## Web Content Display # The Application Of Topology To Phase Diagrams

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One of the few papers on thee metallurgical applications of topology to come from the English-speaking countries is that by Smith'! Although dealing mainly with the shape of grains, Smith includes a brief reference to the topology of phase diagrams. The majority of the literature relating to the topological study of phase diagrams is Russian, and little known outside the U.S.S.R. The basis of the topological method is the division of a given shape into its simplest elements-the simplexes. A surface can be divided into a network of triangles which are simplexes by definition. The given shape is called a complex. By complex is understood a combination of geometrical elements-points, lines, surfaces, volumes, hypervolumes- which together form the given geometric shape. Phase diagrams can be represented as a closed complex of geometrical elements. To produce the phase diagram it is necessary to construct initially the composition diagram and for this purpose use is made of the geometrical diagram, the simplest of which is the co-ordinate simplex. The basic co-ordinates of the composition diagram are the simplest combination of points, lines, surfaces, &c., that will produce the diagram. The composition diagram of a one-component system is a zero dimensional simplex S(l). It consists of a point only (Fig. 1). The composition diagram of a binary system is represented by a linear segment between two points A and B. It is a one-dimensional simplex S(2) and contains two points (a0 = 2) and one edge (a1 = 1). The composition diagram of a ternary system (a two-dimensional simplex S(3)-a triangle) has the following geometric elements: a0 = 3, a1 =3, a2 = 1. A quaternary system has a composition diagram represented by a three-dimensional simplex S(4)-a tetrahedron-with geometric. elements: a0 = 4, a1 = 6, a2 = 4, a3 = 1. A quinary system is represented by the pentatope-a figure having five vertices in fourdimensional space. A schematic model of a pentatope is illustrated in Fig. 1. A quinary system is therefore represented by a four-dimensional simplex S(5) with geometric elements: a0 = 5, a1 = 10, a2 = 10, a3 = 5, a4 = 1. In general, systems with (n + 1) components are represented by polytopes in n-dimensional space. Alternatively, one can say that the basic co-ordinates of the figure representing an n-component system are simplexes of n points in (n - 1) dimensional space. In addition to the co-ordinate simplex, which we have equated to the composition diagram, we also have phase complexes. A phase complex is a combination of geometrical elements defining the phase equilibria in the system. The combination of the co-ordinate simplex with other geometrical elements (e.g. the linear segments representing the temperature axis in a binary system) produces the co-ordinate framework. In alloy systems the co-ordinate framework is usually a composition/ temperature diagram. The co-ordinate framework, together with the phase complex, represents the complete phase diagram of the system. Most of the original work on co-ordinate simplexes and phase complexes was undertaken by Kurnakov. The original papers have recently been gathered together in a publication of Kurnakov's "Collected Works".2 Before considering Kurnakov's work a note of caution is necessary. Phase diagrams must be constructed so that they obey the Phase Rule. Smith notes that a diagram can be topologically correct, even though it violates the phase rule.
• From: International Materials Reviews, Volume 08, Number 30, 1963 (ASM-IOM-Maney)
• Published: 1963
• Pages: 64