinder), numerous simulation works and theoretical studies (mainly Density Functional Theory) have shown that the (equilibrium) pressure for the gas/liquid phase transition in pores greater than 8 nm is correctly predicted by the Kelvin equation provided the pore radius Ro is replaced by the core radius of the gas phase i.e. (Ro -1) [4]. Thirty year ago, Saam and Cole [5] proposed that the capillary condensation transition is driven by the instability of the adsorbed film at the surface of an infinite... 

Phase transitions are central in many industrial problems, for instance, in distillation, absorption, condensation, manufacture of liquid natural gas, and multiphase flow. We shall use the liquid/vapour transition to illustrate the basic hypotheses and usefulness of NET for surfaces, a relatively new application. For applications to homogeneous phases, we refer the reader to the basic literature see Section 2,... 

The obtained distributions of the tetrahedricity measure were used for estimation of the concentration C of the four-coordinated tetrahedrally ordered water molecules. Temperature dependence of this concentration along the liquid-vapour coexistence curve is shown in the upper panel of Fig.5. There is only slight increase of C upon cooling from the liquid-vapour critical temperature to about 350 K (due to the temperature mismatch of ST2 water and real water, about 30 to 35° lower temperature should be expected for real water). The drastic increase of C is evident at lower temperatures, when approaching the liquid-liquid phase transition. At 7 = 270 K, concentrations of the tetrahedrally ordered four-coordinated water molecules in two coexisting phases was found to be about 28% and 46.5%. Such step increase of C is related to a step decrease of density from 0.97 to 0.91 g/cm ... 

Figure 5 Temperature dependence of the concentration C of the tetrahedrally ordered four-coordinated water molecules (upper panel) and of the liquid water density (lower panel) along the liquid-vapour coexistence curve. Vertical dashed line indicates the temperature of the liquid-liquid transition. Dotted lines indicate the densities and concentrations of the coexisting phases. Stars indicate percolation transition of the tetrahedrally ordered four-coordinated molecules.

As we have mentioned in the Introduction, the location of the critical point of the lowest density liquid-liquid transition of real water is unknown and both scenarios (critical point at positive or at negative pressure) can qualitatively explain water anomalies. Recent simulation studies of confined water show the way, how to locate the liquid-liquid critical point of water. Confinement in hydrophobic pores shifts the temperature of the liquid-liquid transition to lower temperatures (at the same pressure), whereas effect of confinement in hydrophilic pores is opposite. If the liquid-liquid critical point in real water is located at positive pressure, in hydrophobic pores it may be shifted to negative pressures. Alternatively, if the liquid-liquid critical point in real water is located at negative pressure, it may be shifted to positive pressures by confinement in hydrophilic pores. Interestingly, that it may be possible in both cases to place the liquid-liquid critical point at the liquid-vapour coexistence curve by tuning the pore hydrophilicity. We expect, that the experiments with confined supercooled water should finally answer the questions, concerning existence of the liquid-liquid phase transition in supercoleed water and its location. 

The shape of the binodal near the critical point is not predicted correctly by the mean-field (Flory-Huggins) theory as demonstrated in Fig. 5.3(a). The difference in the two concentrations coexisting at equilibrium in the two-phase region is called the order parameter. This order parameter is analogous to the order parameter of van der Waals for the liquid-vapour phase transition, that is proportional to the density difference between the two coexisting phases. This order parameter is predicted to vary as a power law of the proximity to the critical point ... 

Critical theories take into account concentration fluctuations and describe phase transitions near the critical point. The critical theory for this phase transition is the Ising model which also describes liquid-vapour transitions near their critical points. The prediction of the three-dimensional Ising model for the critical exponent /3 = 0.3 is in good agreement with experiments as shown in Fig. 5.3(b). Mean-field and cri-... 

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