Two phase coexistence

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]. 

From this equation one concludes that the maximum number of phases that can coexist in a oiie-component system (d = 1) is tliree, at a unique temperature and pressure T = 0). When two phases coexist F= 1), selecting a temperature fixes the pressure. Conclusions for other situations should be obvious. 

While, in principle, a tricritical point is one where three phases simultaneously coalesce into one, that is not what would be observed in the laboratory if the temperature of a closed system is increased along a path that passes exactly tlirough a tricritical point. Although such a difficult experiment is yet to be perfomied, it is clear from theory (Kaufman and Griffiths 1982, Pegg et al 1990) and from experiments in the vicinity of tricritical points that below the tricritical temperature only two phases coexist and that the volume of one slirinks precipitously to zero at T. ... 

Simulations in the Gibbs ensemble attempt to combine features of Widom s test particle method with the direct simulation of two-phase coexistence in a box. The method of Panagiotopoulos et al [162. 163] uses two fiilly-periodic boxes, I and II. 

The alternative to direct simulation of two-phase coexistence is the calculation of free energies or chemical potentials together with solution of the themiodynamic coexistence conditions. Thus, we must solve (say) pj (P) = PjjCT ) at constant T. A reasonable approach [173. 174. 175 and 176] is to conduct constant-AT J simulations, measure p by test-particle insertion, and also to note that the simulations give the derivative 3p/3 7 =(F)/A directly. Thus, conducting... 

Cellulose I. The majority of celluloses in the native state were previously thought to have the same crystal stmcture (Cellulose I), varying only in perfection of the crystaUites. Now, at least two different crystal stmctures are known for these materials, named la and ip. These two phases coexist in... 

First we look at the simpler case of the shrinking of a single cluster of radius R at two-phase coexistence. Assume that the phase inside this cluster and the surrounding phase are at thermodynamic equilibrium, apart from the surface tension associated with the cluster surface. This surface tension exerts a force or pressure inside the cluster, which makes the cluster energetically unfavorable so that it shrinks, under diffusive release of the conserved quantity (matter or energy) associated with the order parameter. 

Barrer s discussion4 of his analog of Eq. 28 merits some comment. Equation 28 expresses the equilibrium condition between ice and hydrate. As such it is valid for all equilibria in which the two phases coexist and not only for univariant equilibria corresponding with a P—7" line in the phase diagram. (It holds, for instance, in the entire ice-hydratell-gas region of the ternary system water-methane-propane considered in Section III.C.(2).) In addition to Eq. 28 one has Clapeyron s equation... 

As described in Section 14-1. when AR and ZlS have the same sign, the spontaneous direction of a process depends on T. For a phase change, enthalpy dominates AG at low temperature, and the formation of the more constrained phase is spontaneous, hi contrast, entropy dominates AG at high temperature, and the formation of the less constrained phase is spontaneous. At one characteristic temperature, A G = 0, and the phase change proceeds in both directions at the same rate. The two phases coexist, and the system is in a state of d Tiamic equilibrium. 

As discussed above, lipid membranes are dynamic structures with heterogeneous structure involving different lipid domains. The coexistence of different kinds of domains implies that boundaries must exist. The appearance of leaky interfacial regions, or defects, has been suggested to play a role in abrupt changes in solute permeabilities in the two-phase coexistence regions [91,92]. 

The quality of the mean-field approximation can be tested in simulations of the same lattice model [13]. Ideally, direct free-energy calculations of the liquid and solid phases would allow us to locate the point where the two phases coexist. However, in the present studies we followed a less accurate, but simpler approach we observed the onset of freezing in a simulation where the system was slowly cooled. To diminish the effect of supercooling at the freezing point, we introduced a terraced substrate into the system to act as a crystallization seed [14]. We verified that this seed had little effect on the phase coexistence temperature. For details, see Sect. A.3. At freezing, we have... 

The determination of the character and location of phase transitions has been an active area of research from the early days of computer simulation, all the way back to the 1953 Metropolis et al. [59] MC paper. Within a two-phase coexistence region, small systems simulated under periodic boundary conditions show regions of apparent thermodynamic instability [60] simulations in the presence of an explicit interface eliminate this at some cost in system size and equilibration time. The determination of precise coexistence boundaries was usually done indirectly, through the... 

