Unstable region

Wlien T < T the graph of H versus m shows a van der Waals like loop, with an unstable region where the... 

Figure A2.5.6. Constant temperature isothenns of redueed pressure versus redueed volume for a van der Waals fluid. Full eiirves (ineluding the horizontal two-phase tie-lines) represent stable situations. The dashed parts of the smooth eurve are metastable extensions. The dotted eurves are unstable regions.

Figure A2.5.7. Constant temperature isothenns of reduced Helmlioltz free energy A versus reduced volume V. The two-phase region is defined by the line simultaneously tangent to two points on the curve. The dashed parts of the smooth curve are metastable one-phase extensions while the dotted curves are unstable regions. (The isothenns are calculated for an unphysical r = 0.1, the only effect of which is to separate the isothenns...

Figure A2.5.9. (Ap), the Helmholtz free energy per unit volume in reduced units, of a van der Waals fluid as a fiinction of the reduced density p for several constant temperaPires above and below the critical temperaPire. As in the previous figures the llill curves (including the tangent two-phase tie-lines) represent stable siPiations, the dashed parts of the smooth curve are metastable extensions, and the dotted curves are unstable regions. See text for details.

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). 

There is tio 7, region, thus no unstable region. The motor c m operate at ttny speed up to the rated one without any stalling region. 

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... 

In another test, the MCC sol at 23°C and 410 ppm was unstable (region II). After decreasing the temperature to 20°C, the MCC sol was stabilized. If the temperature was raised back to 23°C, the MCC sol destabilized and floes formed again. The floes in region II seem to be temperature reversible. 

Separates a stable region from an unstable region. 

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,... 

Non-Random Systems. As pointed out by Cahn and Hilliard(10,11), phase separation in the thermodynamically unstable region may lead to a non-random morphology via spinodal decomposition. This model is especially convenient for discussing the development of phase separating systems. In the linearized Cahn-Hilliard approach, the free energy of an inhomogeneous binary mixture is taken as ... 

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