Scaling laws for two-dimensional dendritic crystal growth in a narrow channel
Here, we investigate analytically and computationally the dynamics of two-dimensional needle crystal growth from the melt in a narrow channel. Our analytical theory predicts that, in the low supersaturation limit, the growth velocity $\textit{V}$ decreases in time $\textit{t}$ as a power law $V ~t^{–2/3}$, which we validate by phase-field and dendritic-needle-network simulations. Simulations further reveal that, above a critical channel width $Λ ≈ 5l_D$, where $l_D$ is 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 $Λ \gg l_D$.