Homogeneous ice nucleation in pure water becomes observable

PHYSICS · Subjective · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Soft condensed matter · Soft condensed matter

Soft Matter at the Thermal Scale, Ch Nucleation and the lifetime of a metastable phase

Problem

Homogeneous ice nucleation in pure water becomes observable, within a watch time tt, once the barrier falls to roughly ΔF∗≈kBTln⁡(I0Vt)\Delta F^* \approx k_BT\ln(I_0Vt), where VV is the droplet volume and I0I_0 is the nucleation-rate prefactor, known only to within I0=1035I_0 = 10^{35} to 1041 m−3 s−110^{41}\,\mathrm{m^{-3}\,s^{-1}}. Using the ice/water interfacial tension γ=0.029 J m−2\gamma = 0.029\,\mathrm{J\,m^{-2}}, molecular volume vm=3.25×10−29 m3v_m = 3.25\times10^{-29}\,\mathrm{m^3}, molar enthalpy of fusion Δh=6.0 kJ mol−1\Delta h = 6.0\,\mathrm{kJ\,mol^{-1}} and melting point Tm=273 KT_m = 273\,\mathrm{K}, estimate the radius of a water droplet that would need to be watched for about t=1 st = 1\,\mathrm{s} to observe homogeneous freezing at the much milder supercooling of −35∘C-35^\circ\mathrm{C}, taking a representative mid-range prefactor I0=1038 m−3 s−1I_0 = 10^{38}\,\mathrm{m^{-3}\,s^{-1}}. Evaluate kBTk_BT at the droplet's actual temperature using the Boltzmann constant kB=1.381×10−23 J K−1k_B = 1.381\times10^{-23}\,\mathrm{J\,K^{-1}}.

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