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Analytic Solution for a Non-Axisymmetric Isothermal Dendrite



Geoffrey B. McFadden, Sam R. Coriell, R F. Sekerka


The Ivantsov solution for an isothermal paraboloidal dendrite growing into a pure, undercooled melt provides a relation between the bulk undercooling and a dimensionless product, the Peclet number, of the growth velocity and tip radius of the dendrite. Horvay and Cahn generalized the axisymmetric solution to one corresponding to non-axisymmetric dendrites with elliptical cross-section, and found an analytical solution for this case as well. They found that as the deviation of the dendrite cross-section from a circle increases, the two-fold symmetry of the interface shape causes a systematic deviation from the undercooling/Peclet number relation of the Ivantsov solution. Analytic solutions are not known for isothermal dendrites with four-fold symmetry that might be used to model dendritic growth in cubic materials. However, we find approximate solutions for four-fold perturbations to the interface that are valid through second order in the perturbation amplitude, and compute self-consistent corrections through this order to the undercooling/Peclet number relation of the Ivantsov solution. For a Peclet number of 0.004, we calculate a correction corresponding to a 9 % increase in the undercooling. This result is in general agreement with the observed deviations in the undercooling/Peclet number relation that have been measured by Glicksman and colleagues for growth of succinonitrile dendrites in microgravity.
- 6308
Report Number


dendrites, dendritic growth, Ivantsov relation, non-axisymmetric dendrites, supercooled liquids


McFadden, G. , Coriell, S. and Sekerka, R. (1999), Analytic Solution for a Non-Axisymmetric Isothermal Dendrite, - 6308, National Institute of Standards and Technology, Gaithersburg, MD, [online], (Accessed April 16, 2024)
Created March 1, 1999, Updated November 10, 2018