Metastable phase equilibria in the lead-tin alloy system: Part II. Thermodynamic modeling
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I.
INTRODUCTION
D U R I N G rapid solidification, the nucleation and/or growth of the thermodynamically stable phases may be difficult. Metastable phases might be favored kinetically; therefore, a knowledge of metastable phase diagrams becomes necessary for the description of solidification paths. Due to lack of suitable experimental techniques, the number of accurate metastable phase diagrams is very limited. In the Pb-Sn alloy system, it was possible for the first time to obtain experimentally the metastable liquidus, solidus, and additional information on metastable crystalline phases over a relatively broad temperature range (80 K) by applying the droplet emulsification technique, m Thermodynamic modeling can be used to check for internal consistency between the thermodynamic and phase diagram data. Furthermore, when appropriate thermodynamic models for all the phases become available, they may be used to calculate many of the possible metastable phase equilibria. These equilibria may often become important for controlling the formation of desired microstructures by rapid solidification or solid-state heat treatment. The objectives of this part of the study are (1) to reassess the lattice stability of Sn in view of the newer pressure-temperature data; (2) to reassess the solution behavior of the various phases using the metastable phase boundaries of a-Pb/a-Pb + L obtained in Part I; and (3) to calculate several possible metastable phase diagrams using the newly obtained models for the various phases in the Pb-Sn binary alloy system.
long to this type. The Margules type of equations used previously tS'6j are used to describe the excess Gibbs energies of the various solution phases in the Pb-Sn system. To specify the thermodynamic expressions, the following subscripts and superscripts are used: Subscript 1: the Pb component Subscript 2: the Sn component Superscript L: the liquid phase Superscript a: the fcc (A1) phase Superscript/3: the bct (A5) phase (4 atoms per unit cell) Superscript 6: the diamond-cubic (A4) phase Superscript Oil: the bct phase (2 atoms per unit cell) A.
The Liquid Phase
The following expressions for the excess Gibbs energy and partial excess Gibbs energy are used for the liquid phase:[5.6t A~SG L = RT{(O.5)XlX2[(w~2 +
+ (Wl~2- w ~ (x2 - Xl)]}
THERMODYNAMIC
A G1 = R T l n y L RT{(O.N)x~[(wL12
q-
H.J. FECHT is with the Keck Laboratory, California Institute of Technology, Pasadena, CA 91125. M.-X. ZHANG, Research Assistant, Y.A. CHANG, Wisconsin Distinguished Professor and Chairman, and J.H. PEREPEZKO, Professor, are with the Department of Metallurgical and Mineral Engineering, University of WisconsinMadison, Madison, WI 53706. Manuscript submitted September 18, 1987. METALLURGICAL TRANSACTIONS A
w2L1)
+ (wE -- wL~)(1 -- 4X01}
[21
xs -- L
A G2 = R T l n T L
MODELS
Metallic solutions may be separated into two different types of behaviors: those deviating slightly from a regular solution behavior and those deviating slightly from a highly ordered s t a t e . [2'3'41 In the first
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