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 Incoming links: Winterbottom construction

Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2017 / Paper 2 / Section A(i) / 13 / ii / Solution 2026-10-20

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The Winterbottom construction extends the Wulff construction to a crystal supported on a substrate. It includes the particle-substrate interfacial energy and the substrate surface energy displaced by contact, so the equilibrium Wulff body is truncated by the support plane. The relevant contact contribution is

(γ_{p s} - γ_{s}) A_{contact}

; it determines wetting and contact geometry. The basic unsupported construction accounts only for orientation-dependent free surface energies.^[ref]

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Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2020 / Paper 2 / Section A(i) / 14 / i / Solution 2026-10-20

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The competition between crystallization, exposed surfaces, adhesion and elastic strain can be summarized by

Δ G = - V Δ g + \sum_{j} γ_{j} A_{j} + (γ_{int} - γ_{support}) A_{contact} + E_{strain} .

(1)

Here

Δ g > 0

measures the bulk driving force from supersaturation, and the

γ

values are interfacial free energies. For a free spherical nucleus,

Δ G = 4 π r^{2} γ - 4 π r^{3} Δ g /3

gives

r_{*} = 2 γ /Δ 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. Equilibrium facet energy and kinetic growth rate are different controls, so a crystal structure does not uniquely determine particle shape.

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Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2023 / Paper 2 / Section B / 20 / ii / Solution 2026-10-20

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The thermodynamic questions can be organized from precursor chemistry to the final supported shape:

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.
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.
Where does it wet and nucleate? Let $γ_{s m}$ , $γ_{p m}$ and $γ_{s p}$ denote support-medium, particle-medium and support-particle interfacial free energies. For an isotropic cap, the contact angle through the particle obeys $γ_{s m} = γ_{s p} + γ_{p m} cos θ$ . This wetting balance determines the extent of substrate contact and the heterogeneous barrier.
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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