An official website of the United States government

Official websites use .gov
A .gov website belongs to an official government organization in the United States.

Secure .gov websites use HTTPS
A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.

Algebraic Representation for the Topology of Multicomponent Phase Diagrams

Published

D Orser

Abstract

A new non-graphical method for representing the topology of phase diagrams in presented. The method exploits the fact that the topological relations between the variously dimensioned equilibria making up the structure of a phase diagram may be treated as a special type of algebraic structure, called an incidence lattice. Corresponding to each topologically distinct phase diagram there is a finite incidence lattice whose elements correspond to the invariant (vertices), monovariant (edges), bivariant (surfaces), etc., transition equilibria of the diagram, and whose operations correspond to moving between these topological elements in a systematic way. Further, we have discovered a method of modeling a given incidence lattice by a family of sets. In this incidence calculus, as we call such a family of sets, the two operations for the incidence lattice are modeled by set intersections. This defines a calculus of phase diagram equilibria specific to that diagram and provides an efficient method for a computer to store and subsequently retrieve the topological relationships between an equilibrium and the rest of the diagram or between any two equilibria. A remarkable mathematical fact is that the incidence calculus may be generated from certain subsets of itself. This is reflective of the fact that knowing a sufficient portion of the topology uniquely determines the remainder. An algorithm exploiting this fact, based on knowing just which n-dimensional phase fields of an n-ary phase diagram are incident on each vertex (point with zero degrees of freedom), is described. Hence, the higher the dimensionality of the diagram, the higher the return from the algorithm. The application of the incidence calculus to a multicomponent data base and its potential for qualitative thermodynamic modeling are discussed.
Citation
Chapter in "Computer Modeling of Phase Diagrams", Larry Bennett, Editor, ASM/TMS, Summer, 1986

Topology

Citation

Orser, D. (1986), Algebraic Representation for the Topology of Multicomponent Phase Diagrams, Chapter in "Computer Modeling of Phase Diagrams", Larry Bennett, Editor, ASM/TMS, Summer, 1986, [online], https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=820251 (Accessed June 14, 2024)