Phase diagram subsystem

Now we consider thermodynamic properties of the system described by the Hamiltonian (2.4.5) it is a generalized Hamiltonian of the isotropic Ashkin-Teller model100,101 expressed in terms of interactions between pairs of spins lattice site nm of a square lattice. Hamiltonian (2.4.5) differs from the known one in that it includes not only the contribution from the four-spin interaction (the term with the coefficient J3), but also the anisotropic contribution (the term with the coefficient J2) which accounts for cross interactions of spins a m and s m between neighboring lattice sites. This term is so structured that it vanishes if there are no fluctuation interactions between cr- and s-subsystems. As a result, with sufficiently small coefficients J2, we arrive at a typical phase diagram of the isotropic Ashkin-Teller model,101 102 limited by the plausible values of coefficients in Eq. (2.4.6). At J, > J3, the phase transition line... 

The phase diagram of the ternary system K3TiCl6-K2ZrCl6-TiCl3 has been investigated 83 three subsystems were formed ... 

Common approaches for the tailoring of nonmetallic (ceramic) materials properties involve topochemical methods (those where the crystal structure remains largely unaffected) and the preparation of phases in which one or more sublattices are alloyed. In principle, such materials are within the realm of CALPHAD. On the other hand, as has already been stated, extrapolation does not really aid the discovery of new or novel phases, with unique crystal structures. Furthermore, assessed thermochemical data for the vast majority of ceramic systems, particularly transition metal compounds, are presently not available in commercial databases for use with phase diagram software. This does not necessarily preclude the use of the CALPHAD method on these systems However, it does require the user to carry out their own thermodynamic assessments of the (n — 1 )th-order subsystems and to import that data into a database for extrapolation to nth-order systems, which is not a trivial task. 

The ternary system was calculated by extrapolation from the binary subsystems (Kasper, 1996) [33]. The calculations cover phase equilibria at one bar and do not assume any solubilities as no experimental evidence for stable sohd solutions between B4+5C and BN or a-BN and graphite exist. The section between graphite and boron nitride including the invariant reactions Uj, Dj and U2 (Fig. 23) is shown in Fig. 22. A calculated potential phase diagram (logpN2-T) can be found in [244], The complete Scheil reaction scheme (P = 1 bar) is shown in Fig. 23. 

The chemical compositions of clays are very complex, and frequently more than three components must be considered. Four-component (quaternary) phase diagrams can be represented by a tetrahedron. As an example. Figure 3.9 shows the quaternary system Ca0-Mg0-Al203-Si02 and several of its subsystems that divide the tetrahedra volume into four-component phase assemblies. 

The first systematic approach to a derivation the global phase diagram of ternary fluid mixture using an analytical investigation of the Van der Waals equation of state with standard one-fluid mixing rules was developed by Bluma and Deiters (1999). Eight major classes of ternary fluid phase diagrams were outlined and their relationship to the main types of binary subsystems were established. 

Figure 1.35 Main types of fluid phase diagrams (T-X projections) for ternary mixtures with one volatile component (A) and immiscibility phenomena of types b, c and d in binary subsystems A-B and A-C (Reproduced by permission of the PCCP Owner Societies).

A selection of phase diagram type for given system from the possible versions obtained by the theoretical derivation (based only on the information about phase behavior in binary subsystems) can be made using an additional experimental data on the ternary phase equilibria. It is clear that the munber of experimental measmements needed for a selection of the right phase diagram type is significantly lower than in the case of experimental way without any theoretical derivations beforehand. 

Similarly to the phase diagrams for binary systems, the main types for fluid phase diagrams of ternary mixtures should not have an intersection of critical curves and inunis-cibUity regions with a crystallization surface in them. Combination of four main types of binary fluid phase behavior la, lb, Ic and Id (Figure 1.2) for constituting binary subsystems gives six major classes of ternary fluid mixtures with one volatile component, two binary subsystems (with volatile component) complicated by the immiscibility phenomena and the third binary subsystem (consisted from two nonvolatile components) of type la with a continuous solid solutions. These six classes of ternary fluid mixtures can be referred as ternary class I (with binary subsystems Ib-lb-la), ternary class II (with binary subsystems Ic-lc-la), ternary class III (with binary subsystems Id-ld-la), or ternary class IV (with binary subsystems Ib-ld-la), ternary class V (with binary subsystems Ib-lc-la) and ternary class VI (with binary subsystems Ic-ld-la). 

T-X diagrams were used for an investigation of ternary fluid phase behavior by the method of continuous topological transformation of ternary monovariant ciuves originated in the nonvariant points of binary subsystems with volatile component. In the case of fluid phase diagrams aU these nonvariant points are the binary critical points and file ternary monovariant curves are the critical curves, which join the binary critical points of the same nature or intersect at ternary nonvariant critical point if they start in file binary critical points of different nature. 

T-X projections of possible fluid phase behavior diagrams in ternary systems with one volatile component and immiscibility phenomena in both binary subsystems consisting of the volatile and nonvolatile components are shown in Figure 1.35. Each ternary class has several types of fluid phase behavior, described by various versions of fluid phase diagrams, where the monovariant critical curves, originating in the same binary critical endpoints, show file different ways of intersection. 

In derivation of ternary fluid phase diagrams (Figure 1.35) the experimental observations of an occurrence of two-phase hole L-G (completely bounded by a closed-loop critical curve Li = L2-G) in the three-phase immiscibility region bormded by a critical curve Li = G-L2 from the high-temperature side (quasi-binary cross-sections of type Id) (Peters and Gauter, 1999) are taken into account. In our derivations it was assumed that this two-phase hole L-G may appear in ternary three-phase immiscibility regions that spread from the binary subsystems of types lb and Ic. 

Complete phase diagrams for ternary systems with one volatile component and immiscibility phenomena in binary subsystems with components of different volatility... 

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