Densities of Pb-Sn alloys during solidification
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I.
INTRODUCTION
LEAD-tin alloys are used extensively for soldering in the microelectronics, construction, and manufacturing industries. In the metallurgical research community, Pb-Sn alloys have been used for solidification studies because of experimental convenience (low melting point) and availability of physical and thermodynamic data. However, even for this relatively simple alloy system, data (including densities) are not given in a form convenient to apply directly to quantitative analyses of dendritic solidification processes. Furthermore, it is often necessary to extrapolate to the solidification temperature range, wherein the solid and liquid phases coexist. This paper utilizes available data for the density of Pb, Sn, and Pb-Sn alloys to develop equations that can be used to calculate either the density of the solid or of the liquid as a function of temperature and its composition. It is important to calculate the density because convective phenomena can play dominant roles in structure development and macrosegregation during solidification, and, in order to quantitatively model the convection, the densities of the dendritic solid, of the interdendritic liquid, and of the "bulk" liquid must be ascribed.
II. SOLUTE PARTITIONING DURING SOLIDIFICATION Either the density of the solid or the liquid depends on temperature and its composition. During dendritic solidification the compositions are calculated by assuming that "Scheil-type" solidification adequately describes the partitioning of solute between the solid and liquid, m In its differential form the solute balance is
dCL
(CL-CD
dfs -
-
-
(1-f~)
[11
where CL = local concentration of solute in the interdendritic liquid, wt pct Sn, C* = local concentration of solute in the dendritic solid at the solid/liquid interface, wt pct Sn, and fs = weight fraction of solid.
D.R. POIRIERis Professor,Departmentof Materials Scienceand Engineering, The Universityof Arizona, Tucson, AZ 85721. Manuscript submittedJuly 22, 1987. METALLURGICALTRANSACTIONSA
The equilibrium partition ratio, k, is defined as C * / Q , and Eq. [1] is often integrated by assuming that k is constant. However, according to the Pb-Sn phase diagram t21for hypoeutectic alloys (Pb-side of the eutectic), k varies substantially. It can be closely approximated with the following polynomial of order n: n
k = ~', a, C~
[21
I=0
where 0 < C, < 61.9 wt pct Sn. The coefficients presented in Table I give a standard error of the regression fit of 1.38 • 10-3. The liquidus for the hypoeutectic alloys is also closely approximated by a polynomial; this is n
r = ~ b, Ck
[3]
t=0 with T in ~ These coefficients are also given in Table I, and they give a standard error of the regression fit of less than 0.5 ~ For the hypereutectic alloys, the equilibrium partition ratio and the liquidus can also be approximated by Eqs. [2] and [3] with 0 < Cc < 38.1 wt pct Pb. Using the coefficients given in Table II, the standard error of the regression fit for k is 8.5 • 10-4; for the liquidus-temperature it is les
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