A Topologically Based Procedure to Identify Invariant Reactions for Isobaric Multicomponent Phase Diagrams
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I.
INTRODUCTION
PHASE diagrams are a graphic representation of phase stabilities under the influence of temperature, pressure, and molar fractions, which provide a powerful tool to tailor material microstructures within processing (e.g., solidification, crystal growth, and soldering) and to predict service performance for materials used at extreme conditions (e.g., reactive diffusion and oxidation). For isobaric c-component systems, the dimension of their phase diagrams is c. To reduce the dimension of phase diagrams, it is imperative to use the techniques of sections and projections to acquire two-dimensional plots.[1] In sectioned phase diagrams, invariant phase equilibria are represented as one-dimensional horizontal lines, indicating the presence of c + 1 phases. In projected phase diagrams, usually liquidus projections, invariant reactions are identified as junctions of c monovariant lines. Principally, invariant reactions have a distinct advantage in designing alloys with characteristic structures, and their topological characteristics have been well established.[2,3] CALPHAD has advanced to its maturity, with wellestablished thermodynamic models and sophisticated algorithm toward global minimization of Gibbs free energies. Now a days, CALPHAD-related databases and software packages have served as a vital tool in designing novel materials and in solving industrial enigma with minimum cost. However, when advancing into multicomponent systems, the software package used may not have the functionality to identify invariant reactions. In addition, new thermodynamic models may be introduced for specific studies, which is especially true for polymer systems in which other models than
those frequently used in metallurgy are demanded. It is thus necessary to design a simple and effective algorithm to identify invariant reactions. In our earlier work,[4] a relevant knowledge basis has been formulated from a fundamental point of view. It has been shown that judging invariant reactions from liquidus projections has its inherent duality, as one cannot locate the monovariant phase region in which all the phases are solids. Subsequently, a procedure that employed reaction coefficients to address this hurdle was presented. Although this technique is simple, one still needs to evaluate reaction coefficients using matrix algebra. There is thus a current demand for a simple procedure that can fully use the functionality of an existing computational package to identify invariant reactions directly, which is the aim of this work.
II.
FUNDAMENTAL BASIS
A. Invariant Reactions and Their Associated Phase Regions Consider one invariant reaction for a fictitious isobaric c-component system, where the numbers of reactant and product phases are r and c þ 1 r, respectively. The total number of phases for this invariant reaction is thus c + 1. Such an invariant reaction can be given by: p1 þ p2 þ þ pr $ prþ1 þ þ pc þ pcþ1
½1
where pi (i = 1 to c þ 1) denotes the ith phase. Alternatively, the invariant reaction can be described
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