Phase extensions, metastable

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...

This method is efficient, and does not require an extensive knowledge of the crystal structures of the phases. It is, however, imperative for each of the phases to be pure. This is sometimes difficult to achieve, particirlarly if one of the phases is metastable. In that case, the first method is used. 

Such data are summarized in the form of vant Hoff plots in which a series of isocompositional lines are plotted as a function of reciprocal temperature and log partial pressure. The results for the above system are presented in Figure 8.8. The dashed lines in the right-hand portion represent the metastable (single phase) extensions into the two-phase region. The enthalpy for the oxidation process, Equation 8.1, can be obtained from the slope of the lines in the single-ph ase region. 

Figure A2.5.1. Schematic phase diagram (pressure p versus temperature 7) for a typical one-component substance. The full lines mark the transitions from one phase to another (g, gas liquid s, solid). The liquid-gas line (the vapour pressure curve) ends at a critical point (c). The dotted line is a constant pressure line. The dashed lines represent metastable extensions of the stable phases.

R is the gas constant per mole, while K is the temperature unit Kelvin). The dashed lines represent metastable extensions of the stable phases beyond the transition temperatures. 

Figure A2.5.4. Themiodynamic fimctions (i, n, and C as a fimction of temperature T at eonstant pressure and eomposition x = 1/2) for the two-eomponent system shown in figure A2.5.3. Note the diflferenee between these and those shown for the one-eomponent system shown in figure A2.5.2. The fiinetions shown are dimensionless as in figure A2.5.2. The dashed lines represent metastable extensions (superheating or supereooling) of the one-phase systems.

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 A2.5.9. (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.

During the last decade knowledge of the ion chemistry of nitro compounds in the gas phase has increased significantly, partly due to the more widespread use of specialized techniques. Thus various ionization methods, in particular electron impact ionization and chemical ionization, have been used extensively. In addition, structure investigations as well as studies on fragmentation pathways have involved metastable ion dissociations, collision activation and neutralization/reionization studies, supplementary to studies carried out in order to disclose the associated reaction energetics and reaction dynamics. In general, the application of stable isotopes plays a crucial role in the in-depth elucidation of the reaction mechanisms. 

Figure 6.1. Gibbs free energy curves for Ti (a) a, j9 and liquid segments corresponding to the stable regions of each phase (b) extrapolated extensions into metastable regions (c) characterisation from Kaufinan (1959a) using equation (6.2) and (d) addition of the G curves for u and f.c.c. structures (from Miodownik...

J The concept of counter-phases. When a stable compound penetrates from a binary into a ternary system, it may extend right across the system or exhibit only limited solubility for the third element. In the latter case, any characterisation also requires thermodynamic parameters to be available for the equivalent metastable compound in one of the other binaries. These are known as counter-phases. Figure 6.16 shows an isothermal section across the Fe-Mo-B system (Pan 1992) which involves such extensions for the binary borides. In the absence of any other guide-... 

There are other factors to be considered. In a number of systems ternary phases exist whose stability indicates that there are additional, and as yet unknown, factors at work. For example, in the Ni-Al-Ta system r, which has the NisTi structure, only exists as a stable phase in ternary alloys, although it can only be fully ordered in the binary system where it is metastable. However, the phase competes successfully with equilibrium binary phases that have a substantial extension into the ternary system. The existence of the ly-phase, therefore, is almost certainly due to... 

It should be noted that it is possible to produce fully stabilized bodies with much higher fracture strengths than listed here but this requires the use of fine particle size, chemically prepared powders (3). The use of this type of material involves a number of penalties both in cost and processability that may be prohibitive for a high volume automotive application. In addition to the type of partially stabilized body described here, two other basic types of partially stabilized bodies have been reported (4, ). Both are classified as transformation toughened partially stabilized zirconias and involve different processing techniques to obtain a body with various amounts of a metastable tetragonal phase. While the mechanical properties of these materials have been studied extensively, little has been reported about their electrical properties or their stability under the thermal, mechanical and chemical conditions of an automotive exhaust system. 

Under ordinary laboratory conditions (at lower temperatures than shown in Fig. 7.5), it is seen that only the red a-sulfur form is stable. The yellow /3-sulfur needles are the most stable phase only in a narrow temperature range around 96-120°C, but they persist as a supercooled metastable phase well below this range (surviving, for example, on a stock-room shelf for indefinite periods). Given a sample of yellow /3-sulfur, it is easy to detect the melting point near 120°C, but is far from easy to detect the enantiomorphic a/13 solid-solid conversion near 96°C, unless a nucleating crystallite or catalyst is introduced. Once produced, the /3-phase tends to persist as a supercooled (undercooled) metastable extension... 

