Interdendritic Spacing: Part II. A Comparison of Theory and Experiment
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I.
INTRODUCTION
SINCE a theoretical analysis of the growth of a dendritic array is so complex, detailed models have been proposed only for the simple case in which the interaction between the neighboring dendrites is negligible. ~,2 Although these models have given us valuable information on the characteristics of dendrite tip, they do not enable us to understand the variation in primary dendrite spacing as a function of growth rate, temperature gradient, and alloy composition. Even though a number of empirical relationships have been proposed in the literature3'4'5 to explain experimental observations on primary dendrite spacings, these relationships are based on experimental data obtained for a very limited range of experimental variables. Consequently, none of these models properly predicts the variation in primary dendrite spacing in different systems over a wide range of experimental conditions. Recently, Hunt6 and Kurz and Fisher7 have proposed theoretical models to predict primary dendrite spacings. These models are examined in this paper to check if they explain the experimental observations reported in Part I. 8 Specifically, we determine whether these models adequately explain the following basic features of primary dendrite spacings observed experimentally:8'9'~~(1) the primary dendrite spacing goes through a maximum as the growth rate is increased, (2) at high growth rates, the variation in primary dendrite spacing with growth rate becomes linear on a log-log plot, and (3) in the high growth rate regime, the temperature gradient affects the primary dendrite spacing significantly, whereas it has no effect on the dendrite tip radius and on the initial secondary dendrite arm spacing. We shall show that current theories do not adequately explain the growth rate dependence of primary dendrite spacing. A simplified theory will be developed which satisfactorily explains important features of the variation of primary dendrite spacing with growth rate.
II.
THEORETICAL MODELS
Two detailed theoretical models have been proposed in the literature6'7 to characterize primary dendrite spacing as a function of growth rate, temperature gradient, and alloy composition. The first significant treatment was presented by Hunt. 6 His model was based on two major assumptions: (1) a dendritic interface with sidearms is approximated as a smooth steady-state interface, and (2) a constant temperature and a constant liquid composition in the direction normal to the primary dendritic growth direction are assumed. Under these assumptions, Hunt derived the shape of the cells by following the procedure developed earlier by Bower, Brody, and Flemings. x~ Since the second assumption of the Hunt model is valid only for the dendrite or cell region which is far behind the tip, the interface shape is not valid near the tip region. Hunt circumvented this problem by fitting part of a sphere to the derived shape at the growing front. Under these assumptions, he obtained the following relationships between the tip radius, p, and the primary
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