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Thin films, surfaces and nanomaterials

In homogeneous nucleation, take the free energy change on forming a spherical nucleus of radius r as ΔG = (4/3)πr³ΔGv + 4πr²γ. For an interfacial energy γ = 0.10 J/m² and a free energy change per unit volume ΔGv = −2.0×10⁸ J/m³, which value is closest to the critical nucleus radius r*?

  1. AAbout 0.5 nm
  2. BAbout 2.0 nm
  3. CAbout 1.0 nm
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Answer: C. About 1.0 nm

Differentiating ΔG with respect to r and setting it to zero gives r* = −2γ/ΔGv, so r* = 2 × 0.10/(2.0×10⁸) = 1.0×10⁻⁹ m = 1.0 nm. About 0.5 nm drops the factor of 2, giving −γ/ΔGv, and about 2.0 nm mistakes the diameter of the critical nucleus (2r*) for its radius. Embryos smaller than r* lower their energy by shrinking and tend to disappear, and only nuclei larger than r* keep growing. The greater the undercooling, the larger |ΔGv|, so r* becomes smaller and the nucleation barrier ΔG* = 16πγ³/(3ΔGv²) also drops sharply, making nucleation easier.

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