On the arrangement of monovariant lines in two-dimensional potential phase diagrams
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On the Arrangement of Monovariant Lines in Two-Dimensional Potential Phase Diagrams TI,
MATS HILLERT T, P diagrams of phase equilibria are fairly simple even if the number of components is high. They are therefore a very useful tool for displaying phase relations. In principle, a T, P diagram is a projection of the complete phase diagram drawn with potential axes, T, P,/xB, lXc, etc. It is obtained by projecting in the directions of I~B, tXc, etc. It shows invariant equilibria as points and monovariant equilibria as lines radiating from the points. The number of lines radiating from a point is equal to the number of phases in the invariant equilibrium, c + 2 according to the Gibbs phase rule. In addition, one should realize that divariant equilibria are represented by surfaces, each one extending between two lines. Schreinemakers ~ showed that the arrangement of lines around a point is related to the compositions of the phases. As a consequence, from information on the T, P diagram one may draw conclusions regarding the compositions of the phases or from information on the compositions one may draw conclusions regarding the T, P diagram. Schreinemakers proved his theorem in an extremely simple way by considering the positions of the divariant surfaces. Morey and Williamson2 later produced a general mathematical proof, and the theorem is sometimes called the Morey-Williamson theorem.3 It seems to have been applied to ternary systems more often than to binary systems. For our present purpose it is sufficient to describe the result for a binary system. The four lines radiating from an invariant point can only be arranged in principle in one way. See the upper part of Figure 1. Each line is identified by giving in parentheses the omitted phase rather than the three phases present. The metastable extensions of the lines are given by dashed lines, and we can see that two extensions fall between two of the lines, here denoted by (/3) and (y). The theorem says that the two omitted phases (in this case /3 and y) are situated between the other two when plotted on the composition axis. The compositions of the other two phases are such that each one is closest in composition to the phase given by the opposite line. The (/3) line is opposite to the (a) line, and a is thus situated outside /3 on the composition axis. The arrangement is shown in the lower part of Figure 1. It should be emphasized that the theorem does not distinguish between mirror images. Either one or both of the parts of Figure 1 may thus be exchanged for its mirror image. When two phases have the same composition, their lines will coincide. Two such cases are illustrated in Figures 2 and 3. Finally, when three phases have the same composition, one line will degenerate to a point, coinciding with the point representing the invariant equilibrium (Figure 4). MATS HILLERT is Professor of Physical Metallurgy, Royal Institute of Technology, S-100 44 Stockholm, Sweden. Manuscript submitted April 30, 1984. METALLURGICAL TRANSACTIONS A
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