Dendrite Tip-Shape Characteristics
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ABSTRACT The assumption that dendrite tips are parabolic bodies of revolution pervades many of the theories and experiments addressingdendritic growth. This assumption, while reasonable, is known to become less valid as regions of interest further from the tip of the dendrite are considered. Experimental measurements were made on pure succinonitrile dendrites at several supercoolings. The equation that describes the dendrite tip profile is extended from a second orderpolynomial (paraboloidal) form to one that includes higher-orderterms. The deviation of a dendrite tip from a parabolic body of revolution can be characterizedby a parameter obtained from the coefficient of the fourth-order term describing the profile. This dimensionless parameter, Q, is found to be a function of the profile orientation only, independent of supercooling.
INTRODUCTION AND BACKGROUND Dendritic Solidification Dendritic microstructures are commonly encountered during the solidification of many engineering materials. It is important to understand the process by which dendrites form because traces of these geometrically complex microstructures persist through subsequent material processing stages and can affect the properties of the finished product. The diffusion controlled process of dendritic solidification is commonly observed during casting and welding. In pure materials, the latent heat released at the solid/melt interface diffuses into the melt ahead of the growing dendrite. A mathematical solution to this heat flow problem was first developed by Ivantsov [1], and may be expressed as St = Pe'Ie- du,(1
where St - AT/(L / C,) defines the Stefan number (dimensionless supercooling), Pe = VR/2a defines the growth P~clet number, AT is the supercooling, L is the molar latent heat, and Cp is the molar specific heat under constant pressure. V is the steady-state dendrite tip velocity, a is the thermal diffusivity of the liquid phase, and R is a length scale, taken here to be the radius of curvature at the dendrite tip, approximated as a paraboloid.
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Mat. Res. Soc. Symp. Proc. Vol. 398 ©1996 Materials Research Society
In addition to the fundamental transport solution of Ivantsov, various theories (see the review by Glicksman and Marsh [2]), suggest a second useful relationship having the form •*=VR 2 -2cd°
(2)
'
where a* is usually referred to as the stability, selection, or scaling constant and do is the capillary length scale (a materials constant). Equation (2) can be re-arranged as VR2 = constant.
(3)
When Ivantsov's diffusion solution, Eq. (1), is combined with the selection rule expressed in Eqs. (2) and (3), it then becomes possible to define the operating state of the dendritic growth process in terms of the tip velocity, V, a length scale, R, and a single process parameter, AT. Recently, Glicksman et al. have shown [3,4] that both the Ivantsov solution, Eq. (1), and the selection rule, Eq. (3), require some modification. Several limiting assumptions were made in the original Ivantsov analysis, including the condition that
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