Soft Matter at the Thermal Scale, Ch Nucleation and the lifetime of a metastable phase
Problem
A supercooled cloud sits at a uniform −10∘C and is to be seeded with silver iodide particles that act as heterogeneous ice-nucleating sites. Take the ice-water interfacial tension γ=0.029Jm−2, the molecular volume of ice vm=3.25×10−29m3, the latent heat of freezing Δh=6.0kJmol−1, and the ice melting point Tm=273.15K, as used earlier for homogeneous ice nucleation. Take the driving force per molecule as Δμ≈ΔhΔT/Tm (per molecule, dividing by Avogadro's constant), where ΔT=Tm−T. Treat the cloud droplets as spheres 1μm across, so each has volume V=5.2×10−19m3, and suppose that a freezing event in a droplet is observable if it occurs within t=100s (a typical cloud lifetime) given a nucleation-rate prefactor I0=1038m−3s−1, using the observability criterion ΔF∗≈kBTln(I0Vt). A silver iodide particle presents a nucleus in the shape of a spherical cap of contact angle θc, so the barrier for freezing on it is reduced to ΔFhet∗=fhet(θc)ΔFhom∗ with fhet(θc)=41(2+cosθc)(1−cosθc)2.
((a)) Find the largest contact angle θc a seeding particle may present if it is to nucleate freezing of a droplet within the cloud's lifetime.
((b)) The cloud occupies 1km3 and contains 108 droplets per cubic metre. Assuming each seeding particle lands in a distinct droplet and, provided its contact angle satisfies part (a), freezes that droplet within the observation time, find how many seeding particles are needed to freeze one droplet in every thousand.