Simulation of paraequilibrium growth in multicomponent systems
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I. INTRODUCTION
THE kinetic theories of diffusional phase transformations in alloys containing both substitutional and interstitial elements are well developed.[1–7] An important feature of various kinetic models is the assumption of local equilibrium of local equilibrium at the interface. Depending on the interface velocity during transformation, it is convenient to classify the kinetics into two distinct modes, as follows. (1) Partitioning local equilibrium is characterized by a low interface velocity while maintaining local equilibrium at the interface. This condition is also referred to as orthoequilibrium (OE). Generally, OE occurs at low supersaturation, and its kinetics is governed by the slow-diffusing species (substitutional elements). For example, the thermodynamic condition for OE between ferrite (␣) and cementite ( ) in ultrahigh-strength (UHS) steels is given by i␣ ⫽ i
[1]
where i is the chemical potential of element i (representing C, Co, Cr, Fe, Ni, Mo, V, and W). (2) Paraequilibrium (PE) is a kinetically constrained equilibrium, in which the diffusivity of the substitutional species is negligible compared to that of interstitial species. Hultgren argued that if carbon diffuses appreciably faster than the substitutional alloying elements, then the growing phase inherits the substitutional alloy contents. Furthermore, if the substitutional alloying elements are not allowed to partition,
G. GHOSH, Research Assistant Professor, and G.B. OLSON, WilsonCook Professor of Engineering Design, are with the Department of Materials Science and Engineering, Robert R. McCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL 60208-3108. Manuscript submitted August 3, 2000. METALLURGICAL AND MATERIALS TRANSACTIONS A
their individual chemical potentials have no physical relevance and, thus, the thermodynamic behavior of these elements can be expressed by one hypothetical element, Z. Then, PE is defined by a uniform carbon potential and a uniform site fraction of substitutional elements across the transforming interface. For example, in the case of the ␣ / transformation, the thermodynamic conditions for PE are given by
C␣ ⫽ C
[2a]
yj␣ ⫽ yj
[2b]
Z␣ (⬅兺 yj j␣) ⫽ Z (⬅兺 yj j )
[2c]
where the yj terms are the site fractions of substitutional element j (representing Co, Cr, Fe, Ni, Mo, V, and W). For a system containing both substitutional (j) and interstitial elements (C or N), the site fractions are related to the ordinary mole fractions (x) as follows. yj ⫽
xj 1 ⫺ xC ⫺ xN
yC or N ⫽
xC or N p q 1 ⫺ xC ⫺ xN
[3a]
[3b]
According to the two-sublattice model[8] used here to express the Gibbs energies, p ⫽ 1 and q ⫽ 3 for ferrite, and p ⫽ q ⫽ 1 for austenite. The schematic concentration profiles across the transforming interface for the aforementioned two distinct modes are shown in Figure 1. The PE growth mode can also be conceived as the complete solute trapping[9] in the substitutional sublattice. The central idea behind solute trapping is that when the int
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