Figure 8: Effect of supersaturation, size dependent surface energy, and cluster excess free energy on classical nucleation barrier.

(a) Dependence of free-energy change ΔG on radius of nucleus R based on the classical expression for heterogeneous ion-by-ion nucleation ΔG=−(2πR³/3ω)·kTσ+2πR²α of hemispherical ACP for α_ACP=40 mJ m⁻² (derived from Fig. 7), values of σ as specified next to the curves, and other symbols as defined in Fig. 7. (See Supplementary Information Analysis of nucleation data: fitting of classical nucleation rate equation for details.) (b) Impact of size-dependent interfacial free energy on the barrier for heterogeneous ion‐by‐ion nucleation of hemispherical ACP, as based on equation 1. Many forms for this dependence have been proposed; however, the exact dependence chosen has only a minor effect on the predicted change in nucleation rates. Here, we assume an exponential dependence, α_ACP(R)=α_∞[1−exp(−(R−R₀)/R_∞)], where α_∞ is the surface energy for the bulk phase, R₀ is on the order of the molecular radius of the nucleating species and R_∞ is the characteristic particle radius at which the interfacial energy approaches the bulk value. Typical literature values for R₀ are ~2 Å, which we use here³⁹. Curves are shown for R_∞=1 nm (blue lines), 2 nm (red lines) and 3 nm (green lines), assuming σ_ACP=0.15 (solid lines) or 0.25 (dashed lines) and α_∞=40 mJ m⁻². All other parameters are as in Fig. 8a. Comparison of the solid curves of (b) with the red curve in (a) shows that the size dependence of surface energy shifts the free-energy barrier to slightly lower values and decreases R_c by a minor amount. (c) Dependence of free energy on nucleus size and ratio of complex to ACP surface energy (α_comp/α_ACP) according to equations 1 and 2 during heterogeneous nucleation of hemispherical ACP through aggregation of 1.1 nm disk-shaped complexes for α_ACP(R)=α_∞[1−exp(−(R−R₀)/R_∞)] with α_∞=100 mJ m⁻², R₀=2 Å and R_∞=1 nm, where values of α_comp/α_ACP and the corresponding ΔG_Ex are specified in the legend. Note that here α_∞ refers to the interfacial free energy of bulk ACP in solution, which we have conservatively taken as 100 mJ m⁻². Heavy solid curves: σ_ACP=0.15, light solid curves: σ_ACP=0.4.