Coexistence lines free energies

The estimate exactly describes the course of the ill-fated series. This suggests another use of the stability analysis, namely, to correct a GDI series if its initial state point is subsequently found to be slightly in error. Elements of this idea are exhibited in the coexistence-line free-energy integration approach recently described by Meijer and Azhar [60]. 

In systems with phase coexistence, the free energy of the system is discontinuous at the interface between the phases. This discontinuity results in an interfacial tension that drives the system toward a minimization of the interface. In three-dimensional systems, the interfacial tension acts on the area (surface tension) in two-dimensional systems, it acts along the line (line tension) that separates the phases. In both cases, if the interfacial tension is large enough, then it may be expected that... 

The free energy of a monolayer domain in the coexistence region of a phase transition can be described as a balance between the dipolar electrostatic energy and the line tension between the two phases. Following the development of McConnell [168], a monolayer having n circular noninteracting domains of radius R has a free energy... 

Fig. 6.31 Results from SCFI calculations for diblock/homopolymer blends (Matsen 1995b). (a) The dimensionless Helmholtz free energy Fu() as a function of homopolymer volume fraction at y X = 12, / = 0.45 and /3 = The dashed line shows the double tangent construction used to locate the binodal points denoted with dots. The dotted line is the free energy of non-interacting bilayers, (b) Phase diagram obtained by repeating this construction over a range of %N. The dots are the binodal points obtained in (a), and the diamond indicates a critical point below which two-phase coexistence does not occur. The disordered homopolymer phase is labelled dis, and the lamellar phase lam.

Figure 9. Coexistence curves for a parent distribution with pf = 0.2. Shown are the values of p, of the coexisting phases horizontal lines guide the eye where new phases appear. Curves are labeled by n, the number of moment densities retained in the moment free energy. Predictions for n = 10 are indistinguishable from an exact calculation (in bold).

Figure 2. Schematic representation of the four conceptually different paths (the heavy lines) one may utilize to attack the phase-coexistence problem. Each figure depicts a configuration space spanned by two macroscopic properties (such as energy, density. ..) the contours link macrostates of equal probability, for some given conditions c (such as temperature, pressure. ..). The two mountaintops locate the equilibrium macro states associated with the two competing phases, under these conditions. They are separated by a probability ravine (free-energy barrier). In case (a) the path comprises two disjoint sections confined to each of the two phases and terminating in appropriate reference macrostates. In (b) the path skirts the ravine. In (c) it passes through the ravine. In (d) it leaps the ravine.

Tie lines are in fact common tangents to the Gibbs free energy surfaces of the phases that coexist in equilibrium. Let s assume that a and p phases, both of which form solid solutions, coexist. The following figure shows two arbitraiy surfaces, a and p ... 

Fig. 1.9 Calculated polymer-solvent phase diagram. The bimodal (continuous line) is the coexistence curve the points below it correspond to thermodynamically unstable states, which undergo phase separation. However, the pints between the bimodal and the spinodal (dashed line) are ki-netically stable, since there is a free-energy barrier to phase separation. C indicates the critical point the collapse temperature. The deviation of the low-concentration branch of the spinodal from the vertical axis below T is an artifact of the mean-field approximation. (From ref. [62])...

The condition of phase stability for such a system is closely related to the behavior of the Helmholtz free energy, by stating that the isothermal compressibility yT > 0. The positiveness of yT expresses the condition of the mechanical stability of the system. The binodal line at each temperature and densities of coexisting liquid and gas determined by equating the chemical potential of the two phases. The conditions expressed by Eq. (115) simply say that the gas-liquid phase transition occurs when the P — pex surface from the gas... 

In conclusion, the deviation from the microhardness additivity law (Fig. 5.4, line 1) can be explained in terms of two distinct contributions (a) a crystallinity depression caused by the coexistence of the PE and t-PP phases, which yields curve 2, and (b) a substantial decrease of the crystal microhardness of the PE and t-PP components caused by an increase of the surface free energy erg with composition, which leads to curve 3. It is suggested that the rise in erg rise is a consequence of the increase in the level of surface defects including entanglements (see Section 4.3). 

Expressions obtained for the free energy, pressure and chemical potentials can be used to study the thermodynamic properties of the electrolyte solutions, in particular, to describe the phase diagram of ionic fluids. Such a possibility is illustrated in Fig. 7, which shows the effect of ion pairing on liquid-liquid coexistence curve in the ion-dipole model as a function of the ion concentration a = ()%/(pi + ps) and reduced temperature T = (Ms)-1/2, bs = Rpl/Rl The solid line corresponds to the ion-dipole model with the parameter of ion association, B = 10. The dashed line corresponds to the ion-dipole... 

For binary mixtures, the binodal line is also the coexistence curve, defined by the common tangent line to the composition dependence of the free energy of mixing curve, and gives the equilibrium compositions of the two phases obtained when the overall composition is inside the miscibility gap. The spinodal curve, determined by the inflection points of the composition dependence of the free energy of mixing curve, separates unstable and metastable regions within the miscibility gap. 

Figure 7.2 Variation of the molar Gibbs free energy G of a binary mixture as a function of the mole fraction of component 2 (solid line). The long-dashed line is the Gibbs free energy for a system of two unmixed pure compounds. The short-dashed line is the tangent line to G at the coexistence compositions 2 =...

Fig. 2. (a) Schematic of Landau phase diagram as a function of the value of parameter b in the development of the critical free energy F as a function of the order parameter p up to sixth order. When b>0, the phase transition is second order. For b< 0, the phase transition is first order. Transition lines are continuous, and for b < 0 the dotted lines show the coexistence region, b — 0 corresponds to a tricritical point. First-order phase transitions may also occur for symmetry reasons when third-order invariant is allowed in the free energy expansion, (b) Schematic representation of the microscopic modification of a variable u(t) = u + p + up(t) in the parent (p — 0) and descendant phases (p/0). Both the mean value < u(t)) — u — p and time fluctuations Sup(t) depend on the phase. 

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