Scaling laws for two-dimensional dendritic crystal growth in a narrow channel

TL;DR

This paper combines analytical and computational methods to study 2D needle crystal growth in narrow channels, revealing how growth velocity scales with time and channel width, with implications for understanding dendritic growth dynamics.

Contribution

It provides a new analytical prediction for growth velocity decay in narrow channels and validates it through phase-field and dendritic-needle-network simulations.

Findings

Growth velocity decreases as t^{-2/3} in low supersaturation.

Needle crystals grow with constant velocity above a critical channel width.

Velocity approaches free-growth velocity as channel width increases.

Abstract

We investigate analytically and computationally the dynamics of 2D needle crystal growth from the melt in a narrow channel. Our analytical theory predicts that, in the low supersaturation limit, the growth velocity $V$ decreases in time $t$ as a power law $V \sim t^{-2/3}$, which we validate by phase-field and dendritic-needle-network simulations. Simulations further reveal that, above a critical channel width $\Lambda \approx 5l_D$, where $l_D$ the diffusion length, needle crystals grow with a constant $V<V_s$, where $V_s$ is the free-growth needle crystal velocity, and approaches $V_s$ in the limit $\Lambda\gg l_D$.