Coexisting phases

On compression, a gaseous phase may condense to a liquid-expanded, L phase via a first-order transition. This transition is difficult to study experimentally because of the small film pressures involved and the need to avoid any impurities [76,193]. There is ample evidence that the transition is clearly first-order there are discontinuities in v-a plots, a latent heat of vaporization associated with the transition and two coexisting phases can be seen. Also, fluctuations in the surface potential [194] in the two phase region indicate two-phase coexistence. The general situation is reminiscent of three-dimensional vapor-liquid condensation and can be treated by the two-dimensional van der Waals equation (Eq. Ill-104) [195] or statistical mechanical models [191]. 

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... 

Figure A2.3.3 P-Visothemis for van der Waals equation of state. Maxwell s equal areas mle (area ABE = area ECD) detemiines the volumes of the coexisting phases at subcritical temperatures.

The van der Waals p., p. isothenns, calculated using equation (A2.5.3), are shown in figure A2.5.8. It is innnediately obvious that these are much more nearly antisynnnettic around the critical point than are the conespondingp, F isothenns in figure A2.5.6 (of course, this is mainly due to the finite range of p from 0 to 3). The synnnetry is not exact, however, as a carefiil examination of the figure will show. This choice of variables also satisfies the equal-area condition for coexistent phases here the horizontal tie-line makes the chemical potentials equal and the equal-area constniction makes the pressures equal. 

A system of interest may be macroscopically homogeneous or inliomogeneous. The inliomogeneity may arise on account of interfaces between coexisting phases in a system or due to the system s finite size and proximity to its external surface. Near the surfaces and interfaces, the system s translational synnnetry is broken this has important consequences. The spatial structure of an inliomogeneous system is its average equilibrium property and has to be incorporated in the overall theoretical stnicture, in order to study spatio-temporal correlations due to themial fluctuations around an inliomogeneous spatial profile. This is also illustrated in section A3.3.2. 

A homogeneous metastable phase is always stable with respect to the fonnation of infinitesimal droplets, provided the surface tension a is positive. Between this extreme and the other thennodynamic equilibrium state, which is inhomogeneous and consists of two coexisting phases, a critical size droplet state exists, which is in unstable equilibrium. In the classical theory, one makes the capillarity approxunation the critical droplet is assumed homogeneous up to the boundary separating it from the metastable background and is assumed to be the same as the new phase in the bulk. Then the work of fonnation W R) of such a droplet of arbitrary radius R is the sum of the... 

Here, A h=hp - is the difference in molar enthalpies of the coexisting phases, and A v is the difference in molar volumes the suffix o indicates that the derivative is to be evaluated along the coexistence line. 

This fomi is called a Ginzburg-Landau expansion. The first temi f(m) corresponds to the free energy of a homogeneous (bulk-like) system and detemiines the phase behaviour. For t> 0 the fiinction/exliibits two minima at = 37. This value corresponds to the composition difference of the two coexisting phases. The second contribution specifies the cost of an inhomogeneous order parameter profile. / sets the typical length scale. 

Each of the coexisting phases will be governed by a Gibbs-Duhem equation so that... 

As for LLE, an expression for G capable of representing hquid—hquid-phase splitting is required as for VLE, a vapor-phase equation of state for computing ( ) is also needed. Moreover, VLLE calculations can in principle and sometimes in practice be carried out with an equation of state vahd for ah. coexisting phases. 

Therefore, one might ask what is meant by the terms hquid and gas. We all know what is the characteristic of a liquid. It has a free surface. However, as soon as we compress the hquid, there is no free surface and the distinction between a gas and liquid is lost. The most logical terminology would be to reserve the terms hquid and vapor for the two coexisting phases and call all other states fluid. A more common terminology is to call the fluid a hquid if its density exceeds the critical density and a gas if its density is lower. Generally speaking, in this chapter we will use the term fluid to describe both the gas and liquid phases and not make any distinction. 

