On unconstrained solidification of spherical metallic drops

TL;DR

This work addresses how curvature and confinement alter solidification of metallic droplets by formulating two idealized growth modes, radial outward (RO) and circumferential (CG), and extending Mullins--Sekerka stability to finite curved domains using a Stefan-type, quasi-steady framework. It derives nucleation criteria, growth kinetics, and stability conditions for RO and CG, revealing a cross-over in dominance governed by undercooling $\Delta$ and droplet size $a$, and predicts a microstructural length scale $\lambda$ arising from the fastest-growing mode. The study connects theory to abrasion-based experiments on Fe particles, explaining hollow morphologies, dendritic and cellular patterns, and coexisting fronts, thereby offering a quantitative, curvature-aware lens on powder production and natural solidification in curved geometries. The findings suggest curvature fundamentally reshapes solidification; the resulting framework enables tailoring of particle morphologies through controlled processing, with practical implications for porosity, surface structure, and powder flowability.

Abstract

The solidification of metallic droplets into powder particles involves a complex interplay between heat diffusion, surface tension, and geometric constraints. In confined, curved systems -- such as those encountered in atomisation, abrasion, and micrometeorite formation -- positive curvature and finite boundaries significantly modify classical solidification dynamics. In this study, we systematically investigate the solidification of metallic spheres, focusing on how curvature and confinement influence nucleation pathways, growth kinetics, and interfacial stability. Two competing growth modes -- radial outward and circumferential -- are analysed using Stefan-type models under a quasi-steady approximation. A generalisation of Mullins--Sekerka stability theory is developed to account for finite spherical domains, revealing that particle size and curvature introduce new destabilising parameters that govern microstructural length scales. Experimental observations of dendritic and cellular morphologies are interpreted through this framework, demonstrating that the interaction between growth fronts, undercooling, and curvature collectively determines the final particle structure. These findings underscore the need to re-evaluate classical solidification theories in the context of curved geometries, with implications for both engineered and naturally occurring metal powders.

Figures (15)

• Figure 1: Occurrences of perfectly spherical particles in nature and the characteristic patterns observed on their surfaces. (a) Original illustrations by Robert Hooke showing magnified views of spherical particles produced by striking steel against flint, revealing early observations of solidification morphologies hooke2007micrographia. (b) Micrometeorite sample exhibiting a near-perfect spherical geometry with dendritic surface features formed by rapid solidification during atmospheric entry tomkins2016ancient. (c) Spherical metallic particles generated via mechanical abrasion, displaying surface patterns indicative of solidification dynamics under curvature and confinement singh2023fiery.
• Figure 2: Comparisons of solidified metallic particles produced via (a) plasma atomization chen2018comparative and (b) abrasion. In the latter, stringy chips (fully or partially un-melted) are also visible alongside spherical particles.
• Figure 3: Scanning electron micrographs of particles obtained by abrasion: (a) Size distribution of particles; (b) Dendritic morphologies observed on spherical particles of various sizes; (c) A hollow particle with a thin outer shell; (d) Distinct cellular morphology on the particle surface.
• Figure 4: (a) Schematic showing a nucleus of radius $r$ growing radially outward inside a sphere of radius $a$. Panel (b) shows a nucleus of geodesic radius $r_g$ on the surface of a sphere of radius $a$. Coordinates of any point on the surface are specified by ($a,\theta,\phi$).
• Figure 5: Variation of dimensionless critical radius of nucleation for the CG mode with undercooling $\Delta$ for various values of sphere radius $a$. Inset shows a magnified view of the critical radius at low undercooling $\Delta$.