The Gibbs Ensemble MC simulation methodology [17-19] enables direct simulations of phase equilibria in fluids. A schematic diagram of the technique is shown in Fig. 10.1. Let us consider a macroscopic system with two phases coexisting at equilibrium. Gibbs ensemble simulations are performed in two separate microscopic regions, each within periodic boundary conditions (denoted by the dashed lines in Fig. 10.1). The thermodynamic requirements for phase coexistence are that each... 

The crucial question is at what value of <)> is the attraction high enough to induce phase separation De Hek and Vrij (6) assume that the critical flocculation concentration is equivalent to the phase separation condition defined by the spinodal point. From the pair potential between two hard spheres in a polymer solution they calculate the second virial coefficient B2 for the particles, and derive from the spinodal condition that if B2 = 1/2 (where is the volume fraction of particles in the dispersion) phase separation occurs. For a system in thermodynamic equilibrium, two phases coexist if the chemical potential of the hard spheres is the same in the dispersion and in the floe phase (i.e., the binodal condition). 

A phase boundary for a single-component system shows the conditions at which two phases coexist in equilibrium. Recall the equilibrium condition for the phase equilibrium (eq. 2.2). Letp and Tchange infinitesimally but in a way that leaves the two phases a and /3 in equilibrium. The changes in chemical potential must be identical, and hence... 

For a single-component systemp and T can be varied independently when only one phase is present. When two phases are present in equilibrium, pressure and temperature are not independent variables. At a certain pressure there is only one temperature at which the two phases coexist, e.g. the standard melting temperature of water. Hence at a chosen pressure, the temperature is given implicitly. A point... 

A two-dimensional illustration of three phases a, ft and % in equilibrium is shown in Figure 6.9. Two phases coexist in equilibrium in planes perpendicular to the lines indicated in the two-dimensional figure and all three phases coexist along a common line also perpendicular to the plane of the drawing. Each of the three two-phase boundaries, which meet at the point of contact, has a characteristic interfacial tension, e.g. ca for the interface, which tends to reduce the area of the... 

We call each solid line in this graph a phase boundary. If the values of p and T lie on a phase boundary, then equilibrium between two phases is guaranteed. There are three common phase boundaries liquid-solid, liquid-gas and solid-gas. The line separating the regions labelled solid and liquid , for example, represents values of pressure and temperature at which these two phases coexist - a line sometimes called the melting-point phase boundary . 

Note A miscibility gap is observed at temperatures below an upper critical solution temperature (UCST) or above the lower critical solution temperature (LCST). Its location depends on pressure. In the miscibility gap, there are at least two phases coexisting. 

On theoretical grounds, the phenomenon of two-phase coexistence at a liquid interface in a complex mixed adsorbing system is certainly not unexpected. So long as the various molecular interactions are of the normally expected form, statistical thermodynamics indicates that interfacial demixing can readily lead to a more favoured stable equilibrium state (Pugnaloni et al., 2004). 

Liquids with equal solubility parameters are miscible, there is no heat of mixing. With increasing difference of <5, two phases coexist, which become miscible at elevated temperature, at the critical consolute temperature Tc. Tc increases with the difference of the <5 s and with the mean molar volume of the two liquids. Another polarity scale was recently introduced by Middleton and co-workers13 based on the bathochromic shift of UV-visible 2max. The obtained spectral polarity index ranks the solvents at one end of the scale is the nonpolar perfluorohexane and at the opposite the highly polar and acidic l,l,2,3,3.3-hexafluoropropan-2-ol. The latter is much more polar than its hydrocarbon analog. 

The Maxwell construction would determine the condition of two phase coexistence or the points on the curves where the first-order phase change occurs [6,7]. It is the condition that the two phases have the same value of g or j d II = 0 from Eq. (2.6) at zero osmotic pressure, v2 and vx being the values of v in the two phases. However, this criterion is questionable in the case Kcritical point). This is because the shear deformation energy has not been taken into account in the above theory. See Sect. 8 for further comments on this aspect. 

On such a two-phase coexistence curve, the system has only a single degree of freedom, so that, for a given T, the pressure P is fixed, and vice versa. For example, if the temperature of a liquid-vapor system is chosen as T = 25°C, the corresponding P (read from the vapor-pressure curve) must be 23.8 Torr, as shown by the dotted line in Fig. 7.1. 

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