Abstract
Dendritic patterns frequently arise when a crystal grows into its own undercooled melt. Latent heat released at the two-phase boundary is removed by some transport mechanism, and often the problem can be described by a simple diffusion model. Its analytic solution is based on a perturbation expansion about the case without capillary effects. The length scale of the pattern is determined by anisotropic surface tension, which provides the mechanism for stabilizing the dendrite. In the case of liquid crystals, diffusion can be anisotropic too. Growth is faster in the direction of less efficient heat transport (inverted growth). Any physical solution should include this feature. A simple spatial rescaling is used to reduce the bulk equation in 2D to the case of isotropic diffusion. Subsequently, an eigenvalue problem for the growth mode results from the interface conditions. The eigenvalue is calculated numerically and the selection problem of dendritic growth with anisotropic diffusion is solved. The length scale is predicted and a quantitative description of the inverted growth phenomenon is given. It is found that anisotropic diffusion cannot take the stabilizing role of anisotropic surface tension.
Highlights
Dendritic microstructures develop from an interface instability of a crystal growing into its melt
The length scale of the pattern is determined by anisotropic surface tension, which provides the mechanism for stabilizing the dendrite
It is found that anisotropic diffusion cannot take the stabilizing role of anisotropic surface tension
Summary
Dendritic microstructures develop from an interface instability of a crystal growing into its melt. The methodology leading to these results is referred to as “microscopic solvability theory” (MST) It is in good agreement with experiments for substances with weak capillary anisotropy such as ammonium bromide solutions [3], whereas for other materials such as pivalic acid the predictions appear to be less accurate [4, 5]. The phenomenon is somewhat counterintuitive, because diffusion removes the latent heat from the interface and enables steady-state dendritic growth, so one might a priori expect growth to proceed preferentially along directions of fast diffusion This is not the case, which may be understood from realizing that globally heat extraction is most efficient, if diffusion is fast perpendicular to the extended flanks of the crystal instead of being fast perpendicular to the smallish tip.
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