 Solution

ID: past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2021/paper-2/section-a-i/14/ii/solution

Let

g_{l} - g_{s}

be the liquid-minus-solid free energy density at the same temperature. Near the bulk coexistence temperature

T_{m}

g_{l} - g_{s} ≃ Δ s_{V} (T_{m} - T) = \frac{Δ H _{V}}{T _{m}} Δ T .

For a sphere, an interface displacement changes area and volume with

d A / d V = 2/ r

. Equality of the bulk driving force and the capillary cost at a curved solid–liquid interface therefore gives

g_{l} - g_{s} = 2 γ / r^{*}

. Equivalently, differentiating

4 π r^{2} γ - (4 π /3) r^{3} (g_{l} - g_{s})

gives the same stationary radius. Combining the expressions yields

Δ T = \frac{2 γ T _{m}}{r ^{*} Δ H _{V}} .

This form of the Gibbs-Thomson effect assumes isotropic interfacial energy, a spherical interface, bulk-like latent heat per volume and a near-coexistence free energy expansion. More detailed free-particle models must distinguish solid–vapor, liquid–vapor and solid–liquid energies.

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