Binodal boundary

Figure 6A is a general representation of the pressure-composition diagrams for systems that display liquid -liquid phase separation. The binodal boundary represents the equilibrium demixing pressures for a monodisperse polymer system. Below the binodal there is another boundary known as the spinodal boundary. The binodal and the spinodal envelopes determine the metastable region (shaded area in between). They... 

When P 2. and at a fixed temperature and pressure, then F =, 4, and only one composition variable is free to be selected, at constant temperature and pressure. This means that the two-phase equilibrium occurs along a unidimen-sional line. i.e.. the binodal boundary. 

FIGURE 8.10 Binodal boundary of PVDF/solvent/water. (Data from Bottino, A. et al. Journal of Membrane Science, 57,1-20,1991. doi 10.1016/S0376-7388(00)81159-X.)... 

Another kinetic jjhenomenon where Calm s critical waves can possibly be visualized and studied is the replication of interphase boundaries (IPB) illustrated in Figs. 8-10. Similarly to the replication of APBs. it can arise after a two-step quench of an initially uniform disordered alloy. First the alloy is quenched and annealed at temperature T in some two-phase state that can be either metastable or spinodally unstable with respect to phase separation. Varying the annealing time one can grow here precipitates ("droplets ) of a suitable size /. For sufficiently large /, the concentration c(r) within A-riched droplets is close to the equilibrium binodal value C(,(T ) (thin curve in Fig. 9). 

Figure 5.7 (a) Theoretical predictions of the unstable regions (miscibility gap) of the solid solutions in the systems AlN-GaN, InN-GaN and AlN-InN [15]. For the system InN-GaN both the phase boundary (binodal) and spinodal lines are shown, (b) Gibbs energy of mixing for the solid solution InN-GaN at 1400 K. 

In contrast to the critical temperature Tc, the spinodal temperature Tsp is well below the binodal temperature for off-critical mixtures and can hardly be reached due to prior phase separation. The diffusion coefficients in the upper left part of Fig. 8 have been fitted by (23) with a fixed activation temperature determined from Dj. The binodal points in Fig. 8 mark the boundary of the homogeneous phase at the binodal. The spinodal temperatures Tsp are obtained as a fit parameter for every concentration and together define the (pseudo)spinodal line plotted in the phase diagram in Fig. 7. The Soret coefficient is obtained from (11) and (23) as... 

The situation is different for c = 0.9, where the PDMS-enriched central part is stabilized and shifted away from the binodal. But now, the regions outside the central area, where PEMS accumulates, cross the phase boundary into the metastable range. The demixing by nucleation and growth is visible in the lower two micrographs in Fig. 16 in the form of a halo of dark droplets around the written structures. 

In Figure 2F-1 the composition where d2( G)/d 22 s equal to zero, or at the inflection point on the Gibbs energy surface, is defined as the spinodal composition. This corresponds to the boundary between an unstable solution and a metastable solution. If the necessary amount of free energy is supplied to the metastable system, the solution will phase separate into two phases with binodal compositions unstable system will always phase separate into the two phases. The temperature where the two points of inflection on the energy surface merge into a single point is defined as the critical solution temperature. 

The thermodynamic definition of the spinodal, binodal and critical point were given earlier by Eqs. (9), (7) and (8) respectively. The variation of AG with temperature and composition and the resulting phase diagram for a UCST behaviour were illustrated in Fig. 1. It is well known that the classical Flory-Huggins theory is incapable of predicting an LCST phase boundary. If has, however, been used by several authors to deal with ternary phase diagrams Other workers have extensively used a modified version of the classical model to explain binary UCST or ternary phase boundaries The more advanced equation-of-state theories, such as the theory... 

Fig. 6. Phase diagram calculated for the ternary system solvent (l)/rigid rod (2)/random coil (3) with X, = 1 and xj = X3 = 100. Numerals on the tie lines that extend below the lower boundary of the diagram denote vi for the conjugate anisotropic phase the binodal for which lies virtually on the 1 — 2 axis (From Ref.

Figure 9.2 Schematic phase diagram of a polymer/solvent mixture, where y is the Flory chi parameter, and xe = 1/2 is x at the theta temperature. The quantity Xe X along the ordinate is a reduced temperature, and is the polymer volume fraction. CP is the critical point, and BL is the binodal line. SSL and KSL are the static symmetry line and the kinetic symmetry line, respectively. These lines define the phase-inversion boundaries during quenches. In quenches that end at the right of such a line, the polymer-rich phase is the continuous phase, while to the left of the line the solvent-rich phase is the continuous one. SSL applies at long times, after viscoelastic stresses have relaxed, while KSL applies at shorter times before relaxation of viscoelas-...

Theoretical diffusion path studies were made with a model system for comparison to the experimentally observed phenomena. A pseudoternary representation was chosen for modeling the phase behavior, and brine and oil were chosen as the independent diffusing species. For simplicity and because their exact positions and shapes were not known, phase boundaries in the liquid crystal region were represented as straight lines. Actually, studies indicate a rather complex transition from liquid crystal to microemulsions as system oil content is increased, especially near optimum salinity (15-16). A modified Hand scheme was used to model the equilibria of binodal lobes (14,17). Other assumptions are described in detail elsewhere (13). 

In the insets of Fig. 15 we show binodal curves for the symmetric blend. Again, we And deviations in the immediate vicinity of the critical point but for larger incompatibilities, xN 2, the mean held predictions provide an adequate description of the phase boundary utilizing the Flory-Huggins parameter extracted from the composition fluctuations in the one-phase region, xN < 2. 

The above equation can be solved for the interaction parameter corresponding to the phase boundary—the binodal (solid line in the bottom part of Fig. 4.8) of a symmetric blend ... 

Section 4.4, the binodal curve that describes the phase boundary was defined. The highest point on the binodal line is the critical point with critical composition [Eq. (4.57)] ... 

The binodal curves (phase boundary) separate the one- and two-phase regions. Below the binodal cnrves are two-phase regions, and above the curves is a single-phase region. At the plait point of the binodal cnrve, all phase compositions are equal. The right plait point is nsnally located very close to the oil apex, and the left plait point is nsnally located very close to the water apex. In a two-phase region, the compositions of phases in eqnilibrium are connected with tie lines, along which may be found all possible proportions of the two phases, as explained in Example 7.1. Before that, we first need to review the lever rale. 

As mentioned earlier, a supersaturated solution is not in the equilibrium condition. Crystallization moves the solution toward equilibrium by relieving its supersaturation. A supersaturated solution is thus not stable. There is a maximum degree of supersaturation for a solution before it becomes unstable. The region between this unstable boundary and the equilibrium (binodal) curve is termed the metastable zone, and it is here that the crystallization process occurs. The absolute limit of the metastable zone, known as the spinodal curve (8), is given by the locus of the maximum limit of supersaturation at which nucleation occurs spontaneously. Thermodynamically, the spinodal curve within the two-phase region is defined by the criterion... 

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