Topological relations in multisystems of more than n+ 3 phases ¹

Abstract

Multisystems of n + k ( k > 3) phases are very complicated and knowledge of them has suffered as a result. The successful solution of the topological relationships in n + 3 phase multisystems ,by Zen (1966, 1967) and Zen & Roseboom (1972) has aroused much interest regarding what will happen in a multisystem of more than n + 3 phases. Since 1979, some important research results on this topic have been published. These results have expounded the substantial rules governing the appearance of phase relations in phase diagrams of n + k ( k > 3) phase multisystems. The most significant conclusions include: ( I ) I t is impossible to incorporate all the possible phase relations in an n + k

A new method for the derivation of the closed nets in the phase diagram space of multisystems. I. The absent phase substitution method

“…According to the combination principle (Guo & Wang, 1982; Guo, 1984), to obtain the closed nets of a high‐level multisystem, it is possible in principle to derive them from the combination of the closed nets of its 1‐level subsystem. Take a unary 5‐phase (2‐level) multisystem as an example.…”

“…If every univariant curve in a net terminates at two invariant points (namely doubly terminating), the net is a completely closed net (Zen, 1966a, p. 403). A closed net has such essential features: there is at least one metastable invariant point in the net (Zharikov, 1961), and the stable part of any univariant curve lies between two adjacent stable invariant points and includes the two points (Guo, 1980a, 1981b). A real phase diagram of a multisystem always has some univariant curves terminating at only one stable invariant point, so it is never entirely closed, but partially or completely open.…”

“…In addition, Guo & Wang (1982) put forward a combination principle for closed nets. It was proved by Cai (1982), and re‐stated by Guo (1984) that any closed net of n + k ( k > 3)‐phase multisystem must be a combination of two or more distinct n + 3 order submultisystem closed nets belonging to the given n + k ‐phase multisystem, if it is not one of submultisystem. This principle suggests a method for the construction of closed nets: all the n + k ( k > 3)‐phase closed nets can be derived from n + 3‐phase closed nets in theory.…”

“…In general, none of the theories or methods in the literature is satisfactory for studying the closed nets of the complex multisystems of many components and phases. For example, the approach of overlapping stability fields can only apply to binary systems, but not to other multisystems (Guo & Cai, 1982; Guo, 1984). The representation polyhedron approach is also impractical for multisystems of many components and phases because of the great difficulty in the plotting, visualisation and theoretical analysis of the representation polyhedra.…”

A simple, universal theory and method for computer plotting of stable equilibrium phase diagrams of a multisystem—SFM method

“…To simplify the discrimination of the metastable curves, Yin [6] also gave a theoretical analysis of the correlation of the topological properties of the curves with the elements in VF, SF and the slopes of the curves. Yin and Han [7] put forward a method to judge the stabilities of invariant points in the light of Guo's theorem [8] with its application to p-T, p-X, T-X phase diagrams. This set of method was called sign function matrix method in refs.…”