On the General Material Balance Equation(S) to Calculate Quasi-Binary Sections of Multi-Component Phase Diagrams

Abstract

Highlights• To calculate quasi-binary sections of multicomponent phase diagrams, special materials balance equations are needed• The general equations to calculate the numerical constants of those equations are derived here• For a k-component system, (k-2) such independent equations are needed. IntroductionTernary and multi-component phase diagrams are important tools in metallurgy and materials science [1][2][3][4][5][6]. They are usually calculated by a Calphad (Calculation of Phase Diagrams) algorithm [7][8]. Material balance equations are vital parts of this Calphad algorithm. The majority of materials balance equations are built within the commercial Calphad codes [9][10][11][12][13][14][15]. However, when quasi-binary sections of ternary or multi-component phase diagrams are of interest [1][2][3][4][5][6][16][17][18][19][20], the user of those codes should create the appropriate materials balance equations. This is usually done in the following format (as in Thermo-Calc):( 1) where I and J present two, arbitrary components in the multi-component system, w I and w J (dimensionless) are the mass fractions or mole fractions or volume fractions of the components, while the Greek letters denote numerical constants. The goal of this paper is to work out simple equations for numerical constants in Eq.(1), being valid for any quasibinary section of any multi-component system. Derivation of the general equationsLet us consider a multi-component system made of A-B-C-….-Z components (generally denoted as I or J). We are interested in one of its quasi-binary sections, having a horizontal concentration axis and a vertical temperature axis, with constant value of pressure. The quasi-binary section connects two alloys with each other: alloy 1 is denoted as A a1 B b1 C c1 …Z z1 , while alloy 2 is denoted as A a2 B b2 C c2 …Z z2 , where a1, b1, c1, …., z1 (generally i1, and j1) and a2, b2, c2,…., z2 (generally i2 and j2) are positive numbers defined between 0 and 1. They can be mass fractions or mole fractions or volume fractions of the components in two alloys. The materials balance requires:It is important to underline that wI and wJ in Eq.(1) and all parameters in Eq-s (2a-b) must have the same definition: they all should be mass fractions, or they all should be mole fractions or they all should be volume fractions.The x-axis of the quasi-binary diagram connects alloy 1 with alloy 2, with fraction x increasing from the value equal to 0 (for alloy 1) to the value equal to 1 (for alloy 2). Then, the component fractions for two arbitrary components I and J can be written as:Arch. Metall. Mater., Vol. 61 (2016) A general form of material balance equations to be used to calculate quasi-binary sections of multi-component phase diagrams is derived here. When this general equation is reduced to ternary systems, it coincides with those, given in the Thermo-Calc manual. For a k-component system, altogether only (k-2) such independent equations should be written from the list of k(k-1)/2 possible equations.