The Unstable Region

Consider first point 8. Since this point does not meet the second stability criterion (Ineq. 12.8.2), it cannot represent a stable homogeneous phase. The system must move to points 2 or 10. (Or, it could break down into two phases represented by points 2 and 10. If these two phases, however, are to be in equilibrium, they must meet the criterion that was developed in Section 12.4 the chemical potential of the fluid must be the same in both phases. For this to occur, according to Problem 9.57, the two areas / and II - below and above the constant pressure line - must be equal. This pressure - not equal to Pq of course - represents the vapor pressure of the fluid at the specified temperature according to the EoS used.) This applies to all states described by the line segment from point 5 to point 9 and they are referred to as unstable. 

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The reason such a large value is obtained can be elucidated as follows Since in the stable region, all the fluctuations are decayed to zero to maintain the electrode surface as flat and stable, the autocorrelation distance tends to approach infinity. On the other hand, in the unstable region, many new fluctuations grow, so that the autocorrelation distance will take a small value. At the critical state (i.e., the boundary between the two regions), therefore, a fluctuation with an extraordinarily large autocorrelation distance appears this value is considered to have a generality... 

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. 

The value of the compactification radius, Rc In the present approach this radius was a free parameter. For demonstration we chose the radius Rc = 0.33 10 13 cm, when the strange A baryon could behave as the first excitation of a neutron. Such an extradimensional object can mimics a compact star with neutrons in the mantle and A s in the core. With smaller Rc the exotic component appears at larger densities - we may run into the unstable region of the hybrid star and the extra dimension remains undetectable. However, with larger Rc the mass gap becomes smaller and the transition happens at familiar neutron star densities. In this way, reliable observations could lead to an upper bound on Rc. 

The effect of adding a lag or a pole is to pull the root locus plot toward the unstable region. The two curves that s ait ats=— Jand.s=—1 become complex conjugates and curve off into the RHP. Therefore this third-order system is closed-loop unstable if is greater than = 20. This was the same result that we obtained in Example 10.5,... 

Note 2 In the unstable region bounded by the spinodal curve, phase domains separation is spontaneous, i.e., no nucleation step is required to initiate the separation process. 

In the unstable region, the concentration fluctuations are delocalized and there is no thermodynamic barrier to phase growth. Thus, separations that take place spontaneously lead to long range phase separation. This process is called spinodal decomposition (SD). In this mechanism, decomposition starts with a co-continuous structure and gradually shifts to a droplet morphology because of the breakdown of the continuous structure [41]. 

Phase separation through NG mechanism cannot be observed for polymer-polymer blend systems that show interfacial tension lying in the range 0.5-11 mN/m. In addition, they predicted that a secondary phase separation could take place inside dispersed rubber particles in the case when the average composition of dispersed domains lies in the unstable region at the end of the phase separation [2], They were not able to observe a phase separation inside dispersed domains with TEM micrographs however, they concluded that there are two phases inside the dispersed domains by the fact that the glass transition temperature of the rubber-... 

Thus the upper end of the oscillatory region lies close to the conditions at which the two stationary-state loci cross, but at slightly lower reactant concentration. The lower end of the unstable region p lies at slightly higher reactant concentrations than that at which the locus ass( p) has its maximum. 

This behavior seems to be measurable by surface dynamic light scattering. Furthermore, in the unstable region 8 < 8Cf we predict a spinodal decomposition on the surface. The growth rate of the surface disturbances takes a maximum at an intermediate wave number km. The balance of the two terms in the last brackets of Eq. (7.22) yields [93]... 

Temperature Dependence. Early experiments with laboratory stills and a variable-speed steam compressor had shown that heat transfer and yield increased with temperature. The No. 5 compressor has a fixed speed and, since the specific volume of the steam decreases as temperature increases, the compressor was evidently starved for steam and operated in the unstable region when run at evaporator temperatures above 125° F. However, a trend toward increasing performance persisted,... 

By this moving jump of the control parameter the system is divided into a stable and an unstable region. Numerical simulations of the ID model in (68) and with the directional quenching (69) show that a periodic solution develops behind the quench interface in the unstable region. Typical examples for large and small values of the... 

Paints, adhesives and lubricants are typically multicomponent polymer systems. The behavior of phase-separated blends in the bulk after quenching into the unstable region of the phase diagram is variable. In the bulk, the concentration fluctuations... 

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