The Metastable and Unstable Regions

We have considered so far the implications of the first requirement for a closed system that reaches equilibrium at a specified T and P its Gibbs free energy must assume its minimum value (Eqs 12.3.8 and 12.3.9). We examine next the implications of the second requirement (Ineq. 12.3.10)  

If we combine our findings with those of the previous Section we conclude that for A = 7  

The values of Xj = 0.092 and 0.908 establish the miscibility limits and the values of Xj = 0.232 and 0.768, the stability limits. (The latter points are also referred to as incipient instability points). 

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Figures 1.1.10-1.1.12 show the CPC and IPPC curves. For the system with PMAA number average molecular weight of 180,000 g/mol, the IPPC has been found to be very small. The spinodal curve is the theoretical boundary between the metastable and unstable regions, where the mutual diffusion coefficient is zero, whereas the IPPC corresponds to the sudden liquid-liquid phase separation under certain experimental conditions. The other difference is that the spinodal has a...

Fig. 4. Schematic representation of free energy of mixing versus composition curves at different temperatures Ti < T2 < T3 < T4 and corresponding phase diagram. In the lower section, I refers to the one-phase region while II and III are the metastable and unstable regions of the phase diagram, respectively.

Thus, we have a temperature interval from the phase boundary at 25 °C down to approximately 22 °C where the droplets are metastable, and a region below 22 °C where the droplets are unstable. Measurements performed on a lower (0 = 0.05) and a higher ( = 0.24) concentration showed the same behavior as the (j> = 0.12 sample. In particular, the temperature separating the metastable and unstable regions was essentially identical (22 °C). 

A phase diagram relative to a polymer solution that phase separates on heating is shown in Figure 4. The solid line in this figure is called the binodal and it separates the stable from the metastable regions of the phase diagram. The dashed line is the spinodal curve, which separates the metastable and unstable regions. The spinodal touches the binodal at the critical point, Tc, which for a polsTner solution is defined by equations 17 and 18. 

For T shaped curves, reminiscent of the p, isothemis that the van der Waals equation yields at temperatures below the critical (figure A2.5.6). As in the van der Waals case, the dashed and dotted portions represent metastable and unstable regions. For zero external field, there are two solutions, corresponding to two spontaneous magnetizations. In effect, these represent two phases and the horizontal line is a tie-line . Note, however, that unlike the fluid case, even as shown in q., form (figure A2.5.8). the symmetry causes all the tie-lines to lie on top of one another at 6 = 0 B = 0). 

Figure 8.5 Conversion vs composition transformation diagram at a constant cure temperature, showing cloud-point curves and spinodal curves that bound stable, metastable, and unstable regions , , and represent the three trajectories, starting from different initial thermoplastic concentrations and leading to different morphologies. (Pascault and Williams, 2000 -Copyright 2001. Reprinted by permission of John Wiley Sons Inc.)...

The driving force for crystallization, the supersaturation, is composed of two zones, the metastable and unstable zones (Figure 7.1). The solid line in Figure 7.1 represents a saturation or solubility curve (S), while the dashed line corresponds to the supersolubility or supersaturation curve (SS). Below S, crystallization is impossible. Above S, in the metastable zone, the system is supersaturated and crystallization is possible with the aid of agitation or seeding. On the other hand, in the unstable region, crystallization is spontaneous and crystals appear after nucleation... 

The spinodal curve constitutes the boundary between thermodynamically metastable and unstable regions. At this critical boundary, the following condition is fulfilled [56]... 

Fig. 11. Binodal and spinodal curves in a conversion vs modifier volume fraction phase diagram. The critical point and the location of stable, metastable and unstable regions are shown...

Figure 9.3 Along subcritical isotherms for pure fluids, the fugacity passes through stable, metastable, and unstable regions just as does the pressure. Here we have plotted the subcritical isotherm TlT = 0.863 for a van der Waals fluid. Each point (a-f) on the fugacity plot corresponds to the point of the same label on the Pv diagram. Points b and e have the same fugacity and pressure (P /= 0.539) and therefore locate the vapor-liquid equUibrium state. Points c and d are on the spinodal. Line segment be locates metastable liquid states segment de locates metastable vapor states segment cd locates unstable states.

Indifferent situations can create problems in the trial-and-error procedures routinely used in calculations for phase and reaction equilibrium. In such calculations, we may start with an T" or T" spedfication that properly doses the problem, but during the course of the trial-and-error search, the algorithm may enter an indifferent situation that couples properties that are otherwise independent. This may occur not only when azeotropes and critical points are encountered, but also when algorithms enter metastable and unstable regions of phase diagrams [20]. 

Only systems with unstable regions in this composition range. The positions of the raetastable and unstable regions are given by the positions of the inflection points C and D in the AG = f [Pg.232]

Figure 1. Schematic phase diagram of a single-component substance showing the region of vapor-liquid coexistence. The full line is the coexistence locus (binodal). The dashed line is the locus of stability limits (spinodal), which separates the stable and unstable regions. Also shown are the destabilizing fluctuations in the metastable (nucleation), and unstable (spinodal decomposition) regions, with denoting the initial uniform density / , the radius of a nucleus, / , the radius of the critical nucleus. A, the wavelength of a density fluctuation, and the critical wavelength [109].

Figure 1. Schematic cuts of the phase diagram of a pure substance, showing the stable, metastable, and unstable regions (a) a low-temperature isotherm with a metastable liquid branch reaching negative pressure, (b) temperature-density cut, and (c) pressure-temperature cut.

The boundary between metastable and unstable regions is given by the second derivative of AGm equal to zero. Then, differentiating Equation 3.3 once more. 

The reaction-induced phase separation may also be described using conversion, x, versus composition, ( )m, transformation diagrams at a constant cure temperature (Fig. 8.5). Cloud point and spinodal curves bound stable, metastable, and unstable regions. Experimental studies of phase separation (Chen et al., 1993 Girard-Reydet et al, 1998), revealed that compositions located close to the critical point (e.g., trajectory 2 in Fig. 8.5) undergo spinodal demixing, while off-critical compositions (e.g., trajectories 1 and 3) exhibit phase separation by a nucleation-growth mechanism. 

Figure 13. 2 Stable, metastable and unstable regions for a liquid-vapor transition. In the region JKL we have (0p/0y) y. > 0, so the system is unstable...

In region BL of Fig. 13.2 the system is a supersaturated vapor and may begin to condense if nucleation can occur. This is a metastable state. Similarly, in region AJ we have a superheated liquid that will vaporize if there is nucleation of the vapor phase. The stable, metastable and unstable regions are indicated in Fig. 13.2. Finally, at the critical point C, both the first and second derivatives of p with respect to Vequal zero. Here the stability is determined by the higher-order derivatives. For stable mechanical equilibrium at the critical point, we have... 

Figure 5.3 Entropy of liquid and crystalline aluminium in stable, metastable and unstable temperature regions [12]. The temperatures where the entropy of liquid and crystalline aluminium are equal are denoted Tf and 7 jm crySt, respectively.

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