The problems of phase transition always deeply interested Ya.B. The first work carried out by him consisted in experimentally determining the nature of memory in nitroglycerin crystallization [8]. In the course of this work, questions of the sharpness of phase transition, the possibility of existence of monocrystals in a fluid at temperatures above the melting point, and the kinetics of phase transition were discussed. It is no accident, therefore, that 10 years later a fundamental theoretical study was published by Ya.B. (10) which played an enormous role in the development of physical and chemical kinetics. The paper is devoted to calculation of the rate of formation of embryos—vapor bubbles—in a fluid which is in a metastable (superheated or even stretched, p < 0) state. Ya.B. assumed the fluid to be far from the boundary of absolute instability, so that only embryos of sufficiently large (macroscopic) size were thermodynamically efficient, and calculated the probability of their formation. The paper generated extensive literature even though the problem to this day cannot be considered solved with accuracy satisfying the needs of experimentalists. Particular difficulties arise when one attempts to calculate the preexponential coefficient. 

Historically, two periods occurred for the determination of the number of hydrate water molecules per guest molecule. In the first century (1778-1900) after the discovery of hydrates, the hydration number was determined directly. That is, the amounts of hydrated water and guest molecules were each measured via various methods. The encountered experimental difficulties stemmed from two facts (1) the water phase could not be completely converted to hydrate without some occlusion and (2) the reproducible measurement of the inclusion of guest molecules was hindered by hydrate metastability. As a result, the hydrate numbers differed widely for each substance, with a general reduction in the ratio of water molecules per guest molecule as the methods became refined with time. After an extensive review of experiments of the period, Villard (1895) proposed Villard s Rule to summarize the work of that first century of hydrate research ... 

Thus obtained results show that the polyamorphic transitions occur not only at compression (Si02, H20, etc.) but at extension as well (C) in the systems having stable or metastable crystal analogs with a different density and a different coordination number z. At the minimal z=2 (chain structures) the transitions may occurs only at compression, at the maximal z=12 (close-packed structures) - only at extension, at the intermediate z (2structure-sensitive properties change and new metastable phases can appear. Amorphization under radiation (crystal lattice extension) can be associated with a softening of phonon frequencies. The transitions in the molecular glasses consisted from the molecules with unsaturated bonds are accompanied by creation of atomic or polymeric amorphous systems. 

Figure 13. Schematic phase diagram of water s metastable states. Line (1) indicates the upstroke transition LDA —>HDA —>VHDA discussed in Refs. [173, 174], Line (2) indicates the standard preparation procedure of VHDA (annealing of uHDA to 160 K at 1.1 GPa) as discussed in Ref. [152]. Line (3) indicates the reverse downstroke transition VHDA—>HDA LDA as discussed in Ref. [155]. The thick gray line marked Tx represents the crystallization temperature above which rapid crystallization is observed (adapted from Mishima [153]). The metastability fields for LDA and HDA are delineated by a sharp line, which is the possible extension of a first-order liquid-liquid transition ending in a hypothesized second critical point. The metastability fields for HDA and VHDA are delineated by a broad area, which may either become broader (according to the singularity free scenario [202, 203]) or alternatively become more narrow (in case the transition is limited by kinetics) as the temperature is increased. The question marks indicate that the extrapolation of the abrupt LDA<- HDA and the smeared HDA <-> VHDA transitions at 140 K to higher temperatures are speculative. For simplicity, we average out the hysteresis effect observed during upstroke and downstroke transitions as previously done by Mishima [153], which results in a HDA <-> VHDA transition at T=140K and P 0.70 GPa, which is 0.25 GPa broad and a LDA <-> HDA transition at T = 140 K and P 0.20 GPa, which is at most 0.01 GPa broad (i.e., discontinuous) within the experimental resolution.

The extensively studied (especially during the recent years) transitions of solids from the metastable amorphous state to the polycrystalline state (see ref. 58 and the references therein) are of autowave character and resemble very much the above regimes of solid-state cryochemical reactions. The action of autodispersion, which facilitates phase transition by allowing it to proceed on the surface of a fracture instead of in the glass volume, cannot be excluded in the case of those processes either. Actually, the two classes of processes are similar in their physical nature both are connected with rearrangement of the solid matrix and are of exothermic character, differing only in the extent of the thermal effect. It should be added that fracturing and autodispersion of the sample are very typical of the autowave destruction of amorphous states and can be seen even by the unaided eye. 

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