Past exam of the materials science course of the University of Cambridge / 2016 / Paper 1 / 6

Table of contents
- a 6
  - Solution a
- b 6
  - Solution b
- c 6
  - Solution c
- d 6
  - i d
    - Solution i
  - ii d
    - Solution ii
  - iii d
    - Solution iii

a

Solution

For a spherical homogeneous nucleus, the capillarity approximation gives

Δ G (r) = 4 π r^{2} γ + \frac{4 π r ^{3}}{3} Δ g_{V} .

The first term penalizes creating an interface; below the melting point the bulk free-energy change per volume

Δ g_{V}

is negative and favors growth. Near

T_{m}

, neglect the temperature dependence of the latent heat. Since the free-energy difference vanishes at

T_{m}

,

Δ g_{V} ≃ Δ H_{V} \frac{Δ T}{T _{m}}, Δ T = T_{m} - T > 0, Δ H_{V} < 0 on freezing .

Set the radius derivative to zero:

\frac{d Δ G}{d r} = 8 π r γ + 4 π r^{2} Δ g_{V} = 0, r^{*} = - \frac{2 γ}{Δ g _{V}} = - \frac{2 γ T _{m}}{Δ H _{V} Δ T} .

The nonzero stationary point is a maximum, since

Δ G^{''} (r^{*}) = - 8 πγ < 0

. Subcritical nuclei lower free energy by shrinking; supercritical nuclei lower it by growing. The positive radius and its inverse-supercooling dependence follow from the negative freezing enthalpy. This derivation assumes an isotropic, size-independent interfacial energy and a bulk-like spherical nucleus; it is not a molecular-scale kinetic theory.

b

Solution

TODO: user-deferred biology. No biological nucleation-agent calculation or biological interpretation has been researched or supplied.

c

Solution

TODO: user-deferred biology. No organism-injury explanation has been researched or supplied.

d

i

Solution

TODO: user-deferred biology. No organism-specific survival mechanism has been researched or supplied.

ii

Solution

TODO: user-deferred biology. No organism-specific survival mechanism has been researched or supplied.

iii

Solution

TODO: user-deferred biology. No organism-specific survival mechanism has been researched or supplied.

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