= Solution
The thermodynamic questions can be organized from precursor chemistry to the final supported shape:
* \b[Can the precursor produce metal?] Evaluate the reaction <Gibbs free energy>, including reduction, ligand exchange, precursor and reductant activities, solvation and byproducts. A nominal metal concentration is not the metal-atom chemical potential.
* \b[What is the driving force for condensation?] Compare the chemical potential of available metal species with that of the condensed phase. The resulting supersaturation supplies the bulk free-energy gain and competes with the surface cost of a critical nucleus.
* \b[Where does it wet and nucleate?] Let $\gamma_{sm}$, $\gamma_{pm}$ and $\gamma_{sp}$ denote support-medium, particle-medium and support-particle interfacial free energies. For an isotropic cap, the contact angle through the particle obeys $\gamma_{sm}=\gamma_{sp}+\gamma_{pm}\cos\theta$. This wetting balance determines the extent of substrate contact and the heterogeneous barrier.
* \b[Which crystal faces survive?] At fixed particle volume, minimize the sum of orientation-dependent surface and interface energies, plus any important strain or defect energy. For a free crystal, the <Wulff construction> gives facet distances proportional to their surface free energies. A <Winterbottom construction> incorporates the support interface and truncates the corresponding equilibrium shape.
Adsorbed ligands and solvent can change different facet energies by different amounts, so the relevant energies belong to the actual chemical environment. Small particles have large surface-to-volume ratios and are particularly sensitive to these terms.
This sequence identifies thermodynamic factors rather than guaranteeing that an equilibrium shape is reached. Facet-specific attachment rates, precursor delivery and limited annealing can preserve a kinetically selected shape even when another shape has lower free energy.
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