The scaling of nucleation rates

The homogeneous nucleation rate,J, forT → T _c can be cast into a “corresponding states” form by exploiting scaled expressions for the vapor pressure and for the surface tension, δ. In the vapor-to-liquid case with δ= δ₀[T _c -T], the classical cluster energy of formation /kT = [16π/3] ? ?³ [T _c -1]³/(lnS)² = [x₀/x]², where ? ≡ δ₀/[k ñ^2/3] and ñ is liquid number density. [1] The ?≈ 2 for normal liquids. (A similar approach can be applied to homogeneous liquid to solid nucleation and to heterogeneous nucleation formalisms using appropriate modifications ofσ and ?.^[2]) The above [x₀/x]² is sufficiently tenable that in some cases, one can use it to extract approximate critical temperatures from experimental data.^[3,4] In this work, we point out that expansion cloud chamber data (for nonane, toluene, and water) are in excellent agreement with lnJ ≈ const. -[x₀/x]² [centimeter-gram-second (cgs) units], and that the constant term is well approximated by ln (Γ_c), whereT _c is the inverse thermal wavelength cubed per second atT =T _c . The ln (Γ_c) is ≈ 60 in cgs units (74 in SI units) for most materials. A physical basis for the latter form, which includes the behavior at smalln, the discrete integer behavior ofn, and a configurational entropy term, τ ln (n), is presented.