Dendritic shapes for various crystalline symmetries: theory, modeling and experiment

Abstract This study develops the Geometrically Morphological Theory (Toropova et al. in Acta Mater 296:121232, 2025) to describe the shapes of dendritic tips, stems, and envelopes for various tip functions. Analytical expressions derived from the theory demonstrate strong agreement with the experimental data for dendrites exhibiting four- and six-fold crystalline symmetries. The analysis highlights the distinct asymptotic behaviors of two shape functions, $$z_\textrm{TIP}(x)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>z</mml:mi> <mml:mtext>TIP</mml:mtext> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and $$z_\textrm{TIP}^\textrm{new}(x)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>z</mml:mi> <mml:mtext>TIP</mml:mtext> <mml:mtext>new</mml:mtext> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , characterizing the tip region.