= Solution
The competition between crystallization, exposed surfaces, adhesion and elastic strain can be summarized by
$$
\Delta G=-V\Delta g+\sum_j\gamma_jA_j+
(\gamma_{\rm int}-\gamma_{\rm support})A_{\rm contact}+E_{\rm strain}.
$$
Here $\Delta g>0$ measures the bulk driving force from <supersaturation>, and the $\gamma$ values are interfacial <free energies>. For a free spherical nucleus, $\Delta G=4\pi r^2\gamma-4\pi r^3\Delta g/3$ gives $r_*=2\gamma/\Delta g$. A support changes the <heterogeneous nucleation> barrier through adhesion and geometry.
Precursor activities and supply rate, <temperature>, time, solvent and ligands control nucleation, attachment, detachment and <surface diffusion>. Many nuclei divide a fixed solute inventory into smaller particles; continued growth or <Ostwald ripening> can enlarge them. Substrate defects determine preferred nucleation sites; <lattice mismatch> and interface bonding constrain orientation and elastic relaxation. Facet-selective adsorption can change both surface energies and growth rates.
Near equilibrium, a free <nanocrystal> follows a <Wulff construction>; a supported particle can follow a <Winterbottom construction>. Under kinetic control, slowly advancing facets remain visible while rapidly growing ones disappear. \b[Equilibrium facet energy and kinetic growth rate are different controls], so a crystal structure does not uniquely determine particle shape.
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