On substitutional element partitioning coefficients of two-phase alloys
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Communications
C␣ ⫽
On Substitutional Element Partitioning Coefficients of Two-Phase Alloys
we have n ⫽
Two-phase solid solution alloys and superalloys are alloys with multicomponents. Because of the complexity of chemical interactions and thermodynamic factors of the constituents, statistical regressions or phase calculations[1–4] are usually needed in order to know the partitioning relations of alloy compositions to the corresponding phase compositions. In fact, the result is partly empirical. However, less is known about the theoretical source of partitioning ratios of substitutional alloying elements of two-phase alloys. Formulas on ideal partitioning coefficients are thus derived with the representation of atomic fraction. The accordance of the measured phase composition data from the published literature with the calculated ones by using the derived equations is verified. To determine the partition of the alloy composition to its phase compositions, let us first take an example of phase ␣ and phase  both with fcc structure and use a unit cell of the crystalline structure to review the concept of phase composition in the alloy system. The atomic concentration of a chemical element in a phase can be defined as the portion occupied by this element in a unit cell composed of four atoms in the phase. The ratio of the portions from the two phases is the concentration ratio of the two phases. In fact, from the concentration ratio, we still cannot see in which way an alloy partitions all its elements to the two phases and makes them reach their chemical concentrations. Therefore, a certain partitioning relationship exists between the alloy and its two phases; i.e., the alloy partitions all the elements to its two basic phases in a certain ratio. We use atomic fraction, atomic percent, as the calculation unit in the following discussion. If there are N atoms of element i per 100 atoms, the number of atoms of element i partitioned to phase  is N, and naturally that to phase ␣ is N␣ ⫽ N ⫺ N. The ratio, N /N␣, is the partitioning ratio of element i between the two phases, designated as Ri. Let n be the number of unit cells of phase  in 25 cells composed of 100 atoms in the fcc lattice. Then, the number of unit cells for matrix phase ␣ is n␣ ⫽ 25 ⫺ n. Also, let X be the number of atoms possessed by element i in a unit cell of phase , and C the atomic fraction of the alloy, as well as C and C␣ the atomic fractions of phases  and ␣, respectively. Then, we have N n X  ⫽ N␣ N ⫺ n X
[1]
n (4C) n C ⫽ ⫽ 100C ⫺ n (4C) 25C ⫺ n C Z.F. PENG, Professor, and Y.Y. REN, Associate Professor, are with the Department of Materials Engineering, College of Power and Mechanical Engineering, Wuhan University, Wuhan 430072, P.R. China. Manuscript submitted March 1, 2002. METALLURGICAL AND MATERIALS TRANSACTIONS A
[2]
100C ⫺ n X 25C ⫺ n C ⫽ ⫽ 4(25 ⫺ n) 25 ⫺ n
Z.F. PENG and Y.Y. REN
Ri ⫽
X␣ N␣ 100C ⫺ N ⫽ ⫽ 4 4(25 ⫺ n) 4(25 ⫺ n)
25(C ⫺ C␣) C ⫺ C␣
[3]
After substituting Eq. [3] into Eq. [1], we obtai
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