Study of Soft Impingement of Diffusion Fields and Its Effect on Cast Microstructure
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THE normal microstructure during solidification of alloys is dendrites. But a curious microstructure develops during Zr addition to Mg alloys—spheroidal grain morphology. Spheroidal grain morphology of cast structure is very predominant during grain refinement of Mg alloys by Zr addition. The absence of dendrites leading to spheroidal grains is of immense scientific interest in the solidification literature. The spheroidal structure reduces porosity and increases strength and ductility of the alloy. Hence, such a microstructure is highly desired in alloys. Kashyap et al.[1] addressed this problem by studying and modeling the soft impingement of diffusion fields and applying constitutional supercooling theory in the two-interface problem. It was shown that as the distance between the two interfaces decreases, there is an absence of constitutional undercooling leading to spheroidal grains. Kattamis et al.[2] showed that spherical grain structure and segregated pattern was produced in Mg alloys where grain size was reduced to that of the dendrite arm spacing. In generalizing a suggestion by Vogel[3] made in context of stir casting, it was pointed out by Doherty[4] that when there is a high density of nuclei growing with full diffusion control, then a stable nondendritic growth form may be expected when the overlapping diffusion fields from adjacent growing crystals interact to decrease the concentration gradient that causes shape instability. Flemings[5] showed that a spheroidal grain morphology occurs during the grain refinement of Mg alloys by Zr addition. The objective of this article is to explain the effect of soft impingement of diffusion fields and its effect on cast microstructure taking into account perturbations at the
solid–liquid interface. The calculations are shown for Al-7Si alloy and can be extended to any other alloy.
II.
MATHEMATICAL MODELING
In 1998, Kurz and Fisher[6] proposed a solution for diffusion field of a moving interface in two-dimensional (2-D) steady-state conduction. By Fick’s law for 2-D diffusion (refer to Table I for definitions of all symbols used) @ 2 C @ 2 C V @C ¼0 þ 2 þ @y2 @z D @z
½1
@ 2 Tl @ 2 Tl V @Tl ¼0 þ 2 þ al @z @y2 @z
½2
@ 2 Ts @ 2 Ts V @Ts ¼0 þ þ as @z @y2 @z2
½3
According to the perturbation theory, DGc Vz C ¼ Co exp þ A 2 sinxyexpðbc zÞ D V Tl ¼ To þ
½4
Gl al Vz 1 exp þ B 2 sinxyexpðbl zÞ al V ½5
Ts ¼ To þ
Gs as Vz 1 exp þ R 2 sinxyexpðbs zÞ as V ½6
K.T. KASHYAP, Professor, and K.B. PUNEETH, Engineer, are with the Department of Mechanical Engineering, PES Institute of Technology, Bengaluru 560085, India. Contact e-mail: ktkashyap@ yahoo.com Manuscript submitted October 30, 2012. Article published online October 12, 2013. METALLURGICAL AND MATERIALS TRANSACTIONS B
Here, V þ bc ¼ 2D
ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 V þx2 2D VOLUME 45B, FEBRUARY 2014—161
Table I. C Tl Ts V D al as Co Gc e x D T0 T+ C+ m Ks Kl
List of Symbols Used and Their Values for Al-7Si Alloy
Concentration Melt temperature (K) Solid temperature (K) Interfac
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