Thermodynamic limit metastability

Infinite Systems The ultimate fate of infinite systems, in the infinite time limit, is quite different from their finite cousins. In particular, the fate of infinite systems does not depend on the initial density of cr = 1 sites. In the thermodynamic limit, there will always exist, with probability one, some convex cluster large enough to grow without limit. As f -4 oo, the system thus tends to p —r 1 for all nonzero initial densities. What was the critical density for finite systems, pc, now becomes a spinodal point separating an unstable phase for cr = 0 sites for p > pc from a metastable phase in which cr = 0 and cr = 1 sites coexist. For systems in the metastable phase, even the smallest perturbation can induce a cluster that will grow forever. 

The presence and width of a metastable zone, in which nucleation is not spontaneous, have been discussed in Chapter 2. The thermodynamic limit of the metastable zone is a locus of points known as the spinodal curve, where spinodal decomposition replaces nucleation and crystal growth as the phase separation. In typical industrial crystallization, nucleation (and release of supersaturation) occurs at much lower supersaturations than the spinodal curve. 

Plots in Fig. 5.16 also indicate that, over certain ranges of /x, u (/x) is a multivalued function where the lowest value of u obviously corresponds to the thermodynamically stable morphology (i.e., phase) the others are only metastable. Metastability ends (i.e., the confined fluid becomes unstable) if the inequality in Eq. (1.82) can no longer be satisfied. The reader should realize that in general metastability in MC simulations is an artifact caused by the limited system size and insufficient length of the Markov chain (i.e., the finite computer time available) [184]. Metastability would not be observed in an infinite system where the evolution of the system could be followed indefinitely. In other words, metastability vanishes in the thermodynamic limit. 

Figure 1. Pure water phase diagram in (P,T) coordinates calculatedfrom the IAPWS-95 equation of state, extrapolated at negative liquid pressures in the superheat domain. The outer lines starting from the critical point are the thermodynamic limits of metastability (spinodal). The dotted line is one of the proposed kinetic metastability limii (see text). Three isochoric lines (950, 900 and 850 kg m ) are also calculated by extrapolation of the IAPWS-95 equation.

The Thermodynamic Limit. In the preceding text (see Fig. 15.2), the limit of the region in which the liquid phase can exist in a metastable state (the liquid spinodal) was introduced this is represented by point C in Fig. 15.2. One view of homogeneous nucleation is that it will occur at the spinodal limit that corresponds (see Fig. 15.2) to the condition (expressed in term of reduced quantities)... 

In Fig. 14a one can see the isotherms of the order parameter, n = N N, against the surface attraction e for different pulling forces,/, which resemble closely those of a conventional first-order phase transition. However, as indicated by the corresponding PDF W(n) (cf. Fig. 14b), the adsorption-desorption first-order phase transition under pulling force has a clear dichotomic namre (i.e., it follows an either/or scenario) in the thermodynamic limit N co there is no phase coexistence The configurations are divided into adsorbed and detached (or stretched) dichotomic classes. The metastable states are completely absent. 

FIG. 1 At point A, bubbles begin to appear. Two pathways to point A are shown. BA represents raising the temperature at constant pressure, whereas CA causes boiling by reducing the pressure at a constant temperature. The portions of the line BA above the full curve, or of CA to the left of the full curve until they reach A are sometimes referred to as the widths of the metastable zone. Ostwald s metastable limit is the kinetic limit of stability, and is shown here as the dotted line. The spinodal represent the thermodynamic limit of instability. 

They define the thermodynamic limits of metastability. For concentrations corresponding to the spinodal points, the system is unstable and demixes spontaneously into two distinct continuous phases which form an interpenetrating system. This type of phase separation characteristic of spinodal regions, is also called spinodal decomposition. 

Traditional solid-state synthesis involves the direct reaction of stoichiometric quantities of pure elements and precursors in the solid state, at relatively high temperatures (ca. 1,000 °C). Briefly, reactants are measured out in a specific ratio, ground together, pressed into a pellet, and heated in order to facilitate interdiffusion and compound formation. The products are often in powdery and multiphase form, and prolonged annealing is necessary in order to manufacture larger crystals and pure end-products. In this manner, thermodynamically stable products under the reaction conditions are obtained, while rational design of desired products is limited, as little, if any, control is possible over the formation of metastable intermediates. ... 

For a nonthinning, unbounded film, Vrij (10) showed via a thermodynamic analysis (i.e., a surface energy minimization) that when 3II/3h is greater than zero the film is unstable. Thus, in Figure 4 the critical thickness limit for metastable films,... 

Shortly after, we recognized that ScCu4Ga2 (Im3) [70] might also be tuned to a QC, but the correct stoichiometry and reaction conditions were not achieved in our limited experiments. Recently, Honma and Ishimasa [71] have reported that i-QC phase forms almost exclusively from a rapidly quenched ScisCu48Ga34 composition, emphasizing a very narrow phase width and its thermodynamic metastability at room temperature. However, the failure turned us to other Ga intermetallics, which led to the pseudogap tuning concepts that follow. 

The temperature dependences of the isothermal elastic moduli of aluminium are given in Figure 5.2 [10]. Here the dashed lines represent extrapolations for T> 7fus. Tallon and Wolfenden found that the shear modulus of A1 would vanish at T = 1.677fus and interpreted this as the upper limit for the onset of instability of metastable superheated aluminium [10]. Experimental observations of the extent of superheating typically give 1.1 Tfus as the maximum temperature where a crystalline metallic element can be retained as a metastable state [11], This is considerably lower than the instability limits predicted from the thermodynamic arguments above. 

Numerous disperse dyes are marketed in a metastable crystalline form that gives significantly higher uptake than the corresponding more stable modification. The molar free enthalpy difference can be used as a criterion of the relative thermodynamic stabilities of two different modifications [53]. Certain dyes can be isolated in several different morphological forms. For example, an azopyrazole yellow disperse dye (3.52) was prepared in five different crystal forms and applied to cellulose acetate fibres. Each form exhibited a different saturation limit, the less stable modifications giving the higher values [54]. 

The determination of individual binary equilibrium diagrams usually only involves the characterisation of a limited number of phases, and it is possible to obtain some experimental thermodynamic data on each of these phases. However, when handling multi-component systems or/and metastable conditions there is a need to characterise the Gibbs energy of many phases, some of which may be metastable over much of the composition space. 

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-... 

Adequacy of Thermodynamic Data. Data on several important aluminosilicates appear to be insufficient for a detailed discussion of all equilibria. Information on the influence of solid solutions or coprecipitated phases on thermodynamic properties appears to be rather limited, as is that for metastable non-stoichiometric oxides (e.g., of manganese) and surface complexes. 

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