Scaling behavior of three-dimensional dendrites

The scaling behavior of geometry parameters in three-dimensional dendritic growth is investigated through a detailed measurement of the morphology of pure succinonitrile dendrites grown on the first microgravity flight of the isothermal dendritic growth experiment M. E. Glicksman, M. B. Koss, and E. A. Winsa, Phys. Rev. Lett. 73, 573 (1993). Measurements are performed of the integral parameters of a sidebranching dendrite, such as the envelope shape, projection area, contour length, volume, surface area, and solid volume fraction. The evidence presented here reveals that unique scaling relations exist between these geometry parameters and the primary tip radius or speed in steady growth. These relations are valid far away from the tip, up to a normalized distance equal to about the inverse of the tip P\'eclet number. For the secondary arm envelope on the sidebranch plane, a self-similar scaling behavior given by Xₓ₈₏/R=0. 668 (Z/R) ^0. 859 is found, where Xₓ₈₏ is the envelope width (or the secondary dendrite tip position), Z is the distance away from the primary tip, and R is the primary tip radius. The normalized projection area F/R^2 and the normalized contour length U/R demonstrate an identical time dependence after some initial transient, which indicates that the interfacial length concentration U/F is time independent and inversely proportional to the tip radius R. The volume V and the surface area A of a dendrite can also be scaled to the primary tip radius R. It is noted that the interfacial area concentration A/V has a similar behavior and the same order value as U/F. The experimental results are compared to analytical predictions E. Brener and D. Temkin, Phys. Rev. E 51, 351 (1995) and generally found to be in good agreement. Finally, the internal solid volume fractions for various envelopes are deduced from the volume measurements and found to be in good agreement with a simple heat transfer model.