Geometric closure of classical nucleation theory for magnetic-field-controlled nanoparticle size across magnetic classes
Controlling nanoparticle size during synthesis remains a central challenge in nanoscience, particularly in systems where external magnetic fields are used as continuous control parameters. Existing descriptions of magnetic-field-assisted nucleation are typically material-specific or rely on computationally intensive atomistic methods. Here, we reformulate classical nucleation theory as a geometrically closed thermodynamic framework by introducing a sphere-packing representation of atomic assembly. This construction establishes a direct link between discrete atomic structure and continuum free-energy contributions under applied magnetic fields, yielding a field-driven evolution equation for the critical nucleus size. The resulting theory provides a unified description of nanoparticle nucleation across superparamagnetic, paramagnetic, and diamagnetic systems within a single formalism. It quantitatively reproduces previously unresolved experimental observations for magnetite and nickel nanoparticles, namely the systematic reduction of mean particle size and narrowing of size distributions with increasing magnetic field. In the diamagnetic limit, the framework recovers our earlier analytical susceptibility-based description of silver nanoparticles, in which the field-dependent critical radius is governed by the induced-magnetization contribution to the nucleation free energy. Beyond modeling the reduction of mean particle size with increasing magnetic field, the framework reveals that the narrowing of size distributions emerges naturally from the curvature of the field-modified free-energy landscape. These results establish magnetic-field-assisted nucleation as a geometrically constrained thermodynamic process, providing a computationally efficient route for controlling nanoparticle size across distinct magnetic material classes.