Models for the isothermal coarsening of secondary dendrite arms in multicomponent alloys
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I.
INTRODUCTION
THE mechanical properties of a casting are largely dependent on secondary dendrite arm spacing,[1] since it determines the solute segregation, the final distribution of foreign phases, and porosity. The finer the secondary dendrite arm spacing, the higher the mechanical properties of a casting. A large number of research has been concentrated on dendrite secondary spacing.[2–6] It has been found that the secondary arm spacing changes as the dendritic structure freezes. Parts of the structure melt as other parts freeze. The melting occurs for surface energy reasons. The process is similar to Oswald ripening. Four physical models proposed for the isothermal coarsening of the secondary arms are shown in Figure 1. Figure 1(a) shows a radial remelt model (model I), Figure 1(b) a neck remelt model (model II), Figure 1(c) an axial remelt model (model III), and Figure 1(d) a coalescence model (model IV). Based on these physical models, theoretical models for the binary alloy system have been developed.[2,3,4] These models have been widely used in predicting the relationship between the secondary dendrite arm spacing and the solidification parameters in binary alloys. However, most of the alloys of industrial importance are multicomponent alloys. Difficulties arise in correlating the concentration of each alloying element to the secondary dendrite arm spacing in a multicomponent alloy. Few attempts have been made toward developing a model for multicomponent alloys.[7–10] The aim of this article is to expand the binary models into multicomponent ones. II.
MATHEMATICAL MODEL
We start from model I shown in Figure 1(a). For model I, it is suggested[2] that during dendritic growth, there is a difference in solute concentration between two secondary dendrite arms of different radius. The concentration is lower at the interface of a small arm than at that of a larger one. During isothermal coarsening, solute diffuses from the larger arm to the smaller one, resulting in the remelting of the smaller arm and the coarsening of the larger one.
QINGYOU HAN, Research Fellow, is with the Department of Materials, University of Oxford, Oxford OX1 3PH, United Kingdom. HANQI HU and XUEYOU ZHONG, Professors, are with the Department of Metallurgy, University of Science and Technology Beijing, Beijing 100083, People’s Republic of China. Manuscript submitted November 12, 1996. METALLURGICAL AND MATERIALS TRANSACTIONS B
Similar to Reference 2, the assumptions are as follows: (a) the partition coefficient, k, of each solute element is a constant; (b) the distribution of the solute element between two arms is linear (straight line); (c) the radius of the larger arm, a, remains constant during isothermal coarsening; and (d) the latent heat, L, and the interface free energy, s, are constant. As illustrated in Figure 1(a), during isothermal coarsening, solute diffuses from a larger arm to a smaller one, and as a result, the radius, r, of the small arm changes from its original value r0, to 0. The flux of the diffusion of solute el
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