Monte Carlo Test of the Classical Theory for Heterogeneous Nucleation Barriers David Winter, Peter Virnau, K. Binder Institut f¨ ur Physik, Johannes Gutenberg-Universit¨ at, D-55099 Mainz, Staudinger Weg 7, Germany (Dated: November 1, 2018) Flat walls facilitate the condensation of a supersaturated vapor: Classical theory of heterogeneous nucleation predicts that the free energy barrier Δ F ∗ het which needs to be overcome for the formation of sphere-cap shaped nucleation seeds is smaller than the barrier Δ F ∗ hom for spherical droplets in the bulk by a factor 0 < f ( θ ) < 1, which only depends on the contact angle θ . In this letter we compute both Δ F ∗ hom and Δ F ∗ het from Monte Carlo simulations and test the theory for the lattice gas model (for which θ can be readily controlled). Even though the theory is only based on macroscopic arguments, it is shown to hold for experimentally relevant nanoscopic nucleation seeds (20 ≤ Δ F ∗ hom /k B T ≤ 200) if (independently estimated) line tension effects are considered. PACS numbers: 64.60.an, 64.60Q-,64.70.F-,68.08Bc Nucleation [1–5] is a ubiquitous process which gov- erns such diverse phenomena as the formation of rain drops or snow flakes in the atmosphere, the crystalliza- tion of proteins and even the formation of industrially relevant polymer foams. Even though considerable em- pirical knowledge has been gathered, e.g., on which ma- terials are suited as seed particles for making clouds rain [6] or as nucleation agents to start precipitation in met- allurgy [7], many questions remain. Reasons for this par- tial shortcoming are manifold: Experiments are very dif- ficult to control. Homogeneous nucleation in the bulk needs to be separated from heterogeneous nucleation at walls or impurities, and small changes in parameters typ- ically lead to large changes in corresponding nucleation rates. Nucleation sites often only consist of a few hundred or thousand atoms and can only be observed indirectly. Unfortunately, theoretical progress is also hampered by these difficulties. Simulations on the other hand offer full control and are at least in principle able to bridge the gap between theory and experiment and, indeed, significant progress has been achieved in recent years [8–10]. According to Classical Nucleation Theory [1, 2], the formation of a (spherical) nucleation seed in the bulk is understood in terms of two competing factors: a volume term which seeks to expand the seed and an opposing surface term F$_{s}$ = 4 πR $^{2}$γ : R is the radius of the droplet (or bubble), γ the in- terfacial tension (of a macroscopically flat interface), Δ µ = µ − µ$_{coex}$ the difference in chemical potential rela- tive to the coexistence value, and ρ$_{l,v}$ the densities of the coexisting phases. The free energy barrier Δ F ∗ can now be obtained by determining the maximum of Eq. (1) at R $^{∗}$. Interestingly, at R ∗ we also obtain Hence, the nucleation barrier can directly be inferred The nucleation barrier in the heterogeneous case is sim- ply reduced by a factor of f ( θ ). Note that this derivation is solely based on macroscopic considerations and has to our knowledge never actually been tested. In this let- ter we will demonstrate with Monte Carlo simulations of an Ising lattice gas model that this formula can be applied to nanoscopic systems if line tension effects are Intriguingly, f ( θ ) = V$_{sphere}$$_{−}$$_{cap}$/V$_{droplet}$ . Therefore, Eq. (2) also holds for the heterogeneous case and we get: Considering Eq. (3) and the geometry of the sphere- cap (see caption of Fig. 1) we obtain Turnbull’s estimate for the surface free energy [13–15]: for incomplete wetting conditions ( γ$_{vℓ}$ > γ$_{wv}$ − γ$_{wℓ}$ ) [12]. Eq. (3) is known as Young’s equation and can be de- rived by considering the forces acting at the three phase boundary. For small droplets, a correction due to the line tension ( τ ) of the contact line arises [13] (which has the length 2 πr , where r = R sin θ ). The presence of a wall facilitates nucleation and re- duces the barrier. For macroscopic droplets (see Fig. 1), the contact angle θ between the droplet and the wall is given by the competition of the vapor-liquid ( γ$_{vℓ}$ ), wall- vapor ( γ$_{wv}$ ) and wall-liquid ( γ$_{wℓ}$ ) interface free energies [11] from the knowledge of the surface free energy of a droplet or bubble. Note that the spontaneous formation of a crit- ical nucleus of the new phase in the bulk requires a free energy barrier Δ F ∗ hom of the order 20 ≤ Δ F ∗ $_{hom}$/k$_{B}$ T ≤ 200. Compared to thermal fluctuations (that one can easily detect e.g. by scattering experiments) this is a rare event. arXiv:1001.3336v1 [cond-mat.stat-mech] 19 Jan 2010 γ$_{vℓ}$ cos θ = γ$_{wv}$ − γ$_{wℓ}$, (3) F$_{s,het}$ = F$_{s,hom}$ · f ( θ ) , f ( θ ) = (1 − cos θ ) $^{2}$(2 + cos θ ) / 4 . (4) Δ F ( R ) = − Δ µ ( ρ$_{l}$ − ρ$_{v}$ ) 4 πR 3 3 + 4 πR $^{2}$γ. (1) Δ F ∗ = 1 3 F$_{s}$ ( R $^{∗}$) . (2) Δ F$_{het}$ = Δ F$_{hom}$ · f ( θ ) . (5)