Modeling of primary spacing selection in dendrite arrays during directional solidification
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I. INTRODUCTION
PRIMARY spacing, , in dendrite arrays is the most important microstructure scale in directionally solidified alloys. The classical theoretical models for primary spacing, such as those proposed by Hunt,[1] Kurz and Fisher,[2] and Trivedi,[3] have been developed for dendrite growth in a steady state where dendrite arrangement is assumed to be uniform. In these models, only a single dendrite is investigated on the assumption that both the geometry and the growth condition of all dendrites are identical. Interdendritic spacing (spacing between adjacent dendrites) is then equated to the lateral size of the isolated dendrite, which is geometrically determined to be proportional to the square of the dendrite tip radius, R. Since both theoretical and experimental studies of R reveal a unique selection process, the corresponding dendrite spacing is then determined to be a unique function of solidification parameters. The preceding models[1,2,3] consider only smooth interface morphology and do not take into account the side branch characteristic of dendrite structure. In the model of Ma and Sahm,[4] primary spacing is taken to be the sum of the dendrite trunk and twice the sidearm lengths. It is only the dendrite trunk, not the entire dendrite, whose curvature radius at the tip is directly related to the dendrite tip radius, R. The sidearm length is calculated by analyzing the lateral branching of secondary arms from dendrite trunk. Since this model remains based on the steady state, it also predicts a unique primary spacing at given solidification conditions, as in other steady growth models.[1,2,3] In actual casting processes, however, the steady state is almost impossible to achieve. Therefore, the steady growth models for primary spacing cannot be applied directly. In DEXIN MA, Research Scientist, is with ACCESS Institute, RWTH, 52056 Aachen, Germany. Contact e-mail: [email protected] Manuscript submitted March 14, 2001.
METALLURGICAL AND MATERIALS TRANSACTIONS B
fact, the experimental results show no unique primary spacing at a given processing condition. Even directional solidification processes, in which a constant temperature, G, and solidification velocity, V, is maintained from beginning to end, reveal a very wide range in spacing size distribution.[5–9] The pioneer work of Huang et al.[6] experimentally identified history dependence in evolution and selection of primary dendrite spacings. It was found that a dendrite array, formed at a lower growth velocity, can remain stable when velocity increases up to tenfold. This delay (sluggish) effect of spacing variation is also observed when growth velocity decreases. Under a given growth condition, there exists a wide allowable range of primary spacing whose upper limit is more than three times the lower one. Even average spacing depends not only on the current growth conditions but also, remarkably, on the way those conditions are achieved.[6,9] Warren and Langer[10,11] carried out a stability analysis of a dendrite array during directio
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