The influence of cylindrical inclusions on the stability of a directionally solidified interface
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12/29/04
10:39 PM
Page 2181
The Influence of Cylindrical Inclusions on the Stability of a Directionally Solidified Interface
handed coordinate system relative to which the solid-liquid interface is at rest at Z 0, i.e., the X-axis. The melt occupies the top region, Z 0 as shown schematically in Figure 1. The equation of the cylindrical inclusion, whose center is at the point (0, Z0), is given by
LAYACHI HADJI It is well known that the presence of an inclusion in the melt near a solidifying front induces a local deformation in the latter provided that the melt’s thermal conductance differs from that of the inclusion.[1–4] This local interfacial deflection is caused by the modification of the thermal gradient in the melt near the particle. The long time evolution of this local disturbance of the interface has been investigated for the case of a spherical particle in a pure substance[5] and in a binary alloy.[6] It is discovered that, provided that the particle-interface distance falls below a critical value, the induced perturbation grows and destabilizes the entire solidliquid interface. This newly uncovered morphological instability is manifested only for some combination of the physical and processing parameters, and its onset is attributed to the reversal of the thermal gradient in the liquid gap between the particle and the interface.[6] This instability, whose characteristic size is of the order of the particle’s diameter, occurs at pulling speeds that are below the threshold for the onset of the Mullins–Sekerka instability. In reality, however, the inclusions have approximately cylindrical shapes. For example, in their study involving a particulate metal matrix composite (PMMC) of magnesium base alloy that is reinforced with SiC particles, Essa et al.[7] state that the inclusions resemble cylinders whose longitudinal sections can be approximated by rectangles of sides a and b (a b). The purpose of this article is to extend the stability analysis that was carried out for a spherical inclusion in Reference 5 to the more realistic case of a cylindrical inclusion. We let the inclusion have the more general shape of an elliptic cylinder; i.e., the longitudinal section is an ellipse of semiaxes a and b, as illustrated in Figure 1. The effect of the aspect ratio c b/a, which emerges in the analysis as an important factor, will be examined. Consider a system in which an inclusion is immersed in a bath of pure melt that is undergoing directional solidification. We ignore the gravitational effects and assume that the inclusion is both solid, i.e., nondeformable, and insoluble in the melt. Let the inclusion’s position be such that the distance from its center to the planar solid-liquid interface is Z0, and its axis of symmetry is parallel to the Y-axis. The problem, being invariant with respect to the Y direction, is thus two-dimensional. The physical process is governed solely by heat diffusion in the melt, solid, and inclusion with appropriate conditions at the interface, at the inclusion’s surface, and at
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