Moreover, the order parameter, which in the case of the gas-liquid transition is defined as the difference between the densities of both coexisting phases, A6 = 62 — 61, approaches zero when the temperature goes to (from below, since above T. the above order parameter is always equal to zero) as... 

The interface free energy per unit area fi,u is taken to be that of a planar interface between coexisting phases. Considering a solution v /(z) that minimizes Eq. (5) subject to the boundary conditions vj/(z - oo) = - v /coex, v /(z + oo) = + vj/ oex one finds the excess free energy of a planar interface ... 

The absence of any degrees of freedom implies that the triple point is a unique state that represents an invariant system, i.e., one in which any change in the state variables T or P is bound to reduce the number of coexisting phases. 

The isotherms represented in Fig. 1 give a general idea of the equilibria in the Pd-H system under different p-T conditions. Most experimental evidence shows, however, that the equilibrium pressure over a + /3 coexisting phases depends on the direction of the phase transformation process p a-p > pp-a (T, H/Pd constant). This hysteresis effect at 100°... 

An example for a partially known ternary phase diagram is the sodium octane 1 -sulfonate/ 1-decanol/water system [61]. Figure 34 shows the isotropic areas L, and L2 for the water-rich surfactant phase with solubilized alcohol and for the solvent-rich surfactant phase with solubilized water, respectively. Furthermore, the lamellar neat phase D and the anisotropic hexagonal middle phase E are indicated (for systematics, cf. Ref. 62). For the quaternary sodium octane 1-sulfonate (A)/l-butanol (B)/n-tetradecane (0)/water (W) system, the tricritical point which characterizes the transition of three coexisting phases into one liquid phase is at 40.1°C A, 0.042 (mass parts) B, 0.958 (A + B = 56 wt %) O, 0.54 W, 0.46 [63]. For both the binary phase equilibrium dodecane... 

The simulation of a first-order phase transition, especially one where the two phases have a significant difference in molecular area, can be difficult in the context of a molecular dynamics simulation some of the works already described are examples of this problem. In a molecular dynamics simulation it can be hard to see coexistence of phases, especially when the molecules are fairly complicated so that a relatively small system size is necessary. One approach to this problem, described by Siepmann et al. [369] to model the LE-G transition, is to perform Monte Carlo simulations in the Gibbs ensemble. In this approach, the two phases are simulated in two separate but coupled boxes. One of the possible MC moves is to move a molecule from one box to the other in this manner two coexisting phases may be simulated without an interface. Siepmann et al. used the chain and interface potentials described in the Karaborni et al. works [362-365] for a 15-carbon carboxylic acid (i.e. pen-tadecanoic acid) on water. They found reasonable coexistence conditions from their simulations, implying, among other things, the existence of a stable LE state in the Karaborni model, though the LE phase is substantially denser than that seen experimentally. The re-... 

It has been shown by FM that the phase state of the lipid exerted a marked influence on S-layer protein crystallization [138]. When the l,2-dimyristoyl-OT-glycero-3-phospho-ethanolamine (DMPE) surface monolayer was in the phase-separated state between hquid-expanded and ordered, liquid-condensed phase, the S-layer protein of B. coagulans E38/vl was preferentially adsorbed at the boundary line between the two coexisting phases. The adsorption was dominated by hydrophobic and van der Waals interactions. The two-dimensional crystallization proceeded predominately underneath the liquid-condensed phase. Crystal growth was much slower under the liquid-expanded monolayer, and the entire interface was overgrown only after prolonged protein incubation. 

In equilibrium the chemical potential must be equal in coexisting phases. The assumption is that the solid phase must consist of one component, water, whereas the liquid phase will be a mixture of water and salt. So the chemical potential for water in the solid phase fis is the chemical potential of the pure substance. However, in the liquid phase the water is diluted with the salt. Therefore the chemical potential of the water in liquid state must be corrected. X refers to the mole fraction of the solute, that is, salt or an organic substance. The equation is valid for small amounts of salt or additives in general ... 

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